Handbook of Nanophysics, Volume 6 ... - Fulvio Frisone

Handbook of Nanophysics Handbook of Nanophysics: Principles and Methods Handbook of Nanophysics: Clusters and Fullerenes Handbook of Nanophysics: Nanoparticles and Quantum Dots Handbook of Nanophysics: Nanotubes and Nanowires Handbook of Nanophysics: Functional Nanomaterials Handbook of Nanophysics: Nanoelectronics and Nanophotonics Handbook of Nanophysics: Nanomedicine and Nanorobotics

Nanoelectronics and Nanophotonics

Edited by

Klaus D. Sattler

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-7550-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Handbook of nanophysics. Nanoelectronics and nanophotonics / editor, Klaus D. Sattler. p. cm. “A CRC title.” Includes bibliographical references and index. ISBN 978-1-4200-7550-2 (alk. paper) 1. Nanoelectronics--Handbooks, manuals, etc. 2. Nanophotonics--Handbooks, manuals, etc. I. Sattler, Klaus D. II. Title. TK7874.84.H36 2010 621.381--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2010001108

Contents Preface........................................................................................................................................................... ix Acknowledgments......................................................................................................................................... xi Editor........................................................................................................................................................... xiii Contributors..................................................................................................................................................xv

Part Iâ•… Computing and Nanoelectronic Devices

1

Quantum Computing in Spin Nanosystems.......................................................................................1-1

2

Nanomemories Using Self-Organized Quantum Dots...................................................................... 2-1

3

Carbon Nanotube Memory Elements................................................................................................. 3-1

4

Ferromagnetic Islands.. ....................................................................................................................... 4-1

5

A Single Nano-Dot Embedded in a Plate Capacitor.......................................................................... 5-1

6

Nanometer-Sized Ferroelectric Capacitors........................................................................................ 6-1

7

Superconducting Weak Links Made of Carbon Nanostructures.. ......................................................7-1

8

Micromagnetic Modeling of Nanoscale Spin Valves................................................................................................8-1

9

Quantum Spin Tunneling in Molecular Nanomagnets..................................................................... 9-1

10

Inelastic Electron Transport through Molecular Junctions............................................................ 10-1

11

Bridging Biomolecules with Nanoelectronics................................................................................... 11-1

Gabriel González and Michael N. Leuenberger

Martin Geller, Andreas Marent, and Dieter Bimberg Vincent Meunier and Bobby G. Sumpter

Arndt Remhof, Andreas Westphalen, and Hartmut Zabel Gilles Micolau and Damien Deleruyelle

Nikolay A. Pertsev, Adrian Petraru, and Hermann Kohlstedt Vincent Bouchiat

Bruno Azzerboni, Giancarlo Consolo, and Giovanni Finocchio Gabriel González and Michael N. Leuenberger Natalya A. Zimbovskaya

Kien Wen Sun and Chia-Ching Chang

v

vi

Contents

Part IIâ•… Nanoscale Transistors

12

Transistor Structures for Nanoelectronics.. ..................................................................................... 12-1

13

Metal Nanolayer-Base Transistor..................................................................................................... 13-1

14

ZnO Nanowire Field- Effect Transistors.................................................................................................14-1

15

C 60 Field Effect Transistors............................................................................................................... 15-1

16

The Cooper-Pair Transistor.. ............................................................................................................ 16-1

Jean-Pierre Colinge and Jim Greer André Avelino Pasa

Woong-Ki Hong, Gunho Jo, Sunghoon Song, Jongsun Maeng, and Takhee Lee Akihiro Hashimoto José Aumentado

Part IIIâ•… Nanolithography

17

Multispacer Patterning: A Technology for the Nano Era................................................................. 17-1

18

Patterning and Ordering with Nanoimprint Lithography.. ............................................................. 18-1

19

Nanoelectronics Lithography........................................................................................................... 19-1

20

Extreme Ultraviolet Lithography.. .................................................................................................... 20-1

Gianfranco Cerofolini, Elisabetta Romano, and Paolo Amato Zhijun Hu and Alain M. Jonas

Stephen Knight, Vivek M. Prabhu, John H. Burnett, James Alexander Liddle, Christopher L. Soles, and Alain C. Diebold Obert R. Wood II

Part IVâ•… Optics of Nanomaterials

21

Cathodoluminescence of Nanomaterials.. .........................................................................................21-1

22

Optical Spectroscopy of Nanomaterials........................................................................................... 22-1

23

Nanoscale Excitons and Semiconductor Quantum Dots................................................................. 23-1

24

Optical Properties of Metal Clusters and Nanoparticles.. ............................................................... 24-1

25

Photoluminescence from Silicon Nanostructures........................................................................... 25-1

26

Polarization-Sensitive Nanowire and Nanorod Optics.. .................................................................. 26-1

27

Nonlinear Optics with Clusters.. .......................................................................................................27-1

28

Second-Harmonic Generation in Metal Nanostructures.. ............................................................... 28-1

Naoki Yamamoto

Yoshihiko Kanemitsu

Vanessa M. Huxter, Jun He, and Gregory D. Scholes

Emmanuel Cottancin, Michel Broyer, Jean Lermé, and Michel Pellarin Amir Sa’ar

Harry E. Ruda and Alexander Shik

Sabyasachi Sen and Swapan Chakrabarti

Marco Finazzi, Giulio Cerullo, and Lamberto Duò

Contents

vii

29

Nonlinear Optics in Semiconductor Nanostructures...................................................................... 29-1

30

Light Scattering from Nanofibers.. ................................................................................................... 30-1

31

Biomimetics: Photonic Nanostructures............................................................................................ 31-1

Mikhail Erementchouk and Michael N. Leuenberger Vladimir G. Bordo Andrew R. Parker

Part Vâ•… Nanophotonic Devices

32

Photon Localization at the Nanoscale.............................................................................................. 32-1

33

Operations in Nanophotonics.. ......................................................................................................... 33-1

34

System Architectures for Nanophotonics.. ....................................................................................... 34-1

35

Nanophotonics for Device Operation and Fabrication.. .................................................................. 35-1

36

Nanophotonic Device Materials....................................................................................................... 36-1

37

Waveguides for Nanophotonics.........................................................................................................37-1

38

Biomolecular Neuronet Devices....................................................................................................... 38-1

Kiyoshi Kobayashi

Suguru Sangu and Kiyoshi Kobayashi Makoto Naruse

Tadashi Kawazoe and Motoichi Ohtsu Takashi Yatsui and Wataru Nomura

Jan Valenta, Tomáš Ostatnický, and Ivan Pelant Grigory E. Adamov and Evgeny P. Grebennikov

Part VIâ•… Nanoscale Lasers

39

Nanolasers......................................................................................................................................... 39-1

40

Quantum Dot Laser.......................................................................................................................... 40-1

41

Mode-Locked Quantum-Dot Lasers..................................................................................................41-1

Marek S. Wartak Frank Jahnke

Maria A. Cataluna and Edik U. Rafailov

Index.. ................................................................................................................................................... Index-1

Preface The Handbook of Nanophysics is the first comprehensive reference to consider both fundamental and applied aspects of nanophysics. As a unique feature of this work, we requested contributions to be submitted in a tutorial style, which means that state-of-the-art scientific content is enriched with fundamental equations and illustrations in order to facilitate wider access to the material. In this way, the handbook should be of value to a broad readership, from scientifically interested general readers to students and professionals in materials science, solid-state physics, electrical engineering, mechanical engineering, computer science, chemistry, pharmaceutical science, biotechnology, molecular biology, biomedicine, metallurgy, and environmental engineering.

What Is Nanophysics? Modern physical methods whose fundamentals are developed in physics laboratories have become critically important in nanoscience. Nanophysics brings together multiple disciplines, using theoretical and experimental methods to determine the physical properties of materials in the nanoscale size range (measured by millionths of a millimeter). Interesting properties include the structural, electronic, optical, and thermal behavior of nanomaterials; electrical and thermal conductivity; the forces between nanoscale objects; and the transition between classical and quantum behavior. Nanophysics has now become an independent branch of physics, simultaneously expanding into many new areas and playing a vital role in fields that were once the domain of engineering, chemical, or life sciences. This handbook was initiated based on the idea that breakthroughs in nanotechnology require a firm grounding in the principles of nanophysics. It is intended to fulfill a dual purpose. On the one hand, it is designed to give an introduction to established fundamentals in the field of nanophysics. On the other hand, it leads the reader to the most significant recent developments in research. It provides a broad and in-depth coverage of the physics of nanoscale materials and applications. In each chapter, the aim is to offer a didactic treatment of the physics underlying the applications alongside detailed experimental results, rather than focusing on particular applications themselves. The handbook also encourages communication across borders, aiming to connect scientists with disparate interests to begin

interdisciplinary projects and incorporate the theory and methodology of other fields into their work. It is intended for readers from diverse backgrounds, from math and physics to chemistry, biology, and engineering. The introduction to each chapter should be comprehensible to general readers. However, further reading may require familiarity with basic classical, atomic, and quantum physics. For students, there is no getting around the mathematical background necessary to learn nanophysics. You should know calculus, how to solve ordinary and partial differential equations, and have some exposure to matrices/linear algebra, complex variables, and vectors.

External Review All chapters were extensively peer reviewed by senior scientists working in nanophysics and related areas of nanoscience. Specialists reviewed the scientific content and nonspecialists ensured that the contributions were at an appropriate technical level. For example, a physicist may have been asked to review a chapter on a biological application and a biochemist to review one on nanoelectronics.

Organization The Handbook of Nanophysics consists of seven books. Chapters in the first four books (Principles and Methods, Clusters and Fullerenes, Nanoparticles and Quantum Dots, and Nanotubes and Nanowires) describe theory and methods as well as the fundamental physics of nanoscale materials and structures. Although some topics may appear somewhat specialized, they have been included given their potential to lead to better technologies. The last three books (Functional Nanomaterials, Nanoelectronics and Nanophotonics, and Nanomedicine and Nanorobotics) deal with the technological applications of nanophysics. The chapters are written by authors from various fields of nanoscience in order to encourage new ideas for future fundamental research. After the first book, which covers the general principles of theory and measurements of nanoscale systems, the organization roughly follows the historical development of nanoscience. Cluster scientists pioneered the field in the 1980s, followed by extensive ix

x

work on fullerenes, nanoparticles, and quantum dots in the 1990s. Research on nanotubes and nanowires intensified in subsequent years. After much basic research, the interest in applications such as the functions of nanomaterials has grown. Many bottom-up MATLAB• is a registered trademark of The MathWorks, Inc. For product information, Please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA, 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

Preface

and Â�top-down techniques for nanomaterial and nanostructure were developed and made possible the development of Â� generation Â�nanoelectronics and nanophotonics. In recent years, real applications for nanomedicine and nanorobotics have been discovered.

Acknowledgments Many people have contributed to this book. I would like to thank the authors whose research results and ideas are presented here. I am indebted to them for many fruitful and stimulating discussions. I would also like to thank individuals and publishers who have allowed the reproduction of their figures. For their critical reading, suggestions, and constructive criticism, I thank the referees. Many people have shared their expertise and have commented on the manuscript at various

stages. I consider myself very fortunate to have been supported by Luna Han, senior editor of the Taylor & Francis Group, in the setup and progress of this work. I am also grateful to Jessica Vakili, Jill Jurgensen, Joette Lynch, and Glenon Butler for their patience and skill with handling technical issues related to publication. Finally, I would like to thank the many unnamed editorial and production staff members of Taylor & Francis for their expert work. Klaus D. Sattler Honolulu, Hawaii

xi

Editor Klaus D. Sattler pursued his undergraduate and master’s courses at the University of Karlsruhe in Germany. He received his PhD under the guidance of Professors G. Busch and H.C. Siegmann at the Swiss Federal Institute of Technology (ETH) in Zurich, where he was among the first to study spin-polarized photoelectron emission. In 1976, he began a group for atomic cluster research at the University of Konstanz in Germany, where he built the first source for atomic clusters and led his team to pioneering discoveries such as “magic numbers” and “Coulomb explosion.” He was at the University of California, Berkeley, for three years as a Heisenberg Fellow, where he initiated the first studies of atomic clusters on surfaces with a scanning tunneling microscope. Dr. Sattler accepted a position as professor of physics at the University of Hawaii, Honolulu, in 1988. There, he initiated a research group for nanophysics, which, using scanning probe microscopy, obtained the first atomic-scale images of carbon nanotubes directly confirming the graphene network. In 1994,

his group produced the first carbon nanocones. He has also studied the formation of polycyclic aromatic hydrocarbons (PAHs) and nanoparticles in hydrocarbon flames in collaboration with ETH Zurich. Other research has involved the nanopatterning of nanoparticle films, charge density waves on rotated graphene sheets, band gap studies of quantum dots, and graphene foldings. His current work focuses on novel nanomaterials and solar photocatalysis with nanoparticles for the purification of water. Among his many accomplishments, Dr. Sattler was awarded the prestigious Walter Schottky Prize from the German Physical Society in 1983. At the University of Hawaii, he teaches courses in general physics, solid-state physics, and quantum mechanics. In his private time, he has worked as a musical director at an avant-garde theater in Zurich, composed music for theatrical plays, and conducted several critically acclaimed musicals. He has also studied the philosophy of Vedanta. He loves to play the piano Â�(classical, rock, and jazz) and enjoys spending time at the ocean, and with his family.

xiii

Contributors Grigory E. Adamov Open Joint-Stock Company Central Scientific Research Institute of Technology “Technomash” Moscow, Russia Paolo Amato Numonyx and Department of Materials Science University of Milano-Bicocca Milano, Italy José Aumentado National Institute of Standards and Technology Boulder, Colorado Bruno Azzerboni Department of Matter Physics and Electronic Engineering Faculty of Engineering University of Messina Messina, Italy Dieter Bimberg Institut für Festkörperphysik Technische Universität Berlin Berlin, Germany Vladimir G. Bordo A.M. Prokhorov General Physics Institute Russian Academy of Sciences Moscow, Russia Vincent Bouchiat Nanosciences Department Centre National de la Recherche Scientifique Néel-Institut Grenoble, France

Michel Broyer Laboratoire de Spectrométrie Ionique et Moléculaire Centre National de la Recherche Scientifique Université de Lyon Villeurbanne, France John H. Burnett Atomic Physics Division National Institute of Standards and Technology Gaithersburg, Maryland Maria A. Cataluna Division of Electronic Engineering and Physics School of Engineering, Physics and Mathematics University of Dundee Dundee, United Kingdom Gianfranco Cerofolini Department of Materials Science University of Milano-Bicocca Milano, Italy Giulio Cerullo Dipartimento di Fisica Politecnico di Milano Milano, Italy

and Institute of Physics Academia Sinica Taipei, Taiwan Jean-Pierre Colinge Tyndall National Institute University College Cork Cork, Ireland Giancarlo Consolo Department of Matter Physics and Electronic Engineering Faculty of Engineering University of Messina Messina, Italy Emmanuel Cottancin Laboratoire de Spectrométrie Ionique et Moléculaire Centre National de la Recherche Scientifique Université de Lyon Villeurbanne, France

Swapan Chakrabarti Department of Chemistry University of Calcutta Kolkata, West Bengal, India

Damien Deleruyelle Centre National de la Recherche Scientifique Institut Matériaux Microélectronique Nanosciences de Provence Universités d’Aix Marseille Marseille, France

Chia-Ching Chang Department of Biological Science and Technology National Chiao Tung University Hsinchu, Taiwan

Alain C. Diebold College of Nanoscale Science and Engineering University at Albany Albany, New York xv

xvi

Lamberto Duò Dipartimento di Fisica Politecnico di Milano Milano, Italy Mikhail Erementchouk NanoScience Technology Center and Department of Physics University of Central Florida Orlando, Florida Marco Finazzi Dipartimento di Fisica Politecnico di Milano Milano, Italy Giovanni Finocchio Department of Matter Physics and Electronic Engineering Faculty of Engineering University of Messina Messina, Italy Martin Geller Experimental Physics and Center for Nanointegration Duisburg-Essen University of Duisburg-Essen Duisburg, Germany Gabriel González NanoScience Technology Center and Department of Physics University of Central Florida Orlando, Florida Evgeny P. Grebennikov Open Joint-Stock Company Central Scientific Research Institute of Technology “Technomash” Moscow, Russia Jim Greer Tyndall National Institute University College Cork Cork, Ireland

Contributors

Akihiro Hashimoto Department of Electrical and Electronics Engineering Graduate School of Engineering University of Fukui Fukui, Japan

Alain M. Jonas Institute of Condensed Matter and Nanosciences Division of Bio- and Soft Matter Catholic University of Louvain Louvain-la-Neuve, Belgium

Jun He Department of Chemistry Institute for Optical Sciences

Yoshihiko Kanemitsu Institute for Chemical Research Kyoto University Uji, Kyoto, Japan

and Centre for Quantum Information and Quantum Control University of Toronto Toronto, Ontario, Canada Woong-Ki Hong Department of Materials Science and Engineering Gwangju Institute of Science and Technology Gwangju, Korea Zhijun Hu Center for Soft Matter Physics and Interdisciplinary Research Soochow University Suzhou, China

Tadashi Kawazoe Department of Electrical Engineering and Information Systems School of Engineering The University of Tokyo Tokyo, Japan Stephen Knight Office of Microelectronics Programs National Institute of Standards and Technology Gaithersburg, Maryland Kiyoshi Kobayashi Department of Electrical Engineering and Information Systems The University of Tokyo Tokyo, Japan and

Vanessa M. Huxter Department of Chemistry Institute for Optical Sciences

Core Research of Evolutional Science and Technology Japan Science and Technology

and

and

Centre for Quantum Information and Quantum Control University of Toronto Toronto, Ontario, Canada

Department of Electrical and Electronic Engineering University of Yamanashi Kofu, Japan

Frank Jahnke Institute for Theoretical Physics University of Bremen Bremen, Germany

Hermann Kohlstedt Christian-Albrechts-Universita¨t Zu Kiel Faculty of Engineering Nanoelectronics Kiel, Germany

Gunho Jo Department of Materials Science and Engineering Gwangju Institute of Science and Technology Gwangju, Korea

Takhee Lee Department of Materials Science and Engineering Gwangju Institute of Science and Technology Gwangju, Korea

xvii

Contributors

Michel Pellarin Laboratoire de Spectrométrie Ionique et Moléculaire Centre National de la Recherche Scientifique Université de Lyon Villeurbanne, France

Jean Lermé Laboratoire de Spectrométrie Ionique et Moléculaire Centre National de la Recherche Scientifique Université de Lyon Villeurbanne, France

and

Michael N. Leuenberger NanoScience Technology Center

Wataru Nomura School of Engineering The University of Tokyo Tokyo, Japan

Nikolay A. Pertsev A.F. Ioffe Physico-Technical Institute Russian Academy of Sciences St. Petersburg, Russia

Motoichi Ohtsu Department of Electrical Engineering and Information Systems School of Engineering The University of Tokyo Tokyo, Japan

Adrian Petraru Christian-Albrechts-Universita¨t Zu Kiel Faculty of Engineering Nanoelectronics Kiel, Germany

and Department of Physics University of Central Florida Orlando, Florida James Alexander Liddle Center for Nanoscale Science and Technology National Institute of Standards and Technology Gaithersburg, Maryland Jongsun Maeng Department of Materials Science and Engineering Gwangju Institute of Science and Technology Gwangju, Korea Andreas Marent Institut für Festkörperphysik Technische Universität Berlin Berlin, Germany Vincent Meunier Oak Ridge National Laboratory Oak Ridge, Tennessee Gilles Micolau Institut Matériaux Microélectronique Nanosciences de Provence Centre National de la Recherche Scientifique Universités d’Aix Marseille Marseille, France Makoto Naruse National Institute of Information and Communications Technology Koganei, Japan

Department of Electrical Engineering and Information Systems School of Engineering The University of Tokyo Tokyo, Japan

Tomáš Ostatnický Faculty of Mathematics and Physics Department of Chemical Physics and Optics Charles University Prague, Czech Republic

Andrew R. Parker Department of Zoology The Natural History Museum London, United Kingdom

Vivek M. Prabhu Polymers Division National Institute of Standards and Technology Gaithersburg, Maryland Edik U. Rafailov Division of Electronic Engineering and Physics School of Engineering, Physics and Mathematics University of Dundee Dundee, United Kingdom

and School of Biological Science University of Sydney Sydney, New South Wales, Australia

André Avelino Pasa Laboratório de Filmes Finos e Superfícies Departamento de Física Universidade Federal de Santa Catarina Santa Catarina, Brazil

Ivan Pelant Institute of Physics Academy of Sciences of the Czech Republic Prague, Czech Republic

Arndt Remhof Division of Hydrogen and Energy Department of Environment, Energy and Mobility Swiss Federal Laboratories for Materials Testing and Research Dübendorf, Switzerland Elisabetta Romano Department of Materials Science University of Milano-Bicocca Milano, Italy Harry E. Ruda Centre for Advanced Nanotechnology University of Toronto Toronto, Ontario, Canada

xviii

Amir Sa’ar Racah Institute of Physics and The Harvey M. Kruger Family Center for Nanoscience and Nanotechnology The Hebrew University of Jerusalem Jerusalem, Israel Suguru Sangu Device and Module Technology Development Center Ricoh Company, Ltd. Yokohama, Japan Gregory D. Scholes Department of Chemistry Institute for Optical Sciences and Centre for Quantum Information and Quantum Control University of Toronto Toronto, Ontario, Canada Sabyasachi Sen Department of Chemistry JIS College of Engineering Kolkata, West Bengal, India Alexander Shik Centre for Advanced Nanotechnology University of Toronto Toronto, Ontario, Canada

Contributors

Christopher L. Soles Polymers Division National Institute of Standards and Technology Gaithersburg, Maryland Sunghoon Song Department of Materials Science and Engineering Gwangju Institute of Science and Technology Gwangju, Korea Bobby G. Sumpter Oak Ridge National Laboratory Oak Ridge, Tennessee Kien Wen Sun Department of Applied Chemistry National Chiao Tung University Hsinchu, Taiwan Jan Valenta Faculty of Mathematics and Physics Department of Chemical Physics and Optics Charles University Prague, Czech Republic Marek S. Wartak Department of Physics and Computer Science Wilfrid Laurier University Waterloo, Ontario, Canada

Andreas Westphalen Department of Physics and Astronomy Institute for Condensed Matter Physics Ruhr-Universität Bochum Bochum, Germany Obert R. Wood II GlobalFoundries Albany, New York Naoki Yamamoto Department of Physics Tokyo Institute of Technology Tokyo, Japan Takashi Yatsui School of Engineering The University of Tokyo Tokyo, Japan Hartmut Zabel Department of Physics and Astronomy Institute for Condensed Matter Physics Ruhr-Universität Bochum Bochum, Germany Natalya A. Zimbovskaya Department of Physics and Electronics University of Puerto Rico Humacao, Puerto Rico and Institute for Functional Nanomaterials University of Puerto Rico San Juan, Puerto Rico

Computing and Nanoelectronic Devices

I

1 Quantum Computing in Spin Nanosystemsâ•… Gabriel González and Michael N. Leuenberger................................ 1-1

2 Nanomemories Using Self-Organized Quantum Dotsâ•… Martin Geller, Andreas Marent, and Dieter Bimberg........2-1

3 Carbon Nanotube Memory Elementsâ•… Vincent Meunier and Bobby G. Sumpter..........................................................3-1

4 Ferromagnetic Islandsâ•… Arndt Remhof, Andreas Westphalen, and Hartmut Zabel.......................................................4-1

5 A Single Nano-Dot Embedded in a Plate Capacitorâ•… Gilles Micolau and Damien Deleruyelle..................................5-1

6 Nanometer-Sized Ferroelectric Capacitorsâ•… Nikolay A. Pertsev, Adrian Petraru, and Hermann Kohlstedt.............6-1

7 Superconducting Weak Links Made of Carbon Nanostructuresâ•… Vincent Bouchiat...............................................7-1

8 Micromagnetic Modeling of Nanoscale Spin Valvesâ•… Bruno Azzerboni, Giancarlo Consolo, and Giovanni Finocchio............................................................................................................................................ 8-1

Introductionâ•‡ •â•‡ Qubits and Quantum Logic Gatesâ•‡ •â•‡ Conditions for the Physical Implementation of Quantum Computingâ•‡ •â•‡ Zeeman Effectsâ•‡ •â•‡ Atom–Light Interactionâ•‡ •â•‡ Loss–DiVincenzo Proposalâ•‡ •â•‡ Quantum Computing with Molecular Magnetsâ•‡ •â•‡ Semiconductor Quantum Dotsâ•‡ •â•‡ Single-Photon Faraday Rotationâ•‡ •â•‡ Concluding Remarksâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Conventional Semiconductor Memoriesâ•‡ •â•‡ Nonconventional Semiconductor Memoriesâ•‡ •â•‡ Semiconductor Nanomemoriesâ•‡ •â•‡ A Nanomemory Based on III–V Semiconductor Quantum Dotsâ•‡ •â•‡ Capacitance Spectroscopyâ•‡ •â•‡ Charge Carrier Storage in Quantum Dotsâ•‡ •â•‡ Write Times in Quantum Dot Memoriesâ•‡ •â•‡ Summary and Outlookâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References Introductionâ•‡ •â•‡ CNFET-Based Memory Elementsâ•‡ •â•‡ NEMS-Based Memoryâ•‡ •â•‡ Electromigration CNT-Based Data Storageâ•‡ •â•‡ General Conclusionsâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Backgroundâ•‡ •â•‡ State of the Artâ•‡ •â•‡ Summary and Discussionâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Studied Configuration: Geometry and Notationsâ•‡ •â•‡ Metallic Dotâ•‡ •â•‡ Semiconductor Dot: Theoretical Approachâ•‡ •â•‡ Finite Element Modeling of a Silicon Quantum Dotâ•‡ •â•‡ Conclusionâ•‡ •â•‡ Appendix A:â•‡ A Numerical Validation of the Semi-Analytical Approachâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Fundamentals of Ferroelectricityâ•‡ •â•‡ Deposition and Patterningâ•‡ •â•‡ Characterization of Ferroelectric Films and Capacitorsâ•‡ •â•‡ Physical Phenomena in Ferroelectric Capacitorsâ•‡ •â•‡ Future Perspectiveâ•‡ •â•‡ Summary and Outlookâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Superconducting Transport through a Weak Linkâ•‡ •â•‡ Superconducting Transport in a Carbon Nanotube Weak Linkâ•‡ •â•‡ Nanotube-Based Superconducting Quantum Interferometersâ•‡ •â•‡ Graphene-Based Superconducting Weak Linksâ•‡ •â•‡ Fullerene-Based Superconducting Weak Linksâ•‡ •â•‡ Concluding Remarksâ•‡ •â•‡ Acknowledgmentâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Backgroundâ•‡ •â•‡ Computational Micromagnetics of Nanoscale Spin-Valves: State of the Artâ•‡ •â•‡ Conclusions and Future Perspectiveâ•‡ •â•‡ References

9 Quantum Spin Tunneling in Molecular Nanomagnetsâ•… Gabriel González and Michael N. Leuenberger............... 9-1 Introductionâ•‡ •â•‡ Spin Tunneling in Molecular Nanomagnetsâ•‡ •â•‡ Phonon-Assisted Spin Tunneling in Mn12 Acetateâ•‡ •â•‡ Interference between Spin Tunneling Paths in Molecular Nanomagnetsâ•‡ •â•‡ Incoherent Zener Tunneling in Fe8â•‡ •â•‡ Coherent Néel Vector Tunneling in Antiferromagnetic Molecular Wheelsâ•‡ •â•‡ Berry-Phase Blockade in Single-Molecule Magnet Transistorsâ•‡ •â•‡ Concluding Remarksâ•‡ •â•‡ Acknowledgmentâ•‡ •â•‡ References

I-1

I-2

Computing and Nanoelectronic Devices

10 Inelastic Electron Transport through Molecular Junctionsâ•… Natalya A. Zimbovskaya........................................ 10-1

11 Bridging Biomolecules with Nanoelectronicsâ•… Kien Wen Sun and Chia-Ching Chang..........................................11-1

Introductionâ•‡ •â•‡ Coherent Transportâ•‡ •â•‡ Buttiker Model for Inelastic Transportâ•‡ •â•‡ Vibration-Induced Inelastic Effectsâ•‡ •â•‡ Dissipative Transportâ•‡ •â•‡ Polaron Effects: Hysteresis, Switching, and Negative Differential Resistanceâ•‡ •â•‡ Molecular Junction Conductance and Long-Range Electron- Transfer Reactionsâ•‡ •â•‡ Concluding Remarksâ•‡ •â•‡ References

Introduction and Backgroundâ•‡ •â•‡ Preparation of Molecular Magnetsâ•‡ •â•‡ Nanostructured Semiconductor Templates: Nanofabrication and Patterningâ•‡ •â•‡ Self-Assembling Growth of Molecules on the Patterned Templatesâ•‡ •â•‡ Magnetic Properties of Molecular Nanostructuresâ•‡ •â•‡ Conclusion and Future Perspectivesâ•‡ •â•‡ References

1 Quantum Computing in Spin Nanosystems 1.1 1.2 1.3 1.4

Introduction.............................................................................................................................. 1-1 Qubits and Quantum Logic Gates......................................................................................... 1-2 Conditions for the Physical Implementation of Quantum Computing........................... 1-3 Zeeman Effects.......................................................................................................................... 1-3

1.5

Atom–Light Interaction...........................................................................................................1-4

1.6 1.7 1.8 1.9

Gabriel González University of Central Florida

Michael N. Leuenberger University of Central Florida

Weak External Fieldâ•‡ •â•‡ Strong External Field Jaynes–Cummings Model

Loss–DiVincenzo Proposal..................................................................................................... 1-7 RKKY Interaction

Quantum Computing with Molecular Magnets................................................................. 1-9 Semiconductor Quantum Dots............................................................................................ 1-12 Classical Faraday Effectâ•‡ •â•‡ Quantum Faraday Effect

Single-Photon Faraday Rotation.......................................................................................... 1-14 Quantifying the EPR Entanglementâ•‡ •â•‡ Single Photon Faraday Effect and GHZ Quantum Teleportationâ•‡ •â•‡ Single-Photon Faraday Effect and Quantum Computing

1.10 Concluding Remarks.............................................................................................................. 1-20 Acknowledgments.............................................................................................................................. 1-20 References............................................................................................................................................ 1-20

1.1â•‡ Introduction The history of quantum computers begins with the articles of Richard Feynman who, in 1982, speculated that quantum systems might be able to perform certain tasks more efficiently than would be possible in classical systems (Feynman, 1982). Feynman was the first to propose a direct application of the laws of quantum mechanics to a realization of quantum algorithms. The fundamentals of quantum computing were introduced and developed by several authors after Feynman’s idea. A model and a description of a quantum computer as a quantum Turing machine was developed by Deutsch (1985). In 1994, Shor introduced the quantum algorithm for the integer-number factorization and in 1997, Grover proposed the fast quantum search algorithm (Grover, 1997). Later on, Wooters and Zurek proved the noncloning theorem, which puts definite limits on the quantum computations, but Shor’s work challenges all that with the quantum error correction code (Shor, 1995). In the last years, the development of quantum computing has grown to enormous practical importance as an interdisciplinary field, which links the elements of physics, mathematics, and computer science. Currently, various physical models of quantum computers are under intensive study. Several types of elementary quantum

computing devices have been developed based on atomic, molecular, optical, and semiconductor physics and technologies. Before contemplating the physical realization of a quantum computer, it is necessary to decide how information is going to be stored within the system and how the system will process that information during a desired computation. In classical computers, the information is typically carried in microelectronic circuits that store information using the charge properties of electrons. Information processing is carried out by manipulating electrical fields within semiconductor materials in such a way as to perform useful computational tasks. Presently it seems that the most promising physical model for quantum computation is based on the electron’s spin. A strong research effort toward the implementation of the electron spin as a new information carrier has been the subject of a new form of electronics based on spin called spintronics. Experiments that have been conducted on quantum spin dynamics in semiconductor materials demonstrate that electron spins have several characteristics that are promising for quantum computing applications. Electron spin states possess the following advantages: very long relaxation time in the absence of external fields, fairly long decoherence time τd ≈ 1â•›μs, and the possibility of easy spin manipulation by an external magnetic field. These characteristics are very promising 1-1

1-2

Handbook of Nanophysics: Nanoelectronics and Nanophotonics

since longer decoherence times relax constraints on the switching speeds of quantum gates necessary for reliable error correction. Typically quantum gates are required to switch 104 times faster than the loss of qubit coherence. Spin coherent transport over lengths as large as 100â•›μm have been reported in semiconductors. This makes electron spin a perfect candidate as an information carrier in semiconductors (Adamowski et al., 2005). The spin of particles exhibiting quantum behavior is specially suitable for the construction of quantum computers. The electron spin states can be used to construct qubits and logic operations in different ways. They can be constructed either directly with the application of a magnetic field or indirectly with the application of symmetry properties of the many-electron wave function (namely by the resulting singlet or triplet spin states). Traditionally, in nanoelectronic devices, the charge of the electrons has been used to carry and transform information, which makes the interaction of spintronic devices with charge-based devices important for compatibility with existing classical computing schemes. Overall, we can say that spin nanosystems are good candidates for the physical implementation of quantum computation.

| ψ〉 = c0 | 0〉 + c1 | 1〉,

(1.1)

where c0 and c1 are complex numbers and the modulus squared of each complex number represents the probability to obtain the qubit |0〉 or |1〉, respectively. Additionally, they must satisfy the normalization condition |c0|2 + |c1|2 = 1. Contrary to the classical bit, the quantum bit takes on a continuum of values, which are determined by the probability amplitudes given by c0 and c1. If we perform a measurement on qubit |ψ〉, we obtain either outcome |0〉 with probability |c0|2 or outcome |1〉 with probability |c1|2. However, if the qubit is prepared to be exactly equal to one of the states of the computational basis, i.e., |ψ〉 = |0〉 or |ψ〉 = |1〉, then we can predict the exact result of the measurement with probability 1. This nondeterministic characteristic between the general state of the qubit and the precise result of the measurement in the basis state plays an essential role in quantum computations. To carry out a quantum computation, we require at least a two-qubit state, i.e., the states of a two-particle quantum system. The two-qubit states can be constructed as tensor products of the basis states |0〉, |1〉. The two-qubit basis consists of the states |00〉, |01〉, |10〉, |11〉, where in the shorthanded notation |0〉 ⊗ |1〉 ≡ |01〉, etc. implied. An arbitrary two-qubit state has the form

(1.2)

where the normalization condition takes the form |c0|2 + |c1|2 + |c2|2 + |c3|2 = 1. Another basic characteristic for quantum computing is the so-called entanglement. The two quantum two-level systems can become entangled by interacting with each other. This means that we cannot fully describe one system independently of the other. For example, suppose that the state (| 01〉 − | 10〉)/ 2 gives a complete description of the whole system. Then a measurement over the first subsystem forces the second subsystem into one of the two states |0〉 or |1〉. This means that one measurement over one subsystem influences the other, even though it may be arbitrarily far away. Qubits can be transformed by unitary transformations (observables) U that play the role of quantum logic gates and which transforms the initial qubit into a final qubit according to

| Ψf 〉 = U | Ψi 〉.

(1.3)

Depending on the type of qubit on which they operate, we deal with either a 2 × 2 or a 4 × 4 matrix. For example, the quantum NOT gate is defined as

1.2â•‡ Qubits and Quantum Logic Gates We know that the information stored in a classical computer can take one of the two values, i.e., 0 or 1, with probability 0 or 1 each. Quantum bits or qubits are the quantum mechanical analogue of classical bits. In contrast, the qubit can be defined as a quantum state vector in a two-dimensional Hilbert space. Suppose |0〉, |1〉 forms a basis for the Hilbert space, then the qubit can be expressed as the superposition of the two states as

| Ψ〉 = c0 | 00〉 + c1 | 00〉 + c2 | 10〉 + c3 | 11〉,

0 UNOT =  1

1 . 0

(1.4)

The effect of the quantum logic gate (4) in the one-qubit state given in Equation 1.1 is to exchange the probability amplitudes i.e.,

 c0   0 UNOT   =   c1   1

1  c0   c1    =  . 0  c1   c0 

(1.5)

An example of an important quantum logic gate that operates on a two-qubit state is the so-called controlled-NOT gate UCNOT, for which the first qubit is the control qubit and the second qubit is the target qubit. The controlled-NOT gate transforms the twoqubit basis states as follows: UCNOT | 00〉 = | 00〉, UCNOT | 01〉 = | 01〉,

UCNOT | 10〉 = | 11〉,

and

UCNOT | 11〉 = | 10〉,

(1.6)

which means that the CNOT (conditionalNOT) gate changes the second qubit if and only if the first qubit is in state |1〉. The matrix representation of the CNOT gate is

UCNOT

1 0 = 0 0 

0 1 0 0

0 0 0 1

0 0 . 1 0

(1.7)

It has been shown that the set of logic operations, which consists of all the one-qubit gates and the single two-qubit gate UCNOT is universal in the sense that all unitary transformations on

1-3

Quantum Computing in Spin Nanosystems

N-qubit states can be expressed by different compositions of the set of universal gates (DiVincenzo, 1995). Using the concepts of superposition and entanglement, we can describe another important characteristic of quantum computation, which is known as quantum parallelism. Quantum parallelism is based on the fact that a single unitary transformation can simultaneously operate on all the qubits in the system. In a sense, it can perform several calculations in a single step. In fact, it can be proved that the computing power of a quantum computer scales exponentially with the number of qubits, whereas a classical computer the scale is only linear.

1.3â•‡Conditions for the Physical Implementation of Quantum Computing The successful implementation of a quantum computer has to satisfy some basic requirements. These are known as the DiVincenzo criteria and can be summarized in the following way (DiVincenzo, 2001).

1. Physical realizability of the qubits. We need to find some quantum property of a scalable physical system in which to encode our information so that it lives long enough to enable us to perform computations. 2. Initial state preparation. It should be possible to precisely prepare the initial qubit state. 3. Isolation. We need a controlled evolution of the qubit; this will require enough isolation of the qubit from the environment to reduce the effects of decoherence. 4. Gate implementation. We need to be able to manipulate the states of individual qubits with reasonable precision, as well as to induce interaction between them in a controlled way, so that the implementation of gates is possible. Also, the gate operation time τs has to be much shorter than the decoherence time τd, so that τ/τd ≪ r, where r is the maximum tolerable error rate for quantum error correction schemes to be effective. 5. Readout. We must be able to accurately measure the final qubit state.

The conditions listed above put certain limitations on the quantum computing technology. For example, the complete isolation of the qubit with respect to the environment disables the read/write operations. Therefore, some slight interaction of the quantum system and the environment is necessary. On the other hand, this interaction leads to decay and decoherence processes, which reduce the performance of the quantum computer. In the decay process, the quantum system jumps in a very short time to a new state, releasing part of its energy to the environment. The decay is characterized by the relaxation time, which for the spin states can be very long. Decoherence is a more subtle process because the energy is conserved but the relative phase of the computational basis is changed. As a result of the decoherence the qubit changes as follows:

| ψ 〉 = c0 | 0〉 + e iθc1 | 1〉,

(1.8)

where the real number θ denotes the relative phase. The appearance of the nonzero relative phase results due to the coupling between the quantum system and the environment and can lead to significant changes in the measurement process. The ratio of the decoherence time to the elementary operation time τs, i.e., R = τd/τs, is an approximate measure of the number of computation steps performed before the coupling with the environment destroys the qubit. This ratio changes abruptly for different quantum computing schemes. For example, R = 103 for the electron states in quantum dots, R = 107 for nuclear spin sates, and R = 1013 for trapped ions. An important factor that should always be kept in mind when constructing quantum computers is the scalability of the device. We should be able to enlarge the physical device to contain many qubits and still fulfill the DiVincenzo requirements described above.

1.4â•‡ Zeeman Effects An external magnetic field superimposed on an atom perturbs its state in a definite way. The Hamiltonian for such an atom may be divided into the operator H0 for the unperturbed atom H′ due to the magnetic and the perturbation operator ⃯ ⃯ ⃯ field. The external magnetic field H causes the vectors L and S to precess about its direction. We shall examine the two cases where the magnetic field is weak and is strong. Let us first write down the perturbation operator explicitly. Consider an electron in the simplest possible atom (hydrogen) rotating around the nucleus. The electron orbit can be regarded as a current loop. The current is the charge per unit time past any fixed point on the orbit, therefore

j=

e( p /me ) e ev = = , T 2πr 2πr

(1.9)

where p and m e are the momentum and mass of the electron, respectively. The magnitude of the magnetic moment of the loop is the current times the enclosed area, i.e., μorbâ•›â•›=â•›â•›πr 2j, therefore

e µB µ orb = r ×p= L, 2me

(1.10)

where we have introduced the Bohr magneton, μB = eħ/2me. We do not have a good semiclassical picture for spin and therefore we can only conclude that the spin magnetic moment is

µ µ sp = g s B S ,

(1.11)

where gs is called the gyromagnetic factor and its value is approximately equal to 2. Therefore, the total magnetic moment is

1-4


µ µ = B (L + 2S ).

(1.12)

The perturbation energy caused by the magnetic field is given then by µ H′ = −µ ⋅ H = B ( J + S ),

(1.13)

1.4.1â•‡ Weak External Field In a hydrogen atom, the electron circles around the proton; but in a system fixed to an electron, the proton circles around the electron and generates an inner magnetic field at the position of the electron given by

µ 0e L. 4πmpr 3

(1.14)

Equation 1.14 gives the following energy for the electron’s spin Hso = µ sp ⋅ H in =

e2 S ⋅ L, 3 2 2 4 πε0r m c

(1.15)

which is called the spin–orbit interaction. We interpret the spin– orbit interaction as an internal Zeeman effect because it splits the energy levels without an external magnetic field. Let the external field be weak compared with the effective internal field. Since the Larmor is proportional to the ⃯ ⃯frequency ⃯ ⃯ magnetic field, the triangle L S J in Figure 1.1⃯rotates about J considerably faster than the around H. Therefore, the coupling of the vectors ⃯ ⃯precession ⃯ L , S , and J in the triangle is not disrupted. Let us now find the correction to the energy due to the ⃯ exter⃯ nal magnetic field. Equation 1.13 involves the vectors J â•›and S , if we consider that the z-axis coincides with the direction of the external magnetic field, then, the only projections of these two vectors that contribute to the energy are the ones along the z-axis. Thus, the mean value of H′ is equal to H

µBH J z  S ⋅ J  1+ 2  ,  J 

(1.16)

and using

S ⋅ J j( j + 1) + s(s + 1) − l(l + 1) = =g 2 j( j + 1) J2

(1.17)

we get

where the plus sign resulted because the charge of the electron is −e.

H in =

〈 H ′〉 =

〈 H′〉 = µ B Hgm j

for m j = − j, − j + 1,…, j.

(1.18)

Thus, the energy levels with a given J⃯are split into as many levels as there are different projections of J â•›on the magnetic field, i.e., 2j + 1. The factor g is called the Landé factor and takes a given value for a given L and S and each corresponding value of J. The Zeeman effect in a weak magnetic field is called anomalous.

1.4.2â•‡ Strong External Field We shall now consider the opposite extreme case, when the external field is strong compared ⃯with ⃯ ⃯the internal field, so that the coupling between the vectors L , S , J is disrupted. ⃯ ⃯ This can be explained by the fact that S precesses twice as fast L ⃯ ⃯ . Then, from the classical analogy, each of the vectors S and L precesses independently about the magnetic field. For this case, the correction to the energy is given by

〈 H ′〉 =

µ B eH (Lz + 2Sz ).

(1.19)

In Equation 1.19, Lz and Sz are the projections of the orbital angular momentum and spin along the z-axis, respectively, and are given by Lz = ml

1 for ml = −l ,…, l , Sz = ms , for ms = ± . 2

(1.20)

The projections Sz and Lz are changed by unity, therefore all the levels in Equation 1.19 are equidistant. Of course, certain values of 〈 H′〉 may be repeated several times if the sum Lz + 2Sz assumes the same value. This Zeeman effect in a strong magnetic field is called the normal Zeeman effect.

1.5â•‡Atom–Light Interaction S

J L

FIGURE 1.1â•… Schematic of the spin–orbit coupling.

Though an atom has infinitely many energy levels, we can have under certain assumptions a two-level atom. These assumptions are (1) the difference of the energy levels approximately matches the energy of the incident photon, (2) the selection rules allow transition of the electrons between the two levels, and (3) all other energy levels are sufficiently detuned in frequency separation with respect to the incoming frequency such that there is no transition to these levels.

1-5


Consider that we apply the two-level approximation to a simple atom where the interaction with visible light involves a single electron. The corresponding wave function of the system is | ψ〉 = c0 | 0〉 + c1 | 1〉,

(1.21)

where |0〉 (|1〉) corresponds to the nonexcited (excited) state and the coefficients c0(c1) represent the probability amplitude to find the system in either state, respectively. The probability amplitudes satisfy the normalization condition |c0|2 + |c1|2 = 1. The wavelength of visible light, typical for atomic transitions, is about a few thousand times the diameter of an atom. Therefore, there is no significant spatial variation of the electric field across an atom, ⃯⃯ ⃯ and E(r ,â•›t) ≈ E(t) can be taken as independent of the position (dipole approximation). Consistent with⃯ this long-wavelength approximation is that the magnetic field H is approximately zero, ⃯ i.e., H ≈ 0, so that the interaction energy between the field and the electron of the atom is given by H′ = −er ⋅ E, hence the total Hamiltonian is H = H0 + H′,

(1.22)

where H0 represents the unperturbed atom system, i.e., H0 | 0〉 = ε 0 | 0〉 and H0 | 1〉 = ε1 | 1〉 , where ε0 and ε1 are the energy values for the nonexcited and excited states, respectively. Seeking a solution to the Schrödinger equation H | ψ(t )〉 = i

∂ | ψ(t )〉 , ∂t

| ψ(t )〉 = c0 (t )e − iε0t / | 0〉 + c1(t )e − iε1t / | 1〉,

dc0 (t ) iΩ*R − i∆t = e c1(t ), dt 2

dc1(t ) iΩR i∆t = e c0 (t ), dt 2

The general solution for Equation 1.29 for strictly monochromatic fields, i.e., ΩR = |ΩR|eiϕ = constant, is given by

(1.23) (1.24)

iΩ  Ωt  c1(t ) = R ei∆t / 2 sin   ,  2  Ω

µ1,2 = − cl (t ) ck (t )e − i( εk − εl )t / 〈l | H′ | k 〉, for l = 0, 1. = dt k = 0 ,1

∑

(1.25)

Representing the light with frequency ν by means of the complex electric field vector E 0 in the form

1 (µ1eiµ2t − µ 2eiµ1t ), Ω

1 E(t ) = (E0e −iνt + E0* eiνt ), 2

(1.26)

and using the fact that the diagonal elements of the interacting Hamiltonian are zero due to the parity of the eigenfunction, i.e., 〈k | H′ | k 〉 = 0, we end up with the following coupled differential equations dc0 (t ) i  E0 ⋅ 〈0 | d | 1〉e − i(ω10 + ν)t + E0* ⋅ 〈0 | d | 1〉e − i(ω10 − ν))t  c1(t ), =   2 dt (1.27)

(1.30)

where

we get i

(1.29)

where ⃯ ⃯ ΩR = E0 . 〈1|d |0〉/ħ is known as the Rabi frequency Δ = (ω10 − ν) is the so-called detuning

c0 (t ) =

in the following form

dc1(t ) i  * = E0 ⋅ 〈1 | d | 0〉ei(ω10 + ν)t + E0 ⋅ 〈1 | d | 0〉ei(ω10 − ν)t  c0 (t ),  2  dt (1.28) ⃯ ⃯ ⃯ where ω10 = (ε1 − ε0)/ħ and 〈0|d |1〉 = 〈1|d |0〉* = 〈0|er |1〉. For the case when ν ≈ ω10, we can drop the terms containing exp[±(ω10 + ν)t] in Equation 1.28 since they oscillate rapidly in time and can be neglected with respect to the near resonant terms, i.e., the terms of the form exp[±(ω10 − ν)t]. This approximation is called the rotating wave approximation (RWA), and the coupled differential equations become

∆ ± ∆ 2 + Ω2R , 2 2

(1.31) 2 R

Ω = µ1 − µ 2 = ∆ + Ω .

Equation 1.31 gives the transition probabilities from the nonexcited to the excited state 2

Ω   Ωt  |c0 (t )|2 =  R  sin2   , |c1(t )|2 = 1− |c0 (t )|2 ,  Ω  2 

(1.32)

which oscillate with frequency Ω (Rabi flopping frequency) between levels ε0 and ε1. Any two-level system can be represented by a 2 × 2 matrix Hamiltonian and hence can be expressed in terms of Pauli matrices. For example, choosing the energy zero to be half way between the excited and nonexcited state |0〉 and |1〉, we have the following matrix Hamiltonian for this system

H0 =

ω  1 2  0

0  ω = σz , −1 2

(1.33)

1-6


where ħω = ε1 − ε0 σz is a Pauli matrix Therefore, we can write the total Hamiltonian of the atom–light interaction in the RWA as H=

ω  2  −Ω*R e − i∆t

−ΩR ei∆t  ω σ z − (ΩR ei∆t σ + + Ω*R e − i∆t σ − ), =  2 2 −ω 

(1.34)

dependent amplitude that varies harmonically with time, i.e., . A k⃯ = −iωk⃯Ak⃯, where ωk = kx2 + k 2y + kz2 = | k | c . Assuming that the field is periodic in space, and the lengths of the periods in three perpendicular directions are equal to the dimensions of the cubic box, then kj = 2njπ/L for j = x, y, z and nj are integers of any sign. Using Equation 1.38 we get

1 0 and σ − =  0 1

0 . 0

H (r , t ) = (1.35)

So far, we have used a semiclassical description of the interaction between a two-level atom and an electric field. A quantum description of the interaction would require a quantization of the radiation field. This description is known as the Jaynes–Cummings model. Consider the sourceless electromagnetic field in a cubic cavity of volume V = L × L × L. Working with the Coulomb gauge for which the vector potential satisfies the requirement ∇ ⋅ A = 0,

(1.36)

and, since there are no sources, it satisfies the homogeneous wave equation 1 ∂2 A ∇ 2 A − 2 2 = 0. c ∂t

(1.37)

Then the fields are fully specified by the vector potential in the form ∂A E=− and B = ∇ × A. (1.38) ∂t We can expand the vector potential in the form

A(r , t ) =

∑  A (t )e

1 0V

k

k

ik ⋅r

+ Ak* (t )e − ik ⋅r . 

(1.39)

⃯ Equation 1.36 implies ⃯that Ak⃯ and Ak* are perpendicular to the vector of propagation k , therefore we can rewrite Equation 1.39 in the following form:

A(r , t ) =

1 0V

∑  A (t )e

k

k

k

ik ⋅ r

∑ ν  A e

k

k

+ k* Ak* e − ik ⋅r  , 

(1.41)

− (k × k* )Ak* e − ir ⋅r . 

(1.42)

ik ⋅r k

k

i 0V

∑ (k × )A e

ir ⋅ r k

k

k

The classical Hamiltonian for the field is given by

1.5.1â•‡ Jaynes–Cummings Model

i 0V

and

where 0 σ+ =  0

E(r , t ) =

+ k* Ak* (t )e −ik ⋅r , (1.40) 

⃯ where ϵ k⃯ = (e1, e2) is called the polarization vector, which is perpendicular to the direction of propagation and Ak⃯(t) is a time

HE − M =

0 2

∫ (| E | +c | H | ) dx dy dz. 2

2

2

(1.43)

V

Substituting Equations 1.41 and 1.42 into Equation 1.43 one gets HE − M =

∑ ω A A* . k

2 k

k

(1.44)

k

If we introduce the variable Ak =

Pk  1   Qk + i ω  , 2 k

(1.45)

then Equation 1.44 becomes HE − M =

1 2

∑ (Q ω k

2 k

2 k

+ Pk2 ).

(1.46)

Equation 1.46 corresponds to the sum of independent harmonic oscillators. This suggests that each mode of the field is dynamically equivalent to a mechanical harmonic oscillator. The canonical quantization of the field consists of the substitution of the variables Qk⃯ and Pk⃯ for operators, which fulfill the commutation relation [Q k , Pk ′ ] = iδ k k ′ ,

(1.47)

this means that [ Ak , Ak†′ ] =

δ . ω k k k ′

(1.48)

Introducing the creation and annihilation operators

ak =

ω k A and ak† = k

ω k † A , k

(1.49)

1-7


we can write the electromagnetic field Hamiltonian in the form HE−M =

∑  ω a a 

† k k k

k

1  + ω k  .  2

(1.50)

The total Hamiltonian that describes the interaction of an atom with the quantized electromagnetic field can be written in the form

H=

∑ ε | i〉〈i | + ∑ ω a a − ∑ i

† k k k

k

i

〈i | d | j 〉 | i 〉〈 j | ⋅E,

i, j

(1.51)

where ϵi is the energy corresponding to the state |i〉 of the atom and we have dropped the constant terms corresponding to the zero point energy. The electric field operator is given by E=i

ω k  ik ⋅r ak e − ak† e −ik ⋅r  . k  0V 

∑ k

(1.52)

⃯⃯ .

Making the dipole approximation, i.e., eik r ≈ 1, and assuming that we are dealing with a two-level atom, i.e., i, j = 0, 1, then we can write Equation 1.51 as H=

ω10 σz + 2

∑ ω a a + ∑ ( g *σ k

† k k k

k

k

+

− g k σ − )(ak − ak† ), (1.53)

where g k = i ω k/0V d01 ⋅ k . The scalar product between the dipole moment and the polarization vector yields a complex number that can be written as d01 ⋅ k = | d01 ⋅ k | eiφ. This allows one to choose the phase ϕ in such a way that g k = g k* . If the atom interacts only with one mode of the electromagnetic field, then we can drop the sum over the vector of propagation to get H=

ω10 σ z + ωak†ak + g k (σ + − σ − )(ak − ak† ) = H0 + H′. (1.54) 2

In the interaction picture, i.e., HI = e − i H0t / H ei H0t / , the Hamiltonian of interaction will have the form

(

HI′ = g k σ +ak e −i(ω10 − ω )t + σ −ak† ei(ω10 − ω )t − σ +ak† ei(ω10 + ω )t

)

−σ −ak e − i(ω10 + ω )t ,

(1.55)

using the RWA, i.e., dropping the rapidly oscillating terms containing e ± i(ω10 + ω)t , we end up with

HI =

ω10 σ z + ωak†ak + g k (σ + ak e − i∆t + σ −ak† ei∆t ), 2

(1.56)

where Δ = ω10 − ω is the detuning. The Hamiltonian of Equation 1.56 corresponds in the Schrödinger picture to

H=

ω10 σ z + ωak† ak + g k (σ + ak + σ − ak† ). 2

(1.57)

Equation 1.57 is the Jaynes–Cummings model.

1.6â•‡ Loss–DiVincenzo Proposal The first proposals for quantum computing made use of cavity quantum electrodynamics (QED), trapped ions, and nuclear magnetic resonance (NMR). All of these proposals benefit from long decoherence times due to a very weak coupling of the qubits to their environment. The long decoherence times have led to big successes in achieving experimental realizations. A conditional phase (CPHASE) gate was demonstrated early on in cavity QED systems. The two-qubit controlled-NOT gate has been realized in single-ion and two-ion versions. The most remarkable realization of the power of quantum computing to date is the implementation of Shor’s algorithm to factor the number 15 in a liquid-state NMR quantum computer to yield the known result 5 and 3. However, these proposals may not be scalable and therefore do not meet the DiVincenzo criteria. The requirement for scalability motivated the Loss–DiVincenzo proposal for a solid state quantum computer based on electron spin qubits (Loss and DiVincenzo, 1998). The spin of an electron in a quantum dot can point up or down with respect to an external magnetic field; these eigenstates, |↑〉 and |↓〉, correspond to the two basis states of the qubit. The electron trapped in a quantum dot, which is basically a small electrically defined box that can be filled with electrons, can be defined by metal gate electrodes on top of a semiconductor (GaAs/AlGaAs) heterostructure. At the interface between GaAs and AlGaAs conduction band, electrons accumulate and can only move in the lateral direction. Applying negative voltages to the gates locally depletes this two-dimensional electron gas underneath. The resulting gated quantum dots are very controllable and versatile systems, which can be manipulated and probed electrically. When the size of the dot is comparable to the wavelength of the electrons that occupy it, the system exhibits a discrete energy spectrum, resembling that of an atom. Initialization of the quantum computer can be achieved by allowing all spins to reach their equilibrium thermodynamic ground state at a low temperature T in an applied magnetic field, so that all the spins will be aligned if the condition |gμBH| ≫ kBT is satisfied (where kB is Boltzmann constant). To perform single-qubit operations, we can apply a microwave magnetic field on resonance with the Zeeman splitting, i.e., with a frequency f = ΔEZ/h (h is Planck’s constant). The oscillating magnetic component perpendicular to the static magnetic field H results in a spin nutation. By applying the oscillating field for a fixed duration, a superposition of |↑〉 and |↓〉 can be created. This magnetic technique is known as electron spin resonance (ESR). In the Loss–Divincenzo proposal, two-qubit operations can be carried out purely electrically by varying the gate voltages between neighboring dots. When the barrier is high, the spins are decoupled. When the interdot barrier is pulsed low,

1-8

Handbook of Nanophysics: Nanoelectronics and Nanophotonics H Hac

e

2DEG

e

e

e

Magnetized layer

Back gate

FIGURE 1.2â•… Theoretical proposal by Loss–DiVincenzo for quantum computing using quantum dots and electric gates.

an appreciable overlap develops between the two electron wave functions, resulting in a nonzero Heisenberg exchange Â�coupling J (see Figure 1.2). The Hamiltonian describing this time dependent process is given by

H(t ) = J (t )Sn ⋅ Sn +1.

called indirect interaction. The basis of this interaction lies in an exchange interaction, which was proposed by Ruderman and Kittel (1954), and extended by Kasuya (1956) and Yosida (1957), and now known as the RKKY interaction. This interaction refers to an exchange energy written as

(1.58)

Equation 1.58 is sometimes referred to as the direct interaction. The evolution of the quantum state is described by the propagator given by U (t ) = T exp[−i ∫ H(t )dt/ ] , where T is the time ordering operator. If the exchange is pulsed on for a time τs such that ∫ J (t )dt / = J 0τ s / = π, the states of the two spins will be exchanged. This is the SWAP operation. Pulsing the exchange for a shorter time τs/2 generates the square root of SWAP operation, which can be used in conjunction with single-qubit operator to generate the controlled-NOT gate. A last crucial ingredient requires a method to read out the state of the spin qubit. This implies measuring the spin orientation of a single electron. Therefore, an indirect spin measurement is proposed. First, the spin orientation of the electron is correlated with its position, via spin to charge conversion. Then an electrometer is used to measure the position of the charge, thereby revealing its spin. In this way, the problem of measuring the spin orientation has been replaced by the much easier measurement of charge. The Loss–DiVincenzo ideas have influenced an enormous research effort aimed at implementing the different parts of the proposal and has been quickly followed by a series of alternative solid state realizations for trapped atoms in optical lattices that may also be scalable. It should also be stressed that the efforts to create a spin qubit are not purely application driven. If we have the ability to control and read out a single electron spin, we are in a unique position to study the interaction of the spin with its environment. This may lead to a better understanding of decoherence and will also allow us to study the semiconductor environment using the spin as a probe.

∑ s ⋅ S,

Hexch = − J (r )

(1.59)

i

i

where The exchange parameter J(r) falls off rapidly with distance r between the center of a localized magnetic ion and elec⃯ tron spin s⃯i is the spin state of a conduction electron S is the spin of a localized magnetic ion The minus sign in Equation 1.59 is related to the Pauli exclusion principle (lowest energy of electron occupancy). Below some critical magnetic ordering temperature, the itinerant or localized spins may condense into an ordered array, i.e., ferromagnetic or antiferromagnetic. As in the case of atomic scattering, any disorder within this array will cause additional electron scattering. This scattering may be elastic, in which case there is no change in energy or spin-flip, or it may be inelastic, in which case the spin state of the conduction electrons changes. Usually RKKY interaction is of significance only in compounds with a high concentration of the magnetic atom. Thus, localized ions may start to interact indirectly via the conduction electrons. Now, consider the ⃯case in⃯ which there exist two localized spins at lattice points Rn and Rm. By the interaction between spin ⃯ Smz localized at Rm and the spin density of conduction electrons ⃯ polarized by spin ⃯Snz localized at R , the following interaction n ⃯ between the spins S n and S m is found:

1.6.1â•‡RKKY Interaction

where

The RKKY interaction is a long-range magnetic interaction that involves nearest-neighbor ions as well as magnetic atoms that are further apart; this interaction is sometimes

Hexch = −9π

2 J 2  Ne  F 2kF | Rn − Rm | Smz Snz ,   εF  N 

F (x) =

(

− x cos( x ) + sin(x ) . x4

)

(1.60)

(1.61)

1-9


Recently, the optical RKKY interaction between two spins was introduced as a means to produce an effective exchange interaction (Piermarocchi et al., 2002). Similar to the Loss–DiVincenzo scheme, in this scheme, the two qubits are defined by the excess electrons of semiconductor quantum dots. Instead of using the direct exchange interaction between the two electrons, the exchange interaction is indirectly mediated by the itinerant electrons of virtual excitons that are optically excited in the host material, which can be made of bulk, quantum well, or quantum wire structures. This scheme has the advantage that twoqubit gates can be performed on the femtosecond timescale due to the possibility of using ultrafast laser optics. The Coulomb interaction between the photoexcited itinerant electrons and the localized electrons in the two quantum dots contains direct and indirect terms. While the direct terms give rise to state renormalization of the localized electrons, the exchange terms lead to an effective Heisenberg interaction of the form

HORKKY = − J12S1 ⋅ S2 ∝ P12

H C2 H X2 P12 , δ3

(1.62)

which is calculated in fourth-order perturbation theory using the diagram depicted in Figure 1.3. P12 is the projection operator on the two-spin Hilbert space of the two localized spins. The control Hamiltonian

HC =

∑ k ,σ

Ω k , σ (t ) −iωPσ † e ck , − σh−† k , σ + h.c. 2

(1.63)

describes the creation of the virtual excitons by means of an external laser field that is detuned by the energy δ from the continuum states of the host material, or more precisely from † † the exciton 1s level. ck,σ and hk,σ are electron and hole creation operators, respectively. The RKKY interaction between a localized electron in the quantum dot and the itinerant electrons in the host material is given in second quantization by ωP , σ+

γ΄

1

ω, ke, – 2

γ

J2 3

ω, k΄c, α

ωP – ω, kh, 2

J1

β΄

1

ωP , σ+

ω, ke, – 2 β

FIGURE 1.3â•… Effective spin–spin interaction for the localized electrons in the dots 1 and 2 (indicated by dotted lines) induced by a photoexcited electron–hole pair (the solid and dashed lines, respectively). The indices β and γ denote the spin states of the electrons localized in the dots. The photon propagator is depicted by a wavy line. (From Piermarocchi, C. et al., Phys. Rev. Lett., 89, 167402, 2002.)

HX = −

1 V

∑

J (k , k ′)Si ⋅ s α , α ′ck† , αck ′ , α ′ .

α ,α ′, k , k ′

(1.64)

The predicted exchange interaction J12(R) (see Figure 1.2) can be of the order of 1â•›meV, which is of the same order as the Heisenberg interaction in the Loss–DiVincenzo scheme.

1.7â•‡Quantum Computing with Molecular Magnets Shor and Grover demonstrated that a quantum computer can outperform any classical computer in factoring numbers (Shor, 1997) and in searching a database (Grover, 1997) by exploiting the parallelism of quantum mechanics. Recently, the latter has been successfully implemented (Ahn et al., 2000) using Rydberg atoms. Leuenberger and Loss (2001) proposed an implementation of Grover’s algorithm using molecular magnets (Friedman et al., 1996; Thomas et al., 1996; Sangregorio et al., 1997; Thiaville and Miltat, 1999; Wernsdorfer et al., 2000); their spin eigenstates make them natural candidates for single-particle systems. It was shown theoretically that molecular magnets can be used to build dense and efficient memory devices based on the Grover algorithm. In particular, one single crystal can serve as a storage unit of a dynamic random access memory device. Fast ESR pulses can be used to decode and read out stored numbers of up to 105, with access times as short as 10−10 s. This proposal should be feasible using the molecular magnets Fe8 and Mn12. Suppose we want to find a phone number in a phone book consisting of N = 2 n entries. Usually it takes N/2 queries on average to be successful. Even if the N entries were encoded binary, a classical computer would need approximately log2 N queries to find the desired phone number (Grover, 1997). But the computational parallelism provided by the superposition and interference of quantum states enables the Grover algorithm to reduce the search to one single query (Grover, 1997). This query can be implemented in terms of a unitary transformation applied to the single spin of a molecular magnet. Such molecular magnets, forming identical and largely independent units, are embedded in a single crystal so that the ensemble nature of such a crystal provides a natural amplification of the magnetic moment of a single spin. However, for the Grover algorithm to succeed, it is necessary to find ways to generate arbitrary superpositions of spin eigenstates. For spins larger than ½, this turns out to be a highly nontrivial task as spin excitations induced by magnetic dipole transitions in conventional ESR can change the magnetic quantum number m by only ±1. To circumvent such physical limitations, it was proposed to use multifrequency coherent magnetic radiation that allows the controlled generation of arbitrary spin superpositions. In particular, it was shown that by means of advanced ESR techniques, it is possible to coherently populate and manipulate many spin states simultaneously by applying one single pulse of a magnetic a.c. field containing an appropriate number of matched frequencies. This a.c. field creates a nonlinear response of the magnet via multiphoton

1-10


absorption processes involving particular sequences of σ and π photons, which allows the encoding and, similarly, the decoding of states. Finally, the subsequent read-out of the decoded quantum state can be achieved by means of pulsed ESR techniques. These exploit the nonequidistance of energy levels, which is typical of molecular magnets. Molecular magnets have the important advantage that they can be grown naturally as single crystals of up to 10–100â•›μm length containing about 1012–1015 (largely) independent units so that only minimal sample preparation is required. The molecular magnets are described by a single-spin Hamiltonian of the form H spin = Ha + V + Hsp + HT (Leuenberger and Loss, 1999, 2000a,b), where Ha = − ASz2 − BSz4 represents the magnetic anisotropy (Aâ•›≫â•›Bâ•›>â•›0). The Zeeman term V = gμBH . S describes the coupling between the external magnetic field H and the spin S of length s. The calculational states are given by the 2s + 1 eigenstates of Ha + g µ B H z Sz with eigenenergies εm = −Am2 − Bm4 + gμBHzm, −s ≤ m ≤ s. The corresponding classical anisotropy potential energy E(θ) = −As cos2 θ − Bs cos4 θ + gμBHzs cos θ is obtained by the substitution Sz = s cos θ, where θ is the polar spherical angle. We have introduced the notation m, m′ = m − m′. By applying a bias field Hz such that gμBHz > Emm′, tunneling can be completely suppressed and thus HT can be neglected (Leuenberger and Loss, 1999, 2000a,b). For temperatures below 1â•›K, transitions due to spin–phonon interactions (Hsp) can also be neglected. In this regime, the level lifetime in Fe8 and Mn12 is estimated to be about τd = 10−7 s, limited mainly by hyperfine and/or dipolar interactions (Leuenberger and Loss, 2001). Since the Grover algorithm requires that all the transition probabilites are almost the same, Leuenberger and Loss (2001) and Leuenberger et al. (2003) propose that all the transition amplitudes between the states |s〉 and |m〉, mâ•›=â•›1, 2, …, sâ•›−â•›1, are of the same order in perturbation V. This allows us to use perturbation theory. A different approach uses the magnetic field amplitudes to adjust the appropriate transition amplitudes (Leuenberger et al., 2002). Both methods work only if the energy levels are not equidistant, which is typically the case in molecular magnets owing to anisotropies. In general, if we choose to work with the states mâ•›=â•›m 0, m 0â•›+â•›1, …, sâ•›−â•›1, where m 0â•›=â•›1, 2, …, sâ•›−â•›1, we have to go up to nth order in perturbation, where n = sâ•›−â•›m 0 is the number of computational states used for the Grover search algorithm (see below) to obtain the first nonvanishing contribution. Figure 1.4 shows the transitions for sâ•›=â•›10 and m 0 = 5. The nth-order transitions correspond to the nonlinear response of the spin system to strong magnetic fields. Thus, a coherent magnetic pulse of duration T is needed with a discrete frequency spectrum {ωm}, say, for Mn12 between 20 and 300â•›GHz and a single low-frequency 0 around 100â•›MHz. The low-frequency field Hz(t) = H0(t)â•›cos(ω0t)ez, applied along the easy-axis, couples to the spin of the molecular magnet through the Hamiltonian

Vlow = g µ B H 0 (t )cos(ω 0t )Sz ,

(1.65)

Energy

|5

ћω5 |6

ћω6 π

|7

σ– ћω7

|8

ћω8

|9 ћω0

ћω9 |10

FIGURE 1.4â•… Feynman diagrams F that contribute to Sm(5,)s for s = 10 and m 0 = 5 describing transitions (of fifth order in V) in the left well of the spin system (see Figure 1.5). The solid and dotted arrows indicate σ− and π transitions governed by Equations 1.66 and 1.65, respectively. We note that Sm( j,)s = 0 for j < n, and Sm( j,)s n.

where ω 0 n. Using rectangular pulse shapes, Hk(t) = Hk, if −T/2 < t < T/2, and 0 otherwise, for k = 0 and k ≥ m0, one obtains (m ≥ m0)

n

∏

k =m

H k e −iΦk H 0m − m0 pm, s (F )

(−1)qF qF !rs (F )! ωn0 −1

s −1

∑ω

s −1

k

k=m

 − (m − m0 )ω 0  , 

(1.68)

where Ωmâ•›=â•›(mâ•›−â•›m 0)!, is the symmetry factor of the Feynman diagrams F (see Figure 1.4), qF = m − m0 − rs (F ), pm, s (F ) =

− t k +1 )

× U (∞, t1 )V (t1 )U (t1 , t 2 )V (t 2 )…V (t j )U (t j , −∞),

(n ) m, s

2π  g µ B  Ωm i  2 

∏

s k =m

〈 k | Sz | k 〉

rk ( F )

,

rk (F ) = 0, 1, 2, …, ≤ m − m0

is the number of π transitions directly above or below the state |k〉, depending on the particular Feynman diagram F , and δ (T ) (ω) =

1 2π

∫

+T /2

−T / 2

eiωt dt = sin(ωT/2)/πω is the delta-function

of width 1/T, ensuring overall energy conservation for ωT >> 1. The duration T of the magnetic pulses must be shorter than the lifetimes τd of the states |m〉 (see Figure 1.5). In general, the magnetic field amplitudes Hk must be chosen in such a way that perturbation theory is still valid and the transition probabilities are almost equal, which is required by the Grover algorithm. According to Leuenberger and Loss (2001), the amplitudes Hk do not differ too much between each other due to the partial destructive interference of the different transition diagrams shown in Figure 1.5. Leuenberger et al. (2002) show that the transition probabilities can be increased by increasing both the magnetic field amplitudes and the detuning energies under the condition that the magnetic field amplitudes remain smaller than the detuning energies. In this way, both high-multiphoton Rabi oscillation frequencies and small quantum computation times can be attained. This makes both methods (Leuenberger

1-12


and Loss, 2001; Leuenberger et al., 2002) very robust against decoherence sources. In order to perform the Grover algorithm, one needs the relative phases φm between the transition amplitudes Sm(n,)s , which is determined by Φ m = Σmk =+s1−1Φ k + ϕm, where Φm are the relative phases between the magnetic fields Hm(t). In this way, it is possible to read-in and decode the desired phases Φm for each state |m〉. The read-out is performed by standard spectroscopy with pulsed ESR, where the circularly polarized radiation can now be incoherent because only the absorption intensity of only one pulse is needed. We emphasize that the entire Grover algorithm (read-in, decoding, read-out) requires three subsequent pulses each of duration T with τ d > T > ω 0−1 > ω m−1 > ω m−1, m ±1. This gives a “clock-speed” of about 10â•›GHz for Mn12, that is, the entire process of read-in, decoding, and read-out can be performed within about 10−10 s. The proposal for implementing Grover’s algorithm works not only for molecular magnets but for any electron or nuclear spin system with nonequidistant energy levels, as is shown by Leuenberger et al. (2002) for nuclear spins in GaAs semiconductors. Instead of storing information in the phases of the eigenstates |m〉 (Leuenberger and Loss, 2001), Leuenberger et al. (2002) use the eigenenergies of |m〉 in the generalized rotating frame for encoding information. The decoding is performed by bringing the delocalized state (1/ n )Σ m | m〉 into resonance with |m〉 in the generalized rotating frame. Although such spin systems cannot be scaled arbitrarily, large spin s (the larger a spin becomes, the faster it decoheres and the more classical its behavior will be) systems of given s can be used to great advantage in building dense and highly efficient memory devices. For a first test of the nonlinear response, one can irradiate the molecular magnet with an a.c. field of frequency ωs−2,s /2, which gives rise to a two-photon absorption and thus to a Rabi oscillation between the states |s〉 and |sâ•›−â•›2〉. For stronger magnetic fields, it is in principle possible to generate

superpositions of Rabi oscillations between the states |s〉 and |s − 1〉, |s〉 and |s − 2〉, |s〉 and |s − 3〉, and so on (see also Leuenberger et al., 2002).

1.8â•‡ Semiconductor Quantum Dots In the following, we mainly focus on quantum dots made of III–V semiconductor compounds with zincblende structure, like GaAs or InAs. The electronic bandstructure of a three-dimensional semiconductor with zincblende structure is illustrated in Figure 1.6. The bands are parabolic close to their extrema, which are all located at the Γ point. The conduction (c) states have orbital s symmetry and are spin degenerate. The valence (v) band consists of three subbands: the heavy-hole (hh), the light-hole (lh), and the split-off (so) band. The v-band states have orbital p symmetry. The bottom of the c band and the top of the v band are split by the band-gap energy Egap. The v-band states with different j ( j = 12 for the so-band, j = 23 for the hh and lh band) are split by Δso in energy due to spin–orbit interaction. The hh states have the angular momentum projections J z = ± 23 and the lh states J z = ± 12 . For finite electron wavevectors k ≠ 0, and the hh and lh subbands split into two branches according to the different curvatures of the energy dispersion, which implies different effective masses of heavy and light holes. The v-band states with spin can be written in terms of the orbital angular momentum basis by using the Clebsch–Gordon coefficients which gives us  3 , 3 = 1,1 ↑  2 2 Heavy hole  3 3  2 , − 2 = 1, −1 ↓

 | 3 , 1〉 =  2 2 Light hole  3 1 | 2 , − 2 〉 =

1 3 2 3

  , 

| 1, 0〉 |↑〉  , | 1, 0〉 |↓〉 + 13 | 1, −1〉 |↑〉 

| 1,1〉 |↓〉 +

2 3

(1.69)

(1.70)

E E

c

c

Egap

Egap K||

Δhh–lh

K||

Δso

(a)

so

lh

hh

(b)

so

lh

hh

FIGURE 1.6â•… Electronic band structure in the vicinity of the Γ point for (a) a three-dimensional crystal and (b) a quantum well. The conduction and valence bands are shown as a function of the wavevector.

1-13


 | 1 , 1〉 =  2 2 Split off  1 1 | 2 , − 2 〉 =

1 3 2 3

2 3

Quantum confinement along the crystal axis quantizes the wavevector component, consequently the hh and lh states of the lowest subband are split by an energy Δhh−lh at the Γ point. Uniaxial strain in the semiconductor crystal can also lift the degeneracy of the heavy and light holes, and thus define the spin quantization axis. If we have a spherically symmetric quantum dot known as colloidal quantum dots, then we can have degeneracy between the heavy and light hole band, i.e., Δhh−lh = 0. Via photon absorption, an electron in a v-band state can be excited to a c-band state. Such interband transitions are determined by optical selection rules. The source or the optical transition rules are due to spin–orbit interaction. The electron–hole pair created with an interband transition is called an exciton. The electron and hole of an exciton form a bound state due to the Coulomb interaction, similar to that of a hydrogen atom. We refer to the system of two bound excitons as a biexciton.

1.8.1â•‡ Classical Faraday Effect Michael Faraday first observed the effect in 1845 when studying the influence of a magnetic field on plane-polarized light waves. Light waves vibrate in two planes at right angles to one another, and passing ordinary light through certain substances eliminates the vibration in one plane. He discovered that the plane of vibration is rotated when the light path and the direction of the applied magnetic field are parallel. In particular, a linearly polarized wave can be decomposed into right and left circularly polarized waves where each wave propagates with different speeds. The waves can be considered to recombine upon emergence from the medium; however, owing to the difference in propagation speed, they do so with a net phase offset, resulting in a rotation of the angle of linear polarization. The Faraday effect occurs in many solids, liquids, and gases. The magnitude of the rotation depends upon the strength of the magnetic field, the nature of the transmitting substance, and Verdets constant, which is a property of the transmitting substance, its temperature, and the frequency of light. The relation between the angle of rotation of the polarization and the magnetic field in a diamagnetic material is

β

| 1,1〉 |↓〉   . (1.71) | 1, −1〉 |↑〉 + 13 | 1, 0〉 |↓〉 

| 1, 0〉 |↑〉 +

β = VHd,

(1.72)

where β is the angle of rotation V is the Verdet constant for the material H is the magnitude of the applied magnetic field d is the length of the path where the light and magnetic field interact (see Figure 1.7)

H

ν d E

FIGURE 1.7â•… The figure shows how the polarization of a linearly polarized beam of light rotates when it goes through a material exposed to an external magnetic field.

A positive Verdet constant corresponds to an anticlockwise rotation when the direction of propagation is parallel to the magnetic field and to a clockwise rotation when the direction of propagation is antiparallel. The Faraday effect is used in spintronics research to study the polarization of electron spins in semiconductors.

1.8.2â•‡ Quantum Faraday Effect The Faraday effect is expected to emerge in low dimensional systems such as semiconductor quantum dots in which the spin states of the electron in the conduction band and the light and heavy hole in the valence band provide a system where different circular polarizations of light couple differently during the process of virtual absorption. The quantum analogue of the Faraday effect does not require an external magnetic field because it is created by selection rules (one circular polarization interacts with the heavy hole band while the other circular polarization interacts with the light hole band) and by the Pauli exclusion principle (the absorption of a right polarized wave is excluded because the allowed transition state between bands have the same spin) (Leuenberger et al., 2005b). In particular, we will be interested in the quantum Faraday effect in a semiconductor colloidal twolevel quantum dot system. We can have a two-level system in a colloidal quantum dot where the heavy hole and the light hole bands are degenerate at the Γ point. This two-level system is achieved by the valence and the conduction band under certain assumptions: (1) the split-off band can be ignored since typical split-off energies are around 102 meV, thus bringing the energy level out of resonance with the single photon; (2) under the appropriate doping and thermal conditions, it can be assumed that the top of the valence band is filled with four electrons, while there is one excess electron in the conduction band; and (3) the energy of the electromagnetic wave is taken to be slightly below the effective band-gap energy, so that the transition from the top of the valence band to the bottom of the conduction band is the strongest transition by far (Leuenberger, 2006).

1-14

Handbook of Nanophysics: Nanoelectronics and Nanophotonics 1,– 1 2 2

1,+1 2 2

hh

σ+ . e iSo

σ+

lh

σ– . e iSo

σ–

(a)

3,– 3 2 2

3,– 1 2 2

3,+ 1 2 2

1, + 1 2 2

1, – 1 2 2

3,+ 3 2 2

σ+

σ+ . e iSo

σ–

σ– . e iSo

hh

lh

3, – 1 2 2

3, – 3 2 2

(b)

3, + 1 2 2

3, + 3 2 2

FIGURE 1.8â•… The solid circles represent filled states and dashed circles represent empty states. The electron in the conduction band interacts with the right or left circularly polarized electromagnetic wave allowing virtual transitions between the conduction and valence band states. After the interaction, the RCP and LCP electromagnetic wave acquire different phases, which produce the rotation of the incident wave.

Interestingly, the idea of the conditional single-photon Faraday rotation first developed by Leuenberger et al. (2005b) and already patented by the authors has been copied (Hu et al., 2008), which demonstrates the importance of this scheme. We now turn our focus to the optical selection rules. In Figure 1.8 we show how different circular polarizations of the photon interact with the two-level system. The conduction band state has one electron with spin either up (| 12 , 12 〉) or down (| 12 , − 12 〉). The presence of a spin up (down) electron in the conduction band forces the virtual transitions of the incident electromagnetic wave to couple | 23 , − 23 〉 (| 23 , − 12 〉) or | 23 , 12 〉 (| 23 , 23 〉) heavy hole and light hole states depending if it is a right or left circularly polarized photon. Examination of the coefficients given in Equations 1.69 and 1.70 shows that the matrix elements of the heavy hole band and the matrix element involving the light hole band are different, leading to different phases and causing a change in the refractive index, which induces the Faraday rotation effect. The Faraday rotation is clockwise if the spin is up, and counterclock̰ wise if the spin is down. The complex refractive index (η) can be calculated as follows:

2 2 = 1 − e N η 0me

∑ (ω i, j

2

f ji , − ω 2ji ) + i γω

(1.73)

where |e| is the electron’s charge N is the electron number density ϵ0 is the permittivity of free space me is the mass of the electron ωji is the transition frequency from the ith state to the jth state γ is the full-width at half maximum (assumed to be much smaller than the detuning energy) ω is the frequency of the incident electromagnetic wave fji is the oscillator strength given by

f ji =

2 | 〈 j | (−e)r ⋅ E | i 〉 |2 . mω ji

(1.74)

This quantum Faraday rotation can be used to measure the spin polarization of charge carriers in quantum wells (Kikkawa et al., 1997; Kikkawa and Awschalom, 1998, 1999). Recently, an experiment based on the quantum Faraday rotation was able to measure the spin state of a single excess electron inside a quantum dot (Berezovsky et al., 2006), which is crucial for the read-out of a qubit in a quantum computing scheme (see below).

1.9â•‡ Single-Photon Faraday Rotation The single-photon Faraday effect (SPFE) is similar to the quantum Faraday effect but involves only the nonresonant interaction of a single photon with the quantum dot two-level system and can result in an entanglement of the photon with the spin of the extra electron (Leuenberger et al., 2005b; Leuenberger, 2006). The SPFE can be described by the Jaynes–Cummings Hamiltonian

(

)

H = ω aσ† + aσ+ + aσ† − aσ− + ω hh σ 3 v + ω hh σ − 3 v 2

2

+ω lh σ 1 v + ω lh σ − 1 v + ω e σ 1 c + ω e σ − 1 c + g 3 v , 1 c

(

2

2

2

2

2

2

× a σ 3 v , 1 c + aσ − σ 1 c , 3 v + a σ − 3 v , − 1 c + aσ + σ − 1 c , − 3 v

† σ−

2

2

(

2

2

† σ+

2

2

2

2

)

)

+ g 1 v , 1 c aσ† − σ 1 v , − 1 c + aσ − σ − 1 c , 1 v + aσ† + σ − 1 v , − 1 c + aσ+ σ 1 c , − 1 v . 2

2

2

2

2

2

2

2

2

2

(1.75)

1-15


Equation 1.76 describes how the right circularly polarized (σ+) and left circularly polarized (σ−) components of a linearly polarized photon field interact with the degenerate quantum dot levels in a cavity. It is assumed that the spin of the excess electron in the conduction band is up (↑). The evolution of the system will be given by the state vector

hh

1 C↑ hh (t ) |↑,hh 〉 + C↑ σ+ (t ) |↑, σ z+ 〉 2  + C↑ lh (t ) |↑, lh 〉 + C↑ σ− (t ) |↑, σ z− 〉  ,

(1.76)

where |↑,â•›hh〉 and |↑,â•›lh〉 are the states in which the quantum dot is in an excited state with a heavy-hole exciton or a light-hole exciton, and the spin of the excess electron in the conduction band is up. |↑, σ +z 〉 and |↑, σ z− 〉 are the states in which the quantum dot is in the ground state with the photon present in the cavity, and the spin of the extra electron in the conduction band is up. In order to solve for the phase accumulated for RCP and LCP components of the linearly polarized photon during the interaction with the quantum dot in the nanocavity, we must find the time evolution of the probability amplitudes C↑σ+(t) and C↑σ−(t), respectively. Assuming the following initial conditions, C↑hh(0) = 0, C↑σ+(0) = 1, C↑lh(0) = 0, and C↑σ−(0) = 1, then the probability amplitudes of interest are given by (Seigneur et al., 2008)

  Ω 3 t  i∆  Ω 3 t  C↑σ+ (t ) = e −i∆t /2 cos  2  + sin  2    2    2  Ω 3   2    Ω 1 t  i∆  Ω 1 t  C↑ σ− (t ) = e −i∆t /2 cos  2  + sin  2   ,  2    2  Ω 1   2 

(1.77)

(1.78)

where Ω23 = ∆ 2 + 4 g 23 v , 1 c and Ω21 = ∆ 2 + 4 g 21 v , 1 c , with g 32 v , 12 c and 2 2 2 2 2 2 g 1 v , 1 c being the coupling strength involving the heavy-hole elec2 2 tron and the light-hole electron, respectively, with Δ = ω − ωph being the detuning frequency and g 23 v , 12 c = 3 g 12 v , 12 c . Rewriting the complex coefficients given in Equations 1.77 and 1.78 in the form C↑ σ+ (t ) = | C↑ σ+ (t ) | exp[i S hh 0 ] and C↑ σ − (t ) = | C↑ σ− (t ) | exp[i S lh0 ], we obtain an expression for the phase accumulated during the interaction of the right and left circularly polarized component with the heavy-hole and the light-hole band, i.e.,  ∆  ∆  Ω 3t    Ω 1t   S0hh = tan −1  tan  2   and S lh0 = tan −1  tan  2   .  Ω3  Ω1  2    2    2  2 (1.79)

To determine the entanglement of the single photon with the electron, we know that the electron–photon state after the interaction reads

(1.80)

where

(

)

1 α |↑〉 | σ(+z ) 〉 + β |↓〉 | σ(−z ) 〉 2

| ψ (hh1) 〉 =

| ψ (t )〉 =

lh

| ψ (ep1) 〉 = eiS0 | ψ (hh1) 〉 + eiS0 | ψ (lh1) 〉,

(1.81)

represents the photon scattering off a heavy hole, and

(

)

1 α |↑〉 | σ(−z ) 〉 + β |↓〉 | σ(+z ) 〉 2

| ψ (lh1) 〉 =

(1.82)

represents the photon scattering off a light hole. Let | ϕ〉 = cos ϕ |↔〉 + sin ϕ |〉

(1.83)

be the photon state with linear polarization that is rotated by φ around the z-axis with respect to the state |↔〉 of linear polarization in x direction. Changing to circular polarization states, Equation 1.83 can be rewritten as | ϕ〉 = =

(

)

(

cos ϕ sin ϕ | σ(+z ) 〉 + | σ(−z ) 〉 + | σ(+z ) 〉 − | σ(−z ) 〉 2 i 2

(

)

1 −iϕ + e | σ(z ) 〉 + eiϕ | σ(−z ) 〉 . 2

)

(1.84)

Using the representation of this linear polarization with ϕ = ±(S0hh − S0lh )/2 = ± S0 /2 , we find that Equation 1.10 becomes

(

hh

lh

| ψ (ep1) (T )〉 = α |↑〉 eiS0 | σ(+z ) 〉 + eiS0 | σ(−z ) 〉

(

hh

lh

)

+ β |↓〉 eiS0 | σ(+z ) 〉 + eiS0 | σ(−z ) 〉

=e

(

)

i S0hh + S0lh / 2

2

)

2

(α |↑〉 | −S /2〉 + β |↓〉 | +S /2〉). 0

0

(1.85)

From this equation, we draw the conclusion that the accumulated CPHASE shift S 0 corresponds to a single-photon Faraday rotation around the z-axis by the angle S 0/2 due to a single spin. Let j be an integer. For S0/2 = (2j + 1)π/4, the electron–photon state | ψ (ep1) 〉 is maximally entangled. If j is even,

| ψ (ep1) 〉 = α |↑〉 |〉 + β |↓〉 |〉.

(1.86)

| ψ (ep1) 〉 = α |↑〉 |〉 + β |↓〉 |〉.

(1.87)

If j is odd,

1-16


where

E

k

FIGURE 1.9â•… The conditional Faraday rotation of the linear polarization of the photon depends on the spin state |ψe〉 = α |↑〉 + β |↓〉. If the spin state is up (down), the linear polarization turns (counter) clockwise. The photon and the spin get maximally entangled if the Faraday rotation angle is S0/2 = π/4. This entanglement between photon polarization and electron spin can be used as an optospintronic link to transfer quantum information from electron spins to photons and back.

For S0/2 = 2jπ/4, the electron–photon state | ψ (ep1) 〉 is not entangled. If j is even,

| ψ (ep1) (T )〉 = α |↑〉 |↔〉 + β |↓〉 |↔〉.

| ψ (T )〉 = α |↑〉 |〉 − β |↓〉 |〉.

(1.88)

(1.89)

This means the entanglement oscillates continually between 0 and 1 if no decoherence is present. The entanglement process between the spin of the electron and the polarization of the photon during the conditional Faraday rotation is visualized in Figure 1.9.

1.9.1â•‡ Quantifying the EPR Entanglement

S S   + β |↓〉  cos 0 |↔〉 + sin 0 |〉 .  2 2 

E[ψ (1) ep ] = − Tre[Tr p ρ ep log 2 (Trp ρep )]

(1.90)

(1.91)

or the normalized linear entropy (Silberfarb and Deutsch, 2004)

2 E L[ψ (1) ep ] = 2[1 − Tre (Trp ρep ) ].

(1.92)

We will use the linear entropy in our calculation. The density matrix is given by

 ρ↑↑ ρep =   ρ↓↑

ρ↑↓ , ρ↓↓ 

2αβ* sin S0  , −αβ* sin2 S20 

 α*β cos2 S0 2 ρ↓↑ =   2α*β sin S0

−2α*β sin S0  , −α*β sin2 S20 

 | β |2 cos2 S20 ρ↓↓ =  2  2 | β | sin S0

2 | β |2 sin S0  , | β |2 sin2 S20 

(1.95)

(1.96) (1.97)

 | α |2 cos2 S0 2  2  −2 | α | sin S0  S  α*β cos 2 20  *  2α β sin S0

−2 | α |2 sin S0 | α |2 sin2

S0 2

αβ* cos 2 S20 −2αβ* sin S

0

−2α *β sin S0 −α *β sin2 S0 2

2

| β | cos

2 S0 2

2 | β |2 sin S0

2αβ* sin S0   −αβ* sin 2 S20   2 | β |2 sin S0   | β |2 sin2 S20 

(1.98)

 | α |2 Trp ρep =   α*β cos S0

αβ* cos S0 .  | β |2

(1.99)

So we obtain the entanglement

The entanglement between the photon and the spin of the excess electron can be calculated by means of the von Neumann entropy (Bennett et al., 1996):

 αβ* cos2 S0 2 ρ↑↓ =   −2αβ* sin S0

(1.94)

in the basis |↑〉 |↔〉; |↑〉 |↕〉; |↓〉 |↔〉; |↓〉 |↕〉. Since the measures for entanglement must be independent of the chosen basis, all of the measures involve the trace of ρep. Taking the trace over the photon states yields

Rewriting Equation 1.85 in the basis states of the linear polarization in x and y directions yields S S   ψ (ep1) = α |↑〉  cos 0 |↔〉 − sin 0 |〉  2 2 

−2 | α |2 sin S0  , | α |2 sin2 S20 

and thus by ρep =

If j is odd, (1) ep

 | α |2 cos2 S20 ρ↑↑ =  2  −2 | α | sin S0

(1.93)

(

)

4 4 2 2 2 EL[ψ (1) ep ] = 2 1− | α | − | β | −2 | α | | β | cos S0 .

(1.100)

With the parametrization α = cos η2 e − iχ /2 , β = sin η2 e − iχ /2 we obtain

η 1   4 η EL[ψ (1) − sin 4 − sin2 η cos2 S0  . (1.101) ep ] = 2  1 − cos   2 2 2

This result is plotted in Figure 1.10.

1.9.2â•‡Single Photon Faraday Effect and GHZ Quantum Teleportation The theory of quantum teleportation was first developed by Bennett et al. (1993) and later experimentally verified by Bouwmeester et al. (1997). This scheme relies on EPR pairs,

1-17


(

1 2

1) | ψ (epe ′ (t C + T )〉 =

α |↑〉 eiS0hh | σ + 〉 |↑′ 〉 y (z ) 

+ eiS0 | σ(−z ) 〉 |↓′y 〉 lh

(

1

lh + β |↓〉 eiS0 | σ(+z ) 〉 ↑′y

EL

0.75 0.5 0.25

0

0

1 η

0

2

S0

|ψ

FIGURE 1.10â•… Entanglement of the photon with the spin of the excess electron.

i.e., pairs of entangled particles, which can be entangled in, for example, the polarization degree of freedom in the case of photons or the spin degree of freedom in the case of electrons or holes. Leuenberger et al. (2005b) show that all the three particles that take part in the teleportation process are entangled in a so-called Greenberger–Horne–Zeilinger (GHZ) state (Greenberger et al., 1989, 1990). We will perform now a GHZ teleportation. In this GHZ quantum teleportation, the photon (qubit 2) interacts first with the electron spin of the destination (qubit 3) and then with the electron spin of the origin (qubit 1). We identify qubit 1, 2, and 3 with the qubits used by Bennett et al. (1993). Qubit 1 is with Alice, qubit 2 with Charlie, and qubit 3 with Bob. We start from the state | Ψpe(1)′ (t A )〉 = |↔〉 |← ′ 〉. After interaction in Bob’s microcavity, we obtain | Ψpe(1)′ (t C )〉 =

=

(

1 | σ +z 〉 |↑′y 〉 + | σ(−z ) 〉 |↓′y 〉 2 1 (|〉 |〉 |↓ ′ 〉). 2

(1) epe ′

(T )〉 =

e

(

)

i S0hh + S0lh / 2

2

+ β |↓〉| + S0 / 2 − π / 4〉|↑′〉

+ β |↓〉| + S0 / 2 + π / 4〉|↓′ 〉 .

(1.104)

Choosing S 0 = π/2, we obtain 1) | ψ (epe ′ (T )〉 =

1 2

(

|〉 −α |↑〉 |↑′ 〉 + β |↓〉 |↓′ 〉 

(

)

)

+ |↔〉 α |↑〉 |↓′ 〉 + β |↓〉 |↑′ 〉  . 

(1.105)

[From this equation, it becomes obvious that we can produce all the four Bell states between qubit 1 (Alice’s electron spin) and qubit 3 (Bob’s electron spin).] Now we change to the Sx representation of Alice’s spin: 1) | ψ (epe ′ (T )〉 =

(

1 |〉 |←〉 −α |↑′ 〉 + β |↓′ 〉  2

(

)

)

+ |→〉 −α | ↑′〉 − β |↓′ 〉  

(1.102)

+ This hybrid photon–spin entangled state corresponds to the shared EPR state of qubit 2 and 3 in the original version of the teleportation (EPR teleportation). Note that there is no way to distinguish between a single-spin or a single-photon Faraday rotation because we do not have any preferred basis defined by α and β. We have not yet used qubit 1 at all. Alice’s photon can be stored for as long as it maintains its entanglement with Bob’s spin. This step is like the distribution of the EPR pair in EPR teleportation. Instead of performing now a Bell measurement on qubit 1 and 2 to complete the EPR teleportation, we let the photon (qubit 2) interact with Alice’s spin (qubit 1), giving rise to a GHZ state in the hybrid spin–photon–spin system. After interaction of the photon in Alice’s microcavity, we obtain

(α|↑〉 | − S0 / 2 − π /4〉|↑′〉

+ α |↑〉| − S0 / 2 + π /4 〉|↓′〉

)

(1.103)

Now we change to the Sz representation of Bob’s spin and to the linear polarization representation of Charlie’s photon, which yields (tC = 0)

–2

3

)

hh + eiS0 | σ(−z ) 〉 |↓′y 〉  . 

2

)

(

1 |↔〉 |←〉 β | ↑′〉 + α |↓′ 〉  2

(

)

+ |→〉 β | ↑′〉 − α |↓′ 〉  . 

)

(1.106)

The difference between our method and the original version of the teleportation (Bennett et al., 1993) is that we have entangled qubit 2 (the photon) with qubit 1 (Alice’s electron spin) by means of the electron–photon interaction, which is not done in the original version. That is why we do not need Bell measurements. So after measuring the photon polarization state (qubit 2) and the spin state of Alice’s electron (qubit 1), the spin state of the electron of the destination gets projected onto −α|↑′〉 + β|↓′〉, −α|↑′〉 − β|↓′〉, β|↑′〉 + α|↓′〉, or β|↑′〉 − α|↓′〉 with equal probability. This corresponds exactly to

1-18


the outcome of the original version of the teleportation. However, we do not use any Bell measurements. Let us check what happens if there is no interaction between Charlie’s photon and Alice’s spin. Then we get 1) | ψ (epe ′ (T )〉 =

1 2

(

|〉 α |↑〉 |↑′ 〉 + β |↓〉 |↑′ 〉 

(

)

r1A

r1B Path 2

)

+ |〉 α |↑〉 |↓′ 〉 + β |↓〉 |↓′ 〉  . 

Photon detectors

Path 1

r2A

r2B Path 3

r3A

(1.107)

In this case, we would have to apply a Bell measurement in order to perform the teleportation.

1.9.3â•‡Single-Photon Faraday Effect and Quantum Computing Recently, the destructive probabilistic CNOT gate for all-optical quantum computing proposed by Knill et al. (2001), which relies on postselection by measurements, has been implemented experimentally (O’Brien et al., 2003; Nemoto and Munro, 2004). Based on a proposal by Pittman et al. (2001), experiments have demonstrated that this destructive CNOT gate and a quantum parity check can be combined with a pair of entangled photons to produce a nondestructive probabilistic CNOT gate (Pittman et al., 2002a,b, 2003; Gasparoni et al., 2004; Zhao et al., 2005). It has been shown theoretically that deterministic quantum computing with postselection is feasible for quantum systems where the qubits are represented by several degrees of freedom of a single photon (Cerf et al., 1998). An interferometric approach to linear-optical deterministic CNOT gate between the polarization and momentum of a single photonic qubit has recently been demonstrated experimentally (Fiorentino and Wong, 2004). A general scheme to teleport a quantum state through a universal gate has been given by Gottesman and Chuang (1999). This idea gave Raussendorf and Briegel the inspiration to develop the one-way quantum computer (Raussendorf and Briegel, 2001; Raussendorf et al., 2003), which performs quantum computations by measurement-induced teleportation processes on an entangled network of qubits. The name one-way quantum computing refers to the irreversibility of the computations due to the measurement processes. Leuenberger developed a scheme for fault-tolerant quantum computing with the excess electron spins in quantum dots based on the SPFE (Leuenberger, 2006). This scheme shares a similarity with the one-way quantum computing scheme in the sense that it makes use of measurement processes to perform quantum computations. Both the one-way quantum computing scheme and Leuenberger’s scheme are fully deterministic and scalable to an arbitrary number of qubits. However, while the oneway quantum computing scheme is irreversible, Leuenberger’s scheme is fully reversible. In addition, the one-way quantum computing scheme needs to maintain the coherence of the whole cluster of qubits and is therefore more sensitive to the environment than single isolated qubits, since in a cluster the decoherence of one qubit leads also to decoherence of neighboring qubits

Single-photon sources

r3B Path 4

r4A Alice’s quantum dots

r4B Bob’s quantum dots

FIGURE 1.11â•… The nonresonant interaction of a photon with Alices and Bobs quantum dot produces the CPHASE gate required for universal quantum computing. This CPHASE gate applies to the case of the noncoded spins, where each microcavity contains only a single quantum dot with a single excess electron.

that are entangled to it. Therefore, Leuenberger’s scheme is more robust in this respect. We describe now Leuenberger’s quantum computing scheme (see Figure 1.11). DiVincenzo showed that the CNOT gate and a single qubit rotation are sufficient for universal quantum computing (DiVincenzo, 1995). In the framework of the conditional Faraday rotation, it is easier to work with the implementation of the CPHASE gate than with the implementation of the CNOT gate. The CPHASE gate

uCPHASE

1 0 = 0 0 

0 1 0 0

0 0 1 0

0 0  0 −1

(1.108)

is equivalent (similar) to the CNOT gate (see Equation 1.7), i.e., they can be transformed into each other by means of a basis transformation. The CPHASE gate shifts the phase only if both spins point down. Let us define two persons Alice and Bob. Both of them have one photonic crystal, in which n noninteracting quantum dots are embedded. The single excess electrons of Alice’s quantum dots are in a general single-spin state |ψ〉A = α|↑〉A + β|↓〉A, where the quantization axis is the z-axis. Bob’s spins are in a general single-spin state |ψ〉B = γ|↑〉B + δ|↓〉B. The photons that interact with both Alice’s and Bob’s quantum dots are initially in a horizontal linear polarization state |↔〉. Since there is no need to detect optically the spin states in transverse direction (Leuenberger et al., 2005a), our quantum dots can be nonspherical. So each photon can virtually create only a heavyhole exciton on each quantum dot (see Figure 1.6). The strong selection rules imply that only a σ(+z ) (σ(−z ) ) photon can interact with the quantum dot if the excess electron’s spin is up (down). This leads to a conditional Faraday rotation of the linear polarization of the photon depending on the spin state of the quantum dot. The photonic crystal ensures that the photon’s propagation direction is always in z-direction, perpendicular to the quantum dot plane.

1-19


to reconstruct the desired two-qubit state, which makes sense, because we perform a two-qubit manipulation, as opposed to a single qubit manipulation in the case of quantum teleportation. Thus, we can produce deterministically the CPHASE gate, which is equivalent to the CNOT gate in the basis {â•›|↑〉A |←〉B,

In Leuenberger’s implementation of the CPHASE gate, a single photon interacts sequentially with Alice’s and then with Bob’s quantum dot. The spin on Alice’s quantum dot is prepared in the state |ψA(0)〉 = α|↑〉A + β|↓〉A. So we start with the electron–Â� photon state |ψAp(0)〉 = (α|↑〉A + β|↓〉A)|↔〉. After the interaction with Alice’s quantum dot, the resulting electron–photon state is

|↑〉A |→〉B, |↓〉A |←〉B, |↓〉A |→〉Bâ•›}, where |←〉 = (|↑〉 + |↓〉)/ 2 and

| ψ Ap (T )〉 = (α |↑〉 A |〉 + β |↓〉 A |〉)/ 2 ,

|→〉 = (|↑〉 − |↓〉)/ 2 in the Sx representation. This can be proven by writing down the mappings

(1.109)

which is maximally entangled. We let the photon interact also with Bob’s quantum dot, which yields

(

| ψ ApB (2T )〉 = |〉 −αγ |↑〉 A |↑〉 B + βδ |↓〉 A |↓〉 B

)

(

+ |↔〉 αδ |↑〉 A |↓〉B + βγ |↓〉 A |↑〉 B =

)

→

(

→ (1.110)

| ψ AB (2T )〉 = −αγ |↑〉A |↑〉B + βδ |↓〉A |↓〉B + αδ |↑〉A |↓〉B + βγ |↓〉A |↑〉B ,

+ αδ |↑〉 A |↓〉 B + βγ |↓〉 A |↑〉 B

)

(1.113)

)

1 |↓〉 A |↑〉B + |↓〉B = |↓〉 A |←〉B , 2

(

)

)

1 |↓〉 A |↑〉B − |↓〉B = |↓〉 A |→〉B , 2

(

)

corresponding to the CPHASE gate in Equation 1.111, and

(1.111) |↑〉 A |←〉B =

| ψ AB (2T )〉 = αγ |↑〉 A |↑〉 B − βδ |↓〉 A |↓〉 B

)

(

(

→

)

1 |↑〉A − |↑〉B − |↓〉B = − |↑〉 A |←〉B , 2

1 |↓〉 A |↑〉B − |↓〉B 2

|↓〉A |→〉B =

If the linear polarization of the photon is measured in the ⤡ axis, we obtain the two results

(

(

+ |〉 αγ |↑〉 A |↑〉B − βδ |↓〉 A |↓〉B

)

(

1 |↓〉 A |←〉B = |↓〉 A |↑〉B + |↓〉B 2

)

)

1 |↑〉 A − |↑〉B + |↓〉B = − |↑〉 A |→〉B , 2

1 |↑〉 A |↑〉B − |↓〉B 2 →

(

+ αδ |↑〉 A |↓〉B + βγ |↓〉 A |↑〉B  . 

(

|↑〉 A |→〉B =

1  |〉 −αγ |↑〉 A |↑〉 B + βδ |↓〉 A |↓〉 B 2 + αδ |↑〉 A |↓〉 B + βγ |↓〉 A |↑〉 B

1 |↑〉 A |↑〉 B + |↓〉 B 2

|↑〉 A |←〉 B =

(1.112)

with equal probability, where the CPHASE shift is π if both spins are up or both spins are down, respectively. For deterministic quantum computing, we need to choose either Equation 1.111 or 1.112 to be the correct phase gate. Let us choose Equation 1.112 to be the correct implementation. The measurement of the photon’s polarization state must be shared through a classical channel between Alice and Bob, in order for the CPHASE gate to be deterministic. Then Equation 1.111 can be transformed into Equation 1.112 by two local single-qubit phase shifts σA,zσB,z applied to Alice’s and Bob’s qubit, where σz is a Pauli matrix. This scheme for implementing the CPHASE gate is similar to the scheme of quantum teleportation in the sense that the measurement outcome of the photon’s polarization needs to be shared between Alice and Bob through a classical channel. This scheme for implementing the CPHASE gate is a generalization of the scheme of quantum teleportation in the sense that it takes not only one local unitary operation but two unitary operations

1 |↑〉 A |↑〉B + |↓〉B 2

(

→ |↑〉 A |→〉 B =

1 |↑〉 A |↑〉 B + |↓〉 B = − |↑〉 A |←〉 B , 2

(

1 |↑〉 A |↑〉 B − |↓〉 B 2

(

→

)

(1.114)

)

1 |↓〉 A |↑〉B − |↓〉B = |↓〉 A |→〉B , 2

(

1 |↓〉 A |↑〉B − |↓〉B 2

(

→

)

(

(

|↓〉 A |→〉B =

)

1 |↑〉 A |↑〉 B − |↓〉 B = |↑〉 A |→〉B , 2

1 |↓〉 A |←〉B = |↓〉 A |↑〉B + |↓〉B 2 →

)

)

)

1 |↓〉 A |↑〉B + |↓〉B = |↓〉 A |←〉B , 2

(

)

corresponding to the CPHASE gate in Equation 1.112.

1-20


For universal quantum computing, the CPHASE gate described above and single-qubit operations are sufficient (Barenco et al., 1995; DiVincenzo, 1995). The single-qubit operations on Alice’s and Bob’s spins can be implemented by means of the optical Stark effect (Gupta et al., 2001). In contrast to Imamoglu et al.’s two-qubit gate (Imamoglu et al., 1999), which relies on a shared mode volume for the exchange of a single virtual photon between the two qubits, this scheme requires only local mode volumes for each qubit separately and is therefore fully scalable to an arbitrary number of qubits. Since the strong interaction between a quantum dot and a cavity inside a photonic crystal has already been experimentally observed (Yoshie et al., 2004), it would be possible to build a fully scalable quantum network inside a photonic crystal hosting qubits, which can be for example spins of excess electrons in semiconductor quantum dots or N-V center states in diamond.

1.10â•‡ Concluding Remarks Quantum computers have excited many researchers because they would perform a kind of parallel processing that would be extremely effective for certain tasks, such as searching databases and factoring large numbers. Also, quantum computers can be used to simulate or model quantum systems and could bring huge advances in physics, chemistry, and nanotechnology. Recently, diamond has become very attractive for solid state electronics. Pure diamond is an electrical insulator, but on doping, it can become a semiconductor. The particular impurity that researchers are interested in is the nitrogen-vacancy (N-V) center. The N-V center has a number of properties and characteristics that make it very promising to build a quantum computer (Awschalom et al., 2007). The N-V center electrons have a spin state that is extremely stable against environmental disturbances. One of the most exciting aspects of the N-V center is that it exhibits quantum behavior even at room temperature; this characteristic is very important because it makes this kind of systems easy to study and easy to turn into a practical technology. Other important aspects of N-V centers come from the fact that they have weak spin–orbit interaction and dipole– dipole interaction, which make the spin state very stable and it can be used to encode quantum information even at room temperatures. The decoherence time of the N-V center spin is about 1â•›ms and the operation time is about 10â•›ns, therefore R = τd /τs = 100,000 operations that can be performed in the millisecond lifetime of the spin quantum system. This rate of decay is well below the threshold and better than any other system of solidstate qubits to date.

Acknowledgments We acknowledge support from NSF-ECCS 0725514, the DARPA/ MTO Young Faculty Award HR0011-08-1-0059, NSF-ECCS 0901784, and AFOSR FA9550-09-1-0450. We would also like to thank Sergio Tafur for his comments on improving the manuscript.

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Zhao, Z., A.-N. Zhang, Y.-A. Chen, H. Zhang, J.-F. Du, T. Yang, and J.-W. Pan. Experimental demonstration of a nondestructive controlled-not quantum gate for two independent photon qubits. Phys. Rev. Lett., 94:030501, 2005.

2 Nanomemories Using Self-Organized Quantum Dots 2.1 2.2

Introduction.............................................................................................................................. 2-1 Conventional Semiconductor Memories.............................................................................. 2-2

2.3

Nonconventional Semiconductor Memories.......................................................................2-4

2.4 2.5 2.6 2.7

Martin Geller University of Duisburg-Essen

Andreas Marent Technische Universität Berlin

Dieter Bimberg Technische Universität Berlin

2.8

Dynamic Random Access Memoryâ•‡ •â•‡ Nonvolatile Semiconductor Memories (Flash) Ferroelectric RAMâ•‡ •â•‡ Magnetoresistive RAMâ•‡ •â•‡ Phase-Change RAM

Semiconductor Nanomemories..............................................................................................2-6 Charge Trap Memoriesâ•‡ •â•‡ Single Electron Memories

A Nanomemory Based on III–V Semiconductor Quantum Dots..................................... 2-7 III–V Semiconductor Materialsâ•‡ •â•‡ Self-Organized Quantum Dotsâ•‡ •â•‡ A Memory Cell Based on Self-Organized QDs

Capacitance Spectroscopy..................................................................................................... 2-10 Depletion Regionâ•‡ •â•‡ Capacitance Transient Spectroscopy

Charge Carrier Storage in Quantum Dots......................................................................... 2-16 Carrier Storage in InGaAs/GaAs Quantum Dotsâ•‡ •â•‡ Hole Storage in GaSb/GaAs Quantum Dotsâ•‡ •â•‡ InGaAs/GaAs Quantum Dots with Additional AlGaAs Barrierâ•‡ •â•‡ Storage Time in Quantum Dots

Write Times in Quantum Dot Memories........................................................................... 2-21 Hysteresis Measurementsâ•‡ •â•‡ Write Time Measurements

2.9 Summary and Outlook..........................................................................................................2-23 Acknowledgments..............................................................................................................................2-23 References............................................................................................................................................2-24

2.1â•‡ Introduction One of the fundamental achievements of today’s information society is providing storage and processing of ever-increasing amounts of data that is based on basic research in physics in combination with fundamental technology developments over the last four decades. As no low-cost universal memory that is suited for all kinds of applications exists, digital data storage is separated mainly into three different lines:

1. The digital versatile disk (DVD) and the compact disk (CD) are nonvolatile storage media based on an optical write and read-out process using laser light at wavelengths of 650 and 780â•›nm, respectively. The newest representative of optical data storage is the Blu-ray disk, using a laser wavelength at 405â•›nm with a maximum capacity of up to 50â•›GB disk−1. 2. The hard-disk drive in a personal computer is a nonvolatile mass storage device with a capacity of up to 1000â•›GB in production year 2008 that uses the magnetization of rapidly rotating platters.

3. Semiconductor memories are the third line of digital storage media and will be the focus of this chapter with a perspective of future nanomemories.

Semiconductor memories can be divided into two groups: volatile memories, like the dynamic random access memory (DRAM), and nonvolatile memories, the so-called Flash. The DRAM presents the main working memory in a personal computer. Flash memories are found, for example, in mp3 players, cell phones, and memory sticks; in automobiles and microwave ovens; and have started to replace the hard-disk drive in notebooks as well as the DVD and CD as easily portable high-capacity nonvolatile memories. Driven by the increasing demand for such portable electronic applications, where nonvolatile data storage with low power consumption is needed, the market for Flash memories is growing rapidly and is replacing the DRAM as the market driver. Up to now, the semiconductor memory industry improved the storage density and performance while simultaneously reducing the cost per information bit just by scaling down the feature size. This leads to an exponentially growing number of 2-1

2-2


components on a memory chip as predicted by Moore in 1965 (Moore 1965) known nowadays as “Moore’s Law.” Contrary to all predictions, Moore’s Law has held remarkably well over the last decades. The feature size for Flash memories has reached 45â•›n m in 2008. The International Roadmap for Semiconductors (ITRS 2008) predicts a further shrinkage down to 14â•›n m in 2020. The problems encountered during this shrinking process were mainly solved by developing new materials, for example, for isolation and interconnects, and for new types of cell structures. Upon further size reduction, quantum mechanics will dominate at least some of the physical properties. In addition, the amount of technological difficulties to realize such structures increases enormously. Therefore, a considerable effort is devoted to the search for alternative memory technologies that are even based on completely new non-semiconducting materials. Phase change access memories (PCRAMs), magnetic random access memories (MRAMs), and ferroelectric RAM (FeRAMs) are just three alternative memory concepts that are explored for having the potential to replace today’s DRAM and/or Flash memory. One completely different technology is the use of selforganized nanomaterials in future nanoelectronic devices. Especially self-organized quantum dots (QDs) based on III–V materials (e.g., GaAs, InAs, InP, etc.) provide a number of important advantages for new generations of nanomemories. Billions of self-organized QDs can be formed simultaneously and fast in a single technological step, allowing for massive parallel production in a bottom-up approach, and offering an elegant method to create huge ensembles of electronic traps without lithography. They can store just a few or even single charge carriers with a retention time depending on the material combination—potentially up to many years at room temperature. With an area density of up to 1011 cm−2, an enormous storage density in the order of 1 TBit per square inch could be possible, if each single QD would represent one information bit and could be addressed individually. The carrier capture process into the QDs is of the order of pico to sub-picoseconds, an important prerequisite for a very fast write time in such memories. The use of self-organized QDs could thus lead to novel nonvolatile memories with high storage density combined with a fast read/write access time. Using self-organized QDs for future semiconductor memory applications is discussed in this chapter. Section 2.2 gives a brief overview of the main semiconductor memories, DRAM and Flash, while the following Section 2.3 concentrates on three nonconventional memories that may replace Flash or DRAM in the future. Section 2.5 presents an alternative nanomemory concept that is based on III–V materials, especially self-organized QDs. By using capacitance spectroscopy, introduced in Section 2.6, the carrier storage time in different QD systems is studied in Section 2.7 and a storage time of seconds at room temperature is demonstrated. Finally, the chapter will show results on fast write times in QD-based nanomemories in Section 2.8 and will close with a summary and outlook in Section 2.9.

2.2â•‡Conventional Semiconductor Memories Presently the semiconductor memory industry focuses essentially on two memory types: the DRAM (Mandelman et al. 2002, Waser 2003) and the Flash (Geppert 2003). Both memory concepts have their advantages and disadvantages in speed, endurance, storage time, and cost. A memory concept that adds the advantages of a DRAM to a Flash would combine high storage density, fast read/write access, long data storage time, and good endurance with low production cost. In addition, for portable applications like mobile phones and mp3 players, low power consumption is demanded. This section describes the two conventional semiconductor memories, while Section 2.3 will focus on nonconventional alternatives that have the potential to replace DRAM and/or Flash.

2.2.1â•‡ Dynamic Random Access Memory Since the invention of the DRAM in the late 1960s (Dennard 1968), its cell structure mainly stayed the same, consisting of a transistor and a capacitor (cf. Figure 2.1a). The capacitor stores the information by means of electric charge. The charge state is defined by the voltage level on the capacitor. The stored charge Bit-line

Bit-line

Word-line Transistor

Transistor

Capacitor

Capacitor

Word-line Transistor

Transistor

Capacitor

Capacitor

(a) Word-line gate Bit-line source n+ (b)

Inversion region Drain

Storage gate Oxide

p

p+

FIGURE 2.1â•… (a) Schematic picture of an array of DRAM cells, where the capacitors act as the storage units. Each cell can be addressed individually by a matrix of word- and bit-lines; it is the so-called randomaccess architecture. (b) Cell layout for a planar DRAM cell.

2-3

Nanomemories Using Self-Organized Quantum Dots

disappears typically in a few milliseconds mainly due to leakage and recombination currents. A DRAM is volatile and requires periodic “refreshing” of the stored charge. After any read process, the information has to be rewritten into the DRAM cell. A chip on the integrated circuit controls the refresh rewrite process automatically and this continuous read and write operation has led to the name “Dynamic.” Figure 2.1b shows schematically the layout of a simple planar DRAM cell (Sze 2002). The storage capacitor uses the inversion channel region as one plate, the storage gate as the other plate, and the gate oxide as the dielectric. The access transistor is a metal oxide semiconductor field effect transistor (MOSFET) with a source, drain, and a gate contact. The drain contact is connected to the storage capacitor. The source contact is connected to the bit-line and the gate contact to the word-line. Via word- and bit-line, a single cell can be addressed in an organized matrix of DRAM cells (Figure 2.1a). This so-called “random access” provides short access times independent of the location of the data (Waser 2003). For a write/read operation, a voltage pulse is passed through the selected bit- and word-line. Only at the crossing point of a given DRAM cell, the access transistor switches to “open” and the capacitor is charged or discharged for a write or read event, respectively. DRAM cells provide fast read, write, and erase access times below 20â•›ns in combination with a very good endurance of more than 1015 write/erase cycles. The endurance is defined as the minimum number of possible write/erase operations until the memory cell is destroyed. DRAM memory cells draw power continuously due to the refresh process and the information is lost after switching off the computer. Another disadvantage is the relatively large number of electrons, presently in the order of 105, needed to store one information bit, leading additionally to permanent high power consumption. Both, the volatility and the power consumption make DRAMs unsuitable for mobile applications. The goal to shrink the feature size, based on the assumption of scalability, of a DRAM (Kim et al. 1998) down to 14â•›nm by 2020 is a major challenge. It is speculated that the leakage currents of the capacitor might inhibit the scalability very soon. A fixed capacitance of the order of 50â•›pF is needed to maintain a sufficiently high voltage signal during the read-out process. Shrinking the area of the capacitor will decrease its capacitance,

Electron energy

Oxide barrier

2.2.2â•‡Nonvolatile Semiconductor Memories (Flash) Nonvolatile memories (NVMs) can retain their data for typically more than 10 years without power consumption. The most important nonvolatile memory is the Flash-EEPROM (electrically erasable and programmable read-only memory); in short just Flash (Geppert 2003, Lai 2008). A Flash offers the possibility of repeated electrical read, write, and erase processes. The Flash memory market is the fastest growing memory market today, since it is the ideal memory device for portable applications. In a mobile phone, it holds the instructions and data needed to send and receive calls, or stores phone numbers. But not only portable applications are ideal for Flash. In each computer, a Flash chip holds the data on how to boot up. Other electronic products of all types, from microwaves ovens to industrial machines, store their operating instructions in Flash memories. Flash is based on a floating-gate structure (Pavan et al. 1997, Sze 1999), where the charge carriers are trapped inside a polysilicon floating gate embedded between two SiO2 barriers (Figure 2.2a). The SiO2 barriers having a height of ∼3.2â•›eV and an average thickness of 10â•›nm guarantee a storage time of more than 10 years at room temperature. However, these barriers are also the origin of the two main disadvantages of a Flash cell: a slow write time (in the order of microseconds) and a poor endurance (in the order of 106 write/erase cycles). The write process is realized by means of “hot-electron injection.” Here a small voltage is applied between the source contact and the bit-line (drain contact) (cf. Figure 2.2b). The word-line (control gate) is set to a high positive bias of 10–20â•›V, the MOSFET is set to “open” and the electrons flow though the inversion channel from the source to drain and are, in addition, accelerated in the direction of the floating-gate due to the high electric field in the order of 107 V cm−1. These “hot-electrons” have sufficient kinetic energy to reach the floating gate over one of the SiO2 barriers, but destroy this barrier step by step by

Floating gate

Gate

Drain

Source

Oxide barrier Inversion channel

3.2 eV n+

Gate (a)

Hot electron injection

the number of stored electrons per DRAM cell, and the amplitude of the read-out signal. Downscaling increases leakage currents due to quantum-mechanical tunneling through thinner dielectrics (Frank et al. 2001). Therefore, there is a search for novel dielectrics.

Floating gate

Ec

n+ p

(b)

FIGURE 2.2â•… (a) Schematic band structure of a Flash memory based on Si as matrix material and SiO2 for the oxide barriers. (b) Schematic cell layout of a floating-gate Flash memory.

2-4


Memory DRAM Flash

Write

Read

~10 ns

~10 ns

~ms

>10,000

>1015

µs–ms

~20 ns

>10 years

~1,000

~106

Storage

Electrons Endurance

FIGURE 2.3â•… Comparison of DRAM and Flash.

creating defects leading eventually to leakage. This is the reason for the poor endurance of a Flash memory cell. The slow write time is due to the low probability for energy relaxation of these hot-electrons into the floating gate. The read-out of the stored information is normally done by measuring the resistance of the two-dimensional electron gas (2DEG) of the MOSFET—the inversion channel. A bias is applied at the gate contact, forming inversion between the source and the drain. Stored electrons in the floating gate reduce the conductivity of the 2DEG and a higher resistance between the source and the drain is measured. Figure 2.3 compares the main properties of the DRAM and Flash. The Flash memory has already replaced the DRAM as a technology driver for the semiconductor memory industry (Mikolajick et al. 2007) in the year 2003. It is leading in storage density and the main innovations are coming from the Flash industry. The feature size of the Flash has decreased from 1.5â•›μm in the year 1990 down to 45â•›nm in the year 2008. Accordingly, the number of electrons decreased from about 105 to less than 1000 per information bit (Atwood 2004) (cf. Figure 2.4). A further shrinkage to 14â•›nm in the year 2020, as predicted, would lead to only 100 electrons per information bit. Flash memory scaling beyond 32â•›nm feature size will become more and more difficult, mainly due to the physical limitations (Atwood 2004). For instance, during the hot-electron injection a voltage of about 4.5â•›V is applied between the source and the drain. Shrinking the gate length while keeping the write voltage

Electrons per cell

105

1990 Year of production

104 2008

103 102

2020

101 1

10

Self-organized quantum dots

100 Feature size (nm)

1000

FIGURE 2.4â•… Electron number per Flash memory cell versus the minimum feature size. The feature size decreases from 1.5â•›μ m in the production year 1990 down to 45â•›n m in the year 2008. Accordingly, the number of electrons decreased from about 10 5 to less than 1000 per information bit.

fixed at 4.5â•›V, leads to an increased electric field in the source– drain channel and in the end to a punch-through. Another limitation for future scaling is the capacitive coupling between different floating gates (cross talk) and the scalability of the floating gate. While the number of electrons decreases to about 200 per cell in year 2020, the main charge leakage mechanisms will remain. Dangling bonds and other defects at the Si/SiO2 interface will dramatically decrease the storage time while scaling to smaller cell sizes.

2.3â•‡Nonconventional Semiconductor Memories The increasing number of difficulties and challenges for further scaling of the DRAM and Flash has led to a continuous search for alternative memory concepts. A large variety of proposed concepts using different physical phenomena to store an information bit have been reported (Burr et al. 2008). This section concentrates on three nonconventional memories, which are the most advanced ones (Geppert 2003, Burr et al. 2008): the ferroelectric RAM (FeRAM), the magnetic RAM (MRAM), and the phase-change RAM (PCRAM).

2.3.1â•‡ Ferroelectric RAM The ferroelectric RAM (FeRAM) (Jones et al. 1995, Sheikholeslami and Gulak 2000) has almost the same structure as a DRAM cell, except for the capacitor. The dielectric inside the capacitor of a DRAM is a non-ferroelectric material like silicon dioxide. When the charge is stored on the metal plates of the capacitor, it leaks away into the silicon substrate within a few milliseconds, causing the nonvolatility of a DRAM cell. In a FeRAM, a ferroelectric film such as zirconate titanate, also known as PZT, replays the dielectric of the capacitor. This ferroelectric material has a remnant polarization that occurs when an electric field has been applied. In PZT the center atom is zirconium or titanium which can be moved by an external electric field into two different stable states, representing a “0” or “1” in a ferroelectric memory. One state is near the top face of the PZT cube, the other one is near the bottom face. Even after removal of the external electric field, these stable states store the information for more than 10 years. When an opposite electric field is applied, the dipoles flip to the opposite direction, that is, the zirconium or titanium atoms are switching into the other stable state. The write operations of a “0” and “1” state in a FeRAM are in principle the same as in a DRAM cell. An electric pulse via a word and a bit-line switches the ferroelectric state of the PZT. To read an information bit, an electrical pulse is applied via the access transistor to the ferroelectric capacitor and a sense amplifier can measure a current pulse. The amplitude depends on the position of the zirconium or titanium atoms in the PZT cube. For instance, if the external electric field from the pulse is pointing in the same direction of the PZT, that is, the atoms

2-5


are already at the top of the cube and a smaller current pulse is detected than for opposite direction. Reading the information in a FeRAM cell destroys the data stored in its ferroelectric capacitor. After a read operation, the information has to be rewritten into the FeRAM cell, which is a major disadvantage in comparison to the Flash memory based on a floating-gate structure. However, the FeRAM is nonvolatile having a random read/write access time below 50â•›ns and an almost unlimited endurance (>1012 write/erase cycles).

2.3.2â•‡ Magnetoresistive RAM The magnetoresistive RAM (Gallagher and Parkin 2006, Wolf et al. 2006, Burr et al. 2008) uses the giant magnetoresistance (GMR) effect that was discovered in the late 1980s by the groups of Albert Fert (Baibich et al. 1988) and Peter Grunberg (Binasch et al. 1989). Grunberg and Fert received the Nobel Prize in Physics 2007 for their discovery. The MRAM is based on a magnetic tunnel junction (MTJ) (see Figure 2.5a), which consists of two magnetic layers separated Free ferromagnetic layer

Bit line

Magnetic tunnel junction

Insulator

Digit line

Fixed ferromagnetic layer

(a) Read-out process Magnetic field

Write current

Magnetic field (b) Write process

FIGURE 2.5â•… Schematic picture of the read (a) and write (b) process of an MRAM cell. During read out a current passes through the MTJ. If the magnetization of the two magnetic layers is parallel to each other a low resistance is measured; if the magnetization is antiparallel the resistance is high. The write process is based on current flow through both the bit and digit line. The sum of both currents is strong enough to flip the magnetic domains inside the free magnetic layer to a “1” or “0” state, depending on the current direction inside the lines.

by a thin insulating layer (like AlOx) with a thickness in the order of 1â•›nm. One of the ferromagnetic layers has a fixed magnetization, while the other layers can flip its magnetization by an external writing event. The read-out of the information in a MRAM cell works as follows: if the access transistor for a certain cell is turned on, a current is driven through the MTJ and the resistance of the MRAM cell is measured. If the two magnetic moments are parallel, the resistance is low, representing a “0” state. If the moments are antiparallel, the resistance is high, representing a “1” state. The difference between these two values can be up to 70%, therefore, GMR. The physical effect that leads to this GMR effect is called tunnel magnetoresistance (TMR). The TMR effect can be understood in terms of spin polarized tunneling of the electrons. The electron spins are polarized inside one magnetic layer and if the spin is conserved during tunneling through the thin insulator layer, an initially spin up electron can only tunnel to a spin up final state. If the magnetic layers have parallel magnetic moments, a higher current is detected than for antiparallel directions. The write mechanism can be understood by looking at Figure 2.5b. During the write mode, a current is passing through two wires: a bit-line that runs over the MTJ and a digit line that is below. The sum of both currents is strong enough to flip the magnetic domains inside the free magnetic layer to a “1” or “0” state, depending on the current direction. Both magnetic states are stable for more than 10 years; hence, an MRAM is a nonvolatile memory device with a random-access architecture and unlimited endurance. In addition, the write/read time has been demonstrated to be below 50â•›ns. However, scaling MRAM to smaller feature sizes is a big challenge as the write current has to remain very high (>1â•›mA), even if the feature size goes below 40â•›nm. Such large currents in very small devices might damage the structure.

2.3.3â•‡ Phase-Change RAM The architecture of the phase-change RAM (PCRAM)—also called ovonic unified memory (OUM)—is again an array of access transistors that are connected to a word- and a bit-line. The capacitor is now replaced by a phase-change material (Geppert 2003, Burr et al. 2008). The PCRAM uses the large resistance change between a (poly) crystalline and an amorphous state in a chalcogenide glass, which is also used in rewritable CDs and DVDs. The chalcogenides used for this type of memory are alloys containing elements like selenium, tellurium, or antimony (GeSbTe, GeTe, Sb2Te3, etc.). The crystalline and amorphous states show a large difference of up to five orders of magnitude in the electrical resistance, representing the binary state “1” for low resistance and the state “0” for high resistance. The read-out can easily be done by a measurement of the resistance of the PCRAM cell. Besides the chalcogenide, the PCRAM cell consists of a top and a bottom electrode that are connected to the word- and bit-line and a resistive heater below the phase-change material (see Figure 2.6). To switch the state of the cell to the crystalline

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Handbook of Nanophysics: Nanoelectronics and Nanophotonics Bit line

2.4â•‡ Semiconductor Nanomemories

Chalcogenide alloy

The semiconductor memory industry has already entered the “nano-world” years ago and has introduced a feature size of 45â•›nm in the year 2008. To improve the device performance of a conventional Flash cell, nanomaterials have been used as replacements for the floating gate. For instance, replacing the Si floating gate with Si nanocrystals—Si particles with diameters in the order of 1–10â•›nm—leads to the advantage that charge leakage from any particular nanocrystal does not discharge the complete floating gate. The ultimate goal would be the usage of only one nanocrystal as one information bit to build a single electron memory. This section will briefly review semiconductor memories based on charge traps and silicon nanocrystals in Section 2.4.1 and a single electron memory based on silicon QD in Section 2.4.2.

Programmable volume (polycrystalline or amorphous) Resistor (heater)

Word line

FIGURE 2.6â•… The PCRAM is based on a chalcogenide that can exist in two different states: in the (poly) crystalline state the resistance is low (representing a “1”), while the resistance is high in the amorphous state (representing a “0”). Switching to the amorphous state is done by heating the programmable volume up to its melting point with the resistive heater and cooling down rapidly. To make it crystalline, the heater is switched on for a short time period (∼50â•›ns) to heat the volume above its crystallization temperature.

one, a short write pulse is applied that heats the programmable volume just below its melting point and holds it there for a certain time. This SET operation limits the write speed of the PCRAM because of the required duration to crystallize the phase-change material. The write speed is in the order of tens of nanoseconds (∼50â•›ns), depending on the used material. In the RESET operation, the memory cell is switched to the amorphous “0” state by applying a larger electrical current (∼mA) in the order of 100â•›ns that heats up the phase-change material just above the melting point. The PCRAM access time is presently longer than for the MRAM. However, the write/erase time improves with scaling, as the active programmable volume is getting smaller and shorter write/erase pulses are sufficient to switch the state of the phasechange material. The read operation shows a very good scaling that gives PCRAM the highest market potential among all nonconventional memory concepts that could replace the Flash memory in the future. A big disadvantage is the endurance of a PCRAM cell (107–1012), being above the conventional Flash cell but several orders of magnitude away from the DRAM (>1015). Control gate

Word-line

2.4.1â•‡ Charge Trap Memories One of the major drawbacks is the limited number of write/erase cycles of a conventional Flash cell, the poor endurance. This will be a major problem during scaling down the feature size and simultaneously the thickness of the SiO2 tunneling barrier. A single defect in the tunneling oxide will always destroy the entire Flash cell. The simplest way to improve the endurance of a conventional Flash memory and, hence, to extend the scalability is to replace the floating gate by a charge trapping material, schematically illustrated in Figure 2.7. A charge trapping memory cell can be realized in different approaches. One possibility is to use a dielectric material that stores the charges in deep traps. Silicon nitride is the most established among such materials and the memory cell is often referred to as oxide–nitride–oxide, short just ONO (Lai 2008). A variation of this type of charge trapping device is the SONOS (silicon–oxide–nitride–oxide–silicon) (White et al. 2000). Figure 2.7a illustrates the floating-gate structure and the SONOS device principle. The thin silicon-nitrite (Si3N4) film is the storage unit where a single defect in the oxide layer will not discharge the complete memory cell. In addition, SONOS cells show a reduced cross talk and lower read/write voltages. A second alternative to create a charge trapping memory is the usage of nanoparticles as the floating gate. The first attempts of such memory devices were made by Lambe and Jaklevic

Silicon nitride

Silicon nanocrystals

Drain Oxide bit-line Source

(a)

n+

n+

n+

p

(b)

n+

p

FIGURE 2.7â•… (a) Schematic picture of a SONOS nonvolatile charge trapping memory that consists instead of a Si floating gate of an oxide–nitride– oxide (ONO) sandwich. The nitride layer can store charges (electrons or holes) in deep traps. (b) A nanocrystal Flash based on silicon nanoparticles.

2-7


(1969) 40 years ago. They used an array of Al droplets—with a diameter in the order of 10â•›nm—embedded in thin oxide and observed a memory effect in capacitance measurements. Tiwari et al. (1996) embedded Si nanocrystals into a floating-gate structure to improve the Flash concept with nanomaterials. The principle of such memory devices is depicted in Figure 2.7b. The Si nanocrystals are fabricated through spontaneous decomposition during chemical vapor deposition onto the tunneling oxide barrier. These particles have a typical size in the order of about 1–10â•›nm. Replacing the floating gate with these Si nanocrystals has the advantage that charge leakage through defect inside the oxide layer will not completely discharge the whole memory cell. Thinner tunneling barriers can be used and an improved endurance is observed.

2.4.2â•‡ Single Electron Memories One charge carrier inside a single nanocrystal or QD for one information bit is the ultimate goal for a memory cell. Such devices were realized with a single polysilicon QD embedded into an oxide matrix (Guo et al. 1997). This device showed a storage time of 5â•›s at room temperature and is schematically depicted in Figure 2.8. It was fabricated by electron beam lithography (EBL) and reactive ion etching (RIE). A source–drain channel with a width between 25 and 120â•›nm was created onto an oxide layer, followed by a second oxide layer and the polysilicon dot. The dot had a width of about 20â•›nm and is covered again by silicon oxide and polysilicon as a gate electrode. By an appropriate bias on the control gate, the single dot is charged and discharged at room temperature and the charge state—the read-out of the information—was possible via a resistance measurement of the source–drain channel. This measurement demonstrates perfectly the possibility of single charge storage and read-out even at room temperature without conductance quantization inside the source–drain channel. However, these structures were fabricated via EBL, which is not suitable for mass production and the storage time is limited to 5â•›s at 300â•›K. ~20 nm Polysilicon dot

Polysilicon control gate

Silicon channel

Oxide layer Substrate

FIGURE 2.8â•… Schematic picture of single electron memory device based on a polysilicon QD above a silicon channel. The silicon channel is connected to a source and drain contact to measure the resistance of the channel. The charge state can be written and erased by the polysilicon control gate. (After Guo, L. et al., Appl. Phys. Lett., 70(7), 850, 1997.)

2.5â•‡A Nanomemory Based on III–V Semiconductor Quantum Dots In this section, a memory concept based on III–V nanomaterials—self-organized QDs—that has the potential to overcome the drawbacks of the current conventional Flash and DRAM is presented (Geller et al. 2006a). Such a nanomemory cell should provide long storage times (>10 years) and good endurance (>1015 write/erase cycles) in combination with even better read/write access time than the DRAM ( 0 V p+

F

i

Capture

δ-Doping Emission barrier

Capture barrier

Ug < 0 V F

EC Emission barrier

EC

Tunneling emission

Quantum dot

QW with 2DEG (a) Storage and read-out

p+ i

(b) Writing a “1”

EC (c) Erase-writing a “0”

FIGURE 2.13â•… Schematic illustration of the (a) storage and read-out, (b) write, and (c) erase process in a possible future QD-based nanomemory.

2.5.3â•‡A Memory Cell Based on Self-Organized QDs Here, a memory concept is presented where the self-organized QDs are embedded in a p–n or p–i–n diode structure (Geller et al. 2006a). The QDs act as storage units, since they can be charged with electrons or holes representing the “0” (uncharged QDs) and “1” (charged QDs) of an information bit. An emission barrier is needed to store a “1” in such a memory concept. The barrier height—which is related to the localization energy of the charge carrier (Figure 2.13a)—can be varied by varying the material and size of the QDs and the material of the surrounding matrix (see Section 2.5.1). If the localization energy is increased, a longer storage time of the charge carriers is expected. Furthermore, a capture barrier is needed to store a “0” (Figure 2.13a). This barrier protects an empty QD cell from unwanted capture of charge carriers. In this concept, the capture barrier is realized by using the band bending of a p–i–n or p–n diode. The major advantage in using a diode structure is the possibility to tune the height of the barriers by an external bias. Therefore, during the write process (Figure 2.13b), the capture barrier can be eliminated by a positive external bias between the gate and source contact (Vg > 0â•›V) and the charge carriers can directly relax into the QD states. This allows to benefit from another advantage of self-organized QDs: the charge carrier relaxation time into the QD states is in the range of picoseconds at room temperature (Müller et al. 2003, Geller et al. 2006b) (see Section 2.7), enabling very fast write times in a QD-based memory. In addition, a very good endurance of 1015 write operations should be feasible. To erase the information, the electric field is increased within the QD layer by a negative external bias between the gate and source (Vg < 0â•›V), such that tunneling of the charge carriers occurs (cf. Figure 2.13c). Figure 2.14 shows schematically the device structure of such a QD-based memory, where the QDs are charged with holes to represent an information bit. The doping sequence of the p- and n-regions would be inverted for electron storage. The distance to the junction is adjusted, such that the QDs are inside the depletion region for zero bias (Vg = 0â•›V in Figure 2.13a). The read-out of the stored information is done by a two-dimensional electron gas (2DEG) for an electron and a two-dimensional hole gas

Gate contact p–n junction

(Al)(Ga)As n+ doped

Source

(Al)(Ga)As undoped

Drain

QW with 2DEG

(In)(Ga)Sb QDs

δ-doping (Al)(Ga)As undoped

Substrate

FIGURE 2.14â•… Schematic picture of the layer sequence for a possible QD-based memory. The QDs are located below the p–i–n junction and a 2DEG is placed below the QD layer to detect the charge state.

(2DHG) for a hole storage device. The 2DEG or 2DHG is situated 10–50â•›nm below the QD layer and is filled with charge carriers, provided by the additional n- or p-δ-doping, respectively. Stored charge carriers inside the QDs reduce the conductivity of the 2DEG/2DHG; hence, a higher resistance is measured between the source/drain contacts, in analogy to the read-out in a conventional Flash cell.

2.6â•‡ Capacitance Spectroscopy To elucidate the potential of nanomaterials like self-organized QDs for future memory applications the localization energy and the carrier storage time at room temperature has to be determined for different III–V heterostructures. The first goal is to find a material combination that yields a minimum storage time of milliseconds at 300â•›K, the benchmark for present DRAM. The capacitance spectroscopy is a powerful tool to study the energy levels and charge carrier emission times in QDs. It has

2-11


been widely used for different device geometries, for instance in p–i–n diodes to investigate the tunneling dynamics and localization energies in self-organized InAs/GaAs QDs (Fricke et al. 1996, Luyken et al. 1999) or to probe single-electron levels in QDs that are laterally confined inside a 2DEG (Ashoori et al. 1992, Ashoori 1996). In this section, capacitance spectroscopy is used to determine storage and emission times as well as localization energies in self-organized QDs. The QDs are embedded in a GaAs matrix nearby an abrupt p–n junction that forms a depletion region. This different capacitance spectroscopy method consists of measurements of the depletion capacitance of a p–n or Schottky diode, it is the “capacitance spectroscopy of a depletion capacitance” (Lang 1974).

2.6.1â•‡ Depletion Region The work function or Fermi energy with respect to the vacuum level of a metal or doped semiconductor differs for different metals or semiconductors having a different doping concentration. If a metal and a semiconductor (or two differently doped semiconductors—a p–n diode) are in electric contact, the free charge carriers are exchanged between the different materials until a thermodynamic equilibrium is reached and the Fermi energy is equal throughout the entire structure. Sometimes only ionized donors and acceptors are present in the vicinity of the interface while all free charge carriers are moved away in the semiconductor (Figure 2.15). The layer depleted from free charge carriers is usually referred to as the “depletion region” (Sze 1985, Blood and Orton 1992). The width of the depletion region depends on the doping concentration and the potential difference between the materials, of which the latter can easily be modified by an externally applied bias. Electron energy

eVs

eVb

s

φm

Vacuum level

Fixed donors

EFm Metal

2.6.1.1â•‡ Schottky Contact A metal-semiconductor contact is usually described in the framework of the Schottky model, which is an acceptable approach to construct the band diagram of the contact (Figure 2.15). Within this model, the barrier height ϕb inside the metal is independent or just weakly dependent on the applied external bias. According to the Schottky model, the energy band diagram is constructed by reference to the vacuum level, defined as the energy of a free electron to rest outside the material. The work function of the metal ϕm and the electron affinity of the semiconductor χs are defined as the energies required to remove an electron from the Fermi level of the metal or the semiconductor conduction band edge, respectively, to the vacuum level. These properties are assumed constant in a given material and it is further assumed that the vacuum level is continuous across the interface. The Fermi levels in the metal and semiconductor must be equal in thermal equilibrium. These conditions result in a band diagram for the interface as shown in Figure 2.15. Since the vacuum level is the same at the interface of the metal and the semiconductor, a step between the Fermi level in the metal and the conduction band edge EC of the semiconductor occurs. This is the barrier height ϕb given by*

+

Free electrons ++ ++ ++++ ++++++++++++++++++ w

EC EF

n-type semiconductor EV

Depletion region

Neutral region

FIGURE 2.15â•… Energy band diagram of a depleted metal-semiconductor Schottky contact. The width of the depletion region w depends on the doping concentration in the semiconductor. A build-in voltage Vb is present without external bias due to the band bending of the depletion region.

φb = φm − χs − eVm .

(2.1)

The band bending in the metal is very small due to the large density of electron states; hence, eVm can be neglected. Therefore, the Schottky barrier height is usually written as

eVm

φb

A metal-semiconductor contact can be described by the Schottky model and is referred to as the “Schottky contact.” A junction of a p-doped and n-doped semiconductor is a “p–n junction.” Both types of contacts provide a depletion region and they are briefly described in the following.

φ b = φm − χ s .

(2.2)

For increasing distance from the interface, the conduction band energy decreases. At the end of the depletion region, it has the same value as in the neutral semiconductor with respect to the Fermi level. The resulting band bending is an effect of the removed free charge carriers, leaving behind a distribution of fixed positive charges from ionized donors. The depletion region ends at that position, where the bands become flat and the associated electric field is zero. The width of the depletion region w is determined by the net ionized charge density according to Poisson’s equation (see Section 2.6.1.3). Since the density of states in the metal is much greater than the doping density in the semiconductor, the depletion width in the metal is much smaller. Therefore, it can be assumed that the potential difference across the metal near the contact (Vm) is negligibly small compared to that in the semiconductor (Vs). * For a Schottky barrier, ϕm must exceed χs otherwise the bands bend in the opposite direction.

2-12


The total zero bias band bending of the Schottky contact, also referred to as “build-in potential” or “build-in voltage” Vb can be written as

where ni is the intrinsic carrier density

eVb ≈ eVs = φm − χs − (EC − EF ) = φb − (EC − EF ).

(2.3)

Experimental values for various metal Schottky contacts on GaAs can be found in Myburg et al. (1998).

Vb =

 Eg  ni = N C N V exp  − ,  2kBT 

2.6.1.2â•‡ p–n Junction The band diagram of an abrupt p–n junction, in Figure 2.16, is considered in a similar manner. Again, two rules are used to construct the diagram: (1) the vacuum level is continuous and (2) the Fermi energy is constant across the junction in thermal equilibrium. For the same p- and n-doped semiconductor the electron affinity χs is the same on both sides of the junction. Therefore, the band bending is caused entirely by the difference in the Fermi level with respect to the conduction band of the two differently doped materials. From Figure 2.16, using subscripts to denote the n and p side, one obtains for zero external bias

(ECp − EF ) + χs = eVb + χs + (ECn − EF ),

(2.4)

hence, the build-in voltage is

eVb = Eg − (EF − EVp ) − ( ECn − EF ),

(2.5)

where E g is the energy gap of the semiconductor. The build-in voltage as the total electrostatic potential difference between the p-side and the n-side is temperature-dependent and a function of the fixed donor and acceptor charges of density Nd and Na (Sze 1985):

kBT  N a N d  ln  2  , e  ni 

N a w p = N dwn

eVb Vacuum level

FIGURE 2.16â•… Energy band diagram of a p–n junction.

–

–––––––––––––– wp

++

++++

–

––––

p-Type

–– –

Free holes

Fixed donors ++

(2.8)

must hold. For similar doping concentrations, the depletion width on the p- and n-side of the semiconductor will be comparable. However, for the purpose of material characterization (such as capacitance experiments) doping concentrations are often chosen such that the depletion region is situated almost entirely on one side of the junction. The depletion region of such an asymmetrical p–n junction resembles the depletion region of a Schottky contact. Such asymmetrical junctions are briefly denoted as p+–n or n+–p junction with Na >> Nd or Nd >> Na, respectively.

s

+

(2.7)

with the effective density of states in the valence NV and conduction band NC, respectively. A typical doping density of 1017 cm−3 yields for GaAs a build-in voltage of 1.3â•›V at 300â•›K. In a p–n junction, depletion regions on each side of the contact exist, where the fixed donor and acceptor charges lead to the band bending. Since the total charge in the p–n diode is zero (charge neutrality), in the depletion approximation (Blood and Orton 1992) of an abrupt depletion layer edge

Electron energy

Fixed acceptors

(2.6)

Depletion region w

s

Free electrons

+ + + + + + + + + + + + + + EC EF Eg

wn

n-Type

EV

2-13


2.6.1.3â•‡ Width of the Depletion Region The entire band bending across the depletion region is defined by the sum of the build-in voltage of the contact Vb (Equation 2.5 or 2.6), and the applied external bias Va in reverse direction: V = Vb + Va. The width of the depletion region and the electric field can be calculated using Poisson’s equation. The electrostatic potential ψ at any point is given by −

∂ 2ψ ∂F ρ(x ) = = , ∂x 2 ∂x εε0

(2.9)

where F is the electric field ε0 is the vacuum permittivity ε is the dielectric constant of the semiconductor material If the donors or acceptors are entirely ionized, the charge density ρ is eNd or eNa, respectively: −

∂ 2ψ eN d = εε0 ∂x 2

for 0 ≤ x ≤ wn .

(2.10)

For a Schottky contact or an abrupt asymmetric p –n junction, integration of Equation 2.9 yields +

F (x ) = F0 +

eN d x εε 0

for 0 ≤ x ≤ wn ,

(2.11)

while F(x) = 0 for x < 0 and x > wn. Hence, using the approximation F(x) = 0 at the edge of the depletion region, gives the boundary condition for the integration constant: eN w F0 = − d n , εε 0

(2.12)

and represents the electric field at the interface F(0), where it has its maximum. Therefore, the electric field F across the depletion region in an n-doped semiconductor with the donor concentration Nd is F (x ) =

eN d (x − wn ) for 0 ≤ x ≤ wn . εε 0

(2.13)

Analogously, the electric field distribution across the depletion region on the p-doped side is F (x ) = −

eN a (x + w p ) for − w p ≤ x ≤ 0. εε 0

x

∫ 0

eN d eN dw n  x2  + ψ(0), (x − wn ) dx = x−  2wn  εε 0 εε 0 

V = Vb + Va =

(2.16)

2 εε 0

wn =

eN d

V.

(2.17)

If the p–n junction is symmetric and the depletion region extends into the p- and n-doped semiconductor, Equation 2.16 is modified to

V = Vb + Va =

eN d 2 eN a 2 wn + wp , 2 εε0 2 εε 0

(2.18)

with the net acceptor doping concentration Na and the width of the depletion region wp in the p-doped side. The total depletion width w is given by

w = (wn + w p ) =

2 εε 0  N a + N d  V. e  N a N d 

(2.19)

The assumption of an abrupt depletion layer edge, is the so-called depletion approximation. A more general Â�description of the depletion region can be found in reference Blood and Orton (1992). 2.6.1.4â•‡ Depletion Layer Capacitance The capacitance that arises from the depletion region is the socalled “depletion capacitance.” In comparison to a plate capacitor, here, the charge is accumulated inside the depletion region of the p–n diode. The number of ionized donors depends linearly on the width w n, while again the width is a function of the square root of the voltage (Equations 2.17 and 2.19). Hence, the capacitance does not depend linearly on the voltage and has to be defined differentially for a small bias signal ΔV:

(2.15)

eN d 2 wn , 2 εε0

or the expression for the depletion width

(2.14)

The potential distribution across the depletion region for a p+–n junction inside the n-doped semiconductor is now obtained by integration of Equation 2.13: ψ(x ) = −

with ψ(0) = 0 as a reference for the potential distribution. For an asymmetric p+–n junction, one notices that the band bending in the present approximation only occurs in the n-doped region of the junction. The contact potential is equal to the total band bending, that means, equal to the build-in voltage plus the external bias ψ(0) = −V = −(Vb + Va). Since the potential on the edge of the depletion region is zero, ψ(wn) = 0, one obtains

C = lim

∆V → 0

∆Q dQ = . ∆V dV

(2.20)

Using Equation 2.17, the entire stored charge inside the depletion region of a Schottky diode or an abrupt asymmetric p+–n junction with area A is

Q = eN d A ⋅ w n (V ) = A 2e εε 0 N dV ,

(2.21)

2-14

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Before bias pulse, Va = Vr QDs empty

During bias pulse, Va = Vp QDs filled

wn p+

After bias pulse, Va = Vr charge carrier emission

wn p+

n

wn p+

n

Electron energy

F

EF

n

F

EC

EC

EF

(a)

(b)

EF

EC

(c)

FIGURE 2.17â•… DLTS work cycle of a p+–n diode with embedded QDs. The upper row schematically shows sketches of the devices for three different bias situations: (a) before, (b) during, and (c) after the bias pulse, respectively. The lower row shows the corresponding band diagrams.

and the capacitance is C(V ) =

εε 0 eN d εε 0 A dQ =A = . dV V wn 2V

(2.22)

The depletion capacitance resembles the capacitance of a plate capacitor with a distance of wn between the plates and a dielectric with relative permittivity ε, although the charge is actually stored in the volume rather than on the edges of the space charge region.

2.6.2â•‡ Capacitance Transient Spectroscopy The depletion width depends on the applied voltage and the doping concentration, that is, on the charge stored inside the space charge region. As a consequence, the measured capacitance is sensitive to the carrier population inside deep levels or QDs situated inside the depletion region (Kimerling 1974). Semiconductor QDs can be considered as deep levels (or traps), storing few or single charge carriers for a certain time. Timeresolved capacitance spectroscopy allows investigating the electronic structure of QDs, since the carrier dynamics can be studied in detail. Historically, time-resolved capacitance spectroscopy was initially used to study and characterize deep levels caused by impurities or defects. In this context it is usually referred to as “deep level transient spectroscopy” (DLTS) or “capacitance transient spectroscopy” (Sah et al. 1970, Lang 1974, Miller et al. 1977, Rhoderick and Williams 1988, Grimmeiss and Ovrén 1981, Blood and Orton 1992). Besides the determination of activation energies and capture cross sections, DLTS also permits to obtain a depth profile of the trap concentration and to investigate the influence of the electric field on the emission process. As QDs act more or less as deep levels, the DLTS has been used successfully

to study thermal emission processes, activation energies, and capture cross sections of various QD systems (Anand et al. 1995, 1998, Kapteyn et al. 1999, 2000a,b, Schulz et al. 2004). 2.6.2.1â•‡ Measurement Principle In the following, the DLTS measurement principle will be described for QDs in a p–n or Schottky diode structure. A more detailed description can be found in Blood and Orton (1992). First, a single layer of QDs with density NQD per area in an n-doped material (with doping concentration Nd) shall be considered. The work cycle of a DLTS experiment of a p+–n structure with QDs embedded is depicted in Figure 2.17. The upper row displays schematically the p–n diode while the lower row depicts the corresponding potential distribution of the conduction band. In an initial step (Figure 2.17a), a reverse bias Vr is chosen such that the depletion region extends well over the QDs. The QDs are completely depleted from charge carriers and the Fermi level is below the QD levels. During the pulse Vp, the depletion region is shorter than the distance of the QD layer from the p–n interface and the QDs are consequently filled with carriers— electrons in Figure 2.17b. The Fermi level is now above the QD states. After switching back to the reverse bias situation (Figure 2.17c), the QDs are again situated inside the depletion region but still filled with electrons. The depletion width is larger for QDs filled with charge carriers, hence, the capacitance after the pulse Vp is smaller and increases again when electrons/holes are emitted due to thermal activation or tunneling emission.* By recording the depletion capacitance as a function of time, transients are observed, which represent the carrier emission * The emission of carriers from QD states is usually slow, whereas the free carriers in the matrix material are considered to follow the change in the external bias instantaneously at the timescale of the experiment.

2-15

Nanomemories Using Self-Organized Quantum Dots Capacitance

ΔC0 Time Time

0 Vp Vr Reverse bias

FIGURE 2.18â•… DLTS work cycle during a DLTS experiment. The lower part displays the external bias on the device as function of time, the upper part the corresponding capacitance.

processes from the QD states. The pulse sequence during the experiment and the measured capacitance transient is schematically depicted in Figure 2.18.

The capacitance transient, recorded from a single emission process is usually mono-exponential having the time-constant τ. Hence, the capacitance transient is given by (Blood and Orton 1992)

S(T , t1 , t 2 ) = C(T , t 2 ) − C(T , t1 ),

(2.23)

where C(∞) is the steady state capacitance at Vr ΔC 0 is the entire change in the capacitance C(t) for t = ∞ (see Figure 2.18) The emission time constant can be determined by using Equation 2.23 and plotting the data on a semilogarithmic scale. However,

(2.24)

or with Equation 2.23   t   t  S(T , t1 , t 2 ) = ∆C0 exp  − 2  − exp  − 1   .    τ(T )   τ ( T ) 

2.6.2.2â•‡Rate Window and Double-Boxcar Method

 t C(t ) = C(∞) − ∆C0 exp  −  ,  τ

in general, the capacitance transient for deep levels and QDs are multi-exponentials due to the ensemble broadening (Omling et al. 1983) and a linear fit is consequently impossible in most cases. In order to obtain the emission time constant, the activation energies and capture cross section from multi-exponential transients, the rate window concept is commonly applied. One investigates the contribution to the observed emission process at a chosen reference time constant τref. By plotting the boxcar amplitude C(t2) − C(t2) for that reference time constant as function of temperature, the relation between temperature and emission rate can be evaluated for a thermal activated process. In the rate window concept, the selection of the contribution for a chosen reference time constant is done by a simple technique: the DLTS signal at a certain temperature S(T,â•›t1,â•›t2) is given by the difference of the capacitance at two times t1 and t2 by

(2.25)

The two times t1 and t2 define the rate window, which has the reference time constant τ ref =

t 2 − t1 . ln(t 2 /t1 )

(2.26)

Plotting S(T, t1, t2) as function of temperature yields the DLTS spectrum, schematically depicted in Figure 2.19. A maximum appears at that temperature, where the emission time constant of the thermally activated process equals almost the applied reference time constant: τ(T) = τref. A maximum appears only for a thermally activated process, a temperature independent tunneling process leads to a constant DLTS signal (Kapteyn

ΔC(T6) ΔC(T5)

ΔC(T3)

C(t2) – C(t1)

Temperature

ΔC(T4)

ΔC(T3) ΔC(T4)

ΔC(T2)

ΔC(T5)

ΔC(T2) ΔC(T1)

ΔC(T6)

ΔC(T1) t1 (a)

Time

t2 (b)

T1

T2

T3

T4

T5

Temperature

T6

FIGURE 2.19â•… The evaluation of capacitance transients (a) for increasing temperature by a rate window, defined by t1 and t2 , leads to a DLTS plot [C(t1) − C(t2)] of a thermally activated emission process (b).

2-16


F

Electron energy

F

EC

EF (a)

EF (b)

F

EC EF

EC

(c)

FIGURE 2.20â•… Work cycle during a charge-selective DLTS experiment (a) before, (b) during, and (c) after the bias pulse, respectively. The filling pulse Vp is chosen, that the emission or capture of approximately one charge carrier per QD is probed.

2001). This relation is also valid for inhomogeneous broadened DLTS spectra of self-organized QDs. The measured emission time constant is the average time constant at the maximum of the Gaussian ensemble distribution (Omling et al. 1983). The thermal emission rate of the charge carriers, eth is now given by (Lang 1974, Blood and Orton 1992)  E  eth = γT 2 σ ∞ exp  − A  ,  kT 

(2.27)

where EA is the thermal activation energy σ∞ is the apparent capture cross section for T = ∞ γ is a temperature-independent constant Knowing the emission time constant τ = 1/eth for different temperatures enables to derive the thermal activation energy and the capture cross section. For varying τref different peak posi2 tions Tmax are obtained. The plot of In(Tmax τref ) as a function of −1 Tmax , cf. Figure 2.22b, is called an Arrhenius plot and is a linearization of Equation 2.27. The slope yields the activation energy EA and from the y-axis intersection, the apparent capture cross section σ∞ at infinite temperature can be obtained. In order to improve the signal-to-noise ratio (SNR), the capacitance transients near t1 and t2 can be averaged over an interval tav (Day et al. 1979). The SNR in this case is found to scale as ~ t av (Miller et al. 1977), and the reference time constant is in good approximation

τ ref =

t 2 − t1 . ln (t 2 + (1/2)t av ) (t1 + (1/2)t av )

(

)

(2.28)

This method to improve the SNR is referred to as “DoubleBoxcar” approach. 2.6.2.3â•‡Charge-Selective Deep Level Transient Spectroscopy As described in the previous section, usually the QDs are completely filled with charge carriers during the pulse bias Vp (Figure 2.17b). Consequently, after the pulse, the emission of many charge carriers from multiply charged QDs is probed.

The activation energy of each emitted charge carrier depends on the actual charge state in such conventional DLTS experiments. The DLTS spectrum is broadened due to different emission processes from different QD states having different emission time constants, for instance, nicely observed in DLTS experiments on Ge/Si QDs (Kapteyn et al. 2000b). In order to study the charge states in more detail, chargeselective DLTS probes the emission of approximately one charge carrier per QD. The principle of this method is schematically illustrated in Figure 2.20. Before the filling pulse, the QDs might be already filled with charge carriers up to the Fermi level. During the filling pulse (Figure 2.20b), the Fermi level is adjusted by the applied pulse bias, such that approximately one carrier per QD will be captured. To probe all QD states from the ground state up to the excited state the pulse bias with respect to the reverse bias is set to a constant height and the reverse bias is decreased step by step. After the bias pulse, the reverse bias is set to the initial condition. Now, the previously captured carrier is emitted again and the emission process of approximately one charge carrier per QD is observed. Narrow peaks appear now in the DLTS spectra, which are due to differently charged QDs (cf. Figure 2.23). Moreover, even for an energy-broadened ensemble of QDs (normally measured in DLTS experiments) the charge-selective DLTS offers a simple method to probe the emission from many charge carriers in different QDs, all having the same activation energy. Decreasing the reverse bias and keeping the pulse bias height fixed (to ensure the emission of only one charge carrier per QDs) gives the activation energy starting at the ground states of the ensemble to the excited states and finishing at completely charged QDs.

2.7â•‡Charge Carrier Storage in Quantum Dots In this section, experimental results from capacitance spectroscopy measurements on different QD material systems are presented. This time-resolved method allows to derive thermal activation energies and capture cross sections of electron and hole states in QDs. In addition, the important storage time at room temperature can be quantified and connected to the localization energy of the charge carriers.

2-17


2.7.1â•‡Carrier Storage in InGaAs/GaAs Quantum Dots Emission of electrons and holes from InGaAs/GaAs QDs was observed using conventional DLTS measurements by Kapteyn et al. (1999, 2000a,b). The electron/hole emission from InGaAs/ GaAs QDs was studied in more detail by using charge-selective DLTS (Geller et al. 2006d) and time-resolved tunneling capacitance measurements (TRTCM) (Geller et al. 2006c). Two samples H1 and E1 were investigated to study the hole and electron emission, respectively, where the QD layer was incorporated in the slightly p-doped/n-doped (∼3 × 1016 cm−3) region of a n+–p or p+–n diode structure. The QD layer was situated 500â•›nm/415â•›nm below the p–n junction for the hole/electron sample. Mesa structures with a diameter of 800â•›μm and ohmic contacts were formed by employing standard optical lithography. The results of these measurements are presented in Figure 2.21. It turns out that phonon-assisted tunneling controls the chargecarrier emission process from self-organized QDs in an electric field. The influence of the tunneling part, however, depends strongly on the effective mass and the strength of the electric field. The observed hole ground state activation energy EAH1 = 120 ± 10â•›meV, hence, underestimates the localization energy. It is the thermal activation part in a phonon-assisted tunneling process: thermal activation into an excited state and subsequent tunneling through the remaining triangular barrier, cf. schematic pictures in Figure 2.21. The true hole localization energy was H1 determined by using the TRTCM method to E loc = 210 ± 20â•›meV. For the electrons, two contributions to the DLTS signal were observed: a DLTS signal with an activation energy of EAE11 = 82 ± 10â•›meV that is in good agreement with the theoretically predicted value for the ground/excited state energy splitting (70â•›meV). In addition, a DLTS signal with a smaller activation energy of E AE12 = 44 ± 10â•›meV is observed. This value is attributed to the first/ second-excited state energy splitting and in satisfactory agreement with the theoretically predicted value of 50â•›meV. The ground

state localization energy was determined by the TRTCM method E1 to Eloc = 290 ± 30â•›meV (Geller et al. 2006c). The electron/hole storage time at room temperature can be estimated by using Equation 2.27 and the capture cross section and the localization energy for sample E1/H1, respectively. An average storage time for electrons of about 200â•›ns and for holes of about 0.5â•›ns is obtained. This means, InAs QDs embedded in a GaAs matrix do not have a sufficiently long storage time to act as storage units in future nanomemories.

2.7.2â•‡Hole Storage in GaSb/GaAs Quantum Dots The storage time can be further increased by changing the material of the QDs and/or the surrounding matrix. A larger difference in the energy bandgap than in InGaAs/GaAs is more promising, for example, InAs/AlAs QDs. Moreover, large band discontinuities and, hence, strong hole localization is expected in type II QD heterostructures (Hatami et al. 1998), for example, GaSb/GaAs or InSb/GaAs. Only holes are confined in GaSb/ GaAs QDs, while a repulsive potential barrier exists for the electrons in the conduction band. Type II material combinations are therefore very attractive for future memory applications. Hole storage in and emission from GaSb0.6As0.4/GaAs QDs was investigated by DLTS and charge-selective DLTS (Geller et al. 2003). The sample was an n+–p diode structure containing a single layer of GaSb QDs. An area density of about 3 × 1010 cm−2, an average QD height of about 3.5â•›nm, and an average base width of about 26â•›nm were determined by structural characterization of uncapped samples grown under identical conditions (Müller-Kirsch et al. 2001). The QD layer was placed 500â•›nm below the n+–p junction in a slightly p-doped (p = 3 × 1016 cm−3) GaAs region. Mesa structures with a diameter of 800â•›μm and ohmic contacts were formed by employing standard optical lithography. 2.7.2.1â•‡ Multiple Hole Emission

Hole sample H1

Electron sample E1

H0 H1 H1 Eloc EA

E1 E2 Eloc E E1 A2 E1 E1 EA1 E0

EC

EV Thermal activation energy

H1

EA = 120 meV H1

E1

EA1 = 82 meV E1

EA2 = 44 meV E1

Localization energy

E loc = 210 meV

Eloc = 290 meV

Storage time (300 K)

t Storage = 0.5 ns

tStorage = 200 ns

H1

E1

FIGURE 2.21â•… Summary of the results from the charge-selective DLTS and tunneling emission experiments on self-organized InGaAs/ GaAs QDs.

In Figure 2.22, conventional DLTS measurements of the QD sample for a reverse bias Vr = 10â•›V and a pulse bias Vp = 4â•›V are displayed. The pulse length was 10â•›ms. For these conditions, the QDs are completely filled during the bias pulse and release all trapped holes after the pulse. The DLTS spectrum of the QD sample (Figure 2.22a) shows three maxima, in the following denoted by “A,” “B,” and “C.” From an Arrhenius plot, an activation energy EAC = 530 ± 20â•›meV and a capture cross section σC∞ ≈ 6 × 10−16 cm2 is obtained for peak C. This signature can be identified as the bulk hole trap H3 in p-type GaAs (Stievenard et al. 1986). Peaks A and B, observed at temperatures T = 140 and 230â•›K, are obviously related to QD formation. An activation energy of about 600â•›meV is found for peak B. This relatively high activation energy is comparable to the unstrained GaSb/GaAs valence band offset (Hatami et al. 1995, North et al. 1998) and could be explained by the existence of relaxed GaSb islands in the QD sample.

2-18

A

600 400

B

200 0

(a)

104

τref = 62.5 ms T 2 τref (K2 s)

DLTS signal ΔC (nF m–2)


C 102

QDs 50

103

100 150 200 250 300 350 Temperature (K)

(b)

EA = 340 meV 6.4

6.8 7.2 1000/T (K–1)

7.6

FIGURE 2.22â•… (a) DLTS signal of the GaSb0.6As0.4/GaAs QD sample, measured at a reverse bias Vr = 10â•›V and a pulse bias Vp = 4â•›V with a filling pulse width of 10â•›ms and a reference time constant τref = 62.5â•›ms. (b) The corresponding Arrhenius plot for peak A, which is related to multiple hole emission from the GaSb QD states. An average activation energy of EA = 340â•›meV is obtained.

τref = 62.5 ms Vp = Vr – 0.5 V

800 DLTS signal ΔC (nF m–2)

Peak A in the DLTS spectrum of the QD sample in Figure 2.22a is attributed to the hole emission from the GaSb QD states, where the activation energy of each successively emitted hole depends on the actual charge state. The DLTS peak is broadened as previously observed for multiply charged Ge/Si QDs (Kapteyn et al. 2000b). From a standard Arrhenius plot in Figure 2.22b, an activation energy of EA = 340â•›meV is obtained. Consequently, this activation energy of peak A represents only an average value for hole emission from completely charged QDs, where 15 holes from different states with different confinement are involved.

Vr = 4.5 V

600

400

200

2.7.2.2â•‡ Charge-Selective DLTS In order to study the charge states in more detail, the QD sample was studied using the charge-selective DLTS method. The pulse bias was always set to Vp = Vr − 0.5â•›V, while the reverse bias was increased from 4.5 to 9.5â•›V. Narrow peaks appear in the DLTS spectra in Figure 2.23. At Vr = 4.5â•›V the DLTS signal shows a maximum at about 80â•›K and for increasing reverse bias the DLTS peak shifts to higher temperature. The activation energy increases accordingly from 150â•›meV at 4.5â•›V to 450â•›meV for 9.5â•›V in Figure 2.24 obtained by standard Arrhenius plots. The DLTS spectra for a reverse bias between 4.5 and 9.5â•›V in Figure 2.23 are attributed to the hole emission from differently charged QDs. All these spectra exhibit a maximum in the temperature range between 80 and 180â•›K, the range covered by peak A in Figure 2.22a. The maximum activation energy of 450â•›meV represents the average hole ground state energy of the QD ensemble, that is, the localization energy. The decrease in the activation energy from 450â•›meV down to 150â•›meV corresponds to an increase in the average occupation of the QDs, see the schematic insets in Figure 2.23. With increasing amount of charge in the QDs, state filling lowers the thermal activation barrier. The completely charged QDs are filled with 15 holes up to the Fermi level at the valence band edge, where Coulomb charging generates the barrier height. In order to compare the storage time for GaSb0.6As 0.4/GaAs with In(Ga)As/GaAs QDs, the observed emission rates were extrapolated to room temperature, using Equation 2.27. A

0

Vr = 9.5 V 50

100

150

200

250

300

Temperature (K)

FIGURE 2.23â•… Charge-selective DLTS spectra of hole emission from GaSb0.6As0.4/GaAs QDs for a reference time constant of τref = 62.5â•›ms. The reverse bias Vr is increased from 4.5â•›V up to 9.5â•›V while the pulse height is fixed at 0.5â•›V for all spectra. The pulse width is 10â•›ms and the data is displayed vertically shifted for clarity.

storage time of about 1â•›μs for localized holes with the ground state energy of 450â•›meV is estimated, three orders of magnitude longer than the hole storage time in InGaAs/GaAs QDs.

2.7.3â•‡InGaAs/GaAs Quantum Dots with Additional AlGaAs Barrier Hole storage in InGaAs/GaAs QDs with an additional AlGaAs barrier is presented in this section. The additional AlGaAs barrier increases the activation energies and a longer storage time at room temperature is observed. Two different samples having an AlGaAs barrier with different aluminum content were studied. The first contains an Al0.6Ga0.4As, the second an Al0.9Ga0.1As barrier below the QD layer. The activation energy in the latter is increased sufficiently to reach a retention time of seconds at room temperature.

2-19


Activation energy EA (meV)

450

1

Average hole number per QD

375 300 225 150 4

5

6

7

8

9

10

Reverse bias (V)

FIGURE 2.24â•… Dependence of the activation energy EA on the reverse bias GaSb 0.6As 0.4/GaAs QDs. For a reverse bias of Vr = 9.5â•›V hole emission from the QD ground states is probed.

2.7.3.1â•‡Storage Time: Milliseconds at Room Temperature


The first sample is an n+–p diode structure, grown by MBE. It contains a single layer of InGaAs QDs embedded in slightly p-doped GaAs (p = 2 × 1015 cm−3). The QDs are placed 1500â•›nm below the p–n junction and an additional undoped Al0.6Ga0.4As barrier of 20â•›nm thickness is situated 7â•›nm below the QD layer to increase the hole storage time. Again, mesa structures with a diameter of 800â•›μm and ohmic contacts were formed. Figure 2.25a shows the charge-selective DLTS spectra with a reference time constant of 5â•›ms (Marent et al. 2006). The pulse bias height was fixed to 0.2â•›V for all spectra (Vp = Vr − 0.2â•›V). For a reverse bias above Vr = 3.2â•›V no DLTS signal is visible, as the Fermi level is energetically above the QD states and no QD states are occupied. By decreasing the reverse bias the Fermi level reaches the QD ground state at Vr = 3.2â•›V and a peak in the DLTS spectrum appears at 300â•›K. This peak is related to thermally activated hole emission from the ground states of the QD ensemble

/5.0

100

Vr = 0.2 V

/1.1

The second sample is also an n+–p diode structure, grown by MOCVD. It contains a single layer of InGaAs QDs embedded in p-doped GaAs (p = 3 × 1016 cm−3). The QDs are placed 400â•›nm below the p–n junction and an additional undoped Al0.9Ga0.1As barrier of 20â•›nm thickness is situated 7â•›nm below the QD layer. Figure 2.26a shows the charge-selective DLTS spectra (Marent et al. 2007). The pulse bias height was fixed here to 0.3â•›V for all spectra (Vp = Vr − 0.3â•›V). At Vr = 5.7â•›V a peak in the DLTS spectrum appears at 380â•›K, related to thermally activated hole emission from the ground states of the QD ensemble across the Al0.9Ga0.1As barrier. As mentioned before, the peak at 380â•›K for τref = 5â•›ms represents an average emission time constant (storage time) of 5â•›ms for hole emission from the ground states of the QD ensemble across the Al0.9Ga0.1As barrier. To obtain the storage time at room temperature, the capacitance transient recorded at

100 200 300 Temperature (K)

EA

500 400 300

n+ p

EA

200

3.4 V

0 (a)

2.7.3.2â•‡ Storage Time: Seconds at Room Temperature

600

Vp = Vr – 0.2 V τref = 5.0 ms

200

across the AlGaAs barrier, see inset in the upper left corner of Figure 2.25b. A further decrease of the reverse bias leads to QD state-filling and emission from higher QD states is observed. A peak in the DLTS appears at that temperature, where the averaged time constant of the thermally activated emission equals the applied reference time constant, cf. Equation 2.26. Therefore, the peak at 300â•›K for τref = 5â•›ms (Vr = 3.2â•›V in Figure 2.25a) represents an average emission time constant (storage time) of 5â•›ms for hole emission from the QD ground states across the Al0.6Ga0.4 As barrier. The thermal activation energies are shown in Figure 2.25b. The highest value of 560 ± 60â•›meV at Vr = 3.2â•›V is related to thermal activation from the hole ground states of the QD ensemble across the AlGaAs barrier. The decrease in the activation energy corresponds to an increase in the average occupation of the QDs. At Vr = 1.4â•›V the energetic position of the Fermi level is at the valence band edge and no further QD state-filling is possible. As a consequence, between Vr = 1.4 and 0.2â•›V carrier emission from the valence band edge is probed. The activation energy remains roughly constant with a mean value of 340â•›meV. This energy represents the energetic height of the Al0.6Ga0.4As barrier.

Activation energy (meV)

15

(b)

0.0

0.6

1.8 1.2 2.4 Reverse bias (V)

3.0

FIGURE 2.25â•… Charge-selective DLTS spectra of thermally activated hole emission from InGaAs/GaAs QDs with an additional Al0.6Ga0.4As barrier below the QD layer (a). Spectra below Vr = 1.6â•›V are divided by a factor from 1.1 up to 5. (b) Dependence of the thermal activation energy on the reverse bias Vr. (Reprinted from Marent, A. et al., Appl. Phys. Lett., 89, 072103, 2006. With permission.)

2-20

Vp = Vr – 0.3 V τref = 5 ms

400

200

Vr = 3.6 V Vr = 5.7 V

0

(a)

Activation energy (meV)



50

150 250 350 Temperature (K)

450

(b)

700

n+ p

650

EA

600 550

EA

500

3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 Reverse bias (V)

FIGURE 2.26â•… Charge-selective DLTS spectra of thermally activated hole emission from InGaAs/GaAs QDs with an additional Al0.9Ga0.1As barrier below the QD layer (a). Dependence of the thermal activation energy on the reverse bias Vr.

in the activation energy corresponds again to an increase in the average occupation of the QDs. At Vr = 3.6â•›V, carrier emission from the valence band edge is probed and the activation energy has a value of 520â•›meV. This energy now represents the energetic height of the Al0.9Ga0.1As barrier.

Capacitance (arb. units)

1

0.1

2.7.4â•‡ Storage Time in Quantum Dots 0.01

T = 300 K Vr = 5.7 V 0

1

2

3

4

5

6

Time (s)

FIGURE 2.27â•… Capacitance transient of hole emission from InGaAs/ GaAs QDs with an additional Al0.9Ga0.1As barrier (for a reverse bias of Vr = 5.7â•›V). An average hole storage time of 1.6â•›s at T = 300â•›K is obtained from a linear fit of the transient on a semilogarithmic scale.

300â•›K is plotted on a semilogarithmic scale in Figure 2.27. From a linear fit an average hole storage time of 1.6â•›s at room temperature is determined. Furthermore, the thermal activation energies are shown in Figure 2.26b. The highest value of 710 ± 40â•›meV at Vr = 5.7â•›V is related to thermal activation from the hole ground states of the QD ensemble across the Al0.9Ga0.1As barrier. The decrease Material System

Carrier emission from different QD systems has been studied in order to determine the carrier storage time at room temperature. In addition, the localization energy was obtained by using time-resolved capacitance spectroscopy (DLTS) and related to the hole/electron retention time. The results of the experiments are summarized in Figure 2.28. An electron/hole localization energy of 290/210â•›meV, respectively, was obtained for InGaAs QDs embedded in a GaAs matrix. Based on these values the storage time at T = 300â•›K was estimated to ∼200â•›ns for electrons and ∼0.5â•›ns for holes. Furthermore, the more promising type II GaSb0.6As0.4/GaAs has been studied in detail. Ground state activation energy of 450â•›meV was determined, which accounts for a room temperature emission time in the order of 1â•›μs. The ground state localization is about twice as large and the retention time at room temperature is about three orders of magnitude longer than in InGaAs/GaAs QDs. Finally, the hole emission from InGaAs/GaAs QDs with an additional AlGaAs barrier was investigated. The activation energy for the QD ground states increases from 210 to 560â•›meV Localization Energy

Storage Time at 300 K

InAs/GaAs

Hole Electron

210 meV 290 meV

~0.5 ns ~200 ns

Ge/Si

Hole

350 meV

~0.1 µs

GaSb0.6As0.4/GaAs

Hole

450 meV

~1 µs

InAs/GaAs with Al0.6Ga0.4 As barrier

Hole

560 meV

5 ms

InAs/GaAs with Al0.9Ga0.1As barrier

Hole

710 meV

1.6 s

Charge Carrier Type

FIGURE 2.28â•… Summary of the measured electron/hole storage time (at 300â•›K) and localization energy for different QD systems.

2-21


Storage time (s) at 300 K

GaSb/AlAs

1M a

InSb/GaAs 1E8

In0.5Ga0.5Sb/GaAs

10 a

GaSb/GaAs

24 h

1

InAs/GaAs + Al0.9Ga0.1As InAs/GaAs + Al0.6Ga0.4As GaAs0.4Sb0.6/GaAs Si/Ge InAs/GaAs

1E–8 0.2

0.4

0.6 0.8 1.0 1.2 Localization energy (eV)

1.4

FIGURE 2.29â•… Dependence of the hole storage time on the localization energy for a variety of QD systems. The solid line is a fit to the experimentally obtained data (full circles). The open circles are estimated storage times for the labeled material systems according to the calculated localization energies and the fit. (Reprinted from Marent, A. et al., Appl. Phys. Lett., 91, 42109, 2007. With permission.)

for an additional Al0.6Ga0.4 As barrier and the hole storage time is in the order of milliseconds. Using an Al0.9Ga0.1As barrier increases the activation energy accordingly from 210 to 710â•›meV and the hole storage time at room temperature increases by nine orders of magnitude to 1.6â•›s. This value is already three orders of magnitude longer than today’s DRAM refresh time, which is in the millisecond range. From the experimental results, material combinations can be predicted to obtain a storage time of more than 10 years at room temperature. Figure 2.29 displays the hole storage time in dependence of the localization energy (full circles). The solid line is a fit to the data and corresponds to Equation 2.27. The storage time shows an exponential dependence on the localization energy as predicted by the common rate equation of thermally activated emission. The storage time increases by one order of magnitude for an increase of the localization energy of about 50â•›meV. From these results, hole localization energies can be estimated, which provide storage times of 24â•›h or 10 years. These storage times are reached for localization energies of 0.96 and 1.14â•›eV, respectively. To find a material system with such large localization energies, the hole localization energies for GaAsx Sb1−x/GaAs and

In x Ga1−x Sb/GaAs QDs have been calculated using eight-band k . p theory (Stier et al. 1999, Schliwa et al. 2007). Based on structural characterization of GaSb/GaAs QDs (Müller-Kirsch et al. 2001) the QDs are modeled as truncated pyramids with a base width of 21â•›nm and a height of 3.9â•›nm. The results are summarized in Figure 2.30 and plotted in Figure 2.29 as open circles. The localization energy in GaAsx Sb1−x/GaAs QDs increases from 350â•›meV up to 853â•›meV for an antimony content of 50% and 100%, respectively. Since the bandgap in III–V materials decreases with increasing lattice constant (Vurgaftman et al. 2001), InSb/GaAs QDs should offer a larger localization energy than GaSb/GaAs. Eight-band k .p calculations for In xGa1−xSb/ GaAs QDs provide a localization energy of 919 and 996â•›meV for an indium content of 50% and 100%, respectively. A hole storage time in InSb/GaAs QDs of more than 24â•›h is predicted (see Figure 2.29). Using an AlAs matrix instead of a GaAs matrix a storage time of more than 10 years can be reached. Since the valence band offset between GaAs and AlAs is about 550â•›meV (Batey and Wright 1986) the entire hole localization energy in GaSb/AlAs QDs is about 1.4â•›eV. This value leads to the prediction of an average hole storage time of more than 1 million years at room temperature (see Figure 2.29), orders of magnitude longer than needed for a nonvolatile memory.

2.8â•‡Write Times in Quantum Dot Memories The carrier capture of electrons/holes from the valence/Â� conduction band into the QD states limits physically the possible write time in a QD-based nanomemory. Normally, carrier capture into QDs is studied by interband absorption optical techniques, like time-resolved photoluminescence (PL) spectroscopy (Heitz et al. 1997, Giorgi et al. 2001). However, such experiments probe the exciton dynamics, that means, the electron–hole capture and relaxation into the QD states is measured simultaneously. A detailed knowledge of either electron or hole capture is of great importance, for future QD-based memory applications as only one sort of charge carriers is stored. Electron or hole capture has been investigated separately, for example, in interband pump and intraband probe experiments of Müller et al. (2003). In addition, the hole capture into GaSb0.6As 0.4/ GaAs self-organized QDs has been studied (Geller et al. 2008) using DLTS experiments. The investigations demonstrate a fast

Charge Carrier Type

Localization Energy

Hole

350 meV

0.1 µs

GaSb/GaAs

Hole

853 meV

13 min

In0.5Ga0.5Sb/GaAs

Hole

919 meV

4h

InSb/GaAs

Hole

996 meV

6 days

GaSb/AlAs

Hole

~1.4 eV

>106 years

Material System GaSb0.5As0.5/GaAs

Storage Time at 300 K

FIGURE 2.30â•… Hole localization energies in different Sb-based QDs calculated by eight-band k.p theory. The storage time is estimated according to the fit of the experimental data in Figure 2.29.


2.8.1â•‡ Hysteresis Measurements As already discussed in Section 2.7, where the charge carrier storage time was determined, the read-out of the charge state (which represents the stored information) inside the QDs is done by the measurement of the capacitance of the p–n diode. The capacitance of a diode structure with embedded QDs depends on the number of holes stored inside the depletion region. A larger capacitance corresponds to unoccupied QDs (“0”) while a smaller capacitance represents a “1,” where the QDs are filled with holes. A fundamental property of storage devices is the appearance of hysteresis when switching between the two information states. Accordingly, a nanomemory device containing self-organized QDs shows a hysteresis in the capacitance measurement after writing (yielding occupied QDs, a “1” state) and erasing (yielding unoccupied QDs, a “0” state) process. Such a hysteresis measurement is shown in Figure 2.31 for InAs/GaAs QDs (a) and GaSb0.6As0.4/GaAs QDs (b). Both memory structures are n-p diodes where holes are stored inside the QDs. The sample description can be found in Section 2.7.1 for the InAs/GaAs QDs and in Section 2.7.2 for the GaSb0.6As 0.4/GaAs QDs. One hysteresis sweep takes a few seconds; hence, the temperature was reduced down to 15â•›K for the InAs QDs where the hole storage time is in the order of minutes. Analogously, for the GaSb0.6As 0.4/ GaAs QDs—having hole localization energy twice as high as the InAs/GaAs QDs—a higher temperature of 100â•›K is already sufficient to obtain hole storage times of several minutes. Figure 2.31 shows the switching between the two information states by a hysteresis curve of the capacitance for both samples. At the reverse bias of 14â•›V/16â•›V (point 1) the charge carriers tunnel out of the QDs (erasing the information). If the reverse bias is now swept from 14â•›V/16â•›V to the storage situation at 8.2â•›V (InAs QDs) or 7.2â•›V (GaSb QDs), respectively, the QDs are empty and a larger capacitance is observed (point 2). If the bias is swept to 0â•›V (point 3), the QDs are charged with holes and a smaller capacitance is observed upon sweeping back to the storage situation (point 4). The maximum hysteresis opening is now defined at

Capacitance (pF)

18

(a)

3

Max. hysteresis opening Cmax

16

14

QDs full

T = 15 K InAs QDs

2

QDs empty

4

1

12

8.2 V 3

T = 100 K GaSb QDs

80

Capacitance (pF)

capture and relaxation process in self-organized QDs in the range of picoseconds at room temperature, more than three orders of magnitude faster than the write time in a DRAM cell. This fast carrier capture should enable very fast write times in a QD-based nanomemory ( 0

EF Metal

(a)

VGS > 0

Metal

Nanotube

Starting p-FET Hole injection for VGS < 0

VGS < 0

EF W

VGS > 0

W

VGS > 0

Nanotube

Annealed n-FET Electron injection for VGS > 0 (b)

(c)

n-Doped No current at any VGS

Highly n-doped Electron tunneling through the barrier (d)

FIGURE 3.2â•… Schematic of the bands in the vicinity of the contacts as a function of the electrostatic gate field for a p-type (a), ambipolar (b), and n-type (c) CNFETs. Annealing the device in vacuum results in a transition from the situation of (a) to that of (c) in case of low doping and (d) for high doping. The transition can be reversed by the introduction of air, clearly showing that the presence of oxygen p-doping of the nanotube. It is clear that the Schottky barrier at the nanotube–metal interface plays a key role. Note also the importance of Fermi-level pinning for the functioning of the device. (Reprinted from Derycke, V. et al., Appl. Phys. Lett., 80(15), 2773, 2002. With permission.)

3-4


channel. The charge causes a threshold potential shift (the threshold potential is defined as the potential at which the majority carriers start flowing) of the nanotube FET. Because the nanotube has very high carrier mobility, the information can be stored in a few (as few as one) electrons configuration. The states can be reversibly written and erased. The reading mechanism is performed via a measurement of the source–drain current, while the writing mechanism is completed using a large bias voltage applied to the gate. A number of groups have obtained convincing results on CNFET-based memory elements. How the information is stored and consequently under which conditions the memory is usable depends on the details of the manufacturing, particularly in the gating material. The first two examples of FET-memory elements were proposed independently by Fuhrer et al. (2002) and Radosavljevic et al. (2002). The functioning of the devices hinges on the operation of a single-electron memory. In this case, the capacitance of the storage node must be small enough so that its Coulomb-charging energy is significantly larger than the thermal energy at the operating temperature, and the readout device must be sensitive enough to detect a single nearby electronic charge. In

Fuhrer’s device, the charge is reversibly injected and removed from the dielectric (placed between the tube and the gate electrode) by applying a moderate (10â•›V) bias between the nanotube and the substrate. The nanotube is ideal as a charge-detecting device due to its high carrier mobility, large geometrical capacitance, and its one-dimensional (1D) nature ensuring that local changes in charge affects the global conductance (due to slow screening). In Fuhrer’s pioneering work, for instance, discrete charge states corresponding to differences of a single, or at most a few, stored electrons are observed. The device is based on the characteristics of a p-type FET (i.e., it conducts at negative gate voltage and becomes insulating at positive gate voltage) and can be operated at temperatures up to 100â•›K. Properties relevant to memory operation are demonstrated by the large hysteresis I–Vg curve that is obtained by sweeping the gate voltage Vg between −10 and +10â•›V. As is shown in Figure 3.3, the threshold voltage is shifted by more than 6â•›V. The mechanism of charge storage is related to the rearrangement of charges in the dielectric or by the injection or removal of charges from the dielectric through the electrodes or the nanotube. Due to the geometry of the device, the electric field at the surface of the

3.0 2.5

Isd (μA)

2.0 1.5 1.0 0.5

(a)

(b)

0.0 –10

–5

0 Vg (V)

5

10

Vg (V)

Isd (μA)

2 1

1

0

0 8

Write

0

Read

–8 (c)

100

Erase 200

300 Time (s)

400

500

FIGURE 3.3â•… (a) Atomic force microscope topographic image of the nanotube device used in Fuhrer et al. (2002). The nanotube extends between the two dark blocks at the top and bottom of the image (i.e., the electrodes). The scale bar represents 1â•›μm. (b) Drain current as a function of gate voltage at room temperature and a source–drain bias of 500â•›mV. As the gate voltage is swept from positive to negative and back, a strong hysteresis is observed, as indicated by the arrows denoting the sweep direction. (c) Series of four read–write cycles of the nanotube memory at room temperature. The upper panel shows the drain current at a source–drain bias of 500â•›mV, while the lower panel shows the gate voltage. The memory state was read at −1â•›V, and written with pulses of ±8â•›V. (Reprinted from Fuhrer, M.S. et al., Nano Lett., 2(7), 755, 2002. With permission.)

3-5

Carbon Nanotube Memory Elements

nanotube is very large, at least large enough to cause the movement of charge in the dielectric (Fuhrer et al. evaluated the field to be comparable to the breakdown field in SiO2). At these high fields the electrons are easily injected into the dielectric from the nanotube and remain trapped in metastable states until the polarity is reversed. In Radosavljevic’s device, the FET used is based on an n-type transistor (obtained from annealing the nanotube in hydrogen gas). Again, it was shown that CNFETs are extremely sensitive to the presence of individual charges around the channel, largely because of the nanoscale capacitance of the CNTs. A technique known as scanning gate microscopy (SGM) (Freitag et al. 2002, Meunier et al. 2004) was used to study the local transport properties across the channel, revealing the key role of nanotube– metal contacts (Schottky barriers) and also the positions of the possible sites where the electrons are trapped (i.e., where the information is physically stored, the storage nodes). Figure 3.4 shows the reproducible hysteresis loop in the I–Vg curve that becomes larger as the range of Vg is increased, indicating that it originates from avalanche injection into bulk oxide traps. The hysteresis can become so large that the device varies from depletion mode (normally on at Vg = 0) and enhancement mode (normally off) behavior. The location and sign of the trapped charge can be determined by examining the directions of the hysteresis loop. After a sweep to positive Vg, the CNFET threshold voltage moves toward more positive gate values, indicating injection of

negative charges into oxide traps. To read the memory, a 1â•›MΩ load resistor is added to create a voltage divider. Read (Vin = 0) and write (Vin = +20 or −20â•›V) are applied to the input terminal (back gate). A logical gate (1 or 0) is defined as Vout = 1 or 0. To write a “1” to the memory cell, Vin is switched rapidly to −20 or +20â•›V and back to “0,” so the CNFET is in the on or off at the read voltage. This memory device was found to be nonvolatile at room temperature and with a bit-storage retention time of at least 16â•›h. The trap charging time limits the speed of the device and the bit was evaluated to be stored in no more than 2, 70, and 200â•›e. In the pioneering works discussed above, it is noteworthy that the storage nodes are not precisely the nanotube itself, but consist of trapping sites in the dielectric layer of the gating material. The functional temperatures, storage density, operating speeds, etc., are very sensitive to the details of the trapping mechanisms. As summarized below, a number of studies have been reported following the seminal studies of Fuhrer and Radosavljevic, where the trapping mechanism was further analyzed and modified.

3.2.3â•‡FET-Based Memory Elements: Further Improvements Charge-storage stability of up to 12 days at room temperature was reported in a 150â•›nm long nanotube channel (Cui et al. 2002). The observed conductance decreases for increasing gate

Current (nA)

5

(a)

4 3 2 1 0 –10

0 Gate (V)

10

(b)

–10

1

0 Gate (V)

0

10

1

1

–20 (c)

00

–10

1

0 10 Gate (V)

0

1

20

0

1

e– Vg

Eg

SWNT

Co e–

Vout (V)

Co

Vin Vout

SiO2

1 MΩ

p++ Si gate (d)

1V

0 (e)

0

200

400 Time (s)

600

FIGURE 3.4â•… (a–c) Current vs. gate voltage data obtained in high vacuum for a source–drain voltage of 0.5â•›mV. The device hysteresis increases steadily with increasing gate voltage due to avalanche charge injection into bulk oxide traps, schematically shown in (d). (e) Demonstration of the CNFET-based nonvolatile molecular memory cell. A series of bits is written into the cell and the cell contents are continuously monitored as a voltage signal (Vout) in the circuit shown in the inset. (Reprinted from Radosavljevic, M. et al., Nano Lett., 2(7), 761, 2002. With permission.)

3-6


potential, a normal feature of p-type, air-exposed single-wall carbon nanotubes (SWCNTs). The hysteresis loop is obtained by sweeping continuously the gate voltage from −3 to +3â•›V. This shows two conductance states at Vg = 0 differing by more than two orders of magnitude, associated with a threshold voltage of 1.25â•›V. The method used by Cui et al. is more technologically straightforward compared to the previously reported techniques, as it does not require separating SWCNT bundles into individual SWCNTs during sample preparation. The sample treatment is likely to be responsible for the modified type of trapping centers, compared to earlier similar devices. The heat treatment and exposure to oxygen plasma cause the metallic nanotubes present in the bundle to be preferentially oxidized, leading to increased gate dependence due to the remaining intact semiconducting tubes. In addition, oxidation-related defects are likely to be formed in the remaining amorphous carbon particles on the bundle surface or at the SiO2 interface. These defects act as charge-storage traps and their close proximity to the surface of the channel accounts for the large threshold voltage shifts. Another type of charge trapping was presented by Kim et al. (2003). In that case, water molecules were shown to be responsible for the hysteresis properties (i.e., responsible for the threshold potential shift) of the I–Vg curves. The water molecules contributing to the trapping could be located on the nanotube surface or on the SiO2 close to the tube. Heating under dry conditions significantly reduces the hysteresis. A completely hysteresisfree CNFET was possible by passivation of the device using a polymer coating, clearly indicating that the storage nodes were removed by the treatment. This work also confirms the central role played by surface chemistry on the properties of the device and that truly robust passivation is needed in order to use CNTbased devices in practical electronic devices, unless it is simply used as a detection device (e.g., of humidity). More recent work of Yang et al. confirmed the role of adsorbed (including water and alcohol) molecules for charge trapping, also showing charge retention of up to 7 days under ambient conditions (Yang et al. 2004).

3.2.4â•‡FET-Based Memory Elements: Controlling Storage Nodes Soon after the publication of the pioneering works of 2002, a number of groups confirmed that the as-prepared nanotube FETs possessed intrinsic charge-trapping centers that were responsible for the shift in the threshold voltage measured in the hysteresis loop of the I–Vg characteristics. As discussed briefly above, the trapping centers can be defects in the SiO2, water, or alcohol molecules adsorbed on the tube or the dielectric, or oxidized amorphous carbon. It was soon understood that a better control of the storage medium was required in order to harness the full potential of CNFET-based memory elements. In 2003, Choi et al. proposed a SWCNT nonvolatile memory device using SiO2-Si3N4-SiO2 (ONO) layers as the storage node (Choi et al. 2003). In that device, the top gate structure is placed above the ONO layer, which is positioned directly above

a few-nanometers-long channel. Charges can tunnel from the CNT surface into the traps present in the ONO layers. The stored charges impose a threshold voltage shift of 60â•›mV, which is independent of charging time, suggesting that the ONO traps present a quasi-quantized state. The choice of SiO2-Si3N4-SiO2 is motivated by the fact that it presents a high breakdown voltage, low defect density, and high charge-retention capability (Bachhofer et al. 2001). Another type of device was demonstrated experimentally in 2005 (Ganguly et al. 2005). In this case, the charge-storage nodes consist of gold nanocrystals placed on the top gate above the nanotube channel. The device was found to have a large memory window with low voltage operations and single-electron-controlled drain currents. The device is based on a Coulomb blockade behavior, in other words on the difficulty of adding supplementary electrons due to the large charging effect of the system with large capacitance and Coulomb repulsion. The Coulomb blockade in the nanocrystals, along with the single-charge sensitivity of the nanotube FET, is suggested to allow for multilevel operations. In this device, the reported retention is rather modest: 6200â•›s at 10â•›K and only 800â•›s at room temperature. Better dielectric and preprocessing should improve these low retention times. More recently, Sakurai et al. proposed a modified design where the dielectric insulator was made up of a ferroelectric thin film (Sakurai et al. 2006). The experimental results show that the carriers in the CNTs are controlled by the spontaneous polarization of the ferroelectric films. Ferroelectric-gate FETs offer potential advantages as nonvolatile memory elements, such as low power consumption, the capability of high-density integration, and nondestructive readable operation. So far, the use of ferroelectric thin films in conventional Si-based FETs for memory applications has been complicated because of the extreme difficulty of fabricating the ferroelectric/Si structure with good interface properties due to the chemical reaction and interdiffusion of Si and ferroelectrics. It was suggested that this problem could be resolved by inserting an insulating layer between the ferroelectric and the semiconductor layers, but at the cost of increased voltage operation and depolarization field. Another method was to use an oxide semiconductor, such as indium tin oxide. However, even though this system also shows good memory operation, the carrier mobility is much lower than in Si, causing problems that are unacceptable for high-performance nonvolatile memory elements. The use of a semiconducting tube as the conducting channel and a ferroelectric as a gate insulator alleviates all of these problems: the new design allows the carriers in the channel to be controlled by the spontaneous polarization of the ferroelectric film, while keeping the high-conduction characteristics of CNTs. In the demonstrated prototypical device, a 400â•›nm thick PZT (PbZr0.5TiO5O3) film was used. The threshold voltage was found to be shifted in the positive direction when Vg was swept from negative to positive bias and was shifted in the negative direction when Vg was swept from positive to negative. This gave a clear hysteresis loop and bistable current values at Vg = 0, due to the spontaneous polarization of the electric field (the effects of charge trapping in the dielectric was found to be

3-7


negligible compared to the effects due to the ferroelectric, at least for small sweeping range of Vg).

3.2.5â•‡FET-Based Memory Elements: Optoelectronic Memory In all of the CNFET-based memory devices mentioned thus far in this chapter, the information is stored by applying a large enough gate voltage to store or remove the charge in, or from, the traps (for the ferroelectric-based FET, the gate is used to flip the polarization field). In Star’s CNFET-memory (p-type) device, the information is still stored as electric charges (Star et al. 2004), however, it differs in that the writing mechanism relies on the use of optical illumination of a photosensitive polymer (which is part of the gate electrode) that converts photons into electric charge stored in the vicinity of the channel. The information is read by measuring the source–drain current, while it is erased (the charge is removed) using a large gate voltage. In other words, this optoelectronic memory is written optically and read and erased electrically. The threshold voltage change compared to the non-illuminated system was measured to be as large as +2â•›V after optical excitation, suggesting a charge transfer of about 300 electrons per micron in the tube.

3.2.6â•‡Redox Active Molecules as Storage Nodes In this experimentally built system, the overall configuration and operating principle consists of a nanotube or nanowire FET functionalized with redox active molecules, where an applied Source

gate voltage or source–drain voltage pulse is used to inject net positive or negative charge into the molecular layer (Duan et al. 2002). The oxide layer, on the surface of the channel, serves as a barrier to reduce charge leakage between the molecules and the channel, and thus maintains the charge state of the redox molecules, thereby guaranteeing nonvolatility. The charged redox molecules gate the FET to a logic on state with higher channel conductance or the off state with lower channel conductance. Positive charges in the redox material act just like a positive gate, leading to an accumulation of electrons and an on state in n-type semiconducting nanowire-based FETs. For p-type FETs, the same effect is obtained with negative charges in the redox layer, which cause an increase in majority carrier density, in this case holes, in the channel (Figure 3.5). Mannick et al. also described a similar device, using SWCNTs as channels, where the conductance is switched on or off, guided by chemical reactivity of a CNT in H2SO4 (Goldsmith et al. 2007, Mannik et al. 2006).

3.2.7â•‡Two-Terminal Memory Devices Transistor devices consist of three electrodes: a source, a drain, and a gate terminal. The source and the drain are good metals (Au, Pt, Co, etc.) and the gate terminal is typically made up of a dielectric layer (e.g., SiO2) sandwiched between the channel and a heavily doped Si-wafer, which is hooked up to the gate battery. The present authors have theoretically proposed a modified version of the FET memory device where the physical gate is replaced by a virtual electrode. The role of the

Drain + + + + + + + + + + + +

Silicon oxide 1 μm

Silicon back gate

+ + + + + + + + + + + + “Write 1”

n-Type on

“Write 0” – – – – – – – – – –– –

Semiconductor

– – – – – – – – – – – – n-Type off

Oxide

Nanowire Redox molecules

FIGURE 3.5â•… Nanowire-based nonvolatile devices. The devices consist of a semiconductor nanowire (NW) configured as a FET with the oxide surface functionalized with redox active molecules. The top-middle inset shows a scanning electron microscope (SEM) image of a device, and the lower circular inset shows a TEM image of an InP NW highlighting the crystalline core and surface oxide. Positive or negative charges are injected into, and stored in, the redox molecules with an applied gate or bias voltage pulse. In an n-type NW, positive charges create an on or logic “1” state, while negative charges produce an off or logic “0” state. (Reprinted from Duan, X.F. et al., Nano Lett., 2(5), 487, 2002. With permission.)

3-8


V (a)

(b)

FIGURE 3.6â•… (a) Schematic representation of the typical setup used to probe the information stored in the nonvolatile memory element based on the encapsulation of a donor–acceptor molecule inside a metallic nanotube. In (a) the system is made up a tetrafluorotetracyano-pquinodimethane molecule encapsulated inside a (10, 10) armchair nanotube. (b) Examples of bistable orientations of the C=C bond of tetracyanoethylene (TCNE) inside a (9, 0) zigzag CNT. (Reprinted from Meunier, V. and Sumpter, B.G., Nanotechnology, 18(42), 424032, 2007. With permission.)

gate is filled by a single donor or acceptor molecule embedded inside a metallic nanotube channel (Meunier and Sumpter 2007, Meunier et al. 2007) (Figure 3.6). The resulting device is a two-terminal system where the conduction properties can be turned on and off by modifying the position of the donor (or acceptor) molecule relative to the nanotube inner core. The intrinsic gating effect works for any nanotube-molecule couple, as long as the molecule possess two stable positions corresponding to a different interacting scheme with the nanotube host (bistability). The gating mechanism works as follows: In one orientation, charge transfer between the molecule and the nanotube imposes a suppression of the current. In another orientation, when the interaction is not only purely electrostatic (charge transfer), but also includes significant directional binding, the gating effect is suppressed and current flow is allowed. It was proposed that the molecule’s orientation can be modified by imposing a lateral stress to the nanotube, changing the local atomic arrangement around the molecule, which in turn results in its flipping. Other optional writing mechanisms include the use of a magnetic field, optoelectronics, etc. This example underlines the importance of developing a memory device with a well-defined and well-characterized storage node (here the storage node is a single molecule). It is also a memory device that lies at the boundary of FET-type system and NEMS device, as it involves both transistor-like technology and “moving parts,” on which NEMS are based, as shown in Section 3.3. Tour and coworkers proposed a memory element, not based on FET operation, but relying on the conformation of a molecule as the storage media (He et al. 2006). The idea is based on the development of a metal-free silicon-molecule-nanotube test bed for exploring the electrical properties of single molecules that demonstrated a useful hysteresis I–V loop for memory storage.

Epoxy HfAlO CNT Metal gate

Metal gate

HfAlO

Control oxide (High-k HfAlO dielectric) CNT

CNT CNT CNT Tunneling oxide (High-k HfAlO dielectric) Si substrate

(a)

Interfacial layer

5 nm

Si substrate

(b)

FIGURE 3.7â•… (a) Schematic structure of the HfAlO/CNTs/HfAlO/Si MOS memory structure of Lu and Dai where a nanotube is used as the Â�storage node. (b) High-resolution transmission electron microscopy (HRTEM) image of CNTs embedded in HfAlO control and tunneling layers. (Reprinted from Lu, X.B. and Dai, J.Y., Appl. Phys. Lett., 88(11), 113104-3, 2006. With permission.)

3-9


Nonvolatile memory for more than 3 days with cyclability of greater than 1000 read and write–erase operations was obtained for a molecular interface between the Si and SWCNTs consisting of π-conjugated organic molecules. In this case, the hysteresis in the I–V curve resulted from a conductivity change at high-voltage bias. The conduction state could be easily switched from high to low conductivity, by using a +5â•›V pulse, and makes it possible for the Si-molecule–SWCNT junction to be used as a nonvolatile memory element. The underlying mechanism for the switching was examined and it was concluded that it was not due to a thermoionic process but likely from a tunneling process across the molecule. Akdim and Pachter suggested that the switching in the device may be driven by conformational changes in the molecule upon the application of an electric field and that the nature of the contact at the interface of the SWCNT mat plays an important role in the switching (Akdim and Pachter 2008).

3.2.8â•‡ Using Nanotubes as Storage Nodes Some authors have reported CNFETs, where nanotubes were used as the storage node. Yoneya et al. proposed a 40â•›K device with a crossed nanotube junction where one of the tubes is used as channel and the other one is used as floating gate (memorystorage node) (Yoneya et al. 2002). In this case it is the large capacitance property of the nanotube that is exploited. The floating gate is charged and discharged using a back gate. The main disadvantage of this approach is the requirement of a low operating temperature. Other reports have focused on room-temperature use of CNTs as storage nodes. Most notably, Lu et al. developed FET-based memory elements (Chakraborty et al. 2008, Lu and Dai 2006) in which the nanotubes are embedded inside the gate oxide in the metal oxide semiconductor structures. The memory structure is made up of an HfAlO/CNTs/HfAlO/Si structure. HfAlO was chosen as the tunneling and control oxides in the memory structures because of its promising performance for high-k gate dielectric applications and floating device applications. The schematic structure of a floating-gate memory device using CNTS as the floating gate is shown in Figure 3.7. The p-type substrate is designed as the current channel and the CNTs are designed to be embedded in the HfAlO film and act as the charge-storage nodes. Excellent long-term charge retention characteristics are expected for the memory structure using CNTs as a floating gate due to their hole-trapping characteristics, as is demonstrated in Lu and Dai’s papers. While short-term charge retention was not found to be excellent in their prototype device, the memory window was found to remain at a reasonably large value over the long term.

3.2.9â•‡ Conclusions CNFET-based devices for memory applications have received tremendous interest as is witnessed by the numerous published works outlined above. In all the examples presented, the working principle is that the channel of the transistor consists of a doped semiconducting CNT. For all the examples presented above,

with the exception of those shown in Section 3.2.8, the storage node is usually located close to the channel but is not the channel itself, and depends dramatically on the surface chemistry of the dielectric, nanotube, and dielectric–nanotube interface. In those cases, it is the charge injection into the defects or charge traps in the dielectrics or interface responsible for the bistability and the memory properties.

3.3â•‡ NEMS-Based Memory 3.3.1â•‡ NEMS: Generalities NEMS are made of electromechanical devices that have critical dimensions from hundreds to few nanometers (Ke and Espinosa 2006a,b). By exploiting nanoscale effects, NEMS offer a number of unique properties, which in some cases can differ significantly from those of the conventional microelectromechanical systems (MEMS). Those properties pave the way to applications such as force sensors, chemical sensors, and ultrahigh frequency resonators. For instance, NEMS operate in the microwave range and have a mechanical quality that allows low-energy dissipation, active mass in the femtogram range, unprecedented sensitivity (forces in the attonewton range, mass up to attograms, heat capacities below yoctocalories), power consumption on the order of 10 attowatts, and high integration levels (Ke and Espinosa 2006a,b). The most interesting properties of NEMS arise from the behavior of the active parts, which is typically in the form of cantilevers or doubly clamped beams with nanoscale dimensions. In NEMS, the charge controls the mechanical motion, and vice versa. The presence of mechanical motion demands that the moving element to possess high-quality mechanical properties, including high strengths, high Young’s moduli, and low density. While limitations in strength and flexibility compromise the performance of Si-based NEMS actuators, CNTs are better candidates to realize the full potential of NEMS, in part due to their one-dimensional structure, high aspect ratio, perfect terminated surfaces, and exceptional electronic and mechanical properties. These properties, now complemented by significant advances in growth and manipulation techniques, make CNTs the most promising building blocks for next-generation NEMS (Bernholc et al. 2002, Chakraborty et al. 2007, and Yousif et al. 2008). A majority of CNT-based NEMS devices exploit the specific propensity of CNTs to respond efficiently to capacitive forces. Capacitive forces can develop between a nanotube and a gate electrode when the presence of a gate potential induces the rearrangement of a net charge on the nanotube’s surface, which in turn causes the appearance of a capacitance force between the nanotube and the gate. Because the nanotube is very flexible, the capacitance force bends the nanotube toward the gate electrode. This phenomenon hinges on two key properties of the nanotube: a large capacitance due to its shape and size and its exceptional mechanical flexibility. In addition to capacitance forces, other forces play a role in nanotube-based NEMS: elastic, van der Waals (vdW), and short-range forces. It is the delicate balance between those forces that makes the


3.3.2â•‡Carbon Nanotube Crossbars for Nonvolatile Random Access Memory Applications Rueckes et al. proposed one of the most promising applications of CNTs for memory applications (Rueckes et al. 2000). The device exploits a suspended SWCNT crossbar array for both I/O and switchable bistable elements with well-defined on and off states. The crossbar consists of a set of parallel SWCNTs on a substrate and a set of parallel SWCNTs that are suspended on a periodic array of supports (Figure 3.8). The storage node is found at each place where two nanotubes cross. The bistability is obtained from a balance between the elastic energy (corresponding to the bending energy of the tube, having a minimum for a non-bent tube), and the attractive vdW energy (which creates a minimum corresponding to a situation where the upper nanotube is deflected downward into contact with the lower nanotube), as shown qualitatively in Figure 3.9. Since one minimum corresponds to a large vertical distance between the two mutually perpendicular arrays (defined roughly by the height of the supports), it corresponds to a situation where no current can flow between the two wires (this is the off state). At the second minimum (roughly 0.34â•›nm, the distance between two tubes in a bundle), tunneling current is possible as the tube–tube distance is small enough to allow wave-function overlap between the two tubes. The “bent” geometry therefore corresponds to the on state.

The device can be switched between the on and off situation by transiently charging the nanotubes to produce attractive or repulsive electrostatic forces. In the integrated system, electrical contacts are made only at one end of each of the lower and upper sets of the nanoscale wires in the crossbar array, which makes it possible to address many device elements from a limited number of contacts. It was found that there is a wide range of parameters that yield a bistable potential for the proposed device configuration. The robustness of the two states Â�suggests: this architecture is tolerant to variations in the structure. The other important feature of this device is the large difference in resistance between the two states (this is the key property of electron tunneling: the

Tube deformation energy Potential energy

functioning of nanotube-based NEMS possible. Here we review a few examples of NEMS implementation for memory applications. We will discuss crossbar nonvolatile RAM, nano-relays, feedback-Â�controlled nano-Â�cantilevers, electro-actuated multiwalled nanotubes, linear-bearing nano-switches, and telescoping nanotube devices.

(a)

Tube-tube distance van der Waals interaction Potential energy

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Potential energy

(b)

FIGURE 3.8â•… NRAM device made from a suspended nanotube device architecture with a three-dimensional view of a suspended crossbar array showing four junctions with two elements in the on state and two elements in the off state. The on state corresponds to nanotubes in contact and the off state to nanotubes separated. The substrate consists of a conducting layer (dark gray) that terminates in a thin dielectric layer (light gray). The lower nanotubes are supported directly on the dielectric film, whereas the upper nanotubes are suspended by periodic inorganic or organic supports (gray blocks). A metal electrode, represented by yellow blocks, contacts each nanotube. (Reprinted from Rueckes, T. et al., Science, 289(5476), 94, 2000. With permission.)

(c)

Tube-tube distance

Total energy

Bistability

Tube-tube distance

FIGURE 3.9â•… Interaction energies between two SWCNTs (one top and one bottom) in the device shown in Figure 3.8. (a) Mechanical strain in the top nanotube and (b) attractive van der Waals interaction, showing a minimum at the nanotube–nanotube distance of about 0.34â•›n m. Plot in (c) shows the resulting bi-stable potential well used for storing information. The first minimum is in the on state (contact) and the second minimum is the off state (physical separation). The existence of the energy barrier between the two minima ensures nonvolatility.

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tunneling resistance decreases exponentially with the separation), which makes the device operation very reliable. The main drawback of the proposed geometry remains however, that at such small dimensions, the junction gap size imposes significant challenges in the nanofabrication of parallel device arrays. The concept proposed by Rueckes et al. was further developed by the Nantero company, which has devoted a particular important effort toward integrating nanotube array-based memory devices for practical applications. In an IEEE communication in 2004, Ward and coworkers improved the initial device by proposing a novel technique to overcome the hurdle of manipulating individual nanotube structures at the molecular level (Ward et al. 2004). This technique allows CNT-based NEMS devices to be fabricated directly on existing production CMOS fabrication lines. The approach relies on the deposition and lithographic patterning of a 1–2â•›nm thick fabric of nanotubes, which retain their molecular-scale electromechanical characteristics, even when patterned with 80â•›nm feature sizes. Because the nonvolatile memory elements are created in an allthin-film process, it can be monolithically integrated directly within existing CMOS circuitry to facilitate addressing and readout. The transfer of the NRAM fabrication process to a commercial CMOS foundry is ongoing, and, when commercialized, will be the first actual application of CNTs for their unique electronic properties. A modified NEMS switch using a suspended CNT was studied by Cha et al. (2005). In that memory element, the device has a triode structure and is designed so that a suspended CNT is mechanically switched to one of two self-aligned electrodes by repulsive electrostatic forces between the tube and the other self-aligned nanotube electrodes. One of the selfaligned electrodes is set as the source electrode and the third is the gate electrode. The nanotube is suspended between the two other electrodes (acting as a drain electrode). As the gate bias increases, the force between the gate and the drain electrode deflects the suspended CNT toward the source electrode and establishes electric contact, which results in current flow between the source and the drain electrodes. The electrical measurements show well defined on and off states that can be changed with the application of the gate voltage. Dujardin et al. exposed another type of memory device based on suspended nanotubes. In their NEMS, multiwall carbon nanotubes (MWCNTs) are suspended across metallic trenches at an adjustable height above the bottom electrode (Dujardin et al. 2005). When a voltage is applied between the nanotube and the bottom electrode, the nanotube switches between conducting and nonconducting states by physically getting closer or farther from the bottom electrode. The device acts as a very efficient electrical switch and can be improved by surface functionalizing the bottom electrode with a self-assembled monolayer. The nanotube bridge was also experimentally studied by Kang et al. who highlighted the importance of the interatomic interactions between the CNT bridge and the substrate and the damping rate on the operation of the NEM memory device as a nonvolatile memory (Kang et al. 2005).

3.3.3â•‡ Nanorelays CNT nanorelays are three terminal devices made up of a conducting CNT positioned on a terrace on a silicon substrate (Figure 3.10). It is connected to a fixed-source electrode (single clamping) and a gate electrode is placed beneath the nanotube. When a bias is applied to it, charge is induced in the nanotube and the resulting capacitance force established between the gate and the tube triggers the bending of the tube, whose end is brought in closer contact to the drain electrode, thereby closing an electric circuit. This device was first proposed by Kinaret et al. (2003), and later demonstrated experimentally by Lee et al. (2004) and by Axelsson et al. (2005). As in other memory devices, the mechanism hinges on the existence of two stable positions (bistability), corresponding to the on and off states, respectively. The advantage of the device is that it is characterized by a sharp transition between the conducting and the nonconducting states. The large variation in the resistance is due to the exponential dependence of the tunneling resistance on the tube-end–drain distance. The transition occurs at fixed source–drain potential, when the gate voltage is varied. In this case, the tube is bent toward the drain and a large current is established, which subsequently disappears when the tube is moved far from it. Aside from possessing the property of a memory element, this device also allows for the amplification of weak signals superimposed on the gate voltage. A number of investigations, theoretical and experimental, have been performed to further characterize and optimize the practical applications of the nanorelay for memory purposes. The fundamental property of the nanorelay is reached for a balance of the vdW, adhesive (close range), electrostatic (capacitance), and elastic forces. The so-called pull-in voltage (voltage required to bring the tube in contact with the drain) was first studied by Dequesnes et al. for a two-terminal device where the entire substrate acted as the gate electrode (Dequesnes et al. 2002). It was found that vdW forces reduce the pull-in voltage, but do not qualitatively modify the on–off transition. The importance of vdW forces decreases with decreasing tube length and increasing terrace height and tube diameter. The transition is, however, very sensitive to short-range attractive forces, which enhance the tendency of the CNT to remain in contact (stiction effect). Stiction makes S

q

x

h G

D

FIGURE 3.10â•… Nanorelay: schematic picture of the model system consisting of a conducting CNT placed on a terraced Si substrate. The terrace height is labeled h, and q denotes the excess charge on the tube. The CNT is connected to a source electrode (S), and the gate (G) and drain (D) electrodes are placed on the substrate beneath the CNT. The displacement x of the nanotube tip is measured toward the substrate. Typical practical lengths are L ~ 50–100â•›nm, h ~ 5â•›nm. (Reprinted from Kinaret, J.M. et al., Appl. Phys. Lett., 82(8), 1287, 2003. With permission.)

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the device unusable as it remains stuck in the on position (Jonsson et al. 2004a,b). The effect of the surface forces can be visualized by means of a stability diagram, which reveals the existence and positions of zero net force (local minima) on the cantilever as a function of gate voltage and tube deflection. A typical curve was computed by Jonsson et al. and is reproduced in Figure 3.11. A stability diagram with more than one local equilibrium for a specific voltage results in a hysteretic behavior in the I–Vg characteristics (more than one position of zero net force means that there actually are three such positions, one of which is not stable). The net force is positive to the right of the curve and negative to the left of the curve, so that, when lowering the voltage for a tube in contact with the drain electrode, it will not release until there is no stable position at the surface. The surface forces exacerbate this effect. The stiction problem corresponds to a stable nanotube position at the surface that can no longer be modified by the application of a gate voltage. The stiction problem is a general issue encountered in many nanotube-based devices. The problem of stiction hinders the use of SWNTs (because they are too flexible) as a bistable NEMS switch. MWNT and SWNT bundles are better candidates for bistable switching (Yousif et al. 2008). This problem can be alleviated, for example, by using stiffer (by avoiding total collapse

B

1

C

A

3.3.4â•‡ Feedback-Controlled Nanocantilevers

x (h)

0.8 With surface forces

0.6

Without surface forces

0.4

0.2

0

of the tube, e.g., by using MWCNTs) or shorter tubes. It can also be avoided by applying a layer of adsorbates. Fujita et al. suggested a very different approach with a modified device design. They proposed using a floating gate that is built between a bundle of CNTs and electrically isolated from the input and output electrodes. The bundle of CNTs is, therefore, bent by the electrostatic force between the floating and control gate, thereby allowing the on–off threshold voltage to be controlled by changing the back-gate voltage, similar to silicon MOSFET (Fujita et al. 2007). A consequence of reducing the possibility of stiction is that a higher pull-in voltage is required, and field emission from the end of the tip can be important. Field emission will become important when the effective potential at the tube end is equal or larger to the work function of the nanotube (i.e., about 4.5â•›V). For a set of design parameters, field emission can be deliberately sought after (noncontact mode). In the contact mode described above, tunneling current passes from the tube to the drain electrode. In the noncontact mode, the device is designed (short tube) in such a way that the tube is never in physical contact with the drain electrodes. In that case, an electron flow is established via a field emission mechanism. The field emission current onsets with a sufficiently large source–drain voltage and then increases nonlinearly as the source–drain voltage (i.e., the applied field) is further increased. The nanotube can be switched very quickly between the on and off states, and the nanorelay operation was evaluated to work in the gigahertz regime (Jonsson et al. 2004a).

0

1

2

3

Vg (V)

4

5

6

7

FIGURE 3.11â•… Nanorelay stability diagram with and without surface forces computed with parameters given in Jonsson et al. (2004b). The curve shows the positions of zero net force on the tube (or local minima) as functions of gate voltage (at constant source voltage = 0.01â•›V) and deflection x (in units of h, see Figure 3.10). The large arrows show the direction of the force on each side of the curves, indicating that one local equilibrium state is unstable. The required voltage for pulling the tube to the surface (pull-in voltage) is given by A (~6.73â•›V). A tube at the surface will not leave the surface until the voltage is lower than the “release voltage,” B and C in the figure. Note that A > B, C, which indicates a hysteretic behavior in the current-gate voltage characteristics, a feature significantly enhanced by surface forces. (Reprinted from Jonsson, L.M. et al., Nanotechnology, 15(11), 1497, 2004a. With permission.)

A variation of the nanorelay device presented in the previous section was proposed (Ke and Espinosa 2004), as shown in Figure 3.12a. It is made of an MWCNT placed as a cantilever over a micro-fabricated step. A bottom electrode, a resistor, and a power supply complete the device circuit. Compared to the nanorelay, this is a two-terminal device, providing more flexibility in terms of device realization and control. When the applied potential difference between the tube and the bottom electrode exceeds a certain potential, the tube becomes unstable and collapses onto the electrode. The potential that causes the tube to collapse is defined as the pull-in voltage, already encountered above (Dequesnes et al. 2002). Above the pull-in voltage, the electrostatic force becomes larger than the elastic forces and the CNT accelerates toward the electrode. At small nanotube-tip electrode distance (of the order of 0.3–1.0â•›nm), substantial tunneling current passes between the tip of the tube and the bottom electrode. The main difference with the nanorelay design is the presence of a resistance R placed in the circuit; with the increase in the current, the voltage drop at R increases, which causes a decrease in the effective tube-electrode potential and the tube moves away from the electrode. It follows that the current decreases and a new equilibrium is reached. Without damping in the system, the cantilever would keep on oscillating at high frequency. However, damping is usually present, and the kinetic energy of the CNT dissipates. The tip ends up at a position where the electrostatic force is equal to the

3-13

Carbon Nanotube Memory Elements CNT H

U i≠0

r

Δ

R

(a) 10–6 10–8

10

Pull-out

i (A)

Δ (nm)

100

Pull-in

(b)

0

10

20

30

0

40

U (V)

(c)

Pull-in

Pull-out

10–12

1 0.1

10–10

0

10

20

30

40

U (V)

FIGURE 3.12â•… (a) Schematic of a feedback controlled nanocantilever device. H is the initial step height and Δ is the gap between the deflected tip and bottom conductive substrate. R is the feedback resistor, as explained in the text. (b,c) Characteristic of pull-in and pull-out processes for a device with L = 500â•›nm, H = 100â•›nm, and R = 1â•›GΩ. (b) The relation between the gap Δ and the applied voltage U. (c) The relation between the current I in the circuit and the applied voltage U. (Reprinted from Ke, C.H. and Espinosa, H.D., Appl. Phys. Lett., 85(4), 681, 2004. With permission.)

elastic one and a stable tunneling current is established. This is the “lower” equilibrium position for the CNT cantilever. If the applied voltage decreases, the cantilever starts retracting. When the applied potential is lower than the so-called pull-out voltage, the CNT leaves the “lower” position and returns back to the upper equilibrium position where the tunneling current significantly decreases and becomes negligibly small. The pullin and pull-out processes follow a hysteretic loop for the applied voltage and current in the device. The lower and upper positions correspond to the on and off states of the switch, respectively. The lower equilibrium state is very robust, thanks to the existence of the tunneling current and the feedback resistor. (The concept of robustness is related to the value of the ratio of current in on and off states. A ratio of 104 is usually needed for such a device to be considered robust.) Representative characteristics are reproduced in Figure 3.12b. This device has been demonstrated experimentally by Ke and Espinosa (2006a,b) and Ke et al. (2005), showing striking agreement between theoretical prediction and experimental realization. The presence of the resistor allows adjusting the electrostatic field to achieve the second stable equilibrium position. This is an advantage compared to the NRAM system developed by Rueckes et al., since it reduces the constraints in fabricating devices with nanometer gaps between the freestanding CNTs and the substrate, thereby increasing reliability and tolerance to variability in fabrication parameters. The main drawback of this device for memory applications is that the memory becomes volatile; when the applied potential is turned off, the tube retracts back to the upper position and the information stored in the position of the tube is erased.

3.3.5â•‡Data Storage Based on Vertically Aligned Carbon Nanotubes Another type of memory cell using a nanotube cantilever is based on the work of Kim et al. on nanotweezers (Kim and Lieber 1999). The nanotweezer consists of two vertically aligned MWCNTs that are brought together or separated by the application of a bias potential between them. Motivated by that realization, Jang et al. proposed a memory system that improves the integration density and provides a simple fabrication technique applicable to other types of NEMS (Jang et al. 2005). The device consists of three MWCNTs grown vertically from predefined positions on the electrodes, as shown in Figure 3.13. The first nanotube is electrically connected to the ground and acts as a negative electrode. Positive electrostatic charges build up in the second and the third tubes when they are connected to a positive voltage. The third tube pushes the second tube toward the first tube, because of the forces induced when the positive bias of the third tube increases, while maintaining a constant positive bias on the second one. Above a threshold bias, the second tube makes electrical contact to the first tube, establishing the on state. The balance of the electrostatic, elastic, and vdW forces involved in the device operation determines the threshold bias. If the attractive force between the first and second tube is larger than the repulsive electrostatic forces between them, they remain held together even after the driving bias is removed (otherwise they would return to the original position). Therefore, the system can act as either a volatile or nonvolatile memory element, depending on the design parameters. Note that a two-terminal memory device can be devised using the same principles. In that

3-14

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Nb contact electrode 1st 2nd 3rd – + + + – – + + – + – + + – + – + – + – + – + + – + – + – + + –

Ni catalyst

1 2 3

SiO2 MWCNT

On-state (a)

(b)

(c)

(d)

FIGURE 3.13â•… A schematic illustration of the CNT-based electromechanical switch device proposed by Jang et al. (a) Schematic of fabrication process: Three Nb electrodes are patterned by electron-beam lithography, followed by sputtering and lift off. Similarly, Ni catalyst dots were also formed on the predefined locations for the growth of MWCNTs. The MWCNTs were then vertically grown from the Ni catalyst dots. (b) Illustration of CNT-based electromechanical switch action. (c)–(d) SEM image of the actual device: The length and diameter of the MWCNTs are about 2â•›μm and 70â•›nm, respectively. The scale bar corresponds to 1â•›μm. (Reprinted from Jang, J.E. et al., Appl. Phys. Lett., 87(16), 163114-3, 2005. With permission.)

a remarkable application of the low-friction-bearing capabilities of MWCNTs to realize a NEM switch as shown in Figure 3.14 (Deshpande et al. 2006). The switch consists of two openended MWCNT segments separated by a nanometer-scale gap. The switching occurs through electrostatically actuated sliding of the inner nanotube shells to close the gap, producing R

4

d I (μA)

case the “gate” electrode is no longer the third tube but a conventional gate that is used to separate the two tubes from their contact position. Jang et al. also reported a nanoelectromechanical-switched capacitor structure based on vertically aligned MWCNTs in which the mechanical movement of a nanotube relative to a CNT-based capacitor defines on and off states. The CNTs are grown with controlled dimensions at predefined locations on a silicon substrate in a process that could be made compatible with existing silicon technology, and the vertical orientation allows for a significant decrease in cell area over conventional devices. It was predicted that it should be possible to read data with standard dynamic random-access memory-sensing circuitry. Jang et al.’s simulations suggest that the use of high-k dielectrics in the capacitors will increase the capacitance to the level needed for dynamic random-access memory applications (Jang et al. 2008).

2

50 nm

0

3.3.6â•‡ Linear Bearing Nanoswitch This type of memory device exploits the friction properties of MWCNTs. In MWCNTs, the intershell resistance force, or friction for the sliding of one shell inside another, is known to be negligibly small, as small as 10 −14 N/atom (Cumings and Zettl 2000, Yu et al. 2000). The inner core of an open MWCNT can therefore be expected to move quasi-freely forward and backward along the tube axis, provided that at least one end of the tube is open. Deshpande et al. presented

0

2

4

6

V (V)

FIGURE 3.14â•… CNT linear bearing nanoswitch. Main panel: abrupt rise in conductance upon sweeping of source–drain voltage (open squares) and subsequent latching in the on state (filled squares). Lowerright inset: SEM image of the device after latching showing that the gap has closed. Upper-right inset: schematic cup and cone model of the system. (Reprinted from Deshpande, V.V. et al., Nano Lett., 6(6), 1092, 2006. With permission.)

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a conducting on state. For double-walled nanotubes, in particular, a gate voltage was found to restore the insulating off state. The device was shown to act as a nonvolatile memory element capable of several switching cycles. The authors indicate that the nanotubes are straightforward to implement, are self-aligned, and do not require complex fabrication, potentially allowing for scalability. The nanotube-bearing device is fabricated in high yield by using electric breakdown to create gaps in a single freestanding MWCNT device, producing an insulating off state. The device is actuated using electrostatic forces between the two ends of the tube that are connected to external electrodes. The force triggers a linear bearing motion that telescope the inner shells in the two multiwall or doublewall segments, so that they bridge the gap (Forro 2000). This restores electrical contact and produces the on state. Adhesion forces between the tube ends maintain the conductive state. The insulating state is controllably restored using a gate voltage, thereby completing the three-terminal nonvolatile memory devices. The gap is closed for a source–drain voltage around 9â•›V, leading to a conducting on state. At a large gate voltage (110â•›V) and small source–drain voltage (10â•›mV), the device snaps back to the zero conductance state. The explanation for the transition from the on state back to the off state is that the gate voltage imposes a bending stress on the nanotube and acts to break the connection. The use of elementary beam mechanics confirms that the force acting upon the two portions of the nanotube is significantly larger than the adhesion forces, allowing for the gap to be reopened. The use of the exceptional low friction between individual walls in MWNTs is also the basic property exploited in a number of theoretically proposed structures for nanotubebased memory elements. In the following paragraphs, we will brief ly review the work of Maslov (2006) and Kang and Jiang (2007). Maslov proposed a concept that uses vertically aligned MWCNTs, that is first opened (or “peeled”) layer by layer, so that the inner core is able to move along the vertical tube axis. By mounting another dielectric above the nanotube at a specific distance from the tube caps, two stable vdW states for the inner core are created and provide for nonvolatile data storage. The two stable states are (1) when the inner tube is away from the top electrode, stabilized by the interaction with the outer shell (off state, where the circuit is open) and (2) when the inner tube is in contact with the top electrode, stabilized by vdW interaction with the top electrode (on state, where the circuit is closed). The switching between the two states is realized by exploiting the electrostatic force resulting from the positive charging of the top (or bottom) electrode and negative charging of the tube inner core, pulling the core toward (away from) the top electrode. For nonvolatility purposes, it is important that the adhesion force is large enough to maintain the tube position, even when the battery potential is turned off. Another interesting theoretical proposal using the friction properties of multiwalled systems was made by Kan and Jiang. The conceptual design and operation principle of the

(a)

–

(b)

(c)

–

V2

V1

+

+

C-C vdW energy CNT-metal binding energy Total energy State 1 (d)

State 0

State 2

d

FIGURE 3.15â•… Electrostatically telescoping nanotube nonvolatile memory device. (a) The initial equilibrium position, (b) the core CNT contacts with the right electrode with V1 and (c) the core CNT contacts with the left electrode with V2. (d) Energetic schematics of the telescoping MWCNT-based memory, showing the presence of two local minima (bistability) and an energy barrier between them (nonvolatility). (Reprinted from Kang, J.W. and Jiang, Q., Nanotechnology, 18(9), 095705, 2007. With permission.)

MWCNT-based storage are illustrated in Figure 3.15 (Kang and Jiang 2007). The main structural element of the devices is composed of a MWCNT deposited on a metallic electrode. The metallic electrode clamps the outer shell of the multiwall tube, while the inner core of the tube is allowed to freely slide back and forth (both ends of the tube outer-shell are opened). This telescoping device possesses three stable positions, resulting from the balance between the wall–wall interactions and the tube–metal electrode interactions. Two minima correspond to inner tube positions close to the electrode (allowing current to flow between that electrode and the bottom one, attached to the outer core of the tube). A third minimum corresponds to the tube positioned completely inside the outer shell. The movement of the inner core is realized using electrostatic forces induced by the voltage differences between the various electrodes. This leads to reversibility and nonvolatility, provided that the electrode material is carefully chosen (large enough vdW force to ensure contact when bias is turned off, while avoiding stiction where the nanotube cannot be separated

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from the electrode anymore due to small range forces, as illustrated in Figure 3.15b).

3.4â•‡Electromigration CNT-Based Data Storage

3.3.7â•‡ Conclusions

During the past decade there have been numerous promising concepts developed, based on utilizing ionized endohedral fullerenes and nanotubes as high-density nonvolatile memory elements. In this section the fundamental concepts underlying the operation and development of these types of systems are described and two specific examples are discussed in detail.

In this section, we have presented the state of the art of fundamental research of the use of CNTs NEMS for memory applications. The CNT-NEMS are particularly interesting because they exploit the two most remarkable properties of CNTs, namely, their electronic and mechanical properties. The viability of CNT-based NEMS switches and their comparison with their CMOS equivalents was recently presented by Yousif et al. (2008). A detailed analysis of performance metrics regarding threshold voltage control, static and dynamic power dissipation, and speed and integration density revealed that apart from packaging and reliability issues (which are the subject of intense current research), nanotube-based switches are competitive in low power, particularly low-standby power, logic, and memory applications.

(a)

3.4.1â•‡A “Bucky Shuttle” Memory Element One of the earliest concepts for a CNT nonvolatile memory element was based on an ionized fullerene that could be electrically shuttled from one energy state to another inside a larger fullerene or CNT, as shown in Figure 3.16 (Kwon et al. 1999).

(b)

Energy (eV)

0

–1

“bit 1”

“bit 0” –2 –8

–6

–4

–2

(c)

0

2

4

6

8

Position (Å) A

B

C

C

D +

“bit 1”

+

+ a (d) Top view

b

c

d

a b Side view

+ “bit 0” c

d

FIGURE 3.16â•… Illustration of the “bucky shuttle” concept as a memory element: (a) Transmission electron micrograph of a synthesized carbon structure showing a fullerene encapsulated inside a short nanotubule. (b) Molecular model depicting the ideal structure. (c) Position-dependent energy of an endohedral K@C60+ within the structure shown in (b). The results (total energy) were obtained from molecular dynamics simulations without an electric field (solid line) and following the application of a small electric field on the system (dashed line). A schematic of the corresponding C 60 position and intrinsic state (“bit 0” or “bit 1”) is shown on the figure. (d) A schematic of the overall “bucky shuttle” memory element concept that corresponds to a high-density memory board. (Reprinted from Kwon, Y.K. et al., Phys. Rev. Lett., 82(7), 1470, 1999. With permission.)

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One advantage of this concept was an appropriate material could be self-assembled from elemental carbon via thermal treatment of diamond power. A unique geometry of a nanotube encapsulating a fullerene, where a small fullerene structure could reside at one of the two ends of a capped nanotube or larger fullerene was experimentally identified (Figure 3.16a), and this structural arrangement became the basis for the original concept of a “bucky shuttle” memory element. In this system the on (bit 1) and off (bit 0) states correspond to the two ends of the nanotube that are local minima in the energy landscape of the endohedral fullerene (Figure 3.16c). The position of the ionized endohedral fullerene (obtained by doping with potassium, e.g., K+@C60) can be manipulated by applying a bias voltage across the ends of the nanotube. The energetics of a switching field of 0.1â•›V/Å generated by applying a voltage of 1.5â•›V is shown in Figure 3.16c. With such a small electric field, integrity problems are not expected since graphitic structures are known not to degrade under fields of VT0 and the “down” 5-1

5-2


Gate (VG)

Oxide

IDS

Nano-dots

IDS

Drain

Source

Gate

Source

Oxide

Drain

Bulk VT0

Channel

VG

FIGURE 5.1â•… Schematic behavior of a MOS transistor. Floating gate

Control gate (VG) Oxide

Source

IDS

IDS

Q>0

FIGURE 5.3â•… Floating nano-dots. Q> a is a/(4t1) ∼ 5.83 × 10−2 for CC(1)/(4πεa) and a/(4(t1 + a)) ∼ 4.73 × 10−2 for CC(2)/(4πεa). In the figure, we see the beginning of this asymptotic behavior. Because the limiting value of t2, in our simulation, is of the same order of t1, the smallest values seen on the figure are approximatively twice the limiting value. For t2/a > 2, the ratio CC(2)/CS remains quite constant, varying from 0.11 to 0.09. The ratio CC(1)/CS varies from 0.15 to 0.1. Finally, in all the range of values of t2/a presented here, only the PPA(2) allows to approximate CS with finite relative error. The difficulty consists in finding a numerical value for the surface S that makes PPA match the capacitance. To end this comparison, the dimensionless coefficients κ1, β(1), and β(2) = α are compared. For great values of t2/a, β(1) tends to 1 while β(2) tends to 0.5. The limiting value of κ1 is also 0.5 (because C2 tends to the capacitance of a single half-sphere). On the other hand, for small t2/a, the limiting value for β(2) is a/(t1 + 2a) ∼ 0.15, while β(1) tends to 0 and C1 also to 0 (for small t2, only C2 contributes to the value of CS). In Figure 5.7, we plot the coefficients κ1, β(1), and β(2) versus t2/a for the same previous configuration. On this figure, it appears that the coefficient β(2) underevaluates κ1 for t2/a > 0.25. The greatest relative difference between κ1 and β(2) is observed for t2/a ∼ 1 and is about 30%. For t2/a > 3, the relative difference is lower than 10%. The coefficient β(1) is far from κ1, even in the range t2/a t2), FEM may have some problems of accuracy for computing those two parameters. To finish this appendix, let us say that a typical calculation of CS by SA takes few seconds, with a non-optimized. MATLAB • -like code, while the same extraction with FEM could takes several minutes.

References 1. Moore G.E., Cramming more components onto integrated circuits, Electronics, 38 (8), 114–117, April 1965. 2. 2007 ITRS ORTC, Public Conference, Makuhari, Japan, http://www.itrs.net. 3. Noguchi M., Yaegashi T., Koyama H., Morikado M., Ishibashi Y., Ishibashi S., Ino, K., et al., A high-performance multi-level NAND Flash Memory with 43â•›nm-node floatinggate technology, in: Proceedings of the International Electron Device Meeting (IEDM), pp. 445–448, Washington, DC, December 10–12, 2007, doi: 10.1109/iedm.2007.4418969. 4. Ishii T., Osabe T., Mine T., Murai F., Yano K., Engineering variations: Towards practical single-electron (few-electron) memory, in: Technical Digest of IEDM’00, San Francisco, CA, December 11–13, 2000. 5. Wang H., Takahashi N., Najima H., Inukai T., Saitoh M., Hiramoto T., Effects of dot size and its distribution on electron number control in metal-oxide-semiconductorfield-effect-transistor memories based on silicon nanocrystal floating dots, Jpn. J. Appl. Phys., 40, 2038–2040, 2001. 6. Molas G., Deleruyelle D., DeSalvo B., Ghibaudo G., Gely M., Perniola L., et al., Degradation of floating gate reliability by few electron phenomena, IEEE. Trans. Electron Devices, 53 (10), 2610–2619, 2006. 7. Redaelli A., Pirovano A., Benvenuti A., Lacaita, A., Threshold switching and phase transition numerical models for change memory simulations, J. Appl. Phys., 103, 111101-1–111101-18, 2008. 8. Sawa A., Resistive switching in transition metal oxides, Mater. Today, 11 (6), 28–36, June 2008. 9. Naruke K., et al., Stress induced leakage current limiting to scale down EEPROM tunnel oxide thickness, in: Proceedings of International Electron Device Meeting (IEDM), pp. 424– 427, San Francisco, CA, December 11–14, 1988. 10. Bez R., et al., Introduction to flash memory, Proc. IEEE, 91 (4), 489–502, 2003. 11. Tiwari S., et al., Volatile and non-volatile memories in silicon with nano-crystal storage, in: Proceedings of the International Electron Device Meeting (IEDM), pp. 521– 524, IEDM 1995, Washington, DC, December 10–13, 1995.

5-21

12. De Salvo B., Gerardi C., Lombardo S., Baron T., Perniola L., Mariolle D., Mur P., et al., How far will silicon nanocrystals push the scaling limits of NVMs technologies? in: Technical Digest of IEDM’03, Washington, DC, December 8–10, 2003. 13. Muralidhar R., Steimle R.F., Sadd M., Rao R., Swift C.T., Prinz E.J., Yater J., et al., A 6â•›V embedded 90â•›nm silicon nanocrystal nonvolatile memory, in: Technical Digest of IEDM’03, Washington, DC, December 8–10, 2003. 14. Choi S., Choi H., Kim T.-W., Yang H., Lee T., Jeon S., Kim C., Hwang H., High density silicon nanocrystal embedded in SiN prepared by low energy ( 0, the spontaneous polarization displays a step-like increase up to 2 a value of Pc = | a11 | /2a111 at Tc = θ + ε 0Ca11 /(2a111 ) . Therefore, a first-order ferroelectric phase transition takes place in this situation, as illustrated in Figure 6.3a. Under an external field directed against the spontaneous polarization, the magnitude of P3(E3) first gradually decreases with increasing field intensity E3. When it reduces down to a certain minimum value Pmin, the “antiparallel” polarization state becomes unstable. As a result, the polarization switches by 180° into the direction parallel to the applied field. In the case of ferroelectrics with a second-order transition, Pmin = Ps / 3 , and the critical switching field equals Eth = −(4/3)a1Pmin ∼ (θ − T)3/2. The field Eth represents the thermodynamic coercive field that corresponds to a homogeneous polarization reversal in the whole crystal. The polarization-field curve resulting from the thermodynamic calculations is hysteretic, as shown in Figure 6.4a. The theoretical hysteresis loop is qualitatively similar to the experimental ones, which differ mainly by a gradual polarization

Ba2+ ac

O2–

ac (a)

G

ct

Ti4+

ac

(b)

T >>Tc

Ps

at

at

Figure 6.2â•… Schematic representation of the unit cell of BaTiO3 in the paraelectric cubic (a) and ferroelectric tetragonal (b) phases.

T > Tc T = Tc T0 < T < Tc T < T0

G T > Tc T = Tc T < Tc

P

(a)

–Ps

+Ps

–Ps

+Ps

P

(b)

Figure 6.3â•… Temperature evolution of the free energy density as a function of polarization shown schematically for ferroelectrics with the first-order (a) and second-order (b) phase transition.

6-4


6.3â•‡ Deposition and Patterning

P3/PS 1

6.3.1â•‡ General Aspects

0.5 E3/Eth –2

–1

1

2

–0.5

–1 (a) 150 nm PbZr0.52Ti0.48O3 film

Polarization (μC/cm2)

50

0

–50 –4 (b)

–2

0

2

4

Voltage (V)

Figure 6.4â•… Theoretical (a) and measured (b) ferroelectric hysteresis loops. The theoretical dependence of the normalized polarization on the normalized applied field corresponds to a homogeneous polarization reversal in the whole crystal. The experimental polarizationvoltage loop was measured for the 150â•›nm thick PZT film.

reversal seen in Figure 6.4b. The measured coercive fields Ec of bulk crystals, however, are typically several orders of magnitude lower than Eth, because the polarization switching develops in reality via the nucleation and growth of ferroelectric domains (Lines and Glass 1977). When the size of a ferroelectric crystal is reduced down to a length scale comparable to the so-called ferroelectric correlation length, its physical properties generally become sizedependent. This feature is due to the fact that ferroelectricity is a collective phenomenon resulting from a delicate balance between long-range Coulomb forces (dipole–dipole interactions), which are responsible for the ferroelectric state, and a short-range repulsion favoring the paraelectric state (Lines and Glass 1977). The scaling of physical characteristics such as the remanent mean polarization Pr = 〈P3(E3 = 0)〉 and coercive field Ec is currently in the focus of experimental and theoretical studies in the field of nanometer-sized ferroelectric capacitors. The most important results of these studies will be discussed in Sections 6.4 and 6.5.

The deposition of thin films belongs to the heart of today’s micro- and nano-electronics. In order to grow thin films with desired properties, several important issues have to be considered. Besides the choice of the deposition method, a number of process parameters are essential, such as the background pressure of the vacuum system, the deposition rate (measured in nm/s), the substrate material and temperature, and the composition of the material source. Furthermore, the process pressure is important, irrespective of the conditions in the chamber, i.e., the state of an ultra-high vacuum or the presence of a noble gas (typically argon) or reactive gases (oxygen or nitrogen). The basic principles of thin-film deposition are described in several textbooks (Chopra 1969, Maissel and Glang 1979, Bunshah 1994). Here we focus on the deposition of complex oxides by means of physical methods. As already mentioned in the introduction, thin-film deposition techniques for the growth of heteroepitaxial oxides have made tremendous progress recently. The purpose of this section is to give a short overview of the current status of the MBE, pulsed laser deposition (PLD), and sputter deposition (SD). For chemical-based methods, such as metal-organic chemical vapor deposition (MOCVD), atomic layer deposition (ALD), and chemical solution deposition (CSD), we refer the reader to the relevant papers (Oikawa et al. 2004, Kato et al. 2007, Schneller and Waser 2007). We also note that ferroelectric polymers are deposited by either a spin-on technique or the LangmuirBlodgett method (Ducharme et al. 2002). Consider a planar ferroelectric capacitor sketched in Figure 6.1. The choice of materials for the substrate, electrodes, and ferroelectric layer strongly depends on the application or the research task. For the epitaxial growth of perovskite ferroelectrics, single-crystalline substrates having small lattice mismatches with these complex oxides are preferable. At present, the commercially available single crystals of SrTiO3 are most popular, although other substrates such as MgO, KTaO3, GdScO3, and DyScO3 have also been successfully employed. When the ferroelectric overlayer is commensurate with a dissimilar thick substrate, it appears to be strained to a certain extent defined by the mismatch in their in-plane lattice parameters. Above some critical thickness, however, these lattice strains start to relax due to the generation of misfit dislocations (see Section 6.4). As for the electrodes, noble metals such as Pt, Ir (also IrO2), and Ru are used in most device applications, e.g., in FeRAMs and MEMS (Kohlstedt et al. 2005a). On the contrary, conducting complex oxides were favored so far as electrode materials in the basic research studies of scaling effects. Prominent examples of such electrode materials are SrRuO3, LaSrxMn1−xO3, and LaCa xMn1−xO3, which are routinely used in the complex-oxide heterostructures (Eom et al. 1992, Sun 1998). A comparison of the advantages and disadvantages of metal and oxide electrodes shows a delicate trade-off. Let us compare, for instance, Pt and SrRuO3.

6-5

Nanometer-Sized Ferroelectric Capacitors

On the one hand, the resistivity of a sputtered thin-film Pt at room temperature (∼10â•›μΩ cm) is much smaller than that of even a high-quality SrRuO3 (≈300â•›μΩ cm). On the other hand, the screening-space-charge capacitance density, which is important for the stabilization of ferroelectricity in ultrathin films (Pertsev and Kohlstedt 2007), is equal to 0.9â•›F/m2 for the SrRuO3 electrode and only to 0.4â•›F/m2 for the Pt electrode (Pertsev et al. 2007). This feature seems to make SrRuO3 electrodes preferable for nanoscale ferroelectric capacitors. In addition, the electrode surface roughness, crystallographic orientation of the ferroelectric layer grown on a particular electrode, and the quality of the electrode–ferroelectric interface must be taken into account. Currently, conducting complex oxides are preferred for the fabrication of the bottom electrode, whereas the top electrode can be made of Pt or other noble metal as well. In the rest of this section, we focus on entirely complex-oxide heterostructures for ferroelectric capacitors.

6.3.2â•‡ Deposition Techniques 6.3.2.1â•‡ Molecular Beam Epitaxy MBE has developed from a simple evaporation technique via the use of ultra-high vacuum (UHV) to avoid disturbances by residual gases and additional incorporation of various effusion (Knudson) cells as material sources. Figure 6.5 schematically shows an MBE system involving several material sources that allow controlled deposition of multi-element compounds.

In contrast to the deposition of most semiconductor materials such as GaN, GaAs, and InP, the growth of oxides by MBE requires relatively high partial pressure of oxygen (∼10−7 mbar) during the deposition. This is necessary to avoid the oxygen deficiency in the final film, which could seriously deteriorate the quality of a ferroelectric capacitor. Partial pumping, the use of reactive oxygen (e.g., ozone), and post-annealing of the films in a high-pressure oxygen atmosphere (several mbar) are used to supply the films with a sufficient amount of oxygen. Owing to the UHV conditions in the MBE chamber, all UHV surface techniques can be employed. This feature, indeed, constitutes the strength of MBE (Haeni et al. 2000). This tool offers the highest degree of freedom to apply sophisticated in situ analytical techniques to study films during the growth and just after the deposition without breaking the vacuum. Complex MBE systems using low-energy electron microscopy (LEEM) and Auger electron spectroscopy (AES), for example, were developed (Habermeier 2007 and Clayhold et al. 2008). The standard technique currently is the reflection high-energy electron diffraction (RHEED), which allows control of the surface chemistry of the last layer during the deposition. This technique provides an opportunity to fabricate oxide films with a definite termination at the surface (e.g., the BaO or TiO2 termination in BaTiO3 films). Many groups successfully demonstrated this approach (Logvenov and Bozovic 2008) with similar oxide materials. An obvious research goal now is to find correlations between atomic terminations at ferroelectric– metal interfaces and the electrical properties of capacitors. QCM

ass te M ome tr ec

sp

Cryo pump

r

Wafer holder

rtz Qua lance roba mic

on Electr gun Source furnaces

RHEED screen

Oxygen

Leak valve

(a)

Mechanical pump

Sources (b)

Figure 6.5â•… Schematic view of MBE system for the growth of multi-element-compound thin films (a) and a photograph of MBE chamber (b). (From Lettieri, J. et al., J. Vac. Sci. Technol. A, 20, 1332, 2002. With permission.)

6-6


An additional important feature of MBE is the low energy of deposited species. Indeed, the temperature of the material source in effusion cells or electron-beam evaporators does not exceed 3500â•›K. Hence, the corresponding thermal energy of the deposited species is about 300â•›meV. This value is an order of magnitude lower than the energies characterizing pulsed laser deposition and typical sputter deposition. Among available deposition techniques, MBE is the most flexible one with respect to the incorporation of analytical tools and offers the highest degree of atomic layer control. On the other hand, MBE systems are difficult to handle, considerable time is necessary for their maintenance, and, last but not least, special methods are needed to supply complex oxides with a sufficient amount of oxygen. Due to the highly complex machinery, experienced researchers working with MBE systems for years translated the acronym MBE as “many boring evenings.” 6.3.2.2â•‡ Pulsed Laser Deposition PLD is a very useful and flexible tool for growing oxide materials (Hubler and Chrisey 1994). A sketch of a PLD system is shown in Figure 6.6. A pulsed laser beam, from a KrF (248â•›nm) or ArF (193â•›nm) excimer laser, for example, is focused on a rotating target made of an oxide material (e.g., BaTiO3, PZT, or SrRuO3). The oxygen gas pressure during ablation can be varied from 10−7 to 0.5â•›mbar. Owing to the intense laser beam, a plasma containing energetic ions, electrons, neutral atoms, and molecules is formed. The energy density is in the range of 2–5â•›J/cm2 at the target surface. As a result, the energy of the ablated material may reach values well above 10â•›eV at the substrate surface. The wavelength of the used laser beam may be 248 or 193â•›nm (at this UV wavelength, the absorption in the oxide target materials is sufficiently large). A repetition rate of several Hz and a pulse length of 25â•›ns represent typical parameters. Initially, a serious problem of the method was the formation of droplets on the substrate, which can easily deteriorate the device properties. Currently, various methods to reduce this effect are known, e.g., the time-of-flight selection of ablated Oxygen inlet Cylindrical lens Laser beam

Quartz window Heater

material. PLD systems are widely used for basic research studies of thin films. The main advantage of this method is that a certain film stoichiometry can be easily achieved by PLD. Many targets (six or more) can be placed on the target carrousel holder. In this way, numerous materials can be deposited without time-consuming rearrangements of the deposition chamber or complicated source exchange procedures, as in the case of MBE or MOCVD. 6.3.2.3â•‡ Sputter Deposition Plasma sputtering is a physical vapor deposition technique that has been known for 150 years since the time when W.R. Grove first observed the sputtering of surface atoms. Different sputtering techniques, such as dc- and rf-sputtering with or without a magnetron arrangement, have been used to grow a variety of materials. Figure 6.7a illustrates the principle of dc-sputtering. A potential of several hundred volts is applied between the target (cathode) and the heater (anode), accelerating positively charged ions toward the target. These accelerated particles sputter off the target material, which finally arrives at the substrate. The discharge is maintained because the accelerated electrons continuously collide with the gas circulating in the chamber and ionize new atoms. For insulating targets such as ferroelectric ones, the dcsputtering is not suitable. Insulating targets have to be sputtered using alternating electric fields to generate the plasma. Typically, an rf-frequency of 13.56â•›MHz is employed. This frequency is not a magic number, rather a frequency that is approved by the government for industrial purposes. A symmetrical arrangement of cathode and anode and the use of a low-frequency alternating field would result in similar sputtering and re-sputtering rates so that the film will not grow. In the case of a high-frequency alternating field, however, light electrons can respond to the field at this frequency, whereas heavy Ar+ ions see only an average electric field (Kawamura et al. 1999). Moreover, the geometrical asymmetry between small cathodes (target side) and large anodes (heater and chamber) leads to a higher electron concentration at the former, resulting in a self-generated dc bias that accelerates Ar+ ions toward the target. The high-pressure sputtering technique of oxide materials was developed by Poppe et al. (1988) and served initially for the

Cathode

Targets

Target

Plasma substrate

e–

n

+

Substrate

Target Substrate

Anode

(a)

To pump

(b)

Figure 6.6â•… Pulsed laser deposition system for the growth of oxide films: PLD setup scheme (a) and a photograph showing the ablation process and the plasma plume (b). (From Schlom, D., Cornell University, Ithaca, Ny.)

(a)

(b)

Heater

Figure 6.7â•… Schematic representation of the dc-sputtering process showing the electrons, positive plasma ions (light gray), and neutral atoms (dark gray) moving toward cathode or anode (a) and a photograph of a high-pressure sputtering system (b). (From Rodríguez Contreras, J. et al., Phd thesis)

6-7


growth of oxide superconductors. A planar on-axis arrangement of the target and substrate is used, as shown in Figure 6.7b. A high sputtering pressure of 2.5–3.5â•›mbar, corresponding to a mean-free-path λmean free = 6 × 10−3 cm at 600°C and exceeding largely the pressure of 10−2 mbar used for conventional sputtering (λmean free = 2â•›cm at 600°C), leads to multiple scattering of the negatively charged oxygen ions accelerated toward the substrate. As a result of the thermalization of ions, the re-sputtering of the deposited films, which is caused by negatively charged ions, is negligible. This technique yields excellent thin films due to the low kinetic energy (as in the case of MBE) of sputtered particles. A disadvantage of the high-pressure sputtering technique could be a low deposition rate of several nanometers per hour, which may lead to interdiffusion at heterogeneous interfaces. To enhance the deposition rate, a low ionization degree of less than 1% of the atoms in the plasma is increased by the use of magnetic fields forcing electrons onto helical paths close to the cathode, which leads to much higher ionization probability. This so-called magnetron sputtering can be employed for highpressure sputtering (Poppe et al. 1988), as well as for conventional, low-pressure sputtering (Fisher et al. 1994). Sputtering is routinely used as a vapor deposition method for the growth of complex-oxide films.

6.3.3â•‡ Patterning The patterning of oxide heterostructures represents an important step in the fabrication of ferroelectric capacitors with small lateral dimensions ranging from a few micrometers to tens of nanometers. As in many other areas of nanoelectronics, two different approaches exist for the device fabrication. A conventional approach relies on the well-established processes used in the modern semiconductor industry: deposition, lithography, and etching. Photo resist

(a)

(b)

Sputtering

Sputtering

Top electrode

Using these techniques sequentially, ferroelectric capacitors can be fabricated. It should be emphasized that this fabrication procedure employs the so-called top-down approach, where external tools are used to create a nanoscale device out of a larger structure. In contrast, the bottom-up approach is based on the self-organization of constituents or their positional assembly necessary for a desired nanodevice. Such techniques recently became very fashionable (Spatz et al. 2000) because they do not require advanced and expensive patterning tools. The bottom-up approach has had considerable success, but improvements are needed to achieve registered arrays of devices, such as those produced by the stateof-the-art complementary metal-oxide-semiconductor (CMOS) technology. In some works, mixed top-down and bottom-up methods were used to produce nanoscale ferroelectric dots and crystals (Kronholz et al. 2006, Szafraniak et al. 2008). One of the simplest ways to fabricate ferroelectric capacitors is the lift-off technique. The main steps of this technique are shown in Figure 6.8a. First, the bottom electrode and the ferroelectric layer are deposited on a substrate. A subsequent photo-lithographic step defines the area of the capacitor. Next, the top electrode is deposited, for example, by the sputtering of Pt. After a lift-off in acetone, the metal with photoresist underneath is removed and the capacitor is ready for electrical characterization. Because the top interface is subjected to photoresist and chemical developer during this procedure, relatively poor electrical properties (e.g., large leakage) are observed here (Rodríguez et al. 2003a and Rodríguez Contreras 2004). The post-annealing of capacitors at high temperatures and in an oxygen atmosphere was successfully used to improve the electrical properties considerably (Schneller and Waser 2007). The aforementioned drawback, however, can be avoided using another method, which involves the fabrication steps shown schematically in Figure 6.8b. Here the whole sandwich (bottom

Ferroelectric

Photolithography + sputtering

Photolithography

Bottom electrode

Lift-off

SrTiO3 substrate

Measurement tips Ar+

Ion beam etching

Measurement tips

Figure 6.8â•… Patterning of capacitors by lift-off technique (a) and ion beam etching (b). (Reprinted from Rodríguez Contreras, J. et al., Appl. Phys. Lett., 83, 126, 2003a. With permission. American Institute of Physics.)

6-8

Photoresist


Pt

PbZr 0.52Ti0.48 O3

Ar+

SrTiO3 substrate

SrRuO3

Ar+ V–

(a)

I+

I–

(b)

Tunneling area (c)

SiO2

(d)

V+

involve combinatorial PLD and MBE systems for the fabrication of epitaxial layers with composition gradients and wedge-like films with a thickness varying across the substrate (Nicolaou et al. 2002, Ohtani et al. 2005, Habermeier 2007). This interesting approach helps to reduce run-to-run uncertainties and allows the variation of certain film parameters in a single deposition run while keeping other conditions fixed. New patterning methods can be developed as well, in particular, a combination of conventional top-down methods and bottom-up techniques. It should be noted that electron beam lithography has been used to produce capacitors with nanoscale dimensions (Szafraniak-Wiza et al. 2008). Focused ion-beam direct-writing techniques already showed their strength, but they suffer from sidewall contaminations at the mesa structure created by etching ions or atoms, e.g., Ga (Hambe et al. 2008).

(e)

Figure 6.9â•… Patterning of tunnel junctions with the aid of photolithography and ion beam etching. (From Rodriquez Contreras, J. et al., Mater. Res. Soc. Symp. Proc., 688, C8.10, 2002. With permission.)

electrode/ferroelectric/top electrode) is deposited without breaking the vacuum. Then the capacitor area is defined by a photolithographic process and dry-etching, typically using Ar ion beam milling. Again acetone is employed to remove the photoresist from the top electrode. Although this method delivers better electrical interface properties than the first one, etch residuals at the sidewalls of the mesa can cause short-circuiting along the sidewall from the top electrode to the bottom one. An etch stop right after reaching the top surface of the ferroelectric can avoid this problem in a simple way. Ion-beam etching is the preferred method for the studies of scaling effects, because it results in the smallest degradation at the top electrode–ferroelectric interface. A more complex procedure should be used to pattern oxide tunnel junctions. As shown in Figure 6.9, three photo-mask steps in total allow the fabrication of tunnel junctions by conventional photolithography and ion-beam etching (Sun 1998). First, the whole layer sequence such as the SrRuO3(20â•›nm)/BaTiO3(2â•›nm)/SrRuO3(20â•›nm) trilayer is deposited in situ. Using a photolithographic step and ionbeam milling, the shape of the bottom electrode is defined. Here the whole trilayer has to be etched down to the substrate. Next, the tunnel junction is defined by the second photo-mask step and an etching down to the bottom electrode. The subsequent deposition of SiO2 is needed to isolate the surroundings of the mesa. A third photo-mask step and the deposition of the wiring layer complete the fabrication of a tunnel junction. Figure 6.9e shows the top view of the junction layout. The four-point probe arrangement, which avoids electrical artifacts during current-voltage measurements, is essential (Rodríguez et al. 2003b and Rodríguez Contreras 2004).

6.3.4â•‡Current Trends in Deposition and Patterning The physical vapor deposition methods described above make it possible to grow complex oxides with a high structural quality and a low surface roughness. Further developments in this field

6.4â•‡Characterization of Ferroelectric Films and Capacitors 6.4.1â•‡Rutherford Backscattering Spectrometry Rutherford backscattering spectrometry (RBS) is an accurate nondestructive technique for measuring the stoichiometry, layer thickness, quality of interfaces, and crystalline perfection of thin films. RBS offers a quantitative determination of the absolute concentrations of different elements in multi-elemental thin films. A collimated mono-energetic beam of low-mass ions hits the specimen to be analyzed. Typically, He+ ions with energy of 1.4â•›MeV are used in RBS experiments. A small fraction of the ions that impinge on the sample is scattered back elastically by the atomic nuclei and are then collected by a detector. The detector determines the energy of the backscattered ions, which provides an RBS energy spectrum. The RBS spectra describe the yield of backscattered particles as a function of their energy. The analysis of RBS spectra is done using modern software. A more detailed description of the technique is given by Chu et al. (1978). In the so-called random experiments, the ion beam is not aligned with respect to the crystallographic directions of the specimen. The energy distribution of the collected ions provides information on the masses of atoms constituting the sample and on the thicknesses of deposited layers. Information on the sharpness of interfaces between these layers is given by the abruptness of the low-energy edge in the “random” spectrum. Epitaxial films usually have the same major channeling axis as the substrate. The degree of epitaxy is determined from ion channeling experiments by a ratio of the elemental signals from the film for the channeled and random sample orientations. This ratio is called the “minimum yield,” χmin, and its value provides information on the crystalline perfection of a film. Defects inside the film lead to higher values of χmin.

6.4.2â•‡ X-Ray Diffraction for Thin-Film Analysis X-ray diffraction (XRD) represents a powerful tool for the Â�characterization of thin films. It can be used to determine whether

6-9


8.2

14 12 10 8 6 4 2 0

1/d 003 (1/nm)

8

7.6 7.4 7.2 7 –3

(a)

STO SRO

7.8

8.2

–2.6

–2.4

–2.2

1/d 001 (1/nm)

7.6 7.4 7.2 7 6.8 –3

–2

73 nm –2.8

–2.6

–2.4

–2.2

–2

1/d 001 (1/nm) In-plane Out-of plane

4.15 Lattice parameter (A)

7.8

(b)

4.20

4.10 4.05

c bulk

4.00

a bulk

3.95 STO substrate

3.90 (c)

14 12 10 8 6 4 2 0

8

BTO strained

17.8 nm –2.8

The amount of strain in an epitaxial film can be nicely visualized by the reciprocal space maps measured, for example, around an asymmetric (103) reflection. These maps can indicate whether the film is fully strained by the substrate or is partially relaxed owing to the generation of misfit dislocations. Representative reciprocal space maps of strained and relaxed epitaxial BaTiO3 films grown on SrRuO3-covered SrTiO3 substrates are given in Figure 6.10a and b. The out-of-plane and in-plane lattice parameters of BaTiO3 films extracted from such maps are plotted in Figure 6.10c as a function of the film thickness t (Petraru et al. 2007). It can be seen that ultrathin films with t < 30â•›nm are commensurate with the substrate, which results in a compressive biaxial in-plane strain and an out-of-plane elongation of the unit cell. The synchrotron x-ray scattering measurements give additional possibilities for the characterization of ultrathin films (Fong et al. 2005). In particular, it was demonstrated that electrode-free PbTiO3 films grown on SrTiO3 remain ferroelectric for thicknesses down to only 3 unit cells (Fong et al. 2004). Finally, we note that the film thickness itself can be measured precisely using the x-ray specular reflectivity method based on interference fringes whose spacing is characteristic for this thickness (Fewster 1996). This method can be applied to films with any structure, crystalline or amorphous, but requires a flat surface over the region studied. It was demonstrated to work even

1/d 003 (1/nm)

the film grown on a crystalline substrate is amorphous, polycrystalline, or single-crystalline (epitaxial growth). Moreover, this technique makes it possible to determine the film thickness, lattice parameters, and the amount of strain in an epitaxially grown film with a high precision. In particular, the 2θ scans performed at a fixed glancing incident angle of the incoming x-ray beam (in the range of 0.5°–2°) are suited for the investigations of polycrystalline films, since the spectrum contains only the peaks coming from the XRD of randomly oriented crystallites. (The single-crystal substrate does not contribute to the XRD because the Bragg condition is not satisfied for this angle of incidence.) For (001)-oriented epitaxial films, the normal θ–2θ scans reveal only the (00l) reflections. Thus, we can distinguish between an epitaxial film on a crystalline substrate and a polycrystalline one. Moreover, from the 2θ position of these reflections, one can precisely determine the out-of-plane lattice parameter. Once this parameter is measured, the in-plane lattice constants could be determined as well by finding the peak positions of the (h0l) reflections, for example. For ultrathin films, however, this becomes difficult because of the overlap with substrate peaks. Therefore, the grazing incidence diffraction, which is characterized by a low penetration depth of the incoming x-ray beam, has to be used to measure the in-plane lattice constants of ultrathin films.

0

50 100 BTO thickness (nm)

150

Figure 6.10â•… X-ray reciprocal space maps around the (103) Bragg reflection obtained for fully strained (a) and partially relaxed (b) BaTiO3 films epitaxially grown on SrRuO3-covered SrTiO3. The film lattice parameters are plotted as a function of the film thickness in panel (c). (From Petraru, A. et al., J. Appl. Phys., 101, 114106, 2007. With permission.)

6-10


104

101

102 0.0

103 102

103

(a)

Experiment Simulation

104 I (a.u.)

I (a.u.)

105

105

SrTiO3(100)

LaNiO3/SrTiO3 Thickness = 27 nm

BaTiO3(001)

106

106

SrRuO3(100)

Thickness BaTiO3 = 7.5 nm

100 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

2θ (deg.)

18

20

22

(b)

24

26

2θ (deg.)

Figure 6.11â•… (a) Interference fringes appearing in an x-ray specular reflectivity scan for the 27â•›nm thick LaNiO3 film deposited on SrTiO3. The film thickness was calculated from the spacing of these fringes. (b) High-angle finite-size oscillations occurring in the θ–2θ scan around the (001) peak of the BaTiO3 film (7.5â•›nm thick) grown on SrRuO3-covered SrTiO3. The solid line shows the measured signal, whereas the dots denote the results of simulations.

in the case of ultrathin films with thicknesses down to 24 Å. The amplitude of oscillations depends mainly on the density contrast between the layers, and the number of oscillations correlates with the roughness of the surface and interfaces involved. In the case of rough surfaces, the average intensity of reflectivity decreases rapidly with an increasing 2θ angle (Nevot and Croce 1980). For epitaxial films, high-angle finite-size oscillations occurring in the θ–2θ scans around the (001) peak allow determination of the number of planes involved in the diffraction, and, therefore, of the film thickness (Schuller 1980). Examples of the low- and high-angle finite-size oscillations are given in Figure 6.11.

Oscilloscope Generator

Cx

C0 (a)

R

I

6.4.3â•‡Ferroelectric Capacitors: P-E Hysteresis Loop Measurements A ferroelectric capacitor usually displays a polarization-field (P-E) hysteresis loop similar to that shown in Figure 6.4b. There are several techniques that are used to measure the P-E loops of ferroelectric capacitors. The simplest method employs a circuit proposed by Sawyer and Tower, which is shown schematically in Figure 6.12a. The circuit consists of a fixed capacitor with known capacitance, the test ferroelectric capacitor, an oscilloscope, and a function generator. The method relies on the fact that two capacitors in a series have the same charge. The ac voltage created by the generator and the potential across the standard capacitor are shown on the x- and y-axes of the oscilloscope. The capacitance of the standard capacitor is chosen to be large enough so that the voltage drop across this capacitor is much smaller than the potential difference across the tested ferroelectric capacitor. Another method uses a fast current-to-voltage converter connected in series with the ferroelectric capacitor (see the circuit shown in Figure 6.12b). In this case, the current-voltage curve is measured as a response of the ferroelectric capacitor to a triangular signal excitation. It is very useful to look at the switching current

FE capacitor

Cx Generator

FE capacitor

– +

U = –I × R

(b)

Figure 6.12â•… Experimental techniques for the measurement of polarization-voltage loops of ferroelectric capacitors: (a) Sawyer-Tower circuit allowing the visualization of ferroelectric hysteresis loops, and (b) an alternative method using a fast current-to-voltage converter, where the loop is obtained via the integration of measured electric current.

peaks that appear in this curve in order to distinguish the ferroelectric switching from artifacts, especially in the case of leaky ferroelectric samples. By numerical integration of the current over the time, the classical P-E hysteresis loop is obtained, from which the remanent polarization and the coercive field can be determined. The remanent polarization of nanoscale SrRuO3/BaTiO3/ SrRuO3 capacitors fabricated on the SrTiO3 substrate is shown in Figure 6.13 as a function of the BaTiO3 thickness t (Petraru et al. 2008). Remarkably, even at t = 3.5â•›nm, the strained BaTiO3 film

6-11

Nanometer-Sized Ferroelectric Capacitors 46 44

77 K 30 kHz

42 40 Pr (μC/cm2)

38 36 34 32 30 28 26

BaTiO3 bulk

24 22 20

3.5

4.0

4.5

5.0

5.5

Film thickness (nm)

Figure 6.13â•… Thickness dependence of remanent polarization in the SrRuO3/BaTiO3/SrRuO3 ferroelectric capacitors measured at 77â•›K. (From Petraru, A. et al., Appl. Phys. Lett., 93, 072902, 2008. With permission.)

remains ferroelectric and has a remanent polarization larger than the spontaneous polarization Ps = 26â•›μC/cm2 of bulk BaTiO3. The coercive field Ec of BaTiO3 capacitors relatively weakly depends on the thickness t (Jo et al. 2006b), which contrasts with a strong increase of Ec (Figure 6.14) in ultrathin PZT capacitors with Pt top electrodes (Pertsev et al. 2003b). Ferroelectric capacitors are often rather leaky, because thin films, especially at small thicknesses, are not perfect insulators. The conduction here results from the Schottky injection or Fowler–Nordheim tunneling through the interfacial barrier followed by the charge transport across the film via the Poole– Frenkel conduction mechanism, space-charge-limited conduction, or variable range hopping (Dawber et al. 2005). The leakage contribution to the total current can be singled out with the aid of the positive-up negative-down (PUND) pulsed method (Smolenskii et al. 1984). It involves the application of a series of voltage pulses from a function generator and the measurement

of the transient current response of a ferroelectric device, which allows the separation of different contributions. As a representative example, we consider the sequence of five train pulses shown in Figure 6.15. The first one (0) is the pre-polarization pulse—it puts the sample into a definite polarization state. The pulse (1) switches the polarization of the sample, and its current response is the sum of the ferroelectric displacement current caused by the switching of spontaneous polarization, the dielectric displacement current, and the leakage current. The pulse (2) has the same polarity but comes after a certain delay time. Therefore, in case of a stable polarization, the current response contains only the components arising from the dielectric response and leakage current. In order to find the switchable polarization (the quantity of primary interest), the current response due to pulse (2) is subtracted from the current created by pulse (1), and the result is numerically integrated over the measuring time. Moreover, this method makes it possible to study the stability of ferroelectric polarization against back-switching. To that end, we can vary the delay time between pulses (1) and (2) and determine the relaxation time of the polarization. A similar analysis can be done for currents resulting from pulses (3) and (4) applied to the capacitor with opposite polarization.

6.4.4â•‡Scanning Probe Techniques: Atomic Force Microscopy and Piezoresponse Force Microscopy Atomic force microscopy (AFM) is one of the most widely used scanning probe microscopy (SPM) techniques (Garcia and Perez 2002). The primary purpose of an AFM instrument is to quantitatively measure the roughness of various surfaces. The lateral and vertical resolutions are typically about 5 and 0.01â•›nm, respectively. An atomically sharp tip is scanned over a surface with feedback mechanisms that enable the piezoelectric scanners to maintain the tip at a constant force (to obtain height information) or height (to obtain force information) above the sample surface. Tips are typically made of Si3N4 or Si and extend down from the end of a cantilever. The AFM head employs an optical 1200

1000

Coercive field (kV/cm)

Coercive field (kV/cm)

1200

800 600 400 200 0

(a)

10

100 Film thickness (nm)

(b)

1000 800 600 400 200 0.00

0.02 0.04 0.06 0.08 0.10 0.12 Inverse of film thickness (nm–1)

0.14

Figure 6.14â•… Coercive field of PZT 52/48 epitaxial films measured at 20â•›k Hz and plotted versus the film thickness t (a) and the inverse of film thickness 1/t (b). The straight line in (b) shows a linear fit to the experimental data, whereas the curve in (a) is a guide to the eyes.

6-12


1

2

2

P1 P2 Subtracted

0

Time (s)

5 × 10–6

0

0 2

–2 0

–4

Switchable polarization

70

3

0 0

–6

5 × 10–3

0

2Pr (μC/cm )

Excitation signal (V)

4

5 × 10–3

0

2 × 10–5

Current response (A)

Excitation Response

Current (A)

6

–5 × 10–3

4

5 × 10–6

Time (s)

4 × 10–5

6 × 10–5

8 × 10–5

1 × 10–4

Time (s)

Figure 6.15â•… Measurements of switchable ferroelectric polarization by the PUND pulsed method. The excitation signal consists of five pulses denoted by thin lines, and the current response is shown by a thick line. The upper inset demonstrates the switching (1) and nonswitching (2) current responses. The integration of their difference gives the switchable polarization plotted in the lower inset.

detection system, in which the tip is attached to the bottom of a reflective cantilever. A laser diode is focused onto the back of this cantilever. As the tip scans the surface of a sample, the laser beam is deflected by the cantilever into a four-quadrant photodiode. In contact mode, feedback from the photodiode difference signal, through the software control from a computer, enables the tip to maintain either a constant force or a constant height above the sample. In the constant force mode, the piezoelectric transducer monitors real-time height variations. In the constant height mode, the deflection force acting on the tip is recorded. The instrument gives a topographical map of the sample surface by plotting the local sample height versus the horizontal probe tip position. For many soft materials like polymers and biological samples, the operation in contact mode often modifies or destroys the surface. These complications can be avoided using the tapping-mode AFM. In tapping mode, the AFM tip–cantilever assembly oscillates at the sample surface during the scanning. As a result, the tip lightly taps the surface while scanning and only touches the sample at the bottom of each oscillation. This prevents damage of soft specimens and avoids the “pushing” of specimens along the substrate. By using a constant oscillation amplitude, a constant tip–sample distance is maintained until the scan is complete. Tapping-mode AFM can be performed on both wet and dry surfaces. Scanning probe microscopy techniques also offer several different possibilities for the investigation of domain patterns in crystals. For imaging domain structures in ferroelectrics, piezoresponse force microscopy (PFM) is most widely used nowadays (Figure 6.16). Introduced in 1992 by Güthner and Dransfeld (1992), the PFM method has been developed by several groups to visualize domain structures in ferroelectric thin films. It became a popular tool in the science and technology of ferroelectrics and is considered to be a main instrument for getting information on ferroelectric properties at the nanoscale. Several reviews on

the SPM-based methods for the characterization of ferroelectric domains are available in the literature (Gruverman and Kholkin 2004, Kholkin et al. 2007). Ideally, when a modulation voltage V is applied to a piezoelectric material, the vertical displacement of the probing tip, which is in mechanical contact with the sample, accurately follows the motion of the sample surface resulting from the converse piezoelectric effect. The applied voltage V generates an electric field E(r) in the ferroelectric film, which creates the lattice strain δu3 = d13E1 + d23E2 + d33E3 in the film thickness direction. Here, dij are the local piezoelectric coefficients of the ferroelectric material, which depend on the polarization orientation. The strain field δu3(r) changes the film thickness at the tip position by an amount δt so that the local piezoresponse signal proportional to δt(V) can be recorded. The amplitude of the tip vibration measured by the lock-in technique provides information on the effective eff piezoelectric coefficient d33 = δt /V . The phase yields information on the polarization direction in a studied ferroelectric domain (Rodriguez et al. 2002). It should be noted that not only the surface electromechanical response but also the electrostatic forces could contribute to the measured signal in the PFM setup, which complicates the analysis of the PFM results. In particular, there exists a nonlocal contribution caused by the capacitive cantilever–sample interaction (Kalinin and Bonnell 2002). Most PFM measurements are performed in a local-excitation configuration where the modulation voltage is applied between the bottom electrode and conductive SPM tip, which scans the bare surface of the film having no top electrode. In this case, the PFM image has a lateral resolution of about 10â•›nm (Gruverman et al. 1998). It should be noted that the electric field generated by the SPM tip in such film is highly inhomogeneous, which makes the quantitative analysis of the field-induced signal extremely difficult. In other words, PFM measurements on a sample

6-13

Nanometer-Sized Ferroelectric Capacitors Four-quadrant photodiode a

Laser

(a + c) – (b + d)

b

(a + b) – (c + d)

c d

Tip (conductive)

Base electrode

Generator

FE film

Out-of-plane

Lock-in 2

Low-pass filter

Cantilever

In-plane

Lock-in 1

Topography

Feedback loop

Syncro

Scanner X, Y, Z

Figure 6.16â•… PFM setup for simultaneous measurements of the surface topography and the out-of-plane and in-plane piezoelectric responses of a ferroelectric sample. The cantilever deflection caused by the voltage-induced surface displacements is detected by a laser beam reflecting into a four-quadrant photodiode. The vertical and horizontal piezoresponses are determined with the aid of two lock-in amplifiers by demodulating the corresponding signals (a + c) − (b + d) and (a + b) − (c + d), respectively. The instrument is operated in a constant force mode.

30 0

60 40

–30

20

–60

0

–3000 (a)

80

d33 (pm/V)

d33 (pm/V)

60

50 A 80 A 120 A 150 A 300 A

–1500

0

1500

3000

Electric field ( kV/cm)

0 (b)

100

200

300

400

500

Thickness (Å)

Figure 6.17â•… Out-of-plane piezoelectric response of SrRuO3/PbZr 0.2Ti0.8O3/SrRuO3 capacitors measured by the PFM technique: (a) piezoelectric hysteresis loops obtained for five different film thicknesses, and (b) the piezoelectric coefficient d33 as a function of the film thickness. (From Nagarajan, V. et al., J. Appl. Phys., 100, 051609, 2006, With permission.)

without extended top electrodes collect signals from a subsurface layer of unknown thickness that is a function of dielectric permittivity and contact conditions (Gruverman et al. 1998). Alternatively, a ferroelectric film with a deposited top electrode may be studied at the expense of a lower lateral resolution. By applying a voltage to the SPM tip contacting the top electrode, a submicron variation of piezoelectric properties in PZT capacitors was demonstrated (Christman et al. 2000, Setter et al. 2006). Under these conditions, a homogeneous electric field is generated in a ferroelectric layer, and the electrostatic tip–sample interaction is suppressed. This approach allows the investigations of domain-wall dynamics and polarization reversal mechanisms in ferroelectric capacitors and quantitative studies of the scaling of piezoelectric properties in ultrathin ferroelectric films. In particular, it was found (Nagarajan et al. 2006) that

the piezoelectric coefficient d33 of epitaxial PbZr 0.2Ti0.8O3 films sandwiched between SrRuO3 electrodes decreases rapidly as the thickness is reduced from 20 to 5â•›nm (see Figure 6.17).

6.5â•‡Physical Phenomena in Ferroelectric Capacitors There are several physical effects that make the phase states and electric properties of thin-film ferroelectric capacitors different from those of bulk ferroelectrics. First, the ferroelectric film is generally subjected to an in-plane straining and clamping due to the presence of a dissimilar thick substrate. Second, an internal electric field exists in the capacitor, which depends on the electrode material and the film thickness. Third, the scaling of

6-14


physical properties may result from the short-range interatomic interactions at the film–electrode interfaces. The current status of the theoretical description of these effects in thin films of perovskite ferroelectrics is given below.

6.5.1â•‡ Strain Effect Owing to the electrostrictive coupling between lattice strains and polarization, the mechanical film–substrate interaction may strongly affect the physical properties of ferroelectric thin films (Pertsev et al. 1998). In a film deposited on a dissimilar thick substrate, the in-plane strains u1, u2, and u6 are totally governed by the substrate, whereas the stresses σ3, σ4, and σ5 are usually equal to zero. (We use the Voigt matrix notation and the reference frame with the x3 axis orthogonal to the film surfaces.) Under such “mixed” mechanical boundary conditions, the equilibrium polarization state corresponds to a minimum of the ∼ modified thermodynamic potential G (Pertsev et al. 1998), but not of the standard elastic Gibbs function G (Haun et al. 1987). In the most important case of a film grown in the (001)-oriented cubic paraelectric phase on a (001)-oriented cubic substrate (u1 = u2 = um, u6 = 0), the stability ranges of different polarization states can be conveniently described with the aid of two-dimensional phase diagrams, where the misfit strain um = (b − a 0)/a 0 and temperature T are used as two independent parameters (a0 is the equivalent cubic cell constant of the free standing film and b is the substrate lattice parameter). Such “misfit straintemperature” diagrams were developed with the aid of thermodynamic calculations for single-domain BaTiO3, PbTiO3, and Pb(Zr1−xTi x)O3 (PZT) films (Pertsev et al. 1998, 2003a). Since the substrate-induced strains lower the symmetry of the paraelectric phase from cubic to tetragonal, the film polarization state may be very different from the ferroelectric phases observed in the corresponding bulk material (see Figure 6.18).

200

Para

150

At large negative misfit strains, films of perovskite ferroelectrics stabilize in the tetragonal c phase with the spontaneous polarization Ps orthogonal to the film–substrate interface, whereas at large positive strains the orthorhombic aa phase forms, where Ps is directed along the in-plane face diagonal of the prototypic cubic cell. At low temperatures, the stability ranges of the c and aa phases are separated by a “monoclinic gap,” where the monoclinic r phase with three nonzero polarization components Pi becomes the energetically most favorable state. These predictions of the thermodynamic theory were confirmed by the first-principles calculations (Bungaro and Rabe 2004, Diéguez et al. 2004). It should be emphasized that the orthorhombic and monoclinic phases do not exist in the bulk PbTiO3 crystals, where only the tetragonal ferroelectric state is stable (Haun et al. 1987). In the case of BaTiO3, the aa phase may be compared with the orthorhombic phase forming in the bulk crystal in the low-temperature range between 10°C and −71°C, whereas the r phase can be regarded as a distorted modification of the rhombohedral phase that exists in a free crystal below −71°C (Jona and Shirane 1962). The most remarkable manifestation of the strain effect appears in thin films of strontium titanate. In a mechanically free state, bulk SrTiO3 crystals remain paraelectric down to zero absolute temperature despite a strong softening of the transverse optic polar mode near T = 0â•›K (Müller and Burkard 1979). The thermodynamic calculations show that this “incipient ferroelectricity” exists in epitaxial SrTiO3 films grown on dissimilar cubic substrates only at small misfit strains ranging from −2 × 10 −3 to −2 × 10−4 (Pertsev et al. 2000). Outside this “paraelectric gap,” the ferroelectric phase transition takes place in the SrTiO3 film at a finite temperature, which rises rapidly with the increase of the strain magnitude. The predicted phenomenon of strain-induced ferroelectricity was observed experimentally in SrTiO3 films grown on (110)-oriented DyScO3, which were found to display ferroelectric properties at room temperature (Haeni et al. 2004).

600 BaTiO3

50

PbTiO3

400 aa-Phase

c-Phase

0 –50

Temperature (°C)

Temperature (°C)

100

Para

r-Phase

–100

c-Phase

aa-Phase

200

0

r-Phase

–150

(a)

–200 –8

–6

0 2 4 –4 –2 Misfit strain um (10–3)

6

–200 –5 (b)

0

5 10 Misfit strain um (10–3)

15

20

Figure 6.18â•… Misfit strain-temperature phase diagrams of single-domain BaTiO3 (a) and PbTiO3 (b) thin films epitaxially grown on (001)-oriented cubic substrates. The second- and first-order phase transitions are shown by thin and thick lines, respectively. (From Pertsev, N.A. et al., Phys. Rev. Lett., 80, 1988, 1998. With permission.)

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The strain-induced increase of the temperature Tc, at which the paraelectric to ferroelectric phase transition takes place, is characteristic of all studied perovskite ferroelectrics (Pertsev et al. 1998, 2003a). In addition, the two-dimensional clamping of the film by a thick substrate may change the order of this transition (Pertsev et al. 1998). Strong dependence of Tc on the misfit strain um explains the very high transition temperatures observed in epitaxial films grown on dissimilar substrates (Choi et al. 2004, He and Wells 2006). The magnitude of the spontaneous polarization is also sensitive to the lattice strains. This effect is especially pronounced in ferroelectric films grown on “compressive” substrates (um < 0), where the polarization Ps is orthogonal to the film surfaces. For fully strained BaTiO3 films grown on SrTiO3 (um = –2.6%), the thermodynamic theory predicts Ps = 35â•›μC/cm2, which is close to the experimental values of 43–44â•›μC/cm2 (Kim et al. 2005, Petraru et al. 2007). Remarkably, the film polarization exceeds the polarization Pb = 26â•›μC/cm2 of bulk BaTiO3 significantly. At the same time, the strain sensitivity of polarization in highly polar Pb-based perovskites, where the ferroelectric ionic displacements are already large in the bulk, is relatively low (Lee et al. 2007). The enhancement of polarization Ps and the decrease of the in-plane permittivity ε11 in the strained c phase (Koukhar et al. 2001) should lead to a considerable increase of the coercive field Ec in thin films (Pertsev et al. 2003b). From the Landauer model of domain nucleation (Landauer 1957) it follows that Ec ~ γ 6 /5 / ε111/5Ps3/5 , where γ ~ Ps3 is the domain-wall energy. Hence, the coercive field Ec ~ Ps3 /ε111/5 of BaTiO3 films grown on SrTiO3 (ε11 ≈ 170) may be about eight times larger than that of the bulk crystal (ε11 ≈ 3600). Although it is certainly a very strong increase, the strain effect alone cannot explain the observed drastic difference between the measured coercive fields Ec ∼ 150–300â•›kV/cm of epitaxial BaTiO3 films (Jo et al. 2006b, Petraru et al. 2007) and the bulk Ec ∼ 1â•›kV/cm. Finally, it should be noted that ferroelectric properties of epitaxial thin films may strongly depend on the orientation of the crystal lattice with respect to the substrate surface. In particular, the phase states and dielectric properties of single-domain PbTiO3 films with the (111)-orientation of the paraelectric phase were found to be very different from those of the (001)-oriented films (Tagantsev et al. 2002).

(

)

6.5.2â•… Depolarizing-Field Effect When the polarization charges ρ = −div P existing at the film surfaces are not perfectly compensated for by other charges, an internal electric field appears in the ferroelectric layer (Figure 6.19). This “depolarizing” field Edep may be significant even in short-circuited ferroelectric capacitors with perfect interfaces because the electronic screening length in metals is finite (Guro et al. 1970, Mehta et al. 1973). For a homogeneously polarized film, the electrostatic calculation gives E dep = −P3/(ε0εb + c it), where P3 is the equilibrium out-of-plane polarization in the film of thickness t, ε0 is the permittivity of the vacuum, εb ∼ 10 is the background

Electrode 1

+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–

P Edep

+

–

+

–

+

–

+

–

+

–

+

– Electrode 2

0

x3

–t

Figure 6.19â•… Imperfect screening of polarization charges in a ferroelectric capacitor. Distribution of the electrostatic potential φ is shown schematically for the case of dissimilar electrodes kept at a bias voltage compensating for the difference of their work functions. (From Pertsev, N.A. and Kohlstedt, H., Phys. Rev. Lett., 98, 257603, 2007. With permission.)

dielectric constant of a ferroelectric material (Tagantsev and Gerra 2006), and ci is the total capacitance of the screening space charge in the electrodes per unit area (Ku and Ullman 1964). When P1 = P2 = 0 (the c phase), the polarization P3(t) can ∼ be calculated from the nonlinear equation of state ∂G/∂P3 = 0 written for a strained film with an internal field E dep. Since the capacitance c i affects the polarization only via the product cit, the dependencies P 3(t) corresponding to different electrode materials can be described by one universal curve P 3(teff ). Here the effective film thickness teff may be defined as teff = (c i/c1)t, where c1 = 1â•›F/m2 . Figure 6.20 shows the dependencies P3(teff ) calculated for fully strained PZT 50/50 and BaTiO3 films grown on SrTiO3 (Pertsev and Kohlstedt 2007). It can be seen that the spontaneous polarization decreases in thinner films and vanishes at a critical film thickness t0. In the case of capacitors with SrRuO3 electrodes (ci = 0.444â•›F/m2), the thickness t0 is about 2â•›nm for PZT 50/50 films and about 2.6â•›nm for BaTiO3 films. However, the size-induced phase transition at t0, also predicted by the first-principles calculations (Junquera and Ghosez 2003), cannot be observed in reality since at a slightly larger film thickness tc < 3â•›nm the singledomain ferroelectric state becomes unstable and transforms into the 180° polydomain state (Pertsev and Kohlstedt 2007). The experimental studies of ultrathin BaTiO3 films (Kim et al. 2005, Petraru et al. 2008) showed that the remanent polarization decreases monotonically with decreasing thickness at t > E created by spikes on the electrodes may also permit the domain formation at small applied fields E. Besides, residual domains probably exist in ferroelectric films, especially in polycrystalline ones. The growth of these domains may play an important role in the polarization switching, as it happens in ferroelectric polymers (Pertsev and Zembilgotov 1991).

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6.5.3â•‡ Intrinsic Size Effect in Ultrathin Films Since the unit cells adjacent to the film–electrode interfaces have an atomic environment different from that of the inner cells, the ferroelectric polarization may depend on the film thickness even in the absence of a depolarizing field. This “intrinsic” size effect can be described with the aid of a modified thermodynamic theory based on the concept of extrapolation length (Kretschmer and Binder 1979). In this theory, the total energy of the ferroelectric layer involves an additional surface contribution, and the polarization distribution across the film, in general, is taken to be inhomogeneous. In the most important case of the (001)-oriented film grown on a compressive substrate (P1 = P2 = 0, P 3 ≠ 0), the polarization profile P 3(x3) can be calculated from the Euler–Lagrange equation (Zembilgotov et. al. 2002). For a film having the same atomic terminations at both surfaces and sandwiched between identical electrodes, the boundary conditions can be written as dP3/dx3 = Pb/δ at x3 = 0 and dP3/dx3 = −Pb/δ at x3 = t, where Pb is the polarization value at the film boundaries and δ is the extrapolation length. The polarization suppression (enhancement) near the film surfaces is described by positive (negative) values of the extrapolation length. In a weakly conducting ferroelectric, such as BaTiO3 or PbTiO3, the inner polarization charges ρ = −dP3/dx3 are largely compensated by charge carriers so that the associated depolarizing field should be negligible. In this case, the spatial scale of polarization variations is determined by the ferroelectric correlation length ξ* = g 11 / a*3 of a strained film, and the strength of the intrinsic size effect is governed by the ratio ξ*/|δ|. (Here g11 and a*( 3 um ) are the coefficients of the gradient term and the renormalized second-order polarization term in the free energy expansion, respectively.) The numerical calculations demonstrate that the polarization suppression in the surface layers reduces the temperature Tc of ferroelectric transition at a given misfit strain um. This reduction, however, is significant only in ultrathin films with thicknesses about a few ξ*(T = 0). Below the transition temperature Tc(um), the mean polarization reduces with decreasing film thickness and vanishes at a critical thickness tc ≈ ξ* (Zembilgotov et al. 2002). Thus, the intrinsic surface effect may lead to a sizeinduced ferroelectric to paraelectric transformation. Since the discussed thermodynamic theory is a continuum theory, it is valid only when the characteristic length ξ* of the polarization variations is larger than the interatomic distances. In the case of a negligible depolarizing field, this condition is satisfied at least near the transition temperature Tc because the coefficient a*3 goes to zero at this temperature (Pertsev et al. 1998). The situation, however, changes dramatically in a perfectly insulating ferroelectric, where the uncompensated polarization charges ρ = −div P inside the film create a nonzero depolarizing field (Kretschmer and Binder 1979, Tagantsev et al. 2008). The characteristic length of polarization variations becomes ξ*d = g 11 / a3* + (ε 0 εb )−1 , which, in contrast to ξ*, does not

increase significantly near Tc. Since ξ*d ∼ 0.1nm only, the continuum approach based on the concept of extrapolation length cannot be used to describe the surface effect in perfectly insulating perovskite ferroelectrics (Tagantsev et al. 2008). In the latter case, however, the ferroelectric film may be assumed to be homogeneously polarized in the thickness direction, and the intrinsic size effect can be described with the aid of a phenomenological approach as well (Tagantsev et al. 2008). To that end, the film free energy is written as the sum of the “bulk” and “surface” contributions, each represented by a polynomial in terms of ferroelectric polarization. In general, the surface contribution should involve not only the even-power terms, but also the odd-power terms, because the surface breaks the inversion symmetry of the ferroelectric (Levanyuk and Sigov 1988, Bratkovsky and Levanyuk 2005). However, when the film–electrode interfaces are identical, the linear term vanishes and the surface energy can be approximated by a quadratic polarization term. The coefficient of this term may be evaluated from the comparison of the phenomenological theory with the results of first-principles calculations performed for ultrathin ferroelectric films. For BaTiO3 capacitors with SrRuO3 electrodes, this procedure reveals that the surface energy is positive (Tagantsev et al. 2008), which implies the polarization suppression at the interfaces. In other metal/ferroelectric/metal heterostructures, however, polarization could be enhanced near the interfaces. Such enhancement was demonstrated by the first-principles-based calculations performed for ultrathin PbTiO3 and BaTiO3 films (Ghosez and Rabe 2000, Lai et al. 2005). The short-range interactions between the film and electrodes must be also taken into account to prove that this effect exists in ferroelectric capacitors as well. The importance of ionic displacements in the boundary layers of SrRuO3 electrodes for the stabilization of ferroelectricity in ultrathin films was revealed by the first-principles investigations (Sai et al. 2005, Gerra et al. 2006).

6.6â•‡ Future Perspective When the thickness of the ferroelectric layer in a biased capacitor becomes as small as a few nanometers, the quantum mechanical electron tunneling across the insulating barrier should become significant (Kohlstedt et al. 2005b, Zhuravlev et al. 2005a). Since the tunnel current exponentially depends on the barrier thickness, the crossover from a capacitor to a tunnel junction takes place near some threshold thickness. The presence of spontaneous polarization in a ferroelectric barrier and its piezoelectric properties are expected to make the current-voltage (I-V) characteristic of a ferroelectric tunnel junction (FTJ) very different from those of conventional tunnel junctions involving nonpolar dielectrics (Kohlstedt et al. 2005b and Rodríguez Contreras 2004). Theoretically, the junction conductance can change strongly after the polarization reversal in the barrier so that the FTJs are promising for the memory storage with nondestructive readout. For the memory applications, asymmetric FTJs with dissimilar electrodes seem to be preferable since such junctions should exhibit


much larger conductance on/off ratios (Kohlstedt et al. 2005b, Zhuravlev et al. 2005a). This feature is due to the fact that here the mean barrier height changes after the polarization reversal by the amount ∆φ ≅ eP3 (t )(cm−12 − cm−11 ) , where cm1 and cm2 are the capacitances of two electrodes and e is the electron charge. The ferroelectric tunnel barrier may also be combined with ferromagnetic electrodes. For such a multiferroic tunnel junction (MFTJ), new functionalities may be expected since the tunneling probability becomes different for the spin-up and spin-down electrons owing to the exchange splitting of electronic bands in ferromagnetic electrodes. In particular, Zhuravlev et al. proposed a new spintronic device, where an electric current is injected from a diluted magnetic semiconductor through the ferroelectric barrier to a normal (nonmagnetic) semiconductor (Zhuravlev et al. 2005b). Their theoretical calculations indicated that the switching of ferroelectric polarization in the barrier may change the spin polarization of the injected current markedly, which provides a two-state electrical control of the device performance. When both electrodes are ferromagnetic, the tunnel current becomes dependent on the mutual orientation of the electrode magnetizations. This phenomenon, which is termed tunneling magnetoresistance (TMR), is important for the applications in spin-electronic devices such as magnetic sensors and magnetic random-access memories. Since the TMR ratio depends not only on the properties of ferromagnetic electrodes, but also on the barrier characteristics (Slonczewski 1989), it may be sensitive to the orientation of the ferroelectric polarization in the MFTJ. This supposition was confirmed by the theoretical calculations performed for junctions involving two magnetic semiconductor electrodes (Zhuravlev et al. 2005b). It was shown that, under certain conditions, the MFTJ works as a device that allows the switching of TMR between positive and negative values. The experimental realization of ferroelectric and multiferroic tunnel junctions, however, is a task with many obstacles because it requires the fabrication of ultrathin films retaining pronounced ferroelectric properties at a thickness of only a few unit cells. Moreover, the ferroelectric state with a nonzero net polarization must be stable at such a small thickness and switchable by a moderate external voltage. In our opinion, reliable FTJs showing resistive switching in the tunneling regime have not been fabricated yet, despite several attempts made in this direction (Rodríguez et al. 2003b, Gajek et al. 2007). The observation of a hysteretic I-V curve or resistance jumps after short-voltage pulses alone is not sufficient to prove the existence of an FTJ (Kohlstedt et al. 2008). Hysteretic I-V curves were also measured for nonferroelectric LaSrxMn1−xO3/SrTiO3/LaSrxMn1−xO3 tunnel junctions and explained by other effects (Sun 2001). Further experiments are necessary to demonstrate the functioning of FTJs unambiguously. In case the above challenges will be overcome in the future, a number of new exciting opportunities for technological applications and interesting physical phenomena are anticipated. Figure 6.21 summarizes the variety of novel functional oxide tunnel junctions. Here, the Josephson tunnel junctions with a ferroelectric or multiferroic barrier are included as well. The

Variety of tunnel junctions Paramagnet Ferromagnet Antiferromagnet Superconductor

Electrode 1

Tunnel barrier

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Electrode 2

Ferroic or multiferroic

Ferromagnet Antiferromagnet

Ferroelectric Antiferroelectric

Figure 6.21â•… A “zoo” of novel tunnel junctions involving multifunctional tunnel barriers and electrodes of various types.

influence of a ferroic tunnel barrier on the Cooper pair and quasiparticle tunneling might lead to interesting new physics. Besides oxide materials, ferroelectric polymers such as PVDF and P(VDF-TrFE) (Xu 1991, Bune et al. 1998) can be incorporated in low-temperature superconducting Josephson junctions [e.g., Nb/Al-AlOx-P(VDF-TrFE)/Nb] (Huggins and Gurvitch 1985) and in magnetic tunnel junctions [e.g., CoxFe1−x/AlOxP(VDF-TrFE)/Ni20Fe80] (Moodera et al. 1995). In principle, one would expect the appearance of physical effects similar to those Nb PVDF AIOx Substrate

Nb

(a) Ni 80 Fe20 PVDF AIOx Substrate

Co50 Fe50

(b)

Figure 6.22â•… Low-temperature superconducting Josephson junctions (a) and magnetic tunnel junctions (b) with the AlOx-ferroelectric polymer composite barriers.


described above for entirely oxide tunnel junctions. On the other hand, metallic (Josephson and magnetic) tunnel junctions are more reliable than oxide ones, and, in addition, ultrathin PVDF films are compatible with AlOx so that composite ferroelectricoxide barriers can be fabricated. The possible structures of superconducting and magnetic metallic tunnel junctions involving PVDF are shown schematically in Figure 6.22. In conclusion of this section, it should be noted that in addition to the planar metal/ferroelectric/metal multilayers discussed above, heterostructures of other geometries may be useful for certain applications in nanoelectronics and may even display specific physical properties. Remarkably, one-dimensional structures in the form of ferroelectric nanowires (Urban et al. 2003) and nanotubes with inner and outer electrodes (Alexe et al. 2006) have been successfully fabricated. Ferroelectric quantum dots are of great interest as well, in particular, for electro-optical devices (Ye et al. 2000).

6.7â•‡ Summary and Outlook The overview presented in this chapter demonstrates impressive achievements in the deposition, characterization, and theoretical description of nanoscale ferroelectric films and heterostructures. Remarkably, capacitors involving only a few nanometers of perovskite ferroelectrics were successfully fabricated, displaying a high remanent polarization. Several advanced analytical tools are now available, showing that high structural quality and sharp interfaces can be retained even in nanoscale capacitors. The basic physical effects in ferroelectric thin films, such as the strain and depolarizing-field effects, are already well understood theoretically, and the influence of short-range interactions at the ferroelectric–metal interfaces is intensively studied by first-principles calculations. Thus, all three constituents of the research triangle shown in Figure 6.1 are functioning effectively, which promises new advances in the physics of nanoscale ferroelectrics and their device applications in the near future.

References Alexe, M., Hesse, D., Schmidt, V. et al. 2006. Ferroelectric nanotubes fabricated using nanowires as positive templates. Applied Physics Letters 89: 172907. Bratkovsky, A. M. and Levanyuk, A. P. 2005. Smearing of phase transition due to a surface effect or a bulk inhomogeneity in ferroelectric nanostructures. Physical Review Letters 94: 107601. Bune, A. V., Fridkin, V. M., Ducharme, S. et al. 1998. Twodimensional ferroelectric films. Nature 391: 874–877. Bungaro, C. and Rabe, K. M. 2004. Epitaxially strained [001]-(PbTiO3)1(PbZrO3)1 superlattice and PbTiO3 from first principles. Physical Review B 69: 184101. Bunshah, R. F. 1994. Handbook of Deposition Technologies for Films and Coatings, Park Ridge, NY: Noyes Publications. Choi, K. J., Biegalski, M., Li, Y. L. et al. 2004. Enhancement of ferroelectricity in strained BaTiO3 thin films. Science 306: 1005–1009.

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Chopra, K. L. 1969. Thin Film Phenomena, New York: McGraw-Hill. Christman, J. A., Kim, S. H., Maiwa, H. et al. 2000. Spatial variation of ferroelectric properties in Pb(Zr0.3,Ti0.7)O3 thin films studied by atomic force microscopy. Journal of Applied Physics 87: 8031–8034. Chu, W., Mayer, J., and Nicolet, M. 1978. Backscattering Spectrometry, New York: Academic Press. Clayhold, J. A., Kerns, B. M., Schroer, M. D. et al. 2008. Combinatorial measurements of Hall effect and resistivity in oxide films. Review of Scientific Instruments 73: 033908. Cohen, R. E. 1992. Origin of ferroelectricity in perovskite oxides. Nature 358: 136–138. Dawber, M., Rabe, K. M., and Scott, J. F. 2005. Physics of thinfilm ferroelectric oxides. Reviews of Modern Physics 77: 1083–1130. Devonshire, A. F. 1954. Theory of ferroelectrics. Advances in Physics 3: 85–130. Diéguez, O., Tinte, S., Antons, A. et al. 2004. Ab initio study of the phase diagram of epitaxial BaTiO3. Physical Review B 69: 212101. Ducharme, S., Palto, S. P., Fridkin, V. M., and Blinov, L. M. 2002. Ferroelectric Polymer Langmuir-Blodgett Films. In Ferroelectric and Dielectric Thin Films, Vol. 3 of Handbook of Thin Films Materials, ed. H. S. Nalwa, San Diego, CA: Academic Press. Eom, C. B., Cava, R. J., Fleming, R. M. et al. 1992. Singlecrystal epitaxial thin films of the isotropic metallic oxides Sr1−xCaxRuO3 (0 < x > D Domain theory Describing the magnetic microstructure of a sample, the shape and detailed spatial arrangement of domain and domain boundaries δ RC

(8.44)

which implies that only the computational cells within the contact area are traversed by the current I, whereas there exists an abrupt cut-off of the current outside the contact. While such an approximation is physically unrealistic, it has been demonstrated that it is able to capture most of the dynamics observed in laboratory experiments. However, more sophisticated shape

Micromagnetic Modeling of Nanoscale Spin Valves

functions can be derived by numerically solving the Poisson equation to get the corresponding current density distribution through the whole computational area.

8.4â•‡ Conclusions and Future Perspective In this contribution, we presented a brief overview of a micromagnetic technique that is used to investigate the behavior of nanoscale spintronic devices. It is based on the numerical integration of the nonlinear LLGS equation. After developing a fullscale FD micromagnetic tool, a question might be asked. Is it possible to develop a framework simpler than the micromagnetic one but that allows for a gain in the equivalent understanding of the complex magnetization dynamics? Actually, magnetization dynamics might be most easily investigated using macrospin (less properly called “single-domain”) approximation. The macrospin approximation assumes that the magnetization of a sample stays spatially uniform throughout its motion and can be treated as a single macroscopic spin. Since the spatial variation of the magnetization is frozen out, exploring the dynamics of magnetic systems is much more tractable using the macrospin approximation than it is using full micromagnetic simulations. The macrospin model makes it easy to explore the phase space of different torque models, and it has been a very useful tool for gaining a zeroth-order understanding of spin-torque physics. On the other hand, it obviously suffers from some intrinsic limitations (e.g., the impossibility of reproducing the nonuniform spatial distribution of magnetization and fields), which sometimes prevents the possibility of mimicking experimental data. It is possible to estimate qualitatively the critical size for which a macrospin approximation could be considered still sufficiently appropriate. It mainly depends on the competition between exchange and magnetostatic energy: the former favoring collinear (uniform) magnetic state, the latter favoring closed-flux configurations. As the exchange energy density of the closedloop configuration obviously increases with decreasing particle size (because magnetization gradients become larger), it leads to the result that only a collinear magnetization state is energetically stable below a certain critical size. It could be demonstrated that such critical length-scale is about—four to eight times the exchange length (about 20–40â•›nm for usual soft magnetic materials). On the other hand, micromagnetic frameworks offer a powerful tool for investigating magnetization phenomena occurring at the mesoscopic scale because of the accurate spatial resolution (few nanometers) and of the theoretically infinite bandwidth (restricted only by the integration time step). Because of that, it is also possible to gain additional information on the investigated problem, which could not be acquired from a laboratory experiment. How can we summarize this issue between an accurate but time-consuming micromagnetic approach and a fast and often oversimplified macrospin model? Let us recall that full-scale simulations are supposed to include more (and not fewer) known features of the investigated

8-15

system than macrospin models and, at the same time, use fewer free (adjustable) parameters. Because of that, if a simplified macrospin approach were able to produce a better agreement with experiment than a micromagnetic approach, it would imply an incomplete comprehension of some crucial properties of the system under study. In other words, it does not necessarily mean that a wrong choice of the values of parameters has been considered, but rather it should recall for further studies to gain a better understanding of the problem, which in turn should bring some corrections to the model. Moreover, because full-scale simulations are quite time-consuming, it is necessary to get an accurate knowledge of the system parameters, together with their spatial dependence (especially near critical regions, such as close to the edges) from independent sources, such as a high-quality experiment. Such information is of crucial importance for the modeling of nano-scale devices due to their extremely small sizes. What about present and future applications of micromagnetic framework? So far, full-micromagnetic frameworks have been able to mimic the experimental behavior of most spintronic devices (pillar, nano-contact, MTJ, exchange-bias systems, phaselocked nano-contacts) as well as to perform predictions on new high-performance setups (Finocchio et al. 2006a,b, 2007, 2008, Choi et al. 2007, Consolo et al. 2007d, 2008, Hrkac 2008). At the same time, micromagnetic tools are successfully used to validate Â�analytical theories as well. For example, the analysis of the disagreement reported between an analytical theory about the excitation of the nonlinear evanescent spin-wave “bullet” mode in in-plane magnetized nanocontact devices (Slavin and Tiberkevich 2005) and earlier results of micromagnetic simulations for the same geometry was intriguing and stimulating (Berkov and Gorn 2006). In fact, while the approximated theory was able to capture the underlying physics and consequently reproduce most of the experimental observations, micromagnetic simulations failed in that attempt. The problem was solved by using a new numerical strategy based on the application of decreasing currents starting from a large supercritical regime (Consolo et al. 2007b), which contains a lot of nonlinear “seeds.” Apart from validating the theory, the micromagnetic approach has also been able to attribute the additional “subcriticallyunstable” nature of these modes, enlarging the knowledge on the spin-wave modes supported by nano-contact devices. From the experimental point of view, there is also an increasing interest in the dynamics (both switching and precession) involving magnetic tunnel junctions because of the discovery of very large TMR values (TMR is the difference in resistance between parallel and antiparallel orientation for the electrode magnetizations of a magnetic tunnel junction) at room temperature. One reason for the interest in spin-torque effects in tunnel junctions is that these devices are better-suited than metallic magnetic multilayers for many types of applications. Tunnel junctions have higher resistances that can often be better impedance-matched to silicon-based electronics, and TMR values can now be made larger than the GMR values in metallic devices.

8-16


Devices based on those effects have already found very widespread applications as magnetic-field sensors in the read heads of magnetic hard disk drives as well as nonvolatile random access memory based on magnetic tunnel junctions. Applications of spin transfer torques are so fascinating as well. Magnetic switching driven by the spin transfer effect can be much more efficient and more industrially intriguing than the usage of static magnetic fields or current-induced magnetic fields. This may enable the production of magnetic memory devices with much lower switching currents and hence greater energy efficiency as well as a larger integration density. The steady-state magnetic precession mode that can be excited by spin transfer is under investigation for a number of high-frequency applications, for example, nanometer-scale microwave sources (tunable by both magnetic field and current), detectors, mixers, modulators, phase shifters, and arrays (or matrixes) of phase-locked devices (whose coupling is used to increase the output power). One potential area of use is short-range chip-tochip or even within-chip communications.

References Alpert, B., Greengard, L., and Hagstrom T. 2002. Nonreflecting boundary conditions for the time-dependent wave equation. J. Comput. Phys. 180: 270–296. Baibich, M.N. et al. 1988. Giant magnetoresistance of (001)Fe/(001) Cr magnetic superlattices. Phys. Rev. Lett. 61: 2472–2475. Berger, L. 1996. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54: 9353–9358. Berkov, D.V. and Gorn, N. 2006. Micromagnetic simulations of the magnetization precession induced by a spin-polarized current in a point-contact geometry. J. Appl. Phys. 99: 08Q701. Berkov, D.V. and Miltat, J. 2008. Spin-torque driven magnetization dynamics: Micromagnetic modelling, J. Magn. Magn. Mater. 320: 1238–1259. Bertotti, G. 1998. Hysteresis in Magnetism. Academic Press, Boston, MA. Brown, W.F. 1940. Theory of the approach to magnetic saturation. Phys. Rev. 58: 736–743. Brown, W.F. Jr. 1962. Magnetostatic Principles in Ferromagnetism, North-Holland Publishing Company, Amsterdam, the Netherlands. Brown, W.F. 1963. Micromagnetics. Wiley, New York. Brown, W.F. 1979. Thermal fluctuations in fine magnetic particles. IEEE Trans. Magn. MAG-15(5): 1196–1208. Choi, S. et al. 2007. Double-contact spin-torque nano-oscillator with optimized spin-wave coupling: Micromagnetic modelling. Appl. Phys. Lett. 90: 083114. Consolo, G. et al. 2007a. Boundary conditions for spin-wave absorption based on different site-dependent damping functions. IEEE Trans. Magn. 43: 2974–2976. Consolo, G. et al. 2007b. Excitation of self-localized spin-wave bullets by spin-polarized current in in-plane magnetized magnetic nanocontacts: A micromagnetic study. Phys. Rev. B 76: 144410.

Consolo, G. et al. 2007c. Magnetization dynamics in nanocontact current controlled oscillators. Phys. Rev. B 75: 214428. Consolo, G. et al. 2007d. Nanocontact spin-transfer oscillators based on perpendicular anisotropy in the free layer. Appl. Phys. Lett. 91: 162506. Consolo, G. et al. 2008. Micromagnetic study of the above-threshold generation regime in a spin-torque oscillator based on a magnetic nano-contact magnetized at an arbitrary angle. Phys. Rev. B 78: 014420. Enquist, B. and Majda, A. 1977. Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 31: 629–651. Finocchio, G. et al. 2006a. Magnetization dynamics driven by the combined action of AC magnetic field and DC spin-polarized current. J. Appl. Phys. 99: 08G507. Finocchio, G. et al. 2006b. Trends in spin-transfer driven magnetization dynamics of CoFe/AlO/Py and CoFe/MgO/Py magnetic tunnel junctions. Appl. Phys. Lett. 89: 262509. Finocchio, G. et al. 2007. Magnetization reversal driven by spinpolarized current in exchange-biased nanoscale spin valves. Phys. Rev. B 76: 174408. Finocchio, G. et al. 2008. Numerical study of the magnetization reversal driven by spin-polarized current in MgO based magnetic tunnel junctions. Physica B 403: 364–367. Garcia-Palacios, J.L. and Lazaro, F.J. 1998. Langevin-dynamics study of the dynamical properties of small magnetic particles. Phys. Rev. B 58: 14937–14958. Gardiner, C.W. 1985. Handbook of Stochastic Methods. Springer, Berlin, Germany. Gilbert, T.L. 2004. A phenomenological theory of damping in ferromagnetic materials. IEEE Trans. Magn. 40: 3443–3449. Higdon, R.L. 1987. Numerical absorbing boundary conditions for the wave equation. Math. Comput. 49: 65–90. Hrkac, G. 2008. Mutual phase locking in high-frequency microwave nano-oscillators as a function of field angle. J. Magn. Magn. Mater. 320: L111–L115. Kim, J.V., Tiberkevich, V., and Slavin, A. 2008. Generation linewidth of an auto-oscillator with a nonlinear frequency shift: Spin-torque nano-oscillator. Phys. Rev. Lett. 100: 017207. Kloeden, P.E. and Platen E. 1999. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, Germany. Krivorotov, I.N. et al. 2004. Temperature dependence of spintransfer-induced switching of nanomagnets. Phys. Rev. Lett. 93: 166603. Landau, L.D. and Lifshitz, E.M. 1935. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8: 153–169. Li, Z. and Zhang, S. 2004. Thermally assisted magnetization reversal in the presence of a spin-transfer torque. Phys. Rev. B 69: 134416. Martinez, E., Torres, L., and Lopez-Diaz, L. 2004. Computing solenoidal fields in micromagnetic simulations. IEEE Trans. Magn. 40: 3240–3243. Newell, A.J., Williams, W., and Dunlop, D.J. 1993. A generalization of the demagnetizing tensor for nonuniform magnetization. J. Geophys. Res. 98: 9551–9555.

Micromagnetic Modeling of Nanoscale Spin Valves

Parkin, S.S., More, N., and Roche, K.P. 1990. Oscillations in exchange coupling and magnetoresistance in metallic superlattice structures: Co/Ru, Co/Cr, and Fe/Cr. Phys. Rev. Lett. 64: 2304–2307. Parkin, S.S., Bhadra, R., and Roche, K.P. 1991. Oscillatory magnetic exchange coupling through thin copper layers. Phys. Rev. Lett. 66: 2152–2155. Preisach, F. 1929. Investigations on the Barkhausen effect. Ann. Physik 3: 737–799. Ralph, D.C. and Stiles, M.D. 2008. Spin-transfer torques. J. Magn. Magn. Mater. 320: 1190–1216. Renaut, R.A. 1992. Absorbing boundary conditions, difference operators and stability. J. Comput. Phys. 102: 236–251. Rippard, W.H. et al. 2004. Direct-current induced dynamics in Co90Fe10/Ni80Fe20 point contacts. Phys. Rev. Lett. 92: 027201. Schad, R. et al. 1994. Giant magnetoresistance in Fe/Cr superlattices with very thin Fe layers. Appl. Phys. Lett. 64: 3500–3502. Slavin, A.N. and Kabos, P. 2005. Approximate theory of microwave generation in a current-driven magnetic nanocontact magnetized in an arbitrary direction. IEEE Trans. Magn. 41: 1264–1273.

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Slavin, A. and Tiberkevich, V. 2005. Spin-wave mode excited by spin-polarized current in a magnetic nanocontact is a standing self-localized wave bullet. Phys. Rev. Lett. 95: 237201. Slavin, A. and Tiberkevich, V. 2008. Excitation of spin waves by spin-polarized current in magnetic nano-structures. IEEE Trans. Magn. 44: 1916–1927. Slonczewski, J.C. 1996. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159: L1–L7. Slonczewski, J.C. 1999. Excitation of spin waves by an electric current. J. Magn. Magn. Mater. 195: L261–L268. Stiles, M.D. and Miltat J. 2006. Spin-transfer torque and dynamics. In Spin Dynamics in Confined Magnetic Structures III. Eds. B. Hillebrands and A. Thiaville, pp. 225–308. Springer, Berlin/Heidelberg, Germany. Van Kampen, N.G. 1987. Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam, the Netherlands.

9 Quantum Spin Tunneling in Molecular Nanomagnets 9.1 9.2 9.3 9.4

Gabriel González University of Central Florida


Introduction.............................................................................................................................. 9-1 Spin Tunneling in Molecular Nanomagnets........................................................................ 9-2 Phonon-Assisted Spin Tunneling in Mn12 Acetate..............................................................9-5 Interference between Spin Tunneling Paths in Molecular Nanomagnets.......................9-5 Berry’s Phase and Path Integrals

9.5 Incoherent Zener Tunneling in Fe8. ...................................................................................... 9-7 9.6 Coherent Néel Vector Tunneling in Antiferromagnetic Molecular Wheels...................9-8 9.7 Berry-Phase Blockade in Single-Molecule Magnet Transistors...................................... 9-11 9.8 Concluding Remarks..............................................................................................................9-12 Acknowledgment................................................................................................................................ 9-13 Referencesï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½9-13

9.1â•‡ Introduction Molecular magnets have attracted considerable interest recently among researchers because they are considered to be ideal systems to probe the interface between classical and quantum physics as well as to study decoherence in nanoscale systems. The advances in nanoscience during the last decade have made it possible to design and fabricate a wide variety of nanosize objects, ranging from a couple of micrometers all the way down to a few tens of nanometers, where quantum effects become important and give rise to new properties. One of the goals of miniaturization was the possibility of observing quantum tunneling effects in mesoscopic systems. One source for such a task is the so-called single-molecule magnets (SMM) and antiferromagnetic molecular wheels, in which the spin state of the molecule is known to behave quantum mechanically at low temperatures. During the last decade, a tremendous progress in the experimental methods for contacting single molecules and measuring the electrical current through them has been achieved. The current through single magnetic molecules like Mn12 and Fe8 has been measured and magnetic excited states have been identified (Jo et al., 2006; Henderson et al., 2007). In a three-terminal molecular single electron transistor, the current can flow between the source and drain leads via a sequential tunneling process through the molecular charge levels, thereby bringing the whole field of Coulomb-blockade physics to molecular systems. Single-molecule magnets are mainly organic molecules containing multiple transition-metal ions bridged by organic ligands. These ions are strongly coupled by exchange interaction, yielding

a large magnetic moment per molecule. The large spin combined with the large magnetic anisotropy provides an energy barrier for magnetization reversal. These systems provided for the first time evidence of quantum tunneling of the magnetization and interference effects as well as oscillations of the tunnel splitting. Ferromagnetic molecular magnets such as Mn12 and Fe8 show incoherent tunneling of the magnetization (Korenblit and Shender, 1978; Enz and Schilling, 1986; van Hemmen and Süto, 1986; Chudnovsky and Gunther, 1988) and allow one to study the interplay of thermally activated processes and quantum tunneling. The spin tunneling leads to two effects. First, the magnetization relaxation is accelerated whenever spin states of opposite direction become degenerate due to the variation of the external longitudinal magnetic field (Friedman et al., 1996; Thomas et al., 1996; Leuenberger and Loss, 1999, 2000a; Leuenberger and Loss). Second, the spin acquires a Berry phase during the tunneling process, which leads to oscillations of the tunnel splitting as a function of the external transverse magnetic field (Wernsdorfer and Sessoli, 1996; Wernsdorfer et al., 2000; Leuenberger and Loss, 2000b, 2001a). Antiferromagnetic molecular wheels are another type of molecular magnets where an even number of transition metal ions with spin form a closed ring in which the exchange interaction between neighbors gives rise to a strong spin quantum dynamics. Antiferromagnetic molecular magnets such as ferric wheels belong to the most promising candidates for the observation of coherent quantum tunneling on the mesoscopic scale (Chiolero and Loss, 1998; Meier and Loss, 2000, 2001). In contrast to incoherent tunneling, in quantum coherent tunneling 9-1

9-2


spins tunnel back and forth between energetically degenerate configurations at a tunneling rate which is large compared to the decoherence rate. The detection of quantum behavior is more challenging in antiferromagnetic molecular magnets than in ferromagnetic systems, but is feasible with present day experimental techniques. Understanding the properties of molecular magnets is only a first step toward achieving technological applications in data storage, data processing, and quantum technologies. A possible next step will be the preparation and control of a well-defined single-spin quantum state of a molecular cluster. Although challenging, this task appears feasible with present-day experiments and would allow one to carry out quantum computing with molecular magnets (Leuenberger and Loss, 2001b).

9.2â•‡Spin Tunneling in Molecular Nanomagnets An effective spin Hamiltonian that captures the basic physics of a molecular nanomagnet is given by the following form:

H = H0 (S ) + HZ + HT + Hsb .

(9.1)

We will now analyze each term in the Hamiltonian given in Equation 9.1 separately. The first term in Equation 9.1 is the dominant term of the Hamiltonian with spin quantum number S = |S⃗ | >> 1 and is called the uniaxial anisotropy, which results from the spin–orbit interactions and generates and easy magnetic axis for the magnetic moment of the nanomagnet. The assumption is that we have a single domain magnetic molecule in which the very strong exchange interactions align the microscopic electronic spins into a parallel or antiparallel configura s j for the ferromagnetic tion, resulting in a total spin S =

∑ ∑ j

(−1) j s j , respectively. moment or the Neél vector with S = j For a nanomagnet like Fe8, H0 (S ) = − ASz2 where A/kB ≈ 0.275â•›K and generates an energy barrier that separates opposite spin projections (see Figure 9.1). Quantum mechanics specifies that a particle in a symmetric double well potential undergoes a tunnel effect that makes the particle go back and forth from one well to the other with a frequency ωT. This implies that the eigenfunctions of the Hamiltonian are delocalized, which means that there is a probability to find the particle either in one well or the other. In a similar fashion, there is a probability for the spin of the nanomagnet to change direction, i.e., if the spin of the nanomagnet points up there is a probability that at a later time the spin will be pointing down. At equilibrium and finite temperature, the probability that the spin is in state |m〉, where Sz|m〉 = m|m〉 for m = −s, −s + 1, …, s − 1, s, is given by

e − Em / kBT Pm = Z

(9.2)

Energy

m

|s − 3

|–s + 3

|s − 2

|–s + 2

|s − 1

|–s + 1

|s

|–s

FIGURE 9.1â•… The graph shows H0 vs. Sz where the horizontal lines indicate the energies of the quantum states.

where Z is the partition function Em = −Dm2 For the case when T → 0 the probability for the spin to go from one well to the other satisfies the following equation: dP (t ) ≈ −ΓP (t ) < 0. dt

(9.3)

Therefore P(t) ∝ e−Γt, where Γ is the rate of decay and Γ−1 is known as the lifetime or characteristic time of the particle. The second term in Equation 9.1 is the Zeeman energy resulting from the interaction of the spin with an external magnetic field, i.e., HZ = µ B gS ⋅ H . A magnetic field applied along the easy axis of the molecule will tilt the potential well and for certain values of the magnetic field (δHz = (m + m′)|A|/gμB), the energy levels on both sides of the anisotropy barrier coincide. Therefore the Hamiltonian given by H = − ASz2 + g µ B δH z Sz is still doubly degenerate (see Figure 9.2). Energy

{

|m

∆

Wm,m+1

|m΄

|A = √21 (|m – |m΄ ) |S =√21 (|m + |m΄ )

δHz

FIGURE 9.2â•… The graph shows (H0 + HZ ) vs. Sz in a constant magnetic field parallel to the easy axis. Δ denotes the tunnel splitting between states |m〉 and |m′〉.

9-3

Quantum Spin Tunneling in Molecular Nanomagnets

  hm H=  ∆ mm′  2

∆ mm′  2  = h (m + m′) 1 0     2 0 1  hm′    1 h + (m − m′)  2  tan θ

tan θ  , −1 

(9.4)

|ψ+ = |m΄

θ θ m + sin m′ , 2 2

ψ − = − sin

θ θ m + cos m′ , 2 2

|ψ– =|S = √12 (|m +|m΄ )

|ψ− =|m΄

h

FIGURE 9.3â•… Anticrossing of the adiabatic eigenvalues E± = 1 h(m + m′) ± h2 (m − m′) + ∆ 2  . The Zener tunneling transition of mm ′  2   the state |ψ+〉 is adiabatic (see Section 9.5).

Then one can calculate the probability that the state |m〉 has tunneled to the state |m′〉 after time t: m′ | ψ(t )

2

= sin

θ θ ψ + | ψ(t ) + cos ψ − | ψ(t ) 2 2

θ θ = sin cos (e −iE+ t / − e −iE− t / ) 2 2

=

2

2

 h2 (m − m′)2 + ∆ 2 mm ′ = sin2 θ sin2  2 

(9.5) (9.6)

Δmm΄

|ψ− = |m

where tan θ = Δmm′/h(m−m′), 0 ≤ θ < π and h is the longitudinal magnetic field with the coupling constant g and Bohr magneton μB absorbed, which makes the double well asymmetric, and Δmm′ is the tunnel splitting between the states |m〉 and |m′〉. The eigenstates of Equation 9.4 read ψ + = cos

|ψ+ = |m

|ψ+ =|A = √ 12 (|m −|m΄ )

Energy

These degeneracies can be removed by the term HT , which corresponds to the transverse anisotropies. For the nanomagnet Fe8, the transverse anisotropy term is given by HT = E(Sx2 − S 2y ), where (E/kB ≈ 0.046â•›K). The anisotropy term lifts the degeneracy, thereby creating an energy gap between the spin states |m〉 and |m′〉, denoted by Δmm′. If the Hamiltonian possesses transverse terms, the pairwise degenerate states get split by Δmm′ amount of energy. If the coupling of our system to an external bath is assumed to be zero, each pair of the degenerate states can be described by means of a two-state Hamiltonian:

 t 

 h2 (m − m′)2 + ∆ 2mm′  ∆ 2mm′ 2 sin t  2 h2 (m − m′)2 + ∆ 2mm′   (9.10)

with eigenvalues

This means that both the tunneling behavior and also the two-level spin resonance behavior are described by a Rabi h (m − m′)  1  oscillation. 2 2 2  E = (m + m′) ± = h(m + m′) ± h (m − m′) + ∆ mm′  .  ± 2  cos θ  2  In reality, the spin of the SMM is interacting with the environ(9.7) ment and this interaction is described by the last term in Equation 9.1. The interaction of the molecular magnet with the environ|ψ+〉 and |ψ−〉 are delocalized as long as h(m − m′) ≤ Δmm′. It is only ment is due to the hyperfine interaction, dipole interaction, spin– in this regime that a spin state can tunnel from one side of the bar- phonon interaction, the interaction between the molecule and rier to the other, which is also relevant for the Zener tunneling (see two contact leads, etc. When the spin interacts with its environSection 9.5). In the limit h → 0, the eigenstates |ψ+〉 and |ψ−〉 are the ment, the wavefunction loses the memory of its phase. This phesymmetrical and antisymmetrical combinations of |m〉 and |m′〉, nomenon is known as decoherence. In this case, the wavefunction 1 1 i.e., | ψ + 〉 = | S 〉 = 2 ( | m〉 + | m′〉 ) and | ψ − 〉 = | A〉 = 2 ( | m〉 − | m′〉 ). is not longer appropriate to describe the system. The problem of a system that interacts with the environment is formulated by For Δmm′ → 0 we are left with |ψ+〉 = |m〉 and |ψ−〉 = |m′〉 for h > 0 means of the density matrix formalism. The spin has a certain and |ψ−〉 = |m〉 and |ψ+〉 = |m′〉 for h < 0 (Figure 9.3). probability to be in state |m〉 at time t, which is described by the Let us now assume that the system starts in the state density matrix ρm(t). Due to the interaction with the environment, the spin undergoes incoherent transitions from a state |m〉 θ θ ψ(t = 0) = m = cos ψ + − sin ψ − , (9.8) to another state |m′〉 at a rate Wm′m until ρm(t), −s ≤ m ≤ s reach 2 2 their equilibrium value. This process is known as relaxation. If these transitions are independent of each other, the density from which one obtains immediately the time evolution matrix evolves according to the Pauli equation or master equation

θ θ ψ(t ) = cos e −iE+ t / ψ + − sin e −iE−t / ψ − . 2 2

(9.9)

dρm (t ) = dt

∑ W

ρ (t ) − Wm′ mρm (t )

mm ′ m ′

m′

(9.11)

9-4


where Wm′m is the transition probability from state |m〉 to state |m′〉. The master equation (Equation 9.11) can also be written as dρm (t ) = dt

∑A

ρ (t )

(9.12)

mm′ m′

m′

where

Amm′

if m ≠ m′

Wmm′  = ′  − Wm ″ m ′  m ″

∑

(9.13)

if m′ = m,

the prime indicates that the term m″ = m is to be omitted in the summation. A formal solution to Equation 9.11 can be obtained in the following manner, let ρm (t ) = pm(n)e −Γnt

∑ m′

 Wmm′ − δ mm′ 

∑ m

 Wmm"  pm(n)′ = 

∑

Amm′ pm(n)′ (9.15)

m′

which implies that −Γn is an eigenvalue of the master matrix defined in Equation 9.13. The master matrix is a square matrix (2s + 1) × (2s + 1), which is not Hermitian. However, the eigenvalue problem can be written in terms of the Hermitian real matrix by making the following transformation

 β(Em − Em′ )  Bmm′ = Amm′ exp   = Bm′ m . 2  

(9.16)

It follows that the general solution of Equation 9.11 is then

 −βEm  ρm (t ) = exp    2 

∑c P

(n) − Γnt n m

e

∑ρ (0)P k

k

(n ) k

 βE  exp  k  .  2 

(9.18)

Equation 9.11 is only valid for times where the correlation time τc in the heat bath is much smaller than the relaxation time of the spin system, necessary for the establishment of irreversibility in a macroscopic system. In fact, in order to describe quantum tunneling, it is necessary to use the generalized master equation.

(9.19)

i∆ ρm = mm′ (ρmm′ − ρm′ m ) − Wmρm + Wmnρn 2 n ≠ m, m ′

∑

(9.20)

and for the off-diagonal elements

i∆ i  ρmm′ = −  ξ mm′ + γ mm′  ρmm′ + mm′ (ρm − ρm′ ),   2

(9.21)

with ξm = −Am2 + gμBδHzm and ξmm′ = ξm − ξm′, similarly for m ↔ m′. Ultimately, we are interested in the overall relaxation time τ of the quantity ρs − ρ−s (see Section 9.7) due to phononinduced transitions. This τ turns out to be much longer than τd = 1/γmm′, which is the decoherence time of the decay of the offdiagonal elements ρmm′ ∝ e −t / τd of the density matrix ρ. Thus, we can neglect the time dependence of the off-diagonal elements, i.e., ρ⋅mm′ ≈ 0. Physically this means that we deal with incoherent tunneling for times t > τd. Inserting then the stationary solution of Equation 9.21 into Equation 9.20, which leads to the complete master equation including resonant as well as nonresonant levels, we get ρm = −Wmρm +

where (Pm(n) , −Γn ) are the (2s + 1) orthonormal set of eigenvectors with the corresponding eigenvalues of the matrix (9.16) and Pm(n) = pm(n) exp(βEm /2). The cn are the constants related to the initial distribution of ρm by cn =

(9.17)

n

∑

i ρmm′ = [ρ, H0 ]mm′ + δ mm′ ρnWmn − γ mm′ρmm′ . n≠m

The difference to the usual master equation is that Equation 9.19 takes also off-diagonal elements of the density matrix ρ(t) into account. This is essential to describe tunneling of the magnetization, which is caused by the overlap of the Sz states. Let us consider the two-state system {|m〉, |m′〉}, which yields the two-state Hamiltonian in the presence of a bias field given in Equation 9.4. Next we insert the two-state Hamiltonian into the generalized master equation (Equation 9.19), which yields for the diagonal elements of the density matrix

(9.14)

where Γn−1 is the characteristic time labeled by the index n = 0, 1, …, 2s. Substituting Equation 9.14 into Equation 9.11, one obtains −Γ n pm(n) =

The generalized master equation that describes the relaxation of the spin due to phonon-assisted transitions including resonances due to tunneling is given by

∑W

ρ + Γ mm′ (ρm′ − ρm ),

(9.22)

mn n

n ≠ m, m ′

where

Γ mm′ = ∆ 2mm′

Wm + Wm′ 4ξ2mm′ + 2 (Wm + Wm′ )2

(9.23)

is the transition rate from m to m′ (induced by tunneling) in the presence of phonon damping. Note that Equation 9.22 is now of the usual form of a master equation, i.e., only diagonal elements of the density matrix ρ(t) occur. For levels k ≠ m, m′, Equation 9.22 reduces to ρk = −Wk ρk +

∑W ρ .

(9.24)

kn n

n

9-5


We note that Γmm′ has a Lorentzian shape with respect to the external magnetic field δHz occurring in ξmm′. It is thus this Γmm′ that will determine the peak shape of the magnetization resonances.

9.3â•‡Phonon-Assisted Spin Tunneling in Mn12 Acetate The magnetization relaxation of crystals and powders made of molecular magnets Mn12 has attracted much recent interest since several experiments (Sessoli et al., 1993; Novak and Sessoli, 1995; Novak et al., 1995; Paulsen and Park, 1995; Paulsen et al., 1995) have indicated unusually long relaxation times as well as increased relaxation rates (Friedman et al., 1996; Hernández et al., 1996; Thomas et al., 1996) whenever two spin states become degenerate in response to a varying longitudinal magnetic field Hz . According to earlier suggestions (Barbara et al., 1995), this phenomenon has been interpreted as a manifestation of incoherent macroscopic quantum tunneling (MQT) of the spin. As long as the external magnetic field Hz is much smaller than the internal exchange interactions between the Mn ions of the Mn12 cluster, the Mn12 cluster behaves like a large single spin S⃗ of length |S⃗ | = 10. For temperatures 1 ≤ T ≤ 10â•›K, its spin dynamics can be described by the spin Hamiltonian given in Equation 9.1, including the coupling between this large spin and the phonons in the crystal (Villain et al., 1994; Garanin and Chudnovsky, 1997; Hernández et al., 1997; Fort et al., 1998; Luis et al., 1998; Leuenberger and Loss, 1999, 2000a; Leuenberger and Loss). The most general spin–phonon coupling reads 2 y

1 + g 3 ( xz ⊗ {Sx , Sz } + yz ⊗ {S y , Sz }) 2

1 + g 4 (ω xz ⊗ {Sx , Sz } + ω yz ⊗ {S y , Sz }), 2

 i t  iϕi (t ) ψ i (R(0)) → e ψ i (R(t )) exp  − Ei (R)dt  .   0  

∫

(9.26)

Plugging Equation 9.26 into Schrödinger’s equation we have dϕ dR = i ψ i | ∇R | ψ i ⋅ , (9.27) dt dt thus, for t = 0 to t = τ we end up with R( τ )

ϕ(τ) − ϕ(0) =

∫

R(0)

ψ i | i∇ R | ψ i dR.

(9.28)

The term of Equation 9.28 is known as Berry’s phase. The overall phase of a quantum state is not observable but the relative phases for a quantum system, which undergoes a coherent evolution, can be detected experimentally. Consider for example two paths R⃗ and R⃗ ′ with the same end points R⃗ (0) = R⃗ ′(0) and R⃗ (τ) = R⃗ ′(τ), the net Berry phase change is given by

1 Hsp = g 1 ( xx − yy ) ⊗ (S − S ) + g 2 xy ⊗ {Sx , S y } 2 2 x

observed oscillations in the tunnel splitting Δm,−m between states Sz = m and −m as a function of a transverse magnetic field at temperatures between 0.05 and 0.7â•›K (see Landau, 1932). This effect can be explained by the interference between Berry phases associated to spin tunneling paths of opposite windings (DiVincenzo et al., 1992; van Delft and Henley, 1992). Usually the Berry phase effect arises in systems that undergo an adiabatic cyclic evolution, and which occurs in such diverse fields as atomic, condensed matter, nuclear and elementary particle physics, and optics. The basic assumption for the derivation of the Berry phase is that, for a slowly varying time-dependent Hamiltonian H(R⃗ (t)) that depends on parameters R1(t), R2(t), …, R N(t) components of a vector R⃗ , the eigenstate evolves in time according to

(9.25)

where gi are the spin–phonon coupling constants ϵαβ(ωαβ) is the (anti-)symmetric part of the strain tensor From the comparison between experimental data (Friedman et al., 1996; Thomas et al., 1996) and calculation, it turns out that the constants gi ≈ A ∀i (Leuenberger and Loss, 1999, 2000a; Leuenberger and Loss).

9.4â•‡Interference between Spin Tunneling Paths in Molecular Nanomagnets Recent experiments have pointed out the importance of the interference between spin tunneling paths in molecules. For instance, measurements of the magnetization in bulk Fe8 have

∆ϕ =

∫ ψ | i∇ i

R

| ψ i ⋅ dR.

(9.29)

This is a line integral around a closed loop in parameter space, which is nonzero in general. The classic example of Berry’s phase is an electron at rest subjected to a time-dependent magnetic field of constant magnitude but changing direction. The Hamiltonian for this system is given by

 Hz H = −µ B σ ⋅ H = −µ B   H x + iH y

H x − iH y  , −H z 

(9.30)

where H⃗ (t) = H(sin θ(t) cos(ϕ(t)), sin(θ(t)) sin(ϕ(t)), cos(θ(t))) and μB is Bohr magneton. It is easy to show that the eigenstate representing spin up along H⃗ (t) for Equation 9.30 has the form

 e −iφ / 2 cos(θ / 2) ↑ =  iφ / 2 .  e sin(θ / 2) 

(9.31)

9-6


C Ω(C)

H(t) e–

FIGURE 9.4â•… The figure shows an electron in a constant magnetic field which sweeps around a closed curve C and subtends a solid angle Ω.

For the case when the magnetic field sweeps out an arbitrary closed circuit C (see Figure 9.4) we can apply Equation 9.29 to get the Berry phase 1 ∆ϕ = − Ω(C ), 2

(9.32)

where Ω(C) is the solid angle described by the field H⃗ along the circuit C.

it is common to get rid of the factor i in the exponential of Equation 9.33 by analytic continuation to imaginary times, through β = it/ħ. Then, the path integral becomes equivalent to the partition function Z by noting β = 1/kBT. Quantum tunneling of a spin is often described with the terminology of a path integral. Consider a general single-spin Hamiltonian Hz ,n = − ASz2 + Bn (S+n + S−n ), with easy-axis (A) and transverse (Bn) anisotropy constants satisfying A >> Bn > 0. Here, n is an even integer, i.e., n = 2, 4, 6, …, so that Hz is invariant under time reversal. Such Hamiltonians are relevant for molecular magnets such as Mn12 (Fort et al., 1998; Leuenberger and Loss, 1999, 2000a) and Fe8 (Leuenberger and Loss, 2000b). The corresponding classical anisotropy energy Ez,n(θ, ϕ) = −As2 cos2 θ + Bnsn sinn θ cos(nϕ) has the shape of a double-well potential, with the easy axis pointing along the z direction. It is obvious that the anisotropy energy remains invariant under rotations around the z-axis by multiples of the angle η = 2π/n. For the following calculation, it proves favorable to choose the easy axis along the y direction where the contour path does not include the south pole. Then the spin Hamiltonian and the anisotropy energy are changed into Hy ,n = − AS 2y + Bn (S+n + S−n ),

(9.36)

where now S± = Sz ± iSx, and

9.4.1â•‡ Berry’s Phase and Path Integrals An alternative approach for quantum mechanics and hence for the calculation of the tunneling frequency is by means of path integrals which were introduced by Richard Feynman (Feynman, 1948). In this section, we will briefly explain the path integral approach to calculate the Berry phase in single molecular magnets. Feynman showed that the probability amplitude 〈 x ′ | exp[i(t ′ − t )H /]| x 〉 for a particle that is at point x′ at time t′ if it was at point x at a time t can be expressed as the path integral x′

 iS  U (x ′ ,t ′ | x,t ) = D[ x ″]exp  E  ,   x

∫

(9.33)

E y ,n (θ, φ) = − As 2 sin2 θ sin2 φ + Bn sn × (cos θ + i sin θ cos φ)n + (cos θ − i sin θ cos φ)n  .

(9.37)

We are interested in the tunneling between the eigenstates |m〉 and |−m〉 of −AS 2y , corresponding to the global minimum points (θ = π/2, ϕ = −π/2) and (θ = π/2, ϕ = +π/2) of Ey,n. For this, we evaluate the imaginary time transition amplitude between these points. For the ground-state tunneling (m = s >> 1), this can be done by means of the coherent spin-state path integral (DiVincenzo et al., 1992)

where t′

∫

SE = dt L[x(t ), x (t )],

(9.34)

t

is the Euclidean action, which involves the classical Lagrangian L[x(t), x⋅(t)] and ∫ D[x ″] is an integral over all paths from x to x′ defined as x′

 m  D[x" ] ≡ lim   N →∞  2 πi∆t  x

∫

N /2

N −1

∫

∏ dxn ,

n =1

(9.35)

where xn = x(tn) and tn ∈ [t, t1, …, tN−1, t′]. For purposes of actual evaluation of quantities using path-integral methods,

−

π −βH π e + = 2 2

π /2, + π /2

∫

π /2, − π /2

DΩ e − SE ,

(9.38)

where β = 1/kBT is the inverse temperature, DΩ = Π τdΩ τ , dΩ τ = [4π/(2s + 1)]d(cos θ τ )dφ τ the Haar measure of the S2 sphere, and SE =

β

∫ dτ[isφ (1 − cos θ) + E 0

y ,n

] the Euclidean action, where

the first term in SE defines the Wess–Zumino (or Berry phase) term, which gives rise to topological interference effects for spin tunneling (DiVincenzo et al., 1992; von Delft and Henley, 1992). We can divide the S2 surface area of integration in Equation 9.38 into n equally shaped subareas An, so that we can factor out the following sum over Berry phase terms without the need of

9-7


4

sin (μ2π)/sin (μ2π/n)

2 μ 2

4

8

6

10

–2

–4

FIGURE 9.5â•… Sum over Berry phase terms for s = 10 and n = 4.

evaluating the dynamical part of the path integral (i.e., the following result is valid for all m), −

π −βHy ,n π e + ∝ 2 2

n

∑e

i (2 π / n)(2 k −1)m

k =1

=

sin(2πm) , sin(2πm/n)

(9.39)

which vanishes whenever n is not a divisor of 2m. Thus, the tunnel splitting energy Δm,−m between the states |m〉 and |−m〉 vanishes if 2m/n ∉ Z. However, if 2m/n ∈ Z, the variable m must be extended to real numbers, i.e., m → μ ∈ R, in order to calculate the limit μ → m of the ratio of the sine functions, which is plotted in Figure 9.5 for a special case. In order to visualize the interference between the Berry phases in Equation 9.39, we select one representative path of each subarea An. Then the vanishing of the amplitude in Equation 9.39 for the case n = 6 (see Figure 9.6) can be thought of as a destructive interference

z

y

θφ –y

FIGURE 9.6â•… Berry phase interference between six different paths for s = 10 and n = 6.

between six different paths. Also, it is important to note that from the semiclassical point of view, the total classical energy of the spin system E must be conserved during the tunneling process. Therefore, the semiclassical paths do not follow the local minima of E with respect to θ alone, except for the case of purely quadratic (i.e., n = 2) anisotropies (DiVincenzo et al., 1992; Garg, 1993). From the above treatment, we conclude that tunneling between two degenerate spin states |m〉 and |m′〉 = |−m〉 is topologically suppressed (i.e., the twofold degeneracy is not lifted) whenever n is not a divisor of 2m. Since n is even this excludes immediately tunneling for all half-odd integer spins s (for all m and n), in accordance with Kramers degeneracy. For s integer, however, tunneling can be either allowed or suppressed, Â�depending on the ratio 2m/n. In the latter case, the twofold degeneracy of spin states is not lifted by the anisotropy, and we can view this result as a generalization of the Kramers theorem to integer spins.

9.5â•‡ Incoherent Zener Tunneling in Fe8 Besides Mn12, there have been several experiments on the molecular magnet Fe8 that revealed macroscopic quantum tunneling of the spin (Barra et al., 1996; Wernsdorfer and Sessoli, 1996; Sangregorio et al., 1997; Ohm et al., 1998; Wernsdorfer et al., 2000). In particular, recent measurements on Fe8 (Wernsdorfer and Sessoli, 1996; Wernsdorfer et al., 2000) lead to the development of the concept of the incoherent Zener tunneling (Leuenberger and Loss, 2000b; Leuenberger et al., 2003). The resulting Zener tunneling probability P inc exhibits Berry phase oscillations as a function of the external transverse field Hx . For many physical systems, the Landau–Zener model (Landau, 1932; Zener, 1932) has become an important tool for studying tunneling transitions (Crothers and Huges, 1977; Garga et al., 1985; Shimshoni and Gefen, 1991; Averin and Bardas, 1995). It must be noted that all quantum systems to which the Zener model (Landau, 1932) is applicable can be described by pure states and their coherent time evolution. In particular, the theory presented in Leuenberger and Loss (2000b) agrees well with recent measurements of Pinc(Hx) for various temperatures in Fe8 (Wernsdorfer and Sessoli, 1996; Wernsdorfer et al., 2000). For the Zener transition, usually only the asymptotic limit is of interest. Therefore it is required that the range over which εmm′(t) = εm − εm′ is swept is much larger than the tunnel splitting Δmm′ and the decoherence rate ħγmm′. In addition, the evolution of the spin system is restricted to times t that are much longer than the decoherence time τd = 1/γmm′. In this case, tunneling transitions between pairs of degenerate excited states are incoherent. This tunneling is only observable if the temperature T is kept well below the activation energy of the potential barrier. Accordingly, one is interested only in times t that are larger than the relaxation times of the excited states. It was shown by Leuenberger and Loss (2000b) that the Zener tunneling can be described by Equation 9.22, where

9-8


2 ∆ mm γ mm′ ′ 2 2 2 2 ε mm ′ (t ) + γ mm ′

is time dependent, in contrast to Equation 9.23. As usual, the abbreviations γmm′ = (Wm + Wm′)/2 and Wm = ΣnWnm are used, where Wnm denotes the approximately time-independent transition rate from |m〉 to |n〉, which can be obtained via Fermi’s golden rule (Leuenberger and Loss, 1999, 2000a). The tunnel splitting (Leuenberger and Loss, 1999, 2000a) is given by

∆ mm′ = 2

∑

m1,…, mN mi ≠ m , m ′

Vm,m1 N −1 Vmi ,mi +1 Vm ,m′ . ∏ ε m − ε m1 i =1 ε m − ε mi +1 N

Vmi ,m j denote off-diagonal matrix elements of the total Hamiltonian Htot. Since all resonances n lead to similar results, Equation 9.22 is solved only in the unbiased case—corresponding to n = 0 (see below)—where the ground states |s〉, |−s〉 and the excited states |m〉, |−m〉, m ∈ [−s + 1, s − 1] of the spin system with spin s are pairwise degenerate. In addition, it is assumed that the excited states are already in their stationary state, i.e., ρ⋅m = 0 ∀m ≠ s, −s. Equation 9.22 leads then to

∫

(9.42)

where Δρ(t) = ρs − ρ−s, which satisfies the initial condition Δρ(t = t0) = 1, and thus Pinc(t = t0) = 0. The total time-dependent relaxation rate is given by Γtot = 2[Γs,−s + Γth], where the thermal rate Γth, which determines the incoherent relaxation via the excited states, is evaluated by means of relaxation diagrams (Leuenberger and Loss, 1999, 2000a). Assuming linear time dependence, i.e., ε mm′ (t ) = α mm′t , in the transition region (Landau, 1932), and with | ε mm ′ | γ mm ′ one obtains from Equation 9.42 t  2∆E 2   α −s s    s, − s t dt − ∆ρ = exp  − arctan ′ Γ  th −s    γ s,− s  − t  α s 

∫

10–3 0.05 K 10–4

0

t   π∆E 2   s,−s dt ′ Γ th  , ≈ exp  − − −s α s   −t

∫

(9.43)

 πE 2   πEs2, − s  ∆ρ = exp  − s ,−−ss  = exp  − .  α s   | ε s , − s (0) | 

(9.44)

0.4

0.8

1.2

Hx [T]

FIGURE 9.7â•… Zener transition probability Pinc(Hx) for temperatures T = 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, and 0.05â•›K . The fit agrees well with data. Note that P inc is equal to 2P. (From Wernsdorfer, W. et al., Europhys. Lett., 50, 552, 2000.)

The exponent in Equation 9.44 differs by a factor of 2 from the Zener exponent (Landau, 1932). This is not surprising since Γtot is the relaxation rate of Δρ, where both ρs and ρ−s are changed in time by the same amount, and not an escape rate like in the case of coherent Zener transition, where only the population of the initial state is changed in time. Equation 9.44 implies Pinc = 1 for |ε⋅s,−s(0)| → 0 (adiabatic limit) and Pinc = 0 for |ε⋅s,−s(0)| → ∞ (sudden limit). After fitting the parameters the incoherent Zener theory is in excellent agreement with experiments (Wernsdorfer and Sessoli, 1996; Wernsdorfer et al., 2000) for the temperature range 0.05â•›K ≤ T ≤ 0.7â•›K if the states |±10〉, |±9〉, and |±8〉 are taken into account. In particular, the path leading through |±8〉 gives a non-negligible contribution for T ≥ 0.6â•›K. One obtains the following from Equation 9.43 for Fe8 in the case n = 0:

Γ tot

where t0 = −t. In the low-temperature limit T → 0 the excited states are not populated anymore and thus Γth, which consists of intermediate rates that are weighted by Boltzmann factors bm (Leuenberger and Loss, 1999, 2000a), vanishes. Consequently, Equation 9.43 simplifies to

0.7 K

(9.41)

  t   1 − Pinc ≡ ∆ρ(t ) = exp  − dt ′ Γ tot (t ′) ,   t0

10–2

(9.40)

Pinc/2

Γ mm′ (t ) =

  −10 = 2  Γ10 +  

8

∑ n=9

2 W10,n

 πE 2 −10 − ∆ρ = exp  − 10−, 10 α 10 

  , 1  + Γ n, −n 

bn

8

∑α n=9

2 n, − n

πE −n n

W10,nbn

2 n, − n

E

+ 2W102,n

 , 

(9.45)

where the approximation γn,−n ≈ W10,n and | ε mm ′ | En, − n , γ n, − n is used. Pinc = 1 − Δρ, which is plotted in Figure 9.7, is in excellent agreement with the measurements (Wernsdorfer et al., 2000).

9.6â•‡Coherent Néel Vector Tunneling in Antiferromagnetic Molecular Wheels Antiferromagnetic molecular clusters are the most promising candidates for the observation of coherent quantum tunneling on the mesoscopic scale currently available (Chiolero and Loss,

9-9


1998). Several systems in which an even number N of antiferromagnetically coupled ions is arranged on a ring have been synthesized to date (Papaefthymiou et al., 1994; Fabretti et al., 1996; Schromm et al., 2001; Gatteschi et al., 2002). These systems are well described by the spin Hamiltonian

Hˆ = J

N

∑ i=1

N

sˆ i ⋅ sˆ i+1 + g µ BH ⋅

∑ i =1

N

sˆ i − kz

∑ sˆ

2 i ,z

,

(9.46)

i =1

where sî is the spin operator at site i with spin quantum number s, sˆN+1 ≡ sˆ1 J is the nearest neighbor exchange H is the magnetic field kz > 0 the single ion anisotropy directed along the ring axis The parameters J and kz have been well established both for various ferric wheels (Papaefthymiou et al., 1994; Fabretti et al., 1996; Kelemen et al., 1998; Cornia et al., 1999; Jansen et al. 1999; Koch et al., 1999; Schromm et al., 2001; Zotos et al., 2001) with N = 6, 8, 10, and, more recently, also for a Cr wheel (Gatteschi et al., 2002). For H = 0, the classical ground-state spin configuration of the wheel shows alternating (Néel) order with the spins pointing along ±ez . The two states with the Néel vector n along ±e z (Figure 9.8), labeled | ↑〉 and | ↓〉, are energetically degenerate and separated by an energy barrier of height Nk zs2. Because antiferromagnetic exchange induces dynamics of Néel-ordered spins, the states | ↑〉 and | ↓〉 are not energy eigenstates. Rather, a molecule prepared in spin state | ↑〉 would tunnel coherently between | ↑〉 and | ↓〉 at a rate Δ/h, where Δ is the tunnel splitting (Barbara and Chudnovsky, 1990; Krive and Zaslavskii, 1990). This tunneling of the Néel vector corresponds to a simultaneous tunneling of all N spins within the wheel through a potential barrier governed by the easy axis anisotropy. Within the framework of coherent state spin path integrals, an explicit expression for the tunnel splitting Δ as a function of magnetic field H has been derived (Chiolero and Loss, 1998). A magnetic field applied in the ring plane, Hx, gives rise to a Berry phase acquired by the spins during tunneling (DiVincenzo et al., 1992; van Delft and Henley, 1992; Garg, 1993; Leuenberger et al., 2003). The resulting interference Z

n=

of different tunneling paths leads to a sinusoidal dependence of Δ on Hx, which allows one to continuously tune the tunnel splitting from 0 to a maximum value which is of order of some Kelvin for the antiferromagnetic wheels synthesized to date. The tunnel splitting Δ also enters the energy spectrum of the antiferromagnetic wheel as level spacing between the ground and first excited state. Thus, Δ can be experimentally determined from various quantities such as magnetization, static susceptibility, and specific heat. Even more information on the physical properties of antiferromagnetic wheels (Equation 9.46) can be obtained from a theoretical and experimental investigation of dynamical quantities, such as the correlation functions N sˆ i or of single spins within the antiferof the total spin Sˆ =

∑

  e −β∆ / 2 e β∆ / 2 sî , z (t )sî , z (0) s 2  e i ∆t / + e −i ∆t /  2 cosh(β∆/2)  2 cosh(β∆/2) 

(9.47)

exhibits the time dependence characteristic of coherent tunneling of the quantity Sî,z with a tunneling rate Δ/h (Meier and Loss, 2001). We conclude that local spin probes are required for the observation of the Néel vector dynamics. Nuclear spins, which couple (predominantly) to a given single spin sî are ideal candidates for such probes (Meier and Loss, 2001) and have already been used to study spin cross-relaxation between electron and nuclear spins in ferric wheels (Lascialfari et al., 1999; Pini et al., 2000). For simplicity, we consider a single nuclear spin Î, I = 1/2, coupled to one-electron spin by a hyperfine contact interaction ˆ = Asˆ ⋅ I. ˆ According to Equation 9.47, the tunneling electron H′ 1 spin sˆ1 produces a rapidly oscillating hyperfine field As cos(Δt/ħ) at the site of the nucleus. Signatures of the coherent electron spin tunneling can thus also be found in the nuclear susceptibility. For a static magnetic field applied in the plane of the ring, Hx, it can be shown that the nuclear susceptibility

χ″I , yy (ω)

n=

FIGURE 9.8â•… The two degenerate classical ground state spin configurations of an antiferromagnetic molecular wheel with easy axis anisotropy.

i=1

romagnetic wheel (Meier and Loss, 2000, 2001). By symmetry arguments, it follows that the correlation function of total spin, 〈Sˆα(t)Sˆα (0)〉, which is experimentally accessible via measurement of the alternating current (AC) susceptibility does not contain a component, which oscillates with the tunnel frequency Δ/h (Meier and Loss, 2000, 2001). Hence, neither the tunnel splitting nor the decoherence rate of Néel vector tunneling can be obtained by experimental techniques which couple to the total spin of the wheel. In contrast, the correlation function of a single spin

π  βγ I H x  δ(ω − γ I H x / )  tanh  4 2 

2   As   β∆  +   tanh   δ(ω − ∆ / ) − [ω → −ω] (9.48)  ∆  2  

9-10


exhibits a satellite resonance at the tunnel splitting Δ of the electron spin system (Meier and Loss, 2001). Here, γIHx is the Lamor frequency of the nuclear spin and the first term in Equation 9.48 corresponds to the transition between the Zeeman split energy levels of I. Because typically As ∼ − 1â•›mK and Δ ≤ 2â•›K in Fe10, the spectral weight of the satellite peak is small compared to the one of the first term in Equation 9.48 unless the magnetic field is tuned such that Δ is significantly reduced compared to its maximum value. The observation of the satellite peak in Equation 9.48 is challenging, but possible with current experimental techniques (Meier and Loss, 2001). The experiment must be conducted with single crystals of an antiferromagnetic molecular wheel with sufficiently large anisotropy k z > 2J/(Ns)2 at high, tunable fields (10â•›T) and low temperatures (2â•›K). Moreover, because the tunnel splitting Δ(H) depends sensitively on the relative orientation of H and the easy axis (Jansen et al., 1999; Zotos et al., 2001), careful field sweeps are necessary to ensure that the satellite peak in Equation 9.48 has a large spectral weight. The need for local spin probes such as NMR or inelastic neutron scattering to detect coherent Néel vector tunneling can be traced back to the translation symmetry of the spin Hamiltonian Hˆ (Meier and Loss, 2000). If this symmetry is broken, e.g., by doping of the wheel, ESR also provides an adequate technique for the detection of coherent Néel vector tunneling. If one of the original Fe or Cr ions of the wheel with spin s = 5/2 or s = 3/2, respectively, is replaced by an ion with different spin s′ ≠ s, this will in general also result in a different exchange constant J′ and single ion anisotropy kz′ at the dopand site, i.e., Hˆ = J

N −1

∑ sˆ ⋅ sˆ i

i+1

+ J ′(sˆ1 ⋅ sˆ 2 + sˆ1 ⋅ sˆ N )

i=2

N

+ g µ BH ⋅

∑ i=1

 sˆ i −  kz′ sˆ12, z + kz 

N

∑ i=2

 sî2, z  . 

(9.49)

Although thermodynamic quantities, such as magnetization, of the doped wheel may differ significantly from the ones of

the undoped wheel, the picture of spin tunneling in antiferromagnetic molecular systems (Barbara and Chudnovsky, 1990; Krive and Zaslavskii, 1990) remains valid (Meier and Loss, 2000). However, due to unequal sublattice spins, a net total spin remains even in the Néel ordered state of the doped wheel (Figure 9.9). This allows one to distinguish the configurations sketched in Figure 9.8 according to their total spin. The dynamics of the total spin ŝ is coupled to the one of the Néel vector (DiVincenzo et al., 1992), and coherent tunneling of the Néel vector results in a coherent oscillation of the total spin. Coherent Néel vector tunneling in doped wheels can hence also be probed by ESR. The AC susceptibility shows a resonance peak at the tunnel splitting Δ,

χ″zz (ω ∆ / ) = π( g µ B )2 e Sˆ z g

2

 β∆  tanh   δ(ω − ∆ /).  2 

(9.50)

with a transition matrix element between the ground state |g〉 and first excited state |e〉, e Sˆz g s ′ − s

8 Jkz s 2 ( g µ B H x )2

(9.51)

for g µ B H x s 8 Jkz . The matrix element in Equation 9.51 determines the spectral weight of the absorption peak in the ESR spectrum. The analytical dependence has been determined within a semiclassical framework and is in good agreement with numerical results obtained from exact diagonalization of small systems (Figure 9.9). In conclusion, several antiferromagnetic molecular wheels synthesized recently are promising candidates for the observation of coherent Néel vector tunneling. Although the observation of this phenomenon is experimentally challenging, nuclear magnetic resonance, inelastic neutron scattering, and ESR on doped wheels are adequate experimental techniques for the observation of coherent Néel vector tunneling in antiferromagnetic molecular wheels.

0.5 0.4 | e|Sˆz|g |

Z

0.3 0.2 0.1 0

(a)

(b)

0

2

4 6 Hx [J/g μB]

8

10

FIGURE 9.9â•… The doped antiferromagnetic molecular wheel acquires a tracer spin which follows the Néel vector dynamics (a). Comparison of results obtained for the matrix element |〈e|Sˆz|g〉| with a coherent state spin path integral formalism (solid line) and by numerical exact diagonalization (symbols) for N = 4, s = 5/2, s′ = 2, J′ = J, kz = kz′ = 0.0055J (b).

9-11


9.7â•‡Berry-Phase Blockade in SingleMolecule Magnet Transistors Single-electron devices show many promising characteristics such as ultimate low power consumption, down scalability to atomic dimensions, and high switching speed. In addition, these devices are supposed to be good candidates for applications in quantum computation and quantum information technologies. Recently there has been a huge interest in using magnetic molecules with large spin in single-electron devices to uncover the effects that a large spin has upon electron transport through single-molecule magnets (SMM) (Sangregorio et al., 1997). During the past few years, recent experiments demonstrated the possibility to place individual molecules between the source and drain leads allowing electron transport measurements (Jo et al., 2006; Henderson et al., 2007). In a three terminal molecular single-electron transistor (SET), the molecule is situated between the source and drain leads with an insulated gate electrode underneath. The insulating ligands on the periphery of the molecule act as isolating barriers and current can flow between the source and drain leads via a sequential tunneling process through the molecular energy levels, which are tuned by the gate electrode (see Figure 9.10). Several experimental and theoretical groups have been trying to predict and prove the effects on electron transport through an individual molecular nanomagnet SET. Heersche et al. reported Coulomb blockade and conduction excitation characteristics in a Mn12 individual molecular nanomagnet SET. Negative differential conductance and current suppression effects were explained with a model that combines spin properties of the molecule with standard sequential tunneling theory. Recently, experiments have pointed out the importance of the interference between spin tunneling paths in molecules and its effects on electron transport scenarios involving SMMs. These results indicate that the Berry-phase interference plays an important role in the transport properties of SMMs in SET devices. Last year, the authors of this book chapter published an article ΓS

ΓD

Source V

Drain CS

CD

(González and Leuenberger, 2007) in which an effect called the Berry-phase blockade was explained. The effect we described is a quantum interference effect that can be detected experimentally by measuring the current through a single molecular electron transistor with oppositely polarized leads and by applying a transverse magnetic field along specific angles. In the following we summarize our results. For weak coupling between the leads and the SMM we used the generalized master equation describing the electronic spin states of the SMM (see Section 9.2). The sequential tunneling rates for absorption of an electron in Equation 9.19 for ground states with spin s and s′ in the case of low temperatures are given by

∑W

(l ) s ′,s

Ws ′,s =

, Ws(′,sl) = w↓(l ) f l (∆ s ′,s ),

l

W− s ′, − s =

∑W

(9.52)

(l ) − s ′, − s

(l ) − s ′, − s

, W

(l ) ↑ l

= w f (∆ − s ′, − s ),

l

and the tunneling rates for the emission of an electron are given by Ws ,s ′ =

∑W

(l ) s ,s ′

, Ws(,sl)′ = w↓(l )[1 − f l (∆ s ,s ′ )], (9.53)

l

W− s ,− s ′ =

∑W

(l ) − s ,− s ′

(l ) − s ,− s ′

, W

= w [1 − f l (∆ − s ,− s ′ )], (l ) ↑

l

(l ) where f l (∆ s ′,s ) = [1 + e ( ∆ s ′,s − µl )/ kT]−1 is the Fermi function. w↓↑ represents the spin-dependent transition rate from the l ∈ [Left, Right] lead to the SMM and are defined in Fermi’s golden rule approximation by w↓(l ) = 2πD ν(↓l ) | t ↓(l ) |2/ and w↑(l ) = 2πD ν(↑l ) | t ↑(l ) |2/ , respectively, where D is the density of states and ν(↑l ) and ν(↓l ) are fractions of the number of spins polarized up and down of lead l such that ν(↓l ) + ν(↑l ) = 1. t ↑(l ) and t ↓(l ) are the tunneling amplitudes of lead l, respectively. Typical values for the tunneling rate of the electron range from around w = 106 s−1 to w = 1010 s−1 (see González and Leuenberger, 2007, and references therein). Solving the generalized master equation for the stationary limit, we obtained the coupled differential equations

I 2

2γ s,− s ∆  ρs =  s , − s   2  ( g µ B H z (s − (− s)))2 / 2 + γ 2s,− s

CG Gate

VG

FIGURE 9.10â•… The sketch shows a three terminal molecular singleelectron transistor. The ligands on the periphery of the molecule act as isolating barriers and electrons can flow between source and drain leads by sequential or cotunneling processes through the discrete energy levels of the molecule.

× (ρ− s − ρs ) + Ws , s ′ ρs ′ − Ws ′,s ρs ,

(9.54)

2

2γ s,− s ∆  ρ− s =  s , − s   2  ( g µ B H z (s − (− s)))2 / 2 + γ 2s,− s

× (ρs − ρ− s ) + W− s, − s ′ρ− s ′ − W− s ′,− s ρ− s .

(9.55)

The other two differential equations are obtained by just replacing s ↔ s′ in the above equations. Solving the set of

9-12


differential equations for ρs , ρ−s, ρs′ and ρ−s′ in the stationary case (t >> 1/Wm,n), we obtain ρs = (Ws , s ′ (W− s ′, − s + Γ s , − s )Γ s ′, − s ′ + W− s , − s ′ (Ws ′, s + Γ s ′, − s ′ )Γ s , − s )/η, ρ− s = (Ws , s ′ (W− s ,− s ′ + Γ s ′ , − s ′ )Γ s ,-s + W− s , − s ′ (Ws ′, s + Γ s ,− s )Γ s ′ , − s ′ )/η, ρs ′ = (Ws ′ , s (W− s ,− s ′ + Γ s ′ , − s ′ )Γ s ,− s + W− s ′ , − s (Ws ,s′′ + Γ s ,− s )Γ s ′ , − s ′ )/η,

where η is a normalization factor such that ∑n ρn = 1. The incoherent tunneling rate is

∆  Γs,−s =  s,−s   2 

2

2γ s,−s

( g µ H ( s − (− s ) ) ) B

z

2

2 + γ 2s , − s

. (9.57)

I = e(W4,7 /2ρ7 /2 + W−4, −7 /2ρ −7 /2 ).

(9.58)

In the case of leads that are fully polarized in opposite directions, i.e., ν(↑L ) = ν(↓R ) = 1 or ν(↓L ) = ν↑( R ) = 1 , we get one of the two following conditions for the transition rates, respectively:

W−4, −7 /2 = W7 /2, 4 = 0 or W4,7/2 = W−7 / 2, −4 = 0.

(9.59)

Choosing the case ν(↑L ) = ν(↓R ) = 1 and using the condition W−4,−7/2 = W7/2,4 = 0, we obtain from Equation 9.58

Δ 4,– 4

2 3

1

2 1 00

] H y[T

FIGURE 9.11â•… The graph shows the log10 I vs. H⊥. The scale varies from I = 0.1â•›n A to 1â•›fA. From the figure we see that the current is completely suppressed at the zeros of the tunnel splitting, i.e., Δ4,−4 = Δ7/2,−7/2 = 0.

9.8â•‡ Concluding Remarks

We now proceed to define the current through the SMM in terms of the density matrix for the case of the single molecule magnet Ni4. In the case of Ni4 we have s = 4 and s′ = 7/2, therefore the current reads

/2

Δ 7/2,–7

[T] Hx

ρ− s ′ = (Ws ′ , s (W− s ′ ,− s + Γ s , − s )Γ s ′ ,− s ′ + W− s ′ , − s (Ws ,s ′ + Γ s ′ ,− s ′ )Γ s , − s )/η, (9.56)

log10 Ip –10 –11 –12 –13 –14 –15

e 2 1 2 1 = + + + . I W−7 /2, −4 Γ 4, −4 W4,7 /2 Γ 7 /2, −7 /2

(9.60)

Equation 9.60 reflects the fact that the current through the SMM depends on the tunnel splittings. The transitions that contribute to the current through the SMM in the case of fully polarized leads ν(↑L ) = ν(↓R ) = 1 are 4 → 7/2 → − 7/2 → −4. Figure 9.11 shows the current as a function of H ⊥ for fully polarized leads. If the tunnel splitting Δ4,−4 or Δ7/2,−7/2 is topologically quenched (see Figure 9.11), Γ4,−4 or Γ7/2,−7/2 vanishes (see Equation 9.57), which leads to complete current suppression according to Equation 9.60. Since this current blockade is a consequence of the topologically quenched tunnel splitting, we call it Berry-phase blockade. Note that the current can also be suppressed by applying Hz, which follows immediately from Equations 9.57 and 9.60.

We have seen how molecular nanomagnets offer an interesting platform to explore the quantum dynamics of mesoscopic systems. The theoretical advantage of using molecular nanomagnets is that molecules are all identical to each other, allowing the performance of experiments on large assemblies of identical particles and detect quantum effects. Molecules can be investigated in solutions or solids depending on the experiment you want to perform, allowing accurate measurements. We have paid special attention to the quantum tunneling of the magnetization of molecular nanomagnets and the different approaches that exist to describe this phenomenon. In particular we have placed emphasis on the interference effects of the quantum tunneling of the magnetization via Berry’s phase and we have included some of our recent results in this field (González and Leuenberger, 2007). Molecular nanomagnets are also the subject of intense research in molecular electronics. Single-electron devices have taken advantage of the small size of nanomagnets to create electronic structures of atomic dimensions. These devices are supposed to be good candidates for applications in quantum computation and quantum information technologies. There has been also a lot of research on single-electron transistors (SET) made of molecular nanomagnets to study quantum effects such as the Kondo effect in this kind of structures (Leuenberger and Mucciolo, 2006). The Kondo effect is a very well known and studied phenomenon in condensed matter physics. This effect arises when a magnetic impurity is placed into a conductor, which causes a dramatic increase in the resistivity of the metal at low temperatures. The Kondo effect can be seen also in a single-electron transistor where the molecular nanomagnet plays the role of the magnetic impurity and the leads of the SET mimics the bulk metal. It has been shown that the Hamiltonian for a molecular nanomagnet placed in a SET device can be mapped onto the Kondo Hamiltonian (González et al., 2008). This result is of great interest because one can apply the poor


man’s scaling theory to describe the Kondo physics at low temperatures and see how the Berry-phase oscillations become temperature dependent. This fact allows us to conclude that the scaling equations can be checked experimentally by measuring the renormalized zero points of the Berry phase. The field of molecular nanomagnets is growing very fast and chemistry is playing a major role concerning the growth of these nanostructures. New nanomagnets with novel and interesting magnetic properties are being produced, which demand more sophisticated theories to explain and describe the dynamics of these magnetic mesoscopic systems. One direction which remains a major area of investigation lately is related to storing and decoding information on a single spin state to be able to create a quantum computer. The implementation of Grover’s algorithm with molecular nanomagnets has already been proposed, however, it is required to perform simultaneous manipulation of many spin phases without losing quantum coherence, which remains still an experimental challenge.

Acknowledgment We acknowledge support from NSF-ECCS 0725514, the DARPA/ MTO Young Faculty Award HR0011-08-1-0059, 0901784, and AFOSR FA 9550-09-1-0450.

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10 Inelastic Electron Transport through Molecular Junctions

Natalya A. Zimbovskaya University of Puerto Rico

10.1 Introduction............................................................................................................................10-1 10.2 Coherent Transport................................................................................................................10-2 10.3 Buttiker Model for Inelastic Transport...............................................................................10-5 10.4 Vibration-Induced Inelastic Effects.....................................................................................10-6 10.5 Dissipative Transport.............................................................................................................10-9 10.6 Polaron Effects: Hysteresis, Switching, and Negative Differential Resistance............10-11 10.7 Molecular Junction Conductance and Long-Range Electron-Transfer Reactions.....10-13 10.8 Concluding Remarks............................................................................................................10-15 References..........................................................................................................................................10-16

10.1â•‡ Introduction Molecular electronics is known to be one of the most promising developments in nanoelectronics, and the past decade has seen extraordinary progress in this field (Aviram et al. 2002, Cuniberti et al. 2005, Nitzan 2001). Present activities on molecular electronics reflect the convergence of two trends in the fabrication of nanodevices, namely, the top-down device miniaturization through the lithographic methods and bottomup device manufacturing through the atom-engineering and the self-assembly approaches. The key element and the basic building block of molecular electronics is a junction including two electrodes (leads) linked by a molecule, as schematically shown in Figure 10.1. Usually, the electrodes are microscopic large but macroscopic small contacts that could be connected to a battery to provide the bias voltage across the junction. Such a junction may be treated as a quantum dot coupled to the charge reservoirs. The discrete character of energy levels on the dot (molecule) is combined with nearly continuous energy spectra on the reservoirs (leads) occurring due to their comparatively large size. When the voltage is applied, an electric current flows through the junction. Successful transport experiments with molecular junctions (Ho 2002, Lortscher et al. 2007, Park et al. 2000, Poot et al. 2006, Reichert et al. 2002, Smit et al. 2002, Yu et al. 2004) confirm their significance as active elements in nanodevices. These include applications as rectifiers (molecular diodes), field-effect transistors (molecular triodes), switches, memory elements and sensors. Also, these experiments emphasize the importance of a thorough analysis of the physics underlying electron transport

through molecular junctions. A detailed understanding of electron transport at the molecular scale is a key step to future device operations. A theory of electron transport in molecular junctions is being developed since the last two decades, and main transport mechanisms are currently elucidated in general terms (Datta 2005, Imry and Landauer 1999). However, a progress of the experimental capabilities in the field of molecular electronics brings new theoretical challenges causing a further development of the theory. Speaking of transport mechanisms, it is useful to make a distinction between elastic electron transport when the electron energy remains the same as it travels through the junction, and inelastic transport processes when the electron undergoes energy changes due to its interactions with the environment. There are several kinds of processes bringing inelasticity in the electron transport in mesoscopic systems including molecular junctions. Chief among these are electron–electron and electron– phonon scattering processes. These processes may bring about significant inelastic effects modifying the transport properties of molecular devices and charging, desorption, and chemical reactions as well. To keep this chapter at a reasonable length, we concentrate on the inelastic effects originating from electron– phonon interactions. In practical molecular junctions, the electron transport is always accompanied by nuclear motions in the environment. Therefore, the conduction process is influenced by the coupling between the electronic and the vibrational degrees of freedom. Nuclear motions underlie the interplay between the coherent electron tunneling through the junction and the inelastic thermally assisted hopping transport 10-1

10-2

Left

Right


(a) μL

(b)

μR

V

Figure 10.1â•… Schematic drawing of a junction including two electrodes and a molecule in between (a). When the voltage is applied across the junction, electrochemical potentials μL and μR differ, and the conduction window opens up (b).

(Nitzan 2001). Also, electron–phonon interactions may result in polaronic conduction (Galperin et al. 2005, Gutierrez et al. 2005, Kubatkin et al. 2003, Ryndyk et al. 2008), and they are directly related to the junction heating (Segal et al. 2003) and to some specific effects such as alterations in both shape of the molecule and its position with respect to the leads (Komeda et al. 2002, Mitra et al. 2004, Stipe et al. 1999). The effects of electron–phonon interactions may be manifested in the inelastic tunneling spectrum (IETS) that presents the second derivative of the current in the junction d 2 I/ dV versus the applied voltage V. The inelastic electron tunneling microscopy has proven to be a valuable method for the identification of molecular species within the conduction region, especially when employed in combination with scanning tunneling microscopy and/or spectroscopy (Galperin et al. 2004). Inelasticity in the electron transport through molecular junctions is closely related to the dephasing effects. One may say that incoherent electron transport always includes an inelastic contribution with the possible exception of the low temperature range. The general approach to theoretically analyzing electron transport through molecular junctions in the presence of dissipative/phase-breaking processes in both electronic and nuclear degrees of freedom is based on advanced formalisms (Segal et al. 2000, Skourtis and Mukamel 1995, Wingreen et al. 1989, 1993). These microscopic computational approaches have the advantages of being capable of providing detailed dynamics information. However, while considering stationary electron transport through molecular junctions, one may turn to the less time-consuming approach based on scattering matrix formalism (Buttiker 1986, Li and Yan 2001a,b), as discussed below.

10.2â•‡ Coherent Transport To better show the effects of dissipation/dephasing on the electron transport through molecular junctions, it seems reasonable to start from the case where these effects do not occur. So, we

consider a molecule (presented as a set of energy levels) placed in between two electrodes with nearly continuous energy spectra. While there is no bias voltage applied across the junction, the latter remains in equilibrium characterized by the equilibrium Fermi energy EF, and there is no current flowing through it. When the bias voltage is applied, it keeps the left and right electrodes at different electrochemical potentials μL and μR . Then the electric current appears in the junction, and the molecular energy levels located in between the electrochemical potentials μL and μR play a major part in maintaining this current. Electrons from occupied molecular states tunnel to the electrodes in accordance with the voltage polarity, and the electrons from one electrode travel to another one using unoccupied molecular levels as intermediate states for tunneling. Usually, the electron transport in molecular junctions occurs via the highest occupied (HOMO) and the lowest unoccupied (LUMO) molecular orbitals that work as channels for electron transmission. Obviously, the current through the junction depends on the quality of contacts between the leads and the molecule ends. However, there also exists the limit for the conductance in the channels. As was theoretically shown (Landauer 1970), the maximum conductance of a channel with a single-spin degenerateenergy level equals:

G0 =

e2 = (25.8 kΩ)−1 π

(10.1)

where e is the electron charge ħ is Planck’s constant This is a truly remarkable result for it proves that the minimum resistance R0 = G0−1 of a molecular junction cannot become zero. In another words, one never can short-circuit a device operating with quantum channels. Also, the expression (10.1) shows that the conductance is a quantized quantity. Conductance g in practical quantum channels associated with molecular orbitals can take on values significantly smaller that G 0, depending on the delocalization in the molecular orbitals participating in the electron transport. In molecular junctions it also strongly depends on the molecule coupling to the leads (quality of contacts), as was remarked before. The total resistance r = g −1 includes contributions from the contact and the molecular resistances, and could be written as (Wingreen et al. 1993)

r=

1  1−T  1+ . G0  T 

(10.2)

Here, T is the electron transmission coefficient that generally takes on values less than unity. The general expression for the electric current flowing through the molecular junction could be obtained if one calculates the total probability for an electron to travel between two

Inelastic Electron Transport through Molecular Junctions

10-3

electrodes at a certain tunnel energy E and then integrates the latter over the whole energy range (Datta 1995). This results in the well-known Landauer expression:

the molecule, the external electrostatic field is screened due to the charge redistribution, and the electron transmission is not sensitive to the changes in the voltage V. Although very simple, this model allows one to analyze the main characteristics of the electron transport through molecular junctions. Within this model one may write down the following expression for the self-energy parts (D’Amato and Pastawski 1990):

I=

e T (E)  f L (E) − f R (E) dE. π

∫

(10.3)

Here, f L,R(E) are Fermi distribution functions for the electrodes with chemical potentials μL,R, respectively. The values of μL,R differ from the equilibrium energy EF, and they are determined by the voltage distribution inside the system. Assuming that coherent tunneling predominates in electron transport, the electron transmission function is given by (Datta et al. 1997, Samanta et al. 1996):

{

}

T (E) = 2Tr ∆ LG∆ RG + ,

(10.4)

where the matrices ΔL,R represent the imaginary parts of the selfenergy terms ΣL,R describing the coupling of the molecule to the electrodes G is the Green’s function matrix for the molecule whose matrix elements between the molecular states 〈i| and |j〉 have the form

Gij = i E − H j .

(10.5)

Here, H is the molecular Hamiltonian including the self-energy parts ΣL,R . When a molecule contacts the surface of electrodes, this results in a charge transfer between the molecule and the electrodes, and in a modification of the molecule energy states due to the redistribution of the electrostatic potential within the molecule. Besides, the external voltage applied across the junction brings additional changes to the electrostatic potential further modifying the molecular orbitals. The coupling of the molecule to the leads may also depend on the voltage distribution. So, generally, the electron transmission T inserted in Equation 10.3 and the electrochemical potentials μL,R depend on the electrostatic potential distribution in the system. To find the correct distribution of the electric field inside the junction one must simultaneously solve the Schrodinger equation for the molecule and the Poisson equation for the charge density, following a self-consistent converging procedure. This is a nontrivial and complicated task, and significant effort was applied to study the effect of electrostatic potential distribution on the electron transport through molecules (Damle et al. 2001, Di Ventra et al. 2000, Galperin et al. 2006, Lang and Avouris 2000a,b, Mujica et al. 2000, Xue and Ratner 2003a,b, 2004, Xue et al. 2001). Here, we put these detailed considerations aside, and we use the simplified expression for the electrochemical potentials:

µ L = EF + η e V ; µ R = EF − (1 − η) e V ,

(10.6)

where the parameter η indicates how the bias voltage is distributed between the electrodes. Also, we assume that inside

(Σ β )ij =

τ*ik ,β τ kj ,β . k ,β + is

∑E−ε k

(10.7)

Here, β ∈ L, R, τik,β is the coupling strength between the ith molecular state and the kth state on the left/right lead εk,β are the energy levels on the electrodes s is a positive infinitesimal parameter Assuming that the molecule is reduced to a single orbital with the energy E 0 (a single-site bridge), Green’s function accepts the form

G( E ) =

1 . E − E0 − Σ L − Σ R

(10.8)

Accordingly, in this case, one may simplify the expression (10.4) for the electron transmission:

T (E) =

4∆L ∆R . (E − E0 )2 + (∆ L + ∆ R )2

(10.9)

To elucidate the main features of electron transport through molecular junctions we consider a few examples. In the first example, we mimic a molecule as a one-dimensional chain consisting of N identical hydrogen-like atoms with the nearest neighbors interaction. We assume that there is one state per isolated atom with the energy E 0, and that the coupling between the neighboring sites in the chain is characterized by the parameter b. Such a model was theoretically analyzed by D’Amato and Pastawski and in some other works (see e.g., Mujica et al. 1994). Based on Equations 10.4 and 10.5, it could be shown that for a single-site chain (N = 1) the electron transmission reveals a well-distinguished peak at E = E 0, shown in Figure 10.2. The height of this peak is determined by the coupling of the bridge site to the electrodes. The peak in the electron transmission arises because the molecular orbital E = E 0 works as the channel/bridge for electron transport between the leads. Similar peaks appear in the conductance g = dI/dV. Assuming the symmetric voltage distribution (η = 1/2), the peak in the conductance is located at V = ±2E 0. As for the current voltage characteristics, they display step-like shapes with the steps at V = ±2E 0. When the chain includes several sites, we obtain a set of states (orbitals) for our bridge instead of the single state E = E 0, and their number equals the number of

10-4


1

Current (μA)

Transmission

20

0.5

0

–1

E (eV)

1

0

10

0

1 Voltage (V )

2

Figure 10.2â•… Coherent electron transmission (left panel) and current (right panel) versus bias voltage applied across a molecular junction where the molecule is simulated by a single electronic state. The curves are plotted assuming ΔL = ΔR = 0.1â•›eV, E 0 = −0.5â•›eV, T = 30â•›K. 1

Current (μA)

Transmission

20

0.5

0

–1

–0.5

0

E (eV)

10

0

1

2

Voltage (V )

Figure 10.3â•… Coherent electron transmission (left panel) and current (right panel) through a junction with a five electronic states bridge. The curves are plotted for ΔL = ΔR = 0.1â•›eV, b = 0.3â•›eV, E0 = −0.5â•›eV, T = 30â•›K.

sites in the chain. All these states are the channels for the electron transport. Correspondingly, the transmission reveals a set of peaks as presented in Figure 10.3. The peaks are located within the energy range with the width 4b around E = E 0. The coupling of the chain ends to the electrodes affects the transmission, especially near E = E 0. As the coupling strengthens, the transmission minimum values increase. Now, the current voltage curves exhibit a sequence of steps. The longer the chain, the more energy levels it possesses, and the more the number of steps in the I–V curves. The second example concerns the electron transport through a carbon chain placed between copper electrodes. In this case, as well as for practical molecules, a preliminary step in transport calculations is to compute the relevant molecular energy levels and wave functions. Usually, these computations are carried out employing quantum chemistry software packages (e.g., Gaussian) or density functional-based software. Also, a proper treatment of the molecular coupling to the electrodes is necessary for it brings changes into the molecular energy states. For this purpose, one may use the concept of an

“extended molecule” proposed by Xue et al. (2001). The point of this concept is that only a few atoms on the surface of the metallic electrode are significantly disturbed when the molecule is attached to the latter. These atoms are located in the immediate vicinity of the molecule end. Therefore, one may form a system consisting of the molecule itself and the atoms from the electrode surfaces perturbed by the molecular presence. This system is called the extended molecule and treated as such while computing the molecular orbitals. In the example considered, the extended molecule included four copper atoms on each side of the carbon chain. The results for electron transport are shown in Figure 10.4. Again, we observe a comb-like shape of the electron transmission corresponding to the set of transport channels provided by the molecular orbitals, and the stepwise I–V curve originating from the latter. Transport calculations similar to those described above were repeatedly carried out in the last two decades for various practical molecules (see e.g., Galperin et al. 2006, Xue and Ratner 2003a,b, Xue et al. 2001, Zimbovskaya 2003, 2008, Zimbovskaya and Gumbs 2002).

10-5

Inelastic Electron Transport through Molecular Junctions ×10–3 200

1

Current (nA)

Transmission

1.5

0.5

0

–4

100

0

–2 E (eV)

4

7 Voltage (V )

10

Figure 10.4â•… Coherent electron transmission (left panel) and current (right panel) through a carbon chain coupled to the copper leads at T = 30â•›K.

10.3â•‡Buttiker Model for Inelastic Transport

b΄1

a1

a2

a΄1

a΄2

a΄4

a4

a΄3

Right

b1

Left

An important advantage of the phenomenological model for the incoherent/inelastic quantum transport proposed by Buttiker (1986) is that this model could easily be adapted to analyze various inelastic effects in electron transport through molecules (and some other mesoscopic systems) avoiding complicated and time-consuming advanced methods, such as those based on the nonequilibrium Green’s functions formalism (NEGF). Here, we present the Buttiker model for a simple junction including two electrodes linked by a single-site molecular bridge. The bridge is attached to a phase-randomizing electron reservoir, as shown in Figure 10.5. Electrons tunnel from the electrodes to the bridge and vice versa via the channels 1 and 2. While on the bridge, an electron could be scattered into the channels 3 and 4 with a certain probability ε. Such an electron arrives at the reservoir where it undergoes inelastic scattering accompanied by phase-breaking and then the reservoir reemits it back to the channels 3 and 4 with the same probability. So, within the Buttiker model, the electron transport through the junction is treated as the combination of tunnelings through

the barriers separating the molecule from the electrodes and the interaction with the phase-breaking electron reservoir coupled to the bridge site. The key parameter of the model is the probability ε that is closely related to the coupling strength between the bridge site and the reservoir. When εâ•›=â•›0 the reservoir is detached from the bridge, and the electron transport is completely coherent and elastic. Within the opposite limit (εâ•›=â•›1), electrons are certainly scattered into the reservoir that results in the overall phase randomization and inelastic transport. Within the Buttiker model, the particle fluxes outgoing from the junctions J′i could be presented as the linear combinations of the incoming fluxes Jk where the indexes i, k label the channels for the transport: 1 ≤ i, k ≤ 4.

J i′ =

∑T J . ik k

(10.10)

k

b΄2

The coefficients Tik in these linear combinations are matrix elements of the transmission matrix that are related to the elements of the scattering matrix S, namely, Tik = |Sik|2. The matrix S expresses the outgoing wave amplitudes b′1, b′2, a′3, a′4 in terms of the incident ones b1, b 2, a3, a4. To provide the charge conservation in the system, the net current in the channels 3 and 4 linking the system with the dephasing reservoir must be zero, so we may write

b2

a3

Reservoir

Figure 10.5â•… Schematic drawing illustrating inelastic electron transport through a molecular junction within the Buttiker model.

J 3 + J 4 − J 3′ − J 4′ = 0.

(10.11)

The transmission for quantum transport could be defined as the ratio of the particle flux outgoing from the system and that one incoming to the latter. Solving Equations 10.10 and 10.11 we obtain

T (E) =

J2′ K ⋅K = T21 + 1 2 , J1 2R

(10.12)

10-6


where

step-like shape to the volt–ampere curve, as was discussed in the previous section. In the presence of dephasing, the peak gets eroded. When the ε value approaches 1 the I–V curve becomes linear, corroborating the ohmic law for the inelastic transport. Within Buttiker’s model, ε is introduced as a phenomenological parameter whose relation to the microscopic characteristics of the dissipative processes affecting electron transport through molecular junctions remains uncertain. To further advance this model one should find out how to express ε in terms of the relevant microscopic characteristics for various transport mechanisms. This should open the way to a making of a link between the phenomenological Buttiker model and the NEGF. Such an attempt has been carried out in recent works (Zimbovskaya 2005, 2008) where the effect of stochastic nuclear motion on electron transport through molecules was analyzed.

K1 = T31 + T41 ; K 2 = T23 + T24 ; R = T33 + T44 + T43 + T34 .

(10.13)

For the junction including the single-site bridge the scattering matrix S has the form (Buttiker 1986)

 r1 + α 2r2  1 αt1t 2 S =  Z βt1   αβt1r2

αt1t 2

βt1

r2 + α 2r1

αβr1t 2

–βr1t 2

β2r1

βt 2

αr1r2 − α

αβt1r2   βt 2  . αr1r2 − α  β2r2 

(10.14)

Here, Z = 1 − α2r1r2, α = 1 − ε , β = ε , and r1,2 and t1,2 are the amplitude reflection and the transmission coefficients for the two barriers. Later, the expression for this matrix suitable for the case of multisite bridges including several inelastic scatterers was derived (Li and Yan 2001a,b). Assuming for certainty the charge flow from the left to the right, we may write down the following expression (Zimbovskaya 2005):

T (E ) =

g (E )(1 + α 2 )  g (E)(1 + α 2 ) + 1 − α 2   g (E)(1 − α 2 ) + 1 + α 2   

2

,

(10.15)

where

g (E ) = 2

∆L∆R

(E − E0 )2 ( ∆ L + ∆ R )2

.

(10.16)

Now, the electron transmission strongly depends on the dephasing strength ε. As shown in Figure 10.6, coherent transmission (ε = 0) exhibits a sharp peak at E = E 0 that gives a

10.4â•‡Vibration-Induced Inelastic Effects The interaction of electrons with molecular vibrations is known to be an important source of inelastic contribution to the electron transport through molecules. Theoretical studies of vibrationally inelastic electron transport through molecules and other similar nanosystems (e.g., carbon nanotubes) have been carried out over the past few years by a large number of researchers (Cornaglia et al. 2004, 2005, Donarini et al. 2006, Egger and Gogolin 2008, Galperin et al. 2007, Gutirrez et al. 2006, Kushmerick et al. 2004, Mii et al. 2003, Ryndyk and Cuniberti 2007, Siddiqui et al. 2007, Tikhodeev and Ueba 2004, Troisi and Ratner 2006, Zazunov and Martin 2007, Zazunov et al. 2006a,b, Zimmerman et al. 2008). Also, manifestations of the electron– vibron interactions were experimentally observed (Agrait et al. 2003, Djukic et al. 2005, Lorente et al. 2001, Qin et al. 2004, Repp et al. 2005a,b, Segal 2001, Smit et al. 2004, Tsutsui et al. 2006, Wang et al. 2004, Wu et al. 2004, Zhitenev et al. 2002). To analyze vibration induced effects on electron transport through molecular bridges, one must assume that molecular orbitals are coupled

1

Current (μA)

Transmission

20

0.5

0

–1

–0.5 E (eV)

0

10

0

0.5

1 Voltage (V )

1.5

Figure 10.6â•… Electron transmission (left panel) and current (right panel) computed within the Buttiker model at various values of the dephasing parameter ε, namely, ε = 0 (dotted lines), ε = 0.5 (dashed lines), and ε = 1 (solid lines). The curves are plotted assuming that the molecule is simulated by a single orbital with E 0 = −0.5â•›eV at ΔL = ΔR = 0.1â•›eV, T = 30â•›K.


10-7

to the phonons describing vibrations. While on the bridge, electrons may participate in the events generated by their interactions with vibrational phonons. These events involve a virtual phonon emission and absorption. For rather strong electron– phonon interaction this leads to the appearance of metastable electron levels that could participate in the electron transport through the junctions, bringing an inelastic component to the current. As a result, vibration induced features occur in the differential molecular conductance dI/dV and in the IETS d2I/dV2. This was observed in experiments (see e.g., Qin et al. 2004, Zhitenev et al. 2002). Sometimes, even current voltage curves themselves exhibit an extra step originating from the electron– vibron interactions (Djukic et al. 2005). Particular manifestations of electron–vibron interaction effects in the transport characteristics are determined by the relation of three relevant energies. These are the coupling strengths of the molecule to the electrodes ΔL,R, the electron–phonon coupling strength λ, and the thermal energy kT (k is the Boltzmann constant). When the molecule is weakly coupled to the electrodes (ΔL,R 1. Within this regime, the characteristic time for the electron bath interactions is much shorter than the typical electron time scales. Consequently, the bath makes a significant impact on the molecule electronic structure. New bath-induced states appear in the molecular spectrum inside the HOMO–LUMO gap. However, these states are strongly damped due to the dissipative action of the bath (Gutierrez et al. 2005). As a result, a small finite density of phonon-induced states appears inside the gap supporting electron transport at a low bias voltage. So again, the environment induces incoherent phonon-assisted transport through molecular bridges. For illustration, we show here the results of

Phonon bath

(10.27)

(

)

ρ2 1 + 1 + ρ2 1 2  E − E0  2 1 4 + 1 + 1 + ρ2  ∆ L + ∆ R  2

where ρ2 = 32kTλ/(ΔL + ΔR)2.

(

)

3

(10.28)

,

Right

ε=

Left

where Γ = ΔL + ΔR + (1/2)Γph(E). Solving the obtained equation for Γph and using Equation 10.19, we obtain (Zimbovskaya 2008)

Figure 10.9â•… Schematic drawing of a molecular junction where the molecular bridge is coupled to the phonon bath via side chains.

10-11

Inelastic Electron Transport through Molecular Junctions 1

T(E)w2/4Δ2

100×g/G0

0.6

0.4

0.5

0

0.2 0

0.5 Voltage (V )

1

–2

–1

0 E/w

1

2

Figure 10.10â•… Left panel: The electron conductance versus voltage for a junction with a single electronic state bridge directly coupled to the phonon thermal bath. The curves are plotted assuming E0 = 0.4â•›eV, ΔL = ΔR = 0.1eV, T = 30â•›K, λ = 0.3â•›eV (solid line), λ = 0.05â•›eV (dashed line). Right panel: Electron transmission through the junction in the case when the bridge state interacts with the phonon bath via the side chain coupled to the bridge state with the coupling parameter w. The curves are plotted assuming ΔL = ΔR = Δ, w/Δ = 20, E0 = 0, λ = 0.3â•›eV (solid line), λ = 0.05â•›eV (dashed line).

calculations carried out for a toy model with a single-site bridge with a side chain attached to the latter. The side chain is supposed to be coupled to the phonon bath. The results for the electron transmission are displayed in Figure 10.10 (right panel). We see that the original bridge state at E = 0 is completely damped but two new phonon-induced states emerge nearby that could support electron transport. An important characteristics of the dissipative electron transport through molecular junctions is the power loss in the junction, that is, the energy flux from the electronic into the phononic system. Assuming the current flow from the left to the right, this quantity may be estimated as the sum of the energy fluxes QL,R at the left and the right terminals (leads):

P = QL + QR .

(10.29)

One may express the energy fluxes in terms of renormalized cur~ rents at the electrodes IL,R(E) that are defined as follows (Datta 2005):

I L, R =

e I L , R (E)dE. π

∫

(10.30)

Then QL,R may be presented in the form

QL , R =

1 EIL , R (E)dE. π

∫

(10.31)

We remark that the current IL,R in Equation 10.30 are related to the corresponding leads, and their signs are accordingly defined. An outgoing current is supposed to be positive whereas an incoming one is negative for each lead. To provide an electric charge conservation in the junction one must require that I = IL = −IR for the chosen direction of the current flow. Therefore, the energy fluxes also have different signs. As for the QL,R magnitudes, they

~

~

may differ only if the renormalized currents IL(E) and IR(E) are distributed over energies in different ways. This cannot happen ~ ~ in the case of elastic transport, for in this case, IL(E) = −IR (E). However, if the transport process is accompanied with the energy ~ ~ dissipation, the energy distributions for IL and IR may differ. In this case, electrons lose some energy while moving through the junction and this gives rise to the differences in the renormalized currents’ energy distributions. For instance, in the case when the electrodes are linked with a single-state bridge, the energy ~ distribution of I L(E) has a single maximum whose position is determined by the site energy E0 and the applied bias voltage V. Assuming that the average energy loss due to dissipation could ~ be estimated as ΔE, the maximum in the IR(E) distribution is ~ shifted by this quantity, so the current I R(E) flows at lower ener~ gies compared to IL(E). This results in the power loss and the Joule heating in the junction (Segal et al. 2003).

10.6â•‡Polaron Effects: Hysteresis, Switching, and Negative Differential Resistance While studying electron transport through molecular junctions, hysteresis in the current–voltage characteristics was reported in some systems (Li et al. 2003). Multistability and stochastic switching were reported in single-molecule junctions (Lortscher et al. 2007) and in single metal atoms coupled to a metallic substrate through a thin ionic insulating film (Olsson et al. 2007, Repp et al. 2004). The coupling of an electron belonging to a certain atomic energy level to the displacements of ions in the film brings a possibility of polaron formation in there. This leads to the polaron shift in the electron energy. It was noticed that multistability and hysteresis in molecular junctions mostly occurred when the molecular bridges included centers of long-living charged electronic states (redox centers). On these grounds, it was suggested

10-12


that hysteresis in the I–V curves observed in molecular junctions appear due to the formation of polarons on the molecules (Galperin et al. 2005). The presence of the polaron shift in the energy of a charged (occupied) electron state creates a difference between the latter and the energy of the same state while it remains unoccupied. Assuming, for simplicity, a single-state model for the molecular bridge coupled to a single optical phonon mode, we may write the following expression for the renormalized energy: λ 2n0 E0 (n0 ) = E0 − , Ω

(10.32)

where the electronic population on the bridge n0 is given by n0 =

f L ( E )∆ L + f R ( E )∆ R 1 dE . 2 2 π  E − E0 (n0 ) + ( ∆ L + ∆ R )  

∫

(10.33)

Epot (arb. units)

Current (arb. units)

So, as follows from Equation 10.32, the polaron shift depends on the bridge occupation n 0, and the latter is related to E˜ 0 by Equation 10.33. Therefore, the derivation of an explicit expression for E˜ 0(n 0) is a nontrivial task even within the chosen simple model. Nevertheless, it could be shown that two local minima emerge in the dependence of the potential energy of the molecular junction including two electrodes linked by the molecular bridge, of the occupation number n 0. These minima are located near n0 = 0 and 1, and they correspond to the neutral (unoccupied) and charged (occupied) states, respectively. This is illustrated in Figure 10.11 (left panel). These states are metastable, and their lifetime could be limited by the quantum switching (Mitra et al. 2005, Mozyrsky et al. 2006). When the switching time between the two states is longer than the characteristic time for the external voltage sweeping, one may expect the hysteresis to appear, for the states of interest live long enough to maintain it. Within the opposite limit, the average

washes out the hysteresis. When the states are especially shortlived this could even result in a telegraph noise at a finite bias voltage that replaces the controlled switching. Further, we consider long-lived metastable states and we concentrate on the I–V behavior. Let us for certainty assume that the bridge state at zero bias voltage is situated above the Fermi energy of the system and remains empty. As the voltage increases, one of the electrode’s chemical potentials crosses the bridge level position, and the current starts to flow through the system. The I–V curve reveals a step at the bias voltage value corresponding to the crossing of the unoccupied bridge level with the energy E 0 by the chemical potential. However, while the current flows through the bridge, the level becomes occupied and, consequently, shifted due to the polaron formation. When the bias voltage is reversed, the current continues to flow through the shifted bridge state of the energy E˜ 0, until the recrossing happens. Due to the difference in the energies of the neutral and the charged states, the step in the I–V curves appears at different values of the voltage, and this is the reason for the hysteresis loop to appear as shown on the right panel of Figure 10.11. One could also trace the hysteresis in the I–V characteristics starting from the filled (and shifted) bridge state. Again, the hysteresis loop in the I–V curves may occur when both occupied and unoccupied states are rather stable, which means that the potential barrier separating the corresponding minima in the potential energy profile is high enough, so that quantum switching between the states is unlikely. This happens when the bridge is weakly coupled to the electrodes (ΔL,R Δi). Therefore ∆i ∆ f ≈ ∆i ∆i + ∆ f

(10.36)

and the coupling to the final states falls out of the expressions for electron transmission and current. Within the chosen model, this is a physically reasonable result for the final states reservoir (the right “electrode” in our junction) was merely introduced to impose a continuum of states maintaining the transfer process at a steady-state rate. Also, considering the current flow, we may suppose that the initial state is always filled [f i(E) = 1], and the final states are empty [f f (E) = 0]. Therefore, the current flow through the “junction” accepts the form

I=

2e dE ∆ i Im(G). π

∫

(10.37)

Both the current and the transfer rate are fluxes closely related to each other, namely, Ket = I/e. So, we may write

K et = −

2 dE ∆ i Im(G). π

∫

(10.38)

Now, Δi could be computed using the expression (10.7) for the corresponding self-energy term. Keeping in mind that the “left electrode” includes a single state with the certain energy ∫i we obtain

2   τi 2 ∆ i = Im   = π τi δ(E − εi ).  E − εi + is 

(10.39)

Accordingly, the expression (10.38) for the transfer rate may be reduced to the form: Figure 10.12â•… Schematic of the model system used in the transfer rate calculations. The initial/final reservoirs correspond to the left/right leads in the molecular junction.

K et = −

2 2 τi Im G(εi ) ,

(10.40)


10-15

where τi represents the coupling between the donor and the molecule bridge. It must be stressed that within the chosen model, τi is the only term representing the relevant state coupling that may be identified with the electronic transmission coefficient HDA in the general expression (10.34) for Ket. This leaves us with the following expression for the Franck–Condon factor:

acceptor subsystems in the standard donor–bridge–acceptor triad are usually complex structures including multiple sites coupled to the bridge. Correspondingly, the bridge has a set of entrances and a set of exits that an electron can employ. At different values of the tunnel energy different sites of the donor and/or acceptor subsystems can give predominant contributions to the transfer. Consequently, an electron involved in the transfer arrives at the bridge and leaves from it via different entrances/exits, and it follows different pathways while on the bridge. Also, nuclear vibrations in the environment could strongly affect the electron transmission destroying the pathways and providing a transition to a completely incoherent sequential hopping mechanism of the electron transfer. All this means that a proper computation of the electron transmission factor HDA for practical macromolecules is a very complicated and nontrivial task. The strong resemblance between the electron-transfer reactions and the electron transport through molecules gives grounds to believe that studies of molecular conduction can provide important information concerning the quantum dynamics of electrons participating in the transfer reactions. One may expect that some intrinsic characteristics of the intramolecular electron transfer, such as the pathways of the tunneling electrons and the distinctive features of the donor/acceptor coupling to the bridge could be obtained in experiments on the electron transport through molecules. For instance, it was recently suggested to characterize electron pathways in molecules using the inelastic electron tunneling spectroscopy, and other advances in this area are to be expected.

1 (FC) = − Im G(εi ) . π

(10.41)

As discussed in Section 10.4, at a weak coupling of the bridge to the leads the electron–vibrionic interaction opens the set of metastable channels for the electron transport at the energies ˜ 0 + nħΩ (n = 0, 1, 2…) where E˜0 is the energy of the bridge En = E state with the polaronic shift included. Green’s function may be approximated as a weighted sum of contributions from these channels: ∞

G(ε i ) =

∑ P(n) ε − E − nΩ + is i

0

n=0

−1

,

(10.42)

where s → 0+, and the coefficients P(n) are probabilities for the channels to appear given by Equation 10.18. Substituting Equation 10.42 into Equation 10.41 we get ∞

(FC) =

∑ P(n)δ(∆F − nΩ). n=0

(10.43)

˜ 0 ≡ ∫i − E0 + λ2/(ħΩ) is the exoergicity of the Here, ΔF = ∫i − E transfer reaction, that is, the free energy change originating from the nuclear displacements accompanied by polarization fluctuations. The effect of the latter is inserted via the reorganization term λ2/(ħΩ) related to polaron formation. The exoergicity in the transfer reaction takes on a part similar to that of the bias voltage in the electron transport through molecular junctions. It gives rise to the electron motion through the molecules. In the particular case when the voltage drops between the initial state (left electrode) and the molecular bridge these two quantities are directly related by |e|V = ΔF. Usually, the long-range electron transfer is observed at moderately high (room) temperatures, so the low temperature approximation (10.43) for the Franck–Condon factor cannot be employed. However, the expression (10.41) remains valid at finite temperatures, only the expression for Green’s function must be modified to include the thermal effects. It is shown (Yeganeh et al. 2007a,b) that within the high-temperature limit (kT > ħΩ) the expression for the (FC) may be converted to the well-known form first proposed by Marcus: (FC ) =

 (∆F − E )p2 1 exp  − 4 E pkT 4πE pkT 

 , 

(10.44)

where Ep = λ2/(ħΩ) is the reorganization energy. While studying the electron-transfer reactions in practical macromolecules, one keeps in mind that both the donor and the

10.8â•‡ Concluding Remarks Presently, the electron transport through molecular-scale systems is being intensively studied both theoretically and experimentally. Largely, unceasing efforts of the research community to further advance these studies are motivated by important application potentials of single molecules as active elements of various nanodevices intended to compliment current siliconbased electronics. An elucidation of the physics underlying electron transport through molecules is necessary in designing and operating molecular-based nanodevices. Elastic mechanisms for the electron transport through metal–molecule–metal junctions are currently understood and described fairly well. However, while moving through the molecular bridge, an electron is usually affected by the environment, and it results in a change of its energy. So, inelastic effects appear, and they may bring noticeable changes in the electron transport characteristics. Here, we concentrated on the inelastic and the dissipative effects originating from the molecular bridge vibrations and thermally activated stochastic fluctuations. For simplicity and also to keep this chapter within a reasonable length, we avoided a detailed description of computational formalisms commonly used to theoretically analyze electron transport through molecular junctions. These formalisms are described elsewhere. We mostly focus on the physics of the inelastic effects in the electron transport. Therefore, we employ very simple models and techniques.

10-16


We remark that along with nonelasticity originating from electron–vibron interactions, which is the subject of the present review, there exist inelastic effects of different kinds. For instance, inelastic effects arise due to electron–photon interactions. Photoassisted transport through molecular junctions was demonstrated and theoretically addressed using several techniques. It is known that optical pumping could give rise to the charge flow in an unbiased metal–molecule–metal junction, and light emission in biased current-carrying junctions could occur. Also, there is the issue of molecular geometry. There are grounds to conjecture that in some cases the geometry of a molecule included in the junction may change as the current flows through the latter for bonds can break with enough amount of current. Obviously, this should bring a strong inelastic component to the transport, consequently affecting observables. It is common knowledge that electron–electron interactions may significantly influence molecular conductance leading to a Coulomb blockade and a Kondo effect. To properly treat electron transport through molecular junctions one must take these interactions into consideration. Corresponding studies were carried out omitting electron–phonon interactions. However, a full treatment of the problem including both electron–electron and electron–phonon interactions has not been completed so far. There exist other theoretical challenges, such as the effect of bipolaron formation that originates from an effective electron– electron attraction via phonons. Finally, practical molecular junctions are complex systems, and a significant effort is necessary to bring electron transport calculations to a result that could be successfully compared with the experimental data. For this purpose, one needs to compute molecular orbitals and the voltage distribution inside the junctions to get sufficient information on the vibrionic spectrum of the molecule, the electron–phonon coupling strengths, and electron–electron interactions. One needs a good quantitative computational scheme for transport calculations where all this information can be accounted for. Currently, there remain some challenges that have not been properly addressed by theory. Therefore, a comparison between the theoretical and the experimental results on the molecular conductance sometimes does not bring satisfactory results. However, there are firm grounds to believe that further efforts of the research community will result in a detailed understanding of all the important aspects of molecular conductance including inelastic and dissipative effects. Such an understanding is paramount to the conversion of molecular electronics into a viable technology.

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Galperin, M.; Nitzan, A., and Ratner, M. A. 2006. Molecular transport junction: Current from electronic excitations in the leads. Phys. Rev. Lett. 96: 166803. Galperin, M.; Ratner, M. A., and Nitzan, A. 2007. Topical review: Molecular transport junctions: Vibrational effects. J. Phys. Condens. Matter 19: 103201. Garg, A.; Onuchic, J. N., and Ambegaokar, V. 1985. Effect of friction on electron transfer in biomolecules. J. Chem. Phys. 83: 4491–4503. Grobis, M.; Wachowiak, A.; Yamachika, M. F. et al. 2005. Tuning negative differential resistance in a molecular film. Appl. Phys. Lett. 86: 204102. Gutierrez, R.; Mandal, S., and Cuniberti, G. 2005. Dissipative effects in the electronic transport through DNA molecular wires. Phys. Rev. B 71: 235116. Gutirrez, R.; Mohapatra, S.; Cohen, H. et al. 2006. Inelastic quantum transport in a ladder model: Implications for DNA conduction and comparison to experiments on suspended DNA oligomers. Phys. Rev. B 74: 235105. Hahn, J. R.; Lee, H. J., and Ho, W. 2000. Electronic resonance and symmetry in single-molecule inelastic electron tunneling. Phys. Rev. Lett. 85: 1914–1917. Ho, J. W. 2002. Single molecule chemistry. J. Chem. Phys. 117: 11033–11061. Imry, Y. and Landauer, R. 1999. Conductance viewed as transmission. Rev. Mod. Phys. 71: S306–S312. Jones, M. L.; Kurnikov I. V., and Beratan, D. N. 2002. The nature of tunneling pathway and average packing density model for protein-mediated electron transfer. J. Phys. Chem. A 106: 200206. Kastner, M. A. 1992. The single-electron transistor. Rev. Mod. Phys. 64: 849–858. Komeda, T.; Kim, Y.; Kawai, M. et al. 2002. Lateral hopping of molecules induced by excitation of internal vibration mode. Science 295: 2055–2058. Kubatkin, S.; Danilov, A.; Hjort, M. et al. 2003. Single-electron transistor of a single organic molecule with access to several redox states. Nature 425: 698–701. Kushmerick, J. G.; Lazorcik, J.; Patterson, C. H. et al. 2004. Vibronic contributions to charge transport across molecular junctions. Nano Lett. 4: 63942. Kuznetsov, A. M. and Ulstrup, I. 1999. Electron Transfer in Physics and Biology. Chichester, U. K.: John Wiley. Landauer, R. 1970. Electrical resistance of disordered onedimensional lattices. Philos. Mag. 21: 863–867. Lang, N. D. and Avouris, Ph. 2000a. Carbon-atom wires: Chargetransfer doping, voltage drop and the effect of distortions. Phys. Rev. Lett. 84: 358–361. Lang, N. D. and Avouris, Ph. 2000b. Electrical conductance of parallel atomic wires. Phys. Rev. B 62: 7325–7329. Li, X.-Q. and Yan, Y.-J. 2001a. Electrical transport through individual DNA molecules. Appl. Phys. Lett. 79: 2190–2192. Li, X.-Q. and Yan, Y.-J. 2001b. Scattering matrix approach to electronic dephasing in longrange electron transfer. J. Chem. Phys. 115: 4169–4174.

Li, C.; Zhang, D.; Liu, X. et al. 2003. In2O3 nanowires as chemical sensors. Appl. Phys. Lett. 82: 1613. Lorente, N.; Persson, M.; Lauhon, L. J. et al. 2001. Symmetry selection rules for vibrationally inelastic tunneling. Phys. Rev. Lett. 86: 2593–2596. Lortscher, E.; Weber, H. B., and Riel, H. 2007. Statistical approach to investigating transport through single molecules. Phys. Rev. Lett. 98: 176807. MacDiarmid, A. G. 2001. Nobel lecture: “Synthetic metals”: A novel role for organic polymers. Rev. Mod. Phys. 73: 701–712. Mahan, G. D. 2000. Many-Particle Physics. New York: Plenum. Marcus, R. A. 1965. On the theory of electron-transfer reactions. VI. Unified treatment for homogeneous and electrode reactions. J. Chem. Phys. 43: 679–701. Mii, T.; Tikhodeev, S. G., and Ueba, H. 2003. Spectral features of inelastic electron transport via a localized state. Phys. Rev. B 68: 205406. Mitra, A.; Aleiner I., and Millis, A. J. 2004. Phonon effects in molecular transistors: Quantal and classical treatment. Phys. Rev. B 69: 245302. Mitra, A.; Aleiner I., and Millis, A. J. 2005. Semiclassical analysis of the nonequilibrium local polaron. Phys. Rev. Lett. 94: 076404. Mozyrsky, D.; Hastings, M. B., and Martin, I. 2006. Intermittent polaron dynamics: Born–Oppenheimer approximation out of equilibrium. Phys. Rev. B 73: 035104. Mujica, V.; Kemp, M., and Ratner, M. A. 1994. Electron conduction in molecular wires. I. A scattering formalism. J. Chem. Phys. 101: 6849–6855. Mujica, V.; Roitberg, A. E., and Ratner, M. A. 2000. Molecular wire conductance: Electrostatic potential spatial profile. J. Chem. Phys. 112: 6834–6839. Nitzan, A. 2001. Electron transmission through molecules and molecular interfaces. Ann. Rev. Phys. Chem. 52: 681–750. Olsson, F. E.; Paavilainen, S.; Persson, M. et al. 2007. Multiple charge states of Ag atoms on ultrathin NaCl films. Phys. Rev. Lett. 17: 176803. Park, H.; Park, J.; Lim, A. K. L. et al. 2000. Nanomechanical oscillations in a single-C60 transistor. Nature 407: 57–60. Persson, B. N. J. and Baratoff, A. 1987. Inelastic electron tunneling from a metal tip: The contribution from resonant processes. Phys. Rev. Lett. 59: 339–342. Prigodin, V. N. and Epstein, A. J. 2002. Nature of insulator–metal transition and novel mechanism of charge transport in highly doped electronic polymers. Synth. Met. 125: 43–53. Poot, M.; Osorio, E.; O’Neil, K. et al. 2006. Temperature dependence of three-terminal molecular junctions with sulfur end-functionalized tercyclohexylidenes. Nano Lett. 6: 1031–1035. Qin, X. H.; Nazin G. V., and Ho, W. 2004. Vibronic states in single molecule electron transport. Phys. Rev. Lett. 92: 206102. Reichert, J.; Ochs, R.; Beckmann, D. et al. 2002. Driving current through single organic molecules. Phys. Rev. Lett. 88: 176804.

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Zimbovskaya, N. A. 2005. Low temperature electronic transport through macromolecules and characteristics of intramolecular electron transfer. J. Chem. Phys. 123: 114708. Zimbovskaya, N. A. 2008. Inelastic electron transport in polymer nanofibers. J. Chem. Phys. 123: 114705. Zimbovskaya, N. A. and Gumbs, G. 2002. Long-range electron transfer and electronic transport through macromolecules. Appl. Phys. Lett. 81: 1518–1520.

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11 Bridging Biomolecules with Nanoelectronics 11.1 Introduction and Background.............................................................................................. 11-1 11.2 Preparation of Molecular Magnets...................................................................................... 11-2 Folding of Magnetic Protein Mn,Cd-MTâ•‡ •â•‡ Protein: A Mesoscopic Systemâ•‡ •â•‡ Quasi-Static Thermal Equilibrium Dialysis for Magnetic Protein Folding

11.3 Nanostructured Semiconductor Templates: Nanofabrication and Patterning.............11-6

Kien Wen Sun National Chiao Tung University

Chia-Ching Chang National Chiao Tung University

Patterned Self-Assembly for Pattern Replicationâ•‡ •â•‡ Fabrication by Direct Inkjet and Mold Imprintingâ•‡ •â•‡ Nanopatterned SAMs as 2D Templates for 3D Fabrication

11.4 Self-Assembling Growth of Molecules on the Patterned Templates.............................. 11-8 11.5 Magnetic Properties of Molecular Nanostructures........................................................ 11-10 11.6 Conclusion and Future Perspectives................................................................................. 11-10 References.......................................................................................................................................... 11-11

11.1â•‡Introduction and Background The field of nanostructures has grown out of the lithographic technology developed for integrated circuits, but is now much more than simply making smaller transistors. In the early 1980s, microstructures became small enough to observe interesting quantum effects. These structures were smaller than the inelastic scattering length of an electron so that the electrons could remain coherent as they traversed them, giving rise to interference phenomena. Studies on the Aharonov–Bohm effect and universal conductance fluctuations led to the field of “mesoscale physics”—between macroscopic classical systems and fully quantized ones. Now the size of the structures that can be produced is approaching the de Broglie wavelength of the electrons in the solids, leading to stronger quantum effects. In addition to interesting new physics, this drive toward smaller length scales has important practical consequences. When semiconductor devices reach about 100â•›nm, the essentially classical models of their behavior will no longer be valid. It is not yet clear how to make devices and circuits that will operate properly on these smaller scales. The replacement for the transistor, which must carry the technology to well below 100â•›nm, has not been identified. It is anticipated that the semiconductor industry will run up against this “wall” within about 10 years. The current very large-scale integrated circuit paradigm based on complementary metal oxide semiconductor (CMOS) technology cannot be extended into a region with features smaller than 10â•›nm.1 With a gate length well below 10â•›nm, the sensitivity of

the silicon field-effect transistor parameters may grow exponentially due to the inevitable random variations in the device size. Therefore, an alternative nanodevice concept of molecular circuits was proposed, which was a radical paradigm shift from the pure CMOS technology to the hybrid semiconductor.2 The concept combines the advantages of nanoscale components, such as the reliability of CMOS circuits, and the advantages of patterning techniques, which include the flexibility of traditional photolithography and the potentially low cost of nanoimprinting and chemically directed self-assembly. The major attraction of this concept is the incorporation of the richness of organic chemistry with the versatilities of semiconductor science and technology. However, before this, one needs to bring directed self-assembly from the present level of single-layer growth on smooth substrates to the reliable placement of three-terminal molecules on patterned semiconductor structures. While physics and electrical engineering have been evolving from microstructures, to mesoscopic structures, and now to nanostructures, molecular biologists have always worked with objects of a few nanometers or less. A DNA molecule, for example, is very long (when stretched out), but it is about 2.5â•›nm wide with base pairs separated by 0.34â•›nm. Since the technology has not existed to directly fabricate and manipulate objects this small, various chemical techniques have been developed for cutting, tagging, and sorting large biological molecules. While enormously successful, these methods utilize batch processing of huge numbers of the molecules and rely on statistical interpretations. It is not possible, in principle, to sequence one particular DNA molecule with these techniques, for example. Ideally, 11-1

11-2


one would like to stretch the DNA straight and simply read the sequence of the base pairs. While we are still far from this goal, nanofabrication techniques promise to bring us much closer. During recent years, self-assembly has become one of the most important strategies used in biology for the development of complex, functional structures. Self-assembly on modified surfaces is one of the approaches to self-assemble structures that are particularly successful. By the coordination of molecules to surfaces, the molecular systems form ordered systems—self-assembled monolayers (SAMs). The SAMs are reasonably well understood and are increasingly useful technologically. A thin film of diblock copolymers can be self-assembled into ordered periodic structures at the molecular scale (∼5–50â•›nm) and have been used as templates to fabricate quantum dots,3,4 nanowires,5–7 and magnetic storage media.8 More recently, in epitaxial assembly of block-copolymer films, molecular level control over the precise size, shape, and spacing of the order domains was achieved with advanced lithographic techniques.9 The development of methods for patterning and immobilizing biologically active molecules with micrometer and nanometer scale control has been proven integral to ranges of applications such as basic research, diagnostics, and drug discovery. Some of the most important advances have been in the development of biochip arrays that present either DNA,10 protein,11 or carbohydrates.12 The use of patterned substrates for components of microfluidic systems for bioanalysis is also progressing rapidly.13–15 Surface modification and patterning at the nanoscale to anchoring protein molecules is an important strategy on the way of obtaining the construction of new biocompatible materials with smart bioactive properties. In fact, surfaces patterned by protein molecules can act as active agents in a large number of important applications including biosensors capable of multifunctional biological recognition. In particular, physical adsorption of proteins onto semiconductor surfaces makes possible to combine the simplicity of the method with the versatility of chemical and physical properties of proteins. In recent years, there has been substantial attention focused on the reactions of organic compounds with silicon surfaces. The major attraction is the incorporation of the richness of organic chemistry with the versatilities of semiconductor science and technology. More recently, it has been demonstrated that TEMPO,2,2,6,6-tetramathylpiperidinyloxy can bond with a single dangling bond on hydrogen-terminated Si(100) and Si(111) surfaces.16 Functional organic molecular layers were found to self-assemble on metal 2 and semiconductor surfaces.17 The technique of self-assembly is one of the few practical strategies available to arrive at one to three-dimensional ensembles of nanostructures. There are many different mechanisms by which self-assembly of molecules and nanoclusters can be accomplished, such as chemical reactions, electrostatic and surface forces, and hydrophobic and hydrophilic interactions. In this chapter, we present most recent advances in the development of techniques in immobilizing a single nanostructure and producing arrays of 3D magnetic protein nanostructures with high throughput on surface modified semiconductor substrates. Well-characterized test nanostructures were prepared

using current state-of-the-art nanofabrication techniques. Both nanoimaging and scanning probe microscopy studies (AFM, SEM, and TEM) on semiconductor nanostructures and these molecular self-assembly systems were performed. The combination of e-beam lithography, scanning probe microscope imaging, spectroscopy, and self-assembly approaches provide not only the high throughput of producing arrays of protein nanostructures but also with highest precision of positioning single nanostructure and/or single molecule.

11.2â•‡Preparation of Molecular Magnets 11.2.1â•‡Folding of Magnetic Protein Mn,Cd-MT Metallothionein (MT) is a metal binding protein that binds seven divalent transition metals avidly via its twenty cysteines (Cys).18 These Cys’ form two metal binding clusters located at the carboxyl (α-domain) and amino (β-domain) terminals of MT.19 The two clusters were identified as α-cluster (M4S11)3− and the β-cluster (M3S9)3− (Figure 11.1),20–22 where M denotes metal ions (Zn2+, Cd2+, or others), according to both x-ray crystallographic and NMR studies.3 MT binds to metals ions via metal-thiol linkages.19 As shown in Figure 11.1, the (M3S9)3− and (M4S11)3− have the zinc-blende like structure that is similar to the “diluted magnetic semiconductor (DMS)” compounds.23 In general, semiconductors are not magnetic. However, a DMS exhibits magnetic properties by doping with Mn and Cd or other II–VI metal ions in certain ratio. The doped metal clusters among the semiconductors are in zinc-blende structures. Meanwhile, these semiconductors possess magnetic property only in low temperatures.23 The magnetic properties may be the result of the d-sp3 orbital hybridization and the alignment of the electron spins. The bridging sulfur atoms may also contribute to the alignment of the spins of the Mn 2+ ions. Thus, by chelating the Mn2+ and Cd 2+ with MT (i.e., Mn,Cd-MT), a “magnetic protein” may be obtained. Recently, the single molecule magnets (SMMs) have attracted much attention.24 However, the Curie temperature of these molecules has to be as low as 2–4â•›K 24–26 to avoid the thermal fluctuation among the electron spin within the molecules.27 To be of practical utilization, it is highly desirable to create a room temperature molecular magnet.28 With this intention in mind, one has to construct and investigate a new metal binding protein, MT, which sustained characteristic magnetic hysteresis loop from 10 to 330â•›K. The protein backbone may restrain the net spin moment of Mn2+ ions to overcome the minor thermal fluctuation. The magnetic-metallothionein (mMT) presented may reveal a possible approach to create high temperature molecular magnet. In order to prepare the Mn,Cd-MT magnetic proteins, the folding mechanism of protein should be introduced.

11.2.2â•‡Protein: A Mesoscopic System Protein is a complex biomolecule that contains a large number of basic residues—amino acids. Therefore, it is not possible to analyze it completely by macroscopic approaches. Meanwhile, it

11-3

Bridging Biomolecules with Nanoelectronics

Cys

Cys

S

Cys

M

S

Cys

Cys

S

M

Cys

M S

Cys Cys

S

Cys S

Cys

S

S

M

S

S

S

Cys

Cys Cys

Cys

S

S

M

S

M

Cys

S

S

Cys M

S

Cys S

S Cys

S

Cys

Cys

β-Cluster

α-Cluster

Figure 11.1â•… Metal binding clusters of MT that was modified from x-ray crystal structure (2). Where the circles denote metal ions Zn 2+, Cd 2+, or Mn2+. Each metal ion was linked with protein via metal-thiol bonds.

is not feasible to describe the dynamics of its polypeptide chain behavior by using conventional statistical approaches either. Thus, a protein can be thought of as a mesoscopic system.29 However, the conformational transition from unfolded state to the native state of a protein may be similar to the conventional phase transition model. In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. In a folding process, proteins follow the thermodynamic theories and transform from the unfolded state to the folded state, where the state is defined as a region of configuration space with minimal potential.30 Therefore, we named this conformational change as “state transition.”31 Due to the complexity of a protein folding system, a single experimental study may reveal only a part of the fact of protein folding. Therefore, we should examine the protein-folding problem with multidimensional approaches and integrate the findings to reveal the true mechanism of protein folding. Protein folding may follow a spontaneous process29 or a Â�reaction-path directed process30 in vitro. A choice between the two may be determined by the intrinsic properties of Â�proteins, for example, the varying folding transition boundaries. However, a general model, named “first-order-like state transition model,” in which the aggregated proteins exist within finite boundaries can encompass both processes without any conflictions.31–36 According to this model, the folding path of the protein may not be unique. It can be folded, without being trapped in an aggregated state, via a carefully designed refolding path circumventing the transition boundary, that is, via an overcritical path.31–36 The intermediates, following an overcritical path, are in a molten globular state, 37 and their behavior is consistent with both a sequential38 and a collapse model.39 However, both soluble (folded) and precipitated (unfolded) proteins can be observed in the direct folding reaction path in vitro. In terms of the “first-order-like state transition model” language, this can be described as stepping across the state transition line in

the protein folding reaction phase diagram.31,32 Since in protein refolding it is important to prevent protein aggregation in vitro, similarly, in biomedical applications, the revelation of the mechanism of the formation of the two states (unfolded and folded) becomes significant. Previous studies have indicated that chemical environment,40 temperature,41 pH,42 ionic strength,43 dielectric constant,44 and pressure49—considered as solvent effects collectively— could affect the fundamental structure, thermodynamics, and dynamics of polypeptides/proteins. The reaction ground state can be expressed as a dual-well potential according to the twostate transition model. The conformational energy, in general, of the unfolded state is relatively higher than that of the native state (Figure 11.1). When the system reaches thermal equilibrium, most protein molecules are found in their native state. No unfolded or intermediate states are observable. However, as described previously,31–36 if a denaturant is added, the reaction potential may change accordingly as indicated in Figure 11.1. Then, an unfolded protein may be stable, as it is now at the lowest energy under the newly established equilibrium. The energy of the system can be expressed as follows:

H T = H p + λH s

(11.1)

where HT, Hp, and Hs denote the potential energy of the interacting protein-solvent total system, the protein, and the solvent, respectively. The factor, λ, is a weighting factor of the solvent environment (0 ≤ λ ≤ 1). It approaches unity when a denaturant is present as pure solvent and decreases in value as the concentration of the denaturant is reduced. When λ of the system is changed drastically, direct folding ensues and leads to the release of some of the bound denaturant. According to the Donnan effect in a macromolecule-counter ions interactive system, the diffusion of the bound denaturant can be expressed by Fick’s first law:

11-4


J = −D∇n

(11.2)

where n denotes the concentration of the denaturant that is dissociated from the protein D denotes the diffusive constant vector J denotes the flux, respectively, of the solute

I 0

According to the Einstein relation kT 6πηRH

–2 –4 –6

(11.3)

where k is the Boltzmann constant T is the temperature in Kelvin η is the viscosity of the solvent R H is the hydration radius of the solutes Due to the intrinsic diffusion process, the solute exchange processes are not synchronous for all protein molecules. Therefore, the folding rate of protein may not be measured directly by a simple spectral technique, that is, the stopped-flow CD,45 Â�continuous-flow CD,46 or fluorescence.47 However, the reaction interval of protein folding can be revealed by the autocorrelation of reaction time from these direct measurements. The detailed mechanism and an example will be discussed later. If we look at the energy landscape funnel model of protein folding,48 it appears that proteins can be trapped in a multitude of local minima of the potential well in a complicated protein system. The native state, though, is at the lowest energy level. When thermal equilibrium is reached, most of the protein molecules are located in the lowest energy state, with a population ratio as low as e−ΔE/kT, according to the Maxwell–Boltzmann distribution in thermodynamics. The ΔE denotes the energy difference between the native state and a local minimum; k and T denote the Boltzmann constant and temperature in Kelvin, respectively. At high concentration (>0.1â•›mg/mL), however, Â�considerable amount of insoluble protein has been observed in protein folding,31,33–37,48 indicating that insoluble proteins are at an even lower energy state than the native protein. Therefore, by considering the intermolecular interactions during the protein folding process, the reaction energy landscape may be expressed as a three-well model (Figure 11.2). As shown in Figure 11.2, the unfolded protein (U) is in the highest energy state; the native protein (N) is in the lower energy state. However, the intermediate (I) that may cause further protein aggregation/Â�precipitation is in the lowest energy state. Although the energy state of the intermediate/aggresome is the lowest energy state, the conformational energy of the individual proteins composing the aggresome may not be lower than the native protein. Namely, in single molecular simulation, this extra potential well of intermediate (I) is nonexistent. Therefore, in the conventional energy landscape model (the single molecule simulation model),

4

U 6

8 6

8

4 2

Figure 11.2â•… Three well model of multi-protein molecules folding reaction. The U denotes the unfolded state. N denotes the native state and I denotes the protein–protein complex (aggresome) intermediate.

the lowest energy state “I” cannot be observed. According to the Zwanzig’s definition of state, the protein molecules in the intermediate (I) belong to an unfolded state.13 Hence, in a direct folding reaction, the soluble (N) and the insoluble parts (U) can coexist and they can be observed simultaneously, which is similar to the situation where the phase transition line is crossed in reactions congruent to the “first-order phase transition” model. Therefore, we named the protein folding reaction as “first-order like state transition model” (as shown in Figure 11.3). The Φ(n1, n2,…) in Figure 11.3 denotes the folding status of protein, where n1, n2,… represent the variables affecting the folding status, such as, temperature, concentration of denaturants, etc. The reaction curve indicates an overcritical reaction path of a quasi-static folding reaction. The gray area in Figure 11.3 indicates the state transition boundary of protein folding. The gray line and dash line indicate the reaction path of direct folding. By combining the three-well model (Figure 11.2) and the direct N

Φ(n1, n2 ,...)

D=

2

N

M5

M4

M3 I΄

M2

Stepwise TED process

I

M1 Directed dilution U

n

Figure 11.3â•… The protein folding phase diagram, where the Φ(n1, n2,…) denotes the folding status (the order parameter) of protein. The n, n1, n2 ,… denote the variables that affect the folding status such as temperature, concentration of denaturants, etc.


folding reaction of the “first-order like state transition model” (Figure 11.3), we realized that those folded protein molecules along the direct folding path might fold spontaneously or form aggregates. Spontaneous folding may be driven by enthalpy– entropy compensation. As indicated previously, the conformation of protein changed with changes of the solvent environment. It seems that the protein may fold spontaneously, such as in Anfinsen’s experiment44 and direct folding reactions. The protein folding reaction, similar to all chemical reactions, reaches its equilibrium by following the fundamental laws of thermodynamics. Although protein folding has been studied extensively in certain model systems for over 40 years, the driving force at the molecular level remained unclear until recently. It is known that polymers and macromolecules may selfassemble/self-organize into a wide range of highly ordered phases/states at thermal equilibrium.49–52 In a condensed solvent environment, large molecules may self-organize to reduce their effective volume. Meanwhile, the number of the allowed states (Ω) of small molecules, such as buffer salt and other counter-ions in solution, increases considerably. Therefore, the entropy of the system, ΔSâ•›=â•›R ln(Ωf/Ωi), becomes large, where i and f denote the initial and final states, respectively. Meanwhile, the enthalpy change (ΔH) between the unfolded and native protein is around hundreds kcal/mol. 53 Therefore, the Gibbs free energy of the system, ΔGâ•›=â•›ΔHâ•›−â•›TdS, becomes more negative in this system when the large molecules self-organize.54 A similar entropy–enthalpy compensation mechanism has been used to solve the reaction of colloidal crystals that self-assemble spontaneously.55,56 According to our studies, 31,33–37 the effective diameter of the unfolded protein is about 1.7–2.5 fold larger than the folded protein. Therefore, with the same mechanism, those macromolecules (proteins) may tend to reduce their effective volumes and increase the system entropy when thermal equilibrium is reached. The increase in entropy may compensate for the change of the enthalpy of the system and enable the reaction to take place spontaneously. This may be the reaction molecular mechanism of spontaneous protein folding reactions. Meanwhile, a similar mechanism can be adopted into the self-assembly process of magnetic protein in nanopore arrays.

11.2.3â•‡Quasi-Static Thermal Equilibrium Dialysis for Magnetic Protein Folding Due to the intrinsic diffusion process, the solvent exchange rate is slow and thus the variation of λ is slow and can be thought of as quasi-static. Therefore, we named this buffer exchanging process as a quasi-static process. We manipulated the reaction direction of the protein folding through this process. Meanwhile, we can obtain stable intermediates in each thermal equilibrium state. These intermediates may help us reveal the molecular folding mechanism of protein that is to be discussed in Section 11.3. The following is an example of the stepwise folding method, 31,33–37 and the buffers used were described in these studies.

11-5

Step 1: The unfolded protein (U) was obtained by treating the precipitate or inclusion body with denaturing/unfolding buffer to make it 10â•›mg/mL in concentration. This solution was left at room temperature for 1â•›h. This process was meant to relax the protein structure by urea and pH (acidic or basic) environments. The disulfide bridges were reduced to SH groups and the protein was unfolded completely. Step 2: The unfolded protein (U) in the denature/Â�unfolding buffer was dialyzed against the folding buffer 1 for 72â•›h to dilute the urea concentration to 2â•›M, producing intermediate 1, or M1. Step 3: M2 was obtained by dialyzing M1 against the folding buffer 2 for 24â•›h to dilute urea concentration to 1â•›M. Step 4: M3, an intermediate without denaturant (urea) in solution, was then obtained by dialyzing M 2 against the folding buffer 3 for 24â•›h. Step 5: M3 was further dialyzed against the folding buffer 4 for 24â•›h, and the pH changed from 11 to 8.8 to produce M4. Step 6: Finally, the chemical chaperonin mannitol was removed by dialyzing M4 against the native buffer for 8â•›h to yield M5. It should be noted that all the equilibrium time of each step is longer than the conventional dialysis time. In general, for the free solvent case, the solute may exchange with the buffer completely within hours. However, it is known that the denaturant molecules interact with protein, similar to the Donna effect, and the solute exchange may be slow and needs more time for the system to reach thermal equilibrium, especially for the first refolding stage. The folding time of each process is relatively longer than the regular solvent exchange process. Therefore, we can obtain the magnetic protein that follows a similar process. Protein microenvironment protects the net electron spin of molecules from thermal fluctuation. The bridging ligands (i.e., sulfur atom, S) between the magnetic ions may be responsible for aligning the electron spin of magnetic ions. As indicated in Figure 11.4, the valance bonding electrons of the bridging Cys may hop between the bonded metal ions, such as Mn2+ and Cd2+; whereas the Cd 2+ in the β metal cluster is rather important in restraining the orientation of the electron spins of the bridging sulfurs and in aligning the spins of Mn 2+ in the metal binding clusters. Therefore, this electron hopping effect may turn the Mn,Cd-MT into a magnetic molecule. However, the protein backbone surrounding the β metal cluster may provide a strong restraining effect to overcome the thermal fluctuations from the environment. Therefore, the magnetization can be observed in room temperature. However, the geometrical symmetry of the spin arrangement in all Mn-MT may cause partial or complete cancellation of detectable magnetization. These results also indicated that the threshold temperature of the molecular magnet might rise to room temperature if the proper prosthetic environment, such as protein backbone, can be linked against the thermal fluctuation of the temperature.

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Handbook of Nanophysics: Nanoelectronics and Nanophotonics Cys

S

Net spin S

Cys Cys

Cys

Mn S

S

Cys S

Cys Mn S

S

Cd S

Cys

S Electron hopping

Cys

Cys

Figure 11.4â•… Proposed electron spin model of Mn2+ in β metal binding cluster of Mn,Cd-MT-2.

Therefore, we have successfully constructed a molecular magnet, Mn,Cd-MT, that is stable from 10 to 330â•›K . The observed magnetic moment can be explained by the highly ordered alignment of (Mn2CdS9)3− clusters embedded in the β-domain in which sulfur atoms serve as key bridging ligands. The discovery of mMT may allude to new schemes in constructing a completely different category of molecular magnets.

11.3â•‡Nanostructured Semiconductor Templates: Nanofabrication and Patterning The rapidly developing of interdisciplinary activity in nanostructuring is truly exciting. The intersections between the various disciplines are where much of the novel activity resides, and this activity is growing in importance. The basis of the field is any type of material (metal, ceramic, polymer, semiconductor, glass, and composite) created from nanoscale building blocks (clusters of nanoparticles, nanotube, nanolayers, etc.) that are themselves synthesized from atoms and molecules. Thus, the controlled synthesis of those building blocks and their subsequent assembly into nanostructures is one fundamental theme of this field. This theme draws upon all of the material-related disciplines from physics to chemistry to biology and to essentially all of the engineering disciplines as well. The second and most fundamental important theme in this field is that the nanoscale building blocks, because of their size being below about 100â•›n m, impart to the nanostructures that are created from them new and improved properties and functionalities that are still unavailable in conventional materials and devices. The reason for this is that the materials in this size range can exhibit fundamentally new behavior when their sizes fall below the critical length scale associated with

any given property. Thus, essentially any material property can be dramatically changed and engineered through the controlled size-selective synthesis and assembly of nanoscale building blocks. The present juncture is important in the fields of nanoscale solid-state physics, nanoelectronics, and molecular biology. The length scales and their associated physics and fabrication technology are all converging to the nanometer range. The ability to fabricate structures with nanometer precision is of fundamental importance for any exploitation of nanotechnology. In particular, cost effective methods that are able to fabricate complex structures over large areas will be required. One of the main goals in the nanofabrication area is to develop general techniques for rapidly patterning large areas (a square centimeter, or more) with structures of nanometer sizes. Presently, electron-beam (e-beam) lithography is capable of defining patterns that are less than 10â•›nm. These patterns can then be transferred to a substrate using various ion milling/ etching techniques. However, these are “heroic” experiments, and can only be made over a very limited area—typically a few thousand square microns, at most. While such areas are immediately useful for investigating the physics of nanostructures, the applications we would like to pursue will eventually require a faster writing scheme and much larger areas. The time and area constraints are determined by the direct e-beam writing. It is a “serial” process, defining single small regions at a time. Furthermore, the field of view for the e-beam system is typically less than 100â•›nm when defining the nanometer-scale structures. One simply cannot position the electron beam with nm precision over larger areas. While e-beam lithographic methods are very general, in that essentially any shape can be written, we will also make use of “natural lithography” (tricks). Similarly, advances in the knowledge of the DNA structure has recently been applied to the fabrication of self-organized nm surface structures. There are many such “tricks” that could prove crucial to the success of the projects in allied fields. A list of current methods toward nanofabrication is given below.

11.3.1â•‡Patterned Self-Assembly for Pattern Replication By exploiting e-beam and focused ion beam lithography, selfassembled monolayers can be patterned into 10–20â•›nm features that can be functionalized with single molecules or small molecular groupings. These patterned areas will then be used as templates to direct the vertical assembly of stacks of molecules or to direct the growth of polymeric molecules. Schematics of the processes and the possible templates used are shown in Figure 11.5. The initial 2D pattern will thus be translated into 3D nanosized objects. With the capability of the full control over the interfacial properties, it will be possible to release the objects from the templates and transfer them to another substrate, after which the nanopatterned surface can be used again to provide an inexpensive replication technique.

11-7


X: 0.14 μm Y: 0.00 μm D: 0.14 μm

(a) Digital Instruments Nanoscope Scan size 4.000 μm Scan rate 0.2999 Hz Number of samples 256 Image Data Height Data scale 300.0 nm

View angle Light angle

1 2 3 μm

x 1.000 μm/div z 300.000 nm/div

0°

051019-400-1.000

(b) Figure 11.5â•… E-beam lithography defined (a) nanopores and (b) nanotrenches on silicon templates. (From Chang, C.-C. et al., Biomaterials, 28, 1941, 2007. With permission. Elsevier.)

11.3.2â•‡Fabrication by Direct Inkjet and Mold Imprinting This part of the technique will draw on recent advances in printing techniques for direct patterning of surfaces. This involves the use of inkjet printing to deliver a functional material (semiconductor or metal) to a substrate, which is then controlled by patterning in the surface free energy of the substrate, to allow very accurate patterning of the printed material. Currently, devices show channel lengths down to 5â•›μm, but indications are that geometries can be reduced to submicron dimensions. The method will be concerned with the limits to resolution that can be achieved by this process, and also the structure and the associated electronic structure at polymer–polymer interfaces, such as that between semiconductor and insulator layers in the field-effect device.

_ _0780 50 kV 2 μm 12000X A+B

2008/03/08

Figure 11.6â•… Nanogratings on silicon templates by using thermal nano-imprinting technique.

In 1995, Professor Stephen Y. Chou of Princeton University invented a new fabrication method in the field of semiconductor fabrication. It is called nano-imprint lithography (NIL).57 Briefly speaking, this technique was demonstrated by pressing the patterned mold to contact with the polymer resist directly. The patterns on the mold will transfer to the polymer resist without any exposure source. Therefore, the diffraction effect of light can be ignored, and the limitation is dependent only on the pattern size of the mold rather than on the wavelength of the exposure light. In Figure 11.6 we show a nano-grating structure made by using the thermal imprinting technique. NIL technology is a physical deformation process and is very different from conventional optical lithography. This technology provides a different way to fabricate nanostructures with easy processes, high throughput, and low cost. Currently, there are three main NIL techniques under investigation, namely, hot-embossing nano-imprint lithography (H-NIL57), ultraviolet nano-imprint lithography (UV-NIL58), and soft lithography.59 Those NIL technologies can be applied to many different research fields, including nano-electric devices,60 bio-chips,61 micro-optic devices,62 micro-fluidic channels,63 etc.

11.3.3â•‡Nanopatterned SAMs as 2D Templates for 3D Fabrication Modern lithographic techniques (e-beam or focused ion beam (FIB), SNOM lithography) are able to generate topographic (relief) patterns in the 10–50â•›nm size ranges. As a further step toward more complicated and functional 3D structures, chemical functionalization at a similar size scale is necessary. By exploiting e-beam, FIB, and near-field optical lithography, it is

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possible to pattern SAMs directly. Using lithography, the SAMs can either be locally destroyed and refilled with other molecules, or the surface of the SAMs can be activated to allow further chemical reactions. The resulting patterned surfaces will be chemically patterned. Patterns can be introduced to incorporate H-bonding, pi–pi stacking, or chemical reactivity. These patterned areas will be used as templates to direct the vertical assembly of stacks of molecules or to direct the growth of polymeric molecules. Large, extended aromatic molecules prefer to stack on top of each other due to pi–pi stacking. These molecules have interesting electronic properties as molecular wires. When surfaces can be patterned to incorporate “seeds” for the large aromatic molecules, the stacking can be directed away from the surface. The initial 2D pattern will thus be translated into 1) Spining coating & baking ZEP-520 Si

2) Exposuring

E-beam ZEP-520 Si

3) Developing ZEP-520 Si

3D nanosized objects. With the full control over the interfacial properties, it will be possible to release the objects from the templates and transfer them to another substrate, after which the nanopatterned surface can be used again to provide an inexpensive replication technique.

11.4â•‡Self-Assembling Growth of Molecules on the Patterned Templates The self-assembling growth of the MT-2 proteins is demonstrated as follows. One mg/mL magnetic MT in Tris. HCL buffer solution was placed onto the patterned surface, and an electric field with an intensity of 100â•›V/cm was then applied for 5â•›min to drive the MT molecules into the nanopores. The sample was then washed with DI water twice to remove the unbounded MT molecules and salts on the surface (the schematic of the process is also shown in Figure 11.7a. Figure 11.8 shows the atomic force microscopy (AFM) image of the template surface with 40â•›nm nanopores after they were filled by the MT-molecules. Keep in mind that most of the Si surface was still protected by photoresist after the etching processes, which has prevented the MT-molecules from forming strong OH bonds with the Si surface underneath. Therefore, the electrical field-driven MT molecules were all anchored on those areas that were not covered with photoresist. The molecules landing in each pore were then self-assembly grown vertically from the bottom of the pore into the shape of a rod (as shown in Figure 11.8). These molecular nanorods have an average height of ∼120â•›nm above the template surface and a diameter equal to the size of the nanopore.

200

4) Reactive ion etching SFG & O2 ZEP-520 Si

150

nm

nm

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100 1.8

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Figure 11.7â•… (a) Flowchart of the lithography, etching processes, and growth of protein molecules, (b) schematics of the patterned templates with nanopores. (From Chang, C.-C. et al., Biomaterials, 28, 1941, 2007. With permission. Elsevier.)

0

Figure 11.8â•… Three-dimensional AFM image of the patterned Â�magnetic molecules. The molecules have self-assembled to grow into a rod shape. (From Chang, C.-C. et al., Biomaterials, 28, 1941, 2007. With permission. Elsevier.)

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a more dense structure in the larger pores compared with the case of the smaller pores. On templates with thinner photoresist and smaller pitch sizes (less than 600â•›nm), we also found that the molecules anchored in the pore can grow laterally toward the neighboring pores (data not shown). By increasing both the e-beam exposure and dry etching time on the Si surface covered with a thin photoresist layer with a thickness of less than 150â•›nm, we were able to create a ring-type area with an exposed Si surface along the periphery of the nanopores. The SEM image of this type of template is shown in Figure 11.10. On this particular template, the molecules not only independently grew inside the pores, but they also grew along the circumference of the pores to form molecular rings on the template. Figure 11.11 shows the two-dimensional (2D) and threedimensional (3D) AFM images of such molecular rings. In order to gain better control of the formation of molecular nanostructures, it is important to uncover the underlying self-assembling growth mechanism. Molecular self-assembly

Figure 11.9â•… Two-dimensional AFM images of the patterned MT-molecules on the template with pore size of 130â•›nm and pitch size of 300â•›nm. (From Chang, C.-C. et al., Biomaterials, 28, 1941, 2007. With permission. Elsevier.)

Height

1.8

60

1.6

However, experiments on the templates with pore sizes larger than 100â•›nm gave quite different results. Figure 11.9 shows the two-dimensional AFM image of the template surface with larger pores, where we can see that the molecules did not grow vertically above the template surface. Therefore, we were not able to generate 3D images of this type of template. However, judging from the AFM phase images, the MT molecules did form

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AA5034 50 kV 0.5 μm 50000X A+B

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Figure 11.10â•… SEM image of the Si template shows a ring shape Si exposed area around the circumference of nanopores. (From Chang, C.-C. et al., Biomaterials, 28, 1941, 2007. With permission. Elsevier.)

1 1.2 1.4 1.6 1.8

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Figure 11.11â•… (a) 2D and (b) 3D AFM images of the patterned MT-molecules. (From Chang, C.-W. et al., Appl. Phys. Lett., 88(26), 263104, 2006. With permission. AIP.)

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can be mediated by weak, noncovalent bonds—notably hydrogen bonds, ionic bonds (electrostatic interactions), hydrophobic interactions, van der Waals interactions, and water-mediated hydrogen bonds. Although these bonds are relatively insignificant in isolation, when combined together as a whole, they govern the structural conformation of all biological macromolecules and influence their interaction with other molecules. The water-mediated hydrogen bond is especially important for living systems, as all biological materials interact with water. We believe that the first layer of proteins anchored inside the nanopores was bonded with the Si surface dangling bonds. They have provided building blocks for proteins that arrived later. With the assistance of spatial confinement from the patterned nanostructures, the rest of the proteins are able to self-assemble via the van der Waals interactions and perform molecular self-assembly.

11.5â•‡Magnetic Properties of Molecular Nanostructures The magnetic properties of the self-assembled molecular nanorods were investigated with magnetic force microscopy (MFM). We monitored the change of contour of a particular nanorod on the template when an external magnetic field was applied. Figure 11.12a shows the MFM image of the nanorod without the external magnetic field. In Figure 11.12b, a magnetic field of 500â•›Oe was applied during the measurement with a field direction from the right to left. The strength of the field was kept at a minimum so as not to perturb the magnetic tip on the instrument. In Figure

11.12 we can see clearly that the contour of the nanorod has changed in shape as compared to the case with no applied field. It indicates that the molecular self-assembly carries a magnetic dipole moment that interacts with the external magnetic field.

11.6â•‡Conclusion and Future Perspectives Success in the synthesis of the magnetic molecules produced from metallothionein (MT-2) by replacing the Zn atoms with Mn and Cd has been demonstrated in this chapter. Hysteresis behavior in the magnetic dipole momentum measurements was observed over a wide range of temperatures when an external magnetic field was scanned. These magnetic MT molecules were also found to self-assemble into nanostructures with various shapes depending on the nanostructures patterned on the Si templates. Data from the MFM measurements indicate that these molecular self-assemblies also carry magnetic dipole momentum. Since the pore size, spacing and shape can easily and precisely be controlled by lithography and etching techniques, this work should open up a new path toward an entire class of new biomaterials that can be easily designed and prepared. The techniques developed in this particular work promise to facilitate the creation of many bio-related nanodevices and spintronics. Magnetic molecular self-assembly may find its use in data storage or magnetic recording systems, as an example. They can also act as spin biosensors and be placed at the gate of the semiconductor spin valve to control the spin current from the source to the drain. More importantly, this work should not be limited to MT-2 molecules

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Figure 11.12â•… MFM images of nanorods (a) without the magnetic field (b) with a 500â•›Oe magnetic field applied with a field direction from right to left. (From Chang, C.-C. et al., Biomaterials, 28, 1941, 2007. With permission. Elsevier.)


and should be extended to other type of molecules and proteins as well. As mentioned in the beginning of this chapter, the various surface patterning techniques developed over the years to interface organic or biological materials with semiconductors have not only provided new tools for controlled 2D and 3D selforganized assemblies, but have also been essential to the creation and emergence of new semiconductor-molecular nanoelectronics as recently discussed by Likharev.64 In the future, it is important to develop techniques for growing and characterizing molecular self-assembly, single nanostructure, and molecule on semiconductor templates to bring a measure of control to the density, order, and size distribution of these molecular nanostructures. Using self-assembly techniques, one can routinely make molecular assembly with precise distances between them. Recent explorations of molecular self-assembly have sought to provide transverse dimensions on the mesoscopic nanometer scale. As a general—although not inviolate—rule, these attempts have led to very good local ordering (e.g., nearest neighbors). We can anticipate, (1) the development of new (supra) molecular nanostructures via the self-assembling method (bottom up technique) and immobilization of single nanostructure or molecule; (2) the design of methods to functionalize molecular self-assembly and devices; (3) an integration of bottom-up and top-down procedures for the nano- and microfabrication of molecularly driven sensors, actuators, amplifiers, and switches; and (4) an increased understanding and appreciation of the science and engineering that lie behind nanoscale processes. All this and more is in the nature of the nanotechnology bonds as it impacts on biology and beyond. In the final analysis, however, the practice of biological synthesis that relies on molecular recognition and self-assembling processes within a very much more catholic framework than is currently being contemplated by most researchers that will dictate the pace of progress in synthesis. The final goal is to understand this whole notion of what selfassembly is. One needs to really learn how to make use of the methods of organizing structures in more complicated ways than we can do now. On a molecular scale, the accurate and controlled application of intermolecular forces can lead to new and previously unachievable nanostructures. This is why molecular self-assembly (MSA) is a highly topical and promising field of research in nanotechnology today. MSA encompasses all structures formed by molecules selectively binding to a molecular site without external influence. With many complex examples all around us in nature (ourselves included), MSA is a widely observed phenomenon that has yet to be fully understood. Being more a physical principle than a single quantifiable property, it appears in engineering, physics, chemistry, and biochemistry, and is therefore truly interdisciplinary.

References 1. Sessoli R., Gatteschi D., Caneschi A., and Novak M.A. (1993), Nature (London), 365, 141–143. 2. Fonticelli M., Azzaroni O., Benitez G., Martins M.E., Carro P., and Salvarezza R.C. (2004), J. Phys. Chem. B, 108, 1898.

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3. Park M., Harrison C., Chaikin P.M., Register R.A., and Adamson D.H. (1997), Science (Washington, DC), 276, 1401–1404. 4. Li R.R., Dapkus P.D., Thompson M.E., Jeong W.G., Harrison C., Chaikin P.M., Register R.A., and Adamson D.H. (2000), Appl. Phys. Lett., 76, 1689–1691. 5. Thurn-Albrecht T., Schotter J., Kästle G.A., Emley N., Shibauchi T., Krusin-Elbaum L., Guarini K., Black C.T., Tuominen M.T., and Russell T.P. (2000), Science (Washington, DC), 290, 2126–2129. 6. Kim H.C. et al. (2001), Adv. Mater., 13, 795–797. 7. Lopes W.A. and Jaeger H.M. (2001), Nature, 414, 735–738. 8. Cheng J.Y. et al. (2001), Adv. Mater., 13, 1174–1178. 9. Kim S.O., Solak H.H., Stoykovich M.P., Ferrier N.J., de Pablo J.J., and Nealey P.F. (2003), Nature, 424, 411–414. 10. Demers L.M., Ginger D.S., Park S.J., Li Z., Chung S.W., and Mirkin C.A. (2002), Science (Washington, DC), 296, 1836–1838. 11. Hodneland C.D., Lee Y.S., Min D.H., and Mrksich M. (2002), Proc. Natl. Acad. Sci., 99, 5048–5052. 12. Houseman B.T. and Mrksich M. (2002), Chem. Biol., 9, 443–454. 13. Chen C.S., Mrksich M., Huang S., Whitesides G.M., and Ingber D.E. (1997), Science (Washington, DC), 276, 1345–1347. 14. Mrksich M., Dike L.E., Tien J.Y., Ingber D.E., and Whitesides G.M. (1997), Exp. Cell. Res., 235, 305–313. 15. Chen C.S., Mrksich M., Huang S., Whitesides G.M., and Ingber D.E. (1998), Biotech. Prog., 14, 356–363. 16. Pitters J.L., Piva P.G., Tong X., and Wolkow R.A. (2003), Nano Lett., 3, 1431. 17. Wacaser B.A., Maughan M.J., Mowat I.A., Niederhauser T.L., Linford M.R., and Davis R.C. (2003), Appl. Phys. Lett., 82, 808. 18. Eaton D.L. (1985), Toxicol. Appl. Pharmacol., 78, 158–162. 19. Robbins A.H., McRee D.E., Williamson M., Collett S.A., Xuong N.H., Furey W.F., Wang B.C., and Stout C.D. (1991), J. Mol. Biol., 221, 1269–1293. 20. Messerle B.A., Schaffer A., Vasak M., Kagi J.H.R., and Wuthrich K. (1992), J. Mol. Biol. 225, 433–443. 21. Boulanger Y., Goodman C.M., Forte C.P., Fesik S.W., and Armitage I.M. (1983), Proc. Natl. Acad. Sci., 80, 1501–1505. 22. Chang C.C. and Huang P.C. (1996), Protein Eng., 9, 1165–1172. 23. Wei S.H. and Zunger A. (1986), Phys. Rev. Lett., 56, 2391–2394. 24. Christou G., Gatteschi D., Hendrickson D.N., and Sessoli R. (2000), MRS Bull., 25, 66–71. 25. Sessoli R., Tsai H.L., Schake A.R., Wang S., Vincent J.B., Folting K., Gatteschi D., Christou G., and Hendrickson D.N. (1993), J. Am. Chem. Soc., 115, 1804–1816. 26. Aubin S.M.J., Dilley N.R., Pardi L., Kryzystek J., Wemple M.W., Brunel L.C., Maple M.B., Christou G., and Hendrickson D.N. (1998), J. Am. Chem. Soc., 120, 4991. 27. Friedman J.R., Sarachik M.P., Tejada J., and Ziolo R. (1996), Phys. Rev. Lett., 76, 3830–3833.

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28. Bushby R.J. and Pailland J.-L. (1995), Molecular magnets, in Introduction to Molecular Electronics, M.C. Petty, M.R. Bryce, and D. Bloor (eds.), Edward Arnold, Hodder Headline, London, U.K. 29. Wolynes P.G. (1997), Proc. Natl. Acad. Sci. U.S.A., 94, 6170–6175. 30. Zwanzig R. (1997), Proc. Natl. Acad. Sci. U.S.A., 94, 148–150. 31. Chang C.-C., Yeh X.-C., Lee H.-T., Lin P.-Y., and Kan L.S. (2004), Phys. Rev. E, 70, 011904. 32. Welker E., Wedemeyer W.J., Narayan M., and Scheraga H.A. (2001), Biochemistry, 40, 9059–9064. 33. Chang C.-C., Su Y.-C., Cheng M.-S., and Kan L.S. (2002), Phys. Rev. E, 66, 021903. 34. Chang C.-C., Tsai C.T., and Chang C.Y. (2002), Protein Eng., 5, 437–441. 35. Chang C.-C., Su Y.-C., Cheng M.-S., and Kan L.S. (2003), J. Biomol. Struct. Dyn., 21, 247–256. 36. Chang C.-C. and Kan L.-S. (2007), Chin. J. Phys., 45, 693–702. 37. Liu Y.-L., Lee H.-T., Chang C.-C., and Kan L.S. (2003), Biochem. Biophys. Res. Commun., 306, 59–63. 38. Levy Y., Jortner J., and Becker O. (2001), Proc. Natl. Acad. Sci. U.S.A., 98, 2188–2193. 39. Creighton T.E. (1993), Proteins, W.H. Freeman, New York. 40. Elcock A.H. (1999), J. Mol. Biol., 294, 1051–1062. 41. Ulrih N.P., Anderluh G., Macek P., and Chalikian, T.V. (2004), Biochemistry, 43, 9536–9545. 42. Despa F., Fernandez A., and Berry R.S. (2004), Phys. Rev. Lett., 93, 228104. 43. Seefeldt M.B., Ouyang J., Froland W.A., Carpenter J.F., and Randolph T.W. (2004), Protein Sci., 13, 2639–2650. 44. Anfinsen C.B., Haber E., Sela M., and White F.H. Jr. (1961), Proc. Natl. Acad. Sci., 47, 1309–1314. 45. Su Z.D., Arooz M.T., Chen H.M., Gross C.J., and Tsong T.Y. (1996), Proc. Natl. Acad. Sci., 93, 2539–2544. 46. Roder H., Maki K., Cheng H., and Shastry M.C. (2004), Methods, 34, 15–27.

47. Royer C.A. (1995), Methods Mol. Biol., 40, 65–89. 48. Misawa S. and Kumagai I. (1999), Biopolymers, 51, 297–307. 49. Dinsmore A.D., Crocker J.C., and Yodh A.G. (1998), Curr. Opin. Colloid Interface Sci., 3, 5–11. 50. Gast A.P. and Russel W.B. (1998), Phys. Today, 51, 24–30. 51. Pusey P.N. and van Megan W. (1986), Nature, 320, 340–341. 52. Ackerson B.J. and Pusey P.N. (1988), Phys. Rev. Lett., 61, 1033–1036. 53. Tsong T.Y., Hearn R.P., Warthal D.P., and Sturtevant J.M. (1970), Biochemistry, 9, 2666–2677. 54. Tokuriki N., Kinjo M., Negi S., Hoshino M., Goto Y., Urabe I., and Yomo T. (2004), Protein Sci., 13, 125–133. 55. Lin K.-H., Crocker J.C., Prasad V., Schofield A., Weitz D.A., Lubensky T.C., and Yodh A.G. (2000), Phys. Rev. Lett., 85, 1770–1773. 56. Lau A.W.C., Lin K.-H., and Yodh A.G. (2002), Phys. Rev. E, 66, 020401. 57. Stephen Y.C., Peter R.K., and Preston J. R. (1995), Appl. Phys. Lett., 67, 3114–3116. 58. Bender M., Otto M., Hadam B., Spangenberg B., and Kurz H. (2000), Microelectron. Eng., 53, 233–236. 59. Xia Y. and Whitesides G.M. (1998), Angew. Chem. Int., 37, 550–575. 60. Lingjie G., Peter R.K., and Stephen Y.C. (1997), Appl. Phys. Lett., 71, 1881–1883. 61. Pépin A., Youinou P., Studer V., Lebib A., and Chen Y. (2002), Microelectron. Eng., 61–62, 927–932. 62. Li M., Tan H., Chen L., Wang J., and Chou S.Y., (2003), J. Vac. Sci. Technol. B, 21, 660–663. 63. Cho Y.H., Lee S.W., Kim B.J., and Fujii T. (2007), Nanotechnology, 18, 465303. 64. Likharev K.K. (2003), Hybrid semiconductor–molecular nanoelectronics. The Industrial Physicist, June/July 2003, Forum: 20–23. 65. Chang C.-C., Sun K.W., Lee S.F., and Kan L.S. (2007), Biomaterials, 28, 1941–1947. 66. Chang C.-C., Sun, K.W., Kan, L.-S., and Kuan, C.-H. (2006), Appl. Phys. Lett., 88, 263104.

Nanoscale Transistors

II

12 Transistor Structures for Nanoelectronicsâ•… Jean-Pierre Colinge and Jim Greer.................................................... 12-1

13 Metal Nanolayer-Base Transistorâ•… André Avelino Pasa......................................................................................... 13-1

14 ZnO Nanowire Field- Effect Transistorsâ•… Woong-Ki Hong, Gunho Jo, Sunghoon Song, Jongsun Maeng, and Takhee Lee........................................................................................................................................................... 14-1

Introductionâ•‡ •â•‡ Evolution of Silicon Processing: Technology Boostersâ•‡ •â•‡ Nonplanar Multi-Gate Transistorsâ•‡ •â•‡ The Rise of Quantum Effectsâ•‡ •â•‡ Carbon Nanotube Transistorsâ•‡ •â•‡ Nonclassical Transistor Structuresâ•‡ •â•‡ Graphene Ribbon Nanotransistorsâ•‡ •â•‡ Sub-kT/q Switchâ•‡ •â•‡ Single-Electron Transistorâ•‡ •â•‡ Spin Transistorâ•‡ •â•‡ Molecular Tunnel Junctionsâ•‡ •â•‡ Point Contacts and Conductance Quantumâ•‡ •â•‡ Atomic-Scale Technology Computer-Aided Designâ•‡ •â•‡ Conclusionâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Band Diagramâ•‡ •â•‡ J−V Characteristicâ•‡ •â•‡ Fabricationâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Backgroundâ•‡ •â•‡ Results and Discussionâ•‡ •â•‡ Summaryâ•‡ •â•‡ Acknowledgmentâ•‡ •â•‡ References

15 C60 Field Effect Transistorsâ•… Akihiro Hashimoto.................................................................................................... 15-1

16 The Cooper-Pair Transistorâ•… José Aumentado........................................................................................................ 16-1

Introductionâ•‡ •â•‡ Fullerene (C60)â•‡ •â•‡ Solid C60â•‡ •â•‡ van der Waals Epitaxy of C60â•‡ •â•‡ Principles of Organic FET Functionsâ•‡ •â•‡ Fabrication of Fullerene Field Effect Transistors (C 60 FET)â•‡ •â•‡ Characterization of C60 FETs on SiO2 or on AlNâ•‡ •â•‡ Summaryâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Theory of Operationâ•‡ •â•‡ Fabricationâ•‡ •â•‡ Practical Operation and Performanceâ•‡ •â•‡ Present Status and Future Directionsâ•‡ •â•‡ Acknowledgmentâ•‡ •â•‡ References

II-1

12 Transistor Structures for Nanoelectronics

Jean-Pierre Colinge University College Cork

Jim Greer University College Cork

12.1 Introduction..........................................................................................................................12-1 12.2 Evolution of Silicon Processing: Technology Boosters..................................................12-2 12.3 Nonplanar Multi-Gate Transistors...................................................................................12-3 12.4 The Rise of Quantum Effects.............................................................................................12-5 12.5 Carbon Nanotube Transistors...........................................................................................12-6 12.6 Nonclassical Transistor Structures...................................................................................12-7 12.7 Graphene Ribbon Nanotransistors...................................................................................12-7 12.8 Sub-kT/q Switch....................................................................................................................12-8 12.9 Single-Electron Transistor..................................................................................................12-8 12.10 Spin Transistor.....................................................................................................................12-9 12.11 Molecular Tunnel Junctions...............................................................................................12-9 12.12 Point Contacts and Conductance Quantum................................................................. 12-11 12.13 Atomic-Scale Technology Computer-Aided Design.................................................... 12-11 12.14 Conclusion..........................................................................................................................12-12 References..........................................................................................................................................12-12

12.1â•‡ Introduction In 1965, Gordon Moore predicted that the number of transistors that could be placed on a silicon chip would double every 18 months. This prediction became the official roadmap of the semiconductor industry and it still is the yardstick by which the progress of microelectronic devices and circuits is measured (Moore 1965). The MOS transistor, also called MOSFET (metal-oxide-semiconductor field-effect transistor) is the workhorse of the semiconductor industry and is at the heart of every digital circuit. Without the MOSFET, there would be no computer industry, no digital telecommunication systems, no video games, no pocket calculators, and no digital wristwatches. Figure 12.1 shows the evolution of the number of MOS transistor per chip vs. calendar year, known as “Moore’s law.” Note that the vertical axis has a logarithmic scale. The effective length of the transistor, L, is the distance between the source and the drain in a region called the “channel region” where the electrostatics and the current flow are controlled by the gate. When the effective gate length becomes too small, the gate loses the control of that region and the so-called short-channel effects appear. These effects increase the OFF current of the device and render the current dependent of the drain voltage. In the most extreme cases, the transistor can no longer be turned off. The effective length of MOSFETs has shrunken from 10â•›μm in

1971 (Intel• 4004 processor) to 1â•›μm in the early 1980s, 0.1â•›μm in 2000, and 10â•›nm dimensions should be reached around 2015. Short-channel effects become so important in classical, planar MOSFETs with dimensions below 5â•›nm that new device structures that improve the electrostatic control of the channel by the gate are being explored. These devices are called multi-gate MOSFETs. They basically are silicon nanowires with a gate electrode wrapped around the channel region. This device architecture maximizes the control of the electrostatics in the channel region and allows one to reduce the effective channel length to sub-10â•›nm dimensions. At those dimensions, quantum transport effects start to kick in, and device simulators need to incorporate the Schrödinger equation to accurately model the transistor’s electrical characteristics. Of course, the reduction of transistor size goes hand in hand with the increase of transistors on a chip. The first 32â•›Gb flash memory chips made using a 40â•›nm technology were announced in 2006. Each of these chips contains over 32 billion transistors. The number of transistor per chip is literally reaching astronomical proportions, considering that the 400â•›Gb mark will be reached around the year 2020. When this milestone is reached, there will be as many transistors on a single chip as there are stars in the Milky Way (Figure 12.1). In 2008, the effective gate length of transistors that are used to fabricate microprocessors is 22â•›nm, and the thickness of the gate insulator is approximately 1â•›nm. Fully functional 12-1

12-2


1012

6 nm

Quantum effects in silicon

1011 Transistors per chip

12.2â•‡Evolution of Silicon Processing: Technology Boosters

Number of stars in our galaxy

1010

10 nm

18 nm

109

Planar MOSFET limit

35 nm L = 100 nm

108 1995

2000

2005

2015

2010

2020

Year

FIGURE 12.1â•… Moore’s law: number of transistors per chip vs. calendar year.

“gate-all-around” transistors made in a silicon nanowire with a diameter of only 3â•›nm, that is, roughly the same diameter as a carbon nanotube, were reported in 2006 (Singh et al. 2006). Although these transistors are not formally called “nanoelectronic devices,” they are clearly of nanometer dimensions. Strictly speaking, the nanoelectronics era has already begun. Section 12.2 describes the evolution of the MOS transistor as its structure reaches nanometer dimensions: new materials are used to optimize electrical conductivity and dielectric constant, semiconductor alloying and the introduction of stress are used to increase carrier mobility, and novel transistor geometries are employed to increase the electrostatic control of the channel by the gate. I 1

In the 1980s, only a handful of elements were used in silicon chips: boron, phosphorus, arsenic, and antimony for doping the silicon, oxygen, and nitrogen for growing or depositing insulators, and aluminum for making interconnections. A few elements, such as hydrogen, argon, chlorine, and fluorine are used during processing in the form of etching plasmas or oxidationenhancing agents. Sulfur is used in sulfuric acid to clean wafers. In the 1990s, a few more elements were added to the list, such as titanium, cobalt, and nickel, used to form low-resistivity metal silicides, tungsten, used to form vertical interconnects known as “plugs,” and bromine, used in a plasma form to etch silicon. The 2000s saw an explosion in the number of elements used in silicon processing: lanthanide (rare earth) metals are being used to form oxides with high dielectric constants (high-k dielectrics), carbon and germanium are used to change the lattice parameter and induce mechanical stresses in silicon, fluorides of noble gases are used in excimer laser lithography, and a variety of metals are used to synthesize compounds that have desirable work functions or Schottky characteristics. Virtually all elements are used, with the notable exception of alkaline metals, which create mobile charges in oxides, and, of course, radioactive elements (Figure 12.2). The use of new elements to obtain new desirable properties is a technology booster that has made it possible to extend the life of CMOS and reduce dimensions beyond barriers that were previously considered insurmountable. For instance, the reduction

II

III

Hydrogen 1

IV

V

VI

VII

1980s

H

He

1.00794(7)

2 3 4 5 6 7

4.002602(2)

Lithium 3

Beryllium 4

Li

Be

6.941(2)

9.012182(3)

Sodium 11

Magnesium 12

Na

Mg

22.989770(2)

24.305(?)

Potasium 19

Calcium 20

K

Ca

Scandium 21

Sc

Titanium 22

Ti

Vanadium 23

V

Chromium 24

Cr

1990s

Boron 5

Carbon 6

Nitrogen 7

Oxygen 8

Fluorine 9

Neon 10

B

10.811(7)

C

12.0107(8)

N

14.00674(7)

O

15.9994(3)

F

18.9984032(5)

Ne

20.1797(6)

2000s

Aluminium 13

Silicon 14

Phosphorus 15

Sulfur 16

Chlorine 17

Argon 18

Manganese 25

Mn

Iron 26

Fe

Cobalt 27

Co

Nickel 28

Ni

39.0983(1)

40.078(4)

44.955910(8)

47.867(1)

50.9415(1)

51.996(6)

54.9380(9)

55.845(2)

58.933200(9)

58.6934(2)

Rubidium 37

Strontium 38

Yttrium 39

Zirconium 40

Niobium 41

Molybdenum 42

Technetium 43

Ruthenium 44

Rhodium 45

Rb

Sr

Y

Zr

Nb

Mo

Tc

Ru

Rh

Ar 39.948(1)

Gallium 31

Germanium 32

Arsenic 33

Selenium 34

Bromine 35

Krypton 36

Ga

Silver 47

Cadmium 48

Ag

Cd

95.94(1)

[98.9063]

101.07(2)

102.90550(2)

106.42(1)

Caesium 55

Barium 56

Lanthanum 57

Hafnium 72

Tantalum 73

Tungsten 74

Rhnium 75

Osmium 76

Iridium 77

Cs

Ba

La

Hf

Ta

W

Re

Os

Ir

132.90545(2)

137.327(7)

138.9055(2)

178.49(2)

180.9479(1)

183.84(1)

186.207(1)

190.23(3)

Francium 87

Radium 88 Promethium 61

Samarium 62

Europium 63

Praseodymium Nedodymium 59 60

Cl 35.4527(9)

Pd

92.90638(2)

Cerium 58

S 32.066(6)

Palladium 46

91.224(4)

Ra

Zn

P 30.973761(2)

69.723(1)

88.90585(2)

[220.0254]

Cu

Zinc 30

Si 28.0855(3)

65.39(2)

87.62(1)

Fr

Copper 29

Al 26.981538(2)

63.546(3)

85.4678(3)

[223.0197]

O Helium 2

Ge

As

Se

Kr

74.92160(2)

78.96(3)

79.904(1)

83.80(1)

Indium 49

Tin 50

Antimony 51

Tellurium 52

Iodine 53

Xenon 54

In

Sn

Sb

Te

I

107.8682(2)

112.411(8)

114.818(3)

Platinum 78

Gold 79

Mercury 80

Pt

Au

Hg

192.217(3)

195.078(2)

196.96655(2)

Gadolinium 64

Terbium 65

Dysprosium 66

Xe

118.710(7)

121.760(1)

127.60(3)

126.90447(3)

131.29(2)

Thallium 81

Lead 82

Bismuth 83

Polonium 84

Astatine 85

Radon 86

Th

Pb

Bi

Po

At

Rn

200.59(2)

204.3833(2)

207.2(1)

208.98038(2)

[208.9824]

[209.9871]

[222.0176]

Holmium 67

Erbium 68

Thulium 69

Ytterbium 70

Lutetium 71

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

140.116(1)

140.90765(2)

144.24(3)

[144.9127]

150.36(3)

151.964(1)

157.25(3)

158.92534(2)

162.50(3)

164.93032(2)

167.26(3)

168.93421(2)

173.04(3)

174.967(1)

Thorium 90

Protactinium 91

Uranium 92

Neptunium 93

Plutonium 94

Americium 95

Curium 96

Berkelium 97

Californium 98

Einsteinium 99

Fermium 100

Mendelevium 101

Nobelium 102

Lawrencium 103

Th

Pa

U

Np

Pu

Am

Cm

Bk

Cf

Es

Fm

Md

No

Lr

232.038(1)

231.03588(2)

238.0289(1)

[237.0482]

[244.0642]

[243.0614]

[247.0703]

[247.0703]

[251.0796]

[252.0830]

[257.0951]

[258.0984]

[259.1011]

[262.110]

FIGURE 12.2â•… Elements used in silicon processing.

Br

72.61(2)

12-3

Transistor Structures for Nanoelectronics

of gate oxide thickness below 1.5â•›nm leads to a gate tunnel current that quickly becomes prohibitively high. Replacing silicon dioxide by a high-k dielectrics such as hafnium oxide (HfO2), which has a dielectric constant of 22 (vs. 3.9 for SiO2), allows one to increase the thickness of the gate dielectric by a factor 22/3.9 = 5.5 without reducing the gate capacitance (i.e., the current drive of the transistor). The use of new gate dielectrics gave rise to the notion of equivalent oxide thickness, EOT, which is defined by the relationship EOTâ•›=â•›td(εox/εd), where td is the thickness of the dielectric layer, and εox and εd are the permittivity of silicon dioxide and the dielectric material, respectively. For example, a 4â•›nm thick layer of HfO2 is electrically equivalent to an SiO2 layer of 0.7â•›nm. To improve the properties of transistors, another technology booster is commonly used: stress. Compressive stress increases hole mobility in (110) silicon (i.e., in the direction of current flow of most transistors), while tensile stress increases electron mobility. Mobility can also be modified by using Si:Ge or Si:Ge:C alloys. Compressive stress can be induced in the channel region of a transistor by introducing germanium in the source and drain. The resulting “swelling” of the silicon in the source and drain compresses the channel region situated between them. Tensile stress can readily be obtained by depositing a silicon nitride contact-etch stop layer on top of the device. Mobility (and thus speed) improvement in excess of 50% can be obtained using stress techniques. The third technology booster deals with the physical geometry of the transistor and is worth being described in detail. As the dimensions of the transistors are shrunk, the close proximity between the source and the drain reduces the ability of the gate electrode to control the potential distribution and the flow of current in the channel region, and undesirable effects, called the “short-channel effects” render MOSFETs inoperable. For all practical purposes, it seems impossible to scale the dimensions of classical “bulk” MOSFETs below 15â•›nm. If that limitation cannot be overcome, Moore’s law would reach an end around year 2012. Short-channel effects arise when electric field lines from source and drain affect the control of the channel region by the gate. These reduce the threshold voltage VTH according to the expression VTH = VTH∞ − ΔVTH − DIBL where VTH∞ is the threshold voltage of a long-channel device, ΔVTH is the “threshold voltage roll-off,” due to the sharing of the space charge region underneath the gate between the gate, and the unbiased source and drain junctions, and DIBL is the “drain barrier lowering” which results from the increase of the share of the space charge region related to the drain junction when the drain voltage is increased. Short-channel effects can be minimized by reducing the junction depth and the gate oxide thickness. They can also be minimized by reducing the depletion depth through an increase in channel doping concentration. In modern devices, however, practical limits on the scaling of junction depth and gate oxide thickness lead to a significant increase of short channel effects and excessively large values of DIBL can quickly be reached.

12.3â•‡ Nonplanar Multi-Gate Transistors The field lines in MOS transistors are illustrated graphically in Figure 12.3. In a bulk device (Figure 12.3A), the electric field lines propagate through the depletion regions associated with the junctions. Their influence on the channel can be reduced by increasing the doping concentration in the channel region. In very small devices, unfortunately, the doping concentration becomes very high (>1019 cm−3), which degrades carrier mobility. The situation can be improved by making the transistor in a thin film of silicon atop of an insulator. The silicon film is thin enough that it is fully depleted of majority carriers when a gate voltage is applied. Such a device is called fully depleted silicon-on-insulator (FDSOI) MOSFET. In FDSOI devices, most of the field lines propagate through the buried oxide (BOX) before reaching the channel region (Figure 12.3B). Short-channel effects can be further reduced by using a thin buried oxide and an underlying ground plane. In that case, most of the electric field lines from the source and drain terminate on the buried ground plane instead of the channel region (Figure 12.3C). This approach, however, has the inconvenience of increased junction capacitance and body effect (Xiong et al. 2002). A much more efficient device configuration is obtained by using the double-gate transistor structure. This device structure was first proposed by Sekigawa and Hayashi in 1984 and was shown to reduce threshold voltage roll-off in short-channel devices (Sekigawa and Hayashi 1984). In a double-gate device, both gates are connected together. The electric field lines from source and drain underneath the device terminate FD SOI

Bulk

S

S

D

D

BOX (A)

(B) FD SOI S

DG S

D

D

BOX Ground plane (C)

(D)

FIGURE 12.3â•… Encroachment of electric field lines from source and drain on the channel region in different types of MOSFETs: (A) Bulk MOSFET, (B) fully depleted SOI MOSFET, (C) fully depleted SOI MOSFET with thin buried oxide and ground plane, (D) double-gate MOSFET.

12-4

Handbook of Nanophysics: Nanoelectronics and Nanophotonics G

G

D

S

G

D

S

1

3

G

D

S

2

D

S 4 Buried oxide

FIGURE 12.4â•… Different multi-gate MOSFET configurations. 1, Â�double-gate; 2, triple-gate; 3, quadruple-gate or gate-all-around; 4, Π or Ω gate.

but smaller than a 4-gate device. The different configurations are shown in Figure 12.4. Short-channel effects are absent if the effective gate length is larger than seven to eight times the natural length, λ. This behavior is common to all gate configurations. Acceptable level of short-channel effects (DIBL < 50â•›mV and subthreshold slope 0). (d) Ib < I0. Phase diffusion. In ultrasmall junctions, thermal ∼ 2π. Since the washboard is tilted, this results in a slow diffusion fluctuations can drive the phase to evolve in a 1D random walk of phase steps Δϕ − ⋅ in one direction, V (= φ0〈δ〉) > 0, although it is still considered to be on the “supercurrent branch.”

switching current Isw, at which the CPT switches from the supercurrent branch to the finite-voltage branch as well as measuring the effective inductance LJ of the CPT when biased at zero current. While the small size of the junctions and internal charge degree of freedom will affect these parameters significantly, it is worthwhile to first look at the single junction case and understand how the dynamics of the phase determine the structure of the I–V curve (see Figure 16.2a). 16.2.1.1.2 Th e Resistively and Capacitively Shunted Junction (RCSJ) Model A more realistic model of a single Josephson junction includes a parallel shunt resistance and capacitance (see inset in Figure 16.2a). The shunt resistance R encapsulates the resistance of the junction to normal/quasiparticle currents through the junction, while the capacitance C is simply the physical capacitance that arises from having two planar electrodes overlapping with an insulating barrier in between. In the junctions comprising a CPT, the intrinsic R > 10â•›k Ω and is considered effectively “unshunted” since R is much bigger than the impedance presented by the Josephson inductance. As noted previously, the Josephson tunnel element is a nonlinear inductance but, for small phase excursions, it looks like a linear inductor. The circuit model presented then looks very much like a parallel LCR circuit or damped simple harmonic oscillator with a natural frequency ωp =

1 2eI 0 . = C LJ 0C

(16.7)

This is usually called the “plasma frequency” and has the significance of simply being the frequency of small oscillations for a Josephson junction.

Although this model can be used to describe the intrinsic Josephson junction dynamics, it is also useful in determining the dynamics when the junction is placed in an arbitrary measurement circuit. If, for instance, we want to current bias the junction, we might add a voltage source Vb in series with a resistor Rb and construct a Norton equivalent that is simply an ideal current bias Ib = V b/Rb in parallel with an output impedance Rb.* In this case, we can roll Rb into the shunt R in our RSJ model. With this simple circuit equivalent, we can derive a set of firstorder differential equations defining the equations of motion for the phase by writing down Kirchoff’s equations for the currents in each branch. We can write this system of equations as a single second-order differential equation

δ + 1 δ + ω2  sin δ −  I b p  RC  I0 

   = 0. 

(16.8)

For small δ at zero current bias, this describes the motion of a particle with coordinate δ oscillating in a quadratic potential with a friction term, viz., it’s a damped simple harmonic oscillator (and is consistent with our earlier statement that LJ is linear for small oscillations). At finite current bias, however, this equation describes the motion of the particle in a tilted sinusoidal (or washboard) potential.† We know from Equation 16.2 that in order for the junction to support a finite dc voltage across * This vast oversimplification has its caveats and must be amended to account for stray capacitance in the bias lines and any other reactances relevant in the frequency range of our junction dynamics (typically ω < 50â•›GHz for small aluminum junctions). † More in depth discussions of the RCSJ model can be found in Tinkham (2004) and Clarke and Braginski (2004).

16-4


it, the particle must have some net motion in the phase, that ⋅ is, 〈δ〉 ≠ 0. Since the particle is trapped at a fixed phase when Ib < I0, we have a well-defined phase and the junction remains in the supercurrent state (see Figure 16.2b). By applying a bias current, we tilt the washboard and, when Ib = I0, the particle is tipped out of its well and begins to run away (Figure 16.2c). In this free-running state, the time derivative of the phase has both an ac and a dc component corresponding to ac and dc voltages across the junction. In a more realistic treatment, Ib includes a stochastic contribution from thermal fluctuations due to the dissipation in the environment and so the current at which the junction switches to the free-running state is lower than I0 since thermal excitations can drive the phase particle over the potential barrier prematurely. Because of this, the measured current is distributed according to the statistics of a thermally excited (Kramers) escape process (Fulton and Dunkleberger, 1974). We will differentiate this switching current Isw from the ideal Ambegaokar–Baratoff value I0 and remember that it is actually a statistical variable. If we reduce the current bias in the free-running, finite voltage state, the particle velocity begins to slow down due to the dissipation in the circuit. Once the dissipation rate can compensate for the ac power generated by the free-running voltage oscillations, the particle “retraps” into a minima in the washboard potential and the junction returns to the zero-voltage supercurrent branch. The current at which this happens, Iret, the “return current” is an indication of the amount of damping or dissipation in the environment at the free-running oscillation frequency (Tinkham, 2004). Roughly, the closer to zero Iret is, the less damping there is. Although the CPT is designed and operated in the unshunted, hysteretic regime, in many practical dc SQUID magnetometers, the junctions are intentionally shunted with damping resistors to eliminate hysteresis in the I–V measurement permitting stable finite voltage operation.

At zero current bias, this results in a one-dimensional random walk in the phase with zero mean. At finite current, the tilt in the washboard biases this diffusive motion in one direction, ⋅ such that the mean phase velocity 〈δ〉 can be positive (negative) at positive (negative) bias (see Figure 16.2d). The result of all of this is that at finite currents below the switching current, we often see a small finite voltage. This slope is fairly uniform near zero current and gives the supercurrent branch a slight resistive contribution that can correspond to 10â•›s or even 100â•›s of Ωs. The downside of this behavior is that the junction or CPT is not truly dissipationless. For electrometer operation, we want Isw to have as narrow a distribution as possible since it is the quantity that will vary with applied gate voltage/polarization charge. It has been demonstrated, however, that one can narrow this distribution and also increase Isw, so that it is closer to I0 by fabricating a larger capacitance (a few picofarad) in the leads shunting the junction to “weigh down” the phase particle (Joyez et al., 1994; Joyez, 1995).

16.2.1.1.3 Concerns Specific to Small Junctions

where

While the picture above is the standard way to look at relatively large junctions (area > 1â•›μm2), the junctions in CPTs are usually about a factor of 100 smaller and have much smaller capacitances. The main consequence of this is that the observed switching currents can easily be smaller than I0 by an order of magnitude. One way to understand this is to return to the harmonic oscillator analogy. We can see that ω 2p ∝ 1/C, so C is an effective mass for our “phase particle.” In CPT junctions, this means that the equivalent phase particle can be relatively light compared with its larger junction cousins and so thermal fluctuations, even at 100â•›mK, can have a more significant effect, kicking the particle into the free-running state much more easily. This has the end effect of reducing Isw far below I0 and also widening the distribution of Isw (Figure 16.2a). A more problematic effect of the light phase particle mass is the fact that the same thermal fluctuations can “kick” the phase particle from well to well in the washboard potential in a process called “phase diffusion” (Kautz and Martinis, 1990).

16.2.2â•‡ Coulomb Blockade While the above discussion of a single junction will eventually be relevant to the CPT measurements we are interested in, we have not yet seriously considered the role of the internal charge degree of freedom in determining the overall critical current and Josephson inductance in this device. This is, afterall, the knob that turns the CPT into a transistor. We can qualitatively recognize the role of the island charge by first determining the energy required to charge the CPT island with a single electron charge e,

EC =

e2 , 2C∑

C∑ ≡ C J 1 + C J 2 + C g + Cstray .

(16.9)

(16.10)

Here, C Σ is the total capacitance as seen by the island, and while it includes the junction and gate capacitances, it also includes a “catch-all” term, Cstray, that is meant to include any contributions from unintentional coupling to ground. In practice, Cstray ~ 1, this approximation fails and one must write down a larger matrix spanning a bigger subspace of |n〉. In this limit, the charge on the island is no longer as well defined as its quantum mechanical state is now described by a linear superposition of several charge states. Likewise, fluctuations in the CPT external phase are then small and the CPT begins to look like a single Josephson junction. 16.2.3.1â•‡The Uncertainty Principle at Work We can numerically diagonalize Equation 16.14 to calculate the eigenvalues at various ng and δ. Figure 16.4 shows the resulting eigenenergy surfaces corresponding to the ground and first excited state energies, ϵ0 and ϵ1. The quantities that we want to measure, Isw(ng) and LJ(ng), are defined by the first and second derivatives of these surfaces in the phase. The size of this modulation is a direct consequence of how well the CPT island localizes Cooper pairs, n2e, through the uncertainty relation between charge number and phase, ΔnΔδ ≥ 1/2. In this sense, the CPT is a strange charge-based device because it relies on the competition between number and phase uncertainty. This is in contrast to the SSET or SET where EC is entirely dominant. In the CPT, we would like EC to be big but would also like EJ to be comparable in magnitude, so that the switching current is not so small that it is difficult to measure well. Similarly, rf measurements of the Josephson inductance (described later in this review) become easier at bigger EJ as larger signal powers can be used while biased on the supercurrent branch. 16.2.3.2â•‡Calculation of the Critical Current Modulation The critical current of the CPT can now be determined from the eigenenergies calculated above. By restricting to the ground state band ϵ(ng,â•›δ), we can compute the critical current (Joyez, 1995) I 0 (ng ) =

1 ∂0 (ng , δ) , ϕ0 ∂δ δmax

(16.15)

16-6

Handbook of Nanophysics: Nanoelectronics and Nanophotonics EJ1 = EJ2 = 80 μeV EC,Σ = 115 μeV 250 200 [μeV]

150

1

100 50 0 –50 2

0

1

ng

0

[e]

–1 –2

–1

–2

0

1

2

δ/π

FIGURE 16.4â•… (See color insert following page 20-14.) The ground (ϵ0) and first excited (ϵ1) states calculated by numerically diagonalizing Equation 16.11 with the energies indicated in the figure. 90 80 I0 [nA]

still 2π periodic, gets “peaky” near odd-integer ng. Despite this, one may still define a Josephson inductance in a similar manner (Sillanpää et al., 2004; Naaman and Aumentado, 2006b), that is,

EJ, jn/EC = 2.78

70 20 0.70

10 0 –2

–1.5

–1

–0.5

0 ng [e]

LJ (ng ) =

0.17

0.5

1

1.5

2

FIGURE 16.5â•… Numerically calculated CPT critical currents for various ratios of EJ,â•›jn/EC . In this example, EC is fixed at 115â•›μeV, while EJ,â•›jn is varied.

where δmax is the phase maximizing the derivative.* In Figure 16.5, we show the calculated critical currents for several ratios of EJ/EC . Note that the calculated currents are 2e periodic in the applied gate charge which is the result of the Josephson coupling of Cooper-pair charge states. In practice, measured switching currents in CPTs are usually significantly smaller than the calculated value due to small-junction effects noted above. However, the total magnitude of the modulation can be on the order of 10â•›nA for practical aluminum device parameters and is easily measured. 16.2.3.3â•‡Calculation of the Effective Inductance Modulation The ground state energy surface ϵ0(ng,â•›δ) corresponds to an effective Josephson energy that is a function of charge as well as phase. When the ratio EJ/EC < 1, the phase dependence, while * Incidentally, one can also take derivatives of the excited state eigenenergies and compute critical currents for those bands. Interested readers can see Flees et al. (1997) for further details.

2 1 ∂ 0 (ng , δ) ϕ20 ∂δ 2

. δ =0

(16.16)

This is similar to Equation 16.5, but since the ϵ0(ng,â•›δ) is not strictly sinusoidal, we do not get the exactly the same result. For typical aluminum device parameters, the inductance modulation can be 10–100â•›nH and, like the switching current, is easily measured.

16.2.4â•‡ Quasiparticle Poisoning Although the “quasiparticle” excitations that are talked about in superconductivity do not explicitly bear a well-defined charge, they do represent the screened excitations presented by an unpaired electron that might exist due to the breaking of a Cooper pair. A quasiparticle can tunnel onto a CPT island by “undressing” itself of its screening cloud and tunneling through a junction barrier alone. At this point, this additional charge presents a full electron charge offset to the CPT island. Since the CPT switching current and inductance are nominally 2e periodic in the gate charge, the addition of an extra electron can present a significant change in the way the CPT is operated, since it fluctuates the charge by e. In the literature, this problem was first discussed in terms of island “parity,” but more recent work has favored the more colorful term “quasiparticle poisoning.” The most important thing about quasiparticle poisoning for electrometry is that it is a stochastic process whose dynamics

16-7

The Cooper-Pair Transistor

can happen on timescales comparable to our measurement time, so its effect on the resulting measurement must be well understood. That being said, quasiparticle poisoning has proven to be an interesting problem in itself, and the CPT has been very useful in its study. To see how quasiparticles enter the picture, we now include the Hamiltonian HQP in our description of the CPT, H QP =

∑ γ γ . j

j

j

† j

(16.17)

The γ j , γ †j are annihilation/creation operators for quasiparticle excitations in the superconductor (Tinkham, 2004), while ϵj is the energy of the excitation. In our case, this can be Δ1 or Δ2, remembering that the island and leads can have different gap energies. These quasiparticles are usually taken to be thermally generated and therefore have an exponentially small probability of existing at low temperatures. The expression for the quasiparticle density is (cf., Shaw et al., 2008),  −∆ j  nqp = D( F ) 2π∆ j kBT exp  .  kBT 

(16.18)

D(ϵF) (=2.3 × 104 μm−3 J−1) is the density of states at the Fermi energy and Δj is the superconducting gap energy in either film 1 or 2. Plugging in Δ ∼ 200â•›μeV (aluminum) and T = 100â•›mK gives us 2 × 10−4 μm−3. This is a very low number considering the volume of a generic CPT island and leads; yet we know, experimentally, that the actual quasiparticle density is typically 10–1000â•›μm−3 (Mazin, 2004; Shaw et al., 2008). The disparity in the thermal prediction versus the experimentally measured numbers is the first indication that the source for the quasiparticles we see at low temperatures is distinctly nonthermal in nature. As yet, no one has determined the source of these nonequilibrium quasiparticles and we are stuck with the problem of understanding them. 16.2.4.1â•‡Energetics of the Nonequilibrium Quasiparticle Poisoning Process Since we are forced to work in an environment filled with nonequilibrium quasiparticles, we must figure out how to include them in the band picture that we constructed above. We have three states of interest (see Figure 16.6) (Aumentado et al., 2004): Even parity

0 state

Odd parity

ℓ state

i state

FIGURE 16.6â•… Three state nonequilibrium quasiparticle model. “0” state: no quasiparticles in vicinity of CPT, even parity. “ℓ” state: quasiparticle in leads even parity. “i” quasiparticle on island, odd parity.

State 0 No quasiparticles in or near the CPT island. Even parity. State ℓ A quasiparticle is in the leads, in the vicinity of the CPT island. No quasiparticles on the island. Even parity. State i A quasiparticle is on the CPT island. No quasiparticles in the leads near the island. Odd parity. These states can all be described with the band structure derived in the previous section by offsetting the modulated energy surfaces by the superconducting gap energy, corresponding to where the quasiparticle lives, viz., E0 (ng ) ≡ 0 (ng , δ = 0), (16.19)

E (ng ) ≡ 0 (ng , δ = 0) + ∆ ,

Ei (ng ) ≡ 0 (ng + 1, δ = 0) + ∆ i .

Here, we include the possibility of different gap energies in the leads and island, Δℓ and Δi, respectively, and confine ourselves to δ = 0. In principle, phase diffusion will smear these levels somewhat, but the average phase on the supercurrent branch will always be localized in the bottom of a potential well in the phase when biased near Ib = 0. In Figure 16.7, we show the energy bands for two different pairs of island + gap energies using typical Coulomb, Josephson, and gap energies for an aluminum device. In the “type H” device (Figure 16.7a), the island gap is greater than the lead gap, and in the “type L” device, the reverse is true. Assuming that nonequilibrium quasiparticles are present, we can limit our discussion to Eℓ and Ei. In the T = 0 limit, we expect the system to relax to the lowest energy state and the problem is reduced to whether the ℓ or i state energy is smallest (denoted in Figure 16.7a and d by the black dotted trace). For this purpose, we introduce the energy difference:*

δEi (ng ) ≡ Ei (ng ) − E (ng ).

(16.20)

At T = 0, the sign of this quantity determines what state we are in, that is,

+1 sgn[δEi ] =  −1

even parity odd parity

(16.21)

* A brief history detour. In early treatments of quasiparticle poisoning, all quasiparticles were assumed to be thermal in nature, so the important energy in the problem is the free energy required for transition from 0 to i state (Tuominen et al., 1992, 1993;Amar et al., 1994; Joyez et al., 1994; Tinkham et al., 1995). This is basically the process of breaking a Cooper pair with thermal fluctuations. Experimentally, this model was more or less verified early on, but the situation was complicated by the anecdotal evidence that this picture failed to explain the numerous unpublished experiments that showed poisoning at low temperatures. In the absence of nonequilibrium quasiparticles, the early free energy theories are still valid and, in any case, when the system is heated sufficiently T > 250â•›mK, the thermal quasiparticle generation rate dominates over the nonequilibrium rate. For a good review of the early theory, see Tinkham (2004).

16-8


300

Type H Δℓ < Δi EC = 115 μeV, EJ,jn = 80 μeV, ΔL = 220 μeV, ΔI = 250 μeV

Δi

100

Δℓ

50

E0 –1.5

–1

1

1.5

2

–1.5

–1

–0.5

0

0.5

1

1.5

2

(e)

Inductance LJ [nH] –1.5

–1

–0.5

0 ng [e]

E0 –1.5

–1

–0.5

0.5

1

1.5

2

(f )

0

0.5

1

1.5

2

1

1.5

2

1

1.5

2

Critical current

15 10 5 –2

–1.5

–1

–0.5

0

0.5

Inductance

60

40 20 –2

Δ

20

10 5 –2

–50 –2 (d)

I0 [nA]

I0 [nA] LJ [nH]

0.5

15

60

(c)

0

Δi

100 0

Critical current

20

(b)

–0.5

150 50

0 (a)

200

Eℓ

150

Ei Eℓ

250 [μeV]

[μeV]

200

–50 –2

300

Ei

250

Type L Δℓ > Δi EC = 115 μeV, EJ,jn = 80 μeV, ΔL = 250 μeV, ΔI = 220 μeV

40 20 –2

–1.5

–1

–0.5

0 ng [e]

0.5

FIGURE 16.7â•… (a) Type H (Δℓ < Δi) energy bands. ℓ state/even parity (light gray), i state/odd parity (medium gray), 0 state/even parity (black solid). Minimum energy state is denoted as the black dotted trace. Corresponding even (light gray) and odd (medium gray) state (b) critical Â�current and (c) effective inductance for a type H device. (d) Type L (Δℓ > Δi) energy bands. Corresponding even (light gray) and odd (medium gray) state (e) Â�critical currents and (f) effective inductances for a type L device. Gray areas mark range in ng where the CPT island is “trap-like,” that is, a potential well for quasiparticles and the parity state is bimodal and can rapidly switch when coupled to thermal excitations. Γℓi Γiℓ

Eℓ

Γ0ℓ

Ei

δEℓi > 0

Γℓ0 (a)

E0

Eℓ

Γ0ℓ Γℓ0 (b)

Γiℓ

Γℓi Ei

δEℓi < 0

E0

FIGURE 16.8â•… (a) Barrier-like configuration of levels. (b) Trap-like configuration of levels.

δEℓi is an effective potential barrier height for quasiparticles. When δEℓi is positive, the CPT island looks like a barrier, and when it is negative, it looks like a trap (Figure 16.8). In reality, the system is at finite temperature, so this qualitative picture must be amended to include the possibility of thermal excitations (phonons) coupling to the quasiparticles and exciting them out of the trap. Since this is a thermal escape process, we must characterize the system in terms of the average lifetimes of the poisoned and unpoisoned states, τo and τe.* Likewise, it is also useful to talk about these lifetimes as rates, Γeo = Γ0ℓ + Γℓi = 1/τo * The “e” and “o” refer to “even” and “odd” parity states. Odd parity corresponds to a single excess quasiparticle on the CPT island.

and Γoe = Γiℓ = 1/τe. Γeo is known as the “poisoning rate” and Γoe is the “ejection rate.” Experimentally, at T < 250â•›mK, the poisoning rate Γeo has little temperature dependence and can be anywhere between 103 and 105 s−1 (Aumentado et al., 2004; Ferguson et al., 2006; Naaman and Aumentado, 2006b; Court et al., 2008b). The fact that this rate is constant in this temperature range points to the notion that the source is nonthermal in nature and the rate for Cooper-pair breaking in the leads determines the poisoning rate, Γeo ∼ Γ0ℓ >> Γℓi. The ejection rate Γoe does, however, have a temperature dependence at these low temperatures since the process of kicking a quasiparticle out of a potential well is a thermal escape process. One can derive the probability of the

16-9


even parity state (no quasiparticle on the island) using detailed balance arguments (Aumentado et al., 2004), 1 Pe = P0 + P = . 1 + β 0 e − δE i / k B T

16.3â•‡ Fabrication 16.3.1â•‡Electron-Beam Lithography and Two-Angle Deposition The majority of CPTs (and single-charge tunneling devices in general) are fabricated from evaporated aluminum using conventional electron beam lithography and electron-gun (or thermal) deposition techniques. The most conventional method of fabrication is to first define the island and lead pattern of the CPT in a special double-layer e-beam resist stack that is spun onto a silicon substrate. The double-layer is constructed such that the upper “image” layer defines * The reason for this is not obvious. Basically, the difference in aluminum gap energies that one can achieve practically is ∼50â•›μeV. In order to see any level of charge localization at T = 100â•›mK, we need EC ∼â•›100â•›μeV. If we want useful switching current modulation (∼10â•›n A) and inductance modulation (∼10â•›nH), then we are stuck with EJ,jn ∼ 50â•›μeV. For these values, it is difficult to get the Eℓ and Ei bands to separate.

1st dep.

1st 2nd

(16.22)

The factor, β0ℓ ≡ Γ0ℓ/(Γ0ℓ + Γℓ0), accounts for the generation and recombination rates for nonequilibrium quasiparticles in the leads. What this all means is that there is not really any strictly quasiparticle-free zone for typical energies that we would use in practice.* In Figure 16.7, we denote the places where the CPT looks trap-like and it is apparent that both the L and H devices are affected. Even with the higher island gap, the H device can trap quasiparticles, albeit in a shallower potential, δEℓi than presented by the L device and also over a smaller range in ng. It is interesting to note that if there is no source for nonequilibrium quasiparticles, Γ0ℓ = 0 (β0ℓ = 0) and Pe = 1 whether or not we have a barrier-like or trap-like profile. In other words, there would be no poisoning regardless of the relative gap energies. The fact that one sees poisoning easily in trap-like devices at low temperatures is the strongest indication that the quasiparticles in these systems are generated by some nonequilibrium process. Surprisingly, the temperature dependence above also predicts that the even state probability can actually be enhanced since, even though the poisoning rate might be fixed, the ejection rate can be increased by heating the CPT and giving quasiparticles energy to escape (Aumentado et al., 2004; Palmer et al., 2007). This model of nonequilibrium quasiparticle poisoning was first roughly outlined and tested in Aumentado et al. (2004) using CPTs with engineered gap energy profiles and measured with the ramped current technique (see below). It was subsequently confirmed in several later experiments in both CPTs and CPBs using rf and dc techniques (Gunnarsson et al., 2004; Yamamoto et al., 2006; Palmer et al., 2007; Savin et al., 2007; Court et al., 2008b; Shaw et al., 2008).

2nd dep.

(a)

dep

. .

dep

(b)

Source

FIGURE 16.9â•… Angle deposition for the CPT. A mask is defined in electron-beam sensitive resist, usually in a bilayer configuration such that the underlayer is overdeveloped and the top (image) layer pattern can cast a shadow on the surface of the substrate from several angles. In this example, the leads for the device are deposited in the first deposition (dark gray) at an angle θ1. After this deposition, oxygen is admitted into the deposition chamber forming an oxide on the surface of the metal. Finally, a second evaporation (light gray) is performed at θ2 depositing the CPT island. In the process, Josephson junctions are formed at the overlap between the island and oxidized leads. (a) Shadow deposition through the top layer “image” resist. (b) Schematic of tilt configuration with respect to source.

the pattern outline, while the lower “ballast” layer is meant to provide a vast “undercut” region underneath the image. This is usually achieved using resist for the lower layer that is much more sensitive to the e-beam exposure than the upper layer. When the pattern is written, the dosage required to generate the image overexposes the lower layer resist such that the pattern is much wider in the lower layer resist, forming an “undercut” region when the sample is developed (cf., Cord et al., 2006). The undercut region allows us to tilt the substrate so that the impinging aluminum atoms can deposit an image through the image resist mask at an angle (Figure 16.9). After the first aluminum deposition, oxygen is released into the chamber that grows a thin (2(Δi + Δℓ). Inset: Simplified measurement circuit schematic. (b) Expanded view of switching current cycle. The current is ramped along the supercurrent branch until the voltage across the CPT switches to the finite valued voltage branch. The current at which this happens is recorded, and the results of many ramp cycles are recorded in a switching current histogram as in (c).

16-11


pulsed current bias techniques. Each has its own merits, but the experimental literature is largely dominated by the ramped technique. 16.4.1.1â•‡Ramped Switching Current Measurement As the name suggests, the bias current through the CPT is ramped linearly in time such that the current exceeds the maximum switching current and is then returned to zero bias for the next cycle. The probability of switching at a particular current can be backed out from the histogram of switching currents accumulated over many cycles (see Figure 16.10a). This ramped bias current measurement was originally done in the late 1970s in relatively large junctions (Fulton and Dunkleberger, 1974), but the methodology is also suitable to small Josephson junction devices such as the CPT (Joyez, 1995). This method can be used for electrometry since it yields a switching current histogram whose mean value changes with gate, but its power lies in the fact that one can obtain the switching or “escape” rate γsw(Ip,â•›ng) through a straightforward transformation of the whole histogram as in the original work by Fulton and Dunkleberger (1974). This is useful since the escape rate contains information about the electron temperature and how well the system is isolated from external noise (Devoret et al., 1987). 16.4.1.2â•‡Quasiparticle Poisoning in the Ramped Current Measurement If quasiparticle poisoning is present, the effective Josephson energy of the CPT flickers between two different values corresponding to even and odd parities. Each of these energies has its own escape rate γsw,e and γsw,o, and the observed switching current in any given ramp cycle is determined by the instantaneous state of the system as the current is ramped. If the ramp rate is fast

enough such that it can span the distance separating the odd and even switching currents faster than the odd/even lifetimes, then the ramped measurement becomes a snapshot of the parity state and we see that the switching currents group around two distinct values as shown in type L device histograms in Figure 16.12. In our example, the modulation of the switching current histograms is similar to that of the type H device, except that it shows a distinct 1e shifted image corresponding to the presence of the odd state, particularly where we predict the island potential to be trap-like. This bimodal behavior is really only evident if the measurement is faster than the state lifetimes. If the current bias ramp rate were slowed down significantly, then the system could flicker back and forth between parity states many times and we would only see switching distributions grouped around whichever parity state had the lowest switching current. That is, we would see a purely 1e modulation in the switching current. In the early literature, the available dc measurement techniques were unable to capture the dynamical nature of the quasiparticle parity states, and the periodicity (1e versus 2e) of the modulation was the only handle with which to gauge whether poisoning was present. Because of the dynamics of the poisoning process, this correlation can be misleading. In the type L example given here, the device is definitely poisoned, particularly where we expect the island potential to look trap-like. However, if we examine the type H device’s switching current modulation, it is distinctly 2e periodic (Figure 16.11). The early picture of poisoning would naively assume that the poisoning is not present when, in fact, it is happening much faster than a simple analysis would indicate. We know, for instance, that the island potential is trap-like over a narrower range and that the trap potential is shallow compared with the operating temperature (this is due to the gap engineering for the H device). Therefore, quasiparticles that get trapped on the island

Type H

10

Isw [nA]

8

ng = 0.99

6

1.26 1.46

4

1.87

2

(a)

–2

–1.5

–1

–0.5

0 ng [e]

0.5

1

1.5

2

(b)

0

50 # counts

100

FIGURE 16.11â•… (See color insert following page 20-14.) (a) Type H Isw histograms versus ng. Histogram height is displayed in grayscale on the right-hand side, whereas all counts are displayed equally on the left-hand side. As in Figure 16.7, the gray box in (a) denotes regions where the island potential is trap like for quasiparticles. (b) Selected histograms corresponding to several gate voltages. Device parameters: Δi = 246â•›μeV, ∼ 115â•›μeV, and EJ1 = EJ2 − ∼ 82â•›μeV. Δℓ = 205â•›μeV, EC −

16-12

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Type L

12

ng = 1

10

Isw [nA]

8 6 4 2

(a)

–2

–1.5

–1

–0.5

0 ng [e]

0.5

1

1.5

2

0 (b)

400 200 # counts

FIGURE 16.12â•… (a) Type L Isw histograms versus ng. Histogram height is displayed in grayscale on the right-hand side, whereas all counts are displayed equally on the left-hand side. As in Figure 16.7, the gray box in (a) denotes regions where the island potential is trap-like for quasiparticles. (b) Histogram at ng = 1. Bimodal structure indicates relatively long-lived even and odd states. Device parameters: Δi = 205â•›μeV, Δℓ = 246â•›μeV, ∼ 115â•›μeV, and EJ1 = EJ2 − ∼ 78â•›μeV. EC −

are rapidly ejected by thermal fluctuations. In this case, quasiparticles can jump in and out of the island before the CPT has time to latch into the free-running voltage state. In other words, the switching current measurement is bandwidth limited and is not able to see short-lived odd-parity events. While the average probability of being in the even state might be close to unity, quasiparticles may constantly be getting trapped and ejected at very rapid rates. In terms of the poisoning and ejection rates,

Pe =

Γoe . Γeo + Γoe

(16.23)

If the ejection rate Γoe is big compared with the poisoning rate Γeo, then Pe approaches one even though Γeo may be arbitrarily large itself. This is an insidious effect since it was previously common for many groups working in CPB qubits to assess their quasiparticle situation from what appeared to be clean 2e modulation characteristics. In fact, 2e modulation is observed in the presence of these fast quasiparticle dynamics when the measurement technique is slow in comparison. Since quasiparticles might have lifetimes far shorter than 1â•›μs, most available methods (dc and rf alike) can have problems with this. In CPTs with deeper trap potentials, such as the type L devices we show here, the ejection rate can be slowed down significantly. Since the measurement shown in Figures 16.12 and 16.13c,d was performed with a ramp rate that was comparable to the quasiparticle ejection rate, it was possible to obtain a bimodal histogram of Isw. In this case, it is easy to correlate the derived escape function with meaningful rates. The two slopes in Figure 16.13d are the even and odd state escape rates, γsw,e and γsw,o, respectively. Thus, the bias current ramp is not infinitely fast, and the parity can flip from even to odd in the time it takes to ramp between the two current values corresponding to the odd and even state switching currents resulting in several switching events in the region between the peaks. These correspond to quasiparticles

jumping into the CPT island during the bias ramp. Since the plateau that connects the even and odd switching rates is derived from the histogram counts between the even and odd peaks, it is a direct indication of the poisoning rate, Γeo. We can apply similar reasoning to the type H data in Figure 16.13a,b and notice that the escape rate for ng = 0.99 is a little curvy, although the other escape rates shown look quite linear (on a semilogarithmic scale). While we do not develop a distinct plateau in this case, the deformation in the form of the escape rate is still an indication of very fast poisoning. This should not be a surprise since we expect the island to look trap-like at this gate voltage as we noted above. Thus, the lesson to be learned here is that despite appearances, the data shown in Figure 16.11 masks the fact that the type H device is poisoned. 16.4.1.3â•‡ Single Shot Measurement In a single shot operation (Cottet et al., 2001), we pulse the current bias up from zero to some value Ip for a time τp. The probability of switching to the voltage state is given by

Psw (ng ) = 1 − e

− γ sw ( I p , n g ) τ

,

(16.24)

where γsw(Ip,â•›ng) is the rate at which the CPT phase escapes to the free-running voltage state when instantaneously biased at Ip and ng. If we are biased at ng and wish to determine whether the polarization charge has shifted by δng, we need to figure out the likelihood of seeing a switching event that is actually due to the shift in switching probability versus just the probability of switching with no charge shift at all,

∆Psw (ng , δng ) = e

− γ sw ( I p ,ng )τ

−e

− γ sw ( I p ,ng + δng )τ

.

(16.25)

If we want to measure this change in polarization charge in a single measurement, then we require that ΔPsw = 1. That is, if we pulse the bias current and see the CPT switch to the voltage state,

16-13


1.26 0.99

0

(c)

106

dI = 1 μA/s dt

105

γsw [s–1]

1.46

103

Trap-like

Odd state

Even state

0 dI = 100 μA/s dt

106

102

(b)

ng = 1

200

Barrier-like

104

101

Type L

400

107

γsw [s–1]

# counts (a)

ng = 1.87

50

# counts

Type H

100

Trap-like γsw,e

105 104

Poisoning rate Γeo

γsw,o

103

0

2

4

6

8

102

10

Isw [nA]

(d)

0

2

4

6

8

10

12

Isw [nA]

FIGURE 16.13â•… (See color insert following page 20-14.) (a,c) Switching current (Isw) histograms and (b,d) derived switching/escape rates for a type H (barrier-like) and type L (trap-like) CPTs. For the type L device, the quasiparticle trapping behavior is evident in the bimodal Isw distribution. In this case, the poisoning rate Γeo can be read directly from the derived escape rate in (d) as shown. Although the type H device is barrier-like for most ng, it still looks like a trap near ng = 1 (see Figure 16.7). This is apparent in the “curvy” structure of the escape curve for ng = 0.99 as compared with the escape rates at other ng in (b).

then the polarization charge has shifted by δng with certainty. This defines a minimum charge shift δng,ss or voltage change δVg,ss (= δng,sse/Cg) on the gate that can be detected with one measurement. If we want to detect a smaller change in gate voltage, then we would have to take multiple measurements until the uncertainty in the shift of 〈Isw〉 is satisfactory. In many experiÂ� ments, this is a completely reasonable approach. It is Â�possible, however, to attempt to reduce the width of the switching current distribution by increasing the shunt capacitance across the CPT (increasing the effective mass of our imaginary phase particle) (Joyez, 1995). In the end, pulsed single shot measurements have never been very popular. This may be because their chief use would have been in charge-based qubits, and other significant measurement schemes were shown to outperform it (Vion et al., 2002). However, it might still be a viable measurement method outside of superconducting quantum computing.

16.4.2â•‡ Zero-Biased rf Electrometry The initial aim of the rf inductance measurements was to demonstrate a dissipationless (or near dissipationless) electrometer that could be used in charge-based quantum circuits such as the CPB, but the most useful application in recent years seems to have been to study the dynamics of quasiparticle tunneling. Initial measurements of the Josephson (or “quantum”) inductance of a CPT were performed by Sillanpää et al. (2004) and soon thereafter by Naaman and Aumentado (2006b). rf measurements of CPTs were also performed by Court and coworkers

(Ferguson et al., 2006) as well, but these measurements focused on a modulation of the dissipation and were not strictly confined to the supercurrent branch.* 16.4.2.1â•‡The rf Measurement Setup Figure 16.14a shows a typical microwave circuit used in rf-CPT electrometry. In this scheme, the CPT is embedded in a tank circuit composed of the parasitic capacitance provided by the leads, the Josephson inductance of the CPT, and an extra surface mount inductor soldered to a printed circuit board near the chip to lower the tank resonance to the range of the microwave amplifier.† Following the circuit path in Figure 16.14a, we reflect an incoming microwave signal (the carrier) off of the CPT resonator circuit and measure the amplitude and phase of the outgoing signal, that is, we measure the scattering parameter S11. As shown * It’s important to note that measurement of the Josephson inductance (the second derivative of the ground state energy in phase) is conceptually linked to the “quantum capacitance” (the second derivative in charge). rf measurements of the capacitance have recently been shown to be an extremely useful measurement of charge-based qubits such as the CPB (Wallraff et al., 2004; Duty et al., 2005) and the transmon (Schuster, 2007). † The word “parasitic” should be a tip-off to the reader that this scheme might be a little hit-or-miss and, indeed, it can be frustrating to target the operating frequency within 10% (this is the typical bandwidth for sub-1â•›GHz low-noise cryogenic amplifiers). While this setup is very similar to that used in typical rf-SET operation (Schoelkopf et al., 1998), it is not necessarily trivial to implement.

16-14


rf

Ib

+35 dBm +58 dBm

In

LC Vg

Cs

Network or spectrum analyzer

Cc

Ls

Out

T = 30 mK

(a) 0

–1

–6

.99

0

–8

(b)

–10 450

fc = 515.2 MHz

–2

0.5

S11 [dB]

0 n g=

5

–4

0

0.7

S11 [dB]

–2

T=4 K

–3 –4 –5 –6 –7

500 f [MHz]

515.2

550

(c)

–1.5 –1 –0.5

0 0.5 ng [e]

1

1.5

FIGURE 16.14â•… (a) Simplified rf-CPT measurement schematic. Incoming microwave power is directed toward CPT+ resonator through a directional coupler. The reflected wave is then amplified and measured at room temperature. In this schematic, several bandpass filters and attenuators have been omitted for simplicity. (b) Reflected power versus frequency at various gate voltages. (c) Reflected power modulation as a function of gate voltage for a fixed carrier frequency, fc = 515.2â•›MHz. (Adapted from Naaman, O. and Aumentado, J., Phys. Rev. B, 73, 172504, 2006b.)

earlier, the gate-dependent Josephson inductance can be many 10â•›s of nano-Henries and, since this inductance provides much of the total inductance available to the resonator, the frequency shift can be on the order of the bandwidth (typical Qs ∼ 20–30). While a purely dissipationless circuit will reflect all of the signal (|S11|2 = 1) (Pozar, 2004), there are, in fact, a number of sources of dissipation, including the lossy traces on the pc board and leads on chip, the losses in the surface mount inductor and wire bonds, and, finally, the intrinsic CPT losses from any phase-diffusion resistance present. The end result is that there is a visible resonance dip in the reflected power amplitude |S11|2 (Figure 16.14b,c). For our purposes then, the frequency shift due to the modulation of the CPT Josephson inductance can be inferred from the reflected amplitude modulation.* That is, since the CPT inductance is a function of island charge, the final output microwave power amplitude and phase modulate with the gate-induced polarization charge. Like the rf-SET, one can characterize the ultimate noise performance of this device as an electrometer in terms of an effective charge resolution in a 1â•›Hz bandwidth. In the rf measurement of the CPT inductance, the best number that has been reported is δQ ∼ 50 µe / Hz (Naaman and Aumentado, 2006b). That is, if we integrate the reflected power at the end of our measurement chain for 1â•›s, we can resolve a change in polarization charge at * In principle, this information is also accessible through the phase of the outgoing signal and can be measured using an IQ mixer. Since loss was present in many of these experiments and the frequency shift was usually very obvious, measurement of the amplitude modulation has ended up being the simplest method.

the gate electrode CgVg of 5.2 × 10−5 electrons. If we compare this with the best rf-SSET numbers δQ < 5 µe / Hz (Brenning et al., 2006), the rf-CPT is more than 100 times slower at resolving the polarization charge. At first blush, this seems to put the rf-CPT at a disadvantage, but the rf-SSET charge resolution comes at the expense of using a relatively complex charge transport cycle that involves both quasiparticles and Cooper pairs. The backacting voltage fluctuations of this cycle presented at the device they are measuring are equally nontrivial and have even been used to cool an electromechanical system that it was intended to measure (Naik et al., 2006). While some would consider this a feature of the rf-SSET, it seems to get away from the notion of simply wanting to use the device as a noninvasive electrometer. In fact, quasiparticle poisoning the rf-CPT has been used as a method to measure the effect of nearby SSETs, demonstrating that the latter emits nontrivial microwave power into its environment when voltage-biased (as they would be when operated as electrometers) (Naaman and Aumentado, 2007). 16.4.2.2â•‡ Operation Beyond 1â•›GHz Although all of the published CPT experiments were operated with carrier frequencies below 1â•›GHz, this is not a fundamental requirement. In fact, this limitation seems to be determined by parasitic self-resonances in the surface-mount inductors, often used in these experiments. In principle, the operating frequency can be raised to many gigahertz using coplanar resonator techniques. The advantage of moving to higher frequencies is that wideband, low-noise HEMT amplifiers are now readily available in the 4–8â•›GHz range and all of the associated passive

16-15


components (directional couplers and isolators) are smaller and much more common (less expensive). 16.4.2.3â•‡ Quasiparticle Poisoning in the rf-CPT As in the switching current measurement, quasiparticle poisoning also has a dramatic effect on rf measurement. Unlike the switching current measurement, a qualitative understanding of the measurement response is very straightforward. Since the inductance of the CPT can switch instantaneously between two different parity states when the gate is biased into a trap-like regime, the reflected power also switches between two different amplitudes. This kind of response is more commonly known as “telegraph noise.” A typical example is shown in Figure 16.15a for a type L device. We note that this is the same bimodal behavior that we observe in the switching current measurements except that we can sit at zero bias and watch quasiparticles jumping in and out of the CPT island. In Figure 16.15c, we histogram the telegraph time traces as a function of gate voltage, we get data that, unsurprisingly, are very similar to our bimodal switching current histograms in Figure 16.12.

1.2

The rates Γeo and Γoe can be derived from an analysis of these time-domain traces but requires a careful characterization of the system measurement bandwidth (Naaman and Aumentado, 2006a). Although it is possible to extract the poisoning rate Γeo from a switching current measurement, it is far more difficult to pull out an ejection rate. This is a direct consequence of the time-ordered nature of the current bias ramp. In contrast, the rf measurement is not burdened by this kind of time ordering in any obvious way, and both the poisoning and ejection rates are available. The availability of the poisoning and ejection rates has provided further verification of the model presented in Aumentado et al. (2004), while confirming that nonequilibrium quasiparticles are generated in the leads at a constant rate below ∼250â•›mK. In addition, the ejection rate Γoe has been used to validate the notion that biased SSETs emit nontrivial levels of microwave power into their environment (Naaman and Aumentado, 2007). This is important as for a long time rf-SSET electrometry had been considered a viable method of measuring CPB qubit charge states.

ng = 1

1 Odd

S11

0.8 0.6 0.4

Even

0.2 0

100

200

(a)

300

400

500

600

700

0

20

40

(b)

t [μs]

60

N × 10–3 800 700 600

S11 [a.u.]

500 400 300 200 100

–1.5 (c)

–1

–0.5

0

0.5

1

1.5

0

ng [e]

FIGURE 16.15â•… (a) Reflected power (linear units) of type L CPT at ng = 1 in the time domain. The even (upper) and odd (lower) state levels are evident in telegraph-noise time traces. (b) Histogram of the full time trace. (c) Histograms versus gate voltage. (Adapted from Naaman, O. and Aumentado, J., Phys. Rev. B, 73, 172504, 2006b.)

16-16


16.4.2.4â•‡Relation to the Cooper-Pair Box Earlier, we alluded to the fact that the CPT was related to the CPB. In fact, when the CPT is biased at zero, it is identical to the CPB if the biasing circuit series impedance is small at the relevant parallel-junction plasma frequency. Since the problem of quasiparticle poisoning is also important to charge-based qubits such as the CPB (Nakamura et al., 1999; Wallraff et al., 2004), quantronium (Vion et al., 2002), and transmon (Koch et al., 2007), the fact that we can study it with relatively high instantaneous bandwidth in the CPT motivated several experiments aimed at studying the dynamics of the poisoning process in detail (Aumentado et al., 2004; Ferguson et al., 2006; Naaman and Aumentado, 2006b; Court et al., 2008b). Ultimately, it was realized that quantum capacitance measurements could achieve the same objective measuring CPBs directly and have yielded the most detailed quasiparticle poisoning studies to date (Lutchyn and Glazman, 2007; Shaw et al., 2008).

16.5â•‡Present Status and Future Directions The CPT is the simplest device in which dc transport characteristics can be correlated to the duality between charge and phase. Although the most prominent recent CPT experiments have focused on the problem of nonequilibrium quasiparticle poisoning, the CPT is also a promising general-purpose lowtemperature, low-backaction electrometer. Since the most reliable techniques for fabricating these devices involve aluminum fabrication with gap, Josephson, and Coulomb blockade energies in the ∼100–300â•›μeV range, operation is restricted to dilution refrigeration, so that the systems that one attaches it to must also be cold. While this seems restrictive at present, there are already several mesoscopic condensed matter systems that might benefit from minimally invasive fast electrometry.

Acknowledgment The author wishes to acknowledge several important conceptual discussions with Ofer Naaman, Michel H. Devoret, and John M. Martinis.

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Clarke, J. and Braginski, A. I. (eds.). (2004). The SQUID Handbook: Volume 1: Fundamentals and Technology of SQUIDs and SQUID Systems. Wiley-VCH, Weinheim, Germany. Cord, B., Dames, C., and Bergren, K. K. (2006). Robust shadowmask evaporation via lithographically defined undercut. Journal of Vacuum Science and Technology B, 24(6):3139. Cottet, A., Steinbach, A., Joyez, P., Vion, D., and Pothier, H. (2001). Macroscopic Quantum Coherence and Quantum Computing, p. 111. Kluwer Academic, Plenum Publishers, New York. Court, N. A., Ferguson, A. J., and Clark, R. G. (2008a). Energy gap measurement of nanostructured aluminium thin films for single cooper-pair devices. Superconductor Science and Technology, 21(1):015013. Court, N. A., Ferguson, A. J., Lutchyn, R., and Clark, R. G. (2008b). Quantitative study of quasiparticle traps using the singlecooper-pair transistor. Physical Review B, 77(10):100501. Devoret, M. H. and Grabert, H. (1992). Introduction to single-charge tunneling. Single Charge Tunneling: Coulomb Blockade Phenomena in Nanostructures (NATO Science Series: B), p. 1. Springer, New York. Devoret, M. H., Esteve, D., Martinis, J. M., Cleland, A., and Clarke, J. (1987). Resonant activation of a brownian particle out of a potential well: Microwave-enhanced escape from the zero-voltage state of a Josephson junction. Physical Review B, 36(1):58. Dolan, G. (1977). Offset masks for lift-off photoprocessing. Applied Physics Letters, 31(5):333. Dolata, R., Scherer, H., Zorin, A., and Niemeyer, J. (2002). Single electron transistors with high-quality superconducting niobium islands. Applied Physics Letters, 80(15):2776. Dolata, R., Scherer, H., Zorin, A., and Niemeyer, J. (2005). Singlecharge devices with ultrasmall nb/alo/nb trilayer Josephson junctions. Journal of Applied Physics, 97:054501. Duty, T., Johansson, G., Bladh, K., Gunnarsson, D., Wilson, C., and Delsing, P. (2005). Observation of quantum capacitance in the cooper-pair transistor. Physical Review Letters, 95(20):206807. Ferguson, A. J., Court, N. A., Hudson, F. E., and Clark, R. G. (2006). Microsecond resolution of quasiparticle tunneling in the single-cooper-pair transistor. Physical Review Letters, 97(10):106603. Flees, D., Han, S., and Lukens, J. (1997). Interband transitions and band gap measurements in bloch transistors. Physical Review Letters, 78(25):4817. Fulton, T. and Dunkleberger, L. (1974). Lifetime of the zero-voltage state in Josephson tunnel junctions. Physical Review B, 9(11):4760. Gunnarsson, D., Duty, T., Bladh, K., and Delsing, P. (2004). Tunability of a 2e periodic single cooper pair box. Physical Review B, 70(22):224523. Josephson, B. (1962). Possible new effects in superconductive tunnelling. Physics Letters, 1(7):251. Joyez, P. (1995). Le Transistor a une Paire de Cooper: un Systeme Quantique Macro-scopique. PhD thesis, L’Universite Paris, Paris, France.


Joyez, P., Lafarge, P., Filipe, A., Esteve, D., and Devoret, M. H. (1994). Observation of parity-induced suppression of Josephson tunneling in the superconducting single…. Physical Review Letters, 72(15):2458. Kautz, R. and Martinis, J. (1990). Noise-affected iv curves in small hysteretic Josephson junctions. Physical Review B, 42(16):9903. Koch, J., Yu, T. M., Gambetta, J. M., Houck, A. A., Schuster, D. I., Majer, J., Blais, A., Devoret, M. H., Girvin, S. M., and Schoelkopf, R. J. (2007). Charge insensitive qubit design from optimizing the cooper-pair box. Physical Review A, 74(4):042319. Lutchyn, R. M. and Glazman, L. I. (2007). Kinetics of quasiparticle trapping in a cooper-pair box. Physical Review B, 75(18):184520. Mazin, B. A. (2004). Microwave kinetic inductance detectors. PhD thesis, California Institute of Technology, Pasadena, CA. Naaman, O. and Aumentado, J. (2006a). Poisson transition rates from time-domain measurements with a finite bandwidth. Physical Review Letters, 96(10):100201. Naaman, O. and Aumentado, J. (2006b). Time-domain measurements of quasiparticle tunneling rates in a singlecooper-pair transistor. Physical Review B, 73(17):172504. Naaman, O. and Aumentado, J. (2007). Narrow-band microwave radiation from a biased single-cooper-pair transistor. Physical Review Letters, 98(22):227001. Naik, A., Buu, O., LaHaye, M. D., Armour, A. D., Clerk, A. A., Blencowe, M. P., and Schwab, K. C. (2006). Cooling a nanomechanical resonator with quantum back-action. Nature, 443:193. Nakamura, Y., Pashkin, Y. A., and Tsai, J. S. (1999). Coherent control of macroscopic quantum states in a single-cooper-pair box. Nature, 398:786. Palmer, B. S., Sanchez, C. A., Naik, A., Manheimer, M. A., Schneiderman, J. F., Echternach, P. M., and Wellstood, F. C. (2007). Steady-state thermodynamics of nonequilibrium quasiparticles in a cooper-pair box. Physical Review B, 76(5):054501. Pozar, D. M. (2004). Microwave Engineering, 3rd edn. Wiley, New York. Savin, A. M., Meschke, M., Pekola, J. P., Pashkin, Y. A., Li, T. F., Im, H., and Tsai, J. S. (2007). Parity effect in Al and Nb single electron transistors in a tunable environment. Applied Physics Letters, 91:063512.

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Schoelkopf, R. J., Wahlgren, P., Kozhevnikov, A. A., Delsing, P., and Prober, D. E. (1998). The radio-frequency singleelectron transistor (rf-set): A fast and ultrasensitive electrometer. Science, 280:1238. Schuster, D. I. (2007). Circuit quantum electrodynamics. PhD thesis, Yale University, New Haven, CT. Shaw, M. D., Lutchyn, R. M., Delsing, P., and Echternach, P. M. (2008). Kinetics of nonequilibrium quasiparticle tunneling in superconducting charge qubits. Physical Review B, 78(2):024503. Sillanpää, M., Roschier, L., and Hakonen, P. (2004). Inductive single-electron transistor. Physical Review Letters, 93(6):066805. Tinkham, M. (2004). Introduction to Superconductivity, 2nd edn. Dover, New York. Tinkham, M., Hergenrother, J. M., and Lu, J. G. (1995). Temperature dependence of even-odd electron-number effects in the single-electron transistor. Physical Review B, 51(18):12649. Townsend, P., Taylor, R., and Gregory, S. (1972). Superconducting behavior of thin-films and small particles of aluminum. Physical Review B, 5(1):54. Tuominen, M. T., Hergenrother, J. M., Tighe, T. S., and Tinkham, M. (1992). Experimental evidence for paritybased 2e periodicity in a superconducting single-electron tunneling transistor. Physical Review Letters, 69(13):1997. Tuominen, M. T., Hergenrother, J. M., Tighe, T. S., and Tinkham, M. (1993). Even-odd electron number effects in a small superconducting island: Magnetic-field dependence. Physical Review B, 47(17):11599. Vion, D., Aassime, A., Cottet, A., Joyez, P., Pothier, H., Urbina, C., Esteve, D., and Devoret, M. H. (2002). Manipulating the quantum state of an electrical circuit. Science, 296:886. Wallraff, A., Schuster, D. I., Blais, A., Frunzio, L., Huang, R.-S., Majer, J., Kumar, S., Girvin, S. M., and Schoelkopf, R. J. (2004). Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature, 431:162. Watanabe, M., Nakamura, Y., and Tsai, J. (2004). Circuit with small-capacitance high-quality Nb Josephson junctions. Applied Physics Letters, 84(3):410. Yamamoto, T., Nakamura, Y., Pashkin, Y. A., Astafiev, O., and Tsai, J. S. (2006). Parity effect in superconducting aluminum single electron transistors with spatial gap profile controlled by film thickness. Applied Physics Letters, 88(21):212509.

Nanolithography

III

17 Multispacer Patterning: A Technology for the Nano Eraâ•… Gianfranco Cerofolini, Elisabetta Romano, and Paolo Amato............................................................................................................................................................17-1 Introductionâ•‡ •â•‡ The Crossbar Processâ•‡ •â•‡ Nonlithographic Preparation of Nanowiresâ•‡ •â•‡ Multispacer Patterning Techniquesâ•‡ •â•‡ Influence of Technology on Architectureâ•‡ •â•‡ Fractal Nanotechnologyâ•‡ •â•‡ Appendixesâ•‡ •â•‡ Abstract Technologyâ•‡ •â•‡ Concrete Technologyâ•‡ •â•‡ References

18 Patterning and Ordering with Nanoimprint Lithographyâ•… Zhijun Hu and Alain M. Jonas................................. 18-1

19 Nanoelectronics Lithographyâ•… Stephen Knight, Vivek M. Prabhu, John H. Burnett, James Alexander Liddle, Christopher L. Soles, and Alain C. Diebold................................................................................................................ 19-1

Introductionâ•‡ •â•‡ Nanoimprint Lithographyâ•‡ •â•‡ Nanoshaping Functional Materials by Nanoimprint Lithographyâ•‡ •â•‡ Controlling Ordering and Assembly Processes by Nanoimprint Lithographyâ•‡ •â•‡ Conclusions and Perspectivesâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Photoresist Technologyâ•‡ •â•‡ Deep Ultraviolet Lithographyâ•‡ •â•‡ Electron-Beam Lithographyâ•‡ •â•‡ Nanoimprint Lithographyâ•‡ •â•‡ Metrology for Nanolithographyâ•‡ •â•‡ Abbreviationsâ•‡ •â•‡ References

20 Extreme Ultraviolet Lithographyâ•… Obert R. Wood II............................................................................................. 20-1 Introductionâ•‡ •â•‡ Historical Backgroundâ•‡ •â•‡ Current State of the Artâ•‡ •â•‡ EUVL Infrastructure Statusâ•‡ •â•‡ Future Perspectivesâ•‡ •â•‡ Abbreviationsâ•‡ •â•‡ References

III-1

17 Multispacer Patterning: A Technology for the Nano Era 17.1 Introduction............................................................................................................................ 17-1 17.2 The Crossbar Process............................................................................................................. 17-2 17.3 Nonlithographic Preparation of Nanowires....................................................................... 17-3 Imprint Lithographyâ•‡ •â•‡ Spacer Patterning Technique

17.4 Multispacer Patterning Techniques..................................................................................... 17-5 Additive Route—S nPT+â•‡ •â•‡ Multiplicative Route—SnPT×â•‡ •â•‡ Three-Terminal Molecules

17.5 Influence of Technology on Architecture......................................................................... 17-10 Addressingâ•‡ •â•‡ Comparing Crossbars Prepared with Additive or Multiplicative Routesâ•‡ •â•‡ Applications—Not Only Nanoelectronics

Gianfranco Cerofolini University of Milano-Bicocca

17.6 Fractal Nanotechnology...................................................................................................... 17-12 Fractals in Natureâ•‡ •â•‡ Producing Nanoscale Fractals via S nPT×

Appendixes........................................................................................................................................ 17-14 17.A Abstract Technology............................................................................................................ 17-14

Elisabetta Romano University of Milano-Bicocca

Bodies and Surfacesâ•‡ •â•‡ Conformal Deposition and Isotropic Etchingâ•‡ •â•‡ Directional Processesâ•‡ •â•‡ Selective Processes

Paolo Amato

17.B Concrete Technology........................................................................................................... 17-18 References.......................................................................................................................................... 17-19

Numonyx and University of Milano-Bicacca

17.1â•‡ Introduction The evolution of integrated circuits (ICs) has been dominated by the idea of scaling down its basic constituent—the metal-oxidesemiconductor (MOS) field-effect transistor (FET). In turn, this has required the development of suitable lithographic techniques for its definition on smaller and smaller length scales. There are several generations of lithographic techniques, usually classified according to the technology required for the definition of the wanted features on photo- or electro-sensitive materials (resists): standard photolithography (436â•›nm, Hg g-line; refractive optics), deep ultraviolet (DUV) photolithography (193â•›nm, ArF excimer laser; refractive optics), immersion DUV photolithography (refractive optics), extreme ultraviolet (EUV) photolithography (13.5â•›nm, plasma-light source; reflective optics), and electron beam (EB) lithography (electron wavelength controlled by the energy, typically in the interval 10−3 to 10−2 nm). The industrial system has succeeded in that, but the cost of ownership has in the meanwhile dramatically increased, because of either the required investment per machine, DUV immersion DUV EUV, or the throughput,

EB DUV.

It is just the dramatic cost escalation that is necessary for the reduction of the feature size that casts doubts on the possibility of continuing the current increase of IC density beyond the next 10 years. Entirely new revolutionary technological device platforms, overcoming the complementary MOS (CMOS) paradigm, must likely be developed to enable the economical feasibility and scalability of electronic circuits to the tera scale intergation (TSI). On another side, the preparation of ICs with bit density as high as 1011 cm−2 seems now possible, with modest changes in the current production process and marginal investment for the fabrication facility, within a different paradigm. The new paradigm is based on a structure, where a crossbar embodies in each of its cross-points a functional material able to perform by itself the functions of a memory cell [1]. The crossbar is indeed producible (via nonconventional lithography or even without any lithographic method) with geometry on the 10â•›nm length scale. Although the crossbar is not yet a circuit, it may nonetheless become a circuit if each cross-point contains a memory cell and each of them can be addressed, written, and read—that requires an external circuitry for addressing, power supply, and sensing. The best architecture for satisfying those functions is manifestly achieved embedding the crossbar in a CMOS circuitry [2]: 17-1

17-2


TSI IC = submicro CMOS IC ∪ nano crossbar ∪ nanoscopic cells.

XB1

This architecture reduces the problem of preparing TSI ICs to that of producing nanoscopic memory cells and inserting them into the cross-points of a crossbar structure. It would be a mere declaration of will were it not for the fact that molecules by themselves able to behave as memory cells have been not only designed [3–5] and synthesized [6,7], but also inserted in hybrid devices [8–12]. This fact opens immediately the possibility of a hybrid route to TSI ICs:

Hybrid TSI IC = submicro CMOS IC ∪ nano crossbar ∪ grafted functional molecules.

XB2

XB3

(17.1)

In this approach, the transport properties of programmable molecules are exploited for the preparation of externally accessible circuits. Because of this, it is usually referred to as molecular electronics. The hypothesized TSI IC has thus a hybrid structure, formed by a nanometer-sized kernel (the functionalized crossbar), linked to a conventional submicrometer-sized CMOS control circuitry (producible with currently achievable technologies) and hosting molecular devices (whose production is left to chemistry).

17.2â•‡The Crossbar Process That self-assembled monolayers on preformed gold contacts may behave as nanoscale memory elements was demonstrated for thiol-terminated π-conjugated molecules containing amino or nitro groups [9]. The first demonstration of nonvolatile molecular crossbar memories employed self-assembled monolayers of thiol-terminated rotaxanes as reprogrammable cells [10]. The molecules were embedded between the metal layers forming the crossbar via a process that can be summarized as follows: XB1, deposition and definition of the first-level (“bottom”) wire array XB2 , deposition of the active reconfigurable molecules, working also as vertical spacer separating lower and upper arrays XB3, deposition and definition of the second-level (“top”) wire array Figure 17.1 sketches the XB process. Although potentially revolutionary, the XB approach, with double metal strips, has been found to have serious limits: • The organic active element is incompatible with hightemperature processing, so that the top layer must be deposited in XB3 at room or slightly higher temperature. This need implies a preparation based on physical vapor

Figure 17.1â•… The first proposed crossbar-architecture fabrication steps. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

deposition, where the metallic electrode results from the condensation of metal atoms on the outer surface of the deposited organic films. This process, however, poses severe problems of compatibility, because isolated metal atoms, quite irrespective of their chemical nature, are mobile and decorate the molecule, rather than being held as a film at its outer extremity [13–15]. • A safe determination of the conductance state of bistable molecules requires the application of a voltage V appreciably larger than kBT/e (with kB being the Boltzmann constant, T the absolute temperature, and −e the electron charge), say V = 0.1–0.2â•›V. Applied to molecules with typical length around 3â•›nm, this potential sustains an electric field, of the order of 5 × 105 V cm−1, sufficiently high to produce metal electromigration along the molecules [16]. • The energy barrier for metal-to-molecule electron transfer is controlled by the polarity of the contact, in turn increasing with the electronegativity difference along the bond linking metal and molecule [17]. The use of thiol terminations for the molecule, as implicit in the XB approach, is expected to be responsible for high-energy barriers because of the relatively high electronegativity of sulfur. Even though the first difficulty can in principle be removed by slight sophistication of the process (for instance, as follows: spin-coating the organic monolayer with a dispersion of metal nanoparticles in a volatile solvent, evaporating the solvent, forming a relatively compact layer via coalescence of the metal particles, and compacting the resulting film by means of an additional amount of PVD metal), other difficulties are more fundamental in nature and require different materials. A solution to the electromigration problem can be achieved preparing the bottom electrodes in the form of silicon wires

17-3

Multispacer Patterning: A Technology for the Nano Era

XB*3 , preparation of a top array of poly-silicon wires crossing the first-floor array XB*4 , selective chemical etching of the spacer XB*5 , insertion of the reprogrammable molecules in a way to link upper and lower wires in each cross-point

XB 1

*

XB 2

* PolySi

XB 3

*

SiO2 Field oxide

The basic idea of process XB*, of inserting the functional molecules after the preparation of the crossbar, is sketched in Figure 17.2. Of the three considered processes (XB, XB+, and XB*), the one based on double-silicon strips is certainly the most conservative one and is thus expected to be of easiest integration in IC processing. For this reason (and for the possibility of using three-terminal molecules, see Section 17.4.3), our attention will be concentrated on the XB* route.

17.3â•‡Nonlithographic Preparation of Nanowires

XB 4

*

PolySi XB5

* SiO2 Field oxide

Figure 17.2â•… The basic idea of XB*: preparing the crossbar before its functionalization.

The preparation of a crossbar requires the use of simple geometries—essentially arrays of dielectrically insulated conductive wires. What is especially interesting is that wire arrays with pitch on the nanometer length scale are producible via nonlithographic techniques (NLTs). Not only is this preparation possible, but also the wire linear density already achieved with the NLTs described in the following is smaller than the one achievable via the most advanced EUV or EB lithographies. Such NLTs exploit the following features: (V), the “vertical” control of film thickness, possible down to the subnanometer length scale provided that the film is sufficiently homogeneous. (V-to-H), the transformation of films with “vertical” thickness t into patterns with “horizontal” width w:

(as done in [12]), and the top electrodes in the form conducting π-conjugated polymers (as suggested in [18]): XB1+ , deposition and definition of the bottom array of polysilicon wires XB2+ , deposition of the active (reconfigurable) element, working also as vertical spacer separating lower and upper arrays XB3+ , deposition and definition of the top array of conducting π-conjugated polymers The use of poly-silicon as material for the top array too seems impossible because it is prepared almost uniquely via chemical vapor deposition at incompatible temperatures with organic molecules.* The only way to overcome this difficulty consists thus in a process, XB*, where the two poly-silicon arrays defining the crossbar matrix are prepared before the insertion of the organic element [11]. Preserving a constant separation on the nanometer length scale is possible only via the growth of a sacrificial thin film on the first array before the deposition of the second one [21]: XB*1 , preparation of a bottom array of poly-silicon wires XB*2 , deposition of a sacrificial layer as vertical spacer separating lower and upper arrays * It is instead possible if the spacer is an inorganic film. This case, although not for molecular electronics, is interesting for memories based on phasechange materials [19] or mimicking the memristor [20].

t NLT  → w.

These techniques are imprint lithography and two variants of the multispacer patterning technique.

17.3.1â•‡ Imprint Lithography Imprint lithography (IL) is a contact lithography where properties (V) and (V-to-H) are exploited for the preparation of the mask. The process is essentially based on the sequential alternate deposition of two films, A and B, characterized by the existence of a preferential etching for one (say A) of them. After cutting at 90°, polishing, and controlled etching of A, one eventually gets a mask formed by nanometer-sized trenches running parallel to one another at a distance fixed by the thickness of B [22,23]. For instance, a contact mask for imprint lithography with pitch of 16â•›nm was prepared by growing on a substrate a quantum well via molecular beam epitaxy, cutting the sample perpendicularly to the surface, polishing the newly exposed surface, and etching selectively the different strata of the well [23]. The potentials of the preparation method based on superlattice nanowire pattern transfer (SNAP) are reviewed in Ref. [24].

17-4


Mold Poly-Si-spacer

Nanoimprint lithography SiO2-spacer Resist

(a) 10 nm 35 nm

Poly-silicon

Mold

(a) (b)

The SPT involves the following steps: SPT0, the lithographic definition of a seed with sharp edge and high aspect ratio SPT1, the conformal deposition on this feature of a film of Â�uniform thickness SPT2, the directional etching of the film until the original seed surface is exposed If the process is stopped at this stage, it results in the formation of sidewalls of the original seed; otherwise, if SPT3, the original seed is removed via a selective etching

(b)

(c)

Figure 17.3â•… Preparation of mold for imprint lithography (left) and its use as contact mask (right); the multilayer has been supposed to be produced with cycles of sequential depositions of silicon and SiO2. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

Actually a number of variants for transferring the pattern to the surface have been developed: molding, embossing, and stamping are the ones most frequently considered [25]. In one of them (molding), after filling the trenches with a suitable polymer, the mask is used as a stamp, pressing it onto the surface; if the polymer has a higher affinity for the surface than for the mask, the pattern is transferred to the surface when the mask is eventually removed [25]. The transfer of the polymer to the surface is possible without loss of geometry only if the trench is sufficiently shallow; this implies that the polymer must sustain a subsequent process where it is used as mask for the definition (via directional etching) of the underlying structure with a high aspect ratio. Another method (embossing), sketched in the right-hand side of Figure 17.3, involves the pressure-induced transfer of the pattern from the mask to a plastic film and its subsequent polymerization. Imprint lithography is generally believed to have potential advantages over conventional lithography because it can be carried out in ordinary laboratory with no special equipments [26]. This situation is expected to make it easy to run along the learning curve to a mature technology. However, very little is known about the overall yield, eventually resulting in production cost, of this process (preparation of mask and stamp, imprint, etching) when the geometries are on the length scale of tens of nanometers. Imprint lithography has been the matter of extended investigation (see for instance, Refs. [25,26]) and will not be discussed here. Rather, this chapter is devoted to describe the multispacer patterning technique and to compare its two variants.

17.3.2â•‡ Spacer Patterning Technique The multispacer patterning technique (SnPT) is essentially based on the repetition of the spacer patterning technique (SPT). In turn, the SPT is an age-old technology originally developed for the dielectric insulation of metal electrodes contacting source and drain from the gate of metal-oxide-semiconductor (MOS) transistors.

what remains is constituted only by the walls of the seed edges. Figure 17.4 sketches the various stages of SPT. This technique has been demonstrated to be suitable for the preparation of features with minimum size of 7â•›nm [27,28] and has already achieved a high level of maturity, succeeding in the definition, with yield very close to unity, of nanoscopic bars with high aspect ratio. The SPT may be sophisticated via the deposition of a multilayered film; Figure 17.5 shows the sidewalls resulting from the deposition of a multilayer and compares it with what is really done in the original application of SPT—the insulation of the source and drain electrodes from the gate [29]. SPT0

SPT1

SPT2

SPT3

20 nm

×140,000 100 nm

Figure 17.4â•… Up: The spacer patterning technique: SPT0 , definition of a pattern with sharp edges; SPT1, conformal deposition of a uniform film; SPT2 , directional etching of the deposited film up to the appearance of the original seed; and SPT3, selective etching of the original feature. Down: Cross-section of a wire produced via SPT. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

17-5


S1PT+

S2PT+

S3PT+

S1PT1+

200 nm

Figure 17.5â•… The original application of the SPT in microelectronics—dielectric insulation of the gate from source and drain electrodes. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

Two SnPT routes have been considered: the additive (SnPT+) and multiplicative (SnPT×) routes. The SnPT+ is recent and was proposed having in mind the preparation of crossbars for molecular electronics [30–32]. The S nPT× is instead much older: the first demonstrators were developed for the generation of gratings with sub-lithographic period [33]; recently, however, this technique has been used for the preparation of wire arrays in biochips also [28]. Since no detailed comparison of the limits and relative advantages of these techniques is known, the following part will try to provide an understanding about them on the basis of fundamental considerations.

17.4.1â•‡Additive Route—SnPT+ The S nPT+ is substantially based on n STP repetitions where the original seed is not removed and each free wall of newly grown bars is used as a seed for the subsequent STP. Each SPT+ cycle starts from an assigned seed and proceeds with the following steps: SnPT+1, conformal deposition of a conductive material SnPT+2, directional etching of this material up to the exposure of the original seed SnPT+3, conformal deposition of an insulating material SnPT+4, directional etching of this material up to the exposure of the original seed The basic idea of the S nPT+ is shown in Figure 17.6: the upper part sketches the process; the lower part shows instead how

S1PT3+ S1PT4+

50 nm

17.4â•‡Multispacer Patterning Techniques

S1PT2+

Poly-Si Poly-Si

Oxide

Figure 17.6â•… Up: the additive multispacer patterning technique. Down an example of S3PT+ multispacer (with pitch of 35â•›nm and formed by a double layer poly-Si|SiO2) resulting after three repetitions of the SPT+. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

poly-silicon arrays separated by SiO2 dielectrics with sublithographic pitch (35â•›n m) can indeed be produced [30–32]. The sketch in Figure 17.6 shows a process in which lines are additively generated onto a progressively growing seed, preserving the original lithographic feature along the repetitions of the unit process. The unit process is based on two conformal depositions of uniform layers (poly-silicon and SiO2) each followed by a directional etching. Figure 17.7 shows, however, that a similar structure could be obtained by a cycle formed by SnPT+′ , conformal deposition of a bilayer film (formed by an insulating layer deposited before the conductive one—the order of deposition is fundamental) SnPT+′′, directional etching of this film up to the exposure of the original seed

Consider an array with pitch P of lithographic seeds, each with width W (and thus separated from one another by a distance P−W). Denote with the same symbols in lower case, p and w, the corresponding sub-lithographic quantities. Starting from the said array of lithographic seeds, after n repetitions of SPT+ any seed is surrounded by 2n lines (an example with n = 4 is shown in Figure 17.8), so that the corresponding effective linear density Kn of spacer bars is given by

17-6


Figure 17.7â•… A variant of the additive multispacer patterning technique able to reduce the number of directional etching by a factor of 2 via sequential deposition of a bilayered film. Two possibilities are considered, consisting in the deposition first of either poly-silicon and then of SiO2 (left), or the same layers in reverse order (right). (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

Poly-silicon CVD SiO2

Poly-silicon

Poly-silicon

170 nm

SiO2

Figure 17.8â•… An example of S 4PT+, showing the construction of four silicon bars per side of the seed. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

Kn =

2n . P

The example of Figure 17.8 shows also that the process can be tuned to preserve the constancy with n of wire width wn, wn = w, and pitch pn, pn = p. On the contrary, the spacer heights sn decrease almost linearly with n,

sn s0 − τn

(17.2)

(with τ the spacer height loss per SPT cycle), at least for n lower than a characteristic value nmax.

This unavoidable decrease is ultimately due to the fact that the conformal coverage of a feature with high aspect ratio results necessarily in a rounding off of the edge shape with curvature radius equal the film thickness and that the subsequent directional etching produces a nonplanar surface. Figure 17.9 explains the reasons for the shape of the resulting bars: even assuming a perfect directional attack, the resulting spacer is not flat and there is a loss of height τ not smaller than t: τ ≥ t. Figure 17.6 shows that the process can actually be controlled to have τ coinciding, within error, with its minimum theoretical value, τ = 1.0t. Figure 17.8 shows however that τ depends on the process, and this can be tuned to have τ = 3.2t. Although the loss of height may seem a disadvantage, in Section 17.5.1 it will be shown how the controlled decrease of sn with n may be usefully exploited. Assuming the validity of Equation 17.2 until sn vanishes, the maximum number nmax of SPT+ repetitions is given by nmax = s0/τ; after nmax SPT+ repetitions, the seed is lost and the process cannot continue further. In view of the availability of techniques for the production of deep trenches with very high aspect ratios, in this analysis s0 (and hence nmax) may be regarded as almost a free parameter. The optimum distance allowing the complete filling of the void regions separating the original lithographic seeds is therefore given by P − W = 2nmaxp. Hence, the maximum number of cross-points that can be arranged in any square of side P is given


17-7

Equation 17.3 gives δ max 8 × 1010 cm −2. The comparison of this + prediction with the lithographically achievable cross-point density (currently of about 2 × 109 cm−2), shows that SnPT+ allows the cross-point density to be magnified by a factor of about 40. This is however achieved only with the construction of 10 consecutive spacers per (bottom and top) layer. The spacer technology is a mature technology with yields close to unity also when employed in more complex geometries than single lines. The increase of processing cost implied by its repeated application in IC processing has been discussed in Ref. [34]. Although this increase is moderate, the integration of so many SPT+ cycles may, however, be not trivial and passes through the development of dedicated cluster tools. It is however noted that the basic idea sketched in Figure 17.7 can be extended to remove this difficulty at least partially: the conformal deposition of a slab with n poly-silicon|insulator bilayers (the insulator being SiO2, Al2O3,…) followed by its directional etching would indeed result in the formation of 2n dielectrically insulated poly-silicon wires. Figure 17.10 sketches the process. The figure however shows that this dramatic simplification of the process can be done only for the deposition of the bottom array; its use for the top array would result in a distance of the top wire from the lower one varying with the order of poly-silicon layer.

Figure 17.9â•… Shape of a sidewall resulting after ideal conformal deposition and directional etching (top; five sketches) and an image of how it results in practice (bottom; magnification of the spacers shown in Fig. 6). (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

by (2nmax)2, and the maximum effective cross-point density δ max + achievable with the SnPT+ is given by  2nmax  δ max = +  P 

2

2

 1  1 = 2 . max  p  1 + W / 2n p 

(17.3)

Equation 17.3 shows that δ max depends on the lithography + (through W) and on the sub-lithographic technique SnPT+ (through nmax and p). Just to give an idea of the maximum obtainable density, Figure 17.6 shows that p = 35â•›nm has already been achieved and nmax ≃â•›10 is at the reach of the SnPT+; for W = 0.1â•›μm (characteristic value for IC high volume production) and p = 30â•›nm,

Figure 17.10â•… Structure resulting after the deposition of a slab of six poly-silicon|SiO2 bilayers (top) in one shot followed by the conformal attack stopped with the exposure of the original lithographic seed, resulting in an array of 12 sub-lithographic wires (bottom). (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

17-8


17.4.2â•‡ Multiplicative Route—SnPT× The SPT allows, starting from one seed, the preparation of two spacers [27]; in principle, this fact allows another, multiplicative, growth technique—S nPT×. The multiplicative generation requires that both sides of each newly grown spacer are used as seeds for the subsequent growth—that is possible only if the original seed is etched away at the end of any cycle. In S nPT× each multiplicative SPT× cycle involves therefore the following steps: SnPT×1, conformal deposition of a film on the seed SnPT×2, directional etching of the newly deposited film up to the exposure of the seed SnPT×3, selective etching of the original seed Figure 17.11 sketches two SnPT× repetitions. Assume that the process starts from a seed formed by an array with pitch P of lithographically defined seeds (lines) each of width W, the linear density K0 of lines being thus given by K0 = P −1. If lower and upper arrays have the same linear density K0, the lithographic cross-point density is K 02 and the repetition of n (bottom) plus n (top) SnPT× results in a sub-lithographic cross-point density δ× given by δ × = 2 2n K 0 .

(17.4) P

W

For any assigned n, this density is optimized maximizing K0, i.e., minimizing P. The minimum value of P is determined by the considered lithography, while W is adjusted to the wanted value controlling exposure, etching, etc. The appropriate ratio W/P is obtained with the following considerations. Let the seeds be formed by a given material A (to be concrete we shall think of it as SiO2) and the SPT× be carried out depositing another material B (again for concreteness, we shall think of B as poly-silicon) forming a conformal layer of thickness t1 = 1W, with 1 < 1. After completion of the multiplicative SPT cycle, the surface will thus be covered by an array of 2K0 wires per unit length each of width w1 = t1 = 1W. Let the process proceed with the deposition of a film of A with thickness t2 = 2w1 = 1 2W, with 2 < 1 (in the considered example, this process could be the oxidation of poly-silicon to an SiO2 thickness t2). After completion of the second SPT× cycle, the surface will be covered by a spacer array of linear density 22K0 each of width w2 = t2 = 1 2W. It is noted that in S nPT× the seed material at the end of each SPT× is inverted from A to B or vice versa, so that the material of the original seed must be chosen in the relation to the parity (even or odd) of n. In the following, the focus is on the search of the mask geometry that maximizes the spacer density. After n reiterations of the SPT×, the spacers will extend both beyond and beneath the original lithographic feature. The zone containing the spacers extends from the edge of the original lithographic feature both into the region separating them and into the region beneath the n original feature by amounts lout and linn given by n lout = w1 + w2 + + wn

S2PT× S1PT×

S1PT1×

S1PT2×

w1

S1PT3×

S2PT1×

S2PT2×

w2

=W

k

k =1

j =1

∑ ∏ , j

(17.5)

linn = w2 + + wn =W

n

k

k =2

j =1

∑∏ .

(17.6)

j

n The estimate of lout and linn requires knowledge of various k values; at this stage, it is impossible to state anything about them. Without pretending to describe the actual technology, but simply to have quantitative (although presumably correct at the order of magnitude) estimates, we assume k independent of k, ∀k(k = ). With this assumption Equations 17.5 and 17.6 become

S2PT3× n lout =W

Figure 17.11â•… Two STP× steps for the formation of a sub-lithographic wire array starting from a lithographic seed array. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

n

n

∑

k

k =1

=W

(1 − n ), 1−

(17.7)

17-9


n in

l =W

n

∑

k

k =2

=W

2 (1 − n −1 ). 1−

(17.8)

The least upper bounds of lout and lin are obtained taking the limit for n → +∞ in Equations 17.7 and 17.8: lout = /(1 − ) and lin = 2/(1 − ); moreover, already for relatively low values of n both n and n−1 are negligible with respect to 1 so that we can reasonably assume

n lout W/(1 − ),

(17.9)

linn W 2/(1 − ).

(17.10)

For any W, the optimum is obtained imposing the condition that all the region beneath the original lithographic feature is filled with nonoverlapping spacers: 2lin = W. Inserting this condition into Equation 17.10 gives

1 . 2

(17.11)

Similarly, the optimum size of the outer region is given by the following condition: 2lout = P − W. Inserting this condition into Equation 17.9 gives

P 3W .

(17.12)

Choi et al. [28] have demonstrated that three SPT× repetitions on a lithographically defined seed result in nanowire arrays of device quality and suggest that very long wires can indeed be produced with a high yield. However, even accepting that the process have a yield so high as to allow the preparation of noninterrupted wires over a length (on the centimeter length scale) comparable with the chip size, if the lines are used as conductive wires of the crossbar its length is so high to have a series resistance larger than the resistance of the molecules forming the memory cell. This problem was considered in Ref. [2], where it was shown that the crossbar memory can conveniently be organized in modules each hosting a sub-memory of size 1–4 kbits. This implies that each module must be framed in a region sufficiently large to allow the addressing of the memory cells. In this way the density calculated with Equation 17.4 is an upper value to the exploitable density.

17.4.3â•‡Three-Terminal Molecules The use of molecules in molecular electronics is essentially due to the fact that they embody in themselves the electrical characteristics of existing devices. The characteristics of nonlinear resistors, diodes, and Schmitt triggers have been reported for two-terminal molecules; their use as nonvolatile memory

cells is possible thanks to the stabilization of a metastable state excited by the application of a high voltage (thus behaving as a kind of virtual third terminal). Three-terminal molecules offer more application perspectives not only because they can mimic transistors but also because they could exploit genuine quantum phenomena like the Aharonov–Bohm effect [35]. The application potentials of three-terminal molecules can however be really exploited only if all terminals can be contacted singularly. This is manifestly impossible using the XB or XB+ routes, but is possible in the XB* framework. The major advantage of poly-silicon in the XB* route is that it does not pose the problem of metal electromigration. However, the multispacer technology can also be adapted for the preparation of nanowire arrays of poly-silicon and metals in arrangements that • Allow the use of three-terminal molecules • Facilitate the self-assembly of functional molecules • Avoid the problem of metal electromigration Assume that the top array defining the crossbar is formed by poly-silicon nanowires whereas the bottom array has a more complicate structure. Assume, as sketched in Figure 17.12a, that each conformal deposition is formed by the following multilayer: SiO2 | polysilicon | SiO2 | metal | SiO2 | polysilicon,

where the metal might be, for instance, platinum obtained via CVD from PtF6 precursor. After an SPT, one gets the structure sketched in Figure 17.12b; a subsequent time-controlled selective etch of the metal electrode will result in the formation of a recessed region, as sketched in Figure 17.12c.

(a)

(b)

(c)

(d)

Figure 17.12â•… The structure resulting after (a) conformal deposition of a multilayer, (b) its directional etching, (c) three repetitions of the above processes, and (d) the time-limited preferential attack of the metal. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

17-10


Si

S

S m

n

Si S

Au

Figure 17.13â•… The three-terminal molecule after the formation of a self-assembled monolayer on the metal and the grafting to the silicon. The larger separation between metal and top poly-silicon electrode than between top and bottom poly-silicon electrodes suggests that the metal surface is subjected to an electric stress much lower than that at the bottom silicon surface.

Now, observe that thiol-terminated molecules self-assemble spontaneously on many metals (like platinum, gold, etc.) forming closely packed monolayers. Therefore, if the considered three-terminal molecules contain two alkyne and one thiol terminations, they can arrange in the cross-points in an ordered way, allowing their covalent grafting by simple heat treatment, as sketched in Figure 17.13. This figure also explains why the electric field at the metal–sulfur interface may be significantly lower than at the silicon–carbon interface.

17.5â•‡Influence of Technology on Architecture Device architecture and preparation procedure are strongly interlocked. This will become especially clear considering that crossbars obtained via SnPT+ may be linked to the external world via methods that cannot be extended to SnPT×.

17.5.1â•‡Addressing If the availability of nanofabrication techniques is fundamental in establishing a nanotechnology, not less vital is the integration of the nanostructures with higher-level structures: once the crossbar structure is formed, it is necessary to link it to the conventional silicon circuitry. This is especially difficult because the nanoworld is not directly accessible by means of standard lithographic methods—“the difficulties in communication between the nanoworld and the macroworld represent a central issue in the development of nanotechnology” [36]. The importance of addressing nanoscale elements in arrays goes beyond the area of memories and will be critical to the realization of other integrated nanosystems such as chemical or biological sensors, electrically driven nanophotonics, or even quantum computers.

In the following the attention will however be limited to the problem of addressing cross-points in a nanoscopic crossbar structure by means of externally accessible lithographic contacts. Several strategies have been adopted to attack this problem: many of them involve materials and methods quite far from, if not orthogonal to, those of the planar technology [37–43]. The consistent strategies with the planar technology are discussed in Ref. [2]. Of them, one can be applied to all crossbars irrespective of their preparation methods. According to this strategy, each line defining the crossbar extends beyond the crossing region and in this zone it is used for addressing. This region is then covered with a protecting cap, which is etched away along a narrow (sub-lithographic) line misoriented with respect to the array by a small angle α. In this way the zones where the bars are not covered are separated by a distance that diverges for α → 0; thus, if α is sufficiently small, the separation between the zones no longer protected makes them accessible to conventional lithography and suitable for contacting the CMOS circuitry. In this method, each line is linked separately from the others to the external circuitry—addressing n2 cross-points requires therefore 2n contacts. The multispacer technique (in particular, the SnPT+) permits, however, novel strategies for the nano-to-litho link in addition to the ones suitable for crossbar prepared with other techniques. In the first of such strategies, the original mask defining the seed is shaped with n indentations with size so scaled that (1) the first indentation is filled, with the fusion of the wires, after the first deposition; (2) the second indentation is filled, with the fusion of the wires, after the second deposition; (3)…; and (4) n, the nth indentation is filled, with the fusion of the wires, after the nth deposition. Taking into account that the minimum distance between the centers of two adjacent fused layers is W + 3p (say 150â•›nm) and that each contact requires the definition of a hole in a region with side 2p (say 70â•›nm), this technique allows the nano-tolitho link. Figure 17.14 shows the cross section demonstrating Multi-spacer Nitride

Poly

Field oxide EHT = 5.00 kV Date: 5 Aug 2004 WD = 4 mm

200 nm

Y419182#07 Mag = 101.72 K X

Figure 17.14â•… Cross-section showing the fusion of the arms of the fourth wire grown on two sides of the indentation. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

17-11


how filling indentation with the fusion of the central wires may render them accessible to lithography. Similarly to the strategy considered above, in this method, each line is linked separately from the others to the external circuitry, so that addressing n2 cross-points requires therefore 2n contacts. The S nPT+ technique results in wires with different heights. This fact can be exploited to inhibit or enable them by means of dielectrically insulated lithographically defined electrodes in the geometry described in Ref. [2]. A crossbar with n × n cross-points can therefore be addressed by controlling the conduction along the wires from one Ohmic contact to the other by means of 2 + 2 electrodes. Therefore, for the addressing of n 2 cross-points, this method needs 3 + 3 electrodes only; however, as discussed in Ref. [2], this architecture requires a complex elaboration of the information involving analog-todigital conversion, with subsequent analysis and elaboration of data.

(a)

(b)

17.5.2â•‡Comparing Crossbars Prepared with Additive or Multiplicative Routes While the repetition in additive way of n SPTs per (bottom and top) layers magnifies the lithographically achievable cross-point density K 02 by a factor of (2n)2, the repetition in multiplicative way gives a magnification of 22n. The magnification factor increases quadratically for the additive way and exponentially with the multiplicative way. To estimate numerically the process simplifications offered by S nPT× over SPT+, consider for instance the case of the 3 SPT× repetitions per layer. This would produce a magnification of the lithographic cross-point density by a factor of 23 × 23. Taking W = 0.1â•›μm, after 3 SPT× repetitions, the spacer width should be of 12.5â•›nm, with minimum separation of 25â•›nm. Taking into account Equation 17.12, the cross-point density achievable with the repetition of 2 × 3 SPT× would thus be almost the same as that obtainable with the repetition of 2 × 10 SPT+ (7 × 1010 cm−2 vs. 8 × 1010 cm−2). Figure 17.15 shows in plan view a comparison between the following crossbars: (a) A 2 × 2 crossbar obtained by crossing lithographically defined lines. (b) A 16 × 16 crossbar obtained via S8PT+ starting from lithographically defined seeds separated by a distance allowing the optimal arrangement of the wire arrays. (c) An 16 × 16 crossbar obtained via S3PT× starting from lithographically defined seeds separated by a distance satisfying Equation 17.12. The figure has been drawn in the following hypotheses: • The lithographic lines in (a) and (b) have width at the current limit for large-volume production, say W = 65â•›nm. • The height loss τ is such that the maximum number of repetitions in the additive route is 8, and the sub-lithographic pitch is the same as shown in Figure 17.6.

(c)

Figure 17.15â•… Plan-view comparison of the crossbars obtained (a) crossing lithographically defined lines, (b) using the lithographically defined lines above as seeds for S 8PT+, and (c) using the lithographically defined lines above as seeds for S3PT×. In each structure the square with dashed sides denotes a unit cell suitable for the complete surface tiling. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

• The lithographic width of (c) is chosen to allow the minimum pitch to be consistent with the one obtained with the additive route (W = 100â•›nm); in this way, producing sublithographic wires with width (12.5â•›nm) has been proved to be producible [28]. For n ≥ 3, this comparison is so favorable to SnPT× to suggest its practical application. The following factors, however, would make S nPT+ preferable to SnPT×:

1. If addressing the wires defining the crossbar are used also as addressing lines, they cannot run along the entire plane; their interruption for addressing reduces the available area. 2. In SnPT× all wires are produced collectively and have the same height and material characteristics. On the contrary, in SnPT+ the wires are produced sequentially, each SPT+ repetition produces wires of decreasing height, and the characteristics of the material (or even the materials themselves) may vary in a controlled way from one cycle to another.

17-12


Whereas at the first glance all the SnPT+ features described the second item may seem detrimental, the analysis of Section 17.5.1 has clarified that they may be usefully exploited for cross-point addressing. Deciding which route, between the additive and multiplicative ones, is actually more convenient for the preparation of hybrid devices depends on the particular application and circuit architecture. In fact, the multiplicative route (certainly less demanding for what concerns the preparation of the crossbar but more expensive for what concerns the nano-to-litho link) is presumably suitable for random access memories; on the contrary, the additive route (more demanding for the crossbar preparation, but much heavier for the elaboration of the signal) seems consistent with nonvolatile memories.

17.5.3â•‡Applications—Not Only Nanoelectronics The proposal of the S nPT is quite recent [30,31], so that it has had only short time to be tested. 17.5.3.1â•‡ Electronics Concerning the preparation of crossbars, the attention has mainly been concentrated on one side on the verification of the possibility of scaling the S nPT+ to large values of n (the productions of arrays with n = 3 [30,31], 4 [44], and 6 [45] have been reported) and on the other on the architectural impact of this technology [46–48]. In the middle, a demonstrator has been prepared showing the feasibility, without stressing the technology, of crossbars with cross-point density of 1010 cm−2 [45]. The electrical characterization of silicon nanowires has been the subject of only a few studies: Ref. [24], addressed to the nanonowires produced via the SNAP technique and Ref. [45], devoted to nanowires built with the S nPT+. The latter paper demonstrates that the SnPT+ is already suitable for the preparation of ultrahigh density memory. The interest of a technology for the cheap production of nanowires is not limited to nanoelectronics but might extend to energetics. 17.5.3.2â•‡ Energetics The large-scale availability of devices able to transform lowenthalpy heat (which would otherwise be dispersed into the environment) into electrical energy without complicate mechanical systems might impact the energy problem on a global scale. In principle the Seebeck effect provides a way for that. The ability of a material to operate as a Seebeck generator is contained in a parameter, ZT—a function of the Seebeck coefficient and of the electrical and thermal conductivities. The possibility of practical application is related to the occurrence of ZT 1. From the technical point of view, Seebeck devices are already known, but materials with high Seebeck coefficient (multicomponent nanostructured thermoelectrics, such as Bi2Te3/Sb2Te3 thin-film superlattices, or embedded PbSeTe quantum dot superlattices) are expensive. This fact has allowed the application of

direct thermoelectric generators only to situations (e.g., space) where cost is not important or due to other factors (e.g., weight in space application). This state of affairs would significantly change only in the presence of highly efficient, low-cost materials. A new avenue to progress in such a direction has been the claim, by two independent collaborations [49,50], that silicon nanowires with width of _ 1. Although silicon is 20–30â•›nm and rough surfaces having ZT ~ certainly a cheap material and a lot of technologies are known for its controlled deposition, this result is of potential practical interest only if the production of nanowires does not involve electron beam or extreme ultraviolet lithographies—hence the relevance of SnPT. It is also noted that the researchers of Refs. [49,50] employed wires with comparable height and width, thus characterized with a very high resistance. The SnPT allows the preparation of wires with much higher aspect ratio (same width and much higher height—actually nanosheets). If it were possible to impart a sufficient roughness to the sidewalls of these sheets, it would result in much more efficient Seebeck generators.

17.6â•‡ Fractal Nanotechnology There is an application where the multiplicative route is manifestly superior: the preparation of fractal structures on the sublithographic length scale.

17.6.1â•‡ Fractals in Nature The volume V of any body with regular shape (spheric, cubic,â•›…) varies with its area A as

V = g 3 A3 / 2 ,

(17.13)

with g3 being a coefficient related to the shape (for instance, g3 = (1/6)3/2 for cubes, g 3 = 1/6 π for spheres, etc.; a well-known variational properties of the sphere guarantees that for all bodies g 3 ≤ 1/6 π ). For bodies with regular shape the ratio A/V diverges for small V and vanishes for large V. Life can be preserved against the second law of thermodynamics only in the presence of a production of negative entropy in proportion to the total mass of the organism. This is achieved via the establishment of a diffusion field sustained by the metabolism inside the organism [51]. Since the consumption of energy and neg-entropy of living systems increases in proportion to V whereas the exchange of matter increases A, small unicellular organisms (like bacteria) have no metabolic problem due to their shape and can grow satisfying Equation 17.13 preserving highly symmetric shapes (the smallest living bacterium, the pleuropneumonia like organism, with diameter 0.1–0.2â•›μm, is spherical). On the contrary, larger organisms (even procaryotic, like the amoeba) may survive only adapting their shapes to have an energy uptake coinciding with what is required to preserve living functions [52,53]:


V = g 2 A,

(17.14)

with g2 being a coefficient characteristic of two-dimensional growth. Moreover, the need to adapt itself to the variable environmental conditions is satisfied only thanks to the existence of an inner organellum, the endoplasmatic reticulum [54], which can fuse to the external membrane, thus allowing an efficient change of area at constant volume [52,53]. This mechanism, however, can sustain the metabolism of unicellular organisms (like the amoeba) only for diameter of at most a few tens of micrometers. Larger organisms require the organization of the constituting cells in tissues specialized to single functions. This specialization, however, requires the formation of a vascular network (to transport catabolites to, and anabolites from, each constituting cell) whose space-filling nature implies a fractal character [55]. Nature prefers to manifest itself with fractal shapes in other situations too, like for the mammalian lung (to allow a better O2–CO2 exchange) [56], the dendritic links of the neuron (to allow a high interconnection degree [57]). That smoothness is not a mandatory feature of the way how nature expresses itself, but rather fractality is of ubiquitous occurrence in a large class of phenomena even in the mineral kingdom has become clear with Mandelbrot’s question about the length of the Great Britain coast [58]. The fractality of nature is manifestly approximate, the lowest length scale being ultimately limited by the atomic nature of matter. That surfaces may continue to have a scale invariance down to the atomic size became however clear only after Avnir, Farin, and Pfeifer’s study of the adsorption behavior of porous adsorbents [59–61]. Surprisingly enough, the artificial production of self-similar structures has remained largely unexplored. This is somewhat disappointing because the use of arrays or matrices with selfsimilar structures is potentially interesting even for applications. For instance, a matrix formed by the Cartesian product of two Cantor sets would provide a way for sensing with infinite probes a surface, leaving it almost completely uncovered. Although the minimum length scale is actually larger than the atomic one, that ideal case suggests the usefulness of the idea. The lowest length scale of biological fractals is determined by cell size, say 104 nm. The preparation of self-similar structures on length scales between 102 and 104 nm is relatively easy via photolithography, but this scale seems inadequate for interesting applications like the highly parallel probing of single cells (as required, for instance, by the needs of systems biology [62]). Rather, for that application the appropriate length scale seems the one characteristic of nanotechnology, 1–102 nm.

17-13

defines a fractal. This fractal, referred to as multispacer fractal set, is self-similar only if the height of each spacer varies with n as 2−n, otherwise, the fractal is self-affine [63]. As mentioned above, the “spontaneous” decrease of height with n, Equation 17.2 renders the fractal self-affine. A self-similar fractal can be obtained at the end of process planarizing the whole structure with a resist and sputter-etching in a nonselective way the composite film until the thickness is reduced to s0/2n. It is however noted that even assuming our ability to scale down the fabrication technology, the atomistic structure of matter limits anyway the above considerations to an interval of 1–2 orders of magnitude, ranging from few atomic layers to the lower limits of standard lithography. Having clarified in which limits the set Sn may be considered a fractal, it is interesting to compare it with other fractal sets. The prototype of such sets, and certainly the most interesting from the speculative point of view, is the Cantor middle-excluded set. Figure 17.16 compares sequences of three process steps Â�eventually leading to the multispacer fractal set S and to the Cantor set C. The comparison shows interesting analogies: consider a multiplicative multispacer with P = 2W; if wn = (1/3)wn−1, the measure at each step of multifractal set coincides with that of the Cantor set. This implies that the multispacer fractal set has null measure. Similarly, it can be argued that the multispacer set, considered as a subset of the unit interval, has the same fractal dimension as the Cantor middle-excluded set—ln(2)/ln(3) [63]. At each step, the multispacer fractal set is characterized by a more uniform distribution of single intervals than the Cantor set; this makes the former more interesting for potential applications than the latter. Once one has one-dimensional fractal sets, two-dimensional fractal sets can be constructed taking their Cartesian products. Although the mixed product C × S is possible, in the following, we concentrate on a comparison of the Cantor and multispacer fractal crossbars. Figure 17.17 compares their plan views showing (A) A 16 × 16 crossbar obtained via S3PT× starting from lithographically defined seeds separated by a distance satisfying Equation 17.12 (B) A 16 × 16 crossbar obtained via S3PT× starting from lithographically defined seeds and arranging the process to generate the Cantor middle-excluded set Although the potential applications of the Cantor set are quite far, trying to reproduce it on the nanometer length scale seems of a certain interest. This is compatible with existing technologies; a possible process would involve

17.6.2â•‡ Producing Nanoscale Fractals via SnPT× Imagine for a moment that, in spite of the atomistic nature of matter and of the inherent technological difficulties, the multiplicative route can be repeated indefinitely. Remembering that the (n + 1)th step generates a set Sn+1− , which is, nothing but the one at the nth, Sn , at a lower scale, the sequence {S0 ,…, Sn ,…}

Figure 17.16â•… Generation of the multispacer set (left) and of the Cantor middle-excluded set (right). (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

17-14


Figure 17.18â•… A process for the generation of Cantor’s middleexcluded set. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

Figure 17.17â•… Plan-view comparison of the crossbars obtained via S3PT×, and a corresponding Cantor middle-excluded set; the width of the lithographic seed in S3PT× has been assumed to be at the technology forefront, whereas in the Cantor set it has been assumed sufficiently large to allow three reiterations of the process. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

(C1) the lithographic definition of the seed (formed, for instance, by poly-silicon) generating the Cantor set (C2) its planarization (for instance, via the deposition of a low viscosity glass and its reflow upon heating) (C3) the etching of this film to a thickness controlled by the exposure of the original seed (C4) the selective etching of the original film (C5) the conformal deposition of a film of the same material as the original seed (poly-silicon, in the considered example) and of thickness equal to 1/3 of its width (C6) its directional etching (C7) the selective etching of the space seed (glass, in the considered example) Figure 17.18 sketches the overall process [64]. The preparation of fractal structures may appear at a first sight nothing but a mere exercise of technology stressing. The following examples suggest however the usefulness of a fractal technology:

1. The highly parallel and real-time sensing of single cells (as required, for instance, by the needs of systems biology [62] of by label-free immunodetection [65]) could be done with a minimum of perturbation (i.e., leaving almost completely uncovered the cell surface) by a matrix formed by the Cartesian product of two Cantor sets [63].

2. Superhydrophobic surfaces may be prepared controlling roughness and surface tension of nonwetting surfaces [66]. Whereas surface tension is a material property, roughness can be controlled by the preparation. For instance, roughness may be achieved by imparting a suitable (fractal) relief on the surface. 3. If the SnPT is used for the preparation of crossbar structures for molecular electronics, the functionalization with organic molecules of the cross-points can only be done after the preparation of the hosting structure. According to the analysis of Ref. [44], this requires an accurate control of the rheological and diffusion properties in a medium embedded in a domain of complex geometry. Understanding how such properties change when the size is scaled and clarifying to which extent the domain can indeed be viewed as a fractal (so allowing the analysis on fractals [67] to be used for their description) may be a key point for the actual exploitation of already producible nanometer-sized wire arrays in molecular electronics.

Appendixes 17.Aâ•‡Abstract Technology The processes considered in this chapter (conformal deposition, directional etching, and selective etching) have been introduced without specifying exactly what they are and which materials they involve. Although the reader certainly has a built-in idea of them, we however believe useful to provide their rigorous definition. Once defined in rigorous mathematical terms, the processes of interest for this work are the building blocks of a nanotechnology that may be seen as the equivalent of Euclid’s

17-15


geometry, where the construction with straightedge and compass is replaced by the construction via conformal, directional or selective, deposition or etchings. Of course, the properties of a figure built with any real straightedge and compass are not exactly the same as those of the same figure constructed with the ideal straightedge and compass. Similarly, the use of real materials and processes produces structures that differ from those achievable via the ideal materials and processes. Nonetheless, it is convenient to develop the theory in abstract terms.

17.A.1â•‡ Bodies and Surfaces The minimal characterization of a body is in terms of its geometry and constituting materials. Definition 17.1 (Simple body)â•… A simple body BX is a threedimensional closed connected subset B of the Euclidean space E3 filled with a material X:

BX = (B, X ).

What is a material is considered here a primitive concept. It may be a substance, a mixture, a solution, or an alloy. The emphasis is on homogeneity—whichever region, however small, of B is considered, the constituting material is the same. When not necessary, the index denoting the constituting material will not be specified. In view of the granular nature of matter, considering the limit for vanishing size is not relevant. Here and in the following when dealing with the concept of vanishing length (as it implicitly happens when considering the frontier B* of B) we mean that the property holds true on the ultimate length scale. Although the ultimate length scale is an operative concept related to the probe used for observing the body, we however have in mind the molecular one. Analogously, we consider a body as indefinitely extended when its size is much bigger than the size variations induced by the considered processes. In this sense, a wafer is a simple body indefinitely extended over two dimensions. Definition 17.2 (Composite body)â•… Consider the N-tuple of simple body (BX1 , BX 2 ,…, BXN ) formed by nonoverlapping sets (B1, B2,…, BN), with

∀I , J (Bi ∩ B j = Bi* ∩ B *j ),

Definition 17.3 (Interface)â•… Let a composite body B be formed by simple bodies BX1 and BX 2 with X1 ≠ X2; the region

defines the interface between BX1 and BX 2. For any x in F1|2 all neighborhoods, however small, contain materials X1 and X2. Definition 17.4 (Total surface)â•… The set

N

∪

Bi .

i =1

If B is connected and the simple bodies (BX1 , BX 2 ,…, BX N ) contain at least two different materials X I and XJ, then the N-tuple is a composite body.

Stot = B *

is the total surface of B.

Proposition 17.1â•… Let a composite body B be formed by the N-tuple of simple body (BX1 , BX 2 ,…, BX N ) . Then Stot =

∪ B* \ ∪ F i

i

j /k

.

j,k

Definition 17.5 (Surface)â•… Let B be a composite body, and B• be the smallest simply-connected (i.e., without holes) set containing B. Then S = B•*

is the outer surface (or simply surface) of B. Definition 17.6 (Inner surface)â•… Let B be a composite body. Then Sin = Stot S

is the inner surface of B.

17.A.2â•‡Conformal Deposition and Isotropic Etching Definition 17.7 (Delta coverage)â•… For any body B ← with surface S, the additive delta coverage D+ δ of thickness δ is the set

and let B=

F1 2 = B1* ∩ B2*

{

}

D + δ = x :| x − y | ≤ δ ∧ y ∈ S ∧ x ∉ B•\S .

The delta coverage can be imagined to result from the application of an operator aδ to B:

D + δ = aδ (B).

17-16


The delta coverage D +δ is the “pod” of thickness δ covering the set B. The set obtained as union of B with its delta coverage is referred to a B+δ:

B+ δ = B ∪ aδ (B).

This operation can be reiterated, and the final set obtained after n : n iteration is referred to as B+δ

B+n δ = B+n δ−1 ∪ aδ (B+n δ−1),

where B1+ δ = B+ δ and B+0 δ = B. The reiteration of a δ is the base for the following definition: Definition 17.8 (Conformal coverage)â•… A conformal coverage C+δ of thickness δ is the following limit*: n −1

C+ δ = lim

n →+∞

∪ a ( B ). δ

i + δ /n

i =0

In other words, the conformal coverage process can be thought of as obtained by applying infinitely many delta coverages; one after another, and each one of infinitesimal thickness. As shown in Figure 17.19, in general conformal coverage and delta coverage of the same thickness can lead to different bodies. Moreover from the same figure it is clear that these transformations are not topological invariants. Definition 17.9 (Conformal deposition)â•… A conformal deposition of a film of material Z of thickness δ is the pair (C+δ , Z). Note that the material Z may be different from the materials Xi forming the original body. The following theorem is trivial: Theorem 17.1 (Planarization)â•… Let dS the diameter of B. Irrespective of the shape of B ← the conformal deposition of a layer for which δ >> dS produces a body whose shape becomes progressively closer and closer to the spherical one as δ increases. If B ← is indefinitely extended in two directions, it undergoes a progressive planarization. Isotropic etching can be defined in a way similar to that used for conformal coverage by introducing an operator sδ , characterized by the following definitions:

* Such a limit is naively clear, but its rigorous specification should require to elaborate many technical details that we prefer to skip in this appendix. The same considerations hold true anytime we introduce limits of this kind.

Figure 17.19â•… The different behavior of conformal coverage (left) and delta coverage (right). (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

sδ (B) = aδ (E3\B• ),

( )

B−n δ = B−n −δ1\sδ B−n δ−1 .

In general, the operator s δ is not the inverse of aδ . In fact, sδ (B) = a−δ (B) only if B is a simply connected set. We refer to sδ as delta depletion. Definition 17.10 (Conformal depletion)â•… A conformal depletion E−δ of thickness δ is the following limit: n −1

E − δ = lim

n →+∞

∪ s (B ) . δ

i − δ /n

i =0

Also in this case, it is true that in general the conformal depletion and delta depletion sδ of the same thickness can lead to different bodies (see Figure 17.20). Definition 17.11 (Isotropic etching)â•… An isotropic (nonselective) etching is a conformal depletion of thickness δ applied to a body B ← regardless of its constituting materials. The etching process just defined is nonselective. However the most part of etching are selective (see Section 17.A.4).

17-17

Multispacer Patterning: A Technology for the Nano Era d d

δ1 d d δ2

Figure 17.21â•… Comparison between delta coverage (left) and directional delta coverage along d (right) of the same thickness δ. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

δ3

Figure 17.20â•… The different behavior of conformal depletion (left) and delta depletion (right). (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

17.A.3â•‡ Directional Processes Definition 17.12 (Shadowed surface along d)â•… Let d be a unit vector in E3. For any point s belonging to the surface S consider the straight line ls starting from s and oriented as d. If l s intersects S, then s is said to be shadowed along d. The set all shadowed points of S is referred to as the shadowed surface Sd along d of the body B. Definition 17.13 (Directional delta coverage along d)â•… The directional delta coverage along d is the set of points lying on the straight lines directed along d and going from the exposed surface to the same surface shifted along d by an amount δ:

D + δ , + d = {x : x = y + ad ∧ a ≤ δ ∧ y ∈ S \ Sd }.

Figure 17.21 shows the results of applying to a given set the delta coverage and the directional delta coverage along d. By using the directional delta coverage along δ instead of the delta coverage, it is possible to define the directional deposition. Definition 17.14 (Directional deposition)â•… A directional deposition C+δ,+d of thickness δ along d is the following limit: n −1

C+ δ , + d = lim

n →+∞

∪a (B δ,+ d

i =0

i + δ / n, + d

),

where aδ,+d is the operator associated with the directional delta coverage and B+n δ , + d = B+n δ−,1+ d ∪ aδ , + d (B+n δ−,1+ d ).

Figure 17.22â•… An example of directional deposition. (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

Figure 17.22 shows an example of directional deposition. This process (as shown in the figure) can create holes inside the bodies. Analogously, to define the directional depletion we introduce the directional delta depletion: Definition 17.15 (Directional delta depletion along d)

{

}

D + δ , − d = x : x = y − ad ∧ a ≤ δ ∧ y ∈ S \ Sd .

17-18


Definition 17.16 (Directional etching along d)â•… A directional etching E−δ,−d of thickness δ is the following limit: n −1

E − δ , − d = lim

n →+∞

∪s

δ, −d

(B−i δ /n, − d ),

i =0

where sδ,−d is the operator associated to the directional delta depletion and B−n δ , + d = B−n δ−,1− d\sδ , − d (B−n δ−,1− d ).

17.A.4â•‡ Selective Processes No indication has hitherto been given about the dependence of the same unit process on the material to which it is applied. In general the effect depends on the material, and this difference is referred to in terms of selectivity. Before introducing selectivity, we note that the surface S of a body B ← is, in general, composed by different materials. In particular, by using Proposition 1 it is possible to define Stot,X, the subset of Stot containing material X: Stot,X =

∪ B* \ ∪ F i

j|k

X i =X

,

j,k

Figure 17.23â•… An example of selective etching (“liftoff”). (Reprinted from Cerofolini, G., Nanoscale devices, Springer, Berlin, Germany, 2009. With permission.)

where

and then the (outer) surface SX composed by material X: SX = Stot,X ∩ S.

Definition 17.17 (Selective delta depletion)â•… For any body B ← with surface S, the selective delta depletion s δ,X(B) of thickness δ is the set

{

}

sδ , X (B) = x :| x − y | ≤ δ ∧ y ∈ SX ∧ x ∉ B•\SX .

In other words, the selective delta depletion is a delta depletion that affects SX only, not the whole surface S. We remark that, in general, s δ,X(B) is not a connected set. Definition 17.18 (Selectivity)â•… Any process is said to be selective with respect to materials X when, applied to a composite body B with frontier S, affects SX only. Definition 17.19 (Selective etching)â•… A selective etching E−δ,X of thickness δ is the following limit (in a sense to be defined): n −1

E − δ , X = lim

n →+∞

∪s i =0

δ, X

(B−i δ /n, X ),

  B− δ /n, X = ( B \ sδ , X (B)) ∪  Bj  ,  j  X ≠X

∪

and the iterative process is defined as in the previous cases. As example of selective etching is given in Figure 17.23.

17.Bâ•‡ Concrete Technology Together with the above operations, one could certainly define other, more complicate (or perhaps even formally more interesting), operations. What is of uppermost importance here is that there already exist combinations of materials and processes of the silicon technology mimicking the ideal behaviors described above. The materials considered in this chapter are well known in the silicon technology: a practical model for wafer is any body extending in two directions exceedingly more than the change of thickness resulting from the considered processes and such that only the processes are actually carried out on only one of its major surfaces. Before undergoing any operations, such wafers have generally homogeneous chemical composition and high flatness and are referred to as substrates. In the typical situations considered herein the substrates are slices of single crystalline silicon. The other materials are polycrystalline silicon (poly-Si), SiO2, Si3N4, and various metals (Al, Ti, Pt, Au,…). The typical processes involved in the silicon technology are lithography (for the definition of geometries), wet or

17-19

Multispacer Patterning: A Technology for the Nano Era Table 17.1â•… Shape Resulting after Etching or Growth Processes Shape Process

Conformal

Attack

Wet etching ← Plasma etching

Growth

CVD

Directional Sputter etching Reactive ion etching → PVD

gas-phase etchings, chemical or physical vapor deposition, planarization, doping (typically via ion implantation), and diffusion. A special class of materials is formed by resists, i.e., photoactive materials undergoing polymerization (or depolymerization) under illumination. The processes of interest here are the following: Lithography requires a preliminary planarization with a resist, its patterning (i.e., the definition of a geometry) via the exposure through a mask to light, the selective etching of the exposed (or unexposed) resist, the selective etching first of the region not protected by the patterned resist, and eventually of this material. Wet etchings are usually isotropic and are used for their selectivity: HFaq etches isotropically SiO2 leaving unchanged silicon and Si3N4; H3PO4 etches isotropically Si3N4 leaving unchanged silicon and SiO2; HFaqâ•›+â•›HNO3 aq etches Si leaving unchanged Si3N4 (but has poor selectivity with respect to SiO2). Sputter etching is produced by momentum transfer from a beam to a target and results typically in nonselective directional etching. Selectivity can be imparted exploiting reactive ion etching. Plasma etching can be tuned to the situation: via a suitable choice of the atmosphere it can be used for the isotropic selective etching of Si, SiO2 or Si3N4; it becomes progressively more directional and less selective applying a bias to the body (“target”) with respect to the plasma. Chemical vapor deposition is the typical way for the conformal deposition that occurs when the growth is controlled by reactions occurring at the growing surface. Poly-Si grows well on SiO2 but requires an SiO2 buffer layer for the growth on Si 3N4; conversely Si3N4 is easily deposited on SiO2. Silicon can be conformally covered by extremely uniform layers of SiO2 with controlled thickness on the subnanometer range via thermal oxidation. Physical vapor deposition is the typical way for the deposition of metal films. The corresponding growth mode is the positive counterpart of directional etching. The conformal deposition of a metal layer can only be done via CVD from volatile precursors (like metal carbonyls or metalorganic monomers). Table 17.1 summarizes the shape (conformal or directional) resulting from the above processes.

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18. Akkerman H. B., Blom P. W. M., de Leeuw D. M., de Boer B., Towards molecular electronics with large-area molecular junctions. Nature, 441 (2006) 69–71. 19. Lankhorst M. H. R., Ketelaars B. W. S. M. M., Wolters R. A. M., Low-cost and nanoscale non-volatile memory concept for future silicon chips. Nature Mater. 4 (2005) 347–352. 20. Strukov D. B., Snider G. S., Stewart D. R., Williams R. S., The missing memristor found. Nature, 453 (2008) 80–83. 21. Cerofolini G. F., Romano E., Molecular electronics in silico. Appl. Phys. A, 91 (2008) 181–210. 22. Natelson D., Willett R. L., West K. W., Pfeiffer L. N., Fabrication of extremely narrow metal wires. Appl. Phys. Lett., 77 (2000) 1991–1993. 23. Melosh N. A., Boukai A., Diana F., Gerardot B., Badolato A., Heath J. R., Ultrahigh-density nanowire lattices and circuits. Science, 300 (2003) 112–115. 24. Wang D., Sheriff B. A., McAlpine M., Heath J. R., Development of ultra-high density silicon nanowire arrays for electronics applications. Nano Res., 1 (2008) 9–21. 25. Gates D. B., Xu Q. B., Stewart M., Ryan D., Willson C. G., Whitesides G. M., New approaches to nanofabrication. Chem. Rev., 105 (2005) 1171–1196. 26. Whitesides G. M., Love J. C., The art of building small. Sci. Am. Rep., 17 (2007) 12–21. 27. Choi Y.-K., Zhu J., Grunes J., Bokor J., Somorjai G. A., Fabrication of sub-10-nm silicon nanowire arrays by size reduction lithography. J. Phys. Chem. B, 107 (2003) 3340–3343. 28. Choi Y.-K., Lee J. S., Zhu J., Somorjai G. A., Lee L. P., Bokor J., Sub-lithographic nanofabrication technology for nanocatalysts and DNA chips. J. Vac. Sci. Technol. B, 21 (2003) 2951–2955. 29. Augendre E., Rooyackers R., de Potter de ten Broeck M., Kunnen E., Beckx S., Mannaert G., Vrancken C., et al., Thin L-shaped spacers for CMOS devices. Eur. Solid-State Device Res., 2003. ESSDERC’03, 2003, pp. 219–222. 30. Cerofolini G. F., Arena G., Camalleri M., Galati C., Reina S., Renna L., Mascolo D., Nosik V., Strategies for nanoelectronics. Microelectron. Eng., 81 (2005) 405–419. 31. Cerofolini G. F., Arena G., Camalleri M., Galati C., Reina S., Renna L., Mascolo D., A hybrid approach to nanoelectronics. Nanotechnology, 16 (2005) 1040–1047. 32. Cerofolini G. F., An extension of microelectronic technology to nanoelectronics. Nanotechnol. E-Newslett., 7 (2005) 5–6. 33. Flanders D. C., Efremow N. N., Generation of (1 − σ)h, a continuous residual film always remains between the substrate and the protruding features of the mold, no matter how hard the mold is pressed in the film (Schift and Heyderman Partial confinement

2003). In this case, information on the ordering process may propagate from recess to recess, which will have significant consequences on the global ordering of the sample.

18.4.1â•‡ Ordering under Partial Confinement The first experimental demonstration of molecular alignment by NIL was obtained for chromophores dissolved in a standard PMMA resist, using line gratings as molds (Wang et al. 2000). Polarized photoluminescence and absorption measurements indicated that the dye molecules are aligned after imprint with their long axis along the grating lines, most probably due to the effect of flow during processing. The fluorescence anisotropy, R = (I// − I⊥)/(I// + 2I⊥), quantifies the degree of alignment by comparing the fluorescence emission polarized parallel to the grating lines (I//) and perpendicular to them (I⊥). R values of 1 correspond to complete alignment, whereas R = 0 for isotropic systems. From the reported dichroic ratios DR = I//â•›/I⊥, R values of 0.1–0.2 can be computed for this specific example, which shows that the polymer flow is able to align the dye molecules only moderately. For semicrystalline polymers crystallized in partial confinement, no or limited preferential orientation of polymer crystals occurs when line gratings are used as molds (Hu et al. 2005, Okerberg et al. 2007). Thin films of poly(vinylidene fluoride) (PVDF, α-phase) crystallize in a normal spherulitic morphology and appear to be insensitive to the presence of the grooves of the NIL mold (Hu et al. 2005). The crystalline lamellae propagate from groove to groove, and the electron diffraction patterns do not show preferred orientation relative to the line grating direction (Figure 18.10). This can be ascribed to fast crystal growth in the thin residual film below the protrusions of the mold, from which crystallization propagates into the linear grooves of the mold. The final morphology is thus not dictated by the mold, but mainly by the processes occurring in the residual film. Similar results were obtained for poly(ethylene oxide) (PEO) (Okerberg et al. 2007), although partial elongation of the crystalline lamellae along the grooves of the mold was sometimes observed. As is readily apparent in Figure 18.11a, the global spherulitic shape is not affected by Full confinement

Substrate

Polymer film

NIL mold

Figure 18.9â•… Two configurations of NIL leading to different microstructures after imprint. For clarity, the mold was placed at the bottom of the drawing.

18-10


alignment by the flow is not sufficient to induce oriented crystal growth and that graphoepitaxy of semicrystalline polymers does not occur in partial confinement. The situation is somewhat different when imprinting a low molar mass liquid crystalline conjugated polymer, poly(9,9-dioctylfluorene-co-benzothiadiazole) (F8BT), in its liquid crystalline nematic mesophase (Zheng et al. 2007, Schmid et al. 2008). This green-light-emitting polymer displayed after imprinting R values as large as 0.97, indicating that most polymer chains are aligned parallel to the line grating direction. The order parameter R was found to increase for thinner films, suggesting that the ordering starts from the surface of the mold and progressively vanishes when going down into the residual film. Polymer-light-emitting diodes (PLEDs) were successfully built based on nanoimprinted F8BT, giving rise to the emission of preferentially linearly polarized light (Figure 18.12). A field-effect transistor was also realized, and the hole mobility was 10–15 times larger parallel to the grating lines than perpendicular to them. These results directly illustrate the beneficial impact of aligning molecules, as far as charge mobility or emission polarization is concerned. It should,

30 nm 20 10 0 1 μm

–10

1 μm

Figure 18.10â•… Microstructure of PVDF (α-phase) crystallized in NIL line grating molds under partial confinement (left: transmission electron micrograph; right: atomic force microscopy topograph). No preferential alignment is obtained, as shown by the arrows that highlight the direction of a few lamellar crystals. (Adapted from Hu, Z. et al., Nano Lett., 5, 1738, 2005.)

the NIL mold, and lamellae are found growing either perpendicular to the grooves (zone 1 in Figure 18.11a and b) or parallel to them (zone 2 in Figure 18.11a and c), depending on the orientation of the radius of the spherulite with respect to the line grating direction. These results clearly indicate that chain

2 1

50 μm (a)

250 nm (b)

(c)

Figure 18.11â•… Microstructure of PEO crystallized in NIL grating molds under partial confinement. Panel (a) is an optical micrograph displaying a spherulite; the vertical stripes indicate the direction of the lines of the grating used to imprint the sample. Panels (b) and (c) are AFM topographic images corresponding to the regions 1 and 2 of panel (a), respectively, and show different relative orientations of the lamellae compared with the line grating direction. (Reprinted from Okerberg, B.C. et al., Macromolecules, 40, 2968, 2007. With permission.) 50

(b)

0°

EL intensity (a.u.)

40 45°

30 20

Ca/Al

90°

Aligned F8BT

10

(a)

0 500

PEDOT:PSS ITO/glass 550

600 650 Wavelength (nm)

700

750

(c)

Figure 18.12â•… (a) Electroluminescence of a PLED made of F8BT imprinted in partial confinement with a line grating. The luminescence is measured through a polarizer parallel (circles) or perpendicular (triangles) to the grating lines. (b) Optical images of the electroluminescent device seen through a polarizer oriented with respect to the grating lines as indicated in the figure. (c) Schematic drawing of the PLED structure. (Reprinted from Zheng, Z. et al., Nano Lett., 7, 987, 2007. With permission.)

Patterning and Ordering with Nanoimprint Lithography

however, be noted that NIL failed to align a similar light-emitting conjugated polymer, poly(9,9-dioctylfluorene) (F8) also displaying a liquid crystalline nematic mesophase (Song et al. 2008). This was tentatively ascribed to the higher molar mass (hence larger viscosity) of this polymer compared with F8BT. Clearly, NIL under partial confinement is able to align liquid crystalline polymers in favorable circumstances; however, further work is required to fully understand the factors which control this alignment that are probably similar to the ones driving the alignment of liquid crystals on buffing layers (Toney et al. 1995).

18.4.2â•‡ Ordering under Full Confinement When the starting film thickness is small enough for the NIL mold to enter in contact with the substrate, full confinement arises (Figure 18.9b). In this case, the propagation of information from groove to groove is suppressed, and complete graphoepitaxial alignment happens as is demonstrated below. It is critical to realize that graphoepitaxial alignment occurs at some stage during the formation of a structure, depending on the details of the processing history. It may thus orient a growing crystal, or a liquid crystalline domain, or even a single molecule, depending on the basic structural element that interacts with the mold. The notion of basic structural element is thus key to understand and reconcile the results published in the literature, and will serve us as guide in the sequel. When a semicrystalline polymer crystallizes from the isotropic melt in a NIL line grating mold, the basic structural element to consider is the nascent crystal. Because polymer crystals are usually elongated along their fast growth axis direction, alignment of this crystal axis along the grating lines is favored for geometrical reasons. This is, for instance, the case for PVDF crystallized in complete confinement in its orthorhombic α-form (Hu et al. 2005), whose fast b-axis is found by electron diffraction to align parallel to the grooves of the NIL mold (Figure 18.13a). Interestingly, when NIL molds containing curved grooves are used, the b-axis follows these curved tracks (Hu and Jonas, unpublished). This can only be possible if the resulting curved nanowires are polycrystalline. Note that graphoepitaxy in linear channels only aligns one crystal axis, leaving open the issue of the rotation of the crystal about this axis. It was found for NIL-imprinted α-PVDF that the chain c-axis is perpendicular to the substrate for films below about 100â•›nm (flat-on lamellae), whereas in thicker films, the crystals have their chain c-axis parallel to the substrate (edge-on lamellae) (Hu and Jonas, unpublished). In both cases, however, the b-axis remains parallel to the grooves. The variation from a flat-on to an edge-on orientation upon increasing film thickness is frequent with polymers and was tentatively rationalized recently (Wang et al. 2008). For polymers imprinted in a liquid crystalline phase, then cooled into their crystalline phase, the basic structural element to consider differs depending on the detailed microstructure of the liquid crystalline phase. Poly(9,9-dioctylfluorene) (F8) was imprinted in its nematic liquid crystalline phase, in which chain axes tend to orient parallel to a director with no other form of longrange ordering (Hu et al. 2007). Thus, the basic structural element

18-11

is simply a rod-like chain, which for entropic and geometrical reasons aligns parallel to the grooves. After cooling and crystallization, the preferential ordering is maintained, and the chain axis (which is the c-axis in the low-temperature crystal phase) remains parallel to the grooves, hence to the axis of the resulting nanowire (Figure 18.13b). The preferential alignment translates into polarized light emission, with an R factor of 0.65–0.7 testifying for a high degree of ordering. This is in stark contrast with the absence of preferential ordering reported when F8 was imprinted in partial confinement (Song et al. 2008), and shows the importance of full confinement for NIL-induced graphoepitaxy. The semiconducting poly(3,3′″-didodecyl-quaterthiophene) (PQT) was also imprinted in its liquid crystalline phase under full confinement (Hu et al. 2007). PQT chains form in the liquid crystalline phase supramolecular rod-like nanostructures, in which extended chains pack with their π-stacking direction along the rod axis. The basic structural element is thus the rodlike nanostructure, whose long axis aligns parallel to the grooves of the mold. As a consequence, the π-stacking direction is parallel to the axis of the imprinted nanowire, and this orientation is kept when the polymer crystallizes on cooling. Therefore, the b-axis of the crystal, which is the π-stacking direction, is similarly aligned at the end of the process (Figure 18.13c), whereas the chain c-axis direction is perpendicular to the nanowire axis. The situation is therefore entirely different from F8, where the chain axis was aligned parallel to the nanowire axis. In the present case, this specific crystal setting is interesting for applications, because the PQT b-axis is the axis of high carrier mobility. A nanowirebased field-effect transistor was thus realized by nanoimprint, and it was demonstrated that the hole mobility is increased by a factor of 1.7 along the nanowire axis compared with an isotropic PQT film. Essentially identical results were reported later for nanoimprinted poly (3-hexylthiophene) (P3HT), which adopts a similar morphology as PQT (Aryal et al. 2009). For rigid conjugated amorphous polymers, which cannot crystallize and do not adopt a liquid crystalline phase, the basic structural element to consider during imprinting is simply their Kuhn segment. The flexibility of a polymer is quantified by its persistence length Lp (Grosberg and Khokhlov 1994), which is the average distance over which the local orientation of the chain persists. Real chains can be modeled as equivalent freely jointed chains having rigid (Kuhn) segments of length b K = 2L p connected by freely rotating junctions (Rubinstein and Colby 2003). Conjugated polymers have large persistence lengths, and therefore long Kuhn segments. These will tend to align parallel to the groove direction during imprinting. The resulting nanowire thus contains chains whose axes are parallel to the nanowire axis, which is favorable for conductivity. For polypyrrole (PPy), for instance, the conductivity was improved by a factor of 1.7 in the imprinted nanowires compared with the isotropic film (Hu et al. 2007), and this was correlated to the preferential chain alignment observed by birefringence measurements (Figure 18.13d). As a final example of graphoepitaxial alignment by NIL in full confinement, the supramolecular assembly of a phase-separated

18-12


(020) (020) = b

=

F C F H C H F C F H C H

250 nm (a) (008) = c (008) =

n C8H17

C8H17 500 nm (b) (300)

(200)

(010) = b

(010)

CH3 (CH2)11

= S

S S

S

n

(CH2)11 500 nm

H3C

(c)

Chain axis re s

=

(d)

of sin

gw

xis

ea

id

ng a

cr

na

no

wi

H N

th

N H

n

200 μm

Lo

De

Figure 18.13â•… Selected examples of functional polymers aligned preferentially by NIL in complete confinement. The left column contains transmission electron microscopy images of (a) nanowires of PVDF, (b) electroluminescent F8, (c) semiconducting PQT, and (d) a polarized microscopy image of birefringent arrays of nanowires of conducting PPy (the nanowire axes are at 45° with respect to the polarizer and analyzer directions). Insets in (a)–(c) are the corresponding electron diffraction, showing the crystallographic axis aligned parallel to the line grating direction of the mold. The right column presents schematic drawings of the basic structural elements aligned during imprinting (a nascent crystal for semicrystalline PVDF, rod-like chains of the nematic liquid crystalline phase of F8, a supramolecular rod of π-stacked chains in the liquid crystalline phase of PQT, and chain segments of the amorphous PPy). (Adapted from Hu, Z. et al., Nano Lett., 5, 1738, 2005; Hu, Z. et al., Nano Lett., 7, 3639, 2007.)


18-13

d2 d1

(a) 200 nm

(b)

Figure 18.14â•… SEM images of a (PS-b-PMMA) diblock copolymer imprinted in full confinement in its hexagonal mesophase. The PMMA cylinders were etched away after mold removal, showing the graphoepitaxial alignment of the mesophase. (Reprinted from Li, H.-W. and Huck, W.T.S., Nano Lett., 4, 1633, 2004. With permission.)

block copolymer is presented (Li and Huck 2004). The selected poly(styrene)-block-poly(methyl methacrylate) (PS-b-PMMA) copolymer consists of flexible amorphous blocks, which do not align by themselves. However, due to the microphase separation of the two incompatible blocks, the PMMA regions form nanocylinders located on the nodes of a hexagonal lattice, within a continuous PS matrix (Hamley 1998). Such microphase-separated copolymers are known to form graphoepitaxially oriented supramolecular crystals (Cheng et al. 2006, Bita et al. 2008). They should thus also align in NIL. Here, the proper structural element is the unit cell of the supramolecular crystal, whose unit cell parameters are in the 10â•›nm range and above. The axes of the block copolymer unit cell were indeed found to be parallel to the direction of the grooves of the mold (Figure 18.14), although which axis is parallel to the grooves depends on the thickness of the starting block copolymer film (Li and Huck 2004). However, the regularity of the packing was limited, most probably because the dimensions of the grooves were not exactly adapted to the natural period of the block copolymer (Li and Huck 2004). Nevertheless, this example illustrates nicely how one can combine two lithographic techniques together, namely, NIL and block copolymer lithography, since the PMMA can be etched away with UV radiation after imprinting, giving rise to a second level of patterning.

So far, all examples given above referred to line gratings as molds. Other feature shapes can be selected, but the success or failure of graphoepitaxy will depend on the match between the intrinsic size of the basic structural element and the shape of the nanofeatures of the mold (Givargizov 1994). When square cavities are selected instead of grooves, it is obvious that one cannot expect anymore preferential alignment in the plane of the film. However, this does not preclude ordering effects from appearing. This was reported for a statistical copolymer poly(vinylidene fluoride-stat-trifluoroethylene) [P(VDF-TrFE)], which was imprinted in its paraelectric liquid crystalline phase, then crystallized by cooling into the ferroelectric pseudohexagonal β-phase (Hu et al. 2009). The mold consisted of square nanocavities about 100â•›nm in lateral size, allowing to shape the polymer film into a dense array of nanosquares (Figure 18.15a). As expected, there is no preferential orientation in the plane of the film after imprint; however, the polar b-axis is aligned almost vertically, and the crystalline perfection is improved. This results in a much lower voltage than usually required to switch the direction of the electrical dipole moment. Figure 18.15b is a map of local polarization acquired by piezoresponse force microscopy (PFM), after writing the word “FeRAM” with a positive or negative bias (5â•›V) in the array of P(VDF-TrFE) imprinted nanostructures. Interestingly,

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nm 80 60 40 20 0

1 μm (a)

nA 20 15 10 5 1 μm

0

(b)

Figure 18.15â•… Array of ferroelectric P(VDF-TrFE) nanostructures nanoimprinted in complete confinement. The orientation of the polar b-axis is close to vertical. (a) AFM topography. (b) Piezoresponse amplitude of the same region after local poling with a positive (bright letters) or negative (dark letters) 5â•›V voltage. The piezoresponse amplitude is proportional to the local electric dipole moment, and was measured by PFM. (Adapted from Hu, Z. et al., Nat. Mater., 8, 62, 2009.)

when nanoimprinting is performed with micro- instead of nanometer-sized cavities, no preferential orientation of the b-axis is noted, and the performance of the device is not improved compared with the macroscopic film (Zhang et al. 2007). This underlines the importance of confining crystallization to dimensions comparable to the intrinsic crystal size for optimal results. This example also illustrates that the combination of shaping and crystalline improvement afforded by NIL is extremely useful for device development. Indeed, high-density arrays of ferroelectric polymer crystals exhibiting a low switching voltage are attractive for the development of cheap organic random-access memories. Such systems are considered to be superior to electrically erasable and programmable read-only memories and to Flash memories in terms of write-access time and power consumption (Sheikholeslami and Gulak 2000).

18.5â•‡ Conclusions and Perspectives NIL is by now a well-established lithographic technique, expected to play a significant role in the development of semiconductor technology. Attractive features are the parallelism afforded by the

methodology, its potential low-cost, its high resolution, its versatility toward a large range of materials, the possibility to mix it with photoresist technology, and its compatibility with integration technology. However, the potential of NIL as a method to control the shape and internal order of functional materials is only being realized currently. NIL was shown in this chapter to be able to shape functional materials into gratings, arrays, nanowires, and other interesting nanoobjects for applications in optics or microelectronics. Far from being limited to polymeric materials, NIL is further opening its application range to biomacromolecules, sol–gel precursors of inorganic ceramics and glasses, and even hard crystalline solids such as silicon. When applied to materials capable to self-organize or crystallize, NIL is also capable to orient graphoepitaxially the basic structural elements of the material, provided it be performed under full confinement. In much rarer cases such as for liquid crystals of low molar mass, partial confinement may also lead to preferential orientation. The control over the internal structure of nanomolded materials translates into improved device performance. Thus, it was shown that NILinduced graphoepitaxy results in polarized light emission for electroluminescent polymers, increased mobility of charge carriers in semiconducting polymers, increased conductivity for conjugated conducting polymers, or easier switching of electrical dipole moments for ferroelectric materials. It was also demonstrated in this chapter that the principles governing graphoepitaxial orientation can be rationalized based on the knowledge of the structure of the material and on the notion of basic structural element. This should help device designers to predict the effects of NIL on the microstructure of materials, and therefore to use fully the power of this promising nanoprocessing method.

Acknowledgments The authors acknowledge the generous financial support from the Wallonia Region (Nanolitho, Nanosens, and Nanotic projects), the French Community of Belgium (ARC Nanorg and Dynanomove), the Belgian Federal Science Policy (IUAP SC2 and FS2), the Belgian National Fund for Scientific Research, the Solvay company through the Fondation Louvain, and the European Commission (STREP Metamos, NoE FAME), which permitted part of the research mentioned here to be performed. Access to the WinFab clean rooms is also gratefully acknowledged, as well as stimulating discussions and collaborations with colleagues and students in UCLouvain and elsewhere, too numerous to be all cited here.

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19 Nanoelectronics Lithography Stephen Knight National Institute of Standards and Technology

Vivek M. Prabhu National Institute of Standards and Technology

John H. Burnett National Institute of Standards and Technology

James Alexander Liddle National Institute of Standards and Technology

Christopher L. Soles National Institute of Standards and Technology

Alain C. Diebold University at Albany

19.1 Introduction............................................................................................................................19-1 19.2 Photoresist Technology..........................................................................................................19-3 Fundamentalsâ•‡ •â•‡ Advances by Material Structureâ•‡ •â•‡ Progress in Resists for EUVâ•‡ •â•‡ Progress in Resists for 193â•›nm Immersion Lithographyâ•‡ •â•‡ Concluding Remarks

19.3 Deep Ultraviolet Lithography...............................................................................................19-8 DUV Lithography Steppersâ•‡ •â•‡ 193â•›nm Immersion Lithographyâ•‡ •â•‡ Double Patterningâ•‡ •â•‡ Ultimate Resolution Limits of DUV Lithography

19.4 Electron-Beam Lithography................................................................................................ 19-14 Resolutionâ•‡ •â•‡ Throughputâ•‡ •â•‡ Overlayâ•‡ •â•‡ Cost of Ownership

19.5 Nanoimprint Lithography...................................................................................................19-17 Variations of NILâ•‡ •â•‡ Fundamental Issuesâ•‡ •â•‡ Overlay Accuracy and Controlâ•‡ •â•‡ Technology Examplesâ•‡ •â•‡ Future Perspectives

19.6 Metrology for Nanolithography.........................................................................................19-24 Advanced Lithographic Processesâ•‡ •â•‡ Metrology for Advanced Lithography Processesâ•‡ •â•‡ Proposed New CD Metrology Methods

Abbreviations....................................................................................................................................19-27 References..........................................................................................................................................19-28

19.1â•‡ Introduction The modern integrated circuit (IC), comprising memory, logic processors and analog function devices, are multicomponent and multilevel nanostructures prepared by a series of patterning and pattern-Â�transfer steps. Figure 19.1 shows the cross-sectional hierarchical structure that starts from the smallest feature, the transistor, to dielectrics and metal contacts that are each well defined and must precisely overlay the previous layer. This three-dimensional nanoelectronics structure is manufactured by a rapid patterning process called lithography. Since the invention of the transistor in 1947 by Bell Labs and Intel’s first microprocessor in 1971, modern lithography has enabled the semiconductor industry to shrink device dimensions. The early progress was first quantified by Gordon Moore in 1965 and is now known as Moore’s law.1 While progress in all process steps are required to achieve the continual shrinking of circuit elements, the advancements in lithography have been the overwhelming driving force. Figure 19.2 shows how Moore’s law has guided the industry’s

systematic increase in the number of transistors per chip since the introduction of the IC. Indeed, now and in the future, the industry will achieve productivity improvement primarily by feature reduction. 2 While a wide variety of lithography technologies have been developed, optical step and repeat lithography technologies are the predominant methods of printing features on the semiconductor surfaces and overlying the interconnect structures. A number of competing technologies have been explored, such as x-ray3,4 and flood electron beam (SCALPEL), 5,6 but optical lithography has dominated through vigorous optical technology and tool developments that have left it the lowest cost solution with highest throughput. Direct-write electronbeam technology provides the highest resolution and remains the leading technique for manufacturing the masks used by optical lithography. A novel way of comparing different lithography strategies is by a plot of resolution versus throughput, also known as “Tennant’s Law,” 7,8 as plotted in Figure 19.3. In this plot, direct-write approaches, such as single-atom placement by scanning tunneling microscopy (STM), atomic-force microscopy (AFM) tip-induced oxidation, and electron-beam

19-1

19-2

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Passivation Dielectric Etch stop layer

Wire Via

Dielectric capping layer Copper conductor with barrier/nucleation layer

Global

Intermediate Metal 1

Pre-metal dielectric Tungsten contact plug Metal 1 pitch

FIGURE 19.1â•… Cross section of an integrated circuit, showing the active semiconductor and multilayer interconnect levels. (Reproduced from International Technology Roadmap for Semiconductors, 2007 edition, SEMATECH, Austin, TX, Figure INTC2, 2007. With permission.)

109

106

10

105

103

102

1

Manufacturing technology (nm)

100

107

Clock speed (MHz)

# of transistors

108

104

1000

104 101 1970

1980 1990 2000 Year of introduction

2010

Transistors Manufacturing technology Clock speed

FIGURE 19.2â•… A composite plot of the scaling of the number of transistors, clock speed, and manufacturing technology versus the year of introduction of Intel Processors. (Data from www.intel.com/ technology/timeline.pdf.)

lithography are compared with optical step and repeat lithography that replicate the features of a mask. The continued advancements in optical lithography have pushed the resolution to smaller features at higher pixel throughput, as analyzed by Brunner.9 The fundamental guide for optical step and repeat lithography is the Rayleigh equation, which provides the scaling criteria for predicting the smallest optically definable image2

R = k1

λ λ ; = k1 NA n sin θ

(19.1)

where λ is the source wavelength n is the medium refractive index n sin θ is the numerical aperture (NA) with the incidence angle (θ) k1 is a process dependent factor While the Rayleigh equation defines the resolution, the ability of the photoresist to replicate the mask features is of critical importance. All approaches and manufacturing tools have taken advantage of the Rayleigh equation to predict transitions from each manufacturing technology node (smallest feature size) up to the diffraction limit. Optical lithography started with the near ultraviolet (UV) g-line (436â•›nm) and i-lines (365â•›nm) of mercuryarc lamps and made way for laser sources in the deep ultraviolet (DUV) from KrF (248â•›nm) and ArF (193â•›nm) excimers. A significant extension, immersion lithography, decreases the 193â•›nm wavelength by using water as the immersion fluid and is on track to produce features down to 32â•›nm. The next-generation photoresist materials, described in Section 19.2, for sub 22â•›nm features expect to image 13.4â•›nm radiation in extreme ultraviolet (EUV) with all reflective lithography imaging tools. While most of this chapter focuses on the leading-edge and next-generation technologies used for defining the finest features in nanoelectronics, it is important to recognize that for most applications it is necessary to connect the nano-world (transistor) to the macro-world (a computer motherboard). Figure 19.1 shows the cross section of an IC. Note that the interconnect sizes increase at subsequent higher levels. Therefore, the previous-generation tools continue to play crucial roles by migrating to the higher levels of interconnect.

19-3

Nanoelectronics Lithography Optical step and repeat reduction printing

1000

Shaped and cell projection e-beam

i-Line

Other Gaussian e-beam lithography using high-speed resists

100 Resolution (nm)

g-Line

248 nm 193 nm 193 nm immersion

AFM using oxidation of silicon (single tip)

10 e-Beam lithography using inorganic resists

Gaussian e-beam lithography with PMMA, poly(methyl methacrylate)

1 Best fit (resolution = 2.3 T0.2) STM (low temp. atom manipulation)

0.1 10–6

10–4

10–2

106 100 102 104 Areal throughput (μm 2/h)

108

1010

1012

FIGURE 19.3â•… The progress of optical step and repeat lithography has driven to higher resolution and throughput as described by “Tennant’s Law,” whereby the resolution (R) versus areal throughput (T) scales as R = 2.3â•›T0.2 (Adapted from Tennant, D., Limits of conventional lithography, in Timp, G.M. (ed.), Nanotechnology, AIP Press, New York, Chapter 4, p. 161, 1999. With permission.)

Microelectronics and now nanoelectronics technology have become a collaborative and competitive worldwide effort, as exemplified by the International Technology Roadmap for Semiconductors (ITRS), available to the public at http://www. itrs.net. The ITRS is updated every two years by subject matter experts from the semiconductor manufacturing industry, the tool and materials supplier industries, the factory automation infrastructure, academia, and government agencies. An important guiding aspect of the ITRS roadmap is the identification of the status of the technology nodes and guidance of the phases of research, development, and pre-production. In particular, the 2007 edition identifies potential lithographic solutions out to 2022, with a predicted dynamic random access memory half pitch of 11â•›nm and FLASH memory half pitch of 9â•›nm as shown in Figure 19.4. In this roadmap, several emerging lithography approaches are highlighted. Double patterning with 193â•›nm DUV water immersion is expected to extend to the 32â•›nm half pitch era with high-volume production in 2013. Alternatively, contending technologies for 22â•›nm half pitch are EUV Lithography, 193â•›nm DUV immersion with higher index fluid and lens materials, maskless lithography (ML2), and nanoimprint lithography (NIL).10,11 NIL is rapidly emerging as a low-cost, high-resolution, and versatile alternative to optical lithography. Below 22â•›nm half-pitch, the likely technology solutions are less clear. All the technologies mentioned above, along with new contenders, such as directed self assembly, have credible paths. However, each would have to surmount numerous technical difficulties while limiting exorbitant cost and loss in throughput.

In the rest of this chapter, we describe some of the challenges facing the photoresist materials (Section 19.2) used in optical lithography including some crucial aspects facing these materials as the feature dimensions are reduced to the length scale of the basic photoresist polymers. In Section 19.3, DUV lithography, the basic optics, advancements in steppers, and approaches to extend to higher-resolution, denser features are described. In Section 19.4, electron-beam lithography is covered with respect to the metrics of resolution, throughput, overlay requirements, and cost. Section 19.5 highlights a nonoptical lithography approach, NIL. In this section, several variations of NIL for transferring mask features into a photoresist are described. Finally, in Section 19.6, we end with an overview of the metrology requirements for nanolithography. Figure 19.1 is a reminder that one must print and measure a wide-range of feature dimensions, from the nanoscale to the macroscale. Lastly, a separate chapter in this handbook is dedicated to EUV lithography.

19.2â•‡ Photoresist Technology The optical image projected from a mask upon a thin film at the semiconductor wafer plane is the first step of photolithography.10,12 The thin films that replicate the mask features are called photoresists with basic process as shown in Figure 19.5. The high sensitivity of photoresists to radiation have consistently met the challenges of smaller, high-fidelity features with increased throughput driven by memory and processor chip performance to feature size gains (Moore’s law). The etch resistance of the

19-4


2007 DRAM 1/2 pitch

65

65 nm

2008 2009

2010 45 nm

2011 2012

193 nm 193 nm immersion with water

2013 32 nm

2014 2015

2016

2019

2017 2018 2020 22 nm 16 nm

2021

2022 11 nm

DRAM half-pitch Flash half-pitch

45

193 nm immersion with water 193 nm immersion double patterning

32

193 nm immersion double patteming EUV 193 nm immersion with other fluids and lens materials ML2, imprint

22

EUV Innovative 193 nm immersion ML2, imprint, innovative technology

16

Innovative technology Innovative EUV, ML2, imprint, Directed Self Assembly

Narrow options

Research required

Narrow options

Narrow options

Development underway

Narrow options

Qualification/pre-production

Continuous improvement

This legend indicates the time during which research, development, and qualification/pre-production should be taking place for the solution.

FIGURE 19.4â•… Roadmap for half pitch scaling. (Reproduced from International Technology Roadmap for Semiconductors, 2007 edition, SEMATECH, Austin, TX, Figure LITH5, 2007. With permission.) Resist substrate

Spin coat/bake

Expose

Positive

Negative

Develop

for example, in a 32â•›n m half-pitch 1:1 dense line, a line with a width of 32â•›n m is followed by a space of equal size. The Rayleigh equation defines the resolution; however, the ability of the photoresist to perfectly replicate the mask features is of critical importance. The line-width variations called line-width roughness (LWR) and line-edge roughness (LER) must be reduced to less than 2â•›n m for 32â•›n m half pitch (HP) as they impact device performance.13,14 Therefore, the photoresist materials chemistry plays a substantial role for both resolution and LER. Methods to extend resist resolution by double patterning methods and new photoresist architectures must be leveraged against meeting the LER requirements for sub-22â•›n m lithography.

19.2.1â•‡ Fundamentals Etch

Strip

FIGURE 19.5â•… Schematic of the lithographic process for positive and negative tone resist.

photoresist allows pattern transfer into the underlying semiconductor wafer. Test structures, such as a line and space pattern as shown in Figure 19.6, must meet criteria as defined by the ITRS roadmap for critical dimension (CD). The CD is the feature size,

The requirements for advanced photoresists are discussed in the 2007 ITRS roadmap.14 Specifically, the “Lithography” chapter highlights difficult challenges: Resist materials at 140°C) provide dimensional stability and wide latitude in post-exposure bake temperature that increases the rates of reaction and photoacid diffusivity. Lastly, polymers have a large degree of lipophilicity in an aqueous base developer that provides a high-development contrast.12 Driven by the CD requirements, it was considered that reducing the photoacid diffusion length would enable smaller CD. The PAG and polymer blend approach may not be the most effective route, since the photoacid could diffuse to lengths longer than the CD. In order to address this viewpoint, an alternative resist structure was devised that covalently bonds the photoacid generator to the polymer as shown in Figure 19.8a.51–54 With this approach, after exposure, the photoacid counter-anion remains covalently bound to the polymer thereby restricting the acidic

PG

PAG bound to molecular resist

PAG

PAG

Polymer bound PAG

PG

(c)

(a)

PG

Ring-like Branched Disk-like

Protecting group

PG

(b)

PG

Core

PG Molecular resist with PAG core

Core

PG

PG

PAG

(d)

PG

PG

FIGURE 19.8â•… Cartoons of photoresist architecture alternative to polymer and PAG blends: (a) Photoresist polymer bound to the PAG, (b) molecular glass resist with variable core structure, (c) molecular glass resist bound to the PAG, and (d) PAG-core molecular resist.

19-8


proton diffusion length. Such approaches remain promising especially to increase the dose sensitivity. Since the CD and LER metrics are approaching the characteristic dimensions of the photoresist polymers, alternative architectures were considered to extend photolithography by using lower molar mass molecules.55 These molecular glass (MG) resists, while smaller, may also improve the uniformity of blends with PAG and other additives since miscibility of polymer blends decreases with increasing molar mass. 56 In general, the molecular glass resist has a well-defined small-molecule core that bears protected base-soluble groups (such as hydroxyls and carboxyls) as shown in Figure 19.8b. With this approach, the core chemistry can vary from calix[4]resorcinarenes (ringlike),57–59 branched phenolic groups,60,61 and hexaphenolic groups (disk-like).62,63 Early approaches with MG led to low glass transition temperatures, however, such problems were resolved by increased hydrogen bonding functionality and the design of the core structure. The MG resists may also benefit from a more uniform development due to the lack of chain entanglements and reduced swelling, when compared with polymers; these are active areas of research. Experimental data using the quartz crystal microbalance method demonstrate that swelling appears during development even with molecular glass resists.64 Most of these alternative resist structures adhere to the chemical amplification strategy. However, nonchemically amplified photoresists are also being considered as they do not contain photoacid generators and hence do not suffer from photoacid diffusion length constraints.65 Two other novel MG variants are a PAG covalently bound to the molecular resist (Figure 19.8c) and the core of the Â�molecule serving as the PAG66 (Figure 19.8d). As designed, there would be no need for the blending of PAG with such resist systems. In the case of Figure 19.8d, photolysis produces a photoacid, which then would deprotect the unexposed acid-sensitive Â�protecting groups of the PAG-core molecular glass. These two approaches (Figure 19.8c and d) are smaller pixel sizes and true one-Â�component systems that in principle eliminate surface segregation and phase separation in cast films. These alternate resist structures, however, typically require additives such as amine base quenchers to limit the diffusion of the photoacid catalyst into unexposed regions.

19.2.3â•‡ Progress in Resists for EUV There has been progress in designing photoresists for EUV lithography in anticipation of the 22â•›nm nodes. The testing and development of new materials typically relies on direct lithographic testing to screen formulations. However, due to the lack of widely available EUV exposure tools, micro-field exposure tools (MET), such as the 0.3â•›NA SEMATECH Berkeley MET have an important role for resist testing. Substantial progress 67 in reaching CD challenges with commercial chemically amplified photoresist were reported with 20â•›nm half-pitch with an EUV dose sensitivity of (12.7–15.2) mJ/cm 2, which is near the theoretical Rayleigh resolution limit of the 0.3-NA system with

a k1 of 0.45. While progress in resolution and sensitivity were observed, LER remains a challenge. In fact, the fidelity of the resist feature may have non-negligible LER contributions from the EUV mask and optical trains as ascertained by modeling. After subtracting an estimated mask contribution, resist LER values appear to approach the 2â•›nm level. Additionally, a pattern collapse of sub-20â•›nm dense features suggests the intrinsic resolution could be better than expected. In fact, unoptimized model EUV polymers and MG photoresists clearly show 20â•›nm features as quantified by the latent image and developed image roughness.68 Alternative development approaches 69 may provide new directions to break the resolution limits (Equation 19.2), which currently do not directly consider development effects, such as swelling and swelling layer collapse.

19.2.4â•‡Progress in Resists for 193â•›nm Immersion Lithography By inspection, the Rayleigh equation (Equation 19.1) directs that the smallest feature may be achieved by employing higher refractive index media (photoresist and immersion fluids). Currently, highly purified water with n = 1.44 is used as the immersion fluid and typical 193â•›nm polymers have n ≈ 1.7. The development of photoresists and immersion fluids with higher n are needed. A strategy to increase the refractive index is by incorporating more polarizable heteroatoms, such as sulfur,70–73 into the polymer structure. Halides such as Cl, Br, and I would increase the refractive index at the expense of absorption and possible photo-induced side reactions. Developer-soluble topcoat barriers74–77 or engineered surface-segregating barrierlayer additives78–83 are typically used to reduce or mitigate the leaching of critical components (PAG and quenchers) into the immersion fluid.

19.2.5â•‡ Concluding Remarks Sub-32â•›nm critical dimensions come with many resist challenges. The challenges of reducing feature size and decreasing LER and sensitivity are inter-related. Current attempts to surpass resolution challenges include novel resist process strategies such as double patterning and double exposure as well as novel architectures. However, it is clear that the quality of the final patterned structures is inter-dependent upon the spatial distribution of the photoacid (optical image quality), the spatial extent of the reaction-diffusion process (latent image), and the development mechanisms. The guidance of the resist resolution metrics in cooperation with improved materials fundamentals will help photoresist materials achieve ever smaller features as suggested by the Rayleigh equation.

19.3â•‡ Deep Ultraviolet Lithography Deep ultraviolet (DUV) lithography, using KrF or ArF excimer lasers with illumination wavelengths of 248 or 193â•›nm, respectively, is now the predominant technology for producing critical

19-9

Nanoelectronics Lithography Excimer laser illumination

Patterned photomask

NA = nfluid sin θ

nfluid (a)

θ

Optical system 4 :1 reduction ratio Resist layer Si wafer

(b)

FIGURE 19.9â•… (a) Schematic of the DUV photolithography exposure process. The numerical aperture is defined by NA = nfluid sin θ, where θ is the half angle of the marginal rays and nfluid is the index of the fluid (gas or liquid) between the final lens element and the resist. (b) Illustration of a commercial 193â•›nm immersion lithography system (ASML TWINSCAN XT:1950i), showing the cylindrical optics barrel in the center, the mask above the optics system, and an illuminated wafer below. (Courtesy of ASML, Veldhoven, the Netherlands. This illustration is based on an artist impression. No rights can be derived from it.)

layers of leading-edge, mass-market semiconductor logic and memory ICs. The technology is based on illuminating a patterned fused silica mask with diffused excimer laser radiation and imaging the transmitted light patterns with 4:1 image reduction onto a photosensitive-resist-coated Si wafer, using a diffraction-limited optical system. The process is indicated schematically in Figure 19.9a. The illuminated wafer is removed from the optical system for chemical processing steps to convert the exposed pattern (latent image) in the resist to a patterned structure on the wafer and put back into the optical system multiple times for further Â�exposure/Â�processing steps. State-of-the-art ICs are built up from about 20–30 layers, a significant fraction of which involve DUV exposure steps. An illustration of a leadingedge 193â•›nm immersion lithography exposure tool for volume production is shown in Figure 19.9b. DUV lithography is an evolution of optical-projection lithography developed in the late 1960s and the 1970s using the strong spectral lines from mercury discharge lamps to illuminate first at 436â•›nm (g-line) and later at 365â•›nm (i-line). In these early systems, the pattern on the mask for the entire circuit was illuminated and imaged onto the wafer for a specified exposure time, and then the wafer was “stepped” to an adjacent unexposed position for another imaging of the circuit pattern. This was repeated until the wafer was filled with as many identical exposures as the wafer size allowed. These wafer “step-and-repeat” systems were termed “steppers.” In modern leading-edge DUV lithography systems, the full circuit illumination of the mask is replaced by a “scanned” illumination. A fraction of the circuit pattern on the mask is illuminated in a narrow strip across its width and the mask is scanned under this strip, while its image is projected with a 4:1 reduction ratio onto a wafer scanning in the opposite direction, at a speed relative to the mask reduced by the factor of 4. When the full circuit pattern has been exposed, the wafer is stepped to the next position and the process repeated. The exposure image is

stepped-and-scanned across the wafer, producing typically over 50 full chip exposures on a 300â•›mm diameter wafer at a rate giving about 100 wafers per hour. These wafer “step-and-scan” systems are referred to as “scanners,” though they are often referred to as “steppers” as well. The first commercially available production optical-projection lithography system, the DSW4800, was introduced by GCA in 1978. It achieved a minimum feature resolution of a little over 1â•›μm and a slightly larger depth of focus.10 At the time, it was fully understood that the resolution of the optical-projection lithography approach was Â�fundamentally restricted by the diffraction limit given roughly by the Rayleigh Â�resolution criterion in Equation 19.1. In DUV lithography, the Â�process-dependent factor k1 is of order 1 Ref. [10,84]. Actually, Equation 19.1 is intended to capture both the limiting effects of diffraction and the impact of the resist processing. Considering just the diffraction effects, Equation 19.1 would refer to the aerial image at the image plane, and k1 would be determined only by the profile of the light intensity. However, the actual size of a feature ultimately created by the lithography process also depends on the chemical processing of the exposed resist as described in Section 19.2. For an isolated feature, this size can be any value, i.e., k1 has no fundamental restrictions. On the other hand, for the dense structures of real circuits, e.g., modeled by a periodic structure, the separation between printed features (the pitch) is fundamentally limited. With the minimum feature size in Equation 19.1 defined as half the pitch for a single exposure, k1 can be rigorously shown to have a minimum value of k1 = 0.25 Ref. [10,84]. This limit holds for any single-exposure process, as long as the process has a linear dose response. Because a functional circuit must have some topology and because imaging control is imperfect, the resist exposure process requires some finite depth of focus (DOF). This also has a diffraction and geometric limitation, which is characterized by Equation 19.3

19-10


λ nλ Small  → k2 2 NA 2   NA NA 2n  1 − 1 − 2  n  

λ = k2 , nsin n2 θ

Borosilicate glass + fused silica glass Critical feature size (half pitch)

(19.3)

where, for small NAs, a process-dependent prefactor k2 is conventionally used.10,84 For large NAs, the exact form of the equation conventionally incorporates a different process-dependent prefactor k3 × 4.85 In the early 1980s, reasonable considerations about the prospect of significantly altering the parameters in Equations 19.1 and 19.3 to improve the resolution and DOF led to the conclusion that optical-projection lithography had an ultimate dense feature size resolution of about 0.5â•›μm. Consequently, it was assumed then that progress in high-volume-production ICs would require switching within a decade or so to alternative lithography techniques with more extendibility potential. These included electron-beam direct-write lithography (with multiple parallel beams), electron-projection lithography, ionprojection lithography, x-ray-proximity lithography, extremeultraviolet lithography, and nanoimprint lithography.10 Few people, if anyone, publicly predicted then that the factors in Equations 19.1 and 19.3 would be relentlessly driven to enable optical-projection lithography to out-complete all the alternative technologies mentioned above, in cost and performance capabilities for volume production, at least into the second decade of this century.

19.3.1â•‡ DUV Lithography Steppers A primary driver for this progress has been the wavelength factor in Equation 19.1. This can be seen in Figure 19.10, which shows a log-linear plot of critical feature sizes of leading-edge ICs as a function of year introduced into large-scale production, along with the illumination wavelength used. Wavelength reduction has been aggressively pursued in part because, as is clear from Equations 19.1 and 19.3, feature size reduction by decreasing wavelength reduces DOF less than that by increasing the NA. However, the switch from Hg-arc lamp illuminators with wavelengths at 436 and 365â•›nm to the DUV excimer laser sources with wavelengths at 248â•›nm (KrF) around 1995 and 193â•›nm (ArF) around 2000 was a very challenging one. Hg-arc lamp sources are compact, relatively inexpensive, reliable, and operate continuously. KrF and ArF excimer lasers are much more complex, are substantially more expensive to purchase and operate reliably, have undesirable laser coherence properties, and the illumination is pulsed. This last characteristic has been particularly troublesome because obtaining the required exposure doses fast enough for acceptable wafer throughputs requires very high pulse peak intensities. This puts very stringent requirements on the durability of lens and window materials in the optical system. Further, the shorter UV wavelengths

UV fused silica glass

1000 436 nm

CaF2 crystal

365 nm

Lithography λ

nm

Depth of focus =

248 nm

100

193 nm 157 nm 135 nm Effective λ

Theoretical minimum half pitch 0.25λ

10

1980

1985

1990 1995 2000 2005 Year of volume production

2010

FIGURE 19.10â•… Plot of critical feature size (half pitch) of leading-edge semiconductor integrated circuits versus year of volume production, on a log-linear plot. Also included is the lithography wavelength used and the theoretical minimum half pitch for single exposures (0.25λ) for this wavelength. 157â•›nm technology has not been brought into production. Effective λ is λ/n for 193â•›nm immersion lithography. Optical materials used for lenses at each λ are also indicated.

substantially limit material options for these optical elements. In particular, at the time of the introduction of 248â•›nm lithography, UV-fused silica glass was the only UV-transmitting material that could meet the tight optical properties specifications for the large lenses required. For 193â•›nm systems, in addition to UV-fused silica, a second high-quality material, crystalline calcium fluoride (CaF2), had to be developed for lenses to correct for chromatic aberrations in the optical system due to the wavelength dispersion of the index of fused silica (see Figure 19.10). Fused silica glass is an amorphous form of SiO2, which minimizes the large birefringence effects of the uniaxial crystal structure of crystalline SiO2 (quartz). However, the amorphous structure makes fused silica a thermodynamically metastable material and it suffers structural changes under exposure to high 193â•›nm laser intensities. These changes result in both volume compaction and rarefaction effects with different intensity dependencies.86 The resulting refractive index and lens geometry changes can substantially degrade lens performance. To minimize these effects, maximum intensities have to be limited throughout the stepper optics, and the more durable CaF2 lens elements generally must be used in the positions with highest intensities, for example, at the final lens position. For wavelengths much below 193â•›nm, UV-fused silica glass has poor transmission, leaving only the cubic structure Group II fluorides as practical lens materials. Of these, only CaF2 has been made with lithography-grade optical quality. Hence, wavelength extension to 157â•›nm with F2 excimer lasers would require the lenses to be made only from CaF2, or possibly from other related fluorides, such as BaF2, if the optical quality could be

19-11

Nanoelectronics Lithography

significantly improved.87 The availability of lithography-quality CaF2 for lens material has been a key issue for the development of both 193 and 157â•›nm lithography technologies, and building a CaF2 manufacturing infrastructure to support the needs of both of these technologies has been a major challenge. A principal difficulty with CaF2 production is that its relatively low thermal conductivity and high thermal expansion coefficient requires it to be cooled very slowly from the melt in order to obtain low-strain, single-crystal material of the sufficiently large sizes needed for lenses. Even many weeks of controlled cooling in elaborate Â�temperature-controlled furnaces gives a low yield of lens blanks meeting specifications. One result is that 193â•›nm stepper systems have been designed to use the minimum number of CaF2 lens elements possible, though CaF2 has not been designed out entirely. A further unexpected complication from the use of CaF2 lens elements in 193 and 157â•›nm lithography systems resulted from a faulty assumption that the cubic crystalline structure of CaF2 would ensure isotropic- and polarization-independent optical properties (for high-quality crystals), as could be “demonstrated” by naive symmetry arguments and measurements at longer wavelengths. In fact, CaF2 turned out to have substantial index anisotropy and birefringence at the short wavelengths of 193 and 157â•›nm.88 This is due to the symmetry-breaking effects of the finite photon momentum at these wavelengths, giving rise to a “spatial-dispersion-induced” or “intrinsic” birefringence. 89 Fortunately, the symmetric nature of this effect has enabled it to be minimized by judiciously orienting the crystal axes of several lens elements to substantially cancel the effects. Beyond 157â•›nm, a few shorter wavelengths have been considered for further resolution extension: 126â•›nm from Ar2 excimer lasers90 and 121.6â•›nm from hydrogen Lyman-α discharge sources.91 At these wavelengths, the only transmissive optical materials known are LiF, which has high extrinsic absorption and poor exposure durability, and MgF2, which has high natural birefringence due to its noncubic crystal structure. These shorter wavelength technologies have not been developed beyond feasibility studies.90,91 Below about 100â•›nm, no practical material is transparent and refractive optics are not possible. For reasons primarily associated with the development of immersion lithography discussed later, 157â•›nm lithography, though demonstrated to be technically feasible,87 has been dropped off of technology roadmaps.14 It now appears likely that the shortest wavelength that will be used for production lithography with refractive optics is 193â•›nm. As of 2008, 193â•›nm steppers are the primary tools for producing critical layers of high-volume, leading-edge ICs. Their ArF excimer laser sources are pulsed (≈6â•›k Hz, 10â•›mJ/pulse) and line narrowed (≤0.25 pm spectral bandwidth).92 The transmission photomasks are made from fused silica with Cr absorbing features and have thin-membrane pellicles to keep uncontrolled particles from collecting on the mask and imaging onto the wafer. The stepper optics are made of fused silica spherical and aspheric lenses, calcium fluoride lenses with clocked crystal axes, mirror elements with aspheric surfaces, and immersion fluids between the last lens element and the wafer, as discussed below.

While the wavelength was progressing down to 193â•›nm, the other two factors in Equation 19.1, k1 and NA, were also being pushed towards their limits. A number of resolution enhancement techniques have been used to drive the k1 factor down to achieve IC structures with half pitches corresponding to a value of k1 approaching the limiting value of k1 = 0.25.93 Nearly all of these have been taken over from established optical techniques in other fields such as microscopy. These have included (1) illumination methods (off-axis illumination and partial-coherence control), (2) mask modifications to engineer desired wavefronts (phase shift masks, sub-resolution mask structures, and other optical proximity corrections), and (3) resist contrast improvements. For the 45â•›nm technology node, logic manufacturers are expected to operate with k1 factors down to about 0.31,94 and further extension with k1 factors down to 0.29 is considered feasible.95 Unfortunately, accompanying these gains, process latitudes have been shrinking to marginally tolerable levels. Clearly, k1 factor improvements for single exposures are nearly tapped out. The last factor left in Equation 19.1 is the numerical aperture, NA = n sin θ. For air between the final lens element and the wafer, the liming value is NA = 1. The GCA 4800 DSW stepper in 1978 had an NA of 0.28.96 The NA has been increased steadily in newer designs by increasing the size and complexity, and consequently the cost of the lens systems. To contain the number and size of lens elements as the NA was increased, the lens system designs had to incorporate aspheric lenses and off-axis mirror elements (catadioptric designs). Further, as the NAs approached 0.9, the polarization effects could no longer be ignored and illumination polarization control became an essential part of the lithography process, further complicating the lithography tools and processes.84,97 State-of-the-art dry 193â•›nm stepper optics have an NA of about 0.93, contain about 30 lenses, with a path length through lens material of about 1â•›m, and weigh 500â•›kg or more.10 With dry NA limit of 1.0, there is very little possible gain left to justify the vastly increased cost and complexity needed for improvement—an exponential increase in cost for an asymptotic gain. The sinâ•›θ factor is now also essentially tapped out. Figure 19.10 shows, along with the critical feature size (half pitch), the limiting value of 0.25λ for each wavelength. Clearly, by 2006, the critical feature size 65â•›nm had approached its limit for 193 or 157â•›nm. In the 1990s, this projection was the reason why it was nearly universally assumed that alternative technologies, such as EUV lithography, would have to take over below this feature size. This has turned out to be incorrect primarily for two reasons: the NA is not limited to 1 if immersion fluids are included and the effective k1 factor can be decreased below 0.25 with multiple exposures and processing on the same layer.

19.3.2â•‡ 193â•›nm Immersion Lithography Equation 19.1 is valid at the image plane, which of course must be in the resist. In principle, the NA = n sin θ should be evaluated in the resist, which typically has a 193â•›nm index in the range

19-12


Lens

nlens

NA = nlens sin θ1

Fluid

nfluid

= nfluid sin θf

nresist

= nresist sin θr

θr

Resist

FIGURE 19.11â•… Schematic of an on-axis ray and a marginal ray through the last elements of an immersion lithography optical system. The angles θl, θf, and θr are the angles of the marginal ray with respect to the surface normal in the lens, fluid, and resist, respectively. For the case shown here, the refractive index of the fluid is less than that of the lens and resist, and nfluid is then the maximum possible NA of the system.

n = 1.6–1.7. However, by Snell’s law (n1 sin θ1 = n2 sin θ2), NA = n sin θ can equally well be evaluated above the resist. Until the recent introduction of the immersion stepper, the space between the final lens element and the wafer was filled with air or N2 gas, with an index near n = 1.0. A consequence is that in spite of the much higher index of the resist, the NA has a maximum value of 1.0. However, as microscopists have known and exploited for centuries, adding a fluid between the image plane and the final lens element allows the maximum NA to be increased.98–100 For a planer final lens element, the NA can theoretically be increased up to the lowest index of the resist, the fluid, or the lens, as indicated in Equation 19.4 and Figure 19.11:

NA max ≤ (nresist , n fluid , n lens )

(19.4)

A nearly ideal immersion fluid for 193â•›nm lithography turned out to be purified water. It has sufficient transparency at this wavelength, is relatively innocuous to the resist and lens materials, is compatible with resist processing, has low enough viscosity to enable rapid wafer scanning, and is inexpensive. These properties enabled remarkably rapid development and implementation of immersion technology.101 From the inception of substantial 193â•›nm immersion lithography efforts in late 2002,102–105 it took only about 4 years for production-worthy water immersion systems to be built and the process brought into production. Immersion imaging does introduce some new issues, however, including bubble formation in the fluid, evaporation residue defects, resist-immersion fluid interactions, and fluid thermal effects. Polarization effects, already issues for dry systems, have been exacerbated at the extreme NAs of immersion lithography, requiring careful polarization control for differently oriented structures, which puts some restrictions on IC design.106 With water as the immersion fluid, having a 193â•›nm index of nwater = 1.437, the theoretical minimum half pitch (HPmin) for a 193â•›nm exposure tool decreases to HPmin = 0.25 × 193.4â•›nm/1.437 = 33.6â•›nm. Furthermore, at HPs achievable for dry systems, Equations 19.1 and 19.3 show that for the same

HP, the DOF is larger for the immersion approach. Note that the minimum HP for 157â•›nm dry systems is HPmin = 0.25 × 157.6â•›nm/1.0 = 39.4â•›nm, higher than that for 193â•›nm immersion systems. Attempts were made to identify high-index 157â•›nm immersion fluids, but only relatively low-index (n [157â•›nm] ≈ 1.35) fluorocarbon liquids were found to have any practical transparency at 157â•›nm,107 and consequently 157â•›nm lithography technology was dropped from technology roadmaps. As of 2008, water-based 193â•›nm immersion lithography systems are being operated at an NA in the range of 1.30–1.35 to satisfy the 45â•›nm technology node with acceptable process latitudes.14 Approaches to extend 193â•›nm immersion lithography technology further with higher-index fluids have been pursued. Practical 193â•›nm transmitting organic fluids with 193â•›nm indices near n = 1.65â•›nm (second generation fluids) have been developed.108,109 However, the last stepper lens element is made of calcium fluoride with a 193â•›nm index of n = 1.50, or fused silica, with an index of n = 1.57. By Equation 19.4, these materials are the bottlenecks for resolution extension. Thus, significant NA gain from higher-index fluids requires higher-index last lens materials as well. Higher-index UV lens materials have been explored for this purpose. Key practical materials identified include Lu3A5O12 (LuAG) [n = 2.21] and polycrystalline MgAl2O4 (ceramic spinel) [n = 1.93].110 LuAG has the highest index, but efforts have not yet succeeded in improving the 193â•›nm transmission to the specifications, which are stringent due to its high thermo-optic coefficient, dn/dT. Also, the large value of the intrinsic birefringence for this material is difficult to compensate for. Ceramic spinel solves this problem with its polycrystalline structure, but the polycrystalline nature also results in a high degree of scattering. A further possible high-index lens material being considered is α-Al2O3 (crystalline sapphire) [n = 1.92]. Its large natural birefringence due to its uniaxial crystal structure has limited its use in precision optics. However, with its crystal optic axis oriented along the optical axis of the system and with the polarization of all rays arranged to be oriented perpendicular to this axis, a sapphire last lens element could be manageable. As of this writing, LuAG is considered the most promising high-index lens material candidate,111 and the industry is still considering whether the practical NA gain to about 1.50 is worth the development effort. If a third-generation immersion fluid with an index near or above n = 1.80 could be developed, then an NA increase to near NA ≈ 1.7 could provide enough incentive to justify the substantial development effort required to implement it, because this would enable the 32â•›nm half-pitch technology node. Pure organic fluids with indices this high cannot have practical 193â•›nm transmissions. However, the indices of the fluids could in principle be increased to this level by loading the liquids with approximately (2–10) nanometer-size suspended particles of high-index oxide crystals, e.g., HfO2 or LuAG, while maintaining acceptable transmission, scattering levels, and viscosities.112–114 This approach is being explored, but it is regarded at best as complementary to multiple patterning approaches to resolution extension discussed next.

19-13

Nanoelectronics Lithography

19.3.3â•‡ Double Patterning The process-dependent k1 factor has been driven down to near the hard limit of k1 = 0.25 for single exposures. However, for multiple exposures and processing steps on the same layer, structures with half pitches corresponding to lower effective k1 values are possible. A number of these multiple patterning approaches are now being pursued and these are expected to satisfy the requirements for the 32â•›nm half-pitch technology node and possibly the 22â•›nm node and below. These approaches differ by the number and sequence of exposure and processing steps used to achieve the desired features. Four basic types are illustrated schematically in Figure 19.12.14 They all use the fact that while the structure pitch created by a single exposure is limited by diffraction, as characterized by Equation 19.1, processing can be used to tailor the line/space ratios in each period, and subsequent exposure and/or processing steps can interleave further structures to increase the structure density. In the Litho-EtchLitho-Etch process, Figure 19.12a, two Litho-Etch processes are done in sequence with the second process shifted by half the period to give a pitch doubling. This general process can be used to create arbitrary structures with the half pitch shrunk by a factor of 2 below the diffraction limit, though this scale shrink requires tighter tolerances on line edge control and overlay, among other issues.

A serious difficulty with this process is that it requires that the wafer be taken out of the stepper track for etching and then realigned before the second exposure, creating overlay challenges and decreasing the throughput for the layer by a factor of about 2. A preferred approach would have the two exposures done in sequence with just one etch step at the end: Litho-Litho-Etch, shown in Figure 19.12b. Unfortunately, it can be shown that for a resist system with a linear dose response, as has been universal in DUV lithography, two exposures cannot generate patterns with pitches below the diffraction limit. However, with a nonlinear dose response system, this is possible. A number of nonlinear response approaches are being pursued, including contrast enhancement layers, 2-photon resists, and positive-and negative-tone threshold response resists.115 One version, Litho-Freeze-Litho-Etch, shown in Figure 19.12c, is, particularly promising.116 In this approach, the latent image captured in the resist from the first exposure is “frozen” by chemical treatment. The first frozen image region is protected from any photo response to a second exposure, then a single etch step can create the pitch-doubled structures. A different approach, known as side wall (or spacer assist) double patterning, requires only one exposure step to create pitch doubling.117 As illustrated in Figure 19.12d, the first exposure and development step is only to create sidewalls at the desired positions. A subsequent thin-film deposition on the sidewalls, chemical-mechanical polishing (CMP) to split the deposited structure,

Litho-Etch-Litho-Etch Resist Si (a)

1st Litho (resist-expose-develop)

Nonlinear resist (b)

1st Etch

2nd Litho (resist-expose-develop)

2nd Etch

Litho-Litho-Etch (double exposure)

1st Litho 2nd Litho (nonlinear resist-expose) (expose-develop)

Etch

Least steps but requires nonlinear response process

Litho-Freeze-Litho-Etch

(c)

1st Litho (resist-expose-develop)

“Freeze” 1st pattern

2nd Litho (expose-develop)

Etch

CMP

Etch

Side wall process (spacer assist)

(d)

Side-wall deposition 1st Litho (resist-expose-develop)

FIGURE 19.12â•… Double patterning approaches: (a) Litho-etch-litho-etch requires 2 exposures and 2 etch steps. (b) Litho-litho-etch (or double exposure) requires 2 exposures and only one etch step, but needs a resist process with a nonlinear dose response. (c) Litho-freeze-litho-etch has intermediate complexity, requiring 2 exposures, a chemical freezing process after the first exposure, and one etch step. (d) Side wall process (spacer assist) requires one exposure and one etch. It uses side wall deposition and chemical-mechanical polishing (CMP) to achieve doubled pitch.

19-14


and elimination of the original pattern enables the desired pitch doubled structures. This approach takes advantage of the well-understood and highly controllable thin-film-Â�deposition technology to obtain the desired linewidths, independent of diffraction limits. Furthermore, the sidewall-derived structures are automatically self-aligned by the original structure. A difficulty with this approach is that restriction to patterning along sidewalls makes design layout much more difficult, and not all structures are topologically possible with single exposures. At least a further exposure/processing step is generally required for arbitrary structures. Still, this method is inherently extendable, and pitch quadrupling, etc. is possible in principle.

19.4â•‡ Electron-Beam Lithography Electron-beam lithography has demonstrated its utility over several decades as both a primary pattern generation tool for the semiconductor industry and as the premiere means of patterning small structures for advanced device development and research in myriad fields. Like any lithographic technology, its suitability for a given application depends on its performance with respect to the four metrics of resolution, throughput, pattern placement, and cost-of-ownership. As we shall see in the following sections, electron-beam systems excel with respect to resolution and pattern placement, but suffer from fundamental limits in terms of throughput.

19.3.4â•‡Ultimate Resolution Limits of DUV Lithography

19.4.1â•‡Resolution

Double patterning methods are already being used in IC production, and they are the declared technology solutions, using 193â•›nm water immersion tools, for several volume semiconductor manufacturers for the 32â•›nm half-pitch technology node projected for about 2013.14 These approaches appear feasible for further resolution shrinking as well, e.g., the 22â•›nm half-pitch technology node. Redesign of 193â•›nm immersion steppers with high-index fluids and lens materials, in combination with double patterning techniques, offer the potential for further resolution extension and relaxation of the k1 factor for a given resolution. When double patterning becomes routine, consideration of triple patterning, quadruple patterning, etc. may become tempting. However, this would likely have to come at the cost of much tighter process control requirements, such as CD and overlay control, than is presently attainable. Improvements to meet the requirements, along with the increased complexity (more exposures and processing steps per layer) will surely drive up costs substantially. There is little doubt that these DUV lithography extension approaches can be made to work technically. It is just a matter of whether they can operate at commercially viable costs and whether any reliable alternatives can be implemented at less cost. For example, rapidly increasing mask costs have made the numerous maskless technologies, such as multi electron–beam direct write schemes, very attractive, at least for low-volume production. For high-volume leading-edge ICs, it is generally agreed that for the 22â•›nm half-pitch technology node and below, a reliable single-exposure EUV solution, with its larger depth of focus and more natural extendibility, would be preferred. However, formidable challenges, repeated introduction delays, and especially rapidly rising projected costs, have made this solution not so inevitable as it once seemed at the beginning of the decade. Cost is the ultimate driver. If or when DUV lithography is displaced, and what its ultimate practical resolution limits are, is now less clear than ever. The technology has thrived on remarkable innovativeness and resourcefulness and its development history does not suggest that these will cease. Previous predictions of its limits and of its imminent replacement have always been wrong.

Resolution in a lithographic process is a rather ill-defined quantity since, by virtue of the fact that it is a process involving many steps, a large number of variables are involved. These can frequently be manipulated to produce isolated features far smaller than might, at first sight, be judged possible. The true test of the process is the minimum pitch of the features that can be fabricated. In what follows, we will consider primarily those factors that are unique to electron-beam exposure and neglect those that are common across the various exposure techniques—those are covered elsewhere in this chapter. The ultimate resolution that can be obtained using electronbeam exposure is determined by the nature of the interactions of the electrons with the material being exposed. There are two parts to be considered: first, the trajectories that the incident, or primary, electrons follow within the material and second, the processes by which the energy deposited by the primary electrons is translated into developable chemistry in the exposed material. The electron trajectories are determined by the combination of incident electron energy and resist/substrate atomic number.118,119 At low energies/high atomic numbers, the electrons scatter strongly within the solid, which leads to substantial broadening of the beam as it penetrates the material. Since most resist materials are organic, their average atomic numbers are not dissimilar from that of carbon, so the beam broadening, or forward scattering in the resist, is determined principally by the electron energy. For this reason, most high-resolution systems operate at energies of 50–100â•›keV. In addition, because the beam broadening increases progressively as the electrons undergo additional elastic scattering events on their way through the resist, thin resists yield higher resolution images. Very low (20 × 25â•›mm2) replication by EUV lithography, Microelectron. Eng. 30, 179–182 (1996). 11. Naulleau, P., Goldberg, K.A., Anderson, E. et al., Status of EUV microexposure capabilities at the ALS using the 0.3NA MET optic, Proc. SPIE 5374, 881– 891 (2004). 12. Naulleau, P., Anderson, C., Chiu, J. et al., Advanced extreme ultraviolet resist testing using the SEMATECH Berkeley 0.3-NA micro field exposure tool, Proc. SPIE 6921, 69213N1–69213N11 (2008). 13. Murakami, K., Oshino, T., Shimizu, S. et al., Basic technologies for extreme ultraviolet lithography, OSA Trends Opt. Photon. 4, 16–20 (1996). 14. Kinoshita, H., Watanabe, T., Niibe, M. et al., Threeaspherical mirror system for EUV lithography, Proc. SPIE 3331, 20–31 (1998). 15. Tichenor, D.A., Ray-Chaudhuri, A.K., Replogle, W.C. et al., System integration and performance of the EUV engineering test stand, Proc. SPIE 4343, 19–37 (2001). 16. Meiling, H., Buzing, N., Cummings, K. et al., EUV system— Moving towards production, Proc. SPIE 7271, 7271021– 72710215 (2009). 17. Murakami, K., Oshino, T., Kondo, H. et al., Development status of projection optics and illumination optics for EUV1, Proc. SPIE 6921, 69210Q1–69210Q7 (2008). 18. Spiller, E., Multilayer optics for x-rays, in P. Dhez and C. Weisbuch (eds.), Physics, Fabrication and Applications of Multilayer Structures, Plenum, New York (1987), pp. 271–309. 19. Barbee, T.W. Jr., Mrowka, S., and Hettrick, M., Molybedenum-silicon multilayer mirrors for the extreme ultraviolet, Appl. Opt. 24, 883–886 (1985).

Extreme Ultraviolet Lithography

20. Stearns, D.G., Rosen, R.S., and Vernon, S.P., Multilayer mirror technology for soft-x-ray projection lithography, Appl. Opt. 32, 6952–6960 (1993). 21. Perera, R.C.C. and Underwood, J.H., Results from our recently delivered automated reflectometer for measurement of reflectivity of EUV lithographic masks, Poster Presentation at the 2006 International EUVL Symposium, Barcelona, Spain, 15–18 October 2006. 22. Sommargren, G., Phase shifting diffraction interferometry for measuring extreme ultraviolet optics, OSA Trends Opt. Photon. 4, 108–112 (1996). 23. Sommargren, G., Phillion, D., Johnson, M. et al., 100-picometer interferometry for EUVL, Proc. SPIE 4688, 316–328 (2002). 24. Goldberg, K., Naulleau, P., Denham, P. et al., EUV interferometry of the 0.3 NA MET optics, Proc. SPIE 5037, 69–74 (2003). 25. Goldberg, K., Naulleau, P., Denham, P. et al., EUV interferometric testing and alignment of the 0.3 NA MET optic, Proc. SPIE 5373, 64–73 (2004). 26. Harned, N., Goethals, M., Groeneveld, R. et al., EUV lithography with the Alpha Demo Tools: Status and challenges, Proc. SPIE 6517, 6517061–65170612 (2007). 27. Tawarayama, K., Aoyama, H., Magoshi, S. et al., Recent progress of EUV full-field exposure tools in Selete, Proc. SPIE 7271, 7271181–7271188 (2009). 28. Meiling, H., Meyer, H., Banine, V. et al., First performance results of the ASML alpha demo tool, Proc SPIE 6151, 6151081–61510812 (2006). 29. Miura, T., Murakami, K. Suzuki, K. et al., Nikon EUVL development progress update, Proc. SPIE 6517, 6517071– 65170710 (2007). 30. Pierson, B., Wallow, T., Mizuno, H. et al., EUV resist performance on the ASML ADT and LBNL MET, Oral Presentation at the 2008 International EUVL Symposium, Lake Tahoe, CA, 28 September 2008. 31. Koh, C., Ren, L., Georger, J. et al., Assessment of EUV resist readiness for 32â•›nm hp manufacturing, and extendibility study of EUV ADT using state-of-the-art resist, Proc. SPIE 7271, 7271241–72712411 (2009). 32. Uzawa, S., Kubo, H., Miwa, Y. et al., Path to the HVM in EUVL through the development and evaluation of the SFET, Proc. SPIE 6517, 6517081–65170811 (2007). 33. Naulleau, P., Anderson, E., Baclea-an, L.-M. et al., The SEMATECH Berkeley microfield exposure tool: Learning at the 22â•›nm node and beyond, Proc. SPIE 7271, 72710W1–72710W11 (2009). 34. Solak, H.H., Ekinci, Y., Kaser, P., and Park, S., Photon-beam lithography reaches 12.5â•›nm half-pitch resolution, J. Vac. Sci. Technol. B 25, 91–95 (2007). 35. Ekinci, Y., Solak, H.H., Padeste, C. et al., 20â•›nm line/space patterns in HSQ fabricated by EUV interference lithography, Microelectron. Eng. 84, 700–704 (2007).

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36. La Fontaine, B., Deng, Y., and Kim, R.-H., The use of EUV lithography to produced demonstration devices, Proc. SPIE 6921, 69210P1–69210P10 (2008). 37. Lorusso, G.F., Hermans, J., Goethals, A.M. et al., Imaging performance of the EUV alpha demo tool at IMEC, Proc. SPIE 6921, 69210O1–69210O11 (2008). 38. Wood, O., Koay, C.-S., Petrillo, K. et al., Integration of EUV lithography in the fabrication of 22-nm node devices, Proc. SPIE 7271, 7271041–72710411 (2008). 39. Park, J.-O., Koh, C., Goo, D. et al., The application of EUV lithography for 40â•›nm node DRAM device and beyond, Proc. SPIE 7271, 7271141–7271149 (2009). 40. Eom, T.-S., Park, S., and Park, J.-T., Comparative study of DRAM cell patterning between ArF immersion and EUV lithography, Proc. SPIE 7271, 7271151–72711511 (2009). 41. La Fontaine, B., Wood, O., and Medeiros, D., The future of EUV lithography, N. Yamamoto (ed.), Proceedings of SEMI Technology Symposium, SEMI Japan, Chiba, Japan, 2008. 42. Ota., K., Watanabe, Y., Banine, V., and Frankin, H., EUV source requirements for EUV lithography, in Vivek Bakshi (ed.), EUV Sources for Lithography, SPIE Press, Bellingham, WA (2006), pp. 27–43. 43. Corthout, M., Apetz, R., Bruederman, J. et al., Sn DPP source-collector modules: Status of Alpha sources, Beta developments, and HVM experiments, Proc. SPIE 7271, 72710A1–72710A10 (2009). 44. Brandt, D.C., Fomenkov, I.V., Ershov, A.I. et al., LPP source system development for HVM, Proc. SPIE 7271, 727031– 72710311 (2009). 45. SEMI P38-1103, Specifications for absorbing film stacks and multilayers on extreme ultraviolet lithography blanks, Semiconductor Equipment and Materials International, San Jose, CA (2003). 46. Levinson, H.J., Principles of Lithography, 2nd edn, SPIE Press, Bellingham, WA (2005). 47. SEMI P37-1102, Specifications for extreme ultraviolet lithography mask substrates, Semiconductor Equipment and Materials International, San Jose, CA (2003). 48. Kearney, P., Lin, C.C., Sugiyama, T. et al., Ion beam deposition for defect-free EUVL mask blanks, Proc. SPIE 6921, 69211X1–69211X7 (2008). 49. Ma, A., Liang, T., Park, S.-J. et al., EUVL blank defect inspection capability at Intel, 2008 International EUVL Symposium, Lake Tahoe, CA, September 29, 2008. 50. Nguyen, K., Attwood, D., Mizota, T. et al., Imaging of EUV lithography masks with programmed defects, OSA Proc. Extreme Ultraviolet Lithography 23, 193–203 (1994). 51. Wallow, T., Higgins, C., Brainard, R. et al., Evaluation of EUV resist materials for use at the 32â•›nm half-pitch node, Proc. SPIE 6921, 69211F1–69211F11 (2008). 52. La Fontaine, B., Deng, Y., Kim, R.-H. et al., Extreme ultraviolet lithography: From research to manufacturing, J. Vac. Sci. Technol. B 25, 2089–2093 (2007).

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53. Van Steenwinckel, D., Gronheid, T., Lammers, J.H. et al., A novel method for characterizing resist performance, Proc. SPIE 6519, 65190V1–65190V12 (2007). 54. He, L., Wurm, S., Seidel, P. et al., Status of EUV reticle handling solutions in meeting 32â•›nm HP EUV lithography, Proc. SPIE 6921, 69211Z1–69211Z10 (2008). 55. Malinowski, M., Grunow, P., Steinhaus, C. et al., Use of molecular oxygen to reduce EUV-induced carbon contamination of optics, Proc. SPIE 4343, 347–356 (2001). 56. Miura, T., Murakami, K., Kawai, H. et al., Nikon EUVL development progress update, Proc. SPIE 7271, 72711X1–72711X11 (2009).

57. Meiling, H., Lok, B., Hultermans, B. et al., EUV alpha demo tools—Stepping stones towards volume production, 2008 International EUVL Symposium, Lake Tahoe, CA, 28 September 2008. 58. Hasegawa, T., Uzawa, S., Honda, T. et al., Development status of Canon’s full-field EUVL tool, Proc. SPIE 7271, 72711Y1–72711Y11 (2009).

Optics of Nanomaterials

IV

21 Cathodoluminescence of Nanomaterialsâ•… Naoki Yamamoto..................................................................................21-1

22 Optical Spectroscopy of Nanomaterialsâ•… Yoshihiko Kanemitsu............................................................................ 22-1

23 Nanoscale Excitons and Semiconductor Quantum Dotsâ•… Vanessa M. Huxter, Jun He, and Gregory D. Scholes.................23-1

24 Optical Properties of Metal Clusters and Nanoparticlesâ•… Emmanuel Cottancin, Michel Broyer, Jean Lermé, and Michel Pellarin.................................................................................................................................................... 24-1

Introductionâ•‡ •â•‡ Fundamentals of Cathodoluminescenceâ•‡ •â•‡ Optical Properties of Semiconductor Quantum Structuresâ•‡ •â•‡ Quantum Structuresâ•‡ •â•‡ Application of TEM-CLâ•‡ •â•‡ Acknowledgmentâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Carbon Nanotubesâ•‡ •â•‡ Nanoparticle Quantum Dotsâ•‡ •â•‡ Multiexciton Generationâ•‡ •â•‡ Summaryâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

Introduction to Nanoscale Excitonsâ•‡ •â•‡ Nonlinear Optical Properties of Semiconductor Quantum Dotsâ•‡ •â•‡ Two-Photon Absorption in Semiconductor Quantum Dotsâ•‡ •â•‡ Conclusionâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Theoretical Descriptionâ•‡ •â•‡ Experimental Results: State of the Artâ•‡ •â•‡ Conclusion and Outlooksâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

25 Photoluminescence from Silicon Nanostructuresâ•… Amir Sa’ar............................................................................. 25-1

26 Polarization-Sensitive Nanowire and Nanorod Opticsâ•… Harry E. Ruda and Alexander Shik............................... 26-1

27 Nonlinear Optics with Clustersâ•… Sabyasachi Sen and Swapan Chakrabarti...........................................................27-1

28 Second-Harmonic Generation in Metal Nanostructuresâ•… Marco Finazzi, Giulio Cerullo, and Lamberto Duò..............28-1

29 Nonlinear Optics in Semiconductor Nanostructuresâ•… Mikhail Erementchouk and Michael N. Leuenberger....... 29-1

30 Light Scattering from Nanofibersâ•… Vladimir G. Bordo.......................................................................................... 30-1

31 Biomimetics: Photonic Nanostructuresâ•… Andrew R. Parker...................................................................................31-1

Introductionâ•‡ •â•‡ Synthesis of Silicon Nanostructuresâ•‡ •â•‡ Luminescence Properties of Silicon Nanostructuresâ•‡ •â•‡ The Vibron Model: The Relationship between Surface Polar Vibrations and Nonradiative Processesâ•‡ •â•‡ Concluding Remarksâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Single Nanowires and Nanorodsâ•‡ •â•‡ Arrays of Nanorodsâ•‡ •â•‡ Conclusionâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References General Introduction: Fundamentals of Nonlinear Optics and Different Nonlinear Optical Effectsâ•‡ •â•‡ Quantum Formulation of Nonlinear Opticsâ•‡ •â•‡ Nonlinear Optics with Materialsâ•‡ •â•‡ Why Are Clusters So Important?â•‡ •â•‡ Review of Nonlinear Optics with Clustersâ•‡ •â•‡ Conclusionsâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Nonlinear Optics and Second Harmonic Generationâ•‡ •â•‡ Second Harmonic Generation in Nanosystemsâ•‡ •â•‡ Leading Order Contributions to the Second Harmonic Generation Process in Nanoparticlesâ•‡ •â•‡ Emission Patterns and Light Polarizationâ•‡ •â•‡ Allowed and Forbidden Second Harmonic Emission Modes: Examplesâ•‡ •â•‡ Second Harmonic Generation in Single Gold Nanoparticlesâ•‡ •â•‡ Perspectivesâ•‡ •â•‡ Conclusionsâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Semiconductor-Field Hamiltonian in k · p-Approximationâ•‡ •â•‡ Quantum Equations of Motionâ•‡ •â•‡ Perturbation Theory: Four-Wave Mixing Responseâ•‡ •â•‡ Nonperturbative Methods: Rabi Oscillationsâ•‡ •â•‡ Conclusionâ•‡ •â•‡ Acknowledgmentâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Backgroundâ•‡ •â•‡ Basic Theoryâ•‡ •â•‡ Some Numerical Resultsâ•‡ •â•‡ Experimental Implementationâ•‡ •â•‡ Discussionâ•‡ •â•‡ Summary and Outlookâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Engineering of Antireflectorsâ•‡ •â•‡ Engineering of Iridescent Devicesâ•‡ •â•‡ Cell Cultureâ•‡ •â•‡ Diatoms and Coccolithophoresâ•‡ •â•‡ Iridovirusesâ•‡ •â•‡ The Mechanisms of Natural Engineering and Future Researchâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

IV-1

21 Cathodoluminescence of Nanomaterials 21.1 Introduction............................................................................................................................ 21-1 21.2 Fundamentals of Cathodoluminescence............................................................................ 21-2 Radiative Recombinationâ•‡ •â•‡ Excitonâ•‡ •â•‡ TEM-Cathodoluminescence Technique

21.3 Optical Properties of Semiconductor Quantum Structures............................................ 21-5 Band Structure of Semiconductorâ•‡ •â•‡ Theory of Light Emissionâ•‡ •â•‡ Polarization

21.4 Quantum Structures..............................................................................................................21-8 Quantum Wellâ•‡ •â•‡ Quantum Wire

21.5 Application of TEM-CL....................................................................................................... 21-15 InP Nanowiresâ•‡ •â•‡ GaAs Nanowireâ•‡ •â•‡ ZnO Nanowires

Naoki Yamamoto Tokyo Institute of Technology

Acknowledgment..............................................................................................................................21-24 References.......................................................................................................................................... 21-24

21.1â•‡ Introduction Dimensionality and size are two factors that introduce new properties into semiconductor materials and enable us to provide new functionality in semiconductor nanoelectrics and photonics. Semiconductor quantum wires have been intensively studied, on one hand, for the theoretical interest in physical properties due to quasi-one-dimensional quantum confinement of an electronic system (Sakaki, 1980; Ogawa and Takagahara, 1991; Tanatar et al., 1998) and, on the other hand, for practical application to future optoelectronic devices such as low-threshold lasers (Arakawa and Sakaki, 1982) and polarization-sensitive devices (Wang et al. 2001). One-dimensional nanostructures have been primarily synthesized by lithography and an epitaxial technique on semiconductor substrates (Arakawa and Sakaki, 1982; Ils et al., 1994; Someya et al., 1995). V- (Kapon et al., 1989) and T-shaped quantum wires (Pfeiffer et al., 1990) were fabricated using epitaxial growth techniques (Wang and Voliotis, 2006), namely, molecular beam epitaxy (MBE) and metalorganic vapor phase epitaxy (MOVPE). However, excitons in the quantum wires were observed to be localized in monolayer step-induced islands at low temperatures (Guilet et al., 2003). The fluctuation of confinement potential along a quantum wire is a serious issue for realizing the characteristic property of a one-dimensional structure. The free-standing nanowires of many semiconductors are recently produced by the growth technique using nanosized liquid droplets of the metal solvent (Hiruma et al., 1995; Gudiksen et al., 2002; Wu et al., 2002). This technique has an advantage

to produce heterogeneous structure along the wire such as p–n junctions (Gudiksen et al., 2002) and superlattices (Wu et al., 2002). The optical properties of the nanowires have been studied by photoluminescence (PL) and absorption spectroscopy, which showed the blue shift of the peak energy due to the quantum confinement effect and giant polarization anisotropy in PL and optical absorption. Those properties will provide a potentiality for new application of the semiconductor nanowires. In spite of many works for synthesizing nanowires, there have been very few experimental works for studying the optical properties of selfstanding nanowires, especially for a single nanowire, because of the limitation in spatial resolution. An isolated single nanowire of InP was studied by PL (Wang et al., 2001), which showed giant polarization anisotropy in PL and photoconductivity measurement. The giant polarization anisotropy is partly due to the excitation process by light, because the electromagnetic field in the nanowire, which generates carriers, depends on the polarization direction of incident light. Polarization anisotropy can also be expected in the light emission process, as in the case of electroluminescence (EL) and cathodoluminescence (CL). We studied the optical properties of semiconductor nanowires, observing cathodoluminescence (CL) spectra and polarized monochromatic CL images of nanowires by using a transmission electron microscope (TEM) combined with a CL detection system (TEM-CL). The TEM-CL technique has an advantage in high spatial resolution and ability to characterize nanoscale structures (Yamamoto, 1984; Mitsui et al., 1996; Yamamoto et al., 2003, 2008; Ino and Yamamoto, 2008). In Section 21.2, a principle of TEM-CL technique is described based on the 21-1

21-2


fundamental knowledge of radiative recombination processes and property of exciton. Luminescence in semiconductor nanowires shows characteristic properties in emission energy and polarization, because the optical transition between electronic states is affected by the quantum confinement effect. For understanding these properties, the theory of the band structure in semiconductors and optical transitions are described in Section 21.3. Quantum confinement effect on the optical properties of semiconductor nanowires and dielectric effect on polarization properties are described in Section 21.4. Finally, the experimental results of the TEM-CL technique for the InP and GaAs nanowires, which show the quantum confinement effect, and for the ZnO nanowires, which show the dielectric effect, are described in Section 21.5.

21.2â•‡Fundamentals of Cathodoluminescence 21.2.1â•‡Radiative Recombination Luminescence in semiconductors is light emission phenomenon due to the recombination of excited electrons in the conduction band with holes in the valence band (Voos et al., 1980). It is called photoluminescence (PL) when those electrons and holes, the carriers, are excited by light, and called cathodoluminescence (CL) when they are excited by fast electrons. The band structures are classified into two types, direct gap type and indirect gap type. In the former type, the conduction band minimum is located at the Γ point (k = 0) in the k-space, whereas it is shifted from the Γ point in the latter type. The valence band maximum is located at the Γ point. Then in the case of the direct gap band structure, the radiative interband transition from the electron states in the conduction band to the hole states in the valence band occur at the Γ point accompanying with emission of photon at high recombination probability. GaAs and InP have band structure of this type. Whereas in the indirect gap type,

the transition occurs with the association of phonon, and the recombination rate is rather small. Si and GaP have band structure of this type. Typical recombination processes in semiconductors are depicted in Figure 21.1 (Pankove, 1971). First group is radiative recombination processes, i.e., (a) interband transition from conduction band to valence band, (b)–(d) annihilation of free exciton (FX) composed of a pair of electron and hole, and related processes of bound excitons (BX), (e)–(g) transitions of carriers trapped by impurity states such as donor and acceptor states. Second group is nonradiative recombinations, i.e., (h) multiphonon scattering, and (i) Auger process. Photon energies in the recombination processes in Figure 21.1 are slightly lower than the band gap energy Eg, though their values are different from each other. The emission intensity by the interband transition from conduction band to valence band is generally small compared with the other processes. Photon energies in the impurity-associated recombination processes (Figure 21.1e and f) are given as follows: (D, h) : E = E g − ED

(e, A) : E = E g − E A ,

(21.2)

where ED and EA are binding energies of the donor level and acceptor level, respectively. Photon energy associated with the donor–acceptor pair transition (D,â•›A) is given by (D, A) : E = E g − ED − E A +

e2 , 4πεR

where R is a distance between donor and acceptor ε is the dielectric constant of the material e is the elemental charge

Conduction band

Excitation

D Phonon

Eg A

e–h pair generation

FX (a)

(b)

(D0, X) (A0, X) (D, h) (c)

(d)

(e)

(e, A)

(D, A)

(f )

(g)

Valence band

Figure 21.1â•… Recombination processes in semiconductors.

(21.1)

Auger process (h)

(i)

(21.3)

21-3

Cathodoluminescence of Nanomaterials Table 21.1â•… Basic Parameters on Exciton in Typical Semiconductors Band Gap (eV) Structure

Band Type

Eg (4â•›K)

Eg (300â•›K)

Si Ge GaAs InP GaN

Diamond Diamond Zinc-blend Zinc-blend Wurtzite

Indirect Indirect Direct Direct Direct

1.1698 0.7454 1.519 1.4236 3.507

1.11 0.669 1.43 1.34 3.39

ZnO

Wurtzite

Direct

3.436

3.37

Dielectric Constant 11.4 15.36 13.1 12.61 10.4 (||c) 9.5 (⊥c) 8.84 (||c) 8.47 (⊥c)

Ex (meV)

FX Radius (nm)

14.7 4.15 4.21 5.12 23

403 11 13 11 3

59

1.4

Note: The data were gathered from many literatures. The values of FX radius are calculated from the data in the other columns.

The last term indicates the Coulomb energy of the electron and hole pair. The impurities locate at lattice sites and R takes many different values. Then the emission peak due to (D,â•›A) is composed of the multiple peaks to become a single broad peak, and dependence of the peak position on temperature and excitation rate is different from that of the other luminescence peaks of a single recombination process.

21.2.2â•‡ Exciton An electron and a hole in semiconductors attract each other to form a FX. FX emission dominates in the radiative recombination process at low temperatures. The binding energy of exciton can be obtained by solving Schrödinger equation for electron and hole system attracted by Coulomb potential (Basu, 1997; Yu and Cardona, 1999). The binding energy of the lowest energy level and Bohr radius of FX in a bulk crystal are expressed by the hydrogen atom model as

(21.4)

m  a X = ε  0  aB ,  µ 

(21.5)

Photon energy of the FX emission is given by E = Eg − EX .

E = E g − E X − Eb ,

(21.7)

where Eb is a binding energy for trapping exciton at impurities. The magnitude of Eb is about 1/3 of ED and EA or less. Basic parameters are listed in Table 21.1 for typical semiconductors. The values of the band gap energy and exciton binding energy are taken after several literatures (Pankove, 1971; Vugaftman et al., 2001). Exciton radius in the last column is calculated by (21.4) and (21.5) using the relative dielectric constants and exciton binding energies listed in the table. The values give only a measure of the exciton radius, because the differences among the reported values for the listed parameters are fairly large.

21.2.3â•‡TEM-Cathodoluminescence Technique

 µ 1 E X = 13.6   2 [eV]  m0  ε

where ε is a relative dielectric constant m0 is the electron mass μ is the reduced effective mass ((1/µ = (1/m*) e + (1 /mh* )) aB is Bohr radius (0.0529â•›nm), respectively

ionized acceptor are expressed by (A0,â•›X) and (A,â•›X), respectively. The photon energy of the recombination is given by

(21.6)

Some of excitons are trapped by impurities to form BX. The recombinations of excitons bounded by a neutral donor and ionized donor are expressed by (D 0,â•›X) and (D,â•›X), respectively. Similarly those of excitons bounded by a neutral acceptor and

TEM-cathodoluminescence (TEM-CL) is a technique to observe luminescence spectra and scanning images using a converged electron beam having a probe size of nm order (Yamamoto, 2002, 2008). The characteristics of TEM-CL are as follows: (1) Inner structures can be observed by TEM using thin samples and comparison between TEM and CL images is possible and (2) high spatial resolution of about 100â•›nm can be achieved. In CL, incident electrons are scattered in a specimen from the incident beam direction, as shown in Figure 21.2. The scattered electrons produce secondary electrons and plasmons, which generate electron-hole pairs in the hatched region (generation region). The excited carriers diffuse into surroundings and some of them recombine radiatively in a surrounding region (diffusion region). The diffusion of the excess minority carriers is important, because majority carriers exist everywhere. The resolution of CL is given by the diameter of the diffusion region, which is determined by the following factors: (1) the electron beam diameter, (2) size of the generation region, and (3) the diffusion length of the minority carrier. The diameter of the electron beam is less than 10â•›nm, and is negligible compared with the other factors.

21-4

Handbook of Nanophysics: Nanoelectronics and Nanophotonics e– Incident beam CL δe (a)

Figure 21.3â•… Monte Carlo simulation of (a) carrier generation by incident electrons and (b) carrier diffusion.

Diffusion Thin film b

Figure 21.2â•… Carrier generation and light emission process in TEM-CL.

The diameter of the beam spread due to elastic scattering in a thin specimen of thickness t (μm) is given by

(b)

 Z  ρ b = 6.25      E   A

1/ 2

t 3/2 [µm],

(21.8)

where Z is the atomic number A is the atomic mass E (keV) is an incident electron energy ρ (g/cm3) is the density (Goldstein et al., 1977) For example, assuming that for GaAs, Zav = 32, Aav = 72.3, ρ = 5.3â•›g/cm3, and a specimen thickness is 0.5â•›μm, we obtain b = 0.24â•›μm. However, Monte Carlo simulation of the electron

scattering showed that the electron density is concentrated on the center of the beam, so the effective beam spread is smaller than that given by Equation 21.8 (see Figure 21.3). Equation 21.8 indicates that the higher accelerating voltage and thinner specimen thickness are advantageous for realizing high spatial resolution. However, when the accelerating voltage exceeds 100â•›kV, CL intensity starts to decrease because of the formation of specimen damage such as vacancies. TEM is typically operated at an accelerating voltage of 80â•›kV for CL measurements. Diffusion length is an effective parameter for determining spatial resolution in CL of semiconductors. In insulators we can ignore carrier diffusion, but in semiconductors, carrier diffusion occurs with diffusion length of about 100â•›nm for GaN and 1â•›μm for GaAs. The diffusion length depends on dopant density and temperature, and reduces by surface recombination effect in thin specimens (Nakaji et al., 2005; Ino and Yamamoto, 2008). In quantum structures of semiconductors, carriers are confined in the localized structures smaller than the diffusion length, and individual nanostructures can be observed in CL images with high resolution as in the case of self-standing nanowires. TEM-CL system based on a 200â•›kV transmission electron microscope (JEM-2000FX) is illustrated in Figure 21.4. Samples

SEM

e–

STEM

CCD camera Monochrometer

Mirror

PMT

Sample Holder CCD detector

TEM

Figure 21.4â•… TEM-CL system.

Photon counter

21-5

Cathodoluminescence of Nanomaterials

are examined in the TEM, both in the stationary beam illumination mode and the scanning beam illumination mode. The scanning electron microscopy (SEM) imaging using a secondary electron detector and the scanning transmission electron microscopy (STEM) imaging using a transmitted electron beam detector are both possible in the TEM. A beam size of 10â•›nm in diameter and a probe current of 1â•›nA was conventionally used. The temperature of a sample can be varied from 20â•›K to room temperature using a liquid-He cooling holder and liquid-N2 cooling holder. Light emitted from a sample is collected by an ellipsoidal mirror or parabolic mirror, being guided through a polarizer and a monochromator, and is detected by a photomultiplier tube (PMT) with a GaAs photocathode for visible light and an InGaAs photocathode for infrared light. A CCD detector is also used for measuring spectra with high sensitivity. Monochromatic CL images can be obtained by scanning the electron beam in the SEM mode. This imaging technique is similar to SEM or EDX mapping, because we use the light intensity collected by the mirror, instead of secondary electron yields or x-ray intensities, as in the case of the SEM. An image is composed of, for example, 100 × 100 pixels, so that the focused electron beam is scanned across the specimen and stays 0.1â•›s for each pixel. Therefore, it takes 1,000â•›s to obtain one complete image. The CL images in this paper are shown with the absolute intensity. In the CL detection system, a linear polarizer is located between the ellipsoidal mirror and monochrometer to select linearly polarized components of the emitted light; here p-polarized (s-polarized) light, which is detected with the polarization direction parallel (perpendicular) to the monochrometer slit, mainly involves linearly polarized light component parallel (perpendicular) to the longer axis of the ellipsoidal mirror or the axis of the parabola. A mask with a small hole is set in front of the CCD detector for the angular resolved measurement using the parabolic mirror. The mask is movable using an X-Y stage in a plane normal to the light path and selects an emission angle.

21.3â•‡Optical Properties of Semiconductor Quantum Structures

1 i k ⋅r e unk (r ), V

u3

=−

3 2 2

u3 2

u3

,

,−

u3 2

u1

=−

,

,−

u1

2

,

,−

1 2

1 |( X + iY ) ↑ 〉 2

1 | ( X + iY ) ↓〉 − | 2Z ↑〉  6

1 | ( X − iY ) ↑〉 + | 2Z ↓〉  = 6

1 2

1 2 2

=−

1 | ( X + iY ) ↓〉 + | Z ↑〉  3

=−

1 | ( X − iY ) ↑〉 − | Z ↓〉  3

(21.11)

where subscripts j and mj in u j ,m j are quantum number of total angular momentum J and the z component Jz, respectively. The functions u 3 3 and u 3 3 express the heavy hole (hh) wavefunc-

and u 1 2

which is a product of a periodic function unk(r) of the crystal lattice and a slowly varying function exp(ikâ•›·â•›r). Subscripts n and k indicate a number of the Bloch band and wavevector, respectively. In many semiconductors of diamond and zincblend structures, the carrier wavefunction at the bottom of the conduction band has the s-like orbital and that at the top of the

(21.10)

1 | ( X − iY ) ↓〉 2

=

3 2

1 2 2

, 2 2

(21.9)

Here, |↑〉 and |↓〉 are the spin orbital functions with up and down spins, respectively. The wavefunctions of the hole states at Γ point in the valence band are given by the kâ•›·â•›p theory using the atomic p-orbital functions |X〉, |Y〉, and |Z〉. Assuming that k is parallel to the z direction, the following functions are used as the valence band wavefunctions,

tions, u 3 1 and u 3

The wavefunction of a carrier in a semiconductor crystal of volume V is of a form of Bloch function, ψ nk =

1  1 = | s ↑〉 and uc  −  = | s ↓ 〉  2 2

uc

, 2 2

21.3.1â•‡ Band Structure of Semiconductor

valence band has the p-like orbital (Basu, 1997; Yu and Cardona, 1999). The function un0(r) has a form of the atomic orbital in the vicinity of the atoms. The band structure of semiconductor has been treated by the kâ•›·â•›p perturbation theory (Luttinger and Kohn, 1955). The wavefunctions of electron states at the Γ point in the conduction band are given by the product of the atomic s-orbital function and spin orbital function as

,−

1 2

1 ,− 2 2

2

,−

2

the light hole (lh) wavefunctions, and u 1

1 , 2 2

are those of the split-off band due to the spin–orbit

interaction. Band structure, energy level, and dispersion can be calculated by orthogonalizing the Luttinger–Kohn Hamiltonian containing 2nd order kâ•›·â•›p perturbation terms (Basu, 1997). The band structure of the direct gap type is schematically illustrated in Figure 21.5a. In the valence band of the zinc-blend structure, the hh and lh bands are degenerated at the Γ point, as seen in the figure, and the split-off band locates below them. Whereas in the wurtzite structure this degeneracy is dissolved, and the A, B, C bands exist separately (Figure 21.5b). The effective mass of hole is determined by the curvature of the dispersion curve at k = 0, i.e., the small curvature gives a large effective mass. If the wavevector of hole k h is inclined from the z-axis, the hole wavefunctions of the valence band change from (21.11). Here we

21-6


E

C.B.

and that of the light hole is expressed as

E

C.B.

Γ7

1 2 1 | ( X ′ − iY ′) ↑〉 + | Z ′ ↓〉 = {(cos θ cos φ + i sin φ) | X 〉 3 6 6

Eg

Eg V.B.

hh

k

V.B.

A (Γ9)

so

C (Γ7)

(b)

Figure 21.5â•… Band structures near the direct gap of (a) zinc-blend structure and (b) wurtzite structure.

take a z′ axis parallel to kh, and x′ and y′ axes to be normal to z′, as shown in Figure 21.6. The Cartesian coordinates (x,â•›y,â•›z) are fixed in space, and the coordinates (x′,â•›y ′,â•›z ′) are related to (x,â•›y,â•›z) by the rotation of angle θ around the y-axis and successive rotation of angle ϕ around the z-axis. The conversion matrix between the wavefunctions in the two coordinates is expressed as  cos θ cos φ T =  cos θ sin φ   − sin θ

− sin φ cos φ 0

sin θ cos φ sin θ sin φ  .  cos θ 

(21.12)

Namely, the wavefunction of the heavy hole is expressed in each coordinates as −

1 2

| ( X ′ + iY ′ ) ↑〉 = −

1 2

2 {sin θ cos φ | X 〉 + sin θ sin φ | Y 〉 + cos θ | Z 〉} |↓〉 3

(21.14)

21.3.2â•‡Theory of Light Emission Photon is emitted when an electron in the higher energy state transfers to the lower state. The transition probability can be given by the Fermi Golden rule,

(21.13)

z z΄ y΄

2π

∑∑| 〈 f | H ′ |i 〉 | δ(E 2

λ

θ

2π

∑∑ λ

f ,i

2

 eA0  − i K ⋅r |ψ i 〉 |2  m  | 〈ψ f |(e ⋅ p) e

2π  eA0   m 

2

Figure 21.6â•… Relation between the coordinates fixed in space and those associated with the wavevector of hole k h.

(21.16)

∑M G(ω) f (E ) 1 − f (E ) δ(E − E − ω). 2

c

c

v

c,v

v

c

v

(21.17)

y

x΄

(21.15)

where K is a wavevector of photon. The first term in (nλ + 1) expresses the stimulated emission and the second term expresses the spontaneous emission. In the following, we are only concerned with the spontaneous emission. The rate of the spontaneous emission at a frequency ω in the transition from electron state in the conduction band to the hole state in the valence band is given by R(ω) =

x

− Ei λ ).

× (nλ + 1)δ(Ei − E f − ω).

kh

φ

fλ

f ,i

Here, H′ is the perturbation term due to the interaction between electron and photon appearing in the Hamiltonian, and is expressed as H′ = −(e/m)Aâ•›·â•›p using a vector potential A and momentum p. The wavefunctions of the initial and final states are denoted by |i〉 and |f 〉, which are given by a product of wavefunctions of electron and photon. The summation over λ in (21.15) is taken over modes of photons. Substituting the above expression for the perturbation term, (21.15) can be rewritten as

{(cos θ cos φ − i sin φ) | X 〉

p

W=

W=

+ (cos θ sin φ + i cos φ) | Y 〉 − sin θ | Z 〉} |↑〉

+

B (Γ7)

lh (a)

+ (cos θ sin φ − i cos φ)| Y 〉 − sin θ | Z 〉 }|↑〉

k

The summation over the photon modes is replaced by the optical density of states G, which is derived from the van Roosbroeck– Shockly relation as

G(ω) =

n3 (ω)2 , πc 2 3 3

(21.18)

21-7


where n is the refractive index of the material. fc(Ec) and f v(Ev) are Fermi distribution functions. It is noted that we should use the pseudo-Fermi energy instead of the equilibrium one, when carriers are excited by outer sources. The product of fc(1 − f v) in (21.17) produces a sharp peak near the band gap energy (Ec − Ev) in luminescence spectra. This is clear difference from the absorption spectrum which shows multiple peaks due to |M(ω)|2 above the band gap energy. The term M is an optical transition matrix element, giving emission intensity of the transition, and is written as M = 〈 ψ ve

− i K ⋅r

(e ⋅ p)ψ c 〉

(21.19)

zero. In other words, the x component of the electric field, Ex, is proportional to 〈uv|px|uc 〉. From symmetry, only the following matrix elements of 〈 Xpxs 〉 = 〈Yp ys 〉 = 〈 Zpzs 〉 = −i

{

∫

}

The second term is zero because of orthogonality of the wavefunctions. The following equation is valid for any periodic function of the lattice translation, f(r) = f(r + Rn),

∫ f (r)e (

i kc − K − kv ) ⋅r

dr =

3 3 1 1 , pxs ↑ = − 〈 Xpxs 〉 = − pcv 2 2 2 2

i k c − K − kv ) ⋅ Rn

n

×

∫ f (r)e (

i kc − K − kv ) ⋅r

dr ∝ δ(kc − K − kv )

Vc

(21.21)

Using this, the integral in (21.20) is found to be non-zero only when k c = kv + K, indicating the conservation of wavevector in the transition. The wavelength of light with band gap energy is relatively long, i.e., K 1), the expected value must be larger than 0.93. The observed value is much smaller than the expected one. This is partly because the polarization mixing occurs due to reflection by the ellipsoidal mirror which corrects light emitted to a large solid angle. Wang et al. (2001) reported a large degree of polarization (P = 0.96) in the photoluminescence from InP nanowires. The PL was excited by a polarized light, so the polarization anisotropy is enhanced in the absorption process. Whereas in the CL experiment, the excitation is performed by the incident electrons, there is no anisotropy in excitation of carriers. Thus, the degree of polarization in the CL measurement is smaller than that in the polarized PL. The CL detection is recently improved by introducing the angler-resolved measurement system, as shown in the later section of ZnO nanowires.

21.5.2â•‡ GaAs Nanowire Gallium arsenide (GaAs) has a zinc-blend structure, with a direct band gap of 1.519â•›eV at 0â•›K. Exciton radius is rather large,

about 15â•›nm, so the quantum confinement effect on photon energy and polarization of the FX emission is expected for thin nanowires. The binding energy of FX in the bulk is about 4â•›meV, and the emission intensity of FX becomes weak at higher temperatures above 200â•›K, even in thin nanowires. Recently, selfstanding nanowires of many semiconductors are produced by a growth technique using nanosized liquid droplets of a metal solvent (Hiruma et al., 1995; Gudiksen et al., 2002; Wu et al., 2002), and optical properties were theoretically studied (Persson and Xu, 2004; Redlinski and Peeters, 2008). Figure 21.20a shows a SEM image of a GaAs nanowire sample covered by an Al0.4Ga0.6As layer, self-standing on a Si(111) substrate. The average outer dimensions of the nanowires are 80â•›nm in diameter and 700â•›nm in height, and the average density of the nanowires is about 10â•›μm−2. The sample was fabricated by a metalorganic chemical vapor phase epitaxy (MOCVPE) technique using Au nanoparticles as the catalyst (Bhunia et al., 2003; Tateno et al., 2004). First, an Al0.3Ga0.7As nanowire was grown to 100â•›nm in height on the Si substrate, and then GaAs nanowire and Al0.3Ga0.7As nanowires were successively grown to 100 and 180â•›nm, respectively. Finally, Al0.4Ga0.6As was deposited as a capping layer, with a thickness of 100â•›nm to reduce the nonradiative recombination on the nanowire surface. Au dots with a radius of about 20â•›nm can be observed on top of the nanowires in Figure. 21.20a. Figure 21.20b schematically represents a diagram of the sample structure. The materials are direct-gap

S-5200 15.0 kV × 80.0 k SE 04/02/10 20:42

500 nm

(a) Au dot Al0.4Ga0.6As 180 nm

100 nm GaAs Al0.3Ga0.7As 100 nm Si (111) (b)

Figure 21.20â•… (a) A SEM image of GaAs/AlGaAs nanowires with a core-shell structure, and (b) schematic diagram of the sample structure.

21-19


semiconductors, with energy gaps of 1.52â•›eV for GaAs, 1.96â•›eV for Al0.3Ga0.7As, and 2.11â•›eV for Al0.4Ga0.6As at 0â•›K (Pavesi and Guzzi, 1994). The Al0.4Ga0.6As capping layer acts as a potential barrier, so a stronger (weaker) confinement structure is formed in the GaAs (Al0.3Ga0.7As) wire region. Carriers excited in the wire by an electron beam are expected to flow into the GaAs wire region, and recombine at the bottom of the band. Therefore, CL only from GaAs nanowires is expected to have sharp emission peaks in the CL spectrum corresponding to their diameters. Figure 21.21a shows a SEM image of the sample observed from the direction normal to the surface. The image was observed in the SEM mode using the TEM operated at an accelerating voltage of 80â•›kV with a scanning area of 1â•›μm2 (Ishikawa et al., 2008). Figure 21.21b shows a panchromatic CL image of the same area as that shown in Figure 21.20a taken at 60â•›K. The localized CL intensity is distributed at the nanowires, and we can specify the emission of each nanowire from the comparison between the CL image and the SEM image. A spectrum from this area is shown

A

in Figure 21.21c, where two broad peaks with a complex shape appear, reflecting the size distribution of the nanowires existing in this area. The emission spectra of similar shapes were frequently observed when the scanning area is of the order of 1â•›μm2, although the peak energies are different. The spectrum taken from the wider area becomes a single broad peak extending over the energy range from 1.7 to 2.0â•›eV. The broadening of the CL peak is due to the superposition of many CL peaks from the nanowires of different diameters. To determine the effect of size distribution, we used a focused electron beam illuminating individual nanowires, A, B, and C indicated by circles in Figure 21.21a. As shown in Figure 21.21d, sharp peaks were observed to appear at different energies because of the difference in diameter of the wire. It is noted that some nanowires such as A and C show double emission peaks. As for the diameters of the nanowires, it is difficult to determine them from the SEM image since the wires are covered by the AlGaAs layer with outer diameters of about 100â•›n m. The

A

B

C

200 nm

B

200 nm

(a)

C

(b) At 60 K

A

CL intensity (a.u.)

CL intensity (a.u.)

At 60 K

B

C

(c)

1.7

1.8

1.9 Energy (eV)

2.0

2.1

(d)

1.7

1.8 1.9 Energy (eV)

2.0

2.1

Figure 21.21â•… (a) A SEM image of the GaAs/AlGaAs nanowires, and (b) a panchromatic CL image taken at 130â•›K. A scan area is 1â•›μm × 1â•›μm. (c) A CL spectrum taken from the 1â•›μm × 1â•›μm area in (b). (d) CL spectra taken from the individual nanowires marked A, B, and C in (a) and (b).

21-20


energy levels were previously theoretically studied by Persson and Xu (2004) for GaAs nanowhisker grown in the [111] direction, on the basis of a tight binding approach. According to the calculation of the transition at the Γ point, the peak energies from 1.7 to 2.0â•›eV in the CL spectrum from the wide area correspond to the diameters from 5 to 2â•›n m. These values are much smaller than that expected from a Au particle size of 20â•›n m. This has been similarly observed in InP self-standing nanowires (Yamamoto et al., 2006), where the peak shift is much larger than that expected from the outer dimensions of the wire. This means that a practical size of carrier confinement region is much smaller than the apparent diameter. Such narrowing of the confinement region may result from the inner stress near the GaAs/AlGaAs interface and interdiffusion of Al into the wire. The temperature dependence of the CL spectra from the single nanowire A shown in Figure 21.21a is shown in Figure 21.22a. The lower energy peak (P1) appears below 140â•›K and the higher energy peak (P2) appears below 100â•›K . Their intensities increase with decreasing temperature. The intensity of the P2 peak increases rapidly at lower temperatures, and exceeds the P1 peak intensity at about 60â•›K. As shown in Figure 21.22b, the P1 peak energy shows a blue shift as temperature decreases, following the band gap shift described by Varshni’s law (solid lines). The P1 emission can be attributed to the radiative recombination of excitons at the ground state of the GaAs wire because the emission peak predominates at low temperatures. The luminescence of excitons remains at rather high

temperatures compared with that in bulk GaAs because the binding energy of excitons becomes large in nanowires owing to the quantum confinement effect (Basu, 1997). The appearance of the two peaks can be explained, assuming that the two parts having different energy states coexist along a single nanowire. The inhomogeneous intensity distribution was frequently observed along the tilted wires in the CL images and the polarization of the emitted light was randomly distributed with respect to the wire axis (Ishikawa et al., 2008). These facts indicate that the quantum-dot-like potential structure is formed along the wire, which does not emit a polarized light parallel to the wire axis. Figure 21.23a shows a SEM image of a GaAs/AlGaAs nanowire with a core-shell structure, as illustrated in Figure 21.20, and Figure 21.23b shows a corresponding panchromatic CL image taken at 130â•›K . In the CL image, discontinuous contrast is seen to appear along the wire, which suggests the existence of a quantum dot-like structure in the GaAs core. Figure 21.23c is a spectrum image taken with the electron beam scanning along the wire axis. A peak line at a wavelength of 830â•›n m is attributed to the bulk luminescence of GaAs. This indicates that the carriers excited in the wire diffuse into the GaAs substrate because this peak vanishes for the nanowire separated from the substrate. It is noted that the emissions from those dots have a common spectral shape in the wavelength range from 700 to 800â•›n m, and the anisotropy in polarization is very small. We still need more consideration about the property of this luminescence.

Al0.4Ga0.6As

2.10

CL intensity (a.u.)

20 K 40 K 60 K

Peak energy (eV)

2.05 2.00 Al0.3Ga0.7As

1.95

P2

1.90

80 K 1.85

100 K

1.80

120 K

1.7

1.8

1.9 2.0 Energy (eV)

2.1

GaAs-like

1.75

140 K

(a)

P1

20 (b)

40

60 80 100 120 140 Temperature (K)

Figure 21.22â•… (a) Temperature dependence of the CL spectrum from the nanowire A in Figure 21.21. (b) Temperature dependence of the peak energies of the two peaks. Solid lines show the temperature dependence of the band gaps of GaAs, Al0.3Ga0.7As, and Al0.4Ga0.6As, respectively, given by the Varshni’s law. The solid curve of GaAs is shifted for easy comparison with the observed one.

21-21


1 μm

Wavelength (nm)

A

(a)

200

s-pol

B

400 600 800 1000

(b)

(c)

1 μm A

Beam position

B

Figure 21.23â•… (a) A SEM image of GaAs/AlGaAs nanowires and (b) a panchromatic CL image of the same area (3â•›μm × 3â•›μm) taken at 130â•›K . (c) A spectrum image taken with the electron beam scanning along a line A-B through the central nanowire in (a).

Zinc oxide (ZnO) is a wide gap semiconductor of wurtzite structure, with a gap energy of 3.44â•›eV at 4â•›K . The valence band structure of bulk ZnO near the BE is composed of three bands (A, B, and C band), and the character of the band has been controversy for many years. The highest energy band (the A band) of many wurtzite-type semiconductors such as GaN and CdSe has a character of heavy hole (the Γ9 symmetry, as shown in Figure 21.5b), whereas many authors assume the Γ7 symmetry character for ZnO (Meyer et al., 2004). The polarization studies showed that the inner emission due to the transition from the conduction band to the A band contain both polarization components parallel and perpendicular to the c-axis. The energy difference between the A and B band was reported to be very small, 4.9â•›meV. The exciton binding energy is about 60â•›meV which is much larger than the thermal energy at room temperature (k BT = 25â•›meV), so the band-edge emission can be observed even at room temperature. However, the exciton radius is very small (a X = 1.3â•›n m), and it is difficult to fabricate a ZnO quantum wire of such a small diameter. The quantum effect of ZnO nanowires has rarely been reported so far. Recently, the optical properties of ZnO nanowires have been studied using self-standing nanowires grown parallel to the c-axis by MOCVD technique. Hsu et al. (2004) measured PL from narrow nanowires at room temperature and observed BE emission and green emission originated from defects. They showed that the BE emission is polarized parallel to the c-axis with the intensity ratio of I||â•›:â•›I⊥ = 1â•›:â•›0.7, whereas the green emission is polarized perpendicular to the c-axis with the ratio of I||â•›:â•›I⊥ = 0.85:1. The polarization dependence of photoconductivity was studied by Fan et al. (2004) using nanowires of 30–150â•›nm in diameter. The photoconductivity had a maximum value for polarization direction parallel to the c-axis. Yu et al. (2003) measured temperature dependence of PL spectrum from ZnO nanowires, showing peaks of BX emission, the P emission, and their phonon replica. The nanowires of wurtzite-type semiconductors such as GaN (Zhang and Xia, 2006; Chen et al., 2008) and CdSe (Lan et al., 2008) have also been studied. Chen et al. (2008) measured the size dependence in polarization of the BE emission from GaN nanowires, and explained the results from

the dielectric effect and the anisotropy in the inner emission. The polarization ratio changed with temperature; the BE emission is polarized parallel to the c-axis at room temperature, and is unpolarized at 20â•›K. They explained this from the change in anisotropy of the internal emission. In the TEM-CL study, ZnO smoke particles were made by burning a Zn wire in air, and were collected by a copper mesh coated by a carbon film with micrometer-size holes. The smoke particle has a tetrapod-like shape, with four legs growing along the c-axis. Figure 21.24 shows CL spectra taken from a thick leg of the ZnO smoke particle at various temperatures. Two peaks at wavelengths of 368â•›nm (3.370â•›eV) and 375â•›nm (3.307â•›eV) appear in the spectrum at 20â•›K. The peak at 368â•›nm can be attributed to the ionized donor BX emission (Meyer et al., 2004), and the peak

×1

20 K

× 1.1

CL intensity (a.u.)

21.5.3â•‡ ZnO Nanowires

30 K

× 1.2

40 K

× 1.3

60 K

× 1.6

80 K

× 1.8

100 K

× 2.2

120 K

× 2.5

140 K

× 3.6

160 K

× 4.4

180 K

× 5.9

200 K

× 11.4

240 K 300 K

360


390

Figure 21.24â•… CL spectra taken from a thick leg of a ZnO smoke particle at various temperatures.

21-22


at 375â•›nm is due to the P emission, which is generated by Augerlike process associated with two excitons (Hvam, 1973; Li et al., 2008). A small peak at 393â•›nm (3.155â•›eV) is a phonon replica of the P emission. The photon energies of those peaks shift to lower energies with temperature, following the Varshni’s law. A broad single peak (a BE emission) at room temperature can be recognized as a mixed peak composed of the BX emission, P emission, and their phonon replica. Figure 21.25a shows a SEM image of a ZnO smoke particle with thick legs, and Figure 21.25b and c are monochromatic CL images taken at the peak wavelength at room temperature for the p and s polarizations, respectively. The BE emission from a leg marked A extending along the y direction is strong in the p polarization, whereas that from a leg marked B extending along

A

B

y

x

100 nm

(a)

p

λ = 383 nm

s (c)

λ = 395 nm

p (d)

d0 z 2 1   I   = 3   − 1 = 0.1 d0 x 2 2   I ⊥  

120

250

100

150 100 50 –60

p=

300

200

60

60 40 P = 0.23

20 90

–90 (b)

(21.74)

80

P = –0.43

0 –30 0 30 Polarization angle (deg)

(21.73)

Whereas the BE emission from a thin nanowire with a diameter of 60â•›nm presented an opposite behavior in polarization, as shown in Figure 21.26b; the intensity has a maximum at θ = 0°. The polarization ratio is estimated to be 0.31 from the intensity ratio of I||câ•›/I⊥â•›c = 1.9. The quantum confinement effect cannot affect the polarization property of the CL emission from the ZnO nanowires because the diameters of those wires are much larger than the exciton radius. Therefore, the parameter p related to the anisotropic internal emission is fixed regardless of the diameter of the nanowires, and should be given by the character of the optical transition in the bulk crystal. If the intensity ratio I||câ•›â•›/I⊥â•›c = 0.40 is considered as a bulk one, the parameter p can be calculated from (21.67) and (21.70) as follows:

Intensity (a.u.)

Intensity (a.u.)

Figure 21.25â•… (a) A SEM image of a ZnO smoke particle with thick legs, and (b) and (c) are monochromatic CL images taken at the peak wavelength (383 nm) at room temperature for the p and s polarizations, respectively. (d) A monochromatic CL image taken at 395 nm.

–90 (a)

I (θ) = (I − I ⊥ )cos 2 θ + I ⊥

(b)

λ = 383 nm

the x direction is strong in the s polarization. These facts indicate that the BE emission is polarized perpendicular to the c-axis. The central part of the particle shows a dark contrast because the emission peak from this part is slightly shifted to the longer wavelength. The central part showed a bright contrast in the monochromatic CL image taken at 395â•›nm (Figure 21.25d). This peak shift may be caused by the band gap narrowing due to strain in the multiple twin structure at the central part. The polarization of the BE emission from ZnO nanowires depends on the diameter of the wire. CL light emitted from a single wire with a large diameter (300â•›nm) elongating perpendicular to the axis of the parabolic mirror was measured with changing polarization angle θ defined in Figure 21.12. The CL intensity as a function of θ is shown in Figure 21.26a. The light was emitted in a direction perpendicular to the thick wire axis. The solid angle of the detection is about 0.05 str. The intensity of the BE emission at 383â•›nm has a minimum at θ = 0°, and shows cos2 θ-dependence. Polarization ratio is estimated to be −0.43 from the intensity ratio of I||câ•›â•›/I⊥â•›c = 0.40, which is deduced using the fitting function of

–60

–30

0

0

30

60

90

Polarization angle (deg)

Figure 21.26â•… CL intensity from (a) a thick wire (diameter of 300â•›nm) and (b) a thin wire (60â•›nm) of ZnO measured by changing an angle θ.

21-23


If the highest energy state in the valence band has a character of the light hole, p = 4 is expected from (21.58) for the transition from the conduction band to valence band. The above value is too small compared to the expected one. In other words, the emission due to the transition to the light hole state should be mainly polarized parallel to the c-axis, but the observed one is rather polarized perpendicular to the c-axis. This fact indicates that the transition to the heavy hole state component in the A band considerably contributes to the CL emission. It was also 1 0.8

Polarization ratio

0.6 0.4 0.2 0 –0.2

0

50

100

150

200

250

300

–0.4

350

400

p = 0.25

–0.6 –0.8 –1

Diameter (nm)

Wavelength (nm)

Figure 21.27â•… Dependence of the polarization ratio on nanowire diameter measured from several nanowires with various diameters. Solid line is a calculated curve using the dielectric constant at the peak wavelength of 4.0 and p = 0.25.

A

B

100 nm JEOL

(a) p-pol

(d)

p s

Polarization ratio

LOAD ACCELHT SM8101 80KV X40K

Wavelength (nm)

(c)

300

(e)

p-pol

400 500 300

s-pol

400 A

B

500 1 0.5 0 –0.5 –1

(b)

observed that the polarization ratio does not change with temperature from 20 to 300â•›K. The CL emission from the thick nanowire is polarized perpendicular to the wire axis (the c-axis), whereas that from the thin nanowire is polarized parallel to the wire axis. We should consider the dielectric effect for explaining this size dependence of the polarization. Figure 21.27 shows the dependence of the polarization ratio on nanowire diameter calculated from (21.71) and (21.72) with the dielectric constant at the peak wavelength to be 4.0 and with p = 0.25. This result well explains the behavior that the polarization ratio P is positive for thin nanowires, and turns to be negative for thick nanowires. The experimental values measured from ZnO nanowires with different diameters are plotted in Figure 21.27. The polarization ratio changes sign around a diameter of 100â•›nm, being fitted with the calculated curve. The spatial variation of polarization can be measured from spectral images taken with scanning electron beam. Figure 21.28a and b shows a SEM image of a ZnO smoke particle with thick legs and a panchromatic CL image, respectively. Change in CL intensity appears along the leg. Figure 21.28c and d are p- and s-polarized spectral images taken with scanning electron beam along the line A-B in Figure 21.28a. The peak wavelength is 383â•›nm at room temperature. Polarization ratio calculated from the division of the peak intensities in the images of Figure 21.28c and d is shown in Figure 21.28e. The polarization ratio is seen to be −0.45 at a thinner part (D = 150â•›nm) and −0.25 at a thicker part along the line. One of the legs is standing vertically near the position B, and the polarization is vanished there.

0

500 Beam position (nm)

1000

Figure 21.28â•… (a) A SEM image of a ZnO smoke particle and (b) a panchromatic CL image, respectively. (c) and (d) are p- and s-polarized spectral images taken with scanning electron beam along the line A-B in (a). (e) Spatial variation of the polarization ratio along the line.

21-24


This behavior is qualitatively explained by the theoretical curve in Figure 21.27. Other types of emission appear under a high excitation condition, when a thin nanowire is illuminated by a strong electron beam. The P emission is one of such types. Emission peak due to the exciton molecules appears near the BE peak, and shifts to the lower energies with increasing electron beam density. The emission peak due to the electron-hole plasma also appears at the lower energy side of the BE peak, and becomes predominant under the high excitation rate. The effect of shape and size of nanowires on the property of these emissions such as threshold condition of lasing has been studied recently (Johnson et al., 2003).

Acknowledgment The author acknowledges Y. Watanabe, K. Tateno, G. Salviati, and L. Lazzarini for the supply of nanowire samples. This work has been supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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22 Optical Spectroscopy of Nanomaterials

Yoshihiko Kanemitsu Kyoto University

22.1 Introduction............................................................................................................................22-1 22.2 Carbon Nanotubes.................................................................................................................22-2 22.3 Nanoparticle Quantum Dots................................................................................................22-4 22.4 Multiexciton Generation...................................................................................................... 22-6 22.5 Summary..................................................................................................................................22-7 Acknowledgments..............................................................................................................................22-7 References............................................................................................................................................22-7

22.1â•‡ Introduction Over the past two decades, there have been extensive studies on the optical properties of semiconductor nanomaterials from the fundamental physics viewpoint and from the interest in the application to functional devices, because they exhibit unique size-dependent quantum properties [1–11]. In this chapter, we discuss optical properties of semiconductor nanomaterials of zero-dimensional (0D) nanoparticle quantum dots and one-dimensional (1D) carbon nanotubes. In optical studies of nanoparticle quantum dots and carbon nanotubes, we would like to point out two important reports opening new active fields: the discovery of room-temperature-visible luminescence from porous silicon in 1990 [12] and the discovery of efficient luminescence from isolated carbon nanotubes in 2002 [13]. These observations of efficient luminescence clearly show that nanoparticles and carbon nanotubes are high-quality crystalline semiconductors. Many different fabrication methods have been developed to obtain stable and efficient luminescence from nanoparticles and carbon nanotubes, e.g., core/shell nanoparticles, suspended isolated nanotubes, and so on [14–20]. These nanomaterials become new materials for optoelectronic devices such as wavelength-tunable light-emitting diodes and lasers, quantum light sources, and solar cell applications. When semiconductor nanoparticles of sizes are comparable to or smaller than the exciton Bohr radius in bulk crystals, the excited state energies and optical properties are very sensitive to their sizes [1,21,22]. Usually, nanoparticle samples are an inhomogeneous system in the sense that they have a distribution of size and shape, and variations of surface structures and surrounding environments [2]. A large nanoparticle has small bandgap energy and a small nanoparticle has large band-gap energy. Furthermore, the nanoparticles have large surface-to-volume

ratios, and then the optical properties of nanoparticles are also sensitive to surrounding environments. The exciton band-gap energy of semiconducting carbon nanotubes is also sensitive to the nanotube diameter and the chiral index. Then, we need to study the intrinsic optical processes in nanoparticle quantum dots and carbon nanotubes hidden by sample inhomogeneity using sophisticated optical spectroscopy. If the laser light excites all nanoparticles or all nanotubes in the sample, the sample shows broad luminescence, reflecting size or diameter distributions. This “global” photoluminescence (PL) or nonresonantly excited luminescence contains contributions from all nanoparticles or all nanotubes in the sample, and the PL spectrum is inhomogeneously broadened, as shown in Figure 22.1a. These sample inhomogeneities are the origin of nonexponential PL decay. In inhomogeneously broadened systems, resonant excitation spectroscopy is a powerful method to obtain intrinsic information from broadened optical spectra. Under resonant excitation at energies within the global PL band, we can observe fine structures in PL spectra at low temperatures, as shown in Figure 22.1b. Resonant excitation at energies within the luminescence band results in a single zero-phonon PL line or a wellresolved phonon progression in PL spectra at low temperatures. In this case, we suppress the inhomogeneous broadening of the luminescence by selectively exciting a narrow subset of nanoparticles. Resonant excitation results in fluorescence line narrowing (FLN) in nanoparticle samples [23–27]. The resonantly excited PL spectra are sensitive to the nature of the band-edge structure and the surface structure [27–29]. Moreover, luminescence hole-burning (LHB) spectroscopy is another resonant excitation spectroscopy. In the LHB experiments, the sample is excited by intense laser at the energy within the PL band [30–32]. After prolonged laser irradiation (burning laser excitation), a spectral hole is formed near the burning laser energy in the luminescence 22-1

(a)

(d)


(b)

(c) Energy

Figure 22.1â•… Luminescence spectra of semiconductor nanoparticles: (a) global luminescence spectrum, (b) resonantly excited luminescence spectrum, (c) LHB spectrum, and (d) single nanoparticle luminescence spectrum.

band. We obtain similar optical information from the FLN and LHB experiments. However, in the resonantly excited PL or FLN spectra, it is difficult to obtain the detailed spectrum near the excitation energy because of the scattering of the excitation laser light. In the LHB experiment, it is comparatively easy to observe the spectral change just at the excitation energy [32]. Single molecule (or nanomaterial) spectroscopy is the most powerful tool for understanding intrinsic optical properties of isolated molecules and nanomaterials [5,33]. Single nanoparticle spectroscopy makes it possible to probe inherent and novel optical properties in nondoped and impurity-doped semiconductor nanoparticles hidden by inhomogeneity, such as size distributions and surrounding environment variations [34–37]. As an example of unique optical phenomena, PL blinking (or PL intermittency) is revealed by single nanoparticle spectroscopy [35]. We have developed many different types of luminescence imaging microscopes. Our systems cover a wide spectral region and wide response times. The spatial resolution is typically about 1â•›μm in our confocal microscopes and about 50–100â•›nm in home-built near-field optical microscopes. In these spaceresolved optical measurements, the most important point is the fabrication of stable and isolated nanomaterial samples. The nanomaterials should be isolated from each other, and the number density is dilute (about 1/μm2 or less). The sample should be stable against oxidation and strong light illumination. Then, chemically synthesized materials are good samples for our optical studies. Here, we discuss the luminescence properties and the mechanism of spectral diffusion and PL blinking of single carbon nanotubes and nanoparticle quantum dots.

22.2â•‡ Carbon Nanotubes A single-walled carbon nanotube (SWNT) with about 1â•›nm diameter and a length greater than several hundred nanometers is a prototype of 1D structures. The recent discovery of efficient PL from semiconducting SWNTs [13,38] has stimulated considerable efforts in understanding optical properties of SWNTs. The

semiconducting SWNTs are 1D direct-gap band structures [8]. Because of the extremely strong electron–hole interactions (excitonic effects) in 1D materials, unique optical properties of SWNTs are determined by the dynamics of 1D excitons [8,39]. In addition, the electronic structure and the PL energy of SWNTs strongly depend on the diameter and the chiral index [38]. The SWNT samples are also inhomogeneous systems, similar to the nanoparticle samples, because many different species of nanotubes exist in the sample. The inhomogeneous broadening and the spectral overlapping of PL spectra cause the complicated PL dynamics of SWNTs. Single nanotube spectroscopy reveals the intrinsic excitonic properties of SWNTs [40–46], such as exciton energy, bright and dark exciton structures, exciton–phonon interaction, and so on. For single nanotube spectroscopy, we synthesized spatially isolated carbon nanotubes on Si substrates using an alcohol catalytic chemical vapor deposition method [45,46]. In our experiments, the Si or SiO2 substrates were patterned with parallel grooves, typically 500â•›nm wide and 500â•›nm deep using an electron-beam lithography technique. The isolated SWNTs grow from one side toward the opposite side of the groove. We used these SWNT samples without matrix and surfactant around the nanotubes to reduce the local environmental fluctuation effect. We show a typical PL spectrum of a single carbon nanotube suspended on the groove [assigned chiral index: (7,6)] at about 40â•›K in Figure 22.2a [46]. Very broad PL bands are observed in the ensemble-averaged spectrum of micelle-wrapped SWNTs dispersed in gelatin. The PL spectral shape of a single carbon

Ensemble

PL intensity (a.u.)

PL intensity

22-2

Single

0.9

1.0

(a)

1.1

1.2

1.3

Photon energy (eV)

(b)

Figure 22.2â•… (a) PL spectrum of a typical suspended single SWNT [assigned chiral index (7,6)] in comparison with the ensemble-averaged spectrum of micelle-wrapped SWNTs dispersed in gelatin. (b) Polar plot of the PL intensity of a typical single SWNT versus the polarized direction of the excitation laser. The PL data (circle) were fitted using cos2â•›θ (solid line). (Reprinted from Matsunaga, R. et al., Phys. Rev. Lett., 101, 147404, 2008. With permission.)

22-3

Optical Spectroscopy of Nanomaterials

nanotube is given by a Lorentzian function, and its linewidth of a few meV reflects homogeneous broadening. Figure 22.2b shows a polar plot of the PL intensity of a typical single carbon nanotube versus the polarization direction of the excitation laser light [46]. Since a 1D dipole moment exists, strong optical absorption occurs when the polarization of the excitation light parallels the nanotube axis. This PL anisotropy is useful for determining the direction of the observed nanotube axis for single nanotube spectroscopy and modulation spectroscopy. The sharp luminescence spectra provide detailed information on the exciton fine structures. We studied the PL fine structure of single SWNTs under magnetic fields at low temperatures. A single sharp PL spectrum arising from bright exciton recombination is observed at zero magnetic field, as shown in Figure 22.2a. When the magnetic field is parallel to the nanotube axis, a new peak appears below the bright exciton peak. Figure 22.3a shows PL spectra of a single carbon nanotube under magnetic fields [46]. These PL spectra are fit well by two Lorentzian functions. With the magnetic field, the lower energy peak shows a redshift and the lower-energy peak intensity increases. We cannot observe these changes when the magnetic field is perpendicular to a single nanotube axis, as shown in Figure 22.3b [46]. The splitting of the PL peak occurs due to the magnetic flux parallel to the nanotube axis. These splitting and magnetic field dependence can be explained by the Aharonov–Bohm splitting of excitons based on the Ajiki and Ando model [8,47]. The singlet

Voigt

Data Fitting Bright Dark

Faraday 0.9

7T Normalized PL intensity

exciton states split into the bonding and antibonding exciton states, and this due to the short-range Coulomb interaction. The bonding state is odd parity (bright) and the antibonding is even (dark). The energy difference between the bright and dark exciton states also depends on the diameter of SWNTs. These experimental observations are consistent with the theoretical calculation. The dark exciton state exists about several meV below the bright exciton state. Studies of 1D dark excitons influencing optical responses of carbon nanotubes [48–52] are very important for optical device applications. The diameter dependence of the exciton energy in single carbon nanotubes is also revealed by single nanotube spectroscopy. At room temperature, the experimentally obtained PL spectra can be approximately reproduced by single Lorentzian functions. The observed PL peaks correspond to the zero-phonon lines of free excitons, and the spectral linewidth of the PL spectra is determined by the homogeneous broadening. We obtained PL spectra from many different isolated SWNTs with a variety of chiral indices. Figure 22.4a shows a distribution of the PL peak energies for the single SWNTs, indicated by diamonds [45]. In Figure 22.4b, we show some of the PL spectra from isolated SWNTs with various emission energies [45]. Only a single sharp peak can be seen in each spectrum. The PL linewidth clearly becomes broader as the diameter decreases. This shows that the exciton–phonon interaction is stronger in smaller diameter tubes. The lowest exciton has fine structures (bright and dark excitons), and the fine structures will determine optical responses and cause unique phenomena. At low temperatures, we observe an interesting phenomenon, spectral diffusion. A few ten percents

RT

6T

1.3

1.4

(6,5) (7,5)

PL intensity (a.u.)

5T

4T

(7,6) (11,3) (9,7) (10,6)

0T 1.12 1.13 1.14 1.15 Photon energy (eV)

Photon energy (eV) 1.1 1.2

(a)

2T

(a)

1.0

(b)

1.02 1.03 1.04 1.05 1.06 Photon energy (eV)

Figure 22.3â•… (a) Normalized magneto-PL spectra of a single (9,4) carbon nanotube at 20â•›K in the Voigt geometry. The split PL spectra are fit by two Lorentzian functions. (b) The normalized magneto-PL spectra of a single (9,5) carbon nanotube at 20â•›K in the Faraday geometry. The PL spectra are fit by a Lorentzian function. (Reprinted from Matsunaga, R. et al., Phys. Rev. Lett., 101, 147404, 2008. With permission.)

(b)

0.9

1.0 1.1 1.2 Photon energy (eV)

1.3

Figure 22.4â•… (a) PL peak energy distribution of obtained PL spectra from about 180 different isolated SWNTs. (b) PL spectra for several species of single SWNTs at room temperature. SWNTs with higher PL emission energy tend to have a larger spectral linewidth. (Reprinted from Inoue, T. et al., Phys. Rev. B, 73, 233401, 2006. With permission.)

22-4


of the nanotubes show spectral diffusion. We consider that spectral diffusion is related to the exciton fine structure, bright and dark excitons. The PL fluctuation due to spectral diffusion is clearly observed at low temperatures. Spectral fluctuation occurs very slowly, the order of several seconds. Figure 22.5a shows a typical temporal evolution of the PL spectrum of SWNT showing spectral fluctuations at 40â•›K [53]. During spectral diffusion, the PL spectra clearly show two peaks. The lower energy is fitted by Gaussian and the higher energy is Lorentzian. From spectral fitting, we can determine the peak positions and linewidth of both peaks of the PL spectra. Temporal changes in two PL peak energies and linewidths of the lower energy PL band are shown in Figure 22.5b [53]. The higher energy peak is almost constant, but the lower energy peak fluctuates. We find a good correlation between the PL peak energy and the PL linewidth of the lower energy band. When the PL peak shows a low energy, the linewidth becomes wider. To clarify the origin

40 K 2s

PL intensity (a.u.)

6s 10 s 28 s 32 s 54 s 0.96

0.98 1.00 Photon energy (eV)

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Energy (eV)

(a)

0.995

Linewidth (meV)

0.990

(b)

8 6 4 2 0

0

50

100 Time (s)

150

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Figure 22.5â•… (a) Temporal evolutions of the PL spectrum of a single SWNT showing spectral fluctuations at 40â•›K. The solid curves indicate the results of fitting analysis assuming Gaussian and Lorentzian functions. (b)Temporal trace of the higher and lower energy peaks of the PL spectra of a single SWNT at 40â•›K. The temporal trace of the linewidth (FWHM) of the lower energy peaks. (Reprinted from Matsuda, K. et al., Phys. Rev. B, 77, 193405, 2008. With permission.)

of the spectral diffusion in the lower energy peak, the PL linewidth of the lower energy peaks is plotted as a function of the emission energy. We obtain the square root dependence of the linewidth on the emission energy [53]. This dependence suggests the quantum-confined Stark effect [54]. The spectral diffusion can be explained by the fluctuation of local electric field. The Stark effect causes a redshift in the exciton energy. A small, fast, local electric field fluctuation results from surface charge oscillations. These observations show that the energy splitting between the bright and the dark exciton states is estimated to be about a few meV [53]. This conclusion is well consistent with the magneto-optic results as mentioned before. Detailed understanding of fine structures of the lowest excitons is important for the optoelectronic applications of carbon nanotubes.

22.3â•‡ Nanoparticle Quantum Dots Similar spectral diffusion and PL blinking phenomena are also observed in nanoparticle quantum dots. PL blinking phenomenon is quite enhanced in 0D nanoparticles rather than 1D carbon nanotubes. PL blinking in single nanoparticles is caused by a random switching between light-emitting “on” and non-lightemitting “off” states under continuous-wave (cw) laser excitation. Since the first blinking observation in nanoparticles [35], the mechanism of nanoparticles PL blinking has been extensively discussed [55–70]. Ionization of nanoparticles due to nonradiative Auger recombination plays an essential role in PL blinking of single nanoparticles [55]. It is well accepted in this field that PL blinking originates from the photoionization and neutralization of nanocrystals under cw light illumination. In CdSe nanocrystals, for example, the non-light-emitting off-time state is due to positively charged nanoparticles, and the light-emitting on-time state is due to the neutral nanoparticle [55,70]. Spectral fluctuations during on-time suggest that both electrons and holes trapped on the nanoparticle surface cause transient and local electric field fluctuations in the light-emitting neutral nanoparticle. In neutral nanoparticles, excitons recombine radiatively. In ionized (or charged) nanoparticles, the photogenerated excitons and excess holes recombine nonradiatively through fast three-body Auger recombination. The PL blinking means that a nanoparticle repeats the neutralization and ionization under cw laser excitation. PL blinking behavior is very sensitive to the local surrounding environments. Figure 22.6 shows the PL intensity time traces of a single CdSe nanoparticle on different substrates: glass, rough, and flat Au surfaces [66]. The samples are excited by cw laser at room temperature. The on-off PL blinking behavior is clearly observed on the glass substrate. The time distribution of the on and off states can be characterized by power law functions [57]. The power law distributions suggest that the PL blinking is caused by a very complicated process. On the metal surfaces, on the other hand, the PL off-time is drastically suppressed. The PL blinking suppression indicates that the very rapid neutralization of ionized nanoparticles occurs through the fast energy transfer. However, the enhancement

22-5


10–4 –1 10 100 101 Time bins (s)

0 600 300

1

(b)

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IPL (a.u.)

50 Counts/50 ms

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10–2

τPL–1 (ns–1)

(a)

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0

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λex = 488 nm λem = 620 nm P (t)

RT

150

ex

40 0.1

20 0

(c) 0

10

20

Time (s)

30

40

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Figure 22.6â•… PL intensity time-traces of single CdSe/ZnS nanoparticles on the glass (a), the rough Au surface (b), and the flat Au surface (c). The inset of (a) is the histogram of the on-time (solid circles) and offtime (open circles) duration of the PL intensity time trace on the glass. (Reprinted from Matsunaga, R. et al., Phys. Rev. Lett., 101, 147404, 2008. With permission.)

and quenching of the PL intensity depend on the roughness of the metal surface. On the rough surface, the PL intensity increases. In this experiment, rough Au surfaces are composed of an assembly of hemispherical particles with lateral sizes of 20–50â•›n m and peaks and valleys of roughly 15â•›n m. On the flat surface, however, the PL intensity decreases. The enhancement of the PL intensity depends on the excitation laser wavelength. The enhancement spectrum agrees well with the absorption spectrum for localized plasmon resonance [66]. Even on rough Au surfaces, we can observe both enhancement and quenching phenomena of the PL intensity. PL intensity on a rough Au surface depends on the polarization angle of the excitation laser [71]. Electric field enhancement depends on the polarization direction. The PL intensity of a single nanoparticle on glass does not change with the polarization angle. We obtain the microscopic structure of semiconductor nanoparticles and rough surfaces from the polarization and wavelength dependences. Thus, we conclude that the PL intensity enhancement is related to the electric field enhancement due to localized plasmon excitation. Studying PL blinking behaviors is a way to understand energy transfer processes between nanoparticles and surrounding environments. Close-packed nanoparticle films or nanoparticle arrays show unique exciton energy transfer and charge carrier transport beyond isolated nanoparticles [72–79]. Many different types of closely packed nanoparticle films, arrayed nanoparticle solids, and nanoparticle suprasolids have been prepared [72–79]. In order to control energy transfer between excitons in semiconductors and plasmons in metals, we fabricate metal-semiconductor hetero-nanostructures and their PL spectrum and dynamics. We fabricated two types of semiconductor-metal nanoparticle heterostructures using the Langmuir–Blodgett technique: closepacked CdSe nanoparticle monolayers on Au substrates [80]

0.01

0.1

Δ–1 (nm–1)

Figure 22.7â•… PL lifetime and PL intensity as a function of the distance between the CdSe nanoparticles monolayer and the Au film. (Reprinted from Ueda, A. et al., Appl. Phys. Lett., 92, 133118, 2008. With permission.)

and mixed CdSe and Au nanoparticle monolayers [81]. In closepacked CdSe nanoparticle monolayers on Au surfaces, the inert polymer thin film was inserted between the nanoparticle monolayer and the Au substrate. The distance between the excitons and plasmons, Δ, is controlled by the polymer thickness. Figure 22.7 summarizes the distance dependence of the PL lifetime and the time-integrated PL intensity [80]. There is a good correlation between the PL decay rate and the PL intensity. PL quenching only occurs when the distance between excitons and plasmons is less than 30â•›nm. In large distance samples, the PL decay rates in close-packed monolayers are much larger than those in isolated nanoparticles in solutions. The PL lifetime in the CdSe monolayers on the glass is governed by the nonradiative recombination of excitons in nanoparticles and the energy transfer from small to large CdSe nanoparticles, which have lower exciton energies. Furthermore, the PL decay increases with a decrease of the distance between the Au surface and the CdSe nanoparticle monolayer. This can be attributed to energy transfer from nanoparticles to surface plasmons of the Au surfaces. The reduction of both the PL lifetime and the PL intensity simultaneously occurs. PL quenching only occurs in the CdSe nanoparticle monolayer in close proximity to the Au films [80]. Close-packed monolayer films composed of CdSe and Au nanoparticles have simple two-dimensional hexagonal lattices [81]. The PL and optical density of the sample films depend on the Au nanoparticle concentration in the film, because of the spectral overlap between the exciton luminescence of CdSe nanoparticles and the plasmon absorption in Au nanoparticle. In the CdSe and Au nanoparticle mixed monolayer samples, the PL decay curves can be reproduced successfully using three exponential decay components. Three kinds of decay channel of excited states exist in the close-packed CdSe and Au nanoparticle monolayer. In mixed monolayer samples, the decay times are classified into three components: 0.2, 1, and 10â•›ns, as summarized in Figure 22.8a [81]. These decay times are almost independent of

22-6


101

Decay time (ns)

τ3

100

τ2 Au: nn-CdSe: τ1

10–1

(ii) 0 (a)

0.2

0.4

0.6

Au NP fraction, x

nnn-CdSe:

0.8 (b)

Figure 22.8â•… (a) Decay time obtained by fitting using the three exponential decays as a function of the Au nanoparticle fraction. The broken lines are guides. (b) Schematic illustration of the configurations of the metal and semiconductor nanoparticles. Direct energy transfer from a nearest-neighbor (nn) CdSe nanoparticle (solid arrow) and from a next-nearest-neighbor (nnn) CdSe nanoparticle [broken arrow]. Stepwise CdSe → CdSe → Au nanoparticle energy transfer [dotted arrow]. (Reprinted from Hosoki, K. et al., Phys. Rev. Lett., 100, 207404, 2008. With permission.)

the Au nanoparticle mixing ratio in the film. The 10-ns decay time originates from radiative recombination within CdSe nanoparticles in the films. As the Au nanoparticle fraction increases, the amplitudes of two 1- and 10-ns components decrease. The amplitude of the fast 0.2-ns decay component becomes dominant. These results indicate that the fast PL quenching is caused by the energy transfer to Au nanoparticles for the CdSe nanoparticles in contact with Au nanoparticles. Here, note that the decay component of about 1â•›ns is a unique characteristic of close-packed mixed nanoparticles solids. Energy transfer between the nearestneighbor CdSe nanoparticles takes part in the slow PL quenching process of 1â•›ns in the mixed film. The 1-ns PL quenching process is the stepwise energy transfer from a CdSe nanoparticle to a CdSe nanoparticle to a Au nanoparticle, as illustrated in Figure 22.8b [81]. Therefore, we conclude that the PL dynamics are explained by three kinds of decay channel. The energy relaxation rate in semiconductor nanoparticles is controlled by changing local surrounding environments. These close-packed nanoparticle heterostructures will show unique exciton energy transfer, and charge carrier transport beyond isolated nanoparticles opens new application fields.

22.4â•‡ Multiexciton Generation Finally, we discuss unique exciton–exciton interactions in nanoparticles and nanotubes. Nanoparticle quantum dots and carbon nanotubes provide an excellent stage for experimental studies of many-body effects of excitons or electrons on optical processes in semiconductors [82–84]. The reduced dielectric screening and the relaxation of the energy–momentum conservation rule in nanostructures enhance the Coulomb carrier–carrier interactions,

leading to multi-carrier processes such as the quantized Auger recombination, multiple exciton generation (MEG), carrier multiplication (CM), and so on [7,85]. The achievement of efficient CM in semiconductors makes it possible to produce highly efficient solar cells with conversion efficiencies that exceed the Shockley– Queisser limit of 32% [86]. Strongly confined electrons and excitons in nanomaterials show unique nonlinear optical properties, compared to semiconductor bulk crystals. Strong interactions between carriers or between excitons cause fast nonradiative Auger recombination of multiple excitons or carriers [82]. Intense interest in Auger recombination in nanoparticles has been stimulated by investigations and searches for new laser and solar cell materials [87–89]. In laser and solar cell applications, nonradiative Auger recombination dominates both the carrier density and the carrier lifetime, determining the device performance. Moreover, in transient absorption, PL, and terahertz conductivity experiments, fast Auger recombination has also been used as a probe in MEG and CM processes [89–96]. The CM efficiencies of nanoparticles are not clear, and the CM mechanism is under discussion. In SWNTs, for example, strong Coulomb interactions enhance the many-body effects of excitons. The fast Auger recombination of excitons has been observed by means of exciton homogeneous linewidth [97] and pump-probe measurements [98–100]. We studied MEG processes in SWNTs at room temperature by temporal change in the carrier density [101]. The fast-decay component grows at increasing excitation intensity. When the photon energy is three times larger than the band-gap energy, Auger recombination occurs efficiently even in the weak intensity region. In our experiment, CM is estimated to be about 1.3 under 4.65â•›eV excitation [101]. We pointed out that a possible mechanism of CM in carbon nanotubes is the impact


ionization. Strong interactions between carriers or between excitons cause unique optical processes in semiconductor nanomaterials. Highly excited semiconductor nanomaterials show new optical functionalities.

22.5â•‡ Summary We briefly discussed luminescence properties of carbon nanotubes and nanoparticle quantum dots by means of single molecular spectroscopy and time-resolved optical spectroscopy. These semiconductor nanomaterials show unique luminescence properties such as spectral diffusion and luminescence blinking. Energy transfer between nanomaterials and surrounding environments affects the PL spectra and dynamics of nanomaterials. Although this chapter is written as a review-type survey of our recent studies, we hope that discussions and many references cited are useful for the readers.

Acknowledgments The author would like to thank many colleagues and graduate students for their contributions and discussions. In particular, Prof. K. Matsuda deserves mention here. Part of this work was supported by a Grant-in-Aid for Scientific Research on Innovative Area “Optical Science of Dynamically Correlated Electrons (DYCE)” (No. 20104006) from MEXT, Japan, and a Grant-in-Aid for Scientific Research (No. 21340084) from Japan Society for Promotion of Science (JSPS).

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23 Nanoscale Excitons and Semiconductor Quantum Dots Vanessa M. Huxter University of Toronto

Jun He University of Toronto

Gregory D. Scholes University of Toronto

23.1 Introduction to Nanoscale Excitons....................................................................................23-1 23.2 Nonlinear Optical Properties of Semiconductor Quantum Dots...................................23-5 23.3 Two-Photon Absorption in Semiconductor Quantum Dots.......................................... 23-6 Biological Imagingâ•‡ •â•‡ Amplified Stimulated Emission and Lasingâ•‡ •â•‡ Optical Power Limitingâ•‡ •â•‡ Two-Photon Sensitizer

23.4 Conclusion...............................................................................................................................23-8 References............................................................................................................................................23-8

23.1â•‡Introduction to Nanoscale Excitons The optical properties of nanoscale excitons are of interest to researchers because their size tunability makes them of practical use; however, understanding the nature of these delocalized electronic states is a challenge (Scholes and Rumbles 2006, 2008). Nanoscale excitons are the electronic excited states formed by the absorption of light by nanoscale systems. A nanoscale system is any chemical or physical system that is in the nanometer-size regime. The kinds of systems that are the subject of this chapter tend to have properties lying between those of molecules and those of bulk semiconductors. Unlike excitons in bulk semiconductors, the energies of nanoscale excitons can often be changed by the size of the system. That property is envisioned, for example, to allow the color of solid-state lasers to be tuned. Well-studied examples of nanoscale systems include semiconductor nanocrystals, carbon nanotubes (CNTs), organic conjugated polymers, and molecular aggregates, shown in Figure 23.1. The excitons in quantum dots and CNTs are closely related to molecular excited states; however, the size of the nanoscale systems has forced researchers to make severe approximations in their quantum-mechanical descriptions. Nonetheless, the excited states of nanoscale systems tend to be more amenable to approximate descriptions than molecules because the wavefunction delocalization reduces the importance of the electron correlation for an accurate calculation of the energies of electronic states. Such electronic excited states are often described as Wannier– Mott excitons (Banyai and Koch 1993; Basu 1997; Gaponenko 1998; Jorio et al. 2008). Other systems, like molecular aggregates, crystals, and certain proteins—namely, those involved in photosynthetic energy transduction—have optical properties that are better described with reference to the molecular building blocks of the aggregate. The lowest electronic excited states of these

aggregates are called Frenkel excitons (Kasha 1976). Conjugated polymers, now used in organic light-emitting diodes and displays (OLEDs), have excited states somewhere between these limits (Sariciftci 1997; Hadziioannou and Malliaras 2006). This relationship is illustrated in Figure 23.2. In the limiting case of Wannier–Mott excitons, it is assumed that the electronic interaction between the building blocks of the system—the atoms or unit cells—is large, akin to a chemical bond. In this case, orbitals that are delocalized over an entire system are a good starting point for describing one-electron states. Photoexcitation introduces an electron into the conduction orbitals, leaving a “hole” in the valence orbitals. The Wannier–Mott description assumes that there are many electrons in the system, so the atomic centers are highly screened from the outermost electrons, that is, the dielectric constant is high. In that case, it is considered reasonable to assume that the electron and the hole move freely in the background dielectric continuum and are weakly bound. The strength of this electron–hole attraction determines the “binding” of the lower energy, optically allowed states compared to the dense manifold of charge carrier states that lie higher in energy. In this model, the electron and the hole move under their mutual attraction in a dielectric continuum, and then the exciton energy levels are found as a series analogous to the states of the hydrogen atom. Note that this model can be useful, but it must be realized that it is highly approximate (Scholes 2008b), and it is only able to give limited insights. The conceptual advantage of this model is that it converges to the free carrier limit where the electron-hole attraction is negligible compared to the thermal energies. It can thereby be seen how the photoexcitation of a bulk semiconductor exciton efficiently produces charge carriers, which is how a typical semiconductor solar cell works. 23-1

23-2


1 PV

Absorbance

B800

2 PV Absorbance

Fluorescence intensity

B850

3 PV

4 PV n

700

800

850

Energy

(b)

Wavelength (nm)

800 Exc itat 700 ion w 600 (nm avele ngt ) h (c)

30,000

20,000

900

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Absorbance

Fluorescence intensity

(a)

750

0.9

1.1 V) nergy (e ission e

Em

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(d)

600

900 1200 1500 1800 Wavelength (nm)

Figure 23.1â•… (See color insert following page 20-14.) Excitons and structural size variations on the nanometer length scale. (a) The photosynthetic antenna of purple bacteria, LH2, is an example of a molecular exciton. The absorption spectrum clearly shows the dramatic distinction between the B800 absorption band, arising from essentially “monomeric” bacteriochlorophyll-a (Bchl) molecules, and the redshifted B850 band that is attributed to the optically bright lower exciton states of the 18 electronically coupled Bchl molecules. (b) The size-scaling of polyene properties, for example, oligophenylenevinylene oligomers, derives from the size-limited delocalization of the molecular orbitals. However, as the length of the chains increases, disorder in the chain conformation impacts the picture for exciton dynamics. Absorption and fluorescence spectra are shown as a function of the number of repeat units. (c) SWCNT size and “wrapping” determine the exciton energies. Samples contain many different kinds of tubes, therefore optical spectra are markedly inhomogeneously broadened. By scanning excitation wavelengths and recording a map of fluorescence spectra, the emission bands from various different CNTs can be discerned, as shown. (Courtesy of Dr. M. Jones). (d) Rather than thinking in terms of delocalizing the wavefunction of a semiconductor through interactions between the unit cells, the small size of the nanocrystal confines the exciton relative to the bulk. Size-dependent absorption spectra of PbS quantum dots are shown. (Adapted from Scholes, G. D. and Rumbles, G., Nat. Mater., 5, 683, 2006.)

In order to further understand the formation of excitons in nanoscale systems, we will examine the case of semiconductors as a model system. The optical properties of semiconductor nanostructures lie in an intermediate regime between a molecular and a bulk description. In a bulk semiconductor, the dimensions of the system are practically infinite compared to the dimensions of the carriers (electrons and holes). In this case,

the wavefunctions, which are standing waves in the material, are spread over an infinite number of unit cells (the repeat unit in a crystal), and the carriers (electrons and holes) are free particles. The density of states for the bulk material is continuous, as shown in Figure 23.3a. Note that in the case of a bulk semiconductor, the density of states for both the hole and the electron are continuous. These two continuous regions are separated by

23-3

Nanoscale Excitons and Semiconductor Quantum Dots Delocalized basis

Localized basis

NH N

NH N

N HN

Wannier excitons

N HN

NH N N HN

Frenkel/Davydov excitons

O

*

* O

Quantum dots

N HN

NH N

n

Conjugated organic materials

Supramolecular assemblies

Density of electron states

(a)

(c)


3D

Energy

1D

Energy

2D

Energy

(b)



Figure 23.2â•… Collective properties of nanoscale materials modify the optical properties, such as the wavelength and the dipole strength for light absorption. The elementary excitations are known as excitons. Excitons are formed through the collective absorption of light by two or more repeat units in a crystal, a molecular assembly, or a macromolecule. Wannier excitons are typical of atomic crystals, semiconductor quantum dots, aromatic molecules, CNTs, conjugated polymers, and so on. Supramolecular assemblies, including J-aggregates and photosynthetic light-harvesting antennae, typify Frenkel excitons—excitons in which the repeat units retain their identity to a significant degree.

(d)

0D

Energy

Figure 23.3â•… Idealized representation of the density of electron states for (a) a bulk semiconductor system (3D) and a semiconductor system confined in (b) one dimension (a 2D system), (c) confined in two dimensions (a 1D system) and (d) confined in all three dimensions (a 0D system). In the bulk case, the energy levels form a continuous band. With increasing confinement, the density of states becomes more discrete, resulting in the delta functional form of the density of states in the 0D case.


the bandgap in which the density of states goes to zero for an ideal system. The bulk system is infinite in all three dimensions. The effect of spatially confining the electrons and holes in one, two, or three of those dimensions is to change the density of states. The spatial confinement of the electrons and holes in one dimension means that the wavefunctions of those particles are quantized in that same one dimension. However, the electrons and holes are not confined and can move freely in the other two dimensions. This is similar to the concept of a plane as a twodimensional object in a three-dimensional space. Examples of these two-dimensional systems with a spatial confinement in one dimension are nanometer-height thin films and quantum wells. The quantization of the electrons and holes in one dimension changes the density of states from the bulk continuum to a step-type function, as shown in Figure 23.3b. Similarly, spatial confinement of the electrons and holes in two dimensions results in the quantization of the particles in those same two dimensions. In these systems, called quantum wires, the electrons and the holes can only move freely in one dimension. The quantization in two of three dimensions results in a further change in the density of states, which becomes more discrete with individual peaks as shown in Figure 23.3c. Finally, the spatial confinement of the electron and the hole in all three dimensions results in quantization in all three dimensions. In this case, the density of states is discrete and takes the form of delta functions, as shown in Figure 23.3d. These systems are called quantum dots and are discussed in more detail later in this chapter. The physical confinement in one, two, or three dimensions changes the boundary conditions imposed on the wavefunctions and quantizes the behavior of the electrons and holes in one, two, or three dimensions. The boundary conditions associated with the spatial confinement of the wavefunctions, in turn, modify the density of states. The resulting change in the density of states from a continuous band in the bulk to discrete levels in a quantum dot is analogous to the particle in a box model in quantum mechanics. The physical dimensions required to spatially confine the electron or the hole depend on the size of the electron and hole wavefunctions. These are determined by the physical properties of the specific material and are usually on the order of one to tens of nanometers. The quantum confinement of the electron and the hole wavefunctions modifies their interaction. The spatially overlapped electron and hole wavefunctions can associate through an energylowering Coulomb interaction due to their relative negative and positive charges, forming an exciton. This association can be described in analogy to a hydrogen atom and, as such, the size of the exciton is described using a Bohr radius. Just like the hydrogen atom, the lowest energy set of states is comprised of closely bound electron-hole pair configurations. These are the bound exciton states, which are the optically active states of the system (those that absorb and emit light). They tend to be clearly distinguished from a band of many higher energy states, which do not absorb light, but are the nanoscale-free carrier states (Scholes 2008b). The energy separation between the lowest energy-exciton states and the onset of the carrier states is called the exciton-binding

Energy

23-4

Figure 23.4â•… Confining the electron and the hole in a potential results in the formation of a hydrogen-like exciton and bound exciton states. The binding of the exciton is mediated through the Coulomb interaction.

energy. We can think of the exciton-binding energy as the energy required to ionize an exciton. Notably, this ionization energy is significantly reduced compared to small, molecular systems because of the many different ways that the electron and the hole can be separated. The exciton-binding energy, meditated by an attractive Coulomb interaction, can also be thought of as the energy reduction associated with forming an exciton as compared to the energies of the free electron and hole confined in the three-dimensional potential. This exciton-binding energy shifts the positions of the energy levels. The exciton in a three-dimensional potential and the associated exciton levels are shown in Figure 23.4. The binding energy of excitons is a topic of widespread interest, especially because of its relevance to photovoltaic science. In high dielectric constant bulk-semiconductor materials the exciton-binding energy is typically small: 27â•›meV for CdS, 15â•›meV for CdSe, 5.1â•›meV for InP, and 4.9â•›meV for GaAs. The small exciton-binding energies of these materials make them well-suited for photovoltaic applications because the optically active exciton states absorb light, then the ambient thermal energy (∼25â•›meV) is sufficient to convert the exciton to free carriers. Thus, the potential energy of the absorbed photon is converted to an electrical potential. On the other hand, in molecular materials, the electron-hole Coulomb interaction is substantial—usually a few eV. In nanoscale materials, we find a middle ground where exciton-binding energies are significant in magnitude—that is, excitons are important. As an example, the valence and the conduction orbitals of a single-wall carbon nanotube (SWCNT) are shown in Figure 23.5. These are typical 1D densities-of-states as were introduced in Figure 23.3. If the electrons were non-interacting, then the lowest excitation energy of a SWCNT would be the same as the energy gap between the highest valence orbital and the lowest conduction orbital. In fact, this is the energy corresponding to the onset of carrier formation. Suitable quantum-chemical calculations introduce interactions between the electrons in these various orbitals, which captures the electron-hole attraction and thus leads to a significant stabilization of the excited states relative to

Energy

Nanoscale Excitons and Semiconductor Quantum Dots

Eg

Figure 23.5â•… The attractive interaction of electrons and holes in a CNT (represented by the model at the top of the figure) results in the formation of excitons. This stabilizes the excited states relative to the orbital energy difference, lowering the energy of the optical gap.

those estimated with the more primitive model of orbital energy differences. That gives the correct energy of the optical absorption bands. The energy difference between the orbital energy difference and the optical gap is the exciton-binding energy. It has been predicted for SWCNTs to be ∼0.3–0.5â•›eV (Zhao and Mazumdar 2004), which was later confirmed by an experiment (Ma et al. 2005; Wang et al. 2005). As a result of the nature of excitons, the optical properties of nanoscale systems provide an interesting link between the properties of extended “bulk” systems and those of molecules. The study of excitons provides the opportunity for new insights into the behavior of nanoscale systems. For the rest of this chapter, we will focus on the properties of a particular nanoscale system: semiconductor quantum dots.

23.2â•‡Nonlinear Optical Properties of Semiconductor Quantum Dots Quantum-confined semiconductor nanocrystals, or quantum dots, have been the focus of intense study over the past three decades due to their size-tunable optical properties and unique physical characteristics. This area of research was founded in the early 1980s when researchers at Bell Laboratories (Rossetti et al. 1983) in the United States and the Yoffe Institute (Ekimov et al. 1980, Ekimov and Onushchenko 1982; Efros and Efros 1982) in Russia (then the U.S.S.R) independently described the properties of nanometer-sized semiconductor quantum dots. This work was quickly followed by studies on colloidal samples (Spanhel et al. 1987), leading to a further understanding of the optical and

23-5

physical properties of quantum dots. Within less than a decade, a basic theoretical framework to describe the observed properties had been proposed and work was underway to explore the fundamental physics of these materials. Much of the early work on quantum dots focused on semiconductor-doped glasses. These materials are characterized by broad size distributions and offer no possibility of control over the shape or the interface characteristics of the particles. The limitations of these doped glasses, particularly the broad size distribution that results in a large static inhomogeneity, masked many of the fundamental physical processes that were occurring in the quantum dots. However, the study of quantum dot systems underwent a revolution in 1993 when it was discovered that nucleation and a controlled growth of colloidal semiconductors could be achieved by injecting highly reactive organometallic precursors into a solvent system that coordinates to the colloid surface (Murray et al. 1993). This coordinating solvent, trioctylphosphine oxide, was an important discovery because it serves to arrest growth and stabilize the nanocrystals. This method allowed a simple and reproducible synthesis of high quality, nearly monodisperse cadmium chalcogenide nanocrystal samples. This development led to an explosion in the field and opened new avenues for research and technology including increased processibility, the possibility of mass production, and chemical manipulation for tailored shape control. In particular, this reliable method of making high-quality samples allowed the exploration of phenomena that had been previously unobservable due to static inhomogeneity associated with broad size distributions. In addition to doped glasses and colloidal samples, there are epitaxially grown quantum dots. While colloidal nanocrystals tend to be sized in the range of 1–10â•›nm in diameter, epitaxial dots, which are grown on solid substrates, may be a couple of nanometers in height but with lateral dimensions of tens of nanometers. The physics of these materials differ from colloidal samples in some fundamental ways; however, the basic properties associated with quantum-confined nanocrystals apply to all three types of quantum dots. Different aspects of these properties have been explored in many comprehensive reviews. For instance, a review of the electronic properties was presented by Yoffe (1993, 2001). The optical nonlinearities of semiconductor nanocrystals were reviewed by Banfi’s group (Banfi et al. 1998). Research on semiconductor quantum dots has evolved from fundamental science (Alivisatos 1996; Empedocles and Bawendi 1997; Klimov 2000; Klimov et al. 2000a,b) to lasing and amplification (Klimov 2006; Klimov et al. 2007), optical power limiting (He et al. 2007a,b), biological imaging (Bruchez et al. 1998; Dubertret et al. 2002; Larson et al. 2003; Michalet et al. 2005) and labeling (Seydack 2005), sensitization (Dayal and Burda 2008), and optical switching (Etienne et al. 2005; He et al. 2005a). One of the defining features of a semiconductor is the bandgap, which separates the conduction band and the valence band. When a semiconductor material absorbs light, an electron is promoted from the valence to the conduction band. The wavelength of light absorbed and emitted from a semiconductor

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material is determined by the width of the bandgap. In semiconductors, when an electron moves from the valence to the conduction band, the resulting gap left in the valence band is called a hole. An exciton can be formed through an energylowering Coulomb interaction between the negative electron and the positive hole. In analogy with the hydrogen atom, the spatial extent of the exciton wavefunction is described by a quantity called the exciton Bohr radius. As a result of the threedimensional spatial confinement of the exciton wavefunction, the density of states becomes discrete, as described earlier in the chapter. The position of these states, and therefore the energy of the gap, depends on the spatial confinement of the exciton, which is determined by the physical size of the nanocrystal. To a first approximation, this quantum-confinement effect can be described using the particle in a box or a simple quantum box model (Efros and Efros 1982; Brus 1984), in which the electron motion is restricted in all three dimensions by impenetrable walls. For a spherical nanocrystal with radius R, the quantum box model predicts that the size-dependence of the energy gap is proportional to 1/R 2, indicating that the energy of the lowest transition increases as the nanocrystal size decreases. In addition, as described above, quantum confinement changes the continuous energy bands of a bulk semiconductor into discrete exciton energy levels. The exciton energy levels produce peaks in the absorption spectrum of quantum dots, which is in contrast to the continuous absorption spectrum of a bulk semiconductor (Alivisatos 1996). The colloidal quantum dots discussed here are composed of a semiconductor core surrounded by a shell of organic ligand molecules (Murray et al. 1993). The organic capping prevents the uncontrolled growth and agglomeration of the nanoparticles. It also allows quantum dots to be chemically manipulated as if they were large molecules, with solubility and chemical reactivity determined by the identity of the organic molecules. The capping also provides an electronic passivation of the nanocrystals by terminating the dangling bonds on the surface. The unterminated dangling bonds can affect the emission efficiency of the quantum dots because they lead to a loss mechanism where electrons are rapidly trapped at the semiconductor surface before they have a chance to emit photons. Using colloidal chemical synthesis, one can prepare nanocrystals with nearly atomic precision with diameters ranging from nanometers to tens of nanometers and a size dispersion as narrow as 5% standard deviation. Because of the quantum-confinement effect, the ability to tune the size or shape of the nanocrystals translates into a means of controlling their optical properties, such as the absorption and emission wavelengths (Scholes and Rumbles 2006; Scholes 2008a). In a quantum dot system, the three-dimensional confinement modifies the Hamiltonian, adding a potential that restricts how far apart the electron and the hole can be. This forced spatial overlap changes the density of states as described earlier in the chapter. The interaction between the confined electron and the hole, mediated by a Coulomb potential, leads to the formation of an exciton, which can be described in analogy to a hydrogen

atom where an electron interacts with a nucleus. As a result of the discrete character of the density of states in quantum dots, as shown in Figure 23.3d, the oscillator strength is concentrated into those few transitions instead of being spread over a continuum of states. This means that the oscillator strength of the states in quantum dots is significantly enhanced. The concentration of the oscillator strength into a few transitions also enhances the nonlinear optical properties of quantum dots as compared to the bulk (Shalaev et al. 1996). As a result of their enhanced nonlinear optical properties, synthetically controllable size tunability and photostability, quantum dots continue to be of great interest both for fundamental research and device applications. One of these areas of research involves nonlinear two-photon absorption (2PA).

23.3â•‡Two-Photon Absorption in Semiconductor Quantum Dots Two-photon absorption in semiconductors is the simultaneous absorption of two photons of identical or different frequencies required to move an electron from the valence to the conduction band. As opposed to the linear intensity dependence of one-photon absorption (1PA), 2PA depends on the square of the light intensity. Therefore, 2PA is a third-order nonlinear optical process and many orders of magnitude weaker than 1PA. The 2PA coefficient, β, is directly related to the imaginary part of the third-order nonlinear susceptibility, χ(3), by β = 3π Im χ(3) /(λn02cε0 ), where n 0 is the linear refractive index, ε0 is the dielectric constant in vacuum, λ the laser wavelength, and c the speed of light in vacuum (Sutherland 2003). Compared to 1PA, 2PA is associated with different selection rules for dipole transitions. For example, in CdSe quantum dots, one-photon transitions satisfy selection rules ΔL = 0, ±2 and ΔF = 0, ±1 while two-photon transitions satisfy ΔL = ±1, ±3 and ΔF = 0, ±1, ±2, where L is the orbital angular momentum of the envelope wavefunction and F is the total angular momentum (Schmidt et al. 1996). Therefore, the comparison between 1PA and 2PA spectra allows a more detailed optical analysis of electronic states in semiconductor quantum dots (Schmidt et al. 1996). For direct bandgap semiconductor quantum dots, a simple theory based on the effective mass approximation was proposed for both 1PA and 2PA (Fedorov et al. 1996). Although this simple model does not consider the mixing between the heavy and light holes bands, it works quite well for describing degenerate 2PA. Recently, this theory has been extended to obtain the analytical expressions for both degenerate and nondegenerate 2PA spectra of semiconductor quantum dots with the parabolic band approximation and k⃗ . p⃗ theory (Padilha et al. 2007). As the incident laser intensity increases, the 2PA in semiconductor quantum dots will be saturated due to the limited density of states of the quantized energy levels (He et al. 2005b). If the allowed 2PA transitions are assumed to occur between the one atomic-like energy level in the conduction band and the other in the

23-7

Nanoscale Excitons and Semiconductor Quantum Dots

valence band, the ensemble of quantum dots can be approximately treated as a two-level system. Since the semiconductor quantum dots are not uniform in size, the saturation of 2PA for such an

Two-photon absorption in semiconductor quantum dots has many important potential applications, some of which are discussed in more detail below.

inhomogeneous system may be derived as β(I ) = β / 1 + I 02 /I s2, 2PA (Sutherland 2003). The saturation intensity, Is,2PA, in quantum dots can be described quantitatively by an inhomogeneously broadened, saturated 2PA model as (He et al. 2005b)

23.3.1â•‡ Biological Imaging

I s2,2PA =

ωπ∆ωp(2ω)N 0 g k  gk  gn β p π /2  1 + g n  

(23.1)

where gk(gn) is the electronic degeneracy of the upper (lower) state τp is the half width at the maximum of the femtosecond laser pulse ħω is the photon energy N0 is the number density of the quantum dots in the sample material p(2ω) is the probability of a homogeneous class of absorbers with a central frequency of 2ω The quantity Δω is related to the dephasing time (T2) of the excitation, i.e., the width of the homogeneous line shape. Two-photon transition does not involve a real intermediate state. The single photon energy is less than that of the quantum dots bandgap. So 2PA is a non-resonant optical nonlinearity. The 2PA coefficients, β, can be determined by two-photon excited fluorescence, nonlinear transmission, Z scan, and pump-probe techniques (Sutherland 2003). For example, the differential equation describing the intensity change in a one-photon transparent but two-photon absorbing material is given by dI 0 /dz = −βI 02, where I0 is the peak intensity of the input laser beam inside the sample. The nonlinear transmission in such a material, excited by a focused continuous-wave (cw) Gaussian beam, can be expressed as follows (Sutherland 2003):

 ln(1 + βlI 0 ) T (I0 ) =  (βlI 0 )

(23.2)

where l is the thickness of the quantum dot sample. A comparison of the expressions for the energy transmittance of a cw tophat, pulsed Gaussian and sech2 beam are available in Sutherland (2003). From Equation 23.2 one can see that at a given input intensity (I0) level, if the nonlinear transmission value is measured, the β value can be readily determined. The 2PA coefficient β is a macroscopic parameter characterizing the quantum dot composite material. The intrinsic 2PA coefficient of quantum dots, βQD, 4 can be derived as βQD = βcompositen02, composite /(n02,QD f v f ), where f v is the volume fraction of the quantum dots in the matrix and f = 3n02, matrix /(n02,QD + 2n02, matrix ) is the local field correction that depends on the dielectric constant of the quantum dots and the matrix material. In addition, the 2PA cross section can be calculated by the use of the definition: σ2 = βħω/N0.

Using quantum dot labeling, 2PA provides a possible way of performing biological imaging that is not possible by traditional one-photon methods as visible wavelengths cannot penetrate human tissue. As tagging materials, semiconductor quantum dots have advantages over fluorescent dyes, such as broad excitation and narrow emission bands, emission wavelength tunability, photostability, and enhanced brightness. The 2PA in robust water-soluble CdSe/ZnS core-shell quantum dots was found to be well-suited for use as fluorescent labels in multiphoton microscopy for biological imaging (Larson et al. 2003). The near-infrared two-photon excitation of such quantum dots in the “tissue optical window” (0.7–1.1â•›μm), in which water and hemoglobin absorb very little light, allows the extensive imaging of living systems, making further developments in medical diagnostics possible.

23.3.2â•‡Amplified Stimulated Emission and Lasing Quantum dot lasers have potential advantages, such as a Â�temperature-insensitive lasing threshold and wide-range color tunability. Due to the quantum-size confinement, the 2PA crosssections in semiconductor quantum dots are enhanced compared to the corresponding bulk material. This allows for a lower lasing threshold with a 2PA pumping mechanism. In addition, quantum dot lasers with 2PA excitation in the “tissue optical window” have important application prospects on laser-assisted biological-medical diagnostics and therapy. Recently, upconverted laser emission from a solution-processed CdSe/CdS/ZnS quantum dot waveguide-resonant cavity has been successfully demonstrated with femtosecond excitation at a wavelength of 800â•›nm (Zhang et al. 2008).

23.3.3â•‡ Optical Power Limiting Optical power limiting and stabilization can be used to protect sensitive equipment or to control noise in laser beams (He et al. 2008). In this case, the output laser intensity approaches a constant value when the input intensity increases beyond a certain threshold, limiting the amount of optical power entering a system. Compared to optical-limiting materials based on organic chromophores, crystals, and polymers, semiconductor quantum dots have a large 2PA cross section and better photostability.

23.3.4â•‡Two-Photon Sensitizer Semiconductor quantum dots are excellent sensitizers for nearinfrared 2PA due to their large and size-tunable 2PA cross sections. For example, quantum dots linked to phthalocyanines

23-8


(Pcs) can be excited using the 2PA of the quantum dots without any significant direct excitation of Pc molecules. In this case, Pcs are electron acceptor molecules, and a near-infrared excitation of the Pc molecules within the spectral therapeutic window (0.7–1.2â•›μm) can be realized via a two-photon sensitization of the quantum dot followed by an energy transfer to the Pc (Dayal and Burda 2008).

23.4â•‡ Conclusion Excitons in nanoscale systems, particularly in semiconductor quantum dots, provide a link between the bulk and the molecular regimes. In quantum dots, the discrete density of states associated with the exciton results in an increased oscillator strength in the optically active levels and an increased nonlinear cross section. The increased nonlinear optical properties of quantum dots provide an opportunity for novel devices and experimental applications including those associated with 2PA. The unique properties of nanoscale excitons and the materials associated with them will provide inspiration and direction to future research.

References Alivisatos, A. P. 1996. Science 271: 933. Banfi, G. P., Degiorgio, V., and Ricard, D. 1998. Adv. Phys. 47: 447. Banyai, L. and Koch, S. W. 1993. Semiconductor Quantum Dots. River Edge, NJ: World Scientific. Basu, P. K. 1997. Theory of Optical Processes in Semiconductors: Bulk and Microstructures. New York: Oxford University Press. Bruchez, M., Moronne, M., Gin, P., Weiss, S., and Alivisatos, A. P. 1998. Science 281: 2013. Brus, L. E. 1984. J. Chem. Phys. 80: 4403. Dayal, S. and Burda, C. 2008. J. Am. Chem. Soc. 130: 2890. Dubertret, B., Skourides, P., Norris, D. J. et al. 2002. Science 298: 1759. Efros, Al. L. and Efros, A. L. 1982. Semiconductors 16: 1209–1214. Ekimov, A. I. and Onushchenko, A. A. 1982. Semiconductors 16: 1215–1219. Ekimov, A. I., Onushchenko, A. A., and Tsekhomskii, V. A. 1980. Sov. Glass Phys. Chem. 6: 511–512. Empedocles, S. A. and Bawendi, M. G. 1997. Science 278: 2114. Etienne, M., Biney, A., Walser, A. D. et al. 2005. Appl. Phys. Lett. 87: 181913. Fedorov, A. V., Baranv, A. V., and Inoue, K. 1996. Phys. Rev. B 54: 8627. Gaponenko, S. V. 1998. Optical Properties of Semiconductor Nanocrystals. New York: Cambridge University Press. Hadziioannou, G. and Malliaras, G. G. eds. 2006. Semiconducting Polymers: Chemistry, Physics and Engineering, Weinheim, Germany: Wiley-VCH. He, J., Ji, W., Ma, G. H. et al. 2005a. J. Phys. Chem. B 109: 4373. He, J., Mi, J., Li, H. P., and Ji, W. 2005b. J. Phys. Chem. B 109: 19184. He, G. S., Yong, K. T., Zheng, Q. et al. 2007a. Opt. Express 15: 12818.

He, G. S., Zheng, Q., Yong, K. T. et al. 2007b. Appl. Phys. Lett. 90: 181108. He, G. S., Tan, L., Zheng, Q., and Prasad, P. N. 2008. Chem. Rev. 108: 1245. Jorio, A., Dresselhaus, M. S., and Dresselhaus, G. 2008. Carbon Nanotubes: Advanced Topics in Synthesis, Structure, Properties and Applications. New York: Springer. Kasha, M. 1976. Molecular excitons in small aggregates. In Spectroscopy of the Excited State, ed. B. DiBartolo. New York: Plenum Press. Klimov, V. I. 2000. J. Phys. Chem. B 104: 6112. Klimov, V. I. 2006. J. Phys. Chem. B 110: 16827. Klimov, V. I., Mikhailovsky, A. A., McBranch, D. W., Leatherdale, C. A., and Bawendi, M. G. 2000a. Science 287: 1011. Klimov, V. I., Mikhailovsky, A. A., Xu, S. et al. 2000b. Science 290: 314. Klimov, V. I., Ivanov, S. A., Nanda, J. et al. 2007. Nature 447: 441. Larson, D. R., Zipfel, W. R., Williams, R. M. et al. 2003. Science 300: 1434. Ma, Y. Z., Valkunas, L., Bachilo, S. M., and Fleming, G. R. 2005. J. Phys. Chem. B 109: 15671–15674. Michalet, X., Pinaud, F. F., Bentolila, L. A. et al. 2005. Science 307: 538. Murray, C. B., Norris, D. J., and Bawendi, M. G. 1993. J. Am. Chem. Soc. 115: 8706–8715. Padilha, L. A., Fu, J., Hagan, D. J. et al. 2007. Phys. Rev. B 75: 075325. Rossetti, R., Nakahara, S., and Brus, L. E. 1983. J. Chem. Phys. 79: 1086–1088. Sariciftci, N. S. 1997. Primary Photoexcitations in Conjugated Polymers: Molecular Exciton versus Semiconductor Band Model. Singapore: World Scientific. Schmidt, M. E., Blanton, S. A., Hines, M. A., and Guyot-Sionnest, P. 1996. Phys. Rev. B 53: 12629. Scholes, G. D. 2008a. Adv. Funct. Mater. 18: 1157. Scholes, G. D. 2008b. ACS Nano 2: 523–537. Scholes, G. D. and Rumbles, G. 2006. Nat. Mater. 5: 683. Scholes, G. D. and Rumbles, G. 2008. Excitons in nanoscale systems: Fundamentals and applications. In Annual Review of Nano Research, eds. G. Cao and C. J. Brinker. Hackensack, NJ: World Scientific. Seydack, M. 2005. Biosens. Bioelectron. 20: 2454. Shalaev, V. M., Poliakov, E. Y., and Markel, V. A. 1996. Phys. Rev. B 53: 2437. Spanhel, L., Haase, M., Weller, H., and Henglein, A. 1987. J. Am. Chem. Soc. 109: 5649–5655. Sutherland, R. L. 2003. Handbook of Nonlinear Optics. New York: Marcel Dekker. Wang, F., Dukovic, G., Brus, L. E., and Heinz, T. F. 2005. Science 308: 838–841. Yoffe, A. D. 1993. Adv. Phys. 42: 173. Yoffe, A. D. 2001. Adv. Phys. 50: 208. Zhang, C. F., Zhang, F., Zhu, T. et al. 2008. Opt. Lett. 33: 2437. Zhao, H. B. and Mazumdar, S. 2004. Phys. Rev. Lett. 93: 157402.

24 Optical Properties of Metal Clusters and Nanoparticles Emmanuel Cottancin Université de Lyon

Michel Broyer Université de Lyon

Jean Lermé Université de Lyon

Michel Pellarin Université de Lyon

24.1 Introduction............................................................................................................................24-1 24.2 Theoretical Description.........................................................................................................24-2 General Considerations on Bulk Metalsâ•‡ •â•‡ Optical Properties of Confined Metallic Systems

24.3 Experimental Results: State of the Art..............................................................................24-12 Nanoparticle Synthesisâ•‡ •â•‡ Techniques of Optical Spectroscopyâ•‡ •â•‡ Experimental Results

24.4 Conclusion and Outlooks....................................................................................................24-21 Acknowledgments...........................................................................................................................24-21 References..........................................................................................................................................24-22

24.1â•‡ Introduction The discovery of fullerenes (Kroto et al. 1985) and the experimental evidence of the so-called magic cluster sizes (Knight et al. 1984, de Heer 1993), more than 25 years ago, were at the origin of unceasing investigations on small clusters and nanoparticles and have contributed to the emergence of nanosciences, at the crossing point of several branches such as physics, chemistry, or even biology. In view of their high surface-to-volume ratio, clusters possess properties, different from those of bulk matter, that are very sensitive to their size and shape, rendering them very attractive from both the fundamental and technological points of view. In particular, the metallic species of a few nanometers in diameter disclose electronic properties intermediate between those of molecular systems and those of bulk matter. Whereas the sparse energy levels are quantized in atoms and molecules and continuously distributed in the energy bands of the crystal (Ashcroft and Mermin 1976), they tend to bunch together in metallic clusters and thus pattern the so-called electronic shells. This shell structure was evidenced experimentally (magic sizes) in the early 1980s in several metals (de Heer 1993) and nicely interpreted in the frame of the jellium model (Brack 1993). However, it was shown that magic sizes can also be correlated with atomic shell closures (Martin 1996) and even that both electronic and geometric structures may compete with each other, depending on the temperature (Martin et al. 1990). The optical properties of metallic clusters being directly underlain by their electronic structure, their optical study is, in this respect, of fundamental interest. For the rest, the fascinating colors of glasses doped with metallic powders have been known for ages even if their origin long remained mysterious. The

synthesis of such materials is utilized since antiquity in the art of making jewels, ornamental glassware, or stained glass in the Middle Ages. One can quote the famous cup of Licurgus (Barber and Freestone 1990) from the fourth century AC (appearing red in transmission and green in reflection) or the stainedglass windows of the Chartres Cathedral in France. The colorful shade of these materials (such as glass ruby) was discovered empirically and various colors were obtained during the seventeenth century by alchemists, metallurgists, or glassworkers. For instance, Glauber mentions that purple can be obtained by precipitating gold from its solution in aqua regia with a solution of tin compound (Hunt 1976). It was however only in the nineteenth century that their optical properties started to become more systematically studied by Faraday who succeeded in producing gold colloids by reducing gold salts (Faraday 1857b) and showed that color effects are intimately correlated to the size and morphology of colloids. In the Bakerian lecture (Faraday 1857a), Faraday spoke about the experimental relations of gold to light in the following terms: “Light has a relation to the matter which it meets with in its course, and is affected by it, being reflected, deflected, transmitted, refracted, absorbed, &c. by particles very minute in their dimensions.” In the lineage of Faraday, Zsigmondy worked on colloidal suspensions and set up the ultra-microscope based on the scattering of particles observed in dark field by illuminating the material to be viewed with a light source placed at right angle to the plane of the objective (Zsigmondy 1926). In 1925, Zsigmondy was awarded the Nobel Prize for Chemistry for his work on the heterogeneous nature of colloidal solutions. On a theoretical point of view, the pioneering experiments by Faraday have been interpreted later by Mie (1908) who solved Maxwell’s equations of a metallic sphere in a homogeneous 24-1

24-2


medium submitted to an external electromagnetic field (plane wave). Mie showed that the original properties of metallic nanoparticles are a consequence of the “dielectric confinement” (limited volume of the material) of particles whose sizes are smaller or of the same order as the wavelength of the excitation field. At the same time, Maxwell-Garnett developed an effective field model (Maxwell-Garnett 1904) in order to describe the optical properties of a medium containing “minute metal spheres.” The main feature in the optical extinction spectra of small metallic clusters is the emergence of a giant resonance in the near UV–visible range, called surface plasmon resonance (SPR) that is related to the collective motion of the conduction electrons induced by the applied field. This resonance is clearly noticeable only in the case of simple metal (alkali, trivalent) and noble metal (gold, silver, and copper) clusters. Its spectral position and width depend on the morphology of the particles (size, shape, and internal structure for alloyed systems), but also on their dielectric environment (medium in which the particles are embedded, local neighborhood) (Kreibig and Vollmer 1995). The development of numerous cluster sources (Sattler et al. 1980, Smalley 1983, Milani and de Heer 1990, Siekmann et al. 1991) in the 1980s enabled to probe the intrinsic properties of the clusters of very small size for which quantum size effects were expected. From a fundamental point of view, alkali clusters constitute a perfect model owing to their simple electronic structure and have been widely studied in the gas phase (Pedersen et al. 1991, Blanc et al. 1992, Bréchignac et al. 1992, de Heer 1993). However, they are immediately oxidized in contact with air once deposited on a surface, and thus are not suitable for applications. In spite of their more complex electronic structure, noble metal clusters in solution or embedded in a transparent matrix are more promising for potential applications because they are more robust toward oxidization. By varying the morphology, structure, or environment of these clusters, their optical response may be more or less controlled, making them attractive in several areas (linear and nonlinear optics [Kreibig and Genzel 1985], nanomaterials, nano-photonics, plasmonics, biosensors [Raschke et al. 2003]). Conversely, as the optical response is closely linked with the electronic structure, the SPR can also be used as a probe of the structure of metallic clusters, especially bimetallic systems or nanoalloys. Furthermore, the exaltation of the electromagnetic field in the vicinity of the particle can be exploited to increase the coupling of molecules with light, for developing biological markers for instance (McFarland and Duyne 2003). Until the past few years, most of the experiments in this field were performed on cluster assemblies, and except the investigation of size-selected clusters in the gas phase, the results are blurred by averaging effects due to the unavoidable cluster size and shape dispersions in samples (Kreibig and Vollmer 1995, Cottancin et al. 2006). To overcome this drawback, new methods of spectroscopy have been developed within the past 10 years, in order to study a single nanoparticle (Tamaru et al. 2002, Raschke et al. 2003, Arbouet et al. 2004, Dijk et al. 2006, Billaud et al. 2007). This has opened up a new field of research, allowing for instance to investigate reliably more complex

nano-objects or to study in detail shape effects on the optical response best than ever. The aim of this chapter is to give some keys to understand the linear optical properties (absorption, scattering, and extinction) of metallic clusters and nanoparticles,* and to present the state of the art of the research in this field. Section 24.2 deals with the theoretical description of the optical response of a metallic particle submitted to an external field. After a brief focus on the optical response of a bulk metal, particular attention is paid to the dipolar approximation that is appropriate for clusters sizes much smaller than the wavelength of excitation. Size, shape, and structural effects that can be expected are then fully discussed. The case of small clusters, for which quantum finite size effects are expected, is concisely sketched, disregarding the case of very small clusters (below 100 atoms per cluster) for which ab initio calculations are necessary (Rubio et al. 1997, Bonacic-Koutecky et al. 2001, Harb et al. 2008). Finally, the broad outlines of the Mie theory required to describe the optical response of larger nanoparticles are given. Section 24.3 is divided into three subsections. The first one briefly sets out various methods for producing clusters together with spectroscopic techniques for probing their optical properties. The second one gives an overview of the major results obtained on simple and noble metal clusters concerning size, shape, and multipolar effects in clusters and nanoparticles. Illustrations are taken equally from results obtained for cluster assemblies or single nanoparticles. Finally, the case of bimetallic clusters is described to illustrate the possibility of using the SPR as a structural probe.

24.2â•‡Theoretical Description 24.2.1â•‡ General Considerations on Bulk Metals In the bulk phase, the close packing of atoms involves an overlap of their outer atomic orbitals that strongly interact. This leads to a broadening of the discrete levels of the free atoms into bands (a continuum of electronic states), of very high density of states in metals. In this last case, the highest occupied energy band is called the conduction band and is filled with electrons originating from the outermost atomic orbitals. These electrons weakly interact with the ionic cores, and can be considered as quasi-free particles. They are delocalized in the metal and responsible for most of electrical and thermal transport properties. The highest occupied energy level in the conduction band (not completely filled in metals) is the Fermi level εF for which the electron velocity is the Fermi velocity vF. Lower-energy atomic orbitals will give rise to deeper energy bands (valence bands) in the solid. In noble metals for instance, the highest of these bands originates from the nd atomic orbitals (n = 3, 4, 5 for Cu, Ag, and Au, respectively). It is narrow owing to its weak hybridization with * The term “cluster” is generally reserved for sizes lower than a few nanometers in diameter, whereas the term “nanoparticle” is used in the range between a few nanometers to a few hundreds of nanometers.

24-3

Optical Properties of Metal Clusters and Nanoparticles

the conduction band (Figure 24.1). When a metal interacts with light, it may absorb a photon of energy hν promoting an electron from an occupied state to an unoccupied state, the latter being located in the incompletely filled conduction band. As for the occupied state, it belongs either to the conduction band or to the valence band. In the first case, the corresponding transition is called “intraband” transition and occurs in the infrared (IR)– visible range. The second case corresponds to “interband” transitions that can occur only if the photon energy is larger than the limiting value ћΩib corresponding to the energy threshold required for reaching the Fermi level from the top of the valence band (see Figure 24.1). In simple metals like alkali or trivalents, this threshold is out of the optical domain (near UV–visible– near IR). Thus, only optical transitions within the conduction band may occur. For noble metals, it lies in the UV–visible range and interband transitions may happen in the visible-UV range (see Figure 24.1). The energy threshold occurs in the UV range for silver (ћΩib ∼ 4â•›eV, λib ∼ 310â•›nm) and in the visible range for gold and copper (ћΩib ∼ 1.9â•›eV, λib ∼ 650â•›nm). The optical response of the metal can be described entirely by its dielectric function ε(ω), which reflects its electronic structure. If a metal is submitted to an external electromagnetic (EM) field E⃗ = E⃗ o cos(ωt) = ℜ(E⃗ oe−iωt), it is polarized such that its polarization (defined as the dipolar moment per volume unit) is P⃗ = εoχE⃗ , where εo is the vacuum permittivity and χ the dielectric susceptibility. The displacement vector can then be written ⃗ = εoE⃗ + P⃗ = εoε(ω)E⃗ , where ε(ω) = 1 + χ(ω) denotes the as D relative dielectric function which is characteristic of the metal. A part of the response may be in phase with the exciting field (α cos(ωt)â•›), whereas the absorption effects induce a response in quadrature phase (α sin(ωt)). Therefore, the dielectric function can be decomposed into a complex form ε = ε1 + iε2. Let us recall that it is directly correlated to the optical index of the medium, nopt = n + iκ = ε , in which n is the refractive index and κ the extinction coefficient (Fox 2001). nopt is real if the medium is non-absorbing.

In the case of simple metals, the Drude model (Drude 1900a,b, Ashcroft and Mermin 1976), developed originally to explain why metals are good conductors of heat and electricity, remains successful to interpret their optical properties (such as the fact that metals are good reflectors for frequencies lower than a threshold frequency, called plasma frequency [Fox 2001]). This model makes the basic assumption that most of the metal properties can be explained in first approximation by those of the conduction electrons if they are considered as independent and quasi-free. In this frame, when an oscillating electric field is applied, the free electrons oscillate and undergo collisions with other particles (electrons, ions, defects) with a characteristic scattering time τ = (1/γo), where γo is the average collision rate of electrons. Since the electrons are independent, their global response is the sum of all individual responses. By applying the principle of dynamics for an electron of effective mass me and charge −e, one obtains:

me

d 2r dr = − − eEoe −iωt , γ m o e 2 dt dt

(24.1)

where ⃗r is the complex position vector of the electron. One can easily solve this equation and deduce the polarization in the metal: P = −ρer =

ε oω 2P me ω (ω + iγ o )

(24.2)

where ρ is the number of electrons per volume unit (electronic density) and

ωP =

ρe 2 me εo

(24.3)

the plasma angular frequency that is introduced to interpret the high reflectivity of metals (Fox 2001) for frequencies lower than ωP. E Intraband transitions

n

εF

Energy

nS

s–p band Interband transitions

Interband transition threshold: hΩib (n – 1)d Atom

d band Solid

hΩib

k

Figure 24.1â•… (Left) Schematic diagram of the transition from discrete electronic levels in atoms to electronic bands in the solid for noble metals. (Right) Schematic density of states for the (n − 1)d and ns–p bands (left) of a noble metal. The ns–p band corresponding to quasi-free electrons in the metal is almost parabolic. As the atoms are brought closer together, their outer orbitals begin to overlap with each other and their strong interaction involves the formation of energy bands. Optical transitions inside the s–p band take place in the IR–visible range whereas interband transitions between the d-band and the conduction band may occur in the visible–near UV range.

24-4

Handbook of Nanophysics: Nanoelectronics and Nanophotonics 10

10 8

8

Silver

6

6

ћΩib ~ 1.9 eV

ε2

ћΩib ~ 4 eV

ε2

Gold

2

2

0

0

–50

–50 ε1

4

ε1

4

–100

–150

–100

0

1

2

3 4 Energy (eV)

5

6

–150

0

1

2 3 Energy (eV)

4

5

6

Figure 24.2â•… Real and imaginary parts ε1 and ε2 of the dielectric function of silver (left) and gold (right). Comparison between the Drude model (gray lines) and the experimental measurements of Johnson and Christy (1972) (squares). The influence of d-core electrons is patent in the imaginary part of ε where an increase of ε2 appears around 1.9â•›eV for gold and 4â•›eV for silver corresponding to the interband transition threshold ћΩib.

The so-called Drude dielectric function can then be written as (see also Figure 24.2) ε D (ω) = 1 −

ω 2P or ω(ω + i γ o )

 D ω 2P ω 2P (if γ o ω) ≈ 1 − ε1 (ω) = 1 − 2 ω + γ o2 ω2   ω 2P γ o ω 2P  D ε ( ω ) = ≈ γ o (if γ o ω) 2  ω(ω 2 + γ o2 ) ω 3 

(24.4)

This model is well suited to reproduce the dielectric function of simple metals, but in the case of noble metals, another term has to be added to the dielectric function in order to account for the polarization of the core electrons. The dielectric function then takes the following form:

ε = 1 + χ D + χ IB = ε D + ε IB − 1,

(24.5)

where εD denotes the component of the dielectric function associated with the conduction electrons and well described by the Drude model (D). It is related to the electron excitations inside the conduction band. εIB is related to interband (IB) transitions and depicts the contribution of the core electrons (d-electrons mainly) to the metal polarizability (Ashcroft and Mermin 1976). Figure 24.2 displays a comparison between the dielectric function calculated with the Drude model and the measured one, for gold and silver. If an almost clear agreement is observed in the IR range (below 1–1.5â•›eV), the experimental values distinctly deviate

from the calculated ones in the visible-UV range, because of the interband transition contribution. The steep increase in ε2 corresponds to the interband transition threshold ћΩib. In the case of copper (not shown here) the increase is steeper than in gold. It occurs at slightly higher wavelengths and is at the origin of the difference of the color of these two metals. As ε1(ω) and ε2(ω) are connected to each other with the Kramers–Kronig relationship (Ashcroft and Mermin 1976), the influence of interband transitions also explains the disagreement observed in the real part of ε(ω) between the Drude model and the experiment, the difference being larger in gold because of the proximity of the IB transitions.

24.2.2â•‡Optical Properties of Confined Metallic Systems 24.2.2.1â•‡ Scattering, Absorption, and Extinction In general, light is absorbed and scattered by a particle (see Figure 24.3). Absorption corresponds to the energy dissipation (heating) in the particle, whereas scattering is an elastic interaction of the wave with the particle. The EM field is reradiated by the particle in the whole space without a change of wavelength (Rayleigh scattering). If one assumes a particle in a homogeneous transparent medium (of dielectric function εm) submit ted to an incoming plane wave Ei = Eoei (kr − ωt ) of average intensity I o = (1/2)c ε o ε m Eo2 (units: W ⋅ m −2 ), the absorption and scattering cross sections (units: m2) are defined as the ratio between the absorbed power Wabs (unit: W) (respectively the scattered power Wsca) and the intensity Io: σabs = Wabs/Io and σsca = Wsca/Io.

24-5

Optical Properties of Metal Clusters and Nanoparticles Ws Wtrans = Wo – Wabs – Wsca

Ei = Eoei(kr–ωt) Io

Wa Detector R

σgeo = πR2

σext

Figure 24.3â•… Schematic view of light absorption, scattering, and extinction of a particle. (Top) A particle excited by an incident plane wave reemits a part of the incident power by scattering (Wsca) and absorbs the power Wa . An in-axis sensor placed far away from the particle detects the incident power minus the lost power related to extinction. The detector is far enough for neglecting a direct collection of the forward scattered light. (Bottom) Naive picture of the extinction cross section σext corresponding to the loss induced by the presence of a spherical particle in the course of the plane wave. σext may be smaller or larger than the apparent surface of the particle πR2 .

The extinction cross section, related to the total power lost in the propagation direction of the incident plane wave far away from the particle, is defined as the sum of both absorption and scattering cross sections: σext = σabs + σsca. This can be viewed naively in terms of perfectly opaque transverse areas shadowing the incoming plane wave (see Figure 24.3). It should be emphasized that, depending on the wavelength, σext may be different from the mere geometrical apparent area of the particle (see Figure 24.3). In the case of metals, in particular simple and noble metals, the dielectric confinement gives rise to a giant resonance (called SPR or Mie resonance) in the UV–visible range due to collective electronic excitations. It will be shown that the dielectric confinement refers to as classical size effects. 24.2.2.2â•‡ Dipolar or Quasi-Static Approximation Let us consider a spherical metallic particle of radius R embedded in a transparent medium of real dielectric function εm submitted to an incident EM field Ei = Eoe i(kr − ωt ), whose diameter is very small as compared to the wavelength of excitation (diameter ϕ 1) the SHG intensity per unit NW length is I v2ω =

π3ω3a 4  Pv2ω  c2

2

=

64π3 γ 2ω5a 4  E0ω 

(26.15)

(k = ω/c). Substitution of Equation 26.15 into Equation 26.14 shows that Pv2ω is determined only by the coordinate-dependent component of E ω (r), directed along the NW axis (z-axis) and has the value

4

c 4 [ε(ω) + 1]

4

.

(26.17)

The radiation is emitted isotropically (independent of a particular direction of light polarization). The surface component of polarization Pv2ω is determined by an E ω discontinuity at the NW interface caused by a dielectric mismatch, and for its determination we can retain only the first, uniform term in Equation 26.15. After simple transformations, we have 2ω s

P

=

4β[ ε(ω) − 1]  E0ω 

2

δ(ρ − a)cos α.

[ε(ω) + 1]2

(26.18)

(ρ,α) are polar coordinates in the xy-plane. The direction of Pv2ω coincides with that of the E 0ω (x-axis). This distribution of polarization has a quadrupole symmetry. Due to the coordinate dependence of Ps2ω , the determination of the electric field of emitted light must contain integration not only over z, but over ρ and α as well. Besides, in this case, we deal with elementary dipoles directed not parallel (as for Equation 26.16) but perpendicular to the NW axis. Due to the image forces, the ac electric field of these dipoles is suppressed by the factor 2/[ε(2ω) + 1] (see Section 26.2.3.1). Under the same conditions as before, we get the intensity of this surface-related SHG component: I

2ω s

4

4 π3β2ω5a 4 [ ε(ω) − 1]  E0ω  2 = 4 4 2 sin (2ϕ) c [ ε(ω) + 1] [ ε(2ω) + 1] 2

ω x

(26.16)

(26.19)

where φ is the angle between the exciting-light wave vector k and the direction to the observation point. As for any quadrupole emitter, the intensity I s2ω has four directional lobes.

Polarization-Sensitive Nanowire and Nanorod Optics

b. k ⊥ z; E 0ω z (transverse polarization) In this case

propagation,

longitudinal

(26.20)

 1 + ε(ω)  E ω = Ezω = E0ω  1 + ikx  2  

and Pv2ω, which is always colinear with the light wave vector, is directed along the x-axis normal to the NW and is equal to 2

Pv2ω = iγk  E0ω  1 + ε(ω)  = const(r).

(26.21)

For such an orientation of elementary dipoles, the SHG is anisotropic with the radiation pattern directed perpendicularly to k, and the ac electric field of these dipoles contains the factor 2/[ε(2ω)â•›+â•›1]. Adding the fields of elementary dipoles in the same manner as before, we get

I

2ω v

=

4 π 3ω 3a 4  Pv2 ω  c 2  ε(2ω) + 1

2

2

sin2 ϕ 4

=

4 π 3 γ 2ω 5a 4  E0ω  1 + ε(ω) c 4  ε(2ω) + 1

2

2

sin2 ϕ.

(26.22)

Since Ps2ω is caused only by the normal component of E 0ω suffering discontinuity at the NW interface, in the case considered it vanishes, and the SHG has only a dipole component given by Equation 26.22 and no quadrupole component. c. k ⊥ z; E 0ω ⊥ z (transverse propagation, transverse polarization)

E ω = Exω =

2 E0ω (1 + iky ).  ε(ω) + 1

(26.23)

This geometry combines characteristic features of the two previous cases. As in case (a), the SHG contains both dipole and quadrupole components. The dipole component of the polarization has the same value as Equation 26.16 but a perpendicular, rather than a longitudinal, orientation. For this reason, its radiation has an anisotropic pattern and is affected by the image force (as in case (b)) resulting in the additional dependence of the SHG on ε(2ω): I v2ω =

4 π3ω3a 4  Pv2ω  c 2 [ ε(2ω) + 1]

2

2

sin2 φ =

256π3 γ 2ω5a 4  E0ω 

4

c 4 [ ε(ω) + 1] [ε(2ω) + 1] 4

2

sin2 φ. (26.24)

26-9

The surface polarization component caused by the first term in Equations 26.15 and 26.23 is exactly the same as in the case (a) (it results from the fact that this component is determined exclusively by light polarization being independent of the value and the direction of k) and hence is described by Equation 26.19. Let us summarize the results obtained thus far. The SHG was shown to consist of volume and surface components. They have, respectively, dipole and quadrupole symmetry and a different spatial distribution of emitted radiation at 2ω. It can be isotropic (Equation 26.17), bidirectional (Equations 26.22 and 26.24), or of a four-lobed shape (Equation 26.19). However, the integral radiation intensity, as in the case of spherical nanodots (Agarwal and Jha 1982), is given by qualitatively similar expressions differing by numerical coefficients and factors depending only on the NW dielectric constant. For this reason, SHG anisotropy is independent of the NW radius a (until the condition aω/c > 1, the ratio of the total SHG intensities I v2ω + I s2ω (symbol Ī means averaging over ϕ) for all three cases considered is 36:ε3:4. It means that the SHG is maximal for the case (b) when the exciting light is polarized along the NW axis. The result is not unexpected since only in this geometry the optical electric field inside the NW is not weakened by NW polarization and is equal to the external field. It correlates with the linear optical phenomena (absorption, photoconductivity, photoluminescence) which are also maximal for this light polarization. Much less trivial is the dependence of the SHG in perpendicularly polarized light in the direction of the light wave vector. The SHG is essentially larger for light propagating along the NW since, for this geometry, the effective dipoles forming Pv2ω are oriented parallel to the NW and their emission is not weakened by NW polarization. It is worth noting that in all three cases, the spatial distribution of the SHG has different characteristics. In case (a), the second harmonic is emitted in all directions perpendicular to the NW (the contribution of the quadrupole component is around 10%), while in case (b), the emission has a dipole and in case (c), a quadrupole symmetry. This fact should be taken into account while interpreting experimental results (see Section 26.2.4.2).


The main distinguishing feature of metal systems is the strong frequency dispersion of ε(ω), which, as in Section 26.2.2, will be approximated by the Drude formula. It is seen that, due to the factors ε(ω) + 1 and ε(2ω) + 1 in the denominators of Equations 26.17, 26.19, 26.22, and 26.24, the SHG is amplified dramatically near the frequencies ω p / 2 and ω p /(2 2 ) when the excitation frequency or its second harmonic coincides with the transverse plasmon frequency in a cylinder. In case (b), the resonance exists only for 2ω since the exciting light is polarized along the NW and does not interact with plasmons. It is interesting to consider SHG anisotropy near these two resonance frequencies when the effect itself is anomalously large. As has been mentioned, at ω ≅ ω p / 2 the SHG increases dramatically for cases (a) and (c) remaining constant in case (b). In other words, for light with parallel polarization, the SHG is negligibly small compared to the case of perpendicular polarization (the situation opposite to semiconductor NWs). The surface quadrupole component (Equation 26.19) is in this case almost two orders of magnitude less than the volume dipole component. The formulae, Equations 26.17 and 26.24, describing the latter for two different geometries, differ by only 12%. This means that for perpendicular polarization, contrary to the semiconductor case, the total SHG intensity is practically independent of the direction of illumination, though in case (a), emission is isotropic and is polarized parallel to the NWs, while in case (c), it has a dipole symmetry and is polarized perpendicular to them. At the plasmon resonance for emitted light, ω ≅ ω p /(2 2 ) , all contributions to the SHG, except Equation 26.17, contain the diverging factor [ε(2ω) + 1]2, so that the SHG for all light directions and polarization increases at ω → ω p /(2 2 ) in a similar manner with the anisotropy remaining constant. Taking into account that ε(ω p /(2 2 )) ≅ −7, it can be seen that in the vicinity of this low-frequency resonance, light with the parallel polarization (case (b)â•›) causes a SHG three orders of magnitude larger than that with perpendicular polarization, qualitatively similar to the situation in semiconductor NWs.

100

SHG intensity (a.u.)

26-10

80 60 40 20 0

0

20

40

60

80

100

θ (deg.)

Figure 26.8â•… Polarization dependence of the SHG intensity (Barzda et al. 2008). The dots represent experimentally measured values, the solid line corresponds to the theoretical formulae (Equations 26.25 and 26.26). (From Barzda, V. et al., Appl. Phys. Lett., 92, 113111, 2008. With permission.)

of an effective dipole parallel to the light wave vector with the intensity distribution Equation 26.22 proportional to sin2 φ, while in the SHG caused by E⊥, the dominating component has a quadrupole character (Equation 26.19) with the intensity proportional to sin 2(2φ). In the experiments, the detector collected radiation from a wide angle −49° < φ < 49° (see Figure 26.9), so that Equations 26.19 and 26.22 should be integrated in these limits. This results in an additional form-factor equal to 0.36 for parallel and to 0.93 for perpendicular light polarization. By taking into account that E∥ = E0 cos θ and E⊥ = 2E0sin θ/ (ε + 1), we obtain the final result for the detected intensity of the SHG:

26.2.4.2â•‡Experiment First experimental investigations of SHG in NWs (Barzda et al. 2008) were performed on ZnSe NWs with a length of 8–10 microns and a diameter of 80–100â•›nm, excited by a laser at a 1029â•›nm wavelength, that satisfies the quasi-static condition ωa/c > 1, Equation 26.25 predicts giant angular oscillations of the SHG with the maximum/ minimum ratio of the order ε3/10. For ZnSe, the value of ε reported by different authors varies from 5.9 to 7.2 (Grigoriev and Meilikhov 1997) but even for the smallest of these numbers, the theoretical ratio exceeds the corresponding experimental value from Figure 26.8, which is close to 20. The most evident explanation is attributed to deviations from an ideal cylindrical NW geometry used in deriving Equations 26.19 and 26.22. Freestanding NWs are inevitably bent, so that the local value of θ changes along the NW. After replacing θ by θ + δθ and averaging over small δθ with Gaussian properties, Equation 26.25 acquires an additional term

δI SH ∝ 3E04 (δθ)2 sin2 θ 2 cos2 θ + (δθ)2 sin2 θ . (26.26)  

As a result, for (δθ)2ε

32

> 2 the maximum/minimum ratio

2

becomes equal to 3(δθ) , independent of the NW dielectric constant. The solid line in Figure 26.8 shows the theoretical angular dependence ISH(θ) given by Equations 26.25 and 26.26 demonstrating a good agreement between theory and experiment. The best fitting of results was obtained for the value of the average NW corrugation (δθ)2 ≅ 0.2.

26.2.5â•‡Core–Shell Nanowires 26.2.5.1â•‡Potential Distribution So far, we have separately analyzed the optical properties of semiconductor and metal nanostructures and demonstrated their essential distinctions. In this connection, it is very interesting to investigate the properties of core–shell structures containing both types of layers, where luminescence in the semiconductor part can be amplified due to the plasmon effects in the metal part. Fabrication of such structures has been demonstrated experimentally (Lauhon et al. 2002, Choi et al. 2003). As a starting point, we must find the formula for the intensity of the optical electric field inside such a structure, which would generalize Equation 26.1 to the case of an NW consisting of a central core with r1 and a dielectric constant ε1, and a shell with radius r2 and a dielectric constant ε2. Calculations show (Ruda and Shik 2005a) that an external electric field E 0 , perpendicular to the NW axis, creates in the core a uniform field

E1 =

4ε 0 ε 2 E 0 ≡ AE 0 2ε2 (ε1 + ε0 ) + p(ε2 − ε1 )(ε2 − ε0 )

E2 =

(26.25)

(26.27)

where p = 1â•›−â•›(r1/r2)2 is the relative shell volume. The field in the shell is a sum of a uniform field and that of a linear dipole:

2ε 0 2ε 2 (ε1 + ε 0 ) + p(ε 2 − ε1 )(ε 2 − ε 0 )  2r(rE 0 )   E × (ε1 + ε 2 )E 0 + 2(ε1 − ε 2 )(1 − p)r12  20 − . r 4   r 

(26.28)

For E0 parallel to the NW axis, E1 = E2 = E0. Formulae (Equations 26.27 and 26.28) can be used to determine the spectra and the polarization dependence of optical absorption in core–shell NWs. We restrict our analysis to the already mentioned case of a metal shell and a semiconductor core, promising luminescence amplification. The specific features of the optical response of metal shells arise from the fact that the plasmon-related absorption maximum for the light with perpendicular polarization is split into two maxima with positions depending on the relationship between r1 and r2. The frequency regions near the plasmon resonances are characterized by an anomalously high electric field strength (light amplification) both in the core and in the shell. As it has been already pointed out for core–shell nanodots (Neeves and Birnboim 1989), this can be used for the enhancement of optical effects in a core (in NWs—only for light with perpendicular polarization). There is also a reciprocal effect: an oscillating dipole d0 in the core is created far from the wire radiation field that may be considerably larger than that in a uniform medium. This effect of light amplification will be considered later in Section 26.2.5.3. In the next section, we investigate the optical absorption near the plasmon resonances caused by the increased electric field E2 in the shell. 26.2.5.2â•‡Spectra and Polarization Dependence of Absorption To calculate the optical absorption in a core–shell NW caused by light with a perpendicular polarization, we must integrate the local absorption Im ε2|E2(r)|2 over the whole shell. Using the Drude formula for ε2 and Equation 26.28 for E2(r), we obtain the absorption spectra shown by solid lines in Figure 26.10. These contain two maxima corresponding to plasmon resonances where the real part of the denominator in Equations 26.27 and 26.28 vanishes and the electric field in the NW becomes anomalously large. With an increasing p, the maxima approach each other and the low-frequency one diminishes so that at p → 1 we get a single peak at ω = ω p / 2 corresponding to the plasma resonance in a metal cylinder. Similar phenomena in core–shell nanodots have been already observed experimentally (Zhou et al. 1994, Oldenburg et al. 1998) and have a good theoretical description (Neeves and Birnboim 1989, Ruda and Shik 2005b). It is important to emphasize that, contrary to nanodots, the described two-peak absorption spectra in NWs are observed only for light polarized perpendicularly to the NWs. For parallel polarization, when the electric field in the NWs is equal to the external one, we have simple monotonically decreasing spectra reflecting the low-frequency absorption given by Im ε2.

26-12


7 2

5

Absorption (a.u.)

Absorption (a.u.)

6

4 3

1

1

2 1 0 0.0

(a)

3

2

2 1

1

1

2

2 0.2

0.4

0.6 ω/ωp

0.8

1.0

0 0.0

1.2

0.2

0.6 ω/ωp

0.4

(b)

0.8

1.0

1.2

3

Absorption (a.u.)

2 2

1

1 1 0 0.0

0.2

0.4

(c)

2 0.6 ω/ωp

0.8

1.0

1.2

Figure 26.10â•… Absorption spectra of NW with a metal shell for p = 0.2 (a), 0.5 (b), 0.8 (c) and ν = 0.1ωp and a semiconductor core with ε1 = 3 (curves 1) and ε1 = 10 (curves 2) for light polarized perpendicularly to NWs. Solid curves correspond to the absorption in a metal shell, dashed curves—in a semiconductor core. (From Ruda, H.E. and Shik, A., Phys. Rev. B, 72, 115308, 2005a. With permission.)

If the core is a semiconductor with the bandgap Eg < ħω then, besides the absorption described above, there is also interband absorption in the core proportional to |E1(r)|2. The spectrum of this absorption can be easily calculated from Equation 26.27 and is shown by dashed lines in Figure 26.10. The spectrum also has a double maximum structure qualitatively similar to a metal shell one, but contrary to that, suffers a dramatic drop with an increasing ε1 due to the extrusion of the electric field from the high-ε core. We emphasize that absorptions in a shell and a core depend on different parameters (i.e., the plasma frequency and the scattering rate in metals, and the bandgap width and the interband matrix element in semiconductors) and thus we can not compare the amplitudes of solid and dashed curves and say a priori what type of absorption is dominating. Moreover, it is also difficult to distinguish experimentally these two absorption components since the absorption spectra in a shell and a core have qualitatively similar characteristics. The problem can be solved by measuring the excitation spectrum for core luminescence, which is proportional only to the partial absorption in the core shown by the dashed lines in the figure. As mentioned above, the dashed lines in Figure 26.10 actually show the spectral dependence of the electric field in a core |E1(r)|2. As we have shown in Section 26.2.1, the semiconductor

and the metal NWs have opposite characteristics of electrical anisotropy: the perpendicular component of the electric field is suppressed as compared with the parallel one, in a semiconductor NW, and enhanced in a metal NW in the vicinity of a plasmon frequency. In composite core–shell structures, both cases can be realized, depending on the shell thickness and the light frequency. The corresponding “phase diagram” using p-ω axes, is shown in Figure 26.11. The areas above the line(s) correspond to the “metal” case E⊥ > E0⊥ while those below the line(s)—to the “semiconductor” case E⊥ < E 0⊥. In other words, these lines correspond to |A| = 1 where the amplification factor A ≡ E⊥ /E 0⊥ is given by Equation 26.27. At a small p, we naturally deal with the “semiconductor” regime. One might be surprised that at p → 1, the regime is not “metal” but should be pointed so that the figure describes the field not in a metal shell, but in a semiconductor core, which in the limit of a pure metal nanowire p → 1 simply disappears. The condition |A| > 1 is satisfied in the vicinity of plasmon resonances, which means that for light-inducing interband absorption in the core, the intensity of the electron-hole pair generation, and hence, the intensity of the luminescence caused by these pairs is larger than in the absence of a metal shell. That is why, for A, we use the term “amplification factor.”

26-13


in frequencies of exciting ωex and emitted ωem light is usually quite large and cannot be covered by a single plasmon peak. However, in core–shell structures, such double amplification can be reached by using both plasmon peaks as shown in Figure 26.10. In this case, the shell parameters should be chosen in such a way that the low-frequency peak corresponds to ωem in a given NW core while the source of excitation is tuned for ωex to coincide with the second high-frequency peak (Ruda and Shik 2005a).

1.0

0.8

p

0.6

0.4

26.3â•‡Arrays of Nanorods

0.2

0.0

26.3.1â•‡Polarization Memory in Random Nanorod Arrays 0.0

0.2

0.4

ω/ωp

0.6

0.8

1.0

Figure 26.11â•… Values of p and ω, for which E⊥ = E 0⊥ at ε1 = 3 (solid line) and ε1 = 10 (dashed lines). The area above the line(s) corresponds to E⊥ > E 0⊥ . (From Ruda, H.E. and Shik, A., Phys. Rev. B, 72, 115308, 2005a. With permission.)

26.2.5.3â•‡Luminescence Amplification Metal shells may cause amplification not only of the ac electric field of exciting light but also of the light emitted by a semiconductor core. The ac electric field emitted by a core, which can be considered as an effective dipole d0 , is disturbed by a metal shell so that the field far from the nanostructure looks like one created by some other effective dipole d. By analogy with the previous section, we may expect that in the vicinity of a plasmon resonance, the condition |d/d0| >> 1 can be realized, which means an effective amplification of the emitted radiation as well. Calculations show that the luminescence of core–shell structures is characterized by exactly the same rules as absorption. The field of dipoles parallel to the NW axis is not disturbed by image forces, while the field of perpendicular dipoles acquires the same amplification factor A as given by Equation 26.27. As a result, for an isotropic distribution of effective dipoles, such as in a cubic semiconductor core in the absence of size quantization, the emission from an NW acquires a partial polarization parallel to its axis at |A| < 1 (as in Section 26.2.3) or perpendicular at |A| > 1. Since A has a strong frequency dependence, the components of the emitted light with a different polarization may have different spectra depending on the metal shell parameters. From the applied point of view, the most important conclusion is as follows: by a proper choice of the metal-shell parameters, we can make one of the plasmon peaks coincide with the luminescence line in the core, which will result in an increase in the luminescence intensity (Ruda and Shik 2005b). It would be even more attractive to enhance the photoluminescence dramatically by a simultaneous amplification of both the exciting and the emitting light. Such a situation often occurs in surfaceenhanced Raman scattering (Moskovits 1985) but in luminescence, its realization is more difficult since the difference

26.3.1.1â•‡Theory The whole sections 26.2 was devoted to the optical properties of individual NWs or NRs and now we will discuss similar effects in NW arrays. All the polarization-dependent effects remain the same in such arrays as a system of parallel NWs (say, in porous dielectric matrices) with a relatively large distance between the NWs, but in the case of randomly oriented semiconductor NWs or NRs (say, in a polymer matrix or solution), the situation changes. Such a system has macroscopically isotropic optical properties but simultaneously possesses a very interesting property of polarization memory (Lavallard and Suris 1995, Ruda and Shik 2005a). If we excite photoluminescence in the system using polarized light, in accordance with Section 26.2.1, nonequilibrium carriers will be generated mostly in the NWs that are oriented close to and parallel to the light polarization. According to Section 26.2.3, light emitted by these NWs will have a preferable polarization parallel to their axes, or in other words, parallel to the polarization of the exciting radiation. In other words, an anisotropic random array of nanostructures “remembers” the polarization of the exciting radiation and emits luminescence in the same direction of polarization. To calculate this effect in a random system of thin NWs, we assume that the NW array occupies the semi-space x < 0 and study the polarization of the net luminescence excited by z-polarized light and measured at some point (x0,â•›0,â•›0) outside the array (that is, x0 > 0). The principle of calculation is as follows. We take some NWs with an orientation characterized by the spherical angles (θ,â•›α) and consider it as a line of emitting dipoles with components d 0 along the NW axis and 2d 0ε0/(ε + ε0) in two other directions. Then, as in Section 26.2.3.1, we find the components (Sx)∥ and (Sx)⊥ of the Poynting vector created by these dipoles at the point (x0,â•›0,â•›0). Since, according to Equation 26.3, the intensity of the absorbed exciting light and hence the intensity of the luminescence depends on θ being proportional to (ε + ε0)2cos2 θ + 4ε0sin2 θ, we integrate (Sx)∥ and (Sx)⊥ over θ and α with this weighting factor obtaining the total intensities I∥,⊥ of light emission with polarizations parallel and perpendicular to the polarization of the exciting light. Referring the reader to Ruda and Shik (2005a) for the details of calculations, we present only the final result:

26-14


I|| = I⊥ 9[(ε + ε0 )2 + 2ε 20 ](ε + ε 0 )2 + 20[(ε + ε 0 )2 + 2ε 20 ] + 21(ε + ε 0 )2 + 336ε20 . 2 2 2 2 2 2 2 3[(ε + ε0 ) + 2ε0 ](ε + ε0 ) + 44[(ε + ε0 ) + 2ε0 ] + 63(ε + ε0 ) + 168ε 0

Section 26.2.1.2. The final answer for the ratio of luminescence intensities with polarization parallel and perpendicular to that of the exciting light is I||e = I ⊥e

(26.29)

2I ⊥a (ω ex )[47d x2 (ω em ) + 10d z2 (ω em )] + 3I||a (ω ex )[13d x2 (ω em ) + 12dz2 (ω em )] . 4 I ⊥a (ω ex )[16dx2 (ω em ) + 11dz2 (ω em )] + 3I||a (ω ex )[23d x2 (ω em ) + 4dz2 (ω em )]

(26.30)

0.4

0.2

0.4 2.4

0.2

1.0 2.6 2.8

2.2

1.6

0.0 0.8

–0.2

2.0 1.8 0.6

–0.4

3.0

0.0

0.5

1.0

1.5 ωem a/c

2.0

2.5

3.0

0.5

1.0

1.5 ωex a/c

2.0

2.5

3.0

(a)

0.5

0.2 0.4 0.6

0.2 Pa

0.0

0.3

–0.2

P

1.2

1.4

0.4

0.4

To clarify, we emphasize the difference in notations between Equations 26.29, 26.30 and 26.10 where I||e and I ⊥e referred to the light polarized parallel and perpendicular to the NW axis. Figure 26.13 shows the dependences of the system polarization ratio P e = ( I||e − I ⊥e ) ( I||e + I ⊥e ) on the frequencies of the exciting

Pe

Equation 26.29 shows that maximal possible polarization, reached at |ε/ε0| >> 1, is equal to I∥/I⊥ = 3. In terms of the polarization ratio P = ( I|| − I ⊥ ) ( I|| + I ⊥ ) it corresponds to P = 0.5. Figure 26.12 shows this polarization ratio for an arbitrary relationship between ε and ε0. For InP NWs in air (ε/ε0 = 12.7), we get P = 0.44. It is worth noting that at ε < ε0 (e.g., in metal NWs near the plasmon resonance), when the electric field in the NWs has a preferably perpendicular orientation, the system also has a polarization memory. This effect is weaker than at a large ε, with the maximal P = 6/68 = 0.088. As we have shown in Sections 26.2.1.2 and 26.2.3.2, for thick NWs, the polarization characteristics for absorption and emission vary dramatically with the light frequency. As a result, the polarization memory in these objects must demonstrate a strong dependence on the frequency of both the exciting (ωex) and the emitted (ωem) light and even change its sign at some intervals of ωex and ωem. The quantitative description of this effect (Ruda and Shik 2006) is based on the same approach as that for thin NWs resulting in Equation 26.29 but with two distinctions. First, the values of the longitudinal and the transverse components of the effective dipole moments in the NW are no longer constant but given by Equations 26.12 and 26.13 with k = ωem/c. Second, the intensity of the absorbed exciting light and hence the intensity of the luminescence (the weighting factor in the angular averaging) is now equal to I||a (ωex )cos2 θ + I ⊥a (ωex )sin2 θ where the intensities of the absorbed light I||a and I ⊥a are determined from Equations 26.5 and 26.6 according to the procedure described in

1.4 0.8 1.6 2.4 1.8 2.8 2.0 1.0 2.2 1.2

0.2

–0.4 0.1 0.0

(b)

0

5

10 ε/ε0

15

20

Figure 26.12â•… Luminescence polarization ratio in a random system of nanowires exited by polarized light. The dot-and-dash line corresponds to the limiting value P = 0.5. (From Ruda, H.E. and Shik, A., Phys. Rev. B, 72, 115308, 2005a. With permission.)

0.0

Figure 26.13â•… Polarization ratio of photoluminescence for a random NW system with ε = 9, ε0 = 1. (a) Luminescence spectra P e(ωema/c) for different ωex. Numbers at the curves indicate the values ωex/c. The curves corresponding to ωexa/c = 2.6 and 2.8 practically coincide. (b) Excitation spectra P a(ωexa/c) for different ωems. Numbers at the curves indicate the values of ωema/c. The curves corresponding to ωema/c = 1.8 and 2.8 practically coincide. For ωema/c = 2.6 polarization is very close to zero and is not presented in the figure.

26-15


and the emitted light, given by Equation 26.30. Since in standard photoluminescence measurements ωex always exceeds ωem, parts of the curves in Figure 26.13 where this condition is violated are given by dashed lines. At ωex, ωem → 0, polarization tends to its low-frequency limit shown in Figure 26.12 and for our choice ε/ε0 = 9 equal to P e ≅ 0.40. It is important to mention that if the NW material is characterized by several luminescence lines with essentially different frequencies, then these lines may have different degrees of polarization, in accordance with Figure 26.6. Since the curve at the figure crosses the level I||e = I ⊥e , there exists even the possibility when one of the luminescent lines may have. The latter case could be called “polarization anti-memory.” 26.3.1.2â•‡Experiment The effect of polarization memory was first observed in porous Si (Kovalev et al. 1995) that has only a distant resemblance to a system of NWs or NRs. Hence, for demonstration of its main experimental features, we have chosen subsequently more detailed measurements in CdSe/ZnS NRs (Kravtsova et al. 2007) with the aspect ratio l/2a ≅ 2.4 dissolved in a liquid and hence having a randomly distributed orientation. Figure 26.14 shows the luminescence spectrum consisting of two peaks corresponding to interband transitions between different sizequantized states. Polarization properties of the emitted light are presented in the form of luminescence spectra for different angular positions of the analyzer related to the direction of the exciting light polarization E0 . The luminescence is seen to be

Luminescence intensity (a.u.)

θ = 0° 2 θ = 90°

1

×10 θ = 0°

θ = 90° 400

450


600

650

Figure 26.14â•… Photoluminescence spectrum of the NR solution for excitation by linearly polarized light with λ =366â•›nm. Different curves correspond to the luminescence components with a polarization in the direction forming the angle θ with the polarization of excitation. θ increases from the upper to the lower curve with the step 10°. The amplitude of the luminescence in the spectral region λ < 520â•›nm is shown with the magnification factor 10. (From Kravtsova, Y. et al., Appl. Phys. Lett., 90, 083120, 2007. With permission.)

polarized mostly parallel to E0 , in qualitative agreement with theoretical predictions. The quantitative treatment of the results meets with some difficulties. According to Equation 26.29, the amplitude of polarization memory in small nanostructures has some universal value depending only on ε/ε0. However, in the experiment, this amplitude differs noticeably for two different spectral lines in the same NRs. Besides, for the short wavelength peak, the amplitude of polarization memory exceeded the theoretically predicted one, while for the long wavelength peak, it was always less. These results could be explained if the matrix element responsible for the ground state 573â•›nm-long wavelength peak is larger for perpendicular light polarization (it is really the case for NRs with a small aspect ratio (Hu et al. 2001)â•›) while that at 470â•›nm is either more isotropic or larger for parallel polarization. This is in agreement with the theoretical (Li and Wang 2003) and the experimental (Thomas et al. 2005) conclusions of essentially different polarization properties of different optical transitions in NRs.

26.3.2â•‡Plasmon Spectra in Self-Assembled Nanorod Structures 26.3.2.1â•‡Self-Assembling of Metal Nanorods In the recent years, an impressive success has been achieved in fabricating various arrays of metal NRs by their self-assembling in solution. The most comprehensive results in this direction were obtained by using Au NRs with the lateral surface covered by a double layer of hydrophilic molecules and the ends terminated with polystyrene molecular chains (Nie et al. 2007, 2008). Such building blocks in solution can form different ordered arrays, such as chains, bundles, raft-like structures, rings, etc. (Figure 26.15), depending on the chemical composition of the solvent. In these systems, the distances between the NRs are very small, compared to their diameter, so that the electric fields of plasmons in neighboring NRs overlap and the plasmon spectrum of the system becomes different from that of individual NRs and depends on the NR array topology. Figure 26.16 shows the plasmon absorption spectra in some of these structures formed from identical NRs. They consist of two peaks (longitudinal and transverse plasmons) where the long wavelength one, corresponding to longitudinal plasmons (see Section 26.2.2.1), demonstrates a strong dependence on the array geometry. Some general qualitative regularities were already formulated in literature and found their experimental confirmation in systems of NRs and nanodots. It was demonstrated (Rechenberger et al. 2003, Su et al. 2003) that by reducing the inter-nanoparticle separation in the direction of light polarization, a red shift in the plasmon resonance occurs, in contrast with a blue shift for the perpendicular direction of polarization. It has a simple qualitative explanation. The end-to-end alignment of NRs increases the effective length of an NR, which results in a decrease of the longitudinal depolarization factor n∥ and, according to Equation 26.7, a decrease in the longitudinal plasmon frequency ω∥, while the side-to-side arrangement increases the effective n∥ and ω∥.

26-16


(a)

(e)

(i)

(o)

(b)

(f )

(l)

(p)

(c)

(g)

(m)

(q)

(d)

(h)

(n)

(r)

Figure 26.15â•… NR arrays obtained by self-assembling in different solvents. (From Fava, D. et al., Adv. Mater., 20, 4318, 2008. With permission.) 1

Absorbance (a.u.)

0.2

2

3

electric field E0. We assume that the NW radius a is many times smaller than both their length l, so that for calculations of longitudinal plasmons, the radial distribution of the induced charges is inessential and we may characterize each NW by a linear charge density q(z) depending on one single coordinate z measured along the NW axis. In the absence of a direct electrical contact between the NWs, their total charges remain zero:

4

0.1

∫

l /2

−l /2

0.0

400

600

800

1000

1200

Wavelength (nm)

Figure 26.16â•… Optical absorption spectra for various NR arrays. (From Nie, Z.H. et al., J. Am. Chem. Soc., 130, 3683, 2008. With permission.)

In the next sections, we present theoretical models allowing one to make some quantitative calculations of plasmon characteristics and their comparison with the experiment.

q(z ) dz = 0. For each metal NW, the charge distribution q(z)

is found from the condition of equipotentiality (Ruda and Shik 2007b). It means that the external field potential −E0zz is exactly compensated by the joint potential created by the charges of all NWs including the given one. To describe the potential created by q(z) of the same NW in a medium with the dielectric constant ε, we use the approximate expression 2q(z)ln(l/a)/ε valid for l >> a (Averkiev and Shik 1996). Potentials of all other NWs can be obtained by a simple Coulomb expression neglecting the finite thickness of NWs. We demonstrate the described procedure for the case of two in-line NWs separated by the interval b and occupying the segments (−l − b/2, −b/2) and (b/2, l + b/2) of the z-axis. Noting the charge densities in the NWs as q−(z) and q+(z), respectively, we obtain the system of two equations:

26.3.2.2â•‡Longitudinal Plasmon Shift Let us consider some arbitrary system of metal NWs or strongly anisotropic NRs (further we will mention them simply as NWs) and find the polarization of each NW in an external uniform

2q± (z )  l  ln   − E0 z z +  a ε

∓b / 2 ∓ l

∫

∓b / 2

q∓ (z ′) dz ′ = C± . (26.31) ε(z ′ − z )

26-17


Instead of q±, we introduce two dimensionless functions η±(ξ) determined from

εE0 z  z + l η± (ξ) 2 ln(l /a) 

where ξ is the dimensionless coordinate measured from the center of the corresponding NWs: ξ = [z ∓ (b/2 + l/2)]/l. It can be easily shown that η−(−ξ) = −η+(ξ). Finding C± from the neutrality conditions:

∫

1/ 2

−1/ 2

η± (ξ) dξ = 0, we reduce Equation 26.31 to one

single equation for η+(ξ):

=

1/ 2

∫

−1/ 2

η+ (ξ′) dξ′ + ξ + ξ′ + 1 + β

1/ 2

 2ξ′ + 3 + 2β 

∫ η (ξ′)ln  2ξ′ + 1 + 2β  dξ′ +

−1/ 2

 2ξ + 3 + 2β   2 + β 1 − (ξ + 1 + β)ln  + ln   2  2ξ + 1 + 2β   1+ β 

+

β  β(2 + β)3  β2  β(2 + β)  ln  .  + ln 2  (1 + β)4  2  (1 + β)2 

(26.33)

This equation was solved numerically for l/a = 10, which corresponds to the experimental results (Nie et al. 2007) to be discussed later. After q(z) is found, the total dipole moment P of an

∫

NW is easily found: P = zq (z ) dz , which, in turn, determines

the depolarization factor n∥ = E0zV/(4πP) (V is the NR volume) (Landau and Lifshitz 1984), connected with the longitudinal plasmon frequency ω∥ by Equation 26.7. Using a similar approach, the frequency ω∥ was found (Ruda and Shik 2007b) for two other systems: the infinite periodic NR chain and a pair of two side-by-side NRs with the distance d > 2a between the axes. The resulting dependences of the longitudinal plasmon frequency ω∥ on inter-NR distances b or d for all three cases are presented in Figure 26.17. It is seen that, in accordance with the above-mentioned qualitative considerations, ω∥ demonstrates a red shift for chains and a blue shift for bunches of NRs. Figure 26.17 also contains experimental points (Nie et al. 2007) partially taken from the curves of Figure 26.16. It is seen that the results for in-line structures (curves 1 and 2) demonstrate a good qualitative agreement but the experimental dependences are steeper than the theoretical. This effect can be caused by the fact that the polystyrene molecules connecting the NR tips are also polarized in the external electric field, so that the effective local dielectric constant of this gap exceeds that of the solvent. This distorts the electric field pattern and increases the intensity of inter-NR interaction and hence, the value of P, as compared to the ideal case considered above. For the side-by-side NR assemblies, practically all the lines of the electric field responsible for the inter-NR interaction are concentrated in the region filled by hydrophilic

3

1.0

1 2

0.9

0.8 0.10

l 2 ln   η+ (ξ) −  a

1.1

(26.32) ω||/(ω||)0

q± (z ) =

1.2

0.15

0.20

0.25

0.30

0.35

b/l, d/l

Figure 26.17â•… Dependence of the longitudinal plasmon frequency ω∥ (in units of ω∥ for individual NRs) on the distance between NRs with l/a = 10. 1, two in-line NRs (experimental points—triangles); 2, chain of NRs (experimental points—squares); 3, two parallel NRs (experimental points—stars).

molecules, so that the dielectric inhomogeneity of the system is irrelevant, and curve 3 demonstrates not only qualitative but also quantitative agreement.

26.4â•‡Conclusion We have described a large variety of polarization-sensitive optical phenomena in anisotropic nanostructures, such as NWs and NRs. The phenomena have a universal character being exclusively due to the difference in the dielectric constants between nanostructures and their environment. They include a strong dependence of linear (absorption, photoluminescence, photoconductivity, etc.) and nonlinear optical properties on polarization of the exciting light, as well as a high degree of polarization of luminescence emitted by these systems. The joint action of absorption and emission anisotropy results in a polarization memory in random arrays of NWs and NRs. In metal nanostructures, all the mentioned phenomena exhibit a dramatic dependence (including sometimes the change of sign) on the light frequency, related to plasmon effects. In composed metalsemiconductor nanostructures, these effects can be used for luminescence amplification. In ordered self-assembled arrays of metal NRs, the plasmon-related optical spectra are sensitive to the array geometry. All the described effects are provided with proper theoretical description and illustrated by experimental observations performed in recent years.

Acknowledgments The authors would like to thank their collaborators, V. Barzda, R. Cisek, D. Fava, E. Kumacheva, Y. Kravtsova, U. Krull, L. Levina, S.F. Musikhin, Z.H. Nee, U. Philipose, and T.L. Spencer from the University of Toronto, whose experimental results are presented in this work.

26-18


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Lauhon, L.J., Gudiksen, M.S., Wang, D., and Lieber, C.M. 2002. Epitaxial core-shell and core-multishell nanowire heterostructures. Nature 420: 57–61. Lavallard, P. and Suris, R.A. 1995. Polarized photoluminescence of an assembly of non cubic microcrystals in a dielectric matrix. Solid State Communications 95: 267–269. Li, J. and Wang, L. 2003. High energy excitations in CdSe quantum rods. Nano Letters 3: 101–105. McIntyre, C.R. and Sham, L.J. 1992. Theory of luminescence polarization anisotropy in quantum wires. Physical Review B 45: 9443–9446. Mei, X., Blumin, M., Sun, M. et al. 2003. Highly-ordered GaAs/ AlGaAs quantum dot arrays on GaAs(100) substrate grown by molecular beam epitaxy using nanochannels alumina masks. Applied Physics Letters 82: 967–969. Moskovits, M. 1985. Surface-enhanced spectroscopy. Reviews of Modern Physics 57: 783–826. Neeves, A.E. and Birnboim, M.H. 1989. Composite structures for the enhancement of nonlinear optical susceptibility. Journal of the Optical Society of America B 6: 787–796. Nie, Z.H., Fava, D., Kumacheva, E. et al. 2007. Self-assembly of metal-polymer analogues of amphiphilic triblock copolymers. Nature Materials 6: 609–614. Nie, Z.H., Fava, D., Rubinstein, M., and Kumacheva, E. 2008. Supramolecular assembly of gold nanorods end-terminated with polymer “pom-poms”. Journal of the American Chemical Society 130: 3683–3689. Oldenburg, S.J., Averitt, R.D., Westcott, S.L., and Halas, N.J. 1998. Nanoengineering of optical resonances. Chemical Physics Letters 288: 243–247. Peng, X., Manna, L., Yang, W. et al. 2000. Shape control of CdSe nanocrystals. Nature 404: 59–61. Philipose, U., Ruda, H.E., Shik, A., de Souza, C.F., and Sun, P. 2005. Light emission from ZnSe nanowires. Proceedings of SPIE 5971: 597116–597128. Qi, J., Belcher, A.M., and White, J.M. 2003. Spectroscopy of individual silicon nanowires. Applied Physics Letters 82: 2616–2618. Rechenberger, W., Hohenau, A., Leitner, A. et al. 2003. Optical properties of two interacting gold nanoparticles. Optics Communications 220: 137–141. Ruda, H.E. and Shik, A. 2005a. Polarization-sensitive optical phenomena in semiconducting and metallic nanowires. Physical Review B 72: 115308. Ruda, H.E. and Shik, A. 2005b. Plasmon phenomena and luminescence amplification in nanocomposite structures. Physical Review B 71: 245328. Ruda, H.E. and Shik, A. 2006. Polarization-sensitive optical phenomena in thick semi-conducting nanowires. Journal of Applied Physics 100: 024314. Ruda, H.E. and Shik, A. 2007a. Nonlinear optical phenomena in nanowires. Journal of Applied Physics 101: 034312. Ruda, H.E. and Shik, A. 2007b. Polarization and plasmon effects in nanowire arrays. Applied Physics Letters 90: 223106.


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26-19

Wang, J., Gudiksen, M.S., Duan, X., Cui, Y., and Lieber, C.M. 2001. Highly polarized luminescence and photodetection from single InP nanowires. Science 293: 1455–1457. Zhou, H.S., Honma, I., Komiyama, H., and Haus, J.W. 1994. Controlled synthesis and quantum-size effect in goldcoated nanoparticles. Physical Review B 50: 12052–12056.

27 Nonlinear Optics with Clusters 27.1 General Introduction: Fundamentals of Nonlinear Optics and Different Nonlinear Optical Effects...................................................................................................... 27-1 27.2 Quantum Formulation of Nonlinear Optics...................................................................... 27-2 27.3 Nonlinear Optics with Materials.........................................................................................27-5 27.4 Why Are Clusters So Important?.........................................................................................27-5 27.5 Review of Nonlinear Optics with Clusters.........................................................................27-5 27.6 Conclusions........................................................................................................................... 27-12 References.......................................................................................................................................... 27-12

Sabyasachi Sen JIS College of Engineering

Swapan Chakrabarti University of Calcutta

27.1â•‡General Introduction: Fundamentals of Nonlinear Optics and Different Nonlinear Optical Effects In recent years, nonlinear optical (NLO) materials have attracted substantial research interests due to their potential applications in information processing and telecommunications [1–4]. NLO processes are particularly useful in photonic devices where photons are used to transmit and process information. Modification in the optical properties of a material by the strong oscillating electric field of a laser beam is in general discussed in the study of NLO. NLO processes have been observed in a wide variety of materials starting from molecules and clusters to bulk materials [5–14]. In molecular systems, the optical nonlinearity depends on the geometrical arrangement of molecules, whereas in case of bulk systems, the electronic characteristics of the system determine the optical nonlinearity. Various experimental as well as theoretical results have reported that NLO properties of small atomic clusters depend on the geometry and the electronic structure property of these materials [15,16]. In general, the electric dipole moment μa of a molecular system is expressed as µ a = µ a (E b (r , t ) = E c (r , t ) = E d (r , t ).... = 0) +

∑α

ab

b

+

1 3!

1 E b (r , t ) + 2!

∑γ

abcd

∑β

abc

E b (r , t )E c (r , t )

bc

E b (r , t )E c (r , t )E d (r , t ) + ,

(27.1)

bcd

In the above equation, Ea (r⃗ ,â•›t), Eb(r⃗ ,â•›t), …, are the external electric fields consisting of a monochromatic and a static

part, and a, b, …. are equal to the Cartesian directions x, y, or z. The first term in Equation 27.1 denotes the dipole moment of the system. αab, βabc, and γabcd represent the elements of the linear polarizability tensor, the first-order hyperpolarizability tensor, and the second-order hyperpolarizability tensor, respectively. The time dependence of the dipole moment leads to various frequency-dependent polarizabilities and hyperpolarizability tensors. The first-order hyperpolarizability tensor is described in terms of static first-order hyperpolarizability [β (0; 0, 0)], second-harmonic generation (SHG) [β (−2ω; ω, ω)], electro-optical Pockels effect (EOPE) [β (−ω; ω, 0)], optical rectification (OR) [β (0; ω, −ω)], sumfrequency generation (SFG), and difference-frequency generation (DFG). Similarly, in the case of the second-order hyperpolarizability tensor, apart from the static part [γ (0; 0, 0, 0)], there exists frequency-dependent components, namely, the third-harmonic generation (THG) [γ (−3ω; ω, ω, ω)], the electro-optical Kerr effect (EOKE) [γ (−ω; ω, 0, 0)], the dc-induced second-harmonic generation (dc-SHG or electric field-induced second-harmonic generation (EFISH)) [γ (−2ω; ω, ω, 0)], and the intensity-dependent refractive index (IDRI) [γ (−ω; ω, −ω, ω)] or the degenerate four-wave mixing (DFWM). Among the different NLO properties, the first-order hyperpolarizability is important because it is the simple third-order molecular property and responsible for phenomena such as the SHG and the EOPE. On the other hand, the second-order hyperpolarizability tensor is related to physical properties like the EOKE. It is also significant in the perspective of quantum optical effects like self-focusing and self-defocusing effects in a material. In the presence of an electric field of field strength

E(t ) = E1e −iω1t + E2e −iω2t + c.c,

(27.2) 27-1

27-2


the second-order nonlinear polarization induced in a crystal is expressed as P (2) (t ) = χ(2) E(t )2 ,

(27.3)

P (2) (t ) = χ(2)[E12e −2iω1t + E22e −2iω2t + 2E1E2e −i(ω1 + ω2 )t + 2E1E2*e −i (ω1 − ω2 )t + c.c] + 2χ(2)[E1E1* + E2 E2* ], (27.4) The first and second terms in Equation 27.4 represent the SHG, which is useful in the perspective of optoelectronic phenomenon such as tuning the frequency of the laser beam. The expression 2 E1E2e −i(ω1 + ω2 )t and 2E1E2*e −i (ω1 − ω2 )t are associated with the sumfrequency generation (SFG) and the difference-frequency generation (DFG) respectively. The SFG is used to produce tunable radiation in the ultraviolet spectral region and the DFG produces tunable radiation in the infrared region. The process of the DFG is also used in the design of the optical parametric oscillator. The last term of Equation 27.4 does not show any frequency dependence and leads to a process known as optical rectification (OR). The third order contribution to the nonlinear polarization is given by P (3) (t ) = χ(3) E(t )3 ,

(27.5)

In the case of the monochromatic applied electric field (E(t) = ξ cos ωt), the nonlinear polarization is expressed as

P (3) (t ) =

1 ( 3) 3 3 χ ξ cos 3ωt + χ(3)ξ3 cos ωt . 4 4

(27.6)

The first term of Equation 27.6 illustrates the third-harmonic generation (THG), which describes the response of the system at the frequency of 3ω. The second term in Equation 27.6 leads to a nonlinear contribution to the refractive index of the electromagnetic wave of frequency ω. This is recognized as the intensity-dependent refractive index (IDRI). In the presence of such nonlinearity, the refractive index becomes

n = n0 + n2 I ,

(27.7)

where n0 is the linear refractive index and

n2 =

12π2 (3) χ , n02c

(27.8)

n2 is an optical constant that characterizes the strength of the optical nonlinearity. I is the intensity of the incident wave expressed as I = (n0 c/8π)ξ2. The intensity-dependent refractive index (IDRI) is related to optical phenomenon, such as the selffocusing of the electromagnetic wave. Self-focusing occurs in a material in which n2 is positive and the electromagnetic waves, with a nonuniform transverse intensity distribution, curve towards each other after passing through such material.

27.2â•‡Quantum Formulation of Nonlinear Optics A large number of quantum chemical techniques have been employed for the evaluation of the NLO properties of materials. Jensen et al. have developed a dipole interaction model. This model successfully describes the response properties of large aggregates of molecular clusters [17]. The density-matrix renormalization group (DMRG)-based formalism was proposed by Pati and coworkers and successfully applied in the description of the frequency-dependent NLO properties of π-conjugated systems [18]. Standard techniques like the time-density functional theory, the time-dependent Hartee Fock technique, as well as the finite-field methods [19] based on DFT are also very successful in describing the NLO responses of several classes of materials. The quantum chemical approach, like the density functional theory, had its limitations in the case of long-range dispersion interaction and has been improved by the introduction of the dispersion-corrected DFT DFD-D [20,21] technique. Higherorder wavefunction-based methods like the coupled cluster approach and the MP2 technique are also efficient but they are excessively expensive. In addition to these, the response properties are also evaluated through semi-empirical methods like ZINDO/MRDCI (multi reference doubles-configuration interaction) formalism with a correction vector. [22]. Another wellcelebrated quantum chemical approach is the use of the two-state model proposed by Oudar and Chemla [23]. This model relates the nonlinear optical coefficients to the excited-state energy, the oscillator strength and the dipole moment of the molecule under consideration. The theoretical results of Zyss [24] and Andrews et al. [25] successfully established the two-state model of Oudar and Chemla. Datta and Pati [26] employed the two-state model to discuss the role of dipolar interaction and H-bonding in tuning the response properties in molecular aggregates. This model is simple and particularly useful in the case of charge-transfer systems [27]. In this two-state model, the expression of β is given by

βtwo-level =

3e 2 ω01 f ∆µ01 2 2 (ω01 − ω2 )(ω201 − 4ω2 )

where f is the oscillator strength ω is the applied frequency ω01 is the frequency of the optical transition between the ground state and the first dipole-allowed state Δμ01 is the difference in moments between the ground state and the first dipole-allowed state In the absence of an applied frequency, the above equation results in a first-order static hyperpolarizability tensor

β(0; 0, 0) = 3π

f ∆µ 01 . ω 301

In two-state formalism, the parameters like f, ω01 and Δμ01 are estimated through standard excited approaches like the

27-3

Nonlinear Optics with Clusters

configuration interaction scheme (CIS) or the time-dependent density-functional theory (TDDFT). Apart from all those techniques described in the above paragraph, another widely accepted quantum chemical technique of response calculation is the analytic approach. In this context, it will be useful to point out some of the basic differences between the finite-difference technique and the analytic approach. In the finitedifference technique, the energy is calculated at different Â�values of the electric field; there after, the finite differentiation is performed to obtain the derivative of energies. The finite-difference method is suitable for any program capable of energy calculation in a perturbed system. However, the main limitation of this method is the prolonged computational time and it is not good enough when higher order derivatives of the energy are involved. On the contrary, in the analytical approach, physical properties are obtained by analytically evaluating the derivatives of the energy and are often preferred in response calculations. The main advantage of this technique is that it gives access to frequency-dependent properties and the data obtained are more accurate. This technique has been successfully applied by van Gisbergen et al. [28] in case of a set of small molecules. Sen et al. employed the same technique in case of CdSe clusters and Al4M4 clusters [11,13]. In the density-functional theoretical formulation of the analytic approach, the starting equation is a variation on the timedependent Kohn–Sham (TDKS) equation. In order to obtain the starting equation of the analytic approach in the time-dependent density-functional theory (TDDFT), the action integral of the TDDFT is considered. t1

∂ A = dt ψ(t ) i − Hˆ (t ) ψ(t ) , ∂t

∫

(27.9)

t0

where ψ is the wave function of the system. In terms of timedependent single-particle orbitals {φj(r⃗ ,â•›t)}, the action integral becomes t1

j

A[{ϕ j }] =

 ∂

∑ ∫ dt ∫ d r ϕ (r ,t )  i ∂t + 3

n

* j

−∞

− dt d 3r ρ(r , t )vext (r , t )

∫ ∫

−∞

1 2

t1

ρ(r , t )ρ(r ′, t ) d t d 3 r d 3r ′ − Axc[{ϕ j }], r − r′

∫ ∫ ∫

−∞

(27.10)

In Equation 27.10, the term vext(r⃗ ,â•›t) is the time-dependent exchange-correlation potential and A xc is the exchange-correlation part of the action functional. On considering the orthonormal Kohn–Sham orbital, the action integral produces the time-dependent Kohn–Sham equation as ∂

 ∇2



∑ ε (t )ϕ (r ,t ) + i ∂t ϕ (r ,t ) = − 2 + ν (r ,t ) ϕ (r ,t ) = F ϕ (r ,t ), ij

j

j

i

s

i

ρ(r , t ) =

occupied

∑

2 ϕi (r , t ) ,

(27.12)

i

When φi(r⃗ ,â•›t) is expanded in a fixed time-dependent basis set of atomic orbitals {χμ(r⃗ )}, the resulting expression becomes

ϕi (r , t ) =

∑ χ (r )C (t ), µ

µ

(27.13)

µ

In Equation 27.13, the coefficient C μ(t) manifests the time dependence of φi(r⃗ ,â•›t). Finally, Equations 27.11 through 27.13 produces the time-dependent Kohn–Sham equation in the matrix form given by

FsC − i

∂ SC = SCε, ∂t

(27.14)

where S is the overlap matrix of the atomic orbitals with

Sµν = dr χ*µ (r )χ ν (r ),

∫

(27.15)

The density matrix (D) is expressed in terms of the coefficient matrix C and the occupation number matrix n and is expressed as

D = CnC †.

(27.16)

The Kohn–Sham equation expressed in Equation 27.14 is the starting point of the perturbative expansion. Equation 27.14 is similar to the starting equation for the NLO calculation in the time-dependent Hartree–Fock technique (TDHF). In Equation 27.14, the term Fs represents the Kohn–Sham matrix or the Fock matrix. This is expressed as

∇2  ϕ j (r , t ) 2 

t1

−

In Equation 27.11, νs(r⃗ ,â•›t) is the time-dependent Kohn–Sham potential and εij(r⃗ ,â•›t) is the Lagrangian multiplier. Different choices of Lagrangian multipliers are allowed. With εij(r⃗ ,â•›t) = 0 we attain the canonical form of the Kohn–Sham equation. In the case of orbitals varying rapidly with time, εij(r⃗ ,â•›t) is suitably chosen to avoid an unphysical divergence in the Kohn–Sham equation. The time-dependent density ρ(r⃗ ,â•›t) is obtained from the relation

s

i

(27.11)

Fs = h + DX 2J + v xc .

(27.17)

In the above equation, h is the one-electron integral matrix comprising kinetic energy, the Coulomb field of the nuclei, and the applied external electric field. J is the four-index Coulomb supermatrices. When all matrices are expanded in different orders of external perturbation and ε is chosen as the time-dependent zero-order matrix, the canonical Kohn–Sham equation for the ground-state DFT is obtained and given by the following expression:

Fs(0)C (0) = S(0)C (0)ε(0) .

(27.18)

27-4


The Lagrangian multiplier matrix [ε(0)] can be chosen at each order of perturbation. Now, if an external electric field of Ea(r⃗ ,â•›t) is applied, the mathematical form of Ea(r⃗ ,â•›t) being

E a (r , t ) = E a (r )X[1 + eiωa t + e −iωa t ],

(27.19)

then in the dipolar approximation, the external perturbation term (H) to the Kohn–Sham Hamiltonian becomes

H = µ ⋅ E(r , t )

(27.20)

where μ is the dipole-moment operator of the electrons. The time dependence of the dipole moment produces various frequency-dependent hyperpolarizablity tensors. Frequencydependent polarizability and hyperpolarizability tensors are obtained from the trace of the dipole moment matrix [Ha] and the nth-order density matrix [D(n)] (where n = 1 for the linear polarizability α, n = 2 for the first-order hyperpolarizability tensor β, n = 3 for the second-order hyperpolarizability γ and so on). If the inducing electric field of frequency ωa, ωb, ωc … acts in the direction a, b, c… the tensors can be represented as

α ab (−ω σ ; ωb ) = −Tr  H a D b (ω b ) ,

(27.21)

βabc (−ω σ ; ωb , ω c ) = −Tr  H a D bc (ωb , ω c ) ,

(27.22)

γ abcd (−ω σ ; ωb , ω c , ω d ) = −Tr  H a D bcd (ωb , ω c , ω d ) , (27.23)

where ωσâ•›=â•›ωbâ•›+â•›ωcâ•›+â•›… These expressions are a set of generalized equations in terms of frequencies. The polarizability and the hyperpolarizability tensors defined above are determined through the iterative solution of the time-dependent Kohn– Sham (TDKS) equations given in Equation 27.14. In ordinary TDDFT, the zeroth-order Kohn–Sham matrix is used, which is given by

Fs(0) = h(0) + D (0) X(2J ) + ν(0) xc ,

(27.24)

where h(0) contains the external potential terms that are of zero order in the external field, kinetic energy and the nuclear Coulomb field. The Coulomb super matrix J is independent of the external electric field and results in a Coulomb term of the form Dab…n(ωa, ωb, …, ωn)X(2J) in the nth-order Kohn–Sham matrix Fsab…n (ωa , ωb ,…, ωn ) . The external perturbation appears only in the first-order Kohn–Sham matrices. Higher-order Kohn–Sham matrices consist of a Coulomb and an exchangecorrelation part. Therefore, the general formula for the higherorder Kohn–Sham matrices becomes Fsab...n (ω a , ω b ,…, ω n ) = D ab…n (ω a , ω b ,…, ω n )X(2 J )

ab...n + v xc (ω a , ω b ,…, ω n ),

(27.25)

The Taylor expansions of Fs, C, ε, and D are inserted in the time-dependent Kohn–Sham equation (Equation 27.14), the normalization condition, and the density matrix (Equation 27.16). This results in a first-order time-dependent Kohn–Sham equation as a (0 ) 0 a 0 a (0 ) a (0 ) (0 ) (0 ) a Fs (ω)C + Fs C (ω) + ωS C (ω) = S C (ω)ε + S C ε (ω). (27.26)

Higher-order coupled equations are obtained by equating the left and the right hand sides of the TDKS equation, the normalization condition and the density matrix. The NLO properties are estimated through the iterative solution of the time-dependent Kohn–Sham equations up to a certain order n. Primarily, the static Kohn–Sham equations are solved that result in matrices Fs(0) , C (0) , ε(0) , D (0), and the converged SCF density ρ(0). These matrices are needed for the solution of the first-order Kohn– Sham equation, which produces the first-order density matrix. The first-order density matrix yields the frequency-dependent polarizability component [αab (−ωσ; ωb)] through Equation 27.21. The solution of the first-order equation provides matrices required for an iterative solution of the second-order equation. In either case, the technique adopted is called the (2n + 1) theorem. The second-order equation is solved to get the second-order density matrix elements from which the frequency-dependent firstorder hyperpolarizability tensor is obtained through Equation 27.23. If the external fields of the frequency zero and a common frequency ω are considered, a number of very important NLO properties become accessible. These are SHG [β (−2ω; ω, ω)], EOPE [β (−ω; ω, 0)], OR [β (0; ω, −ω)], and the static hyperpolarizability [β (0; 0, 0)]. Similar to the two previous cases, the (2n + 1) theorem is also used to compute the higher-order hyperpolarizability tensors like γ, δ (third-order hyperpolarizability tensor), and the rest. When γ is calculated only at frequencies 0 and ω, THG [γ (−3ω; ω, ω, ω)], EOKE [γ (−ω; ω, 0, 0)], dc-SHG [γ (−2ω; ω, ω, 0)], and IDRI [γ (−ω; ω, −ω, ω)] become accessible. In practice, the average polarizability tensor is defined in terms of Cartesian components such as α =

α xx + α yy + α zz , 3

(27.27)

where, αxx, αyy, and αzz are the diagonal elements of the polarizability-tensor matrix. The first-order hyperpolarizability tensor is defined as the third derivative of the energy with respect to the electric field components, and hence, involves one additional field differentiation compared to polarizabilities. The average first-order hyperpolarizability is defined as

 β = 

∑ i

 i  

β β* i

12

; βi = 13

∑ (β j

ijj

+ β jij + β jji ), (27.28)

27-5


where the sums are over the coordinates x, y, z (i, j = x, y, z), and βi * refers to the conjugate of the vector βi. The second-order hyperpolarizability tensor involves one additional field differentiation compared to the first-order analog. The average secondorder hyperpolarizability is defined as

γ =

1 15

∑ (2γ

iijj

+ γ ijji ); (i, j = x , y , z ).

(27.29)

ij

In some of the theoretical approaches second-order hyperpolarizability tensors are determined from a combination of analytical and finite-difference techniques [28]. All components of the γ tensor of interest (dc-SHG/EOKE/static second-order hyperpolarizability) are obtained from the analytical time-dependent calculation of the SHG/EOPE/static first-order hyperpolarizability in the presence of small electric fields. E.g., the relation used in the evaluation of EOKE is

γ abcd (−ω ; ω, 0, 0) = lim d

E →0

βabc (−ω ; ω, 0) E

d

E = Ed

,

(27.30)

Frequency-dependent response properties can also be calculated from the time-averaged quasienergy [29]. In this formalism, the response properties are obtained as the derivative of the quasi energy and the same is estimated by using the variational criterion for the quasienergy and the time-averaged timedependent Hellmann–Feynman theorem. Molecular properties are obtained by using the variational Lagrangian technique in accordance with the 2n + 1 and the 2n + 2 rules. Within this approach, the different frequency-dependent response properties are obtained from the simple extension of the variational perturbation theory to the Fourier component variational perturbation theory.

27.3â•‡ Nonlinear Optics with Materials Much of the present research work in the field of nonlinear optics is motivated by certain nonlinear optical phenomena in suitable materials [14]. Some of these potentially useful phenomena include the ability to alter the frequency or color of light, to amplify one source of light with another, and to alter its transmission features through a medium. The nonlinear optical materials require unusual stability with respect to ambient conditions and high-intensity light sources. These materials can broadly be categorized into two major classes. The first is the class of inorganic materials, which includes semiconducting and metallic clusters, inorganic crystals, bulk materials, etc. The second belongs to the organic or, in general, molecular materials that include mainly organic crystals and polymers. For these systems, the optical nonlinearities are usually derived from their structures. In the present chapter, we will focus only on the first type, i.e., semiconducting and metallic clusters and their NLO behavior.

27.4â•‡ Why Are Clusters So Important? Nonlinear optical processes in cluster materials provide the necessary information about the exact understanding of the quantum confinements and the surface effects in these systems [11]. In general, the properties at the nanoscale are usually nonmonotonic and oscillating in nature due to the quantum size effects, and as such, cluster materials can exhibit properties, which are quite different from those of the bulk [5,11,30]. More specifically, the energy band structure and the phonon distribution of cluster materials may differ from that of the bulk systems. Numerous theoretical as well as experimental investigations have been performed that show that the cluster-assembled materials have novel mechanical, electrical, and optical properties [5–16]. As mentioned earlier in this chapter, these NLO materials find huge applications in high-speed electro-optic devices for information processing and telecommunications [1–4]. In Section 27.5, we present a review on nonlinear optics with various cluster materials.

27.5â•‡Review of Nonlinear Optics with Clusters In this section, semiconducting clusters will be discussed first followed by metallic clusters. An example of a class of materials with a manifestation of unusual physicochemical and optical properties is cadmium selenide clusters [(CdSe)n]. These (CdSe)n clusters are found to be the precursors of a wide range of lowdimensional materials such as quantum dots [7,10], tetrapods [31], and nanowires [32] that exhibit a variety of optical and electronic properties. The electronic properties of cadmium selenide clusters (CdSe)n and quantum dots are semiconducting in nature and these materials find tremendous application in the design and the development of novel nonlinear optical (NLO) materials. There have been a lot of experimental investigations [15,33] on the exploration of the NLO properties in CdSe clusters. The Hyper-Rayleigh scattering technique provides the experimental basis for NLO properties in CdSe clusters and nanoparticles [15]. Aktsipetrov et al. [33] have shown the size dependence of SHG from the surface of a composite material consisting of CdSe nanoparticles embedded in a glass matrix. The first-order hyperpolarizability in CdSe and CdS nanoparticles is reported to be very high [34]. Apart from the experimental works, theoretical ab initio investigations [8,11,12] have also been performed in order to calculate the NLO coefficients in CdSe clusters. Troparevsky and Chelikowsky [35], in their work, reported on the structural and the electronic properties (the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), binding energies, and polarizabilities) of small (CdS)n and (CdSe)n clusters, where n ranges from 2 up to 8 using the pseudopotential method in real space. Karamanis et al. [36] have used DFT for the computation of static polarizability and anisotropy in the static polarizability of small CdSe clusters up to tetramer units (see Figure 27.1).

27-6


(Dipole polarizability per atom)/e2 a02Eh–1

50

TABLE 27.1â•… Average Static First and Second Order Static Hyperpolarizability in CdSe, Cd 2Se2 , Cd3Se3, and Cd4Se4

(CdSe)n [7s6p5d1f/6s5p4d1f ]

45

Sample

SCF MP2 B3LYP

40

30

25

CdSe Cd2Se2

−540.07 (LDA) Absent

Cd3Se3

−0.12258 (LDA) −0.059125 (LB94) Absent

Cd4Se4

35

Average Static Second-Order Hyperpolarizability, 〈γ (0;â•›0,â•›0,â•›0)〉 (a.u.) −556,680 (LDA) 7,306.5 (LDA) 4,398.6 (LB94) 12,200 (LDA) 8,929.3 (LB94) 89,565 (LDA) 50,124 (LB94)

Source: Sen, S. and Chakrabarti, S., Phys. Rev. B, 74, 205435, 2006. With permission.

Bulk

2

Average Static First-Order Hyperpolarizability, 〈β (0;â•›0,â•›0)〉 (a.u.)

3

4

5 2n

6

7

8

FIGURE 27.1â•… Evolution of the mean dipole polarizability with cluster size in a small CdSe cluster, (CdSe)n . (From Karamanis, P. et al., J. Chem. Phys., 124, 071101, 2006. With permission.)

Their investigations suggest that the optical properties of the bulk system can also be predicted from a plot of dipole polarizability per atom. Later on, the same group of Karamanis extended their work on CdSe clusters, highlighting the importance of basis sets and electron correlation in these systems [12]. However, the calculations of the frequency-dependent optical polarizability, as well as that of hyperpolarizability tensors using theoretical methods remained an unresolved issue for CdSe clusters. Recently, Sen and Chakrabarti [11], for the first time, investigated the frequency-dependent nonlinear optical properties (first- and second-order hyperpolarizability tensors) of (CdSe)n clusters up to the tetramer using the time-dependent density functional theory (TDDFT). Within the TDDFT, both the local-density approximation (LDA) and the generalized-gradient approximation (GGA) technique have been used. The more accurate LB94 (van Leeuwen and Baerends 94) [37] functional has been used under the GGA scheme. Response calculations have been implemented in the Amsterdam density functional package (ADF 2004.01) [38]. The analytical approach has been used for the evaluation of all the NLO coefficients. The calculated polarizability and hyperpolarizabilities depend on the choice of the basis set and the exchange-correlation potential. While the results obtained within the DFT are overestimated under the normal LDA and GGA functional, the coupled Hartree–Fock procedure exhibits exactly the opposite trend [39,40]. The static values of the average first-order hyperpolarizability (〈β (0; 0, 0)〉) and the average second-order hyperpolarizability (〈γ (0; 0, 0, 0)〉) of CdSe, Cd2Se2, Cd3Se3, and Cd4Se4 are depicted in Table 27.1. The frequency dependence of the SHG [β (−2ω; ω, ω)], the EOPE [β (−ω; ω, 0)], the EFISH [γ (−2ω; ω, ω, 0)], and the

EOKE [γ (−ω; ω, 0, 0)] was paid premier attention in the investigation of Sen and Chakrabarti [11]. For a complete evaluation of the frequency dependence of NLO properties in different CdSe clusters, a wide range of frequencies (0 to 0.45 a.u.) was considered. Figure 27.2 manifests the frequency behavior of 〈β (−2ω; ω, ω)〉 and 〈β (−ω; ω, 0)〉 in the case of CdSe and Cd 3Se3 respectively. 〈β (−2ω; ω, ω)〉, being a frequency-dependent property, is of paramount interest and has been investigated extensively in numerous previous investigations [15,41,42]. High degrees of fluctuations in 〈β (−2ω; ω, ω)〉 are observed in all the cases. Experimentally, 〈β (−2ω; ω, ω)〉 in CdSe nanocrystals and quantum dots has been observed by the HRS technique and its size dependence was also verified [15]. It is evident from Figure 27.2 that 〈β (−2ω; ω, ω)〉 is negative over a wide range of frequencies. This makes CdSe and Cd3Se3 more significant in the perspective of quantum optics. The frequency variation in 〈β (−2ω; ω, ω)〉 divulges one more significant information, i.e., the abrupt increase in 〈β (−2ω; ω, ω)〉 at specific frequencies. At specific frequencies the magnitude of 〈β (−2ω; ω, ω)〉 becomes very high. It is a common notion that larger values of 〈β (−2ω; ω, ω)〉 are obtained at a near resonance of the input energy [5]. The presence of the near resonance indicates that linear absorption can occur at such frequencies. Similar to the components of the first-order hyperpolarizability tensors, the components of the second-order hyperpolarizability tensors (〈γ (−2ω; ω, ω, 0)〉 and 〈γ (−ω; ω, 0, 0)〉) were reported to be highly sensitive to frequency variation. Figure 27.3 demonstrates the frequency dependence of different γ tensors. It was concluded from both LDA and LB94 results that both 〈γ (−2ω; ω, ω, 0)〉 and 〈γ (−ω; ω, 0, 0)〉 are negative over a wide range of frequencies. The change in the sign of 〈γ (−2ω; ω, ω, 0)〉 and 〈γ (−ω; ω, 0, 0)〉, due to the change in frequency is a significant observation. This led to a frequency selection in assigning the optical activity of the particular cluster. In a separate study, Chakrabarti and coworkers [43] investigated the evolution of electric polarizability and the anisotropy of the polarizability at static as well as dynamic (Nd:YAG laser) frequencies with an increase in the cluster size for small, as well

27-7

CdSe/LDA

β (−2ω; ω, ω) (a.u.)

1,000,000 500,000 0 –500,000 –1,000,000 –1,500,000 –2,000,000 –2,500,000 CdSe/LDA –3,000,000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 (a) Frequency (a.u.)

1,000,000 800,000 600,000 400,000 200,000 0 –200,000

Cd3Se3/LDA

β (−ω; ω, 0) (a.u.)

400,000 200,000 0 –200,000 –400,000 –600,000 –800,000 –1,000,000

β (−2ω; ω, ω) (a.u.)

β (−ω; ω, 0) (a.u.)


100,000 0 –100,000 –200,000 –300,000 –400,000 –500,000 –600,000

Cd3Se3/LDA

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Frequency (a.u.)

(b)

1500

Cd3Se3/LB94 β (−ω; ω, 0) (a.u.)

1000 500 0 –500 –1000

β (−2ω; ω, ω) (a.u.)

5000 Cd3Se3/LB94 4000 3000 2000 1000 0 –1000 –2000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 (c) Frequency (a.u.)

1.00E+008 5.00E+007 0.00E+000 –5.00E+007 –1.00E+008 –1.50E+008 –2.00E+008 –2.50E+008

CdSe/LDA

1.20E+008 9.00E+007 6.00E+007 3.00E+007 0.00E+000 –3.00E+007 CdSe/LDA –6.00E+007 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

(a)

Frequency (a.u.)

γ (−2ω; ω, ω, 0) (a.u.) γ (−ω; ω, 0, 0) (a.u.)

γ (−2ω; ω, ω, 0) (a.u.) γ (−ω; ω, 0, 0) (a.u.)

FIGURE 27.2â•… Variation of 〈β (−2ω; ω, ω)〉 (SHG) and 〈β (−ω; ω, 0)〉 (EOPE) with a frequency at (a) LDA scheme for the CdSe cluster, (b) LDA, and (c) GGA (LB94) scheme for the Cd 3Se3 cluster. (From Sen, S. and Chakrabarti, S., Phys. Rev. B, 74, 205435, 2006. With permission.)

Cd2Se2/LDA

20,000,000 10,000,000 0 –10,000,000 –20,000,000 –30,000,000

1,500,000 1,000,000

Cd2Se2/LDA

500,000 0 –500,000

–1,000,000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

(b)

Frequency (a.u.)

FIGURE 27.3â•… Variation of 〈γ (−2ω; ω, ω, 0)〉 and 〈γ (−ω; ω, 0, 0)〉 with a frequency at LDA. (a) Scheme for the CdSe cluster, (b) LDA and (continued)

27-8

γ (−2ω; ω, ω, 0) (a.u.)

γ (−ω; ω, 0, 0) (a.u.)


1,200,000

Cd2Se2/LB94

900,000 600,000 300,000 0 –300,000 –600,000

400,000 200,000 0

–200,000 –400,000 –600,000

–2,000,000 –4,000,000 –6,000,000 8,000,000 6,000,000

–8,000,000

Cd3Se3/LDA

–10,000,000

Cd3Se3/LDA

4,000,000 2,000,000 0

–2,000,000 0.00

γ (−2ω; ω, ω, 0) (a.u.) γ (−ω; ω, 0, 0) (a.u.)

(d)

0.05

0.10 0.15 Frequency (a.u.)

Cd4Se4/LDA

0.20

0.25

30,000,000 20,000,000 10,000,000 0 –10,000,000 –20,000,000 –30,000,000

20,000,000 15,000,000 Cd4Se4/LDA 10,000,000 5,000,000 0 –5,000,000 –10,000,000 –15,000,000 –20,000,000 0.000.020.040.060.080.100.120.140.160.180.20 (f ) Frequency (a.u.)

γ (−2 ω; ω, ω, 0) (a.u.)

0

Cd3Se3/LB94

8.00E+008 6.00E+008 4.00E+008 2.00E+008 0.00E+000

4,000,000

Cd3Se3/LB94

3,000,000 2,000,000 1,000,000 0

–1,000,000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (e) Frequency (a.u.)

γ (−2ω; ω, ω, 0) (a.u.) γ (−ω; ω, 0, 0) (a.u.)

γ (−2ω; ω, ω, 0) (a.u.)

2,000,000

γ (−ω; ω, 0, 0) (a.u.)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Frequency (a.u.)

γ (−ω; ω, 0, 0) (a.u.)

(c)

Cd2Se2/LB94

(g)

Cd4Se4/LB94

4 × 108 3 × 108 2 × 108 1 × 108 0

2.5 × 108

2.0 × 108 1.5 × 108 1.0 × 108 5.0 × 107 0.0 –5.0 × 107

Cd4Se4/LB94

–1 × 108

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Frequency (a.u.)

FIGURE 27.3 (continued)â•… (c) GGA (LB94) scheme for the Cd 2Se2 cluster, (d) LDA and (e) GGA (LB94) scheme for the Cd3Se3 cluster, (f) LDA and (g) GGA (LB94) scheme for the Cd4Se4 cluster. (From Sen, S. and Chakrabarti, S., Phys. Rev. B, 74, 205435, 2006. With permission.)

27-9


properties of GaAs bulk materials which suggest that GaAs should possess very large static second- and third-order susceptibilities, only a few of them deal with the theoretical evaluation of the hyperpolarizabilities of these cluster materials [50,51]. A recent work by Lan et al. [5] provides a systematic insight into the NLO properties of several small and medium GanAsm clusters, where m + n runs up to 10. They employed the TDDFT method in combination with the sum-over-states (SOS) formalism to calculate the first- and second-order hyperpolarizability tensors. The results indicate the presence of large second- and third-order nonlinear susceptibilities in GaAs cluster materials similar to that of the bulk. These results also suggest that resonance absorption can be avoided in the frequency-dependent NLO experiments. Furthermore, the frequency dependence of second- and third-harmonic generation reveals that these clusters are good candidates for future NLO materials. Besides these two clusters, investigations of the NLO properties have also been carried out on various other III-V atomic nanoclusters. Pineda and Karna [52] estimated the linear and the nonlinear polarizabilities of isolated GaN nanoclusters employing the ab initio time-dependent Hartree–Fock (TDHF) method. Their results suggest a strong dependence of the NLO properties on the size and the shape of the clusters. The linear and the nonlinear optical properties of (CdS)n clusters were analyzed and the merits of the DFT level calculations have been compared with different ab initio results, as well as the basis set dependence of the optical properties of these clusters has been evaluated by Maroulis and Pouchan [53]. Figure 27.4 depicts the variation in the mean polarizability and the mean second-order hyperpolarizability values in small (CdS)n clusters. The values suggest a reduction in the mean

as medium-sized (CdSe)n (n = 1–16) clusters within the density functional theory (DFT). The main motivation of that investigation was

1. None of the theoretical studies reported had gone beyond a cluster size of eight for the polarizability calculations for the CdSe clusters. 2. Except the study by Sen and Chakrabarti [11], no one was aware of any investigations where the dynamic hyperpolarizability for any system size was reported for the CdSe clusters.

Knowing the strong dependence of polarizability on the cluster diameter (the cluster size), they performed polarizability calculations beyond the cluster size of eight and calculated both static and dynamic polarizabilities. Table 27.2 describes computed polarizability values at static and Nd:YAG laser frequencies. An even-odd oscillating behavior is observed in the anisotropy values between the dimer and the heptamer CdSe clusters. Just like the CdSe clusters, GaAs is one such semiconducting cluster, which has gained a lot of importance over the past few years [5]. These cluster materials display long absorption tails in the low energy region [44] owing to the existence of free surfaces in it. Moreover, the static polarizabilities of these systems exceed the bulk value and follow a decreasing tendency with the increase in the cluster size [45]. The variable nature in the polarizabilities also indicates its “metallic-like” behavior. In the experimental investigation [46], it has been observed that the polarizabilities for small and medium-sized GanAsm clusters reside above and below the bulk limit. Albeit, there have been a lot of experimental and theoretical investigations [47–49] based on the NLO

TABLE 27.2â•… Average Values of Electric Polarizability per Atom (〈α〉/2n) and Anisotropy in Polarizability (Δα) against the Cluster Size, 2n at Both Static (0.0 a.u.) and Nd:YAG Laser (0.04283 a.u.) Frequencies At 0.0 a.u. Frequency

At Nd:YAG Laser (0.04283 a.u.) Frequency

Samples

n

2n

Polarizability per Atom (〈α〉/2n) (a.u.)

Anisotropy in Polarizability (Δα) (a.u.)

Polarizability per Atom (〈α〉/2n) (a.u.)

CdSe Cd2Se2 Cd3Se3 Cd4Se4 Cd5Se5 Cd6Se6 Cd7Se7 Cd8Se8 Cd9Se9 Cd10Se10 Cd11Se11 Cd12Se12 Cd13Se13 Cd14Se14 Cd15Se15 Cd16Se16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

34.0349 30.9670 29.8098 29.5301 32.4484 29.1341 29.0036 29.5565 29.4061 29.8171 29.8559 28.4830 30.9808 30.3060 30.2442 33.5302

36.5416 60.5853 99.7527 0.0000 159.8706 43.8215 134.4511 114.3660 49.9355 124.1952 150.1088 0.0165 271.1190 192.7016 158.5999 428.7594

31.3777 32.5664 30.5716 30.8635 33.3488 30.0033 29.9893 30.4782 30.2877 30.7628 30.7738 29.3089 32.0344 31.2651 31.1661 34.9350

Source: Jha, P.C. et al., Comput. Mater. Sci., 44, 728, 2008. With permission.

Anisotropy in Polarizability (Δα) (a.u.) 49.7528 59.0703 102.1434 0.0000 166.2891 43.5338 143.1402 122.1772 54.6105 132.7159 160.6296 0.0169 287.1187 207.4891 170.6002 470.1254

27-10


SCF [7s6p4d/5s4p2d] B3LYP [7s6p4d/5s4p2d] Troparevsky & Chelikowsky

Polarizability per atom/e2a02Eh–1

40

(CdS)n

35 30 25 20 15

45 Hyperpolarizability per atom/103 × e4a04Eh–3

45

10

(a)

SCF [7s6p4d/5s4p2d] B3LYP [7s6p4d/5s4p2d]

40 (CdS)n

35 30 25 20 15 10

2

3

4 5 6 Number of atoms

7

8

(b)

2

3

4 5 6 Number of atoms

7

8

FIGURE 27.4â•… Mean (a) dipole polarizability and (b) second-order hyperpolarizability in (CdS)n clusters. (From Maroulis, G. and Pouchan, C., J. Phys. Chem. B, 107, 10683, 2003. With permission.)

dipole polarizability, as well as in the second-order hyperpolarizability values with an increase in the cluster size. Moreover, the results obtained from B3LYP and other conventional ab initio methods are well in accordance with each other and also with the earlier theoretical results [35]. However, in the case of the monomer geometry, the difference in the hyperpolarizability values between these methods is quite large. Champagne et al. estimated [54] average second-order hyperpolarizabilities of Sin (n = 3–38) clusters using MP2, MP3, MP4, CCSD, and CCSD(T) levels of approximations and demonstrated the variation of polarizability and hyperpolarizability against the cluster size. Apart from these semiconducting materials, cluster science has also made rapid progress in determining the electric properties of various metal clusters [55] including that of mixed metal. At first, a review of the some of the works based on the metal clusters containing only one type of atom is discussed, which is followed by some detailed analysis on mixed metal clusters. There are investigations on lithium [18,56,57], sodium [58], copper [59,60], nickel [61], niobium [62], and zinc [63] clusters. Maroulis and Xenides [57] reported highly accurate ab initio calculations on the polarizability and the hyperpolarizabilities of lithium tetramer, Li4 with specially designed basis sets for Li. The molecule, Li4 has got a very high anisotropic dipole polarizability and quite large second-order hyperpolarizability values. The main feature of this work is the extensive analysis of the basis set and the correlation effects in calculating the electric properties. The results indicate a discrepancy in dipole polarizabilities between theory and experiment, which might be resolved by measuring the anisotropy values experimentally. The same group also presented a thorough analysis of the performance of

several density functional theoretical (DFT) calculations of the (hyper) polarizability of Li4. Their investigations elucidated that of the various DFT methods employed; only the mPW1PW91 and the O3LYP methods produce reliable results. Papadopoulos et al. [63] studied the (hyper) polarizabilities of small and medium-sized zinc clusters employing a hierarchy of basis sets and computational methods. The relativistic effect on the electric polarizabilities has also been investigated by employing the Douglas–Kroll approximation. It has been observed that the relativity contribution is significant in these cluster materials; however, the correlation effect plays a major part in influencing the electro-optic properties. The NLO properties of Ag nanoparticles embedded in a Si3N4 matrix was investigated by Traverse et al. [64]. They performed a two-color sum-frequency generation (2c-SFG) spectroscopy experiment, which exhibited a surface plasmon resonance at a 421â•›nm wavelength in the visible region and absorption in the infrared region. The third-order optical nonlinearity of dielectrics in the presence of nanoparticles has been experimentally investigated by Flytzanis [65]. Chen and coworkers performed [66] high-level ab-initio calculations [HF, MP2, the fourthorder perturbation theory using single, double, quadrupole substitutions, CCSD, and QCISD] on (HCN)n…Li and Li… (HCN)m clusters with electride characteristics. A high value of static first hyperpolarizability is reported in both samples. The static second-order hyperpolarizability of a series of tri-nuclear metal cluster MS4(M′PPh3)2(M′PPh3) (M = Mo, W; M′ = Cu, Ag, Au) have been estimated by [67] Chen and coworkers. They employed a finite-field approach using the hybrid density functional theory (B3LYP) and reported a very high value of secondorder hyperpolarizability. The NLO properties of Aun clusters

27-11

Nonlinear Optics with Clusters TABLE 27.3â•… Average Hirshfeld Charge on the Al4 Ring, Ground State Dipole Moment, Magnitude of Average Static First- and Second-Order Hyperpolarizability in Al4 M4 (M = Li, Na, and K)

Sample Al4Li4 (2D) Al4Li4 (3D) Al4Na4 (2D) Al4Na4 (3D) Al4K4 (2D) Al4K4 (3D)

Average Hirshfeld Charge on the Al4 Ring

Ground-State Dipole Moment (Debye)

Average Static First-Order Hyperpolarizability, 〈β(0; 0, 0)〉 (a.u.)

−0.247 −0.202 −0.258 −0.229 −0.303 −0.297

0.049 0.018 0.003 0.054 0.080 6.513

145.922 33.079 21.537 187.843 316.781 8863.074

Average Static SecondOrder Hyperpolarizability, 〈γ(0; 0, 0, 0)〉 (a.u.) 1.299 × 106 2.406 × 106 1.275 × 106 1.961 ×106 3.380 × 106 1.149 × 107

Source: Sen, S. et al., Phys. Rev. B, 76, 115414, 2007. With permission.

and their alloyed clusters, Aun−mMm (M = Ag, Cu; m = 1, 2) have been theoretically investigated by Xu and coworkers [68]. They used the usual TD-DFT as well as the B3LYP hybrid density functional employing the LANL2DZ basis set and measured the first-order hyperpolarizabilites of such clusters. They suggested that the high degree of NLO properties of these clusters are due to the lack of a centro-symmetric feature in these clusters and an enhancement in the transition electric dipole moment in alloyed clusters. The optical properties of several mixed metal clusters such as LinCum, Na4Pb4, Na6Pb, Al11Fe, and Al18Fe, Al4M4 (M = alkali metals) have also been investigated extensively [69,70]. In this context, we particularly mention the NLO properties of aluminum metal clusters [Al4M4 (M = Li, Na, and K)]. Various investigations have been performed on these Al4M4 systems and their anions due to their unique characteristic features and their structural similarity with that of cyclobutadiene, C 4H4 [6]. These cluster materials are also important since they are better polarized than their organic counterparts. Earlier, Datta and Pati [6] investigated the static linear and nonlinear polarizabilities of Al4M4 (M = Li, Na, and K) systems. They predicted that structural changes through the broken inversion symmetry of Al4M4 could lead to a unique polarization response and large optical coefficients. In their work, both the static linear and nonlinear optical properties of these Al4M4 were estimated through ZINDOmultireference doubles configuration interaction (MRDCI) calculations. They emphasized that the large NLO properties of Al4M4 are due to the charge transfer from alkali metals to the Al4 ring. In spite of the fact that their work exhibits an exceptionally high magnitude of static linear and nonlinear optical coefficients, it lacks the description of its dynamic counterparts. It is to be noted that the importance of a material in nonlinear optics and its applications in device fabrication demands a complete description of the frequency dependence. Hence, it was highly instructive to explore the dynamic (frequency-dependent) NLO properties of these clusters for their proper assessment in optical device fabrication and to explain whether the large NLO properties are due to a charge transfer interaction or due to other excitations. Recently Sen et al. [13] reported various components of first- (SHG, EOPE, static first-order hyperpolarizability) and second- (EFISH, EOKE, and static second-order

hyperpolarizability) order hyperpolarizability tensors at several near resonance frequencies. All these dynamic NLO properties of Al4M4 (M = Li, Na, and K) clusters have been explored using the TDDFT as a prime investigating tool. Within the TDDFT, the generalized gradient approximation (GGA) technique has been implemented with the use of a more accurate LB94 (van Leeuwen and Baerends 94) functional. The magnitude of static 〈β〉 and 〈γ〉 along with an average Hirshfeld charge on the Al4 ring and the ground-state dipole moments are shown in Table 27.3. It is quite clear from Table 27.3 that the magnitude of 〈β(0;â•›0,â•›0)〉 increases significantly from 2D to 3D in the case of Al4Na4 and Al4K4, on the other hand, a reverse trend is observed for Al4Li4. Table 27.3 also suggests that the magnitudes of the average static 〈γ(0;â•›0,â•›0,â•›0)〉 are very high (of the order of 106 and 107 a.u.) for all the clusters under investigation. However, the central issue of the investigation of Sen and coworkers [13] was to explicate the frequency dependence of 〈β(−2ω; ω, ω)〉, 〈β(−ω; ω, 0)〉, 〈γ(−2ω; ω, ω, 0)〉, and 〈γ(−ω; ω, 0, 0)〉 in aluminum metal clusters. These components of the first- and second-order hyperpolarizability tensors were estimated in the vicinity of near-excitation frequencies with a substantial oscillator strength and presented in Table 27.4. A closer inspection of Table 27.4 reveals that 〈β(−2ω; ω, ω)〉 and 〈β(−ω; ω, 0)〉 attain negative values at certain frequencies in all these cluster materials. Exceptionally high values [of the order of 106–108 a.u.] of 〈γ(−2ω; ω, ω, 0)〉 and 〈γ(−ω; ω, 0, 0)〉 were also noticed. A significant observation was in the change in sign of 〈γ(−2ω; ω, ω, 0)〉 and 〈γ(−ω; ω, 0, 0)〉 due to the change in frequency, which lead to a frequency selection in assigning the optical activity of a particular cluster. In order to explain the observed high NLO coefficients in these aluminum metal clusters, an analysis of the magnitude of gross populations of molecular orbitals with the most significant SFO (Symmetrized Fragment Orbitals) was carried out. The SFO study divulged the role of charge transfer interactions. Molecular orbitals with the most significant SFO gross population indicated that except for Al4K4 (2D and 3D) and Al4Li4 (3D) there exists excitations other than a charge transfer interaction at near resonance frequencies. Moreover, it is worth mentioning that the dynamic firstorder hyperpolarizability tensors are functions of the excitation

27-12

Handbook of Nanophysics: Nanoelectronics and Nanophotonics TABLE 27.4â•… 〈β(−2ω; ω, ω)〉, 〈β(− ω; ω, 0)〉, 〈γ(−2ω; ω, ω, 0)〉, and 〈γ (−ω; ω, 0, 0)〉 Measured at Frequencies Nearer to Excitation Frequencies in Al4 M4 (M = Li, Na and K)

Sample Al4Li4 (2D)

Al4Li4 (3D)

Al4Na4 (2D)

Al4Na4 (3D)

Al4K4 (2D)

Al4K4 (3D)

Excitation Frequencies (a.u.)

Oscillator Strength

Frequency at which Data Are Taken (a.u.)

〈β(−2ω;â•›ω,â•›ω)〉

〈β(−ω;â•›ω,â•›0)〉

〈γ(−2ω;â•›ω, ω, 0)〉

0.0205 0.0522 0.0558 0.0377 0.0650 0.0709 0.0238 0.0403 0.0570 0.0271 0.0446 0.0543 0.0244 0.0427 0.0431 0.0215 0.0391 0.0416

0.0042 0.0034 0.0094 0.0441 0.0863 0.3736 0.0068 0.0012 0.0852 0.0301 0.0060 0.0099 0.0066 0.0910 0.0197 0.0010 0.0105 0.0067

0.0200 0.0500 0.0540 0.0360 0.0640 0.0690 0.0220 0.0390 0.0590 0.0260 0.0430 0.0560 0.0230 0.0410 0.0440 0.0200 0.0380 0.0400

−6081.100 224.900 −371.630 4057.900 968.580 −638.610 −429.590 51.085 4641.500 2193.700 −452.340 −7067.500 305.930 −1465.600 −64908 −231620 29403 −120770

12,741 46.143 361.960 −15,252 −902.360 236.140 2697.100 −96.614 −93.815 −13333 −178.82 −168.550 864.090 3477.900 −5213.200 −16,582 150,940 380,220

8.988 × 10 −5.239 × 105 1.226 × 107 6.049 × 106 2.729 ×107 −1.199 ×108 1.266 ×107 4.735 × 104 1.540 × 107 1.675 × 107 4.459 × 106 1.242 × 107 1.286 × 106 −3.240 × 107 −3.143 × 108 1.842 × 107 9.758 × 106 −7.515 × 107

(a.u.)

(a.u.)

(a.u.)

6

〈γ(−ω; ω, 0, 0)〉 (a.u.)

−2.744 × 106 1.155 × 106 −7.558 × 105 1.663 × 107 −8.458 × 104 −1.866 × 107 −7.639 × 107 1.245 × 106 −6.290 × 106 −6.581 × 106 5.213 × 105 1.401 × 106 6.640 × 106 3.698 × 107 2.548 × 107 −1.926 × 107 2.791 × 107 4.444 × 107

Source: Sen, S. et al., Phys. Rev. B, 76, 115414, 2007. With permission.

energy, the dipolar matrix, and the difference between the dipole moment of the ground and the excited states. On the other hand, components of the second-order hyperpolarizability tensor are associated with four dipolar matrices in the numerator and three excitation frequencies in the denominator. Hence, it is very difficult to find all the factors that play a dominant role in the high β and γ values in particular clusters.

27.6â•‡ Conclusions Therefore, it is well established from both the theoretical as well as the experimental standpoint that cluster materials can be considered as promising candidates for future NLO devices due to their high nonlinear optical coefficient. However, the exploration of the fundamentals of the unique physicochemical behavior of these clusters is a challenge to modern physical and chemical research. A theoretical determination of a higher-order nonlinear optical coefficient is still a matter of extensive computation and there is need for further developments to provide a proper physical background for some of the higher-order optical coefficients. There is enough scope for future research work in the field of cluster science and the design of new cluster materials with interesting physical behavior.

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28 Second-Harmonic Generation in Metal Nanostructures 28.1 Introduction............................................................................................................................28-1 28.2 Nonlinear Optics and Second Harmonic Generation......................................................28-2 28.3 Second Harmonic Generation in Nanosystems.................................................................28-3 Spherical Isolated Particles and Plane-Wave Illuminationâ•‡ •â•‡ Low-Symmetry Particlesâ•‡ •â•‡ Nonuniform Illuminationâ•‡ •â•‡ Second Harmonic Generation in Resonant Metal Particlesâ•‡ •â•‡ Random Metal Nanostructures and Rough Metal Surfaces

28.4 Leading Order Contributions to the Second Harmonic Generation Process in Nanoparticles......................................................................................................................28-5 Conservation of Parity and Angular Momentumâ•‡ •â•‡ Long-Wavelength Limit

28.5 Emission Patterns and Light Polarization......................................................................... 28-8 Irradiated Intensityâ•‡ •â•‡ Second Harmonic Polarization

28.6 Allowed and Forbidden Second Harmonic Emission Modes: Examples......................28-9

Marco Finazzi Politecnico di Milano

Giulio Cerullo Politecnico di Milano

Lamberto Duò Politecnico di Milano

Plane-Wave Illuminationâ•‡ •â•‡ Illumination with a Laterally Limited Light Beam

28.7 Second Harmonic Generation in Single Gold Nanoparticles........................................28-11 Near-Field Microscopy Setupâ•‡ •â•‡ Fundamental Wavelength and Second Harmonic Mapsâ•‡ •â•‡ Second Harmonic Polarization Analysis

28.8 Perspectives...........................................................................................................................28-15 28.9 Conclusions...........................................................................................................................28-15 Acknowledgments............................................................................................................................28-16 References..........................................................................................................................................28-16

28.1â•‡Introduction The explosive growth of nanoscience and nanotechnology during the last decade has led to great interest in the investigation of nanoscale optical fields and to the development of experimental tools for their study (Novotny and Hecht 2006). One of the most remarkable effects in light interaction with metal nanostructures is the strong and spatially localized field amplitude enhancement, due to lightning rod effects induced by the sharp curvatures (Novotny and Hecht 2006) and/or resonant excitation of collective electron oscillations (plasma oscillations) in single or coupled particles (Grand et al. 2003, Mühlschlegel et al. 2005). The resonance frequencies associated with such oscillations and thus the optical properties can be tuned in a broad spectral range according to the material and the shape of the nanostructure. Particularly, gold and silver may sustain plasma oscillations at optical frequencies. Field enhancements are best observed by exploiting nonlinear optical effects such as second-harmonic generation (SHG) (Smolyaninov et al. 1997, 2000, Jakubczyk et al. 1999, Zayats et al. 2000, Shen et al. 2001, Biagioni et al. 2007, Breit et al. 2007)

or two-photon photo-luminescence (TPPL) (Jakubczyk et al. 1999, Bouhelier et al. 2003, 2005, Imura et al. 2005, 2006), which depend on the square of the light intensity, thus amplifying local field variations. To observe a significant nonlinear response, the samples are typically illuminated by high peak intensities, such as those associated with ultrashort laser pulses. Nonlinear optical microscopies, either in the far field or in the near field, are thus becoming powerful tools for the study of nanoscale field enhancements. SHG was first experimentally demonstrated in 1961 (Franken et al. 1961) and has since then found many applications, especially in converting laser light to a different color. Another application more relevant to the content of this chapter is surface SHG. Although the sensitivity of the SHG process to boundaries was soon recognized (Bloembergen and Pershan 1962), it was only in the early 1980s that nonlinear optical spectroscopy of surfaces and interfaces started to develop as a well-established analysis tool in a wide area of fields (Shen 1989, Mc Gilp 1996, Bloembergen 1999, Lüpke 1999, Downer et al. 2001). The SHG ability to selectively probe surfaces and interfaces derives from a selection rule (see Section 28.2) that does not allow for dipole 28-1

28-2


emission from the bulk of centrosymmetric materials. At the surface or interface, however, the symmetry of the bulk is broken and dipole generation is no longer forbidden. This peculiar property has determined the enormous spread of SHG-based techniques in the community of surface and interface physics. This technique has the advantages of being highly surface-Â�sensitive, capable of remote sensing and in situ measurement, and applicable to any interface between two condensed media accessible by light, even if buried. Among the applications, surface SHG can be used to monitor the growth of monolayers on a surface or to probe the alignment of absorbed molecules. This chapter presents theoretical and experimental results on SHG from metal nanostructures. It is organized as follows: Section 28.2 contains a brief description of the principles of nonlinear optics and SHG; Section 28.3 presents a survey of the recent literature on SHG from nanostructures; Section 28.4 introduces a theoretical analysis of SHG from nanostructures, allowing to derive selection rules for the process; Section 28.5 describes the light emission patterns and polarization, while Section 28.6 presents experimentally relevant illumination conditions. Section 28.7 reports experimental results on SHG from gold nanoparticles, while Sections 28.8 and 28.9 contain a discussion of possible future developments and conclusions, respectively.

28.2â•‡Nonlinear Optics and Second Harmonic Generation Nonlinear optics (Shen 1984, Boyd 2003) is the branch of physics that describes the phenomena that occur when the matter is illuminated by very intense light fields that have the capability of modifying its optical properties, thus generating an optical response that is a nonlinear function of the light intensity. Since electric fields strong enough to induce a nonlinear optical response can only be obtained by using the spatially and temporally coherent radiations provided by lasers, the birth of nonlinear optics closely coincides with the invention of the laser. The physical origins of the nonlinear optical response of a material can be understood by recalling the Lorentz model, in which the atoms/molecules are modeled as harmonic oscillators that, driven by the external light field, become oscillating dipoles

EFF cos (ωt)

emitting light at the same frequencies as the driving field. In a harmonic oscillator, the restoring force is a linear function of displacement; for large displacements, however, nonlinear terms in the restoring force start to become significant, thus leading to an anharmonic oscillation and emission of new frequency components not present in the driving field. Let us consider an electromagnetic wave crossing a material: the time-varying electric field E(t) induces a time-dependent polarization (dipole moment per unit volume) P(t), which is responsible for light emission. At low light intensities, the polarization is a linear function of the driving field: P (t ) = ε0χ(1) E(t ),

(28.1)

where ε0 is the vacuum permittivity χ(1) is known as the linear susceptibility As the intensity increases, further nonlinear terms in which the polarization is proportional to higher powers of the electric field must be considered:

P (t ) = ε0[χ(1) E(t ) + χ(2) E 2 (t ) + χ(3) E 3 (t ) + ],

(28.2)

where the quantities χ(2) and χ(3) are known as second- and thirdorder nonlinear susceptibilities, respectively. By considering the vectorial nature of E and P, χ(1) becomes a second-rank tensor, χ(2) a third-rank tensor, and so on. The χ(2) term is responsible for second-order nonlinear optical effects, such as sum frequency generation, difference frequency generation, and SHG. The following text briefly describes the SHG process. Let us consider the geometry shown in Figure 28.1a, in which a monochromatic wave at the fundamental frequency ω, E(t) = E0 cos(ωt), impinges on a crystal with nonzero second-order susceptibility. The second-order polarization can then be written as

1 P (2)(t ) = ε 0 χ(2) E 2 (t ) = ε 0 χ (2) E02 cos2 (ωt ) = ε 0 χ(2) E02[1 + cos(2ωtt )]. 2 (28.3)

ESH cos (2ωt) (2)

(a)

(b)

Figure 28.1â•… Scheme of SHG in a nanoparticle: two incoming FW photons are absorbed by the particle and their energy converted in a single photon with double frequency.

28-3

Second-Harmonic Generation in Metal Nanostructures

The nonlinear polarization thus contains, in addition to a constant term (optical rectification), a component at the secondharmonic (SH) frequency 2ω, which irradiates a field at the corresponding frequency. Note that, since the second-order nonlinear polarization depends on the square of the driving field amplitude E0, the SHG efficiency is enhanced by high-peak power, pulsed illumination. The SHG process can also have a simple corpuscular interpretation, according to the energy-level scheme reported in Figure 28.1b: two fundamental wavelength (FW) photons at energy ћω are absorbed to a virtual level of the material, and an SH photon with energy 2ћω is emitted, thus satisfying energy conservation. In a centrosymmetric medium (i.e., a medium with a center of inversion), the χ(2) term vanishes identically. In such a medium, in fact, by changing the sign of the driving field, also the sign of the nonlinear polarization must change. By substituting in Equation 28.3, one obtains

−P 2 (t ) = ε0χ2[− E(t )]2 = ε0χ2 E 2 (t ),

(28.4)

which can only be satisfied if χ(2) ≡ 0. Since gases, liquids, amorphous solids, and also several crystals possess an inversion symmetry, the second-order nonlinear susceptibility in such materials is zero, and thus, SHG is forbidden. At the microscopic level, one can consider the single emitter (atom, molecule) as a nonlinear dipole radiating at the SH frequency 2ω. In order to generate a macroscopic SH signal, the fields emitted by the individual dipoles must interfere constructively, adding their contributions in phase. Since the phase of the nonlinear dipoles depends on the phase of the FW driving field, to achieve constructive interference between the nonlinear emissions at two different positions in the nonlinear medium, one must match the phase velocities of the FW field vpFW = ω/kFW and the SH field vpSH = 2ω/kSH. From vpFW = vpSH, one obtains the condition kSH = 2kFW, also known as “phase-matching” condition, which implies that nSH = nFW, where n is the refractive index of the medium. If the phase-matching condition is satisfied, all the microscopic emissions add in phase and the SH field grows linearly (and the intensity quadratically) with the propagation distance. On the other hand, if Δk = kSH − 2kFW ≠ 0, the SH field depends on the propagation coordinate x as ESH ∝ χ(2) sin(Δk x/2). It thus increases for propagation up to a “coherence length” Lc such that Δk Lc = π, and then starts to decrease due to destructive interference. Achievement of the phase matching condition is made difficult by the fact that most materials display, in the optical frequency range, a refractive index increasing monotonously with frequency (normal dispersion), so that nSH > nFW. One can obtain phase-matching condition exploiting the birefringence of noncentrosymmetric crystals, that is, the dependence of the refractive index on the polarization and propagation direction. By polarizing the SH wave along the direction giving the lower refractive index, one can satisfy the condition nSHâ•›=â•›nFW and thus achieve phase matching. Under such conditions, the SHG process can become so efficient that nearly all the FW power is converted to the SH (Parameswaran et al. 2002).

28.3â•‡Second Harmonic Generation in Nanosystems As discussed in Section 28.2, SHG is forbidden, within the dipole approximation, from the bulk of a centrosymmetric material but is allowed at its surface, where the inversion symmetry is broken. For this reason, SHG has been extensively applied since its discovery to the investigation of the surface properties of centrosymmetric media. In more recent years, thanks to its surface selectivity, SHG has attracted a considerable interest also as a tool to investigate the properties of nonplanar surfaces that characterize nanoscale systems. The following text presents a survey of the recent literature concerning SHG in nanostructures.

28.3.1â•‡Spherical Isolated Particles and Plane-Wave Illumination Perfectly spherical small particles made of isotropic material under FW plane-wave illumination represent the simplest geometry to study SHG in nanostructured systems. Experiments have been conducted on spherical dielectric (Wang et al. 1996, Yang et al. 2001, Shan et al. 2006) or metallic (Vance et al. 1998, Nappa et al. 2005) particles dispersed in dilute suspensions, or on colloidal particles, ordered in a centrosymmetric crystalline lattice (Martorell et al. 1997). The influence of the particle diameter (Yang et al. 2001, Nappa et al. 2005, Shan et al. 2006), concentration (Wang et al. 1996, Vance et al. 1998), and the presence of adsorbates (Wang et al. 1996) on SHG have been investigated, by performing angle- and polarization-resolved measurements (Martorell et al. 1997, Yang et al. 2001, Nappa et al. 2005, Shan et al. 2006). The theory is in good agreement with the experiments, confirming that, when the sphere is small compared with the wavelength of light (Rayleigh limit), the leading-order SH radiation is emitted by a locally excited electric quadrupole and by an electric dipole directed along the direction of the propagation of the incident FW light beam that requires a nonlocal excitation mechanism, in which the phase variation of the pump beam across the sphere needs to be considered (Dadap et al. 1999, 2004). The locally excited electric dipole term, analogous to the source for linear Rayleigh scattering, is absent for the nonlinear case because the FW field induces mutually canceling polarizations at opposite sides of a centrosymmetric spherical particle (Dadap et al. 1999, 2004). The theory predicts the absence of any SHG signal in the forward direction, with both dipole and quadrupole contributions producing SH radiation with an intensity of leading order (ka)6, k being the wave vector and a the particle diameter (Dadap et al. 1999, 2004). Theoretical models have been applied to describe SHG in particles larger than the Rayleigh limit (Dewitz et al. 1996, Pavlyukh and Hübner 2004) and to explicitly account for the electromagnetic response of conducting nanoparticles (Hua and Gersten 1986, Dewitz et al. 1996, Panasyuk et al. 2008).

28-4


28.3.2â•‡Low-Symmetry Particles The fact that SHG is forbidden in the bulk of centrosymmetric materials makes SHG very sensitive to the particle symmetry (Sandrock et al. 1999). For instance, experimental results suggest that SHG can be used to characterize the symmetry and chirality of carbon nanotubes (Su et al. 2008). It has been theoretically (Bachelier et al. 2008) and experimentally demonstrated (Martorell et al. 1997, Nappa et al. 2006) that slight structural deviations from the spherical shape may lead to SH radiation and polarization properties of nanoparticles differing significantly from those of a sphere. For gold particles with a diameter smaller than 50â•›nm, SHG is dominated by a dipole contribution that is not due to defects in the crystalline structure of the particle but to the deviation of the particle shape from that of a perfect sphere (Nappa et al. 2006, Bachelier et al. 2008). For larger sizes, retardation effects in the interaction of the electromagnetic fields with the particles cannot be neglected any longer, and the response exhibits a strong quadrupolar contribution (Martorell et al. 1997, Nappa et al. 2006). The importance of local defects in breaking the symmetry of the particles and, consequently, in influencing the SHG process has been verified also in noncentrosymmetric gold nanostructures lithographed on a glass substrate (Canfield et al. 2004, Canfield et al. 2006). SHG from such particles show a high degree of polarization sensitivity and reveals that responses forbidden to ideal, symmetric particles are not only present but are relatively large compared with the allowed response (Canfield et al. 2004). Indeed, with respect to spherical or ellipsoidal nanoparticles, very different polarization selection rules and SH emission directions are obtained when the symmetry of the particle is lowered (Neacsu et al. 2005, Finazzi et al. 2007). Despite the nonlinear optical properties of a particle are very sensitive to local field enhancement, the presence of very strong fields alone may not be sufficient for efficient SHG from centrosymmetric nanoparticles, as a consequence of the high symmetry selectivity of SHG. An example is SHG from arrays of noncentrosymmetric T-shaped gold nanodimers with a nanogap (Canfield et al. 2007). In this case, SHG arises from asymmetry in the local fundamental field distribution and is not strictly related to the nanogap size, which determines the intensity of the electric field between the dimers. Calculations show that the local field contains polarization components that are not present in the exciting field, which yield the dominant SHG response. The strongest SHG responses occur through the local surface susceptibility of the particles for a fundamental field distributed asymmetrically at the particle perimeters. Weak responses result from more symmetric distributions despite high field enhancement in the nanogap. Nearly constant field enhancement persists for relatively large nanogap sizes (Canfield et al. 2007).

28.3.3â•‡Nonuniform Illumination The shape of particles participating in the SHG process is not the only geometrical parameter that might affect the SH emission

modes and efficiency. Actually, also the spatial distribution of the exciting field can have a great importance to determine the nonlinear response of particles. For instance, for a nonuniform polarizing field, the cancellation of the SH fields generated at opposite sides of a centrosymmetric spherical particle (see Section 28.3.1) is no longer exact and lower order SH emission becomes possible (Brudny et al. 2000). This issue is particularly important since, in practical experimental geometries, the polarizing field is in general neither of a pure longitudinal character nor a simple plane wave, as in the case illumination is performed by focusing the light beam with a microscope objective. In this context, it is worth mentioning an important application of SHG, namely, second-harmonic imaging microscopy (SHIM), which is a technique used for imaging living cells or tissue by detecting the SHG signal generated by noncentrosymmetric molecules (Campagnola and Loew 2003). By exploiting the quadratic intensity dependence of the SHG signal that confines it to the focal volume, SHIM has the capability of three-dimensional imaging deep within the tissue. SHIM offers several advantages: since it does not rely on absorption from a real electronic state, as in fluorescence microscopy, it does not give rise to photobleaching or phototoxicity. In addition, as it exploits the intrinsic hyperpolarizability of noncentrosymmetric molecules, it is a label-free technique that does not require the use of exogenous probes. A general theory for the quadratic nonlinear response of a single small nonmagnetic centrosymmetric sphere illuminated by a nonhomogeneous electromagnetic field was developed by Mochán et al. (2003), while general symmetry-based selection rules for SHG with arbitrary illumination are given by Finazzi et al. (2007) and discussed in Section 28.4. The importance of the nature of FW illumination has also been demonstrated experimentally. In fact, it has been shown that the SH radiation generated in Si spherical nanocrystallites displays a peak along the forward direction with a very narrow angular aperture (Jiang et al. 2001, 2002). As explained in Section 28.3.1, this result contradicts the behavior expected for the SH radiation produced by a single sphere illuminated by a plane wave, which should identically vanish along the forward direction (Dadap et al. 1999, 2004). Nevertheless, this observation can be explained (Brudny et al. 2003) by considering the nonuniform illumination FW field distribution, consisting of a strongly focused Gaussian beam. Similarly, in Si nanocrystals embedded uniformly in an SiO2 matrix, a configuration consisting of two noncollinear, orthogonally polarized FW beams is found to greatly enhance the SHG yield with respect to single-beam illumination (Figliozzi et al. 2005). This effect has been attributed to the strong inhomogeneities in the FW field under double-beam illumination (Figliozzi et al. 2005).

28.3.4â•‡Second Harmonic Generation in Resonant Metal Particles Although SHG is subject to the same geometry-based selection rules in both metal and dielectric nanoparticles (Finazzi et al. 2007), a relevant difference is represented by the fact that, in

28-5


metal particles, the linear and nonlinear optical properties are governed by collective electronic excitations (plasma oscillations), which can be tuned in a broad spectral range by choosing the proper particle geometry. In nanosystems, the quasiparticles associated with the collective electron excitations are often addressed to as localized surface plasmons (LSPs), or as localized plasmon polaritons, to indicate the associated electric field. In particular, depending on the size and the shape, metal particles made of gold and silver may display LSP resonance in the visible region. Thanks to this tunability, the nonlinear optical properties of noble-metal nanoparticles have found applications for in vitro and in vivo imaging (Nagesha et al. 2007), high-resolution analysis of tumor tissues (Durr et al. 2007, Bickford et al. 2008, Park et al. 2008), and enhanced photothermal therapies (O’Neal et al. 2004, El-Sayed et al. 2006). A relevant enhancement of the SHG efficiency is observed when either the FW (Hubert et al. 2007) or the SH (Antoine et al. 1997, 1998, Hao et al. 2002, Johnson et al. 2002, Abid et al. 2004, Russier-Antoine et al. 2004) frequency is set at the LSP resonance of the nanoparticles. When the SH wavelength is tuned in the vicinity of the surface plasmon resonance, the wavelength analysis indicates that the interband transitions do not play a significant role in the total SH response (Antoine et al. 1998). Rather, the nonlinear optical behavior of the clusters is dominated by the free-electron gas-like response of the conduction band, the surface plasmon resonance being much sharper than the interband transition contribution (Antoine et al. 1998, Russier-Antoine et al. 2004). Similarly, the SHG efficiency as a function of the FW frequency shows a curve that reproduces the LSP resonance (Hubert et al. 2007). Both electric dipole and electric quadrupole contributions to the SH-radiated power are observed (Hao et al. 2002).

28.3.5â•‡Random Metal Nanostructures and Rough Metal Surfaces The strong local field enhancements in random metal nanoparticles and rough metal surfaces can be exploited to improve the efficiency of the interaction of light with molecules approached to such nanostructures (Hartschuh et al. 2003), a phenomenon that can be exploited in surface-enhanced spectroscopies to obtain single-molecule sensitivity. Among these, surface-enhanced Raman scattering (SERS) makes use of the excitation provided by LSP and high local fields in such structures to enhance the Raman emission of adsorbed molecules by several orders of magnitudes (generally 106, up to 1012). Surface enhanced spectroscopies such as SERS are, therefore, intimately correlated with the plasmon properties of the nanostructures, offering, at the same time, unique potential for ultrasensitive molecular identification (Nie and Emery 1997). LSP-based nanosensors of biomolecules, based on the shift of the plasmon resonance frequency of thin metallic films, have been developed and are already commercially available. Since SHG is extremely sensitive to both local fields and LSP resonances, both theoretical (Agarwal and Jha 1982, Stockman et al.

2004, Beermann et al. 2006, Singh and Tripathi 2007) and experimental studies performed with either far-field (Bozhevolnyi et al. 2003, Beermann and Bozhevolnyi 2004) or near-field techniques (Smolyaninov et al. 1997, Zayats et al. 2000) have addressed the SHG process in such disordered systems characterized by LSP interacting with local defects. The latter act as plasmon-scattering centers and might also provide highly localized electric fields due to the lightning rod effect. LSP resonances emerge as a consequence of multiple interparticle light scattering (Stockman et al. 2004, Beermann et al. 2006) and exhibit very different strength, phase, polarization, and localization characteristics (Bozhevolnyi et al. 2003, Beermann and Bozhevolnyi 2004). SH emission is experimentally observed from small and very bright spots, whose locations depend on the light wavelength (Bozhevolnyi et al. 2003) and polarization (Zayats et al. 2000, Bozhevolnyi et al. 2003). According to simulations (Stockman et al. 2004), the spatial distributions of the fundamental frequency and SH local fields are very different, with highly enhanced SH hot spots corresponding to areas where FW and SH eigenmodes overlap (Beermann et al. 2006). Another feature of SHG in rough metal surfaces is that SH fields show a very rapid spatial decay and are strongly depolarized and incoherent (Stockman et al. 2004).

28.4â•‡Leading Order Contributions to the Second Harmonic Generation Process in Nanoparticles As illustrated in Section 28.3, the shape of the nanoparticle plays a significant role in determining the conditions for efficient SHG, which can also be significantly influenced by the illumination geometry. The latter represents a particularly important point that needs to be considered when microscopic techniques are employed to locally study field-enhancement processes. In this case, the FW illumination is not a plane wave anymore, and strong field gradients (and even a nonvanishing longitudinal field component when scanning near-field optical microscopy (SNOM) is used) can be present in the illumination area. Although nanoscale SHG has already been discussed in the literature for few particle geometries excited by either far (Zhu 2007, Dadap 2008) or near (Bozhevolnyi and Lozovski 2000, 2002) FW fields, a set of general validity selection rules for SHG might be helpful in addressing more complicated particle geometries and FW field distributions. The following text addresses the concepts that allow one to derive selection rules for SHG in isolated nanoparticles from angular momentum and parity conservation laws, as discussed by Finazzi et al. (2007).

28.4.1â•‡Conservation of Parity and Angular Momentum In the rest reference frame of the particle, the SHG process consists of two photons, belonging to the FW field interacting with the particle, that transfer their energy to a photon in the SH field oscillating at double frequency (see Figure 28.1b). Therefore, the

28-6


energy of the electromagnetic field is conserved and the particle is left in the ground state, that is, the particle initial and final states in the SHG process coincide. This fact has relevant consequences if the particle is symmetric. In this case, the particle ground state is an eigenstate of parity and, possibly, angular momentum operators. In other words, the particle has a welldefined parity (or angular momentum). Since the particle initial and final states are the same, the electromagnetic field cannot transfer parity or angular momentum quanta to the particle, and the total parity or angular momentum of the field will be conserved. These concepts can be expressed more formally by considering the explicit expression of the SHG process cross section σ. By using perturbation theory, σ can be expressed as a third-order term of the form

σ∝

∑ u ,w

TLuω′ ,w,Mω′ ,Lω′′ ,Mω′′ ,L2 ω ,M2 ω

〈ϕ0 | pˆ ⋅A ω | u 〉〈u | pˆ ⋅A ω | w 〉〈w | pˆ ⋅A 2ω | ϕ0 〉 [(Eu − E0 ) − ω + iΓ u ][(Ew − Eu ) − ω + iΓ w ]

∝ 〈ϕ 0 | pˆ ⋅ A(Lω′ , M ω′ )| u 〉〈u | pˆ ⋅ A(Lω′′ , M ω′′)| w 〉〈w | pˆ ⋅ A(L2ω , − M 2ω)| ϕ0 〉, (28.6)

+

〈ϕ0 | pˆ ⋅A ω | u 〉〈u | pˆ ⋅A 2 ω | w 〉〈w | pˆ ⋅A ω | ϕ0 〉 [(Eu − E0 ) − ω + iΓ u ][(Ew − Eu ) + 2ω]

+

〈ϕ0 | pˆ ⋅A2 ω | u 〉〈u | pˆ ⋅A ω | w 〉〈w | pˆ ⋅A ω | ϕ0 〉 , (28.5) [(Eu − E0 ) + 2ω][(Ew − Eu ) − ω + iΓ w ]

2

where φ0 is the particle ground state u and w are eigenfunctions corresponding to excited states of the unperturbed particle E0, Eu, and Ew are the corresponding energies Equation 28.5 consists of the sum of three terms, each corresponding to one of the Feynman diagrams shown in Figure 28.2. In the expression for σ, pˆ is the particle momentum operator, A 2ω the SH vector potential, and Aω the vector potential of the externally applied FW field. Note that the field generated by the particle should not be added to A ω in Equation 28.5. In fact, the particle self-interactions, that is, the interactions between the particle and the electromagnetic field the particle produces are accounted for by u and w, which describe the electronic as well as the electromagnetic excitations of the particle. Retardation 2ω

2ω

0

2ω

0

w

w

u

0

where pˆ ⋅â•›A(L, M) represents the electric or magnetic multipole operator of order (L, M). M2ω is taken with a negative sign since the emitted SH photon carries the corresponding Lz angular momentum component away from the particle. Similar terms can be given for the other Feynman diagrams in Figure 28.2. Each T term corresponds to an SHG interaction involving FW and SH photons characterized by a well-defined total angular momentum. The selection rules that determine the conditions for which a given T term is identically zero depend on how the multipoles in Equation 28.6 transform under the particle symmetry group operations. When the particle has spherical symmetry, its ground as well as excited states are eigenstates of both L2 and Lz. Therefore, the SH photon must carry away a total angular momentum equal to the sum of the angular momenta of the two FW incoming photons. In cylindrical symmetry, only the projection of the photon angular momentum along the symmetry axis (taken as the quantization axis z) needs to be conserved. The photon parity must be conserved in the case the particle displays inversion symmetry. If the particle is invariant under inversion symmetry with respect to a point (S2 symmetry in Schoenflies notation, see Tinkham (1964)), then electric and magnetic multipoles of

0

w

(a)

effects that play an important role in determining SHG in nanoparticles (Dadap et al. 1999, 2004) are therefore implicitly accounted for in Equation 28.5. As far as parity and angular momentum are concerned, it is convenient to expand the FW and SH fields appearing in Equation 28.5 in terms of electric (E) and magnetic (M) multipoles, which represent the photon angular momentum eigenstates. Let L′ω and L′′ω be the L2 quantum numbers of the two FW absorbed photons, M ω′ and M ω′′ their Lz quantum numbers, z being an arbitrary quantization axis, and L2ω and M2ω the respective quantum numbers for the emitted SH photon (see Figure 28.3). The Feynman diagram in Figure 28.2a gives a contribution to Equation 28.5 that can be expressed as the sum of T terms defined as follows:

u ω

ω

(b)

0

u ω

ω (c)

0

ω

ω

Figure 28.2â•… The three Feynman diagrams corresponding to SHG. (Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission.)

28-7

Second-Harmonic Generation in Metal Nanostructures (L΄ω, M΄ω)

(L2ω, M2ω)

(L˝ω, M˝ω)

Figure 28.3â•… Scheme of SHG reporting the L and M quantum numbers that define the angular momentum and parity of the photons involved in the process.

order (L, M) have opposite contributions to the total parity of the electromagnetic field, equal to (−1)L and (−1)L+1, respectively (Jackson 1975). In the case the particle displays inversion symmetry (C2 symmetry) with respect to the quantization axis, electric and magnetic multipoles have the same parity, given by (−1)M (Jackson 1975). Finally, considering inversion with respect to a plane perpendicular to the quantization axis (C1h symmetry), the parity of electric and magnetic multipoles of order (L, M) is given by (−1)L+M and (−1)L+M+1, respectively (Jackson 1975). By considering the electromagnetic field conservation laws imposed by the particle symmetry and contributions to angular momentum and parity associated with each multipole, one can obtain the selection rules listed in Table 28.1, which allow to Table 28.1â•… SHG Selection Rules for the Different Particle Symmetries Illustrated in Figure 28.4 Symmetry and Point Group

Selection Rule  Lω′  M′  ω

Spherical

L2 ω  ≠ 0 and − M 2 ω 

Lω′′ M ω′′

(−1)Lω′ + Lω′′ + L2 ω +m = 1 Cylindrical

M ω′ + M ω′′ − M2 ω = 0

Central, S2

(−1)Lω′ + Lω′′ + L2 ω +m = 1

Axial, C2 Reflection, C1h

28.4.2â•‡Long-Wavelength Limit

(−1)

Mω ′ + Mω ′′ − M2 ω

(−1)

Lω ′ + Mω ′ + Lω ′′ + Mω ′′ + L2 ω − M2 ω + m

=1 =1

Source:â•‡ Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission. Note:â•‡ Conservation of angular momentum for spherical symmetry corresponds to a condition involving a 3-j symbol (Tinkham 1964). The value m indicates the number of magnetic multipole transitions involved in the T term.

z

(a)

z

(b)

z

(c)

z

(d)

determine the nonvanishing T terms that contribute to the SGH process for any combination of FW and SH multipoles (Finazzi et al. 2007). Thus, to apply the selection rules listed in Table 28.1, one has to determine the multipole expansion of the impinging FW electromagnetic field in order to understand which SH multipoles can contribute to the generated SH radiation. If the symmetry of the particle results from a combination of symmetry groups, the corresponding conditions listed in Table 28.1 must be satisfied at the same time. For instance, SHG from particles with C2h symmetry must obey both the selection rules for axial (C2) and reflection (C1h) symmetries with respect to the same quantization axis. For the same reason, when the particle exhibits two or three orthogonal symmetry axes, the corresponding selection rules must be simultaneously satisfied by the field multipole expansions expressed by using each symmetry axis as the quantization axis. Note that, strictly speaking, the spherical and cylindrical symmetry groups would have to be excluded since they are not compatible with any of the symmetry properties of the various Bravais lattices. Moreover, crystallographic defects (such as grain boundaries, dislocations, and atomic vacancies) would disrupt the internal symmetry of the particles. Thus, it would seem that symmetry-based selection rules would not be respected by SHG in real nanostructures. However, such limitations are not expected to play a significant role in the metal structures that are usually investigated in nano-optics (Finazzi et al. 2007), which are typically realized in aluminum, silver, or gold. The electronic and electromagnetic properties of these materials are in fact well described by the free-electron model, so that the particle symmetry properties are independent from the particle crystallography but are just determined by the boundary conditions imposed by the particle surface.

z

(e)

Figure 28.4â•… The particle symmetry groups discussed in the text and referred to the quantization axis z: (a) spherical, (b) cylindrical, (c) central, (d) axial, and (e) reflection symmetries across a mirror plane perpendicular to z. (Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission.)

In general, selection rules allow one to discriminate the transitions that, for symmetry reasons, cannot contribute to the cross section of a particular physical phenomenon, but they do not give any information about the magnitude of the nonvanishing contributions. However, by considering the multipole expansion of the electromagnetic FW and SH fields in the SHG process, it is possible not only to identify the emission channels that are ineffective to SHG but also to estimate a hierarchy among them, identifying the ones that mostly contribute to the SHG process. This is possible in the long-wavelength limit (Rayleigh limit) characterized by ka ≪ 1, with k being the FW wave vector and a the particle lateral size. In fact, matrix elements of the form 〈u | pˆ ⋅â•›A(L,â•›M ) | w〉 rapidly decrease when the multipole order L is increased (Jackson 1975). The ratio R(L) between matrix elements for successive orders, L and (L + 1), of either electric or magnetic multipoles of the same frequency is

R(L) ∼

ka , 2L

(28.7)

28-8


while the magnetic multipoles of order L have cross sections of the same order in ka as electric multipoles of order (L + 1) (Jackson 1975, Dadap et al. 2004). Since the expression of the SHG cross section σ reported in Equation 28.5 depends on the square value of the sum of products involving three matrix elements, we need to consider only the contributions associated with the lowest order multipoles. By restricting the analysis to the lowest order terms, five distinct SHG emission modes can be indicated. Following the notation in Dadap et al. (2004), these are E1 + E1 → E1, E1 + E2 → E1, E1 + M1 → E1, E1 + E1 → E2, and E1 + E1 → M1. In this notation, the first two symbols refer to the nature of the interaction with the FW field, and the third symbol describes the SH emission. For example, the E1 + E2 → E1 interaction represents electric dipole (E1) SH emission that arises through combined FW electric dipole and electric quadrupole (E2) excitations. M1 indicates a magnetic dipole transition. The E1 + E1 → E1 transition corresponds to the leading order contribution to the SHG process, generating an SH field with a magnitude of a factor (ka)−1 higher than the field generated by the other terms, resulting in a (ka)−2 higher irradiated power. The E1 + E1 → E1 interaction violates parity conservation for any incident FW field distribution on centrosymmetric particles since (−1)Lω′ + Lω′′ + L2 ω = (−1)1+1+1 = −1, thus violating the selection rule for S2 symmetry particles reported in Table 28.1. This channel may, however, be allowed for lower symmetry particles.

28.5â•‡Emission Patterns and Light Polarization The emission modes that participate in the SHG process can be experimentally identified by considering SH emission pattern and light polarization. In fact, each multipole contributing to the SHG process is characterized by a well-defined angular dependence of the irradiated intensity and polarization.

28.5.1â•‡Irradiated Intensity Let k 2ω be the SH wave vector and r the position with respect to the particle. According to Jackson (1975), in the far-field limit characterized by k2ω >> r −1, the SH electric fields E(LE2 ω) M2 ω (r) and E(LM2 ω)M2 ω (r), produced by either electric or magnetic multipoles of order (L2ω , M2ω), are given by

E(LE2)ω , M2 ω (r) = −(−i)(L2 ω +1)

ik2 ω r

e (uˆ r × r × ∇YL2 ω, M2 ω ), k2ωr

E(LM2 ω), M2 ω (r) = −(−i)(L2 ω +1)

eik2 ωr (r × ∇YL2 ω , M2 ω ). k2 ωr

(28.8a)

(28.8b)

In Equations 28.8 uˆr is a unit vector in the radial direction, and YL2 ω , M2 ω is the spherical harmonic of order (L2ω, M2ω). Pure electric and magnetic SH multipoles have the same angular

z

z

L2ω = 1, M2ω = ±1

L2ω = 1, M2ω = 0 z

L2ω = 2, M2ω = 0

z

L2ω = 2, M2ω = ±1

z

L2ω = 2, M2ω = ±2

Figure 28.5â•… Graphical representation of emission patterns dP/dΩ(L2ω , M2ω) for dipoles (upper row) and quadrupoles (lower row). The time-averaged radiated power per unit solid angle displays cylindrical symmetry around the quantization axis z. (Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission.)

distribution of the time-averaged radiated power per solid angle dP/dΩ(L2ω,â•›M2ω). A graphical representation of the radiated power is given in Figure 28.5 for dipoles (L2ω = 1) and quadrupoles (L2ω = 2) (Jackson 1975). If more than a single multiplet is responsible for the generation of SH radiation, interference among the SH radiation emitted by the contributing multipoles occurs and the emission pattern can be significantly different from those in Figure 28.5. In such cases, identifying the multipoles that are responsible for the SH emission might be quite complicated. Further information, however, can be extracted by analyzing the polarization dependence of the SH radiation as a function of the polarization of the impinging FW field.

28.5.2â•‡Second Harmonic Polarization From Equations 28.8, one can also obtain the polarization state of the emitted SH light. Table 28.2 reports the direction of the electric far field for pure electric and magnetic multipoles with L2ω ≤ 2. The direction is indicated by a combination of the mutually perpendicular unit vectors uˆθ and uˆ ϕ defined in a spherical coordinate system centered on the particle, in which θ and ϕ are the polar and azimuthal angles, respectively, the former being referred to the z quantization axis. An imaginary (real) ratio between the coefficients of uˆθ and uˆ ϕ corresponds to a π/2 (zero) phase difference between the two field components and hence indicates elliptical (linear) polarization (Jackson 1975).

28-9

Second-Harmonic Generation in Metal Nanostructures Table 28.2â•… SH Electric Field Polarization for Pure Electric or Magnetic Multipoles with L2ω ≤ 2 M2ω = 0

M2ω = ±1

L2ω = 1 (E) L2ω = 1 (M) L2ω = 2 (E)

sin θuˆθ sin θuˆϕ −i sin θ cos θuˆθ

L2ω = 2 (M)

−i sin θ cos θuˆϕ

e (±cos θuˆθ ∓ iuˆϕ) −e±iϕ(iuˆθ ∓ cos θuˆϕ) e±iϕ(∓icos 2θuˆθ + cos θuˆϕ) −e±iϕ(cos θuˆθ + icos 2θuˆ ϕ)

M2ω = ±2

±iϕ

e±2iϕsin θ(icos θuˆθ ± uˆϕ) ±2iϕ e sin θ (±uˆθ + icos θuˆ ϕ)

Source: Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission. Note: For each point in space, the unit vectors uˆ θ and uˆϕ are, respectively, parallel and perpendicular to the plane containing the z (quantization) axis, and are both perpendicular to the radial unit vector uˆ r.

28.6â•‡Allowed and Forbidden Second Harmonic Emission Modes: Examples This section discusses some examples of applications of the arguments discussed above. Two relevant cases of FW illumination are addressed, namely, illumination with a plane wave and illumination with a strongly laterally limited beam, for example, through a high numerical aperture or via a near-field tip.

28.6.1â•‡Plane-Wave Illumination In an arbitrarily polarized FW plane wave propagating along the z direction, the field vector A(z) can be expressed as A(0)eikz, with A(0) being a vector perpendicular to the propagation direction, A(0)â•›⋅â•›uˆ z = 0. By taking the origin of the z axis coincident with the particle position, and truncating the multipole expansion around z = 0 of the matrix element 〈u | pˆ ⋅ A | w〉 to second order in ka, one obtains 〈u | pˆ ⋅A | w 〉

Ew − Eu 〈u | r ⋅ E(0)| w 〉 + 〈u | Lˆ ⋅B(0)| w 〉 ω

+ ik 〈u | pˆ z[xAx (0)) + yAy (0)]| w 〉.

(28.9)

In this expression, E(0) and B(0) are complex vectors representing the amplitude and phase of the electric and magnetic fields,

respectively, at the particle position, while Ax(0) and Ay(0) are the complex components of A(0), and Lˆ = r × pˆ. The three terms on the right-hand side of Equation 28.9 correspond to E1, M1, and E2 transitions, respectively. Let us first discuss the case of circularly polarized light. In this case, it can be shown (Finazzi et al. 2007) that all the three FW absorption transitions described by the matrix elements in Equation 28.9 involve FW photons characterized by an Lz quantum number Mω either equal to +1 or −1 according to the sign of the circular polarization (right or left) of the FW light impinging on the particle. By recalling that dipole terms are characterized by L2 = 1 and quadrupole terms by L2 = 2, the lowest allowed emission multipoles participating in the SHG process can be obtained from Table 28.1 in a straightforward manner. The allowed SH emission modes are listed in Table 28.3. As anticipated above, the lowest order transition E1 + E1 → E1 is forbidden under circularly polarized FW plane-wave illumination for particles displaying central symmetry, but it is allowed for noncentrosymmetric, noncylindrical particles. In this case, SH emission is given by an E1 term with M2ω = 0 or M2ω = ±1 for C2 or C1h symmetry, respectively. This emission mode corresponds to SHG from an electric dipole oriented parallel (for C2 symmetry) or perpendicular (in the case of C1h symmetry) to z. The lowest allowed SHG channel for spherical or cylindrical particles is E1 + E1 → E2 (Dadap et al. 1999, 2004), with M2ω = ±2. This SH emission mode corresponds to an irradiated SH power angular distribution displaying cylindrical symmetry around the propagation axis of the FW field, with a maximum in the plane perpendicular to z and a null in the forward and backward directions (Dadap et al. 1999, 2004), as displayed in Figure 28.5. In the case of linearly polarized FW illumination, it is convenient to consider the selection rules that can be obtained by considering both the propagation direction of the FW plane wave (z axis) and the direction of the FW electric field vector (x axis) as the quantization axis. Therefore, one should calculate two multipole expansions of the FW wave, one for each choice of the quantization axis, and consider the possible values of both the Lz and Lx photon quantum numbers. Moreover, both axes should be taken into account to define the particle symmetry group. Let Mω and M2ω be the FW and SH photon Lz quantum numbers, and Nω and N2ω be the FW and SH photon Lx quantum numbers, respectively. It can be shown (Finazzi et al. 2007) that the multipole expansion

Table 28.3â•… SHG Selection Rules for the Different Particle Symmetries Illustrated in Figure 28.4, under Circularly Polarized Far-Field Plane-Wave Illumination Symmetry and Point Group

E1 + E1 → E1

E1 + E1 → E2

E1 + E2 → E1

E1 + M1 → E1

E1 + E1 → M1

Spherical Cylindrical Central, S2 Axial, C2 Reflection, C1h

Forbidden Forbidden Forbidden Allowed for M2ω = 0 Allowed for M2ω = ±1

Allowed for M2ω = +2 or −2 Allowed for M2ω = +2 or −2 Allowed Allowed for M2ω = 0, ±2 Allowed for M2ω = 0, ±2

Forbidden Forbidden Allowed Allowed for M2ω = 0 Allowed for M2ω = 0



Source: Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission. Note: Every transition multipole is considered separately. The propagation direction (z axis) is chosen as the quantization axis.

28-10


Table 28.4â•… SHG Selection Rules for the Different Particle Symmetries Illustrated in Figure 28.4, under Linearly Polarized Far-Field Plane-Wave Illumination Symmetry and Point Group Spherical Cylindrical Central, S2 Axial, C2 Reflection, C1h

E1 + E1 → E1 Forbidden Allowed for M2ω = 0 Forbidden Allowed for M2ω = 0 Allowed for M2ω = ±1

E1 + E1 → E2

E1 + E2 → E1

E1 + M1 → E1

E1 + E1 → M1

Allowed for M2ω = 0, ±2 Allowed for M2ω = 0, ±2 Allowed Allowed for M2ω = 0, ±2 Allowed for M2ω = 0, ±2

Allowed for M2ω = 0 Allowed for M2ω = 0 Allowed Allowed for M2ω = 0 Allowed for M2ω = 0

Allowed for M2ω = 0 Allowed for M2ω = 0 Allowed Allowed for M2ω = 0 Allowed for M2ω = 0


Source: Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission. Note: The propagation direction (z axis) is chosen as the quantization axis.

Table 28.5â•… SHG Selection Rules for the Different Particle Symmetries Illustrated in Figure 28.4, under Linearly Polarized Far-Field Plane-Wave Illumination Symmetry and Point Group Spherical Cylindrical Central, S2 Axial, C2 Reflection, C1h

E1 + E1 → E1 Forbidden Allowed for N2ω = 0 Forbidden Allowed for N2ω = 0 Allowed for N2ω = ±1

E1 + E1 → E2

E1 + E2 → E1

E1 + M1 → E1

E1 + E1 → M1

Allowed for N2ω = 0 Allowed for N2ω = 0, ±2 Allowed Allowed for N2ω = 0, ±2 Allowed for N2ω = 0, ±2

Allowed for N2ω = ±1 Allowed for N2ω = ±1 Allowed Allowed for N2ω = ±1 Allowed for N2ω = ±1

Allowed for N2ω = ±1 Allowed for N2ω = ±1 Allowed Allowed for N2ω = ±1 Allowed for N2ω = ±1

Allowed for N2ω = 0 Allowed Allowed for N2ω = 0 Allowed for N2ω = 0

Forbidden

Source: Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. Note: The electric field polarization direction (x axis) is chosen as the quantization axis.

of the FW plane wave corresponds to E1, M1, and E2 terms in Equation 28.9 that involve FW photons with an Mω quantum number that can be either +1 or −1 but not 0. On the other hand, the expansion obtained by choosing the x axis as the quantization axis shows that the E1, M1, and E2 terms correspond to transitions involving FW photons with L2 and Lx quantum numbers restricted to (L ω = 1, Nω = 0), (L ω = 1, Nω = ±1), and (Lω = 2, Nω = ±1), respectively. With these figures in mind and the general rules summarized in Table 28.1, one can find the selection rules listed in Tables 28.4 and 28.5. From these, one can see that the E1 + E1 → E1 SHG channel can be excited by linearly polarized FW plane-wave illumination in all noncentrosymmetric particles (Finazzi et al. 2007). In particular, if the particle exhibits C2 symmetry either around the z or the x axis, SH emission will be given by an oscillating dipole parallel to the C2 symmetry axis (Finazzi et al. 2007). In centrosymmetric particles illuminated by linearly polarized light, electric dipole SH emission is always given by a dipole parallel to z, while the quadrupole SH emission pattern is symmetric around x (see Figure 28.4) (Finazzi et al. 2007). M1 emission is forbidden (Dadap et al. 1999, 2004) because the selection rules would lead to M2ω = N2ω = 0, which correspond to a monopole term that cannot radiate (Finazzi et al. 2007).

28.6.2â•‡Illumination with a Laterally Limited Light Beam When the particle is illuminated by a laterally limited FW beam, which might be obtained by focusing with a high numerical-aperture objective or even with near-field illumination, and the beam waist is comparable with the particle size, the beam cannot be

considered as a plane wave any longer. In this case, other emission channels than those listed in Tables 28.3 through 28.5 become available. In these conditions, in fact, the particle experiences large longitudinal components of the FW field, which are absent in the case of illumination with a plane wave. The presence of such longitudinal components can be easily understood for a converging beam focused by an objective, where the FW light wave vector k has a broad angular distribution determined by the objective numerical aperture. Intense longitudinal components should also be expected in the proximity of near-field probes (Novotny and Hecht 2006). The consequence is that SHG can be induced by a combined FW photon absorption from perpendicular field components acting on the particle (Figliozzi et al. 2005), which is not possible for plane-wave illumination. For instance, illuminating with a laterally confined beam, absorption of FW photons with Lz quantum number Mω = 0 from a linearly polarized FW field becomes possible, which is excluded for plane-wave illumination (Finazzi et al. 2007). For the same reason, FW photons with Mω = 0, ±1, ±2 are available for E2 absorption (Finazzi et al. 2007). Another important difference with respect to in-plane wave illumination is represented by the fact that the rapid spatial variations of the field intensity can increase the relative weight of the higher-order multipoles in the SHG process. This is readily understood in far-field illumination when the spot size, which is limited by diffraction to about 1/k, is comparable with the particle size a, resulting in a violation of the long-wavelength approximation. Similarly, strong field gradients are expected at the apex of a near field probe (Novotny and Hecht 2006), again invalidating the long-wavelength approximation and implying that high-order multipoles cannot be a priori neglected.

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kω

Eω

z

kω

z

q2ω x

x (a)

y

q2ω y

(b)

Figure 28.6â•… Allowed E1 second-harmonic emission channels for a particle with C2 symmetry (axial particle over a substrate) excited by (a) a plane wave or (b) a laterally limited beam. The allowed directions of the particle electric dipole q2ω generating the E1 second-harmonic radiation are indicated by the double-headed arrows on the particle. (Reprinted from Finazzi, M. et al., Phys. Rev. B, 76, 125414, 2007. With permission.)

As a relevant example, let us address SHG in lithographed particles on a substrate (see Section 28.7), such as those employed in surface-enhanced spectroscopy techniques. The presence of the substrate breaks the inversion symmetry with respect to the particle center. For ellipsoidal-shaped dots as those described by Grand et al. (2003) and Zavelani-Rossi et al. (2008), the particle symmetry group reduces to C2v (see Figure 28.6). In this case, the selection rules governing SHG in C2 symmetry around the z axis must apply, z being the substrate normal, which is taken parallel to the optical axis. The particles also belong to the C1h symmetry group defined with respect to either the x or y axis, oriented parallel to the in-plane principal axes of the ellipsoids. However, this further symmetry does not provide more restrictive selection rules. When such particles are excited with an FW plane wave with k ∙ z and considering only the lowest-order E1 + E1 → E1 transitions, Mω can only assume the values ±1 both for linearly and circularly polarized light, so parity is conserved only for M2ω = 0, corresponding to SH emission from an electric dipole parallel to both k and z (see Figure 28.6) (Finazzi et al. 2007). When the particle is illuminated by a focused FW beam or by the near field of a tip, photons with Mω = 0 can be absorbed. This results in new SHG channels, namely the ones that correspond to SH radiation from electric dipoles lying in the plane of the substrate, becoming available (see Figure 28.6) (Finazzi et al. 2007). In this case, the details of the SH-radiated intensity and polarization will depend on the particle fine structure, which defines the relative strength of each channel.

28.7â•‡Second Harmonic Generation in Single Gold Nanoparticles As discussed in Section 28.3, the nonlinear optical response of nanostructured systems has been experimentally studied, so far, mainly by far-field techniques. These, however, are

28.7.1â•‡Near-Field Microscopy Setup To observe SHG at the nanoscale, we combine high peak power and high spatial resolution by coupling femtosecond pulses to a hollow-pyramid aperture SNOM (see Figure 28.7). The pulses

30 fs Laser pulses

Lens

Prisms Pre-compressor

Hollow-pyramid probe

Dichroic mirror FW

Microscope objective Sample PZT xyz scanner

SH PMT

PMT

Eω

limited by diffraction and do not allow a direct mapping of the nanoscale field enhancement. These limitations can be overcome by SNOM, which can reach sub-100â•›nm lateral resolution. This technique consists of bringing the sample in interaction with the near field of a source (a tip or a backilluminated aperture in a metal-coated probe). In this case, the bandwidth of spatial frequencies associated with the evanescent waves from the probe is unlimited and the resolution can in principle be arbitrarily optimized. The response of the probe–sample interaction is recorded with standard far-field collection optics. State-of-the-art lateral resolution for aperture probes is around 50â•›nm. Near-field nonlinear optical microscopy is a powerful tool to characterize local field enhancements in metal nanostructures since it combines the great sensitivity of nonlinear optical response with the spatial superresolution of near-field microscopy. However, the technical challenges of combining scanning probe and ultrafast technologies and the typical low peak power available at the output of optical near field probes have so far limited the number of studies of SHG from single nanoparticles in closely packed arrangements (Biagioni et al. 2007, Breit et al. 2007, Celebrano et al. 2007, 2008a,b, Zavelani-Rossi et al. 2008). This section describes SHG from single gold nanoparticles obtained with a SNOM setup. As illustrated in Section 28.6.2, the high lateral confinement typical of near-field illumination enables the observation and exploitation of unusual and peculiar SHG modes. Moreover, the acquisition of SH maps provides complementary information to FW images, which typically results from a complex interplay between scattering, absorption, and reflection, all contributing to light extinction (Novotny and Hecht 2006). The comparison between FW and SH maps therefore allows for a better interpretation of the optical response of metal nanoparticles and discriminating among different light extinction particle behaviors that would not be possible to address by just collecting the FW signals (Biagioni et al. 2007, Celebrano et al. 2007, 2008a,b, Zavelani-Rossi et al. 2008).

Filters

Figure 28.7â•… Schematics of our SNOM, with the ultrashort laser beam coupled to the measurement head: The collection optics, filters, and detector are also indicated. PMT indicates a photomultiplier tube.

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are generated by a long-cavity mode-locked Ti:sapphire oscillator (26â•›MHz repetition rate), producing 20â•›nJ, 27â•›fs pulses at 800â•›nm. The SNOM probe consists of a silicon nitride cantilever with a hollow pyramid tip. The tip is aluminum-coated with a circular aperture at the apex, with diameter ranging between 100 and 200â•›nm. This probe offers several advantages compared with metal-coated tapered optical fibers, such as larger taper angle, lower absorption, preservation of light polarization (Biagioni et al. 2005), and pulse duration (Labardi et al. 2005) at the output. These improvements enable achieving peak powers more than two orders of magnitude higher in the near field, resulting in greatly enhanced nonlinear optical response of the sample. Typical tip throughputs at 800â•›nm range between ∼10−4 for the 100â•›nm tips and 5 × 10−3 for the 200â•›nm ones. The tips with larger aperture were selected for the experiments presented in this section. Tip-sample distance is controlled by an optical lever that is sensitive enough to allow nondestructive contact mode stabilization on the samples. The FW/SH light is collected in the far field by a long working distance microscope objective (numerical aperture = 0.75) and split by a dichroic filter. The FW and the SH are simultaneously detected by photomultiplier tubes. Bandpass filters are inserted in the SH beam to further reject the FW and TPPL from the sample. A 30â•›nm bandpass interference filter centered at 405â•›nm is inserted in the SH beam path to reject TPPL from the sample, which is negligible for wavelength shorter than 450â•›nm (Bouhelier et al. 2005, Imura et al. 2005, 2006). The quadratic dependence of SH intensity on the excitation beam power was verified. A mechanical chopper with a 1:6 duty cycle allows increasing the peak power for a given level of average power, typically 1â•›mW incident on the tip, and lock-in detection improves the signal-to-noise ratio, allowing for faster scans (integration time 30–100â•›ms per point). Hereafter, we present two different types of gold nanoparticles. The first ones are triangles (height 15–25â•›nm) on a glass substrate obtained from a projection pattern (“Fischer pattern” (Fischer and Zingsheim 1981)), in a hexagonal array with 453â•›nm periodicity. Such a sample is chosen as an example of a SNOM study of SHG in a dense network of particles. The second type of particles are well-separated isolated ellipsoids (height about 60â•›nm) produced by electron beam lithography on a quartz substrate (Grand et al. 2003) in square arrays.

28.7.2â•‡Fundamental Wavelength and Second Harmonic Maps Figure 28.8a shows the topography of the Fischer pattern together with its SH optical image (Figure 28.8b). The topography shows the regular array of gold triangles: most of them are well separated, some are in contact with each other, and big defects cover portions of the sample. In the FW image (not shown), triangles appear dark, although the resolution is poor due to the quite large aperture diameter. Nevertheless, contrasted and well-resolved SHG maps from the gold triangles are detected. The background signal is attributed to SHG from both the glass substrate and the tip edges. The inset of Figure 28.8b shows a line profile of the SH image, demonstrating very good signal-to-noise ratio, high contrast (3:1), and good spatial resolution (better than 100â•›nm, see Figure 28.8c). Figure 28.8 highlights the unique capability of nonlinear SNOM to image SHG from closely packed metal nanostructures. It is interesting to note that not all the triangles observed in topography emit SH radiation with the same efficiency: some of them display an intense SH emission, while others are nearly dark. This supports the absence of topographical artifacts (Hecht et al. 1997) in the image, proved by the fact that in further measurements (not shown here), the topographic resolution was missing, yet a clear and well-resolved SHG image could still be collected. The high variability of the SHG signal is addressed further below when SHG in isolated nanoparticles is discussed. The lithographed nanorods are about 60â•›nm high, with a short axis of about 70â•›nm and long axes equal to 100, 150, or 400â•›nm. The array period is 1â•›μm. According to the Mie theory, the plasmon resonances in elongated structures depend on their aspect ratio. Indeed, far-field extinction spectra (see Figure 28.9), with exciting white light beam polarization parallel to the major axis, display a peak around 690â•›nm for the 100â•›nm ellipsoids, around 800â•›nm (i.e., resonant with our FW) for the 150â•›nm ones and more than 1000â•›nm for the 400â•›nm ones (not shown). These three distinct particle geometries thus provide different resonating regimes that correspond to different linear and nonlinear optical behaviors and near-field properties. These are highlighted in Figure 28.10, showing the particle topography, FW near-field extinction, and SH emission obtained with the FW illumination linear polarization parallel to the nanorod long axis. The optical images at both 70

B A

A

500 nm (a)

72

B

0 nm

48

26 fW (b)

(c)

300 600 900 Distance (nm)

SHG (fW)

33

24

Figure 28.8â•… Projection pattern: (a) topography and (b) SH SNOM image (size: 5 × 2.9â•›μm2). The dashed circle (A) shows an SH emitting triangle and solid circle (B) shows a dark one. (c) Cross section from the raw data of the SH image, along the dotted line.

28-13


Normalized extinction

1.0

100 nm 150 nm

0.8 1.0 μm 0.6

(a)

(d) Pol.

0.4

(g) Pol.

Pol.

0.2 1.0 μm 500

550

600

650

700 750 800 Wavelength (nm)

850

900

(b)

950 1000

Figure 28.9â•… Extinction spectra of gold nanorods with a major axis length either equal to 100 (squares) or 150â•›nm (dots). The spectra have been normalized to each other at the maximum extinction. The light polarization is parallel to the particle long axis. The extinction spectra of the 400â•›nm-long nanorods also, discussed in the text, are not reported since the longitudinal plasmon resonance falls off the range of the analyzer. (Courtesy of J. Grand and P.-M. Adam.)

the FW and the SH wavelengths strongly depend on the particle size. In particular, in FW maps, all the 100â•›nm ellipsoids appear bright, while all the 150 and 400â•›nm ellipsoids appear dark. In SH images, (1) most of the 100â•›nm-long particles do not emit, (2) most of the resonant 150â•›nm-long particles emit uniformly with high contrast (see line profiles in Figure 28.10), and (3) the 400â•›nm-long particles appear darker than the substrate. A general picture for these results can be obtained by a combined analysis of the FW and SH maps of the metal nanostructures. As apparent from Figure 28.10, the SH maps provide complementary information with respect to the FW images. For instance, only by comparing the SH emission properties of the 150 and 400â•›nm-long particles, it is possible to recognize that these fall in different optical regimes, due to the FW excitation being on or off resonance, respectively, with respect to the particle LSP excitations. The near-field FW properties of the particles can be understood by recalling that the particle-to-background contrast in the near-field extinction maps results from the interference between an FW wave propagating into the far-field and a nonpropagating near-field distribution (Mikhailowsky et al. 2003). As a general rule, a metal particle should appear brighter than the background when the frequency of the excitation is higher than that of the particle LSP resonance, while it should be darker when the frequency of the FW excitation is set below the resonance (Mikhailowsky et al. 2003). At the LSP resonance, the particle should show no contrast with respect to the background. The presence of these interference effects explains the trend observed in the FW maps displayed in Figure 28.10, where the particles have a well-controlled aspect ratio, but it may hinder the interpretation of FW near-field images in other contexts, especially when the frequency of the LSP resonances of the particles is not known. However, the resonating behavior of the 150â•›nm-long

(e) Pol.

(h) Pol.

Pol.

1.0 μm (c) SH (fW)

0.0

28 20 12

0.5 1.0 1.5 2.0 2.5 Distance (μm)

(f )

0.5 1.0 1.5 2.0 2.5 Distance (μm)

(i)

0.5 1.0 1.5 2.0 2.5 Distance (μm)

Figure 28.10â•… (See color insert following page 20-14.) Nanoparticles: (a), (d), and (g) topography; (b), (e), and (h) FW transmission; and (c), (f), and (i) SH emission SNOM images with corresponding cross sections along the dashed lines, from the raw data. Incident light is polarized parallel to the major axis. The particle major axis lengths are 100â•›nm (a)–(c), 150â•›nm (d)–(f), and 400â•›nm (g)–(i). Image size: 3 × 3â•›μm2 . (Reprinted from Zavelani-Rossi, M. et al., Appl. Phys. Lett., 92, 093119, 2008. With permission.)

nanorods becomes apparent from the SH maps, since these particles efficiently emit SH radiation as a result of the presence of strongly enhanced and localized electric field associated with the particle LSP oscillations, which are effectively excited only for these particles. Instead, most of the 100â•›nm-long particles do not emit SH radiation, and the 400â•›nm-long particles, being larger than the tip aperture, strongly absorb/scatter the SH light generated at the tip and appear darker than the background. The nonlinear optical behavior of off-resonance particles is further highlighted in Figure 28.11, where FW and SH maps are collected by exciting the nanorods with an FW field parallel to the rod minor axis. With this type of excitation, the LSP resonance frequencies are shifted toward the blue region of the spectrum and cannot be excited by the FW light. In this case, SHG is observed at correspondence with the high-curvature regions of the particle, as a consequence of highly enhanced and localized electric fields by the lightning rod effect, in agreement with the theoretical predictions of Beermann and Bozhevolnyi (2004) and Stockman et al. (2004). It is interesting to note that, in Figure 28.11, the bright spots in the SH maps seem to correspond to

28-14


Pol. 85 nW

FW intensity

400 nm

(a)

(d)

(b)

(e)

SHG intensity

22 fW

58 nW

12 fW (f )

(c)

Figure 28.11â•… (a)–(c) FW transmission and (d)–(f) SH emission excited by incident FW light polarized perpendicular to the particle major axis. The particle major axis lengths are 100â•›nm (a) and (d), 150â•›nm (b) and (e), and 400â•›nm (c) and (f). (Reprinted from Zavelani-Rossi, M. et al., Appl. Phys. Lett., 92, 093119, 2008. With permission.)

dark spots in the FW images, that is, the high-Â�curvature regions where strong fields are localized also correspond to areas where larger FW scattering/absorption occurs, giving rise to a larger local linear extinction cross section. We believe that intense localized defects are also responsible for the high variability of the SHG efficiency displayed by nominally identical particles displaying quite similar FW (see Figures 28.8 and 28.10). In this case, local field enhancement would be associated with highcurvature defects in the particles. To summarize, the SHG efficiency depends on the particle fine structure that determines (1) local FW field enhancements, induced either by LSP excitation or by lightning rod effects in areas with high-curvature or local imperfections and (2) the modes contributing to the emission process, which are described in the next section.

p Ti

e

S

pl am

O

ec bj

tiv

e lar Po

ize

r

(b) FW

SH (c)

28.7.3â•‡Second Harmonic Polarization Analysis

Figure 28.12â•… Experimental geometry of the polarization analysis setup in the SNOM microscope. The FW light is polarized parallel to the particle major axis. SH maps have been collected for the two analyzer directions (b) and (c), corresponding to SH linear polarizations parallel and perpendicular to the FW polarization, respectively. The corresponding SH maps are shown in Figure 28.13b and c, respectively.

This section shows how the analysis of the polarization of the SH-emitted light might allow one to recognize which emission modes are contributing to the SHG process for a given particle geometry and illumination conditions. Such analysis will be applied to the same lithographed isolated gold particles, already described in Section 28.7.2. The polarization state of the SH light emitted by the gold nanoellipsoids is analyzed as illustrated schematically in Figure 28.12. The FW light is polarized parallel to the major axes of the 150â•›nm-long particles to excite their strong longitudinal SPT resonance. The SH radiation polarization is analyzed by a polarizer on detection. Figure 28.13 shows two typical SH maps that are obtained with the analyzer parallel (Figure 28.13b) and perpendicular (Figure 28.13c) to the excitation polarization. From these maps, one can see that the SH light emitted by the nanoparticle is polarized parallel to its short axis

(see Figure 28.13b,c,d). This particular polarization pattern of the emitted SH light is incompatible with the lowest allowed emission modes one should expect for such particles for an FW plane-wave illumination. In this case, in fact, SH radiation would be emitted by a dipole oscillating perpendicularly to the substrate sustaining the particle, as illustrated in Figure 28.6a. Such an emission mode would give the same SH maps independently from the direction of the analyzer axis. This is obviously in contradiction with the experimental results shown in Figure 28.13 and is a demonstration that a novel particle emission mode, which is forbidden for plane-wave illumination, becomes accessible. In fact, the presence of a symmetry-breaking substrate together with the strong longitudinal FW field component, typical of near-field illumination, allow for efficient excitation of an SH-emitting electric dipole perpendicular to both the incident FW light polarization and the detection axes,

28-15


SH (fW)

32

b c

24 16 8

200 nm (a)

(d)

100

200 300 400 Distance (nm)

In

(b)

Out (c)

Figure 28.13â•… Nanoellipsoids with 150â•›nm major axis: topography (a), SH emission with a polarizer on detection parallel (b), and perpendicular (c) to the major axis (incident light is polarized parallel to the major axis). Image size: 0.6 × 0.6â•›μm2 . (d) Intensity profiles from the raw data of images (b) (circles) and (c) (triangles).

as illustrated in Figure 28.6b. From the analysis of the SH polarization pattern in Figure 28.13, one can conclude that only the SH-emitting dipole perpendicular to the particle minor axis is significantly excited by the FW field. For this particle geometry, the SHG selection rules discussed in Section 28.4 do not exclude the possibility that SHG could be due to a dipole oscillating in the direction of the long axis of the particles, so there is no simple symmetry-based justification to the fact that such emission is not observed. This behavior has to be ascribed to the particular shape and resonance conditions of the particles. Such characteristics can in effect influence in a complex way the particle SH emission properties. To give a tentative explanation of this behavior, we remark that a much larger portion of gold–air interface is available along the long sides of the ellipsoid, so that a major contribution of SH light oriented parallel to the short axis might actually be expected as a surface contribution to SHG. Finally, we would like to stress that the spatial uniformity of the polarized SHG pattern measured on several particles is a hint of the negligible influence of particle defects, which can also lower the symmetry of the particles and contribute to SHG with modes that would otherwise be forbidden by parity and angular momentum conservation in a particle with a perfectly controlled shape.

28.8â•‡Perspectives A very promising perspective for future nano-optics is the possibility to simultaneously control of the spatial and temporal properties of the optical near field in the vicinity of a nanostructure by

illumination with broadband optimally polarization- and timeshaped femtosecond light pulses. Adaptive shaping of the phase and amplitude of femtosecond laser pulses has been developed into an efficient tool for the directed manipulation of interference phenomena, thus providing coherent control over various quantum-mechanical systems. The interest to extend coherent control methods to nanostructures consists in overcoming the spatial limitation due to diffraction. Coherent control of the spatial and temporal evolution of optical near-fields by amplitude and polarization pulse shaping in plasmonic nanostructures has recently been theoretically proposed (Stockman et al. 2002, Brixner et al. 2005, 2006, Sukharev and Seideman 2007) and experimentally demonstrated (Aeschlimann et al. 2007) by applying an adaptative method based on measuring the local fields with two-photon photoemission electron microscopy. Progress in the coherent control of electric fields in nanostructures thus might open the access to simultaneously ultrafast and nanoscale new physics. Nonlinear near-field optical microscopy might thus play an extremely important role in the development of nanoscale coherent control, since it can be employed as a probe of local field enhancement. In this frame, the temporal dynamics of the nonlinear phenomenon is, however, extremely important. SHG represents an ideal candidate to this task since the conversion of two FW photons into an SH photon is itself a coherent process (Imura et al. 2005), at variance from, for example, TPPL, which is incoherent in noble metal nanostructures (Imura et al. 2005). Indeed, we have recently demonstrated (Biagioni et al. 2009) that the TPPL yield in gold nanowires becomes independent of the pulse duration for laser pulses shorter than about 1â•›ps, while an inverse proportionality with the pulse duration would be expected for a fixed pulse energy. The origin of this behavior consists of the fact that TPPL in gold nanostructures is obtained after a 3d hole is produced by two sequential absorption steps involving a single photon (Imura et al. 2005), and is governed by the relaxation time of the sp conduction band hole generated after the first photon absorption. This temporal limitation would advise against the utilization of TPPL in probing near-field enhancement for coherent control applications, leaving SHG as a better adapted tool.

28.9â•‡Conclusions Thanks to the combination of surface sensitivity and symmetry selectivity, SHG is a very powerful tool to investigate the nanoscale properties of matter. The application of SHG techniques for investigating the structure and various other properties of small particles is a subject of considerable current interest and has stimulated a conspicuous number of both theoretical and experimental works in a broad range of subjects. In particular, being SHG a nonlinear optical process whose yield depends on the square power of the FW field intensity, SHG has found many applications in the study of the electric field enhancement mechanisms in metal nanostructures, where the lightning rod effect associated with high-curvature particles or resonantly excited plasma oscillations can lead to extremely

28-16


intense and highly localized electric fields. Since nanoscience is attracting an exponentially growing attention, SHG in nanostructured systems will be a very hot topic for many years to come.

Acknowledgments We would like to particularly thank Prof. Klaus Sattler for his kind invitation to write this chapter. We are also indebted to many colleagues and friends, whom we had the opportunity to work and discuss with during their activity concerning this work. Among these, we would like to mention M. Allegrini, P.-M. Adam P. Biagioni, M. Celebrano, G. Grancini, J. Grand, M. Labardi, D. Polli, M. Savoini, and M. Zavelani-Rossi. M. F. and L. D. are also affiliated to L-NESS (Laboratory for Nanometric Epitaxial Structures on Silicon and for Spintronics), which is an Interuniversity Centre between Politecnico di Milano and Università di Milano Bicocca. G. C. is also affiliated to ULTRAS (National Laboratory for Ultrafast and Ultraintense Optical Science) research center, which is founded by the National Institute of the Physics of Matter (INFM) of the National Research Council (CNR).

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29 Nonlinear Optics in Semiconductor Nanostructures 29.1 Introduction............................................................................................................................29-1 29.2 Semiconductor-Field Hamiltonian in kâ•›⋅â•›p-Approximation............................................29-2 29.3 Quantum Equations of Motion........................................................................................... 29-6 Electromagnetic Fieldâ•‡ •â•‡ Semiconductor Dynamicsâ•‡ •â•‡ Rotating Wave Approximation

Mikhail Erementchouk University of Central Florida


29.4 Perturbation Theory: Four-Wave Mixing Response.......................................................29-11 29.5 Nonperturbative Methods: Rabi Oscillations..................................................................29-16 29.6 Conclusion.............................................................................................................................29-19 Acknowledgment..............................................................................................................................29-19 Referencesï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½29-19

29.1â•‡ Introduction The physics of semiconductor nonlinear response unites under the same cover a fascinating variety of different physical phenomena, theoretical ideas, and experimental approaches. It provides promising opportunities for technologies, for example, in the form of principally new light sources, and presents a convenient testing field for studying long-standing fundamental questions while raising new ones in fields ranging decoherence to low-energy quantum field theory. It is not therefore, surprising, that the interest of researchers remains steadily persistent for already many decades. There are many books and reviews devoted to the details of different topics. However, as investigations move forward, new types of problems surface and the question of the basic principles of the theory of nonlinear response is raised again and again. Perhaps the most clear demonstration of the recurring process of testing the background is the drastic change of the main theoretical methods, which took place in the 1990s. In the pre1990s era, Green’s functions method was the main technique used in the limit of zero, finite temperature, or nonequilibrium (the Keldysh technique). The main questions were formulated in terms of the spectral characteristics, and using the adiabatical approximation was the usual practice. With the development of new experimental methods and the improvement of the quality of the samples, the attention gradually moved toward the temporal behavior and the transient features of the excitations. As a result, in the post-1990s era, time dependence turned out to be the major question, and the typical theoretical approach became the method of the dynamical equations of motion.

In a sense the whole topic was reinvented, making the question of fundamentals of the theory especially important. In the following sections, we present the derivation of the main dynamical equations for the electromagnetic field and the semiconductor excitations in the general context of quantum field theory. Such an approach is particularly useful because the resonant interaction of light with the semiconductor results in constantly changing the number of elementary excitations in the semiconductor bands. Therefore, the respective description must admit the creation and annihilation of particles on the fundamental level. Also, in order to cover at least partially the various physical situations relevant for the problem of nonlinear response, we avoid using specific approximations such as, for example, the spherical symmetry of the hole’s bands or the translational symmetry of the structures. The latter circumstance compels to formulate the major part of considerations in the coordinate representation. Finally, the main equations are derived for operators that makes them applicable for studying quantum statistics and other complex questions going beyond the single-particle correlation functions (Green’s functions). The structure of the presentation is as follows. In Section 29.2 we derive the quantum-field Hamiltonian in the kâ•›⋅â•›p-Â�approxÂ� imation and briefly review the basic properties of the semiconductor bands. Section 29.3 is devoted to the derivation of the quantum equations of motion for the electromagnetic field and the semiconductor excitations. In Section 29.4, the perturbative approach of analyzing the quantum equations of motion is reviewed. Finally, in Section 29.5 we consider the semiconductor Bloch equation. 29-1

29-2


29.2â•‡Semiconductor-Field Hamiltonian in kâ•›˙â•›p-Approximation The quantum dynamics of the electrons moving in the periodic lattice field in the semiconductor under the action of the external field is described in SI units by the Hamiltonian Hmat =

∑ ∫ dx ψ (x)  2m [p − eA(x)] + U (x) + U 1

† s

2

l

ext

s

+

1 2

 (x ) ψs ( x) 

∑ ∫ dx dx ψ (x )ψ (x )V (x − x )ψ (x )ψ (x ), 1

2

† s

1

† s′

2

1

2

s′

2

s

1

In the following paragraphs we will be interested in establishing the relation with the standard kâ•›⋅â•›p approach, therefore, we will omit both the external potential and the Coulomb interaction and will consider their effect later. The main idea of the kâ•›⋅â•›p approach is to employ the fact that we are interested in the weak and spatially smooth perturbations of the semiconductor near its ground state. Thus, we want to account exactly the effect of the periodic lattice potential forming the semiconductor ground state and to consider the deviation from it in the spirit of the perturbation theory. In order to account for the effect of the periodic potential on the electron dynamics we need to develop an appropriate description of the respective electron single-particle states. The field operator ψ†(x) can always be expanded in terms of the Bloch modes as ψ s (x ) = †

∑ ∫ dk φ* (x)a n, k

n

† n, k

2 (−i∇ + k )2un, k (x ) + U l (x )un⋅k (x ) = n (k )un, k (x ). 2m

(29.2)

where n enumerates the bands and includes the spin index the Bloch wave vector k lies in the first Brillouin zone the wave functions ϕn,k(x) give the solutions of the respective Schrödinger equation in the periodic potential † the operators an,k are the operators, which create electrons in the respective Bloch states According to the Bloch theorem the electron wave function ϕn,k(x) can be presented as

(29.4)

This equation is Hermitian and, hence, functions un,â•›k(x) for a fixed k = k0 constitute a complete set on the space of periodic functions. We can, therefore, represent functions corresponding to other k ≠ k0 as the respective Fourier series un, k (x ) =

∑α

n, m

(k ; k 0 )um, k0 (x )

(29.5)

m

with unitary αn,m(k; k0). The latter follows from orthonormality of the amplitudes un,k(x)

1 1 dx un*′, k (x )un′, k (x ) = δ nn′ , V (2π)3

∫

(29.6)

where  is the volume of the elementary cell. Indeed, substituting Equation 29.5 into Equation 29.6 we obtain

∑ α*

n ′m

(k ; k 0 )α nm (k ; k 0 ) = δnn′ ,

(29.7)

m

which explicitly shows the unitarity of αn,m(k; k0). This motivates the introduction of new (Luttinger–Kohn) function describing the electron states χn,k (x ) = eik⋅x un,k 0(x ).

(29.8)

These functions constitute a complete orthonormal set similarly to the electron wave functions ϕn,k(x). The proof demonstrates the typical line of arguments and, therefore, it is useful to consider it in detail. The completeness of the Luttinger–Kohn functions

∑ ∫ dk χ* (x)χ n, k

,

(29.3)

where un,k(x) is a periodic function solving the equation

(29.1)

s, s ′

where ψs(x) is the electron field operator s and s′ are the electron spin indices A is the vector potential U l(x) is the periodic lattice potential Uext(x) is an external potential caused, for example, by the external static electric field V(x) = e2/4πϵ0ϵb|x|, with ϵb being the background dielectric constant, is the potential of the Coulomb interaction between the electrons

φn, k (x ) = eik ⋅ x un, k (x ),

n, k

(x ′) = δ(x − x ′).

(29.9)

n

follows from the respective relation for the electron wave functions

∑ ∫ d k φ* (x)φ n, k

(x ′) = δ(x − x ′)

(29.10)

n, k

n

if we use Equations 29.5 and 29.7. For orthonormality we need to consider

∫

∫

(n, k|n′, k ′) ≡ dx χn*, k (x )χn′, k ′ (x ) = dx e i ( k

k ′ )⋅ x

un*, k0 (x )un ′,kk 0(x ).

(29.11)

29-3

Nonlinear Optics in Semiconductor Nanostructures

The product un*, k 0 (x )un ′, k 0 (x ) is a periodic function and therefore can be expanded in the Fourier series un*, k 0 (x )un ′, k0 (x ) =

∑B

nn ′

(Q)eiQ⋅ x ,

(29.12)

Q

1 dx e −iQ⋅x un*, k (x )un ′, k (x ). V

∫

(29.13)

Using this representation in Equation 29.11 we obtain (n, k|n′, k ′) = (2π)3

∑B

nn ′

(Q)δ(k − k ′ + Q).

(29.14)

Q

The Bloch-wave-vectors k and k′ are inside the first Brillouin zone and, therefore, the only possibility to reach the singularity point of the δ-function is when Q = 0 leading to (n,â•›k|n′,â•›k ′) = (2π)3Bnn′(0)δ(kâ•›−â•›k ′). Taking into account Equation 29.6 we arrive at (n, k|n′, k ′) = δ nn′ δ(k − k ′).

(29.15)

Such introduced functions χn′,k′(x) provide a general background and are used in different contexts in solid-state physics (Luttinger and Kohn 1955, Johnson and Hui 1993). In what follows we limit ourselves to the case k 0 = 0 and will omit the index corresponding to the Bloch-wave-vector, simplifying notations. In terms of the functions χn,k(x) the electron field operator is presented as ψ †s (x ) =

∑ ∫ dk χ* (x)b n, k

n

† n, k

.

(29.16)

Substituting this representation into Equation 29.1, adopting the Coulomb gauge for the vector potential, ∇â•›⋅â•›A(x) ≡ 0, taking into account Equation 29.4 and using the same trick as for Equation 29.15 we obtain

Hmat =



2 2

n

n,n ′

e − m



∑ ∫ dk  2mk  δ ∑∫ n ,n ′

n,n ′

+

 † k ⋅ pn,n ′  bn, kbn ′, k m 

(29.17)

i dx u*n (x )∇un′ (x ), V

∫

(29.18)

which are responsible for the interband coupling and are the main parameters quantifying the strength of the light–matter interaction. The first term in the r.h.s. of Equation 29.17 describes the dynamics of the semiconductor excitations, and the second term accounts for the interaction with the external field. The specific feature of the “eigen-part” of the Hamiltonian (that is the part which is independent of the external field) is that it is not diagonal with respect to bands. The reason for this is the special status given to the states corresponding to k = k 0 = 0. These states do not exhaust the eigenstates of the semiconductor and, therefore, the effective coupling between different bands should not be surprising. This coupling, however, is proportional to k. Thus, if the relevant excitations are relatively weak and smooth, then the main contribution into the semiconductor dynamics would come from small k and we can treat the respective band coupling as a perturbation. As the first step we could just drop this term. This, however, gives obviously wrong results for the dynamics near the extreme points of the bands. Indeed, the resulting Hamiltonian would predict that the mass of the excitations in the conductance band is the mass of electron in empty space and that the energy of states in the valence band increases with k. Both of these predictions contradict the experiment. Thus, it is necessary to develop a consistent procedure that allows taking the band coupling into account, similarly to the perturbation theory in quantum mechanics. Unfortunately, a straightforward implementation of this approach is significantly complicated by the complexity of the semiconductor band structure. It poses not principal, but technical difficulties, the resolution of which requires developing a special approach employing the symmetry properties of the semiconductor lattice. These questions are beyond the scope of the present consideration (see, e.g., Chapter III in Bir and Pikus (1974)), therefore we limit ourselves to a semiqualitative analysis, which clearly illustrates the main ideas and can be readily adopted for a particular situation, for example for studying the effect of the static magnetic field and so on. In order to eliminate the off-diagonal elements of the eigenpart of the Hamiltonian we perform a unitary Schrieffer-Wolff transformation T = eS bn, k =

e   dk dq  A(q)(pn,n ′ + kδn,n ′ ) − A2 (q)δn,n′  2  

× bn†, k + qbn ′, k ,

pnn′ = −

where the sum runs over the vectors of the reciprocal lattice and Bnn ′ (Q) =

where A2 (q) = (2π)−3/ 2 ∫ dx A2 (x )e −iq⋅ x . The parameters of great importance introduced in Equation 29.17 are the matrix elements of the momentum operator between different bands

∑ ∫ dq(n, k|T|m, q)c m

m, q

, bn†, k =

∑ ∫ dq(m, q|T |n,k)c †

m

† m, q

(29.19)

such that the eigen-part of the transformed Hamiltonian mat = T † HmatT is approximately diagonal. In Bir and Pikus H (1974) a general procedure is discussed, which allows developing

29-4


the respective perturbation theory up to any order. Here, however, we restrict ourselves to the first relevant order, as it is usually done in the k ⋅ p-approximation. In terms of the Â�“generator” S the transformed Hamiltonian is written as

mat = Hmat + [Hmat , S] + 1 [[Hmat , S], S] + H 2

mat = H0 + Hp + HA + [H0, S] + [Hp , S] + [HA , S] H 1 1 1 + [[H0, S], S] + [[Hp , S],, S] + [[HA , S], S] + (29.21) 2 2 2

Since the k ⋅ p-coupling between the bands is small, the transformation diagonalizing the Hamiltonian is close to identical and, hence, the generator S is in some sense small ∼kâ•›⋅â•›p. Therefore, in the lowest order no contributions come from the terms not shown in Equation 29.21 and, moreover, we need to drop the term [[Hp , S], S]. We choose S in such a way that the off-diagonal part of Hp is canceled in the leading order of k ⋅ p. Or, since the diagonal part of Hp is 0, we find such S that

Hp + [H 0, S] = 0.

(29.22)

Resolving this equation with respect to the matrix elements of S yields

(n, k S n′, k ′) = −

k ⋅ pnn ′ δ(k − k ′), mωnn ′

(29.23)

where ωnn′ = ϵn − ϵn′. Using found S for the field related terms in Equation 29.21, we will have among the others the terms of the form A(q)â•›⊗â•›q, where ⊗ denotes the tensor product, so that in a Cartesian basis (kâ•›⊗â•›q)ij = kiq j. Generally the tensor A(q)â•›⊗â•›q is not zero. Its trace vanishes because of the Coulomb gauge imposed on the vector potential, but there are no other general restrictions on its form. Such terms, however, can still be neglected because of the smoothness of the external field. Indeed, because of the slow spatial variation of A(x) the main contribution into A(q) arises from small q, which leads to the essential reduction of the tensor A(q)â•›⊗â•›q comparing to the field terms contained in HA noncommuted with S in Equation 29.21. Thus, we obtain the matrix elements of the transformed Hamiltonian

mat

| n′, k ′

)

 2k 2  =  n + δnn′ δ(k − k ′) 2m   +

(29.20)

where [A,â•›B]â•›=â•›ABâ•›−â•›BA denotes the usual matrix commutator. We want to treat different contributions on the different basis and Â�therefore we present the Hamiltonian as Hmat = H0 + Hp + HA, where H0 stands for the diagonal part of the first term in the r.h.s. of Equation 29.17, Hp is the kâ•›⋅â•›p part and HA is the term related to the external field. Using this presentation in Equation 29.20 we obtain

(n, k | H

2 2m2

∑k⋅p n′′

nn′′

 1 1  k ⋅ pn′′ n′  +  ω nn′′ ω n′n′′ 

e e   dq  A(q) ⋅ (pnn′ + k ′δ nn′ ) − A2 (q)δnn′ δ(k − k ′ − q) − m 2  

∫

−

e m2

 k ⋅ pn′′n′ A(q q) ⋅ pnn′′ k ⋅ pnn′′ A(q) ⋅ pn′′n′  + . ωn′n′′ ωnn′′ 

∑ ∫ dqδ(k − k ′ − q)  n′′

(29.24)

It should be noted that because of the smoothness property discussed above the field terms are symmetric with respect to the change k ↔ k′. The Hamiltonian with the matrix elements given by Equation 29.24 looks quite complex and barely simpler than the initial Hamiltonian. In order to discuss the physical meaning of different contributions into Equation 29.24, we introduce the spatial field operators cn(x) and cn† (x ) according to cn, k =

1 dx cn (x )e −ik ⋅ x (2π)3 /2

∫

(29.25)

and the tensors Mnn′ of effective masses

1 1 1 = 1δ nn ′ + 2 Mnn ′ m m

 pn ′′n ′ ⊗ pnn ′′ pnn ′′ ⊗ pn ′′n ′  + . (29.26) ωnn ′′ ωn ′n ′′ 

∑  n ′′

Additionally we need to analyze the different types of interband, n ≠ n′, and intraband, n = n′, couplings induced by the external field. It can be seen that the main term of interest from the perspective of the semiconductor nonlinear response is ∝A(q)â•›⋅â•›pnn′, while all others can be neglected. Indeed, the terms ∝A ⊗ k describing the interband coupling vanish in the limit k → 0, and thus their contribution is negligible in the region of validity of the k ⋅ p-approximation. In turn, the effect of the terms yielding the intraband coupling is small provided the gap Δ between the valence and the conduction bands is relatively wide. As will be shown below (see Equation 29.56), the strong interaction of light with the semiconductor occurs when the frequency of the electromagnetic field is tuned into the resonance with the gap, ħω ≈ Δ. Thus, the intraband coupling is ∝ħ(ΔT)−1, where T is the duration of the external excitation field, and even at (reasonably) short time scales it can be neglected. Taking into account these considerations the semiconductor Hamiltonian takes a compact form

mat = H

∑∫ n ,n ′

  2 dx cn† (x )  nδnn ′ − ∇ ⋅ ⋅ ∇ − A(x ) ⋅ dnn ′  cn ′ (x ), 2M nn ′   (29.27)

29-5


where dnn′ = pnn′ e/m. This representation clarifies the physical meaning of the operators cn†(x). These are the field operators of quasiparticles corresponding to different bands and characterized by specific tensors of the effective mass. The logic of perturbative treatment of the band coupling implies neglecting the off-diagonal elements Mnn′ for n ≠ n′ as the respective correction are of higher order in k ⋅ p than those kept for derivation of Equation 29.27. This naturally leads to the definition of the effective masses Mnn of the respective bands. The explicit expression for the tensors of the effective mass given by n = n′ elements of Equation 29.26 gives the correct prediction of significant reduction of the masses compared to the electron mass in the empty space and even the negative sign of the mass of the upper valence bands, it also describes correctly the dependence on the width of the gap. However, it lacks more subtle but nevertheless important features, such as the mixing of heavy and light holes. In order to discuss these features, we inspect closely the semiconductor band structure (see, e.g., Chapter 2 in Yu and Cardona (2004)). If the electrons were spinless, the lowest energy bands were the valence and conduction bands. In semiconductors with diamond- or zinc-blende structures (e.g., Ge, GaAs, InP) the valence band would be threefold degenerate. It turns out that under rotations which map the lattice into itself the states belonging to this band transform similarly to the states with the orbital momentum ℓ = 1. Because of this, the band is called* Γ4υ. The states in the conduction band transform trivially, and in group-theoretical terms the band is called Γ1c. In addition to this one needs to take into account the twofold degeneracy of each state due to the electron spin. This makes the conduction band twofold and the valence band sixfold degenerate. Directly it does not imply divergence of the terms in Equations 29.23 and 29.26 corresponding to the degenerate bands because the respective matrix elements of the momentum operator are zero. This, however, does imply certain sensitivity of these bands to the external perturbations. The interaction, which takes advantage of the degeneracy, is the spin– orbit interaction originating from the dependence of the electron energy on the orientation of the spin in spatially inhomogeneous potential (see, e.g., Chapter XII in Schiff (1949)â•›). The spin–orbit interaction leads to partial lifting of the degeneracy forming the semiconductor band structure schematically shown in Figure 29.1. Thus, in order to be able to describe the complex picture of the band coupling in semiconductors, we need to consider the tensors of the effective masses as phenomenological parameters and, in particular, to keep the interband tensors Mn,n′ with n ≠ n′. The great simplification comes from the consideration of restrictions imposed on the form of these tensors due to the lattice symmetry. Since the semiconductor lattice turns into itself under the action of the point symmetry group so does Hamiltonian (29.27). For a general set of tensors of the effective masses Mn,n′ it might not be the case, thus the symmetry implies that these * Strictly speaking, the correct formulation should sound like “the states transform according to irreducible representation Γ4 of the point symmetry group of the lattice.” Detailed and practical introduction into the group theory can be found in Yu and Cardona (2004) and Bir and Pikus (1974).

Conduction Γ1c

Without SO 2

Valence Γ4υ (l = 1)

,

With SO

J= 1 Electrons 2

Holes

6 ,

J= 3 2

J= 1 Split-off 2

me me

–1 +1 –1 +1 3/2 h 1/2 l –1/2

–Mσ,σ΄

–3/2

Figure 29.1â•… Effects of the spin–orbit interaction on the semiconductor band structure. The vertical lines indicate the subbands in the valence and conduction bands, between which the matrix elements of the momentum operator are not zero, together with the helicity of the respective transition. The convention for masses is shown.

tensors must have a special structure. This line of arguments was implemented and clearly explained in the series of papers by Luttinger and Kohn (1955, Luttinger 1956). Studying these restrictions shows that in the conduction band the tensors of the effective masses are decoupled from all bands and are trivial Ms,s′ = 1meδs,s′, where 1 is the unit tensor, and Ms,σ = M σ,s ≡ 0. The tensors of the effective masses of the heavy and light holes are coupled M σ,σ′ ≠ 0 leading to the heavy–light hole hybridization. Additionally, the principal values of the tensors are not equal to each other leading to the anisotropy of the effective masses. It should be noted, however, that the anisotropy usually is not very strong and the mixing of the heavy and light holes is ∝k2 and for small k the effect of hybridization is also small. This supports the popular approximation of spherical, decoupled heavy-hole and light-hole bands widely used for description of the semiconductor weak excitations. We also will use it for the qualitative discussions of some questions while keeping the main equations as general as possible. So far we considered the case of a homogeneous semiconductor. In order to complete the derivation of the effective Hamiltonian we need to consider the spatial variation of the energies of the conduction and the valence bands due to the variation of composition and to include the terms responsible for the effect of the external static field and the Coulomb interaction between the electrons. To this end we take into consideration the last two terms in Hamiltonian (29.1) and substitute ϵn(x) for the variation of the bands edges. Generally, strong coordinate dependence of ϵn(x) would break the whole line of arguments, which allowed us to derive Equation 29.27 and to introduce the effective masses. It turns out, however, that as small as 10 elementary cells are sufficient for the approximation of effective mass to work. In particular, it can be safely applied for the description of the semiconductor quantum dots. Thus, the assumption of smoothness is fulfilled for composition-induced variations of the band edges (quantum heterostructures (Ivchenko 2005) or random interface fluctuations (Langbein et al. 2004)â•›). It also usually holds for the external electric field (e.g., in the case of studying the quantum-confined Stark effect (Schmitt-Rink et al. 1989) on the optical semiconductor response). This assumption is employed in the standard manner. Taking into account that

29-6


the main contribution into the dynamics comes from the small k part of Hamiltonian (29.24), one can show the validity of the important relation

∫ dx f (x)χ* (x)χ n, k

n ′, k ′

( x ) = δ n,n ′

dx

∫ (2π)

f (x )e − i ( k − k ′ )⋅ x

3

(29.28)

for sufficiently smooth f(x). The key element is to observe that if the spatial Fourier spectrum f(q) falls off fast enough, then δ(k′ − k + q = Q), with Q being the vector of the reciprocal lattice, reaches its singularity only when Q = 0, or, in solid-state theory terms that the Umklapp processes can be neglected. Using relation (29.28) makes the transformation to the field operators cn(x), following Equations 29.16 and 29.19, trivial. The Coulomb potential V(x) is singular at the origin, therefore, generally its variation over an elementary cell is not small. However, the effect of the motion of electrons when they are extremely close to each other is not significant for the problem of the optical response. The strongest effect of the Coulomb interaction can be expected when it leads to the formation of bound states (excitons). The typical distance between the particles in such states significantly exceeds the lattice constants and the details of the behavior of the interaction potential at the origin are not important. Therefore, with good accuracy the Coulomb potential can be considered as slowly varying and one can apply relation (29.28). Collecting these considerations, we can write down the full Hamiltonian in terms of the operators cn(x) and cn†(x) Hmat =

∑∫ n, m

+

1 2

  2 dx cn† (x )  n (x )δnm − ∇ ⋅ ⋅ ∇ − A(x ) ⋅ dnm  cm (x ) 2Mnm  

∑ ∫ dx dx c (x )c (x )V (x − x )c (x )c (x ). 1

† 2 n

1

† m

2

1

2

m

2

n

n,m

1

(29.29)

with T + being the standard time-ordering operator. Here we have taken into account that the Hamiltonian may explicitly depend on time because of the presence of the external driving electromagnetic field. The same ideology works also if the initial state is a mixture (e.g., thermal equilibrium) and therefore it should be described by the density matrix ρˆ(0). If there are no decohering processes, then one has

Here we have explicitly spelled out the time dependence of the density matrix in the absence of the incoherent processes ρˆ(t) = U(t;t0)ρˆ(t)U †(t;t0). The situation, however, becomes more complex when the decohering processes, for example, interaction with bath’s phonons, is present. This problem can be treated rigorously using special methods (see, e.g., Chapter 3 in Mukamel (1995)), which are beyond the scope of the present consideration because they are somewhat excessive for the problem of finding the macroscopic electromagnetic field when a simple introduction of a phenomenological decay of the macroscopic polarization usually suffices. The typical structure

ˆ (x ) Φ(t ) = Φ(0) U † (t ; t 0 )A ˆ (x )U (t; t 0 ) Φ(0) , A(x ) = Φ(t ) A

(29.30)

where |Φ(0)〉 is the initial state of the system and the time dependence of the system state |Φ(t)〉 = U(t;â•›t0)|Φ(0)〉 is written in terms of the evolution operator

t   i U (t ; t 0 ) = T + exp  − dt ′H(t ′)    t0 

∫

Oˆ (t ) = U † (t ; t 0 )Oˆ U (t ; t0 )

(29.33)

appearing in Equations 29.30 and 29.32 defines the Heisenberg representation of the field operator Oˆ . Since this representation plays the same role whether the initial state is coherent or not, it is often more convenient to study its time dependence rather than directly the time dependence of the state of the system or its density matrix. It can be easily checked that the time evolution of Oˆ (t) is governed by the Heisenberg equations of motion.

29.3â•‡ Quantum Equations of Motion The main object of interest while studying the nonlinear response is the time dependence of observables such as the macroscopic ˆ (x ) . These are determined by electromagnetic field A(x) = A the state of the whole electromagnetic field–matter system. For example, if the system is in the coherent regime, that is, if the system can be described by the vector of state |Φ(t)〉, then

ˆ (x ) = Tr  ρˆ (0)U † (t ; t 0 )A ˆ (x )U (t ; t 0 ) ρˆ (0). A(x ) = Tr  ρˆ (t )A    (29.32)

i

∂Oˆ = Oˆ , H  . ∂t

(29.34)

Since the explicit time dependence of the Hamiltonian does not change the derivation of the main equations of motion (all operators entering Equation 29.34 are taken at the same instant), we will omit the time argument in what follows.

29.3.1â•‡ Electromagnetic Field We begin the general analysis of the semiconductor optical response from the derivation of the equations of motion of the electromagnetic field. The commutator in this equation is found using the commutation relation for the electromagnetic field in the Coulomb gauge (Cohen-Tannoudji et al. 1992)

(29.31)

∂ ˆ  iδjl ˆ ⊥ ˆ A j (x1 ), 0 ∂t Al (x 2 ) = n2 (x ) δ (x 1 − x 2 ),   1

(29.35)

29-7


where n(x) is the refractive index, which is allowed to be spatially inhomogeneous j and l denote the Cartesian coordinates δˆ ⊥(x ) is the transverse δ-function (Cohen-Tannoudji et al. 1992) As Helmholtz’s theorem (Arfken and Weber 2005) states any vector field V(x) can be presented as the sum of irrotational and solenoidal parts, U(x) = Ul(x) + Ut (x) with ∇ × Ul(x) ≡ 0 and ∇â•›⋅â•›Ut(x) ≡ 0. The convolution of the vector field with the transverse δ-function returns the value of the solenoidal component Vr(x) at the point of singularity

∫ dx ′ δˆ (x − x ′)U(x ′) = U (x). ⊥

(29.36)

t

The Hamiltonian of the electromagnetic field has the standard form HF =

∫

0 dx n2 (x )E 2 (x , t ) + c 2 B 2 (x , t ) , 2

(29.37)

where the transverse electric and magnetic fields are defined in terms of the vector potential as E(x)â•›=â•›−∂A(x)/∂t and B(x) = ∇ × A(x). Substituting into Equation 29.34 H = HF + Hmat, where Hmat is given by Equation 29.29 one obtains the quantum equations of motion for the electromagnetic field

n2 (x) ˆ ˆ +µ A = ∇2 A 0 c2

∑ ∫ d x ′ δˆ (x − x ′)d 3

⊥

c (x ′)cm (x ′). (29.38)

† nm n

n, m

This equation has the standard form of the inhomogeneous wave equation for the vector potential with the integral in the r.h.s. representing the current source. Writing Equation 29.38 we have taken into account that ∇â•›⋅â•›A Â�vanishes in the Coulomb gauge used for Â�derivation of Equation 29.27. Since, generally, the vector field U(x) = µ 0

∑

n, m

dnmcn† (x )cm (x ) is not necessarily solenoidal, the

presence of the transverse δ-function in the r.h.s. of Equation ˆ (x) does not 29.38 is important to ensure that at any instant A violate the Coulomb gauge. Clearly, if one replaces the transverse δ-function by the usual (local) δ-function it implies effective adding to the source the irrotational component of U(x). According to Equation 29.36, the convolution of U(x) with the transverse δ-function is dx ′ δ ⊥ (x − x ′)U(x ′) = U(x ) − U (x ). Equation 29.38

∫

l

ˆ (x) can be presented as a superposition of two is linear, therefore A ˆ ˆ (x) = A(x) ˆ l (x), which satisfy the wave equa+A components, A

tion of the same form as Equation 29.38, with U(x) and −Ul(x) serving as sources. Since Ul(x) is irrotational it can be presented as a gradient of a scalar function Ul(x) = −∇φ(x). Furthermore, following the standard line of arguments as in electrostatics we obtain

ϕ(x ) =

∇ ⋅ U(x ′) 1 . dx ′ x − x′ 4π

∫

(29.39)

Thus φ(x) has the same form as the potential created by the ˆ (x) charge spatially distributed with the density ϵ0∇â•›⋅â•›U(x). If A ˆ ˆ l(x) satisfies the same initial conditions as A(x) the component A is early shown to be irrotational and, hence, it does not contribute to the outgoing radiation field but leads to a modification of the electron–electron interaction. This modification identically vanishes in many important physical situations, such as the excitation by the normally incident wave at a frequency tuned to the response with the heavy-hole excitons. While a rigorous ˆ l(x) on the semiconductor dynamics is study of the effect of A yet to be conducted, it can be conjectured that for the problems which will be studied below the respective correction of the electron–electron interaction is rather small and can be ˆ ˆ (x ) ≈ A (x). neglected; in other words one can approximate A

This significantly simplifies Equation 29.38 since we can use the conventional δ-function instead of δˆ ⊥ (x − x ′) and perform the integration over x′, obtaining

n2 (x ) ˆ ˆ A(x ) = ∇2A(x) + µ0 c2

∑d

c (x )cm (x ).

(29.40)

† nm n

n,m

This equation completely describes the electromagnetic field driven by the semiconductor excitations. The macroscopic ˆ (x) field, as has been discussed above, is found by averaging A over the state of the combined semiconductor-field system. It should be noted, however, that Equation 29.40 can also be used in a more general context to find higher-order moments of the field, for example, single- or two-photon density matrix and so on.

29.3.2â•‡ Semiconductor Dynamics The current, which enters the equation of motion of the electromagnetic field as a source, is determined by the operator of the interband polarization cn†(x1)cm(x 2) taken at the diagonal x1 = x 2. Its time evolution is governed by the Heisenberg equation of motion (29.34). The electron operators obey the canonical anticommutation relation.

{c (x ), c † n

1

m

}

(x 2 ) = δ n,mδ(x 1 − x 2 ),

(29.41)

where {A, B} = AB + BA. The commutator cn† (x 1 )cm (x 2 ), H  is expressed in terms of the anticommutators using the relation, which holds for any four operators [ AB, CD] = A{B, C}D − {A, C}BD + CA{B, D} − C{A, D}B. (29.42)

29-8


Substituting H = HF + Hmat we obtain i

∂ † cn (x 1 )cm (x 2 ) = m (x 2 ) − n (x 1 ) cn† (x 1 )cm (x 2 ) ∂t  2 2 ⋅ ∇1 − δ jn∇2 ⋅ ⋅ ∇2 + δ km∇1 ⋅ 2M jn 2Mmk j,k 

∑

the field operators taken at different cells of the space anticommute to Kronecker’s δ so that υσ, x , υ†σ, x ′ = N δ x ,x ′ with some constant N. This gives for the term proportional to the anticommutator

{

}

cs†1 , x1 cs1 , x 2 N

∑ V

x2 − x ′

x′

− Vx1 − x ′  .

(29.45)

Truncating the Coulomb potential Vx at infinity we see that this term vanishes regardless of the parameters of the truncation and the discretization. Thus, we can conclude that these divergencies do not contribute to the equations of motion. j A more different obstacle is the divergent band-gap renor× c †j (x ′)c j (x ′)cm (x 2 ) (29.43) malization, which enters the equations of motion in the form ϵσâ•›→â•›ϵσ +â•›V(0). This divergence is of the same origin as the prewhere indices at ∇’s denote the number of the spatial variables vious one but it cannot be discarded as easily. The reason is they act on. The form of Equation 29.43 shows why it is not that we have introduced the band edges as the solutions of a enough to consider directly the time dependence of the operator single-Â�electron problem, while the hole sees the band edges in cn†(x1)cm(x 2) at the diagonal x1 = x 2. Because of the presence of the the presence of all the electrons filling the valence band minus spatial derivatives the rate of change of the operator at the diagoone. Therefore, the renormalization is a real physical effect. We nal is determined by the value the operator takes at the neighborincorporate it in the theory in a similar way as the elementarying points requiring, thereby, studying the general case x1 ≠ x 2. charge renormalization is dealt with in quantum field theory. Equation 29.43 is presented in a general form unifying the We will consider from now on that the hole energy offset ϵσ(x) conduction and the valence bands. This makes a constructive in the equations of motion is the genuine one. This can be done analysis difficult because the excitations belonging to different because the actual band gap between the conduction and valence bands have distinctively different properties, for example, the bands is directly observable (see, e.g., Equation 29.56). mass of excitations in the conduction band (electrons) is posiTaking into account these considerations we obtain tive while in the valence band (holes) it is negative. In order to account for these differences it is convenient to introduce spe∂ i υ†σ (x1 )cs† (x 2 ) cial field operators for holes. It is done following the simple ∂t rule—creation of an electron in the valence band corresponds = υ†σ (x1 )cs† (x 2 ) σ (x1 ) + Uˆ∆ (x 1 ) − s (x 2 ) − Uˆ∆ (x 2 ) + V (x1 − x 2 ) to annihilation of a hole and vice versa. As has been discussed  2  1 2 1 +  ∇2 + ∇1 ⋅ ⋅∇1  × υ†σ ′ (x 1 )cs† (x 2 ) above, different subbands in the valence and the conduction 2  me Mσ, σ ′  σ′ bands are enumerated by the projection of the angular momenˆ ˆ tum ( jâ•›=â•›1/2 and jâ•›=â•›3/2 for the conduction and the valence bands, − A (x 1 )⋅ d σ ,s δ(x 1 − x 2 ) + A (x 1 )⋅ d σ , s ′ cs† (x 2 )cs ′ (x 1 ) respectively). Thus, we introduce the hole operators according to s′ υσ† (x) ≡ cυ,σ(x) and υσ(x) ≡ cυ†,â•›σ (x). For the electrons in the conduc+ Aˆ (x 2 ) ⋅ d σ ′, s υ†σ ′ (x1 )υσ (x 2 ), tion band we leave the same notation but will keep only the index σ′ corresponding to the spin of the electron so that c†s (x) ≡ cc†,s(x). ∂ Rewriting Equation 29.43 in terms of the electron and hole i c s†1 (x 1)c s2(x 2 ) ∂t operators is straightforward but is not free from the formal dif= c s†1 (x 1)  s2 (x 2 ) + Uˆ ∆ (x 2 ) − s1 (x 1) − Uˆ ∆ (x 1) cs2(x 2 ) ficulties related to the fact that the holes exist on the background   of all electrons filling the valence band. In a sense, in order to cre2 ate a hole, one first needs to fill completely the valence band and (∇12 − ∇22 )c s†1 (x 1 )c s2(x 2 ) + 2 m e then remove one electron. This naturally leads to divergences of ˆ (x ) ⋅ d s υ† (x )c † (x ) − A ˆ (x1) ⋅ d σ , s cs (x 2 )υσ (x1)  , A the Coulomb energy. In order to demonstrate one kind of diver+ 2 s1 2 1 ′ 1 2 ′ 2, σ ′ σ ′   gence let us consider the Coulomb term, which appears at the σ′ r.h.s. of Equation 29.43 for n = {c, s1}, m = {c, ss} and j = {υ, σ} ∂ i υ†σ2 (x 2 )υσ 1 (x 1) ∂t † † cs1 (x 1 ) dx ′ V (x 2 − x ′) − V (x 1 − x ′) υσ (x ′)υσ (x ′)cs 1 (x 2 ). (29.44) = υ†σ2 (x 2 )  σ2 (x 2 ) + Uˆ ∆ (x 2 ) − σ1 (x 1) − Uˆ ∆ (x 1) υσ 1 (x 1)   ˆ (x 2 ) ⋅ d mk  c †j (x1 )ck (x 2 ) + δ mk Aˆ (x1 ) ⋅ d jn − δ jn A  † + cn ( x 1 ) dx ′ V (x 2 − x ′) − V (x1 − x ′)

∑∫

∑

∑

∑

∑

∫

If one tries to rewrite this expression in the normal form, that is, with all the creation operators collected on the left, one gets an additional term ∝{υσ(x′), υ†σ(x′)} = ħδ(x′â•›−â•›x′), which is infinite if not meaningless. There are different ways to work around this divergence. One of them is to consider Equation 29.44 as a continuous limit of the respective model formulated in a discrete space, where

2 2



δ σ ′′ , σ2

∑  ∇ ⋅ M ˆ (x ) ⋅ d +∑ A  +

1

σ ′ , σ ′′

1

s′

σ ′ , σ1

s ′ , σ1

⋅∇1 − ∇2 ⋅

 δ σ ′ , σ1 ⋅∇2  υ†σ ′′ (x 2 )υσ ′ (x 1) Mσ2 , σ ′′ 

ˆ (x 2 )⋅ d σ , s cs (x 2 )υσ (x 1) , υ†σ2 (x 2 )cs†′ (x 1) − A 2 ′ ′ 1 

(29.46)

29-9


where  Uˆ ∆ (x) = dx ′ V (x − x ′)  

∫

∑ c (x ′)c (x ′) − ∑ υ † s′

s′

† σ′

σ′

s′

 (x ′)υσ ′ (x ′).  (29.47)

Equations 29.40 and 29.46 describe the coupled dynamics of the electromagnetic field and the semiconductor excitations in the endless variety of physical situations. They can be used for studying the modification of the radiative decay of the semiconductor excitations in photonic crystals (Sakoda 2005) or the speckle pattern caused by electron scattering on surface inhomogeneities of quantum wells and so on. In what follows we will be mostly interested in the effect of the many-body correlations on the nonlinear optical response and, therefore, we make the proper simplifications of the problem. We assume that the life time of a photon in the structure is not too long so that the polaritons, which are the coupled states of the photons and the semiconductor excitations, are not formed. In this case one can neglect the reabsorption of the photons emitted in the course of the radiative decay. This significantly simplifies the analysis of Equations 29.40 and 29.46 because now we can consider A(x) in Equation 29.46 as the external classical excitation field, solve Equation 29.46 for cn†(x1)cm(x 2), and then find the field produced by the semiconductor excitation using Equation 29.40.

where the formal summation over μ implies summing over the discrete part and integrating over the continuous part of the spectrum of the operator Lˆ σ, s, and E μ and ϕμ(x 2,x1) are, respectively, the eigenvalues and the eigenfunctions, Lˆ σ, s φµ = Eµ φµ . The operator Lˆ σ , s is (formally) self-adjoint and, hence, its eigenvalues are real and the eigenfunctions form a complete orthonormal set. Using Equation 29.50 in Equation 29.48 and convoluting both sides of Equation 29.48 with ϕμ* we obtain the ordinary differential equation −i

∂ ˆ† Pµ (t ) = Eµ Pˆ µ† (t ) + Aµ(t ), ∂t

(29.51)

where we have introduced

∫

Pˆµ† (t ) = dx 1 dx 2 φµ(x 2 , x 1 )υ†σ (x 2 )cs†(x 1 )

(29.52)

and

∫

A µ(t ) = dx φ µ(x , x )A(x ,t ) ⋅ d σ , s .

(29.53)

The solution of Equation 29.51 is written as the sum of the general solution of the homogeneous equation (with A µ (t ) ≡ 0) and the partial solution of inhomogeneous t

29.3.3â•‡Rotating Wave Approximation The interaction of light with the semiconductor has a strong resonant character when the frequency of the excitation field, Ω, sweeps the vicinity of the gap, ħΩ ∼ Δ. In order to see how the resonances show up in the equations of motion we consider Equations 29.46 in the linear approximation when we need only the equation with respect to the operator of the interband polarization

i

∂ † υσ (x 2 )cs† (x 1 ) = − Lˆ σ , s υ†σ (x 2 )cs† (x 1 ) − A(x 1 ) ⋅ d σ ,s δ(x 1 − x 2 ), ∂t

(29.48)

where the operator Lˆ σ, s is defined as

Lˆ σ, s

 2  1 2 1 = s (x 1) − σ (x 2 ) − V (x 1 − x 2 ) −  ∇1 + ∇2 ⋅ ⋅ ∇2  .  2  me Mσ (29.49)

We have also neglected the heavy-hole–light-hole mixing setting M σ′,σ = δσ′,σ M σ. This does not make principle changes in the character of the following consideration, but greatly simplifies the notation. Next we employ the spectral representation of the operator Lˆ σ, s

Lˆ σ, s f (x 2 , x 1) =

∑ E φ* (x , x )∫ dx ′ dx ′ φ (x ′ , x ′) f (x ′ , x ′), µ µ

µ

2

1

1

2

µ

2

1

2

1

(29.50)

i iE t / iE (t − t )/ Pˆµ† (t ) = Pˆµ† (0)e µ + dt ′ e µ ′ Aµ(t ′).

∫

(29.54)

0

The first term is determined by the initial value of the operators υσ†(x 2)c†s (x1) or, according to the definition of the Heisenberg representation (see Equation 29.33), by their bare values. If, for example, initially the semiconductor is in the vacuum state (empty conduction band and filled valence band) the average value of the first term vanishes (see Equation 29.30) and, hence, it does not contribute to the average interband polarization and, accordingly, to the macroscopic electromagnetic field, which, as follows from Equation 29.40, is driven by υ†σ (x )cs† (x ) . In order to discuss the effect of the external field, we first assume that the external field changes harmonically with time A(x , t ) = A(x , Ω)e −iΩt + A* (x , Ω)e iΩt ,

(29.55)

which yields

e 0 Pˆσ(µ, s)(t ) 0 = Aσ(µ, s) (Ω)

iEµ t /

− e −iΩt eiEµ t − e iΩt + Aσ(µ, s) (Ω) , E µ + Ω E µ − Ω /

(29.56)

where A σ(µ, s) (Ω) and A σ(µ, s) (Ω) are found using in Equation 29.53 A(x, Ω) and A*(x, Ω), respectively. If Ωâ•›≈â•›E μ/ħ, then at the time scale (Ω + E μ/ħ)−1 > a. In such a case, in the vicinity of the nanofiber, Equation 30.4 is reduced to the following* form:

 t2   z 2 cos 2 α  Ei 0 (z , t ) ≈ E 0 exp  − exp  − 2τ 2  . 2σ 2  

(30.5)

On the other hand, to simplify the consideration, we shall assume that the beam radius is much less than the nanofiber length. Then one can neglect the edge effects and consider the nanofiber as infinitely long. The total electric field has the form

α

r

θ

x

if r > a; if r < a,

(30.6)

where E1 and E2 are the electric field vectors of the scattered field and the field inside the nanofiber, respectively. The total magnetic field vector, H, is represented in a similar form. The fields E1, E2, H1, and H2 in turn can be found in terms of the Hertz vectors, ∏1 and ∏2, whose rectangular components satisfy the scalar wave equation. The Hertz vector for an infinite cylinder has the only nonzero component, ∏z ≡ ψ, which can be decomposed into two contributions, ψTM and ψTE, associated with TM and TE polarizations, respectively. Then, the electromagnetic field components in the cylindrical coordinates (r, θ, z) are found as (Stratton, 1941)

z

FIGURE 30.1â•… Model of a nanofiber: dielectric cylinder. y

k1

Ei + E1 E= E 2

α

x

z

FIGURE 30.2â•… Model of a nanofiber: dielectric semicylinder on a perfectly reflecting surface.

E jr =

2 TE ∂2 ψ TM 1 ∂ ψj j − , ∂z ∂r cr ∂t ∂θ

(30.7)

E jθ =

1 ∂2 ψ TM 1 ∂2 ψ TE j j + , r ∂ z ∂θ c ∂ t ∂ r

(30.8)

* This approximation is valid outside the nanofiber region |z| ≤ a tan α.

30-4


E jz =

j ∂2 ψ TM ∂2 ψ TM j j , − 2 2 ∂z c ∂t 2

(30.9)

Hjr =

j ∂2 ψ TM ∂2 ψ TE j j + , ∂z ∂r r ∂ t ∂θ

(30.10)

2

TM j

2

e −iq10 r sin θ =

TE j

∂ψ 1∂ ψ + Hjθ = − j , ∂t ∂r r ∂z ∂ θ

j ∂2 ψ TE ∂2 ψ TE j j , − 2 2 ∂z c ∂t 2

Hjz =

µ j

iβz − iωt

dβdω.

(30.13)

The representation (30.13) can be considered as a superposition µj e iβz − iωt, each of of an infinite number of the elementary waves ψ which should satisfy the wave equation. The Fourier-transformed µj , therefore, can be expanded in the series of the quantities ψ cylindrical functions as follows:

1 TM ψ j (r , θ; β, ω) = 2 qj

1 TE ψ j (r , θ; β, ω) = 2 qj

n

j

n=0

jn

n=0

Zn (q jr ) c jn (β, ω)sin(nθ) + d jn (β, ω)cos(nθ) ,

(30.15)

(1) H n (q1r ) Zn ( X ) =   J n (q2r )

qj =

if r > a ; if r < a ,

ω2 j − β2 , c2

(30.18)

(30.19)

ˆ n Bn = Gn , M

(30.20)

where  J n (q2a)  iβn  2 J n (q2a)  q2 a ˆ Mn =   iωε2 J n′ (q2a)  cq2  0 

iω − cq J n′ (q2a) 2 −

iβn J n (q2a) q22a J n (q2a)

− H n(1) (q1a) iβn − 2 H n(1) (q1a) q1 a −

iω1 (1)′ H (q a) cq1 n 1 0

 0  iω (1)ε′  H q ( n 1a) cq1   iβn (1) H n (q1a)  2  q1 a  − H n(1) (q1a) 

and  a2n   −b2n   d2n   c2n  . An =   , Bn =   a1n   −b1n  d   c   1n   1n 

(30.22)

Here, the prime above the Bessel and Hankel functions denotes differentiation with respect to their argument. The vector functions F⃗ n and G⃗ n depend on the polarization of the incident beam and are determined as

FnTM

Jn and Hn(1) are the Bessel functions of the first kind and Hankel functions, respectively. The choice of the Bessel functions for the cylinder interior ensures finiteness of the solution at its center, whereas the Hankel functions have proper behavior at infinite distance from the cylinder.

0

(30.21)

(30.16)

(30.17)

ˆ n An = Fn , M

(30.14)

∞

∑

−inθ

(β, ω)sin(nθ) + b jn (β, ω)cos(nθ) ,

where the functions Zn(X) are determined as

with

n =−∞

∞

∑ Z (q r) a

10

in Equation 30.3. Then, combining the terms with the same θ-dependence in the boundary conditions and performing the inverse Fourier transform, one obtains the following equations:

−∞ −∞

n

(30.12)

∞ ∞

∫ ∫ ψ (x, y; β, ω)e

∑ J (q r)e

We shall seek the functions ψ µj (μ = TM or TE) in the form of the Fourier integral 1 (2π)2

∞

(30.11)

where c is the speed of light in vacuum the subscript j labels different media: j = 1 corresponds to the surrounding medium, whereas j = 2 denotes the nanofiber interior

ψ µj (x , y , z , t ) =

The continuity of the tangential components of the total fields, Ejθ , Ejz, Hjθ, and Hjz, across the boundary r = a leads to the equations for the coefficients ajn, bjn, cjn, and djn. Taking into account that y = r sin θ, one can use the expansion

GnTM

 −2iσn Ei 0 cos αJ n (q10a)     2σn Ei 0 sin α nJ n (q10a) q10a  = , i 0 J n′ (q10a) 2σn H     0

(30.23)

  −2τn (1 − σn )Ei 0 cos αJ n (q10a)    −2iτn (1 − σn )Ei 0 sin α nJ n (q10a) q10a  = , i 0 J n′ (q10a) −2iτn (1 − σn )H     0

(30.24)

30-5

Light Scattering from Nanofibers

and

FnTE

0     −2iτn (1 − σn )Ei 0 J n′ (q10a)   = ,   τ ( σ ) sin α ( i H nJ q a − 2 1 − ) q a 10 0 10 n n i n       i 0 cos αJ n (q10a) 2τn (1 − σn )H

GnTE

0     −2σn Ei 0 J n′ (q10a)   = ,   σ sin α ( ) − 2 H nJ q a q a 0 10 10 n i n       i 0 cos αJ n (q10a) −2iσn H

(30.25)

(30.26)

where ˆn Dn is the determinant of the matrix M D1n and D4n are the determinants of the matrices obtained ˆ n by replacing the third and fourth columns with from M the column given by F⃗ n, respectively D2n and D3n are the determinants of the matrices obtained ˆ n by replacing the third and fourth columns with from M the column given by G⃗ n, respectively The corresponding Fourier-transformed electric and magnetic field amplitudes can be written in the form E1µ (r , θ; β, ω) = Ei 0 (β, ω)

∞

∑  A

×

µn

with

n= 0

0 σn =  1

if n is even; if n is odd,

(30.27)

∞

and

(30.28)

Here, we have introduced the Fourier-transformed amplitudes of the incident beam

 (ω − ω0 )2 τ2   (β − β0 )2 σ2  2πστ Ei0 (β, ω) = E0 exp  − exp − ,  2 2 cos α    2 cos α 

(30.29)

and a similar expression for H˜ i0(β, ω). Solving Equations 30.19 and 30.20, one obtains the Fouriertransformed electromagnetic field components both inside the nanofiber and outside it. We shall be interested, however, in the field scattered by the nanofiber, which is described by the coefficients a1n, b1n, c1n, and d1n . They are found as

∑ C n= 0

if n = 0; if n ≠ 0.

a1n (β, ω) =

µn

(r ; β, ω)sin(nθ) + Dµn (r ; β, ω)cos(nθ) ,

(30.35)

where the subscript μ runs over the components r, θ, and z, and the functions Aμn, B μn, Cμn, and Dμn are given by the equations

Arn (r ; β, ω) =

Brn (r ; β, ω) =

iβ (1) ′ iωn H n (q1r )a1n (β, ω) − 2 H n(1) (q1r )d1n (β, ω), q1 cq1 r (30.36) iβ (1)′ iωn H n (q1r )b1n (β, ω) + 2 H n(1) (q1r )c1n (β, ω), q1 cq1 r (30.37)

Aθn (r ; β, ω) = −

Bθn (r ; β, ω) =

iβn (1) iω (1)′ H n (q1r )b1n (β, ω) − H n (q1r )c1n (β, ω), 2 cq1 q1 r (30.38)

iβn (1) iω (1)′ H n (q1r )a1n (β, ω) − H n (q1r )d1n (β, ω), cq1 q12r (30.39)

D1n (β, ω) , Dn (β, ω)

(30.30)

D2n (β, ω) , Dn (β, ω)

(30.31)

Azn (r ; β, ω) = H n(1) (q1r )a1n (β, ω),

Bzn (r ; β, ω) = H n(1) (q1r )b1n (β, ω).

b1n (β, ω) = −

D (β, ω) c1n (β, ω) = 3n , Dn (β, ω)

(30.32)

D4n (β, ω) , Dn (β, ω)

(30.33)

d1n (β, ω) =

(30.34)

1µ (r , θ; β, ω) = H i 0 (β, ω) H ×

1 / 2 τn =  1

(r ; β, ω)sin(nθ) + Bµn (r ; β, ω)cos(nθ) ,

Crn (r ; β, ω) =

(30.40) (30.41)

iωε1n (1) iβ H n (q1r )b1n (β, ω) + H n(1)′ (q1r )c1n (β, ω), q1 cq12r

(30.42)

30-6

(30.43)

iω1 (1)′ iβn Cθn (r ; β, ω) = H n (q1r )a1n (β, ω) − 2 H n(1) (q1r )d1n (β, ω), cq1 q1 r

Czn (r ; β, ω) = H n(1) (q1r )c1n (β, ω), Dzn (r ; β, ω) = H n(1) (q1r )d1n (β, ω).

0.2

(30.46)

0.1

(30.47)

0

1 E1µ (r , θ, z , t ) = (2π)2

H1µ (r , θ, z , t ) =

1 (2π)2

∞ ∞

∫ ∫ E

(r , θ; β, ω)e iβz −iωt dβdω,

(30.48)

1µ

−∞ −∞

∞ ∞

∫∫

1µ (r , θ; β, ω)eiβz −iωt dβdω, H

−∞ −∞

(30.49)

one obtains the solution of the problem under consideration. The integrands in Equations 30.48 and 30.49 have the poles given by the zeros of the denominators Dn(β,â•›ω) in Equations 30.30 through 30.33. On the other hand, these zeros determine the allowed values of pairs (β, ω) for the electromagnetic field in a free nanofiber, i.e., its normal modes (Stratton, 1941).

30.3.2â•‡ Nanofiber Normal Modes Among the solutions of the equations

Dn (β, ω) = 0,

(30.50)

there are those which are represented by pairs of real quantities (βnm, ωnm). They give the propagation constants and frequencies of the waveguide (bound) modes. Such modes provide a complete description of the light propagation along the fiber in the steady-state regime far from the light source (Snyder and Love, 1983). Besides that, Equation 30.50 has solutions for which either β or ω, or both, have imaginary parts, i.e.,

β = βr + iβi ,

(30.51)

ω = ωr − iωi .

(30.52)

The contribution of such poles to the integrals (30.48) and (30.49) leads to the transients along the fiber length or in time, or in both length and time, respectively. If βi is small, the modes of the first type can propagate over long distances, and their

SD21

SD32

SD11

0.3

(30.45)

Taking the Fourier transforms of Equations 30.34 and 30.35,

0.4

(30.44)

iω1 (1)′ iβn Dθn (r ; β, ω) = H n (q1r )b1n (β, ω) + 2 H n(1) (q1r )c1n (β, ω), cq1 q1 r

SD31

0.5

TE01 HE11

SD01

SD12 1

HE12 HE31 EH11 TM01 HE21

iωε1n (1) iβ H n (q1r )a1n (β, ω) + H n(1)′ (q1r )d1nn (β, ω), 2 q1 cq1 r

LL

Drn (r ; β, ω) = −

a/λ


LL

SD02

1

SD33 SD13

SD22 0

2

2

βra

SD03 3

4

FIGURE 30.3â•… Dispersion curves of the SD modes. Real parts of the mode propagation constants. The modes with very large values of βi are not shown in the figure. ε1 = 1, ε2 = 2.89. (Reprinted from Bordo, V., J. Phys.: Condens. Matter, 19, 236220-1, 2007. With permission.)

portions localized within or near the fiber core are known as leaky modes (Snyder and Love, 1983). In the following, to distinguish between different modes, we shall call the first-type modes as space-decaying (SD) modes, whereas the second-type ones as time-decaying (TD) modes. Figure 30.3 shows the dispersion curves βr(ω) calculated numerically for the SD modes, SDnm, with n = 0–3 and represented in dimensionless variables. It has been assumed that ε1 = 1 and ε2 = 2.89, which correspond to an isolated nanofiber obtained from an isotropic para-hexaphenyl material immersed in vacuum. The dispersion curves for the bound modes are also shown in Figure 30.3. They are disposed between the light lines ω = cβ 1 (LL1 ) and ω = cβ 2 (LL2 ), and are denoted as it is accepted in the optical waveguide theory (Snyder and Love, 1983). The dispersion curves for the quantities βi(ω) are shown in Figure 30.4. Figure 30.5 represents the dispersion relations ωr(β) for n = 0–3 calculated for the TD modes, TDnm. The corresponding imaginary parts, ωi(β), for the modes that are not strongly decaying are plotted in Figure 30.6. Here, the subscript m numerates the modes with a given n according to the order they approach the axis βr for the SD-modes or the axis ωr for the TD-modes. One can observe that some of these modes, both SD and TD, can be considered as a continuation of waveguide modes beyond cutoff, which coincides with the light line LL1. The imaginary parts of the propagation constants or frequencies for such modes increase with the deviation of the corresponding real parts from cutoff.

30.3.3â•‡ Calculation of the Scattered Field When carrying out the integration in Equations 30.48 and 30.49, one can take into account that the functions E˜i0(β,â•›ω) ˜ i0(β,â•›ω) in the integrands are essentially different from and H

30-7


0.5

TD34

0.10

SD31

TD24 0.08

0.4

TD31

0.3

–ωia/(2πc)

a/λ

TD21

0.2

0.1

0.0

SD22

0

SD02

1

SD33

SD03 SD 12 SD13 2

SD21 3

4

5

6

FIGURE 30.4â•… Same as Figure 30.3 but for the imaginary parts of the mode propagation constants. (Reprinted from Bordo, V., J. Phys.: Condens. Matter, 19, 236220-1, 2007. With permission.)

TD34

0.4 TD03

EH11 TM01 HE TE01

TD23

0.3

TD02 TD01

0.1

1

1.0

1.5

TD12 2.0

2.5

3.0

3.5

4.0

FIGURE 30.6â•… Same as Figure 30.5 but for the imaginary parts of the mode frequencies, which fall into the range |ωia/(2πc)| ≤ 0.1. (Reprinted from Bordo, V., J. Phys.: Condens. Matter, 19, 236220-1, 2007. With permission.)

E1µ (r , θ, z , t ) = Ei 0 (z , t )exp(iβ0 z − iω 0t ) ∞

∑  A

(r ; β0 , ω 0 )sin(nθ) + Bµn (r ; β0 , ω 0 )cos(nθ) , (30.53) H1µ (r , θ, z , t ) = H i 0 (z , t )exp(iβ0 z − iω0t ) ×

µn

n=0

×

TD21

2 βa

3

∑ C n= 0

TD31

4

FIGURE 30.5â•… Dispersion curves of the TD modes. Real parts of the mode frequencies. ε1 = 1, ε2 = 2.89. (Reprinted from Bordo, V., J. Phys.: Condens. Matter, 19, 236220-1, 2007. With permission.)

zero around their maxima at β = β0 and ω = ω0. For typical experimental parameters, the widths of the corresponding peaks, Δβ λ, the

30-8


Hankel functions and their derivatives can be replaced by their asymptotical expansions. Then in the leading order one obtains Sr = SrTM + Sr TE ,

(30.56)

where the TM- and TE-polarized components are given by ω1 S (θ) = 2 4 π2q10 r TM r

n

(−i) [a1n (β0 , ω 0 )sin(nθ) + b1n (β0 , ω 0 ) cos(nθ)]

n=0

(30.57)

and ω S (θ) = 2 2 4 π q10r TE r

2

∞

∑ (−i) [c n

1n

(β0 , ω 0 )sin(nθ) + d1n (β0 , ω 0 )cos(nθ)] ,

n= 0

(30.58)

respectively. The total intensity scattered in all directions per unit length of the nanofiber is determined by the quantity 2π

Stot ≡

∫

Sr (θ)dθ = S

TM tot

+S

TE tot

(30.59)

0

with

TM Stot =

2 ω1  b (β0 , ω 0 ) + 2  10 r 4 πq10 

∞

∑  a

1n

n =1

 2 2  (β0 , ω 0 ) + b1n (β0 , ω 0 )    

(30.60)

and

TE Stot =

The approach developed above can be equally applied to the nanofiber model represented by a circular semicylinder placed on an ideally reflecting plane. When the reflecting plane at y = 0 is present, the total electric field in the half-space y > 0 can be written as

2

∞

∑

30.3.4â•‡Semicylinder on an Ideally Reflecting Surface

2 ω  d (β0 , ω 0 ) + 2  10 4πq10r  

∞

∑ n =1

  c1n (β0 , ω 0 ) 2 + d1n (β0 , ω 0 ) 2   .    

(30.61)

If in an experiment, one varies either the angle of incidence of the light beam at a fixed frequency, or tunes the light frequency at a fixed angle of incidence, the scattered intensity will also change. In the former case, the intensity (30.56) has maxima at r the values β0 = βnm corresponding to the SD modes. In the latter r case, the maxima in intensity occur at ω0 = ωnm and correspond to the TD modes. The widths of those peaks are determined by the quantities βinm and ωinm, respectively.* * Besides the maxima relatively the light frequency discussed here, there may also be maxima originating from resonances in ε2(ω).

E = Ei + Eiref + E1 + E1ref ,

(30.62)

where Eiref and E1ref are the electric fields specularly reflected from the plane y = 0 originating from the incident wave and the field scattered by the nanofiber, respectively. A similar expression is valid for the magnetic field vector. The condition of an ideally reflecting surface implies that the tangential components of the total electric field at it are equal to zero. This requirement will be fulfilled if to introduce, instead of the reflecting plane, an image semicylinder illuminated by the image of the incident wave below the plane y = 0 so that Er = Ez = 0 at both θ = 0 and θ = π (Rao and Barakat, 1994). The same is true for the magnetic field components. The fields scattered by the image semicylinder, E1im and H1im , are determined by Equations 30.34 through 30.47 with the substitutions a1imn = −a1n , b1imn = −b1n , c1imn = −c1n , d1imn = −d1n .

(30.63)

Then the components of the field scattered by the semicylinder in the presence of the reflecting plane are found as E1(θ) + E1im (−θ) and H1(θ) + H1im (−θ). Formally, the field components can be obtained from Equations 30.53 and 30.54 with the following substitutions: Aµn → 2 Aµn , Bµn → 0, Cµn → 2Cµn , Dµn → 0.

(30.64)

Let us assume that the angle of incidence, α, differs from zero. Then the fields Eiref and Hiref do not contribute to the scattered intensity determined by Sr. In the far zone, its TM- and TE-polarized components are given by SrTM (θ) =

ω1 2 π 2q10 r

2

∞

∑

(30.65)

(−i)n a1n (β0 , ω 0 )sin(nθ)

n =1

and ω S (θ) = 2 2 π q10r TE r

2

∞

∑ (−i) c n

1n

n =1

(30.66)

(β0 , ω 0 )sin(nθ) ,

respectively. The corresponding total intensity scattered in all directions per unit length of the nanofiber is determined by the quantity π

∫

Stot = Sr (θ)dθ =

0

ω 2 2πq10 r

∞

∑  a 1

n =1

1n

2 2 (β0 , ω 0 ) + c1n (β0 , ω 0 ) . 

(30.67)

30-9


Note that here the sum over the modes starts from n = 1, indicating that the modes with n = 0 are not excited in the course of light scattering. The suppression of such modes is conditioned by the presence of the reflecting plane. Let us consider the special case of normal incidence with respect to the substrate surface (α = 0). In such a case, β0 = 0 and the equations for TM and TE polarizations are completely separated. This means that for TM incident wave polarization, one can set c1n = d1n = 0. Besides that, the vector F⃗ n, Equation 30.23, is nonzero only for odd indices n. Then the coefficients a1n are nonzero for odd n and only such modes can be observed in scattering. For TE incident wave polarization, a1n = b1n = 0 and the vector G⃗ n, Equation 30.26, is nonzero for odd n. As a result, one obtains the same selection rule as for TM polarization.

120

Let us consider first the dependence of the Poynting vector component Sr, Equation 30.55, calculated in the far zone on the detection angle, θ. This quantity is determined by the coefficients a1n, b1n, c1n, and d1n, as shown in Equations 30.30 through 30.33. Their variation with the nanofiber radius, a, is more rapid for larger values of q10 = (ω 0 /c) 1 cos α and hence for smaller angles of incidence, α. Figures 30.7 and 30.8 show the angular distributions of light scattered by a cylindrical nanofiber for the TM and TE incident light polarizations, respectively, and for α = 45°. One can see that in the case of the TM incident wave polarization, there are two lobes directed along the normal to the plane of incidence (at θ = 0° and θ = 180°). For relatively large nanofiber radius, additional two lobes appear. In the case of the TE incident wave polarization, these lobes are more pronounced. Besides that, for small nanofiber radius, the angular distribution of scattered light is more isotropic as compared to the TM polarization.

120

150

180

90

30

0

180

FIGURE 30.8â•… Same as Figure 30.7 but for TE incident wave polarization.

30.4â•‡ Some Numerical Results

30.4.1â•‡Angular Distribution of Scattered Light

a/λ = 0.1 a/λ = 0.2 a/λ = 0.5

60

30

0

FIGURE 30.7â•… Angular distribution of light scattered from a cylindrical nanofiber of different radii. TM incident wave polarization. α = 45°, ε1 = 1, ε2 = 2.89. All curves are normalized to their maximum values.

a/λ = 0.1 a/λ = 0.2 a/λ = 0.5

60

150

120

We shall illustrate the general theory considered above with some numerical examples. All the calculations have been carried out for ε2 = 2.89 that corresponds to an isotropic para-hexaphenyl film. We shall compare the results obtained for two different models of a nanofiber.

90

90

a/λ = 0.1 a/λ = 0.5

60

150

180

30

0

FIGURE 30.9â•… Same as Figure 30.7 but for a semicylindrical nanofiber on a perfectly reflecting surface.

The same quantity but calculated for a semicylindrical nanofiber placed on a perfectly reflecting substrate and for the TM incident wave polarization is depicted in Figure 30.9. The scattering diagram for the TE incident wave polarization has a similar form. In this case, the lobes directed along the substrate plane are suppressed and light is scattered mainly around the plane of incidence.

30.4.2â•‡Total Scattered Intensity versus Incidence Angle Let us consider the total scattered intensity, Stot, as a function of the incidence angle, α, or, equivalently, the quantity β0 = (ω0 /c) 1 sin α. Figure 30.10 shows this dependence calculated for a cylinder and for the TM incident wave polarization with an account of the terms n = 0–10 in Equations 30.60 and 30.61. The individual contributions from the terms with n = 0, 1, and 2 are also shown in Figure 30.10, whereas the other contributions have much smaller amplitude. The intensity drop at β0 a ≈ 2.2 originates from the approach to the cutoff at which q10 = 0. The curve corresponding to n = 2 exhibits a shoulder around β0 a ≈ 1.75, which is also seen in the total intensity curve. Turning to Figure 30.3, one can identify it as being originating from the mode SD21. The contributions of the other modes are not pronounced because of the large values of the propagation constant imaginary parts, βi, corresponding to them. This plot can be compared with Figure 30.11, which represents the results of calculations for a semicylindrical nanofiber placed on an ideally reflecting surface. In this case, the zeroth term does not contribute to Stot and the contribution of the term with n = 2 is equal

30-10


2.0

Total n=0 n=1 n=2

1.8 1.6

SD21

Stot (arb. units)

Stot (arb. units)

1.4

2.0

1.2 1.0 0.8

1.5

1.0

Total n=0 n=1 n=2 n=3

TD23

TD12

TD33

TD01

0.6 0.5

0.4 0.2 0.0 0.0

0.5

1.0

β0a

1.5

2.0

FIGURE 30.10â•… Light intensity scattered from a cylindrical nanofiber as a function of β0 calculated for ε1 = 1, ε2 = 2.89, and ω0 a/(2πc) = 0.35. TM incident wave polarization.

1.6 1.4

0.8

0.2

0.3 ω0 a/(2πc)

0.6 0.4

0.5

0.4

Total n=1 n=3

TD12

1.2 Stot (arb. units)

Stot (arb. units)

0.1

FIGURE 30.12â•… Light intensity scattered from a cylindrical nanofiber as a function of ω0 calculated for ε1 = 1, ε2 = 2.89. Normal incidence. TM incident wave polarization.

Total n=1 n=2

SD21

1.0

0.0 0.0

2.5

TD33

1.0 0.8 0.6 0.4

0.2

0.2 0.0 0.0

0.5

1.0

β0a

1.5

2.0

2.5

FIGURE 30.11â•… Same as Figure 30.10 but for a semicylindrical nanofiber on a perfectly reflecting surface.

to zero at β0 = 0 (normal incidence) in accordance with the selection rule discussed in Section 30.3.4. As a result, the shoulder in the total intensity originating from the mode SD21 becomes more distinct.

30.4.3â•‡Total Scattered Intensity versus Frequency Another situation when the light beam frequency is scanned at a fixed angle of incidence (α = 0) is illustrated in Figure 30.12 for the case of scattering of a TM-polarized beam by a cylindrical nanofiber. This time the features in the total scattered intensity originate from the TD modes. They can be identified when comparing the positions of maxima, which occur for the curves with different n with the mode frequencies at β0 = 0 in Figure 30.5.

0.0 0.0

0.1

0.2

0.3

0.4

0.5

ω0 a/(2πc)

FIGURE 30.13â•… Same as Figure 30.12 but for a semicylindrical nanofiber on a perfectly reflecting surface. (Reprinted from Bordo, V., J. Phys.: Condens. Matter, 19, 236220-1, 2007. With permission.)

In the case of a semicylinder placed on an ideally reflecting plane (Figure 30.13), only modes with odd n can be excited, as it is expected from the selection rule (see Section 30.3.4). It is worthwhile to note that the modes distinct from those for TM polarization can be observed in the scattering of a TE-polarized light beam (not shown).

30.4.4â•‡ Excitation near Exciton Resonance So far we have implied that the frequency of the incident beam is far from any resonance in the dielectric function of the nanofiber and its frequency dispersion can be neglected. Let us assume

30-11

Light Scattering from Nanofibers ω = 0.9ωT ω = 1.0ωT ω = 1.1ωT

90 120

20 μm

60 HWP P 30

150

Ph

MO

0

180

PMT

F

FIGURE 30.14â•… Angular distribution of scattered light in the vicinity of the exciton resonance. TM incident wave polarization. ε1 = 1, ε∞ = 2.89, a = 4c/ωT, ωL = 1.2ωT, Γ = 10−3ωT, and α = 30°.

2 (ω) =

ω 2L − ω 2 ∞ , ω 2T − ω 2 + iωΓ

(30.68)

where ωT and ωL are the frequencies of the transverse and longitudinal excitons, respectively Γ is the relaxation constant ε∞ is the high-frequency dielectric constant, and the spatial dispersion has been neglected Figure 30.14 shows the angular distribution of the scattered light in this case calculated for a semicylindrical nanofiber on an ideally reflecting surface. It is seen that the polar diagram of scattering changes dramatically in a narrow spectral interval when the frequency of the incident beam scans across the exciton resonance.

30.5â•‡ Experimental Implementation Experimentally, one can observe light scattering from nanofibers in the following setup (Figure 30.15) (Fiutowski et al., 2008). A sample with deposited organic nanofibers is attached to a halfsphere made of fused silica (ns = 1.48 at 325â•›nm) with a radius of 10â•›mm. The sample is oriented so that the nanofiber axes would be parallel to the plane of light incidence. A flat domain of parallelly oriented organic nanofibers is illuminated with a linearly polarized beam of a He-Cd laser (λ = 325â•›nm) from the halfsphere side, and the scattered light intensity is detected along the normal to the sample surface from the backside. The extinction ratio of the polarization system is 105. The light beam passes a pinhole of diameter 400â•›μm and is collimated before entering the half-sphere, which results in a beam divergence of less than 0.7° after leaving the half-sphere. The angles of incidence can be varied between 20° and 65° with respect to the normal axis of the

FIGURE 30.15â•… Experimental setup. HWP, half-wave plate; P, polarizer; Ph, pinhole; MO, microscope objective; F, filter; and PMT, photomultiplier. Inset: Fluorescence microscopy image of a sample domain. (Reprinted from Fiutowski, J. et al., Appl. Phys. Lett., 92, 073302-1, 2008. With permission.)

plane of the half-sphere. A photomultiplier mounted on a goniometric table allows one to detect scattered light intensity distributions from a spot on the sample of a diameter of about 500â•›μm. The acceptance angle of the detection system is 44°. The absolute and relative accuracies of the incident and the detection angles are 1° and 0.5°, respectively. To observe only the wavelengths of interest, one can use band pass or interferometric filters in front of the photomultiplier. The intensity of light scattered perpendicularly to the sample surface as a function of the incidence angle, α, is shown in Figure 30.16. Clear peaks at α = 40° in both TE and TM incident light polarizations are observed. Those peaks originate from the Nanofibers, TE Nanofibers, TM Mica, TE Mica, TM

4 Scattered intensity (arb. units)

now that the frequency ω0 is close to an exciton resonance of the nanofiber. Then its dielectric function can be written as follows:

3

2

1

0 20

30

40 αc 50 Incidence angle (°)

60

70

FIGURE 30.16â•… Intensity of light scattered perpendicularly to the sample surface as a function of the angle of incidence. The incident wave polarization is indicated in the inset. (Reprinted from Fiutowski, J. et al., Appl. Phys. Lett., 92, 073302-1, 2008. With permission.)

30-12


nanofiber array exclusively, as seen from a comparison with the data obtained from a nanofiber-free mica surface. The scattered intensity decreases when the incidence angle exceeds the critical angle for total internal reflection at the quartz semisphere–air interface, αc = 42.5°. To describe the optical properties of a nanofiber grown on a substrate, we use the nanofiber model represented by a semicylinder placed on an ideally reflecting surface and considered above. Figure 30.17 shows the dispersion curves of the nanofiber normal modes plotted in the range of the size parameter, a/λ, corresponding to the experimentally observed typical nanofiber widths (2a) as well as the wavelength of the He-Cd laser used for excitation. The condition of the phase matching between the incident light and the nanofiber normal mode can be written as β = (2π/λ)ns sin α. The corresponding dispersion line for the angle α = 40°, at which the resonance in scattering is observed, is also shown in Figure 30.17. The analysis of the polarization properties of the excited normal modes allows one to specify them. The sharp peak seen in the scattered light intensity at α = 40° has comparable amplitudes in both TE and TM incident wave polarizations. It can therefore be associated with the excitation of the hybrid SD12 mode. The increase in intensity at α < 26° originates probably from a resonance with the hybrid SD21 mode. The mode SD01 is suppressed in scattering due to the presence of the reflecting substrate (see Section 30.3.4). The positions of the maxima in Figure 30.16 correspond to the most probable size parameter within the illuminated spot. The broadenings of the peaks observed in light scattering allow one to estimate the distribution of the nanofibers in the array over their widths. The angular broadening of about 7° corresponds to a scattering of 0.12λ in nanofiber diameters, which HE12 HE31 EH11

0.55 0.50

TM01 HE21 TE01

SD21

0.45 a/λ

HE11 0.40 SD01

0.35 0.30 0.25 1.0

SD12

1.5

2.0

2.5

3.0

3.5

4.0

βa

FIGURE 30.17â•… Dispersion curves of the SD modes plotted in the range of size parameters corresponding to measured nanofiber widths. The inclined dot-and-dash line indicates the dispersion for light incident at the angle α = 40°.

gives approximately 40â•›nm for the relevant wavelength. This value agrees qualitatively with that measured via atomic force microscope (AFM).

30.6â•‡ Discussion We have considered here the simplest theoretical models of nanofibers and applied them for the interpretation of the experimental results on light scattering from an array of organic nanoaggregates deposited on a mica substrate. Although the morphology of nanoaggregates is far from a cylindrical shape, light scattering from them can be similarly described in terms of excitation of their radiative normal modes. This conclusion is also supported by the measurements of photoluminescence excited in the course of light scattering (Fiutowski et al., 2008). In that case, the peaks in the photoluminescence intensity versus the angle of incidence originate from matching with the nanoaggregates’ normal modes, both radiative and waveguiding. Such measurements being carried out at a varying frequency of incident light should allow one to determine the dispersion curves of the actual normal modes of nanoaggregates. Then, using an appropriate model, one could extract information on the morphology of nanofibers. However, it is necessary to keep in mind that the actual normal modes may differ from those of an isotropic cylindrical nanofiber. In particular, the optical anisotropy of nanoaggregates can cause distortion of the dispersion curves as well as their splitting. Another point which has to be taken into account is the different morphology of nanoaggregates within the illuminated spot. As we have already mentioned, this leads to a broadening of the peaks in the intensity of scattered light. On the other hand, the data obtained from the scattering of a focused light beam reflect the local morphology of a sample and, thus, can be used for its local characterization.

30.7â•‡ Summary and Outlook In this chapter, we have considered the theory which describes the scattering of a pulsed Gaussian light beam at an infinitely long dielectric cylinder. The results obtained for this model have been generalized to the model represented by a semicylinder placed on a perfectly reflecting plane. Although being rather simple, these approaches reproduce the main features of light scattering from nanofibers. In particular, this process can be understood in terms of the excitation of nanofiber radiative modes. As a result, the dependence of the scattered light intensity on the angle of incidence exhibits maxima corresponding to matching with the nanofiber SD modes. These conclusions are in agreement with the measurements of light scattering from organic nanoaggregates grown on mica, which have demonstrated experimental evidence of mode launching in this system. Those modes show up as pronounced peaks in light scattering from the nanofibers. Using the model, one is able to identify specific modes that have been excited in the course of light scattering from the sample. The obtained


results provide detailed information about possible electromagnetic mode propagation in nanosized, needle-shaped aggregates and also form the basis for a new way of optically characterizing the morphology of sub-wavelength-sized nanostructures via farfield scattering. The further development of this technique by using a tunable light source should allow one to determine the actual dispersion curves of nanofibers. On the other hand, one can use a broadband light source to illuminate the sample instead of a laser. In such a case, one should expect to observe peaks in the spectrum of scattered light, which originates from the excitation of TD nanofiber modes. The implementation of this technique with the use of a scanning near-field optical microscope for the registration of scattered field would provide an additional advantage. The data obtained in such a setup are not influenced by an averaging over the illuminated spot and, thus, are related to individual nanofibers. Another possible application of light scattering from nanofibers stems from correlation between the positions of peaks in scattered intensity and the dispersion of nanofiber normal modes. Any change in the optical properties of nanofibers caused by either heating, or mechanical tension, or adsorption of molecules from ambient medium would result in a shift of the peaks in light scattering. This opens up new opportunities for the creation of nanosensors of various kinds. Due to its generic nature, the method discussed in this chapter is not limited to organic nanofibers but can be equally well applied to other kind of light-emitting nanoaggregates.

Acknowledgments The author is grateful to Prof. H.-G. Rubahn and Dr. L. Jozefowski for the discussions of the experimental aspects of light scattering from nanofibers.

References Agarwal, R., Barrelet, C., and Lieber, C. (2005). Lasing in single cadmium sulfide nanowire optical cavity. Nano Lett., 5:917–920. Ashkin, A., Dziedzic, J., and Stolen, R. (1981). Outer diameter measurement of low birefringence optical fibers by a new resonant backscatter technique. Appl. Opt., 20:2299–2303. Balzer, F., Bordo, V., Simonsen, A., and Rubahn, H.-G. (2003). Optical waveguiding in individual nanometer-scale organic fibers. Phys. Rev. B, 67:115408–1–8. Balzer, F. and Rubahn, H.-G. (2001). Dipole-assisted self-assembly of light-emitting p-nP needles on mica. Appl. Phys. Lett., 79:3860–3862. Balzer, F. and Rubahn, H.-G. (2005). Growth control and optics of organic nanoaggregates. Adv. Funct. Mater., 15:17–24. Barrelet, C., Greytak, A., and Lieber, C. (2004). Nanowire photonic circuit elements. Nano Lett., 4:1981–1985. Bever, S. and Allebach, J. (1992). Multiple scattering by a planar array of parallel dielectric cylinders. Appl. Opt., 31:3524–3532.

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Birkhoff, R., Ashley, J., Hubbel, H. Jr., and Emerson, L. (1977). Light scattering from micron-size fibers. J. Opt. Soc. Am., 67:564–569. Bordo, V. (2006). Light scattering from a nanofiber: Exact numerical solution of a model system. Phys. Rev. B, 73:205117-1–205117-7. Bordo, V. (2007). Theory of nanofibre excitation by a pulsed light beam. J. Phys. Condens. Matter, 19:236220-1–236220-12. Borghi, R., Santarsiero, M., Frezza, F., and Schettini, G. (1997). Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface. J. Opt. Soc. Am. A, 14:1500–1504. Brewer, J., Maibohm, C., Jozefowski, L., Bagatolli, L., and Rubahn, H.-G. (2005). A 3D view on free-floating, space-fixed and surface-bound para-phenylene nanofibers. Nanotechnology, 16:2396–2401. Chin, A., Vaddiraju, S., Maslov, A., Ning, C., Sunkara, M., and Meyyappan, M. (2006). Near-infrared semiconductor subwavelength-wire lasers. Appl. Phys. Lett., 88:163115-1– 163115-3. Cohen, L., Haracz, R., Cohen, A., and Acquista, C. (1983). Scattering of light from arbitrary oriented finite cylinders. Appl. Opt., 22:742–748. Duan, X., Huang, Y., Agarwal, R., and Lieber, C. (2003). Singlenanowire electrically driven lasers. Nature, 421:241–245. Duan, X., Huang, Y., Cui, Y., Wang, J., and Lieber, C. (2001). Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices. Nature, 409:66–69. Fiutowski, J., Bordo, V., Jozefowski, L., Madsen, M., and Rubahn, H.-G. (2008). Light scattering from an ordered array of needle-shaped organic nanoaggregates: Evidence for optical mode launching. Appl. Phys. Lett., 92:073302-1–073302-3. Haracz, R., Cohen, L., and Cohen, A. (1984). Perturbation-theory for scattering from dielectric spheroids and short cylinders. Appl. Opt., 23:436–441. Huang, M., Mao, S., Feick, H. et al. (2001). Room-temperature ultraviolet nanowire nanolasers. Science, 292:1897–1899. Johnson, J., Choi, H.-J., Knutsen, K., Schaller, R., Yang, P., and Saykally, R. (2002). Single gallium nitride nanowire lasers. Nat. Mater., 1:106–110. Johnson, J., Yan, H., Yang, P., and Saykally, R. (2003). Optical cavity effects in ZnO nanowire lasers and waveguides. J. Phys. Chem., 107:8816–8828. Kerker, M. (1969). The Scattering of Light and Other Electromagnetic Radiation. Academic Press, New York. Kuik, F., de Haan, J., and Hovenier, J. (1994). Single scattering of light by circular cylinders. Appl. Opt., 33:4906–4918. Lord Rayleigh (1881). On the electromagnetic theory of light. Philos. Mag., 12:81–101. Owen, J., Barber, P., Messinger, B., and Chang, R. (1981). Determination of optical-fiber diameter from resonances in the elastic scattering spectrum. Opt. Lett., 6:272–274. Quochi, F., Cordella, F., Mura, A., Bongiovanni, G., Balzer, F., and Rubahn, H.-G. (2005). One-dimensional random lasing in a single organic nanofiber. J. Phys. Chem. B, 109:21690–21693.

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Quochi, F., Cordella, F., Mura, A., Bongiovanni, G., Balzer, F., and Rubahn, H.-G. (2006). Gain amplification and lasing properties of individual organic nanofibers. Appl. Phys. Lett., 88:041106-1–041106-3. Rao, T. and Barakat, R. (1994). Near-field scattering by a conducting cylinder partially buried in a conducting plane. Opt. Commun., 111:18–25. Ruppin, R. (1990). Electromagnetic scattering from finite dielectric cylinders. J. Phys. D Appl. Phys., 23:757–763. Schiek, M., Lützen, A., Koch, R. et al. (2005). Nanofibers from functionalized para-phenylene molecules. Appl. Phys. Lett., 86:153107-1–153107-3. Shepherd, J. and Holt, A. (1983). The scattering of electromagnetic radiation from finite dielectric circular cylinders. J. Phys. A Math. Gen., 16:651–662. Snyder, A. and Love, J. (1983). Optical Waveguide Theory. Chapman and Hall, London, U.K. Stratton, J. (1941). Electromagnetic Theory. McGraw-Hill, New York. Thilsing-Hansen, K., Neves-Petersen, M., Petersen, S., Neuendorf, R., Al-Shamery, K., and Rubahn, H.-G. (2005). Luminescence decay of oriented phenylene nanofibers. Phys. Rev. B, 72:115213-1–115213-7.

Tong, L., Gattass, R., Ashcom, J. et al. (2003). Subwavelengthdiameter silica wires for low-loss optical wave guiding. Nature, 426:816–819. Twersky, V. (1952). Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders. J. Acoust. Soc. Am., 24:42–46. Uzunoglu, N., Alexopoulos, N., and Fikioris, J. (1978). Scattering from thin and finite dielectric fibers. J. Opt. Soc. Am., 68:194–197. Van Bladel, J. (1977). Resonant scattering by dielectric cylinders. IEEE J. Microwave Opt. Acoust., 1:41–50. Wait, J. (1955). Scattering of a plane wave from a circular dielectric cylinder at oblique incidence. Can. J. Phys., 33:189–195. Wang, J., Gudiksen, M., Duan, X., Cui, Y., and Lieber, C. (2001). Highly polarized photoluminescence and photodetection from single indium phosphide nanowires. Science, 293:1455–1457. Zimmler, M., Bao, J., Capasso, F., Müller, S., and Ronning, C. (2008). Laser action in nanowires: Observation of the transition from amplified spontaneous emission to laser oscillation. Appl. Phys. Lett., 93:051101-1–051101-3.

31 Biomimetics: Photonic Nanostructures

Andrew R. Parker The Natural History Museum

31.1 Introduction............................................................................................................................ 31-1 31.2 Engineering of Antireflectors............................................................................................... 31-1 31.3 Engineering of Iridescent Devices....................................................................................... 31-1 31.4 Cell Culture............................................................................................................................. 31-3 31.5 Diatoms and Coccolithophores............................................................................................ 31-5 31.6 Iridoviruses.............................................................................................................................. 31-7 31.7 The Mechanisms of Natural Engineering and Future Research..................................... 31-7 Acknowledgments..............................................................................................................................31-8 Referencesï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½31-8

31.1â•‡ Introduction Three centuries of research, beginning with Hooke and Newton, have revealed a diversity of optical devices at the nanoscale (or at least the submicron scale) in nature.1 These include structures that cause random scattering, 2D diffraction gratings, 1D multilayer reflectors, and 3D liquid crystals (Figure 31.1a through d). In 2001, the first photonic crystal was identified as such in animals, 2 and since then the scientific effort in this subject has accelerated. Now we know of a variety of 2D- and 3D-photonic crystals in nature (e.g., Figure 31.1e and f), including some designs not encountered previously in physics. Biomimetics is the extraction of good design from nature. Some optical biomimetic successes have resulted from the use of conventional (and constantly advancing) engineering methods to make direct analogues of the reflectors and antireflectors found in nature. However, recent collaborations between biologists, physicists, engineers, chemists, and material scientists have ventured beyond merely mimicking in the laboratory what happens in nature, leading to a thriving new area of research involving biomimetics via cell culture. Here, the nano-engineering efficiency of living cells is harnessed, and nanostructures such as diatom “shells” can be made for commercial applications via culturing the cells themselves.

31.2â•‡ Engineering of Antireflectors Some insects benefit from antireflective surfaces, either on their eyes to see under low-light conditions, or on their wings to reduce surface reflections in transparent (camouflaged) areas. Antireflective surfaces, therefore, occur on the corneas of moth and butterfly eyes3 and on the transparent wings of hawkmoths.4

These consist of nodules, with rounded tips, arranged in a hexagonal array with a periodicity of around 240â•›nm (Figure 31.2b). Effectively they introduce a gradual refractive index profile at an interface between chitin (a polysaccharide, often embedded in a proteinaceous matrix; r.i. 1.54) and air, and hence reduce reflectivity by a factor of 10. This “moth-eye structure” was first reproduced at its correct scale by crossing three gratings at 120° using lithographic techniques, and employed as antireflective surfaces on glass windows in Scandinavia.5 Here, plastic sheets bearing the antireflector were attached to each interior surface of triple-glazed windows using refractive-index-matching glue to provide a significant difference in reflectivity. Today the moth-eye structure can be made extremely accurately using e-beam etching,6 and is employed commercially on solid plastic and other lenses. A different form of antireflective device, in the form of a sinusoidal grating of 250â•›nm periodicity, was discovered on the cornea of a 45-million-year-old fly preserved in amber7 (Figure 31.2a). This is particularly useful where light is incident at a range of angles (within a single plane, perpendicular to the grating grooves), as demonstrated by a model made in photoresist using lithographic methods.7 Consequently it has been employed on the surfaces of solar panels, providing a 10% increase in energy capture through reducing the reflected portion of sunlight.8 Again, this device is embossed onto plastic sheets using holographic techniques.

31.3â•‡ Engineering of Iridescent Devices Many birds, insects (particularly butterflies and beetles), fishes, and lesser-known marine animals display iridescent (changing color with angle) and/or “metallic” colored effects resulting from 31-1

31-2


(a)

(d)

(b)

(e)

(c)

(f )

FIGURE 31.1â•… Summary of the main types of optical reflectors found in nature; a–d where a light ray is (generally) reflected only once within the system (i.e., they adhere to the single scattering, or first Born, approximation), and e and f where each light ray is (generally) reflected multiple times within the system. (a) An irregular array of elements that scatter incident light into random directions. The scattered (or reflected) rays do not superimpose. (b) A diffraction grating, a surface structure, from where light is diffracted into a spectrum or multiple spectra. Each corrugation is about 500â•›n m wide. Diffracted rays superimpose either constructively or destructively. (c) A multilayer ref lector, composed of thin (ca. 100â•›nm thick) layers of alternating refractive index, where light rays reflected from each interface in the system superimpose either constructively or destructively. Some degree of refraction occurs. (d) A “liquid crystal” composed of nano-fibers arranged in layers, where the nano-fibers of one layer lie parallel to each other yet are orientated slightly differently to those of adjacent layers. Hence spiral patterns can be distinguished within the structure. The height of the section shown here—one “period” of the system—is around 200â•›nm. (e) Scanning electron micrograph of the “opal” structure—a close-packed array of submicron spheres (a “3D photonic crystal”)—found within a single scale of the weevil Metapocyrtus sp.; scale bar = 1â•›μm. (f) Transmission electron micrograph of a section through a hair (neuroseta) of the sea mouse Aphrodita sp. (Polychaeta), showing a cross section through a stack of submicron tubes (a “2D photonic crystal”); scale bar = 5â•›μm.

photonic nanostructures. These appear comparatively brighter than the effects of pigments and often function in animals to attract the attention of a potential mate or to startle a predator. An obvious application for such visually attractive and optically sophisticated devices is within the anticounterfeiting industry. For secrecy reasons, work in this area cannot be described, although devices are sought at different levels of sophistication, from effects that are discernable by the eye to fine-scale optical characteristics (polarization and angular properties, for example) that can be read only by specialized detectors. However, new research aims to exploit these devices in the cosmetics, paint, printing/ink, and clothing industries. They are even being tested in art to provide a sophisticated color-change effect. Original work on exploiting nature’s reflectors involved copying the design but not the size, where reflectors were scaled up to target longer wavelengths. For example, rapid prototyping was employed to manufacture a microwave analogue of a Morpho butterfly scale that is suitable for reflection in the 10–30â•›GHz region. Here the layer thicknesses would be in the order of 1â•›mm

rather than 100â•›nm as in the butterfly, but the device could be employed as an antenna with broad radiation characteristics, or as an antireflection coating for radar. However, today techniques are available to manufacture nature’s reflectors at their true size. Nanostructures causing iridescence include photonic crystal fibers, opal and inverse opal, and unusually sculpted 3D architectures. Photonic crystals are ordered, often complex, sub-wavelength (nano) lattices that can control the propagation of light at the singlewave scale in the manner that atomic crystals control electrons.9 Examples include opal (a hexagonal or square array of 250â•›nm spheres) and inverse opal (a hexagonal array of similar-sized holes in a solid matrix). Hummingbird feather barbs contain variation ultrathin layers with variations in porosity that cause their iridescent effects, due to the alternating nanoporous/fully dense ultrastructure.10 Such layers have been mimicked using aqueous-based layering techniques.10 The greatest diversity of 3D architectures can be found in butterfly scales, which can include micro-ribs with nano-ridges, concave multilayered pits, blazed gratings, and randomly punctate nano-layers.11,12 The cuticle of many beetles contain

31-3

Biomimetics: Photonic Nanostructures

(a)

(b)

(c)

FIGURE 31.2â•… Scanning electron micrographs of antireflective surfaces. (a) Fly-eye antireflector (ridges on four facets) on a 45-millionyear-old dolichopodid fly’s eye. (Micrograph from Mierzejewski, P. With permission.) (b) Moth-eye antireflective surfaces. (c) Moth-eye mimic fabricated using ion-beam etching. Scale bars = 3â•›μm (a), 1â•›μm (b), 2â•›μm (c). (Micrograph by Boden, S.A. and Bagnall, D.M., Biomimetic subwavelength surfaces for near-zero reflection sunrise to sunset, Proceedings 4th World Conference on Photovoltaic Energy, Conversion, Waikoloa, HI, 2006. With permission.)

structurally chiral films that produce iridescent effects with circular or elliptical polarization properties.13 These have been replicated in titania for specialized coatings,13 where a mimetic sample can be compared with the model beetle and an accurate variation in spectra with angle is observed (Figure 31.3). The titania mimic can be nanoengineered for a wide range of resonant wavelengths; the lowest so far is a pitch of 60â•›nm for a circular Bragg resonance at 220â•›nm in a Sc2O3 film (Ian Hodgkinson, pers. com.). Biomimetic work on the photonic crystal fibers of the Aphrodita sea mouse is underway. The sea mouse contains spines (tubes) with walls packed with smaller tubes of 500â•›nm, with varying internal diameters (50–400â•›nm). These provide a bandgap in the red region, and are to be manufactured via an extrusion technique. Larger glass tubes packed together in the proportion of the spine’s nanotubes will be heated and pulled through a drawing tower until they reach the correct dimensions. The sea mouse fibre mimics will be tested for standard PCF applications (e.g., in telecommunications) but also for anticounterfeiting structures readable by a detector.

The analogues of the famous blue Morpho butterfly (Figure 31.4a) scales have been manufactured.14,15 Originally, corners were cut. Where the Morpho wing contained two layers of scales—one to generate color (a quarter-wave stack) and another above it to scatter the light—the model copied only the principle.14 The substrate was roughened at the nanoscale, and coated with 80â•›nm thick layers alternating in refractive index.14 Therefore the device retained a quarter-wave stack centered in the blue region, but incorporated a degree of randomness to generate scattering. The engineered device closely matched the butterfly wing—the color observed changed only slightly with changing angle over 180°, an effect difficult to achieve and useful for a broad-angle optical filter without dyes. A new approach to making the 2D “Christmas tree” structure (a vertical, elongated ridge with several layers of 70â•›nm thick side branches; Figure 31.4b) has been achieved using focusedion-beam chemical-vapor deposition (FIB-CVD).15 By combining the lateral growth mode with ion-beam scanning, the Christmas tree structures were made accurately (Figure 31.4c). However, this method is not ideal for low-cost mass production of 2D and 3D nanostructures, and therefore the ion-beametched Christmas trees are currently limited to high-cost items including nano- or micron-sized filters (such as “pixels” in a display screen or a filter). Recently further corners have been cut in manufacturing the complex nanostructures found in many butterfly scales, involving the replication of the scales in ZnO, using the scales themselves as templates16 (Figure 31.4d and e).

31.4â•‡ Cell Culture Sometimes nature’s optical nanostructures have such an elaborate architecture at such a small (nano) scale that we simply cannot copy them using current engineering techniques. Additionally, sometimes they can be made as individual reflectors (as for the Morpho structure), but the effort is so great that commercial-scale manufacture would never be cost-effective. An alternative approach to making nature’s reflectors is to exploit an aspect other than design—that the animals or plants can make them efficiently. Therefore we can let nature manufacture the devices for us via cell-culture techniques. Animal cells are in the order of 10â•›μm in size and plant cells up to about 100â•›μm, and hence suitable for nanostructure production. The success of cell culture depends on the species and on type of cell from that species. Insect cells, for instance, can be cultured at room temperature, whereas an incubator is required for mammalian cells. Cell culture is not a straightforward method, however, since a culture medium must be established to which the cells adhere, before they can be induced to develop to the stage where they make their photonic devices. The current work in this area centres on butterfly scales. The cells that make the scales are identified in chrysalises, dissected, and plated out. Then the individual cells are separated, kept alive in culture, and prompted to manufacture scales through the addition of growth hormones. Currently we have cultured blue Morpho butterfly scales in the lab that have identical optical and structural characteristics to natural scales. The cultured scales

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(b) L-SEL EMI = 25.0 kV WD = 14 nm 2.00 μm Curl 1:02 511

PHOTO-2

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FIGURE 31.3â•… (a) A Manuka (scarab) beetle with (b) titania mimetic films of slightly different pitches. (c) Scanning electron micrograph of the chiral reflector in the beetle’s cuticle. (d) Scanning electron micrograph of the titania mimetic film. (Reproduced from DeSilva, L. et al., Electromagnetics, 25, 391, 2005. With permission.)

(a)

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2 μm (e)

FIGURE 31.4â•… (a) A Morpho butterfly with (b) a scanning electron micrograph of the structure causing the blue reflector in its scales. (c) A scanning electron micrograph of the FIB-CVD-fabricated mimic. A Ga+ ion beam (beam diameter 7â•›nm at 0.4â•›pA; 30â•›kV), held perpendicular to the surface, was used to etch a precursor of phenanthrene (C14H10). Both give a wavelength peak at around 440â•›nm and at the same angle (30°). (From Watanabe, K. et al., Jpn. J. Appl. Phys., 44, L48, 2005. With permission.) (d) Scanning electron micrograph of the base of a scale of the butterfly Ideopsis similes. (e) Scanning electron micrograph of a ZnO replica of the same part of the scale in (d). (Reproduced from Zhang, W. et al., Bioinspir. Biomim., 1, 89, 2006. With permission.)

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could be embedded in a polymer or mixed into a paint, where they may float to the surface and self-align. Further work, however, is required to increase the level of scale production and to harvest the scales from laboratory equipment in appropriate ways. A far simpler task emerges where the iridescent organism is single-celled.

31.5â•‡ Diatoms and Coccolithophores Diatoms are unicellular photosynthetic microorganisms. The cell wall is called the frustule and is made of the polysaccharide pectin impregnated with silica. The frustule contains pores (Figure 31.5a through c) and slits that give the protoplasm access to the external environment. There are more than 100,000 different of species of diatoms, generally 20–200â•›μm in diameter or length, but some can be up to 2â•›mm long. Diatoms have been proposed to build photonic devices directly in 3D.17 The biological function of the optical property (Figure 31.5d) is at present unknown, but may affect light collection by the diatom. This type of photonic device can be made in silicon using a deep photochemical-etching technique (initially developed by Lehmann18) (e.g., Figure 31.5e). However, there is a new potential here since diatoms carry the added advantage of exponential growth in numbers—each individual can give rise to 100 million descendents in a month. Unlike most manufacturing processes, diatoms achieve a high degree of complexity and hierarchical structure under mild physiological conditions. Importantly, the size of the pores does not scale with the size of the cell, thus maintaining the pattern.

Fuhrmann et al.17 showed that the presence of these pores in the silica cell wall of the diatom Coscinodiscus granii means that the frustule can be regarded as a photonic crystal slab waveguide. Furthermore, they present models to show that light may be coupled into the waveguide and give photonic resonances in the visible spectral range. The silica surface of the diatom is amenable to simple chemical functionalization (e.g., Figure 31.6a through c). An interesting example of this uses a DNA-modified diatom template for the control of nanoparticle assembly.19 Gold particles were coated with DNA complementary to that bound to the surface of the diatom. Subsequently, the gold particles were bound to the diatom surface via the sequence-specific DNA interaction. Using this method up to seven layers were added showing how a hierarchical structure could be built onto the template. Porous silicon is known to luminesce in the visible region of the spectrum when irradiated with ultraviolet light.20 This photoluminescence (PL) emission from the silica skeleton of diatoms was exploited by DeStafano21 in the production of an optical gas sensor. It was shown that the PL of Thalassiosira rotula is strongly dependent on the surrounding environment. Both the optical intensity and peaks are affected by gases and organic vapors. Depending on the electronegativity and polarizing ability, some substances quench the luminescence, while others effectively enhance it. In the presence of the gaseous substances NO2, acetone, and ethanol, the photoluminescence was quenched. This was because these substances attract electrons from the silica skeleton of the diatoms and hence quench the PL.

(b)

(a) 10000x, 7 kV, 14 mm,

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Date: 17 Aug 2004 EHT = 30.00 kV WD = 7 mm Signal = SE1 Photo no. = 1544 Time: 10:05:13

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FIGURE 31.5â•… (a–c) Scanning electron micrographs of the intercalary band of the frustule from two species of diatoms, showing the square array of pores from C. granii ((a) and (b)) and the hexagonal arrays of pores from C. wailesii (c). These periodic arrays are proposed to act as photonic crystal waveguides. (d) Iridescence of the C. granii girdle bands. (e) Southampton University mimic of a diatom frustule (patented for photonic crystal applications); scanning electron micrograph. (Micrograph by Parker, G. With permission.)

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FIGURE 31.6â•… Modification of natural photonic devices. (a)–(c) Diatom surface modification. The surface of the diatom was silanized, then treated with a heterobifunctional cross-linker, followed by attachment of an antibody via a primary amine group. (a) (i) Diatom exterior surface (ii) APS (iii) ANB-NOS (iv) primary antibody (v) secondary antibody with HRP conjugate. Diatoms treated with primary and secondary antibody with (b) no surface modification (c) after surface modification. (d and e) Scanning electron micrographs showing the pore pattern of the diatom C. wailesii (d) and after growth in the presence of nickel sulphate (e). Note the enlargement of pores, and hence change in optical properties, in (e). (f) “Photonic crystal” of the weevil Metapocyrtus sp., section through a scale, transmission electron micrograph; scale bar: 1â•›μm (see Parker33). (g) A comparatively enlarged diagrammatic example of cell membrane architecture: tubular christae in mitochondria from the chloride cell of sardine larvae. Evidence suggests that preexisting internal cell structures play a role in the manufacture of natural nanostructures; if these can be altered then so will the nanostructure made by the cell. (From Threadgold, L.T., The Ultrastructure of the Animal Cell, Pregamon Press, Oxford, U.K., 1967.)

Nucleophiles, such as xylene and pyridine, which donate electrons, had the opposite effect, and increased PL intensity almost ten times. Both quenching and enhancements were reversible as soon as the atmosphere was replaced by air. The silica inherent to diatoms does not provide the optimum chemistry/refractive index for many applications. Sandhage et al.22 have devised an inorganic molecular conversion reaction that preserves the size, shape, and morphology of the diatom while changing its composition. They perfected a gas/silica displacement reaction to convert biologically derived silica structures such as frustules into new compositions. Magnesium was shown to convert SiO2 diatoms by a vapor phase reaction at

900°C to MgO of identical shape and structure, with a liquid Mg2Si byproduct. Similarly, when diatoms were exposed to titanium fluoride gas, the titanium displaced the silicon, yielding a diatom structure made up entirely of titanium dioxide; a material used in some commercial solar cells. An alternative route to silica replacement hijacks that native route for silica deposition in vivo. Rorrer et al.23 sought to incorporate elements such as germanium into the frustule—a semiconductor material that has interesting properties that could be of value in optoelectronics, photonics, thin film displays, solar cells, and a wide range of electronic devices. Using a two-stage cultivation process, the photosynthetic marine diatom Nitzschia


frustulum was shown to assimilate soluble germanium and fabricate Si-Ge-oxide nanostructured composite materials. Porous glasses impregnated with organic dye molecules are promising solid media for tunable lasers and nonlinear optical devices, luminescent solar concentrators, gas sensors, and active waveguides. Biogenic porous silica has an open spongelike structure and its surface is naturally OH-terminated. Hildebrand and Palenik 24 have shown that rhodamine B and 6G are able to stain diatom silica in vivo, and determined that the dye treatment could survive the harsh acid treatment needed to remove the surface organic layer from the silica frustule. Now attention is beginning to turn additionally to coccolithophores—single-celled marine algae, also abundant in marine environments. Here, the cell secretes calcitic photonic crystal frustules, which, like diatoms, can take a diversity of forms, including complex 3D architectures at the nano- and microscales.

31.6â•‡ Iridoviruses Viruses are infectious particles made up of the viral genome packaged inside a protein capsid. The iridovirus family comprises a diverse array of large (120–300â•›nm in diameter) viruses with icosahedral symmetry. The viruses replicate in the cytoplasm of insect cells. Within the infected cell the virus particles produce a paracrystalline array that causes Bragg refraction of light. This property has largely been considered esthetic to date, but the research group of Vernon Ward (New Zealand), in collaboration with the Biomaterials laboratory at Wright–Patterson Air Force base, is using iridoviruses to create biophotonic crystals. These can be used for the control of light, with this laboratory undertaking large-scale virus production and purification as well as targeting the manipulation of the surface of iridoviruses for altered crystal properties. These can provide a structural platform for a broad range of optical technologies, ranging from sensors to waveguides. Virus nanoparticles, specifically Chilo and Wiseana Invertebrate Iridovirus, have been used as building blocks for iridescent nanoparticle assemblies. Here, virus particles were assembled in vitro, yielding films and monoliths with optical iridescence arising from multiple Bragg scattering from closely packed crystalline structures of the iridovirus. Bulk viral assemblies were prepared by centrifugation followed by the addition of glutaraldehyde, a cross-linking agent. Long-range assemblies were prepared by employing a cell design that forced virus assembly within a confined geometry followed by cross-linking. In addition, virus particles were used as core substrates in the fabrication of metallodielectric nanostructures. These comprise a dielectric core surrounded by a metallic shell. More specifically, a gold shell was assembled around the viral core by attaching small gold nanoparticles to the virus surface using inherent chemical functionality of the protein capsid.25 These gold nanoparticles then acted as nucleation sites for electroless deposition of gold ions from solution. Such nano-shells could be manufactured in large quantities, and provide cores with a

31-7

narrower size distribution and smaller diameters (below 80â•›nm) than currently used for silica. These investigations demonstrated that direct harvesting of biological structures, rather than biochemical modification of protein sequences, is a viable route to create unique, optically active materials.

31.7â•‡The Mechanisms of Natural Engineering and Future Research Where cell culture is concerned it is enough to know that cells do make optical nanostructures, which can be farmed appropriately. However, in the future an alternative may be to emulate the natural engineering processes ourselves, by reacting to the same concentrations of chemicals under the same environmental conditions, and possibly substituting analogous nano- or macro-machinery. To date, the process best studied is the silica cell wall formation in diatoms. The valves are formed by the controlled precipitation of silica within a specialized membrane vesicle called the silica deposition vesicle (SDV). Once inside the SDV, silicic acid is converted into silica particles, each measuring approximately 50â•›nm in diameter. These then aggregate to form larger blocks of material. Silica deposition is molded into a pattern by the presence of organelles such as mitochondria spaced at regular intervals along the cytoplasmic side of the SDV.26 These organelles are thought to physically restrict the targeting of silica from the cytoplasm, to ensure laying down of a correctly patterned structure. This process is very fast, presumably due to optimal reaction conditions for the synthesis of amorphous solid silica. Tight structural control results in the final species-specific, intricate exoskeleton morphology. The mechanism whereby diatoms use intracellular components to dictate the final pattern of the frustule may provide a route for directed evolution. Alterations in the cytoplasmic morphology of Skeletonema costatum have been observed in cells grown in sublethal concentrations of Mercury and Zinc, 27 resulting in swollen organelles, dilated membranes, and vacuolated cytoplasm. Frustule abnormalities have also been reported in Nitzschia liebethrutti grown in the presence of mercury and tin.28 Both metals resulted in a reduction in the length to width ratios of the diatoms, fusion of pores, and a reduction in the number of pores per frustule. These abnormalities were thought to arise from enzyme disruption either at the silica deposition site or at the nuclear level. We grew Coscinodiscus wailesii in sublethal concentrations of nickel and observed an increase in the size of the pores (Figure 31.6d and e), and a change in the phospholuminescent properties of the frustule. Here, the diatom can be “made to measure” for distinct applications such as stimuli-specific sensors. Further, trans-Golgi-derived vesicles are known to manufacture the coccolithophore 3D “photonic crystals.”29 So the organelles within the cell appear to have exact control of (photonic) crystal growth (CaCO2 in the coccolithophores) and packing (SiO3 in the diatoms).30,31 Indeed, Ghiradella9 suggested that the employment of preexisting, intracellular structures lay behind the development of some butterfly scales, and Overton32

31-8


reported the action of microtubules and microfibrils during butterfly-scale morphogenesis. Further evidence has been found to suggest that these mechanisms, involving the use of molds and nano-machinery (e.g., Figure 31.6f and g), reoccur with unrelated species, indicating that the basic “eukaryote” (containing a nucleus) cell can make complex photonic nanostructures with minimal genetic mutation.33 The ultimate goal in the field of optical biomimetics, therefore, could be to replicate such machinery and provide conditions under which, if the correct ingredients are supplied, the optical nanostructures will selfassemble with precision. For further information on the evolution of optical devices in nature, including those found in fossils, or when they first appeared on earth, see references. 34,35

Acknowledgments This work was funded by The Royal Society (University Research Fellowship), The Australian Research Council, European Union Framework 6 grant, and an RCUK Basic Technology grant.

References 1. Parker, A.R. 515 million years of structural colour. J. Opt. A 2, R15–R28 (2000). 2. Parker, A.R., McPhedran, R.C., McKenzie, D.R., Botten, L.C., and Nicorovici, N.-A.P. Aphrodite’s iridescence. Nature 409, 36–37 (2001). 3. Miller, W.H., Moller, A.R., and Bernhard, C.G. The corneal nipple array. In C.G. Bernhard (ed.), The Functional Organisation of the Compound Eye (Pergamon Press, Oxford, U.K., 1966), pp. 21–33. 4. Yoshida, A., Motoyama, M., Kosaku, A., and Miyamoto, K. Antireflective nanoprotuberance array in the transparent wing of a hawkmoth Cephanodes hylas. Zool. Sci. 14, 737–741 (1997). 5. Gale, M. Diffraction, beauty and commerce. Phys. World 2, 24–28 (1989). 6. Boden, S.A. and Bagnall, D.M. Biomimetic subwavelength surfaces for near-zero reflection sunrise to sunset. Proceedings of the 4th World Conference on Photovoltaic Energy, Conversion, Waikoloa, HI (2006). 7. Parker, A.R., Hegedus, Z., and Watts, R.A. Solar-absorber type antireflector on the eye of an Eocene fly (45Ma). Proc. R. Soc. Lond. B 265, 811–815 (1998). 8. Beale, B. Fly eye on the prize. The Bulletin, 46–48 (May 25, 1999). 9. Yablonovitch, E. Liquid versus photonic crystals. Nature 401, 539–541 (1999). 10. Cohen, R.E., Zhai, L., Nolte, A., and Rubner, M.F. pH gated porosity transitions of polyelectrolyte multilayers in confined geometries and their applications as tunable Bragg reflectors. Macromolecules 37, 6113 (2004). 11. Ghiradella, H. Structure and development of iridescent butterfly scales: Lattices and laminae. J. Morph. 202, 69–88 (1989).

12. Berthier, S. Les coulers des papillons ou l’imperative beauté. Proprietes optiques des ailes de papillons (Springer, Paris, France, 2005), 142 pp. 13. DeSilva, L., Hodgkinson, I., Murray, P., Wu, Q., Arnold, M., Leader, J., and Mcnaughton, A. Natural and nanoengineered chiral reflectors: Structural colour of manuka beetles and titania coatings. Electromagnetics 25, 391–408 (2005). 14. Kinoshita, S., Yoshioka, S., Fujii, Y., and Okamoto, N. Photophysics of structural color in the Morpho butterfly. Forma 17, 103 (2002). 15. Watanabe, K., Hoshino, T., Kanda, K., Haruyama, Y., and Matsui, S. Brilliant blue observation from a Morphobutterfly-scale quasi-structure. Jpn. J. Appl. Phys. 44, L48– L50 (2005). 16. Zhang, W., Zhang, D., Fan, T., Ding, J., Gu, J., Guo, Q., and Ogawa, H. Bio-mimetic zinc oxide replica with structural color using butterfly (Ideopsis similis) wings as templates. Bioinspir. Biomim. 1, 89 (2006). 17. Fuhrmann, T., Lanwehr, S., El Rharbi-Kucki, M., and Sumper, M. Diatoms as living photonic crystals. Appl. Phys. B 78, 257–260 (2004). 18. Lehmann, V. On the origin of electrochemical oscillations at silicon electrodes. J. Electrochem. Soc. 143, 1313 (1993). 19. Rosi, N.L., Thaxton, C.S., and Mirkin, C.A. Control of nanoparticle assembly by using DNA-modified diatom templates. Agnew Chem. Int. Ed. 43, 5500–5503 (2004). 20. Cullis, A.G., Canham, L.T., and Calcott, P.D.J. The structural and luminescence properties of porous silicon. J. Appl. Phys. 82, 909–965 (1997). 21. De Stefano, L., Rendina, I., De Stefano, M., Bismuto, A., and Maddalena, P. Marine diatoms as optical chemical sensors. Appl. Phys. Lett. 87, 233902 (2005). 22. Sandhage, K.H., Dickerson, M.B., Huseman, P.M., Caranna, M.A., Clifton, J.D., Bull, T.A., Heibel, T.J., Overton, W.R., and Schoenwaelder, M.E.A. Novel, bioclastic route to selfassembled, 3D, chemically tailored meso/nanostructures: Shape-preserving reactive conversion of biosilica (diatom) microshells. Adv. Mater. 14, 429–433 (2002). 23. Rorrer, G.L., Chang, C.H., Liu, S.H., Jeffryes, C., Jiao, J., and Hedberg, J.A. Biosynthesis of silicon-germanium oxide nanocomposites by the marine diatom Nitzschia frustulum. J. Nanosci. Nanotechnol. 5, 41–49 (2004). 24. Hildebrand, M. and Palenik, B. Grant report Investigation into the Optical Properties of Nanostructured Silica from Diatoms, La Jolla, CA (2003). 25. Radloff, C., Vaia, R.A., Brunton, J., Bouwer, G.T., and Ward, V.K. Metal nanoshell assembly on a virus bioscaffold. Nano Lett. 5, 1187–1191 (2005). 26. Schmid, A.M.M. Aspects of morphogenesis and function of diatom cell walls with implications for taxonomy. Protoplasma 181, 43–60 (1994). 27. Smith, M.A. The effect of heavy metals on the cytoplasmic fine structure of Skeletonema costatum (Bacillariophyta). Protoplasma 116, 14–23 (1983).


28. Saboski, E. Effects of mercury and tin on frustular ultrastructure of the marine diatom Nitzschia liebethrutti. Water, Air Soil Pollut. 8, 461–466 (1977). 29. Corstjens, P.L.A.M. and Gonzales, E.L. Effects of nitrogen and phosphorus availability on the expression of the coccolith-vesicle v-ATPase (subunit C) of Pleurochrysis (Haptophyta). J. Phycol. 40, 82–87 (2004). 30. Klaveness, D. and Paasche, E. Physiology of coccolithophorids. In: Biochemistry and Physiology of Protozoa, 2nd edn., vol. 1 (Academic Press, New York, 1979), pp. 191–213. 31. Klaveness, D. and Guillard, R.R.L. The requirement for silicon in Synura petersenii (Chrysophyceae). J. Phycol. 11, 349–355 (1975).

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32. Overton, J. Microtubules and microfibrils in morphogenesis of the scale cells of Ephestia kuhniella. J. Cell Biol. 29, 293–305 (1966). 33. Parker, A.R. Conservative photonic crystals imply indirect transcription from genotype to phenotype. Recent Res. Develop. Entomol. 5, 1–10 (2006). 34. Parker, A.R. In the Blink of an Eye (Simon & Schuster, London, U.K., 2003), 316 pp. 35. Parker, A.R. A geological history of reflecting optics. J. R. Soc. Lond. Interface 2, 1–17 (2005).

Nanophotonic Devices

V

32 Photon Localization at the Nanoscaleâ•… Kiyoshi Kobayashi.................................................................................... 32-1

33 Operations in Nanophotonicsâ•… Suguru Sangu and Kiyoshi Kobayashi................................................................... 33-1

34 System Architectures for Nanophotonicsâ•… Makoto Naruse................................................................................... 34-1

35 Nanophotonics for Device Operation and Fabricationâ•… Tadashi Kawazoe and Motoichi Ohtsu.......................... 35-1

36 Nanophotonic Device Materialsâ•… Takashi Yatsui and Wataru Nomura................................................................. 36-1

37 Waveguides for Nanophotonicsâ•… Jan Valenta, Tomáš Ostatnický, and Ivan Pelant.................................................37-1

38 Biomolecular Neuronet Devicesâ•… Grigory E. Adamov and Evgeny P. Grebennikov................................................ 38-1

Introductionâ•‡ •â•‡ Backgroundâ•‡ •â•‡ Dressing Mechanism and Spatial Localization of Photonsâ•‡ •â•‡ Summary and Future Perspectiveâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Dissipation-Controlled Nanophotonic Devicesâ•‡ •â•‡ Nanophotonic Devices Using Spatial Symmetriesâ•‡ •â•‡ Summaryâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ System Architectures Based on Optical Excitation Transferâ•‡ •â•‡ Hierarchical Architectures in Nanophotonicsâ•‡ •â•‡ Summaryâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Excitation Energy Transfer in Nanophotonic Devicesâ•‡ •â•‡ Device Operationâ•‡ •â•‡ Nanophotonics Fabricationâ•‡ •â•‡ Summaryâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Nanophotonic Devices Based on Quantum Dotsâ•‡ •â•‡ Nanophotonic AND-Gate Device Using ZnO Nanorod Double-Quantum-Well Structuresâ•‡ •â•‡ References

Introductionâ•‡ •â•‡ Fabrication of Planar and Rib Waveguidesâ•‡ •â•‡ Experimental Techniquesâ•‡ •â•‡ Main Experimental Observations in Active Si-nc Waveguidesâ•‡ •â•‡ Theoretical Description of Active Lossy Waveguidesâ•‡ •â•‡ Application of Active Nanocrystalline Waveguidesâ•‡ •â•‡ Conclusionsâ•‡ •â•‡ Acknowledgmentsâ•‡ •â•‡ References Introductionâ•‡ •â•‡ Some Techniques of Neuro-Molecular and Molecular Information Processing Using Bacteriorhodopsin (Basic Processes, Constructions, Technology)â•‡ •â•‡ Nanostructuring of Bacteriorhodopsin-Containing Molecular Mediaâ•‡ •â•‡ Summaryâ•‡ •â•‡ References

V-1

32 Photon Localization at the Nanoscale 32.1 Introduction............................................................................................................................32-1 32.2 Background..............................................................................................................................32-2 Difficulty in Defining a Wave Function of a Photon in Real Spaceâ•‡ •â•‡ Effective Spatial Wave Function of a Single Photonâ•‡ •â•‡ Canonical Field Quantization: Mode Functions, Field Operators, and Quantum States

32.3 Dressing Mechanism and Spatial Localization of Photons.............................................32-5

Kiyoshi Kobayashi University of Tokyo Japan Science and Technology University of Yamanashi

Virtual Photon Cloud Surrounding a Neutral Source (in Ground State or Excited State) in QEDâ•‡ •â•‡ Electromagnetic Field Correlations and Intermolecular Interactions between Molecules in Either Ground or Excited Statesâ•‡ •â•‡ Effective Near-Field Optical Interaction between Nanomaterials Disconnected but Closely Separatedâ•‡ •â•‡ Localization of a Photon Dressed by Matter Excitation in Nanomaterials at the Nanoscale

32.4 Summary and Future Perspective......................................................................................32-17 Acknowledgments............................................................................................................................32-17 References..........................................................................................................................................32-17

32.1â•‡Introduction You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) field. However, Heisenberg’s uncertainty principle tells us that a state of definite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indefinite in space and time. It suggests the difficulty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, after a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with finite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic falloff of energy density exist, and that

a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999). On the other hand, several authors have claimed that a minimum modification of the naive route leads to a wave-function description of a photon, even though the probability density for the position of a photon and a position-representation wave function cannot be consistently introduced. For example, it has been shown to be possible to introduce a position-representation wave function ψ( r , t ) for a photon, which is the expectation value of the photon energy in a region dâ•›r⃗ about r⃗ (Sipe 1995, Scully and Zubairy 1997, Hawton 1999, Roychoudhuri and Roy 2003). Mandel et al., from another viewpoint, have found that a photon wave function as a probability amplitude is possible in a coarse-grained volume whose linear dimensions are larger than the photon wavelength (Mandel 1966, Mandel and Wolf 1995, Inagaki 1998). The localization of the photon energy density and photodetection rates having an exponential or arbitrarily fast asymptotic falloff have been discussed, as well as causality (Hegerfeldt 1974, Pike and Sarkar 1987, Hellwarth and Nouchi 1996, Adlard et al. 1997, Bialynicki-Birula 1998, Keller 2005). When the interactions with matter were considered, different views opened up. Dressed states and operators, dressed and half-dressed sources in nonrelativistic QED, electromagnetic field correlations, and intermolecular interactions between molecules in either ground or excited states have been discussed (Compagno et al. 1988, 1995, Cohen-Tannoudji et al. 1989, 1992, 32-1

32-2


Power and Thirunamachandran 1993), focusing on the fact that a bare source interacting with a quantum field is surrounded by a cloud of virtual particles. It has been shown that the dressing of the source, or virtual cloud effects, can be detected by a test body (detector) located close to the source. Carniglia and Mandel (1971) proposed a complete basis for electromagnetic fields interacting with a material of refractive index n filled in a half space separated by vacuum in order to quantize the source fields, while Inoue and Hori (2001) discussed the detector modes and the behavior of a photon–atom interacting system near the material surface. Kobayashi et al. (2001) focused on the environmental effects on a nanomaterial interacting with photons and obtained a near-field optical interaction as an effective interaction between nanomaterials electronically disconnected, but closely located, in order to detect the cloud of virtual photons. They have also applied it to a discussion of nanophotonic devices (Sangu et al. 2004). Focusing on the photon degrees of freedom, on the other hand, photon hopping has been employed to discuss a photon–material interacting system (John and Quang 1995, Suzuura et al. 1996, Shojiguchi et al. 2003), and a photon dressing by material excitations has been recently discussed by using the photon-hopping model in real space and a quasiparticle model (Kobayashi et al. 2008). Focusing on light–matter interactions at the nanoscale, we discuss a near-field optical interaction between nanomaterials surrounded by a macroscopic system, a dressing mechanism, and spatial localization of photons in this article, which is organized as follows. Section 32.2 is devoted to background issues for photon localization, in particular, the difficulty of the definition, effective methods, and free-field quantization. In Section 32.3 we discuss a dressing mechanism of photons and their localization in space, including virtual clouds of photons, electromagnetic field correlations and intermolecular interactions, effective nearfield optical interactions, and phonons’ effects on photon localization. Finally, a summary and a future outlook are presented.

32.2.1â•‡Difficulty in Defining a Wave Function of a Photon in Real Space Since there is no probability density for the photon, and thus a position-representation wave function in free space cannot be consistently defined, one has to follow quantum electrodynamics (QED), that is, to redefine one- and a few-photon wave functions in a physically meaningful way in order to obtain fruitful insights into the photon–matter interacting system at the nanoscale. In the following subsections, we will give an overview of both approaches, after pointing out the difficulties involved with a position-representation wave function of a photon in free space. We begin with Maxwell’s equations in vacuum: 1 ∂B(r , t ) ∇ × E(r , t ) = − , c ∂t

(32.1b)

where c is the speed of light E(r, t ) and B(r, t ) are electric and magnetic fields, respectively Let us now define Φ ± (r, t ) as Φ ± (r , t) = E(r , t) ± iB(r , t); then, it Â�follows from Maxwell’s equations that Φ ± (r , t) should satisfy

∂Φ ± (r , t ) = ±c∇ × Φ ± (r , t ). i ∂t

(32.2)

Using the Fourier transform of these functions, we introduce the vectors γ ± ( p, t ) as

Φ ± (r , t ) =

∫

d3 p γ ( p, t )e ip⋅r / , 3/ 2 ± (2π )

(32.3)

which is associated with γ ± ( p, t ) as γ ± ( p, t ) = γ ± ( p, t )e± ( p) with

1 ˆ [e1( p) ± ie2 ( pˆ)], e± ( pˆ) = ∓ 2

(32.4a)

where two unit vectors e⃗ 1(̂p) and e⃗ 2(̂p) are defined such that the unit vectors

(32.4b)

[e1(pˆ), e2 (pˆ), pˆ = p/ | p |]

form a right-handed triad. Then, it is easy to verify that γ ± ( p, t ) satisfy a Schr.Ödinger-like equation in the momentum representation

32.2â•‡Background

1 ∂E(r, t ) ∇ × B(r, t ) = , c ∂t

(32.1a)

∂γ ± ( p, t ) i = cpγ ± ( p, t ), ∂t

(32.5)

which indicates that γ ± ( p, t ) are probability amplitudes for photons of momentum p⃗ , energy E = cp with p = |⃗p|, and positive/ negative helicity. Here note that

γ *+ ( p, t ) ⋅ γ + ( p, t )dp = γ *− ( p ⋅ t ) ⋅ γ − ( p, t )dp

(32.6a)

and

γ *+ ( p, t ) ⋅ γ − ( p, t ) = 0

(32.6b)

32-3

Photon Localization at the Nanoscale

show the probability of detecting a photon of positive helicity and momentum p⃗ between p⃗ and p⃗ + d⃗p, and likewise for a photon of negative helicity. Then the normalization condition is

∫ γ * ( p, t ) ⋅ γ ( p, t) + γ * ( p, t) d p = 1, +

+

3

−

(32.7)

which leads to

+

−

−

3

(32.8)

Φ ± (r , t ) =

∫

d3 p γ ( p, 0)e −icpt / e ip⋅r / . 3/ 2 ± (2π )

(32.9)

Here we note that the sum of Φ± (r , t ) might be regarded as the position-representation wave function of a photon, but it cannot be regarded as such because Newton and Wigner and also Wightman have shown that the photon, being a massless particle, is not localizable in free space, and that there does not exist a probability amplitude and density for the position of the photon in the usual sense (see also (32.15a) and (32.15b)) (Newton and Wigner 1949, Wightman 1962).

32.2.2â•‡Effective Spatial Wave Function of a Single Photon In order to avoid the difficulties mentioned above, Mandel defined an operator representing the number of photons in a volume V as the integral over V of a so-called “detection operator,” which led a simple formula for the probability that n photons are present in V, when the linear dimensions of V are larger than the wavelength of light used (Mandel 1966). Sipe tookanother approach to this issue, seeking a probability amplitude Ψ(r , t ) for the photon energy to be detected d⃗r about of ⃗r (Sipe 1995). Assuming that the integral of Ψ* (r , t ) ⋅ Ψ(r , t )dr over all space is proportional to the photon energy, we normalize it as follows:

∫

Ψ* (r , t ) ⋅ Ψ(r , t )d 3r = cp  γ *+ ( p, t ) ⋅ γ *+ ( p, t )  + γ *− ( p, t ) ⋅ γ *− ( p, t ) d 3 p. 

∫

(32.10)

It is easily shown that if Ψ(r , t ) is set as

Ψ(r , t ) = Ψ+ (r , t ) + Ψ − (r , t ),

(32.11a)

∫

cp γ ( p, t )eip⋅r / d 3 p, 3 /2 ± (2π )

(32.11b)

the normalization condition is satisfied. Here note that γ *+ ( p, t ) ⋅ γ − ( p, t ) = 0 = γ *− ( p, t ) ⋅ γ + ( p, t )

(32.12)

∂ Ψ± (r , t ) i = ±c∇ × Ψ± (r , t ). ∂t

(32.13)

and

and it follows from the dynamical equations that

∫ Φ* (r , t ) ⋅ Φ (r , t ) + Φ* (r , t ) ⋅ Φ (r , t ) d r = 1, +

Ψ± (r , t ) =

At the same time, Ψ ± (r , t ) should be chosen to satisfy an initial condition given by

Ψ± (r , t ) =

cp

∫ (2π)

3 /2

γ ± ( p, 0)e −icpt / e ip⋅r / d 3 p.

(32.14)

Since γ ± ( p, t ) and Ψ± ( p, r ) are not the Fourier transform pairs, the arguments of the photon momentum ⃗p and the position r⃗ associated energy are not conjugate variables. with the photon When Φ ± (r , t ) and Ψ± (r , t ) are related by alocal kernel, it can usually be regarded as a particle. However, Φ ± (r , t ), the Fourier transform of γ ± ( p, t ), is not a reasonable candidate for the position-representation wave function because of the of the photon following relation between Φ ± (r , t ) and Ψ± (r , t )

Φ ± (r , t ) = w(r − r ′ )Ψ± (r ′, t )d 3r ′

∫

(32.15a)

with the nonlocal kernel

w(r − r ′ ) =

∫

′

1 eik ⋅(r − r ) 3 d k. (2π)3 ck

(32.15b)

Nevertheless, Ψ(r , t ) might be meaningful to describe the dynamics of a photon such as a spontaneous emission from an atom and the inverse process, or at least we can detect a photon, within the range of a detector’s precision by placing a detector like an atom close to the source. In other words, it indicates that light–matter interactions near the source play an important role and should be treated consistently.

32.2.3â•‡Canonical Field Quantization: Mode Functions, Field Operators, and Quantum States It is natural to follow the canonical quantization of the electromagnetic field as a starting point for a discussion of light–matter interactions at the nanoscale. Since a lot of famous textbooks on

32-4


QED or quantum field theory have been published, we follow the essence of the theory and restrict ourselves to the free field that is free from charges and currents and whose scalar potential can be set to zero (Sakurai 1967, Roychoudhuri and Roy 2003). In the Coulomb gauge with ∇⋅ A(r , t ) = 0 for the vector potential A(r , t ), three basic equations we work with for the free-field case are B(r , t ) = ∇ × A(r , t ),

(32.16a)

(32.16b)

2 1 ∂ A(r , t ) ∇ A(r , t ) − 2 = 0. c ∂t 2

(32.16c)

∫

(32.18)

where we have for simplicity combined the three components of the wave vector k⃗ and the polarization index α to one index ℓ. Using the normalization constant At and the time-dependent amplitude qt (t), we have

∑  A q (t )u (r ) + A*q*(t)u *(r ) ,

(32.19)

where qt(t) follows from the differential equation of a harmonic oscillator of frequency Ω ≡ ck: d 2q (t ) + Ω2q (t ) = 0. dt 2

2  2 1 1 1 ∂A  ( ) . H= d 3r (E 2 + B 2 ) = d 3r  A + ∇ ×  c ∂t   8π 8π  

∫

′

′

3

2

′

3

∫

(32.21)

3

′

Ω  =   V δ ′  c 

H=

2

′

3

(32.23)

V 2π

∑

2

 Ω  *  c  ( Aq ) ( Aq ).

(32.24)

If we define 1 Q = (q + q* ), c

P = −i

Ω (q − q* ), A = c

(32.25a)

4π , V

(32.25b)

the Hamiltonian can be expressed in terms of a collection of independent and uncoupled harmonic oscillators as H=

∑ 12 (P

2

+ Ω2Q2 ).

(32.26)

Here, Pℓ and Qℓ are seen to be canonical variables:

∂H dP =− , ∂Q dt

(32.27a)

dQ ∂H = . ∂P dt

(32.27b)

(32.20)

The Hamiltonian of the field is

3

we obtain

1 d 3ru*(r ) ⋅ u′ (r ) = δ ′ , V

′

(32.17)

and the boundary conditions set by the shape of a virtual cavity, for example, a box taken to be a cube of side L = V 1/3. It follows from the Coulomb gauge that the direction of uk ,a (r ) has to be ⃗ and thus there are two polarizaorthogonal to the wave vector k, tion degrees of freedom indicated by the index α. Note that the mode functions become more complicated, or even impossible for more sophisticated cavity shapes. The mode functions satisfy the orthonormality condition

∫ (∇ × u ) ⋅(∇ × u* )d r = ∫ ∇ ⋅(u × ∇ × u* )d r + ∫ u ⋅ ∇ × ∇ × u*  d r = u ⋅ ∇(∇⋅ u * ) − ∇ u *  d r = − u ⋅∇ u * d r  ∫  ∫ 2

(∇2 + k )uk ,a (r ) = 0,

A(r , t ) =

and for the B⃗ 2 integration

We expand A(r , t ) into a complete set of mode functions uk ,a (r ) defined by the Helmholtz equation

(32.22)

2

2  1 dq   1 dq*′  * 3  Ω   u (r ) ⋅ u ′ (r )d r =   V q δ ′  c  dt 

∫  c dt   c

1 ∂A(r , t ) E(r , t ) = − , c ∂t

2

â•›⃗ 2 integration Noticing a typical term for the E

The natural method to quantize the field is to replace the variable qℓ and its conjugate momentum pℓ ≡ dqℓ/dt by operators q̂ ℓ and ̂pℓ that satisfy the commutation relations [qˆ , pˆ ] = iδ ′, or

[Q , P ′ ] = iδ ′ . Then, the Hamiltonian operator for the quantized field can be written as follows:

32-5


Hˆ =

∑

1 ˆ2 (P + Ω2Qˆ 2 ). 2

(32.28)

̂ ℓ, given by Next we consider linear combinations of P̂ ℓ and Q

aˆ =

Ω 2h

i ˆ ˆ  Q + Ω P  ,

(32.29a)

aˆ † =

Ω 2h

i ˆ ˆ  Q + Ω P  ,

(32.29b)

where we insert a factor to make âℓ and aˆ dimensionless and thus satisfy the following boson commutation relations: †

i i aˆ , aˆ ′  = _ [Qˆ , Pˆ ′ ] + [P , Q ′] = δ ′ . 2h 2h

(32.30)

Taking care of the order of âℓ and aˆ †′, we obtain the Hamiltonian operator of the electromagnetic field as Hˆ =

∑

1  Ω  nˆ +  ,  2

(32.31)

where nˆ ≡ aˆ †aˆ denotes the number operator. The contribution 1/2 arises from the commutation relations and results in the familiar zero-point energy. Since there are infinitely many modes, the zero-point energy becomes infinite, but in general we drop this contribution by shifting the vacuum energy, which does not influence the dynamics. The quantization procedure is completed by writing the vector potential in terms of âℓ and aˆ † as a field operator

ˆ A (r , t ) =

∑

(

)

2πc 2 aˆ u + aˆ†u * . V Ω

(32.32)

Thus, aˆ † is called the creation operator for a photon specified in ℓ as corresponding to the quantum-mechanical excitations of the electromagnetic field, while âℓ is interpreted as the annihilation operator for a photon in state ℓ. It is important to note ˆ that the ⃗r and t that appear in the quantized field A(r , t ) are not quantum-mechanical variables but just parameters on which the field operator depends, and, in particular, ⃗r and t should not be regarded as the space–time coordinates of the photon. When we adopt a linearly polarized plane wave as the mode function

u (r )= ε kα e ik ⋅r

(32.33)

with the polarization vector εk α, the energy and momentum of ⃗ respectively. Therefore, the the photon are Ω k = k c and ħk,

mass of the photon is zero. In addition, since εk α transforms like a vector, the general theory of angular momentum encourages us to associate with it one unit of angular momentum, which means that the photon has one unit of spin angular momentum. The field operators described above operate on quantum state vectors, and quantum states |Ψ〉 of the electromagnetic field are, in general, multimode states that involve quantum states |ψℓ 〉 for each mode ℓ. One of the useful quantum states is photon number states denoted by |nℓ 〉, which are eigenstates of the number ̂ ℓ operator n

nˆ |n 〉 = n|n 〉

(32.34)

with integer eigenvalues nℓ. At the same time, |nℓ 〉 are eigenstates of the Hamiltonian with eigenenergy nℓħΩℓ, that is, nℓ times the fundamental unit ħΩℓ. It should be noted that nℓ quanta of energy ħΩℓ are in the mode, but the energy is distributed over the entire space, that is, not localized. Other useful quantum states used later are coherent states |α〉, which are eigenstates of the annihilation operator âℓ with eigenvalues α

aˆ |α〉 = α|α〉 .

(32.35)

The phase of the coherent states is completely determined, while the number of photons is completely undetermined. In the subsequent sections, we will employ a quantum electromagnetic field discussed above in order to discuss the nature of light–matter interactions apparently exhibited at the nanoscale.

32.3â•‡Dressing Mechanism and Spatial Localization of Photons 32.3.1â•‡Virtual Photon Cloud Surrounding a Neutral Source (in Ground State or Excited State) in QED A quantum source material system interacting with a quantum field is influenced by virtual processes such as emission and absorption of virtual quanta of the field, and the source can be described as a dressed source, that is, the “bare” source surrounded by a cloud of virtual particles (Compagno et al. 1988, 1995). It is true for a detector. The virtual-cloud effects are responsible for the modification of the values of fundamental constants. For example, in nonrelativistic quantum electrodynamics, the presence of a virtual cloud around a hydrogen atom in its ground state contributes to the Lamb shift. Dressed-source effects can also be seen in different physical systems, such as a nucleon coupled to the meson field, and an electron coupled to the optical phonon modes of a semiconductor (polaron). The virtual cloud around the source also modifies the energy density distribution of the electromagnetic field, and the detailed properties have direct physical significance. The energy density of the virtual photon cloud at a given point is in fact related to the van der Waals interaction experienced by a suitable test

32-6


object as a detector at that point. The presence of a virtual cloud around a source can influence not only its energy levels but also its dynamics. Let us roughly estimate the linear dimensions of the virtual cloud surrounding the source or the detector. Even when the source–field system is in the ground state, a fluctuation of the field leads to the possibility of absorption or the emission of photons, not necessarily to the conservation of energy. Such energy imbalance δE is constrained by the Heisenberg uncertainty principle δE ∼ ħ/τ, where τ is the duration of the fluctuation. Since these fluctuations take place continuously, a steady-state cloud of virtual photons is continuously emitted and reabsorbed. The virtual photon can only attain a finite distance from the source or the detector given by

r ~ cτ ~

c , δE

(32.36)

where c is the speed of light. For a transition corresponding to one of typical visible light, we set δE ∼ 2â•›eV and obtain a typical linear dimension of 100â•›nm. This indicates that dressing effects might be prominent at the nanoscale. We have discussed the virtual clouds of the source or detector in its ground state. From now on, we discuss the virtual cloud of the source or the detector in excited states, which can decay by emission of real photons. The above discussion inclines us to use the perturbation theory, but it fails due to the near degeneracy of states that gives rise to vanishing energy denominators at all the orders of perturbation theory. Another difficulty is the description of decay processes in a consistent way. One of such attempts is based on an extension of the eigenvalue problem to the complex E-plane. The underlying theory, unfortunately, has not yet been established, but it seems to imply time symmetry breaking and irreversibility in the dynamics of the system Section 32.3.2 will be devoted to an approach focusing on field correlations and intermolecular interactions due to the virtual clouds.

32.3.2â•‡Electromagnetic Field Correlations and Intermolecular Interactions between Molecules in Either Ground or Excited States The London–van der Waals interaction between two molecules in their ground states located in free space is attractive with an R−6 power law, where R is the intermolecular separation (Power and Thirunamachandran 1993). When both molecules are excited, the potential energy gives a repulsive force. If one of the pair is excited, the sign of the potential depends on the relative magnitudes of the relevant transition energies of the two molecules. In both cases the power law is R−6 in the near zone. On the other hand, the power law in the far zone tends to R−7 for large intermolecular separations, which is called the Casimir–Polder potential, since the finite speed of propagation (retardation effect) plays an important role in the far zone.

In these kinds of studies, the multipolar quantum dynamics in Coulomb gauge is employed because all the interactions, except for the Coulomb binding within each molecule, are mediated by transverse photons, and at the same time the retarded effects are automatically satisfied (Power and Thirunamachandran 1993). For example, for upward transitions from the ground state |0〉 to an excited state |m〉, the electric–electric spatial correlation expectation value after spatial averaging of the molecular orientation is given by using the second-order perturbation method to include virtual-cloud effects:  µ m0 2  (13δij + 7rî rˆj), for far zone (k0r 1), 7 〈Di (r ) D j (r )〉m←0 ∼  6πk0mr0 2  µ  (δij + 3rî rˆj), for near zone (k0r 1),  3r 6

(32.37)

where Di (r ) is the i-component of the transverse displacement m0 vector field D(r ), which satisfies ∇ ⋅ D(r ) = 0, µ is the electric dipole transition moment for the molecular states |m〉 and |0〉, and k0 ≡ km0 ≡ (Em − E 0)/(ħc) denotes the wave number associated with the m ← 0 transition of the molecule. The i-component of the unit vector r̂ is designated by r̂ i, while the absolute value of the position vector ⃗r is expressed by r. The Kronecker delta is denoted by δij. Similarly, for downward transitions from an excited state |p〉 to the ground state |0〉, the electric–electric correlation function is obtained:  2 p 4 µ 0 p 2  0 (δ ij − rî rˆj), for far zone ( p0r 1), 2  ~  03pr2 Di ( r ) D j (r ) 0← p  µ (δ ij + 3rî rˆj), for near zone ( p0r 1),   3r 6

(32.38)

where p0 ≡ p0p ≡ (Ep − E0)/(ħc) denotes the wave number associated with the 0 ← p transition of the molecule. It should be noted that the r −2 dependence arises from the real photon emission, while the r −6 dependence and the r −7 dependence are due to the virtual photon exchange. The far-zone behavior for the magnetic–magnetic correlation functions due to an electric-dipole source is also described by the same r −2 or r −7 dependence, while the near-zone result is different from its electric analog and the power law is r −5 instead of r −6. We have discussed the electric and magnetic correlation functions leading to the electric and magnetic energy densities associated with electric-dipole transitions in a source molecule, which can be detected by their effect on polarizable test bodies placed in the neighborhood of the source. This situation is analogous to an optical near-field system in which a nanometric material source connected to a macroscopic material and light source

32-7


interacts with a nanometric detector connected to a macroscopic detector system. The difference is that the nanometric detector serving as a test body can disturb the field formed by the nanometric source in the case of the optical near-field system. We will move on to this topic in Section 32.3.3.

cro

sco p sys ic ma tem ter ia

l

Virtual clouds of photons (dressed photons)

Relaxation

32.3.3â•‡Effective Near-Field Optical Interaction between Nanomaterials Disconnected but Closely Separated Several theoretical approaches to optical near-field problems, different from each other in viewpoints, have been proposed in the last two decades (Pohl and Courjon 1993, Ohtsu and Hori 1999). The optical near-field problems, including the application to nanophotonics, are ultimately regarded as how one should formulate a separated (more than two elements) composite system, each of which consists of a photon–electron–phonon interacting system on a nanometer scale as a source or a detector system and, which, at the same time, is connected with a macroscopic light–matter system. These questions must be clearly answered to achieve practical realization of nanophotonics. In order to provide a base for a variety of discussions in this research field, a new formulation has been developed within a quantum theoretical framework, putting matter excitations (electronic and vibrational) on an equal footing with photons (Kobayashi et al. 2001). As discussed in Section 32.2, a “photon,” whose concept has been established as a result of quantization of a free electromagnetic field, corresponds to a discrete excitation of electromagnetic modes in a virtual cavity. Unlike an electron, a photon is massless, and it is difficult to construct a wave function in the position representation that gives a picture of the photon as a spatially localized point particle like an electron. However, if there is a detector near the optical source, such as an atom, to absorb a photon in an area whose linear dimension is much smaller than the wavelength of light, it would be possible to detect a photon with the same precision as the detector size. In optical near-field problems, it is required to consider the interactions between light and nanomaterials surrounded by a macroscopic material system and detection of light by other nanomaterials on a nanometer scale. Then, a more serious question for the quantization of the field is how to define a virtual cavity, or which normal modes are to be used, since there exist more than two systems (nano-source and nano-detector) with arbitrary shape, size, and material in the nanometer region, which are still connected with a macroscopic material system, such as a source or a detector system. In this section, we describe a model and a theoretical approach to address the issue, which is essential to understand the operating principles nanophotonic devices, as well as nanofabrication using optical near fields (Ohtsu et al. 2008). Let us consider a nanomaterial system surrounded by an incident light and a macroscopic material system, which electromagnetically interacts with one another in a complicated way, as schematically shown in Figure 32.1. Using the projection operator method, we can derive an effective interaction between the relevant nanomaterials in which we are interested (nano-source

Ma

t light Inciden

Nano source

Nano detector

Radiation

Figure 32.1â•… Nanometric source and detector system induced by incident light and a macroscopic material system.

and nano-detector—either one is in the excited state), as a result of renormalizing the other effects. It corresponds to an approach to describe “photons localized around nanomaterials,” as if each nanomaterial would work as a detector and light source in a self-consistent way. The effective interaction related to optical near-fields is hereafter called a near-field optical interaction. As will be discussed in detail in this section, the near-field optical interaction between nanomaterials separated by R is as follows:

Veff =

exp(−aR) , R

(32.39)

where a−1 is the interaction range that represents the characteristic size of the nanomaterials and does not depend on the wavelength of the light used. It indicates that photons are localized around the nanomaterials (either of which is in the excited state) as a result of the interaction with matter fields, from which a photon, in turn, can acquire a finite mass. Therefore, we might consider that the near-field optical interaction is produced via localized photon hopping between nanomaterials. On the basis of the projection operator method, we will investigate formulation of an optical near-field system that was briefly mentioned above. Moreover, the explicit functional form of the near-field optical interaction will be given by using the effective interaĉ in a perturbative way. tion mV eff 32.3.3.1â•‡Relevant Nanometric Subsystem and Irrelevant Macroscopic Subsystem As schematically illustrated in Figure 32.1, the optical near-field system consists of two subsystems: One is a macroscopic subsystem including the incident light, whose typical dimension is much larger than the wavelength of the incident light. The other is a nanometric subsystem (nano-source and nano-detector), whose constituents are, for example, a nanometric aperture or a protrusion at the apex of a near-field optical probe, and a nanometric sample. We call such an aperture or a protrusion a probe tip.

32-8


As a nanometric sample we mainly suppose a single atom/ molecule, or quantum dot (QD). Two subsystems are interacting with each other, and it is very important to formulate the interaction consistently and systematically. Here let us call the nanometric subsystem a relevant subsystem n, and the macroscopic subsystem an irrelevant subsystem M. We are interested in the subsystem n; in particular, the interaction induced in the subsystem n. Therefore, it is the key to renormalize the effects originating from the subsystem M in a consistent and systematic way. Now we show a formulation based on the projection operator method, described below. 32.3.3.2â•‡P Space and Q Space It is preferable, for a variety of discussions, to express exact eigenstates |ψ〉 for the total system described by the total ̂ in terms of a small number of bases of as small a Hamiltonian H number of degrees of freedom as possible, which span P space. In the following, let us assume two states as the P-space components: |ϕ1〉 = |s*〉|p〉â•›⊗ |0(M)〉 and |ϕ2〉 = |s〉|p*〉 ⊗ |0(M)〉, both of ̂ 0. Here which are eigenstates of the unperturbed Hamiltonian H |s〉 and |s*〉 are eigenstates of the sample that is isolated from the others, while |p〉 and |p*〉 are eigenstates of the probe tip, which is also isolated. In addition, exciton polariton states, which are a mixture of photons and electron–hole pairs, are used as bases to describe the macroscopic subsystem M, and thus |0(M) 〉 represents the vacuum for exciton polaritons. Note that there exist photons and electronic matter excitations even in the vacuum state |0(M)〉. The direct product is denoted by the symbol ⊗ . The complementary space to the P space is called Q space, which is spanned by a huge number of bases of a large number of degrees of freedom not included in the P space, as schematically shown in Figure 32.2. 32.3.3.3â•‡Effective Interaction Exerted in the Nanometric Subsystem Noticing the relation between a bare interaction Hamiltonian ̂ H ̂ −H ̂ 0 and an effective interaction Hamiltonian operator V ̂ eff, V = Total space spanned by eigenstates of the total Hamiltonian Q space

. . .

. . .

|s*

|p*

|s

|p

. . .

|0(M)

̂ ̂ eff P|ψ〉, we obtain the effective interacnamely, 〈ψ|V|ψ〉 = 〈ψ|PV tion Hamiltonian operator in the P space, given by

ˆ )−1/2 (PJˆ†VJP ˆ ˆ )(PJˆ† JP ˆ )−1/2 , Vêff = (PJˆ† JP

(32.40)

and tracing out the other degrees of freedom gives an effective interaction Hamiltonian of the nanometric subsystem n after renormalizing the effects from the macroscopic subsystem M. Here, P is the projection operator, and Q is the complimentary operator defined by Q = 1 − P, both of which satisfy the following relations:

P = P † , P 2 = P,

[P , Hˆ 0] = 0,

(32.41a)

Q = Q † , Q 2 = Q,

[Q, Hˆ 0] = 0,

(32.41b)

PQ = QP = 0.

(32.41c)

The operator ̂J is defined by

Jˆ = [1 − (E − Hˆ 0 )−1QVˆ ]−1 ,

(32.42)

̂ Using where E are the eigenvalues of the total Hamiltonian H. the effective interaction Hamiltonian, one can forget the subsystem M as if the subsystem n were isolated and separated from the subsystem M. To obtain an explicit expression of the effective interaction ̂ between the Hamiltonian, let us employ the bare interaction V two subsystems, which in the multipolar formalism (Craig and Thirunamachandran 1998) is given by

ˆ   ˆ ˆ Vˆ = − µ s ⋅ D ⊥ (rs ) + uˆ p ⋅ D ⊥ (rp ) ,  

(32.43)

where the canonical momentum of the vector potential operator ˆ A(r ) is proportional to the transverse displacement vector field ˆ operator* D ⊥ (r ), while the electric dipole operator is denoted as ˆ µ(r ). It should be noted that there are no interactions, i.e., ̂ V = 0, without incident photons in the macroscopic subsystem M. The subscripts s and p represent physical quantities related to the sample and the probe tip, respectively. Representative positions of the sample and the probe tip are chosen, for simplicity, by the vectors ⃗rs and ⃗r p, respectively, but may be composed of several positions. In that case the quantities inside curly brackets ˆ in (32.43) should be read as a summation. The operator Π(r ) conˆ ˆ jugate to A(r ) is expressed in terms of D ⊥ (r ) as follows:

P space

Figure 32.2â•… Subdivision of the space spanned by eigenstates of the total Hamiltonian of the system.

* The transverse component is defined by ∇ ⋅ F ⊥ = 0, while the longitudinal || component is defined by ∇ × F = 0, for an arbitrary vector field F(⃗ â•›⃗ r ).

32-9


ˆ 1 ˆ ⊥ 1 ˆ ⊥ 1 ˆ ⊥ Π(r ) = − E (r ) − P (r ) = − D (r ), c 4πc 4πc

(32.44)

ˆ ˆ where E ⊥ (r ) and P ⊥ (r ) are the transverse electric field and the induced polarization field, respectively. With the help of the ˆ ˆ mode expansion of A(r ) and Π(r ), that is, ˆ A(r ) =

{

 2 π c   V ω  eλ (k ) aˆ λ (k )e  k  λ =1

∑∑ k

1/ 2

2

ik ⋅r

+ aˆ λ† (k)e

− ik ⋅r

1/ 2

2

},

k

(32.45)

{

}

we can rewrite the transverse component of the electric displacement operator as ˆ D ⊥ (r ) = i

2

∑∑ k

1/ 2

{

}

 2πω  † − ik ⋅r ik ⋅r ,  V  eλ (k ) aˆ λ (k )e − aˆ λ (k )e λ =1 k

(32.47)

where the plane waves are used for the mode functions, and the creation and annihilation operators of a photon with wave ⃗ angular frequency ωk⃗ , and polarization component λ vector k, are designated by aˆ λ† (k ) and aˆ λ (k ), respectively. The quantization volume is V, and the unit vector related to the polarization direction is shown by e λ(k ). Note that the electric field outside ˆ the material corresponds to D ⊥ (r ). Since exciton polariton states are employed as bases to describe the macroscopic subsystem M, the creation and annihilation operators of a photon in (32.47) are rewritten using the exciton polariton operators ξˆ † (k ) and ξˆ (k ) , and then they are substituted into (32.43). Using the electric dipole operator defined by

(

)

ˆ µ α = Bˆ (rα ) + Bˆ † (rα ) µ α ,

(32.48)

with the creation and annihilation operators of excitation in sub system n, Bˆ † (rα ) and Bˆ (rα ) , and the transition dipole moments μ ⃗ α (α = s, p), we obtain the bare interaction in the exciton polariton picture:

Vˆ = −i

p

∑∑ α=s

k

 2π    V 

1/ 2

( Bˆ (r ) + Bˆ (r ))(K (k )ξˆ(k ) − K *(k )ξˆ (k )). α

†

α

α

α

†

(32.49)

2

∑(

)

µ ε ⋅ eλ (k ) f (k)e ik ⋅rα

λ =1

(32.50)

with

 2πω k  † − ik ⋅r ik ⋅r ,  V  eλ (k ) aˆ λ (k )e − aˆ λ (k )e λ =1 (32.46)

∑∑

K α (k ) =

f (k) =

and ˆ i Π(r ) = − 4πc

⃗ is the coupling coefficient between the exciton polariHere Kα(k) ton and the nanometric subsystem n, given by

ck Ω(k)

Ω2 (k) − Ω2 . 2Ω2 (k) − Ω2 − (ck)2

(32.51)

⃗ is denoted by K α* (k ), while c, Ω(k), The complex conjugate of Kα(k) and Ω are light speed in vacuum and the eigenfrequencies of both exciton polariton and electronic polarization of the macroscopic subsystem M, respectively. The dispersion relation for a free photon, ω k⃗ = ck, is used in (32.51). Note that the wave-number dependence of f(k) characterizes a typical interaction range of exciton polaritons coupled to the nanometric subsystem n. The next step is to evaluate the amplitude of the effective interaction exerted on the nanometric subsystem, for example, the effective sample–probe-tip interaction in the P space after tracing out the polariton degrees of freedom: Veff (2,1) ≡ 〈φ2Vˆ effφ1 〉.

(32.52)

With the first-order approximation Ĵ (1) of Ĵ in (32.42), we can explicitly write (32.52) in the following form: ˆ ˆ (E 0 − E 0 )−1 P φ 〉 + 〈φ P(E 0 − E 0 )−1VQV ˆ ˆ Pφ1〉 Veff (2,1) = 〈φ2PVQV P Q 2 P Q  1 =



ˆ m〉〈 mQVP ˆ φ 〉 ∑ 〈 φ PVQ  E 2

1

m

0 P1

 1 1 . + 0 0 0  − EQm EP 2 − EQm 

(32.53)

The second line shows that a virtual transition from the initial state |ϕ1〉 in the P space to an intermediate state |m〉 in the Q space is followed by a subsequent virtual transition from the intermediate state |m〉 to the final state |ϕ2〉 in the P space. Here 0 E P0 1(EP0 2 ) and EQm denote eigenenergies of |ϕ1〉(|ϕ2〉) in the P space and that of |ϕm〉 in the Q space, respectively. Now we can proceed ̂ to the next step by substituting the explicit bare interaction V in (32.49) with (32.50) and (32.51) into (32.53). First of all, note that the one-exciton polariton state among arbitrary intermediate states |m〉 can only contribute to nonzero matrix elements. Therefore, (32.53) can be transformed into

 K p (k )K s* (k ) K s (k )K *p (k )  3 , Veff (2,1) = − d k + (2π)2  Ω(k) − Ω0 (s) Ω(k) + Ω0 (s)  (32.54)

∫

⃗ where the summation over k⃗ is replaced by k-integration, i.e., V 3 d k , in the usual manner. Excitation energies of the (2π)3 ∫

32-10


sample (between |s*〉 and |s〉) and the probe tip (between |p*〉 and |p〉) are denoted as Es = ħΩ0(s) and Ep = ħΩ0(p), respectively. Exchanging the arguments 1 and 2, or the role of the sample ̂ eff |ϕ2〉: and probe tip, we can similarly calculate Veff(1,2) ≡ 〈ϕ1|V Veff (1, 2) = −

 K s (k )K *p (k ) K p (k )K s* (k )  3 .  d k + Ω(k) − Ω0 ( p) Ω(k) + Ω0 ( p)  (2π)2   (32.55)

2

∑ ∑ ∫ d k (µ

1 Veff (r ) = − 2 4π

3

λ =1 α = p , s

p

)(

)

⋅ e λ (k ) µ s ⋅ e λ (k )  

λi (k ) e λi (k ) = δ ij − ki k j ,

(32.57)

λ =1

)(

(

)

and thus the summation of µ p ⋅ eλ (k ) µ s ⋅ eλ (k ) over λ can be reduced as follows:

2

s

λ

pi λi

λ =1

λ =1

=

sj λj

i, j

∑µ

pi

µ sj (δ ij − kî kˆj )

−

(32.58)

(32.60)

(e ikr − e −ikr ) ˆ ˆ  ri rj ikr 

where r̂ is the unit vector defined by r̂ ≡ r̂ /r, and the jth component is denoted by r̂ j. Hence, the effective interaction can be rewritten as ∞



∫ k dkf (k)∑  E(k) + E 2

2

1

a = s, p

−∞

a

+

 1 E(k) − Ea 

 1 1   1 + 2 2 − 3 3  − (µ s ⋅ rˆ)(µ p ⋅ rˆ)eik ⋅r × (µ s ⋅ µ p )eik ⋅r   ikr k r ik r  

3 3   1 × + −  ikr k 2r 2 ik 3r 3  

(32.61)

where the integration range is extended from (0, ∞) to (−∞, ∞). When the dispersion relation of exciton polaritons, which have been chosen as a basis for describing the macroscopic subsystem M, is approximated as E(k) = Ω +

∑ (µ ⋅ e (k))(µ ⋅ e (k)) = ∑ ∑ (µ e (k ))(µ e (k )) λ

∑e

p

 (eikr − e −ikr )  (eikr + e −ikr ) (e ikr − e −ikr )  − = 2π δij + (δij − 3rî rˆj)   2 2 ikr 3r 3  ikr   kr

(32.56)

where we have set E(k) = ħΩ(k), and E α = ħΩ0(a*) − ħΩ0(α) for α = p and α = s. The summation over polarization λ is performed as 2

 eikr − e −ikr  2π  e ikr − e −ikr  2π (δ ij − kî kˆj)e ±ik ⋅r dΩ = δ ij + ∇ ∇ i j  ik 3   ik  r r

1 Veff (r ) = − 2π

 e ik ⋅r e −ik ⋅r  × f (k)  +   E(k) + Eα E(k) − Eα  2

2

∫

∫

Therefore, the total amplitude of the effective sample–probe-tip interaction is given by the sum of (32.54) and (32.55), which includes the effects from the macroscopic subsystem M. We write this effective interaction for the nanometric subsystem n as Veff(⃗r ) in the following way:

we find

(k)2 (ck)2 = Em + , 2mpol 2 Epl

(32.62)

in terms of the effective mass of exciton polaritons, m pol, or E plâ•›=â•›m pol c2 and the electronic excitation energy of the macroscopic subsystem M, Em = ħΩ, (32.61) is further simplified:

i, j

⃗ with the unit vector k̂ ≡ k/k. Noticing that d3k = k2dkd Ω  = k2dk sin θdθdφ  and

∫

δ ij e ± ik ⋅r dΩ = δ ij

2π 1

∫ ∫e

± ikr cos θ

d(cos θ)dϕ = δ ij

2π ikr (e − e −ikr ), ikr

(32.59a)

0 −1

 e ikr − e −ikr  1 2π − kî kˆ je ± ik ⋅r dΩ = 2 ∇i ∇ j e ± ik ⋅r dΩ = 3 ∇i ∇ j   , k ik r 

∫

∫

(32.59b)

1 Veff (r ) = − 2π

∞

2 Epl

∫ k dkf (k) ∑ (c) 2

−∞

2

2

α = s, p

  1 1 × +  k i k i + ∆ − ∆ ( k i )( k i ) ( )( ) + ∆ − ∆ α+ α+ α− α−    1 1   1 + 2 2 − 3 3  − µ s ⋅ rˆ µ p ⋅ rˆ e ik ⋅r × (µ s ⋅ µ p )e ik ⋅r    ikr k r ik r 

(

)(

)

3 3   1 × + −  ikr k 2r 2 ik 3r 3   Veff,α + (r ) + Veff,α − (r ) ≡ α = s, p

∑

(32.63a)

32-11


with

∆α± ≡

1 2 Epl (Em ± Eα ) , (Em > Eα ). c

(32.63b)

The k-integration can be performed with the residues evaluated at k = iΔα±, and we have 1   (∆ )2 ∆ 1 Veff ,α ± (r ) = ∓ (µ s ⋅ µ p )  α ± + α2 ± + 3 Wα ±e −∆α ± r 2  r r   r

  (∆ )2 3∆ 3 −(µ s ⋅ rˆ)(µ p ⋅ rˆ)  α ± + 2α ± + 3 Wα ±e −∆α ± r , r r    r (32.64a)

where the constants Wα± are defined by Wα ± ≡

Epl Em2 − Eα2 . (32.64b) Eα (Em ± Eα )(Em − Epl ∓ Eα ) − Em2 2

If the angular average of (µ s ⋅ rˆ)(µ p ⋅ rˆ) is taken, the expression

−∆ α + r  (µ s ⋅ µ p ) e −∆α −r  2 e Veff (r ) = − − Wα − (∆ α − )2 Wα + (∆ α + )  r r  3 α=s, p 

∑

(32.65)

is obtained for the effective interaction, or the near-field optical interaction Veff(r), which consists of the sum of the Yukawa functions Y (∆ α ±r ) ≡ e −∆α ± r r with a shorter interaction range Δα+(heavier effective mass) and a longer interaction range Δα−(lighter effective mass). To sum up, we find that the major part of the effective interaction exerted in the nanometric subsystem n is the Yukawa potential after renormalizing the effects from the macroscopic subsystem. This interaction comes from the mediation of massive virtual photons corresponding to the counter-rotating term, or polaritons, where exciton polaritons have been employed in an explicit formulation, but in principle other types of polaritons would be applicable. In this section we have mainly focused on the effective interaction of the nanometric subsystem n, after tracing out the other degrees of freedom. It is certainly possible to have a formulation with a projection onto the P space that is spanned in terms of the degrees of freedom of the massive virtual photons, although this is left as the subject of future work. This kind of formulation emphasizes a “dressed photon” picture, in which photons are not massless but massive, due to light–matter interactions at the nanoscale. Site 1

2

3

32.3.4â•‡Localization of a Photon Dressed by Matter Excitation in Nanomaterials at the Nanoscale In this section, we consider a simple model system, for example, a pseudo one-dimensional near-field optical probe system, to discuss the mechanism of photon localization in space as well as the phonon’s role. In order to focus on the photon–phonon interaction, the interacting part between photon and electronic excitation is first expressed in terms of a polariton, and is called a photon in the model. Then the model Hamiltonian that describes the photon and phonon interacting system is presented. Using the Davydov transformation, we rewrite the Hamiltonian in terms of quasiparticles. On the basis of the Hamiltonian, we present numerical results on the spatial distribution of photons and discuss the mechanism of photon localization due to phonons. 32.3.4.1â•‡Model Hamiltonian We consider a near-field optical probe, schematically shown in Figure 32.3, as an example system where light interacts with both phonons and electrons in the probe on a nanometer scale. Here, the interaction of a photon and an electronic excitation is assumed to be expressed in terms of a polariton basis, as discussed above, and is hereafter called a photon so that special attention is paid to the photon–phonon interaction. The system is simply modeled as a one-dimensional atomic or molecular chain coupled with photon and phonon fields. The chain consists of finite N molecules (representatively called), each of which is located at a discrete point (called a molecular site) whose separation represents a characteristic scale of the near-field system. Photons are expressed in the site representation and can hop to the nearest neighbor sites due to the short-range interaction nature of the optical near fields. The Hamiltonian for the above model is given by

Hˆ =

mi

∑ i =1

 N pˆ 2 N −1 k † k 2  i ωaî aî +  + xˆ i  (xˆ i +1 − xˆ i )2 + 2   i =1 2mi i =1 2 i =1, N 

N

+

∑ i =1

∑

†

N −1

χaî aî xˆ i +

∑ J(aˆ aˆ †

i

i +1

∑

†

+ aî +1aî) ,

(32.66)

i =1

N–2

J

∑

where aî† and âi correspondingly denote the creation and annihilation operators of a photon with energy ħω at site i in the chain x̂ i and p̂ i represent the displacement and conjugate momentum operators of the vibration, respectively

i k

N

k

Figure 32.3â•… Simple one-dimensional model for a light–matter interacting system on a nanometer scale.

N–1

32-12


The mass of a molecule at site i is designated by mi, and each molecule is assumed to be connected by springs with spring constant k. The third and fourth terms in (32.66) stand for the photon–vibration interaction with coupling constant χ and the photon hopping with hopping constant J, respectively. After the vibration field is quantized in terms of phonon operators of mode p and frequency Ωp, bˆ †p , and b̂ p, the Hamiltonian (32.66)

Transforming the Hamiltonian in (32.70) as

N

∑

†

ωaî aî +

i =1

N

+

N

= Hˆ + [Sˆ, Hˆ ] + 1 Sˆ, [Sˆ, Hˆ ] + H  2

∑ p =1

N

∑ ∑ χ aˆ aˆ (bˆ †

ip i

i =1

Ω pbˆ†pbˆ p

i

† p

+ bˆp) +

p =1

N −1

∑ J(aˆ aˆ †

i

i =1

i +1

†

+ aî +1aî), (32.67)

with the coupling constant χip of a photon at site i and a phonon of mode p. It should be noted that the index p designates not the momentum but the mode number, because the translational invariance of the system is broken and the momentum is not a good quantum number. The site-dependent coupling constant χip is related to the original coupling constant χ in terms of the transformation matrix as Pip as χip = χPip

, 2mi Ω p

(32.68)

and the creation and annihilation operators of a photon and a phonon satisfy the boson commutation relation as aî , aˆ †j  = δ ij and bˆ p , bˆq† = δ pq . The Hamiltonian (32.67), which describes the model system, is not easily handled because of the third order of the operators in the interaction term. To avoid this difficulty, this direct photon–phonon interaction term in (32.67) is eliminated by the Davydov transformation in the following subsection. 32.3.4.2â•‡Davydov Transformation Before going into the explicit expression, we discuss a unitary ̂ transformation Uˆ generated by an anti-Hermitian operator S, defined as

Uˆ ≡ exp(Sˆ), with Sˆ † = −Sˆ

(32.69a)

and

(32.69b)

̂ that consists of a diagonalized part H ̂ 0 Assume a Hamiltonian H ̂ and a non-diagonal interaction part V:

Hˆ = Hˆ 0 + Vˆ .

Vˆ = −[Sˆ, Hˆ 0 ],

(32.72)

(32.73)

the Hamiltonian (32.70) is rewritten as

= Hˆ 0 − 1 Sˆ, [Sˆ, Hˆ 0] + , H  2

(32.74)

̂ and can be diagonalized within the first order of V. Now we apply the above discussion to the model Hamiltonian (32.67), Hˆ 0 =

N

∑

N

ωaî†aî +

i =1

Vˆ =

N

∑ Ω bˆ bˆ , † p p p

(32.75a)

+ bˆ p ),

(32.75b)

p =1

N

∑ ∑ χ aˆ aˆ (bˆ † ip i i

i =1

† p

p =1

tentatively neglecting the hopping term. Assuming the antiHermitian operator Ŝ Sˆ =

∑ ∑ f aˆ aˆ (bˆ † ip i i

i

† p

− bˆ p ),

(32.76a)

p

we can determine fip from (32.73) as follows: f ip =

χip . Ωp

(32.76b)

This operator form of Ŝ leads us to not the perturbative but the exact transformation of the photon and phonon operators:

(32.70)

1 = Hˆ 0 + Vˆ + [Sˆ, Hˆ 0] + [Sˆ, Vˆ ] + Sˆ, [Sˆ, Hˆ 0] + .  2

If the interaction V̂ can be perturbative, and if the operator Ŝ is chosen so that the second and third terms in (32.72) are canceled out, then

Uˆ † = Uˆ −1.

(32.71)

we have

can be rewritten as Hˆ =

ˆ ˆ ˆ † = UHU ˆ ˆ ˆ −1 , ≡ UHU H

  αˆ i† ≡ Uˆ †aî†Uˆ = aî† exp  − 

N

∑ ( p =1

χip ˆ † ˆ bp − bp Ωp



) , 

(32.77a)

32-13


 N χ   ip ˆ † αˆ i ≡ Uˆ †aîUˆ = aî exp  (bp − bˆ p ) ,  p =1 Ω p 

∑

N

χip

∑Ω

βˆ †p ≡ Uˆ †bˆ†pUˆ = bˆ†p +

i =1 N

χip

∑Ω

ˆ ˆ pUˆ † = bˆ p + βˆ p ≡ Ub

p

i =1

p

aî†aî ,

(32.77b)

(32.77c)

p

aî†aî .

(32.77d)

These transformed operators can be regarded as the creation and annihilation operators of quasiparticles—dressed photons and phonons—that satisfy the same boson commutation relations as those of photons and phonons before the transformation, namely, ˆ† ˆ  ˆ ˆ †  ˆ †  ˆ ˆ†  ˆ ˆ ˆ† α i , α j  = U [aî , aˆ j ]U = δ ij and β p , βq  = U bp , bq  U = δ pq. Using the quasiparticle operators, we can rewrite the Hamiltonian (32.67) as Hˆ =

N

∑

N

ωαˆ i†αˆ i +

i =1

∑ p =1

N −1

+

Ω pβˆ †pβˆ p −

∑ ( Jˆ αˆ αˆ i

† i

i +1

N

N

N

i =1

j =1

p =1

∑∑∑

χip χ jp ˆ † ˆ ˆ † ˆ αi αiα j α j Ωp

)

+ Jî†αˆ i†+1αˆ i ,

i =1

where a photon at site i is associated with phonons in a coherent state, i.e., a photon is dressed by an infinite number of phonons. This corresponds to the fact that an optical near field is generated from a result of interactions between the photon and matter fields. When βˆ † is applied to the vacuum state |0〉, we have

(32.78a)

∑

(32.78b)

where it is noted that the direct photon–phonon coupling term has ˆ been eliminated, while the quadratic form Nˆ i N â•› j with the number operator of Nˆ i = αˆ i† αˆ i has emerged as well as the site-dependent hopping operator Ĵ i in (32.78b). The number states of quasiparticles are thus eigenstates of each terms of the Hamiltonian (32.78a), except the last term that represents the higher-order effect of photon–phonon coupling through the dressed photon hopping. Therefore, it is a more appropriate form to discuss the phonon’s effect on photon’s behavior as localization.

  αˆ 0 = aˆ exp  −  † i

† i

  = aî† exp  − 

N

∑ ( p =1 N

∑ p =1

χip ˆ† ˆ bp − bp Ωp

1  χip  2  Ω p 

2

  0 , 

)

   χip ˆ †  bp  0 ,  exp  −  Ω p  

(32.79)

   χ 2   ip cos(Ω pt ) − 1  , P(t ) = 1 − exp 2   Ω   p  

(32.81)

where the photon-hopping term is neglected for simplicity. The excitation probability oscillates at a frequency of 2π/Ωp and has the maximum value at t = π/Ωp. The frequencies of the localized phonon modes are higher than those of the delocalized ones, and the localized modes at the earlier time are excited by the incident photons. Figure 32.4 shows the temporal evolution of the excitation probability Pp0 (t ) calculated from     χ 2   ip Pp0 (t ) = 1 − exp 2  0   cos(Ω p0 t ) − 1     Ω   p 0    

32.3.4.3â•‡Quasiparticle and Coherent State In the previous section, we have transformed the original Hamiltonian with the Davydov transformation. In order to grasp the physical meanings of the quasiparticles introduced above, the creation operator αˆ i† is applied to the vacuum state |0〉. Then, it follows from (32.77a) that

(32.80)

and it is expressed by only the bare phonon operator (before the transformation) in the same p mode. Therefore, we mainly focus ˆ i† , α ˆ i ) in the following subon the quasiparticle expressed by (α section. Note that it is valid only if the bare photon number (the expectation value of αˆ i†αˆ i is not so large that the fluctuation is more important than the bare photon number. In other words, the model we are considering is suitable for discussing the quantum nature of a few photons in an optically excited probe system. In the coherent state of phonons, the number of phonons, as well as energy, is fluctuating. This fluctuation allows incident photons into the probe system to excite phonon fields. When all the phonons are in the vacuum state at time t = 0, the excitation probability P(t) that a photon incident on site i in the model system excites the phonon mode p at time t is given by

with  N (χ − χ )   ip i +1 p Jî = J exp  (βˆ †p − βˆ p ) , Ωp  p =1 

βˆ †p 0 = bˆ†p 0 ,

2    χip    cos(Ω pt ) − 1  , (32.82) 2 × exp     p ≠ p0  Ω p 

∑

where a specific phonon mode p 0 is excited, while other modes are in the vacuum state. In Figure 32.4, the solid curve represents the probability that a localized phonon mode is excited as the p0 mode, while the dashed curve illustrates how the lowest phonon mode is excited as the p0 mode. It follows from the figure that the localized phonon mode is dominantly excited at an earlier time.

32-14


Probability

0.07

Hˆ =

N

∑

N −1

(ω − ωi )αˆ i†αˆ i +

0.06

0.05

or in matrix form as

i =1

0.04

0.02

0.01 0.00 0.0

0.5

1.0

1.5

2.0

32.3.4.4â•‡Localization Mechanism of Dressed Photons In this section, we discuss how phonons contribute to the spatial distribution of photons in the pseudo one-dimensional system under consideration. When there are no interactions between photons and phonons, the frequency and hopping constant are equal at all sites, and thus the spatial distribution of photons is symmetric. It means that no photon localization occurs at any specific site. However, if there are any photon–phonon interactions, spatial inhomogeneity or localization of phonons affects the spatial distribution of photons. On the basis of the Hamiltonian (32.78a), we analyze the contribution from the diagonal (the third term) and off-diagonal (the fourth term) parts of the Hamiltonian (32.78a) in order to investigate the localization mechanism of photons. 32.3.4.4.1 Contribution from the Diagonal Part

N

−

N

N

∑∑∑ i =1

j =1 p =1

χip χ jp ˆ † ˆ ˆ † ˆ αi αi α j α j ≡ − Ωp

N

∑ ω αˆ αˆ , i

i =1

† i

∑∑ j =1 p =1

χip χ jp ˆ † ˆ α jα j . Ωp

Aˆ r

In addition, for the moment, we neglect the site dependence of the hopping operator Ĵ i, which is approximated as J. Then the ̂ and α ̂ †) can be Hamiltonian regarding the quasiparticles (α expressed as

(32.85b)

N

∑ E Aˆ Aˆ † r

r

(32.86a)

r

N

∑

N

(Q −1 )ri αˆ i =

i =1

∑ Q αˆ , ir

(32.86b)

i

i =1

and  Aˆ r , Aˆ s†  ≡ Aˆ r Aˆ s† − Aˆ s† Aˆ r = δ rs .  

(32.86c)

Using the above relations (32.86a) through (32.86c), we can write ˆ at down the time evolution of the photon number operator N i site i as follows:  Hˆ   Hˆ  Nˆ i (t ) = exp  i t  Nˆ i exp  −i t     

with

(32.83b)

J

r =1

N

N

(32.85a)

)

Hˆ =

= N

   αˆ J  ω − ω N 

0

where the effect from the phonon fields is involved in the diagonal elements ωi. Denoting an orthonormal matrix to diagonalize the Hamiltonian (32.85a) as Q and the rth eigenvalue as Er, we have

with

ωi =

J ω − ω2

(

(32.83a)

i

(32.84)

+ αˆ i†+1αˆ i ,

ˆ α ≡ αˆ 1† , αˆ †2 ,…, αˆ †N ,

Let us rewrite the third term of the Hamiltonian (32.78a) with the mean field approximation as

)

i +1

with

Time (units of (k/m)–1/2)

Figure 32.4â•… Temporal evolution of a specific phonon mode. The solid curve represents the probability for a localized phonon mode, while the dashed curve shows that of the lowest delocalized phonon mode.

† i

i =1

 ω − ω1  J ˆ Hˆ = α †    0

0.03

∑ J (αˆ αˆ

N

∑ ∑ Q Q Aˆ Aˆ exp{i(E − E )t }. ir

† r

is

r

s

s

(32.87)

r =1 s =1

ˆ (t) is The expectation value of the photon number operator N i then given by

N i (t ) j ≡ ψ j Nˆ i(t ) ψ j =

N

N

r =1

s =1

∑∑Q Q Q Q ir

jr

is

js

cos {(Er − Es )t }, (32.88)

32-15


in terms of one photon state at site j defined by N

ψ j = αˆ †j 0 =

∑Q r =1

jr

Aˆ r† 0 .

(32.89)

ˆ commutes with the Since the photon number operator N i Hamiltonian (32.84), the total photon number is conserved, which means that a polariton, called a photon in this section, conserves the total particle number within the lifetime. Moreover, 〈Ni(t)〉j can be regarded as the observation probability of a photon at an arbitrary site i and time t, initially populated at site j. This function is analytically expressed in terms of the Bessel functions Jj±i:

{

}

2

N i (t ) j = J j −i (2 Jt ) − (−1)i J j + i (2 Jt ) ,

(32.90)

when there are no photon–phonon interactions (ωi = 0) and the total site number N becomes infinite. Here the argument J is the photon hopping constant, and (32.90) shows that a photon initially populated at site j delocalizes to the whole system. Focusing on the localized phonon modes, we take the summation in (32.83b) over the localized modes only, which means that an earlier stage is considered after the incident photon excites the phonon modes, or that the duration of the localized phonon modes that are dominant over the delocalized modes is focused on (see Figure 32.4). This kind of analysis provides us with an interesting insight into the photon–phonon coupling constant and the photon hopping constant, which is necessary for understanding the mechanism of photon localization. The temporal evolution of the observation probability of a photon at each site is shown in Figure 32.5. Without the photon– phonon coupling (χ = 0), a photon spreads over the whole system as a result of the photon hopping, as shown in Figure 32.5a.

χ~N

32.3.4.4.2 Contribution from the Off-Diagonal Part

Probability

0.8

/J)

20

0

5

15 10

of 1

8 10 Site n 12 14 umb 16 er 18

0.4

0.0

un

6

e(

2 4

Ti m

0.0

0.6

0.2

35 30 25 20 its

Probability

0.8

0.4

(32.91)

In the previous subsection, we have approximated J as a constant independent of the sites, in order to examine the photon’s spatial distribution as well as the mechanism of the photon localization. Now let us treat the photon hopping operator ˆJi more rigorously, and investigate the site dependence of the off-diagonal contribution, which includes the inhomogeneity of the phonon fields. Noticing that a quasiparticle transformed from a photon

1.0

0.6

kJ ,

where the localization width seems very narrow.

1.0

0.2

(a)

Here the photon energy ħω = 1.81â•›eV and the hopping constant ħJ = 0.5â•›eV are used in the calculation. Impurities are assumed to be doped at sites 3, 7, 11, 15, and 19, while the total number of sites N is 20, and the mass ratio of the host molecules to the impurities is 5. Figure 32.5b shows a result with χ = 1.4 × 103 fs−1 nm−1; the other parameters used are the same as those in Figure 32.5a. It follows from the figure that a photon moves from one impurity site to another impurity site instead of delocalizing to the whole system. As the photon–phonon coupling constant becomes much larger than χ = 1.4 × 103 fs−1 nm−1, a photon cannot move from the initial impurity site to others, but stays there. The effect due to the photon–phonon coupling constant χ is expressed by the diagonal component in the Hamiltonian, while the off-diagonal component involves the photon hopping effect due to the hopping constant J. The above results indicate that a photon’s spatial distribution depends on the competition between the diagonal and off-diagonal components in the Hamiltonian, i.e., χ and J, and that a photon can move among impurity sites and localize at those sites when both components are comparable under the condition

(b)

2 4

180 160 140 120 /J) 100 of 1 80 6 8 ts 60 uni 10 ( 12 Site n 40 e 14 umb 20 im 16 er T 18 20 0

Figure 32.5â•… (See color insert following page 20-14.) Probability that a photon is found at each site as a function of time (a) without the photon– phonon coupling, and (b) with the photon–phonon coupling comparable to the photon hopping constant.

32-16


J i ≡ 〈 γJîγ 〉.

(32.92)

Here, the coherent state |γ〉 is an eigenstate of the annihilation operator bˆ p with eigenvalue γp and satisfies bˆ p γ = γ p γ .

(32.93)

Since the difference between the creation and annihilation operators of a phonon is invariant under the Davydov transformation, the following relation holds: βˆ − βˆ p = bˆ − bˆ p . † p

† p

0.16 Hopping constants (eV)

operator by the Davydov transformation is associated with phonons in the coherent state (see (32.77a) and (32.77b)), we take expectation values of ˆJi in terms of the coherent state of phonons |γ〉 as

(32.94)

Using (32.78b), (32.93), and (32.94), we can rewrite the sitedependent hopping constant Ji in (32.92) as

 1 = J exp  −  2

6

8

10

12

14

16

18

Site number

Figure 32.6â•… Site dependence of the hopping constant Ji in the case of N = 20. Impurities are doped at sites 4, 6, 13, and 19. The mass ratio of the host molecules to the impurities is 1–0.2, whereas ħJ = 0.5â•›eV and χ = 40.0â•›fs−1 nm−1 are used.

Probability

∑



∑ C  , 2 ip

4

0.4

∑

N

2

0.5

  N   N  Cip ′bˆ †p ′  exp  − Cip ′′bˆ p ′′  γ Cip2  γ exp    p ′ =1   p ′′ =1  p =1

∑

0.10

0

∑ N

0.12

0.08

 N    J i = J γ exp  Cip (b†p − bp ) γ  p =1   1 = J exp  −  2

0.14

(32.95a)

p =1

0.3 0.2 0.1

0.0

where Cip is denoted by

0

Cip ≡

χip − χi +1 p . Ωp

(32.95b)

Figure 32.6 shows the site dependence of Ji in the case of N = 20. Impurities are doped at sites 4, 6, 13, and 19. The mass ratio of the host molecules to the impurities is 5, while ħJ = 0.5â•›eV and χ = 14.0â•›fs−1 nm−1 are used. It follows from the figure that the hopping constants are highly modified around the impurity sites and the edge sites. The result implies that photons are strongly affected by localized phonons and hop to the impurity sites to localize. Here we have not considered the temperature dependence of Ji, which is important for phenomena dominated by incoherent phonons. This is because coherent phonons weakly depend on the temperature of the system. However, there remains room to discuss a more fundamental issue, i.e., whether the probe system is in a thermal equilibrium state or not. In Figure 32.7, we present a typical result that photons localize around the impurity sites in the system as the photon–phonon

5

10 Site number

15

20

Figure 32.7â•… Probability of photons observed at each site. The filled squares, circles, and triangles represent the results for χ = 0, 40.0, and 54.0â•›f s−1 nm−1, respectively. Other parameters are the same as Figure 32.6.

coupling constants χ vary from 0 to 40.0â•›fs−1 nm−1 or 54.0â•›fs−1 nm−1, while keeping ħJ = 0.5â•›eV. As depicted by the filled squares in the figure, photons delocalize and spread over the system without the photon–phonon couplings. When the photon–Â� phonon couplings are comparable to the hopping constants, χ = 40.0â•›fs−1 nm−1, photons can localize around the impurity site with a finite width and two sites at half width and half maximum (HWHM), as shown by the filled circles. This finite width of photon localization comes from the site-dependent hopping constants. As the photon–phonon couplings are larger than χ = 40.0â•›fs−1 nm−1, photons can localize at the edge sites with a finite width, as well as the impurity sites. In Figure 32.7, the photon localization at the edge site, shown by the filled triangles,

32-17


originates from the finite size effect of the molecular chain. This kind of localization of photons dressed by the coherent state of phonons leads us to a simple understanding of phonon-assisted photodissociation using an optical near field: molecules in the electronic ground state approach the probe tip within the localization range of the dressed photons, and can be vibrationally excited by the dressed-photon transfer to the molecules, via a multi-phonon component of the dressed photons, which might be followed by electronic excitation. Thus, it leads to the dissociation of the molecules even if an incident photon energy less than the dissociation energy is used. As a natural extension of the localized photon model, we have discussed the inclusion of the phonon’s effects into the model. The study was initially motivated by experiments on the photodissociation of molecules by optical near fields, whose results show unique features different from the conventional one with far fields (Ohtsu et al. 2008). After clarifying whether the vibration modes in a pseudo one-dimensional system are delocalized or localized, we focused on the interaction between dressed photons and phonons by using the Davydov transformation. We have theoretically shown that photons are dressed by the coherent state of phonons, and found that the competition between the photon–phonon coupling constant and the photon hopping constant governs the photon localization or delocalization in space. The obtained results lead us to a simple understanding of an optical near field itself as an interacting system of photon, electronic excitation (induced polarization), and phonon fields in a nanometer space, which are surrounded by macroscopic environments, as well as phonon-assisted photodissociation using an optical near field.

32.4â•‡Summary and Future Perspective We have briefly outlined the difficulties in defining the positionrepresentation wave function of a photon, followed by several trials to overcome these issues. On the basis of canonical quantization of the electromagnetic fields in free space, we have discussed a dressing mechanism and spatial localization of photons, which is a natural viewpoint from light–matter interactions, or from virtual photon clouds and field correlations. Finally, with the projection operator method, we have shown an effective interaction between nanomaterials electronically disconnected, but closely separated, which are also surrounded by a macroscopic system, in order to detect the virtual clouds. We have discussed the photon dressing by material excitation and pointed out the importance of the phonon’s role for spatial localization of photons at the nanoscale. The pace of development in photonics has accelerated, but most of the underlying science of the field is still Maxwell’s classical electromagnetism, not field-quantized photons. In the near future, however, we are anticipating new breakthroughs in nano- and atom photonics, where the localized and quantized nature of photons, as well as an exact quantization formulation for an optical near-field system, including relaxation processes at the nanoscale, will play a critical role.

Acknowledgments The author is grateful to M. Ohtsu (University of Tokyo) and H. Hori (University of Yamanashi) for stimulating discussions and valuable comments from the early stage of this study. He is also thankful to S. Sangu (Ricoh Co. Ltd.), A. Shojiguchi (NEC Co.), Y. Tanaka (Tokyo Institute of Technology), and A. Sato (Tokyo Institute of Technology) for discussions and collaborations. He greatly acknowledges the valuable guidance given by M. Tsukada (University of Tokyo, emeritus, Tohoku University), K. Kitahara (Tokyo Institute of Technology, emeritus, International Christian University), M. Kitano (Kyoto University), Y. Masumoto (University of Tsukuba), K. Cho (Osaka University, emeritus), and T. Yabuzaki (Kyoto University, emeritus). Finally but not the least, he expresses his gratitude to H. Ito (Tokyo Inst. Technology). T. Kawazoe (University of Tokyo), T. Yatsui (University of Tokyo), T. Saiki (Keio University), K. Matsuda (Kyoto University), H. Nejo (National Institute of Materials Science), M. Naruse (National Institute of Information and Communication Technology), M. Ikezawa (University of Tsukuba), I. Banno (University of Yamanashi), and H. Ishihara (Osaka Prefecture University).

References Adlard, C., Pike, E. R., and Sarkar, S. 1997. Localization of onephoton states. Phys. Rev. Lett. 79: 1585–1587. Bialynicki-Birula, I. 1998. Exponential localization of photons. Phys. Rev. Lett. 80: 5247–5250. Carniglia, C. K. and Mandel, L. 1971. Quantization of evanescent electromagnetic waves. Phys. Rev. D 3: 280–296. Cohen-Tannoudji, C., Dupont-Roc, I., and Grynberg, G. 1989. Photons and Atoms. New York: Wiley. Cohen-Tannoudji, C., Dupont-Roc, I., and Grynberg, G. 1992. Atom-Photon Interactions: Basic Processes and Applications. New York: Wiley. Compagno, G., Passante, R., and Persico, F. 1988. Dressed and half-dressed neutral sources in nonrelativistic QED. Phys. Scr. T21: 33–39. Compagno, G., Passante, R., and Persico, F. 1995. Atom-Field Interactions and Dressed Atoms. Cambridge, NY: Cambridge University Press. Craig, D. P. and Thirunamachandran, T. 1998. Molecular Quantum Electrodynamics. New York: Dover Pub. Hawton, M. 1999. Photon wave functions in a localized coordinate space basis. Phys. Rev. A 59: 3223–3227. Hegerfeldt, G. 1974. Remarks on causality and particle localization. Phys. Rev. D 10: 3320–3321. Hellwarth, R. W. and Nouchi, P. 1996. Focused one-cycle electromagnetic pulses. Phys. Rev. E 54: 889–895. Inagaki, T. 1998. Physical meaning of the photon wave function. Phys. Rev. A 57: 2204–2207. Inoue, T. and Hori, H. 2001. Quantization of evanescent electromagnetic waves based on detector modes. Phys. Rev. A 63: 063805(1)–063805(16).

32-18


John, S. and Quang, T. 1995. Photon-hopping conduction and collectively induced transparency in a photonic band gap. Phys. Rev. A 52: 4083–4088. Keller, O. 2005. On the theory of spatial localization of photons. Phys. Rep. 411: 1–232. Kobayashi, K., Sangu, S., Ito, H. et al. 2001. Near-field optical potential for a neutral atom. Phys. Rev. A 63: 013806(1)–063806(9). Kobayashi, K., Tanaka, Y., and Kawazoe, T. et al. 2008. Localized photon model including phonon’s degrees of freedom. In Progress in Nano-Electro-Optics, vol. VI, ed. M. Ohtsu, pp. 41–66. Berlin, Germany: Springer. Mandel, L. 1966. Configuration-space photon number operators in quantum optics. Phys. Rev. 144: 1071–1077. Mandel, L. and Wolf, E. 1995. Optical Coherence and Quantum Optics. Cambridge, NY: Cambridge University Press. Newton, T. D. and Wigner, E. P. 1949. Localized states for elementary systems. Rev. Mod. Phys. 21: 400–406. Ohtsu, M. and Hori, H. 1999. Near-Field Nano-Optics. New York: Kluwer Academic/Plenum. Ohtsu, M., Kobayashi, K., Kawazoe, T. et al. 2008. Principles of Nanophotonics. Boca Raton, FL: CRC Press. Pike, E. R. and Sarkar, S. 1987. Spatial dependence of weakly localized single-photon wave packets. Phys. Rev. A 35: 926–928.

Pohl, D. W. and Courjon, D. 1993. Near Field Optics. Dordrecht, the Netherlands: Kluwer Academic. Power, E. A. and Thirunamachandran, T. 1993. Quantum electrodynamics with nonrelativistic sources. V. ElectroÂ� magnetic field correlations and intermolecular interactions between molecules in either ground or excited states. Phys. Rev. A 47: 2539–2551. Roychoudhuri, C. and Roy, R. eds. 2003. The nature of light: What is a photon? Opn. Trends 3: S1–S35. Sakurai, J. J. 1967. Advanced Quantum Mechanics. Reading, MA: Addison-Wesley. Sangu, S., Kobayashi, K., Shojiguchi, A. et al. 2004. Logic and functional operations using a near-field optically coupled quantum-dot system. Phys. Rev. B 47: 115334(1)–115334(13). Scully, M. O. and Zubairy, M. S. 1997. Quantum Optics. Cambridge, U.K.: Cambridge University Press. Shojiguchi, A., Kobayashi, K., Sangu, S. et al. 2003. Superradiance and dipole ordering of an N two-level system interacting with optical near fields. J. Phys. Soc. Jpn. 72: 2984–3001. Sipe, J. E. 1995. Photon wave functions. Phys. Rev. A 52: 1875–1883. Suzuura, H., Tsujikawa, T., and Tokihiro, T. 1996. Quantum theory for exciton polaritons in a real-space representation. Phys. Rev. B 53: 1294–1301. Wightman, A. S. 1962. On the localizability of quantum mechanical systems. Rev. Mod. Phys. 34: 845–872.

33 Operations in Nanophotonics 33.1 Introduction............................................................................................................................33-1 33.2 Dissipation-Controlled Nanophotonic Devices................................................................33-2 Basic Principlesâ•‡ •â•‡ Nanophotonic Switchâ•‡ •â•‡ Carrier Manipulation (Up-Converter)

Suguru Sangu Ricoh Company, Ltd.

Kiyoshi Kobayashi University of Yamanashi Japan Science and Technology

33.3 Nanophotonic Devices Using Spatial Symmetries............................................................33-8 Basic Principlesâ•‡ •â•‡ AND- and XOR-Logic Gatesâ•‡ •â•‡ Nanophotonic Buffer Memoryâ•‡ •â•‡ Manipulation of Quantum-Entangled States

33.4 Summary................................................................................................................................33-13 Acknowledgments............................................................................................................................33-13 Referencesï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½33-13

33.1â•‡ Introduction Recently, a vast number of studies on nanophotonics have been published in the field of nanoscale optical measurement and bioimaging with super-resolution (Ohtsu 1998, Maheswari et al. 1999, Hosaka and Saiki 2001, Matsuda et al. 2003) and information processing technologies (Biolatti et al. 2000, Rinaldis et al. 2002, Troiani et al. 2002), owing to the unique characteristics of optical near field, which far exceed the technical limitations of conventional optics (Ohtsu et al. 2008). Especially, nanophotonics has shown promise and is expected to be the technology for next-generation nanoscale devices dealing with large amounts of information resources and low energy consumption (Ohtsu et al. 2002). In such devices, optical near field plays an important role in nanophotonic device operations, since the optical near field is a mixed state between photon and matter excitation, not only breaking the diffraction limit that is dependent on the wave nature of light, but also utilizing interesting characteristics inherent in nanophotonics, such as unidirectional energy transfer, optical forbidden transition (Kawazoe et al. 2002), and operations based on coherence of nanometric matters (Sangu et al. 2004), which have not been used in conventional optical devices. Nanophotonic device is interpreted as a system that consists of several matter systems, localized photon field (optical near field) for driving carriers in the systems, and free photon (radiation) field for extracting some information, as illustrated in Figure 33.1. The important point of nanophotonic system is that the spatial distribution of photons is localized in a nanometric space rather than the matter itself being nanometer-sized. From this point, valuable device operations inconsiderable in conventional optical devices are expected. Relating to a theoretical viewpoint, some restrictions under the long

wavelength approximation is allowed in a nanophotonic device system (Cho 2004). Moreover, quantum nature appears in carrier dynamics. In order to control the carriers, quantum systems with discrete energies, such as quantum dots and molecules, are adapted as matter systems because the processes with matter coherence and with energy dissipation are clearly distinguishable. And also, excitation energies and dissipation dynamics are readily determined by adjusting the sizes of quantum objects. In addition, optical allowed and forbidden transitions for external far-field light are usable to prepare initial excitations. In this chapter, a quantum-dot system is assumed as a matter system of nanophotonic device, and thus the signal carriers correspond to excitons, that is, electron–hole pairs, in the quantum dots. Although such systems resemble a quantum computation device, let us note that a nanophotonic device need not keep quantum coherence in the entire system, and is divided into several quantum coherent parts via dissipative process. In this chapter, operation principles of typical nanophotonic operations are covered by describing exciton population dynamics theoretically and numerically on the basis of the density matrix formalism (Walls and Milburn 1994, Carmichael 1999, Breuer and Petruccione 2002). Basic formulations for energy transfer, dissipation, and exciton excitation are explained, and then, nanophotonic switch and carrier up-converter are numerically demonstrated in Section 33.2. In Section 33.3, nanophotonic device operations, such as logic gates and memory, are described, in which system coherence and spatial symmetries play important roles. As a summary, design concepts for nanophotonic device systems are discussed in Section 33.4.

33-1

33-2

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Near/far-field input

Free photon field Localized photon field Near/far-field input

Guide line

Far-field output

NF-FF converter

Nanometric matters

Figure 33.1â•… Schematic representation of a nanophotonic device system, in which carriers are transferred via localized photon field with interaction length in the order of a matter size.

33.2â•‡Dissipation-Controlled Nanophotonic Devices

2a

a

In this section, nanophotonic devices, where signal carrier in a single input terminal transfers to that in an output one, are formulated and evaluated numerically. Dynamics of signal transfer and control in a nanophotonic device can be described by density matrix formalism. Since quantum mechanical theory expresses an energy-conservative system, dissipation process should be introduced by some sort of approximation because a functional device operation needs to guarantee unidirectional signal transfer. After explanation of basic principles of exciton energy transfer and creation of excitation in a quantum-dot system with dissipation process, some concrete functional operations are explained in the later part of this section.

Ω2

Ω2 ˆ† A

ˆ A

ˆ †2 B

Ω1

ˆ2 B ˆ †1 B

QD-A

ˆ1 B

QD-B

33.2.1.1â•‡Energy Transfer between Two Quantum Dots

Figure 33.2â•… Energy diagram of a two-quantum-dot system, where the sizes of quantum dots are determined to satisfy the resonant condition, and are set as a (QD-A) and 2a (QD-B). There are exciton sublevels in QD-B because of unidirectional energy transfer. Intra-sublevel relaxation originates from coupling between excitons and phonon reservoir.

As the simplest example of carrier dynamics, exciton energy transfer between two quantum dots with radiative and nonradiative relaxations is described by using density matrix formalism (Carmichael 1999, Sangu et al. 2003). Figure 33.2 denotes energy diagram for a two-quantum-dot system, where exciton ground state in QD-A and first exciton excited state in QD-B resonantly couple with each other. This resonant condition can be realized by adjusting the size of quantum dots. For example, in the case of a cubic quantum dot, such as CuCl quantum dot (Masumoto 2002, Kawazoe et al. 2003), the energy with the quantum numbers (nx, ny, nz) is given as

where M and Eg denote the exciton mass and bulk exciton energy, respectively. Obviously, when the sizes of QD-A and QD-B are set as a and 2a, respectively, the (1, 1, 1)-level in QD-A and (2, 1, 1), (1, 2, 1), and (1, 1, 2)-levels in QD-B are the same energy and, thus, they couple resonantly. In the following, the energy levels for three quantum numbers are dealt with collectively and labeled as ħΩ1- and ħΩ2-levels, since degeneracy of several energy levels is not essential. Energy state of the system is described by the following non-perturbative and interaction Hamiltonians:

33.2.1â•‡ Basic Principles

π 2 2 2 E(nx ,ny ,nz ) = nx + n2y + nz2 + Eg , 2 Ma2

(

)

(nx , ny , nz = 1, 2, 3,…),

(33.1)

) ( ˆˆ , = U ( Aˆ Bˆ + AB )

Hˆ 0 = Ω2 Aˆ † Aˆ + Bˆ 2† Bˆ 2 + Ω1Bˆ1† Bˆ1 ,

Hˆ int

†

2

† 2

(33.2)

33-3

Operations in Nanophotonics

where Ω1 and Ω2 are eigenfrequencies of QD-A and QD-B U represents the optical near-field coupling strength, and creation and annihilation operators for excitons are depicted in Figure 33.2 The interaction Hamiltonian Hînt is well known as the Förstertype interaction (Förster 1965), which is often used by describing intermolecular and inter-quantum-dot interaction. Although the actual interaction between QD-A and QD-B occurs via intermediate virtual states of exciton-polaritons (Kobayashi et al. 2000), which are the coupled state of excitons and photons, such exciton-polariton degree of freedom is neglected in Equation 33.2 as is traced out and renormalized in the coupling strength U. Exciton dynamics can be expressed by using density operator ρˆ (t ), which is a projection operator expanding a certain energy state to appropriate bases depending on possible exciton states in a quantum-dot system. There are a three-exciton state, three twoexciton states, three one-exciton states, and a vacuum state in a two-quantum-dot system, as illustrated in Figure 33.3. Equation of motion of the density operator is given by the Liouville equation in the case without dissipation (Breuer and Petruccione 2002). Dynamics with dissipation is often expressed by so-called Lindblad type notation on the bases of the first-order Born approximation (Carmicheal 1999), in which free photon and phonon fields are considered as energy reservoirs as follows:

dρˆ (t ) 1  ˆ = H 0 + Hˆ int , ρˆ (t ) + Dphotonρˆ (t ) + Dphononρˆ (t ), (33.3)  dt i 

where photon (phonon) is a superoperator defined as Dphotonρˆ (t ) =

∑

α = A, B1

Dphononρˆ (t ) =

γα ˆ ˆ ˆ † ˆ † ˆ ˆ 2 αρ(t )α − α αρ(t ) − ρˆ (t )αˆ † αˆ , 2 

Γ (n + 1) 2 × 2 Bˆ 2 Bˆ1†ρˆ (t )Bˆ 2† Bˆ1 − Bˆ 2† Bˆ1Bˆ 2 Bˆ1†ρˆ (t ) − ρˆ (t )Bˆ 2† Bˆ1Bˆ 2 Bˆ1†  +

Γ  ˆ† ˆ ˆ ˆ ˆ† ˆ ˆ† ˆ† ˆ ˆ n 2B2 B1ρ(t )B2 B1 − B2 B1 B2 B1ρ(t ) − ρˆ (t )Bˆ 2 Bˆ1† Bˆ 2† Bˆ1  ,  2  (33.4)

n is the number of phonons in the reservoir, and the number of photons is assumed as zero because of external field being a vacuum state. As you readily understand from the bottom of Equation 33.4, energy transfer between the excited and ground states of an exciton occurs by mediating the intra-sublevel transition with non-radiative relaxation constant Γ, which guarantees unidirectional energy transfer in the two-quantum-dot system. Since analytical solution of Equation 33.3 is complex and the physical meaning is unclear, the following is discussed on the basis of numerical calculations. For far-field light excitation, excitons in a two-quantum-dot system are created via mixed states of several bases, which are shown in Figure 33.3, because the two quantum dots cannot distinguish each other due to the diffraction limit of light. On the other hand, high spatial resolution of optical near field permits direct access for individual quantum dots. For example, an optical fiber probe possesses sub-100â•›nm spatial resolution (Hosaka and Saiki 2001,

e

A

e,e

B

Three-exciton state

e

A

e,g

B

Two-exciton state

e

A

g,e

B

g

A

e,e

B

e

A

g,g

B

g

A

e,g

B

g

A

g,e

B

g

A

g,g

B

One-exciton state

Vacuum state

Figure 33.3â•… Appropriate base states in which excitation and ground states for an isolated quantum dot are used. There are eight bases, which include a three-exciton state, three two-exciton states, three one-exciton states, and a vacuum state.

33-4


coupling strength are applied as U−1 = 10â•›ps (Figures 33.4a and 33.5a) and 50â•›ps (Figures 33.4b and 33.5b). For Figure 33.4a, oscillating behavior still remains because the sublevel relaxation constant Γ is the same order of the coupling strength U. While in the case of Γ > U, coherence in the upper energy levels quickly reduces, and the exciton energy that is stored in the lowest energy level in the system is released as radiative relaxation. The exciton population given in Figure 33.5 is proportional to photoluminescence intensity from the coupled quantum-dot system. Key point to design of a nanophotonic device is to determine energy transfer paths by using resonant coupling among plural energy states, and internal and external dissipation processes. Density matrix formalism discussed above is a useful tool to solve signal dynamics in an exciton–photon coupled or polariton-mediated system. In addition, multi-exciton process is important for functional operations, such as nanophotonic switch, which are given in Sections 33.2.2 and 33.2.3.

1.0

1.0

0.8

0.8 Population

Population

Matsuda et al. 2003) and plasmonic excitation in metallic nanostructures are utilized as an interface in a nanophotonic device (Yatsui et al. 2001, Nomura et al. 2005). As an initial condition to demonstrate energy transfer between the two-quantum-dot systems, an exciton in QD-A is locally prepared that is expressed by density matrix elements of ρin (t ) = A e B g , g ρˆ e A g , g B = 1 and otherwise being zero. Output population is given by the density matrix element of ρout (t ) = A g B g , e ρˆ g A g , e B , which is the lower-level state excited in QD-B, and (|â•›g, e〉B)† = B〈â•›g, e|. In this case, two- and three-exciton bases decouple from the output population dynamics, and it is enough to consider only four base states for calculating the dynamics. In the following, numerical solutions in Equation 33.3 are explained, in which two types of typical dynamics appear depending on the optical near-field coupling strength. Figures 33.4 and 33.5 show the exciton population in the input (QD-A) and output (QD-B) energy levels, respectively, when the optical near-field

0.6 0.4

0.6 0.4

0.2

0.2

0.0

0.0 0

(a)

100

200 300 Time (ps)

400

500

0

(b)

200

400 600 Time (ps)

800

1000

(a)

1.0

1.0

0.8

0.8 Population

Population

Figure 33.4â•… Temporal evolution of exciton population in the ħΩ2-level in QD-A (input). The optical near-field coupling strength are (a) U−1 = 10â•›ps and (b) 50â•›ps, and the other parameters are set as ħΩ2 = 3.22â•›eV (M = 2.3me, Eg = 3.20â•›eV, a = 5â•›nm or 7.07â•›nm), ħΩ1 = 3.21â•›eV (a = 7.07â•›nm), γA−1 = 1â•›ns, γB−1 = 0.5â•›ns, and Γ−1 = 10â•›ps, where a CuCl quantum cube is assumed as a coupled two-quantum-dot system. For simplicity, zero temperature (n = 0) is assumed.

0.6 0.4

0.6 0.4

0.2

0.2

0.0

0.0 0

100

200 300 Time (ps)

400

500

(b)

0

200

400 600 Time (ps)

800

1000

Figure 33.5â•… Temporal evolution of exciton population in the ħΩ1-level in QD-B (output). The optical near-field coupling strength are (a) U−1 = 10â•›ps and (b) 50â•›ps, and the other parameters are the same as in Figure 33.4.

33-5


33.2.1.2â•‡ Creation of Excitation by an External Field Before explanation of multi-exciton process, a way to describe the creation of signal carriers in a quantum-dot system is discussed by means of the density matrix formalism. There are two types of carrier excitations: resonant and nonresonant. In the resonant excitation, energy of external photon field (laser pulse) corresponds to that of the exciton state (Carmicheal 1999), and the equation of motion, which is given in Equation 33.3, is modified with additional contribution of Hˆr as dρˆ (t ) 1  ˆ = H 0 + Hˆ int + Hˆ r , ρˆ (t ) + Dphotonρˆ (t ) + Dphononρˆ (t ),  dt i 

(

)

(

)

Hˆ r = −g Bˆ 1† aˆ + aˆ † Bˆ 1 ≈ −V (t ) Bˆ 1† + Bˆ 1 ,

(33.5)

where â and â† represent annihilation and creation operators of external photon field, and on the basis of the semi-classical approximation, aˆ = aˆ † ≡ V (t ) g . It means that the annihilation and creation operators are renormalized in the parameter (Rabi frequency) V(t). Equation 33.5 expresses the coherent process, which is well known as π-pulse excitation. While in the case of the nonresonant excitation, coherence between an exciton and a photon is already lost because of interaction among a large number of energy states. This situation appears in such a case that an exciton creates mediated by continuous energy levels and surrounding barrier levels. The formulation of exciton excitation is equivalent to exciton creation from photon reservoir with finite temperature. According to Equation 33.4, the following term is added to Equation 33.3 for nonresonant excitation: Pphotonρˆ (t ) =

Vα (t ) ˆ † ˆ ˆ ˆ ˆ † ˆ ˆ ˆ † , 2α ρ(t )α − αα ρ(t ) − ρˆ (t )αα  2 α = A, B1 (33.6)

∑

(a)

33.2.2â•‡ Nanophotonic Switch General switching device consists of three terminals, which correspond to input, output, and control. Similarly, a threequantum-dot system can operate as a nanophotonics switch,

1.0

1.0

0.8

0.8

0.6

0.6

Population

Population

where Vα(t) is pumping rate in the optically allowed state in QD-α.

The above formulations are useful for making initial excited states in a quantum-dot system as well as for applying control signal to select energy transfer paths. In the following, exciton population dynamics for a two-quantum-dot system (Figure 33.2), which is driven by the external photon field, is explained by using numerical results of Equations 33.5 and 33.6. When the bases are restricted less than the one-exciton state, there are four states in this system (See Figure 33.3). Temporal evolution of exciton population in the ħΩ2-levels in QD-A and ħΩ1-level in QD-B are plotted in Figure 33.6a and b, respectively, which is in the case of resonant excitation. The results for nonresonant excitation are also given in Figure 33.7a and b. In both cases, the time width of pumping is set as 50â•›ps, which is denoted by the vertical dashed lines in Figures 33.6a and 33.7a. Temporal evolutions for resonant and nonresonant excitation are similar with each other except for those in the period when an external field is applied. Since the resonant oscillation keeps matter coherence, the population once increases and then decreases beyond π-pulse excitation. Figures 33.7b and 33.8b represent the population in QD-B, and the temporal profiles are similar in both cases, where the decay profiles are determined by the radiative relaxation constant in the lower energy level in QD-B, but total populations (the area of population curves) reflect the population for the early stage in QD-A. Resonant and nonresonant exciton creations are important for projecting input signal to a quantum-dot system as well as for controlling signal carrier transiently in a nanophotonics device. Due to the above formulations for energy transfer, dissipation, and creation of excitation, some functional device operations can be demonstrated numerically, which are discussed in Sections 33.2.2 and 33.2.3.

0.4

0.4

0.2

0.2

0.0

0.0 0

50

100 150 Time (ps)

200

250

(b)

0

500

1000 Time (ps)

1500

2000

Figure 33.6â•… Temporal evolution of exciton population in (a) ħΩ2-level in QD-A (input terminal) and (b) ħΩ1-level in QD-B (output terminal), where ħΩ2-level in QD-A is resonantly excited. The optical near-field coupling strength is U−1 = 50â•›ps, and the other parameters are the same as in Figure 33.4. The solid, dashed, dotted, and dot-dash, dot-dot-dash curves represent the pulse area of 0.5π, 0.75π, 1.0π, 1.25π, and 1.5π, respectively.

33-6

(a)

1.0

1.0

0.8

0.8

0.6

0.6

Population

Population


0.4

0.4

0.2

0.2

0.0

0.0 0

50

100 150 Time (ps)

200

250

0

(b)

500

1000 Time (ps)

1500

2000

Figure 33.7â•… Temporal evolution of exciton population in (a) ħΩ2-level in QD-A (input port) and (b) ħΩ1-level in QD-B (output port), where ħΩ2-level in QD-A is nonresonantly excited. The parameters are the same as in Figure 33.6. The solid, dashed, dotted, dot-dash, and dot-dot-dash curves represent the pulse area of 0.5π, 0.75·π, 1.0π, 1.25π, and 1.5π, respectively. 2a

2a

a

Input Output Control Ω3

Ω3

ˆ† A

ˆ A

ˆ †3 C

ˆ3 C ˆ †2 C

ˆ2 C ˆ †1 C

QD-A

QD-C

Ω2 Ω1 ˆ1 C

Ω3 Ω2 ˆ †2 B

ˆ2 B

ˆ †1 B

ˆ1 B

QD-B

Figure 33.8â•… Energy diagram of a nanophotonic switch that consists of three quantum dots with the size ratio A : B : C = 1 : 2 : 2. Input and control signals are injected in the ħΩ3-level in QD-A and the ħΩ1-level in QD-C, respectively, and output signal is detected from the ħΩ2-level in QD-B. By mediating intra-sublevel relaxations, unidirectional energy transfer is guaranteed in this system.

where three quantum dots correspond to input, output, and control terminals (Kawazoe et al. 2003). In Figure 33.8, energy diagram of the three-quantum-dot system is illustrated where the two resonant paths are labeled ħΩ2 and ħΩ3. The resonance conditions can be realized by choosing the quantum-dot sizes as A : B : C = 1 : 2 : 2 (see Equation 33.1), where the interaction between QD-A and QD-B is not considered because interdot distance between them is assumed large enough for neglecting the interaction. The principle of the switching operation is as follows: when the control signal is injected in the ħΩ1-level in QD-C, the dissipation path toward the ħΩ1-level is blocked because of Fermionic feature of excitons in a system with discrete energies, an initial exciton in QD-A transfers to the second stable state of ħΩ2-levels in QD-B, and annihilates an exciton with a radiative photon. What is important for this type of functional operation,

in which dissipation paths are selectively determined, is to deal with a multi-exciton dynamics in a system. Equation of motion of the nanophotonic switch can be written in accordance with the manner presented in Section 33.2.1, where the considered bases are extended in two-exciton states, and resonant exciton excitations are applied as an input signal in QD-A and a control signal in QD-C. Since output luminescent intensity is proportional to the ħΩ2-level population in QD-B, the population dynamics in QD-B is discussed in the following. Figure 33.9 represents the numerical result of one and twoexciton population dynamics. The dashed curve shows the output population in QD-B, when rectangular optical pulse (π-pulse) with the duration of 10â•›ps is injected in the input terminal of QD-A. The exciton created in QD-A transfers to QD-C

33-7

0.25

0.25

0.20

0.20

0.15

0.15

Population

Population


0.10

0.10

0.05

0.05

0.00

0.00 0

500

1000

1500 Time (ps)

2000

2500

3000

Figure 33.9â•… Temporal evolution of exciton population in the ħΩ2level in QD-B (output terminal). The dashed and solid curves represent the cases where a single rectangular optical pulse is injected in QD-A (input terminal) and rectangular pulses are injected in the ħΩ3-level in QD-A (input terminal) and the ħΩ1-level in QD-C (control terminal). The parameters are set as follows: optical near-field coupling strength −1 −1 U AC = U BC = 50 ps, pulse duration Δt = 10â•›ps (π-pulse amplitude), radia−3 2 tive relaxation time 2 −3 γ A−1 = 2 γ B−1 = γ C−1 = 1000 ps, intra-sublevel −1 relaxation time Γ B,32 = 10 ps, and Γ B−1,21 = Γ C−1,21 = 20 ps .

via optical near-field coupling UAC and then relaxes to the lowest energy level in QD-C because of fast intra-sublevel relaxation. Therefore, output population in QD-B is detected as very small value (OFF-state). While, the solid curve represents the case that two optical pulses with 10â•›ps-pulse duration are injected simultaneously in the lowest energy levels in QD-A and QD-C. In the early stage of the dynamics, a somewhat large signal with oscillation appears (ON-state). This is the result of occupation in the ħΩ1-level in QD-C, which is the optical nutation between lower resonant energy levels in QD-B and QD-C. The rise time or switching time is estimated less than 100â•›ps, which depends on the optical near-field coupling strength (Sangu et al. 2003). The readers may see that the signal falling time from ONto OFF-state is quite slower than the rising time from OFF- to ON-state. This is surely constrained by the radiative relaxation time in the control port of QD-C. However, the falling time is controllable by injection of the third control pulse, because π-pulse excitation can remove the exciton occupied in the ħΩ1level in QD-C. Figure 33.10 represents the same result of exciton population dynamics shown in Figure 33.9, but additional rectangular optical pulse is injected with the delay of 200â•›ps in the control port of QD-C. The dashed curve represents timing of the control pulse. By injecting the third pulse, the optical nutation completely disappears and the dissipation path toward the lowest energy level in QD-C becomes effective. Nanophotonic switches and nanophotonic devices, in which dissipation paths are controlled by the state-filling property of excitons, can realize quite low energy operations. Ideally, the energy loss is determined by intra-sublevel relaxation, which is

0

500

1000

1500 Time (ps)

2000

2500

3000

Figure 33.10â•… Temporal evolution of exciton population in the ħΩ1level in QD-C (output port). The dashed curve represents the profile of the third rectangular optical pulse (π-pulse). The pumping pulse is injected with a delay of 200â•›ps and pulse duration of 10â•›ps. The other parameter values set are the same as in Figure 33.9.

in the order of a few tens of meV. This is a big advantage not only for miniaturizing photonic devices, but also for heat dissipation and wiring in nanometric devices.

33.2.3â•‡ Carrier Manipulation (Up-Converter) Unidirectional energy transfer and switching operation are guaranteed by considering the phonon reservoir system which is coupled with excitons. On the other hand, intense pulse excitation can create the finite phonons in the phonon reservoir system. This leads to blow back an exciton from lower to upper energy levels, instantaneously. Therefore, direction of energy transfer is also controllable by external optical pulse, which has been demonstrated experimentally in the recent publication (Yatsui et al. 2009). This up-conversion dynamics is formulated in a two-quantum-dot system, as shown in Figure 33.2. The intense optical pulse should be dealt with coherent interaction between excitons in the ħΩ1-level in QD-B and external photons. Therefore, the equation of motion described in Equation 33.5 is adopted. Since the atomic lattice in the reservoir system is transiently vibrated by the optical pulse, the phonon number of n(ω0 ) in Equation 33.4 is given as the following function: n(ω 0 ) =

1 , exp ω 0 / kBT (I in ) − 1

(33.7)

where ω0, T, and kB are the angular frequency of incident light, temperature of the system, and the Boltzmann constant, respectively. Equation 33.7 is the Bose–Einstein distribution function. For simplicity, the temperature is assumed to be proportional to the time integration of the optical pulse, Iin. The calculated results of the exciton population dynamics are

33-8

Handbook of Nanophysics: Nanoelectronics and Nanophotonics 0.20

1.0

0.15 Population

Population

0.8 0.6 0.4

0.10

0.05

0.2 0.0 (a)

0.00 0

1000

2000

3000 Time (ps)

4000

5000

6000

0

(b)

1000

2000

3000 Time (ps)

4000

5000

6000

Figure 33.11â•… Temporal evolution of exciton population in (a) the ħΩ1-level in QD-B and (b) the ħΩ2-level in QD-A, when the optical pulse is injected at the time 10â•›ps. The solid, dashed, and dotted curves represent the pulse area of 0.5π, 1.0π, and 1.5π, respectively. The parameters are set as follows: optical near-field coupling strength U−1 = 50â•›ps, pulse duration Δt = 10â•›ps (π-pulse amplitude), radiative relaxation time 2 −3 2 γ A−1 = γ B−1 = 1000 ps, and intra-sublevel relaxation time Γ = 10â•›ps in QD-B. The phonon number in given by [exp (8.0/A 2) − 1]−1 where A denotes the pulse area.

plotted in Figure 33.11. Figure 33.11a and b represent the populations in the ħΩ1-level in QD-B and in the ħΩ2-level in QD-A which is the up-converted energy level. The solid, dashed, and dotted curves correspond to π/2-pulse, π-pulse, and 3π/2pulse, respectively. Owing to coherent excitation process by optical pulse injection, the population in the lower energy level in QD-B maximally increases for π-pulse excitation, and decreases for more intense excitation. Temporal evolution in this energy level obeys solely relaxation time of the lower energy level. The population dynamics in QD-A is more complex. The peak values depend on the population in the ħΩ1-level in QD-B for each optical pulse area while the number of phonons in the phonon reservoir monotonically increases depending on the power of the optical pulse. Therefore, population lifetime depicted in the dotted curve in Figure 33.11b becomes longer than the dashed one because of longer pulse injection period. Such an up-conversion signal cannot observe in an isolated quantum dot, since dipole inactive transition from QD-B to QD-A never occurs without optical near-field interaction. Carrier dynamics driven by both exciton–photon and exciton–phonon interactions has large possibilities for some applications, such as novel light sources, optical near–far field interface devices, efficient photodetectors, and efficient photoelectric conversion devices (Yatsui et al. 2009).

33.3â•‡Nanophotonic Devices Using Spatial Symmetries In Section 33.2, nanophotonic devices, in which dissipation paths are selected by a state-filling condition in multi-exciton excitation states, were discussed. While another type of device using a coherently-coupled state shows interesting features. For such devices, the number of excitons in a system as well as the spatial symmetry of the system determine the device operations. In this section, some logical operations, memory operations, and

operations mediating quantum entangled states are explained with basic theoretical formulations.

33.3.1â•‡ Basic Principles 33.3.1.1â•‡ Symmetric and Antisymmetric States When two identical quantum dots sharing an exciton are coupled with each other, as shown in Figure 33.12, appropriate bases for coupled states instead of those for individual quantum dots give a clear view for system dynamics. There are two coupled states via optical near-field interaction that are described by using mixed states between vacuum and exciton states for individual quantum dots, are as follows:

(

S = e

(

A = e

A

A

g g

B

B

+ g

A

− g

A

e e

B

)

2,

)

2,

B

(33.8)

where the bases for the coupled states, |S〉 and |A〉 are called as symmetric and antisymmetric states as explained later in this section. At the beginning, energies in the symmetric and antisymmetric U 2U e

A

g

A

e

g QD-A

QD-B

B

S A

B

Coupled state

Figure 33.12â•… Energy diagram for two identical quantum dots (left). The system is expressed as a superposition of symmetric and antisymmetric states (right).

33-9


states are evaluated. Since the system Hamiltonian in a twoquantum-dot system with optical near-field coupling is given as Hˆ = Hˆ 0 + Hˆ int , Hˆ 0 = Ω Aˆ † Aˆ + ΩBˆ †Bˆ ,

(33.9)

)

(

ˆ ˆ† , Hˆ int = U Aˆ †Bˆ + AB

where radiative relaxation terms are ignored for simplicity and assuming that the relaxation lifetime is longer than that of optical near-field interaction. The energies for these states are easily derived as S Hˆ S = ( Ω + U ) , A Hˆ A = ( Ω − U ).

(33.10)

Equation 33.10 explains that the symmetric and antisymmetric states have positive and negative energy shift U from the energy in each quantum dot, respectively. In order to extract carrier excitation from the two identical quantum-dot systems, the third (output) quantum dot is set with resonance energy to the symmetric state or antisymmetric state. Detailed explanation of energy transfer properties in the three-quantum-dot system is given in Section 33.3.1.2. In this stage, to examine total dipole moment in such a system is quite instructive. The total dipole moment is expressed by an expectation value of a summation of creation and annihilation operators, such as Sµ

∑ (α

†

α = A, B

Aµ

∑ (α

α = A, B

†

)

+α g

)

+α g

A

A

g g

B

B

33.3.1.2â•‡ Selective Energy Transfer For far-field light, excitation in each quantum dot is not distinguishable and radiation is proportional to total dipole moment, which is discussed in Section 33.3.1.1. However, optical near field can resolve the spatial distribution of exciton occupation in a nanometer scale. In order to access the excited states in a two-identical quantum-dot system by using optical near field, third quantum dot, QD-C, is considered as an output port. Figure 33.13 represents the energy transfer from two-identical quantum-dot system to the third quantum dot. The two identical quantum dots make symmetric and antisymmetric states via optical near-field interaction, and these states also couple to the third quantum dot with optical near-field interaction. Here, to evaluate transition moment between two identical quantum dots to the third quantum dot is useful for understanding the effect of spatial alignment of quantum dots. The interaction Hamiltonian is given as

) ( ( Bˆ Cˆ + Cˆ Bˆ ),

Hˆ int, AC = U AC Aˆ †Cˆ + Cˆ † Aˆ , Hˆ int, BC = U BC

†

(33.12)

†

where UAC and UBC represent optical near-field coupling strength between QD-A and QD-C, and QD-B and QD-C, respectively. Using these interaction Hamiltonians, the transition matrix element from symmetric state to excited state in QD-C reads

A

g

B

g

(

)

e Hˆ int, AC + Hˆ int, BC S g

C

C

=

(U AC + U BC ) , 2 (33.13)

and that from antisymmetric state to excited state in QD-C is

= 2 µ, (33.11)

= 0,

where μ denotes the dipole moment for QD-A and QD-B. Apparently, transition from vacuum state to the antisymmetric state is inactive and, thus, it cannot couple to the external far-field light; this is the reason this state is known as the antisymmetric state. In contrast, optical near field can excite the antisymmetric state because it has a spatial localization character beyond the diffraction limit of light, and can create an exciton in a one-side quantum dot. The state with an exciton is equal to a superposition between symmetric and antisymmetric states. As described above, if the symmetric and antisymmetric states can be manipulated freely, coupling from near- to far-field light becomes controllable, which is useful for some functional operations, such as nanophotonic buffer memory. Furthermore, optical near-field coupling between the two identical quantum dots and the third output dot creates characteristic behaviors depending on spatial symmetry, where the symmetric and antisymmetric states play important roles, as explained in Section 33.3.1.2.

A

g

B

g

(

)

e Hˆ int, AC + Hˆ int, BC A g

C

C

=

(U AC − U BC ) . 2 (33.14)

U΄ S A

g

A

g

U

B

Coupled state

e

C

g

C

QD-C

Figure 33.13â•… Schematic representation of energy transfer between two-identical quantum-dot system (input) and a larger quantum dot (output) than the other two. Antisymmetric state becomes dipoleallowed and dipole-forbidden states, depending on spatial symmetry of the output quantum dot.

33-10


From Equation 33.14, the transition matrix element can take a zero value when the initial state is an antisymmetric state and the coupling strength toward QD-C are set as UAC = UBC ≠ 0. This condition can be realized by spatial symmetric alignment of quantum dots. Therefore, energy transfer mediating antisymmetric state is strongly dependent on spatial symmetry in a quantum-dot system. The feature of selective energy transfer is useful for functional operations of a nanophotonic device, such as logic and memory operations. In the following sections, some functional operations using coherently coupled states, which are symmetric and antisymmetric states, are explained with numerical calculation results.

33.3.2â•‡AND- and XOR-Logic Gates Similar to Section 33.3.1.2, let us consider a three-quantumdot system that consists of two identical input quantum dots (QD-A and QD-B) and the third output quantum dot (QD-C) with energy sublevels. QD-C is located in symmetrical position from QD-A and QD-B. Important points for this system are energy transfer reflecting spatial symmetry and difference of resonance conditions in one- and two-exciton states. By using these points, the principle of logic operations has been proposed (Sangu et al. 2004). First, the case QD-C negatively detuned with energy shift ħU is considered. Figure 33.14 represents energy diagram from initial state to final state in this system. The left (initial state) has no excitation in QD-C, and the right (final state) has an exciton in QD-C. Although negative energy shift creates resonance condition between the states of |A〉|g〉C and |g〉A|g〉B |e〉C for the oneexciton state, the energy transfer mediated by the antisymmetric state is not permitted in the quantum-dot system that is symmetrically aligned, as explained in Section 33.3.1.2. While, for the two-exciton state, the excited state |e〉A|e〉B|g〉C couples resonantly to the symmetric state |S〉|e〉C , which is allowed transition. Therefore, this quantum-dot system permits the energy transfer

2Ω

(Ω + U) Ω (Ω – U)

eA e

B

S g A g

g

U΄

C

gA g

C

Dipole inactive gA g

B

S e A e

C

g

C

Initial state

C

B

e

C

gA g

B

g

A

e

B

S g A g

gA g

B

g

S e A e

C

g

C

g

C

C

ΔΩ = U

U΄

C

C

Initial state

Final state

g

B

e

C

gA g

B

g

C

A

Figure 33.15â•… Energy diagram in three-quantum-dot system, which consists of two identical quantum dots (input port) and a third quantum dot (output port). The third quantum dot is positively detuned against each input quantum dot. The energy transfer occurs only in the case of one-exciton state.

only in the case of two-exciton state. This behavior represents just an AND-logic operation. Second, the energy level in QD-C is set as positive energy shift ħU. The energy diagram is illustrated in Figure 33.15. In this case, two-exciton state is inactive because of antisymmetric state, while symmetric one-exciton state |S〉|g〉C can resonantly couple to the final state of |g〉A|g〉B |e〉C . This means that the energy transfer occurs when either QD-A or QD-B is initially excited, where excitation in a quantum dot in two-identical quantumdot system is equivalent to excite symmetric and antisymmetric states, simultaneously (see Equation 33.8). This corresponds to the XOR-logic operation. Temporal evaluation of the above two logic operations can be derived analytically on the basis of the density matrix formalism. The system Hamiltonian is the same as Equation 33.9 except for additional third quantum dot terms, which include non-perturbed energy and non-radiative intra-sublevel relaxation, and radiative relaxation is ignored because optical near-field coupling and sublevel relaxation process are assumed to be fast enough. Final solution for one-exciton state, which is detected in lower energy level in QD-C, is written as follows (Sangu et al. 2004): Γ

ρ gge , gge (t ) =

1 4U ′ 2 − 2 t + 2 e 2 ω + − ω 2−

×  cos φ + cos (ω +t + φ + ) − cos φ − cos (ω −t + φ − ) , (33.15)

where ρ gge , gge (t ) ≡ C e B g A g ρ(t ) g ing abbreviations have been used:

A

g

B

e C , and the follow-

 2ω  φ± = tan −1  ±  ,  Γ 

C

Figure 33.14â•… Energy diagram in three-quantum-dot system, which consists of two identical quantum dots (input port) and a third quantum dot (output port). The third quantum dot is negatively detuned against each input quantum dot. The energy transfer occurs only in the case of two-exciton state.

e

(Ω + U) Ω (Ω – U)

C

ΔΩ = –U

Final state

2Ω

Dipole inactive

ω 2± =

2 2 2 1 2  2  ( ∆Ω − U ) + W+W− ± ( ∆Ω − U ) + W+  ( ∆Ω − U ) + W−   , 2 

W± = 2 2U ′ ±

Γ , 2

(33.16)

33-11


while the negative detuning for one-exciton state is off-resonant to symmetric state (see Equations 33.15 and 33.16). Therefore, the energy transfer rate is quite small. This is the AND-logic operation. Population dynamics in these devices are plotted in Figure 33.16, where exciton population in QD-C is analytically derived by using Equations 33.15 and 33.17. The solid and dashed curves represent one- and two-exciton states, respectively. Figure 33.16a is the result of the AND-logic gate with negative detuning ΔΩ = −U, where the strength of optical near-field interaction between QD-A and QD-B, and that between QD-A(B) and QD-C are set as U−1 = 10â•›ps and U′−1 = 50â•›ps. In this operation, the two-exciton state in QD-A and QD-B is resonantly coupled with QD-C, while the one-exciton state is not, because of offresonant condition. On the other hand, the XOR-logic operation is shown in Figure 33.16b. In this case, the one-exciton state in QD-A and QD-B is resonant to the energy level in QD-C, where ρSe ,Se (t ) + ρ gg 2e , gg 2e (t ) initial population is prepared in QD-A, which is same as the Γ  1  − t 4U ′ 2 2 = 2 + 2 e cos φ + cos (ω ′+t + φ′+ ) − cos φ − cos (ω ′−t + φ′− )  , coupled state between symmetric and antisymmetric states, and 2  2 ω ′+ − ω ′−  thus, the population reaches only the value of 0.5 because the (33.17) antisymmetric state is a dark state in a spatially symmetric system. Although ON/OFF ratio is not so high, the XOR-logic gate ˆ ρ ( t ) + ρ ( t ) ≡ e S ρ t S e + 2 e g g ρ ( t ) ( ) operation is surely observed. where Se , Se gg 2e , gg 2e C C C B A where ħΔΩ is the energy shift in QD-C against that in QD-A or QD-B. In Equation 33.15, the second term denotes the energy transfer or nutation between two-identical quantum-dot system QD-A and QD-B, and the third of QD-C. Apparently, this term becomes the maximum value in the case of positive detuning ΔΩ = U, because the denominator ω2+ − ω2− becomes minimum. This corresponds to dynamics in the XOR-logic operation, as is mentioned in Figure 33.15. The exciton population can reach maximum value of 0.5 not 1. This is because one-side quantum dot in QD-A or QD-B is excited as an initial excitation, which is same as coupled state between symmetric and antisymmetric states, and the antisymmetric state cannot couple to QD-C due to the symmetrically aligned system. On the other hand, analytical solution for two-exciton state is given in a similar form as

g

A

g

B

2e

QD-C and

C

is defined as the population in the lower level in

33.3.3â•‡ Nanophotonic Buffer Memory

 2ω ′  φ′± = tan −1  ±  ,  Γ 

ω ′±2 =

2 2 2 1 2  2   ( ∆Ω + U ) + W+W− ± ( ∆Ω + U ) + W+  ( ∆Ω + U ) + W−   . 2 

(33.18)

(a)

1.0

1.0

0.8

0.8 Population

Population

From Equations 33.17 and 33.18, negative detuning ΔΩ = −U makes resonance condition for two-exciton state,

As mentioned in Section 33.3.1.2, using energy transfer via a dipole inactive state, which can be controlled by energy detuning and quantum dot alignment or spatial symmetry, various functional operations inherent in nanophotonic devices are expected. In this section, one of interesting operations of a nanophotonic buffer memory is introduced. Quantum-dot alignment for realizing nanophotonic buffer memory is illustrated in Figure 33.17, in which two identical quantum dots (QD-A and QD-B) couple

0.6 0.4

0.6 0.4

0.2

0.2

0.0

0.0 0

100

Time (ps)

200

300

(b)

0

100

Time (ps)

200

300

Figure 33.16â•… Temporal evolution of exciton population in three-quantum-dot systems, which consist of input-side two identical quantum dots (QD-A and QD-B) and an output-side quantum dot (QD-C) with intra-sublevel relaxation. (a) and (b) represent the cases where upper energy level is negatively (AND-logic gate) and positively (XOR-logic gate) detuned. The solid and dashed curves correspond to the one-exciton and twoexciton states, respectively. The calculation parameters in Equations 33.15 through 33.18 are set as U−1 = 10â•›ps, U′−1 = 50â•›ps, and Γ−1 = 10â•›ps.

33-12

Handbook of Nanophysics: Nanoelectronics and Nanophotonics QD-A

Input

1.0 0.8 Population

Ω

U

Input

Ω

0.4 0.2

QD-B

U΄

0.6

0.0 0

ΔΩ = U Output QD-C

Figure 33.17â•… Schematic representation of nanophotonic buffer memory, which consists of two identical quantum dots, and the larger quantum dot with positive detuning ΔΩ = U. The three quantum dots are aligned maximally asymmetric and, thus, the coupling strength between QD-A and QD-C can be neglected.

to the third quantum dot (QD-C) with maximal asymmetric position. The third quantum dot is detuned positively with the energy ħ(Ω + U), that is, ΔΩ = U, which has the same energy diagram as in Figure 33.15, but energy transfer via antisymmetric state is allowed. When both QD-A and QD-B are excited as an initial condition, the two-exciton state in the input side resonantly couples to antisymmetric state with excitation in QD-C and then relaxes to the one-exciton antisymmetric state. This antisymmetric state decouples with outer far-field light because of dipole inactive (dark) state. This is just buffer memory operation. Dynamics of buffer memory operation is numerically evaluated by using density matrix formalism, as explained above. The calculation results are plotted in Figure 33.18, where the vertical axis represents population relating to the two-exciton coupled states, and the coupling strengths via optical near field are set as U−1 = 10â•›ps for QD-A and QD-B, and U′−1 = 50â•›ps for QD-B and QD-C (QD-A and QD-C are decoupled). The solid and dashed curves represent antisymmetric and symmetric states, respectively. In this calculation, radiative relaxation is ignored because the radiative relaxation time is longer than exciton population dynamics via optical near field and, thus, two-exciton antisymmetric state becomes the final state. Actually, the two-exciton state decays into the one-exciton antisymmetric state with a photon radiation from QD-C, and an exciton is retained in the system. In Section 33.3.2, the dark state is used for suppressing energy transfer, leading to logical operations, while it is used to create

100

Time (ps)

200

300

Figure 33.18â•… Temporal evolution of exciton population in threequantum-dot systems, which have maximal asymmetric configuration (see Figure 33.17). The solid and dashed curves represent the populations for the antisymmetric two-exciton state and the symmetric twoexciton state, respectively. The antisymmetric state has a dipole inactive nature and, thus, the excitation is maintained in the three-quantumdot system. The calculation parameters are set as U−1 = 10â•›ps, U′−1 = 50â•›ps, and Γ−1 = 10â•›ps.

excitation in a dark state in this section, which enables us to control far- and near-field conversion. It is one of the characteristic features of nanophotonic devices.

33.3.4â•‡Manipulation of QuantumEntangled States As mentioned in Sections 33.3.1 through 33.3.3, a nanophotonic device system consists of coherent energy transfer process and dissipation process. Since the coherent energy transfer process maintains quantum coherence, this system can be used for quantum information processing as well as interface between quantum and classical computations. As an instructive example, a typical device that identifies quantum mixed states is numerically demonstrated in this section. Figure 33.19 schematically shows spatial alignment of quantum dots, where two identical quantum dots couple resonantly, and the excitations are led to two output quantum dots with positive (QD-C) and negative (QD-D) detuning. Both QD-C and QD-D are set at symmetric and asymmetric positions, respectively. Apparently, QD-C resonantly couples to the symmetric state in the input quantum dots, and QD-D couples to the antisymmetric state. Therefore, this device operates that input signal with arbitral quantum state divides two quantum bases of symmetric and antisymmetric states into two different output signals with different optical frequencies as well as different positions in a nanometric space. Temporal evolution of an input quantum mixed state is numerically calculated in Figure 33.20. When an input state c1|S〉 + c 2|A〉 is injected, where one-exciton state is assumed, the output exciton population in QD-C and QD-D

33-13


33.4â•‡ Summary

QD-A Ω

Input (1)

U΄ ΔΩ = U

U Input (2) U΄

Ω QD-B

ΔΩ = –U

U΄

Output (1) QD-C

Output (2) QD-D

Figure 33.19â•… Spatial arrangement of a four-quantum-dot system, which has symmetric and asymmetric configurations simultaneously. QD-C resonantly couples with symmetric state in QD-A and QD-B, and QD-D couples with antisymmetric state.

1.0

Population

0.8

In this chapter, operation of a nanophotonic device, in which characteristics of the light localized among nanometric objects, that is, optical near fields, are positively used, has been demonstrated theoretically and numerically. Resonant energy transfer between quantum systems mediated by optical near-field interaction and fast energy relaxation in intra-energy levels via exciton–phonon interaction are fundamental for a switching device and excitation carrier manipulations, as discussed in Section 33.2. By using coupled states, which are made from resonant optical near-field interaction between quantum systems, nanophotonic logic-gate, memory, and quantum-classical interface devices can be realized with the help of selective energy transfer depending on energies of the coupled state and a selection rule for optical near and far fields. These nanophotonic devices are taken as extensions of classical devices, in which simple functions are tandemly arranged, to novel-type devices locally utilized the quantum nature. Here, the optical near fields play important roles for controlling quantum coherence and/or decoherence. As a summary, key fundamental design concepts for nanophotonic inherent operations are enumerated in the following: • Externally switchable dissipation paths • Dependence of the number of excitation carriers • Selection rule for optical near field based on spatial arrangement of quantum systems • Selection rule for optical far field based on total dipole moment

0.6 0.4 0.2 0.0 0

100

Time (ps)

200

300

Figure 33.20â•… Population dynamics in four-quantum-dot systems that have two identical quantum dots (QD-A and QD-B) for input ports of an arbitral quantum state, and output quantum dots (QD-C and QD-D) for identifying the input quantum states. The solid and dashed curves represent the output population of QD-C and QD-D, respectively, and the dotted lines correspond to the coefficients of an initial quantum mixed state. The calculation parameters in Equations 33.15 through 33.18 are set as U−1 = 10â•›ps, U′−1 = 50â•›ps, and Γ−1 = 10â•›ps.

reflects the coefficients of c1 and c2 of an input state. In Figure 33.20, the coefficients of initial quantum mixed state are set as c1 = 1/3 and c2 = 2/3, and thus, the output populations approach these values asymptotically. Although general quantum information processing devices are built upon quantum coherent states in whole device systems, nanophotonic devices are characterized by the control of dissipation processes, but keeping the quantum coherence locally. Such a mixed operation of classical and quantum information processing may open up future device technologies.

The above may not explain all concepts for designing nanophotonic devices—additional unique natures may be hidden. Accomplished device systems might be realized, which have possible advantages such as low energy consumption, low heat liberation, large parallelism, miniaturization, and environmental tolerance, by comprehending physical phenomena inherent in nanophotonics correctly and designing nanophotonic devices from a standpoint throughout from nano- to macro-scales.

Acknowledgments The authors are grateful to M. Ohtsu, T. Kawazoe, and T. Yatsui from the University of Tokyo for fruitful discussions. This work was mainly carried out at the project of ERATO, Japan Science and Technology Agency, from 1998 to 2003. The authors would like to thank the persons concerned.

References Biolatti, E., Iotti, R. C., Zanardi, P., and Rossi, F. 2000. Quantum information processing with semiconductor macroatoms. Phys. Rev. Lett. 85: 5647–5650. Breuer, H.-P. and Petruccione, F. 2002. The Theory of Open Quantum Systems. New York: Oxford University Press.

33-14


Carmichael, H. J. 1999. Statistical Methods in Quantum Optics 1. Berlin/Heidelberg, Germany: Springer-Verlag. Cho, K. 2004. Optical Response of Nanostructures: Microscopic Nonlocal Theory (Springer Series in Solid-State Sciences). Tokyo, Japan: Springer-Verlag. Förster, T. 1965. Delocalized excitation and excitation transfer. In Modern Quantum Chemistry. O. Sinanoglu (Ed.), pp. 93–137. London, U.K.: Academic Press. Hosaka, N. and Saiki, T. 2001. Near-field fluorescence imaging of single molecules with a resolution in the range of 10â•›nm. J. Microsc. 202: 362–364. Kawazoe, T., Kobayashi, K., Lim, J., Narita, Y., and Ohtsu, M. 2002. Direct observation of optically forbidden energy transfer between CuCl quantum cubes via near-field optical spectroscopy. Phys. Rev. Lett. 88: 067404-1–067404-4. Kawazoe, T., Kobayashi, K., Sangu, S., and Ohtsu, M. 2003. Demonstration of a nanophotonic switching operation by optical near-field energy transfer. Appl. Phys. Lett. 82: 2957–2959. Kobayashi, K., Sangu, S., Ito H., and Ohtsu, M. 2000. Nearfield optical potential for a neutral atom. Phys. Rev. A 63: 013806-1–013806-9. Maheswari, R. U., Mononobe, S., Yoshida, K., Yoshimoto, M., and Ohtsu, M. 1999. Nanometer level resolving near field optical microscope under optical feedback in the observation of a single-string deoxyribo nucleic acid. Jpn. J. Appl. Phys. 38: 6713–6720. Masumoto, Y. 2002. Persistent spectral hole burning in semiconductor quantum dots. In Semiconductor Quantum Dots: Physics, Spectroscopy and Applications. Y. Masumoto and T. Takagahara (Eds.), pp. 209–244. Berlin/Heidelberg, Germany: Springer-Verlag. Matsuda, K., Saiki, T., Nomura, S., Mihara, M., Aoyagi, Y., Nair, S., and Takagahara, T. 2003. Near-field optical mapping of exciton wave functions in a GaAs quantum dot. Phys. Rev. Lett. 91: 177401–177404.

Nomura, W., Ohtsu, M., and Yatsui, T. 2005. Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion. Appl. Phys. Lett. 86: 181108-1–181108-3. Ohtsu, M. 1998. Near-Field Nano/Atom Optics and Technology. Tokyo, Japan: Springer-Verlag. Ohtsu, M., Kobayashi, K., Kawazoe, T., Sangu, S., and Yatsui, T. 2002. Nanophotonics: Design, fabrication, and operation of nanometric devices using optical near fields. IEEE J. Sel. Top. Quantum Electron. 8: 839–862. Ohtsu, M., Kobayashi, K., Kawazoe, T., Yatsui, T., and Naruse, M. 2008. Principles of Nanophotonics. London, U.K.: Taylor & Francis. Rinaldis, S. D., D’Amico, I., and Rossi, F. 2002. Exciton–exciton interaction engineering in coupled GaN quantum dots. Appl. Phys. Lett. 81: 4236–4238. Sangu, S., Kobayashi, K., Kawazoe, T., Shojiguchi, A., and Ohtsu, M. 2003. Excitation energy transfer and population dynamics in a quantum dot system induced by optical near-field interaction. J. Appl. Phys. 93: 2937–2945. Sangu, S., Kobayashi, K., Shojiguchi, A., and Ohtsu. M., 2004. Logic and functional operations using a near-field optically coupled quantum-dot system. Phys. Rev. B 69, 115334-1–115334-13. Troiani, F., Hohenester, U., and Molinari, E. 2002. Electron–hole localization in coupled quantum dots. Phys. Rev. B 65: 161301-1–161301-4. Walls, D. F. and Milburn, G. J. 1994. Quantum Optics. Berlin/ Heidelberg, Germany: Springer-Verlag. Yatsui, T., Kourogi, M., and Ohtsu, M. 2001. Plasmon waveguide for optical far/near-field conversion. Appl. Phys. Lett. 79: 4583–4585. Yatsui, T., Sangu, S., Kobayashi, K., Kawazoe, T. et al. 2009. Nanophotonic energy up-conversion using ZnO nanorod double-quantum-well structures. Appl. Phys. Lett. 94: 083113-1–083113-3.

34 System Architectures for Nanophotonics 34.1 Introduction............................................................................................................................34-1 34.2 System Architectures Based on Optical Excitation Transfer...........................................34-1 Optical Excitation Transfer via Optical Near-Field Interactions and Its Functional Featuresâ•‡ •â•‡ Parallel Architecture Using Optical Excitation Transferâ•‡ •â•‡ Secure Signal Transfer in Nanophotonics

34.3 Hierarchical Architectures in Nanophotonics.................................................................. 34-6

Makoto Naruse National Institute of Information and Communications Technology The University of Tokyo

Physical Hierarchy in Nanophotonics and Functional Hierarchyâ•‡ •â•‡ Hierarchical Memory Retrievalâ•‡ •â•‡ Design of Unscalable Hierarchical Responseâ•‡ •â•‡ Versatile Functionalities Based on Hierarchy in Optical Near-Fields

34.4 Summary................................................................................................................................34-12 Referencesï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½34-12

34.1â•‡ Introduction To accommodate the continuously growing amount of digital data and qualitatively new requirements demanded by industry and people in society, optics is expected to be highly integrated and to play a wider role in enhancing system performance. However, many technological difficulties remain to be in overcome in adopting optical technologies in critical information and communication systems; one problem is the poor integrability of optical hardware due to the diffraction limit of light (Pohl and Courjon 1993, Ohtsu and Hori 1999). Nanophotonics, on the other hand, which is based on local interactions between nanometer-scale matters via optical near fields, offers ultrahigh-density integration since it is not constrained by the diffraction limit. Fundamental nanophotonic processes, such as optical excitation transfer via optical near fields between nanometer-scale matters, have been studied in detail (Ohtsu et al. 2002, 2008, Maier et al. 2003). Moreover, this higher integration density is not the only advantage that optical near fields have over conventional optics and electronics. From a system architectural perspective, nanophotonics drastically changes the fundamental design rules of functional optical systems, and suitable architectures may be built to exploit this. As a result, it also gives qualitatively strong impacts on information and communication systems. This chapter discusses system architecture for nanophotonics considering the unique physical principles of optical nearfield interactions as well as their experimental verification based on technological vehicles, such as quantum dots and engineered metal nanostructures. In particular, two unique physical

processes in light–matter interactions in the nanometer scale are exploited. One is optical excitation transfer via optical near-field interactions and the other is the hierarchical property in optical near-field interactions, which are explained in Sections 34.2 and 34.3, respectively. The overall structure of this chapter is outlined in Figure 34.1. Through these architectural and physical insights, nanophotonic information and communication systems are demonstrated, which overcome the integration-density limit imposed by the diffraction of light with ultralow-power operation, as well as provide unique functionalities that are only achievable using optical near-field interactions.

34.2â•‡System Architectures Based on Optical Excitation Transfer 34.2.1â•‡Optical Excitation Transfer via Optical Near-Field Interactions and Its Functional Features In this section, optical excitation transfer processes involving optical near-field interactions are reviewed from a system perspective. Here, their fundamental principles are first briefly reviewed and their functional features are introduced for later discussion. The interaction Hamiltonian between an electron and an electric field is given by

ˆ Hˆ int = − ψˆ † (r )µψˆ (r ) ⋅ D(r )dr ,

∫

(34.1) 34-1

34-2


System architecture for nanophotonics

Secure signal transfer

Parallel architecture D1

In

Hierarchical architecture

V

Out

DM

Optical excitation transfer

Optical near-field interactions

Hierarchy in optical near-field interactions

Light–matter interaction in the nanometer scale

Figure 34.1â•… Overview of this section: from light–matter interactions on the nanometer scale to system architectures for nanophotonics.

where µ is a dipole moment ψˆ † (r ) and ψˆ (r ) are, respectively, creation and annihilation operators of an electron at r⃗ ˆ D(r ) is the operator of electric flux density ˆ In usual light–matter interactions, the operator D(r ) is a constant since the electric field of propagating light is considered to be constant on the nanometer scale. Therefore, as is well known, one can derive optical selection rules by calculating a transfer matrix of an electrical dipole. As a consequence, in the case of cubic quantum dots, for instance, transitions to states described by quantum numbers containing an even number are prohibited. In the case of optical near-field interactions, on the other hand, due to the steep electric field of optical near fields in the vicinity of nanoscale matter, an optical transition that violates conventional optical selection rules is allowed. Optical excitations in nanostructures, such as quantum dots, can be transferred to neighboring ones via optical near-field interactions (Ohtsu et al. 2002, 2008). For instance, assume two cubic quantum dots, QDA and QDB, whose side lengths L are a and 2a, respectively (see Figure 34.2a). Suppose that the energy eigenvalues for the quantized exciton energy level specified by quantum numbers (nx, ny, nz) in a QD with side length L are given by

E(nx ,ny ,nz ) = EB +

2π 2 2 (nx + n2y + nz2 ), 2 ML2

(34.2)

where EB is the energy of the bulk exciton M is the effective mass of the exciton According to Equation 34.2, there exists a resonance between the level of quantum number (1, 1, 1) for QDA and that of quantum number (2, 1, 1) for QDB. There is an optical near-field interaction, which is denoted by U, due to the steep electric field in the vicinity of QDA. Therefore, excitons in QDA can move to the (2, 1, 1)-level in QDB. Note that such a transfer is prohibited for propagating light since the (2, 1, 1)-level in QDB contains an even number (Tang et al. 1993). In QDB, the exciton sees a sublevel energy relaxation, denoted by Γ, which is faster than the near-field interaction, and so the exciton goes to the (1, 1, 1)-level of QDB. It should be emphasized that the sublevel relaxation determines the unidirectional exciton transfer from QDA to QDB. Now, several unique functional aspects should be noted in the above excitation transfer processes. First, as already mentioned, the transition from the (1, 1, 1)-level in QDA to the (2, 1, 1)-level in QDB is usually a dipole-forbidden transfer. In contrast, the optical near field allows such processes. Second, in the resonant energy levels of those quantum dots, optical excitation can go back and forth between QDA and QDB, which is called optical nutation. The direction of excitations is determined by the energy dissipation processes. Therefore, based on the above mechanisms, the flow of optical excitations can be controlled in quantum-dot systems via optical near-field interactions. From an architectural standpoint, such a flow of excitations directly leads to digital processing systems and computational

34-3

System Architectures for Nanophotonics

QDB

QDA a

2a

QD3

S1 QD 1

S3

QDC

U (1,1,1)

Γ

(2,1,1) QD2

(1,1,1)

(a)

S2

(c) 0

S2

S1

S3

+

Intensity (a.u.)

CuCl in NaCl 40 K

0.10 0.20

Input (1,1,1)

0.30 0.40

Input (1,0,1)

150 nm

Input (0,0,1) 3.20

Out (b) Summation

(d)

Output

3.25 Photon energy (eV)

OFF state

3.30

ON state

Data

0.24

200 nm (e) Broadcast

200 nm

0

Figure 34.2â•… (a) Optical excitation transfer between quantum dots via optical near-field interactions. (b) Global summation: a basic function for memory-based architectures. (c) Quantum-dot arrangement for summation via an optical near field. (d) Intensity for three different input combinations and the spatial intensity distribution of the output photon energy. (e) Broadcast interconnects for parallel processing. Spatial intensity distribution of the output of 3-dot AND gates.

architectures. First of all, two different physical states appear by controlling the dissipation processes in the larger dot; this is the principle of the nanophotonic switch (Kawazoe et al. 2003). Also, such a flow control itself allows an architecture known as a binary decision diagram, where an arbitrary combinatorial logic operation is determined by the destination of a signal flowing from a root (Akers 1978). Such optical excitation transfer processes also lead to unique system architectures. In this regard, Section 34.2.2 discusses a massively parallel architecture and its nanophotonic implementations. Also, Section 34.2.3 demonstrates that optical excitation transfer provides higher tamper resistance against attacks than conventional electrically wired devices, by focusing on environmental factors for signal transfer.

34.2.2â•‡Parallel Architecture Using Optical Excitation Transfer 34.2.2.1â•‡ Memory-Based Architecture This section discusses a memory-based architecture where computations are regarded as a table lookup or database search problem, which is also called content addressable memory (CAM) (Liu 2002). The inherent parallelism of this architecture is well matched with the physics of optical excitation transfer, and provides performance benefits in high-density, low-power operations. In this architecture, input signal (content) serves as a query to a lookup table, and the output is the address of the data matching the input. This architecture plays a critical role in various

34-4


systems for example in a data router where the output port for an incoming packet is determined based on the lookup tables. All optical means for implementing such functions have been proposed, for instance, by using planar lightwave circuits (Grunnet-Jepsen et al. 1999). However, since separate optical hardware for each table entry is needed if based on today’s known methods, if the number of entries in the routing table is on the order of 10,000 or more, the overall physical size of the system becomes impractically large. On the other hand, by using diffraction-limit-free nanophotonic principles, huge lookup tables can be configured compactly. Then, it is important to note that the table lookup problem is equivalent to an inner product operation. Assume an N-bit input signal S = (s1,…,â•›sN) and reference data D = (d1,…,â•›dN). Here, the

∑

N

si ⋅ di will provide a maximum value inner product S ⋅ D = i =1 when the input perfectly matches the reference data with an appropriate modulation format (Naruse et al. 2004). Then, the function of a CAM is to derive j, which maximizes the S · Dj. 34.2.2.2â•‡Global Summation Using Near-Field Interactions As discussed in Section 34.2.2.1, the inner product operations are the key functionalities of the memory-based architecture. The multiplication of two bits, namely, xi = si · di, has already been demonstrated by a combination of three quantum dots (Kawazoe et al. 2003). Therefore, one of the key operations remaining is the summation, or data gathering scheme, denoted by xi , where all data bits should be taken into account, as schematically shown in Figure 34.2b. In known optical methods, wave propagation in free-space or in waveguides, using focusing lenses or fiber couplers, for example, well matches such a data gathering scheme because the physical nature of propagating light is inherently suitable for the collection or distribution of information, such as global summation. However, the level of the integration of these methods is restricted due to the diffraction limit of light. In nanophotonics, on the other hand, the near-field interaction is inherently physically local, although functional global behavior is required. The global data gathering mechanism, or summation, is realized based on the unidirectional energy flow via an optical near field, as schematically shown in Figure 34.2c, where surrounding excitations are transferred toward a quantum dot QDC located at the center (Kawazoe et al. 2005, Naruse et al. 2005b). This is based on the excitation transfer processes presented in Section 34.2.1 and in Figure 34.2a, where an optical excitation is transferred from a smaller dot (QDA) to a larger one (QDB) through a resonant energy sublevel and a sublevel relaxation process occurring at a larger dot. In the system shown in Figure 34.2c, similar energy transfers may take place among the resonant energy levels in the dots surrounding QDC so that excitation transfer can occur. The lowest energy level in each quantum dot is coupled to a free photon bath to sweep out the excitation radiatively. The output signal is proportional to the (1, 1, 1)-level population in QDB.

∑

A proof-of-principle experiment was performed to verify the nanoscale summation using CuCl quantum dots in a NaCl matrix, which has also been employed for demonstrating nanophotonic switches (Kawazoe et al. 2003) and optical nano-fountains (Kawazoe et al. 2005). A quantum-dot arrangement where small QDs (QD1–QD3) surrounded a large QD at the center (QDC) was chosen. Here, at most three light beams with different wavelengths, 325, 376, and 381.3â•›nm, are irradiated, which excite the quantum dots QD1–QD3 having sizes of 1, 3.1, and 4.1â•›nm, respectively. The excited excitons are transferred to QDC, and their radiation is observed by a near-field fiber probe tip. Notice the output signal intensity at a photon energy level of 3.225â•›eV in Figure 34.2d, which corresponds to a wavelength of 384â•›nm, or a QDC size of 5.9â•›nm. The intensity varies approximately as 1:2:3 depending on the number of excited QDs in the vicinity, as observed in Figure 34.2d. The spatial intensity distribution was measured by scanning the fiber probe, as shown in the inset of Figure 34.2d, where the energy is converged at the center. Hence, this architecture works as a summation mechanism, counting the number of input channels, based on exciton energy transfer via optical near-field interactions. Such a quantum-dot-based data-gathering mechanism is also extremely energy efficient compared to other optical methods such as focusing lenses or optical couplers. For example, the transmittance between two materials with refractive indexes n1 and n2 is given by 4n1n2/(n1 + n2)2; this gives a 4% loss if n1 and n2 are 1 and 1.5, respectively. The transmittance of an N-channelguided wave coupler is 1/N from the input to the output if the coupling loss at each coupler is 3â•›dB. In nanophotonic summation, the loss is attributed to the dissipation between energy sublevels, which is significantly smaller. Incidentally, it is energyand space-efficient compared to electrical CAM VLSI chips (Lin and Kuo 2001, Arsovski et al. 2003, Naruse et al. 2005a). 34.2.2.3â•‡ Broadcast Interconnects For the parallel architecture shown in Section 34.2.2.2, it should also be noted that the input data should be commonly applied to all lookup table entries. In other words, broadcast interconnect is another important requirement for parallel architectures. Broadcast is also important in applications such as matrixvector products (Goodman et al. 1978, Guilfoyle and McCallum 1996) and switching operations, for example, broadcast-andselect architectures (Li et al. 2001). Optics is in fact well suited to such broadcast operations in the form of simple imaging optics (Goodman et al. 1978, Guilfoyle and McCallum 1996) or in optical waveguide couplers; thanks to the nature of wave propagation. However, the integration density of this approach is physically limited by the diffraction limit, which leads to bulky system configurations. The overall physical operation principle of broadcast using optical near fields is as follows. Suppose that the arrays of nanophotonic circuit blocks are distributed within an area whose size is comparable to the wavelength. For broadcasting, multiple input QDs simultaneously accept identical input data carried by

34-5


diffraction-limited far-field light by tuning their optical frequency so that the light is coupled to dipole-allowed energy sublevels. The far- and near-field coupling mentioned above is explained based on a model assuming cubic quantum dots, which was introduced in Section 34.2.1. According to Equation 34.2, there exists a resonance between the quantized exciton energy sublevel of quantum number (1, 1, 1) for the QD with effective side length a and that of quantum number (2, 1, 1) for the QD with effective side length 2a. Energy transfer from the smaller QD to the larger one occurs via optical near fields, which is forbidden for far-field light (Kawazoe et al. 2003). The input energy level for the QDs, that is, the (1, 1, 1)-level, can also couple to the far-field excitation. This fact can be utilized for data broadcasting. One of the design restrictions is that energy sublevels for input channels do not overlap with those for output channels. Also, if there are QDs internally used for near-field coupling, dipole-allowed energy sublevels for those QDs cannot be used for input channels since the inputs are provided by farfield light, which may lead to the misbehavior of internal nearfield interactions if resonant levels exist. Therefore, frequency partitioning among the input, internal, and output channels is important. The frequencies used for broadcasting, denoted by Ωi = {ωi,1, ωi,2,…, ωi,A}, should be distinct values and should not overlap with the output channel frequencies Ωo = {ωo,1, ωo,2,…, ωo,B}. A and B indicate the number of frequencies used for input and output channels, respectively. Also, there will be frequencies needed for internal device operations, which are not used for either input or output, denoted by Ωn = {ωn,1, ωn,2,…, ωn,C}, where C is the number of those frequencies. Therefore, the design criteria for global data broadcasting is to exclusively assign input, output, and internal frequencies, Ωi, Ωo, and Ωn, respectively. In a frequency multiplexing sense, this interconnection method is similar to multi-wavelength chip-scale interconnection (De Souza et al. 1995). Known methods, however, require a physical space comparable to the number of diffraction-limited input channels due to wavelength demultiplexing, whereas in the nanophotonic scheme, the device arrays are integrated on the sub-wavelength scale, and multiple frequencies are multiplexed in the far-field light supplied to the device. To verify the broadcasting method, the following experiments were performed using CuCl QDs inhomogeneously distributed in a NaCl matrix at a temperature of 22â•›K (Naruse et al. 2006). To operate a 3-dot nanophotonic switch (2-input AND gate) in the device, at most two input light beams (IN1 and IN2) were irradiated. When both inputs exist, an output signal is obtained from the positions where the switches exist, as described above. In the experiment, IN1 and IN2 were assigned to 325 and 384.7â•›nm, respectively. They were irradiated over the entire sample (global irradiation) via far-field light. The spatial intensity distribution of the output, at 382.6â•›nm, was measured by scanning a nearfield fiber probe within an area of approximately 1â•›μm × 1â•›μm. When only IN1 was applied to the sample the output of the AND gate was ZERO (OFF state). When both inputs were irradiated the output was ONE (ON state). Note the regions marked by ◾, ⚫, and ♦ in Figure 34.2e. In those regions, the output signal

levels were, respectively, low and high, which indicate that multiple AND gates were integrated at densities beyond the scale of the globally irradiated input beam area. That is to say, broadcast interconnects to nanophotonic switch arrays are accomplished by diffraction-limited far-field light. Combining this broadcasting mechanism with the summation mechanism discussed in Section 34.2.2.2 will allow the development of the nanoscale integration of massively parallel architectures, which have conventionally resulted in bulky configurations.

34.2.3â•‡Secure Signal Transfer in Nanophotonics In addition to breaking through the diffraction limit of light, such local interactions of optical near fields also have important functional aspects, such as in security applications, which particularly tamper resistance against attacks (Naruse et al. 2007a). One of the most critical security issues in present electronic devices is so-called side-channel attacks, by which information is tampered either invasively or noninvasively. This may be achieved, for instance, merely by monitoring their power consumption (Kocher et al. 1998). In this section, it is shown that devices based on optical excitation transfer via near-field interactions are physically more tamper-resistant than their conventional electronic counterparts. The key is that the flow of information in nanoscale devices cannot be completed unless they are appropriately coupled with their environment (Hori 2001), which could possibly be the weakest link in terms of their tamper resistance. A theoretical approach is presented to investigate the tamper resistance of optical excitation transfer, including a comparison with electrical devices. Here, the tampering of information is defined as involving simple signal transfer processes, since the primary focus is on their fundamental physical properties. In order to compare the tamper resistance, an electronic system based on single-charge tunneling is introduced here, in which a tunnel junction with capacitance C and tunneling resistance RT is coupled to a voltage source V via an external impedance Z(ω), as shown in Figure 34.3a. In order to achieve

C (a)

e

h

h

QDA

Z(ω) RT

e

V

U

QDB G

(b)

Figure 34.3â•… Model of tamper resistance in devices based on (a) single charge tunneling and (b) optical excitation transfer. Dotted curves show the scale of a key device and dashed curves show the scale of the environment required for the system to work.

34-6


single charge tunneling, besides the condition that the electrostatic energy EC = e2/2C of a single excess electron be greater than the thermal energy kBT, the environment must have appropriate conditions, as discussed in detail in Ingold and Nazarov (1992). For instance, with an inductance L in the external impedance, the fluctuation of the charge is given by (34.3)

34.3.1â•‡Physical Hierarchy in Nanophotonics and Functional Hierarchy

Therefore, charge fluctuations cannot be small even at zero temperature unless ρ >> 1. This means that a high-impedance environment is necessary, which makes tampering technically easy, for instance by adding another impedance circuit. Here, let us define two scales to illustrate tamper resistance: (1) the scale associated with the key device size and (2) the scale associated with the environment required for operating the system, which are, respectively, indicated by dotted and dashed curves in Figure 34.3. In the case of Figure 34.3a, scale I is the scale of a tunneling device, whereas scale II covers all of the components. It turns out that the low tamper resistance of such wired devices is due to the fact that scale II is typically the macroscale, even though scale I is the nanometer scale. In contrast, in the case of the optical excitation transfer shown in Figure 34.3b, the two quantum dots and their surrounding environment are governed by scale I. It is also important to note that scale II is the same as scale I. More specifically, the transfer of an exciton from QDA to QDB is completed due to the non-radiative relaxation process occurring at QDB, which is usually difficult to tamper with. Theoretically, the sublevel relaxation constant is given by

Γ = 2π | g (ω) | D(ω),

In this section, another feature of nanophotonics, the inherent hierarchy in optical near-field interactions, is exploited. As schematically shown in Figure 34.4a, there are multiple layers associated with the physical scale between the macroscale world and the atomic-scale world, which are primarily governed by propagating light and electron interactions, respectively. Between these two extremes, typically in scales ranging from a few nanometers to wavelength size, optical near-field interactions play a crucial role. In this section, such hierarchical properties in this mesoscopic or sub-wavelength regime are exploited. Such physical hierarchy in optical near-field interactions will be analyzed by a simple dipole–dipole interaction model and an angular spectrum representation of optical near fields, as shown in Section 34.3.2. Before going into the details of the physical processes, functionalities required for system applications are briefly reviewed in terms of hierarchy.

(34.4)

where ħg(ω) is the exciton–phonon coupling energy at frequency ω ħ is Planck’s constant divided by 2π D(ω) is the phonon density of states (Carmichael 1999) Therefore, tampering with the relaxation process requires somehow “stealing” the exciton–phonon coupling, which would be extremely difficult technically. It should also be noted that the energy dissipation occurring in the optical excitation transfer, derived theoretically as E(2,1,1) − E(1,1,1) in QDB based on Equation 34.2, should be larger than the exciton-phonon coupling energy of ħΓ, otherwise, the two levels in QDB cannot be resolved. This is similar to the fact that the condition ρ >> 1 is necessary in the electron-tunneling example, which means that the mode energy ħωS is smaller than the required

(a)

Propagating light

Macro-scale ~λ/10-scale

Optical near-field

Function

~λ/100-scale

Optical near-field

Function

~λ/1000-scale

Optical near-field

Function

Atomic-scale rP/rS = 1

D 3D 6D (b)

Function

Diffraction limit of light

rP

Contrast

where ρ = EC/ħωS ωS = (LC)−1/2 β = 1/kBT

2

34.3â•‡Hierarchical Architectures in Nanophotonics

Macro

e2  β ω S  coth  ,  2  4ρ

Nano

〈 δQ 2 〉 =

charging energy EC . By regarding ħΓ as a kind of mode energy in the optical excitation transfer, the difference between the optical excitation transfer and a conventional wired device is the physical scale at which this mode energy is realized: nanoscale for the optical excitation transfer and macroscale for electric circuits.

rS (c)

0.06 0.05 0.04 0.03 0.02 0.01 0

0 2 4 6 8 10 Ratio of the radius, rP/rS

Figure 34.4â•… (a) Hierarchy in optical near-field interactions. (b) Dipole–dipole interaction. (c) Signal contrast as a function of the ratio of the radius of the sample and the probe.

34-7


One of the problems for ultrahigh-density nanophotonic systems is interconnection bottlenecks, which have been addressed previously in Section 34.2.3 regarding broadcast interconnects. In fact, a hierarchical structure can be found in these broadcast interconnects by relating far-field effects at a coarser scale and near-field effects at a finer scale. In this regard, it should also be mentioned that such physical differences in optical near-field and far-field effects can be used for a wide range of applications. The behavior of usual optical elements, such as diffractive optical elements, holograms, or glass components, is associated with their optical responses in optical far fields. In other words, nanostructures can exist in such optical elements as long as they do not affect the optical responses in far fields. Designing nanostructures accessible only via optical near fields provides additional, or hidden, information recorded in those optical elements, while maintaining the original optical responses in far fields. In fact, a “hierarchical hologram” or “hierarchical diffraction grating” has been experimentally demonstrated, as schematically shown in Figure 34.5 (Tate et al. 2008). Since there is more hierarchy in the optical near-field regime, further applications should be possible, for example, it should be possible for nanometer-scale high-density systems to be gradually hierarchically connected to coarse layer systems. Hierarchical functionalities are also important for several aspects of memory systems. One is related to recent high-density, huge-capacity memory systems, in which data retrieval or searching from entire memory archives is made even more difficult. Hierarchy is one approach for solving such a problem by making systems hierarchical, that is, by recording abstract data, metadata, or tag data in addition to the original raw data. Hierarchical optical element

Macro-scale

Hierarchy in nanophotonics provides a physical solution to achieve such functional hierarchy. As will be introduced below, low-density, rough information is read out at a coarser scale, whereas high-density, detailed information is read out at a finer scale. Sections 34.3.2 and 34.3.3 will show physical mechanisms for such hierarchical information retrieval. Another issue in hierarchical functionalities will be security. High-security information is recorded at a finer scale, whereas less-critical security information is associated with a coarse layer. Also, in addition to associating different types of information with different physical scales, another kind of information can also be related to one or more layers of the physical hierarchy, for instance, traceability, history, or aging of information. Section 34.3.3 will demonstrate a traceable memory as an example.

34.3.2â•‡ Hierarchical Memory Retrieval This section describes a physical model of optical near-field interactions based on dipole–dipole interactions (Ohtsu and Kobayashi 2004). Suppose that a probe, which is modeled by a sphere of radius rP, is placed close to a sample to be observed, which is modeled as a sphere of radius rS . Figure 34.4b shows three different sizes for the probe and the sample. When they are illuminated by incident light, whose electric field is E 0, electric dipole moments are induced in both the probe and the sample; these moments are denoted by pP = αPE 0 and pS = αSE 0, respectively. The electric dipole moment induced in the sample pS , then generates an electric field, which changes the electric dipole moment in the probe by an amount ΔpP = ΔαPE 0. Similarly, pP changes the electric dipole moment in the sample by ΔpS = ΔαSE 0. These electromagnetic interactions are called Common information via optical far-fields Without nanopattern

With nanopattern

Nanopatterns

Diffraction efficiency (a.u.)

500 um

Hierarchical hologram

Near-field scale Near-field probe

0.5 0.4 0.3 0.2 0.1 0.0

500 um

No difference

–1st 0th 1st 2nd

0.5 0.4 0.3 0.2 0.1 0.0

–1st 0th 1st 2nd

Security information via optical near-fields Without With nanopattern nanopattern

100 nm Identity/Brand protection, etc.

Figure 34.5â•… Hierarchical optical elements, such as hierarchical holograms, based on different optical responses obtained in optical far and near fields.

34-8


dipole–dipole interactions. The scattering intensity induced by these electric dipole moments is given by I = p P + ∆p P + pS + ∆pS

2

≈ (α P + α S )2 | E0 |2 + 4 ∆α(α P + α S ) | E0 |2

(34.5)

where Δα = ΔαS = ΔαP (Ohtsu and Kobayashi 2004). The second term in Equation 34.5 shows the intensity of the scattered light generated by the dipole–dipole interactions, containing the information of interest, which is the relative difference between the probe and the sample. The first term in Equation 34.5 is the background signal for the measurement. Therefore, the ratio of the second term to the first term of Equation 34.5 corresponds to a signal contrast, which will be maximized when the sizes of the probe and the sample are the same (rP = rS), as shown in Figure 34.4c. (A detailed derivation is found in Ohtsu and Kobabashi (2004).) Thus, one can see a scale-dependent physical hierarchy in this framework, where a small probe, say rP = D/2, can nicely resolve objects with a comparable resolution, whereas a large probe, say rP = 3D/2, cannot resolve a detailed structure but it can resolve a structure with a resolution comparable to the probe size. Therefore, although a large diameter probe cannot detect smaller-scale structure, it could detect certain features associated with its scale. Based on the above simple hierarchical mechanism, a hierarchical memory system is constructed. Consider, for example, a maximum of N nanoparticles distributed in a region of a subwavelength scale. These nanoparticles can be nicely resolved by a scanning near-field microscope if the size of its fiber probe tip is comparable to the size of individual nanoparticles; in this way, the first-layer information associated with each distribution of nanoparticles is retrievable, corresponding to 2N -different codes. By using a larger-diameter fiber probe tip instead, the distribution of the particles cannot be resolved, but a mean-field feature with a resolution comparable to the size of the probe can be extracted, namely, the number of particles within an area comparable to the size of the fiber probe tip. Thus, the second-layer information associated with the number of particles, corresponding to (N + 1)-different level of signals, is retrievable. Therefore, one can access different set of signals, 2N or N + 1, depending on the scale of observation. This leads to hierarchical memory retrieval by associating this information hierarchy with the distribution and the number of nanoparticles using an appropriate coding strategy, as schematically shown in Figure 34.6a. For example, in encoding N-bit information, (N − 1)-bit signals can be encoded by the distributions of nanoparticles while associating the remaining 1-bit with the number of nanoparticles. The details of encoding/decoding strategies will be found in Naruse et al. (2005c). Simulations were performed assuming ideal isotropic metal particles to see how the second-layer signal varies depending on the number of particles using a finite-difference time-domain simulator (Poynting for Optics, a product of Fujitsu, Japan).

Here, 80â•›nm-diameter particles are distributed over a 200â•›nmradius circular grid at constant intervals. The solid circles in Figure 34.6d show calculated scattering cross sections as a function of the number of particles. A linear correspondence to the number of particles was observed. This result supports the simple physical model described above. In order to experimentally demonstrate such principles, an array of Au particles, each with a diameter of around 80â•›nm, was distributed over a SiO2 substrate in a 200â•›nm-radius circle. These particles were fabricated by a liftoff technique using electron-beam (EB) lithography with a Cr buffer layer. Each group of Au particles was spaced by 2â•›μm. A scanning electron microscope (SEM) image is shown in Figure 34.6b in which the values indicate the number of particles within each group. In order to illuminate all Au particles in each group and collect the scattered light from them, a near-field optical microscope (NOM) with a large-diameter-aperture (500â•›nm) metallized fiber probe was used in an illumination collection setup. The light source used was a laser diode with an operating wavelength of 680â•›nm. The distance between the substrate and the probe was maintained at 750â•›nm. Figure 34.6c shows an intensity profile captured by the probe, from which the second-layer information is retrieved. The solid squares in Figure 34.6d indicate the peak intensity of each section, which increased linearly. These results show the validity of hierarchical memory retrieval from nanostructures.

34.3.3â•‡Design of Unscalable Hierarchical Response The coarse graining process is usually an averaging process, meaning that the signal in the coarser layer is obtained in terms of a mean-field approximation of the fine-grained, lower-layer signals, which is also the case shown in Section 34.3.2. However, it should be noted that the optical near-field amplitude can be distributed independently at different scales of observation. In other words, the coarse graining of the optical near fields in fact provides an optical property independent of the lower-layer feature; such an unscalable hierarchical property of optical near fields will be analyzed below. Here, the angular spectrum representation of the electromagnetic field is used for discussing the hierarchy in optical near fields (Wolf and Nieto-Vesperinas 1985, Inoue and Hori 2005, Naruse et al. 2007b). This allows an analytical treatment and gives an intuitive picture of the localization of optical near fields and represents relevance/irrelevance in optical near-field interactions at different scales of observation since it describes electromagnetic fields as a superposition of evanescent waves with different decay length and corresponding spatial frequency. Suppose, for example, that there is an oscillating electric dipole, d(k) = d(k)(cosϕ(k),â•›0), on the xz plane, which is oriented parallel to the x axis. Now, consider the electric field of radiation observed at a position displaced from the dipole by R = (r||(k )cos ϕ(k ) , z (k ) ). The angular spectrum representation of the z-component of the optical near field is given by

34-9

System Architectures for Nanophotonics Text, tag, less-critical security

Number of particles

… They wanted something glossy to ensure good output dynamic range, and they wanted a human face…

Larger-scale

Distribution

Image, content, high security

Smaller-scale (a)

Intensity (a.u.)

4 3 4

1 mm

1 3

Scattering cross section (a.u.)

4

8.0

8.0

7.0

7.0

6.0

6.0

5.0

5.0

4.0

4.0

3.0

3.0

2.0

2.0

1.0

1.0

0.0

0.0 1

(b)

3

4

5

6

7

Number of particles

(d)

(c)

2

Number of nanoparticle

5

Figure 34.6â•… (a) Hierarchical memory retrieval from nanostructures. (b) SEM picture of an array of Au nanoparticles. Each section consists of up to seven nanoparticles. (c) Intensity pattern captured by a fiber probe tip whose diameter is comparable to the size of each section of nanoparticles. (d) Square marks: calculated scattering cross sections in each section, circular marks: peak intensity of each section in the intensity profile shown in (c). ∞

 iK 3  s|| Ez (R) =  ds|| f z (s|| , d(1) ,…, d(N ) ) sz  4πε 0 

∫

1

(34.6)

where N

f z (s|| , d(1) ,…, d( N ) ) =

∑d k =1

(k )

(

(

s|| s||2 − 1 cos(φ(k ) − ϕ(k ) )J1 Kr||(k )s||

)

× exp − Kz (k ) s||2 − 1 .

)

(34.7)

Here, s ∙ is the spatial frequency of an evanescent wave propagating parallel to the x axis, and Jn(x) represents Bessel functions of the first kind. Here, the term fz(s ∙ , d(1),…, d(N)) is called the angular spectrum of the electric field. In the following, a two-layer system is introduced where (1) by observing very close to the dipoles, two items of first layer information are retrieved, and (2) by observing relatively far from the dipoles, one item of second-layer information is retrieved.

Suppose that there are two closely spaced dipole pairs (so there are four dipoles in total). The dipoles d(1) and d(2) are oriented in the same direction, namely, φ(1) = φ(2) = 0, and another dipole pair, d(3) and d(4), are both oriented in the opposite direction to d(1) and d(2), namely, φ(3) = φ(4) = π. These four dipoles are located at positions shown in Figure 34.7a. Here, at a position close to the x axis equidistant from d(1) and d(2), such as at the position A1 in Figure 34.7a, the electric field is weak (logical ZERO) since (1) the angular spectrum originating from d(1) and d(2) vanishes, and (2) the electric field originating from d(3) and d(4) is small because they are far from the position A1. In fact, as shown by the dashed curve in Figure 34.7b, since the angular spectrum at position A1 oscillates, the integral of the angular spectrum, which is correlated to the field intensity at that point, will be low. For the second layer retrieval, consider the observation at an intermediate position between the dipole pairs, such as the position B in Figure 34.7a. From this position, the four dipoles effectively appear to be two dipoles that are oriented in opposite directions to each other. As shown by the solid curve in Figure 34.7b, the angular spectrum involving relatively low

34-10


B 1/6

A1

1/4

1/12 d (1)

(a)

d (2)

A2 1/4 1/16

d (3)

d (4)

Angular spectrum (a.u.)

40 30

B

20 10 0 –10 –20

A1, A2 10

(b)

20 30 Spatial frequency

40

50 (V/m)2 102

A1

B 0

A2

1

0 90 nm

B

A1 1

0

A2 1

1

10–2 10–4 10–6

60 nm

10–8 (c)

(d)

10–10

Figure 34.7â•… Examples of unscalable hierarchy in optical near fields. (a) The positions and orientations of four electric dipole systems. (b) And the corresponding angular spectrum observed at positions A1, A2, and B, indicating that the first-layer signals (obtained at A1 and A2) are logical ZEROs, while the second-layer signal (obtained at B) is logical ONE. (c) Calculated electric field intensity distribution agreed with theoretical predictions shown in (b). (d) Another electric field distribution where logical ONEs are retrieved at the first layer (A1 and A2) and logical ZERO is retrieved at the second layer (B).

spatial-frequency components shows a single peak, indicating that the electric field in the xy-plane is localized to the degree determined by its spectral width so that a logical ONE is retrievable at position B. Meanwhile, the angular spectrum observed at position A2, shown in Figure 34.7a, is identical to that observed at position A1, meaning that the electric field at A2 is also at a low level. To summarize the above mechanisms, a logical level of ONE in the second layer can be retrieved even though the two items of information retrieved in the first layer are both ZEROs; therefore, an unscalable hierarchy is achieved. As described above, one of the two first-layer signals, the electric field at A1, primarily depends on the dipole pair d(1) and d(2), and the other, the electric field at A2, depends primarily on the dipole pair d(3) and d(4). The second-layer signal is determined by all of those dipoles. Concerning such a hierarchical mechanism, it was shown that a total of eight different signal combinations were achieved by appropriately orienting the four dipoles (Naruse et al. 2007b). Numerical simulations were performed based on finite-difference time-domain methods to see how they agree with the theoretical analysis based on the angular spectrum. Four silver nanoparticles (of radius 15â•›nm) containing a virtual oscillating light source were assumed in order to simulate dipole arrays. Their positions are shown in Figure 34.7c. The first and the second layers were located 40 and 80â•›nm away from the dipole plane, respectively. The operating wavelength was 488â•›nm. The electric fields obtained at A1, A2, and B agree with the combinations of the

first- and second-layer signals to be retrieved, as shown in Figure 34.7c. As another unscalable hierarchy example, Figure 34.7d represents a situation where logical ONEs are obtained at the first layer, while logical ZERO is obtained at the second layer. These agree with the theoretical analysis based on the angular spectrum.

34.3.4â•‡Versatile Functionalities Based on Hierarchy in Optical Near-Fields The hierarchical nature has been further exploited by combining other physical mechanisms in nanophotonics. For example, we can associate one of the hierarchical layers with energy dissipation processes. Specifically, a two-layer system is demonstrated where (1) at smaller scale, called Scale 1, the system should exhibit a unique response, and (2) at a larger scale, called Scale 2, the system should output two different signals. Such a hierarchical response can be applied to functions like the traceability of optical memory in combination with a localized energy dissipation process at Scale 1 (Naruse et al. 2008c). Optical access to this memory will be automatically recorded due to energy dissipation occurring locally in Scale 1, while at the same time, information will be read out based on Scale 2 behavior. Therefore, such hierarchy enables the traceability of optical memory, which will be important for the security (confidentiality is ensured) and management of digital content. Shape-engineered metal nanostructures can achieve the hierarchy required for traceable memory (Naruse et al. 2008e). Here, two types of shapes are assumed. The first one (Shape I) has two

34-11


triangular metal plates aligned in the same direction, and the other one (Shape II) has them facing each other, as shown in Figure 34.8a. The metal is gold, the gap between the two apexes is 50â•›nm, the horizontal length of one triangular plate is 173â•›nm, the angle at the apex is 30°, and the thickness is 30â•›nm. An incident uniform plane wave with a wavelength of 680â•›nm is assumed for input light. The polarization is parallel to the x axis in Figure 34.8b. Now, Scale 1 is associated with the scale around the gap of the triangles, and Scale 2 is associated with the scale covering both of the triangles, as shown in Figure 34.8a. Regarding the optical response at Scale 1, as shown in Figure 34.8b, the electric field near the surface (1â•›nm away from the metal surface) shows an intensity nearly five orders of magnitude higher than the surrounding area. It should also be noted

that nearly comparable electric-field enhancements are observed near the apexes of Shapes I and II, which are, respectively, denoted by the squares and circles in Figure 34.8b. On the other hand, Shapes I and II exhibit different responses at Scale 2. As shown in Figure 34.8c, Shape I exhibits larger scattering cross section compared to Shape II. This indicates that a digital output is retrievable by observing the scattering from the entire structure (Scale 2), where, for example, digital 1 and 0 are associated with Shape I and Shape II, respectively. In order to experimentally demonstrate the principle, Shapes I and II were fabricated in gold metal plates on a glass substrate by a liftoff technique using EB lithography. A NOM in an illumination collection setup was used with an apertured fiber probe having a diameter of 500â•›nm, as shown in Figure 34.8d. The light

Shape I

[Scale 2] Information

“ONE”

[Scale 1] Trace

“ZERO” (a)

Shape II

173 nm

104

(b)

Scale 2: Data retrieval

Shape I Shape II

100 nm

Near-field intensity (a.u.)

Scale 1: Traceability

30 Shape I

–60

y

60

x 0

Shape II –90

–60 –50 –40 –30 –20 –10 0 Horizontal position (nm)

90

0

(c) (I1)

(II1) (I1) (II2) (I2) (II3)

(I2)

3.0

375 nm

2.5 Intensity (a.u.)

Apertured probe D = 500 nm LD

2.0 1.5 1.0

PMT l = 780 nm (d)

0

–30

0.5 (e)

(II1) 0

(II2)

1

2

(II3)

100 nm 3 4 5 6 7 8 9 Horizontal position (mm)

Figure 34.8â•… (a) Hierarchy in optical near fields by engineering the shape of metal plates at nanometer scale. (b) In Scale 1, both shapes exhibit comparable electric-field enhancement. (c) In Scale 2, they exhibit different system responses. (d) Experimental setup for Scale 2 signal retrieval. (e) Electric field intensity for Scale 2 signals for both Shapes I and II.

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source used was a laser diode with an operating wavelength of 780â•›nm. The distance between the substrate and the probe was maintained at 375â•›nm. Figure 34.8e shows the electric field intensity depending on the shape of the metal plates, where the Shape I series exhibited larger values compared to the Shape II series, as expected. Hierarchical nature in optical near-field interactions provides other unique functions. For example, the near-field photoluminescence of semiconductor quantum dots exhibits a hierarchical nature and its spectra diversity is maximized at an optimal scale of the optical near fields. This leads to a novel non-pixelated memory architecture, which can simultaneously retrieve a sequence of bits, as opposed to conventional bit-sequential pixelated architecture. The principles have been experimentally verified by InAs quantum dots (Naruse et al. 2008a). Also, a nanophotonic lock-and-key system has been demonstrated for the authentication of applications based on shape-engineered nanostructures and their associated optical near fields (Naruse et al. 2008b,d).

34.4â•‡ Summary In this chapter, fundamental nanophotonic system architectures were presented along with two principal physical features of nanophotonics: one is optical excitation transfer and the other is hierarchy in optical near-field interactions. Both of these physical features originate from light–matter interactions on the nanometer scale. It should be emphasized that those basic features provide versatile applications and functionalities besides the example demonstrations shown in the above sections. Also, there are many other degrees-of-freedom in the nanometer scale that need to be deeply understood for systems. Further exploration and attempts to exploit nanophotonics for future devices and systems will certainly be exciting.

References Akers, S. B. 1978. Binary decision diagrams. IEEE Trans. Comput. C-27:509–516. Arsovski, I., Chandler, T., and Sheikholeslami, A. 2003. A ternary content-addressable memory (TCAM) based on 4T static storage and including a current-race sensing scheme. IEEE J. Solid-State Circuits 38:155–158. Carmichael, H. J. 1999. Statistical Methods in Quantum Optics I. Berlin, Germany: Springer-Verlag. De Souza, E. A., Nuss, M. C., Knox, W. H. et al. 1995. Wavelengthdivision multiplexing with femtosecond pulses. Opt. Lett. 20:1166–1168. Goodman, J. W., Dias, A. R., and Woody, L. M. 1978. Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms. Opt. Lett. 2:1–3. Grunnet-Jepsen, A., Johnson, A. E., Maniloff, E. S. et al. 1999. Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA. Electron. Lett. 35:1096–1097.

Guilfoyle, P. S. and McCallum, D. S. 1996. High-speed low-energy digital optical processors. Opt. Eng. 35:436–442. Hori, H. 2001. Electronic and electromagnetic properties in nanometer scales. In Optical and Electronic Process of NanoMatters, ed. M. Ohtsu, pp. 1–55. Tokyo: KTK Scientific/ Dordrecht, the Netherlands: Kluwer Academic. Ingold, G.-L. and Nazarov, Y. V. 1992. Charge tunneling rates in ultrasmall junctions. In Single Charge Tunneling, eds. H. Grabert and M. H. Devoret, pp. 21–107. New York: Plenum Press. Inoue, T. and Hori, H., 2005. Quantum theory of radiation in optical near field based on quantization of evanescent electromagnetic waves using detector Mode. In Progress in Nano-Electro-Optics IV, ed. M. Ohtsu, pp. 127–199. Berlin, Germany: Springer-Verlag. Kawazoe, T., Kobayashi, K., Sangu, S. et al. 2003. Demonstration of a nanophotonic switching operation by optical near-field energy transfer. Appl. Phys. Lett. 82:2957–2959. Kawazoe, T., Kobayashi, K., and Ohtsu, M. 2005. Optical nanofountain: A biomimetic device that concentrates optical energy in a nanometric region. Appl. Phys. Lett. 86:103102 1–3. Kocher, P., Jaffe, J., and Jun, B. 1998. Introduction to differential power analysis and related attacks. Cryptography Research. http://www.cryptography.com/resources/whitepapers/ DPATechInfo.pdf Li, B., Qin, Y., Cao, X. et al. 2001. Photonic packet switching: Architecture and performance. Opt. Netw. Mag. 2:27–39. Lin, P.-F. and Kuo, J. B. 2001. A 1-V 128-kb four-way set-associative CMOS cache memory using wordline-oriented tag-compare (WLOTC) structure with the content-addressable-memory (CAM) 10-transistor tag cell. IEEE J. Solid-State Circuits 36:666–675. Liu, H. 2002. Routing table compaction in ternary CAM, IEEE Micro 22:58–64. Maier, S. A., Kik, P. G., Atwater, H. A. et al. 2003. Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides, Nat. Mater. 2:229–232. Naruse, M., Mitsu, H., Furuki, M. et al. 2004. Terabit all-optical logic based on ultrafast two-dimensional transmission gating. Opt. Lett. 29:608–610. Naruse, M., Miyazaki, T., Kawazoe, T. et al. 2005a. Nanophotonic computing based on optical near-field interactions between quantum dots. IEICE Trans. Electron. E88-C:1817–1823. Naruse, M., Miyazaki, T., Kubota, F. et al. 2005b. Nanometric summation architecture using optical near-field interaction between quantum dots. Opt. Lett. 30:201–203. Naruse, M., Yatsui, T., Nomura, W. et al. 2005c. Hierarchy in optical near-fields and its application to memory retrieval Opt. Express 13:9265–9271. Naruse, M., Kawazoe, T., Sangu, S. et al. 2006. Optical interconnects based on optical far- and near-field interactions for high-density data broadcasting. Opt. Express 14:306–313.


Naruse, M., Hori, H., Kobayashi, K. et al. 2007a. Tamper resistance in optical excitation transfer based on optical nearfield interactions. Opt. Lett. 32:1761–1763. Naruse, M., Inoue, T., Hori, H. 2007b. Analysis and synthesis of hierarchy in optical near-field interactions at the nanoscale based on Angular Spectrum. Jpn. J. Appl. Phys. 46:6095–6103. Naruse, M., Nishibayashi, K., Kawazoe, T. et al. 2008a. Scaledependent optical near-fields in InAs quantum dots and their application to non-pixelated memory retrieval. Appl. Phys. Express 1:072101 1–3. Naruse, M., Yatsui, T., Hori, H. et al. 2008b. Polarization in optical near- and far-field and its relation to shape and layout of nanostructures. J. App. Phys. 103:113525 1–8. Naruse, M., Yatsui, T., Kawazoe, T. et al. 2008c. Design and simulation of a nanophotonic traceable memory using localized energy dissipation and hierarchy of optical near-field interactions IEEE Trans. Nanotechnol. 7:14–19. Naruse, M., Yatsui, T., Kawazoe, T., Tate, N. et al. 2008d. Nanophotonic matching by optical near-fields between shapeengineered nanostructures. Appl. Phys. Express 1:112101 1–3. Naruse, M., Yatsui, T., Kim, J. H. et al. 2008e. Hierarchy in optical near-fields by nano-scale shape engineering and its application to traceable memory Appl. Phys. Express 1:062004 1–3.

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Ohtsu, M. and Hori, H. 1999. Near-Field Nano-Optics. New York: Kluwer Academic/Plenum Publishers. Ohtsu, M. and Kobayashi, K. 2004. Optical Near Fields. Berlin, Germany: Springer-Verlag. Ohtsu, M., Kobayashi, K., Kawazoe, T. et al. 2002. Nanophotonics: Design, fabrication, and operation of nanometric devices using optical near fields. IEEE J. Sel. Top. Quantum Electron. 8:839–862. Ohtsu, M., Kobayashi, K., Kawazoe, T. et al. 2008. Principles of Nanophotonics. Boca Raton, FL: Taylor & Francis. Pohl, D. W. and Courjon, D. eds. 1993. Near Field Optics. Dordrecht, the Netherlands: Kluwer Academic. Tang, Z. K., Yanase, A., Yasui, T. et al. 1993. Optical selection rule and oscillator strength of confined exciton system in CuCl thin films. Phys. Rev. Lett. 71:1431–1434. Tate, N., Nomura, W., Yatsui, T. et al. 2008. Hierarchical hologram based on optical near- and far-field responses. Opt. Express 16:607–612. Wolf, E. and Nieto-Vesperinas, M. 1985. Analyticity of the angular spectrum amplitude of scattered fields and some of its consequences. J. Opt. Soc. Am. A. 2:886–890.

35 Nanophotonics for Device Operation and Fabrication 35.1 Introduction............................................................................................................................35-1 35.2 Excitation Energy Transfer in Nanophotonic Devices.....................................................35-1 35.3 Device Operation....................................................................................................................35-3

Tadashi Kawazoe The University of Tokyo

Motoichi Ohtsu The University of Tokyo

Nanophotonic AND Gateâ•‡ •â•‡ Nanophotonic NOT Gateâ•‡ •â•‡ Interconnection with Photonic Devices

35.4 Nanophotonics Fabrication...................................................................................................35-7 Nonadiabatic Near-Field Optical CVDâ•‡ •â•‡ Nonadiabatic Near-Field Photolithography

35.5 Summary................................................................................................................................35-11 References..........................................................................................................................................35-12

35.1â•‡ Introduction The optical near field is an electromagnetic field that mediates the interaction between nanometric particles located in close proximity to each other. Nanophotonics utilizes this field to realize novel devices, fabrications, and systems. That is, a photonic device with a novel function can be operated by transferring the optical near-field energy between nanometric particles and subsequent dissipation. In such a device, the optical near field transfers a signal and carries the information. Novel photonic systems become possible by using these novel photonic devices. Furthermore, if the magnitude of the transferred optical nearfield energy is sufficiently large, structures or conformations of nanometric particles can be modified, which suggests the feasibility of novel photonic fabrications. Note that the true nature of nanophotonics is to realize “qualitative innovation” in photonic devices, fabrications, and systems by utilizing novel functions and phenomena caused by optical near-field interactions, which are impossible as long as conventional propagating light is used. On reading this note, one may understand that the advantage of going beyond the diffraction limit, that is, “quantitative innovation,” is no longer essential, but only a secondary nature of nanophotonics. In this sense, one should also note that optical near-field microscopy, that is, the methodology used for image acquisition and interpretation in a nondestructive manner, is not an appropriate application of nanophotonics because the magnitude of the optical nearfield energy transferred between the probe and sample must be extrapolated to zero to avoid destroying the sample. Quantitative innovation has already been realized by breaking the diffraction limit. Examples include the optical–magnetic

hybrid disk storage systems, nanophotonic devices and systems, and photochemical vapor deposition and photolithography. However, it is important to note that these examples also realize qualitative innovation. Details of these examples are described in this chapter.

35.2â•‡Excitation Energy Transfer in Nanophotonic Devices Kagan et al. observed the energy transfer among CdSe quantum dots (QDs) coupled via a dipole–dipole inter-dot interaction [1]. Crooker et al. also studied the dynamics of the exciton energy transfer in close-packed assemblies of monodisperse and mixedsize CdSe nanocrystal QDs and reported the energy-dependent transfer rate of excitons from smaller to larger dots [2]. These examples are based on the optical near-field interaction. The physical model for the unidirectional resonant energy transfer between QDs via the optical near-field interaction has been presented, and the optically forbidden energy transfer among randomly dispersed CuCl QDs has been demonstrated experimentally using optical near-field spectroscopy [3]. The theoretical analysis and temporal evolution of the energy transfer via the optical near-field interaction were discussed in Ref. [4]. This section reviews the principles of nanophotonic devices and experimental works involving the direct observation of energy transfer from the exciton state in a CuCl QD to the optically forbidden exciton state in another CuCl QD using optical near-field spectroscopy. Cubic CuCl QDs embedded in an NaCl matrix have the potential to be an optical near-field coupling system that exhibits 35-1

35-2


the optically forbidden energy transfer. This is made possible because for this system, other forms of energy transfer, such as carrier tunneling and Coulomb coupling, can be neglected as the carrier wave function is localized in the QDs; this occurs because the potential depth exceeds 4â•›eV and the Coulomb interaction is weak due to the small exciton Bohr radius of 0.68â•›nm in CuCl [3]. The energy transfer via a propagating light is also negligible, since the optically forbidden transition in nearly perfect cubic CuCl QDs is used; that is, the transition to the confined exciton energy levels has an even principal quantum number [5]. So far, this type of energy transfer has not been observed directly because such a nanometric system is usually extremely complex. However, CuCl QDs in a NaCl matrix is a very simple system. The translational motion of the exciton center of mass is quantized due to the small exciton Bohr radius for CuCl QDs, and CuCl QDs become cubic in a NaCl matrix [6–8]. The potential barrier of CuCl QDs in a NaCl crystal can be regarded as infinitely high, and the energy eigenvalues Enx,ny,nz for the quantized Z3 exciton energy level (nx, ny, nz) in a CuCl QD with side length L depends on the values of quantum numbers nx, ny, nz, and the effective side length d = (L − aB) found after considering the dead layer correction [6], where aB is an exciton Bohr radius. The exciton energy levels with even quantum numbers are dipole-forbidden states, which are optically forbidden [5]. However, the optical near-field interaction is finite for such coupling to the forbidden energy state [9]. Figure 35.1 shows schematic drawings of different-sized cubic CuCl QDs (A and B) and the confined exciton Z3 energy levels. Here, d and 2d are the effective side lengths of cubic QDs A and B, respectively. The quantized exciton energy levels of (1,1,1) in QD A and (2,1,1) in QD B resonate with each other. Under this Optical near field 2d d

QD A QD B

(a)

(1,1,1)

τsub

τet n

io iss

(b)

Em

(2,1,1) (1,1,1)

τex

FIGURE 35.1â•… (a) Schematic drawings of closely located cubic CuCl QDs A and B with effective side lengths (L – aB) of d and 2d, respectively, where L and aB are the side lengths of the cubic quantum dots and the exciton Bohr radius, respectively. (b) Their exciton energy levels. nx, ny, and nz represent quantum numbers of an exciton. EB is the exciton energy level in a bulk crystal.

resonant condition, the coupling energy of the optical near-field interaction is given by the following Yukawa function [9,10]:

V (r ) =

 π 3m p A exp  −  a me r

 r , 

where A is a proportional constant r is the separation between the two QDs a is the size of the QD mp is the effective mass of an exciton–polariton me is the effective mass of an electron Assuming that the two CuCl QDs in the NaCl matrix have side lengths 5 and 7â•›nm (a size ratio of 1 : 2 ) and the inter-dot distance is 6.1â•›nm, then the coupling energy V(r) is 5.05â•›μeV. This corresponds to an energy transfer time of 130â•›ps due to the optical near-field coupling, which is much shorter than the exciton lifetime of a few nanoseconds. In addition, the inter-sublevel transition τsub, from higher exciton energy levels to the lowest, as shown in Figure 35.1, is generally less than a few picoseconds [11] and is much shorter than the transfer time τet. Therefore, most of the energy of the excitation in a cubic CuCl QD with a side length of d is transferred to the lowest exciton energy level in a neighboring QD with a side length of 2d and recombines radiatively in the lowest level. The CuCl QDs embedded in NaCl matrix used experimentally were fabricated using the Bridgman method and successive annealing, and the average size of the QDs was found to be 4.3â•›nm. The sample was cleaved just before the near-field optical spectroscopy experiment to keep the sample surface clean. The cleaved surface of a 100â•›μm-thick sample was sufficiently flat for the experiment; that is, its roughness was less than 50â•›nm, at least within a few microns squared. A 325â•›nm He–Cd laser was used as the light source. A double-tapered fiber probe was fabricated using chemical etching and a 150â•›nm gold coating was applied [12]. A 50â•›nm aperture was fabricated using the pounding method [13]. The spatial distributions of the luminescence intensity, that is, near-field optical microscope images, clearly show anti-correlation features in their intensity distributions. This anti-correlation feature can be clarified by noting that these spatial distributions in luminescence intensity represent not only the spatial distributions of the QDs, but also some kind of resonant interaction between the QDs. This interaction induces energy transfer from QDs A (L = 4.6â•›nm) to QDs B (L = 6.3â•›nm) because most of the 4.6â•›nm QDs located close to 6.3â•›nm QDs cannot emit light, but instead transfer the energy to the 6.3â•›nm QDs. As a result, in the region containing embedded 6.3â•›nm QDs, the luminescence intensity from 4.6â•›nm QDs is low, while the intensity from the 6.3â•›nm QDs is high at the corresponding position. This anti-correlation feature originates from the nearfield energy transfer, which appears for every pair of QDs with

35-3

Nanophotonics for Device Operation and Fabrication

different sizes to satisfy the resonant conditions of the confinement exciton energy levels. This is the first spatially resolved observation of energy transfer between QDs via an optical near field. This evidence of the near-field energy transfer between QDs can give rise to a variety of applications, as shown in the following sections.

QDinput A d Input A

35.3.1â•‡ Nanophotonic AND Gate Operation of a nanophotonic AND gate using cubic CuCl QDs embedded in a NaCl matrix has been demonstrated [16,17]. When closely spaced QDs with quantized energy levels resonate with each other, near-field energy is transferred between them, even if the transfer is optically forbidden, as noted in

√2d

True

Output False

Input B

35.3â•‡ Device Operation Optical fiber transmission systems require increased integration of photonic devices for higher data transmission rates. Since conventional photonic devices, for example, diode lasers and optical waveguides, have to confine light waves within their cavities and core layers, respectively, their minimum sizes are limited by the diffraction of light [14]. Therefore, they cannot meet the size requirement, which is beyond this diffraction limit. An optical near field is free from the diffraction of light and enables the operation and integration of nanometric optical devices. That is, by using a localized optical near field as the carrier, which is transmitted from one nanometric element to another, the above requirements can be met. Based on this idea, nanometer-sized photonic devices have been proposed, which are called nanophotonic devices [15]. A nanometric all-optical AND gate (i.e., a nanophotonic switch) is one of the most important devices for realizing nanophotonic integrated circuits, and the operation of a nanophotonic AND gate has been already demonstrated using a coupled QD system. A logic gate, for example, an AND gate and a NOT gate, is a block in a digital system. Logic gates have some inputs and some outputs, and every terminal is under one of two binary conditions, low (0) or high (1), given by different optical intensities for the optical device. The logic state of the input terminal is controlled by the optical input signal, and the logic state of the output terminal changes depending on the logic state of the input terminals. An intensity of approximately zero and a much higher intensity are preferable in the low and high logic states, respectively, such that the ratio of high to low intensity exceeds 30â•›db. For nanophotonic devices, the high state (1), which is called “true,” is defined simply as the higher intensity state and the low state (0), which is called “false,” is defined as the lower intensity state. Section 35.3.1 presents the operation of nanophotonic AND gates using three CuCl QDs [16,17], while a nanophotonic NOT gate using CuCl QDs is outlined in Section 35.3.2. The optically forbidden energy transfer between neighboring nanostructures via the optical near-field interaction, which was reviewed in Section 35.2, is a key phenomenon for these operations.

QDoutput

False

n

o ati

la x Re

QDinput B 2d

(a)

QDinput A d Input A True

QDoutput √2d Output True

Input B True (b)

QDinput B 2d

FIGURE 35.2â•… Principle of AND-gate operation. (a) and (b) The “false” and “true” states of the nanophotonic AND gate, respectively.

Section 35.2. The output is “true” when both inputs are “true”; otherwise, the output is “false.” Figure 35.2a and b explains the “false” and “true” states of the proposed nanophotonic AND gate. Three cubic QDs, QDinput A, QDinput B, and QDoutput, are used as the two inputs and output ports of the AND gate, respectively. Assuming an effective size ratio of 1 : 2 : 2 , the quantized energy levels (1,1,1) in QDinput A, (2,1,1) in QDoutput, and (2,2,2) in QDinput B resonate with each other. Furthermore, energy levels (1,1,1) in QDoutput and (2,1,1) in QDinput B also resonate. In the “false” state operation (Figure 35.2a), for example, input A is “true” and input B is “false,” almost all of the exciton energy in QDinput A is transferred to the (1,1,1) level in the neighboring QDoutput, and then to the (1,1,1) level in QDinput B. Therefore, the input energy escapes to QDinput B, and consequently no optical output signals are generated from QDoutput. This means that the output is “false.” In the “true” state (Figure 35.2a) when inputs A and B are both “true,” the escape route to QDinput B is blocked by the excitation of QDinputB due to state filling in QDinput B on applying the input B signal. Therefore, the input energy is transferred to QDoutput and an optical output signal is generated. This means that the output is “true.” These operating principles are realized with the condition τex > τet > τsub, where τex, τet, and τsub are the exciton lifetime, energy transfer time between QDs, and inter-sublevel transition time, respectively. Since τex, τet, and τsub are a few nanoseconds, 100â•›ps, and a few picoseconds, respectively, for the CuCl QDs used in a NaCl matrix, the condition of operation described in Section 35.2 is satisfied.

35-4


In an experiment using CuCl QDs embedded in a NaCl matrix, a double-tapered UV fiber probe was fabricated using chemical etching and coated with 150â•›nm-thick aluminum (Al) film. An aperture less than 50â•›nm in diameter was formed by the pounding method [13]. To confirm the AND-gate operation, the fiber probe was used to search for a trio of QDs that had an effective size ratio of 1 : 2 : 2. Since the homogeneous linewidth of a CuCl QD increases with the sample temperature [18,19], the allowance in the resonatable size ratio is 10% at 15â•›K. The separation of the QDs must be less than 30â•›nm for operation of the proposed AND gate because the energy transfer time increases with the separation; moreover, it must be shorter than the exciton lifetime. It is estimated that at least one trio of QDs that satisfies these conditions exists in a 2 × 2â•›μm scan area. To demonstrate AND-gate operation, a QD trio had to be found in the sample, as shown in Figure 35.2. Near-field photoluminescence (PL) pump-probe spectroscopy was used to find the QD trio. Figure 35.3 shows the PL spectrum obtained at the position where the QD trio exists. In this figure, peaks appear at the positions of the (1,1,1) levels in the 4.6 and 3.5â•›nm QDs. The appearance of the satellite peaks means that the AND-gate system proposed in Figure 35.2 was present in the area under the probe. In other words, a trio of cubic QDs with sizes of 3.5, 4.6, and 6.3â•›nm existed. Since their effective respective sizes L − aB were 2.8, 3.9, and 5.6â•›nm (aB: an exciton Bohr radius of 0.7â•›nm in CuCl), the size ratio was close to 1 : 2 : 2 and they could be used as QDinputA, QDoutput, and QDinputB, respectively. The pumping to the 6.3â•›nm QD blocks the energy transfer from the 3.5 and 4.6â•›nm QDs to the 6.3â•›nm QD due to state filling of the 6.3â•›nm QD, and the 3.5 and 4.6â•›nm QDs emit light that results in the peaks in Figure 35.3. Therefore, a QD trio for a nanophotonic AND gate was found. The PL peak from the 4.6â•›nm QD corresponds to the output signal in Figure 35.2b. 7 6.3 nm QD

6

PL intensity (a.u.)

4.6 nm QD 5

The PL intensity from the 4.6â•›nm QD was 0.05–0.02 times the PL intensity from the 6.3â•›nm QD, which was obtained with the probe laser only. This value is quite reasonable considering the pumping pulse energy density of 10â•›μW/cm 2 because the probability density of excitons in a 6.3â•›nm QD is 0.1–0.05 [18], which is close to the PL intensity from the 4.6â•›nm QD. This result indicates that the internal quantum efficiency of the AND-gate system is close to unity. In the experiment examining the AND-gate operation, the second harmonics of Ti:sapphire lasers (wavelengths 379.5 and 385â•›nm), which were tuned to the (1,1,1) exciton energy levels of QDinputA and QDinputB, respectively, were used as the signal light sources for inputs A and B. The output signal was collected by the fiber probe, and its intensity was measured using a cooled microchannel plate after passing through three interference filters of 1â•›nm bandwidth tuned to the (1,1,1) exciton energy level in QDoutput at 383â•›nm. Figure 35.4a and b shows the spatial distribution of the output signal intensity in the “false” state (i.e., with one input signal only) and in the “true” state (i.e., with both input A and input B signals) using near-field spectroscopy at 15â•›K. The insets in this figure are schematic drawings of the existing QD trio used for the AND gate, which was confirmed by the nearfield PL spectra. Here, separation of the QDs by less than 20â•›nm was estimated theoretically from time-resolved PL measurements, as explained in the next paragraph (see the illustrations in Figure 35.4). In the “false” state, no output signal was observed because the energy of the input signal was transferred to QDinputB and swept out as PL at 385â•›nm. To quench the output signal in the “false” state, which was generated by accumulating excitons in QDinputB, the input signal density to QDinputA was regulated to less than 0.1 excitons in QDinputB. In the “true” state, a clear output signal was obtained in the dashed circle of Figure 35.4b. The output signal was proportional to the intensity of the control signal, which had a density of 0.01–0.1 excitons in QDinputB. Next, the dynamic properties of the nanophotonic AND gate were evaluated using the time-correlation single-photon counting method. As a pulse-input B signal source, the 385â•›nm second harmonic of a mode-locked Ti:sapphire laser was used. The repetition rate of the laser was 80â•›MHz. To avoid cross talk originating from spectral broadening of the pulse duration between input signals

3.5 nm QD 4 3 2 1 375

380

385 Wavelength (nm)

390

395

FIGURE 35.3â•… Near-field differential PL spectra measured at the position of a QD trio acting as a nanophotonic AND gate.

(a)

20 nm

(b)

20 nm

FIGURE 35.4â•… Spatial distribution of the output signal from the nanophotonic AND gate in the “false” (a) and “true” (b) states measured using near-field microscopy.

35-5


Intensity (a.u.)

3

Output signal

QDIN

Optical power

√2d

2

Output True

Input

1

QDOUT (1 + α) d

False (a)

0

Input B signal pulses QDIN 0

10

Time (ns)

20

√2d

30

FIGURE 35.5â•… Temporal evolution of the output signal from the nanophotonic AND gate located in the dashed circle in Figure 35.4b. The duration and repetition rate of the control pulse were 10â•›ps and 80â•›MHz, respectively.

A and B, the pulse duration of the mode-locked laser was set to 10â•›ps. The time resolution of the experiment was 15â•›ps. Figure 35.5 shows the temporal evolution of the input B pulse signal (lower part) applicable to QDinputB and the output signal (upper part) from QDoutput. The output signal increases synchronously with the input B pulse within less than 100â•›ps, which agrees with the theoretically expected result based on the Yukawa model [9]. As this signal rise time is determined by the energy transfer time between the QDs, the separation between the QDs can be estimated from the rise time as being less than 20â•›nm; the rise time can be shortened to a few picoseconds by decreasing the separation of the QDs. Since the decay time of the output signal is limited by the exciton lifetime, this nanophotonic AND gate can be operated at a few hundred megahertz, and the operating frequency can be increased to several gigahertz by exciton quenching using plasmon coupling [20]. The output signal ratio between “true” and “false” was about 10, which is sufficient for use as an all-optical AND gate, and can be increased using a saturable absorber and electric field enhancement of the surface plasmon [21]. The advantages of this nanophotonic device are its small size and high-density integration capability based on the locality of the optical near field. The figure of merit (FOM) of an optical AND gate should be more important than the switching speed. Here, the FOM is defined as F = C/VtswPsw, where C, V, tsw, and Psw are the “true”–“false” (ON–OFF) ratio, volume of the device, switching time, and switching energy, respectively. The FOM of the nanophotonic AND gate is 10–100 times higher than that of conventional photonic gates because its volume and switching energy are 10−5 times and 10−3 times those of conventional photonic gates, respectively.

35.3.2â•‡ Nanophotonic NOT Gate A nanophotonic NOT gate is a key device for realizing a functionally complete set of logic gates for nanophotonic systems, and its

Optical power

Output Input True (b)

R

on

ati

x ela

QDOUT (1 + α) d

False

FIGURE 35.6â•… A nanophotonic NOT gate. (a) and (b) Schematic explanation of the “true” and “false” states using cubic QDs.

operation is demonstrated in this section using CuCl QDs [22]. Figure 35.6 shows a schematic explanation of the nanophotonic NOT gate. QDIN and QDOUT correspond to the input and output terminals of the NOT gate, respectively. Assuming a pair of QDs with a size ratio of 1 + α : 2 (α 40â•›K, the unidirectional energy transfer is obstructed by the thermal activation of excitons in the QDs. A 325â•›nm He–Cd laser was used as the excitation light source. A double-tapered UV fiber probe with an aperture 20â•›nm in diameter was fabricated using chemical etching and coated with a 150â•›nm-thick Al film to ensure sufficiently high detection sensitivity [12]. Figure 35.9b shows the typical spatial distribution of the PL from QDs operating as an optical nanofountain. The bright spot inside the dashed circle corresponds to a spurt from an optical nanofountain, i.e., the focal spot of the nanometric optical condensing device. The diameter of the focal spot was less than 20â•›nm, which was limited by the spatial resolution of the nearfield spectrometer. From the Rayleigh criterion (i.e., resolution = 0.61â•›.â•›λ/NA) [26], its numerical aperture (NA) was estimated to be 12 for λ = 385â•›nm. To demonstrate the detailed operating mechanism of the optical nanofountain, we show the size-selective PL intensity distribution; that is, the photon energy is shown in Figure 35.9c. Here, the collected PL photon energy, Ep, was 3.215â•›eV ≤ Ep ≤ 3.350â•›eV, which corresponds to the PL from QDs of size 2.5â•›nm ≤ L ≤ 10â•›nm. The area scanned by the probe are

equivalent to those in Figure 35.9b. The PL intensity distribution is shown using a gray scale. This device can also be used as a frequency selector based on the resonant frequency of the QDs, which can be applied, for example, to frequency domain measurements, multiple optical memories, multiple optical signal processing, and frequency division multiplexing. Practical nanophotonic devices for room-temperature operation are under development using III–V compound semiconductor QDs [27] and ZnO nanorods [28].

35.4â•‡ Nanophotonics Fabrication This section presents the nonadiabatic processes involved in optical chemical vapor deposition (CVD) and photolithography. These methods have realized qualitative innovation in nanofabrications by utilizing the spatially localized nature of optical near fields.

35.4.1â•‡ Nonadiabatic Near-Field Optical CVD Conventional optical CVD utilizes a two-step process: photodissociation and adsorption. For photodissociation, a propagating light must resonate the reacting molecular gases to excite molecules from the ground state to an excited electronic state. The Franck–Condon principle holds that this resonance is essential for excitation. The excited molecules then relax to the dissociation channel, and the dissociated atoms adsorb to the substrate surface. However, a nonadiabatic photodissociation process is observed in near-field optical chemical vapor deposition (NFO-CVD) under the nonresonant condition of the electronic transition, which violates the Franck–Condon principle. This section discusses the nonadiabatic NFO-CVD of nanometric Zn dots and presents experimental results based on the exciton–Â� phonon–polariton (EPP) model. Figure 35.10 shows the cross-sectional profiles of the shearforce topographical images after NFO-CVD for photon energies of 5.08â•›eV (λ = 244â•›nm; broken curve) and 2.54â•›eV (λ = 488â•›nm; solid curve) of Zn dots deposited on a sapphire substrate in

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244 nm

488 nm Atomic-level step

5

45 nm

2

1

Height (nm)

Height (nm)

10

(a)

50 nm 0

0 –100

–50

0 Position (nm)

50

100

(b)

FIGURE 35.10â•… Cross-sectional profiles of the shear-force topographical images of the deposited Zn patterns.

atomic-level steps [29]. In the experiment, diethylzinc (DEZn) was used as the CVD gas source. For the broken curve, the laser power was 1.6â•›μW and the irradiation time was 60â•›s. Before carrying out NFO-CVD, 0.4â•›nm-high atomic-level step structures were clearly observed on the sapphire substrate. After NFOCVD, they disappeared and a deposited Zn dot less than 50â•›nm in diameter was seen at the center of the image. This occurred because the optical near field deposited the Zn dot directly under the apex of the fiber probe. Furthermore, since high-intensity propagating light leaks from a bare fiber probe, that is, one without a metallic coating, and is absorbed by the DEZn, a Zn layer was deposited on top of the atomic-step structures. For the solid curve, the laser power was 150â•›μW and the irradiation time was 75â•›s. The photon energy (2.54â•›eV) was higher than the dissociation energy of DEZn, but it was still lower than the absorption edge of DEZn [30]. Therefore, it was not absorbed by DEZn. However, a Zn dot less than 50â•›nm in diameter appears at the center. While using conventional CVD with propagating light, a Zn film cannot be grown using a light source with a photon energy lower than the absorption edge (4.13â•›eV: λ = 300â•›nm) [31]. Deposited Zn dots were observed on the substrate just below the apex of the fiber probe using NFO-CVD. The atomic-level steps in this figure are still observed, despite the leakage of the propagating light from the bare fiber probe. These curves confirm that Zn dots with a full-width at half-maximum (FWHM) of 30â•›nm were deposited in the region where the optical near field is dominant. The dashed curve has 4â•›nm-high tails on both sides of the dot, which represent the deposition caused by the leaked propagating light. This deposition process is based on the conventional adiabatic photochemical process. In contrast, the solid curve has no tails; therefore, it is clear that the leaked 488â•›nm propagating light did not deposit a Zn layer. Note that a Zn dot 30â•›nm in diameter without tails was deposited under nonabsorbed conditions (λ = 488â•›nm). Figure 35.11 shows shear-force topographical images of the sapphire substrate after NFO-CVD using an optical near

40 nm

(c)

FIGURE 35.11â•… Shear-force topographical images after NFO-CVD at wavelengths of λ = 325â•›nm (a), 488â•›nm (b), and 684â•›nm (c). The laser output power and irradiation time for deposition were 2.3â•›μW and 60â•›s (a), 360â•›μW and 180â•›s (b), and 1â•›mW and 180â•›s (c), respectively.

field with photon energies of 3.81â•›eV (λ = 325â•›nm) (a), 2.54â•›eV (λ = 488â•›nm) (b), and 1.81â•›eV (λ = 684â•›nm) (c). The respective laser power and irradiation time were (a) 2.3â•›μW and 60â•›s, (b) 360â•›μW and 180â•›s, and (c) 1â•›mW and 180â•›s. The high quality of the deposited Zn was confirmed by x-ray photoelectron spectroscopy. Furthermore, PL was observed from ZnO dots, which were fabricated by oxidizing the Zn dots deposited by NFOCVD [8]. In Figure 35.11a, the photon energy (ħω) exceeds the dissociation energy (Ed) of DEZn, and is close to the absorption band edge (Eabs) of DEZn, i.e., ħω > Ed and ħω ≅ Eabs [30]. The diameter (FWHM) and height of the topographical image were 45 and 26â•›nm, respectively. The small tail (shown by the dotted curve) represents a Zn layer less than 2â•›nm thick, and was deposited by the propagating light leaking from the bare fiber probe. This deposition is possible because the DEZn absorbs some of the propagating light at ħω = 3.18â•›eV. The very high peak suggests that the optical near field enhances the photodissociation rate at this photon energy because its intensity increases rapidly at the apex of the fiber probe. In Figure 35.11b, the photon energy still exceeds the dissociation energy of DEZn, but is lower than the absorption band edge of DEZn, i.e., Eabs > ħω > Ed [30]. The diameter and height of the image were 50 and 24â•›nm, respectively. This image has no tail because Zn was not deposited by the high-intensity propagating light leaking from the bare fiber probe. This confirmed that the photodissociation of DEZn and Zn deposition occurred only with an optical near field of ħω = 2.54â•›eV. Figure 35.11c represents the cases ħω < Ed and ħω < E abs. Zn dots were deposited successfully at these low photon energies.

35-9

Nanophotonics for Device Operation and Fabrication Photon flux (×1011 photons/s) 101

102

Potential energy

Deposition rate (atoms/s)

108

106

104

Eabs

Molecular vibration levels 3 2

Dissociation energy Eg (2.26 eV)

1

Ground state Inter-nuclear distance

102

100 100

Triplet state

Excited states

Absorption

100

101

102

Photon flux (×1014 photons/s)

103

FIGURE 35.12â•… The relationship between the photon flux and the rate of Zn deposition. The dotted, solid, and dashed curves represent the calculated values fitted to the experimental results.

The topographical image showed dots with a diameter of 40â•›nm and a height of 2.5â•›nm. The experimental results in Figure 35.11 demonstrate dissociation based on a nonadiabatic photochemical process that violates the Franck–Condon principle. To discuss this novel dissociation process quantitatively, Figure 35.12 shows the relationship between the photon flux (I) and the deposition rate of Zn (R). For ħω = 3.81â•›eV (▲), R is proportional to I. For ħω = 2.54â•›eV (■) and 1.81â•›eV (●), higher order dependencies appear and R is fitted by the third-order function R = aħω I + bħωI2 + cħωI3. The respective values of aħω , bħω , and cħω are a3.81 = 5.0 × 10−6, b3.81 = 0, and c3.81 = 0 for ħω = 3.81â•›eV, a2.54 = 4.1 × 10−12, b2.54 = 2.1 × 10−27, and c2.54 = 1.5 × 10−42 for ħω = 2.54â•›eV, and a1.81 = 0, b1.81 = 4.2 × 10−29, and c1.81 = 3.0 × 10−44 for ħω = 1.81â•›eV. The results of fitting are shown with the solid, dashed, and dotted curves in Figure 35.12. Since no conventional photochemical processes, for example, the Raman process and two-photon absorption, can explain these experimental results, the discussion below uses a unique theoretical model based on the discussion in Ref. [32]. Figure 35.13 shows the potential curves of an electron in a DEZn molecular orbital drawn as a function of the internuclear distance of the C–Zn bond, which is involved in photodissociation [30]. The relevant energy levels of the molecular vibration mode are indicated by the horizontal broken lines in each potential curve. When a propagating light is used, photo-absorption (indicated by the white arrow) triggers the dissociation of DEZn [33]. With an optical near field nonresonant to the electronic state, there are three possible origins of photodissociation [34]: (1) the multiple-photon absorption process; (2) a multiple-step transition process via the intermediate energy level induced by the fiber probe; and (3) multiple-step transition via an excited state of the molecular vibration mode. Case (1) is negligible because the optical power density was less than 10â•›kW/cm2.

FIGURE 35.13â•… Potential curves of an electron in DEZn molecular orbitals. The relevant energy levels of the molecular vibration modes are indicated by the horizontal broken lines.

Case (2) is also negligible because the DEZn was dissociated by ultraviolet–near-infrared light, although DEZn does not have relevant energy levels over such a broad wavelength region. As a result, the experimental results strongly support Case (3). That is, the photodissociation is caused by a transition to an excited state via a molecular vibration mode, which involves three multiple-step excitation processes, as shown in Figure 35.13. Since the system is strongly coupled with the vibration state, it must be considered a nonadiabatic system. For this consideration, an exciton–phonon polariton (EPP) model was presented in Ref. [32]. The EPP model holds that the optical near fields excite the molecular vibration mode due to the steep spatial gradient. Since the optical near-field energy distribution is spatially inhomogeneous in a molecule due to its gradient, the electrons respond inhomogeneously. As a result, the molecular vibration modes are excited because the molecular orbital changes and the molecule is polarized as a result of the inhomogeneous response of the electrons. The EPP model describes this excitation process quantitatively. The EPP is a quasiparticle, which is an exciton–polariton carrying the phonon (lattice vibration) generated by the steep spatial gradient of the optical field energy distribution. In contrast, since the propagating light energy distribution is homogeneous in a molecule, only the electrons in the molecule respond to the electric field of the propagating light. Therefore, the propagating light cannot excite the molecular vibration. Zinc-bis(acetylacetonate) (Zn(acac)2) has never been used for conventional optical CVD due to its low optical activity. With NFO-CVD, however, the optical near field can activate the molecule nonadiabatically and the dissociated Zn atom is adsorbed under the fiber probe. Figure 35.14a shows a shear-force topographical image of Zn deposited on a sapphire substrate. The laser power and irradiation time were 1â•›mW and 15â•›s, respectively. The Zn dot was 70â•›nm in diameter and 24â•›nm high [35,36]. The chemical stability of Zn(acac)2 keeps the substrate surface clean and helps to fabricate an isolated nanostructure. Figure 35.14b shows the shear-force topographical image of a deposited Zn dot that is among the smallest ever fabricated using

35-10

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Photomask 24 nm

70 nm

Photoresist Si substrate

(a)

(a)

(b) 0.3 nm

5 nm

(c) 0

(b)

FIGURE 35.14â•… Shear-force topographical images after NFO-CVD using Zn(acac)2 with a 457â•›nm-wavelength light source. (a) A deposited Zn dot with a diameter of 70â•›nm and height of 24â•›nm. (b) A deposited Zn with a diameter of 5â•›nm and height of 0.3â•›nm.

NFO-CVD (5â•›nm in diameter and 0.3â•›nm high). The deposition conditions consisted of Zn(acac)2 at a pressure of 70â•›mTorr in the CVD chamber and a laser wavelength, power, and irradiation time of 457â•›nm, 65â•›μW, and 30â•›s, respectively.

35.4.2â•‡Nonadiabatic Near-Field Photolithography Section 35.4.1 reviewed a unique nonadiabatic photochemical reaction, which was explained using the EPP model. According to this model, the nonadiabatic photochemical reaction can be considered a universal phenomenon and is applicable to several photochemical processes. This section reviews the application of the nonadiabatic photochemical reaction to photolithography, which can be called nonadiabatic photolithography [36,37]. For the mass production of photonic and electronic devices, nonadiabatic photolithography can be used because conventional photolithographic components can be applied to this system. The wave properties of propagating light cause problems for high-resolution photolithography due to diffraction and the dependence on the coherency and polarization of the light source. To fabricate high-density corrugations, the optical coherent length is too long compared to the separation between adjacent corrugation elements, even when a Hg lamp is used. In addition, the absorption by the photoresist is insufficient to suppress interference of scattered light. Furthermore, since the intensity of the propagating light transmitted through a

2000 4000 Position (nm)

6000

FIGURE 35.15â•… Experimental results of nonadiabatic photolithography. (a) A schematic of the photomask and Si substrate on which the photoresist (OFPR-800) was spin-coated. (b) Atomic force microscopy images of photoresist OFPR-800 exposed to the g-line of a Hg lamp. (c) AFM images of photoresist OFPR-800 developed after a 4â•›h exposure with a 672â•›nm laser.

photomask strongly depends on its polarization, the photomask must be designed while considering these dependences. In contrast, the outstanding advantage of nonadiabatic photolithography is that it is free from these problems. Figure 35.15a shows a schematic configuration of the photomask and the Si substrate on which the photoresist (OFPR-800: Tokyo-Ohka Kogyo) was spin-coated. They were used in contact mode. Figure 35.15b and c shows atomic force microscopy (AFM) images of the photoresist surface after development. Figure 35.15b shows the result obtained using conventional photolithography. The g-line (436â•›nm) from a Hg lamp was used as the light source. The fabricated pattern of corrugation was an exact replica of the photomask. Conversely, with nonadiabatic photolithography using a 672â•›nm-wavelength light source, the grooves on the photoresist appeared along the edges of the Cr mask pattern, as shown in Figure 35.15c. The corrugated pattern was 30â•›nm deep. The line width was 150â•›nm, which was narrower than the wavelength of the light source. On the photomask, a steep spatial gradient of optical energy distribution is expected due to optical near fields, while direct irradiation with 672â•›nm light cannot expose the photoresist. This demonstrated that the photoresist was patterned using a nonadiabatic process. Figure 35.16 shows AFM images of another photoresist surface (TDMR-AR87 for the 365â•›nm-wavelength i-line from a Hg lamp: Tokyo-Ohka Kogyo) after development. Figure 35.16a shows the corrugated pattern fabricated using linearly polarized g-line light. Two-dimensional arrays of circles and T-shapes have also been fabricated successfully on this photoresist

35-11


(a)

1000 nm

(b) 1000 nm

(c)

1000 nm

FIGURE 35.16â•… Experimental results of nonadiabatic photolithography. (a) AFM images of photoresist TDMR-AR87 exposed to the linearly polarized g-line of a Hg lamp for 3â•›s. (b) AFM images of photoresist TDMR-AR87 exposed to the linearly polarized g-line of a Hg lamp for 10â•›s using a circle-shaped array photomask. (c) AFM images of photoresist OTDMR-AR87 developed after a 40â•›s exposure to the g-line of a Hg lamp using a T-shaped array photomask.

(see Figure 35.16b and c). This would be impossible using adiabatic photolithography due to its polarization-dependent nature and interference effects. An optically inactive electron beam (EB) resist film (ZEP520: ZEON) can also be patterned nonadiabatically. Figure 35.17 shows an AFM image of the developed EB resist surfaces. The light source was the third harmonic of a Q-switched Nd:YAG laser and the exposure time was 5â•›min. A two-dimensional array of 1â•›μm-diameter disks was fabricated successfully, even on the EB resist, which would be impossible using propagating light. The developed pattern had a depth of 70â•›nm, which is sufficient for the subsequent etching of the substrate. Since the EB resist film has an extremely smooth surface, the homogeneity in the contact with the photomask was improved. This suggests that a smooth organic or inorganic thin film can be used as a photoresist irrespective of its optical inactivity.

1000 nm

FIGURE 35.17â•… AFM image of the surface of an electron beam resist exposed for 5â•›min using a Q-switched laser (355â•›nm) and a circle-shaped (1â•›μm diameter) array photomask.

35.5â•‡ Summary After nanophotonics was proposed by M. Ohtsu in 1993 [38], it now exists as a novel field of optical technology in nanometric space. However, the name “nanophotonics” is occasionally used for photonic crystals [39], plasmonics [40], metamaterials [41,42], silicon photonics [43], and QD lasers [44] using conventional propagating lights. For example, plasmonics utilizes the resonant enhancement of the light in a metal by exciting free electrons. The letters “on” in the word “plasmon” represent the quanta, or the quantum mechanical picture of the plasma oscillation of free electrons in a metal. However, plasmonics utilizes the classical wave optical picture using conventional terminology, such as the refractive index, wave number, and guided mode. Even when a metal is irradiated with light that obeys the laws of quantum mechanics, the quantum mechanical property is lost because the light is converted into the plasma oscillation of electrons, which has a short phase relaxation time. Furthermore, the energy transferred via this interaction must be dissipated in the nanometric particles or adjacent macroscopic materials to fix the position and magnitude of the transferred energy. Since plasmonics does not deal with this local dissipation of energy, it is irrelevant for quantitative innovation by breaking the diffraction limit, or for qualitative innovation. Local energy transfer and its subsequent dissipation have become possible only in nanophotonics by using optical near fields [45,46]. Here, we should consider the stern warning by C. Shannon on the casual use of the term “information theory,” which was a trend in the study of information theory during the 1950s [47]. The term “nanophotonics” has been used in a similar way, although some work in “nanophotonics” is not based on optical near-field interactions. For the true development of nanophotonics, one needs deep physical insights into the virtual exciton– polariton and the nanometric subsystem composed of electrons and photons.

35-12


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32. Kobayashi, K., Kawazoe, T., and Ohtsu, M., 2008. Localized photon model including phonons’ degrees of freedom. In M. Ohtsu (ed.) Progress in Nano-Electro-Optics VI, SpringerVerlag, Berlin, Germany, pp. 41–66. 33. Kawazoe, T., Kobayashi, K., Takubo, S., and Ohtsu, M., 2005. Nonadiabatic photodissociation process using an optical near field. J. Chem. Phys. 122: 024715-1–024715-5. 34. Kawazoe, T., Yamamoto, Y., and Ohtsu, M., 2001. Fabrication of a nanometric Zn dot by nonresonant near-field optical chemical vapor deposition. Appl. Phys. Lett. 79: 1184–1186. 35. Kawazoe, T., Kobayashi, K., and Ohtsu, M., 2006. Near-field optical chemical vapor deposition using Zn(acac)2 with a nonadiabatic photochemical process. Appl. Phys. B 84: 247–251. 36. Kawazoe, T. and Ohtsu, M., 2004. Adiabatic and nonadiabatic nanofabrication by localized optical near fields. Proc. SPIE 5339: 619–630. 37. Yonemitsu, H., Kawazoe, T., Kobayashi, K., and Ohtsu, M., 2007. Nonadiabatic photochemical reaction and application to photolithography. J. Lumin. 122–123: 230–233. 38. Ohtsu, M., (ed). 2006. Progress in Nano-Electro-Optics V, Spinger-Verlag, Preface to Volume V: Based on “nanophotonics” proposed by Ohtsu in 1993, OITDA (Optical Industry Technology Development Association, Japan) organized the nanophotonics technical group in 1994, and discussions on the future direction of nanophotonics were started in collaboration with academia and industry.

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36 Nanophotonic Device Materials 36.1 Introduction............................................................................................................................36-1 36.2 Nanophotonic Devices Based on Quantum Dots..............................................................36-1 Discrete Energy Levels in Spherical Quantum Dotsâ•‡ •â•‡ The Observation of Dissipated Optical Energy Transfer between CdSe QDsâ•‡ •â•‡ Controlling the Energy Transfer between Near-Field Optically Coupled ZnO QDs

Takashi Yatsui University of Tokyo

36.3 Nanophotonic AND-Gate Device Using ZnO Nanorod Double-Quantum-Well Structuresï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½36-7 Referencesï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½å°“ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½36-11

Wataru Nomura University of Tokyo

36.1â•‡ Introduction Systems of optically coupled quantum structures should be applicable to quantum information processing (Bayer et al. 2001, Stinaff et al. 2006). Additional functional devices, i.e., nanophotonic devices (Ohtsu et al. 2002, Kawazoe et al. 2003, 2005, 2006), can be realized by controlling the exciton excitation in quantum dots (QDs) and quantum-well structures (QWs). This chapter reviews the recent achievements with nanophotonic devices based on colloidal QDs (Section 36.2) and nanorod QWs (Section 36.3).

36.2â•‡Nanophotonic Devices Based on Quantum Dots 36.2.1â•‡Discrete Energy Levels in Spherical Quantum Dots The translational motion of the exciton center of mass is quantized in nanoscale semiconductors when the size is decreased so that it is as small as an exciton Bohr radius. If the QDs are assumed to be spheres with radius R, with the following potential

0 for x ≤ R  V (x ) =  ∞ for x > R

(36.1)

the quantized energy levels are given by a spherical Bessel function as

r  Rnl (r ) = Anl jl  ρn, l   R

(36.2)

Figure 36.1 shows the lth order of the spherical Bessel function. Note that an odd quantum number of l has an odd function and it is a dipole-forbidden energy state. To satisfy the boundary conditions as

Rnl (R) = Anl jl (ρn,l ) = 0

(36.3)

the quantized energy levels are calculated using

E(n, l) = EB +

2π2 2 ξn, l 2mR 2

(36.4)

where πξn,l = ρn,l is the nth root of the spherical Bessel Â�function of the lth order. The principal quantum number n and the Â�angular momentum quantum number l take values n = 1,â•›2,â•›3,…, and l = 0,â•›1,â•›2,…, respectively. ξn,l takes values ξ1,0 = 1, ξ1,1 = 1.43, ξ1,2 = 1.83, ξ2,0 = 2, and so on (see Table 36.1) (Sakakura and Masumoto 1997). Figure 36.2 show schematic drawings of different-sized spherical QDs (X and Y) and confined exciton energy levels. Here, R and 1.43R are the radii of spherical QDs X and Y, respectively. According to Equation 36.4, the quantized exciton energy levels of E(1,0) in QD X and E(1,1) in QD Y resonate with each other. Although the energy state E(1,1) is a dipole-forbidden state, the optical near-field interaction is finite for such coupling to the forbidden energy state (Kobayashi et al. 2000). In addition, the intersublevel transition, τsub, from higher exciton energy levels to the lowest one, is generally less than a few picoseconds and is much shorter than the transition time due to optical near-field coupling (τET) (Suzuki et al. 1996). Therefore, most of the energy of the exciton in a QD X with radius R transfers to the lowest exciton energy level in the neighboring QD Y with a radius of 1.43R and recombines radiatively at the lowest level. In this manner, unidirectional energy flow is achieved. 36-1

36-2


QD X

QD Y

0.8 R

0.6

1.43 R

0.4 0.2 –10

–5

5

10

x

τET

–0.2

(a)

0.4

FIGURE 36.2â•… Schematic drawings of different-sized spherical QDs (X and Y) and the confined exciton energy levels.

–5

5

10

x

–0.2 –0.4

(b)

0.3 0.2 0.1 –10

–5

5

10

x

–0.1

(c)

τsub E(1,0)

0.2

–10

E(1,1)

E(1,0)

FIGURE 36.1â•… lth order of spherical Bessel function. (a) l = 0 (j0(x) = (sin x)/x), (b) l = 1 (j1(x) = (1/x)(((sin x)/x) – cos x)), and (c) l = 2 (j2(x) = (1/x) (((3 – x2)/x2) sin x) – ((3/x) cos x)). Table 36.1â•… Calculated ξn,l to Satisfy the Condition of jl(πξn,l) = 0 l=0

l=1

l=2

l=3

l=4

n=1 n=2 n=3

1 2 3

1.43 2.46

1.83 2.90

2.22 3.31

2.61

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

36.2.2â•‡The Observation of Dissipated Optical Energy Transfer between CdSe QDs To evaluate the energy transfer and the energy dissipation, we used CdSe/ZnS core–shell QDs from Evident Technologies. As described in Section 36.2.1, assuming that the respective diameters, D, of the QDa1 and QDa2 were 2.8 and 4.1â•›nm, the ground energy level in the QDa1 and the excited energy level in the QDa2 resonate (Trallero-Giner et al. 1998). A solution of QDsa1 (D = 2.8â•›nm) and QDsa2 (D = 4.1â•›nm) in 1-feniloctane at a density

of 1.0â•›mg/mL was dropped on mica substrate (see Figure 36.3a), such that areas A and C consisted of QDsa1 and QDsa2 , respectively, while there were both QDsa1 and QDsa2 in area B. Using transmission electron microscopy (TEM), we confirmed the mean center-to-center distance of each QD was maintained at about less than 10â•›nm in all areas, due to the 2â•›nm-thick ZnS shell and surrounding ligands (2â•›nm-length long chain amine) of the QDs (Figure 36.3b). In the following experiments, the light source used was the third harmonic of a mode-locked Ti:sapphire laser (wavelength 306â•›nm, frequency 80â•›MHz, and pulse duration 2â•›ps). The incident power of the laser was 0.6â•›mW and the spot size was 1 × 10−3 cm2. The density of QDs was less than 3.5 × 1012 cm−2 and the quantum yield of CdSe/ZnS QD was 0.5. Under these conditions, the probability of exciton generation by one laser pulse in each QD was calculated to be 1.6 × 10−2. Therefore, we assumed single-exciton dynamics in the following experiments. The energy transfer was confirmed using micro-photoluminescence (PL) spectroscopy. Temperature-dependent micro-PL spectra were obtained. In the spectral profile of the PL emitted from area A, we found a single peak which originated from the ground state of QDa1 at a wavelength of λ = 540â•›nm, from room temperature to 30â•›K (broken line in Figure 36.4). From area C, the single peak, which originated from the ground state of QDa2, was found at λ = 600â•›nm (dotted line in Figure 36.4). By contrast, area B had two peaks at room temperature, as shown by solid line in Figure 36.4. This figure also shows that the PL intensity peak at λ = 540â•›nm decreased relative to that of at λ = 600â•›nm on decreasing the temperature. This relative decrease in the PL intensity originated from the energy transfer from the ground state in the QDa1 to the excited state in the QDa2 and the subsequent rapid dissipation to the ground state in the QDa2. This is because the coupling between the resonant energy levels becomes stronger due to the increase in the exciton decay time on decreasing the temperature (Itoh et al. 1990). Furthermore, although nanophotonic device operation using CuCl quantum cubes (Kawazoe et al. 2003, 2005, 2006) and ZnO quantum wells (Yatsui et al. 2007) has been reported at 15â•›K, we observed the

36-3

Nanophotonic Device Materials

Area A

Area B

QDa1 D = 2.8 nm

Area C

QDa2 D = 4.1 nm

~10 nm

(a)

50 nm (b)

FIGURE 36.3â•… (a) Schematic images of CdSe QDs dispersed substrate. Areas A, B, and C are covered by QDsa1, both QDsa1 and QDsa2, and QDsa2, respectively. (b) TEM image of dispersed CdSe/ZnS core–shell QDs in area B.

RT

PL intensity (a.u.)

130 K

60 K

30 K

520

540 Area A

560 580 Wavelength (nm) Area B

600

620 Area C

FIGURE 36.4â•… Temperature dependence of the micro-PL spectra. Broken, solid, and dotted lines show the spectra from areas A, B, and C, respectively.

decrease in the PL intensity at λ = 540â•›nm at temperatures as high as 130â•›K, which is advantageous for the high-temperature operation of nanophotonic devices. To confirm this energy transfer from QDa1 to QDa2 at the temperature under 130â•›K, we evaluated the dynamic property of the energy transfer using time-resolved spectroscopy with the time correlation single photon counting method. Circles Aa1, squares

Ba1, and triangles Ba2 in Figure 36.5a represent the respective time-resolved micro-PL intensities (60â•›K) from ground energy level in QDa1 (D = 2.8â•›nm) in area A, QDa1 (D = 2.8â•›nm) in area B, and QDa2 (D = 4.1â•›nm) in area B. The peak intensities at t = 0 were normalized to unity. Note that Ba2 decreases faster than Aa1, although these signals were generated from QDs of the same size. In addition, although the exciton lifetime decreases on increasing the QD size, owing to the increased oscillator strength, B a2 decreased more slowly than Aa1 over the range t < 0.2â•›ns (see the inset of Figure 36.5a). Furthermore, as we did not see any peak in the power spectra of Aa1, Ba1, and Ba2 , we believe that the temporal signal changes originated from the optical near-field energy transfer and subsequent dissipation. Since the QDsa1 in area B were near QDsa2 whose excited energy level resonates with the ground energy level of the QDa1 (see Figure 36.5b), near-field coupling between the resonant levels resulted in the energy transfer from the QDa1 to the QDa2 and the consequent faster decrease in the excitons of the QDa1 in area B compared with area A. Furthermore, as a result of inflow of the carriers from the QDa1 to the QDa2 , the PL intensity from the QDa2 near the QDa1 decayed more slowly than that of the QDa1. For comparison, we also obtained time-resolved PL profiles of different pairs of CdSe/ZnS QDs. Their diameters were D = 2.8â•›nm (QDa1) and 3.2â•›nm (QDb1), which means that their energy levels did not resonant with each other. Figure 36.6a shows a schematic image of a sample named area D, where QDsa1 and QDsb1 are mixed with a mean center-to-center distance of less than 10â•›nm. Circles Aa1 and diamonds Da1 in Figure 36.6b show the time-resolved PL intensity (30â•›K) from the ground energy level in QDsa1 from areas A and D, respectively. There is no difference in the decay profile. This indicates that the excited carriers in QDsa1 did not couple with QDsb1 due to their off-resonance and, consequently, no energy was transferred (Figure 36.6c). This supports the postulate that Figures 36.4 and 36.5 demonstrate energy transfer and subsequent dissipation due to near-field coupling between the resonant energy levels of the QDsa1 and QDsa2.

36-4


1

PL intensity (a.u.)

: Aa1 0.3

0.1

0.2

0

0.4

QDa1

: Ba1 0.01 (a)

2

Area A 4 t (ns)

6

8

QDa2

Ba1

Aa1

: Ba2 0

QDa1

Ba2 Area B

(b)

FIGURE 36.5â•… (a) Time-resolved PL intensity profiles from QDsa1 in area A (circles Aa1), QDsa1 in area B (squares Ba1), and QDsa2 in area B (triangles B a2). The peak intensities were normalized at t = 0. (b) Schematic of the respective system configurations in areas A and B.

Area D

PL intensity (a.u.)

1

QDa1 D = 2.8 nm

QDb1 D = 3.2 nm

: Da1 QDa1

(b)

QDb1

0.1

0.05

(a)

: Aa1

0

2

4 t (ns)

6

8

Da1

Area D

(c)

FIGURE 36.6â•… (a) Schematic image of CdSe QDs dispersed substrate area D. (b) Time-resolved PL intensity profiles from QDsa1 in area A (circles Aa1) and QDsa1 in area D (diamonds Da1). The peak intensities were normalized at t = 0. (c) Schematic of the system configuration in area D.

To discuss the exciton energy transfer from QDa1 to QDa2 quantitatively, we investigated the exciton dynamics by fitting multiple exponential decay curve functions to curves Aa1, Ba1, and B a2 (Bawendi et al. 1992, Crooker et al. 2003):

 −t   −t  Aa1 = RS1 exp   + RS2 exp   ,  τ s1   τ s2 

 −t  Ba1 = RS ⋅ Aa1 + RtS exp   ,  τt 

(36.5) (36.6)

and

 −t   −t  Ba2 = RL1 exp  + RL2 exp  .  τ L1   τ L2 

(36.7)

We used double exponential decay for Aa1 and Ba2 (Equations 36.5 and 36.7), which corresponds to the non-radiative lifetime (fast decay: τS1 and τL1) and radiative lifetime of free-carrier recombinations (slow decay: τS2 and τL2). Given the imperfect homogeneous

distribution of the QDsa1 in area B, some QDsa1 lack energy transfer routes to QDa2. However, we introduced the mean energy transfer time τt from QDsa1 and QDsa2 in Equation 36.6. In these equations, we neglected the energy dissipation time of about 1â•›ps (Guyot-Sionnest et al. 1999) because that is much smaller than exciton lifetimes and energy transfer time. Figure 36.7 shows the best-fitted numerical results and experimental data. Here, we used exciton lifetimes of τS2 = 2.10â•›ns and τL2 = 1.79â•›ns. The mean energy transfer time was τt = 135â•›ps, which is comparable with the observed energy transfer time (130â•›ps) in CuCl quantum cubes (Kawazoe et al. 2003) and ZnO QWs (Yatsui et al. 2007). Furthermore, the relation τt < τS2 agrees with the assumption that most of the excited excitons in QDsa1 transfer to excited exciton energy level in a QDa2 before being emitted from QDa1.

36.2.3â•‡Controlling the Energy Transfer between Near-Field Optically Coupled ZnO QDs ZnO is a promising material for room-temperature operation of a nanophotonic device because of its large exciton binding energy (Ohtomo et al. 2000, Huang et al. 2001, Sun et al. 2002). Here, we

36-5

Nanophotonic Device Materials 1 Aa1: τS1 and τS2

PL intensity (a.u.)

Ba1: τS1, τS2 and τt 0.1

0.01

Ba2: τL1 and τL2 0.001

0

2

4 t (ns)

6

8

FIGURE 36.7â•… Experimental results (circles, squares and triangles) and fitting curves (broken, solid, and short broken lines) using Equations 36.5 through 36.7 for the PL intensity profiles. The fitting parameters are R S1 = 0.560, τS1 = 2.95 × 10−10, RS2 = 0.329, τS2 = 2.10 × 10−9, RS = 0.740, RtS = 0.330, τt = 1.35 × 10−10, R L1 = 0.785, τL1 = 2.94 × 10−10, RL2 = 0.201, τL2 = 1.79 × 10−9.

used chemically synthesized ZnO QD to realize a highly integrated nanophotonic device. We observed the energy transfer from smaller ZnO QDs to larger QDs with mutually resonant energy levels. The energy transfer time and energy transfer ratio between the two QDs were also calculated from the experimental results (Yatsui et al. 2008). ZnO QDs were prepared using the sol-gel method (Spanhel and Anderson 1991, Meulenkamp 1998).

1. A sample of 1.10â•›g (5â•›mmol) of Zn(Ac)2â•›.â•›2H2O was dissolved in 50â•›mL of boiling ethanol at atmospheric pressure, and the solution was then immediately cooled to 0°C. A sample of 0.29â•›g (7â•›mmol) of LiOHâ•›.â•›H2O was dissolved in 50â•›mL of ethanol at room temperature in an ultrasonic bath and cooled to 0°C. The hydroxide-Â�containing solution was then added dropwise to the Zn(Ac)2 suspension with vigorous stirring at 0°C. The reaction mixture became transparent after approximately 0.1â•›g of LiOH had been added. The ZnO sol was stored at 0°C to prevent particle growth. 2. A mixed solution of hexane and heptane, with a volume ratio of 3:2, was used to remove the reaction products (LiAc and H2O) from the ZnO sol. 3. To initiate the particle growth, the ZnO solution was warmed to room temperature. The mean diameter of ZnO QD was determined from the growth time, Tg.

Figure 36.8a shows a TEM image of synthesized ZnO dots after the second step. Dark areas correspond to the ZnO QD. This image suggested that monodispersed single crystalline particles were obtained.

To check the optical properties and diameters of our ZnO QD, we measured the photoluminescence (PL) spectra using He–Cd laser (λ = 325â•›nm) excitation at 5â•›K. We compared the PL spectra of ZnO QD with Tg = 0 and 42â•›h (solid and dashed curves in Figure 36.8b, respectively). A redshifted PL spectrum was obtained, indicating an increase in the QD diameter. Figure 36.8c shows the growth time dependence of the QD diameter. This was determined from the effective mass model, with peak energy in the PL spectra, Egbulk = 3.35 eV, me = 0.28, mh = 1.8, and ε = 3.7 (Brus 1984). This result indicated that the diameter growth rate at room temperature was 1.1â•›nm/day. Assuming that the diameters, D, of the QDS and the QDL were 3.0 and 4.5â•›nm, respectively, ES1 in the QDS and EL2 in the QDL resonated (Figure 36.9a). An ethanol solution of QDS and QDL was dropped onto a sapphire substrate. The mean surfaceto-surface separation of the QD was approximately 3â•›nm. The spectra S and L in Figure 36.9b correspond to the QDS and the QDL, with spectral peaks of 3.60 and 3.44â•›eV, respectively. The curve A in Figure 36.9b shows the spectrum from the QDS and QDL mixture with R = 1, where R is the ratio (number of QDS)/(number of QDL). The spectral peak of 3.60â•›eV, which corresponded to the PL from the QDS, was absent from this curve. This peak was thought to have disappeared due to energy transfer from the QDS to the QDL, because the first excited state of the QDL resonated with the ground state of the QDS. Our hypothesis was supported by the observation that when R was increase by eightfold, the spectral peak from the QDS reappeared (see spectrum C in Figure 36.9b). To confirm this energy transfer from the QDS to the QDL at 5â•›K, we evaluated dynamic effects using time-resolved spectroscopy with the time-correlated single photon counting method. The light source used was the third harmonic of a mode-locked Ti:sapphire laser (photon energy 4.05â•›eV, frequency 80â•›MHz, and pulse duration 2â•›ps). We compared the signals from mixed samples with ratios R = 2, 1, and 0.5. The curves TA (R = 2), TB (R = 1), and TC (R = 0.5) in Figure 36.10 show the respective time-resolved PL intensities from the ground state of the QDS (E S1) at 3.60â•›eV. We investigated the exciton dynamics quantitatively by fitting multiple exponential decay curve functions (Bawendi et al. 1992, Crooker et al. 2003):

 −t   −t  TRPL = A1 exp   + A2 exp    τ1   τ2 

(36.8)

We obtained average τ1 and τ2 values of 144â•›ps and 443â•›ps, respectively (see Table 36.2). Given the disappearance of the spectral peak at 3.60â•›eV in the PL spectra, it is likely that these values corresponded to the energy transfer time from the QDS to the QDL and the radiative decay time from the QDS, respectively. This hypothesis was supported by the observation that the average value of τ1 (144â•›ps) was comparable with the observed energy transfer time in CuCl quantum cubes (130â•›ps) (Kawazoe et al. 2003).

36-6


5 nm (a) 1

Radius (nm)

PL intensity (a.u.)

3

0.5

0 3.2 (b)

3.3

3.4

3.5

3.6

3.7

1 –10

3.8

Photon energy (eV)

2

0

(c)

10

20

30

40

50

Growth time (h)

FIGURE 36.8â•… (a) TEM image of the ZnO QD. The dark areas inside the white dashed circles correspond to the ZnO QD. (b) The PL spectra observed at 5â•›K. The solid and dashed curves indicate growth time Tg = 0 and 42â•›h, respectively. (c) The growth time dependence of the mean ZnO QD diameter.

QDL D = 4.5 nm

QDS D = 3.0 nm

C (R = 8) A (R = 1) S

EL2

ES1: 3.60 eV 3.0 nm

(a)

PL intensity (a.u.)

B (R = 4)

EL1: 3.44 eV

L

(b)

3.2

3.3

3.4 3.5 Photon energy (eV)

3.6

3.7

FIGURE 36.9â•… (a) Schematic of the energy diagram between a QDS and QDL. (b) The PL spectra observed at 5â•›K. The labels S and L indicate QDS and QDL, respectively. The labels A, B, and C indicate mixes with R ratios of 1, 4, and 8, respectively.

36-7


36.3â•‡Nanophotonic AND-Gate Device Using ZnO Nanorod DoubleQuantum-Well Structures

τ1

PL intensity (a.u.)

TA

τ2

TB

TC

0

0.5

1

1.5

Time (ns)

FIGURE 36.10â•… Time-resolved PL spectra observed at 5â•›K. The values of R were 2, 1, and 0.5 for curves TA, TB, and TC, respectively. Table 36.2â•… Dependence of the Time Constants (τ1 and τ2) on R as Derived from the Two Exponential Fits of the Time-Resolved PL Signals and the Coefficient Ratio A1/A2 R = QDS/QDL 2 1 0.5 Average

τ1 (ps)

τ2 (ps)

A1/A2

133 140 160 144

490 430 410 443

12.4 13.7 14.4

We also investigated the value of coefficient ratio A1/A2 (see Table 36.2); this ratio was inversely proportional to R, hence proportional to the number of QDL. This result indicated that an excess QDL caused energy transfer from QDS to QDL, instead of direct emission from the QDS. We observed the dynamic properties of exciton energy transfer and dissipation between ZnO QD via an optical near-field interaction, using time-resolved PL spectroscopy. Furthermore, we successfully increased the energy transfer ratio between the resonant energy state, instead of the radiative decay from the QD. Chemically synthesized nanocrystals, both semiconductor QD and metallic nanocrystals (Brust and Kiely 2002), are promising nanophotonic device candidates, because they have uniform sizes, controlled shapes, defined chemical compositions, and tunable surface chemical functionalities.

ZnO/ZnMgO nanorod heterostructures have been fabricated and the quantum confinement effect has been observed from single QW structures (Park et al. 2003, 2004). In this section, we review the time-resolved near-field spectroscopy to demonstrate the switching dynamics that result from controlling the optical near-field energy transfer in ZnO nanorod double-quantumwell structures (DQWs). We observed nutation of the population between the resonantly coupled exciton states of DQWs, where the coupling strength of the near-field interaction decreased exponentially as the separation increased (Yatsui et al. 2007). To evaluate the energy transfer, three samples were prepared (Figure 36.11a): (1) Single-quantum-well structures (SQWs) with a well-layer thickness of Lw = 2.0â•›n m (SQWs), (2) DQWs with Lw = 3.5â•›n m with 6â•›n m separation (1-DQWs), and (3) three pairs of DQWs with Lw = 2.0â•›n m with different separations (3, 6, and 10â•›n m), where each DQW was separated by 30â•›n m (3-DQWs). ZnO/ZnMgO quantum-well structures (QWs) were fabricated on the ends of ZnO nanorods with a mean diameter of 80â•›n m using catalyst-free metalorganic vapor phase epitaxy (Park et al. 2002). The average concentration of Mg in the ZnMgO layers used in this study was determined to be 20â•›atm %. The far-field PL spectra were obtained using a He–Cd laser (λ = 325â•›nm) before detection using near-field spectroscopy. The emission signal was collected with an achromatic lens (f = 50â•›mm). The near-field photoluminescence (NFPL) spectra were obtained using a He–Cd laser (λ = 325â•›nm), collected with a fiber probe with an aperture diameter of 30â•›nm, and detected using a cooled charge-coupled device through a monochromator. Blueshifted PL peaks were observed at 3.499 (IS), 3.429 (I1D), and 3.467 (I3D) eV in the far- and near-field PL spectra (Figure 36.12a). These peaks originated from the respective ZnO QWs because their energies are comparable to the predicted ZnO well-layer thicknesses of 1.7 (IS), 3.4 (I1D), and 2.2 (I3D) nm, respectively, calculated using the finite square-well potential of the quantum confinement effect in ZnO SQWs (see Figure 36.12b) (Park et al. 2004). To confirm the near-field energy transfer between QWs, we compared the time-resolved near-field PL (TR NFPL) signals at the IS, I1D, and I3D peaks. For the time-resolved near-field spectroscopy, the signal was collected using a micro-channel plate through a band-pass filter with 1â•›nm spectral width. Figure 36.13 shows the typical TR NFPL of SQWs (TR S), 1-DQWs (TR1D), and 3-DQWs (TR3D). We calculate the exciton dynamics using quantum mechanical density-matrix formalism (Coffey and Friedberg 1978, Kobayashi et al. 2005), where the Lindblad-type dissipation is assumed for the relaxation due to exciton–photon and Â�exciton–phonon couplings:

36-8

Handbook of Nanophysics: Nanoelectronics and Nanophotonics SQWs Lw: 3 nm

1-DQWs

3-DQWs

6 nm L : 2 nm w

Lw: 3.5 nm

3 nm

50 nm ZnMgO

Lw: 2 nm

6 nm

Lw: 2 nm

10 nm

30 nm c

Zn0.8Mg0.2O

ZnO QWs

ZnMgO

ZnO QWs

ZnMgO

ZnO stem

ZnO QWs ZnMgO

80 nm

(a)

(b)

FIGURE 36.11â•… ZnO/ZnMgO nanorod quantum-well-structures. c: c-axis of the ZnO stem. (a) Schematic of ZnO/ZnMgO SQWs, DQWs (1-DQWs), and triple pairs of DQWs (3-DQWs). (b) Z-contrast TEM image of 3-DQWs clearly shows the compositional variation, with the bright layers representing the ZnO well layers. Scale bar: 50â•›nm. I2ZnO

3.65

IS

3.6

NFS I1D

NF1D

Photon energy (nm)

PL intensity (a.u.)

FFS

FF1D I3D

3.55 IS

3.5 3.45

I3D

FF3D

3.4

NF3D

(a)

3.35

3.40 3.45 3.50 Photon energy (eV)

3.55

3.35 (b)

I1D

1

2

3 4 5 Well width (nm)

6

FIGURE 36.12â•… Near-field time-resolved spectroscopy of ZnO nanorod DQWs at 15â•›K. (a) NFS, NF1D, NF3D: near-field PL spectra. FFS, FF1D, FF3D: far-field PL spectrum of ZnO SQWs (Lw = 2.0â•›nm), 1-DQWs (Lw = 3.5â•›nm, 6â•›nm separation), and 3-DQWs (Lw = 2.0â•›nm; 3, 6, and 10â•›nm separation). (b) Well width dependent on the exciton ground state and the first excited state. SQWs: open triangle, 1-DQWs: open circle, 3-DQWs: open square.

i ρ = − [ H , ρ] +

γn 2 AnρAn† − An† Anρ − ρAn† An 2

∑ ( n

)

(36.9)

where ρ is the density operator H is the Hamiltonian in the considered system An† and An are creation and annihilation operators for an exciton energy level labeled n γn is the photon or phonon relaxation constant for the energy level The exciton population is calculated using matrix elements for all exciton states in the system considered. First, we apply the

calculation to a three-level system of SQWs (Figure 36.14a), where the continuum state ħΩC is initially excited using a 10â•›ps laser pulse. Then, the initial exciton population in ZnO QWs is created in ħΩ1S, where the energy transfer from ħΩC to ħΩ1S is expressed phenomenologically as a Gaussian input signal with a temporal width of 2σ1S (an incoherent excitation term is added in Equation 36.9), because non-radiative relaxation paths via exciton–phonon coupling make a dephased input signal, statistically. Finally, an exciton carrier relaxes due to the electron– hole recombination with relaxation constant γ1S. Figure 36.14b shows a numerical result and experimental data. Here, we used 2σ1S = 100â•›ps, and γ1S was evaluated as 460â•›ps. A similar calculation was applied for DQWs. We used two three-level systems, coupled via an optical near-field with a

36-9


PL intensity (a.u.)

TRS

TR1D

TR3D

0

1.0 Delay (ns)

2.0

FIGURE 36.13â•… TR S, TR1D, and TR 3D show TR NFPL signal obtained at IS, I1D, and I3D, respectively, using the 4.025â•›eV (λ = 308â•›nm) light with a pulse of 10â•›ps duration to excite the barrier layers of ZnO QWs.

ħΩC

Α0, 2σ1s

ħΩ1s 10 ps

γ1S QWs1

ħΩC

Α1, 2σ1D

ħΩ1D 10 ps

Α2, 2σ2D ħΩ2D

U12

γ2D

γ1D (c)

QWs1

QWs2

Population

(a)

coupling strength of U12 (Figure 36.14c). Figure 36.14d shows the numerical results for the exciton population in QWs1 and the experimental data. Here 2σ1D and 2σ2D were set at 200â•›ps, which is twice the value for SQWs, because the relaxation paths extend the barrier energy state in the two quantum wells (QWs). γ1D and γ2D are evaluated as 200â•›ps. We believe that the faster relaxation for DQWs compared with SQWs reflects the lifetime of the coupled states mediated by the optical nearfield. Furthermore, the characteristic behavior that results from near-field coupling appears as the oscillatory decay in Figure 36.14d. This indicates that the timescale of the nearfield coupling is shorter than the decoherence time, and that coherent coupled states, such as symmetric and antisymmetric states (Sangu et al. 2004), determine the system dynamics. Furthermore, nutation never appears unless unbalanced initial exciton populations are prepared for ħΩ1D and ħΩ2D. In the far-field excitation, only the symmetric state is excited because the antisymmetric state is dipole-inactive. Then, the exciton populations of the two quantum wells are equal and they have the same decay rate. By contrast, in the near-field excitation, both the symmetric and antisymmetric states are excited due to the presence of a near-field probe. Since the symmetric and antisymmetric states have different eigenenergies, the interference of these states generates a detectable beat signal. The unbalanced excitation rate is given by A1/A2 = 10 here. From

0 (b)

1.0 Delay (ns)

2.0

0 (d)

1.0

2.0

Delay (ns)

FIGURE 36.14â•… The schematic depict the (a) SQWs and (c) 1-DQWs system configurations. ħΩC: barrier energy state with a central energy. Theoretical results on the transient exciton population dynamics (solid curves) and experimental PL data (filled squares) of (b) SQWs (same as curve TRS in Figure 36.13) and (d) 1-DQWs (same as curve TR1D in Figure 36.13).

36-10

Handbook of Nanophysics: Nanoelectronics and Nanophotonics 3 nm

6 nm

10 nm

10

Frequency (GHz)

Power spectrum

PSS

PS1D

∞ exp(-ar)

PS3D (a)

0

2

4 6 8 Frequency (GHz)

10

(b)

1 12

4

6 8 10 Separation (nm)

12

FIGURE 36.15â•… Evaluation of the nutation frequencies between the QWs. (a) PSS, PS1D, and PS3D show the power spectrum of TR S, TR1D, and TR 3D, respectively. (b) Separation D dependence of frequency of the nutation.

the period of nutation, the strength of the near-field coupling is estimated to be U12 = 7.7â•›ns−1 (= 4.9â•›μ eV). We evaluated nutation frequencies using Fourier analysis. In Figure 36.15a, the power spectral density of SQWs (PSS) does not exhibit any peaks, indicating a monotonic decrease. By contrast, the power spectral density of 1-DQWs (PS1D) had a strong peak at a frequency of 2.6â•›ns−1. Furthermore, that of 3-DQWs (PS3D) had three peaks at 1.9, 4.7, and 7.1â•›ns−1. Since, the degree of the coupling strength, which is proportional to the frequency of the nutation, increases as the separation decreases, the three peaks correspond to the signals from DQWs with separations of 10, 6, and 3â•›n m, respectively. Since the coupling strength ħU [eV] is given by ħπf (f: nutation frequency), ħU is estimated as 4.0, 9.9, and 14.2â•›μ eV for DQWs with respective separations of 10, 6, and 3â•›n m. These values are comparable to that estimated above (U12 = 4.9â•›μ eV). Furthermore, the peak intensity for the DQWs with 3â•›n m separation is much lower than for those with 10â•›n m separation, Input

QW1 (Lw: 3.2 nm)

QW1 (Lw: 3.8 nm)

which might be caused by decoherence of the exciton state due to penetration of the electronic carrier. Considering the carrier penetration depth, the strong peak of DQWs with 10â•›n m separation originates from the near-field coupling alone. The solid line in Figure 36.15b shows the separation dependence of the peak frequency. The exponentially decaying dependence represented by this line supports the origin of the peaks in the power spectra from the localized near-field interaction between the QWs. Next, we performed the switching operation. Figure 36.16a and b explains the “OFF” and “ON” states of the proposed nanophotonic switch, consisting of two coupled QWs. QW1 and QW2 are used as the input/output and control ports of the switch, respectively. Assuming Lw = 3.2 and 3.8â•›nm, the ground exciton state in QW1 and the first excited state in QW2 resonate. In the “OFF” operation (Figure 36.16a), all the exciton energy in QW1 is transferred to the excited state in the neighboring QW2 and relaxes rapidly to the ground state. Consequently, no output

Input Output

EA1 Relaxation (a)

QW2

QW1

EB2: 3.435 eV Control

EB1: 3.425 eV

(b)

FIGURE 36.16â•… The switching operation by controlling the exciton excitation. Schematic of the nanophotonic switch of (a) “OFF” state and (b) “ON” state.

36-11


EA1

PL intensity (a.u.)

NFON

0 NFControl

NFOFF

3.40

Luminescence intensity (a.u.)

EB1

3.45 Photon energy (eV)

3.50

FIGURE 36.17â•… (a) NFON, NFControl, and NFOFF show NFPL signal obtained with the illumination of input laser alone, control laser alone, and input and control laser, respectively.

signals are generated from QW1. In the “ON” operation (Figure 36.16b), the escape route to QW2 is blocked by the excitation of QW2 owing to state filling in QW2 on applying the control signal; therefore, an output signal is generated from QW1. Figure 36.17 shows the NFPL for the three pairs of DQWs with Lw = 3.2 and 3.8â•›nm with different separations (3, 6, and 10â•›nm). Curve NFOFF was obtained with continuous input light illumination from a He–Cd laser (3.814â•›eV). No emission was observed from the exciton ground state of QW1 (EA1) or the excited state of QW2 (EB2) at a photon energy of 3.435â•›eV, indicating that the excited energy in QW1 was transferred to the excited state of QW2. Furthermore, the excited state of QW2 is a dipole-forbidden level. Curve NFControl shows the NFPL signal obtained with control light excitation of 3.425â•›eV with a 10â•›ps pulse. Emission from the ground state of QW2 at a photon energy of 3.425â•›eV was observed. Both input and control light excitation resulted in an output signal with an emission peak at 3.435â•›eV, in addition to the emission peak at 3.425â•›eV (curve NFON), which corresponds to the ground state of QW2. Since the excited state of QW2 is a dipoleforbidden level, the observed 3.435â•›eV emission indicates that the energy transfer from the ground state of QW1 to the excited state of QW2 was blocked by the excitation of the ground state of QW2. Finally, the dynamic properties of the nanophotonic switching were evaluated. We observed TR NFPL signals at 3.435â•›eV with both input and control laser excitation (see Figure 36.18). The

0.5 Time (ns)

1.0

1.5

FIGURE 36.18â•… Near-field time-resolved PL signal with “ON” state.

decay time constant was found to be 483â•›ps. The output signal increased synchronously, within 100â•›ps, with the control pulse. Since the rise time is considered equal to one-quarter of the nutation period τ (Sangu et al. 2003), the value agrees with those obtained for DQWs with the same well width in the range from τ/4 = 36â•›ps (3â•›nm separation) to τ/4 = 125â•›ps (10â•›nm separation). We observed the nutation between DQWs and demonstrated the switching dynamics by controlling the exciton excitation in the QWs. Examination of the electronic coupling between QWs is now in progress to analyze the detailed switching dynamics. For room-temperature operation, since the spectral width reaches thermal energy (26â•›meV), a higher Mg concentration in the barrier layers and narrower Lw are required so that the spectral peaks of the first excited state (E2) and ground state (E1) do not overlap. This can be achieved by using two QWs with Lw = 1.5â•›nm (QW1) and 2â•›nm (QW2) with a Mg concentration of 50%, where the energy difference between E2 and E1 in QW2 is 50â•›meV (Park 2001).

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Crooker, S. A., Barrick, T., Hollingsworth, J. A., and Klimov, V. I. 2003. Multiple temperature regimes of radiative decay in CdSe nanocrystal quantum dots: Intrinsic limits to the dark-exciton lifetime. Appl. Phys. Lett. 82: 2793–2795. Guyot-Sionnest, P., Shim, M., Matranga, C., and Hines, M. 1999. Intraband relaxation in CdSe quantum dots. Phys. Rev. B 60: R2181–R2184. Huang, M. H., Mao, S., and Feick, H. 2001. Room-temperature ultraviolet nanowire nanolasers. Science 292: 1897–1899. Itoh, T., Furumiya, M., Ikehara, T., and Gourdon, C. 1990. Sizedependent radiative decay time of confined excitons in CuCl microcrystals. Solid State Commun. 73: 271–274. Kawazoe, T., Kobayashi, K., Akahane, K., Naruse, M., Yamamoto, N., and Ohtsu, M. 2006. Demonstration of nanophotonic NOT gate using near-field optically coupled quantum dots. Appl. Phys. B 84: 243–246. Kawazoe, T., Kobayashi, K., and Ohtsu, M. 2005. Optical nanofountain: A biomimetic device that concentrates optical energy in a nanometric region. Appl. Phys. Lett. 86: 103102. Kawazoe, T., Kobayashi, K., Sangu, S., and Ohtsu, M. 2003. Demonstration of a nanophotonic switching operation by optical near-field energy transfer. Appl. Phys. Lett. 82: 2957–2959. Kobayashi, K., Sangu, S., Itoh, H., and Ohtsu, M. 2000. Near-field optical potential for a neutral atom. Phys. Rev. A 63: 013806. Kobayashi, K., Sangu, S., Kawazoe, T., and Ohtsu, M. 2005. Exciton dynamics and logic operations in a near-field optically coupled quantum-dot system. J. Lumin. 112: 117–121. Meulenkamp, E. A. 1998. Synthesis and growth of ZnO nanoparticles. J. Phys. Chem. B 102: 5566–5572. Ohtomo, A., Tamura, K., Kawasaki, M. et al. 2000. Roomtemperature stimulated emission of excitons in ZnO/(Mg, Zn)O superlattices. Appl. Phys. Lett. 77: 2204–2206. Ohtsu, M., Kobayashi, K., Kawazoe, T., Sangu, S., and Yatsui, T. 2002. Nanophotonics: Design, fabrication, and operation of nanometric devices using optical near fields. IEEE J. Sel. Top. Quantum Electron. 8: 839–862. Park, W. I., Yi, G.-C., and Jang, M. 2001. Metalorganic vaporphase epitaxial growth and photoluminescent properties of Zn1-xMgxO (0 < x < 0.49) thin films. Appl. Phys. Lett. 79: 2022–2024. Park, W. I., An, S. J., Long, Y.-J. et al. 2004. Photoluminescent properties of ZnO/Zn0.8Mg0.2O nanorod single-quantumwell structures. J. Phys. Chem. B 108: 15457–15460.

Park, W. I., Kim, D. H., Jung, S.-W., and Yi, G.-C. 2002. Metalorganic vapor-phase epitaxial growth of vertically well-aligned ZnO nanorods. Appl. Phys. Lett. 80: 4232–4234. Park, W. I., Yi, G.-C., Kim, M. Y., and Pennycook, S. J. 2003. Quantum confinement observed in ZnO/ZnMgO nanorod heterostructure. Adv. Mater. 15: 526–529. Sakakura, N. and Masumoto, Y. 1997. Persistent spectral-holeburning spectroscopy of CuCl quantum cubes. Phys. Rev. B 56: 4051–4055. Sangu, S., Kobayashi, K., Kawazoe, T., Shojiguchi, A., and Ohtsu, M. 2003. Quantum-coherence effect in a quantum dot system coupled by optical near fields. Trans. Mater. Res. Soc. Jpn. 28: 1035–1038. Sangu, S., Kobayashi, K., Shojiguchi, A., and Ohtsu, M. 2004. Logic and functional operations using a near-field optically coupled quantum-dot system. Phys. Rev. B 69: 115334. Spanhel, L. and Anderson, M. A. 1991. Semiconductor clusters in the sol-gel process; quantized aggregation, gelation, and crystal growth in concentrated ZnO colloids. J. Am. Chem. Soc. 113: 2826–2833. Stinaff, E. A., Scheibner, M., Bracker, A. S. et al. 2006. Optical signatures of coupled quantum dots. Science 311: 636–639. Sun, H. D., Makino, T., Segawa, Y. et al. 2002. Enhancement of exciton binding energies in ZnO/ZnMgO multiquantum wells. J. Appl. Phys. 91: 1993–1997. Suzuki, T., Mitsuyu, T., Nishi, K. et al. 1996. Observation of ultrafast all-optical modulation based on intersubband transition in n-doped quantum wells by using free electron laser. Appl. Phys. Lett. 69: 4136–4138. Trallero-Giner, C., Debernardi, A., Cardona, M., MenendezProupin, M., and Ekimov, A. I. 1998. Optical vibrons in CdSe dots and dispersion relation of the bulk material. Phys. Rev. B 57: 4664–4669. Yatsui, T., Jeong, H., and Ohtsu, M. 2008. Controlling the energy transfer between near-field optically coupled ZnO quantum dots. Appl. Phys. B 93: 199–202. Yatsui, T., Sangu, S., Kawazoe, T., Ohtsu, M., An, S. J., Yoo, J., and Yi, G.-C. 2007. Nanophotonic switch using ZnO nanorod double-quantum-well structures. Appl. Phys. Lett. 90: 223110.

37 Waveguides for Nanophotonics 37.1 Introduction............................................................................................................................ 37-1 Basic Types of Waveguides for Nanophotonicsâ•‡ •â•‡ Silicon Nanophotonics

37.2 Fabrication of Planar and Rib Waveguides........................................................................ 37-2 37.3 Experimental Techniques......................................................................................................37-4

Jan Valenta Charles University

Tomáš Ostatnický Charles University

Ivan Pelant Academy of Sciences of the Czech Republic

Techniques to Study Internal Photoluminescence Propagation in Waveguidesâ•‡ •â•‡ Techniques to Study Propagation of Light from External Sourcesâ•‡ •â•‡ Techniques to Study Losses and Optical Amplification in Waveguides

37.4 Main Experimental Observations in Active Si-nc Waveguides...................................... 37-7 37.5 Theoretical Description of Active Lossy Waveguides....................................................... 37-9 37.6 Application of Active Nanocrystalline Waveguides....................................................... 37-12 Amplification of Lightâ•‡ •â•‡ Other Applications

37.7 Conclusions........................................................................................................................... 37-14 Acknowledgments............................................................................................................................ 37-14 References.......................................................................................................................................... 37-14

37.1â•‡ Introduction

37.1.2â•‡ Silicon Nanophotonics

The term nanophotonics may be understood as abbreviation from “photonics of nanostructures.” This rapidly evolving research area deals with light interaction in nanostructured materials and their applications in photonic devices like light sources, modulators, detectors, etc.

The recent years can be characterized by the association of microelectronics with optoelectronics or photonics. While the microelectronics is based almost exclusively on silicon (indirect band-gap semiconductor), photonic light sources are currently made out of direct band-gap III–V compounds (family of GaAs materials). In order to reduce the material diversity, an effort to develop an efficient, electrically pumped silicon-compatible light-emitting device is becoming very strong (Pavesi and Lockwood 2004). This tendency to “siliconize photonics” is driven also by the need to reduce the overheating of silicon integrated circuits due to the ohmic resistance of excessively long multilevel metal interconnects (Pavesi and Guillot 2006). Supplementary advantages like the reduction in charging (RC) time constant (speeding up the circuit performance) and prevention from crosstalks can be obtained as a bonus. Since bulk silicon, as an indirect band-gap semiconductor, is a very bad light emitter, nanometer-sized silicon nanocrystals, brightly luminescent at room temperature, represent one of the possible solutions (Pavesi and Lockwood 2004). It is then expected that the light signal originating in Si-nc and carrying the required information propagates in a low-loss medium, which assures, at the same time, a good directionality of the radiation. It has been discovered recently that slabs of fused quartz SiO2 “doped” with Si-nc are able to accomplish both the role of a light emitter and that of a waveguide (Khriachtchev et al. 2001, Valenta et al. 2003a, Khriachtchev et al. 2004). We shall call this type of nanophotonic waveguides (which generate the luminous signal themselves and therefore there is no need to

37.1.1â•‡Basic Types of Waveguides for Nanophotonics Optical waveguides are structures that are able to confine and guide optical electromagnetic field. Classical waveguides (optical fibers, Figure 37.1a) use refractive index difference between the core and the cladding layer to guide light by total internal reflection. When the size of waveguide approaches the wavelength of light, the confinement decreases and losses increase. Therefore, the size of practical classical waveguides is limited to several hundreds of nanometers. Somewhat better confinement can be achieved in more complex waveguides with guiding properties determined by the formation of photonic bands due to their regular spatial structure (Figure 37.1b and c): (1) photonic crystal waveguides using “defects” in periodic structures of photonic crystals to confine light and (2) plasmonic waveguides based on metallic nanostructures with plasmon resonance (see, e.g., Pavesi and Guillot 2006). Due to limited space, we are going to describe only a special type of the “classical” waveguides formed by luminescing silicon nanocrystals (Si-nc).

37-1

37-2


n2 n1

Cladding Core (a)

Classical refractive waveguide

(c)

(b)

Photonic waveguide W1

Plasmonic waveguide

FIGURE 37.1â•… Nanophotonic waveguides: (a) “Classical” waveguide based on total internal reflection using the core with higher refractive index than the cladding. (b) Photonic crystal waveguide formed by a “defect” in the regular structure—here is the W1 waveguide formed by a row of missing holes. (c) Plasmonic waveguide formed by a row of metallic nanocrystals.

37.2â•‡Fabrication of Planar and Rib Waveguides There have been many various methods of how to grow thin sheets of luminescent Si-nc embedded in an optically transparent matrix: (1) Implantation of fused quartz (SiO2) slabs with Si+ ions (Figure 37.2) (Cheylan et al. 2000), (2) reactive Si deposition onto fused quartz (Khriachtchev et al. 2002), (3) plasma-enhanced chemical vapor deposition (PECVD) of substoichiometric silicon oxide SiOx thin films on a Si substrate (Iacona et al. 2000), (4) co-sputtering of a Si wafer and a piece of glass using fused quartz plates or Si wafers as a substrate (Imakita et al. 2005),

(5) crystallization of Si/SiO2 superlattices with nm-thick amorphous Si or SiO layers (Riboli et al. 2004), and (6) embedding of porous silicon grains into sol-gel derived SiO2 layers (Luterová et al. 2004), to list at least the most frequently used. Most of these techniques comprise high-temperature (1100°C–1200°C) annealing of deposited SiOx films in order to achieve the phase separation between Si-nc and the SiO2 matrix. The thickness of the resulting sheets containing Si-nc can vary from hundreds of nanometers to tens of micrometers. Due to the difference in refractive index between the matrix (n silica ≈ 1.45) and silicon (n Si ≈ 3), such sheets act as planar or rib waveguides. Interestingly, the attractive waveguiding features that we are going to discuss, namely, the wavelength selective guiding of light (which can be also called “spectral filtering”) and microcavity-like behavior, are critically dependent upon the preparation method: Till now, they have been discovered only in samples fabricated using the first two methods. We describe them in detail. Implantation of accelerated Si+ ions can be applied either to fused quartz (silica) slabs or to a thin SiO2 layer thermally

400 keV 1–6 × 1017 cm–2

Δn

Air

0

Substrate

0.4

3, 4, 5, 6 ×1017 cm–2

Annealing

z (μm)

couple the light to them from an external source) “active waveguides.” In this chapter, we shall describe the preparation methods of active nanocrystalline waveguides, their experimentally observed properties, and relevant theoretical description. We shall also briefly mention their various application possibilities.

0.8 1100°C in N2 ambient for 1 h Hydrogen passivation 1 h in forming-gas (95% N2/5% H2) at 500°C (a) Implantation

1

1.4

n(z)

1.8

(b)

FIGURE 37.2â•… (a) The schematics of a Si-nc planar waveguide preparation. (b) Refractive index profiles (n as a function of the depth z beneath the surface) of silicon nanocrystalline waveguides (implanted with 400â•›keV Si+ ions and different implant fluences), extracted from optical transmission measurements. (After Valenta, J. et al., J. Appl. Phys., 96, 5222, 2004.)

37-3

Waveguides for Nanophotonics

grown on a silicon wafer. Implantation energy varies usually between 30 and 600â•›keV, and implant fluences (ion doses) are of the order of 1016 –1017 cm−2. The attractive waveguide properties of fabricated planar waveguides, i.e., pronounced emission line narrowing and high output beam directionality in the visible region, have been found in waveguides prepared from 1â•›mm thick silica (Infrasil) slabs with optically polished surface and edges, using the implantation energy of 400â•›keV and the implant fluences 1 × 1017 cm−2 to 6 × 1017 cm−2. Because the Si+ ions are almost monoenergetic, their stopping distance beneath the silica slab surface has only a small dispersion, which results in narrow and slightly skewed implant distribution, as reflected in the resulting refractive index profiles shown in Figure 37.2 (Valenta et al. 2004). Peak excess Si concentration was up to 26 atomic%. Spherical silicon nanocrystals with diameter distribution roughly from 4 to 6â•›nm were formed by annealing the implanted samples at 1100°C in a nitrogen ambient for 1â•›h. To enhance several times the luminescence intensity of Si-nc, an additional anneal at 500°C in forming gas (N2 + H2) for 1â•›h was applied (Cheylan and Elliman 2001). The last operation promotes enhanced hydrogen passivation of non-radiative defects. The reactive silicon deposition has yielded substoichiometric SiOx films on circular 1â•›mm-thick substrate silica plates (Khriachtchev et al. 2001, 2004). Electron beam evaporation and radio frequency cells were used as silicon and oxygen sources. The SiOx layer thickness was ∼2â•›μm, and the value of the x parameter varied around x = 1.70. The as-grown material contained amorphous Si inclusions. Annealing of these “suboxide” films at 1100°C in nitrogen atmosphere resulted in the formation of well-defined Si-nc with diameters of 3–4â•›nm, as evidenced by Raman spectroscopy. Extensive investigations

of this type of silica waveguides containing Si-nc discovered a small effective optical birefringence inside the layers, however, without radically influencing the properties of the waveguides (Khriachtchev et al. 2007). Refractive index profile across the thickness of these waveguides is flat, unlike the preceding case of the implanted layers. The reason why Si-nc waveguides fabricated using other techniques do not show the spectral filtering effect are not fully clear at present. Of course, the condition sine non qua is asymmetry of the waveguide, i.e., the waveguiding layer must not be sandwiched between two materials (cladding layers) having the same effective refractive index, as we shall see in the next section. One can speculate about several further critical parameters: suitable composition and optical quality of the matrix; suitable density of Si-nc (in the order of 1018 cm−3); favorable layer thickness (around 1â•›μm); excellent flatness and parallelism of both interfaces; and, last but not least, certain optimum value of propagation losses (several dB/cm). The dominant losses in the waveguides under discussion are probably due to self-absorption and Mie light scattering on Si-nc clusters, not due to surface roughness (Pelant et al. 2006). The above discussion has dealt implicitly with planar waveguides. Rib-loaded waveguides containing nanocrystals seem, in principle, even more desirable for nanophotonic applications. They can be also prepared using the ion implantation technique in either thin SiO2 films thermally grown on Si or in polished fused quartz slabs, as described above. The rib structure can be formed (using standard photolithography and etching) by two possible means: either before or after the implantation procedure. Figure 37.3a represents schematics of such a rib waveguide structure.

Rib profile Rib width 2–5 μm Implant. layer

Rib height ~1 μm

Substrate (a)

(b) Selective excitation of ribs Total internal refl.

Laser

(c)

Optical fiber without cladding

Evanescent wave

FIGURE 37.3â•… (a) Sketch of a rib waveguide structure with silicon nanocrystals, prepared on a polished silica slab. The ribs were fabricated by optical lithography before Si+ implantation. (b) Microscopic (reflection) image of a part of the structure. The spacing between the ribs is 100â•›μ m. (c) Methods to selectively excite the rib waveguides through the TIR prism or the evanescent field of a fiber. (After Skopalová, E., Mode structure in the light emission from planar waveguides with silicon nanocrystals, Diploma thesis, Charles University, Prague, Czech Republic, 2007.)

37-4


Lateral confinement of the light in a narrow rib is expected, in comparison with the planar waveguides, to have some advantages: saving considerable space in photonics circuits and offering much wider variability in their design. On the other hand, the transition from the two-dimensional to the one-dimensional case can degrade the optical quality of the waveguide and introduce additional losses, such as those due to sidewall roughness or even due to considerably deformed rib cross section. Indeed, measurements of propagation losses in similar waveguides have given values above 10â•›dB/cm, and this loss coefficient even increases for rib widths below 4â•›μm (Pellegrino et al. 2005). Therefore, in what follows, we discuss predominantly the twodimensional nanocrystalline waveguides, and limited space only will be given to the rib structures.

37.3â•‡ Experimental Techniques 37.3.1â•‡Techniques to Study Internal Photoluminescence Propagation in Waveguides One of the most important advantages of active waveguides is that light sources are embedded in the waveguide. In our material, Si nanocrystals forming the waveguide core can efficiently emit luminescence in the orange-red-infrared spectral range when excited with the UV-blue light (usually focused laser beam). Such internally produced light is automatically coupled to all possible modes of the waveguide: radiation, substrate, and

Microscope

Imaging spectrograph

(a)

CCD camera

Objective lens Sample

+

PL

guided modes (see Section 37.5). The emission modes may be distinguished by measuring their propagation direction, spectra, and polarization, which imposes requirements to experimental setup. Two types of setups can be conveniently applied to study luminescence from active waveguides: (1) The micro-imagingspectroscopy setup is based on a microscope connected to an imaging spectrograph with a CCD camera (Figure 37.4a). For good angular resolution, low numerical aperture (NA) objective lenses should be used (in our experiments, we used mostly the lens with 2.5 × magnification and NA = 0.075, i.e., an angular resolution of about 8.6°). The sample is fixed to an x−y−z table with a rotating holder and excited by a laser beam (325â•›n m from the cw He–Cd laser), approximately perpendicular to the objective axis. (2) In order to achieve better angular resolution, an experimental arrangement based on a goniometer is employed (Figure 37.4b). Here, the sample is fixed to the center of the goniometer, and the photoluminescence emission is collected by a silica optical fiber (core diameter 1â•›m m) rotated around the sample at a distance of 50â•›m m, giving an angular resolution slightly less than 1° (NAâ•›∼â•›0.01). The output of the fiber can be coupled to the same detection system (a spectrograph with a CCD camera), as described above. Typical images observed with the microscopic setup (Figure 37.4a) are illustrated in Figure 37.4c and d. Here, the diameter of the excitation spot, located about 1â•›mm from the sample edge, is roughly 1â•›mm. One can easily recognize PL emission from the excited spot as a bright ellipsoid. However, there is also a second

Excitation To spectrograph ~1°

Excitation beam

–

–

(b)

Substrate

+

Optical fiber (light collection) +15°

–15°

(c)

Sample

(d)

Substrate

FIGURE 37.4â•… (a) Micro-spectroscopy setup for studying spatially resolved emission from waveguides, (b) collection of signals from a waveguide using optical fiber mounted on a goniometer. (c,d) Luminescence microimages of the active planar waveguide (obtained with the setup illustrated in panel a), where the elliptical spot is the spot excited by the laser beam and the light line is emission emanating from the sample facet. The inclination of the sample with regard to optical axis is −15° and +15°, respectively. (Adapted from Valenta, J. et al., J. Appl. Phys., 96, 5222, 2004.)

37-5


contribution emanating from the facet of the sample. This light is obviously guided in the implanted layer or close to it. The images in Figure 37.4c and d were collected for sample inclination angles of −15° and +15°, respectively, i.e., in a geometry for which the excited spot was observed either directly (Figure 37.4c) or through the substrate (Figure 37.4d). The experimental arrangement shown in Figure 37.4a enables the detection of the PL either from the excited spot or from the edge of sample by positioning the entrance slit of the spectrograph to different locations of the PL image. All experiments described in this chapter were performed at room temperature.

37.3.2â•‡Techniques to Study Propagation of Light from External Sources The coupling of external light to narrow submicrometer waveguides is a difficult task. The most used approaches are as follows: (1) Prism coupling of light from the surface of the sample (Figure 37.5a). Light from Xe lamp, halogen lamp, or LED is collimated into a prism in contact with waveguide. For better optical contact, an immerse liquid should be dropped between the prism and the sample (the best optical contact—i.e., minimizing reflections on interface—is achieved when refractive index of the immersion liquid is between the values for the materials to be connected). A second prism may be placed on the opposite surface of waveguide to couple out the passing beam to avoid total internal reflection. (2) Direct coupling into the facet (Figure 37.5b, sometimes called end-fire coupling). The edge of the sample can be polished at some angle (here 70°) in order to separate light refracted to the higher index waveguide from light White light

(a)

γ White light (b)

(c)

Prism

Shield α0

Detection

Optical fiber on goniometer

Prism

α0

70° Shield

(d)

FIGURE 37.5â•… External light coupling into a narrow waveguide: (a) prism coupling, (b) end-fire coupling, (c) grating coupling, and (d) evanescent-wave coupling. (Adapted from Janda, P. et al., J. Lumin., 121, 267, 2006; Pavesi, L. and Guillot, G. (Eds.), Optical Interconnects. The Silicon Approach, Springer-Verlag, Berlin, Germany, 2006.)

entering lower index substrate. For both the external light coupling setups, the signal can be collected by an optical fiber and guided to the entrance slit of the imaging spectrometer with a CCD detector. The other coupling methods described in literature are grating coupling and evanescent wave coupling (sometimes called waveguideto-waveguide coupling) (see Figure 37.5c and d).

37.3.3â•‡Techniques to Study Losses and Optical Amplification in Waveguides The losses in waveguide are divided into insertion (coupling) losses and propagation losses, which contain scattering, radiation, and absorption (Figure 37.6a). If the waveguide material can be pumped to reach population inversion, the guided light may be amplified by stimulated emission. In this case, the absorption coefficient α becomes negative and it is called the gain coefficient g(g = −α). The most used methods to measure losses in waveguides are (Figure 37.6b through e) (1) “cutback” method detecting light output from waveguide with different length; (2) scattering detection from different points of waveguide surface; (3) Fabry– Perot resonance method, which can be applied to waveguides with good facets; and (4) shifting excitation spot (SES) method; which can be applied to active waveguides. Optical gain is measured by the variable stripe-length (VSL) method or by the pump-and-probe (PP) method. The principle of the VSL method consists in excitation of a narrow stripe-like region at the edge of a sample (Figure 37.7a) (Shaklee and Leheny 1971, Valenta et al. 2003b, Dal Negro et al. 2004). If the pumping is able to induce population inversion in studied material, photons passing through the excited region can be amplified by stimulated emission. It corresponds to a single-pass optical amplifier. If the emission from the sample edge IVSL(l,â•›λ) is detected as a function of the stripe length l, the net optical gain G(λ) is calculated from a simple equation

I VSL (l , λ) =

I sp (λ) exp(G (λ) ⋅ l ) − 1 , G( λ ) 

(37.1)

where Isp is the intensity of spontaneous emission G(λ) = [g(λ)â•›−â•›K] is net optical gain (K stands for losses and g(λ) is material gain coefficient, i.e., negative absorption coefficient) (Valenta et al. 2002) Equation 37.1 is valid only when Isp(λ) and g(λ) are constant over the whole excited stripe. It means that the exciting power density and properties of sample must be uniform. Also, the coupling of the output emission to a detector must be constant, independent of x, for any part of the stripe. In reality, the geometry and emission properties of the sample are never perfectly constant, and the coupling of light is influenced by directionality of emitted light, limited collection angle (NA), the confocal effect of imaging optics, etc. (Valenta et al. 2003b). Therefore, the SES method has been proposed to correct the VSL method (Valenta et al.

37-6

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Scattering Coupling

Output Guided light

Absorption losses Insertion losses (a)

Radiation

Detection

Output

Input

Coupling

Detection I(ℓ)

ℓ

ℓ

(b)

ℓ

Reflections

I(ℓ) ℓ

(c) Excitation

Reflections

Detection Detection

Coupling I(λ)

Active waveguide

ℓ I(ℓ)

λ

(d)

ℓ

(e)

FIGURE 37.6â•… (a) Schematic illustration of losses in a waveguide. Methods to measure losses: (b) cutback, (c) scattering light detection, (d) Fabry– Perot resonance, and (e) SES methods.

Moving shield

Distance of the excited spot from the edge Moving slit

Pumping beam

Pumping beam

Excited length

Edge emission

(a)

Saturation

Excited length (cm)

log Ilum (a.u.)

Ex po ne nt ial inc re a

log IASE (a.u.)

se

Edge emission

(b)

Detection I SES n

Excitation 1 2 3 4 X0

Detection

Ex po

ne

nt ia

ld

ec

re a

I VSL n

se

n

dn

Excitation 1 2 3 4 X0

n

l = n · X0

Distance from edge (cm)

FIGURE 37.7â•… The principle of the (a) VSL and (b) SES methods and typical dependence of the detected signal as function of the excited length and the distance from edge, respectively (lower panel). (Adapted from Park, J.H. and Steckl, A.J., Appl. Phys. Lett., 85, 4588, 2004.)

2002). The principle of the SES method takes advantage of the fact that instead of the whole stripe, only small segments are excited and measured separately (Figure 37.7b). Figure 37.8 illustrates the procedure to compare the VSL and SES results. The most transparent approach consists of SES

FIGURE 37.8â•… Schematic illustration of the principle of combining the SES and VSL experiments.

measurement with the spot width equal to the shift step. Let us assume that the VSL measurement is done using the same elemental steps x0 as the SES experiment (obviously, also the stripe width and excitation density must be the same for both SES and VSL). Then we can compare the VSL signal I nVSL for a stripe consisting of n elemental steps (l = nâ•›.â•›x0) with integrated SES signal IniSES, i.e., sum of SES signals from n steps

37-7

Waveguides for Nanophotonics Test beam νS Pumping νP

Spatial filtration

Beam splitter

Sample

Detector (measuring I0)

d Pumping

Active layer Substrate

Probe beam

Spectrometer Detector (ION , IOFF)

(a)

(b)

Detection

d

FIGURE 37.9â•… The basic arrangement of the PP experiment for a (a) planar and (b) rib waveguide.

n

I niSES =

∑

I kSES .

k =1

Integrated SES signal should contain the same amount of spontaneous emission as the VSL measurement (including losses and effects of imperfect experimental conditions), the only difference being that in VSL experiment some photons go through the excited area and might be amplified by stimulated emission or affected by induced absorption. Which effect takes place is clear from plotting both I nVSL and I niSES in one figure. Similarly, we can calculate the differential VSL intensity I ndVSL , i.e., the difference of signals obtained for stripe lengths l = nâ•›.â•›x0 and l = (n − 1)â•›.â•›x0

I ndVSL = I nVSL − I nVSL −1

and compare it with I nSES. Again, the detected signal comes from spontaneous emission in the nth excited spot that is passing through the excited or unexcited area for the dVSL and SES, respectively. In the ideal case, we can use equations I ndVSL = const exp[(g − K )dn ] and InSES = const exp(− Kdn ) to derive the net optical gain g(λ)

g (λ ) =

(

ln I ndVSL (λ) / I nSES (λ) dn

),

where dn is the distance of the center of the nth spot from the edge. The PP technique is based on the application of two beams: the first one is pumping the active sample and the second one is testing the state created by the pumping beam. In case of studying stimulated emission, the test beam may be amplified by stimulated

emission. In general, the test beam spot must be situated inside the pumping spot in order to test area pumped as homogeneously as possible. In case of pulsed beams, the temporal coincidence of both pumping and testing pulses must be controlled (the test beam may be delayed after pumping pulses to study decay of the effect). If the thickness of the active layer is small, the induced effect on the test beam may be difficult to detect. Therefore, for waveguide samples, the test beam is often coupled to the waveguide, which is excited from the surface (Figure 37.9b). Then, changes of the outcoupled light with and without pumping are detected.

37.4â•‡Main Experimental Observations in Active Si-nc Waveguides The most important observations for active planar waveguides are (1) spectral filtering; (2) TE (transverse electric) and TM (transverse magnetic) mode splitting; and (3) high directionality of TE, TM mode emission. These effects are illustrated in Figure 37.10; the panel (a) shows that the PL leaving the edge of waveguide layer is substantially different from the PL perpendicular to the layer. The perpendicular PL is formed by a single band that corresponds to the well-known emission of Si nanocrystals, but the edge PL contains two narrow PL bands. Both narrow lines are linearly polarized, one in the direction of the waveguide layer (TE mode) and the second one perpendicular to the layer (TM mode). The spectral position of the TE and TM modes depends on the exact profile of the refractive index, it means on the implantation fluence (see also Figure 37.16, which shows spectra of another set of samples). The panels (b) and (c) illustrate that the TE and TM modes are emitted only in an extremely narrow angle (a few degrees), basically in the direction of the waveguide plane (only slightly inclined to the substrate, i.e., under positive angles in our notation).

37-8

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Photon energy (eV) 2

1.9

1.7

1.8

PL intensity (a.u.)

Perpendic. PL

1.6

5.0 4.5 Perpendic. PL

4.0

Exc.

PL intensity (a.u.)

650

700


TM

700 800 900

Edge PL 850

900

6 × 1017 cm–2

Edge PL 6 × 1017 cm–2

–30°

–5° –10°

PL

0°

+5°

+10°

Wavelength (nm)

0

+30°

Integ. PL int. 700–900 nm (a.u.)

600

TE

(b)

1.4

Impl. dose 6 × 1017 cm–2

5.5

Edge PL Det. angle 5°

(a)

1.5

(c)

FIGURE 37.10â•… PL spectra of five fused silica slabs implanted to the fluences of 4 × 1017 cm−2 to 6 × 1017 cm−2. (a) Upper curves (a single wide band) correspond to PL emitted in a direction perpendicular to the waveguide, while lower spectra with doublet peaks are facet-PL detected in a direction α = 5° (a sketch of the experimental arrangement is shown in the inset). (b) Angle-resolved facet PL spectra of the sample 6 × 1017 cm−2. (c) Polar representation of integrated PL intensity of the angle-resolved facet spectra from the panel b. Most of the PL intensity is emitted in a direction close to 0°. (Adapted from Janda, P. et al., J. Lumin., 121, 267, 2006.)

The coupling and propagation of external light in active Si-nc waveguides were studied by the direct (end-fire) and the prism coupling (Figure 37.5a and b). The results for one sample are illustrated in Figure 37.11. For prism coupling, two broad transmission bands (in the blue and red spectral region) are observed in the measured spectral range. The positions of both bands coincide with those of the PL modes (Figure 37.10a). Our calculations show that the red and blue bands corresponds to the second and third order of substrate modes (the first order being in the infrared region), see Section 37.5 and Figure 37.14. Broadening of the mode structure may be a consequence of the very low number of reflections undertaken by coupled light before escaping to the substrate. Coupling of external light through a truncated facet (Figure 37.5b) gives the best result for a coupling angle γ ∼ 20°, as expected (Figure 37.11, upper curves). In this configuration, the narrow and polarizationsplit peaks at an output angle αâ•›∼â•›2° are detected. The peaks

are, however, not transmission but absorption peaks. This can be understood if it is assumed that the detected light comes not from substrate modes (which represent a small portion of transmitted light) but from filtered transmitted light propagating almost parallel to the Si-nc waveguide from which a part of power escaped to the substrate modes that are, however, gradually absorbed in the waveguide core. The blue third order modes are much stronger compared to second order because of higher absorption in the blue spectral region. Experimental observation of rib waveguides indicates (Figure 37.12) that normal guiding of light is improved—see increased intensity of the long-wave edge of PL spectra in Figure 37.12a and b. This is due to the introduction of confinement in the second lateral direction of waveguide (in this case, the confining refractive index profile is symmetrical and the contrast higher (compared to the implanted layer profile) as the surrounding medium is air). On the other hand, roughness of

37-9


37.5â•‡Theoretical Description of Active Lossy Waveguides

No polar

Transmission (a.u.)

Edge coupling γ = 20°, α = 2° TE

5.5 × 1017 cm–2

TM

Prism coupling α = 7°

400

450

500

550

600

650

700

750

800

Wavelength (nm)

FIGURE 37.11â•… Comparison of transmission spectra of a sample 5.5 × 1017 cm−2 obtained by the direct facet-coupling (upper curves, solid line—no polarization, dashed and dotted lines correspond to TE and TM polarization, respectively) and by the prism coupling (lower spectrum). (Adapted from Janda, P. et al., J. Lumin., 121, 267, 2006.) 3 × 1017 cm–2

PL intensity (a.u.)

(a)

4 × 1017 cm–2

(b)

5 × 1017 cm–2 Rib waveguide Planar waveguide

(c)

6 × 1017 cm–2

Light coupling to a waveguide and its further propagation may be understood in terms of coupling between the waveguide modes and nanocrystals emitting photons. Light field inside the waveguide may be expressed as a coherent superposition of excited waveguide modes, which arise as a solution of a homogeneous wave equation with proper boundary conditions [the waveguide modes are an orthonormal basis of the waveguide optical field (Snyder and Love 1983)]. It is therefore necessary to distinguish between various types of the waveguide modes in order to explain different types of energy transport in the waveguides. Let us consider a common model structure with three transparent layers (see Figure 37.13): the top cladding layer with the refractive index n1, the bottom substrate with the refractive index n3, and the core with the refractive index n2 > n3 > n1 and thickness d. We may imagine the modes as rays emerging from inside the waveguide core and propagating toward one of the core boundaries. At the boundary, the ray may be either totally reflected back to the core or it may be partially reflected and partially refracted. The ray then travels toward the other boundary where it totally or partially reflects again. Only those modes that are totally reflected twice during one round-trip may propagate in the core without losses: these are the common guided modes. There are, in addition, lossy modes that are called the substrate radiation modes (they are refracted only into the substrate) and the radiation modes refracted both to the substrate and the cladding. In many applications, considering traveling of light at large distances compared to its wavelength (and thus working in the far-field limit), we may consider only the guided modes as efficient carriers of energy between a source and a detector. In silicon-based nanophotonics devices, however, the situation may be even more complicated. The typical distances between components on a chip may be comparable to the light wavelength and the far-field approach fails. It is therefore important to analyze an influence of the substrate radiation modes on the response. We may rule out the radiation modes at this stage: compared to the substrate modes, they play only a negligible role in the overall system behavior. z

n(x)

600 (d)

700

800

θ2

900

Wavelength (nm)

d

n2

FIGURE 37.12â•… Comparison of the edge PL spectra from planar and rib waveguides (dashed and continuous line, respectively) in samples implanted to a fluence of (a) 3 × 1017 cm−2, (b) 4 × 1017 cm−2, (c) 5 × 1017 cm−2, and (d) 6 × 1017 cm−2.

the rib edges can introduce significant losses due to scattering. The influence of rib structure on the substrate (leaky) modes is less significant (see Figure 37.12c and d). This is because these modes propagate mostly in substrate, where the effect of side walls is negligible.

n1

α n3 x

Side view

FIGURE 37.13â•… Schematic representation of propagation and decoupling of substrate modes.

37-10


   4πin2 d cos(θ2 )  Emode (λ, θ2 ) = E0 (λ)  1 + r21r23 exp  +   λ    

TE

0.8

TM TE

TM TE

0.6

TM

0.4 0.2

(37.2)

Here, r 21 and r23 are the reflection coefficients at the respective boundaries, which depend upon the propagation angle θ2. The wavelength is λ and E0(λ) is an effective emission amplitude of a nanocrystal to the waveguide mode. Considering the guided modes and thus |r21| = |r23| = 1, the electric field intensity reveals sharp resonances implying there are well-defined discrete guided modes. The energy carried by each mode is finite, indeed. Unlike the guided modes, the electric field intensity is finite for the substrate modes and there are no sharp resonances in the spectra. We may, however, resolve some weak resonances, depending on the phase factor of the term r21r23 exp[4πin2d cos(θ2)/λ]: the electric field is maximum if this term is positive and real. It is clear from the ray optics that there are substrate modes that refract to the substrate at the angles near π/2, which means that they propagate nearly parallel to the core/substrate boundary; these modes may be understood as a crossover between the guided modes and the substrate radiation modes which decouple rapidly from the core, possessing mixed characteristics of the both types of modes. As the modes may be assigned to a distinct guided mode of the nth order (energy of the substrate mode is slightly below the cutoff energy of the nth guided mode), we denote each series of the substrate modes to be of the nth order (see Figure 37.14). The Fresnel formulae give us |r21| = 1 and |r23| ≈ 1 in such a case (Figure 37.15), and it is then clear from Equation 37.2 that these substrate modes reveal sharp resonances in the spectra (for a fixed angle θ2 < arcsin(n2/n3)). Energy carried by the substrate modes may be therefore comparable to the energy carried by guided modes. The most important point here is that this type of substrate modes cannot be experimentally distinguished from the guided modes at short distances from the source as they propagate near the core/substrate boundary. The substrate modes therefore significantly contribute to the system response and they may play an important role in many experiments as well as in real on-a-chip devices. Although the guided and the substrate modes propagate in a similar direction, they may behave in very different ways because they propagate in two different environments. In our particular case, the guided modes are mostly localized inside the waveguide core doped by Si-nc, providing a possibility of reabsorption or optical amplification. The substrate modes, on the contrary, propagate in a transparent substrate and they are therefore neither amplified nor attenuated. Rigorous derivation of PL spectra in real waveguides accounting for losses or

(a)

Relative width (%)

E0 ( λ ) . = 1 − r21r23 exp[4πin2 d cos(θ2 ) / λ]

1.0 Refr. index contrast

Once light is emitted by a nanocrystal inside the waveguide mode, it undergoes internal reflections at the core/cladding and the core/substrate boundaries. The overall intensity of electric field of an excited mode inside the waveguide core may be written as superposition of the partial waves:

(b)

200

TE

TM

TE

TM


900

150 100 50 TM

TE

0 300

TM TE

FIGURE 37.14â•… (See color insert following page 20-14.) Calculated spectral positions of the substrate modes as a function of (a) the refractive index contrast Δn = n2 − n3 and (b) the relative thickness of the waveguide core compared to the sample 5 × 1017 Si cm−2. Gray scale indicates intensity from black up to white for the highest intensity. Several orders of modes are seen starting from the first one in infrared region. Lower interface

Upper interface

G

S

π

1

TM

R

φ TE

TE TM 0

θ

θc

0 90°

FIGURE 37.15â•… Reflectance R = |rij|2 and phase shifts ϕ on the planar boundary between two dielectric media plotted for TE and TM modes versus incident angle θ. S and G stand for substrate and guided modes, respectively. (Adapted from Pelant, I. et al., Appl. Phys. B, 83, 87, 2006.)

amplification, decoupling of the light from the waveguide and real detection geometry is presented elsewhere (Ostatnický et al. 2008); here, we summarize only the main results. Considering lossy waveguide core, the substrate modes become dominant in the optical response of a waveguide when detected in the direction parallel to the core/substrate boundary using a

37-11


3 × 1017 cm–2

TM TE

4 × 1017 cm–2 TM

5 × 1017 cm–2 TE

PL intensity (a.u.)

TE

TM

6 × 1017 cm–2 TE

(a)

600 (b)

700

800

900 (c) Wavelength (nm)

700

800

TM

900

FIGURE 37.16â•… (See color insert following page 20-14.) (a) Photograph of the edge of a set of Si+ ion implanted layers with direction of PL indicated by arrows, the edge is on the left. (b) Measured PL from samples implanted to different Si ion fluences in standard (the broadest curves) and waveguiding geometry (black lines, the slightly lighter gray lines stand for TE and TM resolved polarizations). (c) Theoretically calculated PL spectra. We note that these results were obtained on different set of samples than in Figure 37.10. The mode positions are not exactly the same for samples with identical implantation dose because the annealing conditions were slightly different. Therefore, the refractive index profiles are not identical. (Adapted from Pelant, I. et al., Appl. Phys. B, 83, 87, 2006.)

detector with a small numerical aperture. This situation is clearly illustrated through the comparison of our experimental data with the theoretical estimates in Figure 37.16 (we considered a continuous refractive index profile in our calculations). The broad part of the emission spectra originates from the (attenuated) guided modes and the narrow peaks are due to the substrate modes. The double-peak structure is further resolved as two single peaks with the respective TE and TM polarizations. In order to explain this point, we should remark that the coefficient r23 is real for the substrate modes while the coefficient r21 is in general complex. Considering a fixed angle of incidence θ2, r21 gives different values of the phase shift for the respective TE and TM polarizations (see Figures 37.14 and 37.15). Therefore, the resonance condition is met at different wavelengths for different polarizations. From these considerations, the necessity of waveguide asymmetry for the appearance of the TE–TM polarization–resolved modes follows. The TE–TM splitting was also reported in Khriachtchev et al. (2004), but the respective peaks have an opposite order in the spectra compared to our experiments. Our numerical

calculations have shown that the TE–TM splitting depends strongly on the refractive index profile of the waveguide core. For the three-layer structures, the TE mode is positioned always at the high-energy side of the PL spectra. It is, nevertheless, possible to fabricate a structure with multiple layers or with a graded refractive index profile, which provides an arbitrary TE–TM splitting (including both negative and positive values and also their degeneracy); this feature is well illustrated in Figure 37.14b, where the respective TE and TM modes interchange depending on the thickness of the guiding layer. The appearance of the substrate modes is well controllable by the parameters of the particular layers of the waveguide. Clearly, the optical thickness of the core d determines the resonance condition through the exponent in the denominator in Equation 37.2 and the ratio n2/n1 has influence on the phase of the r21 term. The latter may be used in detectors integrated on a chip—the cladding can be made of a material that changes its refractive index due to the changes of the surrounding environment conditions, and the position of the emission peak may then

37-12

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Stripe length (cm) 0.00

0.02

0.04

0.06

0.08

0.10

0.12

(a)

640

G = 35 cm–1 15° 11° 6° 4° 2° 0° –5°


15°

880

11° 6° 4° 2° 0° (b)

TM G = 28 cm–1 PL 825 nm

–5° 720 800 880 Wavelength (nm)

FIGURE 37.17â•… (a) Measured PL spectra at different detection angles α relative to the waveguide axis. (b) Numerical simulation of the measurements from the panel a. The sample with an implantation fluence of 5 × 1017 cm−2. (Adapted from Ostatnický, T. et al., Guiding and amplification of light due to silicon nanocrystals embedded in waveguides, in Khriachtchev, L. (ed.), Silicon Nanophotonics, World Scientific, Singapore, 2008, pp. 267–296.)

be correlated with some external variable (temperature, pH, etc., see Figure 37.20) (Luterová et al. 2006). As the sharp resonance in the spectra of the substrate modes is restricted only to the region where r 23 ≈ 1, we may expect disappearance of the substrate modes at large detection angles, i.e., when the substrate modes do not propagate parallel to the core/ substrate boundary. This feature is illustrated in Figure 37.17 in comparison to the experimental and the theoretical data. We see a very good agreement justifying the correctness of our model. An important aspect of nanodevices is the magnitude of the decoupling length for the substrate modes, i.e., the distance at which the substrate modes completely leak out from the waveguide core. Our calculations (Ostatnický et al. 2008) show that this distance may be few micrometers but also it may be as large as several hundreds of micrometers. As a consequence, the appearance of the substrate modes and their decoupling from the waveguide core may be of a big importance in nanodevices on the contrary to the fiber optics where the optical response to any excitation is provided solely by the guided modes at the typical distances of the order from meters to kilometers.

37.6â•‡Application of Active Nanocrystalline Waveguides 37.6.1â•‡Amplification of Light The significant narrowing of substrate leaky mode emission suggests immediately that optical amplification could be responsible for this observation. The VSL method (described in Section 37.3.3) has been applied to the planar asymmetric Si-nc waveguides in several laboratories (Khriachtchev et al. 2001, Ivanda et al. 2003, Luterová et al. 2005). It is tempting to interpret the frequently observed initial weak exponential growth of the VSL curve (Figure 37.18a) as a manifestation of

(a)

0 TE

PL intensity (lin.u.)

PL intensity (a.u.)

TE

(b)

TM

PL 825 nm 0

TE TM G = 29 cm–1 G = 22 cm–1 PL 825 nm

(c)

0 0.00

0.02

0.04 0.06 0.08 0.10 Distance from edge (cm)

0.12

FIGURE 37.18â•… VSL and SES measurements on a sample implanted to the dose of 4 × 1017 cm−2 under continuous wave excitation 325â•›nm, 0.26â•›W/cm2: (a) VSL measurement at the peak of TE and TM modes and for non-guided PL around 825â•›nm. The fits (lines) give values of G = 35 and 28â•›cm−1 for TE and TM modes, respectively, and losses of 11â•›cm−1 for non-guided PL. (b) Results of the SES measurement performed under identical conditions as the VSL. (c) Integration of data from panel (b). The gain fits (lines) give values of G = 29 and 22â•›cm−1 for TE and TM modes, respectively. (Adapted from Valenta, J. et al., Appl. Phys. Lett., 81, 1396, 2002.)

the occurrence of optical gain. High output directionality of the substrate mode emission apparently supports such interpretation. However, it has turned out that, as shown above, it is very difficult if not impossible to evaluate correctly the optical gain magnitude of the substrate modes because the nonlinear growth can originate in the mode leaking itself. Figure 37.18 demonstrates that the evaluation of the VSL experiments can yield a false optical gain if proper comparison between the VSL and SES results is not taken into account. On the other hand, VSL measurements employing a high pulsed laser excitation, performed on a similar sample, revealed a characteristic switch from the light attenuation (G < 0) to amplification (G > 0) with increasing excitation energy density (Figure 37.19). It may be also of interest here to call the reader’s attention to recent articles reporting firmly positive optical gain on leaky

37-13


substrate modes in asymmetric thin-film organic waveguides (Nakanotani et al. 2007, Yokoyama et al. 2008). At present, checking the reliability of the VSL method in the case of active asymmetric thin waveguides represents an issue requiring further investigation.

40

ASE intensity (a.u.)

λdet = 760 nm TM polarization 30

Pumping fluence 86 mJ/cm2 Pumping fluence 5 mJ/cm2

20

37.6.2â•‡ Other Applications

G = –6 cm–1

10 G = 12 cm–1 0

0

0.5

1.0 1.5 Stripe length (nm)

2.0

2.5

FIGURE 37.19â•… Time-resolved VSL measurement of optical amplification at position of the TM mode (760â•›nm) under pumping by 6â•›ns pulses, 355â•›nm. The fit with Equation 37.1 gives the net gain coefficient of (−6 ± 6) and (12 ± 2) cm−1 for pumping fluence of 5 and 86â•›mJ/cm2 , respectively. The threshold for positive gain is about 50â•›mJ/cm 2. Sample was implanted to a dose of 4 × 1017 cm−2. (After Luterova, K. et al., Phys. Status Solidi (c), 2, 3429, 2005.) α-Bromineraphtalene n = 1.657

Although the benefit of the waveguide spectral filtering for nanocrystalline thin-film laser design is still controversial, the substrate modes may be attractive for other nanophotonic applications. Firstly, they provide a way how to generate easily the spectrally narrow, polarization resolved and directional emission in the wavelength range 650–950â•›nm without the necessity of building optical (micro)cavities. Moreover, this emission can be spectrally tuned simply by engineering the silicon excess content, as shown in Figures 37.10 and 37.16. Potential feasibility of these properties for demultiplexing optical signal in photonic circuits is evident, but improvement in quality of the rib waveguides is anticipated. Secondly, spectral sensitivity of the radiative substrate modes to surrounding (organic) compounds can be utilized in photonic sensing (Figure 37.20): Magnitude of the refractive index a

Liquid drop

b

x z

Immersion oil n = 1.515

SiO2 substrate Excitation

Ethanol n = 1.361 Acetone n=1.359

~7 mm 900 Exper.

Air n = 1

TE TM

Theory

(a)

860

820

PL maximum (nm)

PL intensity (a.u.)

CCI4 n = 1.46

780

740 (b)

700


820

1.0 (c)

1.2 1.4 Refractive index

1.6

FIGURE 37.20â•… The influence of a drop of liquid on the PL spectra: (a) the drop is on the excited spot and (b) the drop is out of excited spot (between the spot and the sample edge). (c) Comparison of experimental (points) and calculated (line) positions of PL maxima for different refractive indices of liquid drops. Sample is implanted to a dose of 5 × 1017 cm−2. (Adapted from Pelant, I. et al., Appl. Phys. B, 83, 87, 2006; Luterová, K. et al., J. Appl. Phys., 100, 074307, 2006.)

37-14


n1 of the compound (playing role of the cladding layer) affects markedly the spectral position of the TE–TM doublet. When n1 exceeds the magnitude of the core refractive index (n2 ≈ 1.45), the condition of total reflection |r21| = 1 is canceled and the TE– TM doublet completely disappears. A more systematic research in this direction is, however, missing. Finally, the active waveguides give the chance to circumvent the coupling problem connected with discrete photonic elements, i.e., the problem of how to inject efficiently light from an external source in a waveguide (Orobtchouk 2006). The active waveguides eliminate the need of any optical couplers and enable it to design integrated photonic circuits. Theoretical analysis of active Si-nc devices integrating optical emission and waveguiding, compatible with silicon VLSI processing technology, have been submitted quite recently (Milgram et al. 2007, Redding et al. 2008).

37.7â•‡ Conclusions Active waveguides formed by densely packed silicon nanocrystals represent a promising type of nanophotonic waveguides. In contrast to other types of nanowaveguides (based on photonic waveguides, plasmonic structures, etc.), it can be prepared by various technological approaches available and well mastered in many research and industrial laboratories. Its potential applications range from optical amplifiers and filters to optical sensors. The descriptions of characterization techniques and theory of waveguide modes in this chapter are applicable to many other waveguide structures.

Acknowledgments This work was supported by the Czech Ministry of Education, Youth and Sports through the research center LC510 and research plans MSM0021620835 and 60840770022, the project 202/07/0818 of the Grant Agency of the Czech Republic, and the projects IAA101120804 and KAN401220801 of the Grant Agency of the Academy of Sciences. Research carried out in the Institute of Physics was supported by the Institutional Research Plan AV0Z10100521. The authors thank Prof. R.G. Elliman (Australian National University, Canberra) and Ing. V. Jurka for the fabrication of waveguides.

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Iacona, F., Franzò, G., and Spinella, C. 2000. Correlation between luminescence and structural properties of Si nanocrystals. Journal of Applied Physics 87: 1295–1303. Imakita, K., Fujii, M., Yamaguchi, Y., and Hayashi, S. 2005. Interaction between Er ions and shallow impurities in Si nanocrystals. Physical Review B 71: 115440-1–115440-7. Ivanda, M., Desnica, U. V., White, C. W., and Kiefer, W. 2003. Experimental observation of optical amplification in silicon nanocrystals. In L. Pavesi, S. Gaponenko, and L. Dal Negro (Eds.), Towards the First Silicon Laser, NATO Science Series, II: Mathematics, Physics and Chemistry, Vol. 93, pp. 191–96. Dordrecht, the Netherlands: Kluwer Academic Publishers. Janda, P., Valenta, J., Ostatnický, T., Skopalová, E., Pelant, I., Elliman, R. G., and Tomasiunas, R. 2006. Nanocrystalline silicon waveguides for nanophotonics. Journal of Luminescence 121: 267–273. Khriachtchev, L., Räsänen, M., Novikov, S., and Sinkkonen, J. 2001. Optical gain in Si/SiO2 lattice: Experimental evidence with nanosecond pulses. Applied Physics Letters 79: 1249–1251. Khriachtchev, L., Novikov, S., and Lahtinen, J. 2002. Thermal annealing of Si/SiO2 materials: Modification of structural and photoluminescence emission properties. Journal of Applied Physics 92: 5856–5862. Khriachtchev, L., Räsänen, M., Novikov, S., and Lahtinen, J. 2004. Tunable wavelength-selective waveguiding of photoluminescence in Si-rich silica wedges. Journal of Applied Physics 95: 7592–7601. Khriachtchev, L., Navarro-Urios, D., Pavesi, L., Oton, C. J., Capuj, N. E., and Novikov, S. 2007. Spectroscopy of silica layers containing Si nanocrystals: Experimental evidence of optical birefringence. Journal of Applied Physics 101: 044310-1–044310-6. Luterová, K., Dohnalová, K., Švrček, V., Pelant, I., Likforman, J.-P., Crégut, O., Gilliot, P., and Hönerlage, B. 2004. Optical gain in porous silicon grains embedded in sol-gel derived SiO2 matrix under femtosecond excitation. Applied Physics Letter 84: 3280–3282. Luterová, K., Navarro, D., Cazzanelli, M., Ostatnicky, T., Valenta, J., Cheylan, S., Pelant, I., and Pavesi, L. 2005. Stimulated emission in the active planar optical waveguide made of silicon nanocrystals. Physica Status Solidi (c) 2: 3429–3434. Luterová, K., Skopalová, E., Pelant, I., Rejman, M., Ostatnický, T., and Valenta, J. 2006. Active planar optical waveguides with silicon nanocrystals: Leaky modes under different ambient conditions. Journal of Applied Physics 100: 074307. Milgram, J. N., Wojcik, P., Mascher, P., and Knights, A. P. 2007. Optically pumped Si nanocrystal emitter integrated with low loss silicon nitride waveguides. Optics Express 15: 14679–14788. Nakanotani, H., Adachi, C., Watanabe, S., and Katoh, R. 2007. Spectrally narrow emission from organic films under continuous-wave excitation. Applied Physics Letters 90: 231109-1–231109-3.


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Shaklee, K. L. and Leheny, R. F. 1971. Direct determination of optical gain in semiconductor crystals. Applied Physics Letters 18: 475–477. Skopalová, E. 2007. Mode structure in the light emission from planar waveguides with silicon nanocrystals. Diploma thesis. Prague, Czech Republic: Charles University. Snyder, A. W. and Love, J. D. 1983. Optical Waveguide Theory. London, U.K.: Chapman & Hall. Valenta, J., Pelant, I., and Linnros, J. 2002. Waveguiding effects in the measurement of optical gain in a layer of Si nanocrystals. Applied Physics Letters 81: 1396–1398. Valenta, J., Pelant, I., Luterová, K., Tomasiunas, R., Cheylan, S., Elliman, R., Linnros, J., and Honerlage, B. 2003a. Active planar optical waveguide made from luminescent silicon nanocrystals. Applied Physics Letters 82: 955–957. Valenta, J., Luterová, K., Tomašiunas, R., Dohnalová, K., Ho″ nerlage, B., and Pelant, I. 2003b. Optical gain measurements with variable stripe length technique. In L. Pavesi, S. Gaponenko, and L. Dal Negro (Eds.), Towards the First Silicon Laser, NATO Science Series, II: Mathematics, Physics and Chemistry, Vol. 93, pp. 223–242. Dordrecht, the Netherlands: Kluwer Academic Publishers. Valenta, J., Ostatnický, T., Pelant, I., Elliman, R. G., Linnros, J., and Ho″ nerlage, B. 2004. Microcavity-like leaky mode emission from a planar optical waveguide made of luminescent silicon nanocrystals. Journal of Applied Physics 96: 5222–5225. Yokoyama, D., Moriwake, M., and Adachi, C. 2008. Spectrally narrow emissions from edges of optically and electrically pumped anisotropic organic films. Journal of Applied Physics 103: 123104-1–123104-13.

38 Biomolecular Neuronet Devices 38.1 Introduction............................................................................................................................38-1 Bacteriorhodopsin Is an Advanced Material for Molecular Nanophotonics

38.2 Some Techniques of Neuro-Molecular and Molecular Information Processing Using Bacteriorhodopsin (Basic Processes, Constructions, Technology).....................38-2

Grigory E. Adamov Central Scientific Research Institute of Technology “Technomash”

Evgeny P. Grebennikov Central Scientific Research Institute of Technology “Technomash”

Basic Process of Classic Optical Neural Network in Bacteriorhodopsin Mediumâ•‡ •â•‡ Neuronets Based on Multilayered Optical Structures Including Polymeric Bacteriorhodopsin-Containing Layers

38.3 Nanostructuring of Bacteriorhodopsin-Containing Molecular Media.......................38-10 Procedure of Complex Estimation of Functional Characteristics of BacteriorhodopsinContaining Materialsâ•‡ •â•‡ Functional Characteristics of Nanostructured Bacteriorhodopsin Filmsâ•‡ •â•‡ Metal Nanoparticles and Bacteriorhodopsin-Based Hybrid Nanostructures

38.4 Summary................................................................................................................................38-17 References..........................................................................................................................................38-17

38.1â•‡Introduction Due to the constant development of nanotechnologies and functional nanomaterials used in information systems, researchers need to look for new ideas and radically new constructive solutions to ensure efficient device operation, record-breaking high speed, and integration of elements. On the other hand, functional elements being formed in nanometer range require landmark approaches. It refers especially to the formation of informationlogical devices at the molecular level. That is, the formation of nanodevices by direct impact on separate molecules or atoms is not efficient in the long run. The total time taken for operations with separate molecules leads to tremendous amount of time taken to form the device and huge costs. Another techno-systematic problem is the necessity to use “macro–nano” and “nano–macro” interfaces to put in the original information and deduce final results. The advantages of the small-size molecular elements cannot help to ensure access to them. Probably the only effective way to arrange the bonding with molecular system is to use optical interaction. Under the circumstances, it seems quite reasonable to refer to the most competent specialist, whose experience in creating molecular systems cannot raise any doubts, that is, to the nature itself and try to apply bionic principles to engineering and design. The solution seems to be connected with hierarchic structural and functional self-organization of molecular systems that use assemblies of molecules as a basis for molecular information-logical appliances. Among all the information systems, bionic approach seems to be the most effective, harmonic, and holistic one; it is important

to point out molecular neural network appliances. Both adaptive self-organization principles of data processing and self-Â� organization principles of functional nanostructures can be applied to those systems. Below you will find a detailed description of techno-Â�systematic approaches aimed at implementation of biomolecular neural network appliances. Mentioned below, biological material—Â�protein “bacteriorhodopsin”—has unique technological potential and its optical properties allow the use of optical input–output devices and create molecular appliances based on self-organization principles.

38.1.1â•‡Bacteriorhodopsin Is an Advanced Material for Molecular Nanophotonics Bacteriorhodopsin (BR)—light-sensitive protein—is similar to the optic rhodopsin of the human eye. BR is obtained from halobacteria containing BR in cellular membranes (so-called purple membranes). During the separation from the bacterial cells, purple membranes save the entire structure (Vsevolodov, 1988; Oesterhelt et al., 1991). The typical size of purple membranes is 500–1000â•›nm. It is the unique biocrystalline structure capable of saving its permanent properties for a number of years, which consists of dry and polymer films with the thickness from 5â•›nm (monolayer) up to a few tens of micrometers. The fundamental BR property is the photochemical cycle availability: after the light quantum absorption, BR molecule passes through the sequence of states and spontaneously returns to the primary form (Figure 38.1). At that point, in compliance with the cycling of BR molecule state, the light-induced cycling 38-1

38-2

Handbook of Nanophysics: Nanoelectronics and Nanophotonics Ground state

hν

Q380

hν

P490

Br570

hν hν

K610

O640

hν N520

L550

M412

Figure 38.1â•… BR photocycle.

Molar absorptivity (M–1*cm–1*10–4)

of optical characteristics (refraction and absorption indices) occurs (Haronian and Lewis, 1991; Zeisel and Hampp, 1992). Each of the interstitial states is identified as the intermediate according to its absorption spectrum. The existence of branched photocycles is typical for some BR types (Birge et al., 1999). The main BR function in purple membranes is light-Â�dependent proton transfer (H+) over the purple membrane that results in electrochemical hydrogen potential formation on halobacteria membrane. The potential energy is utilized by cell. Ejection of H+ occurs outside the cell membrane, and the H+ is captured inside the cell (from cytoplasm). Supposedly, it happens during the formation and the disappearance of intermediate M412 (Siebert et al., 1982; Haronian and Lewis, 1991). BR spectral sensitivity lies in the optical band (Figure 38.2). Absorption maximum in primary state of BR570 corresponds to wave length 570â•›nm. M412 in the main intermediate state reaches its absorption maximum at wave length 412â•›nm (Birge et al., 1999). The absorption of optical emission by BR-containing medium is characterized by certain peculiarities. That happens due to the change of adsorption sites concentration (molecules in form BR570), which is the result of light quantum absorption at wave Br

6 5 4

Q

M

2 1 400

450


38.2â•‡Some Techniques of NeuroMolecular and Molecular Information Processing Using Bacteriorhodopsin (Basic Processes, Constructions, Technology) 38.2.1â•‡Basic Process of Classic Optical Neural Network in Bacteriorhodopsin Medium BR properties listed above allow us to illustrate one of the available basic processes of data transformation in BR-containing media. It is based on reversible light-sensitive changes of absorption index and appears in optical effects considered below. 38.2.1.1â•‡Nonlinear Absorption of Optical Radiation: Medium Bleaching The absorption of optical radiation in substance is described by some known classic equations. Generalized Bouguer–Beer law associates intensities of the incident light and the light transmitted through the substance layer with the thickness of the layer and molecular concentration of absorption agent:

I = I 0 ⋅ e − D = I 0 ⋅ e − αd = I 0 ⋅ e − εcd ,

(38.1)

where I is the transmitted light intensity I0 is the incident light intensity D is the optical density of the substance α is the absorption index of the substance d is the thickness of the substance layer c is the molar concentration of absorbing substance molecules ε is the extinction coefficient, characteristic feature of absorbing substance molecule

3

350

length 570â•›nm by these sites and transition to form M412 with a low absorption at wave length 570â•›nm. As a result, absorption in the yellow range reduces; medium becomes more transparent— bleached. The intensity of BR-containing medium bleach effect depends particularly on the time of intermediate M412 molecules relaxation to form BR570. Relaxation time is characterized by half-value period of intermediate M412 molecules. The light effect at wave length 412â•›nm results in fast coercive transition of molecules to the primary state. The values of photocycle time parameters lie in the range from microseconds up to tens of seconds. Thus, BR behaves as a photochromic medium with a quick time of information storage. Optical and dynamic BR characteristics change in wide range by production conditions and matrix composition (medium).

600

650

700

Figure 38.2â•… Absorption spectra of BR and of the main photocycle intermediates.

The values of ε, α, and D depend on the wave length of incident light. When the values of coefficients in Equation 38.1 are invariable, the transmitted light intensity is in direct proportion to the incident light intensity.

38-3

Biomolecular Neuronet Devices

Nonlinear absorption of optical radiation by BR-containing media is connected to the change of absorption centers concentration (molecules in form BR570) as a result of light quanta absorption by these centers at wave length 570â•›nm and transformation to M412 with low absorption at wave length 570â•›nm. Finally, as was mentioned above, absorption in the yellow range (λ = 570â•›nm) decreases, medium becomes more transparent—bleached. 38.2.1.2â•‡I ndirect Interaction of Optical Radiation Fluxes in BacteriorhodopsinContaining Media Mediated interaction of optical radiation fluxes appears during the sequential or combined transmission through the same part of BR-containing medium and is at its clearest for monochromatic radiation with wave length 412 and 570â•›nm. As a result of interaction between BR and the radiation with wave length 570â•›nm during the transmission through the medium, the energy of the light flux is absorbed, and in BR-containing medium, the photo-induced allocation of the absorption index forms. The light-induced allocation of the absorption index variation corresponds to energy distribution along the surface of the transmitted light wave front (Figure 38.3A). Non-modulated along the front surface, the light pulse with the wave length of 570 or 412â•›nm (as actuating or inhibiting signal) absorbs spatially and nonuniformly in compliance with the changed value of the absorption index (Figure 38.3B and C). Thus, the energy distribution along the preceding pulse front surface, indirectly, over the BR-containing medium, modulates the energy distribution along the following pulse front surface. As a result, BR-containing medium is capable of accumulating (summing up) effects signed “+” and “−,” correspondingly, at wave lengths 570 and 412â•›nm. Predicating upon the indirect interaction of optical radiation in BR-containing media, the method of formal neuron creation in such media is available to offer. At that point, all the main functions can be realized by optical technique.

38.2.1.3â•‡One of the Possible Methods of Formal Neuron Main Functions Realization in Bacteriorhodopsin Medium Realization of the main operations of neuronet algorithms— weighing of input signal vector according to the matrix of weighing coefficients of synaptic bonds; composition of weighed values of input signals; realization of activation (threshold) function by optical method without optoelectronic buffering—permits to simplify the construction and the technological realization of multilayered optical neuronet, to increase the integration of neuro-like elements in device, and to solve the problem of areal density limitation inherent in microelectronic elements and electric connections. 38.2.1.4â•‡Formation of the Neuro-Like Element The neuro-like element is formed under the exposure of optical emission with the spectrum, corresponding to the absorption spectrum of BR molecules’ initial state and BR-containing material medium. This element is a part of the BR-containing medium with photo-induced absorption index. The threshold properties of such neurons are defined by the concentration ratio of molecules in primary and photo-induced forms, and the interaction of neuro-like elements is provided by optical emission. The example of the similar neuronet realization, based on information conversion of basic process in BR-containing media (Figure 38.4), is considered below. Construction for optical neuronet formation includes the following:

1. The source of the plane light front (transparent for normal incident light flux) providing the signal formation and transfer to neurons. 2. Photo-detecting layer based on BR-containing material for imaging of the input optical information by photoinduced variation (according to light energy distribution along the surface of input light front) of absorption/transmission in BR-containing medium.

λ = 570 nm λ = 412 nm

(A)

(B)

(C)

(D)

λ = 412 nm

Figure 38.3â•… The indirect interaction of optical radiation fluxes in BR-containing media: (A) green light pulse (modulated along the front surface) acts on a layer of BR-containing medium (nontransparent for green light); (B) modulated allocation of molecules in form M412 (transparent for green light) and BR570 (nontransparent for green light), (C) the action of unmodulated blue light pulse, (D) modulated blue pulse as a result of transmission via the modulated medium (BR-containing medium layer recovered the primary state).

38-4


2

4

5

6

7

Plain light front from the source 1, modulated by intensity according to the contour of absorption index of photo-detecting layer 2 by lens 3, allocates on the surface of weighing coefficients layer 4. 38.2.1.5â•‡Realization of Weighing Function The input vector weighing function comes around during the transfer process of input light signal—the components of input vector—over the matrix of weighing coefficients 4. By the component of input vector, we mean the quantity of the light energy (the intensity multiplied by exposure—exposition) affecting the matrix section. Weighing coefficient (by which the input vector component is multiplied) is a transmission coefficient of the corresponding section of BR-containing matrix:

Light

3

Realization of weighting function

Realization of the activation function

Realization of the weighted signals composition function

Figure 38.4â•… Neuronet organization based on the basic process: 1, 6—plane waveguides, including gitters for emission input/output; 2, 4, 7—BR-containing layers (photo-detecting layer, layer of weighting coefficients and neuron layer); 3, 5—cylindrical lenses.

3. Light flux expanding lens. 4. Layer of synaptic bond synthesis (matrix of weighing coefficients) made of BR-containing material. 5. The lens, focusing the light flux on layer 7 and forming the combination of input optical signals based on neuro-like elements in that manner. 6. The source of flat light front (transparent for normal incident light flux) for parallel comparison with thresholds, formation, and transfer to other layers of output neuron signals in layer 7 as light front, modulated by intensity along front surface according to the values of absorption/ transmission area of BR-containing medium (corresponding to neuro-like elements). 7. The layer of neuro-like optical elements (being obtained on the basis of BR-containing material) composes input optical signals, traversing matrix 4 and realizing the activation function during the light front traverse from the source 6, and forms thereby output signals.

The input optical information in the form of light flux (input vector) effects the photo-detecting layer 2 that results in the absorption of light flux energy in BR-containing medium of photo-detecting layer and the distribution of the photo-induced absorption index forms. Distribution of the photo-induced BR-containing medium absorption index along the surface and in depth corresponds to the power distribution along the surface of the effective light front.

I w ij ⋅ t = ω ij ⋅ I in i ⋅ t ,

(38.2)

where Iw ij is the light intensity transmitted over the ij matrix section or weighed component of the input vector Iin i is the light intensity of the light front section or a component of the input vector t is the exposition time of the corresponding component of the input vector ωij is the weighing coefficient or transmission of the corresponding ij-section of the BR-containing matrix Transmission of the ij matrix section is defined according to Bouguer–Beer law

ω ij = e

− εdci / f ij

,

(38.3)

where ε is the BR absorption factor d is the thickness of BR-containing layer ci/f ij is the concentration of the BR molecules in the initial state in the ij matrix section The properties of the input vector weighed components are formed by lens 5 in a light flux that gets to the inputs of the corresponding neuro-like elements of the layer 7. 38.2.1.6â•‡Realization of the Weighed Signals Composition Function Composition function of the input signals is realized in the BR-containing medium in layer 7 by the converging cylindrical lens 5 as a result of the combined effect at the same area of BR-containing medium of the light energy exposition by the corresponding components of the input vector. Every component of the input vector contributes to the formation of the molecules ensemble in photo-induced state in proportion to the intensity and exposure time:

∆c j p /i = k ⋅

∑I

w ij

⋅ t,

(38.4)

38-5


where Δcj p/i is the concentration of BR molecules in photo-induced spectral state k is the coefficient of proportionality depending on the concentration of BR molecules in the initial state, crosssection interaction, photo response of BR-molecules transition from the initial to the photo-induced form, and the effecting light wave length Iw ij is the intensity of the input vector weighed ij-component t is the exposure time of the input vector i-component Thus, Δcjâ•›p/i contains information about the value of the weighed input interactions sum on the j-neuron. 38.2.1.7â•‡Realization of the Activation Function

∑

Magnitude I w ij ⋅ t (the total dose of light energy effecting i on the j-area of BR-containing medium) assigns the point at the graph (dependence of the transmission value on the sum of weighed input effects) and defines the transmission of the light signal over the j-neuro-like element. The changed transmission ωn/e value of the BR-containing medium j-section according to the j-neuro-like element assigns the value of the activation function and the output signal of the neuro-like element in layer 7, according to Figure 38.5. This magnitude depends on the number of molecules possessing the changed spectral properties of the medium section corresponding to the neuro-like element of the layer 7 according to ωn/e j = e

− εd (ci / f − ∆c j p/i )

.

(38.5)

0.9 0.8 0.7

Pout /Pin

0.6 0.5 0.4 0.3 0.2

I out n / e ⋅ t active = Wn / e ⋅ I active ⋅ t active .

(38.6)

The minimum value of the output signal is fixed by the transmission of non-firing neuron (the input signals sum value is close to zero) and corresponds to the initial transmission of photochromic medium, and the maximum value is close to the value of active front energy and corresponds to the saturation area of the curve (Figure 38.5) and to the maximum excited state (transmission) of firing neuron. Output signals of neuro-like elements form the continued light front being modulated according to the activation function at every point of BR-containing layer 7. System learning (formation of matrix weighing coefficients) corresponds to the formation of adequate values of transmission coefficients of matrix sections based on BR-containing medium that can be achieved by the inverse transformation method. The optical version of the inverse transformation method can be simply and effectively realized (failing optoelectronic transformations) by combined presentation of learning pair: the input image in its usual direction and the ordered output as the light front in the counter direction. Due to the reversibility of the light passing, both of light fronts will affect the matrix made of BR-containing material and will change the matrix transmission corresponding to the intensity distribution.

38.2.2â•‡Neuronets Based on Multilayered Optical Structures Including Polymeric Bacteriorhodopsin-Containing Layers

0.1 0

The graphic chart on the dependence of the BR-containing layer relative transmission on the emission energy consists of the area with the initial transmission value (unequal to zero), the area of almost linear transmission change, and the saturation area (Figure 38.5). In general, the curve corresponds to the activation function proposed for the neuronet realization by Grossberg (Wasserman, 1989). The similar compressive function automatically provides the output signal range from 0 to 1 and corresponds to the necessary requirements for realization of reconversion algorithm during the neuronet learning, for example, according to scheme (Wasserman, 1989). The output signal formation of neuro-like element (threshold comparison and realization of the activation function) is carried out by the light front of the specified intensity and duration induced by the source-former 6. The output signal of the neuro-like j-element is formed as an energy portion of the active light signal being transferred over to the corresponding section of BR-containing medium in the layer of neuro-like elements 7 (according to the transmission of the section considered) in conformity with the formula

1

20

40

60

Pin (mW/sm2)

80

100

Figure 38.5â•… Transmission (Pout/Pin) of BR-containing layer depending on the effective emission energy (P in).

The considered approach of neuro-computer element base formation takes into account the cyclicity of processes in living systems appearing, for example, in spontaneous activity of pacemaking (assigning the rhythm of functioning) neurons. As was conclusively shown in Prigogine (1980), the cyclicity proceeds

38-6


as a result of the processes’ self-organization in open nonlinear nonequilibrium systems and the origination of stable dissipative structures due to which the coordination of trophic processes (providing cell nutrition) is possible in living systems. The similarity of cyclic processes in the living cell membranes (including neurons) and the processes under optical emission exposure in isolated purple membranes is obvious, particularly, taking into consideration that in that case one of the halobacteria trophic cycles is reproduced. If the neuron is represented as a structure population (purple membranes) allocated in media and the interactions between the neurons are carried out by light fluxes, the “neurolike medium” in question can be considered as the alternative to the net of neuro-like elements (Grebennikov, 1997). In the interpretation being stated, the problem of neuronet formation is in simulation of nonequilibrium nonlinear systems in BR-containing media with allocated parameters by optical emission of two different wave lengths corresponding to the absorption maxima of two “long-living” intermediates. It can be solved in the following consequence:

1. Using the spectral sensitivity of BR in optical range to initiate the cyclic light-dependent processes by the light flux and to control the processes by the application of emission with various wave lengths. 2. Stationary and dynamic structuring of BR-containing films by modulated light fluxes to create the conditions of formation and development of neuro-like relations in neuro-like medium and to correlate the multitude of cyclic processes.

Data input and processing in that case is “deformation” of the correlated cyclic processes in BR-containing medium, and the commands of system behavior control are the transient processes originating in the organized neuro-like medium. On the other hand, analyzing the development tendencies of neuronet technology, the best perspective can be emphasized, and their combination in the same device can provide essential expansion of neuronet data processing:

1. Optical mechanism of data transmission and processing, which will allow to construct three-dimensional nets that function simultaneously at high speed. The problem of wide application is in large dimensions, peculiar to optical systems, and the inevitable efficiency losses due to multiple intermediate optoelectronic conversions and the finite size of the optoelectronic elements.

The application of continuous photochromic media with high resolution close to molecular level of data representation and processing can become a solution to the problem. BR is the most available and well-investigated (at present) photochromic material with sufficiently high cyclicity (>106) and is suitable for the considered purposes. Using BR enables to carry out data processing in optical mode without intermediate optoelectronic conversions and to increase the areal density of neuro-like elements value comparable to the native neuron systems.

2. Neuro-like structure construction on the principles of selforganization of the processes in the open nonlinear allocated dissipative systems like biological objects. Although in model version, systems require substance transfer proceeding for nonequilibrium conditions maintenance.

BR application can solve the problem of substance transfer that complicates the construction and limits the continuous working life of neuro-like media processing. Light-dependent properties of that material allow the open nonlinear allocated dissipative systems to simulate effectively. Optical exposure in the range of 520–650â•›nm is like the input flux and dissipative properties, and in that case can be provided by the component of trophic cycle that remains invariable in composition of artificial medium (for example, in polymeric matrix) and also by emission exposure in blue light range (λ = 400–420â•›nm).

3. Element base adaptability to losses of some elements during the preparation and exploitation compensated by self-organizing and self-modification of neuro-like structures. Application of continuous media based on BR in conjunction with optical methods permits to form neurolike elements, and the bonds between it suit requirements according to the light energy allocation.

Classic methods of the light fluxes formation by lens systems result in the loss of advantages expected from significant density of neuro-like elements and dimension restrictions of elements and systems, peculiar to optical computers. Moreover, when the volume of BR-containing medium is greater, the probable process integration is higher. However, the emission access to all the molecule groups is more unfavorable as the medium absorption increases. In the field of optoelectronic technology, efforts are made to deflate the construction dimension of optical neuro-computers by the application of multilayered structures. Multilayered structures forming neuronets and containing the layer of spatial light modulators as the liquid crystal matrix, the layer of Â�photoconducting material (Engel, 1990) or material with the photovoltaic effect (Akiyama et al., 1995), and the layer of electric transducer-amplifiers (forming and transducing the commands to liquid crystal panels) are proposed. Essential disadvantage of the proposed constructions is the facility of intermediate conversion of light exposure to electric current or voltage used for the following changes of optical transmission modulator. Furthermore, neuronet function realization requires the application for these purposes of optoelectronic and microelectronic elements. Consequently, the application of metallic conductors in inter-cell links for realization of neuro-systems with a great number of neurons results in delays in communication lines and neuronet processing deceleration due to the capacity influence of inter-cell communication lines. The existing areal density restrictions of optoelectronic elements and electrical bonds inevitably limit the spatial resolution of constructions.

38-7


38.2.2.1â•‡Multilayered Constructions, Including Bacteriorhodopsin-Containing Films We proposed multilayered structures including layers based on the BR for the realization of neuronet medium in nonequilibrium nonlinear dynamically allocated dissipative systems. Supposedly, the multilayered structures would allow the data processing to continue at the level of BR molecules groups by forming neuro-like elements using optical methods in BR-containing media. The basic processes in BR medium are defined by lightdependent changes in absorption index allocation profile along the surface of BR-containing polymeric films. In multilayered structures, many light fluxes circulate without interaction. This property is usually proved as an advantage of optical methods enabling data processing and transmission in the three-dimensional space. At the same time, the information arrays in the form modulate indirectly by intensity light fronts over the reciprocal fluctuation of absorption index local value of BR-containing media sections, and the local intensity value of the light front sections realize concurrent information interactions in the three-dimensional space of the multilayered structure. We would like to consider a possibility of neuro-like element net organization in BR medium by optical method using multilayered structures (Figure 38.6) including BR-based layers, wave guide layers, and reflecting layers. To allocate light fluxes, the system of waveguides, transparent in the optical range, is formed in BR-containing medium. It is possible to input the controlled emission in the form of light front in BR-containing medium, activating at that point the groups of neuro-like elements, by producing the sections with the disturbed conditions of total internal reflection in waveguides. It is expected that the multilayered structure will provide not only functioning and interaction of neurons and their ensembles but also the generation of new neurons and links (emission output from one layer and penetrating to the other layers) between single neurons and neuronets, and will permit, according to the information (image) at system output, to connect and to correlate the cyclical processes originating in BR medium. At that

7 6 5 4 3 2 1

9

point, the process of self-organization of data processing system will continue. The adaptability principles realization of data processing elements and system self-organization will permit to essentially reduce the requirements of the elements and facility as a whole to provide reliable functioning in case of the single-element failure. Reduction of technological requirements is achieved by that neuro-like elements and links that are formed in continuous (uniform, i.e., not divided into constructive matrix elements) transparent (without optical dispersion) layer of photochromic material according to the light energy allocation. 38.2.2.2â•‡Expected Parameters of the Element Base According to the traditional criteria, it is acceptable to evaluate the number of the neuro-like elements in medium containing BR at area 10â•›mm2 in quantity not less than 106. At the area in question, not less than 1011 bonds per second are realized (circuit time 0.1â•›ms, coefficient of bond formation 10). 38.2.2.3â•‡The Basic Elements of the Multilayered Structures The multilayered structures (Figure 38.6) for the realization of the basic neuronet data processing include: the system of flat waveguides, the elements of optical emission input as gitter, and the devices of the surface light front formation (Figure 38.7). BR-containing polymeric films are meant for neuro-like elements formation by the change of the absorption/transmission surface geometry by modulated light flux effect. The elements of optical emission input in flat waveguides and the elements of output are made as diffraction lattices (gitters) (Zlenko et al., 1975; Unger, 1980). The angles of radiation input and output depend on different layer indexes and glitter spacing; therefore, those angles can be different for various multilayered waveguides.

Θ2

n2 n0 n1

8

Θ1 BR

Л1

11 10

BR L

Figure 38.6â•… Fragment of multilayer structure: 1—substrate (glass K-8); 2, 10—layers, containing BR; 3, 5—boundary layers of flat waveguide; 4—guide layer of flat waveguide; 6—emission input area; 7—input emission; 8—output emission; 9—output emission gitter, 11— input emission gitter; L is the length of the output emission gitter.

Θ2

Θ1

Л2

n2 n0 n1

Figure 38.7â•… Device for surface light front formation: Θ1, Θ2—angles of emission input; Λ1, Λ2—diffraction lattice spacing; n 0, n1, n2—refraction indices.

38-8


The device for surface light front formation (Figure 38.7) generates directed light fluxes for effective allocation of the light energy in BR-containing polymeric films for neuro-like elements concurrent formation, their concurrent interaction, and output of the data processing results in neuro-like element medium as optical signals. 38.2.2.4â•‡Requirements of BR-Containing Polymeric Films The following are the requirements of optical and geometrical properties of BR-containing polymeric films. For photochromic effects (induced changes of refraction and absorption indexes) to significantly appear during functioning process, high optic density and consequently substantial BR concentration in polymeric films are needed. Optical density of such films should be 0.8–1.3. In those conditions, basic characteristics of BR-containing media are utilized under optimum light flux density so that the media could be used for technical purposes. Exposure to light of the fluxes with radiation density equal to 10–100â•›mW/cm2 induces the films to experience changes in their absorption and transmission characteristics as much as 10%–50% from the original indexes in as long as 0.1–10â•›s. Functioning of films under induced changes in absorption level requires quite homogenous distribution of BR concentration, as optical heterogeneity infringes the conversion of optical information. Besides, we must ensure repeatability of the main structural parameters: film thickness, surface finish, homogenous BR distribution throughout film surface (1–10â•›cm 2), etc. Requirements to physical and chemical properties of polymeric matrix. Selection of matrix material. To minimize the influence of diffraction divergence on information conversion processes, the thickness of BR-containing films must be 6–14â•›μm. To reach the specified optical density with that thickness, the volume of BR content in polymeric films should be 40%–50%. Far from all, the polymers transparent in optical range can meet the requirements above. Moreover, only water-soluble polymers can be used to form BR-containing polymeric films. The comparative studies held to form polyvinyl alcohol- and gelatine-based BR-containing polymeric films have proved that gelatin-based polymeric matrixes have obvious advantages. Gelatin-based films could apparently have the highest possible BR concentration (up to 50 vol. %) without aggregation of PM fragments due to thermodynamic peculiarities of gelatin polymerization process. Gelatin properties allow us to avoid destruction of BR protein structure while making polymeric mixture and further polymerization. PM fragments embedded into a gelatin matrix are long-lasting and resistant to many technological factors. Polymerized gelatin creates optimum conditions for BR to function while retaining enough water needed for photochromic cycle. For the same reason, gelatin matrices make it possible to place environmentmodifying water-soluble components and to change the photo cycle time frame.

38.2.2.5â•‡Obtaining of Bacteriorhodopsin-Containing Polymeric Films for Multilayered Structures It is important for processing and conversion of optical information to take into consideration the dispersion of optical emission in BR-containing media conditioned by purple membranes size (500–1000â•›nm)—comparable to the wave length of the optical range emission. Therefore, the primary suspensions of PM and BR-containing films based on them are optically nonhomogeneous, which results in the functional properties loss. The dispersion demagnification is reached consequently: at the step of preparation of purple membranes suspension—by PM fragments separation; at the steps of polymeric mixture preparation and polymerization—by elimination of the aggregation process of the purple membranes fragments. 38.2.2.5.1 Preparation of Bacteriorhodopsin Suspension For the preparation of PM suspension, the triple centrifugal purification (3000â•›rpm, 5â•›min) was carried out; pH value and BR concentration in suspension were measured. The pH value is significant for the following polymeric solutions and film obtaining, since the investigations showed that at pH less than 4.1 PM aggregated, the optical transparency of the suspension was not achieved. It was determined that the ultrasound exposure results in the decrease of pH value in suspension at 0.2–0.4. The control of pH value was managed by the addition of 0.01â•›M borax buffer solution Na2B4O7â•›·â•›10H2O, pH = 9.18. During the ultrasound treatment, the suspension temperature must not exceed 36°C. As the result of technological investigations of ultrasound treatment, optically transparent homogeneous BR suspensions were obtained without detergent addition. The side effect can be the partial melting of protein. Optically transparent PM suspensions were obtained with BR concentration up to 15â•›mg/mL. Size evaluation (8.7 ± 0.5â•›μm) of PM fragments in treated suspensions was carried out by the intensity of Rayleigh scattering and showed that the applied technological mode of suspension treatment permits to separate PM into naturally minimum Â�fragments—trimers without BR protein destruction and the principle possibility of BR-containing medium optical resolution at the level of a few thousand lines per millimeter can be considered. 38.2.2.5.2 Preparation of Bacteriorhodopsin-Containing Mixture During the preparation of polymeric mixture based on BR, the last exhibited the property of aggregation on the polymer molecules that lead to optical heterogeneity of films. As the result of technological experiments combining thermal parameters and operating pH of components, the conditions selected under that aggregation were not observed and transparent optically homogeneous BR-containing polymeric mixtures were obtained. At the step of polymeric mixture preparation, pH control of gelatine solution was carried out because pH value in gelatine solution depends either on the obtaining method or on gelatine concentration. The component stirring in the mixture was also carried out by the ultrasound exposure. All the modifying components were put into the polymeric mixture at the last step

38-9


under condition: the final pH value of the polymeric mixture had to be >4.1. 38.2.2.6â•‡The Properties of BR-Containing Polymeric Films for Multilayered Structures According to the elaborated technology, there are Â�transparent and optically homogeneous BR-containing polymeric films (thickness 6–14â•›μm with optical density 0.8–1.3â•›D at λ = 570â•›nm) on substrates of glass K-8 and fused quartz (area up to 60 × 48â•›mm2) and also on Si plates (76â•›mm in diameter). 38.2.2.6.1 Evaluation of PM Size Embedded to Film Since during the film polymerization from the polymeric mixture fragment aggregation is possible, the evaluation of PM fragments size is being embedded into film. The placement of films with the embedded PM fragments in detection system between crossed polarizers resulted in no changes in the initial zero signal of photodetector. It means either no rotation of polarization in the light-pass direction, or no significant dispersion that confirms the PM fragments size to be much less than 0.63â•›μm. Optical heterogeneity specified by surface geometry and allocation of BR concentration to the film surface is given below. The distribution of optical absorption heterogeneity coefficient of BR-containing films is specified by the product of two values: the allocation heterogeneity of BR bulk concentration to the film surface and quality of the film surface as the local dilatation from the average thickness value. In the aggregate it results in the local dilatation of so-called surface concentration and, correspondingly, the optical density. The optical homogeneity of the film being specified only by the surface quality is provided comparatively easily both for the films being realized by glazing method and for the films being obtained by centrifugation method. It was determined that the typical thickness deviation of BR-containing films being obtained equals less than 50â•›nm at length 10â•›mm, that, for example, at film thickness >5â•›μm equals 0 the phase difference of EM wave at those points remains ϕ0, we say the EM wave has perfect coherence between the two points. If this is true for any two points of the wave front, we say the wave has perfect spatial coherence. Now, consider a fixed point on the EM wave front. If at any time the phase difference between time t and time t + Δt remains the same, where Δt is some time delay, we say that the EM wave has temporal coherence over a time Δt. If Δt can be any value, we say the EM wave has a perfect temporal coherence. If this happens only in a range 0 < Δt < t0, we say it has partial temporal coherence, with a coherence time equal to t0. Laser light is highly coherent, and this property has been widely used in measurement, holography, etc. Laser beam is also highly directional, which implies that laser light has very small divergence. This is a direct consequence of the fact that laser beam comes from the resonant cavity, and only waves propagating along the optical axis can be sustained in the cavity. The directionality is described by the light beam divergence angle. 39-1

39-2


The brightness of a light source is defined as the power emitted per unit surface area per unit solid angle. Brightness is inversely proportional to the square of laser divergence; therefore laser light is much brighter than normal light source.

E2

hν E1

(a)

39.2â•‡ Overview of Lasers In gas lasers, the active medium can be regarded effectively as an ensemble of absorption or amplification centers (like, e.g., atoms or molecules) with only electronic energy levels, which couple to the resonant optical field. Other electronic states are used to excite or pump the system. The generic laser structure is shown in Figure 39.1. It consists of a resonator (cavity) formed by two mirrors and gain medium where the amplification of electromagnetic radiation (light) takes place. A laser is an oscillator analogously like an oscillator in electronics and requires a resonator, which provides feedback. Feedback is provided by two mirrors. Mirrors confine light and provide optical feedback. One of the mirrors is partially transmitting, which allows the light to escape from the device. There must be an external energy provided into the gain medium (process known as pumping). The most popular (practical) pumping mechanisms are by optical or electrical means. The gain medium can be created in several ways. Conceptually, the simplest one is the collection of gas molecules. The pumping process excites these molecules into a higher energy level. The popular visualization of such collection of molecules is as twolevel systems (TLS) (see Figure 39.2). Only two energies (out of many in the case of a molecule) are selected and the transitions are considered within these two energies. As illustrated, three basic processes are possible: absorption, spontaneous emission, and stimulated emission. Such TLS are found very often in nature. Generally, for an atomic system, for the case under consideration, we can always separate just two energy levels: upper level and ground state, thereby forming TLS. Pump

Gain medium

Resonator (two mirrors)

Figure 39.1â•… Generic laser structure: two mirrors with a gain medium in between. The two mirrors form a cavity, which confines the light and provides the optical feedback. One of the mirrors is partially transmitted and thus allows light to escape. The resulting laser light is directional, with a small spectral bandwidth.

E2 hν > Eg

hν

E2

hν

E1

(b)

E1

hν ≈ Eg

(c)

Figure 39.2â•… Possible electronic transitions in two-level system. (a) Absorption, (b) Stimulated emission, and (c) Spontaneous emission.

Electron can be excited into upper level due to external interactions (for lasers through a process known as pumping). Electrons can lose their energies radiatively (emitting photons) or nonradiatively, say by collisions with phonons. For laser action to occur, the pumping process must produce population inversion meaning that there are more molecules in the excited state (here upper level with energy E2) than in the ground state. If population inversion is present in the cavity, the incoming light can be amplified by the system (see Figure 39.2b) where one incoming photon generates two photons as the output. The way how TLS is practically utilized results in various types of lasers, like gaseous, solid state, or semiconductor. Also, different types of resonators are possible as will be discussed in subsequent sections.

39.3â•‡ Semiconductor Lasers A significant percentage of today’s lasers are fabricated using the semiconductor technology. Those devices are known as semiconductor lasers. Over the last 15 years or so, several excellent books describing different aspects and different types of semiconductor lasers have been published [2,3]. The operation of semiconductor lasers as sources of electromagnetic radiation is based on the interaction between EM radiation and the semiconductor. Typical semiconductor laser structures are shown in Figure 39.3. These lasers can be classified as in-plane laser where light propagates in a parallel direction [4] and vertical cavity surface emitting laser (VCSEL) [5]. The largest dimension of in-plane structures is typically in the range of 250â•›μm (longitudinal direction) and as such cannot be considered as a nanolaser. The structure of interest to us is the one where light propagates perpendicularly to wafer’s surface and it is known as VCSEL (see Figure 39.3b). The typical diameter of VCSEL cylinder is about 10â•›μm. The basic semiconductor laser is just a p–n junction (see Figure 39.4) in which the cross-section along the lateral–transversal directions is shown. Current flows (holes on p-side and electrons on n-side) along the vertical direction, whereas the light travels horizontally and leaves the device on both sides. Light propagation with amplification is illustrated in Figure 39.5. Mathematically it is described by assuming that there is no phase change on reflection at either end (left and right). The left end is defined as z = 0 and right end as z = L. At the right facet, the forward optical wave has a fraction rR reflected (amplitude

39-3

Nanolasers Output light Active regions

Transversal direction Output light

Bragg mirrors

Longitudinal direction Lateral direction

(a)

(b)

Figure 39.3â•… In-plane laser (a) and VCSEL (b).

The wave traveling one full round will be

Holes in

p-type Light out

Active region

Feed back

n-type

Electrons in

Figure 39.4â•… The basic p–n junction laser. rL

β

(39.1)

The above terms are interpreted as follows. In the first bracket, there is an original forward propagating wave at z, in the second bracket, there is wave traveling from z to L, third bracket describes wave propagating from z = L to z = 0, and the last one contains wave traveling from z = 0 to the starting point z. At that point, the wave must match original wave and thus one obtains condition for stable oscillations:

rR

β

{E0 e gz e − j z } {e g ( L − z )e − jβ( L − z ) }{rR e gL e − jβL } × {rL e gz e − j z }

rR rL e2 gL e −2 jβL = 1

(39.2)

That condition can be split into amplitude condition

rR rL e2( g m −αm )L = 1

(39.3)

e −2 jβL = 1.

(39.4)

and phase condition

From the amplitude condition, the following relation is obtained: z=0

z

1 1 ln 2L rR rL

(39.5)

z=L

Figure 39.5â•… Schematic illustration of the amplification in a Fabry– Perot (FP) semiconductor laser with homogeneously distributed gain.

reflection) and after reflection the fraction travels back (from right to left). In order to form a stable resonance, the amplitude and phase of the single round trip must match the amplitude and phase of the starting wave. At arbitrary point z inside the cavity (see Figure 39.5) the forward wave is

g m = αm +

E0 e gz e − j βz

where we have dropped ejwt common term and g = gm − αm. Here rR and r L are, respectively, right and left reflectivities, g is gain (and loss), L length of the cavity, and β the propagation constant.

From the phase condition, it follows that

2β L = 2πn

(39.6)

where n is an integer. The last equation determines wavelengths of oscillations since

β=

2π λm

(39.7)

with λm being the wavelength. In VCSEL, the cavity is formed by the so-called Bragg mirrors and an active region typically consists of several quantum well layers separated by barrier layers (see Figure 39.3). Bragg mirrors consist of several layers of different semiconductors, which have different values of refractive index. Due to the Bragg reflection,

39-4


(a)

Gain spectra (also other properties) strongly depend on size of the active region. Different gain media are possible, consisting of 0D, 1D, 2D, 3D (bulk) structures (see Figure 39.6). The typical thickness of a quantum well is about 10â•›nm. The dimensionality of these structures have profound consequences on laser properties. An illustration of gain spectra obtained with gain media for structures of different dimensionalities is shown in Figure 39.7. As can be seen, the shape of spectrum becomes sharper with increasing quantization dimension. This is due to the variation of the density of states.

(c)

(b)

(d)

Figure 39.6â•… Illustration of various active regions with different dimensionality. (a) three-dimensional (3D) or bulk structure, (b) twodimensional (2D) structure known as quantum well, (c) one-dimensional (1D) structure known as quantum wire, and (d) zero-dimensional (0D) structure known as quantum dot. (Adapted from Arakawa, Y. and Sakaki, H., Appl. Phys. Lett., 40, 939, 1982.) 104

39.4â•‡Rate Equation Approach In a typical laser, there exists two types of subsystems: photons and carriers. A quantitative description of a laser system is given in terms of rate equations. One introduces the number of photons S inside the cavity and number of carriers N (can be also a number of excited molecules for some systems). The rate equations describe the time evolution of S and N, as follows:

Dot

Gain (1/cm)

Wire Well

103 Bulk

2 × 102 1.2

1.3

1.4 1.5 Wavelength (μm)

1.6

1.7

Figure 39.7â•… Calculated gain spectra for quantum dot, quantum wire and quantum well. For comparison, results for bulk crystal are also shown as dotted line. (Adapted from Asada, M. et al., IEEE J. Quantum Electron., 22, 1915, 1986.)

such a structure shows a very large reflectivity (around 99.9%). Such large values are needed because a very short distance of propagation of light does not allow to build enough amplification when propagating between mirrors. From the electromagnetic analysis of the optical cavity, the quality factor Q of the cavity [6] can be determined as

Q=

∆ω ω0

where Δω is the cavity linewidth ω0 the resonant frequency of the cavity

(39.8)

dN I N = ηi − − v g g (N )S dt qV τ

(39.9)

dS S = Γv g g (N )S − + ΓβRsp τp dt

(39.10)

We have explicitly indicated that gain g depends on the carrier’s concentration. In Equation 39.9 the first term is responsible for pumping (in this case electrical), with I being the current, the second term accounts phenomenologically for losses, and the last one describes coupling to the photon system. The last term in Equation 39.10 accounts for spontaneous emission (with the coefficient β). The β coefficient describes the amount of spontaneous emission that contributes to the lasing mode. The β factor is inversely proportional to the number of available modes into which the gain medium can spontaneously emit photons. In typical in-plane lasers, it is a small number (like 10 −4) [3]. The value of β is between 0 and 1. The typical power–current characteristic of semiconductor laser is shown in Figure 39.8 for two extreme cases of β. When β = 1, all the spontaneously emitted photons end up in the lasing mode. For small values of β, the laser has a well-defined threshold. At threshold, the optical gain compensates for the losses. Above the threshold, the laser operates in a stimulated emission mode and below the threshold, spontaneous emission dominates. A large β is the key factor in single-photon laser sources [7,8]. The (still hypothetical) case of β = 1 is often referred to as a thresholdless laser [9]. In the thresholdless laser, all photons participate in the stimulated emission. Such a device would require a small amount of energy to operate. Its dimensions should be very small, say ∼λ.

39-5

Nanolasers

Output power β=1

β=0

Current

Figure 39.8â•… Current–power characteristics for two values of spontaneous emission factor.

Several issues must be addressed before one could fabricate thresholdless laser [10,11]. Those include: (a) optical modes that induce undesired spontaneous emission should be suppressed where possible, (b) creation of a single-cavity mode with a sufficiently high Q factor and a small modal volume is essential, and (c) excited carriers should be concentrated to emit light coupled to the single-mode cavity.

39.5â•‡ Definition of Nanolaser The need to reduce the size of the semiconductor laser is one of the most active and challenging areas of modern optoelectronics. On the theoretical side, size reduction is important for understanding the basic laser concepts and fundamental light–matter interactions. On the practical side, smaller lasers will find various applications as light sources in integrated optical systems. If photonics should be compatible with VLSI as for as lasers are concerned, photonic devices must shrink in size to 100â•›nm (or less) length scales. As indicated in the previous section, the smallest lasers available commercially today are VCSELs. However, in the last few years, the new type of even smaller devices is emerging, namely, nanowire lasers. Their cylindrical dimensions range from few tens up to hundreds of nanometers, whereas their lengths are typically within a few hundreds of microns. The typical volume of conventional miniature lasers such as VCSELs is determined by the volume of the cavity mode. The effective wavelength in the dielectric should be of the order of the characteristic length of the device. This leads to the existence of the effective modal volume Vâ•›>â•›(λ/2n)3, a condition known as the diffraction limit. Reduction of the volume of the active region is the potential factor for lowering the threshold current. Recent advances in technology allow the fabrication of optical nanoscale devices where the wave nature of photons becomes one of the most critical variables. It provides the challenges in the realization of a tiny coherent photon source. The localization of the wave is difficult when wavelengths of photons become much larger than the spatial variation of the confinement structure.

As the dimensions of optical nanodevices scale down, devices can be fabricated with effectively only one optical emission mode. These structures could be termed nanolasers (see [8]). Alternatively, nanolasers can be defined as structures having dimensions smaller than the wavelength of light in all three dimensions [12]. Recent summaries of research on nanolasers can be found in Refs. [12,13]. In the following sections, we will discuss examples of recently the developed nanolasers (and related) structures: • • • • •

Nanowire lasers Plasmonic lasers Photonic crystal lasers Scattering lasers Organic lasers

We conclude this chapter with a brief discussion of applications of nanolasers in sensing and medicine.

39.6â•‡ Nanowire Lasers Various types of nanocavities have been fabricated and a coherent laser emission from such structures has been observed. Among others, lasing has been demonstrated in droplets [14,15], silica [16], and polystyrene spheres [17], semiconductor microdisks [18–20], micropillars [21], photonic crystal cavities [22], nanoribbons [23], ZnO arrays [24], GaN nanowires [25], and single-crystal ZnO nanowires [26]. Several methods of the synthesis of semiconductor nanowire heterostructures have been developed, including chemical vapor deposition and the vapor–liquid–solid growth of crystalline semiconductor nanowires. Recent progress in the development of semiconductor nanowires was reviewed by Lauhon et al. [27]. The schematics of a typical cylindrical nanowire laser is shown in Figure 39.9 and the possible schemes of current injection are schematically shown in Figure 39.10. These structures provide photon confinement in volumes of a few cubic wavelengths. A typical lateral dimension of a nanowire is between 20 and 400â•›nm, with a length in the range of 2–40â•›μm. As was shown in the journal articles cited earlier, it is possible to grow nanowire arrays with tight control over size (diameter R

(39.12)

where Jn and Yn are the Bessel functions of the first and second kind, respectively. H n = H n(1) = J n + iYn is the Bessel function of the third kind (Hankel’s function). Also, the following definitions of transverse wave numbers were introduced: κ12,2 = ε1,2 ω 2 / c 2 − h2 . The eight unknown coefficients A, B, C, D, F, G, N, and M should be determined from the boundary conditions at the interfaces r = r′ and r = R. The numerical approach is based on solving Maxwell’s equations using finite difference time-domain (FDTD) method in cylindrical coordinates [35]. Typical computational window is shown in Figure 39.13. The modes supported in such nanowire structures are similar to those of optical fibers (for a discussion of modes in fiber, see [39]) but are more localized due to high refractive index contrast between the nanowires and the surrounding air. The Maslov–Ning analysis suggests that the natural facets of the nanowires provide very low quality factors (of the order of hundreds) for nanowires of about 10â•›μm in length. These factors are very sensitive to the mode type and nanowire radius. Far-field patterns of the emitted radiation were also discussed [34,35]. It was determined how the radiation pattern depends on the lasing mode. The radiation is emitted in a very broad range of angles with respect to the nanowire axis. Also, the directionality weakens with an increase of the nanowire radius.

PML Nanowire Incident wave packet

r

Figure 39.13â•… Schematic of the FDTD computational domain. (Adapted from Maslov, A.V. and Ning C.Z., Opt. Lett., 29, 572, 2004.)

Using their methods, Maslov and Ning recently reported on the numerical analysis of semiconductor nanowire covered with a metal as a possible laser waveguide [40]. Their analysis opens the possibilities of fabricating even smaller nanowire-based lasers. Maslov and Ning analyzed the possible advantages of using a semiconductor nanowire encased in a metal as a laser waveguide. They showed that despite large Joule loss, such structure can be a good candidate for subwavelength laser operating in TM01 mode. Coupled drift–diffusion simulations of nanowire lasers have been recently reported by Chen and Towe [38]. They extended the FDTD approach by including carrier effects. Their method is based on numerical solution of the steady-state 2D drift–diffusion carrier transport equations, which are coupled with the photon generation rate equations. The basic system of equations is as follows:

∇⋅  − ε ∇ψ(x , y ) = q  p(x , y ) − n(x , y ) + N D+ − N A− 

(39.13)

∇ ⋅ Jn = −q G(n, p) − Rsp (n, p) − Rst (n, p) − RAuger (n, p) − RSRH (n, p)

∇⋅ J p = q G(n, p) − Rsp (n, p) − Rst (n, p) − RAuger (n, p) − RSRH (n, p)

Jn = q Dn ∇n (x , y ) − q µnn (x , y ) ∇ ψ (x , y )

(39.16)

J p = − q D p ∇p ( x , y ) − qµ p p ( x , y ) ∇ ψ ( x , y )

(39.17)

(39.14)

(39.15)

Gm Sm −

Sm + Rsp,m = 0 τ p ,m

(39.18)

where G(n, p) is the carrier generation rate, R sp(n, p) is the local spontaneous recombination rate, R st(n, p) is the

39-8


local stimulated recombination rate, R SRH(n,â•›p) represents Shockley–Read–Hall (SRH) dark recombination rate, Gm, Sm, τp,m, R sp,m are, respectively, the modal gain, the photon number in the cavity, photon lifetime and for the mth lasing mode, and the spontaneous emission rate that couples to the mth lasing mode.

39.6.3â•‡Recent Work on Nanowire Lasers We conclude the section on nanowires with a description of recent works that go beyond single cylindrical nanowire. First, we describe nanowires with metal coating and then the combination of nanowires and quantum wells. The dimensions of semiconductor nanolasers can be further shrunk by using metal-coated nanowires. Such a structure was first fabricated in 2007 by Hill et al. [41] and this structure was shown to be the smallest electrically pumped nanolaser. They coated a thin semiconductor heterostructure post made of In/InGaAs/InP with a layer of gold as shown in Figure 39.14. The diameter was 210â•›nm. Electrons are injected through the top of the pillar and holes are injected through a large-area lateral contact. The device lased in the near-infrared at a wavelength of 1408â•›nm. The mode was an HE11-like mode oscillating near the InGaAs region cutoff frequency. All previously described studies have concentrated on fabrication of nanolasers from homogeneous semiconductors such as gallium nitride (GaN). This means that the laser wavelength is determined by the band gap of the used material. In such a design, there is no possibility to tune the properties of the laser. However, with proper technological tools, structures in which gain and cavity functionalities are decoupled can be designed and fabricated. The purpose of decoupling the gain medium and the cavity is not to separate but to combine the best properties of both subsystems. Thus quantum wells provide the optical gain medium whereas the nanowire acts as the optical cavity.

n-InGaAs n-InP

Au

500 nm

Au

InGaAs Ti p-InP p-InGaAsP

Pt Ti

InP (a)

(b)

Figure 39.14â•… (a) Structure of the cavity formed by a semiconductor pillar encapsulated in gold. (b) Cross-section of the pillar. (Adapted from Hill, M.T. et al., Nat. Photonics, 1, 589, 2007.)

InGaN i-GaN p-GaN p-GaN (a)

n-AlGaN

n-GaN

InGaN

i-GaN p-AlGaN p-GaN (b)

(c)

Figure 39.15â•… Cross-sectional view of a (a) GaN-based core/multishell nanowire structure and (b) the corresponding energy band diagram. (c) InGaN multiple quantum well (MQW) nanowire structure. (Adapted from Qian, F. et al., Nat. Mater., 7, 701, 2008; Qian, F. et al., Nano Lett., 5, 2287, 2005.)

This new type of nanolasers was described by Qian et al. [42]. The structure is shown in Figure 39.15. It consists of a GaN nanowire core that acts as the optical cavity surrounded by InGaN/GaN multiple quantum well (MQW) shells that serve as a composition-tunable gain medium. By varying the indium content, the emission wavelength can be tuned between 365 and 494â•›n m, with all devices operating at room temperature. Their nanowire heterostructure contains 3 to 26 quantum wells. Typical nanowires were 200–400â•›n m in “diameter” and 20–60â•›μ m in length. Although the nanolasers were optically pumped, the authors believe that electrical injection is possible.

39.7â•‡ Plasmonic Lasers Surface plasmon mode confinement has been used to achieve subwavelength modal dimensions at the expense of optical loss [43–45]. Several structures that involve plasmons were proposed. A comprehensive introduction to the main physical aspects involved in plasmonic devices were recently summarized by Dragoman and Dragoman [46]. A summary of the analysis of the plasmonic nanoresonators has recently been provided by Maier [47]. Plasmon cavities can confine electromagnetic energy into both physical and effective mode volumes far below the diffraction limit. Manolatou et al. [48,49] proposed and analyzed a family of nanoscale cavities for electrically pumped surface-emitting semiconductor lasers that use surface plasmons to provide optical confinement. The analyzed cavities have radii between 100 and 300â•›nm and the heights of the dielectric part of the cavity between 100 and 250â•›nm. The metal layers were assumed to be thick enough to prohibit light transmission. The typical circular nanopatch laser analyzed by Manolatou and Rana [49] is shown in Figure 39.16.

39-9

Nanolasers

Metal

p Gain n

Figure 39.18â•… The geometry of a bowtie plasmonic nanolaser. Typical separation between metallic nanotriangles is 20â•›nm. (Adapted from Chang, S.-W. et al., Opt. Express, 16, 10580, 2008.)

Substrate

Figure 39.16â•… Circular nanopatch laser.

39.7.2â•‡ Dipole Nanolaser

y x

z Quantum dot

Figure 39.17â•… Horseshoe optical nanoantenna. (Adapted from Sarychev, A.K. and Tartakovsky, G., Phys. Rev. B, 75, 085436, 2007.)

Sarychev and Tartakovsky [50,51] proposed plasmonic nanolaser where the metal nanoantenna operates similarly to a resonator. The structure is shown in Figure 39.17. In this type of laser, metallic horseshoe-shaped nanoantenna interacts with a two-level amplifying system (TLS). TLS can represent quantum dot and can be pumped optically or electrically. The size of the proposed plasmonic nanolaser is much smaller than the wavelength.

39.7.1â•‡ Bowtie Structures Bowtie structure consists of two opposing tip-to-tip metallic nanotriangles separated by a gap with an active element in the form of quantum dot in-between or quantum wells below it. The interaction of a single quantum dot with a bowtie antenna was demonstrated by Farahani et al. [52] for visible light. The enhancement of the electromagnetic field in such a structure was described by Sundaramurthy et al. [53]. They also conducted FDTD analysis. The theory of the electrically pumped plasmonic nanolaser based on bowtie structure was recently reported by Chang et al. [54]. They considered both quantum dot and quantum well. The geometry of a bowtie nanolaser is shown in Figure 39.18. It consists of a metallic bowtie separated (typically) by about 20â•›n m. Multiple quantum wells are located below the metallic bowtie, which at optical frequencies have negative dielectric constant. The bowtie tips reduce the effective volume of the cavity mode and lead to the field enhancement around the bowtie tips. This results in an increase of the stimulated and spontaneous emission rates and significant decrease of the threshold current.

Diple nanolaser (DNL) has been proposed by Protsenko et al. [55,56]. It consists of a metallic nanoparticle of size r0 and a twolevel system (TLS) of size r 2 formed by a quantum dot, separated by a small distance r (see Figure 39.19). The device does not need an optical cavity and may have a volume much smaller than λ, the lasing wavelength. An incoherent pump provides population inversion in the TLS. The system can be pumped optically or electrically, in which case pumping is provided by injection of carriers from the bands of a semiconductor material surrounding an embedded quantum dot. The transition frequency of TLS ω2 is close to the plasmon resonance frequency ωp of the metallic nanoparticle. There is a strong interaction (a dipole interaction) between those two systems through a near field. It is known that such coupled interaction significantly modifies optical emission. To understand the operation of DNL, let us remind ourselves that the usual lasing mechanism involves stimulated emission into the mode of a cavity (for the single mode operation) from a medium in which there is a population inversion. Above the threshold condition, when stimulated emission (gain) exceeds absorption plus internal losses, the energy from incoherent pump is transferred into coherent laser radiation with a narrow lasing spectrum. In DNL, instead of electromagnetic field, one deals with linear oscillations of polarization of a medium. Such a medium can be excited in such a way that the total energy flux into polarization exceeds losses. This leads to polarization oscillations with a narrow spectrum. The analysis of dipole nanolaser conducted by Protensko et al. [55] is based on equations of motion identical to Maxwell–Bloch equations for a TLS in the electromagnetic field of a cavity. Lasing conditions are created when the population r2

r0

x 2

r

0

Figure 39.19â•… Dipole nanolaser. Nanoparticle (here on the right) is labeled with “0,” TLS (here on the left) has label “2.” (Adapted from Protsenko, I.E. et al., Phys. Rev. A, 71, 063812, 2005.)

39-10


inversion D of energy states of TLS is maintained by optical pumping or injection current to be [56]

D > Dth

(39.19)

where Dth = (r/rcr)6, rcr = (4|α0||α2|)1/6 and α0 and α2 are polarizabilities of the nanoparticle and the TLS. When D exceeds the threshold value D th, the macroscopic dipole moment of two particles appears spontaneously and they emit coherent dipole radiation at the frequency ω ≈ ωplr, the nanoparticle plasmon resonance frequency. Condition (39.19) is fulfilled if the distance between particles is small, which means a strong interaction between them through the near field.

Recently, scientists from the Yokohama National University in Japan [61] have reported interesting results concerning PC nanolasers. They demonstrated high-performing room-temperature nanolaser in the form of PC slab. The laser is made of a GaInAsP. This ultrasmall laser has a modal volume close to the diffraction limit. When operating in a high-Q mode (about 20,000), it will be useful for optical devices in optical integrated circuits. In a moderate-Q (1500) configuration, the nanolaser needs only an extremely small amount of external power to bring the device to the threshold of producing laser light. In this near-thresholdless operation, it might permit the emission of very low light levels, even single photons.

39.8.1â•‡Edge-Emitting Photonic Crystal Nanolaser

39.8â•‡ Photonic Crystal Lasers Photonics crystal (PC) structure consists of a drilled repeating pattern of holes through the laser material. This pattern is called a photonic crystal. One can deliberately introduce an irregularity, or defect, into the crystal pattern, for example, by slightly shifting the positions of two holes. The photonic crystal structure and the defect prevent light of most frequencies from existing in the structure, with the exception of a small band of frequencies that can exist in the region near the defect. Photonic crystal defect microcavities can provide extremely small mode volume [22,57]. The schematic of the laser proposed by Painter et al. [22] is shown in Figure 39.20. The interhole separation is 515â•›nm. It was fabricated from InGaAsP grown by MOCVD on an InP substrate. The active region consists of four (here only two are shown) 9â•›nm 0.85% compressively strained InGaAsP quantum wells. Two-dimensional photonic crystal hexagonal lattice was formed by etching, resulting in air holes that penetrate through the active region and into an underlying sacrificial InP layer. Over the last few years, various devices have been fabricated and characterized [58–61]. Such structures are associated with singlephoton sources, which are required for quantum computing and quantum communications. Single-photon sources are recent applications of the Purcell effect in quantum-dot microcavities. Etched air holes

Active region

Tiny resonators and waveguides based on photonic crystals provide a promising approach to fabricating high-density photonic integrated circuits. Recently, Yang et al. [62] reported the first demonstration of an edge-emitting photonic crystal nanocavity laser integrated with a photonic crystal waveguide. The structure is based on a double-heterostructure photonic crystal nanocavity with a InAs quantum dot active region. The device consists of four photonic crystal sections, each with a slightly different lattice constant. Five layers of InAs quantum dots, each with a quantum dot density of about 2 × 1010/cm2, were embedded in the 220â•›nm-thick GaAs membrane. An output waveguide is butt coupled to the mirror with the fewer number of periods. The device was optically pumped with a semiconductor laser diode at 850â•›nm wavelength. The threshold peak pump power absorbed by the cavity was estimated to be 12â•›μW.

39.8.2â•‡ Silicon Nanocrystals Compatibility with CMOS materials have stimulated research on nanoscale silicon laser by Jaiswal and Norris [63]. They have theoretically analyzed and numerically simulated various designs based on photonic crystal concept and utilizing Si nanocrystals embedded in SiO2. Their studies were motivated by observations of optical gain in silicon nanocrystals [64]. Pavesi et al. [64] observed light amplification in silicon itself. Silicon nanocrystals in the form of quantum dots were dispersed in a silicon dioxide matrix. Net optical gain was seen in both waveguide and transmission configurations, with the material gain being of the same order as that of direct band gap quantum dots. Their findings open the possibility for the fabrication of a silicon laser.

39.9â•‡ Scattering Lasers Defect region

Figure 39.20â•… Cross-section through the middle of the photonic crystal microcavity. A defect is formed (shown in the middle) by removing a single hole. (Adapted from Painter, O. et al., Science, 284, 1819, 1999.)

In scattering lasers, the feedback is provided through scattering of light instead of a cavity [65]. These lasers are of two basic types: Mie lasers consisting of only one sphere and random lasers that are formed by many scattering particles. In Mie lasers, the surface of the sphere serves as multiple scatterer, while in random lasers, scattering is provided by randomly distributed particles.

39-11

Nanolasers

In both types of scattering lasers, the average particle size used in typical experiments was about 5–10â•›μm [66,67]. They are, therefore, not of the size of true nanolasers.

Gain and scattering medium Gain medium

39.9.1â•‡ Mie Nanolasers In a Mie nanolaser, the mirrors of a conventional laser are replaced by the boundary of a microsphere. Light is multiply scattered at the boundary, and along the boundary, whispering gallery modes (WGMs) at a certain wavelength exist for specific sizes of the sphere. WGMs occur at particular resonant wavelengths of light for a given droplet size (see Figure 39.21). At these wavelengths, the light undergoes total internal reflection at the particle surface and, after one roundtrip, interferes constructively. It becomes trapped within the particle for timescales of the order of nanoseconds. The nomenclature of these modes derive from the observation of Lord Rayleigh in the dome in St. Paul’s Cathedral in London. He observed sound (“whispers”) propagating along the walls and circling around the dome several times. Theoretical and experimental study of spherical resonators, which form the basis of Mie lasers, is still an open field of research. This interest stems from the analysis of fundamental processes such as scattering, energy propagation through disordered media, and cavity quantum electrodynamics, and from the large number of applications in photonics, chemistry, meteorology, astronomy, and sensing.

39.9.2â•‡Random Lasers In random lasers, the conventional optical cavity is replaced by light scattering from many particles. They do not have mirrors or optical elements. In random lasers, the feedback is provided by multiple scattering of light at many scattering points. Random laser does not have a clear feedback mechanism like a conventional laser. It is nonresonant, even disordered and works in qualitatively different way than feedback by a resonant cavity (see [68] for a recent review). A comparison of conventional and random laser is shown in Figure 39.22. The wavelengths of random lasers span from the ultraviolet to the mid-infrared region. The materials used in random lasers include inorganic dielectrics, semiconductors, polymers, and liquids. The size of random lasers can vary from a cubic micrometer to hundreds of cubic millimeters.

Figure 39.21â•… Whispering gallery mode inside a microsphere.

(a) Conventional laser

(b)

Random laser

Figure 39.22â•… Comparison of conventional and random laser. (Adapted from Wiersma, D., Nature, 406, 132, 2000.)

A short history of the development of random lasers is as follows: • In 1968, Letokhov [69] described an idea of scattering with “negative absorption.” • In 1994, Lawandy et al. [70] conducted experiments on scatterers in laser dye and observed threshold and line narrowing. • In 1995, Wiersma et al. [67] conducted multiple scattering experiments in Ti:Al2O3 random lasers. Recently a definite reference on solid state random lasers was published by Noginov [71] who tried to answer the question of what is a random laser by reviewing many types of random lasers. On the theoretical side, we mention two approaches to describe scattering lasers [72,73]. Fratalocchi et al. [72] derived a 3D vector set of Maxwell–Bloch equations, which were solved numerically by employing FDTD method. They performed a series of numerical experiments by investigating the process of laser emission from a single nanosphere covered by a layer of active material for different pumping rates. Novel theoretical approach to diffusive random lasers has recently been developed by Tuereci et al. [73]. They developed a model by considering all the possible ways in which light can reflect back and forth in the medium. In their approach, they considered coexistence of gain, nonlinear interactions, and overlapping resonances.

39.10â•‡ Organic Nanolasers Ease of fabrication make organic materials attractive for various optoelectronic devices, including lasers. O’Carroll et al. [74] used the flexibility of structuring organic materials to fabricate subwavelength optical nanowire lasers. Using a melt-assisted template wetting method, they synthesized arrays of semicrystalline nanowires with diameters in a range of 150–400â•›nm, and with typical length of ∼6â•›μm. Their structures are cylindrical wires with optically flat end facets, which form optical cavities. Lasers were optically pumped by an external pump laser. The observed that lasing wavelength (about λ = 460â•›nm) was determined by a standard standing wave condition. Organic lasers can potentially have various applications [75] as inexpensive and lightweight sources of coherent radiation.

39-12


39.11â•‡Applications To conclude this chapter, we briefly discuss a few recent applications of nanolasers focused on nanowire and photonic crystal-based devices.

39.11.1â•‡Nanolaser Device Detects Cancer in Single Cells Investigators at Sandia National Laboratories in New Mexico reported a method to rapidly detect cancer in a single cell using nanolaser techniques [76,77]. Their technique, which allows the investigator to distinguish between malignant and normal cells, has the potential of detecting cancer at a very early stage, a development that could change profoundly the way cancer is diagnosed and treated. The method of determination of malignant cells is based on a technique that rapidly assesses the properties of cells flown through a nanolaser. They observed biophotonic differences in normal and cancer mouse liver cells by using intracellular mitochondria as biomarkers for disease. This difference arises from the fact that mitochondria, the internal organelles that produce energy in a cell, are scattered in a chaotic, unorganized manner in malignant cells, while they form organized networks in healthy cells. This difference produces a marked change in the way that malignant cells scatter laser light.

39.11.2â•‡ Biological and Chemical Detection Optofluidic integration of a new type of photonic crystal nanolaser incorporated into a microfluidic chip was described by Kim et al. [78]. The proposed nanolasers are an ideal platform for highfidelity biological and chemical detection tools in micro-totalanalytical or lab-on-a-chip systems. Its operation is based on the wavelength tunability induced by the refractive index variation of the fluid. A record high sensitivity utilizing photonic crystal nanolaser was recently reported by Kita et al. [79]. The index resolution limit of their sensor could be (t ′− t ) −[1 − f α (t ′) fβ (t ′)Dαβ (t ′ − t )] . × f α (t ′) 1 − fβ (t ′) Dαβ

Gαr (ω) =

1 , ω − ε α + i η

(40.10)

with the frequency ω has a pole at the free-carrier energy with η > 0, η → 0. Quasiparticle renormalizations are obtained from the solution of the Dyson equation: τ

>

< Dαβ (τ ) =

{

and the Fourier transform

interaction processes. In Equation 40.11, the so-called randomphase approximation has been used to formulate the interaction term with the phonon propagator

 ∂  − ε α  Gαr (τ) = δ(τ) + dτ ′Gαr (τ ′) i τ ∂  

∫

0

∑ G (τ − τ′)D r β

β

> αβ (τ − τ ′),

(40.11)

where the sum involves all available states β. The Dyson equation contains a self-energy for the many-body description of

(40.13)

In comparison to Boltzmann scattering rates, the delta-functions are replaced by convolutions of the phonon propagator with retarded Green’s functions. The latter account for the polaronic renormalizations of the initial and final carrier states. Furthermore, memory effects are included by the time dependence of the population factors on the past evolution of the system. Boltzmann scattering integrals follow as a limiting case, when the Markov approximation is applied (population functions fα(t′) are taken at the external time t) and when quasiparticle renormalizations are neglected with the use of free-carrier retarded Green’s functions (40.9). Examples for ultrafast carrier scattering processes in the polaron picture are displayed in Figure 40.5. Even if the level

40-10


0.8 0.6

0.2

0.4

0.1

0.2

Population f e

Population f e

0.3

0 s-Shell

0.4 0.3

0.1 0

1

Time (ps)

2

0 s-Shell

0.6 0.4

∆E = 1.0 ħωLO ∆E = 1.2 ħωLO ∆E = 1.4 ħωLO

0.2

0

∆E = 1.0 ħωLO ∆E = 1.2 ħωLO ∆E = 1.4 ħωLO

p-Shell

p-Shell

0.2 0

3

0

5

10 Time (ps)

15

20

Figure 40.5â•… Temporal evolution of the QD electron population due to scattering of carriers with LO-phonons. Calculations have been performed in the polaron picture for the InGaAs/GaAs material system at room temperature. Left: carrier relaxation from the initially populated p-shell (top) into the initially empty s-shell (bottom). Right: carrier capture from the WL into the initially empty p-shell (top) and s-shell (bottom). Different energy spacings ΔE between the s- and p-shell in units of the LO-phonon energy are compared. (From Seebeck, J. et al., Phys. Rev. B, 71, 125327, 2005. With permission.)

spacing between s and p-shell and between p-shell and WL bandedge exceeds the LO-phonon energy by 40%, efficient carrier relaxation and carrier capture are observed, while Boltzmann scattering integrals would predict vanishing carrier transitions for the interaction with LO-phonons.

40.4.1â•‡ Optical Susceptibility From Maxwell’s equations, a wave equation for the optical field E(r,â•›t) can be derived, which describes how the emitted field from the sample can be traced back to the macroscopic polarization P(r,â•›t) inside the sample:

40.4â•‡ Optical Gain of the Active Material Of central importance for the design process and for various emission properties of lasers are the optical gain of the active material and the corresponding refractive index changes. Both quantities are linked via the optical susceptibility, which is introduced in this section. In the following, we give some examples for the role of the optical gain. The threshold current of a laser is determined by the transparency carrier density at which the active material switches from absorption to optical gain. The carrier-density dependence of the gain influences the modulation bandwidth, and the temperature dependence of the optical gain is the main factor for the temperature stability of the laser operation. It is also important to study the physical mechanisms of gain saturation, which limits the achievable optical gain. For the edge-emitting laser structures, the optical mode propagates in the QD plane and refractive index changes, which are induced by excited carriers, directly influence the mode properties like frequency chirp or filamentation, as well as the laser linewidth. In the following, we establish a microscopic picture to determine these gain and refractive index properties in QD systems.

 1 ∂2  ∂2  −∇ × ∇ × − c 2 ∂t 2  E(r , t ) = µ 0 ∂t 2 P(r , t ).

(40.14)

The polarization represents the macroscopic dipole density in the medium, which is induced by the field itself, as expressed by the optical susceptibility

∫ ∫

P(r , t ) = d 3r ′ dt ′χ(r , r ′, t, t ′)E(r ′, t ′).

(40.15)

In order to fulfill Equations 40.14 and 40.15 simultaneously, a selfconsistent solution is necessary: the field determined from the wave equation should be the same function entering the calculation of the polarization. For simplicity, we neglect the tensorial character of χ and consider given polarization directions for P and E. From Equation 40.15 can be inferred that the susceptibility is a response function describing the answer of the medium to the optical field. Mathematically this can be expressed with the functional derivative

χ(r, r ′, t, t ′) =

δP(r, t ) . δE(r ′, t ′)

(40.16)

40-11

Quantum Dot Laser

The optical absorption spectrum is commonly defined via the linear response of the medium. In this case, χ depends only on the difference of the time arguments (as can be shown from the explicit quantum mechanical calculation of the macroscopic polarization). Then one obtains in Equation 40.15 a convolution in time, which translates into a product in Fourier space and to give

χQD (ω) =

PQD (ω) . EQD (ω)

(40.17)

For notational simplicity, we have no longer written the space dependence of the functions. In the considered case of an active material of QDs, the optical susceptibility is given as the macroscopic QD polarization Pqd divided by the linear optical test field at the QD position Eqd. The absorption spectrum α(ω) as well as the refractive index changes of the medium due to resonant excitation δn(ω) of the QD system is given by

−K δn(ω) + iα(ω) =

ω χQD (ω), ε0nBcL

(40.18)

where c and ε0 are the speed of light and permittivity in vacuum nB is the background refractive index L is the thickness of the QD sheet K is the wave number of the optical field

40.4.2â•‡Interband Transitions and Macroscopic Polarization

the statistical operator ρ. Since d is a single-particle operator, a partial trace over the statistical operator can be performed in a way that only the single-particle statistical operator with the matrix elements ρν1 , ν2 contributes. The sum involves all singleparticle states of the chosen basis, as discussed for QD systems in Section 40.2. In the following, we further specify the single-particle density-matrix elements. The diagonal elements represent the population of the state ν and have already been abbreviated by fν in Section 40.3. The off-diagonal elements Ψ ν1,ν2 are transition amplitudes between the corresponding states. Then the singleparticle density matrix has the form

Ψ ν1 , ν2  . f ν2 

PQD (t ) = nQD 〈d 〉 = nQD

ν1 , ν2

Ψ ν1 ,ν2(t ),

(40.20)

ν1, ν2

40.4.3â•‡ Optical Gain Calculations A consistent microscopic theory for the calculation of coherent optical properties (related to the interaction of the active material with a classical optical field) can be formulated on the basis of equations of motion for the single-particle density matrix elements. These equations can be derived from the Hamiltonian of the QD system, which contains the contributions of free-carriers, their dipole interaction with the optical fields, as well as further relevant interaction processes like the Coulomb interaction of carriers and the carrier–phonon interaction. When only the free-carrier Hamiltonian and the dipole interaction with the optical field are considered, the density matrix elements obey the optical Bloch equations (Meystre and Sargent III, 1991). We obtain coupled equations for the interband transition amplitude Ψν1, ν2(t ) and for the occupation probabilities f ν1,2 (t ).

i

 ∂ i  Ψν ,ν = ν − ν2 −  Ψν1,ν2 + ( f ν1 − f ν2 )dν1,ν2EQD , ∂t 1 2  1 T2 

(40.21)

i ∂ fν = ∂t 1

∑[d

ν1, ν2

EQD Ψ *ν1,ν2 − c.c.] −

f ν1 − Fν1 , T1

(40.22)

EQD Ψν*1, ν2− c.c.] −

f ν2 − Fν2 . T1

(40.23)

ν2

i ∂ fν = − ∂t 2

∑[d

ν1, ν2

ν1

A phenomenological way to include the influence of carrier scattering and dephasing is via time constants T1,2, respectively. For the last term in Equations 40.22 and 40.23, a relaxation-time approximation has been used, which describes the evolution of the occupation probabilities f ν1,2 toward quasi-equilibrium functions Fν1,2 on a time-scale T1. Damping of the interband transition amplitude Ψ ν1, ν2, which determines the linewidth of the interband transitions, is accounted for with the dephasing time T2 in Equation 40.21. The stationary solution of Equation 40.21 together with Equations 40.17 and 40.20 allows to calculate the optical susceptibility in the form

(40.19)

χQD (ω) = nQD

For the calculation of the optical polarization, we use that the diagonal dipole-matrix elements vanish exactly. Also, we consider only interband transition contributions to the dipole coupling. For the final calculation of the macroscopic QD

∑d

with the (sheet) density of QDs nQD.

The macroscopic optical QD polarization can be determined from the quantum mechanical expectation value of the dipole operator 〈d 〉 = Tr{dρ} = Σ ν1,ν2 dν1,ν2 ρν2,ν1, which is calculated with

 f ν1 ρν1 , ν2 =   Ψ *ν1 , ν2

polarization, we assume that the optical field averages over sufficiently many QDs to obtain

∑| d

ν1 ,ν2

|2( f ν1 − f ν2 )

ν1,ν2

 1 1 − ×  ω − ε ν + i ω − ε ν12 + i 12 T2 T2 

 .  

(40.24)


40.4.4â•‡Gain Saturation, Refractive Index, and α-Factor A microscopic theory for gain calculations in QD systems, which includes excitonic effects as well as excitation-induced energy renormalization and dephasing due to the screened Coulomb interaction and carrier–phonon interaction, has been developed in Lorke et al. (2006a). An example for the transition of a QD system from absorption to gain with increasing excitation density is shown in Figure 40.6. The calculations are performed at room temperature and assume a quasi-equilibrium distribution of the carriers over the QD and WL states due to efficient carrier-scattering processes. For a low excitation density (dotted line), QD s-shell and p-shell transitions as well as the WL exciton resonance at E − EG ≈ −138, −80, and −17â•›meV, respectively, are broadened due to carrier–phonon and carrier–carrier scattering processes. With increasing carrier density, the transition lines are bleached (due to phase-space filling) and further broadened (due to increased scattering efficiency). Note that bleaching reduces the oscillator strength, which needs to be distinguished from the increasing the linewidth (broadening). With population inversion of the QD states, optical gain is realized. Just above the transparency carrier density, only the s-shell transition exhibits optical gain. At higher carrier densities, the optical gain of the p-shell and of the WL are taking over. The WL gain is due

1.5

0.06 1 0.03

[arb. units]

The imaginary part of χ qd(ω) describes the absorption spectrum of the system and reveals discrete lines for interband transitions between the QD states ν1,2 with a linewidth ∝1/T2 at the interband energies ε ν12 = ε ν1 − ε ν2. When the population in the upper state exceeds that in the lower state, the population factor f ν1 − f ν2 changes its sign and the system switches from absorption to optical gain. It should be noted that the optical Bloch equations (Equations 40.21 through 40.23) and the resulting susceptibility (40.24) represent only a free-carrier picture, which is usually a poor approximation for a quantitative analysis. Important is the additional inclusion of Coulomb interaction effects in the optical Bloch equations for semiconductor systems (Haug and Koch, 2004), which is responsible for different types of effects. First of all, the interband Coulomb exchange interaction between electrons and holes accounts for excitonic effects at low excitation densities and interband Coulomb enhancement of the optical transitions. Secondly, energy renormalization shows up as excitation-dependent shifts of the transition energies, of which the band-gap renormalization is the most prominent signature. A closer inspection reveals, however, nonrigid shifts of all states. As a third modification, the microscopic replacement for scattering terms, expressed by the relaxation approximation in Equations 40.22 and 40.23, has been provided in Section 40.3. For consistency, similar terms need to be used in Equation 40.21 to describe the excitation-induced dephasing, which replaces the constant dephasing time T2. Furthermore, screening of the Coulomb interaction contributes to a nonlinear dependence of the optical properties on the excitation conditions.

Im

40-12

0 0.5 –0.03 –150

–100

–50

0 –100 E – EG [meV]

–150

1 × 1010 cm–2 10

–2

11

–2

5 × 10 cm 1 × 10 cm

–50

0

2 × 1011 cm–2 5 × 1011 cm–2 1 × 1012 cm–2

Figure 40.6â•… Imaginary part of the optical susceptibility, representing the absorption spectrum, versus energy relative to the band edge of the WL for a InGaAs/GaAs QD system. The spectrum contains two interband transitions due to confined states, which are magnified in the inset. (From Lorke, M. et al., Phys. Rev. B, 73, 085324, 2006. With permission.)

to the increased population of the corresponding quantum-welllike states. The combination of band-gap shrinkage and band filling positions the WL gain near the low-density WL exciton resonance. The related small excitation-dependent shifts of the quantum well gain are discussed in Chow and Koch (1999). When the QD states are completely populated with electrons and holes, a further increase of the total carrier density in the system does not lead to a larger optical gain at the corresponding transitions. This is clearly visible at the s-shell transition for the largest carrier densities in Figure 40.6. A higher carrier density in the assumed quasi-equilibrium situation only increases the population of the higher QD states and the WL states. The complete filling of the QD states is the origin for the observed gain saturation. Adding even more carriers to the system finally starts to reduce the QD ground-state gain due to further increasing dephasing (Lorke et al., 2006b). At low carrier densities, the QD interband transition lines can be associated with QD excitons. The terminology might be viewed with reservations, since the electron–hole pairs are restricted to the QDs by the confinement potential and not—like quantumwell or bulk-semiconductor excitons—bound by the Coulomb interaction. The interplay of direct and exchange Coulomb interaction even allows for “anti-bound” states; a biexciton transition

40-13

Quantum Dot Laser

with an energy larger than that of two exciton transitions is the most prominent example. With the term “QD exciton,” one emphasizes the role of the Coulomb interaction for characterizing the states. For quantum well and bulk semiconductor excitons, it is known that the delicate balance of energy renormalization (mainly band-gap shrinkage) and reduction of the exciton binding energy due to phase-space filling and screening leads to a nearly constant energetic position of the exciton resonance, which is bleached and broadened for increasing carrier density. The same behavior is observed for the WL exciton in Figure 40.6. However, the QD excitons show a pronounced red shift, which persists as a shift of the gain peaks at elevated carrier densities. When the energy renormalization is determined within a manybody theory formulated in a single-particle basis, it is the result of a partial compensation of state-diagonal and state-nondiagonal self-energies. It can be shown that this partial compensation is strongly reduced in QD systems (Lorke et al., 2006a). The physics of QD gain spectra is expected to play an important role for QD-microcavity lasers with a small number of dots. In edge-emitting devices, inhomogeneous broadening can mask the discussed effects until the sample quality improves or the laser behavior (saturation effects, gain dynamics) is examined in greater detail. The excitation-induced refractive index changes are directly linked to the corresponding absorption spectra via the Kramers– Kronig relation (Haug and Koch, 2004), since both can be derived from the complex optical susceptibility according to Equation 40.18. Refractive index changes directly influence the mode properties of edge-emitting lasers, including filamentation, as well as frequency chirp and emission spectra. In the past, the α-factor (linewidth-enhancement factor, antiguiding parameter) has been used to characterize the importance of the excitation-densityinduced refractive index changes. In the free-carrier picture outlined above, the discrete nature of the QD states leads to symmetric absorption lines and vanishing refractive index changes at the absorption peak. The resulting value of zero for the α-factor has led to the prediction of many beneficial properties of QD lasers. Calculations based on microscopic semiconductor models show deviations from this simple picture, but also smaller α-factors than for quantum-well lasers (Lorke et al., 2007).

40.5â•‡ Laser Emission Properties The optical gain characterizes the active material itself, while the steady-state and dynamical emission properties of lasers depend more generally on the interplay of the photon and carrier systems. The optical gain determines the rate of stimulated emission of photons into the laser mode. Together with the spontaneous emission rate, which is related to the photoluminescence spectrum, and the cavity losses, these processes govern the laser field inside the laser resonator and the light output. The carrier dynamics is determined by the interplay of the pump process, the transfer of carriers into the laser levels, as well as the optical processes. The most intuitive approach to the steady-state and dynamical properties of lasers is provided by rate equations, which

will be introduced in Section 40.5.1. Such a theory is based on restricting the information about the laser output to the mean photon number in the laser mode, as well as on a convenient parametrization of the rates for various processes. A generalization, which allows the systematic inclusion of semiconductor effects, is outlined in Section 40.5.2. The statistical properties of the light emission as well as the control of light–matter interaction in microcavity lasers are addressed in Section 40.5.3.

40.5.1â•‡Rate Equations for Quantum-Dot Systems In a rate-equation description (Yokoyama and Brorson, 1989; Rice and Carmichael, 1994), one uses two coupled dynamical equations for the number of photons in the laser mode n and the number of excited emitters N, which are QDs for the present purpose:

d nN N − , N =P− τl τsp dt

(40.25)

d (n + 1)N n = −2κn + . dt τl

(40.26)

The pump rate P increases the emitter number, while the cavity loss rate 2κ reduces the photon number. The loss rate is directly connected to the Q-factor of the laser mode, Q = ħωl/2κ, with the laser frequency ωl. The total rate of spontaneous emission N/τsp, which includes emission into all available lasing and nonlasing modes, reduces the number of excited emitters, while only the spontaneous emission directed into the laser mode N/τl contributes to the increase of the respective photon number. The simulated emission is additionally proportional to the mean photon number n, and the corresponding rate nN/τl appears in both rate equations with opposite signs. An important parameter to characterize laser resonators is the β-factor, which is defined as the ratio of the spontaneous emission rate into the laser mode to the total spontaneous emission according to β=

1 / τl . 1/ τsp

(40.27)

In edge-emitting laser diodes, the large number of nonlasing modes leads to small values of β = 10−6 … 10−5, while in microcavity lasers the spontaneous emission into nonlasing modes can be strongly suppressed and β ≈ 1 is approached. A general evaluation of the rate equations can be readily performed by means of a direct numerical solution in time for a given pump rate P. If the initial condition is the unexcited system and the pump rates are switched on to a constant value, the solution either shows a smooth evolution to the steady-state or damped relaxation oscillations (Milonni and Eberly, 1991), depending on the chosen parameters. The results for the steady-state solution and various β-factors are shown in Figure 40.7. A set of parameters referring to

β × photon number in laser mode

40-14


underscores the importance of semiconductor models for a more detailed microscopic description of photoluminescence and gain in QD systems.

10 1 10

–1

10–2 10–3 10

–4

10–5 10–6 10–4

β= 1

40.5.2â•‡Microscopic Generalization: Semiconductor Effects

β= 0.1 β= 0.01 β= 10–3 β= 10–4 β= 10–5 10–3 10–2 –1 Pump rate P [ps ]

10–1

Figure 40.7â•… Calculated input–output curves from the laser rate equations for β = 1 to 10−5 from top to bottom. The photon number and the pump rate are scaled with β in order to have the thresholds appear at equal pump intensities for better comparison. (From Gies, C. et al., Phys. Rev. A, 75, 013803, 2007. With permission.)

QD-microcavity lasers as been used: τsp = 50â•›ps and κ = 20â•›μeV. The corresponding cavity lifetime is about 17â•›ps, yielding a Q-factor of ≈30,000. The curves show a typical intensity jump ∝ β−1 from below to above threshold. In the limit β = 1, the kink in the input–output curve disappears. The above discussed form of the rate equations is based on various assumptions. Only one optical mode is subject to stimulated emission; otherwise separate rate equations for the photon numbers in different laser modes need to be used and their modal gain determining τl needs to be distinguished. The derivation of the rate equations from a microscopic theory is based on the adiabatic elimination of transition amplitudes as well as on the factorization of carrier and photon correlations (Rice and Carmichael, 1994). Furthermore, in the rate equations, one assumes that the spontaneous and stimulated recombination rates are proportional to number of excited emitters N. Our discussion of the optical gain in Section 40.4.3 has already revealed that this is only a rough estimate neglecting optical nonlinearities and saturation effects. Furthermore, each QD is assumed to possess only two possible configurations, the excited and de-excited state. In reality, QDs often contain more than two localized states that can be occupied by several charge carriers. This can lead to various modifications of the simple “atomic approach.” The form of the spontaneous recombination term in the laser rate equations predicts an exponential decay of the photoluminescence after pulsed excitation and in the absence of stimulated emission. Recent experiments with self-assembled InGaAs/ GaAs QDs embedded in GaAs-based micropillars (Schwab et al., 2006) showed, however, a nonexponential decay of the time-resolved photoluminescence. Furthermore, this decay was shown to be accompanied by a strong dependence on the excitation intensity. These two effects make it impossible to assign a single spontaneous emission lifetime to the QDs and question the simple picture used in the rate equations. This further

A semiconductor theory for QD lasers can be formulated on the basis of equations of motions for the carrier occupation probabilities of the QD levels and the photon number in the laser mode. For a microscopic derivation, one starts from the Hamiltonian describing the quantized electromagnetic field, the carrier system in second quantization, and the carrier–photon interaction in dipole approximation. Further interaction processes, like the Coulomb interaction of carriers and the carrier–phonon interaction, can be included systematically. Heisenberg equations of motion for the carrier and photon operators can be used to derive dynamical equations for the mean photon number (nâ•›=â•›b†b) and the carrier occupation functions f νe = 〈cν†c ν 〉, f νh = 1 − 〈v ν†v ν 〉. In these expectation values, b† and b are photon creation and annihilation operators, cν† and cν are creation and annihilation operators for conduction-band carriers in the states ν and v ν†v ν are the corresponding valence-band operators, respectively. The contribution of the interaction of carriers with the laser mode then leads to

2 d  2  dt + 2κ  n = | g |

∑ Re 〈b v c

† † ν′ ν′

〉,

ν′

(40.28)

d e ,h 2 = − | g |2 Re 〈b†vv†cv 〉 , fν opt dt

(40.29)

with the light–matter coupling strength g. It can be seen that the corresponding dynamics of the photon number and carrier populations are determined by the photon-assisted polarization 〈b†v ν†cν 〉, that describes the expectation value for a correlated event, where a photon is created in connection with an interband transition of an electron from the conduction to the valence band. The sum over ν involves all possible interband transitions from various QDs. The time evolution of the photon-assisted polarization follows from its equation of motion, d  † † e h  dt + κ + Γ + i(ε ν + ε ν − ω l)  〈b v νc ν〉 = f νe f νh − (1 − f νe − f νh)n

∑V

+ i (1 − f νe − f νh )

νανα

〈b†vα† cα 〉

α

+

∑C α

x αννα

+ δ〈b†bc ν†c ν〉 − δ〈b†bv ν†v ν 〉.

(40.30)

Here, the free evolution of 〈b†v ν†c ν 〉 is determined by the detuning of the QD transitions at the renormalized energies εeν,h from

40-15

Quantum Dot Laser 2 β = 0.1

β = 0.001

g (2)(0)

β=1 1.5

1

β × photon number

the optical mode ωl. Cavity losses κ and dephasing processes represented by Γ lead to a damping of the time evolution and to a broadening of the spectral components of the optical processes, which are spontaneous and stimulated emission. When deriving the equation of motion for 〈b†v ν†c ν 〉, one finds that the source term of spontaneous emission is described by an expectation value of four carrier operators 〈cα† vαv ν†c ν 〉. For uncorrelated carriers, the Hartree–Fock factorization of this source term leads to f νe f νh, which appears as the first term on the right hand side of Equation 40.30. It describes a spontaneous recombination probability proportional to the occupation probabilities of electrons and holes, as expected in a free-carrier picture, but opposed to the situation in an atomic system, where the recombination depends only on the electron population. However, electrons and holes will not just contribute as independent carriers. Correlation contributions x = δ〈cα† v ν†c νv α 〉. to the spontaneous emission are included in Cαννα These correlations describe the joint probability of a two-particle process, where two interband carrier transitions between the states α and ν take place. As mentioned above, the indices include not only the electronic states but also the QD position. Consequently, one must distinguish between correlated transitions within one QD, which can be due to an excitonic population, and correlations of transitions within two separate QDs. The latter case is connected to the phenomenon of superfluorescence or superradiant coupling of different emitters. The Coulomb interaction contributes to the excitonic correlations of electrons and holes, which, in turn, are weakened by screening, phase-space filling, and dephasing effects of the excited carriers. The detailed microscopic description of these correlations is an intricate problem of many-body theory, and for further details we refer the interested reader to Baer et al. (2006). In a similar fashion, stimulated emission or re-absorption of photons is represented by (1 − f νe − f νh )n in the absence of carrier–photon correlations. The latter are represented by δ〈b†bc ν†c ν 〉 and δ〈b†bv ν†v ν 〉, which require their own equations of motion. Furthermore, the interband Coulomb–exchange interaction with the matrix elements Vνανα is responsible for excitonic effects similar to its appearance in the semiconductor Bloch equations (Haug and Koch, 2004). While the semiconductor theory offers the potential to account for various interaction effects in a much more detailed way, this also complicates a numerical analysis significantly. Results from the semiconductor theory for the input–output curves are shown in the lower part of Figure 40.8. Deviations from the rate-equation results involve a different height in the intensity jump at the laser threshold and saturation effects. The semiconductor theory also opens the possibility to calculate parameters entering rate equation models based on single-particle properties and interaction effects.

1

10–2

β=1 β = 0.1 β = 0.01

10–4 β = 0.001 10–6 10–4

10–3

10–2 10–1 Pump rate [ps–1]

1

Figure 40.8â•… Input–output curves and auto-correlation functions g(2) (τ = 0) for various β-factors calculated with the semiconductor model. (From Gies, C. et al., Phys. Rev. A, 75, 013800, 2007. With permission.)

close to unity (Strauf et al., 2006; Ulrich et al., 2007). The strongly increased cavity-Q (corresponding to a long lifetime of photons in the cavity) allows to fabricate QD-based lasers with a small number of dots in the active region. The strongly enhanced light-matter coupling and the operation with a small number of photons in the laser mode leads to novel emission properties that are directly related to the quantum-mechanical nature of light. In particular, the photon statistics of the emitted radiation does no longer exhibit the transition from thermal to coherent radiation for increasing pumping, and “nonclassical” properties on the level of few photons can be realized. To characterize these systems, one needs to study the coherence properties of the emitted light and the statistical properties of the photons. Following Glauber, the quantum states of light can be characterized in terms of photon correlation functions. Coherence properties of the electromagnetic field itself are reflected by the (normalized) correlation function of first order, g (1) (τ) =

〈b† (t )b(t + τ)〉 . 〈b† (t )b(t )〉

(40.31)

40.5.3â•‡Quantum-Dot Microcavity Lasers: Modifications of the Photon Statistics

Latest advances in the growth and design of semiconductor-QD microcavity lasers have now attained the regime of β-values

Its decay in τ is determined by the coherence time of the emitted light τc = | g (1) (τ) |2 d τ. Here, b† and b are again the creation

∫

40-16


and annihilation operators for photons in the laser mode. Information about the statistical properties of the emitted light can be deduced from the correlation function of second order at zero delay time g (2) (τ = 0) =

〈 n2〉 − 〈n〉 〈b†bb†b〉 − 〈b†b〉 = . 〈n〉2 〈b†b〉2

(40.32)

The function g(2)(τ = 0) reflects the possibility of the correlated emission of two photons at the same time. Often discussed are the limiting cases of light emission from a thermal, a coherent, and a single-emitter light source. Thermal light is characterized by an enhanced probability that two photons are emitted at the same time (bunching), reflected in a value of g(2)(0) = 2. For coherent light emission with Poisson statistics, one finds g(2)(0) = 1. An ideal single-photon emitter exhibits antibunching with g(2)(0) = 0. This, the second-order correlation function, can be used to characterize the emission and to analyze the transition from thermal (or even sub-Poissonian) to coherent light emission. For atomic systems, various methods have been established to analyze photon correlation functions. A master equation can be used to describe an ensemble of emitters interacting with a single high-Q laser cavity mode and a bath of non-lasing modes (Rice and Carmichael, 1994). For a single-atom laser, a direct analysis of the von Neumann equation for the statistical operator including the coupling to dissipative systems is possible (Mu and Savage, 1992). Semiconductor effects can be included in generalized equations of motion for higherorder carrier and photon correlation functions (Gies et al., 2007). Results of a semiconductor theory for the secondorder photon correlation function are displayed in the upper part of Figure 40.8. For smaller β-factors referring to conventional lasers, the output-intensity jump at the laser threshold is accompanied with a sudden change of the correlation function between values representing thermal and coherent light. For increasing values of β, the gradually disappearing jump in the photon number is connected with a smoother transition of g(2)(0) and the presence of correlations for small pump rates.

Acknowledgments The author thanks P. Gartner and J. Wiersig for the intense and productive collaboration on the covered subjects. The contributions from our PhD students T. Nielsen, N. Baer, J. Seebeck, M. Lorke, and C. Gies are also gratefully acknowledged. This work was financially supported by the Deutsche Forschungsgemeinschaft. We also thank P. Gartner and C. Gies for the critical reading of the manuscript and A. Beuthner for the preparation of Figure 40.1.

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41 Mode-Locked Quantum-Dot Lasers 41.1 Introduction............................................................................................................................ 41-1 41.2 Ultrafast Laser Diodes........................................................................................................... 41-2 Basics of Mode Lockingâ•‡ •â•‡ Mode-Locking Techniques in Semiconductor Lasersâ•‡ •â•‡ Passive Mode Locking: Physics and Devicesâ•‡ •â•‡ Requirements for Successful Passive Mode Lockingâ•‡ •â•‡ Self-Phase Modulation and Dispersion

41.3 Quantum Dots: Distinctive Advantages for Ultrafast Diode Lasers..............................41-5 The Role of Dimensionality in Semiconductor Lasersâ•‡ •â•‡ Quantum Dots: Materials and Growthâ•‡ •â•‡ Broad Gain Bandwidthâ•‡ •â•‡ Ultrafast Carrier Dynamicsâ•‡ •â•‡ Low Absorption Saturation Fluenceâ•‡ •â•‡ Low Threshold Current and Low Temperature Sensitivityâ•‡ •â•‡ Low Linewidth Enhancement Factor

41.4 Mode-Locked Quantum-Dot Lasers: State of the Art......................................................41-8

Maria A. Cataluna University of Dundee

Edik U. Rafailov University of Dundee

Pulse Durationâ•‡ •â•‡ Toward Higher Pulse Repetition Ratesâ•‡ •â•‡ Temperature Resilience of Mode- Locked QD Lasersâ•‡ •â•‡ Mode Locking Involving Excited-State Transitions

41.5 Summary and Outlook........................................................................................................ 41-10 Critical Discussionâ•‡ •â•‡ Future Perspectives

Acknowledgments............................................................................................................................ 41-10 References.......................................................................................................................................... 41-10

41.1â•‡ Introduction Over the last three decades, laser physics has advanced dramatically. Starting from lasers operated in a continuous wave (cw) regime, scientists have developed techniques for generating periodic sequences of optical pulses with ultrashort durations— between a few picoseconds (1 × 10 −12 s) and a few femtoseconds (1 × 10−15 s). To put this into perspective, 1â•›fs compared to 1â•›s is the same as 1â•›s compared to 32 million years! Such ultrafast lasers have important applications in medicine, micromachining, optical communications, spectroscopy, and anything else that requires studying physics at extremely high powers or extremely short timescales. For instance, these lasers have been successfully adapted in eye surgery, because ultrashort pulses can make extremely precise cuts with minimum thermal damage. However, despite the wide range of important areas that can benefit from ultrafast lasers, the use of these lasers is constrained due to several limitations. The ultrafast lasers currently available are often bulky, expensive, and difficult to operate. The ideal ultrafast laser would be a low-cost, handheld, and turnkey laser—features which could be offered by semiconductor lasers. Semiconductor lasers cannot yet directly generate the sub-100â•›fs pulses routinely available from crystal-based lasers, but they represent the most compact and efficient sources of picosecond

and sub-picosecond pulses. Furthermore, the bias can be easily adjusted to determine the pulse duration and the optical power, thus offering, to some extent, electrical control of the characteristics of the output pulses. These lasers also offer the best option for the generation of high-repetition rate trains of pulses, owing to their small cavity size. Ultrafast diode lasers have thus been favored over other laser sources for high-frequency applications such as optical data/telecommunications. Being much cheaper to fabricate and operate, ultrafast semiconductor lasers also offer the potential for dramatic cost savings in a number of applications that traditionally use solid-state lasers. The deployment of high-performance ultrafast diode lasers would therefore have a significant economic impact, by enabling ultrafast applications to become more profitable, and even facilitate the emergence of new applications. Novel nanomaterials such as quantum dots (QDs) have enhanced the characteristics of semiconductor lasers, greatly improving their performance. QDs are tiny clusters of semiconductor material with dimensions of only a few nanometers. At these small sizes, materials behave very differently, giving QDs distinctive physical properties of quantum nature—for instance, the emission wavelength or “color” depends on the size of the dot! These nanomaterials afford major advantages in ultrafast science and technology, and they can form the basis for very 41-1

41-2


compact and efficient lasers delivering short pulses of the order of hundreds of femtoseconds (Rafailov et al., 2007). In this chapter, we show how QDs have enabled the generation of ultrashort pulses from compact optical sources based on semiconductor laser diodes. In Section 41.2, the necessary background information is presented on ultra-short-pulse generation from diode lasers. The concept of mode locking is introduced, and an overview of the mode-locking techniques available for semiconductor lasers is provided. The unique properties of QD materials and their suitability for ultra-shortpulse diode lasers are explained in Section 41.3. Finally, a summary of the state of the art in the field of QD mode-locked laser diodes is provided in Section 41.4. The chapter is finalized by a summary and an outlook on the future perspectives of this fascinating field.

41.2â•‡ Ultrafast Laser Diodes 41.2.1â•‡ Basics of Mode Locking Mode locking is a technique that involves the locking of the phases of the longitudinal modes in a laser. This results in the generation of a sequence of pulses with a repetition rate corresponding to the cavity round-trip time. This well-established technique enables the production of the shortest pulse durations and the highest repetition rates available from ultrafast lasers, whether they are semiconductor or crystal-based laser systems. In a standing-wave resonator, the pulse repetition rate f R is given by

fR =

c 2nL

where c is the speed of light in vacuum n is the refractive index L is the length of the laser cavity In terms of Fourier analysis, there is an inverse proportionality between the duration of a mode-locked pulse and the corresponding bandwidth of its optical spectrum. The product of both the pulse duration Δτ and the optical frequency bandwidth Δν is called the time-bandwidth product (TBWP). For a given frequency bandwidth, there is a minimum corresponding pulse duration—if this is the case and the optical spectrum is symmetrical, then the pulse is said to be transform-limited, and the TBWP equals a constant K, whose value depends on the shape of the pulse, whether it is Gaussian, hyperbolic, secant, squared, or Lorentzian. By measuring the full-width at half maximum from an optical spectrum Δλ, it is easy to calculate the TBWP of a given pulse:

∆ν ⋅ ∆τ = K ⇒

c ∆λ ⋅ ∆τ = K λ2

Another important property of mode-locked lasers is that the energy that was dispersed in several modes while in cw

operation, is now concentrated in short pulses of light. This implies that although the output average power Pav may be low, the pulse peak power Ppeak can be significantly higher:

Ppeak =

Ep 1 Pav ⇒ Ppeak = ⋅ ∆τ ∆τ f R

where Ep is the pulse energy.

41.2.2â•‡Mode-Locking Techniques in Semiconductor Lasers In recent years, mode-locked laser diodes have been at the center of a quest for ultrafast, transform-limited, and high-repetitionrate lasers. To achieve these goals, a variety of mode-locking techniques and semiconductor device structures have been demonstrated and optimized (Vasil’ev, 1995). The three main forms of mode locking can be described as active, passive, and hybrid techniques, as outlined below. Active mode locking relies on the direct modulation of the gain with a frequency equal to the repetition frequency of the cavity, or to a sub-harmonic of this frequency. The main advantages of this approach are the resultant low jitter and the ability to synchronize the laser output with the modulating electrical signal. These features are especially relevant for optical transmission and signal-processing applications. However, high repetition frequencies are not readily obtained through directly driven modulation of lasers because fast RF (radiofrequency) modulation of the drive current becomes progressively more difficult with increase in frequency. The frequency limitation imposed by electronic drive circuits can be overcome by employing passive mode-locking techniques. This scheme typically utilizes a saturable absorbing region in the laser diode. In a saturable absorber, the loss decreases as the optical intensity increases. This feature acts as a discriminator between cw and pulsed operation and can facilitate a self-starting mechanism for mode locking. Most importantly, saturable absorption plays a crucial role in shortening the duration of the circulating pulses, as will be explained, thus providing the shortest pulses achievable by all three techniques and the absence of a RF source simplifies the fabrication and operation considerably. Passive mode locking also allows for higher pulse repetition rates that are determined solely by the cavity length. Inspired by active and passive mode locking, the technique of hybrid mode locking meets the best of both worlds because the pulse generation is initiated by an RF current imposed in the gain or absorber section, while further shaping and shortening is assisted by saturable absorption. The next section explores in more detail the physical mechanisms behind passive mode locking.* * From this point, mode locking will implicitly mean passive mode locking, unless otherwise stated.

41-3

Mode-Locked Quantum-Dot Lasers

41.2.3â•‡Passive Mode Locking: Physics and Devices

Reverse bias

So far, a simple frequency-domain picture for mode locking has been provided, where the relative phases are of primary relevance. A physical model for passive mode locking can alternatively and equivalently be described in terms of the temporal broadening and narrowing mechanisms. Upon startup of laser emission, the laser modes initially oscillate with relative phases that are random such that the radiation pattern consists of noise bursts. If one of these bursts is energetic enough to provide a fluence that matches the saturation fluence of the absorber, it will bleach the absorption. This means that around the peak of the burst where the intensity is higher, the loss will be smaller, while the low-intensity wings become more attenuated. The pulse generation process is thus initiated by this family of intensity spikes that experience lower losses within the absorber carrier lifetime. The dynamics of absorption and gain play a crucial role in pulse shaping. In steady state, the unsaturated losses are higher than the gain. When the leading edge of the pulse reaches the absorber, the loss saturates more quickly than the gain, which results in a net gain window, as depicted in Figure 41.1. The absorber then recovers from this state of saturation to the initial state of high loss, thus attenuating the trailing edge of the pulse. It is thus easy to understand why the saturation fluence and the recovery time of the absorber are of primary importance in the formation of mode-locked pulses. This temporal scenario can be connected to the previously described frequency domain description of mode locking. The burst of noise is the result of an instantaneous phase locking Gain

Mirror

Absorber

Mirror

Laser diode cavity length

Loss and gain

Loss Net gain

Gain Time

Intensity

Time

FIGURE 41.1â•… A schematic diagram of the main components that forms a two-section laser diode (top). Loss and gain dynamics that lead to pulse generation (bottom).

Forward bias

e rat bst u S

L abs

orb

er

L gain

FIGURE 41.2â•… A schematic of a two-section semiconductor laser diode.

occurring among a number of modes. The self-saturation at the saturable absorber then helps to sustain and strengthen this favorable combination, by discriminating against the lower power cw noise. In practical terms, a saturable absorber can be integrated monolithically into a semiconductor laser, by electrically isolating one section of the device (Figure 41.2). By applying a reverse bias to this section, the carriers that are photogenerated by the pulses can be more efficiently swept out of the absorber, thus enabling the saturable absorber to recover more quickly to its initial state of high loss. An increase in the reverse bias serves to decrease the absorber recovery time, and this will have the effect of further shortening the pulses.

41.2.4â•‡Requirements for Successful Passive Mode Locking Ultrafast carrier dynamics are fundamental for successful mode locking in semiconductor lasers, particularly in the saturable absorber, because the absorption should saturate faster and recover faster. Indeed, the absorption recovery time is one of the determining factors for obtaining ultrashort pulses. In particular, for high-repetition-rate lasers, the absorber recovery time should be much shorter than the cavity period so that the absorber can return to a state of total attenuation prior to the incidence of each incoming pulse. The fast absorption recovery also prevents the appearance of satellite pulses within the window of the net gain. On the other hand, the gain recovery time should be shorter than the cavity round-trip time. Thus, for lasers operating at pulse repetition rates of 20â•›GHz or more, this means that the recovery times of both gain and absorption should be much shorter than 40â•›ps. The saturation dynamics represents another crucial aspect for successful mode locking, as shown schematically in Figure 41.3. Such dynamics can be translated in terms of saturation fluence Fsat or saturation energy E sat = A . Fsat, where A is the optical mode cross-sectional area. The saturation energy is an indication of how much energy is necessary to saturate the absorption or the gain. Indeed, to achieve robust mode locking, the saturation

41-4


dg/dN

Gain (g) Transparency

Carrier density (N) Absorption (a)

da/dN

E(t , x ) = E0 exp iΦ(t ) = E0 exp i(ω 0t − kx ), k =

ω0 n( t) c

where Φ(t) is the time-varying phase k is the wave vector ω0 is the optical carrier frequency c is the speed of light n(t) is the time-varying refractive index The instantaneous frequency is the time derivative of the phase and thus can be written as

a energy of the absorber E sat should be as small as possible and g smaller than the saturation energy of the gain E sat :

a E sat =

hνA hνA g < E sat = ∂a ∂g ∂N ∂N

where h is Planck’s constant ν is the optical frequency ∂a/∂N and ∂g/∂N are the differential loss and gain, respectively The special dependence of the loss/gain with carrier density in a semiconductor laser allows ∂a/∂N > ∂g/∂N, as shown in Figure 41.3. This condition implies that the absorber will saturate faster than the gain for a given pulse fluence, thus enabling the creation of the net gain window as already mentioned. The ratio between saturation energies should also be as large as possible to ensure that the losses saturate more strongly than the gain.

41.2.5â•‡ Self-Phase Modulation and Dispersion In a semiconductor material, both the refractive index and gain (or loss) depend on the carrier density and are thus strongly coupled. As the pulse propagates in the gain section,* the carrier density and thus the gain is depleted across the pulse, as the carriers recombine through stimulated emission. This leads to a dynamic increase of the refractive index, which then introduces a phase modulation on the pulse, changing the instantaneous frequency across the pulse. This phenomenon is called self-phase modulation (SPM) and is one of the main nonlinear effects associated with pulse propagation in semiconductor media. To understand the mechanism of SPM, consider the simple and illustrative example of a plane wave E(t, x): * In the following discussion, reference is made mostly to gain to simplify the description. However, all this reasoning can be applied equally well to the absorber.

ω(t ) =

∂ ω ∂n(t ) Φ(t ) = ω 0 − 0 x ∂t c ∂t

From this expression, it is clear that if the refractive index varies with time, then the instantaneous frequency of the plane wave will vary relative to ω0 and in a manner proportional to the temporal derivative of the index. The time dependence of this instantaneous frequency is called the frequency chirp. An up-chirp† (down-chirp) means that the frequency increases (decreases) with time. An example of a frequency up-chirped pulse is illustrated in Figure 41.4. SPM is not dispersive in itself, but the pulse will not remain transform-limited when it propagates in a dispersive material such as the laser medium. The effect of dispersion manifests itself in the variation of refractive index for different wavelengths which means that different spectral components will travel at different speeds. For an up-chirped pulse, the frequency is higher in the trailing edge than in the leading edge. When the pulse propagates through a material exhibiting positive (normal) dispersion, the trailing edge of the pulse propagates more slowly than the leading edge of the pulse and so this results in a temporal broadening of the pulse. In a monolithic two-section mode-locked semiconductor laser, where a saturable absorber and a gain section coexist, the resulting chirp is a balance between the effects caused by the absorber and the gain. In the gain section, a frequency up-chirp results, while the saturable absorber helps to further

Electric field

FIGURE 41.3â•… Dependence of absorption/gain with carrier concentration in a semiconductor laser.

Time

FIGURE 41.4â•… Illustration of an electric field of a strongly up-chirped pulse, where the instantaneous frequency increases with time. †

Up-chirp is also known as blue-chirp or positive chirp.

41-5


shape the pulse by contributing with a negative chirp. With a suitable balance between both sections, the chirp can be close to zero thereby leading to transform-limited pulses. Unfortunately, this is the exception rather than the rule, because this usually only occurs for a limited set of bias conditions and/or for given ratios of absorber/gain lengths. Therefore, up-chirp prevails for passively mode-locked lasers, leading to significant pulse broadening as the pulse propagates. The combined effect of SPM and dispersion impose the strongest limitation in the achievable shortest duration of pulses from mode-locked semiconductor diode lasers. The mechanism of SPM implies that in addition to the original frequency ω0, there are now more frequencies inside the pulse envelope. This richer spectral content is not necessarily unhelpful because it can provide bandwidth support for shorter pulses, if the chirp of a pulse can be removed by provision of a suitable dispersion-induced chirp of the opposite sign. For up-chirped pulses, a dispersion compensation setup can be configured such that a negative (anomalous) group velocity dispersion is able to slow down the leading edge of the pulse and speed the blueshifted trailing edge to such an extent that at a certain point both edges propagate simultaneously and the pulse is shorter. To routinely generate pulses that are nearly transformlimited, an alternative could be found in the choice of a material that exhibits lower coupling between refractive index and gain, as described by linewidth enhancement factor (LEF), or α-factor:

α=−

4π dn/dN λ dg /dN

A higher α-factor implies a more significant coupling between gain and refractive index changes with carrier concentration and thus the possibility for higher levels of SPM and frequency chirp.

41.3â•‡Quantum Dots: Distinctive Advantages for Ultrafast Diode Lasers 41.3.1â•‡The Role of Dimensionality in Semiconductor Lasers The history of semiconductor laser materials has been punctuated by dramatic revolutions. Everything started with the proposal of p-n junction semiconductor lasers in 1961, followed by experimental realization on different semiconductor materials (Basov et al., 1961; Basov, 1964). However, the lasers fabricated at that time exhibited an extremely low efficiency due to high optical and electrical losses. In fact, until the mid-1960s, only bulk materials were used in semiconductor devices, which were functionalized by introducing a doping profile. At the time, pioneers like Alferov and Herbert Kroemer independently considered the hypothesis of building heterostructures, consisting

of layers of different semiconductor materials (Alferov, 2001). The classic heterostructure example consists of a lower bandgap layer surrounded by a higher bandgap semiconductor material. Such design results in electronic and optical confinement, because a higher bandgap semiconductor also exhibits a higher refractive index. The enhanced confinement improved notably the operational characteristics of laser diodes, in particular the threshold current density, which decreased by two orders of magnitude. But another revolution was about to come when it was realized that the confinement of electrons in lower dimensional semiconductor structures translated into completely new optoelectronic properties, when compared to bulk semiconductors. And how small should this confinement be? In order to answer this question, let us recall the concept of the de Broglie wavelength of thermalized electrons, λB:

λB =

h = p

h 2m* E

where h is the Planck’s constant p is the electron momentum m* is the electron effective mass E is the energy In the case of III–V compound semiconductors, λB is typically of the order of tens of nanometers (Saleh and Teich, 1991). If one of the dimensions of a semiconductor is comparable or less than λB, the electrons will be strongly confined in one dimension, while moving freely in the remaining two dimensions—this is the case of a quantum well (QW). A quantum wire is a one-dimensional confined structure, while a QD is confined in all the three dimensions. QDs are thus tiny clusters of semiconductor material with dimensions of only a few nanometers, surrounded by a semiconductor matrix that has a higher bandgap. The spatial confinement of the carriers in lower dimensional semiconductors leads to dramatically different energy–momentum relations in the directions of confinement, which results in completely new density of states, when compared to the bulk case, as depicted in Figure 41.5. As dimensionality decreases, the density of states is no longer continuous or quasi-continuous but becomes quantized. In the case of QDs, the charge carriers occupy only a restricted set of energy levels rather like the electrons in an atom, and for this reason, QDs are sometimes referred to as “artificial atoms.” For a given energy range, the number of carriers necessary to fill out these states reduces substantially as the dimensionality decreases, which implies that it becomes easier to achieve transparency and inversion of population—with the resulting reduction of threshold current density. In fact, this reduction has been quite spectacular over the years, with sudden jumps whenever the dimensionality is decreased (Alferov, 2001).

41-6

(a) EC

E

(b) EC

E

(c) EC

D(E)

D(E)

D(E)

D(E)


E

(d) EC

E

FIGURE 41.5â•… Schematic structures of bulk and low-dimensional semiconductors and corresponding density of states. The density of states in different confinement configurations: (a) bulk; (b) quantum well; (c) quantum wire; and (d) quantum dot.

41.3.2â•‡ Quantum Dots: Materials and Growth The group of QD materials that has shown particular promise is based on III–V QDs epitaxially grown on a semiconductor substrate. For instance, InGaAs/InAs QDs on a GaAs substrate emit in the 1–1.3â•›μm wavelength range, which could be extended to 1.55â•›μm. Alternatively, InGaAs/InAs QDs can be grown on an InP substrate that covers emission in the 1.4–1.9â•›μm wavelength range (Ustinov et al., 2003). The remarkable achievements in QD epitaxial growth have enabled the fabrication of QD lasers, amplifiers, and saturable absorbers offering excellent performance characteristics. To date, the most promising results have been achieved using the spontaneous formation of three-dimensional islands during strained layer epitaxial growth in a process known as the Stranski–Krastanow mechanism (Goldstein et al., 1985; Ustinov et al., 2003). In this process, when a film A is epitaxially grown over a substrate B, the initial growth occurs layer by layer, but beyond a certain critical thickness, three-dimensional islands start to form—the quantum dots. A continuous film lies underneath the dots, and is called the wetting layer. The most important condition in this technique is that the lattice constant of the deposited material is larger than the one of the substrate. This is the case of an InAs film (lattice constant of 6.06 Å) on a GaAs substrate (lattice constant of 5.64 Å), for example. In spite of being an extremely complex process, the Stranski– Krastanow mode is now widely used in the self-assembly of QDs. An advantage of this technique is that films can be grown using the well-known techniques of molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD), and therefore the science of QDs growth has benefited immensely from all the previous knowledge gained with this technology. These are also good news for commercialization, because manufacturers do not have to invest in new epitaxy equipment to fabricate these structures. Due to the statistical fluctuations occurring during growth, there is a distribution in dot size, height, and composition but, at the moment, epitaxy techniques have evolved to such an extent

that the amount of fluctuations can be reasonably controlled, and can be as small as a few percent. If the dots are grown on a plane surface, their lateral positions will be random. An example of such structure is shown in Figure 41.6. In the self-assembly process, there is no standard way of arranging the dots in a planar ordered way, unless they are encouraged to grow at particular positions in a pre-patterned substrate. At present, the densities of QDs lie typically between 109 cm−2 and 1011 cm−2. The sparse distribution of QDs results in a low value of optical gain. Thus, the levels of gain and optical confinement provided by a single layer of QDs may not be enough for the optimal performance of a laser. In order to circumvent this problem, QDs can also be grown in stacks, which allows an increase in the modal gain without increasing the internal optical mode loss (Smowton et al., 2001), where the various layers are usually separated by GaAs barriers. The GaAs separators are responsible for transmitting the tensile strain from layer to layer, inducing the formation of ordered arrays of QDs aligned on top of each other. Further optical confinement is enabled through the cladding of such arrays within layers of higher refractive index and bandgap energy, therefore forming a heterostructure.

25 nm

100 nm (a)

(b)

FIGURE 41.6â•… Photographs of an InGaAs quantum dots grown on GaAs substrate: (a) A TEM image of a single sheet of quantum dots. (b) A TEM image of a cross section of an 8-layer thick stack of quantum dots in GaAs layers.

41-7


41.3.3â•‡ Broad Gain Bandwidth A QD laser was proposed in 1976 (Dingle and Henry, 1976) and the first theoretical treatment was published in 1982 (Arakawa and Sakaki, 1982). The main motivation was to conceive a design for a low-threshold, single-frequency, and temperature-insensitive laser, owing to the discrete nature of the density of states. In fact, practical devices exhibit the predicted outstandingly low thresholds (Kovsh et al., 2004; Liu et al., 2005), but the spectral bandwidths of such lasers are significantly broader than those of conventional QW lasers (Rafailov et al., 2007). This results from the self-organized growth of QDs, leading to a Gaussian distribution of dot sizes, with a corresponding Gaussian distribution of emission frequencies. Additionally, lattice strain may vary across the wafer, thus further affecting the energy levels in the quantum dots. These effects lead to the inhomogeneous broadening of the gain—a useful phenomenon in the context of ultrafast applications, because a very wide bandwidth is available for the generation, propagation, and amplification of ultrashort pulses. The effects of inhomogeneous broadening on the density of states are schematically illustrated in Figure 41.7. However, it is important to stress that a highly inhomogeneously broadened gain also encompasses a number of disadvantages, because it partially defeats the purpose of a reduced dimensionality, by broadening the density of states. Indeed, the fluctuation in the size of the QDs has the effect of increasing the transparency current and reducing the modal and differential gain (Qasaimeh, 2003; Dery and Eisenstein, 2005). Therefore, much effort has been put into improving the dots uniformization by engineering the growth and post-growth processes (Ustinov et al., 2003). The extremely broad bandwidth available in QD mode-locked lasers offers potential for generating sub-100â•›fs pulses provided all of the bandwidth can be engaged coherently and dispersion effects suitably minimized. EES

D(E)

EGS

(a)

EC

E EES

D(E)

EGS

(b)

EC

E

FIGURE 41.7â•… Schematic morphology and density of states for charge carriers in (a) an ideal quantum-dot system and (b) a real quantum-dot system, where inhomogeneous broadening is illustrated.

CB

ES GS

(a)

(b)

(c)

VB

FIGURE 41.8â•… Schematic of the energy levels in a QD material (a), and radiative transitions via GS—ground state (b) and ES—excited state (c). CB—Conduction band; VB—valence band.

Indeed, it has been shown that there is usually some gain in narrowing/filtering effects in mode-locked QW lasers (Delfyett et al., 1998). With the inhomogeneously broadened gain bandwidth exhibited by QDs, there is support for more bandwidth and this can oppose the effect of pulse broadening that may arise from spectral narrowing. Additionally, due to the particular nature of QD lasers, many possibilities open up in respect of the exploitation of ground-state (GS) and excited-state (ES) bands,* as schematically represented in Figure 41.8. Such versatility has been successfully exploited in a multiple-wavelength-band switchable mode locking (Cataluna et al., 2006c). On the other hand, the interplay between GS and ES can be deployed in novel mode-locking regimes (Cataluna et al., 2006a). Using an external cavity, it is possible to set up tunable mode-locked sources that can operate in the wavelength range that extends from the GS to the ES transition bands (Kim et al., 2006a).

41.3.4â•‡ Ultrafast Carrier Dynamics In the initial studies of QD materials, it was thought that their carrier dynamics would be significantly slower than those in QW materials due to a phonon bottleneck effect (Mukai et al., 1996). Interestingly, experiments have demonstrated quite the opposite. As a consequence of access to a number of recombination paths for the carriers, QD structures exhibit ultrafast recovery both under absorption and gain conditions (Borri et al., 2006). In two evaluations, the absorber dynamics of surface and waveguided QD structures were investigated by using a pump-probe technique (Borri et al., 2000; Rafailov et al., 2004b). This showed the existence of at least two distinct time constants for the recovery of the absorption. A fast recovery of around 1â•›ps is followed by a slower recovery process that extends over 100â•›ps (Rafailov et al., 2004b). More recently, sub-picosecond carrier recovery was measured directly in a QD absorption modulator when a reverse bias was applied (Malins et al., 2006). Absorption recovery times ranged from 62â•›ps down to 700â•›fs and showed a decrease by nearly two orders of magnitude when the reverse bias applied to the structure was changed from 0â•›V to −10â•›V. This important observation provides significant promise for ultrafast modulators that can operate * Ground and excited states are also available in quantum wells. However, the δ-like density of states associated with quantum dots enables an easier access to the ES, owing to the faster saturation of the GS in quantum dots.

41-8


above 1â•›THz and for the optimization of saturable absorbers used for the passive mode locking of semiconductor lasers at high repetition rates, where the absorption recovery should occur within the round-trip time of the cavity. Crucially, the shaping mechanism of the fast absorption recovery also enhances the shortening of the mode-locked pulses, and thus QD lasers have the potential for generating shorter pulses than their QW counterparts.

41.3.5â•‡ Low Absorption Saturation Fluence QD-based saturable absorbers exhibit lower saturation fluence than QW-based materials due to their delta-like density of states. For example, in a QD one electron is enough to achieve Â�transparency and two to achieve inversion. This characteristic facilitates the self-starting of mode locking at modest pulse Â�energies. This feature is particularly important in high-repetition-rate lasers where the optical energy available in each pulse is small. Indeed, it has also been observed that the saturation power is at least 2–5 times smaller for a QD saturable absorber than for a QW-based counterpart when integrated in a monolithic mode-locked laser (Thompson et al., 2004a). In this paper, the authors pointed out that saturation would further depend on the density of dots, reverse bias, and inhomogeneous broadening.

41.3.6â•‡Low Threshold Current and Low Temperature Sensitivity As devices, QD diode lasers have the advantage of requiring a very low threshold current to initiate lasing (Ustinov et al., 2003). This attribute applies also to operation in the modelocking regime, because most QD lasers exhibit mode-locked operation right from the threshold of laser emission. (Bistability between the non-lasing state and the onset of lasing/mode locking might be present, as has been shown experimentally (Huang et al., 2001; Thompson et al., 2006b) and numerically (Viktorov et al., 2006).) A low threshold current is clearly advantageous because this can represent a device that is compatible as an efficient and compact source of ultrashort pulses where the demand for electrical power can be very low. Furthermore, having a low threshold avoids the need for higher carrier densities for pumping the laser and this implies less amplified spontaneous emission and reduced optical noise in the generated pulse sequences. Due to the discrete nature of their density of states, QD lasers also exhibit low-temperature sensitivity (Mikhrin et al., 2005), making them excellent candidates for applications where resilience to temperature effects is important.

41.3.7â•‡ Low Linewidth Enhancement Factor One of the main motivations for the enthusiastic investigation of QD materials in the last few years has been the theoretically predicted potential for very low values of LEF, owing to the symmetry of the gain associated with QD structures. The possibility of a low LEF is very attractive for a number of performance aspects, such as lower frequency chirp in directly modulated lasers,

lower sensitivity to optical feedback effects, and suppressed beam filamentation. The potential of a lower effect of SPM in QD lasers also held a promise for the generation of transformlimited pulses. However, disparate reports have been published in the last 3 years, with some reports of LEF values of nearly zero (Newell et al., 1999), and others with values of LEF similar (Ukhanov et al., 2004) or significantly higher than in QW structures (Dagens et al., 2005). Ultimately, the LEF is a characteristic that is highly dependent on the operation conditions of the laser, and as such, its meaning always needs to be contextualized for a set of particular conditions. This is a topic that is currently under intense investigation.

41.4â•‡Mode-Locked Quantum-Dot Lasers: State of the Art 41.4.1â•… Pulse Duration The first demonstration of a QD mode-locked laser was reported in 2001, with pulse durations of ∼17â•›ps at 1.3â•›μm and repetition rate of 7.4â•›GHz, using passive mode locking (Huang et al., 2001). Hybrid mode locking at the same wavelength was demonstrated in 2003 by the Cambridge University group (Thompson et al., 2003); they reported an upper limit estimation of 14.2â•›ps for the shortest pulses measured at a repetition rate of 10â•›GHz. Later in 2004, the same group demonstrated Fourier-transform-limited 10â•›ps pulses at 18â•›GHz repetition rate, using passive mode locking (Thompson et al., 2004b). In 2004, we demonstrated the generation of sub-picosecond pulses directly from a QD laser where the shortest pulse durations were measured to be 390â•›fs, without any form of supplementary pulse compression (Rafailov et al., 2004a, 2005). These pulses were generated by a two-section passively mode-locked QD laser and this was the first time that subpicosecond pulses were generated directly from such a monolithic laser. The generation of sub-picosecond pulses was reported later by several groups (Laemmlin et al., 2006; Thompson et al., 2006b). In one of these reports (in 2006), Thompson and coworkers demonstrated the generation of pulses as short as 790â•›fs, by using a flared waveguide configuration in a two-section QD laser (Thompson et al., 2006b). Because the beam mode size in the saturable absorber section was much smaller than that in the gain section, the ratio of saturation fluences in the absorber and gain sections was increased. This enhanced the pulse formation mechanisms and allowed for better pulse shaping and shortening. There has also been much effort in designing QD-based mode-locked sources that could be deployed in the 1.55â•›μm band (Lelarge et al., 2007). Ultra-short-pulse generation has been achieved from single-section lasers based either on InAs QDs (Renaudier et al., 2005) or quantum dashes (Gosset et al., 2006) grown on an InP substrate. These authors have suggested that there are no fundamental differences between the QDs and dashes in the context of mode-locked laser sources.

41-9


41.4.2â•‡Toward Higher Pulse Repetition Rates To achieve higher repetition rates in mode-locked lasers, it is necessary to decrease the cavity length. This poses a significant challenge to QD lasers because of their lower gain and the operation in short cavities may shift the emission to the ES band (Markus et al., 2003). To avoid this problem, a higher number of QD layers should be deployed in the active region. Using this simple approach, the highest repetition rate directly generated from a passively mode-locked QD two-section laser was 80â•›GHz (Laemmlin et al., 2006), when a 15-layer structure was used. Another method to boost the repetition rate of mode-locked lasers is to use colliding pulse mode locking. This technique is similar to passive mode locking, but the saturable absorber region is placed at the precise center of the gain section. Two counter-propagating pulses from each outer gain section therefore meet in the saturable absorber region, bleaching it much more efficiently than if just one pulse was present. This process can also result in shorter and more stable pulses. Owing to the device geometry, mode locking is achieved at the second harmonic of the fundamental (round-trip) frequency, and the pulse repetition rate is doubled. A variation of colliding pulse mode locking is harmonic mode locking, where more than two pulses circulate in the cavity, the number being equal to the harmonic. Collidingpulse mode-locking was first demonstrated for QD lasers in 2005 (Thompson et al., 2005), resulting in a modest repetition rate of 20â•›GHz. Harmonic mode locking has also been demonstrated with repetition rates of approximately 40, 80, 120, and 240â•›GHz (Rae et al., 2006).

41.4.3â•‡Temperature Resilience of Mode-Locked QD Lasers Due to their delta-function-like density of states, QDs offer great potential for designing temperature-resilient devices. If their high-speed performance is also proven to be resilient to temperature, QD lasers can become the next generation of sources for ultrafast optical telecoms and datacoms, because the constraint of using thermoelectric coolers can be avoided, thus decreasing cost and complexity. In this context, we have demonstrated stable passive mode-locked operation of a two-section QD laser over an extended temperature range (from 20°C to 80°C) at relatively high output average powers (Cataluna et al., 2006b). Additionally, to meet the requirements for high-speed communications, it is important to investigate the temperature dependence of the pulse duration. For instance, in communication systems with transmission rates of 40â•›Gb/s or more, the temporal interval between pulses is less than 25â•›p s and so the duration of the optical pulses should be well below this value at any operating temperature. We have shown that the pulse duration and the spectral width decrease significantly as the temperature is increased up to 70°C (Cataluna et al., 2007). The combination of all these effects resulted in a sevenfold

decrease of the time-bandwidth product (the pulses were still highly chirped due to the strong SPM and dispersion effects in the semiconductor material). To account for the decrease in pulse duration with temperature, a model for mode locking in QD lasers was used. It was found that the pulse durations are determined principally by the escape rate of the carriers in the absorber section, which lead to a decrease of absorber recovery time with increasing temperature, thus inducing a decrease in the pulse durations. This has been verified recently using ultrafast spectroscopy to probe the absorber recovery time as a function of temperature (Malins et al., 2007).

41.4.4â•‡Mode Locking Involving Excited-State Transitions It has been observed that laser emission in QD lasers can access the transitions in GS, ES or both (Markus et al., 2003), as represented in Figure 41.8. Furthermore, sub-picosecond gain recovery has been demonstrated for both GS and ES transitions in electrically pumped QD amplifiers (Schneider et al., 2005). In this reported work, the LEF was shown to decrease significantly for wavelengths below the GS transition, even becoming negative at ES thereby implying a potential for chirp-free operation for the range of wavelengths involved. Laser emission in the ES is also characterized by a higher differential gain than GS, with associated benefits for ultrafast QD lasers. We have demonstrated an optical gain-switched QD laser, where pulses were generated from both GS and ES, and where the ES pulses were shorter than those generated by GS alone (Rafailov et al., 2006). The potential for shorter and chirp-free pulses from ES transitions motivated us to investigate the mode-locked operation of QD lasers in this band. We demonstrated, for the first time, passive mode locking via GS (1260â•›n m) or ES (1190â•›n m) in a QD laser, at repetition frequencies of 21 and 20.5â•›GHz, respectively (Cataluna et al., 2006c). The switch between these two states in the mode-locking regime was easily achieved by changing the electrical biasing conditions, thus providing full control of the operating spectral band. It is important to stress that the average power in both operating modes was relatively high and exceeded 25â•›mW. In the range of bias conditions explored in this study, the shortest pulse duration measured for ES transitions was ∼7â•›ps, where the spectral bandwidth was 5.5â•›n m, at an output power of 23â•›mW. These pulse durations are similar to those generated by GS mode locking at the same power level. Although pulse durations from ES spectral band have been below expectations so far, it is our opinion that exploitation of the ES transitions—a unique feature of QD lasers—can lead to a new generation of high-speed sources, where mode locking involves electrically switchable GS or ES transitions that are spectrally distinct. This could enable a range of applications extending from time-domain spectroscopy, through to optical interconnects, wavelength-division multiplexing, and ultrafast optical processing.

41-10


41.5â•‡ Summary and Outlook 41.5.1â•‡ Critical Discussion In this chapter, we have presented the physics and reported on the progress of ultrafast laser diodes based on QD materials. The results presented in the literature show that monolithic passively mode-locked QD lasers can currently surpass the performance of similar QW lasers in terms of pulse duration (Rafailov et al., 2005, Thompson et al., 2006b). There are other particular features where QD lasers have already been shown to have a superior performance, notably in the case of pulse timing jitter where record low values have been reported (Choi et al., 2006, Thompson et al., 2006a). We strongly believe that the appeal of QD lasers also resides in the novel functionalities that are distinctive of QDs. These are the exploitation of an ES level as a means to achieve novel modelocking regimes; the temperature resilience offered by the quantized density of states; lower threshold and higher output power levels; and access to the enlarged spectral bandwidths associated with the inhomogeneously broadened gain features. These characteristics are not only useful from an operational point of view, but also provide some insights into a more comprehensive understanding of the underlying physical mechanisms of mode locking in QD lasers.

41.5.2â•‡ Future Perspectives Although there have been many advances in the control of the growth of QD laser having ultra-low threshold current and temperature resilience, it is not yet understood what is the most advantageous QD structure layout to be used in the regime of mode locking. In particular, it is not clear what is the optimum level of inhomogeneous broadening that results in shorter and higher peak power pulses. Therefore, it is relevant to investigate if and how the inhomogeneously broadened spectral modes are engaged coherently in the generation of ultrashort pulses and how that effect could be used to improve the performance of the lasers toward sub-picosecond pulse durations. Exploiting novel QD materials based on p-doped and tunnel injection structures could also bring advantages in minimizing the effect of any deleterious SPM effects in mode-locked lasers. Comparison between theory and experiment of undoped and p-doped lasers has shown how this technique can improve the LEF (Kim and Chuang, 2006b). By tuning the level of doping, lasers can exhibit zero and even negative LEF at low current densities (Alexander et al., 2007). Tunnel injection QD structures can also be of great interest for use in mode-locked lasers, as the injection of cold carriers may bring many benefits to the operation of mode-locked lasers (Delfyett, 2006). The LEF has been recently calculated and has been demonstrated to be much less than that reported for other lasers (Mi and Bhattacharya, 2007). Selective excitation of population in these lasers has been demonstrated, which could lead to a mitigation of the inhomogeneous broadening effects and contribute to a narrower spectrum and the production of transform-limited pulses (Bret and Gires, 1964).

The ES spectral band can also be exploited in tunable lasers using QD materials where the inhomogeneous broadening is controlled so as to maximize the overlap between GSs and ESs. While in cw operation, QD lasers have been demonstrated with tunability ranges up to 200â•›nm (Varangis, 2000), by exploiting the gain available from the GSs and ESs. Combining this tunability with the possibility of generating ultrashort pulses, it will be possible to achieve a new generation of versatile lasers emitting pulses across a wide range of wavelengths—as if we had compressed many different lasers into a single laser! Such disruptive characteristics will offer endless possibilities and enable applications never seen before in science and technology.

Acknowledgments We wish to thank Dr. D. Livshits, Dr. I. Krestnikov, and Dr. A. Kovsh from Innolume GmbH (Dortmund) for helping to prepare samples and for stimulating discussions. This work was supported in part by the European Community’s Seventh Framework Programme FAST-DOT under grant agreement 224338.

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Index A AFM, see Atomic force microscopy Aharonov–Bohm effect, 11-1 Aluminum and gap engineering, 16-9 thru 16-10 Ambegaokar–Barato relation, 16-2 AND and XOR logic gates, 33-10 thru 33-11 Andreev bound states, 7-3 thru 7-4 Andreev reflection, 7-2 Atomic force microscopy (AFM), 6-11 thru 6-12, 35-10 Atomic-scale technology computer-aided design (TCAD), 12-11 thru 12-12 Atomistic models, 40-5

B Bacteriorhodopsin (BR) absorption spectra, 38-1 thru 38-2 basic process activation function realization, 38-5 formal neuron main functions realization, 38-3 medium bleaching, 38-2 thru 38-3 neuro-like element formation, 38-3 thru 38-4 optical radiation fluxes, indirect interaction, 38-3 weighed signals composition function realization, 38-4 thru 38-5 weighing function realization, 38-4 nanostructure complex estimation procedure, 38-10 thru 38-11 functional characteristics, 38-12 thru 38-14 hybrid nanostructures, 38-13 thru 38-17 optical neuronets basic elements, multilayered structures, 38-7 thru 38-8 cyclicity processes, 38-5 thru 38-6 data processing, 38-6 element base parameters, 38-7 light fluxes formation, 38-6 multilayered constructions, 38-7

polymeric film property, 38-9 thru 38-10 polymeric film requirements, 38-8 polymeric mixture preparation, 38-8 thru 38-9 suspension preparation, 38-8 photocycle, 38-1 thru 38-2 Berry’s phase interference definition, 9-5 electron in constant magnetic field, 9-6 path integrals, 9-6 thru 9-7 in single-molecule magnet transistors, 9-11 thru 9-12 Biomaterial shaping, 18-6 thru 18-7 Biomimetics, photonic nanostructures antireflector engineering, 31-1, 31-3 cell culture, 31-3, 31-5 diatoms and coccolithophores frustule, 31-5 Nitzschia frustulum, 31-6 thru 31-7 photochemical-etching technique, 31-5 photoluminescence (PL) emission, 31-5 thru 31-6 photonic device, 31-5 thru 31-6 porous glass, 31-7 eukaryote cell, 31-8 iridescent device engineering Aphrodita sea mouse, 31-3 2D christmas tree structure, 31-3 thru 31-4 metallic colored effect, 31-1 thru 31-2 Morpho butterfly scale, 31-2 thru 31-4 optical characteristics, 31-2 photonic crystal, 31-2 iridoviruses, 31-7 optical reflector types, 31-1 thru 31-2 silica deposition vesicle (SDV), 31-7 trans-Golgi-derived vesicle, 31-7 Biomolecular neuronet devices, see Bacteriorhodopsin Bose–Einstein distribution function, 10-8 Bowtie plasmonic nanolaser, 39-9 Bragg mirrors, 39-3 Bragg MOKE technique, 4-8 thru 4-9 Break junction experiment, 12-9 Bucky shuttle memory element, 3-16 thru 3-17 Buttiker model, 10-5 thru 10-6

C Cantor’s middle-excluded set, 17-13 thru 17-14 Capacitance spectroscopy capacitance transient spectroscopy (see Deep level transient spectroscopy) depletion region depletion capacitance, 2-13 thru 2-14 p–n junction, 2-12 Schottky contact, 2-11 thru 2-12 width of, 2-13 Carbon nanotube (CNT) memory elements CNFET-based memory band structure, 3-3 charge-storage stability, 3-5 thru 3-6 controlling storage nodes, 3-6 thru 3-7 Fuhrer’s device, 3-4 nanotube storage nodes, 3-9 optoelectronic memory, 3-7 Radosavljevic’s device, 3-5 redox active molecules, 3-7 schematic cross section, 3-2 two-terminal memory devices, 3-7 thru 3-9 electromigration CNT, 3-16 thru 3-17 NEMS-based memory capacitive force response, 3-9 feedback-controlled nanocantilevers, 3-12 thru 3-13 linear bearing nanoswitch, 3-14 thru 3-16 nanorelays, 3-11 thru 3-12 nonvolatile random access memory, 3-10 thru 3-11 properties, 3-9 vertically aligned carbon nanotubes, 3-13 thru 3-14 Carbon nanotube transistors, 12-6 thru 12-7 Carbon nanotube weak link experimental realizations, 7-6 thru 7-7 Fabry–Perot cavity, 7-6 high-frequency irradiation, 7-9 thru 7-10 nanotube quantum dot CNT superconducting transistor, 7-7 thru 7-8 gate dependence, 7-8 thru 7-9

Index-1

Index-2 Carrier–carrier interaction capture and relaxation time, 40-7 Coulomb interaction matrix element, 40-7 Markov approximation, 40-6 quantum-kinetic equation, 40-7 screening effects, 40-7 thru 40-8 Carrier manipulation (up-converter), 33-7 thru 33-8 Carrier–phonon interaction perturbation theory vs. polaron picture, 40-8 thru 40-9 polaron states and kinetics, 40-9 thru 40-10 Cathodoluminescence band structure, types, 21-2 donor–acceptor pair transition, 21-2 exciton, 21-3 recombination process, 21-2 TEM-CL (see Transmission electron microscope-cathodoluminescence) C60 field effect transistors Bell Laboratory misconduct, 15-2 thru 15-3 characterization on SiO2/AlN I–V characteristics, 15-19 sub-threshold region, 15-19 thru 15-20 fabrication of C60 epitaxy, 15-18 thru 15-19 device process, 15-18 performance measurements, 15-19 SiO2/AlN substrates, 15-18 field effect doping, 15-3 operation principles, 15-3 thru 15-4 solid C60 and insulator films interface, 15-3 superconductivity, 15-4 Charge-selective deep level transient spectroscopy, 2-16 Charge trap memories, 2-6 thru 2-7 Chemically amplified resist (CAR), 20-12 Classical Faraday effect, 1-13 CNT field-effect transistor (CNTFET), 12-7 band structure, 3-3 charge-storage stability, 3-5 thru 3-6 Fuhrer’s device, 3-4 optoelectronic memory, 3-7 Radosavljevic’s device, 3-5 schematic cross section, 3-2 storage nodes control, 3-6 thru 3-7 nanotube storage nodes, 3-9 redox active molecules, 3-7 two-terminal memory devices, 3-7 thru 3-9 Coherent electron transport electrochemical potential expression, 10-3 self-energy parts expression, 10-3 through a carbon chain, 10-4 via HOMO, 10-2 via LUMO, 10-2 Coherent Néel vector tunneling doped antiferromagnetic molecular wheel, 9-10

Index satellite resonance, 9-10 two degenerate classical ground state, 9-9 Computational micromagnetics boundary element (BE) method, 8-7 finite difference (FD) method Ampère field, 8-10 anisotropy field computation, 8-10 exchange field computation, 8-9 exchange length, 8-8 Langevin equation, 8-11 magnetostatic field computation, 8-10 mesh for, 8-8 spatial distribution field, 8-11 thermal field computation, 8-12 vs. FE method, 8-8 finite element (FE) method, 8-7 thru 8-8 pillar vs. point-contact, 8-12 thru 8-15 spin-transfer torque and giant magnetoresistance, 8-12 thru 8-13 Confined metallic systems chemical structure influence, 24-10 thru 24-11 cluster shape influence, 24-8 thru 24-9 dielectric function, 24-6 dipolar approximation, 24-10 thru 24-12 quantum size effects, 24-6 thru 24-8 quasi-static approximation/dipolar, 24-5 thru 24-6 scattering, absorption, and extinction, 24-4 thru 24-5 Conformal deposition and isotropic etching, 17-15 thru 17-17 Contact arc method, 15-5 Control ordering and assembly processes crystal growth process, 18-8 crystallization process, 18-8 full confinement factor, 18-11 thru 18-14 partial confinement factor, 18-9 thru 18-11 semicrystalline polymers morphology, 18-8 Cooper-pair transistor (CPT) aluminum and gap engineering, 16-9 thru 16-10 band structure critical current modulation calculation, 16-5 thru 16-6 effective inductance modulation calculation, 16-6 uncertainty principle at work, 16-5 Coulomb blockade, 16-4 thru 16-5 current switching electrometry quasiparticle poisoning, 16-11 thru 16-12 ramped switching current measurement, 16-11 single shot measurement, 16-12 thru 16-13 switching current measurement, 16-10 electron-beam lithography and two-angle deposition, 16-9 quasiparticle poisoning, 16-6 thru 16-9

single Josephson junction Ambegaokar–Barato relation, 16-2 concerns specific to small junctions, 16-4 field emission micrograph, 16-2 I–V curve, 16-2 thru 16-3 RCSJ model, 16-3 thru 16-4 zero-biased rf electrometry operation beyond 1â•›GHz, 16-14 thru 16-15 quasiparticle poisoning, 16-15 relation to the cooper-pair box, 16-16 rf measurement setup, 16-13 thru 16-14 Coulomb blockade, 16-4 thru 16-5 Coulomb carrier–carrier interaction, 22-6 Crossbar process, 17-2 thru 17-3 C60 superconducting transistor, 7-18 Current switching electrometry quasiparticle poisoning, 16-11 thru 16-12 ramped switching current measurement, 16-11 single shot measurement, 16-12 thru 16-13 switching current measurement, 16-10

D Davydov transformation, 32-12 thru 32-13, 32-16 Deep level transient spectroscopy (DLTS) charge-selective deep level transient spectroscopy, 2-16 measurement principle, 2-14 thru 2-15 rate window and double-boxcar method, 2-15 thru 2-16 Deep ultraviolet (DUV) lithography CaF2 lens elements, 19-11 depth of focus (DOF), 19-9 thru 19-10 double patterning, 19-13 thru 19-14 fused silica glass, 19-10 193â•›nm immersion lithography, 19-11 thru 19-12 numerical aperture, 19-11 optical-projection lithography system, 19-9 photolithography exposure process, 19-9 plot of critical feature size, 19-10 ultimate resolution limits, 19-14 Depletion capacitance, 2-13 thru 2-14 Dipolar approximation expansion coefficients, 24-11 extrinsic size effects, 24-12 incident, scattering, and internal fields, 24-11 quadrupolar mode, 24-12 vector Helmholtz wave equation, 24-10 Dipole nanolaser, 39-9 thru 39-10 Discharge-produced plasma EUV source, 20-9 thru 20-10 Dissipated optical energy transfer CdSe QDs, schematic images, 36-2 thru 36-3

Index-3

Index dispersed substrate area, 36-3 thru 36-4 dynamic property of, 36-3 experimental results, 36-4 thru 36-5 exponential decay curve functions, 36-4 temperature dependent micro- (PL) spectroscopy, 36-2 thru 36-3 Dissipation-controlled nanophotonic devices carrier manipulation (up-converter), 33-7 thru 33-8 energy transfer between two quantum dots density matrix formalism, 33-2 thru 33-3 energy diagram, 33-2 excitation creation, 33-5 thru 33-6 exciton dynamics, 33-3 thru 33-4 resonant coupling, 33-4 nanophotonic switch energy diagram, 33-6 exciton population, 33-6 thru 33-7 ON- to OFF-state, 33-7 switching operation, 33-6 spatial symmetry, 33-8 thru 33-13 Dissipative electron transport contact time, 10-9 electron conductance vs. voltage, 10-10 thru 10-11 junction power loss, 10-11 phonon bath, 10-9 thru 10-10 through large DNA molecules, 10-10 Domain theory, 8-2 Donnan effect, 11-3 Double-boxcar method, 2-15 thru 2-16 Double exposure method, 19-24 thru 19-25 Double patterning, 19-13 thru 19-14 Drude expression, 26-4 Drude model, 24-3 thru 24-4 Dynamic random access memory (DRAM) cell array structure, 2-2 endurance, 2-3 planar cell layout, 2-2 thru 2-3 vs. Flash memory, 2-4 Dyson equation, 40-9

E Edge emitter laser diodes microcavity lasers, 40-3 thru 40-4 optical gain, 40-2 types, 40-2 Effective-mass approach, 40-5 Effective near-field optical interaction effective interaction Hamiltonian operator, 32-8 effective sample–probe-tip interaction, 32-10 electric displacement operator, 32-9 electromagnetic field correlations and intermolecular interactions, 32-6 thru 32-7 exciton polariton, 32-9 irrelevant macroscopic subsystem, 32-7 thru 32-8

nanomaterial system, 32-7 photon–electron–phonon interacting system, 32-7 P space and Q space, 32-8 Yukawa functions, 32-11 Electromigration CNT, 3-16 thru 3-17 Electron-beam lithography cost of ownership, 19-17 overlay, 19-16 thru 19-17 resolution, 19-14 thru 19-15 throughput, 19-15 thru 19-16 Electron–phonon interactions, 10-2, 10-7, 10-10 Electron-transfer reactions, 10-13 thru 10-15 Envelope-function approximation, 40-5 EPR entanglement, 1-16 EUV imaging system development, 20-2 thru 20-3 EUVL, see Extreme ultraviolet lithography Excitation energy transfer, 35-1 thru 35-3 Exciton–phonon–polariton (EPP) model., 35-7 Extended boundary condition method (EBCM), 30-2 Extreme ultraviolet lithography (EUVL) alpha demo tool (ADT), 20-3 aspheric mirrors, 20-4 components, 20-1 defect-free blanks, 20-14 DPP EUV source-collector module, 20-9 thru 20-10 EUV device integration, 20-8 thru 20-9 2008 EUV focus areas, 20-9 EUV masks blanks, 20-10 thru 20-11 defectivity, 20-11 thru 20-12 round-trip shipment, 20-13 SEMI P27-1102 specifications, 20-10 S-pod, 20-13 EUV resists, 20-12 exposure tools ASML, 20-5 thru 20-6 CAD model, 20-6 thru 20-7 interference lithography, 20-7 thru 20-8 scanning electron microscope (SEM) image, 20-7 SEMATECH Berkeley MET, 20-6 thru 20-7 six-mirror imaging system, 20-6 high-power pulsed laser, 20-1 immersion lithography, 20-15 intermediate focus, definition, 20-9 LPP EUV source-collector module, 20-10 multilayer (ML) reflective coating, 20-1, 20-3 thru 20-4 normal-incidence reflective optics, 20-2 optics and mask contamination, 20-13 thru 20-14 phase-shifting point-diffraction interferometer (PS/PDI), 20-5 projection optics, 20-2 thru 20-3 Rayleigh equation, 20-14 thru 20-15

F Fabry–Perot cavity, 7-6 Feedback-controlled nanocantilevers, 3-12 thru 3-13 Fermi Golden rule, 21-6 Ferroelectricity fundamentals ferroelectric correlation length, 6-4 ferroelectric hysteresis loops, 6-4 perovskite oxide BaTiO3 unit cell, 6-3 phenomenological definition, 6-2 second-order ferroelectric phase transition, 6-3 thermodynamic theory, 6-3 Ferroelectric RAM (FeRAM), 2-4 thru 2-5 Ferromagnetic islands experimental techniques, 4-5 ferromagnetic domains, 4-2 thru 4-3 ferromagnetic dots longitudinal Bragg MOKE, 4-8 thru 4-9 micromagnetic simulation results, 4-7 thru 4-8 nucleation and annihilation field, 4-7 permalloy dots array, 4-7 ferromagnetic rings, 4-9 ferromagnetic spirals, 4-10 thru 4-11 ferromagnetism, 4-1 thru 4-2 hysteresis loops, 4-3 magnetostatic interactions honeycomb structure, 4-13 onion state, 4-12 open frame system, 4-11 Py dipole arrays, 4-12 thru 4-13 spin ice state, 4-11 symmetrical stable remanent states, 4-12 micromagnetic simulations, 4-3 thru 4-5 noncircular rings, 4-9 thru 4-10 rectangular and elliptical islands, 4-6 thru 4-7 sub-micrometer-sized island preparation, 4-5 Ferromagnetic molecular magnets, 9-1 Ferromagnetic rings, 4-9 Ferromagnetic spirals, 4-10 thru 4-11 Fick’s first law, 11-3 thru 11-4 Figure of merit (FOM), 35-5 Finite difference (FD) method, LLGS Ampère field, 8-10 anisotropy field computation, 8-10 exchange field computation, 8-9 exchange length, 8-8 Langevin equation, 8-11 magnetostatic field computation, 8-10 mesh for, 8-8 spatial distribution field, 8-11 thermal field computation, 8-12 vs. FE method, 8-8 Flash memories electron per cell vs. feature size, 2-4 schematic band structure, 2-3 vs. DRAM, 2-4

Index-4 Floating nano-dots, 5-2 Fluctuation-dissipation theorem, 8-11 Fokker–Planck equation, 8-11 FOM, see Figure of merit Förster type interaction, 33-3 Fourier transform, 32-2 Fowler–Nordheim tunneling, 6-11 Fractal nanotechnology fractals in nature, 17-12 thru 17-13 producing nanoscale fractals via SnPT×, 17-13 thru 17-14 Franck–Condon factor, 10-13, 10-15 Fuhrer’s device, 3-4 Fullerene (C 60) discovery of, 15-4 thru 15-5 FET Bell Laboratory misconduct, 15-2 thru 15-3 characterization on SiO2/AlN, 15-19 thru 15-20 fabrication of, 15-18 thru 15-19 field effect doping, 15-3 operation principles, 15-3 thru 15-4 solid C60 and insulator films interface, 15-3 superconductivity, 15-4 MBE, 15-10 thru 15-11 micro-clusters, 15-1 properties of, 15-5 thru 15-7 solid crystals, 15-1 thru 15-2 absorption and luminescence spectra, 15-8 crystal structure, 15-7 electronic structures, 15-7 thru 15-8 optical measurements results, 15-9 photo-conducting spectra, 15-10 photoluminescence spectrum, 15-8 structural phase transition, 15-7 third nonlinear susceptibility, 15-9 van der Waals interaction, 15-7 synthesis of, 15-5 van der Waals epitaxy (see van der Waals epitaxy) Fullerene-based superconducting weak links, 7-17 thru 7-18 Full-width at half-maximum (FWHM), 35-8 Fully depleted silicon-on-insulator (FDSOI) devices, 12-3

G Gas evaporation method, 15-5 Giant magnetoresistance (GMR), 8-12 thru 8-13, 13-5 Graphene-based superconducting weak links proximity effect, 7-15 thru 7-17 SQUIDs devices, 7-17 Graphene ribbon nanotransistors, 12-7 thru 12-8 Grover algorithm, 1-9 thru 1-12

Index

H Half pitch scaling, 19-4 Hamiltonian model, 32-11 thru 32-12 Heisenberg’s uncertainty principle, 32-1 Hellmann–Feynman theorem, 27-5 Helmholtz equation, 32-4 Hierarchical memory retrieval, 34-7 thru 34-8 Horseshoe optical nanoantenna., 39-9 Hot-carrier diode, 13-1 Hybrid nanostructures absorption spectrum, 38-14 thru 38-15 AFM characterization, gold nanoparticle, 38-14 thru 38-15 borohydride method, 38-13 bridge molecules, 38-15 dialyze, 38-14 factual spectrum, 38-17 gold nanoparticles spectrum, 38-14 schematic composition, 38-13 spectral characteristics, 38-17 water colloidal suspension, silver nanoparticles, 38-15 thru 38-16

I III–V semiconductor quantum dots self-organized quantum dots based nanomemories, 2-10 carrier emission process, 2-9 electron and hole states, 2-9 fabrication, 2-8 semiconductor material hetereostructure, 2-7 thru 2-8 193â•›nm Immersion lithography DUV lithography, 19-11 thru 19-12 photoresist technology, 19-8 Imprint lithography (IL), 17-3 thru 17-4 Incoherent Zener tunneling, 9-7 thru 9-8 Inelastic electron transport Buttiker model, 10-5 thru 10-6 coherent transport curves, 10-4 thru 10-5 electrochemical potential expression, 10-3 self-energy parts expression, 10-3 through a carbon chain, 10-4 via HOMO, 10-2 via LUMO, 10-2 dissipative transport contact time, 10-9 electron conductance vs. voltage, 10-10 thru 10-11 junction power loss, 10-11 phonon bath, 10-9 thru 10-10 through large DNA molecules, 10-10 molecular junction conductance and longrange reactions, 10-13 thru 10-15 polaron effects, 10-11 thru 10-13 vibration-induced inelastic effects electron transmission vs. energy, 10-7 electron–vibron interaction, 10-6 thru 10-07

inelastic electron tunneling spectrum, 10-8 thru 10-9 polaronic shift, 10-7 Inelastic electron tunneling spectrum (IETS), 10-8 thru 10-9 InGaAs/GaAs quantum dots carrier storage, 2-17 with additional AlGaAs barrier, 2-18 thru 2-20 Intensity-dependent refractive index (IDRI), 27-2 Interband transitions, 40-11 Intrinsic size effects dielectric function, 24-6 quantum size effects absorption cross section vs. energy, 24-7 jellium model, 24-7 Mie theory, 24-6 plasmon angular frequency, 24-8 Iridoviruses, 31-7

J Jaynes–Cummings model, 1-6 thru 1-7 Josephson weak link, 7-2 thru 7-3

K Kohn–Sham equation, 24-7 Kohn–Sham matrix, 27-4 Kuhn segment, 18-11

L Landau–Ginzburg–Devonshire theory, 6-1 thru 6-2 Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation general form, 8-6 solution based on FD method Ampère field, 8-10 anisotropy field computation, 8-10 exchange field computation, 8-9 exchange length, 8-8 Langevin equation, 8-11 magnetostatic field computation, 8-10 mesh for, 8-8 spatial distribution field, 8-11 thermal field computation, 8-12 vs. FE method, 8-8 Landau–Zener model, 9-7 Langevin equation, 8-11 Laser emission properties microscopic generalization, semiconductor effects, 40-14 thru 40-15 photon statistics modifications, 40-15 thru 40-16 rate equations description, 40-13 thru 40-14 Laser-produced plasma EUV source, 20-10

Index-5

Index Laser-vaporization cluster beam mass spectroscopy, 15-4 Lawrence Berkeley National Laboratory (LBNL), 20-3 Lift-off technique, 6-7 Linear bearing nanoswitch, 3-14 thru 3-16 Liouville equation, 33-3 Localized surface plasmons (LSPs), 28-5 Loss–DiVincenzo proposal RKKY interaction, 1-8 thru 1-9 using quantum dots and electric gates, 1-7 thru 1-8 Low-dimensional sp2 carbon structures, 7-4 thru 7-5 Low-energy cluster beam deposition (LECBD), 24-13 Low linewidth enhancement factor, 41-8

M Macroscopic polarization, 40-11 Magnetically ordered materials, 8-1 thru 8-2 Magnetic-metallothionein (mMT), 11-2 Magnetic protein folding Donnan effect, 11-3 first-order-like state transition model, 11-3 of Mn,Cd-MT, 11-2 protein folding phase diagram, 11-4 quasi-static thermal equilibrium dialysis, 11-5 thru 11-6 self-assemble, 11-5 three well model, 11-4 Magnetocrystalline anisotropy, 4-4 Magnetoresistive RAM, 2-5 Mask Blank Development Center (MBDC), 20-11 Maxwell–Garnett theory, 24-14 MBE, see Molecular beam epitaxy Metal and semiconductor nanowire dipole orientation, 26-8 frequency dispersion, 26-10 Gaussian properties, 26-11 linear optical phenomena, 26-9 nonlinear polarization, 26-8 plasmon resonance, 26-10 radiation pattern, 26-9 second harmonic generation (SHG), 26-7 thru 26-8 SHG angular distribution, 26-10 Metal clusters and nanoparticles alkali clusters, 24-15 thru 24-16 bimetallic cluster annealed alloy measurement, 24-19 dielectric function, 24-19 gold–silver alloy nanoparticle, 24-19 thru 24-20 low energy ion scattering (LEIS) measurement, 24-20 Ni–Ag cluster optical absorption spectra, 24-20 thru 24-21 bulk metals conduction band, 24-2 thru 24-3 dielectric function, 24-3

Drude dielectric function, 24-4 intraband and interband transition, 24-3 Kramers–Kronig relationship, 24-4 plasma frequency, 24-3 cluster source, 24-2 confined metallic systems, optical properties chemical structure influence, 24-10 thru 24-11 cluster shape influence, 24-8 thru 24-9 dielectric function, 24-6 dipolar approximation, 24-10 thru 24-12 quantum size effects, 24-6 thru 24-8 quasi-static approximation/dipolar, 24-5 thru 24-6 scattering, absorption, and extinction, 24-4 thru 24-5 dielectric confinement, 24-2 electronic shells, 24-1 minute metal spheres, 24-2 nanoparticle synthesis chemical methods, 24-12 thru 24-13 physical methods, 24-13 thru 24-14 noble metal clusters discrete dipole approximation, 24-19 gold clusters, 24-17 jellium model, 24-18 silver clusters, 24-17 SPR, 24-16 thru 24-17 TDLDA, 24-18 optical spectroscopy, 24-14 thru 24-15 single nanoparticle, 24-2 spectroscopic techniques, 24-2 Metallic dot capacitance and coupling coefficient, 5-3 thru 5-4 influence of polarization coupling coefficient analytical expression, 5-6 thru 5-7 dielectric medium potential, 5-7 method of images, 5-4 thru 5-5 numerical implementation invertible matrix, 5-7 vs. classical approximation, 5-7 thru 5-9 semi-analytical expression for capacitance B1 coefficients expression, 5-4 thru 5-5 capacitance matricial expression, 5-6 partial capacitance expression, 5-6 semi-analytical expression for potential, 5-6 Metal nanolayer-base transistor band diagram band banding conditions, 13-1 Co/Si-n Schottky diode characteristics, 13-2 hot-carrier diode, 13-1 MS contacts, 13-1 thru 13-2 TE theory, 13-2

fabrication atomic arrangement, 13-4 thru 13-5 dependency of the current gain, 13-7 epitaxial growth, 13-4 glow discharge-deposition process, 13-4 magnetic multilayers, 13-5 thru 13-6 organic semiconductor as top layer, 13-6 test of dependency, 13-7 J–V characteristic, 13-3 thru 13-4 Metal nanostructures allowed and forbidden second harmonic emission modes laterally limited light beam, 28-10 thru 28-11 plane-wave illumination, 28-9 thru 28-10 emission patterns and light polarization, 28-8 thru 28-9 field enhancements, 28-1 lightning rod effects, 28-1 second harmonic generation nanoparticles, 28-5 thru 28-8 nanosystems, 28-3 thru 28-5 nonlinear optics, 28-2 thru 28-3 single gold nanoparticles, 28-11 thru 28-15 two-photon photo-luminescence (TPPL) yield, 28-15 Metal-semiconductor (MS) junction, 13-1 Microcavity lasers, 40-3 Micromagnetic theory, 8-2 Mie nanolasers, 39-11 Mie theory, 24-6, 24-10, 24-12 Mode-locked quantum-dot lasers advantages, 41-5 thru 41-8 basics, 41-2 excited-state transitions, 41-9 passive mode locking, physics and devices, 41-3 pulse duration, 41-8 pulse repetition rates, 41-9 requirements, 41-3 thru 41-4 self-phase modulation and dispersion, 41-4 thru 41-5 semiconductor lasers, 41-2 temperature resilience, 41-9 Molecular beam epitaxy (MBE), 6-5 thru 6-6 C60, 15-10 thru 15-11 Molecular junction conductance, 10-13 thru 10-15 Molecular magnets preparation magnetic properties, molecular nanostructures, 11-10 thru 11-11 magnetic protein folding Donnan effect, 11-3 first-order-like state transition model, 11-3 of Mn,Cd-MT, 11-2 protein folding phase diagram, 11-4 quasi-static thermal equilibrium dialysis, 11-5 thru 11-6 self-assemble, 11-5 three well model, 11-4

Index-6 nanostructured semiconductor templates direct inkjet and mold imprinting, 11-7 nanopatterned SAMs for 3D fabrication, 11-7 thru 11-8 patterned self-assembly, 11-6 thru 11-7 self-assembling growth patterned magnetic molecules, AFM, 11-8 patterned MT molecules, AFM, 11-9 Si template, SEM image, 11-9 Molecular nanomagnets coherent Néel vector tunneling doped antiferromagnetic molecular wheel, 9-10 satellite resonance, 9-10 two degenerate classical ground state, 9-9 incoherent Zener tunneling in Fe8, 9-7 thru 9-8 phonon-assisted spin tunneling in Mn12 acetate, 9-5 single-molecule magnet transistors, 9-11 thru 9-12 spin tunneling anticrossing of adiabatic eigenvalues, 9-3 Berry’s phase and path integrals (see Berry’s phase interference) generalized master equation, 9-4 Pauli or master equation, 9-3 thru 9-4 Rabi oscillation, 9-3 spin Hamiltonian, 9-2 uniaxial anisotropy, 9-2 Molecular transistors, 7-18 Molecular tunnel junctions, 12-9 thru 12-11 Monte Carlo simulation, 24-20 Moore’s law, 12-1 thru 12-2 Multiferroic tunnel junction, 6-18 Multilayer polymeric structures bacteriorhodopsin suspension preparation, 38-8 basic elements, 38-7 thru 38-8 containing mixture, 38-8 thru 38-9 cyclicity processes, 38-5 thru 38-6 data transmission and processing, 38-6 element base adaptability, 38-6 expected parameters, 38-7 fragments, 38-7 property, 38-9 thru 38-10 requirements, 38-8 Multilayer (ML) reflective coating, 20-1, 20-3 thru 20-4 Multispacer patterning technique abstract technology bodies and surfaces, 17-15 conformal deposition and isotropic etching, 17-15 thru 17-17 directional processes, 17-17 thru 17-18, 17-18 additive route—SnPT+, 17-5 thru 17-7 addressing cross-point, 17-10 thru 17-11 concrete technology, 17-18 thru 17-19

Index crossbar-architecture fabrication steps, 17-2 crossbar comparison, 17-11 thru 17-12 electronics application, 17-12 energetics application, 17-12 fractal nanotechnology fractals in nature, 17-12 thru 17-13 producing nanoscale fractals via SnPT×, 17-13 thru 17-14 multiplicative route—SnPT×, 17-8 thru 17-9 nonlithographic nanowire preparation imprint lithography (IL), 17-3 thru 17-4 spacer patterning technique (SPT), 17-4 thru 17-5 three-terminal molecules, 17-9 thru 17-10

N Nano-dot, see Spherical dot in plate capacitor Nanoelectromechanical systems (NEMS) based memory capacitive force response, 3-9 feedback-controlled nanocantilevers, 3-12 thru 3-13 linear bearing nanoswitch, 3-14 thru 3-16 nanorelays, 3-11 thru 3-12 nonvolatile random access memory, 3-10 thru 3-11 properties, 3-9 vertically aligned carbon nanotubes, 3-13 thru 3-14 Nanoelectronics lithography advanced lithographic processes CD-SAXS, 19-27 CD-scanning electron microscopy (CD-SEM), 19-26 double exposure method, 19-24 thru 19-25 ellipsometry measurements, 19-26 line edge roughness measurement, 19-27 overlay measurements, 19-27 scatterometry measurements, 19-26 spacer double patterning method, 19-25 DUV lithography CaF2 lens elements, 19-11 depth of focus (DOF), 19-9 thru 19-10 double patterning, 19-13 thru 19-14 fused silica glass, 19-10 193â•›nm immersion lithography, 19-11 thru 19-12 numerical aperture, 19-11 optical-projection lithography system, 19-9 photolithography exposure process, 19-9 plot of critical feature size, 19-10 ultimate resolution limits, 19-14

electron-beam lithography cost of ownership, 19-17 overlay, 19-16 thru 19-17 resolution, 19-14 thru 19-15 throughput, 19-15 thru 19-16 NIL defect inspection and dimensional metrology, 19-22 functional materials, 19-19 nanoimprint materials development, 19-21 overlay accuracy and control, 19-23 residual layer control, 19-22 thru 19-23 technology examples, 19-23 thru 19-24 template fabrication and availability, 19-21 thru 19-22 thermal NIL, 19-18 thru 19-19 tool types, 19-19 thru 19-20 transfer printing, 19-19 UV NIL, 19-19 photoresist technology base quencher additives, 19-6 thru 19-7 EUV lithography, 19-8 line and space pattern, 19-4 thru 19-5 material structure advances, 19-7 thru 19-8 193â•›nm immersion lithography, 19-8 photolysis process of photoacid generator, 19-5 thru 19-6 positive and negative tone resist, 19-4 residual swelling fraction, 19-7 resist materials, 19-4 thru 19-5 Nanofibers, light scattering angular distribution, 30-9 cylindrical nanofiber diffraction Bessel function, 30-4 electromagnetic field components, 30-3 Fourier-transformed amplitudes, 30-5 thru 30-6 incident beam polarization, 30-4 thru 30-5 incident wave electric field, 30-3 inverse Fourier transform, 30-4 definition, 30-1 dispersion curves, 30-12 exciton resonance excitation, 30-10 thru 30-11 Fresnel reflection coefficient, 30-2 Hankel functions, 30-1 He-Cd laser, 30-11 ideal surface reflection, 30-8 thru 30-9 inorganic semiconductor materials, 30-1 multiple-scattering linear equation, 30-2 normal modes, 30-6 optical process, 30-1 photoluminescence measurement, 30-12 photomultiplier, 30-11 polarization properties, 30-12 pulsed Gaussian light beam, 30-12 scattered field calculation, 30-6 thru 30-8 scattered light intensity, 30-12 scattering efficiency, 30-2

Index-7

Index T-matrix method, 30-2 total scattered intensity vs. frequency, 30-10 total scattered intensity vs. incidence angle, 30-9 thru 30-10 Nanoimprint lithography (NIL) cavity fill process, 18-4 thru 18-5 control ordering and assembly processes crystal growth process, 18-8 crystallization process, 18-8 full confinement factor, 18-11 thru 18-14 partial confinement factor, 18-9 thru 18-11 semicrystalline polymers morphology, 18-8 defect inspection and dimensional metrology, 19-22 functional materials, 19-19 functional materials nanoshaping antireflective structure creation, 18-6 biomaterial shaping, 18-6 thru 18-7 interface shaping, 18-5 nanopore generation, 18-5 optical grating creation, 18-5 thru 18-6 imprinting resists, 18-4 molds in, 18-4 nanoimprint materials development, 19-21 overlay accuracy and control, 19-23 residual layer control, 19-22 thru 19-23 resolution limit, 18-4 technology examples, 19-23 thru 19-24 template fabrication and availability, 19-21 thru 19-22 thermal NIL, 19-18 thru 19-19 for thermoplastic polymers, 18-3 tool types, 19-19 thru 19-20 transfer printing, 19-19 UV NIL, 19-19 Nanolasers applications, 39-12 definition, 39-5 laser radiation property, 39-1 thru 39-2 nanowire, 39-5 thru 39-8 organic nanolasers, 39-11 overview, 39-2 photonic crystal lasers, 39-10 plasmonic lasers, 39-8 thru 39-10 rate equation approach, 39-4 thru 39-5 scattering lasers, 39-10 thru 39-11 semiconductor lasers, 39-2 thru 39-4 Nanomaterials cathodoluminescence band structure, types, 21-2 donor–acceptor pair transition, 21-2 exciton, 21-3 recombination process, 21-2 TEM-CL (see Transmission electron microscope-cathodoluminescence) kinetic energy, 21-8 thru 21-9 optical spectroscopy global photoluminescence, 22-1

luminescence hole-burning (LHB) spectroscopy, 22-1 luminescence spectra, 22-1 thru 22-2 multiexciton generation, 22-6 thru 22-7 nanoparticle quantum dots, 22-4 thru 22-6 optical properties, 22-1 semiconductor nanoparticles, 22-1 single molecule spectroscopy, 22-2 SWNT (see Single-walled carbon nanotube) photoluminescence (PL), 21-1 thru 21-2 quantum well, 21-9 thru 21-10 quantum wire density of states, 21-10 electron wavefunction, 21-10 hole wave-functions, 21-11 homogeneous electric field, 21-13 intensity and polarization ratio, 21-14 thru 21-15 Luttinger–Kohn Hamiltonian, 21-11 optical absorption, 21-14 photoluminescence excitation (PLE) spectrum, 21-13 polarization anisotropy, 21-11 quantum confinement effect, 21-12 thru 21-13 transition matrix element, 21-12 thru 21-13 semiconductor quantum structures, optical properties Bloch function, 21-5 conversion matrix, 21-6 light emission theory, 21-6 thru 21-7 Luttinger–Kohn Hamiltonian, 21-5 polarization, 21-7 thru 21-8 semiconductor quantum wires, 21-1 transmission electron microscope (TEM), 21-1 thru 21-2 Nanomemories based on III–V semiconductor QD, 2-7 thru 2-10 (see also Quantum dot memories) semiconductor charge trap memories, 2-6 thru 2-7 single electron memories, 2-7 Nanometer-sized ferroelectric capacitors characterization of films and capacitors P-E hysteresis loop measurements, 6-10 thru 6-11 RBS, 6-8 scanning probe techniques, 6-11 thru 6-13 XRD, 6-8 thru 6-10 deposition techniques combinatorial PLD and MBE systems, 6-8 MBE, 6-5 thru 6-6 PLD, 6-6 sputter deposition, 6-6 thru 6-7 ferroelectricity fundamentals ferroelectric correlation length, 6-4

ferroelectric hysteresis loops, 6-4 perovskite oxide BaTiO3 unit cell, 6-3 phenomenological definition, 6-2 second-order ferroelectric phase transition, 6-3 thermodynamic theory, 6-3 low-temperature superconducting Josephson junctions, 6-18 thru 6-19 multiferroic tunnel junction, 6-18 novel functional oxide tunnel junctions, 6-18 patterning electron beam lithography, 6-8 lift-off technique, 6-7 photo-mask steps, 6-8 physical phenomena depolarizing-field effect, 6-15 thru 6-16 intrinsic size effect, ultrathin films, 6-17 strain effect, 6-14 thru 6-15 “research triangle,” 6-1 thru 6-2 tunneling magnetoresistance (TMR), 6-18 Nanoparticle quantum dots Au nanoparticles, 22-6 CdSe/ZnS nanoparticles, 22-4 thru 22-5 close-packed monolayer films, 22-5 decay times, 22-5 thru 22-6 enhancement and quenching phenomena, 22-5 light-emitting neutral nanoparticle, 22-4 PL blinking phenomena, 22-4 unique exciton energy transfer, 22-5 Nanophotonic buffer memory, 33-11 thru 33-12 Nanophotonic devices AND-gate, ZnO nanorod electronic carrier penetration, 36-10 exciton–phonon couplings, 36-7 thru 36-8 exciton populations, 36-9 input laser llumination, 36-11 micro-channel plate, 36-7 near-field spectroscopy, 36-7 thru 36-8 nutation frequency evaluation, 36-10 schematic depict, SQW, 36-9 single-quantum-well structures(SQW), 36-7 thru 36-8 switching operation, 36-10 thru 36-11 dissipation-controlled nanophotonic devices carrier manipulation (up-converter), 33-7 thru 33-8 energy transfer between two quantum dots, 33-2 thru 33-5 nanophotonic switch, 33-5 thru 33-7 operation excitation energy transfer, 35-1 thru 35-3 nanophotonic AND gate, 35-4 thru 35-5 nanophotonic NOT gate, 35-5 thru 35-6 photonic devices interconnection, 35-6 thru 35-7

Index-8 quantum dot materials controlling the energy transfer, 36-4 thru 36-6 discrete energy levels, 36-1 thru 36-2 dissipated optical energy transfer, 36-2 thru 36-4 near-filed optically coupled ZnO QDs, 36-4 thru 36-7 schematic representation, 33-2 using spatial symmetries AND- and XOR-logic gates, 33-10 thru 33-11 nanophotonic buffer memory, 33-11 thru 33-12 quantum entangled states manipulation, 33-12 thru 33-13 selective energy transfer, 33-9 thru 33-10 symmetric and antisymmetric states, 33-8 thru 33-9 Nanophotonics devices (see Nanophotonic devices) fabrication NFO-CVD (see Nonadiabatic nearfield optical CVD) nonadiabatic near-field photolithography, 35-10 thru 35-11 hierarchical architectures memory retrieval, 34-7 thru 34-9 physical and functional, 34-6 thru 34-7 unscalable design, 34-8 thru 34-10 optical excitation transfer (see Optical excitation transfer) parallel architecture broadcast interconnects, 34-4 thru 34-5 global summation, 34-4 memory-based architecture, 34-3 thru 34-4 secure signal transfer, 34-5 thru 34-6 switch energy diagram, 33-6 exciton population, 33-6 thru 33-7 ON- to OFF-state, 33-7 switching operation, 33-6 versatile functionalities, 34-10 thru 34-12 waveguides (see Nanophotonic waveguides) Nanophotonic waveguides active lossy waveguides cladding layer, 37-9 double-peak structure, 37-11 electric field intensity, 37-10 Fresnel formulae, 37-10 substrate modes, 37-12 TE–TM splitting, 37-11 active Si-nc waveguides direct facet-coupling, 37-8 thru 37-9 PL spectra, 37-7 thru 37-8 prism coupling, 37-8 refractive index, 37-7 external source light propagation, 37-5

Index internal photoluminescence propagation, 37-4 thru 37-5 losses and optical amplification insertion (coupling) loss, 37-5 loss measurement methods, 37-5 thru 37-6 propagation loss, 37-5 pump-and-probe (PP) method, 37-5, 37-7 shifting excitation spot (SES) method, 37-5 thru 37-6 variable stripe-length (VSL) method, 37-5 thru 37-7 nanocrystalline waveguides light amplification, 37-12 thru 37-13 optical couplers, 37-14 photonic circuits, 37-13 PL spectra, 37-13 planar and rib waveguide fabrication advantages, 37-4 cladding layers, 37-3 fused quartz implantation, 37-2 thru 37-3 optical transparent matrix, 37-2 reactive silicon deposition, 37-3 SiO2 film, 37-3 spectral filtering, 37-2 silicon nanophotonics, 37-1 thru 37-2 Nanorelays, 3-11 thru 3-12 Nanoscale excitons definition, 23-1 density of states, 23-2 thru 23-3 electron-hole attraction, 23-1 electron quantum confinement, 23-4 exciton-binding energy, 23-4 extended bulk system, 23-5 Frenkel excitons, 23-1 optical properties, semiconductor nanostructure, 23-2 semiconductor quantum dots nonlinear optical properties, 23-5 thru 23-6 two-photon absorption (2PA), 23-6 thru 23-8 single-wall carbon nanotube (SWCNT), 23-4 thru 23-5 spatial electron confinement, 23-3 thru 23-4 Wannier–Mott excitons, 23-1 Nanoscale spin valves anisotropic effects, 8-4 computational micromagnetics (see Computational micromagnetics) energy balance and free energy functional formulation, 8-5 equation of motion, 8-5 thru 8-6 exchange interactions, 8-4 external bias field, 8-5 magnetization vector, 8-3 thru 8-4 magnetostatic interactions, 8-4 thru 8-5 spin-transfer torque effects, 8-6 thru 8-7 Nanoshaping functional materials antireflective structure creation, 18-6

biomaterial shaping, 18-6 thru 18-7 interface shaping, 18-5 nanopore generation, 18-5 optical grating creation, 18-5 thru 18-6 Nanotube-based superconducting quantum interferometers magnetometry applications, 7-13 thru 7-14 quantum information applications, 7-14 thru 7-15 quantum interference with weak links, 7-10 thru 7-11 tunability of the phase shift, 7-11 thru 7-12 Nanowire lasers basic properties, 39-6 cavity structure, 39-8 GaN nanowire cross section, 39-8 ring resonator, 39-6 thru 39-7 schematic representation, 39-5 thru 39-6 N2 atmospheric train-sublimation method, 15-5 thru 15-6 Near-field optically coupled ZnO QDs energy diagram, 36-6 growth time dependence, 36-5 TEM image, 36-5 thru 36-6 time constants dependence, 36-7 time-resolved PL intensity, 36-5, 36-7 Near-field photoluminescence (NFPL), 36-7 Negative differential resistance (NDR), 10-13 Nonadiabatic near-field optical CVD (NFO-CVD), 35-7 electron potential curves, 35-9 photodissociation, 35-7 photon flux relationship, 35-9 sapphire substrate, 35-8 shear-force topographical images, cross section, 35-7 thru 35-8 Nonadiabatic photolithography, 35-10 Noncircular rings, 4-9 thru 4-10 Nonconventional semiconductor memories FeRAM, 2-4 thru 2-5 magnetoresistive RAM, 2-5 PCRAM, 2-5 thru 2-6 Nonlinear optics average Hirshfeld charge, 27-11 cadmium selenide clusters ((CdSe)n ), 27-5 cluster materials, 27-5 density functional theory (DFT), 27-9 Douglas–Kroll approximation, 27-10 electric dipole moment, 27-1 excitation frequency, 27-11 thru 27-12 frequency-dependent optical polarizability, 27-6 frequency variation, 27-6 thru 27-8 generalized-gradient approximation (GGA) technique, 27-6 hyperpolarizability tensor, 27-1 inorganic materials, 27-5 intensity-dependent refractive index (IDRI), 27-2 lithium tetramer, 27-10 mean dipole polarizability, 27-5 thru 27-6 mean second-order hyperpolarizability, 27-9 thru 27-10

Index-9

Index metallic cluster, 27-5 mixed metal cluster, 27-11 nanocluster, 27-9 Nd:YAG laser frequency, 27-9 organic materials, 27-5 quantum formulation average polarizability tensor, 27-4 density matrix, 27-3 dipole interaction model, 27-2 frequency-dependent response properties, 27-5 Kohn–Sham equation, 27-3 Lagrangian multiplier matrix, 27-4 Taylor expansion, 27-4 time-dependent density-functional theory (TDDFT), 27-3 two-state model, 27-2 second-order hyperpolarizability tensor, 27-12 second-order nonlinear polarization, 27-2 semiconducting cluster, 27-5 of semiconductor nanostructures (see Semiconductor nanostructures) symmetrized fragment orbitals (SFO), 27-11 time-dependent Hartree–Fock (TDHF) method, 27-9 Nonlithographic nanowire preparation imprint lithography (IL), 17-3 thru 17-4 spacer patterning technique (SPT), 17-4 thru 17-5 Nonplanar multi-gate transistors, 12-3 thru 12-4 Nonvolatile memories (NVM), see Flash memories

O Optical excitation transfer parallel architecture broadcast interconnects, 34-4 thru 34-5 global summation, 34-4 memory-based architecture, 34-3 thru 34-4 secure signal transfer, 34-5 thru 34-6 via optical near-field interactions global summation, 34-2 thru 34-3 light–matter interactions, 34-2 Optical nanofountain, 35-6 thru 35-7 Optical near-field (ONF), 34-10 thru 34-12 coupling strength, 33-4 thru 33-7 ONF system, 32-6 thru 32-7 Optical neuronets basic elements, multilayered structures, 38-7 thru 38-8 cyclicity processes, 38-5 thru 38-6 data processing, 38-6 element base parameters, 38-7 light fluxes formation, 38-6 multilayered constructions, 38-7 polymeric film property, 38-9 thru 38-10 polymeric film requirements, 38-8

polymeric mixture preparation, 38-8 thru 38-9 suspension preparation, 38-8 Optical-projection lithography system, 19-9 Optical spectroscopy luminescence hole-burning (LHB) spectroscopy, 22-1 luminescence spectra, 22-1 thru 22-2 multiexciton generation, 22-6 thru 22-7 nanoparticle quantum dots, 22-4 thru 22-6 optical properties, 22-1 semiconductor nanoparticles, 22-1 single molecule spectroscopy, 22-2 SWNT (see Single-walled carbon nanotube) Optical transition matrix element, 21-7 Organic FET am-bipolar operations, 15-17 thru 15-18 device structures, 15-16 field effect doping, 15-17 metal/organic layer interface, 15-16 thru 15-17 Organic nanolasers, 39-11 Ovonic unified memory (OUM), see Phase-change RAM

P Passivation, 14-7 thru 14-9 Passive mode locking physics and devices, 41-3 requirements, 41-3 thru 41-4 Pauli equation, 9-3 thru 9-4 Phase-change RAM (PCRAM), 2-5 thru 2-6 Phase-matching condition, 28-3 Phase-shifting point-diffraction interferometer (PS/PDI), 20-5 Phonon-assisted spin tunneling, Mn12 acetate, 9-5 Photolithography, 18-1 thru 18-3; see also Nonadiabatic photolithography Photoluminescence (PL) energy-band diagram, 25-1 thru 25-2 photon momentum, 25-1 porous silicon (PS), 25-2 quantum confinement (QC) model, 25-2 quantum size effects, 25-3 silicon nanostructures Arrhenius plot, 25-11 confinement energy, 25-8 co-sputtered SiNC, 25-7 thru 25-8 Coulomb exchange interaction, 25-11 dynamical characteristics, 25-10 excitons, 25-8 f-sum rule/Thomas–Reiche–Kuhn sum rule, 25-13 nanocrystal size distribution, 25-12 thru 25-13 nonradiative process, 25-14 thru 25-15 optical characterizations and luminescence bands, 25-7 oscillator strength vs. confinement energy, 25-13 thru 25-14

PL decay time, 25-10 PL yield vs. silicon volume content, 25-9 radiative process, 25-14 relaxation process, 25-11 spin-orbit interaction, 25-12 surface passivation, 25-9 synthesis, 25-3 thru 25-7 SiO2 , 25-1 surface phenomena, 25-3 surface-to-volume (STV) ratio, 25-2 thru 25-3 vibron model adiabatic approximation, 25-17 conduction-ISL (CISL), 25-20 core nanocrystal, 25-22 electron–phonon interaction, 25-18 inter-sub-level (ISL) optical transition, 25-19 laser pyrolysis technique, 25-21 nanocrystal passivation, 25-16 noncoherent surface vibration, 25-19 phonon dispersion relation, 25-16 photoinduced absorption (PIA) spectra, 25-19 thru 25-20 polaron formation mechanism, 25-18 polar optical phonon, 25-19 resonant coupling condition, 25-17 silicon–oxygen bond, 25-16 thru 25-17 silicon–silicon dioxide (SiO2) interface, 25-15 thru 25-16 Si–O vibration, 25-20 thru 25-21 valence-ISL (VISL), 25-20 Photolysis process, 19-5 thru 19-6 Photo-mask steps, 6-8 Photonic crystal lasers, 39-10 Photonic crystal waveguide, 37-1 thru 37-2 Photon localization canonical field quantization eigenstates, 32-5 Hamiltonian operator, 32-4 thru 32-5 harmonic oscillator, 32-4 Helmholtz equation, 32-4 light–matter interaction, 32-3 quantization procedure, 32-5 dressing mechanism and spatial localization Davydov transformation, 32-12 thru 32-13, 32-16 effective interaction Hamiltonian operator, 32-8 effective sample-probe-tip interaction, 32-10 electric displacement operator, 32-9 electromagnetic field correlations and intermolecular interactions, 32-6 thru 32-7 exciton polariton, 32-9 Hamiltonian model, 32-11 thru 32-12 irrelevant macroscopic subsystem, 32-7 thru 32-8 mean field approximation, 32-14 nanomaterial system, 32-7

Index-10 phonon-assisted photodissociation, 32-17 photon–electron–phonon interacting system, 32-7 photon hopping operator, 32-15 thru 32-16 photon number operator, 32-14 thru 32-15 photon–phonon coupling constant, 32-15 pseudo one-dimensional near-field optical probe system, 32-11 P space and Q space, 32-8 quasiparticle and coherent state, 32-13 thru 32-14 virtual photon cloud, 32-5 thru 32-6 Yukawa functions, 32-11 effective spatial wave function, 32-3 Heisenberg’s uncertainty principle, 32-1 position-representation wave function, 32-1 quantum electrodynamics (QED), 32-1 virtual cloud effects, 32-2 wave function definition, 32-2 thru 32-3 Photoresist technology base quencher additives, 19-6 thru 19-7 EUV lithography, 19-8 line and space pattern, 19-4 thru 19-5 material structure advances, 19-7 thru 19-8 193â•›nm immersion lithography, 19-8 photolysis process of photoacid generator, 19-5 thru 19-6 positive and negative tone resist, 19-4 residual swelling fraction, 19-7 resist materials, 19-4 thru 19-5 Piezoresponse force microscopy (PFM), 6-12 thru 6-13 Plasmon amplification, 26-5 Plasmon frequency, 26-4 Plasmonic lasers bowtie structures, 39-9 dipole nanolaser, 39-9 thru 39-10 horseshoe optical nanoantenna., 39-9 surface plasmon mode confinement, 39-8 Plasmonic waveguides, 37-1 thru 37-2 PLD, see Pulsed laser deposition Point contacts and conductance quantum, 12-11 Poisson–Schrödinger resolution, 5-13, 5-18 Polarization-sensitive nanowire and nanorod optics absorption coefficient, 26-4 anisotropic nanostructure, 26-1 core–shell nanowires luminescence amplification, 26-13 potential distribution, 26-11 spectra and polarization dependence, 26-11 thru 26-13 depolarization factors, 26-2 electrodynamic effects, 26-2 first-kind Hankel function, 26-3 Helmholtz wave equation, 26-3

Index light-induced phenomena, 26-1 metal and semiconductor nanowires, nonlinear phenomena dipole orientation, 26-8 frequency dispersion, 26-10 Gaussian properties, 26-11 linear optical phenomena, 26-9 nonlinear polarization, 26-8 plasmon resonance, 26-10 radiation pattern, 26-9 second harmonic generation (SHG), 26-7 thru 26-8 SHG angular distribution, 26-10 metal nanostructures, 26-4 thru 26-5 optical properties, 26-3 photoconductivity, 26-2 thru 26-3 planar interface, 26-1 plasmon spectra longitudinal plasmon shift, 26-16 thru 26-17 self-assembling, 26-15 thru 26-16 polarization memory, random nanorod arrays luminescence polarization ratio, 26-14 polarization antimemory, 26-15 polarization-dependent effects, 26-13 polarization ratio, 26-4 semiconductor nanostructures, 26-2 intensity ratio, 26-6 isotropic interband matrix element, 26-5 Poynting vector, 26-6 radiation characteristics, 26-7 refractive index, 26-5 ZnO NW luminescence spectrum, 26-7 Power spectral density (PSD), 20-4 Proximity effect, 7-15 thru 7-17 Pulsed laser deposition (PLD), 6-6

Q Quantum computing atom–light interaction Jaynes–Cummings model, 1-6 thru 1-7 in RWA, 1-4 thru 1-6 Loss–DiVincenzo proposal RKKY interaction, 1-8 thru 1-9 using quantum dots and electric gates, 1-7 thru 1-8 physical implementation conditions, 1-3 qubits and quantum logic gates, 1-2 thru 1-3 semiconductor quantum dots classical Faraday effect, 1-13 electronic band structure, 1-12 quantum Faraday effect, 1-13 thru 1-14 SPFE conditional Faraday rotation, 1-16 GHZ quantum teleportation, 1-16 thru 1-18 Jaynes–Cummings Hamiltonian description, 1-14 thru 1-15

quantifying EPR entanglement, 1-16 quantum computing, 1-18 thru 1-20 with molecular magnet double-well potential, 1-11 Feynman diagrams, 1-10 Grover’s algorithm implementation, 1-12 single-spin Hamiltonian description, 1-10 Zeeman effects strong external field, 1-4 weak external field, 1-4 Quantum dot laser edge emitter laser diodes (see Edge emitter laser diodes) electronic state continuum approaches and atomistic models, 40-5 envelope-function approximation, 40-5 localized and delocalized, selfassembled structures, 40-4 thru 40-5 laser emission properties microscopic generalization, 40-14 thru 40-15 photon statistics modifications, 40-15 thru 40-16 rate equations, 40-13 thru 40-14 material systems, 40-1 thru 40-2 optical gain gain saturation, refractive index, and α-factor, 40-12 thru 40-13 interband transitions and macroscopic polarization, 40-11 optical gain calculations, 40-11 thru 40-12 optical susceptibility, 40-10 thru 40-11 transparency carrier density, 40-10 scattering process carrier–carrier interaction, 40-6 thru 40-8 carrier–phonon interaction, 40-8 thru 40-10 Quantum dot (QD) memories; see also Selforganized quantum dots hysteresis measurements, 2-22 write time measurements, 2-22 thru 2-23 Quantum entangled states, 33-12 thru 33-13 Quantum Faraday effect, 1-13 thru 1-14 Quantum spin tunneling anticrossing of adiabatic eigenvalues, 9-3 Berry’s phase and path integrals (see Berry’s phase interference) generalized master equation, 9-4 Pauli or master equation, 9-3 thru 9-4 Rabi oscillation, 9-3 spin Hamiltonian, 9-2 uniaxial anisotropy, 9-2 Quantum teleportation, 1-16 thru 1-18 Quasiparticle poisoning energetics of the nonequilibrium process, 16-7 thru 16-9

Index-11

Index in ramped current measurement, 16-11 thru 16-12 in rf measurement, 16-15 Quasi-static thermal equilibrium dialysis, 11-5 thru 11-6

R Rabi oscillation, 9-3, 29-16 thru 29-19 Radosavljevic’s device, 3-5 Random lasers, 39-11 Rate window method, 2-15 thru 2-16 RBS, see Rutherford backscattering spectrometry Rectangular and elliptical islands, 4-6 thru 4-7 Relevant nanometric subsystem, 32-7 thru 32-8 “Research triangle,” 6-1 thru 6-2 Resistively and capacitively shunted junction (RCSJ) model, 16-3 thru 16-4 Richardson’s constant, 13-2 Rotating wave approximation (RWA), 1-4 thru 1-6 Rutherford backscattering spectrometry (RBS), 6-8

S Sawyer-Tower circuit, 6-10 Scattering lasers, 39-10 thru 39-11 Schottky barrier, 13-1 Schottky contact, 2-11 thru 2-12 Second harmonic generation nanoparticles crystallographic defects, 28-7 Feynman diagram, 28-6 fundamental wavelength (FW) field interaction, 28-5 long-wavelength limit, 28-7 thru 28-8 perturbation theory, 28-6 retardation effects, 28-6 scanning near-field optical microscopy (SNOM), 28-5 SHG selection rules, 28-7 nanosystems low-symmetry particles, 28-4 nonuniform illumination, 28-4 random metal nanostructures and rough metal surfaces, 28-5 resonant metal particles, 28-4 thru 28-5 spherical isolated particles and plane-wave illumination, 28-3 nonlinear optics, 28-2 thru 28-3 SH electric field polarization, 28-9 single gold nanoparticles Fischer pattern, 28-12 lithographed nanorods, 28-12 thru 28-13 near-field FW properties, 28-13 near-field nonlinear optical microscopy, 28-11

nonlinear optical behavior, 28-13 thru 28-14 second harmonic polarization analysis, 28-14 thru 28-15 SHG efficiency, 28-14 SNOM, 28-11 thru 28-12 Second harmonic maps, 28-12 thru 28-14 Self-organized quantum dots based nanomemories, 2-10 carrier emission process, 2-9 charge carrier storage hole storage in GaSb/GaAs QD, 2-17 thru 2-18 in InGaAs/GaAs QD, 2-17 InGaAs/GaAs QD with AlGaAs barrier, 2-18 thru 2-20 storage time, 2-20 thru 2-21 electron and hole states, 2-9 fabrication, 2-8 localization energy and carrier storage time determination (see Capacitance spectroscopy) Self-phase modulation (SPM), 41-4 thru 41-5 Semi-analytical approach numerical validation convergence, 5-19 FEM approach, 5-19 thru 5-21 Semiconductor lasers Bragg mirrors, 39-3 in-plane laser, 39-2 thru 39-3 vertical cavity surface emitting laser (VCSEL), 39-2 Semiconductor nanomemories charge trap memories, 2-6 thru 2-7 single electron memories, 2-7 Semiconductor nanostructures four-wave mixing response, perturbation theory electron–hole pair, 29-11 excitation field possess, 29-13 excitons, definition, 29-11 Green function, 29-16 macroscopic exciton polarization, 29-12 multiwave mixing response, 29-14 Pauli blocking spectrum, 29-15 phase factor, 29-14 quantum well, 29-11 semiconductor nonlinear response, 29-11 S-matrix, 29-12 Green’s functions method, 29-1 quantum equations Coulomb energy, 29-8 Coulomb gauge, 29-6 density matrix, 29-6 divergent band-gap renormalization, 29-8 electromagnetic field–matter system, 29-6 electron–electron interaction, 29-7 Heisenberg equation, 29-7 Heisenberg representation, 29-6

Helmholtz’s theorem, 29-7 radiative decay, 29-9 rotating wave approximation, 29-9 thru 29-10 quantum field theory, 29-1 rabi oscillations, nonperturbative methods exciton polarization, 29-19 Fourier series, 29-18 Hartree–Fock approximation, 29-17 integro-differential operator, 29-17 interband polarization, 29-18 spacial decoupling scheme, 29-16 semiconductor-field hamiltonian, kâ•›⋅â•›p-approximation Bloch mode, 29-2 Coulomb gauge, 29-4 effective mass, 29-5 electron field operator, 29-3 Fourier series, 29-3 heavy–light hole hybridization, 29-5 Hermitian equation, 29-2 interband coupling, 29-3 intraband coupling, 29-4 Luttinger–Kohn function, 29-2 quantum dynamics, 29-2 Schrieffer−Wolff transformation, 29-3 semiconductor quantum dot, 29-5 spatial field operators, 29-4 spatial Fourier spectrum, 29-6 spin-orbit interaction, 29-5 Semiconductor quantum dots nonlinear optical properties, 23-5 thru 23-6 two-photon absorption (2PA) amplified stimulated emission and lasing, 23-7 CdSe, 23-6 continuous-wave (cw) Gaussian beam, 23-7 incident laser intensity, 23-6 thru 23-7 optical power limiting and biological imaging, 23-7 two-photon sensitizer, 23-7 thru 23-8 Silica deposition vesicle (SDV), 31-7 Siliconize photonics, 37-1 Silicon nanostructures Arrhenius plot, 25-11 confinement energy, 25-8 co-sputtered SiNC, 25-7 thru 25-8 Coulomb exchange interaction, 25-11 dynamical characteristics, 25-10 excitons, 25-8 f-sum rule/Thomas-Reiche-Kuhn sum rule, 25-13 nanocrystal size distribution, 25-12 thru 25-13 nonradiative process, 25-14 thru 25-15 optical characterizations and luminescence bands, 25-7 oscillator strength vs. confinement energy, 25-13 thru 25-14 PL decay time, 25-10

Index-12 PL yield vs. silicon volume content, 25-9 radiative process, 25-14 relaxation process, 25-11 spin-orbit interaction, 25-12 surface passivation, 25-9 synthesis co-sputtering method, 25-4 electrochemical etching, 25-3 high angle annular dark field (HAADF) detector, 25-6 thru 25-7 laser pyrolysis, 25-5 molecular beam epitaxy (MBE) system, 25-6 silicon nanocrystals (SiNCs), 25-3 thru 25-4 silicon/SiO2 serial sputtering, 25-6 transmission electron microscope (TEM), 25-5 Silicon quantum dot boundary conditions, 5-14 charge density in the nanocrystal, 5-17 DOF vs. maximum mesh size, 5-16 mesh algorithm, 5-15 meshing optimization, 5-16 quarter-structure in FEM, 5-14 structure for FEM simulation, 5-14 total energy of system, 5-17 vs. metallic approximation, 5-17 thru 5-18 Single electron memories, 2-7 Single-electron transistor (SET), 12-8 thru 12-9 Single-molecule magnet transistors, 9-11 thru 9-12 Single-photon Faraday rotation (SPFE) conditional Faraday rotation, 1-16 GHZ quantum teleportation, 1-16 thru 1-18 Jaynes–Cummings Hamiltonian description, 1-14 thru 1-15 quantifying EPR entanglement, 1-16 quantum computing Alice’s quantum dot, 1-19 Bob’s quantum dot, 1-19 Leuenberger’s scheme, 1-18 optical Stark effect, 1-20 Single-walled carbon nanotube (SWNT) 1D direct-gap band structure, 22-2 exciton states, 22-3 Lorentzian function, 22-3 PL spectrum, 22-3 thru 22-4 Si/SiO2 substrate, 22-2 spectral diffusion, 22-4 Stark effect, 22-4 Small angle x-ray scattering (CD-SAXS), 19-27 Spacer double patterning method, 19-25 Spacer patterning technique (SPT), 17-4 thru 17-5 Spatial symmetry nanophotonic buffer memory, 33-11 thru 33-12 quantum entangled states manipulation, 33-12 thru 33-13

Index selective energy transfer, 33-9 thru 33-10 symmetric and antisymmetric states, 33-8 thru 33-9 AND and XOR logic gates, 33-10 thru 33-11 Spherical dot in plate capacitor geometry and notations, 5-3 metallic dot Bl coefficients expression, 5-4 thru 5-5 boundary conditions for potential, 5-4 capacitance and coupling coefficient, 5-3 thru 5-4 capacitance matricial expression, 5-6 method of images, 5-4 thru 5-5 numerical implementation and results, 5-7 thru 5-9 partial capacitance expression, 5-6 polarization influence, 5-6 thru 5-7 semi-analytical expression for potential, 5-6 semiconductor dot multiple electrons, 5-12 thru 5-13 quantization effects, 5-9 thru 5-10 quantum dot with single electron, 5-10 thru 5-11 second electron addition, 5-11 thru 5-12 silicon quantum dot boundary conditions, 5-14 charge density in the nanocrystal, 5-17 DOF vs. maximum mesh size, 5-16 mesh algorithm, 5-15 meshing optimization, 5-16 quarter-structure in FEM, 5-14 structure for FEM simulation, 5-14 total energy of system, 5-17 vs. metallic approximation, 5-17 thru 5-18 Spin-transfer torque effect, 8-3 for both directions of current, 8-6 and giant magnetoresistance, 8-12 thru 8-13 LLGS equation, 8-6 spin-diffusion length, 8-7 Spin transistor, 12-9 Spintronics, 1-1, 8-3, 12-9 Spin-valve system active region, 8-3 nanoscale (see Nanoscale spin valves) Sputter deposition, 6-6 thru 6-7 Static random access memory (SRAM) unit cell, 20-8 thru 20-9 Stratonovich interpretation, 8-11 Sub-kT/q switch, 12-8 Substrate radiation mode, 37-9 Superconducting Josephson junction Ambegaokar–Barato relation, 16-2 concerns specific to small junctions, 16-4 field emission micrograph, 16-2 I–V curve, 16-2 thru 16-3 RCSJ model, 16-3 thru 16-4

Superconducting quantum interferometers (SQUID), see Nanotube-based superconducting quantum interferometers Superconducting weak links carbon nanotube weak link experimental realizations, 7-6 thru 7-7 Fabry–Perot cavity, 7-6 high-frequency irradiation, 7-9 thru 7-10 nanotube quantum dot, 7-7 thru 7-9 coherent transport, 7-2 electronic state density, 7-2 fullerene-based links, 7-17 thru 7-18 gate control, 7-4 thru 7-5 graphene-based links proximity effect, 7-15 thru 7-17 SQUIDs devices, 7-17 Josephson effect, 7-2 thru 7-3 low-dimensional sp2 carbon structures, 7-4 thru 7-5 nanotube-based SQUID magnetometry applications, 7-13 thru 7-14 quantum information applications, 7-14 thru 7-15 quantum interference with weak links, 7-10 thru 7-11 tunability of the phase shift, 7-11 thru 7-12 normal electron transport, 7-3 transport through Andreev bound states, 7-3 thru 7-4 Surface-architecture-controlled transistors depletion width and effective diameter, 14-14 electrode CS, 14-12 thru 14-13 IDS–VG curves, 14-12 surface band bending, 14-13 Surface plasmon resonance (SPR)/Mie resonance, 24-2, 24-6 Surface-roughness effects, 14-7 thru 14-9 Sweep rate effect, 14-9 thru 14-11

T Thermal NIL, 19-18 thru 19-19 Thermionic emission (TE) theory, 13-2 Thomas–Reiche–Kuhn sum rule, 25-13 Three-dimensional carrier confinement, 40-4, 40-5 Three-terminal molecules, 17-9 thru 17-10 Time-bandwidth product (TBWP), 41-2 Time-dependent Kohn–Sham equation, 27-3 Time-dependent local-density approximation (TD LDA), 24-7 thru 24-8, 25-15 TMOS, schematic behavior, 5-1 thru 5-2 Transfer printing, 19-19 Transistor structures atomic-scale TCAD, 12-11 thru 12-12 carbon nanotube transistors, 12-6 thru 12-7 graphene ribbon nanotransistors, 12-7 thru 12-8

Index-13

Index molecular tunnel junctions, 12-9 thru 12-11 Moore’s law, 12-1 thru 12-2 nonclassical transistor structures, 12-7 nonplanar multi-gate transistors, 12-3 thru 12-4 point contacts and conductance quantum, 12-11 quantum effects, 12-5 thru 12-6 SET, 12-8 thru 12-9 silicon processing, technology boosters, 12-2 thru 12-3 spin transistor, 12-9 sub-kT/q switch, 12-8 Transmission electron microscopecathodoluminescence (TEM-CL) carrier generation and light emission process, 21-3 thru 21-4 characteristics, 21-3 diffusion length, 21-4 gallium arsenide (GaAs) nanowire CL spectra, temperature dependence, 21-20 GaAs/AlGaAs nanowire, SEM image, 21-18 thru 21-19, 21-21 zinc-blend structure, 21-18 InP nanowires broad peak blue-shifts, 21-16 CL spectra, 21-17 thru 21-18 degree of polarization, 21-18 electron confinement energy, 21-16 emission energy, 21-15 metal-organic vapor phase epitaxy (MOCVPE) technique, 21-15 phonon scattering process, 21-17 SEM image, 21-15 thru 21-16 Monte Carlo simulation, 21-4 photomultiplier tube (PMT), 21-5 resolution factors, 21-3 scanning electron microscopy (SEM), 21-5 zinc oxide (ZnO) nanowires CL intensity, 21-22 CL spectra, 21-21 emission peak, 21-24 fitting function, 21-22 optical properties, 21-21

polarization ratio, 21-23 smoke particles, 21-22 thru 21-23 wurtzite-type semiconductor, 21-21 Tunneling magnetoresistance (TMR), 6-18 Tunnel magnetoresistance (TMR) effect, 2-5 Two-level amplifying system (TLS)., 39-9

U Ultrafast laser diodes broad gain bandwidth, 41-7 dimensionality role, 41-5 thru 41-6 low absorption saturation fluence, 41-8 low threshold current and low temperature sensitivity, 41-8 materials and growth, 41-6 ultrafast carrier dynamics, 41-7 thru 41-8 Ultrahigh data storage, 39-12 UV NIL, 19-19

V van den Berg’s ferromagnetic islands, 4-3 van der Waals epitaxy bottom contact structure, 15-11 initial growth surface hexagonal AlN (0001) surface, 15-14 thru 15-16 hexagonal GaN (0001) surfaces, 15-12 thru 15-14 H-terminated Si (111) surfaces, 15-11 thru 15-12 van Roosbroeck thru Shockly relation, 21-6 thru 21-7 Vapor–liquid–solid (VLS) mechanism, 14-3 Vapor transport method FESEM images, vertically well-aligned nanowires, 14-5 thru 14-6 furnace system schematic, 14-4 thru 14-5 Vertical-cavity surface-emitting lasers (VCSELs), 40-3

W Wess–Zumino phase, see Berry’s phase

X X-ray diffraction (XRD) reciprocal space maps, 6-9 specular reflectivity method, 6-9 thru 6-10 synchrotron x-ray scattering measurements, 6-9

Z Zeeman effects, 1-4 Zero-biased rf electrometry operation beyond 1â•›GHz, 16-14 thru 16-15 quasiparticle poisoning, 16-15 relation to the cooper-pair box, 16-16 rf measurement setup, 16-13 thru 16-14 ZnO nanorod double quantum-well structures, 36-7 thru 36-11 ZnO nanowire field-effect transistors crystal structure of ZnO, 14-2 fabrication and characterization back gate and top gate configurations, 14-7 passivation and surface-roughness effects, 14-7 thru 14-9 sweep rate effect, 14-9 thru 14-11 FET, 14-3 thru 14-4 general growth methods, 14-2 thru 14-3 N-channel depletion-mode and enhancement-mode, 14-11 physical parameters of ZnO, 14-2 surface-architecture-controlled transistors depletion width and effective diameter, 14-14 electrode CS, 14-12 thru 14-13 IDS–VG curves, 14-12 surface band bending, 14-13 vapor transport method FESEM images, vertically well-aligned nanowires, 14-5 thru 14-6 furnace system schematic, 14-4 thru 14-5

FIGURE 16.4â•… The ground (ϵ0) and first excited (ϵ1) states calculated by numerically diagonalizing Equation 16.11 with the energies indicated in the figure.

FIGURE 16.11â•… (a) Type H Isw histograms versus ng. Histogram height is displayed in grayscale on the right-hand side, whereas all counts are displayed equally on the left-hand side. As in Figure 16.7, the gray box in (a) denotes regions where the island potential is trap like for quasiparticles. ∼ 115â•›μeV, and EJ1 = EJ2 − ∼ 82â•›μeV. (b) Selected histograms corresponding to several gate voltages. Device parameters: Δi = 246â•›μeV, Δℓ = 205â•›μeV, EC −

FIGURE 16.13â•… (a,c) Switching current (Isw) histograms and (b,d) derived switching/escape rates for a type H (barrier-like) and type L (trap-like) CPTs. For the type L device, the quasiparticle trapping behavior is evident in the bimodal Isw distribution. In this case, the poisoning rate Γeo can be read directly from the derived escape rate in (d) as shown. Although the type H device is barrier-like for most ng, it still looks like a trap near ng = 1 (see Figure 16.7). This is apparent in the “curvy” structure of the escape curve for ng = 0.99 as compared with the escape rates at other ng in (b).

FIGURE 20.6â•… (a) Artist illustration of ASML’s EUV ADT showing a DPP EUV source on the left, illumination optics in the middle, and mask, projection optics, and wafer on the right. (Courtesy of ASML, Veldhoven, the Netherlands.)

FIGURE 20.18â•… Artist illustration of ASML NXE: 3100 preproduction EUV exposure tool showing the LPP EUV source on the right, the illuminator optics in the center, and the mask and the projection optics on the right. (Courtesy of ASML, Veldhoven, the Netherlands.)

Figure 23.1â•… Excitons and structural size variations on the nanometer length scale. (a) The photosynthetic antenna of purple bacteria, LH2, is an example of a molecular exciton. The absorption spectrum clearly shows the dramatic distinction between the B800 absorption band, arising from essentially “monomeric” bacteriochlorophyll-a (Bchl) molecules, and the redshifted B850 band that is attributed to the optically bright lower exciton states of the 18 electronically coupled Bchl molecules. (b) The size-scaling of polyene properties, for example, oligophenylenevinylene oligomers, derives from the size-limited delocalization of the molecular orbitals. However, as the length of the chains increases, disorder in the chain conformation impacts the picture for exciton dynamics. Absorption and fluorescence spectra are shown as a function of the number of repeat units. (c) SWCNT size and “wrapping” determine the exciton energies. Samples contain many different kinds of tubes, therefore optical spectra are markedly inhomogeneously broadened. By scanning excitation wavelengths and recording a map of fluorescence spectra, the emission bands from various different CNTs can be discerned, as shown. (Courtesy of Dr. M. Jones). (d) Rather than thinking in terms of delocalizing the wavefunction of a semiconductor through interactions between the unit cells, the small size of the nanocrystal confines the exciton relative to the bulk. Size-dependent absorption spectra of PbS quantum dots are shown. (Adapted from Scholes, G. D. and Rumbles, G., Nat. Mater., 5, 683, 2006.)

Figure 24.9â•… Evolution of the absorption cross section of a nanoshell of silver (top) or copper (bottom) in the dipolar approximation versus energy (left) or versus wavelength (right) for various thicknesses of the shell. The core is filled with water and the external medium is also water (n = 1.33). The dielectric functions of copper and silver have been extracted from Palik (1985–1991). The correspondence between the energy in eV and the wavelength in nm is E (eV) = 1239.85/λ (nm). The total radius of the cluster is always 15â•›nm and the thickness takes the following values: e = R − Rc = 5; 4; 3; 2; 1â•›nm corresponding to ratios between the shell thickness and the total cluster radius: (e/R) = 0.33; 0.27; 0.2; 0.13; 0.07. The spectra in black correspond to the fully homogeneous cluster.

Figure 24.15â•… (Top) Color image of a typical sample of silver nanoparticles as viewed under the dark-field microscope. The brightness of the particles increases from blue to red due to both the intrinsic optical scattering cross section and the spectral output of the light source (the red particle is overexposed). This image is correlated to its electron microscopy image (Bottom). (Reprinted from Mock, J.J. et al., J. Chem. Phys., 116, 6755, 2002. With permission.)

Figure 24.16â•… (Left) Normalized UV–visible spectra of Au–Ag alloy nanoparticles with varying composition. (Insert) Location of the SPR maximum as a function of the gold content. (Right) Corresponding solutions whose colors vary from the red (pure gold nanoparticles) to the yellow (pure silver nanoparticle). (From Russier-Antoine, I. et al., Phys. Rev. B, 78, 35436, 2008. With permission.)

Figure 25.2â•… The PL spectrum from porous silicon with a maximum PL at a wavelength of about 675â•›nm. The photograph at the inset demonstrates the red color of the emitted PL from a circular layer of porous silicon (the sample has been illuminated with a UV lamp).

Figure 25.6â•… (a) A photograph showing the PL variation along the substrate from silicon nanocrystals deposited by the laser pyrolysis technique. (b) The normalized PL spectra from different positions along the substrate. (Reprinted from Ledoux, G. et al., Appl. Phys. Lett., 80, 4834, 2002. With permission.)

Figure 28.10â•… Nanoparticles: (a), (d), and (g) topography; (b), (e), and (h) FW transmission; and (c), (f), and (i) SH emission SNOM images with corresponding cross sections along the dashed lines, from the raw data. Incident light is polarized parallel to the major axis. The particle major axis lengths are 100â•›nm (a)–(c), 150â•›nm (d)–(f), and 400â•›nm (g)–(i). Image size: 3 × 3â•›μm2 . (Reprinted from Zavelani-Rossi, M. et al., Appl. Phys. Lett., 92, 093119, 2008. With permission.)

Figure 29.3â•… The imaginary part of the two-dimensional Fourier spectrum, P(Ω, ω). (a) The spectrum calculated using Equation 29.92. (b) The experimental results of Zhang et al. (2005).

Figure 32.5â•… Probability that a photon is found at each site as a function of time (a) without the photon–phonon coupling, and (b) with the photon–phonon coupling comparable to the photon hopping constant.

FIGURE 37.14â•… Calculated spectral positions of the substrate modes as a function of (a) the refractive index contrast Δn = n2 − n3 and (b) the relative thickness of the waveguide core compared to the sample 5 × 1017 Si cm−2. Gray scale indicates intensity increasing from black up to white for the highest intensity. Several orders of modes are seen starting from the first one in infrared region.

FIGURE 37.16â•… (a) Photograph of the edge of a set of Si+ ion implanted layers with direction of PL indicated by arrows, the edge is on the left. (b) Measured PL from samples implanted to different Si ion fluences in standard (the broadest curves) and waveguiding geometry (black lines, the slightly lighter gray lines stand for TE and TM resolved polarizations). (c) Theoretically calculated PL spectra. We note that these results were obtained on different set of samples than in Figure 37.10. The mode positions are not exactly the same for samples with identical implantation dose because the annealing conditions were slightly different. Therefore, the refractive index profiles are not identical. (Adapted from Pelant, I. et al., Appl. Phys. B, 83, 87, 2006.)