Plasmons and magnetoplasmons in semiconductor

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Plasmons and magnetoplasmons in semiconductor heterostructures

Manvir S. Kushwaha Instituto de FõÂsica, Universidad AutoÂnoma de Puebla, Apdo. Post. J-48, Puebla 72570, Mexico

Amsterdam±London±New York±Oxford±Paris±Shannon±Tokyo

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Contents 1. A kind of introduction 2. Historical survey of plasmons in composite systems 2.1. Conventional (classical) structures 2.1.1. Single surface/interface 2.1.2. Thin ®lms 2.2. Systems with diminishing dimensions 2.2.1. Inversion layers 2.2.2. Quantum wells and superlattices 2.2.3. Quantum wires and lateral superlattices 2.2.4. Quantum dots and antidots 2.3. Some variants of quantum structures 2.3.1. Finite-size 2DEG 2.3.2. Periodically modulated structures 2.3.2.1. Electrostatic potential modulation 2.3.2.2. Magnetic ®eld modulation 2.3.3. Hofstadter's butter¯y spectrum 3. Methodologies for inhomogeneous plasmas 3.1. Macroscopic model theories 3.2. Microscopic model theories 3.3. The strategy of working within the RPA 4. Plasmons on a semiconductor surface 4.1. Zero magnetic ®eld 4.2. Non-zero magnetic ®eld 4.2.1. Voigt con®guration 4.2.2. Perpendicular con®guration 4.2.3. Faraday con®guration 5. Plasmons in thin semiconducting ®lms 5.1. Zero magnetic ®eld 5.1.1. Supported ®lms 5.1.2. Unsupported ®lms 5.2. Non-zero magnetic ®eld 5.2.1. Voigt con®guration 5.2.2. Perpendicular con®guration 5.2.3. Faraday con®guration 5.2.4. Resonance splittings 6. Plasmons in double inversion layers 6.1. DILs with zero thicknesses 6.1.1. Zero magnetic ®eld 6.1.2. Non-zero magnetic ®eld 6.1.3. Effect of optical phonons in the spacer layer 6.2. DILs with ®nite thicknesses 6.2.1. Zero magnetic ®eld 6.2.2. Non-zero magnetic ®eld 7. Plasmons in binary compositional superlattices 7.1. Plasmon polaritons in metal±dielectric superlattices

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8. 9.

10. 11.

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7.1.1. Collective excitations 7.1.2. Electron energy loss functions 7.2. Plasmon polaritons in semiconductor±dielectric superlattices 7.2.1. Zero magnetic ®eld 7.2.2. Non-zero magnetic ®eld 7.3. Plasmon polaritons in semiconductor±semiconductor superlattices 7.3.1. Zero magnetic ®eld 7.3.2. Non-zero magnetic ®eld Plasmons in periodic multi-heterostructures 8.1. Semiconducting multi-heterostructures 8.2. Metallic multi-heterostructures Plasmons in doping (n±i±p±i) superlattices 9.1. Zero magnetic ®eld 9.2. Non-zero magnetic ®eld 9.2.1. Voigt con®guration Ð non-reciprocal propagation 9.2.2. Perpendicular con®guration Ð collective excitations Finite-size effects in superlattices 10.1. Compositional superlattices 10.2. Doping (n±i±p±i) superlattices Quantum structures Ð intrasubband excitations 11.1. Periodic systems of 2D layers with zero thicknesses 11.1.1. Zero magnetic ®eld 11.1.1.1. In®nite type-II superlattice 11.1.1.2. Truncated type-II superlattice 11.1.1.3. In®nite type-I superlattice 11.1.1.4. Truncated type-I superlattice 11.1.1.5. Numerical examples 11.1.2. Non-zero magnetic ®eld in the perpendicular geometry 11.1.2.1. In®nite type-II superlattice 11.1.2.2. Truncated type-II superlattices 11.1.2.3. In®nite type-I superlattices 11.1.2.4. Truncated type-I superlattice 11.1.2.5. Numerical examples 11.2. Periodic systems of 2D layers with ®nite thicknesses 11.2.1. Zero magnetic ®eld 11.2.2. Non-zero magnetic ®eld in the Voigt geometry 11.2.3. Coupling to the optical phonons Quantum structures Ð intra and intersubband excitations 12.1. Quantum wells 12.1.1. Collective CDEs 12.1.2. Inelastic light scattering Ð theory and experiment 12.1.2.1. Theory 12.1.2.2. Experiment 12.1.3. Inelastic electron scattering from a quantum well 12.2. Quantum wires 12.2.1. Characterization of narrow channels 12.2.2. Collective CDEs

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12.2.3. Inelastic light scatteringÐtheory and experiment 12.2.3.1. Theory 12.2.3.2. Experiment 12.2.4. Inelastic electron scattering from a quantum wire 12.3. Quantum dots 12.3.1. Magneto-optical spectrum in single quantum dots 12.3.1.1. Single-particle spectrum 12.3.1.2. Fermi energy 12.3.1.3. Optical transitions Ð selection rules 12.3.1.4. Straight-line representation of single-particle spectrum 12.3.2. Collective excitations in 2D arrays of quantum dots 13. Magnetic-®eld effects in some variants of quantum structures 13.1. EMPs in ®nite-size geometries 13.2. Commensurability oscillations in periodically modulated structures 13.3. Hofstadter's butter¯y spectrum 14. Epilogue Acknowledgements References

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Plasmons and magnetoplasmons in semiconductor heterostructures Manvir S. Kushwaha* Instituto de FõÂsica, Universidad AutoÂnoma de Puebla, Apdo. Post. J-48, Puebla 72570, Mexico Manuscript received in final form 6 June 2000

Abstract The purpose of this review is to survey the status of the theory and experiment which can contribute to our knowledge of plasmon excitations in synthetic semiconductor heterostructures. With increasing sophistication of experimental probes, plasmons are now often used as a diagnostic tool to characterize the electronic structure of new materials. The surface plasmon becoming known as plasmon±polariton is simply an electromagnetic (EM) wave that propagates along the interface separating the two media. The strength of the ®elds associated with the wave is localized at and decays exponentially away from the interface. In preparing this review we feel that, beyond the presentation of the results achieved, there is a need to examine carefully the methodologies employed to obtain such results. By methodologies, we do not mean to go into the details of the mathematical machinery (this is carefully avoided!). The time has come, however, to look with a critical eye at the development of the principles, tools, and applications of the model theories applied to diverse geometries of heterostructures. Attention is largely given to the non-radiative modes, scrutinizing the effects of an applied magnetic ®eld, carrier collisions, retardation, and interaction with optical phonons. We have mostly con®ned our attention to the intrasubband modes, which could be investigated through the use of classical as well as quantal approaches. Abiding by the central theme of the review, numerous theoretical results on plasmons and magnetoplasmons in several geometries of practical interest have been gathered and reviewed. This survey is preceded by the basics of several methodologies, both classical and quantal, applicable to a wide variety of systems of varying interest. The report concludes addressing brie¯y the anticipated implications of plasmon observation in the respective composite systems under a variety of circumstances. Our satisfaction in writing this review, like any other review which covers a considerably longer period,was in bringing together the results of primary research papers and presenting some uni®ed account of progress in the ®eld. The background provided is believed to make less formidable the task of future writers of reviews in this ®eld and hence enable them to deal more readily with particular aspects of the subjects, or with recent advances in those directions in which notable progress may have been made. # 2001 Elsevier Science B.V. All rights reserved.

* Tel.: ‡52-22-45-7645; fax: ‡52-22-44-8947. E-mail address: [email protected] (M.S. Kushwaha).

0167-5729/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 2 9 ( 0 0 ) 0 0 0 0 7 - 8

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Some stars just twinkle, while others burn bright, And none of us really know why, Although some stars just twinkle, while others burn bright, Each brings a light to the sky. Ð Paula J. Welter from ``Each brings a light'' 1. A kind of introduction Human beings crave legends, heroes, and epiphanies. All three run through the history of solid state electronics like special effects in one of the Hollywood's summer blockbusters. The current and the future status of the solid state electronics, born with the ®rst transistor of Bardeen, Brattain, and Schockley, now rests on the two thoughtful aspects: speeding up the charge carriers carrying the signal and diminishing the size of the key device. The electronic industry and integrated circuits share inverse destiny. The industry grows as the circuits shrink. How far would the theory and practice allow us to go? To begin with, we recall the following remarks from one of the Latour's essays: In the last century and a half, scienti®c development has been breathtaking; but the understanding of this progress has dramatically changed. It is being characterized by the transition from the culture of ``science'' to the culture of ``research''. Science is certainty; research is uncertainty. Science is believed to be cold, straight, and detached; research is warm, involving, and risky. Science puts an end to the vagaries of human disputes; research creates controversies. Science produces objectivity by escaping as far as possible from the shackles of ideology, passions, and emotions; research feeds on all of those to render objects of inquiry familiar. There is a philosophy of science, but unfortunately there is no philosophy of research. People have been curious about the sciences for millennia, but the term ``physics'' was only coined in the 19th century. In 1850 Cardinal Newman de®ned it to be: ``that family of sciences which is concerned with the sensible world, with the phenomena which we see, hear, and touch; it is the philosophy of matter''. Gradually, physicists have narrowed their focus over the past century, perhaps because they have realized that the span of knowledge has been growing so great and so fast that few, if any, can encompass it all. Nevertheless, if there is any subject that can still claim to lie at the heart of knowledge of the natural world, it is physics. However, there are no boundaries within science except those that we have created from our partial understanding of the world. This remains true Ð notwithstanding the increasingly debatable ``fragile wall'' between the basic and applied research in essentially all sciences. The world's renowned thinkers, movers, and shakers from the past and present believe that the transformation of the society has produced, to be sure, many beautiful ruins, but not a better society. The physics and particularly the solid state physics that has an exceptionally poignant creation myth is the keystone behind such a contention. This then raises a burning and wider (than life) question: what is science and hence physics for? All of us are sensitive to nature's beauty. It is not unreasonable that some aspects of this beauty are shared by the natural sciences. But one may ask the question as to the extent to which the quest for beauty is an aim in the pursuit of science. On this issue, Poincare is unequivocal. In one of his essays he has written: The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it; and he takes pleasure in it because it is beautiful. If nature were not beautiful,

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it would not be worth knowing and life would not be worth living . . . I mean the intimate beauty which comes from the harmonious order of its parts and which a pure intelligence can grasp. Embarking on writing a history of events in the systematic development of any science, which have accompanied its growth toward major importance in determining the course of affairs of everyday world, poses a delicate problem of choice Ð the uncertainties being possibly comparable to those of a traveler making acquaintance of a new land. One thing is clear: the farther we are from the subject, in space and/or time, the fewer details we acquire; and we rarely use binoculars to enhance our pleasure when looking at natural scenery, more often satisfying ourselves with a synthetic view which gives at a glance an esthetic impression of harmony. Here, then, we content ourselves with a bird's eye view of the evolution of the solid state physics, which is comparatively the vastly growing branch of physics. The references must then play the role of binoculars in our comparison. Having enjoyed its ``golden age'' in 1950s when it was armored with all the adequate ``¯esh and bones'' to survive both classical and quantal attacks, the solid state physics that has changed its name into condensed matter physics to demonstrate its momentum can safely be regarded as the discipline of -ons. These -ons include the ``real particles'', such as electron, quanta of collective excitations, such as phonon, and ``dressed particles'', such as polaron. Often times, the name and the underlying concept associated with almost all the -ons (anion, boson, electron, exciton, fermion, helicon, hoctron, magnon, neutron, phonon, photon, plasmon, plasmeron, polariton, polaron, proton, roton, skyrmion,. . .) were the work of separate people, far removed in time. These -ons have wave propagation characteristics in the conventional (as well as arti®cial) materials as are the steel girders and concrete that support the modern structures. A review on plasmons may appropriately begin with plasma that bears the parentage of plasma physics which in turn gave birth to plasma effects in solids. The word plasma is the brainchild of Irving Langmuir, the great General Electric engineer-turned scientist, who, while experimenting with gas discharges in the early 1920s, noticed a fancied resemblance between the oscillations in his electron tubes and the seething movement in living cells. Plasma oscillations can be compared to the trembling of jelly in a bowl when the bowl is shaken. In living cells there is a jelly-like medium which contains many particles. Langmuir recalled the original name for the cellular jelly, protoplasma, and simply borrowed a portion of it to label the medium he was observing. He himself gave the ®rst mathematical analysis of plasma oscillations in natural (gaseous or space) plasma. So, perhaps for once the historians are right and it is reasonable to regard modern plasma physics as beginning with Langmuir [1,1680] who wrote: We shall use the name plasma to describe this region containing balanced charges of ions and electrons. Plasma existed in the universe long before man began to adapt his environment to his needs. Early man was familiar with two kinds of natural plasmas, lightning and the auroras, although he regarded them omens or acts of gods. Scienti®c revelation has lifted both lightning and auroras from their shrouds of superstition and has placed them within man's known framework of natural events. As we leave our puny earth, nearly everything we meet is in the plasma state. The Kennely±Heaviside layer, which sends our radio transmission back to us; the auroras we see as northern lights; the Van Allen belts, which cause concern to the astronaut; the solar ¯ares, and spots are familiar forms of plasma. The sun itself, as all the stars, is in the plasma state. Indeed, the whole interstellar space may be treated as plasma. Now it is ®rmly believed that 99.9% of the material universe is plasma. Even as man has

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gradually harnessed one natural phenomenon after another for his welfare, he is seeking ways of using plasmas in their natural state, and he has already learned to generate them arti®cially. Arti®cial plasmas are becoming more and more commonplace on earth each day. They are found in the blinding light of a welding torch, in the pale glow of the ¯uorescent lamps, and in the electron tubes that serve our TV sets. Plasmas are employed to cut and coat metals, and are being carefully examined for roles in future power generation and space engines. Apart from such technological interests, the study of plasmas is paramount for understanding diverse phenomena in space and astrophysics. A plasma is a collection of relatively mobile, charged particles that interact with one another via Coulomb forces. It commands a glamorous status in the physics literature by representing the fourth state of matter. Intuitively, one thinks of such a system as electri®ed gas or ¯uids, characterized by its great ability to respond to electric and magnetic perturbations. The responsiveness of a plasma is a direct consequence of the mobility of its constituents. Gas discharges, electrons in metals, and electrons and holes in semiconductors are examples of systems that exhibit plasma-like behavior. However, it should be stressed that not all charged-particle systems have this character. A NaCl crystal, e.g., is also made up of positively and negatively charged …Na‡ and Clÿ † ions, but these are too tightly bound to one another by electrostatic forces to move about freely and hence to show the typical plasma behavior. It is interesting to note that any ¯uid moves according to three basic physical principles: (i) the law of conservation of mass; (ii) the law of conservation of momentum; (iii) the law of conservation of energy. The conservation laws are expressed mathematically in terms of the classical equations of ¯uid mechanics. Because they govern all ¯ow situations, the equations of ¯uid mechanics are also used for analyzing electrically conducting ¯uids such as mercury, the blood in veins and arteries of mammals, and plasmas. However, electrically conducting ¯uids are often subject to a magnetic ®eld. For that reason, their behavior is quite different from that of ordinary ¯uids. Whenever the equations of ¯uid mechanics are used in conjunction with electrically conducting ¯uids, they must be so adjusted as to include in them terms from Maxwell's ®eld equations and Ohm's law. The resultant equations are then the governing equations of magneto¯uid mechanics Ð often also referred to as magnetohydrodynamics and in good bureaucratic style abbreviated as MHD. The plasma physics that deals with this large part of the universe is an offshoot of MHD. It is worth mentioning that a plasma may be analyzed according to the macroscopic approach or the microscopic approach. Within the macroscopic approach, the motion of a plasma is described without regard for individual particles. On the other hand, within the microscopic approach the various particles that constitute the plasma are closely examined. In the light of this background, it does not seem to be surprising that most readers associate the word ``plasma'' solely with the phenomena taking place in gaseous (or space) plasma. However, the existing categorical international publication media has already minimized such a possibility, because the two subjects Ð plasma physics and solid state plasma physics Ð sing different notes at different places. Historical records reveal that plasmon Ð the subject matter of this review Ð is the progeny of two David's (David Bohm and David Pines), who introduced quantum mechanics to plasma oscillations in solids. The ®rst paper of their series, published in 1951 [2], states the following: In order to obtain a collective description of the electrons in a metal, we must use a quantum mechanical treatment, since the electron gas is highly degenerate. Previous treatments . . . were not extensible to quantum theory. In the later work in 1952, Pines and Bohm [3] have concluded that plasma oscillations should be wellde®ned elementary excitations in metals whose conduction electrons were believed to be nearly free,

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such as Mg, Be, Al, etc. Evidence to support this view came from the experiments of Ruthemann and Lang in 1948 on the scattering of a kilovolt beam of electrons by thin metallic ®lm [4,1681]. The explanation was provided by Neville Mott at the 1954 Solvay Congress [5]; he showed that in cases where the valence electron plasma frequency is large compared to the majority of the valence electron interband excitation frequencies, oscillations near the free-electron plasma frequency will take place with only little damping associated with resonant interband excitations. By 1956 both the theoretical arguments and the experimental evidence for plasma oscillations as the dominant long-wavelength mode of excitations in most solids had progressed to the point that at the Maryland Conference on Quantum Interactions of Free-Electrons, David Pines proposed the term ``plasmon'' to describe the associated quantum of elementary excitations and demonstrated that characteristic-energy-loss experiments provided information on plasmon energies, lifetimes, and dispersion; as well as the critical wave vector beyond which plasmons could no longer be regarded as well-de®ned excitations [6]. The term solid state plasma (SSP) gives an impression that the medium represented is similar to the space plasma. This is true only to a certain extent, but it should be recognized that there are large quantitative differences between the two types of systems. For instance, in space plasmas the mean-free paths of the various particles are often comparable to the dimension of the container, while in SSP this is rarely true. The SSP is effectively an in®nite medium where boundary conditions make no sense, whereas in space plasma they can be very crucial. Equally important difference between space and SSPs concerns the ease with which the two types of systems can be driven out of thermal equilibrium. The SSP is a uniform, equilibrium system, tightly bound to the lattice, which can be only strongly excited by relatively drastic perturbations. Small mean-free paths for electron±lattice scattering are responsible for this state of affairs. On the contrary, space plasma is already a highly excited state of matter in which particle collisions are usually negligible. Nonequilibrium velocity distributions can easily be generated in such plasmas, which, in turn, give rise to a host of instabilities. The SSP is a stable system ideally adapted to study the wave propagation; space plasmas are inherently unstable, often non-uniform, best suited to the study of the instabilities to which they are so susceptible. The con®nement of space plasma and the existence of instabilities therein resist each other, whereas SSP does not pose any such problem. The SSP is characterized by two quite different modes of behavior, depending upon the wavelength of the probe used to study the plasma. If it is perturbed by a short wavelength (large wave vector) disturbance, the Coulomb energy is relatively small and the plasma behaves as a collection of noninteracting particles giving rise to the single-particle excitations (SPEs). On the other hand, if the perturbation has a suf®ciently long-wavelength (small wave vector), the Coulomb energy is dominant and forces the plasma to behave in a collective way thereby suppressing the particle-like aspects of the system and giving rise to the collective excitations. The two domains of behavior of a plasma are separated by a critical length (lc ) called Debye length (screening length) in classical plasma (quantum plasma). For l > lc (l < lc ) a plasma responds collectively (single-particle-like). The critical length has another equally important meaning. It is the screening length of the plasma, i.e., the extent to which an external electrostatic ®eld will penetrate it before being counter-balanced by the induced ®eld due to polarization of the medium. Every plasma has a characteristic frequency Ð the plasma frequency op Ð which sets the scale of its response to time-varying perturbations. The plasma frequency stands for the frequency of the basic collective mode Ð the longitudinal charge-density wave (CDW) Ð of a charged medium.

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It is well known that both classical and quantum plasma exhibit collective behavior due to the longrange Coulombic interaction between the electrons. The collective behavior has, as alluded before, two important manifestations: screening and collective oscillation. They are interrelated and may be explained qualitatively as follows. If there is an imbalance in the charge distribution at some point in the plasma, e.g., an excess of positive charge, electrons will be attracted to that region. Since electrons are mobile, they will move in and try to screen the in¯uence of the charge neutrality of the plasma. In doing so, the electrons will, in general, overshoot their mark because of their inertia. They are then pulled back toward that region, overshoot again, etc., in such a way that an oscillation is set up in the process about the state of charge neutrality. These are the oscillations characterized by the plasma frequency op, which is a function of the free charge density and the effective mass of the charge carriers in the plasma. Additional important features show up when the plasma is subjected to a magnetostatic ®eld. This gives rise to another characteristic frequency called cyclotron frequency oc, which is a function of the strength of the applied magnetic ®eld and the effective mass of the charge carriers. Physically speaking, it is frequency of the electron's helical motion about the magnetic lines of force. The presence of a magnetic ®eld greatly alters the response of a plasma to low frequency (o  oc ) perturbations. This is, in fact, a dramatic effect which makes it possible for magnetized plasma to support low-frequency, propagating EM waves, whereas a non-magnetized plasma cannot. One of the important consequences of magnetizing the plasma is that its polarizability becomes highly anisotropic Ð even though the medium without applied magnetic ®eld is isotropic. Many observable peculiarities of magnetized plasma arise solely due to this fact. It is important to note that these low-frequency EM waves are quite different in character from the plasma waves discussed above. The low-frequency EM waves are transverse waves and their properties depend strongly upon the direction of propagation relative to the applied magnetic ®eld. In contrast, plasma waves are high-frequency excitations and are longitudinal in nature. These facts also force one to use entirely different experimental probes in studying the two types of modes. The low-frequency EM waves have been investigated using microwave or radiowave measurements. The plasma waves, on the other hand, can only be excited by means of high energetic probes, such as inelastic electron or light (Raman) scattering. It is a bit surprising that while investigation of waves and instability phenomena in space plasma (such as in earth's ionosphere) had occupied many researchers for many decades, the analogous studies in SSPs were undertaken only in 1960. I believe it was Konstantinove and Perel [7], who ®rst pointed out that certain low-frequency, weakly damped, EM waves could be allowed to propagate through noncompensated metals subjected to a magnetostatic ®eld. Aigrain [8] (then at Ecole Normal Superieure, Paris) popularized these excitations naming them helicons in honor of their circular polarizations; he preferred semiconductors to metals. It was immediately followed by the proposal of Alfven wave propagation in certain compensated conductors, such as bismuth by Buchsbaum and Galt [9]. Then the ¯uid gates really opened with the observation by Bowers et al. [10] of standing helicons in Na. Within less than a decade the ®eld of helicons and Alfven waves in SSP culminated into hundreds of research articles published in prestigious international journals and a few text books by Pioneers in the subject (see, e.g., Refs. [11±32]. We would like to shed some light on the excitation mechanism of helicons and Alfven waves in SSP that aroused so much interest world-wide in the ®eld. The existence of plasma sets restrictions on the type of wave that can be propagated through it. These restrictions are basically determined by the motions of the charged carriers that constitute the plasma; in a sense the wave must be ``in tune'' with these motions. The motion of an electron in plasma can be

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in¯uenced by four forces. First, it will be accelerated by an electric ®eld that can be an externally applied or an electric component of the ambient wave propagating through the plasma. Second, the electron will be decelerated by frictional forces such as might arise from collisions with other particles. Third, the electron will be de¯ected by the Lorentz force that acts at right angles to both the electron's motion and the ®eld itself. Fourth, the electron's net acceleration will be resisted by the inertial force, which is proportional to the electron's mass and acceleration. The charm behind the plasma effects in solids is that in some materials friction can be brought to be negligible. Charge carriers of high mobility, largely undamped by the friction, can be attained in some semiconductors and in some very pure metals by cooling them to lower temperatures, which reduces the thermal vibrations of the atoms in crystal lattice of the material. Assuming that the friction can be neglected, the electric force on the electron will be balanced by the sum of the Lorentz force and the inertial force. If one calculates the relative amount of these forces for the waves of moderate frequencies, one ®nds that for electrons the Lorentz force is far greater than the inertial force; hence the magnetic ®eld dominates the electron's motion. Because of the nature of the Lorentz force (see above), the electron is obliged to rotate around the magnetic lines of force. It turns out that the only kind of wave that can impart to the electron a permissible type of motion, and therefore propagate, is the one whose electric ®eld and associated current rotate about the magnetic ®eld at a certain frequency. Clearly, the wave will propagate if and only if its sense of rotation coincides with the sense of rotation that electrons wish to follow in a magnetic ®eld. Such a wave is characterized by a circular polarization and its frequency is inversely proportional to the square of its wavelength. The latter fact gives the wave an unusual property: its velocity increases as its frequency increases. A wave with just these characteristics is known as helicon wave; this is the solid state analog of whistler wave in space plasma. The helicon waves can be excited in uncompensated metals and extrinsic semiconductors. The reader is referred to Ref. [22] for an exhaustive list of references during the climax period of investigation on helicons. The second type of wave that could be allowed to propagate in SSPs has a character quite different from that of the helicon wave. It is the dominant mode in a compensated plasma that has equal number of positive and negative charge carriers, both of which are highly mobile. In such a material medium the electric currents that are due to the electric ®eld and the Lorentz force can balance out to zero. Even though there is no net ¯ow of electric charge, there is nonetheless a ¯ow of particles. As a consequence, certain waves can be excited in such a system due to the interplay between magnetic and kinetic energies. Such a wave motion is called an Alfven wave after the great astronomer Hannes Alfven, who ®rst predicted its existence in sunspots [33]. Analyzing the factors that determine the frequency of an Alfven wave reveals that the wave is characterized by a linear dispersion relation; and the two circular polarizations are seen to be degenerate, that means that the velocity of an Alfven wave, unlike that of helicon wave, is constant. Some elegant experiments (see, e.g., Ref. [22]) have demonstrated the propagating Alfven wave in Bi, which is a semimetal with some attributes of both metals and semiconductors and serves the nice purpose of a compensated SSP. Semimetals and intrinsic semiconductors are good candidates for realizing the Alfven waves. Summarizing the wave phenomena in SSPs leads us to stress two important points: (i) Both helicons and Alfven waves can be viewed as a dynamical Hall effect Ð they follow at once when the appropriate Hall constant and conductivity are inserted into Maxwell's ®eld equations. Therefore, the study of wave propagation provides complete knowledge of the magnetoconductivity tensor. (ii) Helicons are essentially undamped as long as oc t  1 (i.e., large Hall angles); t being the phenomenological

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relaxation time; if this condition is met, they can be observed at arbitrarily low frequencies. On the other hand, Alfven waves only exist if ot  1, and thus require experiments at very high frequencies. For that reason, most of the available experimental work was concerned with helicons. Our description of helicon and Alfven waves is restricted to a simple model of SSP in which effects of velocity distribution function are neglected. In that model the wave attenuation is brought about by carrier collisions. For metals, which are degenerate and hence obey Fermi±Dirac statistics, there is much more important loss mechanism due to Doppler-shifted cyclotron resonance (DSCR). The difference between a simple model and a real metal can be best expressed in terms of the wave vector k and the mean-free path l. In a simple model (the so-called local regime) kl  1, and there is helicon propagation of all frequencies o < oc , provided that oc t  1. When o ˆ oc , the CR takes place and the wave is strongly absorbed. However, if the non-local effects become important the upper frequency can be drastically decreased. In that case, if the wave of frequency o is propagating along the magnetostatic ®eld ~ B0 (k ^z axis) in a metal, the frequency that an electron moving with a velocity component v z experiences is the Doppler-shifted value o ‡ kv z . When o ‡ kv z ˆ oc , we should expect the wave to be strongly damped due to the resonant energy transfer from the wave to the electron that would now be capable of entering the next higher Landau level (LL). Such damping is called cyclotron damping. Since the Fermi velocity v f is the highest velocity an electron can attain in a (non-drifted) metal, the condition for the onset of DSCR can be speci®ed as: o ‡ kv f ˆ oc . For helicons, if we restrict to the regime o  oc , we obtain a critical wave vector k ˆ oc =v f . Since l ˆ v f t and oc t  1, it follows that kl  1. This de®nes the non-local regime when DSCR is of crucial importance. Both helicons and Alfven waves suffer from cyclotron damping. The critical value of the magnetic ®eld Bc when DSCR sets in is called Kjeldaas edge after Kjeldaas [34], who had ®rst proposed the same mechanism for the damping of ultrasonic shear waves in metals. In a collisionless medium both helicons and Alfven waves also suffer from Landau damping. This collisionless damping always exists for both kinds of waves propagating at an angle (y) to the magnetic ®eld. For non-spherical Fermi surfaces, Landau damping can also occur even when the propagation is along the magnetic ®eld. From a classical point of view, Landau damping comes about due to the energy transfer from the wave to the charge carriers whose average velocity (in the direction of propagation) equals the phase velocity of the wave. Kaner and Skobov [14] ®rst considered the theory of Landau damping for helicon waves. Subsequent work by Buchsbaum and Platzman [35] has shown that magnetic Landau damping, brought about by RF magnetic ®eld parallel to the magnetostatic ®eld, is more important than the usual Landau damping due to the RF electric ®eld parallel to the magnetostatic ®eld. Since the Landau damping occurs when o ˆ kv f cos…y† and o  oc for helicons, it is quite likely that Landau damping takes place before the DSCR sets in. In general, any system with in®nite degrees of freedom and continuous energy spectrum, or classically a system with continuous spectrum of k for a given o, will exhibit Landau damping. This is attributed to the fact that energy at a given state will couple to all other states and will not come back to its original state for a long time (in®nite Poincare period). A characteristic which has been identi®ed with plasmas since their initial study is the time variation of their physical properties, i.e., the instabilities associated with the existence of plasmas. The motivation behind investigating instabilities is aimed at devising schemes Ð geometrical and physical Ð which may be helpful in eliminating the major instabilities or at least in limiting their growth rate to tolerable levels. This was the idea concerned with the space plasma where the con®nement is resisted by the instability phenomenon. In contrast, the major practical interest of the instabilities in SSP is the

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possibility of adjusting geometries and physical conditions to maximize their growth rate, and thus derive from the plasma useful intensities of high frequency power. In a lossless or slightly lossy systems, we distinguish two kinds of modes: (i) passive modes (or positive-energy carrying waves) and (ii) active modes (or negative-energy carrying waves). The interaction (or coupling) of a pair of modes (one active, one passive) leads to instabilities. If the group velocities of the interacting modes are in the same direction, the instability is classi®ed as convective type (representing growth in space). On the other hand, if the group velocities are in the opposite directions, the instability is classi®ed as absolute type (representing growth in time). In order to correctly identify these two major classes of instabilities, it is necessary to investigate carefully the behavior of the complex o and complex k. The scheme involves the mapping of the complex o-plane onto the complex k-plane. We intend to omit, for the reasons of space, an extensive discussion of the numerous types of instabilities (each of which ®nally belongs to one of the above-mentioned two classes) and, instead, refer the reader to Ref. [22]. Since helicons are such slow waves, it occurred to many people that their velocity could be made to match the velocity of sound waves in metals. An interesting bunch of studies has since focused on the interaction of helicon waves with other excitations, such as phonons, magnons, plasmons (see, e.g., Ref. [22]). The ®rst of these to excite interest was the helicon±phonon interaction observed in potassium by Grimes and Buchsbaum [36]. Rest of the excitations interacting with helicons remain as yet theoretical predictions rather than experimental veri®cations. Such studies of well-characterized helicons interacting with other elementary excitations should be of meaningful assistance in understanding the properties of solids and of the other excitations involved. While most of the above discussion regarding the early work on helicons (as well as Alfven waves) is concerned with the conventional solids (supposedly in®nite in extent), it is noteworthy that a considerable attention has also focused on the realistic, ®nite SSP and instabilities which can be induced in them [37±52,1682]. In such composite plasmas one essentially deals with a surface wave instability that does not strongly depend on the plasma thickness. The ®rst pioneering idea of amplifying helicon waves in only one-component, bounded plasma (a sandwich structure) was put forward by Baraff and Buchsbaum [37]. The most exciting feature of their novel instability mechanism was that the drift velocity needed to manifest the instability could be far less than the phase velocity of the helicon wave propagating in one-component SSP. This idea was later extended by the present author to study the instabilities associated with helicon wave propagation in layered structures made up of ntype II±VI or III±V semiconductors with isotropic energy bands [51] as well as those made up of IV±VI semiconductors with anisotropic energy bands [52]. The con®guration studied was a three-layer structure with metallized surfaces, which was later generalized to n-layer structures. It was demonstrated that the anisotropy of the effective masses plays a signi®cant role in achieving the greatest ef®ciency of the system at the smallest value of the threshold drift velocity [52]. In addition, the author stressed that the choice of IV±VI polar semiconductors [52] over the II±VI or III±V compounds [51] gives an additional advantage in enhancing the wave ampli®cation. The latest, to our knowledge, review paper that intended to revive some interest in the related subject is by Wallis and Martin [53]. We now introduce the EM waves, which propagate in a wave-like manner along the interface between two media having different dielectric properties and decay exponentially away from the interface. Their role in the propagation of RF EM waves along the earth's surface and in the skin effect of metals was already recognized at the beginning of the century. In the pioneering work of Cohn [54], Uller [55], Zenneck [56], Sommerfeld [57], and Weyle [58] it was shown that the surface EM waves

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occur at the boundary separating two media when one of the two media is either a lossy dielectric (or metal) and the other is a lossless dielectric. It is the imaginary part of the dielectric constant (or the real part of the conductivity) which is held responsible for localizing the EM waves to the interface. It was Fano [59] in 1941, who recognized that the surface EM modes at the lossy dielectric±air interface (Zenneck modes), at the metal±air interface (now known as Fano modes), and at the lossless dielectric± air interface (Brewster modes) had features in common Ð namely, that they ``represent for media of different electrical properties the same singular case de®ned by the same mathematical equation: they are pairs of plane waves one on each side of the boundary between two media which are able to ful®ll the boundary conditions by themselves.'' In the process of popularization of the ideas, Schumann [60] in 1948 calculated the surface plasma excitations in a slab of space plasma in free space. The existence of surface plasmons at a metal surface in air was demonstrated theoretically by Ritchie [61] in 1957. In 1958, Stern (as quoted by Ferrell [62]) while deriving the dispersion relation for surface EM waves noticed that the surface EM waves at the metal±air interface involved the coupling of EM radiation with surface plasmons. It was this same year that the term ``polariton'' was coined by Hop®eld [63] who wrote: The polarization ®eld ``particles'' analogous to photons will be called ``polaritons''. (Excitons will be shown to be one kind of polaritons in Section 4. Optical phonons are another example of polaritons.) To be more explicit, the surface polaritons are EM surface waves, i.e., they are the solutions of Maxwell's ®eld equations in which the effects of retardation Ð the ®niteness of the speed of light Ð are included. Surface plasmons [61,1683] are an important subclass of surface polaritons. These are viewed as the limiting case of surface polaritons when the speed of light is allowed to become in®nite. Notwithstanding the fact that surface plasmons are a limiting case of surface polaritons, in an operational sense these can be said to have an existence of their own, since there are experiments that can measure them directly. These include, for instance, external probes with longitudinal electric ®eld associated with them, such as (fast) electron beam whose wavelength is shorter than the vacuum wavelength of the dipole active excitations in dielectric medium. In the literature, the expression ``surface plasmon±polariton'' is often used which is identical to the term surface plasmon. It is worth mentioning that when a more speci®c designation is desired for a particular situation such as, e.g., in frequency regions where EM waves in a medium are predominantly dipole excitations in character, the modes are termed as exciton±polaritons, magnon±polaritons, phonon±polaritons, plasmon±polaritons, etc. Whether or not the quali®er ``surface'' is used with them, the polaritons are invariably meant to be the surface polaritons in any n-layered structures (n  1); until and unless speci®ed otherwise. In 1958, the existence of surface plasma excitations on a slab of space plasma was established experimentally by Gould and Trivelpiece [64]. This experiment aroused considerable research interest and stimulated analogous investigations on the solid surfaces. Consequently, Powell and Swann [65] made the ®rst observation of the existence of surface plasmons in Al and Mg with the aid of fast-EELS. The existence of (bulk) polaritons was ®rst demonstrated experimentally by Henry and Hop®eld [66] in GaP using Raman spectroscopy (RS). The 1960s saw the ®eld of surface polaritons growing at a moderate pace until a major breakthrough occurred in 1968. It was then that Otto [67] devised the attenuated total re¯ection (ATR) method to probe the surface polaritons. The ATR Ð a non-destructive and sensitive (as compared to the RS and EELS) method Ð was soon developed by Kretschmann [68] and Abeles [69] into alternative con®gurations. It was Marschall and Fischer [70] who ®rst used ATR to observe surface polaritons in GaP. The interest in surface polaritons, following the spur given by Otto,

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gathered momentum and broadened throughout the 1970s. Resulting progress in the ®eld can be seen through numerous extensive review papers published after the mid-1970s (see, e.g., Refs. [71±77]). This progress accounts for the fact that an external magnetic ®eld, non-local effects, and the complexity of the band structure of the solid can give rise to a large variety of the group of polaritons. Finally, we touch brie¯y the wonderful subject of the systems of reduced dimensionality. The origin of the development of lower dimensional systems dates back to 1957 when Schrieffer [78] anticipated the quantization of energy levels in inversion layers. But the two-dimensional (2D) nature of electron gas Ð when only the lowest electric subband is occupied Ð was ®rst con®rmed experimentally by Fowler et al. [79,1684] in 1966. This experiment, performed on n-type inversion layers on a (1 0 0) surface of Si in the presence of an applied magnetic ®eld oriented perpendicular to the surface, laid the foundation of what we know today as the ``®eld of low-dimensional systems''. However, it was in 1970 that Esaki and Tsu [80] of IBM Thomas J. Watson Research Center, New York, heralded a genuine research interest in 2D (superlattice) systems by predicting the possibility of designing entirely new type of devices, such as Bloch oscillator. At the same time the advent of a new crystal-growth technique, molecular beam epitaxy (MBE), opened the way to the growth of the semiconductor heterostructures with almost ideal heterointerfaces as envisioned by Esaki (see, e.g., [81]). In 1974, two interesting experiments were performed: Esaki and Chang [82] reported the oscillatory behavior of the perpendicular differential conductance due to resonant-tunneling across potential barriers; and Dingle and coworkers [83] showed directly, through optical measurements, the quantization of energy levels in quantum wells, the well-known elementary concept of quantization in standard text books on quantum mechanics. In 1980, von Klitzing's observation of what has now become known as integral quantum Hall effect (IQHE) in Si-MOSFET proved to be a ``fuel to the ®re''. This discovery was soon followed by a new surprise Ð the observation of fractional quantum Hall effect (FQHE) in GaAs±Ga1ÿx Alx As heterojunctions in 1982 [84,1685,1686]. The investigations on 2D electron gas (2DEG) systems realizable in semiconductor heterojunctions and superlattices have since then proliferated at an explosive rate. The early development of the subject can be seen in a monumental review published by Ando et al. [85]; a special issue of IEEE Ð Quantum Electronics on Semiconductor Quantum Wells and Superlattices [86]; and the proceedings of biennial conferences on electronic properties of 2D systems (EP2DSs) [87].1 During the past decade the ability to make progressively smaller microstructures, developed for electronic industry, has allowed physicists to study how the charge carriers behave when con®ned to still lower dimensions. Thus the advancement of research on electron systems has been predominantly toward more con®nement Ð from quantum well (2 degrees of freedom) to quantum wire (1 degree of freedom) to quantum dot (0 degree of freedom). The basic principle behind quantum wells, wires, and dots is the same: con®ne electrons in a restricted region of semiconductor by sandwiching it within another semiconductor with a larger band-gap, a measure of the amount of energy needed to be pumped into the material to get electrons ¯owing (see, e.g., Fig. 1). Like water seeking a lower point, electrons will naturally tend to ¯ow in the con®ned region where the band-gap is lower. In quantum well that Ê thick slice of GaAs embedded in a larger band-gap Ga1ÿx Alx As. Con®ned region is often a 100±200 A

1

The Proceedings provide very good overview of the field at various times. The title of the conferences ``Electronic Properties of Two-dimensional Systems'' is a bit misleading because equally useful information is also available on the 1D and 0D systems.

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Fig. 1. Schematics of the quantum con®nement (top panel) for achieving quasi-n-dimensional (n ˆ 3; 2; 10) systems. The regions in purple refer to the desired quasi-n-dimensional systems, whose dimensionality can be reduced by sandwiching it between two layers of another material that has higher energy gaps. The con®nement changes the density of states (DOS) (bottom panel), or speci®c energy levels, that will be ®lled by incoming electrons. The current conducted by a quantum-well device peaks when the energy level of the incoming electrons matches (or is in resonance with) an energy level of the quantum well.

in the slice, the electrons have so little headroom that their energy states are forced to cluster around speci®c peaks. Quantum wires are produced by con®ning electrons on four sides rather than two. The result is that electrons are squeezed into a linear channel and thus provide even sharper clustering of energy states. At the end of this quantum rainbow are quantum dots (or arti®cial atoms), which are obtained by con®ning electrons on all six sides; their energy levels are discrete and not continuous. The most exciting thing about these quantum structures is that their electronic properties can be tuned at the will of the designer. This is done, e.g., by appropriate choice of the semiconductors and of the widths of their layers. These nanostructures turn the broad energy bands of conventional semiconductors into more sharply de®ned energy levels. And that is a transformation that promises greater speed and ef®ciency for the resulting electronic and optical devices. Let us focus on our concern with plasmons in 2D systems. Plasma oscillations of a 2DEG differ in a non-trivial way from the corresponding oscillations in a 3D system. This difference stems from the fact that the electric ®elds (responsible for providing restoring force for the charge-density oscillations) remain 3D while the induced charge densities attain a 2D character. This has a number of important consequences that have been explored in great detail from a theoretical point of view [88± 95,1687,1688]. There are two key features: (i) the frequency of a 2D plasmon goes to zero as the wave vector goes to zero; and (ii) the frequency is perturbed by the shape and dielectric properties of matter in the immediate vicinity of the 2DEG. It should be pointed out that the 2D behavior for the plasmons

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does not require that electron's motion be essentially 2D at microscopic level. It merely requires that the wavelength of the plasmons be large compared to the thickness of the sheet of 2DEG. In marked contrast to the considerable theoretical efforts devoted in the early 1970s, the ®rst experiment that directly probed the 2D plasmon on electrons bound to a helium surface appeared in 1976 [96] followed by the second one on Si-inversion layers [97] and the third one on GaAs± Ga1ÿx Alx As heterostructures [98]. The ®rst two of these experiments con®rmed the theoretical predictions of the characteristic square-root dependence; whereas the third one established the linear dependence on the in-plane wave vector. The latter experiment was seen to be in agreement with theory under the circumstances of strong coupling (i.e., vanishingly small separation between 2D layers) and the non-zero Bloch vector. The later advances in miniaturization, both theoretical and experimental, that followed after mid-1980s have added a remarkable dimension to the fundamental as well as device physics. The ability to manipulate matter on atomic scale, create unique materials, and design exotic devices with tunable electronic and optical properties has a universal appeal. It marks a triumph of human intellect and imagination over the natural rules that dictate the formation of materials. There is one trait that many theoretical physicists share with philosophers. In both cases the interest in a ®eld of study seems to vary in inverse proportion to how much one must learn to qualify as an expert. There is, we believe, an important core of truth to the fact that ``expertness is what survives when what has been learnt has been forgotten''. True learning leading someone to become an expert is a long-term project Ð ideally life-long. However, present-day learning of any sub®eld of science, particularly the vastly growing physics, motivated to gain mastery is unimaginably hazardous for survival in the existing competitive world of science. In this relatively short space we shall try to review brie¯y what we have been able to learn from the theoretically motivated analysis of the plasmons and magnetoplasmons in semiconductor heterostructures. All possible efforts would be made to do justice with the experimental side as well. We intend to cover both classical and quantum structures Ð from single interface to multi-heterointerfaces. The efforts have also been made to give ¯avor to the topics concerned with the microstructures where classical physics breaks down and only quantum physics survives. After Section 3, the mathematical details will be kept to a minimum and only indispensable expressions will be given; the lengthy equations will be recalled referring to the original articles. Finally, we appeal to the interested reader to digest only what he should and what he can and blame the author for the rest. 2. Historical survey of plasmons in composite systems History begins with senior people telling younger ones what it was like back in the old days, and this remains by far the most popular kind of history. For many people the past is merely what they remember themselves, perhaps supplemented by what acquaintances remember of their own lives. The hasty competition and team work these days do not provide even the senior (invited) people enough time during the usual conferences to speak on how they made their famous discoveries. In this limited space we ®rst wish to substantiate brie¯y the common belief why the 1950s are known as the ``golden age'' for the condensed matter physics. The important ideas and concepts that led a few pioneers to lay the foundation of the modern condensed matter physics during the 1950s are the following: metal±insulator transition (MIT) [99], Feynman diagrams for perturbation theory [100,1689], super¯uidity [101], electronic transport in 1D

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systems [102], Nyquist's ¯uctuation±dissipation theorem [103,104], Wick's theorem [105], Fermi hole [106], theory of small systems Ð the size effects [107], theory of Bloch electrons in a magnetic ®eld [108±112], EMA [113], biblical account of the dynamical theory of crystal lattices [114], ®rst reliable and representative band structure calculations [115], quantum theory of plasma oscillations and manyparticle systems [116±159], quantum theory of electrical transport [160,162±164], Friedel oscillations [165±167], van Hove's pair-correlation functions [168], BCS (microscopic) theory of superconductivity [169], Anderson localization of electron waves [170], Kubo's correlation functions [171,172], Fermiliquid theory [173,174], Azbel±Kaner CR [175], Landauer formula for resistance [176], Aharonov± Bohm (AB) effect [177], temperature dependent Green functions [178], MoÈssbauer effect [179], Kohn anomalies in phonon dispersion [180], early classic work on electron±phonon interactions [181±184], double-time Green functions [185], among others. The early years of the following decade also provided some extraordinary tools to the understanding of diverse phenomena in solids. These include, e.g., Kohn theorem on CR [186], Josephson effect [187], Luttinger liquid model [188], GW approximation (GWA) in many-body theory [189], densityfunctional theory (DFT) [190±192], etc. This brief list refers, in particular, to the formal theories, effects, and theorems which became the cornerstones for the later development of the condensed matter physics. With the advances in technology (e.g., MBE and electron lithography), the fabrication of potential nanoscale devices is becoming possible. Such mesoscopic and nanosystems have opened the door to a rich new ®eld of the fundamental as well as experimental condensed matter physics. However, the basic understanding of the physical phenomena in these emerging systems is not really seen to be based on drastically different formal tools than those put forward by our founding fathers during the golden age of the condensed matter physics. The apparent newness in the current methodologies is largely attributable to the varying size and dimensions of the system at hand. 2.1. Conventional (classical) structures 2.1.1. Single surface/interface In my view of the evolution of condensed matter physics, the 1960s were the decade of bulk phenomena. We made enormous strides in understanding crystals Ð we mapped their energy bands, developed clever computational schemes, and became experimentally and theoretically adept at characterizing the bulk solids. At the same time, the practically real issues regarding the precise knowledge of atomic arrangements on or near the surfaces were starting to become everyday puzzles for physicists as well as chemists. The provocative curiosities such as, e.g., the formation of surfaces, stability of interface structure, surface/interface defects, dependence of electronic properties on surface geometry, trap states at surfaces, systematics of Schottky barriers, heterointerfaces and superlattices, effects of interfaces in submicron structures, a few to name, were posing thoughtful questions to be addressed. That was the time when the immortal words of the great Italian scientist, Galileo Galilei: As scientists, should we be satis®ed to belong to that class of less worthy workmen who procure from the quarry the marble out of which, later, the gifted sculptor produces those masterpieces which lay hidden in this rough and shapeless exterior? (No, we must) . . . belong to a higher science . . . (for) there may be some great mystery hidden in these true and wonderful results. were echoing all around those who were anxious to explain such curiosities as mentioned above. The amazing progress made in understanding the surface phenomena in the 1970s was

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rigorously assessed at the 2nd ICTP±IUPAP Semiconductor Symposium held at Trieste (Italy) in 1982. Today, the world-wide drive to develop eversmaller microelectronic devices, exotic multicomponent composites, complex catalysis, and biological implants is increasing the need to understand interfaces. An interfacial zone is a transition region between two different materials. It might be the boundary formed when a metal (semiconductor) is deposited on a semiconductor (metal), or the boundary between two different metals (or semiconductors), or the boundary between a metal (or semiconductor) and a dielectric, etc. Interface research focuses on the unique properties of such boundary regions. It seeks to understand how the properties of the boundary regions differ from those of the bulk solids on either side and how these regions in¯uence the behavior of the resultant composite. The properties of the interface region are in¯uenced by, e.g., dimensional constraints (quantum effects), disorder, defects, the formation of compounds, heterogeneity, and kinetics. Interfaces are therefore fascinating regions rich in both scienti®c challenge and technological importance. The recent great advances in solid state electronics can be traced back to a unique combination of basic conceptual advances, the perfection of new materials, and the development of new device principles Ð which all can be attributed to our understanding of the physics of heterointerfaces. A great variety of the fundamental electronic properties of a solid can be successfully described by the concept of single electrons moving around the periodic array of atoms. Another quite different approach to derive the properties of a solid emerges with the concept of plasma: the free-electrons in a metal (or doped semiconductor) are treated as an electron liquid of high density, ignoring the lattice background in the ®rst approximation. In this approach, it turns out that longitudinal density ¯uctuations, plasma oscillations, propagate the pthrough   bulk solid. The quanta of these oscillations Ð the plasmons Ð have an energy  hop ˆ h 4pne2 =me, where n is the free-electron density and me the effective mass. They are excited by external electrons which are shot into the solid. This exciting phenomenon has been studied in detail both theoretically and experimentally with electron energy loss spectroscopy (EELS) [193]. An important aspect of the age-old plasma physics has been accomplished by the concept of surface plasmons, which can, under appropriate physical conditions, propagate along the free surface or an interface between two dielectric media. Their EM ®elds decay exponentially into the space perpendicular to the surface and have their maximum in the surface, a characteristic of the surface waves. Their dispersion relation lies to the right of the light line (in the rarer medium of the composite structure) which means that surface plasmons have longer propagation vector than the (bare) light waves of the same energy propagating along the surface. That is why they are called ``non-radiative'' surface plasmons, which describe ¯uctuations of the surface electron density. It is worth mentioning that the non-radiative surface plasmons in n-layered structures (n  2), can also be accompanied by the ``radiative'' surface plasmons, which are seen to propagate to the left of the corresponding light line. In the simplest situation (with absorption neglected), the existence of bona®de surface plasmons only requires an interface between an active (with the dielectric constant Ea < 0) and an inactive (with the dielectric constant Ei > 0); together with the condition that Ea ‡ Ei < 0. It should be pointed out that exchanging the active and inactive media in the physical space does not matter at all because the dispersion relation is symmetrical in Ea and Ei . This then justi®es the use of either Otto- or Kretschmann-con®guration for their excitation. A convenient classi®cation of surface plasmons into Fano, Brewster, Zenneck, and evanescent modes was suggested by Otto and Burstein (see, e.g., Table 1 in Ref. [194]).

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Early experimental investigations focused on the re¯ectivity measurements were motivated to diagnose the surface plasmon dispersion in simple metals and a few semiconductors: n-type InSb [195], n-type Ge [196], Ag [197], Au [198], Cu [199], Al [200], Li and Na [201]. The ATR experiments performed for this purpose were found to be, in general, good agreement with the theoretical calculations. Subsequent works, theoretical as well as experimental, were embarked on with a motivation to either characterize the geometrical structures and/or to study the in¯uence thereof on the dispersion characteristics of surface plasmons. For example, Agranovich and Malshukov [202] demonstrated that there is a resonance of the transition layer (within which the dispersion is only due to phonons) vibrations with the substrate surface plasmons, the latter exhibits a splitting. The magnitude of this gap was shown to be of the order of …l=l†1=2 , where l (l) is the layer thickness (surface plasmon wavelength). This theoretical prediction was immediately con®rmed by Yakovlev et al. [203] for thin LiF ®lm on sapphire (Al2 O3 ) and rutile (TiO2 ) substrates, using ATR. Similar resonance-induced splitting of the surface plasmon was also later discussed by Lopez-Rios [204], albeit the proposed mechanism for the splitting was different. He stressed on the absorption in the metal (K) ®lm on the metallic (Al) substrate; the splitting (or gap) was shown to be strongly dependent on the absorption in the ®lm, the gap vanishing for strong absorption. It is worth mentioning that while most of the theoretical work on surface plasmons was based on the Maxwell theory, there was no dearth of the quantal (microscopic) description of the problem. The latter type of work based on the self-consistent (or random-phase approximation (RPA)) started becoming available just after the mid-1960s (see, e.g., Refs. [205±217]). However, it should be pointed out that while most of such quantal approaches were framed to include a non-local dynamic dielectric function, the retardation effects were excluded. In that sense the modes studied were truly surface plasmons and not the surface polaritons. An extensive review of such microscopic approaches is given by Beck and Dasgupta [218]. By the mid-1970s, the existence and propagation characteristics of the surface plasmons in diverse (semi-in®nite) geometries were becoming well understood [219,220,1690]. However, the possibility of other experimental probes, such as RS [221±225] and EELS [226±230], for the excitation and detection of surface plasmons were being constantly explored. The effects of damping, retardation, anisotropy of the material media, spatial dispersion, surface roughness, and the magnetostatic ®elds on the dispersion characteristics of surface plasmons were the emerging issues at that time. A resurgence of interest in the excitation and detection of surface plasmons on metal surfaces by means of inelastic EELS was seen in the mid-1980s. Better sample preparation techniques and highresolution equipments stimulated tremendous interest for detailed analysis of the electronic excitations. The classical local optics picture was abandoned in favor of highly sophisticated ®rst-principle theoretical schemes. The reason being that the theoretical description of these modes is very sensitive to the electronic structure of the surface region. Hence, approximate treatments of the ground state electronic properties or response at ®nite frequencies can easily give qualitatively erroneous spectra. The key issues motivating the ongoing theoretical efforts in an attempt to seek better agreement with the observed spectra are the following: (i) an overall overestimate of the frequencies, (ii) the observed dispersions with qk (the parallel (to the surface) propagation vector) are ¯atter than theoretical, and (iii) the measured plasmon-line widths are considerably larger than theoretical. There are three possible reasons for these persistent discrepancies between theory and experiment: (i) the (simple) jellium model neglects any effects due to crystal potential (core polarization, interband transitions), (ii) the use

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of LDA (local-density approximation) in the ground state does not reproduce image potential, and (iii) the theoretical models are after all approximate: the TDLDA (time-dependent local-density approximation) incorporates a local adiabatic exchange-correlation correction (in a region with rapidly varying density), while the RPA ignores exchange-correlation effects in the response entirely. It is noteworthy that these issues are of concern both with ``monopole'' and ``multipole'' surface plasmons. Even though the spatial dispersion is accounted for in a simplest manner (as is the case with, e.g., hydrodynamic model (HDM)), the surface modes at ®nite qk , reveal two important features not contained in the local classical picture: (i) at small qk , the plasma frequency os …qk † shifts linearly downwards (a negative dispersion); after reaching a minimum near qk  0:15, os …qk † increases, and (ii) a second surface mode, om …qk †, appears at slightly higher frequencies; it shows a positive dispersion even at small qk . Both features are sensitive to the width a of the surface region with inhomogeneous equilibrium electron density. If the width a is diminished, the upper mode ceases to exist and only the lower, ordinary surface plasmon exhibiting a positive dispersion (even at small qk ) persists. In the direction normal to the surface, the distribution of the ordinary surface plasmon consists of a simple peak, i.e., it has a monopole character Ð hence, the name monopole surface plasmon. The charge distribution of the upper mode has a node, i.e., it has a dipole character. Because of this spatial form perpendicular to the surface, this mode is referred to as multipole surface plasmon. To be more explicit, the integrated weight of the ¯uctuating charge density of the monopole (dipole) mode is ®nite (zero). The multipole modes have been justi®ably recognized as the resonances in the electron±hole pair spectrum and hence highly Landau damped. It is conceivable that higher order multipole modes exist at metal surfaces. These would certainly be even more heavily damped than the dipolar surface plasmon; and therefore dif®cult to observe. Clearly, then, the shape of the charge density pro®le plays a decisive role on the excitation spectra of metal surfaces. This is especially true for the multipole mode; its shorter effective wavelength normal to the surface implied by the extra node makes it extremely sensitive to details of the density pro®le. It should be stressed that the distinction between monopole and dipole distribution along the direction normal to the surface is valid rigorously only to the true surface eigenmodes, not to the externally driven modes generated in response to the applied ®eld. Driven modes at real frequencies are always dominated by the induced part of the surface charge density. The monopole and dipole character of these driven modes are partly lost and obscured by the Friedel oscillations [231]. In the recent years there have been numerous experimental observations of the surface plasmons on several metal surfaces: Na and K [232,233], Cs and Al [234], Li and Mg [235], Na and K overlayers onto Ag (1 1 1) [236], Hg-®lm onto Cu (1 0 0) [237], Pd [238], and Ag [239±246]. Both monopole and multipole surface plasmons have been claimed to be successfully observed, except on Al. In the case of high density metals such as Al, identifying the multipole mode is dif®cult, since the surface polarizability is so weak that it (the multipole mode) remains hidden in the tail of the ordinary (monopole) surface plasmon. However, if this main loss feature is suppressed by using, e.g., photons instead of electrons, the multipole mode in Al can also appear. This is because the light does not couple to the monopole mode for clean ¯at surfaces; consequently, the monopole surface plasmon mode is removed from the spectrum leaving the multipole mode observable. All these measurements on simple metals show a negative dispersion of the monopole surface plasmon at small qk , con®rming Feibelman's microscopic calculations in the framework of LDA based-RPA [247,1691]. The most striking deviation from this general behavior is seen on Ag [239±246], where the occupied 4d bands not

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only affect the overall frequency of the surface plasmon but also its dispersion characteristics with propagation vector Ð the slope of the monopole mode at small qk is positive, just as for the multipole mode. Noticeable effects due to shallow core levels are also found for Hg. Other kinds of modi®cations arise if the metal under scrutiny is charged or if it is in contact with a dielectric medium or metallic adsorbate. The negative dispersion of the monopole surface plasmon at small qk can be understood in terms of a simple physical picture [231]. Since the centroid of the plasma surface charge lies outside the jellium edge and the induced potential associated with this charge decays exponentially towards the interior of the metal, this potential extends over a region of lower average density as qk increases. A lower density implies a lower plasma frequency. (On the other hand, if the centroid was located inside, the frequency would increase because the induced potential then extends over a region of increasingly higher density.) Towards larger qk, the shorter wavelength parallel to the surface increases the kinetic energy of the plasma oscillation and ultimately leads to an upturn of the dispersion curve. Most of the current theoretical models such as RPA, LDA based-RPA, and TDLDA [248±260,1692] substantiate the negative dispersion of the monopole surface plasmon at small qk and the positive dispersion of the multipole mode(s) for the whole range of qk . However, none of these model theories gives an overall quantitative agreement with the observed spectra. This remark is rather true in the case of almost every metal for which observed spectrum is available. Since there is no unique way of including band structure and/or exchange-correlation effects in the dynamic response of the metal surfaces Ð and the magnitude of such inclusions could have varying effects in different metals Ð much thoughtful theoretical schemes which could interpret the observed spectra over larger range of the propagation vector are still needed to be investigated. For the reasons of space, we do not want to expand on the related investigations, theoretical or experimental, on the metallic (small and large) clusters where the principal objective has been to explore the relationship between the dispersion of the plasmons and the size dependence of the Mie resonance in clusters. Next major step in the investigation of surface plasmons was the consideration of the effect of an applied magnetostatic ®eld ~ B0 . The reason for this was evidently the fact that the magnetic ®eld effect (in the bulk or on the solid surfaces) is very striking and easily observable in the experiments. Apart from some numerous individuals involved in the subject (see, e.g., Refs. [261±267]), the magnetic ®eld effects on the surface plasmons have been thoroughly investigated theoretically by Wallis and coworkers [268±271], Quinn and coworkers [272±277], Nakayama [278], Omura and Tsuji [279,280], and Uberoi and coworkers [281±291]. Three principal con®gurations with respect to the orientation of the applied magnetic ®eld have been considered: ~ B0 parallel to the surface and to the propagation vector q (Voigt geometry), and ~ B0 Re~ q (Faraday geometry), ~ B0 parallel to the surface and perpendicular to Re~ perpendicular to the surface and to Re~ q (perpendicular geometry). The application of an external magnetic ®eld on the surface plasmons has been shown to cause interesting qualitative changes in the behavioral characteristics of these surface excitations. In general, there is an excellent agreement between the theoretical predictions and the existing experimental observations [292±298]. A notable exception is, however, with the experimental results of Baibakov and Datsko [299±302], which have not, to our knowledge, yet seen a satisfactory explanation. On the contrary, evidence for the nonexistence of the low-frequency surface modes reported by Laurinavichyus and Malakauskas [303] has already been published. A reader is referred to Ref. [194] for the logical dispute regarding the validity of the experimental results in Refs. [299±302]. It is noteworthy that while most of the theoretical studies of magnetic ®eld effects on the surface plasmons in semiconductors considered the medium

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with free charge carriers in a spherical energy band, Wallis [304] also generalized the corresponding investigations to the case of anisotropic energy bands. Let us mention brie¯y some very exclusive peculiarities concerned with the Voigt geometry. In this geometry, the dispersion relation becomes non-reciprocal with respect to the direction of propagation and the spectrum in the frequency±wave vector space exhibits a magnetic-®eld dependent gap within which no magnetoplasmons are allowed to propagate. The most important and exceptional character of the Voigt geometry is that it has an advantage of preserving the TM (p-polarized) nature of the plasma waves, just as in the absence of an applied magnetic ®eld. The non-reciprocal behavior in the Voigt geometry has already been observed for the surface plasmons [294,296,298], for surface magnons [305,306], and for surface-acoustic-waves (SAWs) [307]. Interestingly, the non-reciprocity has also been demonstrated in a simple re¯ection experiment on InSb slab [308] and in uniaxial antiferromagnet MnF2 [309]. It was argued that non-reciprocal re¯ection can only occur in the presence of (nonreciprocal) absorption in the material medium and that it disappears for negligible damping. It is worth mentioning that such an effect was proposed for magnetic media, where it was named as equatorial Kerr effect, almost three decades ago [310,311]. An extensive account of the non-reciprocal behavior in diverse situations can be seen in an excellent review by Camley [312]; where one can ®nd simple symmetry arguments that help in order to understand the observable symmetry breaking leading to the non-reciprocity in frequency, localization, and optical re¯ectivity. 2.1.2. Thin ®lms After the mid-1970s, the major efforts were made to investigate surface plasmons in two-interface (®lm) geometries. In particular, the attention was drawn to study the lifetime and propagation length of the surface plasmons in the symmetrically or asymmetrically bounded metallic ®lms [313±318]. The motivation behind the quest for long-lived and long-range surface plasmons (LRSPs) was the possibility of their potential applications in non-linear optics. In the course of these investigations, some typical geometries where, for instance, part(s) of the dielectric ®lm bounded by the perfectly conducting sheets were treated as surface-active region(s) (and the rest of the ®lm remains surface inactive) were also considered [319±322]. In such a model structure the grounding sheets cause the propagating modes to be discrete rather than continuous. The goal was to predict the surface plasmon generation ef®ciency for a semi-in®nite surface-active dielectric medium in the process of focusing light onto the end face of a sample. Such an ``end±®re coupling'' was shown to yield promising conversion ef®ciency. Some theoretical studies have also been undertaken in trying to ®nd a con®guration, which would give a longer propagation length with its associated strongly enhanced local ®eld [323,324]. In all these studies it was imagined that the active medium must have a large negative real part to its dielectric constant; and that its absolute value should be greater than the dielectric constant of the surface-inactive medium. Surprisingly, a later work has indicated that an LRSP may be supported by a thin ®lm which has a large imaginary part to its dielectric constant, the real part may be even almost negligible [325,326]. The optical excitation of surface plasmons by the ATR technique was ®rst demonstrated by Otto [67]. In this originally proposed method, which is the most widely employed one, the light beam encounters ®rst a dielectric layer and then the plasma layer. The second alternative con®guration of ATR ®rst proposed by Kretschmann [68] is such that the light beam encounters ®rst the plasma layer and then the dielectric. The third alternative con®guration, which is virtually a combination of the previous two, was suggested by Abeles and Lopez-Rios [69]. This allows one to excite and detect the decoupled optical

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excitation of surface plasmons at two surfaces of the plasma ®lm. The characteristic mechanism in these experiments is such that the total internal re¯ection of monochromatic light polarized parallel to the plane of incidence is measured as a function of angle. A sharp dip observed in the re¯ectance indicates the excitation of a surface plasmon. Under this condition the energy of the incident beam is fed into the surface wave and the total re¯ection is frustrated. An extensive use of ATR technique was made by Kovacs and Scott [327,1693] in order to optically excite surface plasmons in two different layered structures. The ®rst system has two ®lms (MgF2 ±Ag) between the glass prism and air, and the second has three ®lms (Ag±MgF2 ±Ag) between the prism and air. This scheme was claimed to be very sensitive to the surface roughness of the evaporated MgF2 ®lms. A calculation of the Poynting vector ®eld shows the localization of the excitation corresponding to each resonance. Knowing the region of concentrated energy ¯ux allows understanding the effect the surface roughness will have on the observed resonances in re¯ectance. The calculation of current distributions and surface charge densities induced by the incident monochromatic plane wave shows the different modes (of a de®nite symmetry) corresponding to each resonance. To be brief and conclusive, Ref. [327] presents the most complete study of excitation and detection of surface plasmons (and guided modes in thick dielectric ®lms) ever made using the ATR technique. Subsequently, Holm and Palik [328] embarked on a systematic observation of surface plasmons in a two-interface system of an active ®lm on an active substrate. Both ®lm and substrate were taken to be n-type GaAs with different carrier concentrations. They found a reasonably good agreement between the experimentally observed and theoretically calculated spectra. Generally, surface plasmon resonances represent highly sensitive probes for metallic surfaces. In the late 1970s, several experiments have demonstrated the high sensitivity of ATR spectroscopy for systems with thin inorganic coatings on Ag and Au surfaces [329,330]. Later this method has, however, also been applied to transparent organic monolayers deposited by the Langmuir±Blodgett (LB) technique onto the Ag and Au ®lms in the Kretschmann con®guration [331±334]. The above said con®guration was argued to be favorable in that it is easier to control the thickness of the deposited metallic ®lms (in contact with the prism) than to control that of an air gap (in the Otto con®guration). Such a scheme was extensively explored by Swalen and coworkers [331±334] in order to determine the optical constants of the coated LB layers and in turn study the effects of such transition layers on the surface plasmons. They suggest that although, in general, surface plasmon resonance measurement alone is not suf®cient for a complete quantitative description of anisotropic thin coatings, surface plasmon spectroscopy represents, if combined with other experimental methods, a powerful tool for the optical characterization of molecules on metal surfaces. After the 1980s, the experimental efforts were dedicated to explore several thin ®lm geometries which could give rise to longer range of propagation for surface plasmons [335,336], to antisymmetrically coupled surface plasmons in LB/metal/LB structure [337], and to waveguide modes with substantial shift in the ranges of existence and dispersion dependencies due to interaction with the ®lm and substrate phonons [338]. The most dramatic aspect that an LRSP may exist at an interface between two dielectrics Ð one of which has a large absorption coef®cient Ð was demonstrated theoretically as well as experimentally [339]. This was done for a thin vanadium ®lm on a quartz substrate Ð with the air gap between the prism and the active medium replaced by a ¯uid, a mixture of hexachloro-1,3-butadiene, and tetrachloroethylene, that has negligible absorption at the wavelength used (3:391 mm). The concentration of the mixture was so adjusted that the dielectric constant was very close to that of the quartz substrate; the coupling ¯uid acting also as the upper dielectric in the nearly

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symmetrically bounded vanadium ®lm. The existence of this LRSP is advocated to delimit the exclusive use of metals or non-absorbing dielectrics in the non-linear optical studies. Closely associated with the experimental side of surface plasmons is the proposal of scanning plasmon near-®eld microscope (SPNM) by Specht and coworkers [340]. The SPNM, which is based on the interaction of surface plasmons with a sharp metal tip placed close to the surface of the object, provides a means of investigating the sample surface with subwavelength super-resolution. This is speculated that elastic plasmon scattering and radiationless energy transfer from the tip to the sample are the main physical processes responsible for the high resolution achieved in the SPNM. The magnetic ®eld effect on the surface plasmons in a ®lm geometry was, to our knowledge, ®rst studied by DeWames and Hall [341]. They considered an unsupported ®lm in the Voigt con®guration. Besides the non-reciprocal behavior, they found the pileup of the waveguide modes at the hybrid cyclotron±plasmon frequency and the existence of the surface magnetoplasmons above this frequency. The usual situation in the ®lm geometry considered theoretically and/or experimentally by far was when a metal or semiconductor ®lm bounded by dielectric media was treated. A complementary to this is the symmetric (asymmetric) strip transmission line (STL) made up of two identical (unidentical) semi-in®nite semiconductor plasmas separated by a transparent ®lm. Nakayama and collaborators [342,343] investigated the surface plasmon propagation in such a symmetric (and asymmetric) STL in the presence of an applied magnetic ®eld. Interesting shifts in the dispersion characteristics of the surface magnetoplasmons propagating at the two surfaces of STL, due to both the interaction due to the thin ®lm and the magnetic ®eld, were reported. Subsequently, Sarid [344], investigated surface plasmons in a thin metallic ®lm bounded by two semi-in®nite identical semiconductors in the Voigt geometry. The author has shown that the application of a magnetic ®eld modi®es the symmetry of the (otherwise identical) bounding media, as a result of which both modes, particularly the long-range one, experience a dramatic decrease in their propagating lengths. The thin-®lm geometry studied by Sarid corresponds to the symmetric STL, a well-established magnetospectroscopic technique in the submillimeter range [345]. Kushwaha and Halevi (KH) [346±349] embarked on an extensive systematic investigation of the magnetic ®eld effects on the surface plasmons in a thin semiconducting ®lm asymmetrically bounded by two dielectrics. They derived exact general dispersion relations for the magnetoplasmons in all the three (Faraday, Voigt, and perpendicular) geometries. KH examined their general analytical results in a number of interesting special cases, which include, e.g., the NRL, the thin-®lm approximation, the effect of the substrate optical phonons, the magnetized transition layer, etc. In particular, the thick-®lm approximation led KH to reproduce numerous well-established results for the surface plasmons under diverse situations, both with and without an applied magnetic ®eld. Most of these results were reviewed brie¯y in Ref. [350]. The work of KH in the Voigt geometry was later extended by Tilley and coworkers [351,1694] to the two-component plasma ®lms. Regrettably, no experiment is to date, to the best of our knowledge, available on the magnetoplasmons in the ®lm geometry. 2.2. Systems with diminishing dimensions 2.2.1. Inversion layers Essential to our understanding of the myriad of exotic phenomena associated with the systems of lower dimensions is a knowledge of the ``surface quantization'' (at the clean, perfectly ordered semiconductor surface) whose seeds were disseminated by Schrieffer [78] during the golden age of the

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condensed matter physics. From the point of view of its electronic structure, the surface may be roughly divided into three separate regions: (i) the space-charge layer, referring to the surplus or de®cit of free charge carriers (with the thickness of a few hundred nm); (ii) the surface proper, consisting of ®rst few atomic planes of semiconductor; and in the case of a real surface, (iii) an adsorbed layer of foreign material. The latter two are usually the sites of localized electronic states Ð surface states. The energy levels of such surface states may, a priori, be either discretely or continuously distributed, and may lie within the gap or within the allowed bands. Of course, only those within the gap are readily accessible to detection; the others remain hidden by the dominant large density of band states. The existence of the surface alone, which is virtually a giant defect in itself, can give rise to as many states as one per surface atom. Such states are known as either Tamm states or Shockley states (after their early investigators [352,353]), depending upon the nature of the perturbations to which they owe their origin. The essence of their existence was brought to the limelight by Bardeen [354] in 1947 in a farreaching hypothesis that the potential barrier at the semiconductor surface (responsible in accounting quantitatively for the basic features of the recti®cation and photovoltaic effects) was produced by surface states rather than by contact potential (the difference in the work functions) between the semiconductor and the metal in contact with it. Bardeen's idea was that electrons from the semiconductor bulk could be trapped in the surface states, leaving behind an equal and opposite charge to maintain an overall neutrality. Thus a space-charge region and associated with it a potential barrier form near the surface. If the density of surface states is suf®ciently large, the potential barrier remains essentially unaltered, when placed in contact with a metal or another semiconductor. However, this is undesirable. What is desirable is to cause the potential barrier to vary in a controllable manner (usually as a periodic function of time) and make simultaneously the measurements of a variety of surfacesensitive properties feasible. Only if the density of surface states is negligibly small, or if the another solid in contact is in some manner able to nullify the effect of the surface states, can the contact potential modify the space-charge region and thus control, to any appreciable extent, the relevant characteristics. An estimate of the density of surface states on a free sample was made possible from the contact potential measurements [355]. These and similar studies, combined with careful measurements of point contact recti®cation characteristics, brought to light the important role played in surface phenomena by minority carriers, even though their concentration in the bulk is very small. It becomes evident that the space-charge region Ð produced beneath a conducting solid by an external source or by the proximity of another solid with a different work function Ð consisted not only of the immobile donor or acceptor ions constituting the Schottky barrier, but more importantly of free minority carriers. In a p-type semiconductor, e.g., the conductivity type may change from p-type in the interior to the n-type at the surface, giving rise to what is known as an inversion-type space-charge layer. In a classical paper, Brattain and Bardeen [356] made it pretty clear that it is such an inversion layer that governs the current±voltage characteristics observed on a (Ge) point contact recti®er. The discovery of the junction transistor [357], pioneered principally by Shockley, followed shortly afterwards. This was undertaken largely with the view of minimizing the undesirable surface effects. The surface phenomena of predominantly electrical nature involve the free carriers in the spacecharge layer, the surface states, and the mutual interaction between the two. The most fundamental variable controlling the electronic processes at the surface is the height of the potential barrier, i.e., the drop in the electrostatic potential between the surface and the underlying bulk [358]. In any surfacesensitive measurement of a quantitative nature, it is essentially important to vary the barrier height and

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Fig. 2. The cross-section of a silicon n-channel MOSFET. The current between the ground source and drain at a positive potential is controlled by the potential on the gate. An ohmic contact to the substrate (not shown) is also used to apply a small potential difference between the substrate and the source.

determine its magnitude throughout its variation. This is accomplished most successfully by, e.g., ®eldeffect technique. (A cross-section of n-channel Si-MOSFET (metal-oxide semiconductor ®eld-effect transistor) is illustrated in Fig. 2.) The term ``®eld-effect'' refers to the change in sample conductance or capacitance as a result of a capacitively applied ®eld normal to the surface. The early attempts in device technology based on ®eld-effects were not successful; it turned out that most of the induced charge got trapped in surface states [359,1695,1696].2 Subsequently, the means have been devised to nullify the effects of the surface states by, e.g., performing the experiments on clean surfaces in ultra-high vacuum (UHV) conditions. As such, it has been pointed out that semi-classical physics of space-charge layer in surfaces of high perfection gives suf®cient background to realize the quantum size effects in the presence of a strong electric ®eld perpendicular to the surface [360,1697,1698]. The manifestation of the space-charge layer is a long-range potential disturbance, usually referred to as band bending, that can extend a few hundred nanometers (nm) into the solid. Depending upon the type of doping and relative carrier concentration, the space-charge layers are classi®ed as follows [360]. De®ne a dimensionless energy scale u ˆ …Ef ÿ Ei †=kB T, where Ei is the Fermi level position for the intrinsic semiconductor, and the rest of the symbols have their usual meanings. ub and us give the values for u deep inside the bulk and at the surface, respectively. Further, we assume that donors and acceptors are fully ionized, so that the electron density n and hole density p in a non-degenerate semiconductor are given at every point by: n ˆ ni exp…‡u† and p ˆ ni exp…ÿu†; with ni the intrinsic electron concentration. The condition u > 0 implies that at every point n > p, and u < 0 signi®es that at every point n < p, etc. Now, we are in a position to classify the space-charge layer in a, say, p-type semiconductor into three groups: (i) accumulation layer Ð ps > pb , us < ub < 0; (ii) depletion layer Ð ps < pb , ns < pb , ub < us < ÿub ; and (iii) inversion layer Ð ns > pb , 0 < ÿub < us . Strictly speaking, achieving inversion layer from the corresponding depletion layer only requires a stronger electric ®eld normal to the surface Ð causing band bending up above the Fermi level. The analogous classi®cation of space-charge layer in n-type semiconductor is de®ned just by interchanging n and p; and reversing the sign of inequalities with respect to the potentials us and ub . That means that the characteristic band bendings in p-type material are obtained by turning them upside down in n-type material (see, e.g., Fig. 3). It should be remarked that it is the inversion layer (typically of width  10 nm) which has mostly been the subject matter of extensive theoretical as well as experimental investigation during the past three decades. The electric ®eld associated with an inversion layer is strong enough to produce a 2

The early work on semiconductor surfaces is reviewed in the classic monographs.

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Fig. 3. The energy bands at the surface of a p-type semiconductor for: (a) the ¯at-band case with no surface potential; (b) accumulation of holes at the surface to form an accumulation layer due to a small negative potential applied to the gate; (c) depletion of the holes due to a small positive potential applied to the gate leading to the formation of depletion layer of width  100±1000 nm; and (d) band bending strong enough such that the conduction band is bent down below the Fermi level leading to the formation of inversion layer of width  1±10 nm. This inversion layer, where the minority carriers reside, is an n-channel where the electrons can ¯ow from the source to the drain, depending on the voltage applied. If the semiconductor is n-type, the sign of the applied voltage and hence the bending of the bands have a story contrary to the one illustrated in this ®gure (see Section 2.2.1 for details). The subscripts on E such as c, A, F, and v refer, respectively, to the conduction band, acceptor level, Fermi level, and valence band. The surface potential fs measures the band bending.

potential well of width (in the direction, perpendicular to the surface) smaller than the wavelengths of the carriers. Thus the electronic motion exhibits quantized energy levels (also termed as electric subbands) for motion perpendicular to the surface, with continuum for the motion in the plane parallel to the surface. Such quantization achieved through the application of an electric ®eld is called ``electric size effect''. If the applied electric ®eld is strong enough to create an inversion layer so thin ( 1 nm) that only the lowest electric subband is occupied, the system behaves practically as a 2DEG exhibiting what is known as ``electric quantum limit''. Thinking of inversion layers capable of exhibiting quantum size effects leads us to imagine that the charge carriers are localized at the semiconductor surface. The localization of electrons at the surface may also be achieved through the application of a magnetostatic ®eld. If the magnetic ®eld is oriented parallel to the surface, a free-electron (e.g., in a metal or an n-type semiconductor) intending to leave the surface toward the inside of the sample will be de¯ected and will eventually be forced (by the Lorentz force) to reach the surface again, if no scattering occurs in the process. After specular re¯ection at the surface, it will again go on the section of a circular path and in this way always be made to stay close to the surface. The thickness of the sheet (beneath the surface of the sample) containing such (virtually) localized electrons could be made thin enough (through the stronger applied ®eld) so as to observe the ``magnetic size effects''. Evidently, the underlying mechanism does not parallel the localization of charge carriers (in the inversion layers) through the applied electric ®eld. It is important to note that the effect of a parallel magnetic ®eld is such that every 2D subband (which is characterized

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by a single subband quantum number) is affected by the ®eld and that the subbands with higher quantum numbers would be more ®eld-dependent. In addition, several magnetic levels will originate from a single subband in this case, and these can quite likely interact with each other. This type of surface transport is noted to have been observed on the metal surfaces by optical investigations [361]. Before we proceed further, it is worth to mention that the size effects were, in fact, known in the literature even before Schrieffer envisioned the surface quantization (in particular, due to the existence of the space-charge layers). For instance, it was well known since the late 1930s that the apparent resistivity of a specimen increases well above that of the bulk material when the dimensions of the specimen become comparable to, or smaller than, the electronic mean-free path. The ®rst rigorous theoretical treatments of this size effect were given by Fuchs [362] for the case of a thin ®lm, and by Dingle [363] for a thin wire. Also, when suf®ciently large magnetic ®elds reduce the electronic orbital radius to a size comparable to, or smaller than, the dimensions of the specimen, the mechanism of the surface scattering would be modi®ed, yielding a ®eld-dependent resistance change. Theoretical treatments of this ``magnetic size effect'' were given by Sondheimer [364,1699], MacDonald and Sarginson [365], Koenigsberg [366], and Azbel [367] for thin ®lms, and by Chambers [368] for thin wires. The magnetic size effect was interpreted to cause the distribution function to be a ¯uctuating function of the distance from the surface. Although no observation of this effect in thin ®lms was reported, a related phenomenon had been observed in Na wires by Babiskin and Siebenmann [369]. The early ®eld-effect experiments on the transport properties [370±372] and the simultaneously ongoing theoretical efforts [373±375] dedicated to the understanding of diverse characteristics of inversion layers on Si reveal that IBM Thomas J. Watson Research Center, New York, is really the birth place of the research activities in the ®eld of 2D electron systems (2DESs). Most of the outstanding growth of the ®eld was, in one way or the other, fathered by Frank Stern. A closer look at the published work (at least until the mid-1970s) on semiconductor space-charge layers gives an impression as if this were a Si-oriented science. Apart from the fact that Si has since long been at the forefront of microelectronics applications, the availability of the technical procedures for preparing a suitable surface free from unwanted ingredients projected it also to a leading role in the physics research. This is true notwithstanding the fact that wide and varied world of semiconductor parameters Ð different energy band-gaps, the direct and indirect band-gap materials, single- and multi-valley bands with various degrees of non-parabolicity, polar and non-polar semiconductors Ð is available to one who dares. The 1970s have seen a tremendous growth of research activities on the space-charge layers realized on Si-MOSFET and some other III±V compound semiconductors, including the lately extensively studied GaAs±Ga1ÿx Alx As heterojunctions [85]. An open secret responsible for initiating a ¯urry of theoretical as well as experimental activity is the fact that the electron concentration can be varied continuously over a wide range simply by varying the strength of the con®ning electric ®eld (or the gate voltage). Therefore, these systems offer a better testing ground for the model theories accounting for the many-body effects. Experimental observation of, e.g., unexpected dependencies on surface carrier density, magnetic ®eld, temperature, frequency, and depletion charge of the cyclotron resonance (CR) parameters (such as effective mass, electron g-factor) [376±384], subharmonic structure in the CR line shape [385,386], evidence to support the concept of a mobility edge [387±390], negative magnetoresistance [391±394], Wigner crystallization [395,396], magnetophonon resonance in GaAs±

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Ga1ÿx Alx As heterojunction [397], intersubband transitions [398±408,1700,1701], 2D plasmon dispersion [409±416], and LL spectroscopy [417±421] stimulated considerable theoretical work. The curiously controversial issues such as the observed shift in the effective mass m , g-factor, and the subharmonic structure in CR drew immediate attention. This was because the experimentalists attributed such effects to the many-particle interactions of the inversion layer electrons. Such an explanation appeared to theorists a bit perplexing, because Kohn [422] had already proved, with simple arguments, that the position of CR is not affected by many-particle interactions in homogeneous systems. Actual systems are, however, no longer homogeneous because of the physical existence of the surface alone (that amounts to scatterers such as impurities, and surface roughness). Therefore, one can expect to sense the effects of many-body effects on CR. Within the different frameworks of the model theories accounting for the many-body effects (e.g., Hartree±Fock-, Hubbard-, random-phase-, Rice[423], GW- [424,1702] approximations), Quinn and coworkers [425,1703,1704], Vinter [426,1705], and Ando [427] presented a qualitative explanation of the shift in m . The g-factor in the bulk deduced from oscillations in the magnetoconductivity of electrons arises from spin±orbit coupling. Earlier measurements demonstrating the dependence on surface carrier density of g-factor [372] deduced effective g-factor g to be enhanced by 25±65% over the free-electron value of g ˆ 2:0023. Theoretical calculations of the self-energy in the RPA led Janak [373] to derive g values which were found to be even higher (than the experimental values) over the whole range of the density of inversion layer electrons. He proposed that the exchange interactions among the surface electrons were the cause of the observed shift in g. A later experiment [377] studied the magnetic ®eld dependence of g-factor by making use of Shubnikov±de Haas (SdH) oscillations. It was argued that just as the change in carrier density affects the self-energy, so is the change in applied magnetic ®eld. In fact, it was conjectured that the effects of the magnetic ®eld may be more pronounced at high ®elds when LLs are well resolved. The experimental results demonstrated that the maxima attained by the gfactor are a function of the magnetic ®eld, particularly for ns > 1  1012 cmÿ2. The observed shift in g was ascribed to the many-body (exchange) effects among the surface electrons. This was also in apparent contradiction to the simple argument due to Yafet [428], who has shown that, whatever the quasi-particle spin splitting, the electron system (in the absence of spin±orbit coupling or skin effect) can only absorb RF power when the frequency is very near the free-electron resonance. In the nonhydrogenic many-electron systems (describable by L±S coupling) a total escape from the spin±orbit coupling is unexpected. Therefore, one is bound to sense the (many-body) exchange effects. Note also that since the Coulomb interaction becomes stronger as the electron density decreases (it is proportional to r ÿ1 , but the kinetic energy is proportional to r ÿ2 , where r is the interelectron distance; so the latter dominates at high densities), we expect to ®nd larger shift in g for smaller densities, which is what observed. The g-factor was calculated within a framework of a static approximation by Suzuki and Kawamoto [429] and by Ando and Uemura [430]. Later, Quinn and coworkers [425] gave a qualitatively better estimate (and closer agreement with the experiment) of the observed g-shift using dynamic RPA. The most unexpected and striking result is the observation of subharmonic structure in CR. In the presence of a strong magnetic ®eld (perpendicular to the surface) the energy spectrum becomes discrete because of the complete quantization of the orbital motion (see Fig. 4). Consequently, the singular nature of the DOS plays a dramatic role in CR; causing quantum oscillation of the CR line shape at low temperatures. In the absence of e±e interactions, impurity scattering was shown to lead to the occurrence of ``harmonics'' at exact multiples of bare cyclotron frequency [431]. A model theory

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Fig. 4. The magnetic ®eld effect in 2D systems: (a) energy levels and DOS of a heterojunction in the absence of a magnetic ®eld; (b) energy levels and DOS of a single quasi-2D level in a magnetic ®eld (scaled up compared to (a)). Electrons occupy LLs up to some last partially ®lled LL.

incorporating electron±electron (e±e) interactions in the presence of a random array of impurity scatterers led Ando and Uemura [430] and Quinn and coworkers [432,1706] to explain qualitatively the surprising results on the observed subharmonics in CR. It would be interesting to recall the experimental observation of the mobility edge Ð a notion often also termed as the Mott±Anderson localization, or minimum metallic conductivity. The story begins with the classic paper of Anderson [433] in which he considered a tight-binding band of width W and introduced a random potential V on each well such that ÿ 12 V0 < V < ‡ 12 V0 . He showed that if V0 =W is greater than some critical value then electrons would not freely move through the lattice in extended (Bloch) states, but would be localized in space Ð the electrons then move from one site to another by exchanging energy with phonons, a process called ``thermally activated hopping''. When the states are localized the electron wave functions will decay exponentially as eÿar , where a stands for the decay constant of the wave function. Subsequently, Mott [434] suggested that if the Anderson criterion is not ful®lled the states will be localized in the tail of the band and an energy Ec Ð the so-called mobility edge Ð separating the localized from extended states exists. Similar argument was used by Licciardello and Thouless [435] to discuss the minimum metallic conductivity (rather conductance, h) in 2D systems. Mott [436,1707] propounded that because of the potential ¯uctuations smm ˆ 0:12e2 = (which arise primarily because of the immobile charges in the adjacent insulator, at the surface, and in the semiconductor itself) tails of localized states might exist below the 2D subbands in the semiconductor inversion layers. Stern [437] scrutinized the (by then) published experiments to conclude that the mobility edge or ``Swiss cheese'' model can reasonably well interpret the observed data, but he did not exclude alternative explanations thereof. All the experiments [387±390] provide overwhelming evidence to support the concept of a mobility edge, and show that sweeping the gate voltage can push the Fermi energy through the localized states into the extended states and let the Anderson transition from activated to inactivated conduction observable.

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In the late 1970s, mysteries in the theory of 2D systems prevailing in the inversion layers (and on the surface of liquid helium) involved fundamental issues, such as the conceptual adequacy and generality to account for, e.g., the spin and valley splitting, non-zero temperature, impurity scattering, surface roughness, image potential, spatial extent of inversion layers, multi-subband occupancy, and most importantly, the many-body effects. A crucial test of the attempted model theories was their degree of success in explaining the CR experiments and spectroscopic measurements of intersubband transitions. As a matter of fact, there has been considerable progress in our understanding of, e.g., the spectroscopic measurements. However, various additional (to exchange-correlation) effects were demanded, since these were expected to shift the resonance energy from the corresponding subband spacings. These additional effects are called resonance screening (or depolarization) effect [438,1708] and vertex correction (or excitonic effect) [439,1709]. The physical origin of the depolarization effect is attributed to the fact that each electron feels a ®eld, which is different from the external ®eld by the mean Hartree ®eld of other electrons polarized by the external ®eld. It has been frequently noted that the Hartree approximation overestimates the Coulomb repulsive force of other electrons, and the exchangecorrelation effect greatly underestimates the effective repulsive potential. In view of this it is expected that the exchange-correlation effect on the depolarization effect greatly reduces the shift of the resonance energy. The local-®eld correction to the depolarization effect is often called the excitonic effect, which is associated with the interaction of the excited electrons in the higher subband with the holes left in the lower subband. It has been suggested by Ando [440] that at low (high) carrier concentration the excitonic (depolarization) effect dominates and the resonance energy becomes smaller (larger) than the subband separation. He also pointed out that in a static local approximation the excitonic effect almost exactly cancels the depolarization effect for typical inversion-layer electronic densities. A large number of theoretical schemes accounting for the many-body effects aimed at searching a better agreement with the measurements on the intersubband transitions and providing convincing conceptual arguments have been reported during the past two decades. These have been based on almost all the available basic frameworks (e.g., Hartree±Fock approximation (HFA), RPA [127], Hubbard approximation [142], Fermi-liquid theory [173,174], HKS (Hohenberg±Kohn±Sham) density functional theory [190±192], GWA [424,1702], and STLS (Singwi±Tosi±Land±SjoÈlander) approximation [441,1710], plasmon-pole approximation (PPA) [442,1711]) depending upon the taste and convenience of the authors (see, e.g., Ref. [85] for the early important development of many-body theories for quasi-2DEG, in particular, due to Ando, Vinter, Quinn, Stern, Jonson, and Das Sarma, among others). However, there seems to be really no conceptual consensus on the generality and validity of any of the existing schemes pertaining to their applicability to a wide variety of systems Ð they have all been reasonably successful as well as unsuccessful (!) This is not surprising given the fact that a proper inclusion of the many-body effects on the optical transitions is really a ``hard nut to crack''. Many interesting effects accompanied by the inherent complications due to the multivalley structure of the conduction band of Si (and Ge) remain controversial and unsolved problems. The most unexpected and interesting fact is the valley degeneracy factor gv which can attain various values on different surfaces. A simple effective mass theory [443,1712] predicts gv ˆ 4 in the (1 1 0) and gv ˆ 6 in the (1 1 1) surface [373]. Experimentally, however, gv turned out to be 2 in both surfaces [444]. Further, the observed conductivity is seen to be isotropic in the (1 1 1) surface [445] which seems to contradict gv ˆ 2. Extensive efforts have focused on the problem of valley degeneracy by Dorda and

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coworkers [446,1713,1714]. In order to explain the observed reduction of the valley degeneracy factor on the (1 1 0) and (1 1 1) surfaces, Kelly and Falicov [447,1715±1717] proposed a charge-density wave (CDW) ground state resulting from a large phonon-mediated intervalley coupling. Later, various experimental results have appeared which contradict previous ones and are also seen unfavorable to Kelly±Falicov theory (see, e.g., Ref. [85]). Equally important and controversial is the issue of the Wigner crystallization in 2D systems (of semiconductor inversion layers and on the surface of liquid helium). As ®rst pointed out by Wigner [448,1718] (see also [1719]), the electron system will intend to adopt a con®guration of the lowest potential energy and crystallize into a lattice when the potential energy dominates the kinetic energy. In the Wigner crystal (or lattice, solid, etc.) each electron becomes localized about a lattice site. The most fundamental aspect Ð irrespective of the system's dimensions Ð is that of the behavior of the system in the low-density limit, when we should expect the Wigner crystallization of the electrons (or holes). A crucial parameter is rs ˆ r0 =aB , where r0 is the intersite spacing and aB ˆ h2 =me2 the 3D Bohr radius. In 3D systems the Wigner crystal is believed to have a body-centered-cubic lattice, since it has the lowest static lattice energy. The literature reveals that for more than three decades, after the original proposal [448,1718,1719], numerous researchers have made a number of different estimates of the critical rs , denoted by rsc, at which the Wigner transition is expected to occur at absolute zero temperature. The most common mechanism of estimating rsc is by utilizing the Lindemann [449] melting criterion, which states that the melting will occur when the root-mean-square (electron) displacement hu2 iav is some fraction d (< 1) of the square of the intrinsic intersite separation r0 . The estimates of rsc have varied over a wide range, say from 5 to 700. Such a wide variation of predicted critical rs re¯ects the inherent dif®culty of the problem, and the whole subject of the transition density remains rather questionable. The situation with 2D system is even more dif®cult, since the meaning of a 2D crystal is not well understood beyond doubt. This is, in part, attributed to the fact that in 2D systems, at any ®nite temperature, the mean square displacement at a ®xed site, averaged over the phonon spectrum, diverges logarithmically. This then prevents us to use the Lindemann melting criterion to simply determine the melting curve. In fact, the divergence of the ¯uctuations has been used to substantiate the decades-old conclusion of Peierls [450] that the thermal motion of long-wavelength phonons will destroy the longrange order in two and one dimensions, except perhaps at zero temperature [451,452,1720]. Several workers, notably Mermin, has, however, suggested that such a thermal motion does not necessarily destroy the angular correlations responsible for in®nitely long-wavelength transverse mode at any ®nite temperature. Since solids have the transverse modes and liquids do not, it is the existence of a longwavelength transverse mode that is used to justify the existence of a Wigner solid. The possibility of 2D Wigner crystallization, with and without an applied magnetic ®eld, has been studied by several workers [453±472,1721,1722]. The most important and interesting arguments in favor of the existence of a 2D Wigner crystal are the machine experiment of Hockney and Brown [457] and the computer calculations of Meissner and collaborators [460]. It was in the machine experiment that the authors, considering a system of 104 particles moving in one plane and interacting via Coulomb's law, established that at suf®ciently low temperature the particles are indeed arranged at the sites of a triangular lattice. Inspite of the long-standing theoretical objections to the existence of longrange order in 2D systems, this machine experiment suggests that actual 2D systems overcome these objections by forming ordered domains whose size varies with temperature. Meissner and colleagues, using a three-layer (metal±SiO2 ±Si) structure, calculated the energy of the electrons in an inversion

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channel and demonstrated that a triangular lattice is the most favorable structure Ð the square lattice was shown to have energy only 0.5% larger than that of the triangular lattice but proved to be dynamically unstable. The presence of a perpendicular magnetic ®eld is shown to help solidi®cation even further and enhance the melting temperature monotonically [464]. It has been suggested that, if the quantization of the orbital motion and the discrete energy spectrum are considered explicitly, the electrons belonging to each LL may form a kind of crystal when their concentration is suf®ciently small, although not all the electrons can form a Wigner solid. Such a possibility was considered theoretically by Tsukada [465,1722], who embarked on hexagonal lattice in the Si inversion layer, ignoring the inter-LL interactions. He determined vibrational frequencies in the self-consistent harmonic approximation (SCHA) and showed that the electrons could form a lattice when the occupation ratio of each LL p h=eB0 , was suf®ciently small. The possibility of a CDW ground state similar n ˆ 2pns l2c , with lc ˆ c to the Wigner crystal in origin but quite different from the one suggested by Kelly and Falicov [447,1715±1717] has also been discussed by Fukuyama and colleagues [466±472]. Ever since their conception, the inversion layers in a wide variety of semiconductors have provided a testing ground for theories representing dynamical consequences of the 2D systems through the appropriate dielectric function. By dynamical consequences we mean that these systems are not 2D in a strict sense, both because wave functions have a spatial extent in the third dimension and EM ®elds are not con®ned to a plane but spill out into the third dimension. Among the properties that have attracted comparatively greater theoretical and experimental interest are plasma modes (the plasmons). The most important aspect of a 2D system is its response to EM ®elds, which act to polarize the plasma medium. The polarization of the plasma converts the bare Coulomb interaction into an interaction of ®nite range and hence cures the otherwise appearance of logarithmic divergences. This is the well-known plasma screening, all plasmas exhibit it. In fact, it could be well stated that a medium is not a plasma unless it does screen external electrostatic ®elds. A considerable body of theoretical work on plasmon dispersion in space-charge layers accumulated during the past three decades has been seen to aid in the interpretation of experiments [473±524,1723±1726]. This includes the work done with both simple semiclassical and quantal approaches Ð considering the in¯uence of, e.g., retardation effects, magnetic ®elds, carrier collisions, optical phonons, etc. We would like to recall brie¯y some important contributions that led to advance our theoretical understanding of the electronic and optical properties of 2D systems, which played a remarkably pivotal role in the development of the later emerging more sophisticated 2D, 1D, and 0D systems. Theoretical work, directly or indirectly related to the plasmon dispersion in space-charge layers, was triggered by a classic paper of Stern [88] in which he calculated the 2D polarizability function within the self-consistent ®eld approximation (SCFA). This one as well as many subsequent works [90±92] have based on the assumption that the system is ideally 2D, which makes some sense only in the limit of low carrier concentration and zero temperature. Note, however, that at high temperatures more than one subband can be populated with carriers, and the situation demands drastic theoretical modi®cations. Such generalizations were considered by Stern and Howard [374] and more fully by Siggia and Kwok [474]. The realistic calculations require a self-consistent solution of SchroÈdinger and Poisson equations. This was also accomplished ®rst by Stern [475] and later by others [504,513,514,517,607]. An alternative way of treating the response of 2D systems is to use the HDM for an electron gas, with appropriate charge-density pro®le. A great advantage of the HDM over SCFA is that one is allowed to do a reasonable amount of analytical work giving one a qualitative feel of the step-by-step calculations.

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Fetter [92] was the ®rst to make extensive use of HDM to investigate the nature of plasmons in a 2DEG. The importance of retardation effects on the plasmon dispersion in the long-wavelength was stressed by many authors [477±481,489,497]. Interesting feature of the plasmon dispersion arises if the geometry and properties of the media surrounding 2DEG are varied. That is, e.g., if the insulator is bounded by another semiconductor or semimetal. Opposite charges will be induced on the opposite side of the insulator and there will be two coupled modes for moderately thin insulator. The dispersion relation simpli®es considerably if the bounding semiconductors are identical but oppositely doped (i.e., one is n-type and another is p-type). This problem was considered by several authors [481,486,490,505,512,1726]. Takada [490] pointed out the existence of an acoustic plasmon in a single inversion layer (with a single type of charge carrier) if more than one subband is occupied. In an ideally 2D system there is one collective mode, i.e., a plasmon in which all the electrons in the system oscillate in phase. The multi-subband system, on the other hand, can give rise to an additional mode Ð the socalled acoustic plasmon Ð in which electrons of each subband oscillate out of phase. Beck and Kumar [483,1723] calculated the plasmon dispersion taking correlation effects and the ®nite spatial extent of the inversion layer into account. Rajgopal, in his paper on the longitudinal and transverse dielectric functions of a 2DEG [488], pointed out an erroneous term in the original result of Beck and Kumar; just as he uncovered several errors in Ref. [476]. The effect of three different (Hubbard, RPA, and STLS) approximations to the many-body effects was considered by Jonson [485] to report the model-dependent result, although the general trend predicted by each many-body theory was shown to be unaffected. An interesting coupling between surface phonons and 2D plasmons leading to a new mode of excitation Ð the so-called surface plasmeron Ð was investigated by Gersten [496,1724]. The possibility of using 2D plasmons as a source of generation of the bulk and surface hypersound waves at the frequency of the plasmons was suggested by Chaplik and Krasheninnkov [498]. The effects of an applied magnetic ®eld on the plasmon dispersion in 2DEG have also been considered by many authors [478,480,482,493,499]. Interestingly, a consideration of the non-local effects in the presence of a magnetic ®eld leads to resonance structure at o ˆ noc (n > 1). This gives rise to additional branches in the spectrum of magnetoplasmon dispersion. Precise results have been reported by Chiu and Quinn [478] and Horing and Yildiz [482] using RPA and by Lee and Quinn [480] using Fermi-liquid theory. Considering a realistic system of quasi-2DEG, a few authors have made an extensive study of the electrodynamics of the inversion layers in the RPA [479,489,491]. It was demonstrated that at wave vectors larger than the Fermi vector, a variety of interesting physical effects, such as mixing of intersubband transitions and plasmons, should become available. In some respect, Ref. [491] extended the hypothesis of Burstein and coworkers [438,1708]. The present author embarked on a systematic study of the ®nite, double-inversion layer structures with metallized surfaces, both with [508±510] and without [505] an applied magnetic ®eld. All the three (perpendicular, Voigt, and Faraday) con®gurations were considered to investigate the dependence of the coupled magnetoplasma modes on the insulator thickness and size of the layered structure. It is worth mentioning that the model developed was suf®ciently realistic in that it does take 3D effects in the bulk into account, besides the 2DEG and 2DHG resting at the opposite sides of the insulator. Numerous qualitative changes in the behavior characteristic of plasma modes, due to both the magnetic ®eld and its orientation, were reported. In Ref. [512,1726], the author also investigated a doubleinversion layer system in the presence of a perpendicular magnetic ®eld, in the local approximation. The motivation was to study resonance splittings due to optical phonons in the insulator. Almost at the

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same time, Kosevich and associates [511] investigated a simpler geometry of a single 2DEG embedded in two dissimilar dielectrics with perfectly grounding surfaces in the local approximation. Let us ®nally recall some important contents from Refs. [521±523]. Merkt has shown that, in the quantum limit, there are two distinct dipole excitations of a strongly interacting, disordered 2D system [521]. One mode is CR with a quadratic shift (because e±e interactions in a disordered system overcome Kohn theorem) that re¯ects the disordered potential, the other is a disorder-induced mode with a dispersion inversely proportional to the magnetic ®eld strength. Liu and colleagues [522] have investigated a bilayer (e±e or electron±hole) system in the absence of any external magnetic ®eld using STLS approximation generalized for layers of unequal densities. They ®nd that static interlayer correlations have no effect on the intralayer correlations. For a system of electron±hole layers, they report that decreasing the spacing between the layers gives rise to a divergence in the static susceptibility of the liquid that signals an instability towards a charge-density ground state. Further, they claim to have seen evidence of a precursor to the onset of excitonic bound state which is preempted by the charge-density-wave instability. The acoustic plasmon is shown to exhibit a crossover in behavior from a coupled mode to a mode that is con®ned to a single layer. For the electron±hole case, they ®nd that proximity to the charge-density-wave instability has an unusual effect on the dispersion of the optical plasmon mode. A short note by Price [523] shows that the condition for the number of electric (from the R 1subbands 1=2 V dz converges, quantization in an accumulation or inversion layer) to be ®nite is that the integral where V…z† is the potential as a function of depth z. 2.2.2. Quantum wells and superlattices The literature reveals that the idea of a synthetic superlattice dates back to the early 1960s, when Keldysh [525] suggested that a periodic potential could be produced arti®cially by a periodic deformation of a sample by the ®eld of a high-power standing ultrasonic wave. The major breakthrough in further pursuit of the idea of a superlattice came in 1970 with the announcement of the possibility of detecting negative differential conductance in a crystalline structure with a built-in periodic variation in the lattice potential by Esaki and Tsu [526,1727]. The idea of superposing a periodic potential immediately led DoÈhler [527] to propose a semiconductor structure composed of ultrathin layers with alternating n- and p-doping, where the superlattice potential is due to the space-charges which are variable. To put the three proposals in a nutshell, a periodic potential could be produced by a sound wave [525], a composition variation [526,1727], or doping modulation [527]. So, a superlattice is formed when a periodic spatial modulation is imposed and diminishes the effective dimensionality of a resultant structure (from three to quasi-two). The present discussion will be con®ned to the compositional and doping (or n±i±p±i) superlattice (see Fig. 5). The presence of this additional periodic potential (with a period greater than the original lattice constant) alters drastically the band structure of the host materials and let the superlattices acquire some exotic electronic and optical properties uncommon to the homogeneous (bulk) samples. Since the superlattice period is much greater than the original lattice constant, the Brillouin zone is divided into a series of minizones, giving rise to narrow allowed minibands separated by minigaps (see Fig. 6). These minibands have a non-zero band-width and allow electron motion along the whole length of the superlattice. Subbands may be regarded as a special case of minibands with zero band-width. If the characteristic dimensions, such as the superlattice period and layer thicknesses (or widths of potential

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Fig. 5. Schematics of the spatial variation of the conduction and valence band edges in two most widely studied superlattices: (a) doping superlattices; (b) compositional superlattice.

wells or barriers), in the semiconductor heterostructure are reduced to less than the electron mean-free path, the entire electron system enters into a quantum regime, where the classical physics affords failingly. To the understanding of sophisticated but interesting behavior of a superlattice system, the knowledge of band-gap engineering is paramount. Band-gap engineering consists of the tailoring of an extremely large number of combinations of materials in order to custom-design the resulting structure for some desired properties unattainable in the host constituents. This allows one to design a large variety of new energy band diagrams through the use of band-gap grading. One of the most dramatic consequences of band-gap engineering is the ability of independently tuning the transport properties of the charge carriers (both electrons and holes), using the difference between the conduction and valence band discontinuities in a given heterojunction. Semiconductor heterointerfaces exhibit an abrupt discontinuity in the local band structure, usually associated with a gradual band bending in their vicinity, which re¯ects space-charge effects. According to the character of such discontinuity, known heterointerfaces are classi®ed into four kinds: type-I, type-IIA, type-IIB, and type-III, as illustrated in Fig. 7. The conduction band discontinuity DEc is equal to the difference in the electron af®nities of the two semiconductors. Type-I applies to the GaAs±Ga1ÿx Alx As system, where their energy-gap difference DEg ˆ DEc ‡ DEv . Type-IIA refers to the In1ÿx Gax As±GaSb1ÿy Asy system, whereas typeIIB applies to InAs±GaSb system, where their energy-gap difference DEg ˆ jDEc ÿ DEv j and electrons

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Fig. 6. Potential pro®le of a superlattice (upper panel) and electron energy versus wave vector in the minizones (lower panel).

Fig. 7. Discontinuities of band edge energies at four kinds of heterojunctions: band offsets (left), band bending and carrier con®nement (middle), and superlattices (right).

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and holes are con®ned in the different semiconductors at their heterojunctions. Type-III is exempli®ed by HgTe±CdTe system, where one constituent (HgTe) is a zero-gap semiconductor. This type of superlattice cannot be made from III±V semiconductors. The advancements in crystal growth techniques such as MBE [528,1728] and metalorganic chemical vapor deposition (MOCVD) [529,1729] have made possible the growth of semiconductor heterostructures with controllable changes of composition and doping on a reliably ®ne scale. The ®rst man-made semiconductor superlattice reported by Blakeslee and Aliotta [530] was GaAs1ÿx Px system Ð with alternating values of x (the mole fraction of phosphorous) Ð grown by using vapor phase epitaxy (VPE). Heterostructures which exhibit quasi-2D behavior have also been made using liquid phase epitaxy (LPE) [531]. For the status of the existing semiconductor heterostructures and the epitaxial techniques used, the reader is referred to several useful review articles [532,1731±1734] and similar upcoming proceedings of the EP2DS [87]. In this section, I would like to cover some space on the fundamental concepts associated with compositional and doping superlattices, before discussing the status of the theory and experiment related to plasmons and magnetoplasmons in such systems. A compositional superlattice is a 1D periodic array of ultrathin layers of two different semiconductors in alternation. The two semiconductors are chosen to have their band-gaps that differ in width. The periodic alternation of layers gives rise to a periodic variation in electric potential. Each layer of a semiconductor with narrower band-gap produces a potential well. Inside each potential only certain energy states are available to conduction band electrons; again each state is split into a miniband (of generally a non-zero width). The situation is quite similar to a crystalline solid, where there is a natural periodicity due to a regular pattern of atoms. Each atom creates its potential well wherein only certain energy states are available to electrons. This similarity is, however, accompanied by a drastic difference. The electronic properties of a crystalline solid are predetermined by nature, whereas those of superlattice can be designed by, e.g., the appropriate choice of the host semiconductors and their layer thickness. Moreover, the width of the miniband (determined by the strength of the interaction between the neighboring potential well) can also be tailored Ð the width increases as the layer thickness of the wider band-gap semiconductor decreases. The minibands provide an ideal environment for realizing the important phenomenon of solid state physics known as Bloch oscillations, which seems to be undetectable in a natural crystalline solid. It relies on the fact that a high electric ®eld tilts the bands of a semiconductor. The slope of the tilt is given by eV=L, where V is the voltage applied and L the length of the crystal; thus, for a given length of the crystal, the slope is proportional to the voltage. The applied voltage drives the electrons in the conduction band toward the upper edge of the tilted band. In a conventional semiconductor the bands are broad and the electrons lose the energy they have gained from the ®eld long before they reach the upper edge (of the tilted band) by emitting phonons, i.e., by exciting thermal vibrations in the crystal. In the superlattice the situation is different. The minibands are narrower, and hence the probability of electrons reaching the upper edge of the tilted band is higher. At the upper edge the electrons are re¯ected, in essence because they cannot pass through the minigaps into the next miniband. After such a re¯ection (the quantum phenomenon called Bragg re¯ection) they turn back toward the bottom edge of the miniband. In fact, they may be re¯ected repeatedly so that they shuttle back and forth between the upper and the lower edge of the miniband, performing many of these Bloch oscillations before they lose energy, emit phonons, and fall to the lower energy. A remarkable thing about the Bloch oscillations in a compositional superlattice is that Esaki and Tsu predicted them before the superlattice existed.

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The average distance by which the emission of a phonon shifts the position (the CM) of an electron along the conduction band decreases with increasing tilt of the band. Thus an increase in the voltage applied to a superlattice can have a curious effect of making the current ¯owing through it decrease. In other words, the crystal can exhibit a negative resistance: it can refrain from consuming energy like a resistor and instead can feed energy into an oscillating circuit. A superlattice might therefore serve as an active element of EM wave generator. Unlike other negative-resistance devices it would react almost instantaneously to a voltage; and hence it could generate microwave radiation with wavelengths of less than 1 mm. The aforesaid discussion is pertinent to the case where the charge carriers in the layers are generated by the absorption of light. Such charge carriers, however, disappear rapidly owing to the electron±hole recombination when the excitation of the crystal has stopped. Alternatively, one can introduce a permanent population of electrons or holes by doping the semiconductor with impurity atoms (donors or acceptors). A doped semiconductor is an electron or hole conductor depending upon whether it was doped with donors or acceptors Ð it is no longer an insulator even at the temperature of absolute zero. Unfortunately, the donors and acceptors do more than just provide the charge carriers. They reduce the mobility of the charge carriers. The reason is that a donor, having given up an electron, or an acceptor, having gained one, becomes an ion and therefore gets an electrostatic ®eld that impedes the motion of the charge carriers by scattering them. This is disadvantageous for the engineer, because the charge carriers would not move as fast as one would wish. This is also disadvantageous for the scientist interested in studying quantal aspects of many-body problem, because the interaction between electrons will be obscured by the interaction between electrons and impurity atoms, which could be of comparable strength. In a compositional superlattice one can get rid of such a disadvantage by a simple trick: con®ne the doping of donor atoms to the layers of the semiconductor that has a larger band-gap. Each donor will release a free-electron to the crystal, and such electrons will seek the states that have lowest energy. These states will, however, be available in the layers of other semiconductor (with smaller band-gap). The electrons will thus achieve spatial separation from the donors, and the conductivity (rather conductance) of the system may be made many times greater than it would be in a conventional semiconductor with the same concentration of the charge carriers. This trick was ®rst used by Dingle and coworkers [533] in 1978 to con®rm the theoretical prediction. There are some well-known similarities between type-I and type-II compositional superlattices. For thin layers, the subbands of electrons and holes tend to be farther apart in energy; subband widths also widen with an increasing degree of non-linearity in the dispersion relation. In the opposite case, the subbands shrink to discrete states and the electronic system becomes virtually 2D with a DOS independent of energy except at the onset of a subband where it takes a step rise. Important differences exist, however, between these two types of superlattices. Unlike type-I superlattice where both electron and hole states are primarily con®ned in GaAs layer, electrons mainly exist in InAs (or In1ÿx Gax As) while holes in GaSb (or GaSb1ÿx Asx ). This spatial separation of electron and hole states has obvious consequences in the optical properties such as absorptions and carrier lifetimes. Another difference, perhaps more important, is the existence of energy range between Ec1 and Ev2 (subscripts c and v refer to the conduction and valence bands and 1 and 2 to the ®rst and second host semiconductors) in InAs±GaSb and In1ÿx Gax As±GaSb1ÿy Asy superlattices of low compositions, where both electron and hole states can be present simultaneously. As the layer thickness is increased, Egs (the effective band-gap) decreases and may become zero and eventually negative, in contrast to the situation in

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GaAs±Ga1ÿx Alx As system where Egs is limited to the energy gap of GaAs. The crossover of E1e and E1h in the band structure of InAs±GaSb, seen to occur at a critical width of the ®rst layer, was suggested [534] to correspond to a semiconductor-to-semimetal transition, beyond which electrons ¯ow from the valence band of GaSb to the conduction band of InAs (and leave behind an equal number of holes). The situation is quite remarkable in that a semimetallic superlattice can be created out of two host semiconductors, and large densities of carriers, spatially free from impurities, can be generated. However, there remain some unanswered questions related with this semiconductor-to-semimetallic transition. For instance, the theoretical claim by Altarelli [535,1735] that the system can never become semimetallic, except for extrinsic doping effects due to residual impurities, raises a serious objection against the experimental ®nding of the said effect. Let us now have a cursory look at the fundamental characteristics of the doping superlattices. A doping superlattice usually refers to the system of a periodic array of n- and p-doped layers, separated, perhaps, by undoped layers of the same semiconductor. The latter layers are ``i'' layers, where ``i'' stands for intrinsic, and hence the superlattice is a n±i±p±i crystal. The resulting residual charge (positive in the n-layer and negative in the p-layer) of the ionized donors and acceptors would create a periodic electrostatic potential which would in turn modulate the conduction and valence bands as much as it does in a compositional superlattice. One great advantage of such a doping superlattice is the ¯exibility with which it can be tailored. For one thing, any semiconductor can be chosen to be the host material for the doping superlattice, provided that both n and p doping are allowed. (A single source of GaAs containing an amphoteric group IV element (Ge) as a dopant is a good example. The dopant on either Ga sites resulting in n-type material or on As sites resulting in p-type material is regulated by an intentional change of the growth composition.) In contrast, the choice of two semiconductors for a compositional superlattice turns out to be strongly constrained by the requirement of their being latticematched. Lattice-mismatched or strained-layer superlattices have also interested some workers. See, for instance, a considerable body of work done by Osbourn and colleagues [536,1736]. The effective band-gap of a doping superlattice can be given any value from zero to the band-gap characteristic of undoped host material. For suf®ciently large doping and layer thickness, the superlattice behaves like a semimetal. It should be pointed out that the electronic and optical properties of a doping superlattice can be designed even after the fabrication, say by a weak optical excitation of the crystal or the application of a low electric current. This tunability, which differentiates doping superlattices from their compositional counterparts, arises from the spatial separation between electrons and holes that lack partners for recombination. If the separated charge carriers obeyed the laws of classical physics, they would never recombine, unless, say, thermal energy in the crystal happens to push them to the same location. The recombination by this mechanism is thus likely only at high temperature. Abiding by the laws of quantum physics, however, another means of recombination exists: the particles can tunnel through a potential barrier. Still, the probability of recombination by tunneling decreases dramatically with increasing height and width of the barriers. The theoretical calculations, therefore, predict that the doping superlattices can be tailored so that the free charge carriers can have lifetimes ranging from a few nanoseconds up to several hours. The unlikelihood of recombination (of electrons and holes) seemed at ®rst disadvantageous, because emission of light by a crystal requires that electrons and holes recombine; which is inconsistent with longer lifetime of the charge carriers. Actually, this is not the case Ð the possibility of recombination comes into being automatically. Since the recombination is unlikely, the concentration of the charge carriers rise; thus the potential ¯attens and the effective band-gap widens. The ¯attening increases the

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likelihood of tunneling, leading to the shortening of the lifetime of the charge carriers. This happens when the n±i±p±i crystal is subjected to the strong illumination. The testing of the optical properties enabled DoÈhler and coworkers to verify the predicted tunability of doping superlattices. The most impressive proof was provided by the luminescence experiments. At increasing intensities of the incident light, the carrier concentration, the effective band-gap, and consequently, the energy of the photons emitted by the superlattice proved to increase by as much as 20% [537]. Similar increases were observed in doping superlattices excited by the injection of charge carriers. It was, therefore, suggested that doping superlattices were worth examining for their usefulness as modulators of light in, e.g., electronic systems that process light beams. All these phenomena predicted for doping superlattices were quite unique and although they appeared quite appealing with respect to the fundamental physics as well as device applications, their experimental veri®cation came relatively much later than their compositional counterparts. The list of intriguing properties of doping superlattices predicted by DoÈhler [537] in the early 1970s through quantum physics and veri®ed by experiments must also include the experimental veri®cation of the energy differences between minibands. It should be pointed out that previous few paragraphs in this section heavily rely on, in my opinion, the best informative article ever written on the semiconductor superlattices [1805]. A practical guide to the formation of superlattices is as follows. A heterojunction made up of, say, Ga1ÿx Alx As, with the band-gap discontinuity that arises in the presence of an abrupt composition change, creates, in general, a quasi-triangular potential well that can con®ne electrons and lead to the quantum effects similar to those seen in inversion layers. Two such heterojunctions can be combined to produce a quantum well, e.g., a thin layer of GaAs sandwiched between two layers of GaAs± Ga1ÿx Alx As. Such structures, with central layers too thick to show quantum effects, are already known to have long-standing applications in double heterostructure lasers to con®ne both charge carriers and light. As the thickness of the central layer is made comparable to the Fermi wavelength 2p=kF , quantum effects become important and the energy spectrum of the system is modi®ed. If many such wells with adjoining barriers are combined in a regular fashion, a superlattice is formed. The ability of making combination of materials from which heterojunctions Ð and therefore quantum wells and superlattices Ð can be made has stimulated a tremendous research interest, both experimentally and theoretically, in these systems of generally quasi-two dimensional character. Initial experimental efforts were made to observe the superlattice formation in a semiconductor heterostructure. This reminds us a classic paper by Dingle and coworkers [538] that constituted the ®rst extensive report on the optical characteristics of a ®nite MBE-grown GaAs±Ga1ÿx Alx As superlattice. They reported on the optical-absorption measurements that gave a clear evidence of the coupling of both electron and hole states in closely spaced potential wells. By monitoring the evolution of GaAs absorption spectrum as the number of coupled wells was increased from 1 to 10, an unequivocal evidence for the tunneling of electrons and holes through Ga1ÿx Alx As barriers was reported. Structures with 10 or more coupled wells were shown to approximate the superlattice regime, whereas structures with fewer wells were well described in terms of interacting single wells. The coupling behavior of the wells (with widths in the range of 5±20 nm) proved that synthetic superlattices could indeed be formed. Subsequently, Manuel et al. [539] reported on the observation of enhancement in the RS crosssection for photon energies near electronic resonance in GaAs±Ga1ÿx Alx As superlattices of a variety of con®gurations. Both the energy positions and the general shape of the resonance curves were shown to

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Fig. 8. Schematics of the energy band diagram of an MD GaAs±Ga1ÿx Alx As heterostructure before (left panel) and after (right panel) the charge transfer has taken place. The relative energy bands are, as usual, measured relative to the vacuum level situated at an energy f (the electron af®nity) above the conduction band. The Fermi level in the GaAs±Ga1ÿx Alx As bulk material is supposed to be pinned on the donor level, which implies a large donor binding energy (x  0:25).

be in good agreement with those derived theoretically based on the two-dimensionality of the quantum states in such superlattices. This was, perhaps, for the ®rst time that the strong enhancement of resonant RS arising from 2D nature of the electronic states had been demonstrated. Esaki and associates [540] observed magnetoquantum oscillations (SdH effect) in a few exemplary GaAs±Ga1ÿx Alx As superlattices. The observed SdH oscillations manifest the electronic subband structure, which becomes increasingly 2D in character as the band-width is narrowed. The excellent agreement between the experimental observations and the theoretical calculations Ð not only in the Fermi energy or the period of oscillations but also in their dependence on the ®eld orientation Ð had demonstrated and brought into focus the central feature of a superlattice system, that of control of subband dimensionality and anisotropy. Dingle and coworkers [541] embarked on the growth and properties of MBE-grown heterojunction superlattices of GaAs±Ga1ÿx Alx As in which the independent notion of modulation doping was introduced (see Fig. 8). They made a detailed investigation of uniformly doped (UD) as well as modulation-doped (MD) superlattices. Typical deliberate uniform doping can produce superlattices with electron concentration n  1018 cmÿ3 . In this doping regime, mobilities signi®cantly greater than m  103 cm2 Vÿ1 sÿ1 are dif®cult to achieve. Such mobilities, which are considerably below the BHD (Brooks±Herring±Dingle) limit for uncompensated GaAs at this concentration, severely limit the usefulness of these structures. The MD superlattice is achieved by synchronization of Si and Al source ¯uxes so that only the Ga1ÿx Alx As layers are deliberately doped with Si impurities Ð with the intention that all the mobile carriers (electrons con®ned to the GaAs layers) and their parent donor impurities (in the Ga1ÿx Alx As layers) are spatially separated from each other in an irreversible manner. This is the

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most important feature of the MD structures, which can usually lead to achieve mobilities greater than the upper limit set by BHD theories. This is because there the Coulomb interaction with ionized impurities Ð responsible for the T 3=2 scattering-law Ð should be greatly reduced by the segregation of carriers and impurities. This yields a more metallic-like behavior; the mobilities showing a smooth increase with decreasing temperature. Dingle and coworkers described MBE-grown MD superlattices in which low-temperature and room-temperature electron mobilities can be signi®cantly higher than those in equivalent GaAs grown in other ways. These four experiments Ð together with those reported in Refs. [82,83] Ð triggered unbeatable interest and enthusiasm for the later development of the ®eld. During the past two decades, many research groups have performed several kinds of experiments [542±636,1737±1740], preceded or followed by the extensive theoretical work [637±833,1741±1760], in order to explore the fundamental and device physics associated with the quantum con®nement of the charge carriers in the single and/or multiple quantum wells (MQWs). The distinction between the MQW and superlattice depends on the relative magnitude of the barrier width b and of the wave function penetration depth (d) in the barrier. In the multi-quantum wells, b  d ) b=d  1, so most of the physical properties are those of a series of uncoupled wells. Conversely, in superlattices, b < d ) b=d < 1, and the tunnel-coupling among wells signi®cantly modi®es the physical properties of the system. Even though we cannot afford, for the reasons of space, to discuss each and every work referred to in the review, we will intend to highlight some of them, particularly, those that bear some direct or indirect relevance to the collective excitations in the system. Just as in the inversion layers, free charge carriers in the semiconductor heterostructures reveal characteristic many-body effects in the 2DEG realized at the heterojunctions. The MD structures are particularly suitable for studying fundamental many-particle interactions because extremely high mobilities and negligible scattering rates can be achieved for the free carriers con®ned in the well. The Coulomb interactions, reduced dimensionality, and coupling of the wells (in the MQW system) manifest themselves in the spectrum of elementary excitations. The interacting electron gas sustains collective charge-density excitations (CDEs) and spin-density excitations (SDEs); as well as singleparticle excitations (SPEs) that correspond to the weakly interacting quasi-particles or electron±hole pairs [637±639]. Each of the three types of excitation can be divided into two groups: intrasubband modes, which correspond to the charge-density oscillations in the x±y plane of con®nement; and intersubband modes, which correspond to the charge-density oscillations along the z-direction (perpendicular to the plane of con®nement). The separation of the elementary excitation spectra into intrasubband and intersubband modes is, in general, valid in the long-wavelength limit. The distinction also remains approximately valid, however, even at ®nite (but not too large) wave vectors, because the coupling between the two types of excitations is usually small in the non-resonant situations. More speci®c comments on this coupling will be made later. In the dispersion spectrum of (both intra and intersubband) CDE, SDE, and SPE, the shifts of the collective excitation energies from the single-particle transition energies are a measure of the strength of Coulomb interactions [85]. It has been realized that these interactions have consequences that are unique to 2DEG. In a dense 2DEG, such shifts from subband spacings are understood to take place due to two effects: the depolarization effect and the excitonic effect (see the preceding section). The upward shift of the CDE with respect to SPE is due to depolarization effect, while the downward shift of the SDE from the SPE is due to excitonic effect. The need to understand fundamental interactions stimulated intensive theoretical and experimental research as discussed in what follows.

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The collected list of experimental works includes RS, infrared emission, and transmission spectroscopy used to observe electronic, optical, and transport properties of heterojunctions, quantum wells, and superlattices, both with and without the applied magnetic ®eld. We ®rst consider the experiments on single quantum well and MQW [542±620,1737], excluding some speci®c work on DQWs [621±632,1738±1740] and on the spin-density waves [633±636] for the subsequent discussion. The ®rst category of experiments focuses mostly on the observation of elementary excitations using diverse forms of RS [542,544,546±548,551,554,560,561,566,569,580±582,594,601,604,606,612,616, 618,620,1737]. Inelastic light (or Raman) scattering had proved to be a very successful tool to study single-particle and collective excitations spectra of 3D plasmas much before the era of reduced dimensionality began. Initial breakthroughs in the application of RS to probe the systems of lower dimensions were undertaken by Aron Pinczuk (at Bell Laboratories, Murray Hill, NJ), who is known to have re®ned and developed new methods of using this technique accordingly. Pinczuk et al. [542] reported, perhaps, the ®rst observation of intersubband excitations of the multilayered 2DEG in the MD GaAs±Ga1ÿx Alx As heterojunction superlattices, using resonant RS. Their ®ndings included intersubband spacings, spectral line shape with interesting energy dependence, and resonant enhancement curves with width comparable to Fermi energies. Unexpected and by then unexplained effects in the polarization spectra were speculated to be related to the Coulomb interactions between charge carriers in the system. RS technique is specially powerful because it is capable of measuring both single-particle and collective excitations, including SDE which involve spin-¯ip (SF) processes. Pinczuk and associates [546] made the ®rst direct observation and evaluation of the depolarization effect associated with intersubband transitions of a 2DEG in the type-I superlattices. The evaluation of the two many-body effects was hypothesized as follows. The polarization selection rules [542,544] allow the observation of intersubband transitions which are either pure spin-density (SF) or pure charge-density (non-SF) excitations. Because SDEs are not subjected to the depolarization effect, they have single-particle character with energies which may be shifted from subband spacings by excitonic effect. The differences between the two types of spectra reveal directly the depolarization effects. The excitonic effects can be investigated by comparing the energies of the intersubband SDE with the calculated subband spacings. The collective character of CDE was established by the observations of their coupling with GaAs LO phonons. This is known to be the characteristic signature of the depolarization effect in a polar semiconductor. As regards the excitonic effect, they found that measured energies of SDE were systematically lower than the calculated subband spacings. This indicated that excitonic effect in GaAs, although quite likely smaller than in Si, may not be negligible. These ®ndings proved to be the real ``stamp'' for further use of the resonant RS with the aim of elucidating many-body effects in 2D systems. Pinczuk and colleagues [547] later reported on the resonant RS by photoexcited electrons in GaAs± Ga1ÿx Alx As (with x ˆ 0:2) MQWs to discover that a variable-density 2D electron±hole plasma is created in a controlled manner. Surprisingly enough, they found that energy level structure remained unaltered by photoexcitation, indicating that for densities as high as  3  1012 cmÿ2 there is no spatial separation between electrons and holes. The collective excitations, broadened by the variable plasma density in different wells, display a complex behavior due to coupling among different well excitations and also with LO phonons. In a 2DEG with spatially modulated charge density, Mackens et al. [551] observed a splitting of the 2D plasmon dispersion. The existence of minigaps in a charge-density modulated MOS capacitor was attributed to the superlattice effect, which comes into being due to the

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lateral periodic potential variation. The excitation of plasmons in such periodic lateral microstructures was investigated by transmission spectroscopy. It was suggested that if one is able to prepare such periodic structures with grating period comparable to the electron mean-free path (by, e.g., using smaller periodicity), the superlattice effects will have a strong and controlled in¯uence on the SPE spectrum. In the MD GaAs±Ga1ÿx Alx As quantum well superlattices (with x ˆ 0:23), Sooryakumar et al. [554] performed RS measurements to observe collective intersubband excitations over a wide in-plane wave vector (qk ) range. Their results revealed the importance of dispersion terms to order q2k and provided the ®rst veri®cation of theoretical predictions over a large wave vector range. While a theorist has the option to treat a superlattice system as perfectly periodic, semi-in®nite, or ®nite structure, an experimentalist has no other choice but to use a ®nite system, even though he can approximate the superlattice regime by increasing the number of periods (N) in the system. During the course of experiment, one can always approximately determine the minimum number of periods (Nmin ) required to observe the superlattice behavior. For N < Nmin, the experiment should reveal the discrete states as predicted by theory for a ®nite superlattice system Ð realized by truncating the ideal superstructure on both (top and bottom) sides. Pinczuk et al. [560] observed discrete plasmons in layered 2DEG with a ®nite number of periods (N ' 15). They studied several MD multiple Ê ), with x ˆ 0:24, quantum well heterostructures using RS. In the Ê )±Ga1ÿx Alx As(520 A GaAs(258 A 11 ÿ2 system with ns ˆ 4  10 cm and m ˆ 9  104 cmÿ2 Vÿ1 sÿ1 , they determined a lowest subband spacing of 13 meV, with ns < 0:2  1011 cmÿ2 in the ®rst excited subband. The observed spectra are characteristic plasmon doublets. These are explained by the lifting of the degeneracy of pairs of modes with wave numbers q? and (2p=d ÿ q? ) in the ®rst Brillouin zone of the superlattice, where d is the period and q? …ˆ 2pn=Nd; n ˆ 1; 2; 3; . . . ; N† is the normal component of the wave vector. In the perfectly periodic superlattice these modes are degenerate because each 2DEG is at a plane of mirror symmetry. The lifting of the degeneracy in ®nite superlattice is due to the loss of these symmetries. The observed plasmon dispersion was found to be in agreement with the theoretical results of Jain and Allen [662]. RS measurements on an MD ®nite MQW GaAs±Ga1ÿx Alx As system (with x ˆ 0:3) with ®ve periods were also performed by Fasol et al. [561] to report the discrete plasmons manifesting the coupling of the electrons in different layers by Coulomb interaction. Comparing the observed spectra with Jain's theory [662], they were able to ®nd good agreement for the modes n ˆ 1±3. The theoretical dispersion curves for the lowest mode (n ˆ 5) deviate somewhat from the measured values. The calculated mode for n ˆ 4 does not correspond to a measured mode. These deviations were attributed to the fact that the calculations [662] assumed in®nitely thin electron layers, while in realistic samples the layers had a considerable spatial extent perpendicular to the wells. The same group later reported the RS measurements of the qk dependence and temperature dependence of the SPEs [566], using the same sample as in Ref. [561]. The Raman spectra were well explained by the Lindhard response function, using a phenomenological relaxation time tsp whose temperature dependence was also determined. The ratio of tm (the impurity scattering component of the mobility relaxation rime determined from the Hall measurements) and tsp was shown to be greater than 1 (tm =tsp ˆ 4:0  0:8). Such a high value of tm =tsp is characteristic of the remote impurity scattering in MD samples, where small-angle scattering events are particularly important. The photoexcitation of an electron±hole plasma in semiconductors is a powerful tool to probe the interaction of the carriers among themselves and with the lattice. Oberli et al. [569] employed a time-resolved RS technique to investigate the mechanism of intersubband scattering of electrons

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Ê )GaAs± by longitudinal phonons in a 2DEG con®ned to an MBE-grown 30 periods of (215 A Ê )Ga1ÿx Alx As MQWs (with x ˆ 0:3). The lifetime and the intersubband scattering time of the (100 A photoexcited electrons on the lowest subband are uniquely determined separately. These measurements have revealed a relatively long lifetime of the electrons that are photoexcited on the second subband of Ê MQWs. Intersubband scattering by the deformation potential of the acoustic phonons 215 A substantiates this result in the wide-well limit, while intersubband scattering by longitudinal optical phonons becomes dominant in the narrow-well limit. Fasol et al. [580] embarked on an extensive experimental-cum-theoretical investigation of intrawell and interwell coupling of plasmons in an MD MQW GaAs±Ga1ÿx Alx As system (with x ˆ 0:3). The observed spectra were well substantiated by full RPA theory to demonstrate that with N quantum wells and M occupied subbands per well, the intrawell interactions split the plasmon dispersion into M groups of modes. The interwell Coulomb interactions split each group further into N eigenmodes and differ strongly between the different groups of plasmon modes. Considering speci®cally a sample with ®ve periods (N ˆ 5) with three occupied subbands (M ˆ 3) per well, they concluded that splitting of the plasmon modes is a measure of the strengths of intrawell and interwell interactions in an MD quantum wells. This work shows how the RS can be used to characterize MQWs with multiple subband occupancy. At small in-plane wave vectors the energies of the SDE are shifted from single-particle transition energies by the exchange Coulombic interactions (the excitonic effect). The CDE have energy shifts due to direct (depolarization effect) as well as exchange terms. However, since exchange interactions were expected to be small in GaAs, SDE were previously interpreted as the energy spacings of quantum well states and referred to as SPE. Similarly, the shift of CDE from single-particle transition energies was considered in terms of direct Coulomb interactions and coupling to optical phonons. In an eyeopening RS experiment, Pinczuk and colleagues [581] showed that such widely used interpretations of light-scattering measurements required a careful scrutiny. In the small-wave vector intersubband excitation spectra, they found unexpected single-particle transitions, besides the peaks of collective CDE and SDE. Analysis of the energies of the three excitations led them to make quantitative determinations of Coulombic interactions in which the exchange and direct terms were seen to be of comparable strength. A time-resolved antistokes RS measurement of intersubband relaxation time (t21 ) due to the emission of LO phonons in an MD 60-period GaAs(14.6 nm)±Ga1ÿx Alx As(15.7 nm) multiple quantum-well structure is worth mentioning [582]. In these structures, the energy spectrum displays subbands whose spacing increases with increasing con®nement of the charge carriers. For strongly con®ned carriers, where the subband splitting is greater than LO phonon energy hoL, rapid relaxation by LO phonon emission is allowed. Earlier measurements of this process in GaAs quantum wells had given values of the C2 ±C1 intersubband relaxation time t21 > 10 ps (see, e.g., Refs. [1,2] in Ref. [582]). On the other hand, theoretical calculations (see, e.g., Refs. [3,4] in Ref. [582]) found that t21  1 ps for wells with in®nite potential barriers. This discrepancy (of more than an order of magnitude) between theory and experiment has been attributed variously to poor con®nement of electrons in the C2 subband, or to the combined effects of screening, intervalley scattering, and non-equilibrium phonons. The experiment in Ref. [582] provided the ®rst direct measurement of t21 . The measured relaxation time of t21  1 ps was found in excellent agreement with theoretical intersubband scattering via 2D con®ned phonons. In order to investigate the effects of non-parabolicity on the intersubband excitations, Brozak et al. [594] performed the RS measurements on MBE-grown 20 periods of GaSb±AlSb strained-layer

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superlattices. The density of the photoexcited 2D electron±hole plasma was controlled by varying the incident laser power. A comparison of the experiment with the theoretical calculations, within TDLDA, reveals that a quantitative agreement can only be made if the non-parabolicity of the conduction band of the narrower band-gap GaSb is explicitly included. They suggested that the non-parabolicity of the GaSb conduction band can be accounted for in the response functions simply by allowing the qk dependence of the masses. The mass thus varies not only from subband to subband but also within a subband with increasing in-plane wave vector. It is shown that non-parabolicity, in addition to allowing the resonance energies of the collective excitations to decrease, provides a mechanism for observing SPEs at the zero momentum transfer. By means of the resonant RS, Kirchner et al. [601] observed a characteristic shift of holeintersubband transitions with excitation energy in two p-type MD GaAS±Ga1ÿx Alx As MQWs: well Ê ) and x ˆ 0:43 (0.32) for sample 1 (sample 2). The non-parabolicity of the width Lz ˆ 100 A (200 A valence subbands as well as ¯uctuations of the well-width are assigned the possible reasons for the shift. It is, however, suggested that both contributions can be separated practically by the application of an external magnetic ®eld along the superlattice axis, which quantizes the in-plane motion of the carriers. It is shown that the shifts observed at zero magnetic ®eld are mainly caused by subband nonparabolicity. This interaction is con®rmed by simulation of single-particle Raman spectra through a subband dispersion based on Luttinger's 4  4 Hamiltonian. Conventional theories (as well as experiments) predict an inevitably coupled collective motion of the intra and intersubband transitions in the long-wavelength limit in wide quantum wells; but this is clearly not the case for narrow quantum wells. This has been demonstrated by Peng and Fonstad [604] in an experiment using polarization-resolved infrared technique used to study the quasi-2DEG in d-doped AlAs±In0:7 Ga0:3 As±AlAs (3±4±3 nm) single quantum well. It is suggested that distinct excitations of inter and intrasubband transitions are feasible through the use of quantum wells designed to increase their energy difference and thus diminish the likelihood of any coupling between these two processes. Moreover, it is shown that by careful examination of their polarization characteristics, the collective and single-particle natures of intra and intersubband transitions can be clearly resolved. These results should have important implications for the design of future infrared detectors and modulators based on the intersubband spectroscopy. A novel many-body manifestation of fundamental e±e interactions in a very dilute 2DEG is revealed by RS in an MD GaAs quantum well [606]. The 2D density obtained from Photoluminescence (PL) data is continuously tuned in a unique way by applying hydrostatic pressure. A quantitative analysis of the energies of the elementary excitations shows, for the ®rst time, the crossing between SPE and collective CDE in the low-density regime of  2  1010 cmÿ2 . This is an indication that at these densities excitonic shift overcomes the depolarization shift. This effect, which does not exist within the RPA which neglects the vertex corrections entirely, was anticipated within the TDLDA (see the brief discussion related with Ref. [790] below). Optical spectroscopy of intersubband transitions in quantum wells is reliably powerful technique to study the subband structure in general, and the effect of the band non-parabolicity on the subband energies in particular. For conventional GaAs quantum wells, as the fundamental and effective bandgaps involve G-conduction-band minimum, where the electrons are located, inelastic light scattering by both collective and SPEs can be observed. For InAs quantum wells, however, only E1 and E1 ‡ D1 band-gaps lie in the spectral range accessible in resonant RS. Thus only coupled intersubband plasmon±LO phonon modes, which may couple to light via electro-optic, deformation potential, and

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FroÈhlich mechanism, can be observed in RS from InAs quantum wells. For optical excitations in resonance with E1 band-gap of InAs in AlSb±InAs±AlSb quantum wells, the spacing between the ®rst and second subbands was deduced as a function of well-width from the Raman data [612]. The observed results were well substantiated by the theory including the effects of strain and nonparabolicity. Well-de®ned plasma oscillations have also been observed in a superlattice miniband even though the Fermi energy lies in the minigap [616]. Direct observation of the effect of a superlattice on the optical observation of a parabolic quantum well was carried out through CR measurements, along with C±V (capacitance±voltage) measurements to extract the 3D carrier density. Despite the complex band structure, the resonance shows a remarkable insensitivity to changes in the number of electrons in the parabolic well in which the superlattice is placed Ð a feature of the generalized Kohn theorem (GKT) [806] that is expected to hold only in the limit that Fermi energy lies near the bottom of the lowest miniband. This is a consequence of the pinning of the Fermi energy by the parabolic potential. The authors state that such bare resonances as abide by the GKT were observed only in the Faraday geometry and that they could not observe similar resonance in the Voigt geometry. Intersubband resonances in InAs±AlSb quantum wells were also considered by Kotthaus and associates [618], using Fourier-transform spectroscopy. InAs±AlSb systems are very well suited for investigation of band coupling effects for a number of reasons. First and foremost, InAs has a small band-gap so that the mixing of conduction-band s and valence-band p states is much larger than in conventional GaAs system. This mixing is responsible, for instance, for the large conduction-band nonparabolicity observed in this system. Intersubband resonances are described by essentially three properties: the energy, the matrix element hz12 i, and the line shape. It was found that only the transition energy is strongly in¯uenced by the non-parabolicity. The one band estimate of hz12 i differs only by  10% as compared to multiband calculation in the k  p approximation. Moreover, it was shown that the non-parabolicity does not strongly in¯uence the line shape because of the very pronounced depolarization effect. The results on the narrow line shape are argued to be important in a device context; as a broad response would be undesirable for detectors and lasers based on the intersubband transition scheme. RS measurements of coupled intersubband plasmon±LO phonon modes in InAs±AlSb quantum wells, highlighting the importance of InAs conduction-band non-paraboliciity, were also undertaken by Richards et al. [620,1737]. They demonstrated that under strong illumination conditions, electron densities can be as high as  4  1012 cmÿ2 , resulting in a signi®cant population of the second subband. The observed plasmon mode in such condition is shown to be due to an intersubband transition between the second and the third con®ned subbands, not between the ®rst and the second subbands as had been assumed previously [612]. The Raman spectra also provided the clear evidence of Landau damping of 1 ! 3 intersubband plasmon by SPEs, which can exist over a wide energy range due to the large differences in the effective masses of the quantized subbands. Besides the plasmons, phonon excitations in quantum wells and superlattices has been the next major topic to draw attention of a wide class of researchers. Narayanmurti and colleagues [543] observed, for the ®rst time, selective transmission of high-frequency phonons due to narrow band re¯ection determined by the superlattice period. Colvard et al. [545] reported on the observation of RS Ê )±AlAs(11.4 A Ê ) superlattice. The from the folded longitudinal acoustical phonons in a GaAs(13.6 A imposed periodicity along the superlattice axis leads to Brillouin-zone folding (as mentioned before), resulting in the appearance of gaps in the phonon spectrum for wave vectors satisfying Bragg condition

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Ð l0 ˆ 2d; d being the superlattice period. Furthermore, RS revealed con®ned optical phonons [555], interface vibrational modes [556], interface phonon-polaritons in an asymmetrically bound GaAs±AlAs heterostructure [575], and folded acoustic phonons in ®nite n…ˆ 3; 4†-layered superlattices [602]. It is worth mentioning that besides RS, other experimental techniques, such as high-resolution electron energy-loss spectroscopy (HREELS) [559] and FIR±ATR [619], have also been used to probe the interface phonons and related characteristics, particularly in GaAs±Ga1ÿx Alx As superlattice systems. While the optical properties of quantum wells have been the subject of increasing research interest, some studies have also concentrated on the electric-®eld dependence of the linear and non-linear properties, particularly those associated with the exciton absorption resonances near band-gap energy [86]. For electroabsorption with electric ®elds parallel to the interfaces, the effects are similar to those of bulk semiconductors, except that the Coulomb screening is relatively more pronounced and that room-temperature measurements are easily made. However, the large variety of speculative devices using room-temperature optical properties relies on the consequences of electric ®elds oriented perpendicular to the interfaces. Some effects, such as Stark ladder [552,567,573], photon-mediated sequential resonant-tunneling [603], Bloch oscillations [610,613], and dynamic localization and absolute negative conductance [617], are seen in this con®guration that are of device interest related with, e.g., a variety of modulators, and optical switches. The eigenstates of a periodic potential of period d are distributed in bands. If an electron is in a state of a band of width D, an electric ®eld would induce, in the absence of scattering and intersubband transitions, an oscillatory motion of frequency n ˆ eEd=h, restricted to a spatial region D=eE (the Bloch oscillation). The energy spectrum is given by E0 ‡ neEd (n ˆ 0; 1; 2;), where E0 is the eigenenergy of an isolated quantum well (Stark ladder). An essential feature of the above-mentioned effects is Bloch oscillation in the presence of an intense high-frequency electric ®eld suf®cient to drive carriers beyond the minizone boundary, into a region of k-space with negative velocity. It had been suggested by Esaki and Tsu [526,1727] that, were these effects to exist, their observation would be much easier in superlattices because of the reduced widths of their minibands. In close analogy to semiclassical Bloch oscillations in solid state [834,835], the electronic wave packets perform oscillations in real space and k-space under the combined in¯uence of a periodic potential of the superlattice and the static electric ®eld. The frequency of such Bloch oscillations can be tuned by the applied electric ®eld. The electric-®eld-induced effects also include the observation of Tamm states in GaAs±Ga1ÿx Alx As superlattices [588], interaction between extended and localized states in a superlattice in which periodicity is destroyed by a wider well at the end of an otherwise ideal superlattice [589], and the observation of miniband formation in a graded-gap superlattice [607,614]. While all these investigations were made in the situation where the applied electric ®eld was considered perpendicular to the interfaces, an interesting experiment had also been performed with the applied electric ®eld parallel to the interfaces of p-type MD MBE-grown 20 periods of GaAs±Ga1ÿx Alx As quantum wells [558]. Shah and colleagues [558] observed heating of minority electrons in the presence of a cool, highdensity hole plasma in GaAs quantum wells to demonstrate the existence of non-equilibrium between electrons and holes under the in¯uence of strong electric ®elds. This non-equilibrium is drived by high electric ®elds in combination with lattice room temperature, where the energy-loss rates of electrons to the lattice and to the majority holes become comparable. This was, perhaps, the ®rst experimental determination of the energy transfer by the electron±hole Coulomb scattering.

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Out of all the external probes used to study the response of a system, the magnetostatic ®eld is relatively more interesting. This is because the effect of the magnetic ®eld on the band structure is more striking and is easily observed in experiments. A number of interesting phenomena originate from the alteration in the band structure by magnetic ®elds; these have been extensively studied and serve as diagnostic tools for characterizing the materials. The important point about the application of the magnetic ®eld is that although the general characteristics of the band structure remain unaltered, the electron motion (in each band) transverse to the magnetic ®eld becomes quantized. It is this quantization which leads to a host of interesting transport phenomena (e.g., Landau diamagnetism, de Haas±van Alphen effect, SdH effect, CR, quantum Hall effects, etc.). During the past 18 years ever since the discovery of quantum Hall effects, there has accumulated a vast literature on the magnetic®eld effects, particularly in the systems of reduced dimensionality. While we ®nd it quite impertinent to discuss considerably extensive literature related with the quantum Hall effects, we would like to refer to some works associated with the magnetic-®eld effects on optical transitions in 2DEG, among others. In the presence of an applied magnetic ®eld (~ B0 ), there are three main con®gurations (as discussed in ~ Section 2.1) out of which the case where B0 is perpendicular to the plane of con®nement (and parallel to the con®ning electric ®eld) is a very special one. In this case, the magnetic ®eld allows each discrete (electric) subband to split into LLs and leads therefore to a complete quantization, which has been extensively studied in the recent past. In this situation, the Hamiltonian can be separated in an electric part leading to subbands and a magnetic part leading to LLs. For any other orientation, this separation is not possible anymore and the band structure is relatively more complicated. However, there is no dearth of experiments (or even theory, for that matter) where the effects of magnetic ®elds parallel to the interfaces in particular, and tilted with respect to the sample normal in general, have been explored (see, e.g., Refs. [836±838]). The experiments exploiting magnetic-®eld effects include, e.g., the effective mass determination of heavily doped InAs±GaSb superlattices using helicon wave propagation in an FIR transmission [549], band-gap optical recombination of 2DEG in MD GaAs±Ga1ÿx Alx As quantum wells in the extreme quantum limit [574], observation of roton DOS in 2D inter-LL excitations in strong perpendicular magnetic ®elds using RS [576], measurements of Hall resistance of a 2DEG in GaAs±Ga1ÿx Alx As heterostructures in order to determine the spatial extent of equilibration among the N current carrying states in the quantum Hall regime [585], a novel temperature-dependent, magnetic-®eld-induced energy gap characterizing the suppression of tunneling that occurs only in the presence of a perpendicular magnetic ®eld [586], FIR measurements indicating the quenching of collective phenomena in combined intersubband-CR in the MD GaAS±AlAs heterojunction in magnetic ®elds tilted with respect to the sample normal [591], evidence of a strong dependence on energy of electron effective mass and gfactor in the narrow-gap InAs quantum wells, which makes the spin-split CR of a 2DEG observable [598], and the observation that the in-plane magnetic ®eld (in the Voigt con®guration) mediates the intersubband CDE, whereas the bulk inversion asymmetry of InAs induces SDE in the InAs±AlSb MQWs [611]. A recent detailed account of the mechanism adopted to investigate intersubband excitations in the quantum wells and superlattices is given in Ref. [839]. From the initial discussion on the heterojunction, quantum wells, and superlattice formation, it is by now clear how selective or modulation doping makes it feasible to have mobile charge carriers spatially separated from the parent ionized impurities. This has resulted in very high mobility quasi-2D electron and hole gases, and the development of the high-electron-mobility transistor (HEMT). This has stimulated a number of interesting transport and magnetotransport experiments which were performed

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with diverse motivations [550,553,562,564,571,572,577,578,583,584,587,592,593,596,605]. Such experiments were undertaken with the purpose of observing, for instance, miniband width [553], variation of the electron effective mass with the barrier height [562], breakdown of the effective mass p h=eB0 , the magnetic length) is smaller approximation (EMA) in strong magnetic ®eld when lc (ˆ c than d (the superlattice period) [572], the evolution of the CR with the magnetic-®eld strength leading to the breakdown of CR when lc  d [577], the evidence of negative differential velocity (NDV) in the superlattices, supported (speculatively) by the non-parabolicity mechanism for NDV [583], evidence of thermal saturation of the band transport in superlattices [587], miniband transport by (electrical) timeof-¯ight experiments as a function of temperature to reaf®rm the NDV and to show the existence of mobility gap in the superlattices [593], the enhancement of cyclotron mass in InAs quantum well, sandwiched by AlSb barriers, due to the electron-wave function penetration into AlSb barriers [596], among other characteristics. A few remarks on the CR and the saturated response are in order. It is well documented that EMA is the cornerstone of our understanding of electron transport in solids in periodic potentials. The validity of EMA requires that the characteristic length of the probe be much larger than the wavelength of the periodic potential of the system. In the case of the CR, which is known to be the most powerful technique for measuring the effective mass in solids, this requires that the cyclotron diameter be large compared to the period of the potential. Although most solids invariably satisfy this requirement, the character of electron states in a magnetic ®eld when this condition fails to hold was ®rst discussed by Harper [110]. In the intervening years, this problem has attracted repeated theoretical interest, but only with the discovery and perfection of semiconductor superlattices do we have the opportunity of addressing the relevant issues experimentally. Most of the above-mentioned experiments took advantage of this to show how, why, and where the CR takes place when the effective miniband mass approximation breaks down. It has by now become well-known that semiconductor superlattices can be fabricated with precisely engineered band structures which allow the investigation of electron dynamics in the regimes which are not accessible in the materials existing in nature. An important, and largely unexplored, feature of any such narrow-conduction-band system is saturated response. The superlattice systems have made possible the observation of magnetic saturation of band transport, which occurs when the cyclotron energy exceeds the miniband width [553,562,572,577,584]. Another long-sought-after regime, where electrons are accelerated by electric ®elds to gain energies, which span the miniband widths and Bloch oscillations are expected to occur, has also been partially investigated [583]. Next comes the thermal saturation of miniband transport in a superlattice that takes place when the temperature of the sample is increased so that thermal energy kB T becomes larger than the miniband width. Under these conditions, the electric ®eld cannot cause current to ¯ow and the conductance is expected to vanish. The miniband conductance has been observed to quench signi®cantly [587]. In a sense the phenomena of electric, magnetic, and thermal saturation of miniband transport can be exempli®ed by ``overfeeding'' of a human being. A man, for instance, needs to feed himself in order to persist his dynamism; but what if he is made to overfeed himself to the extent that he starts feeling unable even to breath properly. The stage when he does so could be referred to the ``feeding saturation''; no length scales are, perhaps, required to de®ne such an equivalently ``overdrunk status''. Apart from numerous optical and transport measurements, a few efforts have also been made to probe the superlattice systems by high-resolution EELS (HREELS) (see, for instance, Refs.

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[599,600,608,609]). However, the tragedy is that while optical spectroscopies (such as RS technique) have been successfully applied to study the subband structure, HREELS is seen to be limited to characterize only plasmon loss features. There is a sensitivity mechanism associated, unfortunately, with both RS and electron scattering techniques. While RS has high resolution and polarization selection rules, and sees excitations deep in the bulk of the system, electron scattering has high sensitivity to the surface excitations and its cross-section is determined by the strength of the electric ®eld induced by the surface modes. It is therefore inferred that for a precise characterization of the plasma modes of a system, the combination of the advantages of RS and electron scattering is worthy of recommendation. This was done by Ushioda and coworkers [599] to study the strained-layer superlattices. More precise comments on, e.g., the resolution of EELS will be made in the later part of this review. All the experiments on optical transitions, transport properties, and electric and magnetic ®eld effects we have referred to by now were concerned with either ideal quantum wells or perfectly periodic superlattice systems. Some contemporary efforts have, however, also been made to investigate the 1D quasi-periodic systems, which are known with various names, such as aperiodic, incommensurate, nonperiodic, quasi-periodic, or Fibonacci superlattices. The interest here stems from the fact that the Bloch theorem is inapplicable and also that this problem represents, in some sense, an intermediate case between periodic and disordered 1D solids. Quasi-periodicity has been predicted to lead to some interesting, rich physics in these systems, such as the Cantor set spectrum and the existence of localized, extended, and singular continuous states. The main idea is that of Anderson localization and of possible transitions (mobility edges) between localized and extended states. Purposely disordered quasi-1D systems, besides providing a controllable kind of non-periodicity, are self-similar, meaning that their properties are similar at different length scales. Ever since the quasi-periodic superlattices have ®rst been grown and their quasi-periodic ordering con®rmed [557,563], a considerable research interest, both theoretical and experimental, has focused on such systems. The MBE-grown GaAs±Ga1ÿx Alx As system still remains the best documented structure. The experimental efforts include, e.g., the observation of carrier localization and inhibition of vertical transport in intentionally disordered GaAs±Ga1ÿx Alx As systems using photoluminescence (PL) [565], RS from acoustic phonons in Si±Gex Si1ÿx strained-layer [568] and GaAs±Ga1ÿx Alx As [570] Fibonacci superlattices, selective-excitation PL measurements for studying the energy dependence of the motion of resonant excitons in purposely disordered GaAs±Ga1ÿx Alx As superlattices [579], observation of the self-similarity of LLs in a GaAs±Ga1ÿx Alx As Fibonacci superlattice in a parallel magnetic ®eld [590], resonant RS by plasmons in a GaAs±Ga1ÿx Alx As Fibonacci superlattice [595], measurements of effective elastic constants and phonon spectrum in metallic Ta±Al Fibonacci superlattice by Brillouin scattering [597], and PL investigation of a ®nite GaAs±Ga1ÿx Alx As aperiodic superlattice in an electric ®eld to demonstrate the exciton resonance occurring between strongly coupled Stark ladder states [615]. The 2DESs particularly the ones existing in MD GaAs±Ga1ÿx Alx As heterostructures offer several advantages over the 3D electron systems in terms of a systematic study of many-body effects. First, reduced dimensionality typically enhances interaction effects. Second, the charge-density can be varied over (almost) two orders of magnitude in MD 2D systems. Third, these man-made systems can be made ultrapure because the mobile carriers are spatially separated from the parent ionized dopants. Fourth, arti®cial structuring allows the introduction of an additional degree of freedom into the problem, e.g., separation between the neighboring layers, which are unavailable in purely 2D or 3D systems,

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thus allowing the possibility of further tuning the interaction effects. Finally, and perhaps most importantly, the application of a strong magnetic ®eld perpendicular to the layers quenches the kinetic energy of the system because of the Landau quantization, thereby increasing the importance of interaction effects. Because of these reasons, as well as for the obvious reason of technological relevance, there has been a tremendous research interest in the (many-particle) interaction-induced effects in these systems. Perhaps, the simplest extension of the standard 2DES that promises to afford interesting new phenomena is the double quantum wells (DQWs), ®rst proposed by Shevchenko [840] and Lozovik and Yudson [841]. In this structure two 2D electron or hole gases are established parallel to each other separated by a potential barrier thick enough to prevent particles from tunneling across it while thin enough to allow for strong interlayer Coulomb interaction. In such systems many-body correlations due to Coulomb interactions are the crucial ingredients of the detailed description of their behavior [770±787,1756]. For example, 2DEG have proven to be fertile systems in which correlations reveal themselves in a striking multitude of phenomena. In these systems, the kinetic energy can be quenched to the extent that correlations dominate, giving rise to effects such as the fractional quantum Hall effect and Wigner crystallization. In electron±hole double layer systems, the carriers of opposite change occupying adjacent layers attract each other and may form excitions, which condense into a super¯uid phase. In DQW systems, it has been shown that Coulomb correlations lead to interesting phenomena such as new states in the fractional quantum Hall regime. It has also been proposed [842,843] that in double-layer systems (DLSs) interlayer e±e interactions create a frictional force, which drags current through one layer when a current is driven through the other. The magnitude of this drag force is a convenient, direct, and sensitive probe of these important e±e interactions. Collective modes, which are known to be strongly affected by the Coulomb correlations, represent another signi®cant area for the study of many-body effects in reduced dimensions. Interesting experiments performed with the aim of estimating the Coulomb correlations in DQW include, e.g., the measurements of Coulomb drag effect between 2DEG and 3DEG [621], between two 2DEG [623], between 2DEG and 2DHG [624], and between two 2DEG in the presence of a perpendicular magnetic ®eld [630]. In the absence of any magnetic ®eld, the contribution of e±e interactions to the drag rate tÿ1 D has generally been attributed to Coulomb processes; phonon-mediated e±e interactions were also speculated to be potentially important, however [627]. In the presence of a magnetic ®eld and at low temperatures, where the drag is shown to be more sensitive to the spin splitting of the LLs than the SdH oscillations, an enhancement of the drag by two orders of magnitude over the zero-®eld case is claimed [630]. The magnetotransport experiments in DQW systems reveal a high magnetic-®eld regime evidenced by missing quantum Hall states at ®lling factors n (ˆ hNtot =eB† ˆ odd [622]. This regime is explained to depend critically on the additional degree of freedom (in the third dimension) provided by the DQW structure. Similar studies later evidenced a new FQH state (resembling the original Laughlin 13 state) that has no known counterpart in a single layer 2DEG system [625,1738]; its existence was established due to an interplay between intralayer and interlayer Coulomb interactions. Related studies of tunneling in DQW has suggested that the magnetic ®eld qualitatively alters the tunneling density of states (DOS), creating a wide gap at the Fermi level [626,1739,1740]; the origin of this effect was attributed to the strong Coulomb correlations characteristic of Landau quantized 2DEG. Subsequently, n ˆ 1 manybody IQHE, as a result of an interplay between single-particle tunneling and many-body effects, was observed in DQW structure [628]. This IQHE state exhibits a phase transition to a compressible state at

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large thick barrier; at thin barrier, an intriguing second transition to incompressible state, driven by an in-plane magnetic ®eld, was shown. Other experiments in DQW structure include the 2D±2D tunneling spectroscopy used to determine the density and temperature dependencies of thermal e±e scattering rate [629]. In zero magnetic ®eld, the correlation effects on collective modes have been investigated experimentally by both optical [581] and compressibility [626,1739,1740] measurements. Transport measurements, while being a good probe of single-particle behavior, are generally much less sensitive probe of the correlations. However, one speci®c transport experiment, the Coulomb drag in DQW systems, is predicted [780] to be a sensitive probe of the double-layer plasmons. Since the plasmons are sensitive to correlations, this allows the unique possibility for a transport measurement to probe many-body correlations in a 2D system. Such an experiment measuring plasmon enhanced Coulomb drag in a GaAs±Ga1ÿx Alx As DQW system has recently been performed [631]. A crossover, in both temperature and magnetic ®eld, from singleparticle behavior to coupled plasmon enhancement was observed. Quite recently, RS measurements have been made to probe the plasmon dispersion in a DQW structure as a function of wave vector [632]. Both in phase (the optical plasmon) and out of phase (acoustic plasmon) modes have been observed (up to  0:2kF ). The modeling of the observed temperature dependence of the Landau damping of the acoustic plasmon, within the RPA using Hubbard approximation (at T ˆ 0 K), was shown to result only in a qualitative agreement; the need to have better theoretical framework at higher temperature is pointed out. It is widely known that electron correlations become increasing important as either the dimensionality or the effective charge density is lowered. These enhanced correlations have been known to give rise to new states of matter, of which FQHE is the best known example. Similar important consequences of electron correlations, particularly associated with the existence of collective SDEs in quasi2DEG, have been investigated, both experimentally [633±636] and theoretically [788±792]. For instance, it has been shown [791] that the exchange-correlation-induced many-body excitonic effects may lead to an instability in the normal ground state of a DQW by suppressing the symmetric± antisymmetric intersubband gap at a critical density nc  0:7  1011 cmÿ2, which at still lower density (ns  0:1  1011 cmÿ2 ) may have reentrance back into normal state. Here the signature of the instability is the collapse of the energy of the long-wavelength (q ˆ 0) intersubband SDE mode of the symmetric to antisymmetric transitions. While this zero-®eld instability has not been observed in GaAs DQW, RS measurements in perpendicular magnetic ®eld have uncovered marked softening of the longwavelength intersubband SDE of dilute 2DEG in DQW at even n quantum Hall states [636]. Their softening was attributed to the enhanced excitonic effect in quantum Hall states. The collapse of the SDE mode to an energy close to the Zeeman splitting was suggested as the existence of unstable SF intersubband excitations. The overwhelming majority of theoretical investigation of plasmons and magnetoplasmons in superlattices have been on GaAs±Ga1ÿx Alx As systems. The foundation of such theoretical work was laid much before the ®rst RS measurements [98] became available. Early work on plasma oscillations or dielectric screening [640±646,1741±1743] mostly directed towards the layered electron gas (LEG), suggestive of graphite structure, e.g., was formal within the framework of self-consistent (or random phase) approximation [126,158]. The ®rst illustrative example on the magnetoplasmons within the RPA and on cyclotron waves within a semiclassical theory in a model system were given by Yokota and coworkers [644]. The real superlattice systems as proposed by Esaki and coworkers were ®rst considered for gradually sophisticated electronic band structure calculations by several workers after the mid-1970s [647±653,1744].

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The theory of plasmons and magnetoplasmons in real superlattice systems underwent dramatic development after 1980s. The ®rst extensive formal paper by Das Sarma and Quinn [658,1748] and simultaneous series of short papers by Bloss and Brody [654] and Bloss [655,1745,1746] stimulated tremendous theoretical work on these systems. In their ®rst paper, Das Sarma and Quinn [658,1748] considered purely intrasubband plasmons, in both type-I and type-II superlattice systems, both with and without the applied magnetic ®eld in the perpendicular geometry, in the framework of RPA as well as hydrodynamical model. The later development included sophisticated model theories, both microscopic [661±705,1749±1751] and macroscopic [706±733,1752,1753], and their applications for studying plasmons and magnetoplasmons in superlattices in diverse situations. In addition, several authors have embarked on the theory of, e.g., EELS [734±742,1754,1755], plasmons and magnetoplasmons in n±i±p±i superlattices [743±752], ®nite-size effects in compositional as well as doping superlattices [753±759], plasmons and magnetoplasmons and related effects in quasi-periodic superlattices [760±769], plasmons and Coulomb drag effects in DQWs [770±787,1756], and many-body vertexcorrection-driven effects on SDEs in quantum wells [788±792]. While the aforesaid development resulted from the continued efforts of various research groups dedicated to the respective topics, numerous contemporary occasional authors have also appeared on the scene to add remarkably important contribution to the subject [793±833,1757±1760]. The microscopic theories developed to study the plasmons and magnetoplasmons in superlattice systems include, e.g., the theory of RS from bulk and surface plasmons in the layered 2DEG within RPA by Jain and Allen [661±663], and Jain [664], prediction of surface plasmons in both type-I and type-II superlattices by Quinn and coworkers [669,670], prediction of a wide variety of 2D and 3D plasmons and magnetoplasmons in type-I superlattices within RPA by Quinn and coworkers [672], a uni®ed theory of collective modes of an in®nite superlattice including miniband structure and manybody effects leading to depolarization and excitonic shifts using self-consistent linear response by Quinn and coworkers [674], a theory of magnetoplasmons in an ideal 2DES where an integral number of LLs are ®lled and Coulomb energy e2 =Elc is smaller than the cyclotron energy hoc [682], magnetoroton theory of collective excitations in FQHE [685±687,1749], inverse dielectric functions (IDFs) for intrasubband plasmons in type-I and type-II superlattices [689,690], generalized RPA theory to obtain the dispersion relation of the coupled plasmon±photon modes in type-I superlattices [692], RPA theory of longitudinal plasmons in ultrathin metallic ®lms accounting for a large number of subbands [693], a modi®ed dielectric continuum model for electron±phonon interactions in layered quantum wells [697], a linear response theory for collective intra and intersubband excitations in HgTe± CdTe superlattice, including the effects of an applied magnetic ®eld and electron±phonon interactions [701,1751], an RPA-based dielectric response theory of 2D Bose and Fermi plasmas in the presence of a magnetostatic ®eld [703,704], and the derivation of the general exact analytical expressions for the non-local dynamic IDF, which knows no bound with respect to the subband occupancy, for quasi-ndimensional (n ˆ 0; 1; 2) systems [705]. The macroscopic approaches seem to begin with Yariv and coworkers [706,707], who modeled each type of semiconductor layer in the superlattice by a different classical local dielectric function and solved the set of Maxwell equations for the system. This approach could be well termed as the transfer matrix method (TMM). Within this approach one ®nds a set of transverse electric (TE) (s-polarization) and transverse magnetic (TM) (p-polarization) modes corresponding to photon modes in the device. This model theory is unable to support any type of charge-density oscillations, and consequently is unable to describe the dispersion relation of the plasmons in the system. However, the formulation

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allows the liberty to include the retardation effects, the magnetostatic ®eld effects, the coupling to optical phonons, the phenomenological relaxation time, and even the spatial dispersion effects through the hydrodynamical model. This remains true whether the superlattice system is ideally periodic (or in®nite along the growth axis), or semi-in®nite (i.e., truncated on one side), or ®nite (i.e., truncated somewhere both on the top and bottom layers). In an in®nite superlattice, the application of Bloch theorem allows one to obtain bulk continuum bands. In the semi-in®nite superlattice, where the Bloch theorem is inapplicable, one obtains surface plasmons. In the ®nite systems, the bulk band (of an ideally periodic system) splits into the discrete modes. In an n-layered ®nite superlattice (n being the number of layers in the unit cell), the total number of discrete modes is 2m, where m ( n) is the number of active layers in the unit cell Ð an active layer (of dispersive medium) supports, in the simplest situation, two bona®de modes propagating along its surfaces. The ®rst extensive work on bulk and surface plasmons and EELS, within such macroscopic approach as mentioned above, was carried out for a model binary metal±insulator superlattice in a classic paper by Camley and Mills [708,1752]. Subsequently, Camley and coworkers [710,711] applied similar approach to study the magnetic ®eld effects (in the Voigt geometry) on the binary semiconductor superlattices. Apart from obtaining the non-reciprocal propagation of surface magnetoplasmons, it was demonstrated that, for arbitrary direction of propagation in the ^x±^y plane, the surface polariton modes that were pure s- and p-polarized (in the zero magnetic ®eld) become mixed (in the non-zero magnetic ®eld). A linear Green-function theory was developed to study the plasmon±polaritons and RS intensity in binary semiconductor superlattices by Babiker et al. [712±715]. Wallis and Quinn introduced retardation effects [717] and later studied the magnetic ®eld effects (in the Voigt geometry) to investigate the bulk and surface plasma modes in binary metallic [718] and semiconductor [719] superlattices, using standard EM theory and appropriate boundary conditions. The bulk and surface plasmons in three-layered multi-heterostructures were investigated by Kushwaha [720], using fully retarded TMM. The effect of an applied magnetic ®eld in the perpendicular geometry on the bulk and surface plasmons in binary semiconductor [721] and metallic [722] superlattices were also investigated by Kushwaha. It should be pointed out that even though the dispersion relations for bulk and surface plasmon±polaritons obtained through solving boundary conditions or through TMM are implicitly identical, it is impossibly ``hard nut to crack'' to transform exactly the ®nal formal results from one form to the other. This can be seen, e.g., by comparing the analytical results in Ref. [721] with those in Ref. [723]. Ideally 2D type-I as well as type-II superlattices have also been investigated within the framework of macroscopic theory, both with and without an applied magnetic ®eld. For example, Kushwaha has studied them using fully retarded theory with Dirac-d-function representing the charge-density pro®les along the superlattice axis, both in the absence [724] and in the presence [726] of a perpendicular magnetic ®eld. Subsequently, Kushwaha and Djafari-Rouhani (KD) have extended the interface response theory (IRT) [799,1759] to study the magnetoplasmons in binary semiconductor superlattices in the Voigt [727], perpendicular [728], and Faraday [729] con®gurations. Quite recently, KD have also investigated perfectly 2D type-I and type-II superlattices within the framework of IRT both in the absence [730] and in the presence of an external magnetic ®eld in the Voigt [731], perpendicular [732], and Faraday [733] con®gurations. Some speci®c comments on such theoretical developments will have to await until we discuss the numerical results in the later sections of this paper. Finally, we take a leap to recall some important works on electronic and optical properties of superlattices presented by some scattered authors [793±833,1757±1760]. Ando and Mori [793,1757]

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considered the electronic subband structure of heavily doped n-type GaAs±Ga1ÿx Alx As superlattices, including band bending effects caused by charge transfer from barriers to wells, within self-consistent HFA. The theory of magnetophonon effect Ð also called Gurevich±Firsov oscillations Ð for 2DEG in semiconductor heterostructures in the presence of a quantizing perpendicular magnetic ®eld was developed by Lassing and Zawadzki [795,1758]. Klein [798] presented extensive theoretical models to explain RS measurements on phonons in most widely studied type-I superlattices. Next came the IRT by Dobrzynski [799] that allows one to investigate almost all quasi-particle excitations (such as plasmons, phonons, magnons, etc.) on an equal footing; the analytical results within IRT are obtained in an elegantly compact form. This (the IRT) is, however, a semiclassical theory and hence excludes the miniband structure from its formulation. Ever since its inception, the IRT has been extensively applied to study a large variety of physical properties in heterostructures and superlattices. The IRT allowed Djafari-Rouhani and Dobrzynski [800] to obtain general expression for different types of collective excitations in the n-layered superlattices. Palmer [801] identi®ed a new type of EM oscillator which utilizes the negative differential conductance (NDC) of a quantum-well-tunnel junction to drive surface plasmons on the outer boundaries of the junction. Que and Kirczenow [804,1760] predicted that in semiconductor superlattices with electron tunneling between different quantum wells, the energy of the plasmon associated with the motion of electrons along the growth axis should exhibit strong oscillations as a function of magnetic ®eld. This collective phenomenon is suggested to be useful to measure the carrier density in the superlattice and also to probe the LL broadening and many-body enhancement of the electron g-factor. Fertig [805] and Chakraborty [807] investigated the collective excitation spectrum, respectively, in two/three-layer electron system and in a superlattice with alternating densities in the FQHE regime. Brey et al. [806] showed that a parabolic quantum well absorbs FIR radiation at the bare harmonic-oscillator frequency independent of the e±e interaction and the number of electrons in the well. This was a direct consequence of the GKT [186]. The collective intersubband CDE and SDE in quantum wells were studied by Yu and Hermanson [809], using non-local response of the electrons in the RPA, indicating that direct- and exchange-correlation interactions have comparable strengths in GaAs±Ga1ÿx Alx As single-well heterostructures. A brief critique of the continuum model theories of con®ned optical phonons and polaritons in superlattices by Ridley and Babiker [811] is worth mentioning. A macroscopic theory of optical phonons in superlattices by Enderlein [812] and a microscopic theory of (magneto)plasmon±phonon excitations and dynamical screening of the polar optical phonons in quantum wells by Gurevich and Shtengel [815] make interesting reading on the subject. Shi and Grif®n [816] formulated a uni®ed theory of the longitudinal and transverse plasmons, including retardation effects, in the RPA in a momentum-space, just as the standard approach used in bulk metals. Miniband conduction transport properties in semiconductor superlattices have also been studied using balance-equation theory [818]. Mantica and Mantica [819] developed a general method, based on the classical theory of moments, for computing the spectra of periodic SchroÈdinger operators, with an application to the electronic states of modulated superlattices in the envelope-function approximation. This allows one to obtain appropriate information on the effects of variable effective mass and relative-band-offset ratio and hence to master the important physical effects (on the electron dynamics) taking place in these structures. Schaich and MacDonald [820] considered the bound energies of plasmons con®ned within 3D slabs, 2D strips, and 1D line segments, using a hard wall potential. Comparing the RPA results with naive estimates led them to conclude that the in¯uence of the

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surface is more important in 2D and 1D than in 3D due to different form of Coulomb interactions. In addition, it was noted that corrections to the simple predictions are required in 1D and 2D (in any D) if the semiclassical approximations are used for the response functions of the system (if the fully quantum theory is used). Golden and Kalman [821] demonstrated that interlayer correlations bring about important qualitative changes in the simple RPA collective mode structure of superlattices. The RPA acoustic plasmon is shown to cease to exist and is converted into an optic-like mode where the qk ˆ 0 frequency is a sensitive function of the strength of interlayer interactions and of qz . A theoretical study of strained AlSb±InAs±AlSb quantum wells reported that the biaxial tensile strain in the InAs well increases the subband energies and lowers the electron cyclotron mass due to the reduction of the direct gap and the splitting of the heavy-hole±light-hole bands [823]. Dumelow and Tilley [825] demonstrated that uniaxial nature of the superlattice causes their FIR spectra to exhibit a range of unusual optical phenomena. Many of the polariton modes that result are particularly useful for characterization purposes; in particular, a range of phonon con®nement parameters unavailable to other techniques can be measured. A self-consistent theory using TDLDA was proposed for plasma excitations in an LEG by Beck [826]. In this model theory, whose introduction is motivated by the success of the jellium model in describing the dynamic response in the bulk and at the surface of alkali metals, the ionic cores are replaced by parallel sheets of uniform positive charge. These charge sheets provide the neutralizing charge and the external potential which con®nes the interacting electrons in the local-density functional calculation of the ground state system. The proposed model, for low-density systems in which only the lowest subband is occupied, provides an adequate description of the electronic response. However, in higher density systems where more subbands are occupied, there are features which are admittedly not seen to be well covered by the model using the envelope treatment of the electrons. Kim and Sohn [827] considered a ®nite type-II superlattice system concentrating mainly on the effects of overlapping between two adjacent layers within the RPA. In a model system with only two subbands occupied in the electron layers and only the lowest subband occupied in the hole layers, they studied the dispersion relations both for bulk and surface plasmons; as well as the RS intensities. Tilley and coworkers [828] investigated the surface magnetoplasmons in the perpendicular and Faraday con®gurations with an effective medium theory. In the region of high frequency dispersion, which is that of most interest, the free-space wavelength is much greater than the superlattice period d (the effective medium limit). This implies that the wave vector qk appearing in the dispersion relation is small compared with d ÿ1 (i.e., qk d  1). In view of this, Tilley and coworkers carried out systematic Taylor expansions to order q2k d2 of the general TMM results of Kushwaha [723] to obtain their Eq. (14), which was used to compute the dispersion relation for surface magnetoplasmons. Wu [829] studied the electronic structure of Ga1ÿx Alx As±GaAs±Ga1ÿx Alx As single quantum wells subjected to the in-plane magnetic ®elds within a self-consistent scheme. To be speci®c, he developed a simple model in which the con®nement potential energy, the energy and the wave function of the discrete levels, the Fermi energy, the electron density in different energy levels, and the depletion lengths can be calculated as a function of the known material parameters, growth parameters, and experimental conditions. Rudin and Reinecke [830] studied the scattering of 2D plasmons by the carrier density non-uniformity created by a ®xed charge impurity. The plasmons are described by an integral equation obtained within the hydrodynamical model of 2D plasma containing a ®xed-point charge. They obtained the energy of the scattered plasmons and the scattering cross-section in the Born

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approximation. They also employed RPA to treat both correlation and ®nite thickness effects in quantum wells. Analogous to their earlier work on type-II superlattices [827], Kim and Sohn [831] also investigated the intra and intersubband plasmons in a ®nite HgTe±CdTe superlattices. Finally, we mention an interesting series of papers by Golden and coworkers [832,833], who have extensively investigated the collective plasma modes in correlated superlattices in the framework of what they call quasi-localized charge approximation (QLCA). The term ``correlated'' refers to the effects of both intralayer and interlayer correlations on the plasma dynamics of the system. The QLCA approach has been seen to be reliably successful in the recent studies of collective excitations in bilayer systems as well as in superlattice systems, both with and without the applied magnetic ®eld, including the effects of retardation. While most of their work has concentrated on correlated semiconductor superlattices, the QLCA model is suggested to be equally well suited to metallic superlattices consisting of an alternating array of thin metallic layers and thick insulating slabs [833]. Analyses of the dispersion relations at long wavelengths compared with the spacing between adjacent layers has shown that RPA collective mode structure is substantially modi®ed by particle correlations [832]. 2.2.3. Quantum wires and lateral superlattices Scientists have long thought about the way the basic notions would change in systems with dimensionality different from that of the 3D space we are so accustomed to. The efforts to see the consequent changes began with the reduction in the dimensionality of the system from three to two. A tremendous progress achieved in 2DES, which provided us with new fundamental physics and potential devices, encouraged the human curiosity to extend this trend to continue diminishing system's dimensionality. So the next step naturally was 1D systems Ð semiconductor structures in which conduction electrons are constrained in two dimensions or are allowed free motion in the third dimension. Such systems are the so-called quantum wires, 1DES, or quasi-1DEG for broader range of physical understanding and will be the subject matter of this section. We will not intend to expand on the structures that have been made or proposed for con®ning charge carriers to wire-like structures, and fabrication methods will be alluded to only brie¯y during the discussion of the existing experiments. The con®ning mechanisms are grouped into several categories such as electrostatic, compositional, geometrical, procedural (physical or chemical), selective growth, etc., although actual structures usually involve more than one mechanism. These have been discussed in several exhaustive review papers (see, e.g., Ref. [844]). The systems in which the charge carriers are con®ned to one dimension of free motion (quantum wires) are providing materials with remarkable new properties (for instance, quantization of electrical conductance in ballistic quantum channels [845]). But the challenge of fabricating these wire-like structures are greater than those for making 2D layered structures, and extensive improvements in the fabrication techniques are required. A starting point for the fabrication of quantum wires has often been 2D layered structures that are lithographically processed to achieve lateral con®nement. But higher performance will require fabrication of smaller structures for which it will probably be necessary to actually control the motion of atoms during the growth of the materials. This presents a major challenge in the growth technology for the next generation of quantum structures [846,1761±1763]. It has long been recognized that the behavior of the 1DEG is expected to differ dramatically from that of its 3D counterpart. The in¯uence of ¯uctuations [847], expected possibility of giant conductivities [848], the Peierls instability of the metallic state [849], and the effect of disorder [850] have been the

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objects of early occasional speculations. This interest ceased to be merely academic with the development of a class of anisotropic conductors whose structures consist of parallel linear chains, along which conduction electrons propagate essentially in one dimension [851]. These include, e.g., organic compounds such as tetrathiofulvalene±tetracyanoquinodimethane (TTF±TCNQ). Because of limitations on crystal size and quality, reliable experimental data on these systems were rather scarce until the mid-1970s, and were con®ned largely to the measurements of static and low-frequency characteristics. Subsequently, it was found that at suf®ciently high frequencies, the materials appear metallic in one dimension. This encouraged interpretation of the data in terms of a simple Drude metal even at high frequencies well above the nominal width of the conduction band. Even though a few scattered works citing approximate plasmon dispersion relations are available [852±854], a thorough study of the dielectric response of the 1DEG was scarce until 1974. Much of the fundamental theoretical understanding of electron dynamics in 1D systems have been emerged from the work on the Tomonaga±Luttinger liquid model (TLLM) [855,1764±1767]. The TLLM makes some drastic simplifying assumptions which allow one to solve the interacting problem completely. One of the surprising results that is obtained from the solution of TLLM is that even the smallest interaction results in a disappearance of the Fermi surface, leading to a system which is describable as a non-Fermi liquid Ð in the sense that the elementary excitations are very different from those of the non-interacting system. Therefore, one would expect that the experimental properties of the semiconductor quantum wires should be quite different from any predictions based on the assumption that 1DEG is a Fermi liquid. However, as will be discussed below, the contrary observations have been rather ®rmly established. An early motivation behind the proposal of semiconductor quantum-wire structures was the suggestion [856] that 1D k-space restrictions would severely reduce the impurity scattering, thereby substantially enhancing the low-temperature electron mobilities. As a result, the technological promise that emerges is the routes to faster transistors and optoelectric devices fabricated out of the quantum wire structures. Research interest burgeoned in quantum wires owes not only to their potential applications, but also to the fundamental physics involved. For instance, they have offered us an excellent unique opportunity to study the real 1D Fermi gases in a relatively controlled manner. Such quantum-well wires (QWWs) of GaAs surrounded by Ga1ÿx Alx As with dimension as small as 20 nm  10 nm in cross-section were ®rst fabricated by Petroff et al. [857], using MBE and a combination of photolithography and chemical etching. Initial theoretical studies, particularly the transport properties, of 1DES were triggered with the attempts to observe the ®nite-temperature manifestations of the localized behavior predicted by Thouless [858]. He argued that electronic states should be localized in any wire whose impurity resistance is greater than  10 kO. At suf®ciently low-temperatures this will lead to a T ÿ2 increase in resistance because 1D phonons or excited electrons are needed to cause transitions between localized states. In particular, as Thouless predicted, the ®nite-temperature corrections to conductivity scale with the diffusion length between inelastic collisions, i.e., those which destroy the coherence of an electron wave packet. Subsequently, Arora [859] assessed the importance of quantum size effects on the conductivity under the condition that ld (the de Broglie wavelength) of a thermal electron becomes comparable to the transverse dimension of a thin wire (of rectangular cross-section). In the ultrathin limit, the ratio of longitudinal resistivity to bulk resistivity was shown to be proportional to Al2d , where A is the area of cross-section. Lee and Spector [860] investigated the impurity-limited mobility of a thin wire under the conditions where the impurities are distributed inside the wire, or outside the wire, or

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both inside and outside the wire. They came to the conclusion that when the distribution of the ionized impurities outside the wire is uniform, the mobility is independent of the wire radius; if the impurities are separated from the wire by a ®xed distance, Sakaki's results (that mobility increases exponentially) are recovered; and if the impurities are distributed uniformly both inside and outside the wire, the mobility is again seen to be size independent. The subsequent development of the phenomenon of electron transport in 1DES can be seen in numerous proceedings of the international workshops (see, e.g., Refs. [87,846,1761±1763]). The long-range Coulomb interactions in metals and semiconductors give rise to the plasmon mode, which is a collective mode and which is not present in models where the Coulomb interactions are neglected. In metals it is known since long that plasmons are the most prominent feature of the interaction between electrons. In 2DES, the width of the well can always be neglected without losing the main singular behavior of the Coulomb potential. In an ideal 1DES (zero width), the Fourier transform of the Coulomb interaction is divergent [861]. However, if we use a more realistic ®nitewidth model, the 1D Fourier transform of the Coulomb interaction turns out to be non-singular and the matrix element thereof is ®nite for a ®nite wave vector q. Consequently, the interaction potential in 1DES depends on the wire radius. Another interesting aspect of the 1DES is that there is no Landau damping except on two isolated lines E ˆ Eq  hqv f (Eq ˆ h2 q2 =2m and v f ˆ hkF =m is the Fermi velocity). This is quite different from the 2D and 3D cases, where there is a continuous Landau hqv f  E  Eq ‡  hqv f ) in the energy±momentum space. Landau damping, as damping region (Eq ÿ  proposed by Landau [862], is a damping mechanism in which a collective mode decays rapidly by exciting a single-particle±hole pair. In general, once Landau damping comes into being, the collective mode has such a short lifetime that it no longer represents a well-de®ned excitation mode of the system. One of the most important feature of the 1D plasmon dispersion is its strong dependence on the wire dimension. This is very different from its 2D counterpart where information about the subband quantization shows up only in the higher-order corrections. This leads one to infer that plasmon spectroscopy (even in the q ! 0 limit) can be used as a characterization tool to obtain quantitative information about the 1D quantization in the wire systems. The crucial ®rst step in developing the theory of plasmon dispersion in quasi-1DEG was taken by Williams and Bloch [863] using RPA. They presented not only the formal results, but also the extensive numerical examples for intrasubband single-particle and collective modes. It was found that the plasmon modes remain free from Landau damping and that for long-wavelengths these modes have eigenfrequencies ranging from the usual 3D plasma frequency for propagation along the chain axis to zero for propagation perpendicular to it. Subsequently, a number of authors considered the similar problem of the plasmon dispersion in quasi-1DES [864±868]. Some of them also considered the effects of exchange-correlations, within STLS scheme, on the simple RPA plasmon dispersion [865,866]. Lee and associates [867] predicted the existence of multiple non-Landau-damped acoustic plasmons in a slender wire. These slender acoustic plasmons (SAP) arise as a result of the collective, longitudinal oscillations of the electrons grouped in one of the discrete transverse-motion levels against those grouped in neighboring levels. It was commented that earlier workers [866] failed to notice these SAP modes mainly because they considered only a single transverse level and, in that sense, their system was purely 1D. The analytical diagnosis [868] of the dielectric function E…q; o† obtained within SCFA [158], in the static limit and in the limit of wire radius goes to zero reaf®rmed, respectively, the logarithmic singularity of E…q; o† indicative of the Peierls transition and logarithmic divergence of E…q; o† indicative of the divergence of 1D Fourier transform of the Coulomb potential.

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Even though they still remain regrettably unnoticed, a number of interesting optical experiments characterizing indirectly the plasmon dispersion in 1D organic conductors had been performed around the mid-1970s [869±872]. The ®rst direct measurements of plasmon dispersion and damping in TTF±TCNQ were made by Ritsko et al. [873], using high-energy inelastic electron scattering in thin ®lms at 300 K. While a model calculation based on the RPA [863] was seen to be qualitatively consistent with the observed spectrum, there was no detailed agreement. In particular, the ®rst known observation of a negative dispersion for bulk plasmons was seen to be in qualitative agreement with the theory of Williams and Bloch [863]. The observed momentum dependence of the plasmon width remained unexplained, since no available theoretical treatment agreed with the experimental measurement. Because of the early predictions that semiconductor quantum wires could form the basis of very high mobility transistors due to the reduced phase space for carrier scattering in 2D-con®ned systems, a large variety of experiments were performed in quasi-1D systems [874±938,1769]. These include, e.g., the early transport measurements [874±884,886,888,889,895,899,907], the optical spectroscopic measurements [885,887,890±894,896±898,900±906,908±912,914±920,923±928,930±935,937, 938,1769], fabrication techniques for producing unusual shaped quantum wires [913,921,936], and the experiments reporting surface-acoustic wave (SAW) scattering from quantum wires [929]. The group efforts [939±992,1770] and scattered authors [993±1006] on the theoretical side have made possible the emergence of some unique features [1007±1022] investigated in 1DES. The magnetic ®eld effects on the plasmon dispersion and transport properties [1023±1043] include, e.g., the experimental evidence [1044±1049] and theoretical explanation [1050±1058,1771] of the quenching of the quantum Hall effect. Analogous to DQW systems, several theoretical efforts have been made to report some novel phenomena in double quantum well wire (DQWW) [1059±1070]. Some occasional authors, sensing the signi®cance of studying diverse optical and transport properties in 1D systems, have also added a remarkable contribution to the subject [1071±1103,1772±1774]. In what follows, we would shed some light on the important, from our point of view, experimental as well as theoretical aspects of the quantum wire systems. At the outset, Fowler et al. [874] studied the conductance in an electrostatically squeezed SiMOSFET accumulation layer samples. A temperature-dependent conductance s ˆ s0 exp‰ÿ…T0 =T†n Š was observed, where n ˆ 12 …13† for smaller (larger) channel widths. While short of any convincing theory at hand, they believed that this behavior arose from a transition from 1D to 2D variable-range hopping in the sample. In a similar electrically con®ned  100 nm wide Si-inversion layers, Skocpol et al. [875] demonstrated the simultaneous presence of localization and interaction effects in good agreement with the 1D theory using parameters estimated from 2D results. Wheeler et al. [876] extracted the localization and e±e interaction contributions to the temperature-dependent differential conductance in a narrow Si-MOSFET. In a narrow Al wire of width 200 nm < w < 600 nm, Santhanam et al. [878] demonstrated clearly the 1D localization effect predicted by Thouless [858]. Extensive studies of the temperature, gate voltage, and electric-®eld dependencies of the conductance peaks in narrow Siinversion layers were made by Fowler and coworkers [879], in order to distinguish between a resonanttunneling model [1071] and hopping model [1072]. They conclude that many of the peaks in their experiment are consistent only with a hopping model, whereas some could be consistent with an early resonant-tunneling model. They also suspect that none of their structure is consistent with resonant tunneling if the formulation of Stone and Lee [1073] is believed to be correct. Licini et al. [880] observed 1D localization effects, ®rst predicted by Thouless, in Li ®lms of varying widths; the

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localization effects were shown to be dominant and clearly differentiable from other contributions to the resistivity. In a narrow channel within a split gate GaAs±Ga1ÿx Alx As heterojunction ®eld-effect transistor, Pepper and coworkers [881] have demonstrated that there are both quantum interference and interaction corrections to the conductance. At temperatures such that the inelastic length (an average distance carriers diffuse between inelastic collisions) Lin ˆ …2Dtin †1=2 and the interaction length Lint ˆ … hD=2kB T†1=2 are greater than the channel width, the quantum corrections to the conductivity become 1D. Warren et al. [882] reported the fabrication and measurement of devices in which periodic gate was employed to produce 250 ultranarrow Si-inversion lines in parallel. For the narrowest lines, the transconductance exhibited a regular oscillation that was consistent with the calculated quasi-1D DOS. Kaplan and Hartstein [883] studied the aperiodic conductance ¯uctuations in narrow channel SiMOSFET as the magnetic ®eld, its orientation, and gate voltage were varied. It was demonstrated that such magnetoconductance ¯uctuations depend on the perpendicular component of applied magnetic ®eld. The magnitude of the ¯uctuations was found to be roughly the same, independent of the variation of the magnetic ®eld or the gate voltage. These observations imply that both types of conductance ¯uctuations arise from a common quantum interference phenomenon. Berggren et al. [884] observed transverse magnetoconductance in the range 0:3 T < B < 8 T of a narrow variable-width channel in a GaAs±Ga1ÿx Alx As heterojunction. They showed that as the channel width is decreased below 0:25 mm, the structure in the magnetoconductance changes from SdH oscillations to the magnetic depopulation of 1D subbands. Such a magnetic depopulation is unique to 1D and the effect occurs in 2D only in the presence of a parallel magnetic ®eld. Similar measurements of the resistance in 1D wires in GaAs±Ga1ÿx Alx As heterostructures by Timp et al. [886] showed the quantum-interference-induced ¯uctuations in the resistance as a function of magnetic ®eld (in the range 0  oc t  300). It was tentatively proposed that these ¯uctuations in the resistance of the wire are due to the AB effect, and that the change in the typical frequency (of the oscillation) is indicative of the change (with magnetic ®eld) in the width of the distribution of the electron trajectories across the waveguide. Pepper and coworkers [888] showed that a 1D system in which transport is ballistic (i.e., without collisions of any kind) possesses a quantized resistance h=ie2 , where i is the number of occupied 1D subbands. They claim that further quantization is enhanced by a parallel magnetic ®eld which lifts the spin degeneracy and brings about a quantization of resistance at values h=2…i ‡ 12†e2 , but does not alter the DOS as does the transverse ®eld. A transverse magnetic ®eld causes, as above-mentioned, depopulation of levels and the formation of hybrid subbands. It was argued that because scattering (of any kind) does not occur in the channel, ¯uctuation effects are not important despite the small area ( 10ÿ10 cm2 at the smallest) resulting in as few as 20 electrons being present in the channel. Similar results, but in the zero magnetic ®eld, were obtained by van Wees and coworkers [889], who showed that conductance changes in quantized steps e2 =ph when the channel width controlled by a gate on top of the heterojunction is varied. In this experiment, up to 16 steps were claimed to have been observed when the point contact is widened from 0 to 360 nm. An explanation was proposed, which assumed quantized transverse momentum in the point contact region. By the use of X-ray lithography, Kastner and associates [895] fabricated quasi-1D inversion layers in Si-MOSFET in order to study how their conductance behaves as the gate voltage or carrier density is varied. They discovered that at low temperatures a narrow disordered channel might exhibit strikingly regular conductance oscillations, which are periodic in carrier density. Based on a theoretical model

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(see Ref. [10] in Ref. [895]), they inferred that this would lead to thermally activated conductance because of the pinning of the CDWs by impurities in the narrow channel. The activation energy would be determined by the most strongly pinned segment of the sample, and periodic conductance oscillations would re¯ect the condition that an integer number of electrons is contained between two impurity centers delimiting that segment. It should be pointed out that such periodic conductance oscillations as discovered in this experiment are in contrast to the aperiodic conductance ¯uctuations usually observed in such structures (see the preceding discussion). Subsequently, Kastner and associates [899] reported a new set of experiments in which they investigated narrow channels interrupted by two controlled potential barriers and having a tunable electron density in GaAs±Ga1ÿx Alx As heterojunctions. Reproducible conductance oscillations periodic in electron density were found to correspond to the sequential addition of single electrons to the segment of the channel between the two barriers. The shape of the conductance peaks was shown to be proportional to dF=dE, where F…E ÿ m; T† is the Fermi±Dirac distribution function, E the singleparticle energy in an electron gas, and m the electrochemical potential. It was argued that a line shape proportional to dF=dE is theoretically expected when the conductance between two fermion baths is limited by tunneling through a narrow transmission resonance if the natural width of the resonance is  kB T. Consideration of correlations and disorder were speculated to be essential for explaining their observations. Beenakker and coworkers [907] reported on the experimental study of the periodic conductance oscillations as a function of gate voltage in split-gate disordered quantum wires in the GaAs± Ga1ÿx Alx As heterostructure. Periodic conductance oscillations as a function of gate voltage were found in both, intentionally (Be-doped) and unintentionally disordered, systems, in the regime where only a few hundred electrons are present in the wire. The dominant oscillations were seen to be very regularly spaced, with a period that is quite insensitive to a strong magnetic ®eld, and persist up to a few kelvin. A strong magnetic ®eld was found to enhance the amplitude of the oscillations up to values approaching e2 =h. The experimental data were analyzed in terms of the theory for Coulomb blockade (CB) oscillations in the conductance of a quantum dot in the regime of comparable level spacing DE and charging energy e2 =C, based on the presence of conductance-limiting segment in the wire. Good agreement with the experiment led them to infer that these are the CB oscillations. The appearance of additional periodicities and the onset of irregular conductance ¯uctuations at very low temperatures in some of the wires were attributed to the presence of multiple segments in these wires. At low temperatures, a crossover from the classical regime kB T0DE to quantum regime kB T9DE was found. In contrast to the traces of the conductance as a function of gate voltage, no magnetoconductance oscillations were observed, in support of the predicted CB of the AB effect. Before embarking on the discussion of diverse optical experiments, let us brie¯y de®ne the new term introduced in the preceding paragraph Ð the CB. Charge transport in 3D con®ned microstructures is inhibited if the energy necessary to add an additional electron to the mesoscopic (Coulomb) island exceeds the electrochemical potential of the reservoirs and the thermal energy kB T. This is known as the CB (of tunneling). The CB occurs because no current can ¯ow through the microstructure until a threshold voltage Vth is reached that is suf®cient to supply the energy needed to move an electron on to or off from the region. Although, in principle, both quantum dots (see the next section) and Coulomb island are 3D-con®ned regions connected to two Fermi reservoirs through penetrable barriers, there are some basic differences in experimentally realizable devices. While the single-electron charging is

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dominant in Coulomb islands (the charging energy Uc  DE), the size quantization and single-electron charging can be of the same order in quantum dots. Immediately after the mid-1980s, spectroscopic techniques were seen to allow a deep understanding of the man-made quasi-1D semiconductor quantum wires. Numerous experimental efforts have since been devoted to the probing of quasi-1DEG energy spectrum using different techniques such as farinfrared spectroscopy [885,890,891,894,898,902,912,925,927,933,937,938], PL [900,903,904,906,908± 911,915,918,922,923,926], RS spectroscopy [897,901,905,906,914±917,919,920,924,928,930±932, 934,935,937,1769], capacitance measurements [887,892], and transport measurements [890,891, 893,938]. In transport studies, quantization into 1D subbands has been inferred from the observation of oscillatory conductance structure in multiwire Si-MOSFET [882], as well as from magnetoconductance oscillations in narrow channel GaAs±Ga1ÿx Alx As heterojunction FET [884]. These 1D subbands are also sometimes called the ``channels'', just as in the multichannel Landauer formulation. Because the spacing between these 1D subbands are usually of the order of a few meV, the FIR spectroscopy was considered to be a suitable experimental probe to study electronic excitations in these 1DES structures, either with or without an applied magnetic ®eld. The term ``1D'' means that, starting from original 2DES in quantum wells, a single-particle-energy spectrum E…k† ˆ  h2 q2x =2m ‡ Eyi ‡ Ezj , consisting of energetically 1D subbands, is formed, due to lateral con®ning potential. The ^z direction is perpendicular to the original 2D plane. The discrete 1D subband energies Eyi …i ˆ 0; 1; 2; . . .† arise from the lateral con®nement, which is assumed to act in the ^y direction. The propagation vector qx characterizes the free motion in the ^x direction and Ezj …j ˆ 0; 1; 2; . . .) stands for the 2D subband energies in the original 2DES. In the samples usually investigated, both experimentally and theoretically, only the lowest subband Ez0 is assumed to be occupied. In other words the width of quasi-1DEG along ^z direction is treated to be vanishingly small (or practically zero). In most of the systems studied so far, the 1D con®nement is achieved by split-gate con®gurations; in some structures charge carriers are con®ned by shallow (or deep) mesa etching of the doped Ga1ÿx Alx As layer. Most of the early experimental reports until 1990 involved the fabrication of QWW and their characterization by, e.g., demonstrating the quantization into 1D subbands and determining the relevant parameters such as subband spacings, Fermi energy, carrier concentration, and subband occupancy, etc. This is true irrespective of the techniques used to probe the system and whether or not the system was subjected to an external magnetic ®eld. It is also noteworthy that MBE-grown, MD GaAs±Ga1ÿx Alx As quantum wells, which have been the base of almost all fabricated QWW structures, remain the most documented systems to date. We recall the magnetotransport measurements on single and multiple QWW structures by Demel et al. [893], who showed well-resolved SdH oscillations, which, in contrast to 2DES, exhibited characteristic deviations from a linear Bÿ1 dependence (B denotes the applied magnetic ®eld perpendicular to the interface). These deviations are a signature of the formation of 1D subbands [884]. The exact 1D subband energies Eyi depend, of course, on the con®ning potential and the magnetic ®eld, and should, in principle, be calculated self-consistently. However, a simple analysis is possible provided that the con®ning potential is approximated by a harmonic oscillator model: V…y† ˆ …12†m O20 y2 . This model should yield a good approximation to the actual (self-consistent) potential since the con®nement arises from the electric ®elds of the distant ionized donors. In this model the magnetic-®eld-dependent h2 q2x =2m …B† ‡ hO…n ‡ 12†, with O2 ˆ o2c ‡ O20 and m …B† ˆ …O2 = energy is given by En …q; B† ˆ  2  O0 †m . The depopulation of the 1D subbands by a magnetic ®eld within this model was calculated

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using O0 and the total 1D carrier density N1D as ®tting parameters. The experimental fan chart (see Fig. 2 in [893]) was found to be best describable for hO0 ˆ 2:3 meV and N1D ˆ 4:5  106 cmÿ1 . For these values, six 1D subbands were found to be occupied at B ˆ 0. De®ning the width w of the electron channel by the amplitude at the Fermi energy by Ef …B ˆ 0† ˆ V…y ˆ 12 w†, they found w ˆ 160 nm, which was smaller than the geometrical width t ˆ 250 nm. Therefore, they de®ned a ``lateral edge depletion region'' on either side of length wdl ˆ 12 …t ÿ w†  45 nm. In conclusion, they demonstrated the formation of 1DES in ultra®ne mesa-etched single and multiple QWW. These values are cited here to give an estimate of the relevant parameters in the QWW systems of interest. By the time QWW structures captured attention, it had become well known that the 2D intersubband resonance energy is shifted from the subband spacings due to depolarization and excitonic shifts, which were also known to partially cancel each other [85]. Similar shifts had been expected in 1D subband spectroscopy, but could not be initially reliably estimated due to the unavailable appropriate many-body theory. A rough estimate was made by Alsmeier et al. [890] who determined single-particle 1D subband spacings from oscillations in static magnetoresistivity and compared them with intersubband transition energies obtained by FIR spectroscopy on narrow inversion channels in InSb. They reported that depolarization and excitonic shifts account for about 20% of the intersubband energies at an electron density N1D ˆ 3  106 cmÿ1. These estimates, however, did not account for non-parabolicity whose in¯uence was approximated through energy-dependent mass. The ®rst extensive experimental observations of plasmon excitations for a lateral superlattice of quantum wires in an MD GaAs±Ga1ÿx Alx As MQW systems were made by Egeler et al. [901], using resonant RS. Five layers of thin (6.3 nm) GaAs quantum wells were separated by thick (49 nm) centered-doped Ga1ÿx Alx As (x ˆ 0:35) barriers. 2D electron density determined from SdH oscillations was 7:2  1011 cmÿ2 , measured after illumination of the sample. Only the lowest 2D subband was occupied. The geometrical wire-width was w ˆ 400 nm. They estimated 1D subband spacing of 1±2 meV and 15 occupied subbands. The Raman spectra exhibited a set of resonances with a strongly anisotropic dispersion. For momentum transfer parallel to the wires a signi®cant dispersion was observed; perpendicular to the wires the dispersion was nearly ¯at. The ¯at dispersion perpendicular to the wires indicates that the Coulomb coupling between the wires is very weak. This is the reason why the experimental results have largely been interpreted in terms of excitations in an isolated wire, even though the experiments are performed, for the sensitivity reasons, on the arrays of wires. A very interesting FIR experiment in deep-mesa-etched GaAs±Ga1ÿx Alx As quantum wires was performed by Demel et al. [902]. In the MD samples, the quantum con®nement was estimated to induce a 1D subband spacing of 1.7 meV. The linear charge density was N1D ˆ 4  106 cmÿ1 , the width of the electronic system w ˆ 200 nm, and six 1D subbands were occupied at B ˆ 0. By means of metallic grating coupler oriented perpendicular to the wires, they were able to couple FIR radiation to 1D plasmons propagating parallel to the wires. They carefully balanced the width w of the wires, the distance between the wires, and the period b of the grating coupler, which determines the plasmon wave vector q ˆ 2p=b, to approach the conditions for isolated 1D plasmons. An external magnetic ®eld oriented perpendicular to the sample surface was applied. The most prominent feature of their observation is the wire-width-dependent negative B dispersion of 1D plasmon (o0;q ), a behavior which has only been observed for edge magnetoplasmons (EMPs), whose frequencies are determined by the circumference of the samples. The slope of the dispersion depends on the wire width and density pro®le. Additional higher frequency modes with positive B dispersion were observed. These modes arise from a mixing between a con®ned plasma oscillation perpendicular to the wires and a freely

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propagating 1D plasmon along the wire, and show, for q > 0, an anticrossing behavior. It was pointed out that the interesting resonance o0;q was not observable without the grating coupler. No such unique feature was, however, observed in PL experiments by Plaut et al. [904,909]. In the 1D quantum limit, the intrasubband plasmon displays the linear dispersion, which is a characteristic of 1D free-electron behavior. Clear signatures of 1D behavior are expected in the quantum limit, when the Fermi energy is comparable to the subband spacing and only the lowest subbands are occupied by the electrons. Such a clear signature of 1D behavior was demonstrated by GonÄi et al. [905], who performed resonant RS measurements on GaAs±Ga1ÿx Alx As multiple quantum well wires. The 1D pattern consisted of  70 nm-wide channels with a period of d ˆ 200 nm. At power density of  1 W=cm2 , they obtained Fermi energy Ef ˆ 5:8  0:5 meV and intersubband spacing E01 ˆ 5:2  0:5 meV. The electron gas was therefore very nearly in a 1D quantum limit with only a slight occupation of the ®rst excited subband. They observed a plasmon dispersion with an almost linear q dependence predicted by the RPA for 1DES. In contrast, collective intersubband excitations were found to be dispersionless having energy E01 as expected for a parabolic potential well. The spectral width of the observed intersubband CDE exhibits the unique features of being free from Landau damping at ®nite wave vectors. The same research group had later observed large optical singularities at the Fermi level Ð usually referred to as Fermi-edge-singularity (FES) Ð both in optical absorption and emission, in MD QWW with only one or two occupied subbands [906,910]. The FES arising from the response of the Fermi sea to a hole in the valence band were shown to disappear at temperatures compared to Fermi energy. The increased strength of FES in 1D, its dominance in the optical spectra, and its persistence at high carrier concentrations are understood in terms of a suppression of hole-recoil effects in 1D. Kotthaus and collaborators [912] studied high-frequency polarizability of quantum wires in a novel GaAs±Ga1ÿx Alx As FET heterojunction induced by positive gate voltage beneath grating stripes, which were considerably narrower than in Si-MOSFET devices, in the FIR regime. While in conventional narrow quantum wires only fundamental mode of the dimensional resonance is observed, the quantum wires investigated by them exhibited higher harmonics. In addition, at high-electron densities and intermediate magnetic ®elds, they observed unexpected splitting of the fundamental mode. The strength of the splitting and the magnetic ®eld range were seen to be independent of the radiation polarization with respect to the wires. This behavior was only observed if the gate bias exceeds a critical value that decreases with decreasing array period. The non-parabolicity of the bare con®nement and interwire interaction were thought of as the possible origins of the splitting. Since interwire Coulomb interaction increases with decreasing wire separation and increasing carrier density, this would be in accordance with their observations. However, it would imply a sudden change of the interaction at a critical magnetic ®eld, and this leaves one with a puzzle: Why should the magnetic ®eld have such a dramatic effect on the interaction potential? GonÄi et al. [914,1769] reported the ®rst RS spectra of 1DEG in GaAs quantum wires subjected to the perpendicular magnetic ®eld. They observed the features associated with a roton minimum in the wave vector dispersion of 1D magnetoplasmons, in qualitative agreement with time-dependent HFA. Roton features are among the most signi®cant manifestations of e±e interactions. They result from an interplay between the direct and exchange-correlation terms in the Hamiltonian. The depth of the minimum is determined by the strength of the exchange vertex corrections, which represent an excitonic binding that is expected to be strongly enhanced by con®nement to 1D. At intermediate and higher magnetic ®eld, when the Fermi energy is comparable to or smaller than the Zeeman splitting of

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the lowest magnetic subband, they observed spin-¯ip (SF) excitations that are blushifted from the longwavelength magnetoplasmons. The observation of the blueshift of SF modes is interpreted as evidence of the spin-polarization of 1DEG and suggests a substantial enhancement of the spin gap of the lowest magnetic 1D subband. Most of the experiments discussed hitherto are the FIR probes, which are incapable to provide a clue to the many-body effects whose observation is hampered by three inherent limitations. Firstly, due to GKT [1082] only the lowest intersubband CDE couples to the FIR radiation for a bare parabolic potential well. Secondly, a different experiment, usually magnetotransport measurements, has to be performed in order to be able to compare the observed collective CDE energy with the SPE spectrum. Thirdly, different kinds of collective excitations involving changes in spin density (SDE) cannot be observed. The latter (the SDE) are of special interest, because, unlike the CDE, they are only affected by the exchange and correlation parts of the e±e interactions (excitonic shift), and therefore their observation allows a clear distinction between direct and exchange-correlation interactions. These limitations are overcome by measuring the RS by electrons, which is well suited to observe both collective CDE and SDE, as well as SPE, which are not affected by many-body correlations. All these three excitations can be identi®ed by simple polarization selection rules. Pinczuk and colleagues [916,917] made the ®rst clear-cut observation of 1D intersubband SDEs through resonant RS. The energetic position of the intersubband SDE was seen to be shifted from SPE by a correction due to excitonic effects. It was argued that, similar to the 2DES, this excitonic shift is comparable to the depolarization shift in 1D GaAs quantum wires. For the relevant experimental parameters, they estimated the ratio of the shifts Wxc =Wdep to be  55%. Bayer et al. [918] performed PL measurements on GaAs±Ga1ÿx Alx As modulated quantum wires in a magnetic ®eld up to B ˆ 9:0 T, with varying orientations. By using high optical excitation, emission from three quantum wire subbands was observed. For ®elds perpendicular to the wires, they observe a magnetic-®eld-induced transformation of quantization from quasi-1D quantum wire states at low ®elds to fully quantized quasi-0D states at high ®elds. This occurs as the magnetic con®nement energies become comparable to the lateral quantization energies and simultaneously the widths of the wave functions become comparable to the wire sizes. For the ®elds along the wires, only a weak dependence on the ®eld strength was found. Physically, in this case the con®ning potential from the magnetic ®eld is in the plane perpendicular to the wire axis. For all ®elds, the magnetic length is larger than the quantum-well-width (5 nm). Therefore, the energies of the lateral subband transitions are given primarily by the size quantization, and a transformation to a fully quantized Landau state is not observed. Major part of the above-mentioned experiments was focused on the collective charge-density and spin-density excitations. Apart from these collective excitations, SPEs are a dominant feature of the excitation spectrum in the con®nement regime, where exchange-correlation contributions to the e±e interactions are negligibly small. However, no considerable attention was paid to observe the wave vector dispersion of SPE in quasi-1DES with several occupied subbands. Strenz et al. [920] reported the RS by SPE in shallow etched GaAs±Ga1ÿx Alx As quantum wires and dots. They observed up to three intersubband peaks in depolarized scattering geometries, which showed a clear blueshift with decreasing lateral con®nement length. In dot structures the transitions appear as dispersionless, whereas in wires they show strong broadening and an intensity dip at the energetic center of the excitation with increasing wave vector parallel to the wires, as expected for SPEs. Their line shape correlates with linear wave vector dispersion of additionally observed intrasubband excitations. In the case of

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intersubband excitations jn >! jm >, there are two different types of transitions. The ®rst type ends at a subband which is unoccupied. For wave vector small compared to the Fermi wave vector of the starting level, these transitions have a full linewidth of hqvf;n and are centered at the subband spacing Enm ˆ En ÿ Em . If the level m is partially occupied, some transitions close to the subband spacing are forbidden because of the Pauli exclusion principle. This forbidden region becomes larger with increasing occupation of the level m. This results into a splitting of the intersubband excitation into two peaks (see Ref. [920] for further discussion). Characteristic many-body effects on the optical properties of 1DEG have been mostly investigated in QWW created by electrostatic con®nement or by shallow processing of MD quantum wells. Among the latter, wires fabricated by low-energy ion bombardment have a moderate potential modulation, which for narrow wires result in one or two well-de®ned 1D subbands. Optical transitions in these shallow wires tend to be quasi-direct in real space because of the weak potential modulation. This is the origin of strong optical singularities observed in these systems [906,910]. In contrast, samples prepared by reactive-ion etching or wet etching have spatially indirect optical transitions as a result of the strong potential modulation. Consequently, in these systems the wave function overlap decreases, and the observation of optical singularities is hampered. Pinczuk and coworkers [923] extended their previous work [906,910] to report on the evolution of optical FES with magnetic ®elds in the shallow GaAs±Ga1ÿx Alx As QWW with only one subband occupied. PL and photoluminesence excitation (PLE) measurements were performed on a multiple QWW structure with wire-width of 100 nm and period of 200 nm at temperatures between 1.4 and 24 K. The observed spectra show a strong FES that narrows as the Fermi energy in the wires is continuously reduced by the enhancement of the electron effective mass in a perpendicular magnetic ®eld. At high magnetic ®eld, when the cyclotron energy is larger than the 1D subband spacing, the lowest optical transition is shown to split into a doublet due to a spatial modulation of the LLs by the wire potential. Coupling of electrons in the occupied and empty subbands is stated to play a minor role on the FES intensity in their quantum-wire samples. In various papers on wires [902,912] some different types of interactions has been observed, which occurs at or near the crossing of wire modes with the harmonics noc (n  2) of cyclotron frequency. These interactions resemble the so-called Bernstein modes (see Ref. [20] in Ref. [925]), which have been observed in 3DES (see Ref. [21] in Ref. [925]) and 2DES (see Refs. [22,23] in Ref. [925]). Excitations of harmonics of the CR are strictly forbidden in an isotropic translationally invariant 3DES or 2DES with parabolic band structure. However, the dynamic spatial modulation of the charge density of a plasmon breaks the isotropy and causes a strong interaction. This leads to an anticrossing when the magnetoplasmon dispersion crosses harmonics noc of the cyclotron frequency. An interaction of magnetoplasmon resonance with harmonics of CR noc was observed with FIR spectroscopy by Heitmann and coworkers [925]. They substantiated their observations with self-consistent calculations of the magnetoplasmon dispersion to show similar effects when the lateral con®nement potential has a slight deviation from a parabolic form. Thus it was established that the Bernstein modes are a general phenomena observable not only in 3DES and 2DES but also in 1DES and 0DES. Bayer et al. [926] performed PL experiments on modulated barrier GaAs±Ga1ÿx Inx As (x ˆ 0:13) quantum wires in perpendicular magnetic ®elds up to 10.5 T. By using high excitation powers they observed emission from several lateral occupied subbands. They found a clear-cut downward shift of the transition energies as a function of plasma density due to band-gap renormalization (BGR), which is a manifestation of many-body effects. An estimation of BGR is of paramount importance in

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determining the emission wavelength of coherent emitters as used in semiconductors. The estimated BGR in the magnetoplasma in quantum wires was claimed to be enhanced by about 50% as compared with 2DES. This is not unexpected since the lateral con®nement leads to an increased overlap of the particle wave function and hence to the enhanced Coulomb interactions. Heitmann and associates [928] investigated deep-mesa-etched GaAs±Ga1ÿx Alx As quantum wires (and dots) using resonant RS to report on the observation of quasi-1D (and quasi-0D) con®ned plasmons. Interestingly, they observed, at higher frequencies, not only the original 2D intersubband excitations, but also additional modes ok in the polarized scattering geometry. Their experimental ®nding is that the frequencies of these modes obey the relation o2k  o22D ‡ o21D=0D , where o2D is the frequency of the vertical intersubband CDE and o1D (o0D ) is a lateral quasi-1D (quasi-0D) con®ned plasmon frequency in wires (dots). These additional modes are stated to arise from a coupling of lateral and vertical electron motion, which result in combined 2D±1D intersubband and 2D±0D interlevel excitations of collective charge-density type. Perez et al. [930] demonstrated, through RS measurements, that 1D plasmons in doped quantum wires are either extended or longitudinally con®ned when the lateral width of the wires is periodically modulated. Steinebach et al. [931] observed a strong splitting of the con®ned plasmon modes in quantum wires in a perpendicular magnetic ®eld, by using RS measurements. These splittings are attributed to the coupling of con®ned magnetoplasmon modes to Bernstein modes of the 1DES. The observed splittings and anticrossing are shown to depend sensitively not only on the wave vector but also on the shape of the con®ning potential as well as on the mode index of the con®ned plasmon mode. SchuÈller et al. [932] investigated all types of elementary excitations Ð SPE, CDE, and SDE Ð in one and the same quantum wire (dot) sample. The interesting point that singles out their investigation from the earlier ones is that the systems with several occupied subbands were targeted. It is shown that scattering by SPE is strongly enhanced under conditions of extreme resonance which indicates that energy-density ¯uctuations are responsible for the excitation of unscreened SPE. The high intensity of the SPE allowed them a detailed analysis of many-body effects in both 1D and 0D systems. Their studies yield a typical ratio of excitonic and depolarization shifts Wxc =Wdep  0:1, which holds for both wire and dot samples with several occupied subbands or levels. Kotthaus and coworkers [933] fabricated and studied ®eld-effect-induced quantum wire lattices with a unit cell that contains two closely spaced electron channels in a perpendicular magnetic ®eld. The FIR transmission spectra are shown to contain an additional mode that is not present in lattices with a simple unit cell investigated in earlier experiments. It was shown with a classical model as well as with a self-consistent time-dependent Hartree approximation that this additional mode (occurring between the CR frequency and the fundamental intrawire mode of the single wires) can be explained by an interwire interaction that is characteristic of double-wire system. The work by Perez et al. [934] on the folded and con®ned 1D plasmons in periodically modulated doped wires is similar to their earlier work in Ref. [930]. Heitmann and associates [935] performed resonant RS measurements to investigate wave vector and magnetic ®eld dependence of plasmons propagating along quantum wires. Such experiments were by then carried out with FIR spectroscopy [902], which, however, gives only access to CDE and does not allow a continuous investigation of wave vector dependencies. A particular feature of RS is that one can distinguish by polarization selection rules between collective CDE and SDE. If the polarization directions of incoming and scattered photons are parallel to each other (polarized spectra) CDE can be observed and if they are perpendicular (depolarized spectra) SDE are observed. This holds for

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vanishing magnetic ®eld (B ˆ 0) and unstructured samples. In the case of B 6ˆ 0, these selection rules may no longer hold. Moreover, one has to distinguish carefully between SD and SF excitations, which are degenerate in the case B ˆ 0. For laterally structured samples, deviations from these selection rules can also occur due to near-®eld effects within the grating formed by the wires themselves. In their studies, Heitmann and associates [935] observed a well-pronounced negative B dispersion that is stated to arise from the skipping-orbit motion of the individual electrons within the collective excitations. At higher frequencies, they found a set of additional modes that represent a superposition of freely propagating oscillations along the wire and con®ned oscillations perpendicular to the wires. These modes exhibit strong interaction with harmonics, n ˆ 2; 3; . . ., of the cyclotron frequency noc, and resemble, in that sense, with Bernstein modes. Etiene and coworkers [937] investigated dry, deep-etched GaAs±Ga1ÿx Alx As quantum wires of width down to 45 nm by Raman scattering (RS) and down to 180 nm by FIR magnetotransmission. From the laterally-con®ned plasmon and magnetoplasmon dispersions and intensities they have deduced the con®nement potential and the free and trapped electron densities. Free-electrons are observed down to a critical width wc , signi®cantly smaller under strong illumination (RS: wc ˆ 50 nm) than in dark conditions (FIR: wc ˆ 130 nm). The induced changes in the external con®ning potential and lateral electron distribution were analyzed in terms of a semiclassical electrostatic approach. This work demonstrates the capability of RS and FIR experiments to accurately determine the lateral electron distribution in nanostructures. Using a specially designed sample, Lefebvre et al. [938] performed electrical transport measurements in both con®gurations (parallel and perpendicular to the wires) and FIR measurements on the same quantum wire array, while keeping the sample in identical conditions of temperature and illumination. By controlling the modulation strength of the 2DEG concentration, using an electrostatic grating technique, the sample properties have been studied in three regimes of modulation: weak, intermediate, and strong modulation. FIR transmission data show that each of these regimes is characterized by plasmon modes with a distinctive behavior. These behaviors were analyzed further with the use of transport data, which allows to determine the electron concentration in the structure for every condition of gate voltage. Main conclusions drawn from their observations are the following. In the weak modulation regime, a quantitative analysis shows that the collective mode energy is consistent with that of a classical 2D plasmon at q ˆ 2p=a, where a is the period of the split gate. In the intermediate regime, the collective modes are con®ned plasmons. The observation of ``con®ned Bernstein modes'' indicates that the bare con®nement potential is non-parabolic in this regime. In the strong modulation regime, the observation of a FIR resonance energy which does not depend on the modulation amplitude, while the effective 2D electron concentration (within each wire) varies with gate voltage, shows that the collective mode is a Kohn mode. This implies that the bare con®ning potential is parabolic and hence Bernstein modes are then no longer allowed. This work essentially embarks on how a unique design of samples can allow one to establish the correspondence between the transport and optical measurements; although in some cases a more detailed quantitative analysis would be needed for a better understanding of their results. Nash et al. [929] have investigated anisotropic SAW scattering from GaAs±Ga1ÿx Alx As quantum wire arrays as a function of carrier density and magnetic ®eld. With the SAW propagating parallel to the wires, they observed a strong suppression of SAW scattering rates at low electron concentrations in narrow wires. Surprisingly, at high carrier concentration, the scattering rate is shown to increase sharply and exceed that of the unstructured 2DEG by a factor of 3. They speculate that this could possibly be

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due to the excitation of 2D intrasubband plasmon-like modes. With the SAW propagating perpendicular to the wires, when one naively expects no interaction classically, they observed strong oscillations in the transmitted intensity indicating the magnetic depopulation of 1D subbands as described by full quantal description of SAW-electron interactions. This is the ®rst experiment of its kind where contactless measurements ideal for studying large arrays with good signal-to-noise ratio were made. Finally, some interesting proposals for fabrication of quasi-1DEG in, e.g., arrowhead-shaped [913], sawtooth-shaped [921], and ¯at-ridge-shaped [936] quantum wires are worth mentioning. The ¯atridge-shaped quantum wires represent virtually a p±n±p structure, characterized by a tunable, strongly non-parabolic, and non-symmetric con®ning potential that would allow an easy way to observe the Bernstein modes as brie¯y discussed above. The current understanding of the theoretical aspects of the plasmon and magnetoplasmon dispersion in semiconductor quantum wires owes much to the intensive research efforts made by Das Sarma and coworkers [939±951] in setting up the mathematical machinery and predicting and explaining various special features of the experimental data. One of the most obvious reasons behind the pursuit of theoretical studies in quantum wires has been the dilemma whether these systems are describable within the framework of a Tomonaga±Luttinger liquid model or a Fermi-liquid theory. Several people raised their eyebrows at the outset of the theoretical development, particularly when the early experimental results were seen to be explicable on the basis of the normal Fermi-liquid theory. The widespread belief was that 1DES are singular Tomonaga±Luttinger liquids where the standard Fermiliquid theories are inapplicable. Das Sarma and coworkers [945±947] resolved the puzzle in the early 1990s (see the following). Much that we wish to describe here involves, in large part, the well-plowed arguments of Das Sarma and coworkers [945] and Hu and Das Sarma [946,947], who addressed the issue of how and why the applicability of a Fermi-liquid theory in describing real quantum wire systems is convincing and unquestionable; despite by the well-accepted theoretical claims that both disorder and interaction effects are singularly non-perturbative in 1D and should lead to ground states which are drastically different from the normal Fermi liquids. They begin by mentioning three important ways in which an ideal 1DEG is theoretically expected to be strikingly different from its higher dimensional counterparts. In each case, a perturbation to the system, which in higher dimension tends to leave the system in a Fermi-liquid state, theoretically drastically changes the behavior of the system in 1D (i.e., the Fermiliquid behavior is a highly unstable ®xed point in 1D). They then argue that actual semiconductor quantum wires may behave quite differently from the theoretical zero temperature ideals because of the effects of ®nite temperature, ®nite size, and scattering, which may serve to stabilize Fermi-liquid behavior in real quantum-wire systems. First, the presence of any electron±phonon coupling, which is invariably present, in 1D system should theoretically result in a lattice Peierls distortion [849], accompanied by a charge-density-wave ground state at absolute zero. However, in actual semiconductor quantum wires, the electron±phonon interaction via deformation potential is so weak that even at zero temperature, at which experiments on these systems are performed, the Peierls distortion does not occur. Second, the disorder-induced Anderson localization [433] dictates that in 1D (unlike higher dimensions) the presence of any degree of disorder localizes all electronic states. The currently fabricated semiconductor quantum wires are obviously not disorder-free, and hence, in the electric quantum limit, all the quantum-wire electronic states are expected to be exponentially Anderson localized, and the notion of an electron gas should not apply here. However, it is argued that in the state-of-the-art high-quality MD quantum wires, typical

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localization lengths are long enough (many microns) and therefore the electrons may be considered to be extended for all practical purposes. Third, the paradigm for interacting 1D Fermi systems is the strongly correlated Luttinger liquid [855,1764±1767]. This implies that in the electric quantum limit semiconductor quantum wires should behave as Luttinger liquids. In experiments involving RS, FIR spectroscopy, PL, and capacitance studies, quantum wires have shown no obvious sign of Luttingerliquid behavior. For instance, an essential feature of Luttinger liquid is that it is devoid of Fermi surface (i.e., the momentum distribution function nk is continuous through the Fermi momentum kF ); and yet luminescence experiments have already shown strong Fermi-edge singularities [906,910,923]. Thus, just as the Peierls instability and Anderson localization, the effects of the strong correlations of the Luttinger liquids may convincingly be negligible in real quantum wires. The ``heart of the issue'' is that the actual quantum-wire systems are pure enough not to abide by the notion of Anderson localization while impure enough to suppress the Luttinger-liquid behavior. Besides, it has been shown that the long-wavelength RPA plasmon dispersion and the (Tomonaga± Luttinger) boson dispersion of the 1DES are equivalent by virtue of the vanishing of vertex corrections in the irreducible polarizability of the 1DEG [945]. While the earlier work by Das Sarma and coworkers [939±944] was mostly con®ned to one-subband model, they also later generalized [948] the RPA theory to two occupied subbands in an effort to reproduce the three/four CDE modes found in the RS measurements of GonÄi et al. [905]. Subsequently, they advocated the validity of PPA [442,1711] which, as it is argued [950], becomes more accurate approach in quantum wires than in higher dimensional systems because of severe phase-space restrictions on particle±hole continuum in 1D. Quite recently, they have also investigated collective CDE spectra of two distinct types of experimentally realizable two-component quasi-1D quantum plasmas con®ned in semiconductor quantum-wire nanostructures: spatially separated double quantum wire structures and photoexcited homogeneous 1D electron±hole plasmas in a single wire [951]. In this work, they found, as expected in a two-component plasmas, two plasmon modes: the in phase optical plasmon with o / qj ln…qa†j1=2 at long-wavelengths and the out of phase acoustic plasmon with o / q, where q is 1D propagation vector and a the characteristic wire width. Although potentially interesting from the perspective of 1D collective excitation spectra, no experiment has, to the author's knowledge, yet been attempted at to probe such a system. The next crucial steps in the theoretical development of plasmon dispersion in single and multiple quantum wire systems were taken by Que and Kirczenow [952±954], Gold and Ghazali [955,1770], Gold [956,957], Mendoza and Lee [958], Mendoza and Schaich [960], Horing and coworkers [961,962], Yu and Hermanson [963±965], Leburton and associates [966±968], Hu and O'Connell [969± 972], Vasilopoulos and coworkers [973±975], MacDonald and coworkers [976±978], Reboredo and Proetto [979,980], Tanatar [981], Constantinou and coworkers [982,983], Wendler and colleagues [984,985], Wu and colleagues [986,987], Borges and associates [988,989], Kushwaha and Zielinski [990±992], and several occasional authors [993±1006]. These works were mostly aimed at studying the excitation spectra in quantum wire systems in the absence of an applied magnetic ®eld and, generally, within the framework of the RPA. We would like to discuss brie¯y some of these works in what follows. Que and Kirczenow [952±954] developed the theory of plasmons in lateral multiwire superlattices, in the quantum regime where the wires are electrically insulated from each other, and where a weak tunneling between the wires is allowed. The theory indicates that the observed FIR resonance in the 1D regime should be due to collective intersubband plasmon [952]. It was stressed that because of the

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Coulomb coupling of the wires intersubband plasmons (with arbitrary wave vectors perpendicular to the wires) can exist even if the wires are electrically insulated from each other. This theory was then extended to the general case of arbitrary number of subbands [953]. An interesting conclusion drawn from this study is that if the con®ning potential of a single wire is parabolic, then for intersubband plasmons that involve virtual transitions between two adjacent subbands, a system of an arbitrary number of subbands is equivalent to a system of two subbands in an approximate way. This result is non-trivial because for systems with n occupied subbands, there are n intersubband transitions involving different pairs of adjacent subbands. Note that occupied subbands should only be partially occupied. The equivalence to a two-subband system means that all n intersubband transitions interfere constructively to produce a dominant intersubband plasmon, with its energy described by a formula analogous to that for a system with only two subbands. The energy of this intersubband plasmon can have a large shift from the subband separation. The other intersubband plasmons in the system can have energy close to the subband separation, and they are only weakly coupled to the experimental probes, and make their observation impossibly dif®cult. The analysis of the coupling between intrasubband and intersubband plasmons, with the same spirit as stated above, led Que and Kirczenow [954] to disagree, on a number of issues involving analytical as well as numerical results, with the results obtained by Li and Das Sarma [943]. Gold and Ghazli [955,1770] investigated a cylindrical quantum wire within a two-subband model to study several electronic properties, such as plasmon dispersion, shallow impurity states, and mobility limits in quantum wires. Their results have been derived within the RPA (for plasmons), separablepotential (for shallow impurity states), and Born approximation (for mobility). They conclude that their model calculation for cylindrical wires does not represent any strong restriction to more general geometries, that the shape effects are not very important, and that the leading effects in quantum wires are subject to the area of the wire. This work was latter generalized to multiple cylindrical quantum wires [956], where tunneling effects between the wires were neglected. Within a two-subband model, they predicted a coupling between the intrasubband and intersubband plasmons and analyzed the depolarization shift for the intersubband plasmon. Subsequently, Gold [957] investigated, for the ®rst time, a square lattice made up of circular wires within a one-subband model and presented analytical (using the RPA including local-®eld effects) and numerical results for intrasubband plasmons. Such structures as studied by Gold [957] represent bulk quantum liquids in a reduced dimension. In these works, Gold [956,957] has studied both fermionic and bosonic quantum wires. Mendoza and Lee [958] stressed on the existence of multiple non-Landau-damped acoustic plasmon modes in a quasi-1D wire at ®nite temperatures. It was argued that these plasmons are of two basic types: the ®rst one is made up by collective, longitudinal oscillations of the electrons (essentially of a given transverse energy level) oscillating against the electrons in the neighboring transverse energy level, and are the so-called slender acoustic plasmons (SAP). The other mode is the quasi-1D acoustic plasmon, in which all the electrons oscillate together in phase among themselves, but out of phase against the positive background. It was demonstrated that even for ®nite temperature comparable to the mode separation D ho, the SAP and quasi-1D plasmon persist. As it is well known, in the long…m† wavelength limit (q ' 0), the plasma frequency op / …q…3ÿm† †1=2, where m is the dimensionality of the system. Therefore, the quasi-1D plasmon is also acoustic in nature. Mendoza and Schaich [959] employed a hydrodynamical model to study the intrasubband plasmons in quasi-1DES; a comparison with the RPA results showed an excellent agreement in the small-q limit, but, at large q, the HDM was shown to miss several subtle behavior characteristics. Subsequently, the same authors [960] stressed on

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the existence of ``gap plasmons'' (identical to SAP), if both subbands involved in the transition are partially occupied. To aid the search for gap plasmon that disperses downward with increasing q, they suggested the idea of FIR grating-coupler transmission experiment [885,890]. Horing and coworkers [961,962] considered the 2D arrays of quantum wires (wires repeated in two perpendicular directions), where the electrons on adjacent wires couple not only through their mutual Coulomb repulsion, but also through the overlap of their wave functions. Such a quasi-1DES has properties intermediate between 1DEG and 2DEG, bypassing the 2D stage, by controlling the potential barrier height, the superlattice period, and the carrier density. Within a tight-binding scheme they solved the RPA integral equation to study the plasmon excitation in¯uenced by the ®nite wave function overlap between neighboring quantum wires. Both intra and intersubband excitations were considered, to scrutinize how the symmetry of the Wannier wave functions would affect the existing plasma modes. Yu and Hermanson [965] reported self-consistent calculations of the subband structure of GaAs± Ga1ÿx Alx As quantum wires, using the sample parameters of Weiner et al. [897]. The SchroÈdinger and Poisson equations were solved simultaneously in a rectangular waveguide geometry, using Fermi statistics at ®nite temperature. The calculated energy levels, wave functions, charge-density, and con®ning potential were shown to differ from those from the simple square well and parabolic potentials, because of the in¯uence of accumulation and depletion regions. Based on the selfconsistently determined subband structure, the dynamical structure factor was evaluated using a nonlocal description of the dielectric response in the RPA at ®nite temperature. Their results show that, in the long-wavelength limit, the resonance frequencies of the intersubband plasmons are shifted dramatically, in good agreement with the observations. By monitoring the maxima of the dynamical structure factor S…q; o†, the dispersion relations of the plasmons were obtained. They anticipated that at higher temperatures the intrasubband plasmon modes would still be well de®ned. Leburton and coworkers [966] presented a Monte Carlo simulation, including polar optical phonon and inelastic acoustic phonon scattering, of multisubband quasi-1D GaAs±Ga1ÿx Alx As structures, in order to analyze the in¯uence of intersubband transitions on the transport properties. They pointed out that working in the extreme quantum limit and neglecting the in¯uence of multiple subbands (as is done in most of the transport models) is precarious, particularly when the subband spacing is comparable to kB T; stressing that ability to achieve subband spacing comparable to kB T at room temperature makes 1D structures well suited for comparison to magnetically con®ned systems. Further, they demonstrated [967] that the linear approximation to the Boltzmann transport equation in 1D quantum structures breaks down for ®elds as low as 50 V/cm at room temperature. Wang and Leburton [968] have also solved the Maxwell equations for cylindrical dielectric waveguide including retardation in order to derive plasmon dispersion relation of a quantum wire. At T ˆ 0 K and in the long-wavelength limit, their waveguide model provides a ®nite group velocity, given by the Fermi velocity. As it is, by now, well known in the existing literature, there are some dif®culties encountered in the study of the electronic properties of quasi-1DES: (i) the polarization function, which serves as the core of any kind of microscopic transport theory, has a divergent problem in a strictly 1D system at absolute zero; (ii) the Coulomb interaction in a strictly 1D is not well de®ned and is hard to calculate when quasi-1D nature is taken into account; (iii) the DOS in the free-electron model is divergent near the bottom of each of the subbands. In a series of papers, Hu and O'Connell [969±972] have tackled various issues of these problems associated with the electronic properties of quasi-1DES. In particular, they have shown that in the harmonic potential model, the e±e interactions can be studied analytically. Also, when ¯uctuation effects (originating from the e±e, electron±impurity, and electron±phonon

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interactions) are included, they obtained the polarizability function free from divergence and convenient for calculations. These efforts served as basic ingredients in developing a systematic transport theory [971] and a reliable formalism for studying collective excitation spectra for quasi1DES [969,970,972], within the generalized RPA. Regarding the transport along a quantum wire, most of the theoretical treatments consider a d-function or model impurity-scattering potentials, starting from a linear response theory or from the Boltzmann transport equation without a thorough treatment of screening, and the electron transport is considered only in the ballistic regime. Given the importance of these quasi-1DES, a rigorous treatment of transport should include a proper account of screening in restricted geometries, and should be valid for arbitrary potentials including e±e interactions. Using the so-called dielectric formalism, Vasilopoulos and coworkers [973,974] developed the quantum kinetic equation for a quasi-1DEG in a single quantum wire with several subbands. They modeled the lateral con®nement with a square or parabolic well and the vertical one with a triangular or square well; screening was treated dynamically; and the electrons were assumed to interact with each other as well as with an external system taken to be impurities or phonons. The theory was also later generalized to the case of a superlattice of quantum wires [975]. Most of the theoretical work has focused on the theories which approximate non-local effects in the response kernel but which were expected to be valid in the long-period limit for the multiple quantum wires. The objective of the experimental work, on the other hand, has been to push toward the regime where quantum effects of the lateral structure become dominant and to identify the resulting features. In this regime the semiclassical approaches are not applicable and microscopic quantum theories which properly treat the non-local effects are required. Some authors have employed fully quantum schemes but have usually assumed both that the modulation is strong and the period is short so that only one or a few subbands are occupied. The response behavior is then dominated by distinct intersubband transitions within a single well. MacDonald and coworkers [976] have reported a scheme which extends the fully quantum theory to the regime wherein the lateral period is still large than the Fermi wavelength and many subbands are partially occupied. They distinguished four different regimes Ð plasmon band structure, Landau-broadened plasmons, con®ned plasmons, and depolarization-shifted intersubband transitions Ð in which the optical and transport properties are qualitatively different: crossovers occur when the lateral period becomes shorter than the Fermi wavelength and/or when the modulation potential becomes larger than the Fermi energy. They reported the changes which take place in the density response [976] and in the fractional change of transmission in the FIR absorption [977] of a laterally structured 2DEG as these crossovers occur. Quite recently, ZuÈlicke and MacDonald [978] have studied the qualitative differences in the plasmon dispersion and the ground state correlation functions in quantum wires and the Hall bars. The microscopic calculations led them to infer the following. While the plasmon energy in the quantum wire case has the contributions only from kinetic-energy gain and exchange energy loss in the underlying electron system, the energy of EMPs in quantum Hall bars has additional contributions arising from electrostatic (Hartree) and external potential terms but no contribution due to kineticenergy gain. Despite these differences, the low-energy effective Hamiltonian for both systems is p identical for qW  1, where W ˆ 2kF l2 is the sample width and lc ˆ hc=eB is the magnetic length. A common disadvantage of most of the theoretical works (except for Refs. [963±965]) on the singleparticle and collective excitations is that the potential shape is assumed from the very beginning to be of

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a soluble form and tailored so as to embark on matching the results. Reboredo and Proetto [979,980] have noted that, although giving general insight into the properties of the quasi-1DEG, the geometry of the model used by Yu and Hermanson [963±965] seems to respond to the needs of the numerical work and is not in direct correspondence with the experimental samples aimed at. As such, it is dif®cult to relate their results to the nature of the microscopic mechanism that actually lies behind the actual con®ning potential. In view of this, Reboredo and Proetto [979] reported self-consistent calculations of the con®ning potential, wave function, and subband structure of the MD multiple quantum wires. As is generally the case, the price of paying the realistic modeling is an increasing complexity of the numerical work. They kept as close as possible to the actual experiments of their interest [897± 899,990,901±905]; quite good agreement with experimental data is stated to be obtained when a distribution of negatively charged centers (presumably induced by the ion milling in the real samples) is included. Exchange and correlations were included within the LDA and shown to renormalize the intersubband splittings by a fraction of a meV. On the basis of this self-consistent model, they studied elementary excitation spectra within the RPA [980] to conclude that the corrections due to exchangecorrelations should be small in the quantum-wire systems. Two-component plasmas (or more generally multi-component plasmas) as in an electron±hole liquids can also support, other than the usual plasmon mode, a new collective mode called the acoustic plasmon mode. Dispersion and damping of such acoustic plasmons, which have been studied in 3D [120] and 2D [770] systems, were investigated in quasi-1DES by Tanatar [981] within the RPA, including local-®eld corrections using Hubbard approximations. The effects of local-®eld corrections were shown to decrease the frequency and maximum wave vector for propagation of ordinary as well as acoustic plasmons, and increase the damping. Later, Constantinou and coworkers [982] investigated the effects of cross-sectional geometry on the plasmon modes of quantum wires, within the assumption of an in®nite barrier in a two-subband RPA. They considered circular, elliptical, and rectangular wires; all having the same cross-sectional area and linear carrier concentration. It was shown that for ®xed crosssectional area and carrier density, the intrasubband plasmon energy is only marginally dependent on the wire geometry, whereas the intersubband plasmon energy may vary considerably due to its dependence on the electronic subband energy difference. Wendler and Grigoryan [984] studied cylindrical quantum wires within a two-subband RPA. They calculated both intra and intersubband plasmons both with and without including the in¯uence of the image forces. The spatial symmetry of the cylindrical quantum wires was shown to guarantee that each of the two (intra and intersubband) modes splits into two. The upper (lower) intrasubband plasmon was classi®ed as an optical (acoustic) plasmon mode; the intersubband plasmon branches were both typi®ed as acoustic plasmons. The image forces were shown to increase the eigenfrequencies of 1D plasmons. However, the branches of intrasubband and intersubband plasmons are stated to have different dependence on the image potential as the wave vector varies. Subsequently, Wendler and Haupt [985] investigated the effects of non-parabolic con®ning potential on the dispersion relation of quasi-1D plasmons; reaf®rming the violation of GKT. Sun et al. [986] and Sun and Yu [987] investigated collective excitations and FIR absorption in GaAs±AlAs parallel quantum wires. It was shown that when lateral period Lx  20 nm, the FIR optical absorptions are dominated by single-particle direct transitions between different transitions. For Lx  30 nm, the optical absorptions are due to intersubband transitions. Between these two regimes, both single-particle direct and intersubband transitions are stated to make important contributions to the FIR absorptions.

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Borges and colleagues [988,989] investigated GaAs±Ga1ÿx Alx As parabolic quantum wires within the self-consistent STLS scheme for the response function of the quasi-1DES, using a two- and threesubband model with only the lowest one occupied. Quantities such as the effective potential, static structure factor, pair-correlation function, and the plasmon dispersion were calculated as a function of subband spacing and electron density. They report signi®cant differences due to the local-®eld correction included in their model calculation when compared to the corresponding RPA results. In particular, the collective intra as well as intersubband RPA results were shown to be higher energy modes, as compared to those due to STLS scheme, over the whole range of propagation vector. Kushwaha and Zielinski [990] have investigated GaAs±Ga1ÿx Alx As QWW within a two-subband full RPA. Main purpose of this work was to study the fast-particle energy loss to a quasi-1DEG in quantum wires. For this purpose, they used an exact expression for the inverse (nonlocal and dynamic) dielectric function [705] and parabolic potential well to characterize the lateral con®nement. Three geometries were considered: the fast-particle (i) moving parallel to, (ii) being specularly re¯ected from, and (iii) shooting through the quasi-1DEG. The illustrative numerical examples led them to infer that the dominant contribution to the loss peaks, in all the three geometries, comes from the collective intra and intersubband excitations. Furthermore, it was argued that the HREELS could prove to be a potential alternative of the existing optical techniques for probing the quantum wires. Generalization of this work for a realistic self-consistent model calculations corresponding to actual experiments in the real sense [991] and for a model with multiple (occupied and unoccupied) subbands, including the effects of an external perpendicular magnetic ®eld [992] is in progress. Several occasional workers [993±1006] have made similar efforts, as above stated, to investigate the plasmon dispersion in single and multiple quantum wires system, with and without including manybody exchange-correlations effects. Notably distinguishable is the semiclassical approach by Zhu and Zhou [993], just as the other three previous efforts [941,959,968], which neglects the subband structure, but can accommodate the retardation effects. Next worth mentioning is the theoretical prediction of novel carrier-acoustic instability for a quasi-1D plasma in semiconductor quantum wires with nonequilibrium carrier distribution [1000]. Equally interesting is the work by Thakur and Neilson [1004], who embarked on the realistic quantum wires that have both impurities and surface roughness, within the STLS scheme to include the many-body correlations. There the disorder was introduced by replacing the neat irreducible susceptibility w0 …q; o† with a particle conserving expression, given by Mermin [1104], which captures essential physics of impurity collisions on w0 …q; o†; and the surface roughness was accounted for within the scheme suggested by Gold and Ghazali [955,1770]. It was found that strong correlations signi®cantly depress the plasmon dispersion and also boost the spectral strength of the SPE, so that the plasmon at ®nite wave vector does not saturate the spectral strength. Recent progress in submicron semiconductor technology has aroused growing interest in the transport properties of 1DES, motivated by the belief that these systems are ideal for high performance electronic devices. Impurities, which are inevitably present in real (howsoever clean) systems, play an important role in electronic transport in solids. This is particularly true of semiconductors where they serve as donors or acceptors which supply the system with mobile charge carriers, and also act as the source of scattering which limit the carrier mobility. In the systems of reduced dimensionality, the role of impurities changes dramatically with the application of a strong magnetic ®eld. Theoretical explanation of an IQHE in 2DES within the framework of edge states, introduced by Halperin in 1982, shed light on the relevant issues concerned with the impurity scattering. A remarkable feature of such explanation is that impurities do not backscatter the edge-state electrons, which results in

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dissipationless transport and quantization of Hall conductance. The impurities are important, however, in controlling the width of the quantum Hall plateaus and transitions between them. The QHE is also observed in narrow conducting channels or quantum wires, but there the resonant states bound to an impurity can lead to its breakdown, as was predicted by Jain and Kivelson [1050] and observed experimentally by several authors [1044±1049]. Moreover, in mesoscopic physics, many of the novel effects rely on electron trajectories shaped by the nanostructure's potential pro®le, and therefore impurities can have a major effect. Motivated by such aspects, several authors [1007±1022] have brought to attention some unique features associated with the electronic transport in 1D quantum wires. These include, e.g., FES [1007], 1D Wigner crystal whose pinning leads to the non-linear transport properties, characteristic of CDWs [1009], conductance resonances due to magnetically bound impurity states [1012], negative transport lifetime, whenever forward scattering due to acoustic phonon absorption is the dominant electron interaction [1014], Friedel oscillations for interacting 1D fermions [1017], modi®ed Wiedemann±Franz law for interacting 1DEG [1018], non-universal conductance quantization [1020], charging effects due to both the ®nite range of Coulomb interactions and the long-range of the Friedel oscillations [1021], bosonic excitations induced anomalous thermal transport in quantum wires [1022], among others. Numerous workers [1023±1043] have investigated the magnetic ®eld effects on the interacting as well as non-interacting 1DEG in quantum wires, in different contexts. Experimentalists often apply an external magnetic ®eld in their optical and transport measurements. This is not simply because the magnetic ®eld is a convenient tunable parameter, but also because there may occur some additional phenomena and possibly some applications that arise in the presence of magnetic ®elds. The emergence of quantum Hall effects is a live example that has stimulated tremendous interest in the many-body physics of interacting electrons in the presence of a magnetic ®eld. Most of the theoretical works we have mentioned are concerned with magnetic-®eld effects on the plasmon dispersion in quasi-1DEG in single and multiple quantum wires. Explicit analytical and numerical results have been reported, mostly in the case where con®ning potential is parabolic and the GKT [1082,1774] is abided by. Exception to the parabolic potential well are: hydrodynamical model [1024] that does not account for the subband structure, sinusoidal modulation potential of arbitrary strength, which covers the whole range between 2D and 1D behavior [1029], parabolic-Hartree±Fock self-consistent potential when GKT is violated [1033], strong periodic potential used to study giant oscillations in the Hall conductivity in coupled quantum wires [1036], in®nite potential well employed to study magnetoplasmons of a cylindrical quasi-1DES in the presence of an axial magnetic ®eld [1037], 1D Dirac-d potential used to study FIR absorption spectrum of narrow antiwires [1038], shifted-parabolic potential well giving rise to the evolution of Bernstein modes in the presence of a perpendicular magnetic ®eld [1042], and a 1D periodic potential well used to investigate the FIR absorption spectra of short-period parallel quantum wires in a perpendicular magnetic ®eld [1043]. Early model calculations of magnetoconductance in quasi-1DEG with a parabolic con®ning potential well [1023,1025,1026] had, in fact, explained the essential physics that was seen to be re¯ected in the later theoretical work on the magnetic-®eld effects [1027,1028,1030±1032,1034,1035,1039±1041] in parabolic quantum wires. It was established that energy levels for the case of parabolic potential turn out to be hybrids of the bare LLs and the quasi-1D states; the hybrid subbands depopulate with increasing magnetic ®eld. In the high-®eld limit, o approaches the cyclotron frequency. The result is that because of the electrostatic con®nement, a plot of subband index n versus magnetic ®eld is gradually non-linear. This non-linearity is used to extract electron concentration and channel width of

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the system. While most of the studies employ quantizing magnetic ®eld perpendicular to the original 2D sheet, some authors have also studied the axial geometry [1027,1034] and tilted magnetic ®eld [1035]. It is worth mentioning that inspite of the extensive, theoretical as well as experimental, work on magnetoplasmons, our understanding of the magnetoplasmon excitations in quantum wires is quite limited. This is so because no efforts are seen to have been made on the search of many-body effects in the presence of a magnetic ®eld on excitation spectra, since most of the theoretical works are con®ned to the simple RPA (see Ref. [1033] where HFA was used); besides, not much attention has been paid to where bare con®ning potential is different from parabolic potential. Two common features that have been established are: the negative B dispersion of intrasubband magnetoplasmons and almost qindependent dispersion of intersubband magnetoplasmons. The magnetization as a function of chemical potential is shown to have two superimposed oscillations [1034,1035]: one with a large period given by ho‡ and the other with a small period given by hoÿ, o being the hybrid (of cyclotron and con®ning)  frequencies. The magnetization oscillations are attributed to the differing signs in do =dB. Without quantum con®nements no such oscillations can occur in an axial magnetic ®eld [1034]; special care has to be taken for the survival of magnetization oscillations in a tilted magnetic ®eld [1035]. The experimental observation of quantization of the Hall effect and simultaneous vanishing of the magnetoresistance in an electronic system which exhibits spatial dispersion in 3D has already demonstrated (see, e.g., [1804]) that strict two-dimensionality is not a prerequisite for the observation of QHE. Yet the report by Roukes et al. [1044] of the quenching of the Hall voltage across a quasi1DES at low magnetic ®elds came as a surprise and was met with some initial skepticism; in retrospect, it appears that certain anomalies in the data of Timp et al. [886] were precursors of this phenomenon. Subsequently, several experiments performed in diverse physical situations [1045±1049] demonstrated that it was generic in the sense that it occurred for differently fabricated ballistic microstructures over a wide range of carrier densities. Interestingly, Ford et al. [1047] demonstrated that near vanishingly small magnetic ®eld the Hall resistance could be quenched, enhanced over its classical values, or even negative. Theoretical explanations [1050±1058,1771] hinge, at least, on one common belief that the observed low-magnetic-®eld anomaly (of generic quenching of Hall resistance) is a multiterminal junction effect, namely, the widening of the wires near junction region; where the typical scattering of electrons brings about the effect. This widening, which is typically present in quasi-1D microstructures but need not be, acts like an acoustic horn and sets up a non-equilibrium distribution in which modes with high longitudinal momentum are preferentially populated. The resulting collimation of the electrons injected into the junction suppresses RH (the Hall resistance) since the electrons in these modes contribute very little to RH . Together all experimental and theoretical results indicate conclusively that junction geometry, and speci®cally the collimation effect, is the basic mechanism for the observed generic quenching of the Hall resistance. There is an exception, however: Beenakker and van Houten [1051] had argued semiclassically that the observed quenching is due to the suppression of edge states in the channel at low magnetic ®elds. Having been convinced by the later contrasting theoretical explanations based on the exact microscopic theories [1052±1058,1771], the authors of Ref. [1051] have abandoned the edge-suppression theory (see, e.g., [1057]). The interest in the interaction-induced energy transfer between two Coulomb coupled quantum wires was triggered by Price [1059]. The model of a double quantum-wire structure was envisioned by Gold [1060] in the context of a charge-density-wave instability. Since then a number of authors have

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appeared on the scene to report diverse interesting phenomena investigated in DQWW [1061±1070]. Sun and Kirczenow [1061,1062] reported, within HKS DFT [190±192], that energy levels of a pair of parallel quantum wires lock together when the wire widths are similar and their separation is not too small. This energy-level locking is a joint effect of Coulomb interactions and of the DOS singularities that are characteristic of quasi-1D fermionic systems. It was argued that this level locking in similar wires persists to quite a large interwire separations, but is gradually suppressed by the interwire tunneling when the separation becomes too small. In dissimilar parallel wires, level locking was expected to be much less likely to occur. The tendency of a DQWW towards charge-density-wave instability was examined by Wang and Ruden [1063], within the linear response theory to include local-®eld effects in the STLS scheme. It was found that the DQWW system at low temperatures is unstable subject to the conditions of low density and suf®ciently close proximity of the quantum wires. It was argued that because the local-®eld correction in the Hubbard approximation [1060] can never exceed the factor of half, the tendency towards an instability at large q is underestimated in the Hubbard approximation. FIR absorption, with and without a perpendicular magnetic ®eld, was studied within the RPA by Shahbazyan and Ulloa [1066]. Thakur and Neilson [1067] investigated many-body correlations of coupled electron±hole quantum wires using an STLS approach. The asymmetry of the system allows the acoustic plasmon mode to survive as a sharp peak after it enters the SPE region; the acoustic collective mode, but not the optical mode, was shown to be sensitive to the separation between wires. Heitmann and coworkers [1070] studied the FIR response within a self-consistent RPA. It was demonstrated that the oscillator strength of the acoustic mode is strongly enhanced if the con®ning potential in the two wires is different. In a perpendicular magnetic ®eld and a shifted-parabolic potential they found Bernstein modes occurring in the optical as well as in the acoustic branch. Furthermore, for small separation, the acoustic magnetoplasmon was shown to exhibit oscillations which are related to the magnetic depopulation of 1D subbands. Several interesting phenomena such as the Coulomb drag effect, BGR, and energy transfer rate in DQWW structure have been studied by Tanatar and colleagues [1064,1065,1068] and Tanatar [1069]. Their ®nding reports that the temperature dependence of the drag rate is signi®cantly enhanced when a dynamically screened effective interwire interaction is used. It was argued that the local-®eld corrections employed within the Hubbard approximation decrease the BGR at low densities especially when the lateral width of the quantum wires is small. The local-®eld effects describing the correlations beyond the simple RPA have been advocated to be very important for the low densities altering the energy transfer rate signi®cantly. The occasional authors have also added a remarkable contribution to the electronic, optical, and transport properties of quasi-1DES. These include, e.g., the binding energies of the bound states [1074,1076], ionized-donors-limited and phonon-limited mobilities [1075], the effect of lifetime broadening on the conductivity and thermopower [1077], the decisive self-consistent calculations of SchroÈdinger±Poisson equations dictating the nature of con®ning potentials in quantum wires [1078], the magnetotransport (SdH effect) in arrays of quasi-1D inversion structures [1079], the eigenstates in crossbar geometry [1080,1081], GKT for quantum wire structures [1082], phonon dispersion in a 2D arrays of quantum wires [1084], universal and non-universal conductance ¯uctuations in quantum wires [1085], interaction effects on transport in a single-channel Luttinger liquid [1087], prediction of a large negative differential resistance in a quasi-1DEG in quantum wires [1088], microscopic theory of phonon dispersion in an array of thin quantum wires [1090], the CB of tunneling into a quasi-1D

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quantum wire [1091], the recovery of quantized ballistic conductance in a periodically modulated channel [1092], self-consistent calculation of ionized impurity scattering in semiconductor quantum wires [1094], the temporal non-linear evolution of unstable modes in a 1D quantum plasma [1096], the electron density and wire width dependent attraction between a negative test charge and an electron, which is strongly enhanced by many-body effects [1098], the energy relaxation via con®ned and interface phonons [1099,1100], the multiple-subband-induced electrophonon resonances and a splitting and shifting of magnetophonon resonances in quantum wires [1101], the con®ned-phonon effects on BGR [1102], and the a.c. response of a quantum wire with short-range interactions [1103]. 2.2.4. Quantum dots and antidots The quest for diminishing dimensions of a system has reached a point where researchers have been able to create quantum dots, in which charge carriers are con®ned in all the three directions to have zero degree of freedom. Quantum dots, more fancifully dubbed as arti®cial atoms, are small boxes 100 nm on a side, holding a number of electrons that may be varied at will. These heterostructures, made possible by modern nanofabrication technologies such as epitaxial growth, lithography, and etching techniques, consist of nanoislands of one type of semiconductor either embedded into a different type of semiconductor or free-standing on a suitable substrate. Success with the lithography and etching techniques has, however, been quite limited because of the technological requirements of producing ultrasmall structures that are defect-free and that exhibit an abrupt carrier con®ning potential. An alternative approach that uses direct epitaxy for the fabrication of quantum dots is the so-called selfassembling technique, in which both the growth thermodynamics and kinetics are used to direct the atoms to a speci®c region on the surface. The method also relies on choosing the proper growth mode, i.e., island growth regime for the epitaxy of quantum dots. By sandwiching these islands between two epitaxial layers of wider band-gap materials, self-assembled quantum dots are directly produced. Historically, the ®rst hints that zero-dimensional quantum con®nement was possible, came in the early 1980s, when A.I. Ekimov and his colleagues at the Ioffe Physical±Technical Institute in St. Petersburg noticed unusual optical spectra from samples of glass containing CdS or CdSe. The samples had been subjected to high temperature; Ekimove suggested tentatively that the heating had caused nanocrystallities of the semiconductor to precipitate in the glass and that quantum con®nement of electrons in these crystallites caused unusual optical behavior. Explicit concept for 3D con®nement of charge carriers, as now being realized in semiconductor quantum dots, was proposed by Arakawa and Sakaki in 1982 [1105], who showed that, in a 3D-con®ned quantum-well lasers, the thermal broadening should vanish because the electron DOS is d-function-like. In order to grasp the chain of reasoning, imagine an electron trapped in a box. Quantum mechanics dictates that electron being a subatomic particle has wave properties (de Broglie duality), like the ripples on water or the vibrations of a violin string. Just as a violin string is tied down at both ends, so is the electron wave bounded by the walls of the box. And as the wavelength of the string's vibrations must ®t within its two ends, so will the electron wave's within those con®nes. In the case of a violin string, the point at which it is tied down changes as the violinist's ®nger slides up the ®ngerboard. The length of the allowed waveform shortens, and the frequency of the string's vibrations increases, as does that of all its harmonic overtones. With the same token, if the electron's con®ning box is made smaller, the electron's lowest energy level (the analog of the fundamental pitch of the violin) will increase. For semiconductor nanocrystallities, the fundamental ``pitch'' is the threshold energy for optical absorption, and the harmonic overtones correspond to new absorption features to be seen at higher energies.

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The fabrication and physics of nanostructures have been largely separated activities until recently, but engineers and physicists have lately started joining forces, both because the techniques of fabrication involve a larger element of physics and because the structures are entering the length regime where interesting physical phenomena are greater rather than minor perturbations. Once two or more of the length scales of a structure are 100 nm or smaller, the mode of operation of any device becomes qualitatively different from that of the larger devices in current use. Miniaturization, as such, seems to be reaching an end, and while optical and other radical alternatives to information processing may emerge, there is a possibility of total crystal engineering in the near future. The ability to tailor 3D nanostructures in a wide range of materials may lead to synthetic solids with more desirable device characteristics than those provided by nature. The combination of MBE crystal growth and lithography is the current research frontier en route to the desired fully synthetic materials. For the moment, diminishing dimensions and miniaturization continue occupying a large majority of condensed matter physicists in the search of exciting consequences of the nanotechnology [1106±1120]. As with quantum wires, the obvious way to make dots is to start with an original 2DES. If the original 2DES de®nes the x±y plane, a narrow con®ning potential acting in the x direction produces a quantum wire. Additional con®nement along y direction yields quantum dot, an arti®cial atom created by lateral con®ning potentials in both x and y directions. Electrons in quantum dots can exhibit astounding behavior. Such structures, coupled to electrical leads through tunnel junctions, have been given various names: 0 (or quasi-0) DEG, arti®cial atoms, single-electron transistors (SETs), Coulomb islands, besides quantum dots. For intuitive as well as practical purposes, these are all regarded as arti®cial atoms Ð atoms whose effective nuclear charge is controlled by metallic electrodes. Like natural atoms, these small electronic systems contain a discrete number of electrons and have a discrete energy spectrum. Arti®cial atoms, however, have a unique and spectacular property: the current through such an atom or the capacitance between its leads can vary by many orders of magnitude when its charge is changed by a single electron. As in real atoms, electrons are attracted to a central location. In a natural atom, this central location is a positively charged nucleus; in an arti®cial atom, electrons are typically trapped in a bowl-like parabolic potential well in which electrons tend to fall in toward the bottom of the well. The connotation is that the electron wavelength (in arti®cial atom) is of the same length scale as the con®nement so that the quantum effects are important. As such, one can consider the arti®cial atom as a tiny laboratory where to test the textbook quantum mechanics. Planes, lines, and dots are mathematical constructs: they have no physical extent. How then is it possible to make them out of a real, 3D material? The answer lies in Heisenberg's uncertainty principle, a tenet of quantum mechanics. The position of an object (e.g., an electron) and its momentum cannot be known to arbitrary precision. More closely an electron is con®ned, more uncertain its momentum must be. This wider range of momenta translates to a higher average energy. In general, the energy of electrons in a semiconductor is limited by their temperature and by the properties of the material. When the electrons are con®ned in a small enough region, however, the requirements of the uncertainty principle in effect override other considerations. As long as the electrons do not have enough energy to break out of the con®nement, they become virtually trapped in that small region. This locution is not just an approximation. Electrons con®ned in a plane have freedom of motion in two dimensions, those con®ned in a quantum wire are free only in one dimension, and those con®ned in a quantum dot are not free in any dimension. For a conventional semiconductor, the length scale for a free conduction electron is about 10 nm. An electron inside a cube of semiconducting material 10 nm on a side is essentially con®ned to a point. In arti®cial atoms, electrons are con®ned in structures about 100 nm in diameter.

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Thus, an arti®cial atom in a crystal comprises many real atoms. The quantum theory of solids explains why the electrons do not get trapped on the real atoms of the crystal and how the electrostatic potentials of the atoms superimpose to provide a medium in which some electrons behave as free electrons, albeit with a different mass. The ``effective mass'' of the electron is thus less than its actual mass. For example, electrons in GaAs (Si) appear to carry a mass that is only 7% (14%) of the mass of free electrons. The physical characteristics of an arti®cial atom differ considerably from that of a natural atom for one important reason: arti®cial atoms are typically much larger than real ones. The electron orbits do not simply scale with the size. Imagine an atom containing many electrons whose size is continuously variable. As it becomes larger, the Coulomb energy arising from the repulsion between electrons orbiting around the nucleus decreases because average spacing between electrons increases. There is another energy scale in the problem, however, the separation in energy of the different orbits of electrons in the arti®cial atom. As the atomic size increases, the differences in the orbital energies decrease faster than the Coulomb energy. It follows that in a large atom, the effects of e±e interactions are more important than in a small atom. The present view of small electronic devices depicts them as controlling the motion of small but classical ``seas'' of a large number ( a few thousand) of electrons. As the devices shrink, this view no longer holds. In fact, it is already feasible to create electronic devices small enough to have device characteristics sensitive to the motion of single electrons within them, even at room temperature. In natural atoms, con®nement of electrons is caused by the radially directed electrostatic force of the nucleus, and the electron wave functions are radially symmetric. In quantum dots the shape of the gate electrodes controls the size, shape, and symmetry of the con®ning potential, and so ``wave function engineers'' may eventually be able to study atomic physics previously inaccessible in nature, such as the wave functions or electrons in an at-will shaped atoms. The realization of a well-de®ned quantum dot has required a precise control over damaging surface effects in, e.g., lithographically designed quantum dots. Groups at IBM, AT & T, and Max Planck in Stuttgart have managed to eliminate surface effects entirely. They make quantum dots by placing tiny gate electrodes on top of a buried layer that con®nes electrons to two dimensions. The top electrodes squeeze the electrons into quantum-con®ned islands. One advantage of this approach is that it is possible to put as many or as few electrons in the dot as desired. Electrostatic squeezing produces dots whose con®nement can be controlled more easily than can that of dots produced by other techniques. Thus, produced structure offers the researcher control over many of the variables that de®ne a dot, including size, number of electrons, and transparency of the con®ning barriers. The Delft and M.I.T. groups have discovered that energy levels of these small dots are determined not only by quantum rules based on the size quantization but also by those relying on charge quantization. The energy levels in a quantum dot depend in part on its capacitance and on the amount of charge contained within it; the amount of charge must of course be a multiple of e. The coexistence of these two kinds of quantization causes a subtle interplay of effects. To understand which one will be most important, one needs to know not only the wavelength and effective mass of the electron in the dot but also its electrical capacitance. If a dot is metallic, it has many more conduction electrons than does a semiconductor; the wavelength of the conduction electrons is very small (1 nm). As a result, in a 10 nm metallic dot, charge quantization exerts a much stronger effect than size quantization. The capacitance of the metallic dot, however, is not so different from that of the semiconductor dot of the same size, and in the semiconductor the energies of the two effects may be

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approximately the same. The most challenging obstacle is to achieve essentially perfect control over the size and purity of these nanostructures. The ``top-down'' approach to fabrication Ð carving, dicing, or squeezing semiconductors Ð may not be suf®cient without revolutionary advances in materials and fabrication. Indeed, it may be necessary to develop novel materials and synthesis techniques that blend conventional semiconductor technology with alternative approaches. The most important threat that researchers face, however, is not so much learning how to build quantum-con®ned devices; instead, it is to design useful circuits that exploit their potential. By stringing two dots together to form an arti®cial molecule, one can investigate coupling between the states of adjoining quantum dots. And, as Kouwenhoven of Delft University has already demonstrated, it is even possible to string many dots together in a pearl-necklace fashion to generate an arti®cial 1D crystal and to watch how the energy band structure of crystal forms. A group at Max Planck in Stuttgart, as well as a team at IBM and M.I.T., has made large, periodic arrays of dots by fabricating a gridlike gate electrode Ð the nanostructure equivalent of a window screen. Voltage applied to the grid forms a regular lattice of quantum con®nement in the underlying material. The size and number of electrons in each dot can be controlled, as can the height and thickness of the barrier between the dots. The appearance of regular peaks in the optical absorption spectra of these structures testi®es to the precision with which the arrays Ð some containing more than a million dots Ð have been made, as any variation in size would smear out the harmonic spectra. These results open up the possibility of making a planar arti®cial lattice in which virtually all the properties of the constituent atoms can be controlled. Just as individual quantum dots display energy levels analogous to those of atoms, an arti®cial lattice would possess an energy band structure analogous to that of a crystalline semiconductor, under certain polarization selection. No one, however, seems to have made a planar arti®cial lattice and unambiguously demonstrated its band structure as yet. As Mark Reed remarked, success will require not only exciting precision in fabricating the electrode grid but also heroic control of defects in the underlying 2DES. In natural crystalline systems, engineers can rely on the fact that all real atoms are identical, but in an arti®cial lattice, they will have to impose this uniformity by craft. An array of such quantum dots may be envisioned as a periodic potential landscape ®lled with electrons up to some Fermi level. Because the z component is usually much stronger than the in-plane con®nement, the electron distribution in a quantum dot is pancake-like rather than spherical. An intriguing twist in this genre is the topologically complementary structure of the so-called antidot lattice. If the voltage on the grid is reversed, the islands that attracted electrons now repel them. Electrons are then forced to reside in the intervening space, bouncing off the antidots as they move through the array in what is probably the smallest ``pinball machine'' ever made. Some people visualize the antidot lattice as a reversed structure with respect to quantum dots, where geometrical holes are ``punched'' into an original 2DES. Just as the FET, which is a switch that turns on when the electrons are added and off when they are removed, a new kind of transistor is created when electrons are con®ned to a small volume of space (such as in quantum dots) between the source and the drain, separated from the two contacts by thin insulating barriers. These barriers allow the electrons to enter or exit only by quantum tunneling. According to quantum mechanics, electrons can penetrate regions of space from which they would classically be excluded. This is called tunneling. Because of the (quantum) con®nement, both the charge and energy of the electrons become quantized, so the small volume behaves like an arti®cial atom. One way to learn about natural atoms is to measure the energy required for adding or removing

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electrons. This is usually done by photoelectron spectroscopy. For instance, the minimum energy required to remove an electron is the ionization potential, and the maximum energy of photons emitted when an atom captures an electron is the electron af®nity. To learn about arti®cial atoms, we also measure the energy needed to add or subtract electrons. However, it is done by measuring the current through the arti®cial atom. The results are truly amazing [1112,1113]. The conductance displays sharp resonance that are almost periodic in Vg (the gate voltage). Most remarkably, the period of the oscillations has been shown to be the voltage necessary to add one electron to the con®ned pool of electrons. In other words, for a speci®c value of Vg , the energy of the transistor with N con®ned electrons is equal to that with N ‡ 1 con®ned electrons. For this value of Vg , the charge on the dot can ¯uctuate and current can ¯ow. Because the transistor conducts at one value of Vg for every value of N, it turns on and off every time an electron is added to it. That is why it is called a single-electron transistor (SET). SETs have been made in a number of ways using lithographic techniques [1115]. But most of them operate only at very low temperatures, typically less than 10 K (low temperatures reduce thermal ¯uctuations that disturb electronic motion). The temperature of operation is proportional to the energy required to add an extra electron to the dot; and because the latter results primarily from the Coulomb repulsion between electrons, it varies as the inverse radius of the con®ned region. So, to make SETs that operate at higher temperatures, one needs to con®ne electrons to smaller regions than are accessible with current lithographic techniques (50 nm). Conductance through a small con®ned region in a quantum dot, e.g., is activated due to the Coulomb interactions which impose an energy barrier for changing the number of electrons in the dot. This phenomenon is known as the CB of single-electron tunneling. Curiously, the phenomenon of CB was ®rst reported by Gorter [1121] almost ®ve decades ago. Subsequently, Zeller and Giaever [1122,1775] noticed such a blockade as they were measuring current±voltage (I±V) characteristics of tunnel junctions containing isolated metal grains within the insulating barrier. To have a feel of the phenomenon, think about how an electron in the dot tunnels from one lead onto the dot particle and then onto the other lead. Suppose the particle is neutral to begin with. To add a charge Q to the particle requires energy Q2 =2C, where C is the total capacitance between the particle and the rest of the system. Since you cannot add less than one electron the ¯ow of the current requires a Coulomb (or charging) energy Ec ˆ e2 =2C. This energy barrier is called the CB. A fancier way to say this is that charge quantization leads to an energy gap in the spectrum of states for tunneling. For an electron (hole) to tunnel onto the particle, its energy must be larger (smaller) than the Fermi energy of the contact by Ec. Consequently, the energy gap has a width 2Ec (the gap in the tunneling spectrum is the difference between the ionization potential and the electron af®nity of the dot). If the temperature is low enough that Ec  kB T, neither electrons nor holes can ¯ow from one lead to the other. A key assumption of this ``orthodox model'' of CB is that GB (the conductance of the barrier between the dot and the leads) < e2 =h (the conductance quantum), so that number of electrons in the quantum dot is sharply de®ned classical variable that can take on only integer values. For GB > e2 =h no CB is expected classically. The situation is, however, quite different if the dot is subjected to a strong magnetic ®eld, in the quantum Hall regime (see some remarks later in what follows). The CB model accounts for the charge quantization but ignores the energy quantization resulting from the size quantization of the arti®cial atom. This con®nement of the electrons makes the energy spacing of levels in the atom relatively large at low energies. If one thinks of quantum dot as a small box (of side a), at the lowest energies the level spacings are of the order of h2 =ma2 . At higher energies,

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the level spacings decrease for a 3D dot because of the large number of standing electron waves possible for a given energy. In the controlled-barrier atom (as is the case with quantum dots), where the number of electron can be as much or as few as desired, the energy spectrum for adding an extra electron to the dot is discrete, just as it is for natural atoms. In a natural atom, one has little control over the spectrum of energies for adding or removing electrons. There the electrons interact with the ®xed potential of the nucleus and with each other, and these two kinds of interaction determine the spectrum. In the quantum dot, however, one can change this spectrum by altering the dot's geometry and composition. In spite of the dramatic differences in the behavior characteristics of natural and arti®cial atoms, incentive for the research pursuit of quantum dots truly comes from the analogies with the natural atoms. Performing optical (FIR) experiments, which can directly induce optical transitions between the meV levels con®ned in the quantum dot, seem to be much more direct way to determine their energy level spectra. To achieve a detectable signal requires dot arrays with active areas on the order of 10 mm2 . Fabricating such a surface, with  108 nearly identical quantum dots, poses a challenge. Typically, such spectroscopic experiments are carried out at liquid helium temperatures to eliminate thermal broadening. Experimenter often applies a magnetic ®eld (usually) normal to the original 2DES. One can thus exploit the interplay between electric and magnetic con®nements very effectively to characterize lengths, energies, and interactions in the quantum dots. In the presence of a magnetic ®eld, the cyclotron energy of conduction electrons may become comparable to the con®nement energy; and equivalently, the magnetic length may become comparable to the typical con®nement length. It should be pointed out that even in the most relevant experiment such as FIR, quantum effects are indicated very indirectly. The reason for this is that, in most of the laterally con®ned quantum dots, the con®nement is electrostatic and, as shown in a reliable self-consistent band structure calculations by Stern and coworkers [1123,1776], has a nearly parabolic form for the external con®ning potential. Nearly parabolic con®nement is seen to be a common feature of many known quantum dot structures, and this has important consequences for the observed optical response. In the model calculations, one is bound to choose a speci®c form for the electrostatic con®nement, although, in the end, the results depend only weakly on the geometry. For the case of exactly harmonic con®nement (parabolic potential) the analytical diagnoses of single-particle energy spectra, in the presence of a perpendicular magnetic ®eld, were made by Fock [1124], Darwin [1125], and Dingle [1126,1777] many decades before the modern era of quantum dots began. In the absence of a magnetic ®eld, a unique spectroscopic characteristic of parabolic potential is that the spectrum shows only one resonance at the bare con®nement energy. Most strikingly, this result holds for an arbitrary number of electrons in the dot. This is easily understandable: the Hamiltonian for an interacting, but parabolically con®ned, many-electron system is separable into two parts describing, respectively, the center-of-mass (CM) motion of the whole system and the relative internal (RI) motions of the electrons. In addition, because lFIR  2r (r being the radius of the dot), the dipole approximation is very well ful®lled and the exciting electric ®elds couple only to the CM motion. Thus, the optical response of a quantum dot with parabolic con®nement, in the absence of a magnetic ®eld, represents a rigid collective CM motion at the frequency of the bare con®ning potential. This makes FIR spectroscopy of con®ned electrons insensitive to both the number of electrons and to their interactions leading the formation of the compressible and incompressible states. This result is a generalization of the famous Kohn theorem [1127]. If the experiment demonstrates any (additively) higher frequency modes, it is, for sure, that the external potential deviates somewhat from the purely parabolic shape and senses the RI motions among

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the electrons. RI motions among the electrons can become more easily visible when either the con®ning potentials become non-parabolic or when the materials with non-parabolic band structure come into use. A non-parabolic potential shape can be achieved by bringing the charges that control the con®nement closer to the con®ned electrons. The application of an external magnetic ®eld, usually oriented perpendicular to the original MD 2DEG, to the quantum dots offers interesting possibilities to study few-electron systems. Since the cyclotron energy  hoc can readily be made much larger than con®nement energy ho0, one can examine the transition from electrically bound states to magnetic (Landau-type) levels. In natural atoms, observation of transition between magnetic levels (e.g., quasi-Landau resonances) only is feasible when the electrons are excited to high Rydberg states or when the atoms are exposed to megatesla ®elds present near pulsars. More closely, the quantum-dot system is related to the one of shallow donors in semiconductors. In contrast to donor atoms, however, one can adjust not only the size of the dots but also their electron population. Because the FIR response of electron system in a quantum dot re¯ects primarily the rigid CM motion, it is appealing to approach these excitations by way of a classical model of collective modes. The dispersion equation Ð Do ˆ …o20 ‡ 14 o2c †1=2  12 oc , with o0 as the characteristic frequency of the con®ning parabolic potential Ð one gets from quantum approach can then also be derived classically as describing a plasma oscillation, in which, in the absence of a magnetic ®eld, a disk of conduction electrons moves back and forth rigidly with respect to the ®xed donor ions. As the resonance splits and separates with increasing magnetic ®eld, the cyclotron orbit eventually becomes much smaller than the dot diameter. Then the classical model describes two resonant frequencies: the high-frequency mode is a magnetoplasma oscillation with o ! oc . In this mode all electrons orbit coherently around individual centers with the cyclotron frequency and hardly feel the con®nement of the dot potential. The low-frequency mode re¯ects a collective motion of the centers of all the cyclotron orbits around the center of the dot. In this mode each cyclotron-orbit center drifts around the center of the dot under the joint effect of the con®ning electrostatic ®eld and the applied magnetic ®eld, with a drift velocity vd. The resulting frequency at large magnetic ®elds, vd =r ˆ o20 =oc , thus decreases as Bÿ1 . It is known that for a hard-wall potential, this mode behaves like a surface magnetoplasmon and becomes localized at the edge of the dot in a strong magnetic ®eld. Therefore, it is often called an EMP (see Section 2.3.1 below). At low temperatures, electrons fall into discrete quantum levels of these quantum dots. Transitions never observed in the spectra of natural atoms are readily seen for arti®cial atoms. The electrons in quantum dots can also act lethargically, reluctant to displace themselves to make way for another electron, even on a time scale of milliseconds. Most strangely, the large Coulomb repulsions can make it seem that electrons attract one another. This is realizable only through the most sophisticated scheme such as HF technique, which is beyond the self-consistent model, and which approximates the electron repulsion using average ®elds of all of the electrons in their orbits. However, HF does take proper account of the exchange interaction that leads to a remarkable phenomenon in quantum dots. The HF scheme demonstrates an apparent short-range attraction between electrons, and produces an interesting dilemma for electrons in a quantum dot in the presence of a magnetic ®eld, where spin orientations play an important role. However, even HF scheme has certain limitation; it ignores the effect of electron correlations on the Coulomb repulsion. Consideration of other methodologies that do take these correlations into account suggest even richer behavior in the spectra of quantum dots as we move to higher magnetic ®elds and more detailed measurements.

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The nature of charging effects and e±e interactions are central to much of the recent experimental [1128±1190,1778] and theoretical [1191±1300,1779,1780]3 work on semiconductor quantum dots. A key question to be answered by spectroscopic studies on quantum dots is the role of e±e interaction in modifying the dot's electronic level structure. Bryant [1191] has addressed this question for quantum dots containing just two electrons. He found a continuous evolution of the level structure, from singleparticle-like states in the limit of a very small dot to a level structure dominated by the e±e interaction in larger dots. Since the con®nement potential in quantum dots can be controlled at will, a large range of this continuum which is not accessible in atomic physics can be examined. Crucial to the system containing more than one dot is an understanding of how ``coupled'' dots interact. It is becoming known that isolated-dot arrays show strong CB conductance peaks (versus gate voltage) which split into two (double dot) and three (triple dot) peaks as the coupling increases. Many interesting phenomena have been predicted for coupled quantum dot arrays in the tunneling regime, including conductance peak splitting, peak suppression, single-electron solitons, and quasiperiodicity, among others. Experiments on quantum dots (single or multiple) are implicitly motivated by the quest to observe the consequences of either the charge or the energy quantization Ð both of which are brought about by the quantum con®nement Ð on the electronic transport at low temperatures. In the electrical measurements, the change in the electrostatic potential of the dot when a single electron is added, df ˆ e=C, results in CB oscillations in the conductance versus gate voltage, with each oscillation corresponding to the addition of one more electron to the dot. The CB oscillations thus permit a direct determination of a change in the number of electrons in the dot. The energy quantization can also become accessible in the electrical transport. When the spacing DE between individual quantum levels, or 0D states, of the dot becomes larger than kB T, these states can also be seen in the transport measurements. They are most directly observed when a large source±drain bias Vds > DE is applied. Features are then observed when the Fermi energy of the source or the drain aligns with the energy levels of the dot. The determination of energy spectrum from the transport experiments is, however, dif®cult because of the charging effects. Measurements of the energy levels of electrons in quantum dots yield a wealth of information about a new physical regime. A large number of experimental probes, such as infrared absorption, resonant tunneling, and capacitance spectroscopy, have been employed for this purpose. These experiments which probe electrons in quantum dots with nearly parabolic con®nement, with or without an applied magnetic ®eld, remain rather insensitive to the many-body Coulomb interaction effects, due, in fact, to the GKT [1127]. Capacitance and transport measurements are also not favorable for the study of isolated dots since they require coupling to external contacts; particularly for small dots their interpretation is additionally hampered by the CB. Despite these dif®culties, much information on the single-particle spectra could be deduced from transport data. Such a limitation of these probes has, however, suggestively been remedied by using, e.g., PL spectroscopy, in which electron±hole (or exciton) states are probed rather than electron states. The PL experiment is suggested to be sensitive to the many-body effects, independently of the shape of the con®ning potential. Even better and relatively more powerful tool is the resonant Raman scattering in which novel applications emerge from the comparatively large momentum transfer that correspond to the radii of quantum dots. In this wave vector regime, the GKT, which limits the reach of other long wavelength optical probes, is no longer 3

Ref. [3] in [1241] is noteworthy.

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valid, and the Raman scattering is found to be effective in giving access to excitations that exhibit the combined effects of the con®ning potentials and e±e interactions. Experimental techniques used to probe the semiconductor quantum dots, mostly in the presence of an external magnetic ®eld usually oriented perpendicular to the original 2DEG, include, e.g., electrical measurements of the I±V curve [1128,1141,1146,1150,1171,1173,1176,1180,1185,1186], capacitance spectroscopy [1129,1131,1135,1167], FIR spectroscopy [1130,1133,1134,1136,1138,1144,1145,1155, 1160,1163,1167,1178,1188], transport measurements [1132,1137,1139,1140,1143,1151,1153,1154, 1156,1158,1161,1166,1172,1182,1189], resonant tunneling [1142,1147,1149,1152,1164,1169,1174], single-electron capacitance spectroscopy (SECS) [1148,1159,1162,1187], PL spectroscopy [1157, 1175,1177,1179,1184], microwave photoconductivity technique [1165], photon-assisted tunneling (PAT) [1168], single-electron tunneling spectroscopy (SETS) [1170,1183], resonant Raman scattering [1181], and EELS [1190]. Here the SECS is shown to allow the direct measurement of the energies of quantum levels of an individual dot as a function of magnetic ®eld. The method is based on observation of capacitance signal resulting from single electrons tunneling into discrete quantum levels [1148]. Later, the SECS was also used to measure the ground state energies of a single quantum dot containing N (ˆ 0 to 50) electrons. The experimental spectra have been shown to reproduce many features of a non-interacting electron model with an added ®xed charging energy [1159]. While most of the above-mentioned experiments have focused practically on the single quantum dot structures, some of them have also considered double [1172±1174,1180,1182,1185] and triple [1174] quantum dots, 1D crystal of quantum dots [1137,1143], 2D array of quantum dots [1130,1144, 1145,1170], 2D array of antidots [1144,1165], 2D array of 3D quantum dots [1160,1163], vertically aligned and electrically coupled quantum dots [1177,1185], self-assembled quantum dots [1184], and a square lattice of circular quantum dots [1188]. The coupling between two (or more in series or arrays) dots can be primarily Coulombic, implying that the charge state of one dot in¯uences the potential and hence tunneling on or off the other dot. The effects of this coupling have been extensively studied in semiconductor quantum dots, particularly where quantum level spacing is very ®ne () DE  kB T). Such effects are now largely understood, with the exception of the effects of tunneling on interdot Coulomb interaction [1174]. Transport through coupled quantum dots is also expected to be strongly in¯uenced by the quantization of the energy levels of the individual dots. This is because the tunneling between the two dots is primarily elastic, requiring the energy state of one dot to align with the energy state of the other for conduction. Interdot tunneling, and hence transport through the entire system, should be a sensitive function of the alignment of the quantum levels of the dots. Although several studies on transport through quantum dots in magnetic ®elds exist, it is not well understood how the few-electron system behaves in magnetic ®elds corresponding to ®lling factors n  1, where the spin effects become important. Recently, von Klitzing and coworkers [1189] have discussed, for the ®rst time, the electron-spin resonance in a single quantum dot at n  2. They show that the exchange interaction leads to an enhanced effective g factor, which in turn gives rise to spatially separated regions of compressible and incompressible states with different spin orientations in the dot. It was argued that this enhanced g has considerable in¯uence on the transport properties of a quantum dot, since it re¯ects a certain spin texture in the dot. Equally exciting is the use of EELS [1190], where energy-®ltered transmission electron microscopy (TEM) has allowed the simultaneous study of low-loss EEL spectra for a number of individual CdS nanoparticles of different sizes. It was shown that the plasmon peak energy increases with decreasing particle size and that the features in the spectrum appear to broaden in proportion to the magnitude of the shift.

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In a nutshell, transport through quantum dots is dominated by CB effects and single-electron tunneling, as has been demonstrated by several experiments on electrical and transport measurements. Apart from these electrostatic phenomena, quantum effects become observable in nanoscale devices (see, e.g., Refs. [1140,1159]). In these systems, the quantization of the charge and energy of the electron gas has important implications for transport. Charge quantization is important, since it means that adding an extra electron to the dot can cost a ®nite, measurable charging energy. Transport is suppressed if this charging energy Ec > kB T, creating a Coulomb island Ð a small region of electron gas electrically insulated from the leads by Coulomb interactions. While charge and energy quantization effects are often taken separately, the regime in which both are important has already begun to be explored. Some theoretical works are known to predict that the characteristics of the CB conduction peaks are affected by the quantized single-particle eigenstates of the dot. Kastner [1115] has already demonstrated that the conductance peaks in a small dot, in the presence of moderate magnetic ®elds, re¯ect dramatic, distinct structure as a function of B. This was shown to result from the B dependence of the quantized single-particle energy states of the dot. Recently, this view also seems to have gained support from an exciting experiment by Kouwenhoven and coworkers [1183], who used SETS to probe electronic states of a tunable, few-electron, vertical quantum dot. It is well-said that science has periods and ®elds in which our efforts are well channeled. The channels are de®ned by conceptual tools, theoretical schemes, experimental techniques, and a technology which allows us to look at a certain range of phenomenon. Using nanofabrication technology developed during only the last 10 years, experimenters have now been able to make nanoscale structures that can show strong effects when just a single electron is added or subtracted. This sensitivity to individual electrons has led people to talk of ``single electronics'', when the Coulomb energy controls the electron number with single-electron precision and of arti®cial atoms when the discreteness of the quantum energy levels is important. It is the prospect of developing such systems to the point where a single electron can carry one bit of information that has inspired talk of a revolution in computer technology (the quantum computer!). The theoretical development we discuss below is a result of almost simultaneous (to nanofabrication) efforts dedicated to explain the experimental results and/or predict new hypotheses in the ®eld of nanophysics. Theoretical development revolves in large part around two distinct phases: electrically isolated quantum dots and tunnel (or Coulomb)-coupled quantum dots. Because of con®nement in all three dimensions, isolated quantum dots contrast the single quantum-well and quantum-wire structures for one important reason: no collective (plasmon) modes are allowed in the isolated dots, due to utter lack of any degree of freedom. Theoretical understanding of the isolated quantum dots is then left to the study of a complete quantum-con®ned systems containing an arbitrary (but ®nite) number of electrons and/or holes, where only intradot few-body effects will be important. In the coupled quantum dots, on the other hand, quantum con®nement, intradot exchange-correlations, and interdot tunneling, as well as long-range Coulomb forces can all play important roles. As such, even if the quantum dots in 2D array (or 1D chain) are considered electrically insulated from each other in the sense that electrons cannot transfer from one dot to another, the long-range Coulomb force couples the quantum dots; and this coupling can lead to the collective excitations in the system. This has been demonstrated by some authors, as will be discussed a little later. If the number of electrons N is large, it is reasonable to expect that their electronic properties may be qualitatively described by approximating to known properties of the in®nite (N ! 1) electron gas. However, for small number of electrons (say, N < 10) the electronic properties of the system are likely

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to be strongly N dependent, particularly in the quantum regime where the nanophysics is relevant. The large variation in chemistry of the ®rst few elements in the periodic table suggests that this will really be the case. More so because allowance for control over the number of electrons in the dot is like having a knob that tunes an atom to different elements of the periodic table. Moreover, according to the geometrical shape of the quantum dots, the con®nement length scales in three spatial directions (Lx ; Ly , and Lz ) can be different. Within a simple particle-in-a-box picture, the level splitting scales as DE  Lÿ2 , with L as the con®nement length. It follows that for Lx  Ly ; Lz, the electrons will be stuck in the lowest y, z subbands; y, z degrees of freedom will be frozen out. The nanostructure is then a quasi-1D in the sense that electrons only have some signi®cant freedom along x direction. Similarly, if Lx  Ly  Lz , the nanostructure is quasi-2D; if Lx  Ly  Lz , the nanostructure is quasi-3D. Given that the trend in the targeted quantum dot structures is toward ever smaller devices (and hence smaller N), and given that the quantum, few-body problem has relevance for a wide range of nanostructures, we will con®ne the following discussion on the theoretical development of the subject to systems containing few particles and with con®nement in all three spatial directions. It is noted that energy spectrum of a few-electron semiconductor quantum dots is extremely rich, since the con®nement energy, cyclotron energy for moderate magnetic ®elds, and e±e interaction energy can all be of comparable magnitude (typically of a few meV); none of the three energy scales can a priori be thought of as a small perturbation. While special attention will be paid to the (theoretical) work on plasmons and magnetoplasmons in quantum dot systems, other relevant discussion would be kept, just as before, to a ``collection of observations''; a relatively personal perspective. This is because the time as well as space are preventing me to give a broad, critical review of so very rapidly and extensively developing ®eld of nanostructure physics. Theoretical formulation of few-electron quantum dots represents a real challenge to theorists since it marks a departure from the typical many-electron systems studied in the other systems of reduced dimensionality. Systems such as quantum wells and quantum wires are not strictly nanostructures because they still possess at least one degree of freedom; the energy spectrum thus comprises bands of allowed energies, as in a conventional semiconductor. Such structures can, in principle, be treated as the ones containing an in®nite number of electrons and it only makes sense to talk about an electron density. Many-body effects can therefore be tackled with considerable con®dence using traditional mean-®eld (i.e., large N) theories. Such approaches are likely to be suspect for quantum dots where N is small. It might appear that correct description of a quantum dot could possibly be borrowed from atomic physics. However, this is also expected to be problematic. This is because in systems such as real atoms, where the two competing energy scales are the kinetic energy (which scales as the inverse square of the con®nement length) and the e±e interaction energy (which scales as the inverse of the interelectron separation), the con®nement-length scale is small, and kinetic energy tends to dominate. Perturbative treatments of e±e energy starting from a shell model work reasonably well and have had much success for atoms. Such a simpli®cation for the quantum dot systems is not generally possible since the kinetic energy and potential energy are typically comparable, both being of the order of a few meV. A further complication arises from the fact that even moderate electric and magnetic ®elds will introduce perturbations of this order, which should also be treated on equal footing. Conclusively, quantum dots represent interacting, few-body systems with a response to external probes which may be intractable within the framework of a linear-response theory. Most of the theoretical work aimed at calculating N-particle energies in quantum dots has made use of the EMA for the electrons. Within the EMA, the exact SchroÈdinger equation for N-particles in a

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d-dimensional quantum dot, subjected to a con®ning potential Vc …r† and an applied magnetic ®eld B oriented along z-axis, is given by HC ˆ EC with H ˆ Hr ‡ Hs ; Hr and Hs being the parts of H that depend on spatial and spin coordinates, respectively. The spin±orbit coupling, which can be shown to be small in, e.g., GaAs samples, is usually ignored in almost all theoretical work known to us (and so will be done here). Explicitly, " #  X 1  X eAi 2 p ‡ ‡V …r † ‡ VI …ri ÿ rj † (2.1) Hr ˆ i c i 2m c i i6ˆj and

X Hs ˆ g mB B Si;z ;

(2.2)

i

where pi ; Ai ; Vc …ri †; VI …ri ÿ rj †; g ; mB , and Si;z are, respectively, the momentum associated with the ith particle, vector potential, con®ning potential, binary e±e interaction potential, effective electron g factor,PBohr magneton, and spin components along the z-axis. The z-component of total spin Sz ˆ i Si;z , and represents a good quantum number for the system. It is generally assumed that VI …ri ÿ rj † represents a translationally invariant system, which is not a priori true due to the presence of image charges in the surrounding dielectric materials and gates. However, such a complication has been ignored in all the existing theoretical work to date. In addition, if Vc …ri † is assumed to be harmonic, then the shape and magnitude of VI …ri ÿ rj † is immaterial (see remarks on GKT). However, the wave functions of electrons in 2D (and 1D) quantum dots have a small but ®nite extent in the remaining strongly con®ned directions. This results in a slight smearing of the electron charge, modifying the pure Coulomb form, which is appropriate to the point charges. As remarked implicitly before, the translationally invariant VI …ri ÿ rj † only affects the energy spectrum associated with the RI motion of the electron system, and adds really nothing if the con®ning potential is parabolic. The analytically solvable model calculations in d ( 3) dimensional dots employ the Coulomb interaction VI …r† ˆ ar ÿ2 (inverse square interaction) and VI …r† ˆ dV0 ÿ m o20 r2 =2 (harmonic interaction), where a, V0 , and o0 are all non-negative parameters; while the numerical results take the form VI …r† ˆ br ÿ1 . For discussion of some microscopic model calculations in d ( 3) dimensional quantum dots, we refer to Ref. [1118]. It seems that the theoretical interest in quantum dots was sparked by Bryant's early calculations of electronic structure for N ˆ 2 electrons in a long narrow square box (L  Ly ˆ 10Lx ) with in®nite barriers and zero magnetic ®eld [1191]. He investigated the interplay of the kinetic energy ( Lÿ2 ) and the potential energy ( Lÿ1 ) as a function of the box-size. It was found that the energy spectrum depends strongly on the aspect ratio Lx =Ly . For smaller box, subband spacing becomes dominant and the carriers become frozen in the lowest subbands. For suf®ciently large box, the correlation effects become dominant and the possible precursor of a Wigner crystal has been seen. Subsequently, Bryant's major interest focused on studying excitons and biexcitons in GaAs quantum boxes [1192±1199], stressing the importance of Coulomb effects in determining the extent to which the electron and hole can correlate in forming an exciton. He showed that con®nement has little effect on the exciton for L  100 nm. For L  100 nm, the con®nement energies become comparable to the exciton energies, and Coulomb interaction becomes less effective at mixing single-particle states to form a correlated exciton. At the length scale of complete quantization (L  10 nm), the box and exciton are the same size, the exciton energy is dominated by con®nement energy, and there is no level mixing so

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electron±hole pair occupy the pair of lowest-energy single-particle states. More closely, as the box shrinks, there is a larger reduction in electron±hole separation and an enhancement of the exciton oscillator strength. These were argued to be the con®nement effects, not effects due to the increase in direct Coulomb interaction brought about by the con®nement. Later, Bryant has also studied energy spectra of vertically coupled quantum dots in the absence of a magnetic ®eld [1197]; emphasizing on how the quantum con®nement, intradot correlation, and interdot tunneling determine the charge transfer between the dots. Kirczenow and coworkers [1200±1205] have made a very interesting contribution to the understanding of e±e interactions in quantum dot systems, both with and without an applied magnetic ®eld. In fact, Que and Kirczenow [1200] were, to our knowledge, the ®rst to study the collective excitations in 2D arrays of quantum dots, before any optical (FIR) experiments on quantum dot systems were attempted. For this purpose, they chose a 2D periodic array of quantum dots lying in the x±y plane, with z motion frozen out, implying that it was only necessary to consider the lowest subband in the z direction. Furthermore, they hypothesize that even though the quantum dots are electrically insulated from each other in the sense that electrons cannot transfer from one dot to another, the longrange Coulomb force couples the quantum dots and this coupling can lead to collective excitations in the system. Disallowing the wave function overlap, they studied the collective excitations within the RPA, using tight-binding wave functions in the x and y directions. Assuming that each quantum dot has two energy levels, they simpli®ed the general equation to compute the collective mode dispersion. The two modes (longitudinal and transverse) were explicitly shown to be non-degenerate in the three principal symmetry directions, with the exception at high symmetry points G and M where the two modes remain degenerate. In this way, they indicated that a system with a dispersionless single-particle energy spectrum can have a dispersive collective energy spectrum, and that degeneracy in the singleparticle spectrum can be lifted in the collective spectrum. It was thus demonstrated that while a single quantum dot containing only one electron can only have SPEs, an array of such dots can also support collective excitations with energies shifted signi®cantly from SPE energies. Subsequently, Kirczenow and coworkers [1201] investigated the quantum Hall effect in a single quantum dot coupled to Hall and current leads by narrow constrictions and in a ®nite ballistic 2D periodic arrays of coupled quantum dots in transverse magnetic ®elds using diverse formal strategies such as scattering-matrix approach [1202], Halperin's edge-state scheme [1203], and Buttiker± Landauer transport theory [1204]. They also scrutinized the two largely used binary e±e interaction energies in quantum dots: the Coulomb and harmonic interaction [1205]. These two models have fundamentally different ground states for quantum dots in high magnetic ®elds. The Coulomb ground state can have large total angular momentum, while the harmonic ground state is always at minimum angular momentum. Johnson and Kirczenow [1205] bridge the two models, and show that harmonic interaction is a good approximation to the Coulomb interaction for strongly con®ned quantum dots. They also demonstrate that while the Laughlin wave function is an exact eigenstate of the harmonic interaction, the latter does not exhibit the FQH ground state. This implies that harmonic interaction is not a reasonable approximation to systems with strong con®nement, where the FQHE is quenched. The collective excitations in 2D arrays of coupled quantum dots were extensively studied by Huang and colleagues [1206±1209]. Their work was, in a sense, an extension of that presented in Ref. [1200] by allowing tunneling between neighboring quantum dots. In general, the effects due to tunneling can be accounted for in two aspects: the energy dispersion in the proper polarizability function and the wave function overlap between different quantum dots. In their earlier work [1206±1208], the tunneling

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was included through energy dispersion, and they neglected the wave function overlap from different quantum dots. This approximation is stated to be valid provided that the ratio of the lateral width of the dots to the barrier width between them is not too large. They ®rst allowed tunneling in only one direction [1206±1208], considered the hard-wall potential, included only the lowest ®lled subband (T ˆ 0 K), neglected vertex correction, replaced the Wannier function by a Gaussian function, and expanded the RPA polarizability function in the limit of in®nitesimally small band-width. Further, within the ®rst two excited states, they obtained two sets of decoupled intersubband excitations, analogous to those for 1D quantum wire. The splitting of each mode into two is attributed to the coupling between the transverse motion of the electrons con®ned in 0D quantum dots and the resultant longitudinal plasmon propagating along the quantum dot string. There they also considered the case of intrasubband plasmon excitation [1207]. Later the same group [1209] also investigated the plasmon excitations in a 2D periodic array of quantum dots by allowing tunneling (via energy dispersion) in both x and y directions. The dispersion relations of these modes were derived using tight-binding approximation between the adjacent dots and the RPA, just as before. They reported that in the presence of tunneling, both the depolarization shift and the splitting between longitudinal and transverse modes are greatly enhanced. This was suggested to be detectable in the Raman scattering experiments. In most experiments, an electron is tightly bound in a quantum well in the z direction, and its motion is frozen out in this direction. It is then quite reasonable to consider the electron to be bound laterally in the quantum well in the x±y plane by a potential Vc ˆ …1=2†m o20 r2 ; r 2 ˆ x2 ‡ y2 measures the distance from the center of the quantum dot. The simple mechanical analog of this system is a ball in the bowl whose cross-section is a parabola, and o0 is the frequency that a classical particle would exhibit while oscillating in this bowl. This case of a circularly symmetric parabolic potential approximates well with the actual case in experiments and turns out to be rather simple for analytical diagnoses [1123]. Suppose now that a magnetic ®eld is also applied perpendicular to the 2D (x±y) plane. The corresponding single-particle states were ®rst derived by Fock [1124] and Darwin [1125] in the late 1920s, and are recognized as Fock±Darwin states. The single-particle energy levels (the Fock±Darwin levels (FDLs)) of such a system are given by hO ‡ 12 l hoc ; En;l ˆ …2n ‡ jlj ‡ 1† 12 

(2.3)

where O ˆ …o2c ‡ 4o20 †1=2 , and n…ˆ 0; 1; 2; . . .† and l…ˆ 0; 1; 2; . . .† are, respectively, the radial and angular quantum numbers. These two quantum numbers correspond to the fact that the electron moves in two dimensions. In the absence of a magnetic ®eld (oc ˆ 0), Eq. (2.3) reduces to ho0 . n is a positive integer which corresponds to the number of nodes in the wave En;l ˆ …2n ‡ jlj ‡ 1† function as one moves radially out from the dot center, and 2jlj gives the number of nodes seen in moving circumferentially about the dot center. States of negative (positive) l correspond to electron wave functions moving clockwise (anticlockwise) with time as viewed looking down from the positive z-axis. These stand for a series of circular orbits, with states of larger jlj orbiting farther away from the dot center. The situation in the presence of a magnetic ®eld is different. The energy levels now depend not only on the absolute value of l, but also on l; clearly, then, the electron states with negative l have values lower than those with positive l. In addition, the ®rst term in Eq. (2.3) ensures that the energies (in the presence of a magnetic ®eld) are higher, and this is because the magnetic ®eld has enhanced the electron con®nement in the dot. Another important change in the quantum states due to an applied magnetic ®eld is that states of all l values shrink in radius as the magnetic ®eld is increased. For high

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magnetic ®elds (such that oc  o0 ), the radius of a particular l state shrink as Bÿ1=2 . In the limit of o0 ! 0, the single-particle energies reduce to En;l ˆ ‰n ‡ 12 …jlj ‡ l† ‡ 12Šhoc and depend only on the quantum number N ˆ ‰n ‡ 12 …jlj ‡ l†Š; physically N is the FDL index. Without the con®ning potential, the energies of the negative l states would be independent of l. This is the principal difference in the behavior of free and con®ned electrons, and it is responsible for much of the new physics uncovered in the quantum dot systems. The magnetic ®eld also creates an energy difference between the spin-up and spin-down electrons in the state. The electron spins have an associated magnetic moment, and so in an externally applied magnetic ®eld the electrons act like bar magnets tending to align their moments with the external ®eld (Zeeman effect). The energy difference between spin-up and spin-down states is usually much smaller than any other energy splittings in the quantum dot. In most of the theoretical work, a major assumption until the early 1980s was that in a system of more than one electron a strong magnetic ®eld keeps them spin-polarized, so that only one spin state is present. In 1983, Halperin [1806] ®rst pointed out that in GaAs systems, the effective electron g factor is one quarter of the free-electron value. Therefore, the Zeeman energy is approximately 60 times smaller than the cyclotron energy. This seems to be the reason that most of the theoretical work since then has been carried out ignoring the Zeeman term in the Hamiltonian (see Eq. (2.2)). For more than one electron in the dot, the interactions between electrons make understanding of the spectra substantially more complicated, but richer, than one might conclude from the above discussion. This can be seen by considering the basic problem of a quantum dot containing just two electrons. In this case, because of the Pauli exclusion principle, the two electrons must exist in quantum states that have at least one quantum number that differs. In zero magnetic ®eld, the electron energy does not depend on the spin orientation. The two electrons thus fall into the lowest available energy states with quantum numbers n ˆ 0, l ˆ 0, and (spin quantum) s ˆ  12. Such a state with oppositely pointing spins is known as singlet state. At suf®ciently high ®elds, the electron which in zero magnetic ®eld had its spin magnetic moment pointing against the (now) applied magnetic ®eld will, owing to the Zeeman effect, ¯ip its spin so that the moment points along the ®eld. When this happens, both electrons will have their spins pointing the same way and they will have the same value of s. In this Ð the so-called triplet state Ð one of the electrons moves from an l ˆ 0 state to l ˆ 1 state. Thus, this singlet to triplet crossing arises when the Zeeman energy exceeds the energy needed to promote an electron to the l ˆ 1 state. Since the spatial and spin part of the two-electron wave function decouple, the total spin determines the symmetry of the spatial wave function under particle permutation: singlet states belong to spatial wave function that are symmetric, while for triplet states the spatial part of the wave function is antisymmetric. It has been noted that when Coulomb interaction between the electrons is added to the problem, it actually drives the singlet to triplet crossing. In fact, the singlet to triplet crossing is predicted to occur even in the absence of Zeeman interaction. The next puzzle is: what happens to more (than two) electrons in a quantum dot? The spectrum can no longer be predicted exactly and one has to resort to some approximate schemes. There are several methods which have been attempted at in deriving the energy spectra in dot systems (as will be commented later); the most accurate is known as ``exact diagonalization''. This method in which one typically approximates the many-particle wave functions using combinations of a ®nite number of single-particle wave functions, is very much computer-intensive. However, in the lack of any better alternative, method of exact diagonalization has very frequently been used in the more-than-twoelectron dot systems.

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P The eigenstates of the system are the eigenstates of the total angular momentum J ˆ l, which is conserved by the e±e interaction. The main difference between the low-®eld and high-®eld behavior is in the ground state angular momentum. The general trend is that the energies increase with J, because the single-particle energies increase with l. In contrast, the interaction term decreases because electrons with higher angular momenta move in orbitals of larger radii, thereby decreasing the Coulomb energy. The net result is that the total energy as a function of J has a minimum. At low ®elds, this minimum occurs at the lowest available J, i.e., the smallest angular momentum compatible with placing all the electrons in N ˆ 0 state. At high ®elds, the minimum occurs at a higher J. The ground state of electrons in a magnetic ®eld therefore occurs only at certain magic values of J, where there are basis states in which electrons are sustained very effectively. The ground state always occurs at one of these values and the competition between interaction and con®nement determines the optimum J. The major effect of the interaction is that the ground state J increases with the magnetic ®eld. That is to say that variation in the magnetic ®eld should have dramatic effect on the excitation spectrum as well as the ground state. A pioneering contribution to the subject of quantum dots in a magnetic ®eld was made by Chakraborty and coworkers [1210±1217]. They were the ®rst to report the magnetic-®eld dependence of the eigenstates of interacting electrons, to generalize the Kohn theorem to quantum dots, and to explain the earlier FIR-experiments' insensitiveness to the e±e interactions in the few-electron (Nˆ3, 4) quantum dots con®ned by a parabolic potential [1210]. The possibility of observing the effects of e±e interactions was suggested through either the non-parabolic potential con®nement or the measurements of thermodynamic properties such as electronic heat capacity Cv, which is found from the temperature derivative of the mean energy. Subsequently, they studied magneto-optical transitions in a Coulombcoupled double quantum dots, where the GKT is violated and e±e interaction effects become observable [1211], excitons in a parabolic quantum dot in magnetic ®elds [1212], and observable e±e interaction effects on the magnetization of parabolic quantum dots [1213]. Later they compared the Hartree, HF, and exact-diagonalization methods on the calculation of energies, pair-correlation functions, and particle densities of the quantum-dot helium (i.e., N ˆ 2) in a magnetic ®eld [1214]. They found that exact and HF results for the triplet state agree reasonably well, which indicates the importance of exchange interaction for the few-electron systems. The results for the singlet state were shown to differ considerably; this was attributed to the neglect of electron correlations in the HFA. That is, the exact singlet ground state contains products of single-particle states with different angular momenta but these products are not accounted for in the HF ground state. As expected, the results with the Hartree approximation showed strong discrepancies from both the exact and HF treatments. The evolution of electronic energy levels as a function of magnetic ®eld were studied in an anisotropic quantum dot (with N ˆ 1; 2) characterized by a con®nement potential Vc …x; y† ˆ 2 2 2 2 1  2 m …ox x ‡ oy y † [1215]. In this work, they also derived the selection rules for the dipole transitions to higher energy, which dictate just two modes as in the case of isotropic parabolic con®nement. The only major difference in the case of anisotropy is that at B ˆ 0, the two modes split, DE ˆ h…ox ÿ oy †, in accordance with the experimental observation (see, e.g., Ref. [10] in Ref. [1215]). Maksym et al. [1216], while studying various quantum dot models, suggested that screening effects alone could have a very signi®cant effect on quantum dot energetics. They add that in a real system these effects would be combined with the effects of ®nite thickness which tend to modify the interaction at short range, so the interaction potential for real dots is likely to be quite different from a pure Coulomb potential. Recently, Maksym and coworkers [1217] have studied the optical-absorption spectrum in vertically coupled

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quantum dots (with a total of 3 spin-polarized electrons) in the strong magnetic ®eld. They report that the electron correlation in the double dot leads to the magic numbers, which are intrinsic to the vertically coupled dots, and have shown that they could be probed experimentally. In addition, it was demonstrated that the absorption energy of the double dot should exhibit discontinuities at the B values where J changes from one magic number to another. Broido and coworkers [1218±1222] switched their interest to studying EM response of the quantum dots. The calculation of the EM response, with the standard mean-®eld theories, consists of two steps. First, from the initial con®ning potential the ground state and the ®nal con®ning potential of the system are determined through a self-consistent solution of the coupled SchroÈdinger±Poisson equations. Second, the response of the ground state to EM radiation is obtained by employing the RPA or TDLDA, if the exchange-correlation effects are to be accounted for. In general, both steps are to be included in order to obtain a reliably accurate response. Many works involving quantum wells, wires, and dots are seen to have omitted the self-consistent ground state calculation. In the case of quantum wires and dots, this is, to a large extent, because the exact form of the initial con®ning potential is not easily known, and it may be different from the ®nal potential. Broido and coworkers [1218] studied the EM response and stressed that if a mean-®eld approach is used for this purpose and for making quantitative comparison with experiment, it is important that it include both the ground state and the response calculation. With a slightly different (from that in Refs. [1127,1210]) ideology they also established the GKT [1219]. Later, they reported the self-consistent theory of the FIR response for electrons con®ned in a quantum dot [1220]. They found that for small N, the FIR absorption spectrum corresponds to that associated with parabolic con®nement; whereas for large N, an upward shift in the response frequency occurs as the electron density probes the increasingly non-parabolic curvature of the dot potential. Moreover, additional effects of non-parabolic con®nement such as multiple mode absorption and mode coupling were suggested to occur if the dot structure possessed strongly reduced symmetry and higher electron occupancy that could probe the edge asymmetries. Subsequently, they reported on the dipole± dipole interactions leading to a spontaneous polarization of quantum dots [1221], and showed that if the interdot spacing is suf®ciently small and the dots contain enough electrons, such spontaneous polarization should occur, provided that the resonance frequency of a dot does not increase signi®cantly with the number of electrons in the dot. The resulting phase transition is of second order, and, in the absence of any external electric ®eld, the antiferroelectric phase is the most stable. Theoretically, spontaneous polarization occurs for a given wave vector k when the denominator of the susceptibilities vanishes. Broido et al. [1222] also investigated the single-particle spectrum of holes in parabolic quantum dots in a magnetic ®eld, where the coupling between heavy-hole (HH) and light-hole (LH) states causes the Kohn theorem to be violated, even when the con®ning potential is parabolic. Darnhofer and RoÈssler [1223] studied the band-structure and spin±orbit-coupling effects on the dipole spectrum of a single- and two-particle InSb quantum dots in the magnetic ®eld. They report that for a single-particle dot, the non-parabolicity of the conduction band leads to lowering of the o‡ mode and a splitting of the oÿ mode; this splitting is attributed to the spin±orbit-induced level anticrossing of the (0, 0, ÿ 12) and (0, ÿ1, ‡ 12) states. For quantum-dot helium, the coupling of CM and RI motions leads to an additional splitting of o‡ mode. They claim that replacing InSb parameters by those for GaAs, all deviations from the EMA are negligible. Later, they investigated extensively the dipole excitations and FIR response for holes in quantum dots in the magnetic ®eld, particularly by accounting for the mixing of HH and LH states leading to a substantial shift of the otherwise resonance frequencies

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and giving rise to new features at transition energies comparable to the HH±LH subband separation [1224±1226]. It is generally accepted that electrostatic con®nement, through the state-of-the-art nanolithography of the 2DEG, produces quantum dots where the con®ning potential is more likely represented by parabolic potential. An alternative method is based on etching techniques which produce quasi-0DEG where the con®nement may be represented by hard-wall potential. Peeters and coworkers [1227] initially investigated the electronic states, the energy levels, and the optical transitions of a collection of noninteracting electrons in a single quantum dot with hard-wall potential, in the presence of a transverse magnetic ®eld B. The results were compared with the ones with parabolic con®ning potential. There are important differences in the transition energies of the magneto-optical spectrum: (i) in contrast to the parabolic case where only two transition energies are found, in the hard-wall case there are many transitions possible which have different energies; only a small number of them, however, have suf®cient oscillator strength to be observable; (ii) with an increasing magnetic ®eld the energies approach the 2D result much faster for hard-wall case than for the parabolic one. In quantum dots with many electrons they calculated the Fermi energy as a function of magnetic ®eld. Since the electronic spectrum is quite complex, the Fermi energy is not a smooth function of the magnetic ®eld. For suf®ciently large B, however, the Fermi energy approaches 12 hoc , i.e., the energy of the lowest LL in the ideal 2DEG. Matulis and Peeters [1228] have reported a renormalized perturbative approach, which is based on the l-expansion technique (with H ˆ H0 ‡ lV), including e±e correlation in the calculation of the energy levels of a parabolic quantum dot. They illustrated the application of this technique for a quantum-dot helium for which exact results are available in the literature. They claim that their numerical results are adequate for l < 1 for the ground state and is valid over a large l-range for the excited states. In the range where l > 1, the l-expansion tends to diverge, which was remedied by using asymptotic expansion l ! 1 through quasi-classical approximation. This led them to reconstruct the renormalized l-expansion that agrees within 1% with the exact results for all e±e interaction coupling-constant values. They expect that this technique should give reasonable results even for systems with larger number of electrons; however, they suspect the similar l-expansion technique's validity for the case incorporating the non-parabolic con®ning potential and/or the extent of the wave function in the z direction. Subsequently, Peeters and Schweigert [1229] investigated the energy levels of a quantum disk containing one or two electrons, in the presence of a magnetic ®eld. Using the hard-wall potential of the ®nite height, they have been able to describe the CR data of Ref. [1167] reasonably well without involving any ®tting parameters, showing that CR transition energy exhibits jumps each time the ground state undergoes a singlet $ triplet transition. Later, Peeters and coworkers [1230] reported the numerical studies of the ground-state con®guration, its energy, and the spectrum of the normal modes of the classical arti®cial molecules consisting of two parabolic 2D atoms laterally separated by a distance d, demonstrating several d-dependent structural transitions. They also studied the magnetoplasma excitations of two vertically coupled quantum dots, with different parabolic potentials so that the GKT is no longer valid, with a large number of electrons which justi®es their use of classical hydrodynamical model [1231]. Being unaware of the development of the GKT for quantum wells, wires, and dots with parabolic con®nement, Que [1232,1779] erroneously commented on Ref. [1130] that FIR experiment's insensitivity to the number of electrons is an intrinsic property of quantum dots. Later, Que et al. [1233] generalized their earlier basic formalism on the collective excitations [1200] to include the

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effects of a weak magnetic ®eld B. This led them to explain the experimental observation of B dependent multiple branches and anticrossings; speci®cally, they predicted that anticrossings can occur if N > 2. For N  2, they found one branch with positive B dispersion, and one branch with negative B dispersion. The proposed use of left (right)-circularly polarized light was suggested to result in signals predominantly from positive (negative) B dispersion branches. They also argued that excitonic effects should be small in quantum-dot arrays because they represent the local-®eld corrections, while the depolarization shift in quantum-dot arrays is mainly due to interdot coupling. Subsequently, Que reported exact solution for excitons in quantum dots with parabolic con®nement potential, within an EMA [1234,1780]. Our major concern is their reporting of the tunneling plasmons in a 2D array of quantum dots [1235]. While it was becoming well-known, through experiments as well as theories, that the zero-dimensionality of the quantum dots in a 2D array is re¯ected in the measured optical spectra, the intriguing question Que raised was: does the two-dimensionality of the array manifest in the spectra as well? By then answer to this question was very controversial. In order to shed some light on this issue, he considered a rectangular array of quantum dots with small enough spacing between the dots in the x direction so that electrons can tunnel (weakly), but no tunneling was allowed in the y direction. Thus, there are three different dimensionalities involved in the system: 0D of the dots, 1D tunneling motion, and 2D array dimension. Let us recall that if the bare con®ning potential deviates from the parabolic shape, the GKT breaks down, but the collective plasmon picture remains valid. In the absence of tunneling, only discrete energy levels of the dots occur. In the presence of tunneling, each of the energy levels broadens into a miniband, similar to the formation of minibands in semiconductor superlattices. Known in the case of superlattices is that the presence of tunneling can allow a new branch of plasmon in the forbidden energy range of a non-tunneling system; this new branch of plasmon has been dubbed ``tunneling plasmon'' Que's analysis shows that although the dots are 0D and the tunneling is 1D, the tunneling plasmon, in the long wavelength limit, exhibits the characteristic 2D behavior; i.e., a square-root dependence on the wave vector. It is widely known that only in the long wavelength limit is the dimensionality dependence of the characteristics of plasmons clearly known. For example, their dispersion relations in the long wavelength limit (q ! 0) are given by the following expressions:  1=2 4pne2 2 2 3D ‡b q ; (2.4) o …q†  m  1=2 2pne2 q 2D ; (2.5) o …q†  m  1=2 ÿ2pne2 q2 ln…qt† 1D o …q†  ; (2.6) m where n is the electron density per unit volume, area, or length for 3D, 2D and 1D; and e…m † is the electron charge (effective mass). The constant b describes the leading wave vector dependence, and t stands for the ®nite extent in the transverse directions; the latter's non-zeroness is useful to avoid the logarithmic divergence of the Coulomb interactions. Note that only for 3D does the plasmon frequency tend to a non-zero limit as q ! 0; 2D and 1D plasmon frequency goes to zero as q ! 0. The numerical results in Ref. [1235] also reveal that for a wide range of wave vector along the GX direction the dispersion appears linear, which is a common feature in periodic systems. For instance, a linear

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dispersion has been observed in quantum-well superlattices [98] as wells as in quantum-wire superlattices [905]. The ¯at dispersion along the XM direction and the disappearance of the tunneling plasmon along the GX0 direction are both attributed to the lack of tunneling in the y direction. Besides the foregoing propagation characteristics of the tunneling plasmons associated with the wave vector dispersion, the energy of the tunneling plasmon was shown [1235] to be an oscillatory function of the electron density, in contrast to thep monotonic dependence on the electron density of plasmons in  3D, 2D, and 1D electron gases (o  n). This oscillatory dependence can be understood in the following way. In the systems such as considered by Que, since the tunneling is weak, at least some minibands are separated by energy gaps; and charge oscillation involving electron tunneling (in the x direction) can occur if there are partially ®lled minibands. If the Fermi level lies in an energy gap between minibands, such charge oscillation is inhibited; this is re¯ected in the corresponding polarization function becoming zero (see Eq. (4) in Ref. [1235]) in this case. As the electron density changes, the Fermi level sweeps through the regions where the tunneling plasmon is allowed and the points where it is forbidden. The tunneling plasmon energy vanishes when the electron density is such that there are only completely ®lled/empty minibands. This oscillatory behavior is, in a sense, similar to the magneto-oscillations in the plasmon energy in the tunneling superlattices [804,1760], with the difference that in the quantum dot arrays this effect occurs in the absence of any magnetic ®eld. Bockelmann and Bastard [1236] embarked on the dimensionality effects on the phonon scattering and associated energy relaxation properties of 2D, 1D, and 0D electron gases based on lattice matched InP±Ga1ÿx Inx As quantum wells, to conclude that for quantization energies above 2 meV, the energy relaxation by LA phonons is generally slower in 0D than in 1D and 2D structures; the LA phonon scattering in 0D systems becomes increasingly quenched with an increasing spacing of the discrete energy levels. The resulting strongly reduced relaxation of hot carriers is expected to limit the luminescence ef®ciency of small quantum dots in an intrinsic manner. However, when phonon scattering becomes weak does an alternative, ef®cient relaxation mechanism exist? Experimentally it has been shown that Coulomb interaction is important for the relaxation of hot carriers at high densities in bulk GaAs. Bockelmann and Egeler [1237] dealt with electronic transitions in InP±Ga1ÿx Inx As quantum dots mediated by the interaction with an electron±hole plasma, to discuss the in¯uence of dot size, plasma density, and temperature, within the RPA. They ®nd that for sizable plasma density, Auger processes (the two electron interactions) are ef®cient and represent the dominant relaxation mechanism for hot electrons, even in small quantum dots where the relaxation by phonon scattering is weak. Later, Bockelmann [1238] studied relaxation and radiative recombination of quantum-dot excitons, using rate-equation analysis, and neglecting HH±LH mixing and excited subbands which is justi®ed for near-band-gap excitons in quantum dots fabricated from the narrow quantum wells. His results indicate that the radiative lifetimes of excitons which derive from the ground state (excited states) of the relative-internal coordinates increase (decrease) with increasing con®nement. Exciton±phonon scattering rates strongly decrease with increasing energy-level spacing. For intermediate quantization, the analysis of luminescence suggests the possibility of strong excited transitions. In the case of strong con®nement, a slowed relaxation by LA-phonon emission was expected to lead to a very weak groundstate luminescence. He also investigated the electron±phonon and exciton±phonon scattering in GaAs± Ga1ÿx Alx As parabolic quantum dots in the presence of a magnetic ®eld [1239]. In the Born approximation, the rates of electron scattering between the ®rst excited state and the ground state were shown to be comparable in pure spatial and pure magnetic quantizations. Increasing spatial (magnetic) con®nement leads to a decreasing (increasing) electron±phonon scattering rate. With a sizable spatial

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con®nement, an increasing magnetic ®eld yields an increase in the scattering rate, followed by pronounced oscillations, and ®nally a decrease at high ®eld. Exciton relaxation by phonon emission is enhanced by magnetic ®eld in this system. For exciton relaxation dynamics, spatial quantization is fundamentally different from pure magnetic (or Landau) quantization. Halperin and coworkers [1240] have studied collective excitations in a model array of quantum dots with 2D parabolic con®nement, with and without an applied magnetic ®eld. Their formal technique treats the e±e interactions on the same dot exactly and those between electrons on different dots in a simple but reasonable approximation. The result is that the Hamiltonian separates into two parts, where the ®rst part depends on the CM motions of different dots and the second part involves only the RI motions within each dot. Diagonalizing the CM part of the Hamiltonian, they have shown that light couples only to the CM modes to an excellent approximation. They argue that even though the modes show dispersion at large wave vector q, it would be very dif®cult to observe this dispersion in the FIR optical absorption Ð ®rst, because q in such an experiment is always smaller than the reciprocal lattice vectors of the dot array, and second because the effects of retardation serve to reduce still further the dispersion in the optical region. When the 2DEG is further con®ned in the plane by a potential barrier, the energy of the states near the boundary will be altered. Moreover, it has long been recognized that states near the boundary produce a current, which ¯ows in a direction opposite to the circulation of inner orbits. These the socalled edge states, and their importance as a paramagnetic correction to the Landau diamagnetism, were discussed by Darwin [1125] in his seminal paper in which he considered electrons in a parabolic con®ning potential for which analytical solutions are possible. Subsequently, Dingle [1126,1777] dedicated to the investigation of the effects of such edge states in the ``small systems''. Recent upsurge of interest in the small systems, speci®cally the quantum dots, has been stimulated by the fabrication of single dots and quantum-dot arrays in semiconductors. Aimed at establishing the precise nature of edge states and their relationship to the classical ``skipping'' orbits, Lent [1241] studied the wave functions and currents in a circular quantum dot in a transverse magnetic ®eld. He considered both hard-wall and parabolic potentials, within the EMA, neglecting e±e interactions and ignoring spin effects. It was found that at low magnetic ®elds, the edge states are re¯ected by a single clockwise current ¯ow near the dot perimeter. As the ®eld increases, a counterclockwise current appears at the perimeter and the clockwise current is squeezed inward; in this intermediate regime, clockwise (``skipping'') current is not an edge current but rather a central current. At high ®elds, both rings of opposing current are localized in the central region of the dot and the states become similar (asymptotically) to bulk LLs. The qualitative discussion of the transformation of the states into LLs, with respect to the hard-wall and parabolic potentials, remains almost unchanged. Chaplik and coworkers studied the discrete energy spectra of a parabolic quantum-dot helium in a perpendicular magnetic ®eld, within the EMA, stressing on the Zeeman splitting as a function of dot size [1242]; investigated the collective excitations of Coulomb-coupled 2D periodic arrays of quantum dots both in the square and hexagonal lattices [1243]; predicted oscillations between spin-singlet and spin-triplet ground states as a function of magnetic ®eld [1244]; and showed that if a full Hamiltonian is solved, a sharp peak in the susceptibility and an additional feature in the speci®c heat occur, the role of e±e interactions were stated to be crucial [1245]. Most importantly, they suggested how FIR spectroscopy could be used to gather information about RI motions and hence observe the effects of e±e interactions; thereby circumventing the GKT in parabolic potentials [1246]. For this purpose, they proposed to study the quadrupole interaction of FIR radiation with quantum dots by using grating

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couplers asymmetric in, at least, one planar direction. They argue that the sensitivity for detecting these quadrupole transitions could be further enhanced (compared to ordinary dipole transitions) by subtracting the signals of two measurements employing differently polarized FIR light. In their initial work, Gerhardts and coworkers [1247,1248] emphasized that even the small deviations from the exceptional case of a strictly parabolic con®nement allow the coupling of CM and RI motions and make the FIR observation of Coulomb interactions feasible. This was demonstrated through a circular symmetric correction (e.g., / r4 ) within the framework of self-consistent RPA as well as the exact diagonalization schemes. Many experimental results, notably on FIR magnetospectroscopy of quantum dot arrays, cannot be satisfactorily understood without a good knowledge of con®nement potential, and this has its source in the spatial distribution of remote charges in the doped barrier and is not well known. Motivated by this fact, Lier and Gerhardts [1249] performed self-consistent calculation for the spatial distribution of inhomogeneously ionized deep Si donors in the barrier of a nanostructured GaAS±Ga1ÿx Alx As heterostructure with a corrugated top gate. In contrast to previous work by Stern and coworkers [1123,1776], calculating ground state of a single dot with a few electrons within Hartree approximation, they considered a periodic array of quantum dots, which simpli®es the electrostatic boundary conditions and facilitates comparison with experiment. A Thomas±Fermi treatment of 2DEG was used, which allowed them to cover the whole range from a weakly modulated 2DEG to isolated dots. Their data support the arguments of Ref. [1248] stressing on the importance of symmetrybreaking effects of deviations from a strictly parabolic con®nement potential. Subsequently, Gerhardts and coworkers [1250] investigated FIR response of quantum dots with a variable N and a non-parabolic con®nement in a magnetic ®eld, comparing the results of a Hartree and HFAs with those obtained by an exact diagonalization of a few-particle Hamiltonian. A good qualitative agreement was found between the HFA and exact calculation. The few-particle spectra are ®ngerprints of the ground state of the electron system and depend on both N and B. They found that even though the ground state computed by both methods differ considerably, the FIR resonances agree very well. Differences, of course, appeared in the results for the oscillator strength of resonances, since this quantity is much more sensitive to the features of initial and ®nal wave functions. With the increasing number of electrons, the Hartree and HF results show new features, which are stated to resemble the non-local mode coupling effects observed in the magnetoplasmon dispersion in 2D and 1D systems. While classical models have been quite successful describing the FIR absorption of large-period arrays of antidots, most of the quantum models have either oversimpli®ed the con®nement potential or neglected the periodicity in an attempt to model the short-period arrays. In arrays of quantum dots, the interaction between electrons in different dots has typically been approximated to a lower order than the intradot interactions [1240], again rendering the models inappropriate to describe a short-period lattice of dots, which can actually be strongly coupled electrically and/or even have an overlapping of wave functions. Gudmundsson [1251] has recently reported a model for the FIR absorption spectra of a shortperiod system that can equally be applied to an array of dots or antidots. In this model, the square array of quantum dots or antidots is described by a static external potential Vex ˆ V‰ sin…gx=2† sin…gy=2†Š2 , g2 ˆ 2p^y=L, and L is where g is the length of the fundamental reciprocal lattice vector: ~ g1 ˆ 2p^x=L, ~ ~ g1 ‡ ny~ g2, with …nx ; ny † 2 Z. The e±e interactions are period; the reciprocal lattice is spanned by G ˆ nx~ treated self-consistently within the Hartree approximation on equal footing in the ground state and the absorption spectrum, both for intradot and interdot case. The commensurability condition between the magnetic length lc and the period L requires the magnetic ®eld to assume the values B ˆ pqF0 =L, with

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(p, q) 2 Z, and F0 ˆ hc=e (the magnetic ¯ux quantum). How the placement of the system in a weak or strong coupling limit is governed by ns (the electron density), L, B, and V, all individually or hand in hand, is noticeable. For weak perpendicular magnetic ®elds (lc  L) the band structure manifests itself in the absorption. In the high magnetic ®eld both systems (of dots and antidots) show the characteristic two-peak picture, associated with CM motion of the dots, and the edge and bulk modes of antidots. In the intermediate region the magnetoplasmon splits into Bernstein modes, made visible owing to a certain shift in the parabolic potential shape. Theory of Raman scattering for 1D chain of parabolic quantum dots in a perpendicular magnetic ®eld was reported by Gumbs and coworkers [1252]. Within an oversimpli®ed d-function-like con®ning potential, the same group later reported the commensurability oscillations in the cyclotron mass [1253] and magnetoplasmon excitations [1254] in a 2D square lattice of antidots. Subsequently, they studied magnetotransport properties in a 2D square array of dots and antidots [1255], stressing on the quenching of Hall effect for antidots and the AB oscillations in quantum dot arrays; besides the observation of a threshold magnetic ®eld above which the system behaves like a homogeneous 2DES for antidots, and insulating behavior (manifesting through the vanishing of the longitudinal and Hall conductances) for quantum-dot arrays. They also presented a self-consistent theory for intersubband absorption coef®cient of a 2D square array of GaAs±Ga1ÿx Alx As quantum dots with parabolic con®nement potentials for electrons and holes [1256]. This work was aimed at making a qualitative comparison of the theoretical results with those observed in the (contactless) photore¯ectance experiment, wherein modulation of the built-in electric ®eld in the sample is caused by photoexcited electron±hole pairs created by a pump source such as lasers. While most of the theoretical work hinges on the quantum approaches, classical model theories, which were encouraged by Lorke et al. [1155], are not scarce either. For instance, Shikin and coworkers [1257,1258] have calculated the equilibrium properties and dynamic response of quantum-dot system (speci®cally an electron disk) with parabolic con®nement potential, within a local theory, both with and without an applied magnetic ®eld. The quantum corrections were estimated through the Thomas±Fermi approximation; it was shown that quantum corrections are negligibly small if aB =R  1, where h2 =m e2 is the Bohr radius and R the radius of the disk. In some experimental samples this ratio aB ˆ E is really small,  10ÿ2 for a dot with N ˆ 200 and R ˆ 160 nm (see, e.g., Ref. [1138]). So, in the ®rst approximation, the expected quantum corrections can be neglected. With this ideology, their classical results could explain the experimental results on mode spectrum of quantum dots containing  200 electrons in a disk of radius R  150 nm. It is quite clear from the foregoing discussion in this section that e±e interaction can (and does) play a dramatic role in the performance of the devices made out of such nanostructures. In particular, the capacitance of these devices are so small ( 10ÿ16 F) that any change in the number of electrons changes the total energy signi®cantly. For example, novel resonances have been observed in the conductances of these devices as a function of gate voltage in the presence of a magnetic ®eld [1140]. Recent studies have clearly demonstrated the importance of Coulomb interactions, which have two conceptually different effects in these small electronic systems. First, they lead to a phenomenon known as CB, which is a manifestation of charge quantization due to the small capacitance of the system. Second, they introduce non-trivial correlations in many-particle wave function, which should, in principle, reveal some interesting effects on the fractional quantum Hall (FQH) states in such small samples. (Note that electron correlations are responsible for the formation of FQH states in macroscopic 2D samples [84,1685,1686].) A crystal-clear signature of possible in¯uence of e±e

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interaction was noticed in the observation of conductance oscillations well above e2 =h [1151], that demonstrates that in the quantum Hall regime CB effects are considerably more robust. In a series of papers, Wingreen and associates [1259±1264] have theoretically shown that the persistence of the CB above the quantum conductance, and its exotic features, are consequences of the electrostatics of a quantum dot in the quantum Hall regime. In particular, it has been shown, by exact diagonalization of the Hamiltonian describing up to eight electrons in a parabolic quantum dot in a strong magnetic ®eld so that only lowest level is occupied, that at the simple ®lling factor n ˆ 1=p all conductance peaks are suppressed by a factor of N ÿ…pÿ1†=2 ; whereas at n ˆ 23 odd and even peaks are lowered differently [1261,1263]. Further, e±e interaction has been found to be a function of LL index and magnetic ®eld, and that Coulomb interaction strongly in¯uences the evolution of addition spectrum with B [1262]. Similar efforts have been made by Beenakker and coworkers [1265±1267] arguing in support of the dramatic effects that Coulomb interactions can have on the conductance in a quantum dot in the FQH regime. For instance, they have shown that AB effect (which in a singly connected geometry, such as a point contact or a quantum dot, is a result of transport via edge states) in the quantum Hall regime requires the incremental charging of the dot by single electrons; and that if the hoc, the AB conductance oscillations are virtually blocked by the Coulomb electrostatic energy e2 =C0 repulsion Ð a phenomenon termed as the ``CB of the AB effect'' [1266]. Subsequently, they studied the CB oscillation in the conductance of a quantum dot in the quantum Hall regime, within a selfconsistent model of McEuen et al. [1262] generalized to include extended as well as localized edge states [1267]. They found that a CB could exist for the transfer of an electron from an extended to a localized edge state, in accordance with the experiment [1151]. They have shown the role played by the incompressibility of the extended edge states, and predict that the conductance oscillations will be suppressed at lower temperatures when an odd, rather than an even, number of extended edge channels is present. As stated above, a full understanding of the experimental results requires knowledge of the manyelectron energy spectrum for a quantum dot. The theoretical treatments discussed above have been computationally intensive and have employed either the Hartree approximation that neglects the exchange-correlation effects, or the exact diagonalization which is limited by the convergence problems to treating just a few electrons (N  6). Johnson and Payne [1268] reported analytically solvable model, which describes a quantum-dot system that may contain an arbitrary number of interacting particles. There the many-body problem is solved exactly, yielding analytical expression for the energy spectrum as an explicit function of external magnetic ®eld and the particle number. The exact solutions imply that the effects of exchange-correlations are automatically included in the treatment. The two features of this model that allows exact analytical solution of the many-body problem are: (i) the bare (unscreened) 2D con®ning potential V…ri † for the ith particle is taken to be parabolic; and (ii) the interaction potential V…ri ; rj † between particles i and j moving in the 2D con®ning potential is taken to saturate (decrease quadratically) at small (with increasing) interparticle separation. ri ÿ~ r j j 2 ; V0 More closely, their model is de®ned by the interaction potential V…ri ; rj † ˆ 2V0 ÿ 12 m o20 j~ and o0 being the positive parameters, which can be chosen to model dots of different sizes and materials. The same authors later applied their exact analytical results in [1268] to study the many-body effects in resonant tunneling through quantum dots [1269]. It was shown that (i) many-body effects dramatically alter the resonant tunneling energies, (ii) the B dependence of these resonance energies differs signi®cantly from the single-particle results, (iii) the resonance energies of the dot cannot be

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written as the sum of the single-particle con®nement energies and constant, B-independent, Coulomb charging energies, and (iv) the semiclassical description of Coulomb charging effects (e.g., CB) in terms of capacitance is incorrect. Moreover, it was noted that the proximity of leads and gates can also introduce a complicated electric-®eld distribution at the dot, thus perturbing the N-electron wave functions and energies. Subsequently, the same exactly solvable model was used to study the excitation spectrum of an interacting electron gas in a quantum dot in the strong magnetic ®eld regime [1270]. It was demonstrated that the energy spectrum exhibits complex crossings as a function of the strength of e±e interaction. The corresponding eigenfunctions are generalizations of the Laughlin wave functions, but include signi®cant inter-LL mixing. Further, it was claimed that the Laughlin quasi-particles represent a poor description of the elementary excitations. Recently, Benjamin and Johnson [1271] have presented an analytically solvable model of P (the number of dots) collinear, 2D quantum dots, each containing two electrons. It is shown that the interdot coupling via e±e interaction gives rise to the sets of entangled ground states, which have crystalline interplane correlations and arise discontinuously with increasing magnetic ®eld. Their ranges and stabilities are found to depend on the dot size ratios, and to increase with P. Investigation into the physics of semiconductor quantum dots has received tremendous impetus ever since the prediction that the use of quantum dots would produce semiconductor lasers with high ef®ciency due to their discrete DOS [1105]. However, some authors have cast suspicion on this view [1272]; reasoning that although the dot's discrete energy levels are undoubtedly advantageous for optical purposes, they might hinder considerably the carrier relaxation for level separation DE as small as  1 meV. This issue was raised in [1272], suggesting that reduced relaxation could pave the way to an intrinsic mechanism for the poor luminescence yields of quantum dots. In the process of light emission, electrons and holes initially trapped into the excited states of the active region, relax in cascade to the band bottoms, emitting phonons, and ®nally recombine to produce light. Thus, any realistic discussion of light emission frequency is bound to entail a discussion of the relaxation process. In a quantum dot, the LO-phonon emission, which is the dominant relaxation path in higherdimensional systems (such as wells and wires), exactly matches the zone-center LO-phonon energy hoLO . Deformation potential interaction with LA-phonons, which is already weak in the bulk, weakens  further as the dot size is reduced due to decreasing form factor. Thus, the electrons are compelled to remain at the excited levels. Hole relaxation is expected to be much faster due to smaller DE, and, therefore, one ends up with electrons at the excited levels and holes at the ground levels. Orthogonality prevents radiative recombination between an excited-level electron and a ground-level hole, resulting in poor ef®ciency, i.e., phonon bottleneck. Since the hole relaxation is faster than that of electrons, the problem boils down to the search of some possible mechanism that would allow electron relaxation with the same pace (as that of holes). Several relaxation schemes have been proposed in speci®c situations: plasma-assisted relaxation [1237], exciton relaxation [1238], statistical approach based on the distribution of dot-level separation [1273], and interaction with speci®c phonons [1274,1275]. The last work dealt with a second-order perturbation calculation of the relaxation rate to conclude that twophonon processes LO  LA (‡ and ÿ refer to phonon emission and absorption, respectively) give rise to rapid ( 10 ps) relaxation if DE is near, but not too close to, the LO-phonon energy. The electron levels were assumed to have vanishing width, and inclusion of level broadening that requires a selfconsistent theory, was not pursued. In addition, coherence effects were neglected altogether. Recently, Sakaki and coworkers [1276] have improved upon the earlier work by investigating the effects of both coherence and multiple-phonon interactions taking into account the self-consistent level broadening.

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However, a reliably satisfactory solution to the problem of ef®cient electron relaxation still seems to be evading. Hawrylak [1277] embarked on the magnetoluminescence which, in contrast to the FIR, was shown to be a direct probe of e±e interactions in parabolic quantum dots. He demonstrated that the energy of the emitted photons, in an ensemble of quantum dots, undergoes discontinuous jumps as the dots are driven through a sequence of Laughlin states by increasing magnetic ®eld. The size and sign of the discontinuity is a direct measure of e±e and ®nal-state interactions. Subsequently, he studied the evolution of charging energies of few-electron quantum dots in the magnetic ®eld [1278]. The evidence of magnetic-®eld-induced spin and angular momentum transitions in strongly interacting dot was presented. Later, he calculated the excitation spectrum, given by the dynamical structure factor S…q; o†, of interacting electrons in a quantum dot in a strong magnetic ®eld [1279]. The wave vector and frequency dependence of S…q; o†, measured in electronic Raman scattering, is shown to be dominated by collective excitations of the CM motions at small momentum transfers, but shows a signi®cant oscillator strength due to transition between incompressible states of the dots at large wave vectors. He suggested that by sweeping the magnetic ®eld, one should be able to measure the opening and closing of the energy gaps tuned by the magnetic ®eld. Latge et al. [1280] have studied the wave vector and frequency dependent dielectric response of a 2D square lattice of quantum dots, using the lattice periodic potential within the framework of a nearestneighbor tight-binding model and the RPA. The calculations are restricted to the long wavelength regime and only intrasubband electronic excitations are considered. Both the dispersion relation and energy-loss function Im‰E…q; o†Šÿ1 were obtained for different carrier concentration ns . For low values of ns , the results are well approximated by the EMA; whereas for high ns the Fermi level moves to regions in k-space where band structure strongly deviates from that of nearly free electrons, and the loss spectrum exhibits observable structures. As for the plasmon excitation, it was shown that, for all values of ns , the long wavelength behavior of the plasmons can be described by a 2D free-electron model provided that a renormalized effective mass is introduced. The formal technique of Latge et al. seems to be almost the same as the one adopted by Que [1235]. It is well known that most of the research, both theoretical and experimental, on quantum-dot systems has been focussed on electronic properties in strong magnetic ®eld Ð sometimes also termed as the quantum Hall regime. Much of the work on quantum dots in this regime is restricted to the existence of maximum-density-droplet (MDD) states [1169], which are the quantum-dot analogs of the incompressible states responsible for the FQHE in bulk systems, and to the edge reconstructions which occur when these states become unstable. Some experiments have demonstrated the possibility of measuring the chemical potential mN of a droplet of N-electrons created by lateral con®nement of a 2DEG [1140,1159]; such con®ned systems can be thought of as quantum dots. Abrupt shifts of mN occur at those values of magnetic ®eld B where the ground state of the droplet changes. The quantity which is measured in such experiments is the B-dependence of the ``addition spectrum'', i.e., the energy needed to add one more electron to a dot; this is given by mN ˆ EN ÿ ENÿ1, where EN is the ground state energy of an N-electron quantum dot. The measured addition spectra have often been interpreted in terms of constant interaction (CI) models in which e±e interactions are considered by including a charging energy (Ec ) which is characterized by a ®xed self-capacitance, or when this fails, by using Hartree or HFAs. These aspects have stimulated considerable theoretical work on the quantum dots in the quantum Hall regime. For instance, MacDonald and coworkers [1281] have investigated parabolically con®ned

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quantum dots to demonstrate that a quantum dot in the FQH regime exhibits mesoscopic oscillations in the thermodynamic properties with a period which is a multiple of the period for free (uncon®ned) electrons; the observation of the period of these oscillations is suggested to be an important con®rmation of the theory of FQHE. Moreover, several authors [1278,1282±1284] have embarked on the B±N phase diagram, in which each phase is designated by the quantum numbers of the ground state; the shifts in mN …B† takes place at the phase boundaries. These studies also include the double-layer quantum-dot systems investigated by MacDonald and coworkers [1284], with the purpose of identifying parameter regimes where electrons form MDD states corresponding to the bulk incompressible quantum Hall states at n ˆ 1 and 2. Ground states and bulk collective excitations were approximated using HF and time-dependent HFAs, respectively, and the relationship between the edge excitations of the dots and the EMPs of bulk double-layer systems was discussed. By direct diagonalization of the many-particle Hamiltonian for two model-con®nement potentials, Das Sarma and coworkers [1285] studied the dipole-allowed magneto-optical spectra for N…ˆ 6†electron quantum dot subjected to a strong magnetic ®eld B. The non-parabolic con®nement potential V…r† ˆ V0 …r ÿ r0 †2n y…r ÿ r0 † is speci®ed as n ˆ 2 and r0 ˆ 0 (n ˆ 1 and r0 6ˆ 0) de®ning quartic (¯atsurface parabolic) con®nement for the ®rst (second) model. In contrast to the widely studied case of parabolic con®nement, the cusplike structures in the ground-state energy and the sharp structures in the magnetization were noticed; these non-monotonic structures occurring at some speci®c values of B are attributed to the pure electron correlations for the interacting systems. Subsequently, Stafford and Das Sarma [1286] investigated the collective Coulomb blockade (CCB) of a 1D and 2D-square array of tunnel-coupled, N…> 10†-electron quantum dots with parabolic con®nement in a weak applied magnetic ®eld. The e±e interactions in a single dot were considered within a self-consistent Hartree approximation, and parametrized by a capacitive charging energy Ec ˆ e2 =C…N†, in accordance with the CB picture [1113]. The collective phenomena were modeled by introducing a tunneling matrix element ta between equivalent single-particle states in nearest-neighbor approximation, and ta 's between non-equivalent states were neglected; this is then a usual tight-binding approximation. In this way, they studied the addition spectrum of a 2D square (1D linear) array of four quantum dots with three (four) single-particle energy levels per dot and level spacing DE ˆ 0:3Ec . It was concluded that the interplay of quantum con®nement, interdot tunneling, and intradot Coulomb interactions leads to a sequence of two phase transitions separating a region of CB of individual dots, CCB, and the breakdown of CB altogether; this happens, respectively, at weak, intermediate, and strong interdot tunneling in 2D square array. For a linear chain of four quantum dots, Mott±Hubbard MIT was evidenced, in possibly close accordance with the (factually unnoticed) experimental ®nding on a 1D crystal of 15 dots [1137]. Kim and Ulloa [1287] studied the collective excitations and optical response of 2D square arrays of quantum dots by allowing both the tunneling (through energy dispersion) between nearest-neighbor dots and the overlap of wave functions coming from different quantum dots. Considering the latter aspect of the tunneling led them to go one step beyond the earlier work [1209,1235,1280] on the plasmon excitations, where the wave function overlap between different dots was neglected. Con®ning themselves to a one-subband model within a self-consistent Hartree scheme and Wannier representation, described in the usual tight-binding approximation, they calculated the plasmon dispersion in the three principal symmetry directions. Drastic changes in the shape of the dispersion curves were noticed when the lattice constant was allowed to vary; the energy spacing between the two bands (plotted in the non-tunneling limit) was seen to increase with decreasing lattice constant. This is

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quite likely an artifact of the enhanced Coulomb ®elds at smaller interdot separation. The allowance for tunneling causes strong effects on the intersubband mode dispersion. They show that including the suitable wave function overlap in the polarization function creates bands of modes near the intersubband SPE region and a tunneling plasmon at low energies; the latter was found to have a nonvanishing oscillator strength. Pfannkuche and Ulloa [1288] studied the transport properties of a single parabolic quantum dot containing one, two, and three electrons Ð fashionably termed as quantum-dot hydrogen, helium, and lithium, respectively Ð both with and without an applied magnetic ®eld. They embarked on the Coulomb interaction between the electrons that enters the part of the Hamiltonian that corresponds to the RI degrees of freedom (remember, such a separation into CM and RI parts is always allowed in truly parabolic con®nement potential). They obtain the matrix elements of the interaction for the usual (/ 1=r) Coulomb potential. The role of electron correlations they stress upon comes into the picture somewhat indirectly through the following strategy. In the RPA, i.e., neglecting phase correlations between the initial state of the electron (in the lead) and its ®nal state (in the dot), tunneling rate Faa0 factorizes into an effective tunneling rate for a non-interacting electron traversing barrier and an overlap matrix element [1263]. They emphasized the latter that describes the extent to which the compound state, built by an incoming electron and …N ÿ 1†-electron state a0 in the dot, overlaps the Nelectron dot state a. While for an uncorrelated system this overlap is known to evaluate to unity, correlations reduce it considerably [1263]. Because of the summation over all the single-particle dot states with equal weight, this quantity remains insensitive to any features of the incoming electron; it only re¯ects the correlations in the dot state a and a0 . The analysis of the overlap matrix elements, which govern the tunneling rate for an electron traversing a dot, revealed that the strong correlations present in a few-electron quantum dot states are extremely important. It was argued that they strongly reduce the probability of tunneling through channels involving excitations of RI degrees of freedom, leading to a dominance of CM excitations which remain virtually unaffected by correlation effects. Since most interactions in nature can be expressed in the form of a two-body interaction, it is natural to suppose that the motion of particles in an N-body system is largely correlated on a pair-wise basis. rij †, which This is the hypothesis behind the adoption of a Jastrow type trial wave function: F ˆ Pi 0, and screening decreases the induced dipole moment only. A dramatic effect is seen in the case when o > oc , where E…q; o† changes its sign, and vanishes at the 2D bulk magnetoplasma (BMP) frequency omp ˆ ‰o2c ‡ o2p …q†Š1=2 ; op …q† is the 2D-plasma frequency at wave vector q and B ˆ 0. When o of the oscillating dipole moment coincides with omp …q†, the 2D bulk magnetoplasmon with wave vector q is emitted into the surrounding 2D medium. The collective modes of a single antidot therefore have a strong nondissipative (emissive) damping at o > oc . Emissive damping of 2D bulk magnetoplasmons by a single antidot in a 2D plane is very similar to the radiation of transverse EM waves by an oscillating dipole in a 3D plasma. In an array of dots, an interdot interaction can be taken into account in the dipole approximation, and is normally negligible provided that the interdot separation is much larger than the dot radii. In an array of antidots, interantidot interaction can be neglected only if o  oc , when the charge density ¯uctuation is localized near the antidots. When o0oc , the overlapping of antidot potentials is considerably enhanced due to the 2D bulk magnetoplasmon emission. This then leads us to infer that interantidot interaction is signi®cant even in a scarce lattice of antidots. Theory of collective excitations in a single antidot and in a 2D square array of antidots was presented by Mikhailov and Volkov [1296], within the framework of classical electrodynamics neglecting the non-local and retardation effects. It was shown that the collective excitation spectrum in a single antidot consists of two branches: the ®rst mode coincides with the single-particle CR o ˆ oc , and the second one is the EMP mode which has the vanishing damping for o < oc ; for o > oc it decays due to the emission of 2D bulk magnetoplasmons into the surrounding 2D medium. It was argued that induced electrical potential and charge density of the EMP have the form of outgoing cylindrical waves at o > oc . As such, neglect of interantidot interactions in an array of antidots at o > oc can be dangerous. Subsequently, Mikhailov [1297] studied the FIR response and collective excitations in a 2D

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square array of quantum dots in the self-consistent-®eld approximation using the full set of the Maxwell equations, stressing on the radiative decay of collective modes existing in such structures. Retardation effects are usually neglected in the calculation of FIR response of low-DES, and it is generally justi®able since the said effects are determined by the parameter …a=l†2, which is several orders of magnitude smaller than unity in real structures; here a is the lattice period (typically smaller than 1 mm) and l is the wavelength of light (typically 50±200 mm). However, the retardation effects can result in a radiative decay of plasma modes and hence in¯uence the linewidth of the observed resonances. In fact the author of [1297] has shown that radiative decay of collective excitations in the dot lattices can be of the order of or larger than the damping due to collisions. Let us leap back to the phenomenon of CB in order for describing brie¯y the recent work on the electronic transport in a pair of quantum dots connected by a tunnel junction. Transport of electrons in such devices is strongly affected by the Coulomb interactions. The most striking manifestation of these interactions is this phenomenon of CB of conductance between two leads coupled to a small conducting (dot) region by tunnel barriers. This phenomenon is observed in both metallic devices and GaAs heterostructures [1113] at low temperatures ( 1 K). The CB in such structures manifests itself as periodic peaks in conductance as a function of gate voltage, and is due to the discreteness of the charge transferred in each tunneling event. The probability that electrons are transmitted from the source (through the dot and) to the drain is approximately proportional to the conductance GB ' …e2 =h†T [176]; it is known that one must have GB < e2 =h for each of the barriers in order to observe the conductance resonances. An equivalent argument is to show that electrons in a disordered conductor are localized for GB < e2 =h. Also, this condition is equivalent to requiring that the level separation is greater than their width G   h=t, where t is the duration the electron spends on the dot during the tunneling event [1115]. More closely, the charge on the dot is quantized, and therefore the CB exists as long as GB < e2 =h; the charge quantization is completely destroyed at GB  e2 =h. Inspired by the experimental results of Waugh et al. [1174] on two (and three) tunnel-coupled dots of equal and widely disparate charging energies, Baranger and coworkers [1298±1300] investigated the effects of interdot tunneling on the CB oscillations in a double quantum-dot system. They summarize their ®nding as follows. The ¯uctuating electron charge between the two dots strongly affects the tunneling through such a composite structure. The positions and amplitudes of peaks in their linear conductance are directly related, respectively, to the ground state energy and to the dynamics of charge ¯uctuations. If the interdot tunneling is weak, the period of the peaks doubles; a dramatic feature of the experimental observation. In the strong tunneling limit, they predict a striking fractional power-law temperature dependence of the peak amplitudes. For a single dot at relatively high temperatures,when the conduction is achieved through thermal activation of electrons over the tunnel barriers, they ®nd that CB oscillations persist, peaks generally are asymmetric, and are shifting with increasing temperature. With the similar spirit, Halperin and coworkers [1301] studied the relation between the barrier conductance and the CB peak splitting for a double quantum-dot system. They propose to write the fractional peak splitting f  F…Nch ; gB †, where Nch and gB are, respectively, the number of channels and the dimensionless barrier conductance per channel; assuming that d < U2 < W, where d, U2 , and W are, respectively, the level spacing, interdot charging energy, and bandwidth of states over which the amplitudes for transmission through the barrier are roughly constant. Here F…x; y† is a function of x and y. However, this proposal led them to achieve only a qualitative agreement with the experiment [1174]. These results were later extended to a similar system of a pair of quantum dots connected by an arbitrary number of tunneling channels with W > U2 [1302].

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In the discussion of the experimental and theoretical achievements in quantum-dot systems, the principle ``from particular to general'' is preferred to the principle ``from general to particular''. Though the latter makes the review paper more compact and helps to avoid repetition, the former is more convenient for readers who have no intention of entering deeply into a speci®c direction and who would like to have a thorough feel of the subject. In what follows, we would like to adopt the latter principle in order to list some important contributions made by the occasional authors [1303±1330] to the scenery. These include, e.g., the electron energy spectrum of a quantum-dot hydrogen and helium in an asymmetric 2D harmonic con®nement potential in both transverse and tilted magnetic ®eld [1303], the effects of edge states on de Hass±van Alphen oscillations in a singly connected dot in the quantum Hall regime [1304], the charging effects in double-barrier dot structures [1305,1307], the miniband dispersion in 1D open quantum-dot superlattices [1306], optical absorption due to interface phonons and plasmons in a quantum dot [1308], the effects of Coulomb interactions on the magnetoconductance oscillations in quantum dots explicable in terms of the interaction between trapped states and fully transmitted edge states [1309], the breakdown of the phase coherence and suppression of AB oscillations due to e±e scattering [1310], the splitting of cyclotron mass in a parabolic quantum dot into two (m‡ and mÿ ) Ð m‡ …mÿ † being lower (higher) than the bare band mass [1311], the size and N dependence of the chemical potential and differential capacitance [1312], the spectral behavior of plasmons in cluster type compounds and quantum dots [1313], the magnetic response of disordered ballistic quantum dots [1314], the suppression of AB oscillations in a ballistic quantum dot due to a non-equilibrium population drived by a time-dependent magnetic ¯ux [1315], CB of the resonant electron tunneling in a pair of coupled quantum dots [1316], the RPA theory of quantum effects on the Raman spectrum of a quantum dot [1317], the role of quantum symmetry constraints in elucidating the origin of magic numbers [1318], the temperature and gate-voltage dependence of CB oscillations in the strong tunneling regime [1319], unsuccessful theoretical efforts dedicated to the understanding of the experimentally observed [1148,1187] pair-tunneling states leading possibly to the exchange-induced attractive interactions [1320,1326], the AB oscillations in a mesoscopic ring with a quantum dot included in one of its arms [1321], the symmetry-breaking instabilities of the electron distribution in a pair of coupled quantum dots for Coulomb charging energies larger than a certain threshold value [1322], the visible effects of e±e interactions on the CR of a quantum dot with disorder [1323], the many-body effects on the electronic transport through a quantum dot [1324], the local-potentialinduced Breit±Wigner type resonances in spherical quantum dots [1325], proposal of a shell model for 3D quantum dots in a tilted magnetic ®eld demonstrating the N dependence of the FIR spectrum [1327], a linear chain model of coupled quantum dots studied under the condition N (the number of electrons in a dot)  NQD (the number of quantum dots in 1D crystal) [1328] and explaining qualitatively the experimental results of Ref. [1137], e±e interaction-induced disintegration of a quasiparticle in a quantum dot [1329], and the electron self-energy and the spectral density characterizing the electron relaxation between the excited and the ground states [1330]. 2.3. Some variants of quantum structures 2.3.1. Finite-size 2DEG The current status of the nanofabrication technology leads us to imagine formation of not only 2D, 1D, and 0D structures such as quantum wells, wires, and dots, but also more complicated structures such as quantum pipes, snakes, balls, rings, and ribbons (see, e.g., Fig. 9) where electrons are con®ned

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Fig. 9. Schematics of some uncommon quantum structures Ð the concept of quasi-dimensionality: quantum pipe (a), quantum snake (b), quantum ball (c), quantum ring (d), and quantum ribbon (e). The fabrication and investigation of such essentially ®nite-size geometries could lead to dramatic control of the electronic and optical properties of solids. Electron dynamics in the skin region (shaded areas) is expected to be dramatically in¯uenced by any variation (of size and/or properties) in the core and surrounding materials.

in the shaded regions with quasi-dimensionality between 3D and 0D. The fabrication of essentially arbitrary geometries could lead to dramatic control of the electronic and optical properties of solids. The skin region of such structures has a larger electron af®nity than those of the core and surrounding materials, where core and surrounding media have not to be necessarily identical. Impurity doping into one or all of the three regions affects the localization characteristics of the charge carriers. The size (and shape) of the core through which con®ned charge carriers can tunnel is the most important parameter. Role of the boundaries Ð the inner and outer perimeters Ð in understanding several electronic and transport phenomena in such quantum nanostructures has been much appreciated during the past decade. We refer, in particular, to the importance of the edge states in understanding, e.g., the magnetotransport in quantum Hall regimes in a broad range of mesoscopic and macroscopic systems. The notion of edge states forming at the boundaries of the sample was introduced by Halperin in the context of IQHE [1331]. The concept of magnetotransport along such edge states now allows one to interpret a number of experiments performed in both integral and fractional quantum Hall regimes [1332±1352,1781±1784]. That the transport properties of the quantum Hall states are governed by the edge excitations at low temperatures was fairly well supported by many experiments [1353±1359] shortly after the initial theoretical development of the edge state picture [1332±1337,1781]. Major key steps in the edge transport theory were put forward by MacDonald, BuÈttiker, Wen and their coworkers.

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Even though it is undesirable, a review paper sometimes compels the author to brief on the subject he is not an expert in. This is the case with the ongoing discussion on the edge states, which were introduced as supplementary to the Laughlin's gauge invariance concept in order to have a better insight into the quantum Hall effects. The main features of the integral and fractional quantum Hall effects are the quantization of the Hall conductance and the vanishing of the longitudinal resistance at the integral and fractional ®lling factors correspondingly. The two discoveries generated an enormous amount of activity, which resulted in understanding the magnetotransport in 2DEG in a strong magnetic ®eld. From a theoretical point of view, the conductance quantization was understood in terms of the magnetic quantization for integral ®lling factors and in terms of the formation of the incompressible quantum liquid for the fractional ®lling factors [84]. The basic picture is such that in a strong enough magnetic ®eld, at a ®lling factor between the quantum Hall plateaus, the electron system breaks up into the compressible and incompressible regions corresponding to the integral or fractional states. Those regions are separated by edge channels which form a percolating network. Depending upon the value of b, a parameter determined by the typical gradient of the electron density ¯uctuations, these edge channels can be either wide or narrow. It turns out that the magnetotransport depends on the width of these edge channels. The incompressibility (i.e., a discontinuity in the chemical potential at a magnetic®eld dependent density ns …B†) implies a gap for charged and neutral excitations in the bulk of the system. But the ns …B† at which the gap occurs requires the existence of gapless excitations which are localized at the edge of a ®nite sample. The concept of the edge states is thus based on the fact that the 2DEG can support gapless excitations only at the edge. Hence the transport properties can be understood without explicitly considering the bulk of the sample. The edge of a 2DEG is created by the con®nement potential which bends the LLs up in energy. The intersection of each LL with the Fermi energy gives origin to a chiral 1D edge channel. Naturally, the total number of channels within the framework of a single-particle picture is given by the bulk ®lling factor. It has been shown that in a realistic con®nement potential edge channels have ®nite width owing to the e±e interactions [1341]. The edge channels not being strictly 1D brings up the problem of their internal structure describable within a quantal treatment. The two limits where the problem can be tackled are: very sharp and very smooth con®nement. When the con®nement potential is very sharp it dominates over the e±e interactions and the single±particle theory must be valid. However, it is dif®cult to assess how to implement this situation experimentally. In the case of a very smooth potential, a compressible strip breaks up into a number of fractional states with fractional edge channels in between. This explains the observation of fractionally quantized Hall conductance for integer bulk ®lling factor in devices with non-ideal contacts [1354]. Understanding the phenomena of quantum Hall effects is the problem of electrons in a strong magnetic ®eld that has largely been studied from a geometrical point of view. The gauge invariance discussed by Laughlin and the role of edge states emphasized by Halperin are central to these investigations. Making use of the gauge arguments, it has been shown that, just as integral quantum Hall (IQH) states, FQH states also support the gapless edge excitations [1335±1341,1781,1782]. However, while the IQH states are describable within a single-particle picture, the description of FQH states is a strongly correlated many-body problem. Although the electronic states in the bulk are localized, the electronic states at the edge of the sample are extended (i.e., the electron propagator along the edge is long ranged). Therefore the non-trivial transport properties of the quantum Hall states arise from the gapless edge excitations. Many attempts have been made in order to understand the edge excitations. Electron spin is shown to have played little role in the theories of edge states, which have

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considered two limits: the IQH regime, where the two spin states are taken to be degenerate, and the FQH regime, where the system is generally assumed to be fully spin-polarized. The role of e±e interactions in the IQH edge states was emphasized by Halperin and coworkers [1342] within the HFA. They demonstrated that, in the absence of Zeeman splitting, the outermost edge state undergoes a spontaneous transition between the spin-unpolarized and spin-polarized ground states at a critical value of bulk ®lling factor. The transition leads to a separation between edge states of opposite spin on the order of magnetic length that should be observable in a variety of magnetotransport experiments. The later theoretical development of edge states has covered typical sample geometries and concepts like a square lattice in a rational magnetic ®eld [1343,1345,1784], a narrow strip of ®nite width in a weak periodic potential and a strong magnetic ®eld [1344], the effects of e±e interactions on the transition between sharp and smooth density distributions at the edges of quantum Hall liquids [1346], a generalized n ˆ 23 state which uni®es the sharp edge picture of MacDonald and the soft edge picture of Beenakker [1347], the edge states and topological quantum numbers of a periodically modulated 2DES in a magnetic ®eld [1348], an effective Chern±Simons theory of FQH liquids treating bulk and edge properties in a uni®ed fashion [1349], the structure of fractional edge states using Jain's compositefermion theory [1350], the microscopic numerical test of the chiral Luttinger liquid theory for FQH edge states [1351], and a microscopic picture of chiral Luttinger liquid using composite fermion theory of edge states [1352], to name a few. It is now widely accepted that the edge states of the IQHE are well understood to be a chiral Fermi liquid, whereas the edge states of the FQHE are believed to be a chiral Luttinger liquid [1339]. Due to the chiral structure of the edge excitations, there is no backscattering (i.e., carriers moving along the edge of the sample in a high magnetic ®eld cannot effectively reverse direction if scattered at an impurity or by an inelastic event) and a static current on a single edge cannot cause a voltage drop along the edge. The voltage drop can appear only if (a) the edge excitations interact with bulk excitations or (b) edge excitations tunnel from one edge to another. The effect (a) can safely be ignored at low temperatures at which the quantum Hall experiments are usually performed. The principal results which emerge from the generalized edge state theory are shown to describe the dynamical properties of IQH edge states in terms of 1D Fermi-liquid theory, while for FQH states they are described by the U(1) Kac±Moody algebras [1339]. There are a minimum of two edge excitations corresponding to the leftgoing and right-going states localized on the opposite edges of the sample, although the gapless edge excitations may have many branches. Here we are interested to cover only the macroscopic electrodynamics of edge states which lead to the appearance of gapless EMPs which have been detected experimentally [1353±1394] and studied theoretically [1395±1420] by many workers since their ®rst observation in 1985. The discovery of quantum Hall effects has generated much interest in the excitation spectrum of 2DES in a magnetic ®eld. The nature of the spectrum depends on the region of the phase diagram explored. There are two experimentally observable regimes: the classical regime, kB T  hoc ; h2 ns =m , h2 ns =m . Both of these regimes have been explored using different and the quantum regime, kB T   experimental techniques on the diverse sample geometries. The resonant edge excitations of a bounded 2DES is a perpendicular magnetic ®eld, the EMPs, are the modes which propagate along and characterized by the sample perimeter (s). These EMP are now well understood as the 2D analogs of the 3D surface magnetoplasmons, and exhibit features which are manifestations of the dynamical Hall effect. The initial studies by Allen et al. [1360] on 2D array of disk-shaped 2DEG in GaAs±Ga1ÿx Alx As

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heterojunctions, by Mast et al. [1361] and by Glattli et al. [1362] on the surface of liquid helium, and by Andrei et al. [1363] on GaAs±Ga1ÿx Alx As heterojunctions have stimulated a large body of experimental and theoretical work on the EMPs in 2D, 1D, and 0D electron systems. Theoretically, the reduced degrees of freedom allow detailed and often exact calculations. Experimentally, new, unexpected phenomena have been observed. While 2D systems display analogs of characteristics of 3D systems, the lower dimensionality modi®es their properties dramatically. The Lorentz force acting on a bounded system of charge carriers in a magnetic ®eld can induce charge accumulation at the boundaries. The Hall resistance, which is a stationary manifestation of this phenomenon, has traditionally been exploited to measure the charge carrier density. However, the dynamics of a plasma in a magnetic ®eld in the presence of a density inhomogeneity (manifested by the boundaries of a ®nite sample) does possess physical aspects, which are more general than the Hall resistance. The early experimental observation of the EMP revealed two sets of resonant absorption spectra. One set is the normal plasmon±cyclotron coupled modes with positive B dispersion. An unusual feature is the existence of other set of modes whose frequencies decrease with magnetic ®eld (i.e., a negative B dispersion). This decrease is one of the dynamical manifestations of the Hall effect, as was established by the early experiments [1360±1363]. EMPs represent collective excitations where the individual electrons perform skipping cyclotron orbits along the boundary of the 2DES. With increasing B, the charge-density oscillations are concentrated in a very small region close to the edge. The interesting point of the 2DES is that the oscillatory charge cannot be concentrated in a local d-function line (which would lead to logarithmic divergence in the potential and electrostatic energy) [1401,1402]. This is a unique property of a 2DES and in contrast to surface plasmons at the surface of a 3DES (neglecting the spatial dispersion). Thus, the oscillatory charges of edge plasmons are inherently distributed over a certain region perpendicular to the edge and respond in a non-local manner. Nonlocal effects are of the order of …qv f =op †2 , where q, v f , and op are, respectively, the propagation vector, the Fermi velocity, and the local plasmon frequency. Large non-local effect is a unique property of a ®nitesize 2DES when the edges become important. The non-local effect arises from a signi®cant in¯uence of the diagonal conductivity sxx on the EMP excitations; it can be characterized by a length l / sxx , which is related to the spatial extent in the direction perpendicular to the edge of a 2DES. It represents a kind of ``transverse'' non-locality and also occurs for very small value of q. Experimentally, systems have been investigated with sizes that differ by many orders of magnitudes. At one extreme are spatially quantized quasi-0DES where edge excitations are interpreted as singleparticle transitions between states of different angular momentum [1364]. There the resonance frequencies o are found in the FIR with ot  1, where t is the B ˆ 0 momentum relaxation time. At the other extreme are large-area 2DES where surprisingly narrow EMP resonances can be observed at radio frequency (rf), even though ot  1 [1363]. In this limit, sharp EMP resonances occur whenever the diagonal conductivity sxx is much smaller than the Hall conductivity sxy, and have frequencies that, for a given sample, are essentially proportional to sxy . Thus rf EMPs are observed on large samples only on Hall plateaus. Theoretically, it is emphasized that the EMP should exist for frequencies ot  1 [1402], and it is now clear why in the early experiments [1361] at ot  1, the EMP damping was so anomalously small. A key element behind the low damping is that the EMP is carried by the Hall currents, which in a strong magnetic ®eld are almost perpendicular to the electric ®elds. This explains qualitatively why low-frequency EMP exist at (fractional) quantum Hall plateaus, where the Hall angle is very close to 90 , but rapidly damp out beyond the plateaus [1365]. The EMP damping is much more sensitive to the details of the edge charge distribution than that of EMP frequency.

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The observation of EMPs was one of the most unexpected and intriguing discoveries in the physics of 2DES. In this collective mode the charge density ¯uctuation exists only in the close vicinity of the boundary of the charge sheet within a narrow strip whose thickness decreases with increasing magnetic ®eld. For 2DES in the quantum Hall regimes, boundary phenomena are of extreme importance, because all electrical transport is con®ned to the so-called edge channels [1334]. EMP form the lowest-energy excitations of such 2D systems. Therefore there is tremendous current interest in using EMP spectroscopy to investigate fundamental issues regarding the integral and fractional quantum Hall effects [1365,1367,1372,1377,1380,1385,1389,1390,1393,1394]. Basically, however, the EMP excitations are entirely classical phenomena and can be investigated in the non-degenerate 2DES on the surface of liquid helium [1361,1362,1368,1382,1383,1386,1387,1390,1392]. Most of the experimental studies have been performed in the frequency domain. However, the propagation of EMPs can be directly recorded in the time domain, as was ®rst suggested by Ashoori et al. [1369]. An advantage of the time-domain technique is the ability to determine the direction of the EMP travel. It has been proposed [1337] that the EMP response of the 2DEG may be an effective probe of edge effects in the FQHE especially if the direction of the plasmon propagation can be determined. ^ drift of electrons at the edge of the sample; The EMPs travel only in the direction given by the eB  N ^ N being the outward normal at the edge of the 2DEG [1369]. Later, time-domain measurements have been performed by several workers [1377,1379,1388,1389,1391], using diverse magnetospectroscopies. Out of this long list of experimental investigations of the EMP on the GaAs±Ga1ÿx Alx As heterojunctions and on the liquid helium surface [1360±1394], of particular interest are the observation of interedge magnetoplasmons (IEMPs) [1383,1392], which were ®rst predicted theoretically from general and classical grounds [1403]. The IEMPs propagate along the boundary of two contacting 2D regions with different conductivities. The EMP propagating along the outer edge of a single 2D layer is a special case of IEMP. The experimental observation of IEMP on the two 2DES on liquid helium surface [1383,1392] has revealed, in agreement with the theoretical prediction [1403], that the frequency of the IEMP is proportional to the difference of the Hall conductivities at either side of the boundary. More explicitly, o ˆ ajdns jeq=Eh B, where a, dns , and Eh are, respectively, a parameter that depends on the shape of the density pro®le, the difference in the electron density, and the dielectric permittivity of liquid helium. The direction of propagation of IEMP is determined by the sign of dns . The damping is determined by the average diagonal conductivity sxx. Experimental observation is that the linewidth of the IEMP increases on cooling below the melting temperature of the high-density region. All data are consistent with the disappearance of IEMP when the electrons in the region where the IEMP is located are crystallized. Qualitatively, the temperature dependence is consistent with the known behavior of sxx upon crystallization. Moreover, it is argued [1392] that the drive voltage suggests that this might be a coincidence and that the melting explicitly controls the IEMP behavior. It is thus concluded that IEMP might become an interesting tool for studying Wigner crystallization of a 2DES in a magnetic ®eld. A theory of EMP excitations which was developed in parallel with the early experimental research [1360±1363] has succeeded in deriving many interesting results on the properties of EMP in a variety of limiting cases. We refer to the edge plasmons on a lateral surface of a semiconductor superlattice studied within the electrostatic limit by Quinn and coworkers [1395±1397]. The basic physics of the problem was dealt within terms of three simple equations, which must be solved self-consistently. These are the Poisson equation, the equation of continuity, and the constitutive equation relating surface current density and the electric ®eld. Such a theoretical analysis enabled Quinn and coworkers to obtain

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the spatial dependence of the form of either propagating or evanescent waves. The latter were associated with propagating edge modes discovered in 2D electrons on liquid helium [1361,1362]. For periodic array of 2D strips [1395±1397], the modes in different strips were seen to interact and form bands. Almost at the same time, Fetter developed a scheme in the framework of a HDM [1398±1400] in order to explain the EMP in a classical 2DEG bounded on the liquid helium [1361,1362] in both the half-plane and the disk geometry. Subsequent theoretical development has provided us with relatively more sophisticated schemes for understanding the observed EMP in 2DEG con®ned within different shapes. These include, e.g., a rigorous theory of EMP by Volkov and Mikhailov [1402], which applies to all types of 2D systems with sharp conductivity pro®les near a boundary. They had explicitly shown that EMPs propagate in a narrow strip along the boundary of the 2D system, have a gapless excitation spectrum (for half-planes), and have a small damping. In the light of this fact, they con®ned their attention to the properties of EMP in the quantum Hall regimes. They also pointed out that excitations analogous to EMP also exist in the in®nite but inhomogeneous 2D systems in strong magnetic ®elds. An approximate analytical expression for the EMP was derived by Nazin and Shikin [1405], who took a systematic account of the effect of real equilibrium electron density n0 pro®le on the EMP spectrum, in contrast to the sharp yfunction pro®le considered by Fetter [1398±1400] and by Volkov and Mikhailov [1402]. Huang and Gumbs [1406] considered a 1D chain of quantum dots forming a superlattice with a magnetic ®eld along the z-axis and assumed a parabolic con®nement potential in each x±y plane. Within the linear-response theory (the RPA) and lowest partially occupied level, they found anticrossing of the EMPs (at low B) and splitting of the tunneling magnetoplasmon branch. Their general conclusion is that a single-particle picture is inadequate for accounting for the anticrossing and that split branches for a quantum-dot array are due to many-body effects and tunneling. Proetto [1407] analyzed the dynamical response of a classical 2DEG con®ned in a ring geometry in a perpendicular magnetic ®eld, within the hydrodynamical theory. He obtains two edge magnetoplasma modes: one (the dot EMP) is associated with the charge density circulating around the perimeter of the ring and whose frequency decreases with the magnetic ®eld, while the second one (the antidot EMP) represents a charge density which circulates along the inner boundary of the ring and its frequency increases with the magnetic ®eld. The theoretical result pertaining to the second mode was soon followed by its experimental detection in a ring-shaped 2DEG in GaAs±Ga1ÿx Alx As heterojunction [1378] and on the surface of liquid helium [1386]. The authors, however, realized that the theoretical results were only in qualitative agreement with experiments, and the differences arise from the simplifying assumption made in Ref. [1407] that the rings were fully screened by metallic electrodes (the so-called localcapacitance approximation (LCA)). By relaxing the LCA, Reboredo and Proetto [1408] later obtained excellent quantitative agreement with the experiments. For each value of the angular momentum, the spectrum can be characterized as a set of high-frequency bulk magnetoplasmons and two low-lying EMPs related to the density ¯uctuations circulating along the two ring boundaries, in just opposite directions. Fessatidis et al. [1409] developed a hydrodynamical theory to study the collective excitations of a 2DEG con®ned to move about a circular, in®nite potential barrier (i.e., an antidot structure). The theory reveals, in qualitative agreement with the experiments [1366,1370], three low-frequency branches in the spectrum: a high-frequency mode that is basically the principal 2D magnetoplasmon, a mode at the CR frequency, and a low-frequency mode that is unique to this antidot system (corresponding to the charge density circulating along the inner boundary of the antidot). As the authors admit [1409], the

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de®ciency of their model is its inability to incorporate the B dependence of the 2D electron density ns, which leads to the negative B dispersion at low magnetic ®eld. Wu and Zhao [1410] considered a 2D square array of antidots and framed a theory Ð the so-called Wigner±Seitz (WS) model Ð to study the collective excitations and FIR response and obtained a quantitative agreement with the experiments [1366,1370], regarding the new resonances o‡ and oÿ , in addition to the bare CR. The WS model is argued to reduce the complicated problem to essentially that of a (circular) WS cell that allows to treat intracell and intercell Coulomb interactions on equal footing. It is known from the solved theoretical models (see, e.g., Ref. [1402]) that an EMP has a gapless spectrum of the form of o…k† / k ln…1=k†. This result is obtained within a simple model in which the electron density ns near the boundary of the sample drops abruptly to zero in a stepwise fashion. Such a model predicts the existence of only a single branch of the EMP. This result follows directly from the assumption that the charge accumulation can occur only within an in®nitesimally narrow region around the periphery of the system. However, for a realistic con®ning potential the electron density varies smoothly in the vicinity of the edge. The smooth density variation leads to the appearance of additional modes [1405]. The analytical results for the properties of the modes propagating along a smooth boundary of a 2D electron liquid con®ned to a half-plane in a strong magnetic ®eld were obtained within an exactly solvable model by Aleiner and coworkers [1411,1412]. They noticed that, for a smooth con®nement, the dispersion relation for the EMP is very similar to the one for the abrupt edge:     2ns e2 1 k ln ; (2.7) o…k†   jkaj Em oc where ns is the electron density far from the edge, E is the background dielectric constant, and a is the characteristic width of the narrow strip around the boundary. The logarithmic function appears due to the long-range nature of the Coulomb interactions. The origin of this mode can be understood as follows. The charge dq accumulated at the edge produces an electric ®eld Ey / k ln…1=jkaj†dq parallel to the boundary. This electric ®eld generates currents, which are perpendicular to the boundary, because of the strong magnetic ®elds. These currents tend to recharge the edge; the net effect is the displacement of the charge along the boundary. This process repeats, and the density pattern moves along the edge. Aleiner et al. [1411] noticed that, in addition to such EMP as discussed above, other edge modes exist, which have linear dispersion relations. The excitation spectrum of these acoustic edge modes (AEMs) is given by oj …k† ˆ sj k;

sj ˆ

2ns e2 ; Em oc j

(2.8)

where j ˆ 1; 2; 3; . . . . The difference between the AEM and the usual EMP originates from their charge distribution pattern; in EMP charge varies monotonically across the narrow strip around the boundary, whereas the charge oscillates j times in the jth AEM. At a given wave vector k, the frequencies of the AEM are lower than those of the usual EMP by a factor of 1=j ln…ka†j. It was shown in [1411] that the oscillator strengths and damping for the AEM, with low j, differ from the corresponding values for the EMP by powers of the same small parameter. Moreover, it was suggested that the observability condition for these AEM is optimal at a relatively weak magnetic ®eld, with the external ac electric ®eld perpendicular to the strip. The theoretical prediction of the existence of AEM was soon con®rmed by their experimental observation [1382,1389].

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These studies are motivated by the fact that edge excitations, different from the bulk modes, are gapless and are, as a result, particularly relevant for the thermodynamic behavior at low temperature. The macroscopic picture underlying many of the available theoretical models is the hydrodynamic description of an incompressible 2D electron liquid characterized by the propagation of edge modes with drift velocity v d ˆ cE=B, where E is the electric ®eld generated by the charge carriers at the edge of the sample. This picture presents, however, a series of dif®culty since the electric ®eld generated by 2D charge ¯uctuations exhibits a logarithmic enhancement at the boarder. Moreover, considerable attention has been paid to models where electrons interact through effective short-range forces and where the concept of electric ®eld does not hold (see, e.g., Ref. [1340,1783]). Giovanazzi et al. [1413] developed a microscopic description of edge excitations for ®nite dots in the quantum Hall regime. In this scheme, which is analogous to the Feynman's theory of super¯uids, they derived explicit dispersion law for edge excitations. In the limit of large N (the number of particles), the dispersion law was shown to be proportional to k ln…1=k†, just as before. For short-range interactions, it was demonstrated to instead behave as k3 . The exact equivalence between the microscopic and macroscopic results proved in this work con®rms in a clear way the general statement that electrons in a strong magnetic ®eld exhibit a behavior typical of Bose super¯uids and that their dynamics is properly described within the framework of classical hydrodynamics. It was argued that in both the long-range and the short-range cases the dispersion law differs from the linear law widely considered in the literature on Bose liquids and reveals interesting features characteristic of the edge excitations in quantum Hall effects. In addition to the regular surface plasmons, an electron gas with diffuse-density pro®le can support higher multipole surface plasmon modes. These modes appear only when the electron density decreases suf®ciently slowly from its bulk value to zero through the surface. The name higher multipole is appropriate for these modes because, in contrast to the regular surface plasmon, the integral (along the normal to the surface) of their charge density vanishes for every point on the surface. Within a simple classical approximation, Xia and Quinn [1415] studied the EMP excitations near the edge of a 2DEG with smooth density pro®le. Two distinct groups of EMP were found. One group consists of two lowfrequency modes whose frequencies o…k† approach zero as the wave vector k goes to zero. They exist only for ®nite values of B and propagate in only one direction along the edge for a given direction of B. The lower (higher) of these two modes is identi®ed as interedge (usual) magnetoplasmon mode. The other group comprises of high-frequency modes which emerge from the 2D BMP mode. They occur only for (higher) ®nite values of the wave vector along the edge, and are identi®ed as higher multipole edge modes, corresponding to the multipole edge plasmons previously discussed by the same authors [1414]. No effort has, to our knowledge, yet been made on the experimental observation of higher multipole edge modes. On the contrary, physical situation studied in [1414] has been termed [1387] as an unrealistic case when the density pro®le falls off linearly to zero at the edge. Kuchma and Sverdlov [1416] derived the dispersion relation for the EMP in a 2DEG in a half-plane geometry using quantum kinetic-equation approach. This allowed them to describe the edge electron states in a self-consistent way. It was shown that these edge states play an essential role in the formation of the EMPs. The result, as they state, is unexpected: in the limit of strong magnetic ®eld EMP may be described in terms of the edge states alone. It is argued that in the leading order in kD (D being the width of the narrow strip of edge electron states), the EMP potential has a constant value within the narrow strip where the edge states are located; in this approximation the damping of the EMP modes due to the impurity scattering of electrons is absent. Consequently, the SCFA seems to be suf®cient to derive exact dispersion law within the logarithmic accuracy. Their ®nal expression for the dispersion

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relation is similar to the one obtained within the classical description of an electron gas in Ref. [1402], even though the nature of the origin of EMP in the classical and self-consistent descriptions is different. Finally, we discuss brie¯y a variety of formal microscopic model theories for EMPs developed by Vasilopoulos and coworkers [1417±1420]. Their scheme is based either on the microscopic evaluation of local current density [1417,1418] or framed within the RPA [1419], and is valid in the quantum Hall regime for ®lling factor n ˆ 1 or 2 and at low temperatures when the dissipation is localized near the edge. The lateral con®ning potential is generally assumed to be smooth on the magnetic length scale lc , but suf®ciently steep at the edges so that the LL ¯attening and formation of compressible and incompressible strips can be discarded. LL coupling, dissipation (related essentially to LL mixing), screening by edge states, and inhomogeneity of the current density near the edges were taken into account. For wide channels and n ˆ 1 and 2, they report the existence of independent EMP modes that are either spatially symmetric or antisymmetric with respect to the edge. Certain of these modes can propagate nearly undamped, even when the dissipation is strong, and were thus termed edge helicons. Recalling the theoretical results of [1402,1411], they argue that the results of these papers are valid, respectively, for in®nitely sharp and smooth density pro®les that are independent of n, and miss an important quantal aspect, the LL structure. They demonstrate that combining their density pro®le with the localization of the dissipation near the edge leads to strong modi®cations of the EMP results [1417]. In addition to the fundamental EMP mode with o  k ln…1=k† and the AEM with o  k, they reported [1419] an additional mode Ð the so-called dipole EMP with o  k3 Ð which is known to directly signal the non-local responses that were neglected in previous studies. Here it is noteworthy that a similar dispersion law was obtained in [1413] for short-range forces, i.e., for an interaction that is essentially different from the Coulomb type. In this series of papers [1417±1420], the authors have been tempted to make close comparisons of their results with those of [1402] and [1411]. It was noticed that despite their partial similarities, signi®cant differences exist even when the dissipation accounted for in [1417±1420] is very weak. 2.3.2. Periodically modulated structures The present understanding of device physics has advanced to the point where quantum interactions can be used to develop novel device concepts. Not only must quantum effects be taken into account when characterizing next-generation devices in tight arrays or structures such as MOSFET or MODFET, they can lead to new conceptual phenomena. It is generally believed that nearly all the postwar discoveries, both fundamental and technological, have their roots in the ideas originally put forward by the founders (of modern physics) in the prewar time, that gave this era of quantum-solidstate history its character. For instance, the beginning of the age of ®rst systems of reduced dimensionality (i.e., of the 1D periodic superlattice systems) [80] was motivated by the idea of designing structures that would make feasible the observability of Bloch oscillations. It was originally suggested by Zener [835] that, given a suf®ciently low scattering rate, the electrons would cycle through the minizone in k-space and thereby oscillate in real space, an effect now known as Bloch oscillations (see Section 2.2.2). If this would occur, then current would decrease with increasing electric ®eld leading to negative differential conductance (NDC) [1421]. Such a condition could be utilized to produce Bloch oscillator, a source of terahertz EM radiation. There are, however, reasons why this phenomenon is not thought to have been observed in practice. If the electrons are scattered parallel to the superlattice planes then they can be scattered into higher minibands, and thus miss the zone-edge effects, since there is no quantization of electronic motion in the lateral directions.

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If a periodic potential could be applied in all three directions with a suf®ciently small period, then electrons would be completely con®ned to the minizones. The situation might be realized by a 3D structure of alternating blocks of GaAs in Ga1ÿx Alx As, leading to a truly 3D periodic superlattice of quantum wells. Since the technology for this does not exist, it becomes necessary to omit periodicity in one direction. The lateral surface superlattice (LSSL), ®rst proposed by Sakaki and coworkers [1422], satis®es this criterion when the electrons are con®ned to a thin layer perpendicular to the surface. Depending upon the dimensionality of the in-plane periodic potential imposed, the LSSL could represent a periodic multiwire system or a 2D array of quantum dots or antidots, as the case may be. This scheme extinguishes the continuum of states in the direction perpendicular to the surface (i.e., the 2DEG) and leaves gaps in the remaining quantized directions along the surface. Warren et al. [1423] used a dual-stacked-gate con®guration in Si MOSFET device in order to investigate the transport properties of such a LSSL structure. Kotthaus and coworkers [1424] embarked on the existence of an arti®cial miniband structure in LSSL on the MOSFET devices. Bernstein and Ferry [1425] implemented a similar idea in grid-gated GaAs±Ga1ÿx Alx As MODFET devices and observed the NDC which was attributed to the onset of Bloch oscillations. Ismail et al. [1426] reported LSSL effects on the conductance measurements in a grating-gate GaAs±Ga1ÿx Alx As MODFET in the low-®eld regime. Although the fabrication of LSSL constitutes a great technological challenge, they are appealing under both practical and fundamental considerations. They provide for model systems in which basic solid state properties can be studied on a variable length scale. Moreover, they allow one to apply external electric and magnetic ®elds which can compete with the built-in potentials, a situation which is not, or only poorly, achievable in naturally existing solids. Theoretical aspects of optical and transport properties of 2DEG under the in¯uence of a periodic potential had been fascinating a few workers in the mid-1980s, with the emphasis on effects originating from the band structure of the lateral superlattices [1427±1429]. Kelly [1428] demonstrated a strong anisotropy in the plasmon dispersion of a 2DEG in the presence of a 1D periodic potential. The authors of [1429] reported conductance oscillations of a 2DEG modulated by a weak 1D periodic potential in a proposed device called washboard transistor. Such oscillations in the transconductance (as a function of the gate voltage) were attributed to the electron-velocity modulation caused by a quantum interference effect, rather than to the DOS modulation. In this section, we would discuss the optical and transport properties of the systems achieved under the in¯uence of a periodic electrostatic potential modulation and also the systems achieved under the in¯uence of a periodic magnetic ®eld modulation. The role of periodic (electric and magnetic) perturbations has been known to provide a wealth of information on the optical and transport properties of quantum nanostructures during the past decade. In a seminal paper (without reference!), Chaplik [1430] had anticipated that a 2DEG under a 1D periodic potential should exhibit some unique magnetotransport properties. It is interesting to note that the existence of a periodically modulated carrier density in such structures had already been con®rmed by Sakaki and coworkers [1431], who measured the anisotropy of the resistivity parallel and perpendicular to the interference fringes. They stated that the smallest possible period for ®nding an anisotropic resistivity is  four/®ve times as large as the thickness of Ga1ÿx Alx As barriers. Subsequently, Kelly [1432] described a variety of effects that follow from the introduction of a weak periodic potential into a 2DEG. He noticed that the electronic DOS, optical properties, and transport coef®cients are all modi®ed, and indicated that several experiments based on the size of the periodic potential should be quite feasible.

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2.3.2.1. Electrostatic potential modulation. At low temperatures the magnetoresistance of a degenerate 2DEG exhibits the well-known SdH oscillations periodic in Bÿ1 , with period DSdH …c=B† ˆ e=hkF2 , where hkF is the Fermi momentum. The 2D motion of electrons subjected to both a periodic potential and a  perpendicular homogeneous magnetic field leads to some interesting commensurability problems, owing to the presence of two length scales, the period a of the potential and the magnetic length lc . Drastic effects on measurable quantities, such as magnetotransport properties of a quasi-2DEG, are expected if a and lc are roughly of the same order of magnitude. Since transport is mainly due to electrons at the Fermi energy EF, the Fermi wavelength lF ˆ 2p=kF , which is related to the average carrier density kF ˆ …2pns †1=2 of the 2DES, becomes important as a third length scale, which complicates the situation a bit. A superimposed 1D periodic potential lifts the degeneracy of the LLs and leads to a hkF , novel type of magnetotransport oscillations also periodic in Bÿ1 , but with a period D…c=B† ˆ ea=2 which is determined by three length scales a, lc , and kFÿ1 , and is much larger than DSdH …c=B†. Such commensurability oscillations becoming known as Weiss oscillations were first observed by Weiss et al. [1433] at Max Planck Institute in Stuttgart in 1989. The experiments by Weiss et al. [1433] were performed on conventional MBE-grown GaAs± Ga1ÿx Alx As heterojunctions with carrier densities ranging from 1.51011 to 4.31011 cmÿ2 and lowtemperature mobilities ranging from 0.23106 to 1106 cm2 /V s. In order for analyzing the magnetotransport parallel and perpendicular to the interference fringes, an L-shaped sample geometry was chosen. The samples were illuminated at liquid-helium temperatures with well-de®ned fringe pattern of two interfering laser beams. Exploiting the persistent photoconductivity effect, a periodic modulation of the positive background charge in the doped Ga1ÿx Alx As layer was produced with a period that was well known from laser wavelength and the geometry of the setup. The advantage of this kind of ``microstructure engineering'' is its simplicity and the achieved high mobility in the microstructured sample owing to the absence of defects introduced by the usual pattern transfer techniques [846,1761±1763]. After short holographic illumination of the sample, the resistance parallel (rk ) and perpendicular (r? ) to the interference fringes were measured at low temperatures as a function of the magnetic ®eld oriented perpendicular to the plane of the 2DEG. While pronounced oscillations of this new type were found to dominate r? ˆ rxx at low magnetic ®elds (B  0:3 T), weaker oscillations with a phase shift of 180 relative to the r? data were visible in the rk ˆ ryy measurements; here x stands for the direction of 1D periodic modulation. No additional structures were found in the Hall resistance. At higher magnetic ®elds (B  0:3 T) both r? and rk were seen to exhibit usual SdH oscillations with a periodicity inversely proportional to 2D carrier density ns. As the temperature was increased the SdH oscillations were seen to strongly damp whereas the additional (Weiss) oscillations remained virtually unaffected. Weiss et al. [1433] found that apart from a phase factor f, maxima in the magnetoresistance arise whenever the classical cyclotron diameter 2Rc …ˆ 2kF l2c † at the Fermi energy is a multiple of the period a such that 2Rc ˆ … j ‡ f†a;

j ˆ 1; 2; 3; . . . :

(2.9)

Eq. (2.9) can be arranged to obtain the periodicity D…c=B† ˆ ea=2hkF, and to calculate the carrier density ns using ns ˆ …e=p h†D…c=B† and the relation kF ˆ …2pns †1=2 . The validity of Eq. (2.9) had been con®rmed by performing these experiments on different samples, by changing the carrier density with an applied gate voltage, and by using two laser wavelengths in order to vary the period a [1433±1436]. Before discussing the theoretical understanding of the Weiss oscillations, we would like to give a rough

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sketch of our collection of observations on the wide variety of experiments stimulated by this discovery of low-®eld commensurate oscillations. In the mean time, it is worth mentioning that another type of novel magneto-oscillatory effect, whose origin is closely related to that of Weiss oscillations, had been found in quasi-2D organic conductors, y(BEDT±TTF)2 I3 , by Kajita et al. [1437] in 1989. These oscillations were dictated to arise in the angular dependence of magnetoresistance in these systems whose Fermi surface has a weakly corrugated cylindrical shape. The essence of this angular dependent magnetoresistance oscillation (ADMRO) also lies in a periodic quenching of kz dispersion as a function of magnetic ®eld. Yagi et al. [1438] reported a semiclassical interpretation for the ADMRO and stressed that ADMRO effect has much in common with the commensurability oscillations observed in spatially modulated 2DEG [1433±1436]. Later, Yagi et al. [1439] experimentally demonstrated that ADMRO effect can also be observed in arti®cial semiconductor superlattices which provide some advantage over organic conductors, in that superlattice period as well as modulation doping offers a greater degree of freedom for Fermi surface tailoring. On the other hand, achievable carrier density is rather limited and at the expense of carrier mobility. However, they suggested that it would be interesting to study the ADMRO effect in systems with higher carrier concentration; for which superlattices and graphite intercalation compounds are suitable candidates. Interest in the magnetotransport properties of spatially modulated 2DES grew so fast that numerous experiments have since been dedicated to the observation of Weiss oscillations in diverse achievable physical conditions. These include, e.g., the investigations on 2DES periodically modulated in one [1433±1436,1444,1445,1448,1449,1466,1468,1469,1472±1475] or both [1440±1443,1446±1448,1450± 1465,1467,1470,1471] lateral directions on a submicron scale. These experimental ®ndings were soon followed by distinct extensive model theories, both classical and quantal, in an attempt to understand the existence of such Weiss oscillations [1476±1511,1785]. Based on the early experimental observations and their theoretical explanations, these oscillations were understood to possess the following features: (i) they are periodic in Bÿ1 like SdH oscillations, (ii) the periodicity depends on carrier density like …ns †1=2 , while SdH have a ns dependence, (iii) these oscillations have almost no temperature dependence unlike SdH oscillations, ruling out the possibility that these oscillations are due to an occupation of ®rst excited subband, (iv) they show up most clearly at low magnetic ®elds, because at higher ®elds they are obscured by SdH oscillations, (v) Weiss oscillations in ryy are much weaker (than those in rxx ), are out of phase with the oscillations in rxx , and their magnitude is such that jDryy j  jDrxx j, (vi) while oscillations in rxx are explicable within a semiclassical picture: they are due to guiding-center-drift resonance [1476], those in ryy have a quantum origin: they arise due to the peculiar oscillatory dependence of modulation-broadened bandwidth of LLs on the band index [1481], (vii) the observed magnetocapacitance oscillations [1436] have the same origin as the novel magnetoresistance oscillations. The above statements were drawn from the early experimental-cum-theoretical understanding of the Weiss oscillations in the 2DEG spatially modulated by 1D periodic potential. The later theoretical developments, however, revealed that a quantum theory which treats the collisionbroadening effects (on the modulation-broadened Landau bands) and the current relaxation in a consistent manner, is capable of explaining all the observed oscillatory effects as resulting from the same origin, the oscillatory dependence of the bandwidth on the band index, and that no additional mechanism has to be invoked to explain the oscillatory behavior of ryy [1481±1483,1486, 1487].

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Subsequent experimental work on 2DES with 1D periodic potential embarked on, e.g., the quenching of the magnetoresistance oscillations in the strongly modulated 2DEG [1444], the temperature dependence of the magnetoresistance oscillations demonstrating the quenching of these oscillations with increasing temperature [1445], the magnetic breakdown of the 2DEG in a 1D periodic potential [1449], a crossover from positive to negative magnetoresistance with increasing potential strength unveiling the important role of anisotropic relaxation process [1466], the importance of small-angle scattering for both the low-®eld magnetoresistance and amplitude of Weiss oscillations [1468], the commensurability oscillations in magnetothermopower [1469], the observation of commensurability oscillations in the magnetoresistance oscillations of a weakly modulated 2D hole gas [1472], the suppression of the conductance with increasing electric ®elds and with increasing temperature, demonstrating a remarkably clear signature of e±e scattering in the LSSL [1473], the SAWs measurements revealing a transition at n ˆ 12 from a ®lled Fermi-sea of composite fermions to a condensed state of bosonic quasi-particles, as observed at odd denominator ®lling factors [1474], and the dc transport measurements on 2DEG with 1D and 2D periodic potentials indicating that modulation triggers an unexpected strong response for composite fermions [1475]. Almost contemporary efforts were also made on the experimental observation of the effects of 2D periodic potential modulation on the magnetotransport in 2DEG, as referenced above. When a 2D potential modulation is increased such that the electron density vanishes in some areas of the sample in general two complementary situations can be realized. One is the formation of an array of quantum dots, i.e., isolated electron systems con®ned in all three spatial dimensions. The other is an array of antidots, i.e., an array of voids in 2DEG. These two patterns transform into each other in a situation of weakly coupled quantum dots [1136]. Quantum dots have been intensively studied both in the FIR and dc transport experiments in order to obtain insight into the quantum level structure both with and without an applied magnetic ®eld (see Section 2.2.4). Antidots have been investigated through magnetotransport measurements that show the magnetoresistance anomalies in low magnetic ®elds where the cyclotron diameter commensurates with the period of the antidots. The array of antidots, which behave like high-resistive scattering centers in the 2DEG, can be generated by various technologies including gate electrode with dielectric material [1440], shallow (or deep) etching for drilling holes into 2DEG [1441], and focus-ion-beam implantation [1442]. These antidots, forming impenetrable barriers for electrons, are characterized by their (normalized) cross-section d^ ˆ d=a (where d is the diameter of an antidot and a the period of the structure) and the steepness of the imposed repulsive potential which depends on the lithographic diameter and the depletion region around the antidots. The effective diameter of the antidot is increased by the depletion length. Even though the quantum length scales such as electron mean free path (le ) and phase coherence length (lf ) may be greater than the period (a) of the array, the commensurability of the cyclotron orbit diameter and the antidot lattice period, which is one of the remarkable results revealed in the experiments, is basically a macroscopic effect. This is because the dimensions of the antidot array are larger than le and lf . Experimentally antidot lattices with hexagonal [1441,1464], triangular [1454], and rectangular [1461,1464,1470] symmetries have been investigated following the extensive studies on square lattices [1440,1442,1443,1446±1448,1450±1453,1455±1460,1462±1465,1467±1471]. Lorke [1457] has schematically displayed some of the different situations that could be realized when the density of a 2DEG is spatially modulated by a 2D periodic potential. When the modulation amplitude is higher than the Fermi energy an array of small isolated quantum dots is formed. With increasing Fermi energy, or

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equivalently decreasing modulation amplitude, the dots will eventually merge and a situation is created where there is an array of isolated voids present in the 2DEG. The latter pattern, which is considered to be complementary to the former, is labeled as antidot array. The situation of very low amplitude is termed as ``weakly modulated 2DEG'' in the literature. Initial transport experiments [1440±1443] performed on 2D periodically modulated 2DEG in the presence of a perpendicular magnetic ®eld concentrated on the distinguishable (low-®eld) commensurability and (high-®eld) SdH oscillations and claimed that the low-®eld series of oscillations periodic in Bÿ1 has the same origin and can be understood by using theoretical arguments similar to those as used in the 1D case. In gated structures, the density modulation can be tuned from a weakly modulated 2DEG to a situation with a periodic arrays of areas which are completely depleted of electrons Ð the case also referred to as a strongly modulated 2DEG. Interesting new optical and transport properties have been observed on such systems, especially in a magnetic ®eld perpendicular to the formerly 2DEG. For strongly modulated systems with a grid in a multiply connected 2DEG, magnetotransport anomalies such as the existence of AB oscillations and the quenching of Hall effect had been reported by Smith et al. [1446]. These two effects generated considerable research interest in the later experimental (and theoretical) investigation on 2D array of antidots [1447,1450,1452,1453,1456,1459±1461,1463±1465,1467]. Third very important issue concerned with such systems is the fractal energy spectrum of Bloch electrons in a magnetic ®eld [1448,1455,1458,1464,1471], or better known as Hofstadter's butter¯y spectrum whose experimental observation still remains elusive (see the next section). The most striking experimental result is that the additional modulation in the second lateral direction leads to a drastic suppression of the band conductivity. It has been demonstrated [1448,1455] that, once the peculiar single-particle energy spectrum for this situation is taken into account correctly, this suppression is explained by straightforward application of conventional transport theory with due consideration of collision broadening effects and the characteristic subband splitting of LLs resulting in a Hofstadter energy spectrum. The 2D motion of electrons in the presence of both a 2D periodic potential of period a and perpendicular magnetic ®eld B introducing the magnetic length lc leads to intricate commensurability problems. As a function of B, a complicated self-similar energy spectrum (Hofstadter's butter¯y) has been obtained in the two complementary, but mathematically equivalent limits: ®rst, a strong potential modulation and a weak magnetic ®eld in the tight-binding approximation, and second, a weak potential modulation and strong (quantizing) magnetic ®eld. In the second case, which has been discussed in the context of quantum Hall effect [84,1685,1686], one ®nds that each LL splits into p subbands provided that a ˆ Ba2 =f0 ˆ a2 =2pl2c ˆ p=q, i.e., if the ¯ux F ˆ Ba2 per unit cell is a rational multiple of the ¯ux quantum f0 ˆ hc=e. Each of these subbands is predicted to contribute an integer multiple of e2 =h to the Hall conductivity sxy, in such a manner that the total contribution per LL (and per spin) sums to e2 =h. Until the nineties, this subband splitting could not be veri®ed experimentally; only recently has it been demonstrated that this subband splitting is the very key for the understanding of the suppression of the band conductivity observed in the experiments [1455,1458,1465,1471]. Magnetoresistance oscillations periodic in B in the mesoscopic conductors are usually considered a manifestation of AB effect. The AB effect has been commonly observed in magnetoresistance both of mesoscopic metal and semiconductor rings. This is attributed to consecutive constructive and destructive interference due to a change in phase of the electron wave function in one branch relative to the other. Two electron trajectories passing through opposite sides of the ring acquire a relative phase shift proportional to a magnetic ¯ux threading through the ring [177]. According to the modulation of

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the phase shift, the quantum transmission coef®cient of the ring oscillates with period h=e. Similar effects have been found in systems with quantum dots [1132], where loops of propagating edge states are formed along the boundary of the quantum dots in high magnetic ®elds. The edge states are restricted to have discrete energy levels because the phase difference acquired after a revolution along the loop should be 2p times an integer. As the magnetic ¯ux through the loop shifts the phase difference, the discrete energy levels successively intersect the Fermi energy and the magnetoresistance oscillates with the period h=e [1460]. To observe B periodic oscillations, with period D…B=c† ˆ h=ea2 corresponding to the addition of  one ¯ux quantum through antidot unit cell, in the multiple connected rings the electron wave function needs to be coherent across the entire array. Antidot superlattices consisting of a periodic array of holes drilled through a 2DEG are closely related to such a large, multiply connected ring geometry. Initial investigations [1450,1460] ascribed B periodic features in the magnetoresistance of the lateral superlattices to the AB effect. Subsequently, it was shown that B periodic oscillations observed in antidot arrays with dimensions large compared to lf result from a modi®ed electron energy spectrum [1463,1464,1467]. It was argued that the spectrum can be interpreted in terms of a few quantized periodic orbits which takes into account the chaotic nature of the phase space. In contrast to the AB effect, this latter mechanism requires phase coherence only on a length scale given by the circumference of the cyclotron orbit but not across the entire lattice. The oscillations periodic in B dominate only the low B regime (2Rc > a ÿ d); at high B, the system behaves as if unpatterned, and rxx displays Bÿ1 periodic SdH oscillations. A crossover to B periodic oscillations at low B discloses the in¯uence of the imposed potential. Before going on to the experimental results on quenching of the Hall effect, it is useful to specify some characteristic quantities of the system. Distinct transport anomalies stem from the commensurability between cyclotron radius Rc and the period a. In a 2DEG the cyclotron radius (Rc ˆ …2pns †1=2 l2c ) is directly related to the carrier density ns and the magnetic ®eld B. In order to observe these effects, it is necessary that the electron mean free path le ˆ v F t, dependent on the Fermi velocity and the Drude relaxation time, is much larger than the period a. The Fermi wavelength lF ˆ …2p=ns †1=2 , which is a measure of the extent of the wave function at zero ®eld, is, however, still smaller than the period a, which is typically on the order of  200 nm in the currently available antidot systems. Hence electron transport can be treated in a semiclassical picture where electrons bounce like balls ballistically through the antidot lattice. Classically, the motion of a particle in an antidot type of potential is known to be chaotic [1487]. It is argued that this classical electron dynamics leaves distinct marks in the experiments. Although there exist by now several experiments [1446,1450,1452,1453,1456,1459,1461,1464] which have directly or indirectly demonstrated the quenching of the Hall effect in antidot arrays, we discuss below only the striking relevant results of Weiss et al. [1450,1464] who justi®ed the effect for three different samples with periods ranging from 200 to 400 nm, at low temperatures (T 1.5 K). It was demonstrated that pronounced low-®eld anomalies dominate the rxx and rxy trace of the antidot array in a regime where the transport coef®cients are usually described by Drude expressions rxx ˆ m =…ns e2 t†, and rxy ˆ B=ens c. New peaks at low B are accompanied by (nonquantized) steps in rxy , and the Hall effect is quenched about B ' 0; such features are characterized by the ®eld positions where Rc  0:5a and 1.5a. When Rc < 0:5a, rxx drops, SdH oscillations commence, and rxy begins to display accurately quantized Hall plateaus. In this (quantum Hall) regime traces from patterned and unpatterned segments become essentially identical indicating that the electron gas between the etched holes preserves its initial high electron mobility. To understand the low-B magnetotransport in the

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patterned samples, Weiss et al. [1450] envisioned the transport as involving three distinct ``pools'' of carriers: pinned, scattered, and drifted ones. In a patterned sample, each contingent contributes to the total resistivity, which is obtained from the inverted sum of the individual conductivity tensors. The simple semiclassical description Ð the so-called ``pinball model'' Ð used by them while provided a good explanation of the low-®eld commensurability features in rxx , it failed to account for the quenching, involving correlated multiple re¯ections. This was attributed to the omission of an essential ingredient in their model: the ®nite slope of the potential around each antidot which causes deviations from a circular cyclotron motion. It was later argued that the pinball model works as long as the length over which the antidot potential varies is small compared to (a ÿ d); then deviations from strictly circular trajectory occur only in the immediate locale of each antidot [1464]. For larger d^ the validity of the pinball model becomes more and more questionable. In other words, the electron dynamics in a ``soft'' antidot potential cannot be described in terms of circular cyclotron motion characteristic of conventional electron gases in the limit oc t  1. It was later pointed out by Schuster et al. [1459] that low-B rxx reveals maxima that are related to physical situation where a fraction of electrons follows pinned trajectories that encircle an integer number of antidots. Furthermore, they suggested that the plateau-like structures in rxy that arise when rxx displays maxima allows for an estimation of the number of carriers that do not contribute to transport. Theoretical developments in an attempt to explain the Weiss oscillations in a 1D modulation potential of 2DEG in the presence of a perpendicular magnetic ®eld emerged soon after the initial experimental observation of the effect. Some of these theoretical works have already been discussed brie¯y in what precedes. The present list of references includes work on optical and transport properties of 1D [1476±1484,1488,1491,1496,1501,1503,1504,1506±1511,1785] and 2D [1485±1487,1489,1490,1492± 1495, 1497±1500,1502,1505] periodically modulated 2DEG subject to a magnetic ®eld. These studies on both kinds of systems were aimed at explaining (and sometimes predicting) the experimental observation of diverse physical phenomena such as magnetoresistance oscillations, Hall resistance; thermodynamic quantities like thermopower, the thermal conductivity, and the thermal resistivity; the position and width of the CR line, to mention a few. The major attention was focused on the quantum theoretical treatments of the experimentally discovered magnetoresistance oscillations in a periodically, weakly modulated 2DEG. The ®rst step in trying to explain the Weiss oscillations in a weak 1D periodic modulation was taken, as mentioned before, by Beenakker [1476]. He showed these oscillations in (rxx ) can be explained as being classical effect: they are due to guiding-center-drift resonances. However, the weak oscillations in ryy remained unexplained in his theoretical scheme. This was suggestively remedied by Gerhardts and Zhang [1479] as explained above. The most successful and uni®ed quantum theories were developed by Peeters and coworkers [1477,1483,1491] and by Gerhardts and coworkers [1480,1481,1486, 1501,1508], among others. By uni®ed theory we mean a quantum theory capable of explaining the observed magnetoresistance both in rxx and ryy simultaneously. The essence of such a theory is that the antiphase oscillations observed for the resistivity components rxx and ryy have as a common origin: the oscillating bandwidth of the modulation-broadened Landau bands, which re¯ects the commensurability of the period of the spatial modulation and the extent of the wave functions. It was demonstrated by Gerhardts and coworkers [1481] that even the magnetocapacitance experiments [1436] are well understood within the framework of this theory. In order to give a clear ¯avor to the aforesaid statement, we consider a 2DEG in the x±y plane, and a 1D weak periodic modulation U…x† along the x direction. Using the Landau gauge ~ A ˆ …0; Bx; 0† for the vector potential, we write the single-particle

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Hamiltonian as 1  e 2 Hˆ  ~ A ‡U…x†; p‡ ~ (2.10) 2m c where ~ p is the momentum operator. In the absence of the modulation, i.e., for U…x† ˆ 0, the normalized, unperturbed eigenfunctions of Eq. (2.10) are given by the plane-wave structure: cnky ˆ …1= …Ly †1=2 †Fn …x ‡ x0 †exp…iyky †, where Fn …x† are the Hermite functions centered at x0 ˆ l2c ky with n being the LL index, and Ly the length of the 2DEG in the y direction. The corresponding eigenvalues are Enky ˆ …n ‡ 12†hoc , which are degenerate with respect to ky . The modulation potential is usually approximated by the ®rst Fourier component of the periodic potential, i.e., U…x† ˆ Vx cos…2px=a†, which is expected to be a good approximation, in particular for the electrostatically induced 1D periodic potentials. As a further simpli®cation, we implicitly assume that the effective modulation potential seen by electrons is independent of the magnetic ®eld. This is argued to be a poor approximation for strong magnetic ®elds at which SdH oscillations are well resolved, since then the screening of the external electrostatic potential by the 2DEG becomes strongly dependent on the ®lling of the LLs [1481]. In the presence of the modulation potential U…x† the exact eigenstates of Eq. (2.10) are dif®cult to obtain in closed form. However, in the limit of weak modulation potential, one can evaluate the correction to the energy levels by ®rst-order perturbation theory using the unperturbed wave functions given above. In Ref. [1434] an exact diagonalization of Eq. (2.10) was shown to approve this to be an excellent approximation for n > 4, while in the experiment the relevant n is about 10. For U…x† ˆ Vx cos…Kx†, K ˆ 2p=a, we obtain hoc ‡ Vn cos…Kx0 †; En;ky ˆ …n ‡ 12†

(2.11)

where Vn ˆ Vx exp…ÿ 12 u†Ln …u† with u ˆ 12 K 2 l2c , and Ln …u† is a Laguerre polynomial. Since x0 ˆ l2c ky we notice that the presence of the modulation lifts the ky degeneracy: the electron energy depends on the position of the center of cyclotron orbit x0 . This result should be valid as long as the separation between the LLs is greater than the broadening of the levels. With this the formerly sharp LLs are now broadened into minibands, called Landau bands. The width of the Landau band given by ' jVn j, for the corresponding level index n, oscillates as a function of n, because Ln …u† is an oscillatory function of its index n. For the system at hand the magnetic ®eld is small, and in order to make an estimate of the position of the minima and maxima of the bandwidth we take large n limit of the Laguerre polynomial, i.e., Ln …u†  exp…‡ 12 u†…p2 nu†ÿ1=4 cos…2…nu†1=2 ÿ p=4† ‡ O…1=n3=4 †, which is zero when u ˆ …1=n† ‰…p=2†… j ‡ 34†Š2 , j ˆ 0; 1; 2; . . . and maximum when u ˆ …1=n†‰…p=2†… j ‡ 14†Š2 . Given that the cyclotron radius Rc ˆ lc …2nF ‡ 1†1=2 , with nF the LL index at the Fermi level, we obtain 2Rc =a ˆ …j ‡ f†, where f ˆ 14 …34† for maximum (minimum) bandwidth. The minimum bandwidth is known to be accompanied by a maximum height of the DOS peak contributed by this LL. The amplitude oscillation of the DOS related to this bandwidth oscillation has been observed directly in the magnetocapacitance experiments [1436]. Another direct proof of the quantum origin of the phenomenon is the fact that at lower temperatures the Weiss oscillations occur as an amplitude modulation of SdH oscillations [1455,1458]. As further consequences of these bandwidth oscillations, Weiss-type oscillations have been predicted for the collective magnetoplasmon excitations [1478], for the thermodynamic transport coef®cients [1483,1491], for the energy-loss of a fast-particle [1488], and in the SAW propagation [1510]. Recent theoretical developments related to the Weiss oscillations in 1D modulation potential include, e.g., the

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role of screening, exchange, and lateral modulation on the band structure [1501], the quenching of the Weiss oscillations in a strong 1D periodic potential [1504], the light scattering from a 1D periodically modulated 2DEG with partially ®lled LLs [1506], the composite fermions in modulated structures [1509], the Weiss oscillations in SAW propagation [1510], and the effect of small-angle scattering processes on the Weiss oscillations [1511]. With an additional modulation potential U…y† ˆ Vy cos…Ky† applied along the y direction, the potential matrix elements hnky0 jVy cos…Ky†jnky i ˆ 12 Vy Ln …dky0 ;ky ‡K ‡ dky0 ;ky ÿK †

(2.12)

couple LL states with center coordinates differing by integer multiples of Kl2c . Since all potential matrix elements in the nth LL have the common factor Ln, the bandwidth oscillations are the same for the 2D (Vx ˆ Vy ) and for the corresponding 1D (Vy ˆ 0) case, and besides the scaling factor Ln, the internal energy structure is the same for all LLs. If the commensurability condition (a ˆ a2 =2pl2c ˆ p=q, see above) holds, the energy eigenvalues are de®ned on the magnetic Brillouin zone k†… j ˆ 1; 2; . . . ; p†, which are q-fold (jkx j  p=aq, jky j  p=a) and form p subbands (per LL) En; j …~ degenerate. Plotting these values as a function of q=p, one obtains the self-similar pattern known as Hofstadter butter¯y [1493]. Contrary to the unmodulated case, the velocity now has non-zero intra-LL k but not in the subband index j. The matrix elements hn; k0 ; j0 jVm jn; k; ji which are diagonal in ~ assumption of a weak lateral potential is essential for the stated approach. It guarantees that, for the range of magnetic ®elds one is usually interested in, the lateral superlattice does not lead to a mixing of LLs, and that it can be treated in the lowest-order perturbation expansion. Then, the most important effect of 2D superlattice is to split each LL into subbands. An important consequence of the common factor Ln and hence the similar oscillatory dependence of the bandwidth on n in 2D (and 1D) case is that the magnetoresistance oscillations observed on samples with 2D superlattice have the same period in Bÿ1 as those with 1D superlattice of the same lattice constant and with the same carrier density. Experimentally, such a comparison between 1D and 2D modulation has been made most directly by a successive two-step holographic illumination (with rotation of the same by 90 after ®rst step) of the sample, keeping all other parameters ®xed [1448,1455]. Experimentally, a dramatic reduction of large-amplitude oscillation is observed after the second illumination, i.e., after the modulation in the second lateral direction, and only the weaker antiphase oscillations related to the scattering-rate oscillations survive. To understand such a dramatic suppression of the band conductivity observed in the experiment, the peculiar subband splitting of the energy spectrum must be taken seriously. If the energy separation between adjacent subbands j and j0 is large than the collision broadening of these bands, the corresponding spectral functions do not overlap [1490]. Thus, the non-diagonal velocity matrix elements between these subbands do not contribute to the band conductivity Dsmm …E† arising from intra-LL terms (n0 ˆ n). Then the band conductivity of the 2D system is seen to be considerably smaller than that of the corresponding 1D one. It has been argued [1493] that by changing the relative strength of collision broadening and 2D superlattice potential, one can switch between situations in which the band conductivity dominates the Weiss oscillations and those in which the scattering-rate oscillations dominate, in agreement with the experiment on gated samples [1458]. The suppression of the band conductivity is witnessed as a genuine quantum effect, which is not justi®ed by quasi-classical treatments [1486]. For further details on the quantum magnetotransport in 2D modulated systems we refer the reader to interesting

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theoretical work done by Gerhardts and coworkers [1448,1455,1458,1486,1490,1493]. Several other occasional authors have also added important contribution to the subject; see, e.g., the quantum treatments of magnetic minibands in lateral superlattices [1492], the quenching and negative Hall resistance in antidot arrays [1495], the band conductivity in weak and strong lateral modulation potential [1494,1497], a quantitative description of magnetotransport experiments via a fully quantum theory of realistic antidot potential and a self-consistent treatment of impurity scattering [1498], the phase coherence effects [1499] and the periodic-orbit theory of quantum transport [1500] in antidot arrays, the magnetotransport studies within the self-consistent Born approximation [1502], and the observability of the magnetic band structure in a strongly modulated 2DEG in a perpendicular magnetic ®eld [1505]. Up to this point, we have focused mostly on the magnetotransport properties of uni- and bidirectionally modulated 2DEG, discussing both experimental and theoretical work. We now concern with shading some light on the existing work on magnetoplasmons in periodically modulated 2DEG subjected to a homogeneous, perpendicular magnetic ®eld. To the best of our knowledge, no direct experimental effort has yet been made to observe the magnetoplasma modes in such systems. However, our preceding Sections 2.2.4 and 2.3.1 partially serve to ®ll this gap. Theoretical work on the magnetoplasmons in 1D modulated 2DEG was triggered by Horing and coworkers [1478], who reported novel oscillatory structure in the intra-Landau-band magnetoplasma spectrum within the RPA. These low-energy plasmon excitations again exhibited the novel commensurability oscillations due to the modulation-induced broadening of the Landau bands. The existence of this plasmon mode and its periodic oscillations have the same origin as in the magnetoresistance [1433±1435] and the magnetocapacitance [1436] of a modulated 2DEG. In Ref. [1478], the authors stressed that the intra~ vanishes with the modulation potential as jVn j1=2 and that there are ®nite Landau-band frequency o ~ This work met an immediate criticism by Zhang [1484] who gaps between the Landau bands in o. commented that while the former (part of their preceding statement) is correct the latter is qualitatively ~ are not physically meaningful. He further added that if one wrong. He pointed out that the gaps in o treats the Fermi energy self-consistently, the spectrum should turn out to be gapless regardless of how small the modulation potential is. Subsequently, Gerhardts and coworkers [1480] reported self-consistent dynamical response theory of the dielectric response of a unidirectionally modulated 2DEG in a strong magnetic ®eld. They claimed that their methodology was capable of describing systems with arbitrary modulation, including the limiting cases of a purely 2DEG and a completely decoupled array of quantum wires. To test the ability of this method, they had calculated the interband magnetoplasmon dispersion along the directions parallel and perpendicular to the modulation. Their results are typical for a system in the interim regime between 2D and 1D behavior, which previous quantum theories [939,953] have not been able to describe. Experiments on laterally modulated GaAs±Ga1ÿx Alx As heterostructures (see, e.g., [1808]) indicate that the magnetoplasmon resonances at qx ˆ 0 and 2p=a are shifted towards, respectively, higher and lower frequency, once the modulation strength is increased. The results presented in [1480] are qualitatively in accordance with this effect. Horing and coworkers [1488] predicted the representative commensurability oscillations in the energy loss of particle trajectories parallel and perpendicular to the modulation of a 2DEG in a quantizing magnetic ®eld. This is not an explicit calculation of magnetoplasmon dispersion, but the IDF which is central to the problem of energy loss does implicitly account for the magnetoplasma modes which are de®ned by the poles of the IDF. The novel oscillations in the energy loss problem are also

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oscillations due to the commensurability between the length scales of the cyclotron orbit at the Fermi energy and the modulation period. In the quantizing magnetic ®eld, the energy loss is subject to the SdH-type oscillations associated with the passage of LLs through the Fermi level as the magnetic ®eld is varied. The authors argue that these energy loss results exhibit close correspondence with that of the linear magnetoresistance studies of the modulated 2DEG. Stewart and Zhang [1503] presented the dynamical response function of a 1D spatially modulated 2DEG under a perpendicular magnetic ®eld within the framework of the RPA. It was found that the dynamical response function is not only broadened by the periodic potential, it also contains a series of singularities at the band edges. The maximum number of such singularities (at the cyclotron frequency and its harmonics) is shown to be 2nmax , where nmax is the number of occupied LLs, and the last level may be partially ®lled. It is further proposed that these singularities should be observable in the light-scattering or EM absorption experiments. This work bears direct relevance to the magnetoplasmon spectrum in that the authors computed the results on the imaginary part of the dielectric function, Im[E…q; o†], and its inverse, Im[Eÿ1 …q; o†]. The latter quantity is of direct relevance to the study of Raman scattering cross-section as well as the EM absorption in a system at hand. Similar conclusions were drawn from a closely related investigation on light scattering from a 1D modulated 2DEG in a perpendicular magnetic ®eld by Zhang and coworkers [1506]. In this work, the authors stress that the partial ®lling of the last Landau band may be detected in the optical spectroscopy either in the SDE spectra or in the CDE spectra, and that such spectra will provide information about the modulation potential and the magnitude of the partial ®lling. Fessatidis and Cui [1485] investigated the collective CDEs of a 2DEG subject to both a perpendicular magnetic ®eld and an in-plane potential modulation within the hydrodynamical model theory. Particular attention was paid to the long-wavelength limit of the magnetoplasmon spectrum, which was shown to have two branches instead of only one for an unmodulated 2DEG. The potential modulation was held responsible for the creation of a mode which exhibits commensurability oscillations periodic in Bÿ1 in addition to its concurrent modi®cation to the principal mode. The origin of the magnetoplasma oscillations is attributed to the resonant drift of the cyclotron-orbit center, just as was pointed out by Beenakker [1476] in an attempt to explain the corresponding magnetoresistance oscillations in unidirectionally modulated 2DEG. Several geometries such as 2D rectangular, square, and hexagonal superlattices were considered. A 2D rectangular superlattice with different degrees of modulation in the x and y directions was shown to be particularly interesting in that its plasma frequency turned out to be dependent on the direction of propagation. Such a directional dependence was, however, found to be absent for the square and hexagonal superlattices, both with equal modulation strengths in the x and y directions. Closely related to the magnetoplasma excitations are the investigations on the quantum treatments of magnetic minibands, showing the effect of band structure on the transport properties, of a bilaterally modulated 2DEG [1492,1498,1505]. Rotter et al. [1505] emphasized that for lattice periods of the order of Fermi wavelength the picture of commensurate orbits breaks down and the magnetic band structure gains considerable in¯uence. As a characteristic feature, the longitudinal magnetoresistance shows oscillations, with a leading period of one ¯ux quantum per unit cell, which have their origin in the magnetic band structure. It was demonstrated that they do only weakly depend on temperature and electron density. The authors have also discussed the necessary conditions for their experimental observation.

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2.3.2.2. Magnetic ®eld modulation. High mobility 2DEG, formed in GaAs±Ga1ÿx Alx As heterostructures, allow ballistic electron motion over distances on the order of several microns. Using techniques such as electron beam lithography, 1D and 2D arrays of electrostatic gates with a feature size as small as 100 nm have been fabricated on the surface of the heterostructures, providing a periodic potential through which the electrons move. A variety of effects have been observed in the magnetoresistance of the 2DEG, depending upon the nature of the externally applied electrostatic potential. The first effects of this type were seen with 1D or 2D weak periodic potential modulations, where the Weiss oscillations were observed in magnetoresistance (see the preceding section). Following and in parallel with this work there was significant interest in the behavior of magnetically modulated 2DEG, though it remained confined only to the theoretical subject until 1993. The physics of 2DEG subjected to a uniform magnetic field forms a large part of contemporary semiconductor physics. However, despite some theoretical work available before 1990s [1512,1513], the experimental observation of the electron dynamics in a periodic magnetic potential proved elusive. One of the possible reasons for this has been the practical difficulty in realizing such a situation. Most emerging proposals have involved the use of patterned gates made out of ferromagnets or superconductors in a layer a few hundred angstroÈms above the plane of the 2DEG [1514±1517]. Such a scheme is, however, known to have two disadvantages: (i) the variation in the magnetic ®eld is very small and it dies away rapidly a short distance below the gate, and (ii) the patterned gate layers act like electrical gates and induce an unwanted electric ®eld (or scalar potential) modulation of the 2DEG, which is, in general, stronger than the magnetic ®eld (or vector potential) modulation; this makes it hard to attribute the observed effects unambiguously to the magnetic ®eld modulation. However, most of the recent experiments [1518±1531,1786] have been considerably successful in effectively compensating this built-in (scalar) potential modulation by an appropriate choice of the gate bias. It turned out that for the by-now studied devices, the positive bias voltage worked so as to reduce the total (electric plus strain) scalar potential. For the reasons of distinction, we would refer to, in what follows, the effect of electric (magnetic) ®eld modulation on the magnetoresistance as electric (magnetic) Weiss oscillations. Transport properties of electrons in a 2DEG subjected to a periodic magnetic ®eld have attracted considerable theoretical interest, both preceding [1532±1542] and following [1543±1561] the ®rst experimental efforts to observe the effect on the energy spectra and the magnetotransport [1518]. Subsequent experiments were performed mainly by four research groups, namely, Iye and coworkers [1518,1521,1526,1786], Beaumont and coworkers [1519,1525,1529,1530], von Klitzing and coworkers [1520,1522,1527,1531], and Pepper and coworkers [1523,1524,1528]. While the strategy of the experimental setup of the ®rst three groups has been similar (i.e., creating spatially modulated magnetic ®eld by depositing ferromagnetic or superconducting stripes on the surface of the heterostructure with 2DEG), the last group adopted a different scheme. Pepper and coworkers relied on the fact that the development of regrowth technology, using in situ cleaning techniques, means that one can now investigate the effects of varying the topography of an electron gas in addition to varying the dimensionality [1545,1547]. That is to say that regrowth of, e.g., III±V semiconductors on patterned or etched substrates opens the possibility of investigating not only the behavior of electrons in a curved quasi-2D space, and the effects of varying that curvature, but also presents a novel way of studying electron transport in a transverse magnetic ®eld that varies with the position depending upon the angle between the ®eld and the local facet. This offers a ¯exible technique to investigate the electron dynamics in a non-homogeneous magnetic ®eld, even though the originally applied magnetic ®eld is

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homogeneous. It has been argued [1545] that a semi-in®nite 2DEG subjected to a resultant nonhomogeneous magnetic ®eld, has, in addition to current carrying edge states, 1D states which lie within the interior of the gas, and also have a ®nite dispersion Ð an effect which may be used to create quantum wires or other structures. It is also shown that, in the absence of a magnetic ®eld, the curvature of 2DEG gives rise to a potential variation which is inversely proportional to the square of the radius of curvature, an effect which may also be used to con®ne the electronic motion to 1D. Let us consider a 2DEG in the x±y plane, subject to the magnetic ®eld: ~ B ˆ …B0 ‡ Bm cos…Kx††^z, where K ˆ 2p=a, and a the modulation period, with jBm j  jB0 j. In the Landau gauge the vector potential ~ A that describes this periodic magnetic ®eld is de®ned as ~ A ˆ ‰0; B0 x ‡ …Bm =K† sin…Kx†; 0Š. In the effective mass (m ) approximation the corresponding single-particle Hamiltonian can be written as Hˆ

1 o0 m o20 2 2 ‰ p ‡ … p ‡ eB x† Š ‡ ‡ eB x† sin…Kx† ‡ …1 ÿ cos…2Kx††; … p y 0 y 0 2m x K 4K 2

(2.13)

where pm (m  x; y) is the momentum operator, o0 …ˆ eBm =m c†  oc …ˆ eB0 =m c†. The ®rst term on the right-hand side of Eq. (2.13) is the Hamiltonian of a free electron in a magnetic ®eld. The corresponding normalized eigenvectors and eigenvalues are just as de®ned in the preceding section. Because of the inequality jBm j  jB0 j one can consider the last two terms on the right-hand side of Eq. (2.13) as perturbation and evaluate the energy correction to the unperturbed eigenenergies Enky by the ®rst-order perturbation theory using the unperturbed eigenstates cnky . The result is, to the ®rst order in o0 , hoc ‡ Fn …u† cos…Kx0 †; Enky ˆ …n ‡ 12†

(2.14)

where u ˆ 12 K 2 l2c , and Fn …u† ˆ 12  ho0 exp…ÿ 12 u†‰L1nÿ1 …u† ‡ L1n …u†Š; Lam being the associated Laguerre polynomial. Therefore, under magnetic ®eld modulation, LLs broaden into minibands whose widths oscillate with band index n. This is the basic reason behind the commensurability oscillations observed in the transport experiments [1518±1531,1786]. For the sake of comparison of electric and magnetic modulations, we make use of the large n limit of the associated Laguerre polynomials, i.e., L1n  exp…12 u†…p2 nu†ÿ1=4 uÿ1=2 n1=2 sin ‰2…nu†1=2 ÿ p=4Š ‡ O…1=n1=4 †; and L1n …u†  L1nÿ1 …u† for large n. One can immediately notice that the factor Fn is zero when u ˆ …1=n†‰…p=2†… j ‡ 14†Š2 and maximum when u ˆ …1=n†‰…p=2†… j ‡ 34†Š2 ; j ˆ 0; 1; 2; . . . . Given that Rc ˆ lc …2nF ‡ 1†1=2 , we obtain 2Rc =a ˆ … j ‡ f†, where f ˆ 34 …14) for maximum (minimum) bandwidth; the minimum bandwidth is also termed as a ``¯at-band condition'' in the literature. Comparing Vn (see the preceding section) and Fn leads us ho0 ; and that (i) bandwidths are out of phase, and (ii) the amplitude in the case of to note that Vx  12  magnetic modulation is larger by a factor of …n=u†1=2 ' …2nF †1=2 =Klc ' Rc =Kl2c ˆ akF =2p ˆ …a2 ns =2p†1=2 than the corresponding case of electric modulation. Thus rxx of a 2DEG with a weak electric modulation displays minima at B ®eld, whereas in a weak magnetic modulation (of the same period a and carrier density ns ) maxima are expected. These oscillations have been long predicted [1532] and re¯ect the commensurability between the cyclotron radius Rc and the period of the modulation a. Just as in the case of electric modulation, it was expected [1532] that weaker novel oscillations in ryy would be present which would be in antiphase with those of rxx . However, this still remains to be con®rmed, since no experiment is, to our knowledge, known to have embarked on the observation of weaker oscillations in ryy in the case of magnetic modulation.

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Theoretical predictions [1532±1542] of the existence of magnetic Weiss oscillations and modi®ed energy spectrum due to 1D magnetic modulation had all been con®rmed experimentally by the abovementioned four groups in 1995 alone [1519±1524], providing necessary details required for their successful observation. This also includes the only experiment by von Klitzing and coworkers [1522] that exploited the effect of 2D modulation and showed how the semiconductor±ferromagnet hybrid system can be used to study the magnetotransport properties of nanoscale particles. Subsequent experiments added only little to the existing experience in the subject. For instance, Beaumont and coworkers [1525], using type-II superconducting stripes on the surface of the heterostructure in order to create a 1D periodic magnetic ®eld, stressed that their low-®eld magnetoresistance was dominated by the built-in electric modulation. They estimated that in order to have the magnetic modulation as the dominant effect in their samples, the amplitude of the electric modulation would have to be as low as 0.2 meV for 2 mm-period sample. They suggested that a 2DEG closer to the surface may make the magnetic modulation relatively stronger, since its amplitude is exponentially dependent on the distance from the surface ( exp…ÿKx†), but the built-in potential is not. Similarly, Iye and coworkers [1526], using ferromagnetic stripes for magnetic modulation, emphasized on adjusting gate bias for suppressing built-in electric modulation. They argue that while large positive gate bias helps reducing the unwanted built-in potential, decreasing the gate bias restores the electric modulation. By changing the relative strength of the two modulations with the gate bias, they observed continuous phase shift of the oscillatory magnetoresistance in accordance with the theory [1536]. Later von Klitzing and coworkers [1527,1531] showed that amplitude of the periodic magnetic modulation can be remarkably enhanced if the externally applied magnetic ®eld B0 is tilted towards the plane of the 2DEG (at an angle y) while still kept normal to the ferromagnetic gratings. Such an enhanced amplitude of the periodic magnetic ®eld is of interest, e.g., for studying tunneling through magnetic barriers or the investigation of chaotic electron motion [1540,1543,1560] in suf®ciently strong modulated magnetic ®eld. The impact of the increased magnetic modulation is huge as the amplitude of the magnetoresistance oscillation in rxx increases with B2m . The huge angle-dependent magnetoresistance observed in these experiments obviously belongs to the last commensurability maximum. The strong modulation leads to overlapping, modulation-broadened, Landau bands. This could complicate the analysis of the SdH oscillations, sitting on top of the magnetoresistance maximum, whose amplitude decays with increasing magnetic modulation. The spin splitting at about B? (ˆB0 cos y) ' 2 T was shown to disappear with increasing modulation and can be ascribed to overlapping spin-split Landau bands. Note that this behavior contradicts the tilted-®eld experiments on conventional 2DEG where spin-split is enhanced by increasing the tilt angle as the Zeeman energy depends on the total magnetic ®eld, but not on the normal component. As regards the temperature dependence, at low temperatures a rich oscillatory structure sitting on top of the last commensurability maximum was shown. At about 18.5 K the commensurability oscillations were found nearly washed out. Similar angle-dependent measurements were performed by Beaumont and coworkers [1529,1530] where the origin of the observed giant low-®eld magnetoresistance was attributed solely to the signalternation of the magnetic modulation. The anomalous temperature dependence of this magnetoresistance was argued to be an evidence of e±e scattering contribution to the resistivity possibly due to the formation of minibands. Early theoretical predictions of modi®ed energy spectrum [1513] and magnetic Weiss oscillations [1532] in a 1D magnetic modulation of 2DEG spurred further theoretical interest due in part to the experimental availability [1518±1531,1786] of such systems. Vast theoretical efforts on 2DEG in a non-

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homogeneous magnetic ®eld range from electronic transport to dynamical response in the limit of a weak magnetic modulation, added, generally, to an underlying homogeneous magnetic ®eld. The impact of slightly non-homogeneous magnetic ®eld on the LLs of a 2DEG strip was considered by MuÈller [1533]. He considered a perpendicular magnetic ®eld that varies linearly along one direction (across the strip) and showed that single-particle electronic structure consists of states that have a remarkable time-reversal symmetry. Xue and Xiao [1534] studied electronic and transport properties of 2DEG under a periodic magnetic ®eld. In the perturbation framework, the magnetoresistances (rxx and ryy ) were shown to behave similarly to their electric counterparts, although the classical picture of cyclotron motion differs from ~ E ~ B drift. Consequently, the ¯at-band condition or the position of the rxx minima differ from those in the case of electric modulation, and the Hall conductivity acquires an additional term that is absent in the electric modulation case. Wu and Ulloa [1535] studied the collective excitations of a 2DEG subjected to a periodic magnetic modulation, within the RPA. They showed that the collective excitation spectrum undergoes a dimensional crossover, as either the relative spatial period p…ˆ a=lc † or ®eld strength s…ˆ Bm =B0 † is changed, and suggested that such a crossover should be observable in the FIR experiments. The interest in non-uniform (rather periodic) magnetic ®elds, with spatial variations on a nanoscale, gained tremendous momentum after the comprehensive quantum treatment of Weiss oscillations by Peeters and Vasilopoulos [1536]. They studied the magnetotransport in the presence of both an electric and a magnetic modulation in one lateral direction. This study had been con®ned to purely sinusoidal modulation in two special cases: (i) the cosine magnetic modulation is in phase with that of the electric modulation, and (ii) there is a phase difference of p=2 between them. It has been argued that the position of the resulting magnetoresistance oscillations depends critically on the parameter ho0 †, the ratio of the two modulation strengths. Their analytical diagnosis clearly d ˆ …K=kF †…Vx = reveals that when the modulations are out of phase the position of the Weiss oscillations is not in¯uenced with varying d, but the amplitude is; and for a critical value of d the oscillations are washed out. On the other hand, when the two modulations are in phase the external positions of Weiss oscillations are shifted continuously with increasing electric modulation. Subsequent progress in the subject is extensive and we would discuss only some interesting works on the energy spectrum and magnetotransport in what follows. Wu and Ulloa [1537] studied the electron-density distribution and collective excitations in a 1D magnetic modulation of a 2DEG. They found that magnetic modulation gives rise to an electron-density modulation, which is reduced solely due to the counteracting induced electric ®eld. The effect of a uniform magnetic ®eld with a 2D electric modulation is known to produce a fractal spectrum Ð the socalled Hofstadter butter¯y (see the following section). As demonstrated by Wu and Ulloa [1538], the fractal spectrum also exists for a 2D magnetic modulation. They concentrated in the regime where the relative ®eld strength s ' 1 and the relative magnetic ¯ux Fr ˆ F=F0 > 1. They also studied the collective excitations for this 2D modulation within the RPA, and found that the excitation spectrum does not map out the butter¯y spectrum exactly. Although it would be interesting to observe the excitation spectrum of a 2DEG with 2D magnetic modulation, there is still no effort devoted to an experimental realization that we know of. The 2DEG with a 2D periodic magnetic modulation has also been studied in the context of chaotic and ballistic electron dynamics [1540,1543]. The wealth of new and interesting information characterizing the behavior of a 2DEG subjected to a perpendicular non-homogeneous magnetic ®eld has stimulated works to explore many important avenues in the subject. Peeters and coworkers [1541] have, e.g., investigated a 2DEG under the

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in¯uence of a magnetic step, magnetic well, and magnetic barrier; and predicted an interesting wave vector dependence of electron tunneling in magnetic barriers [1542]. Modeling the magnetic ®eld by a series of equally spaced Dirac-d function with alternating sign such that the average magnetic ®eld is zero, they proposed a magnetic analog of the classic Kronig±Penney (KP) model [1548]. The presence of the magnetic ®eld was shown to make this model an inherently 2D contrary to the KP model which describes the electron motion in 1D. This difference in dimensionality was demonstrated to result in several interesting effects. The work in this series of their papers [1541,1542,1548] was later extended a step further to the case where a 2DEG was subjected to a perpendicular magnetic ®eld of arbitrary strength and which is periodic in one direction [1550,1554]. The difference from their Ref. [1536] is that here there is no homogeneous background ®eld and consequently the average magnetic ®eld is zero. This implies that electron states can be extended. Four different situations were considered: (i) a magnetic KP system with the magnetic-®eld pro®le consisting of a periodic array of Dirac-d functions with alternating sign, (ii) a periodic array of magnetic-®eld steps, (iii) a sinusoidal magnetic-®eld pro®le, and (iv) a saw-tooth magnetic-®eld pro®le. In contrast to the usual potential modulation case, these systems are argued to retain 2D character. The computed energy spectra were shown to be consisting of minibands, which allow to access the associated classical trajectories of electrons. Extensive computation of DOS and the different components of conductivity tensor were reported to reveal rich structure due to the presence of the magnetic minibands. Chang and Niu [1544] studied the energy spectrum of a 2DEG in a 2D periodic magnetic ®eld. The excitation spectra are expressed as a function of quasimomenta that are good quantum numbers of the system. Both square and triangular magnetic lattices were considered, within the perturbation scheme (jBm j  jB0 j). They showed that the general feature of the band structure is the bandwidth oscillation as a function of the band index. Conditions were derived under which effects of weak modulation may be reproduced by an effective electric potential modulation, and vice versa. These effective potentials are shown to differ from band to band, and do not always exist. Foden et al. [1545] investigated energy band structure in a curved 2DEG. In such a system, the application of a homogeneous magnetic ®eld, as a function of the curvature of 2DEG, results in an effective non-homogeneous magnetic ®eld. The authors argue that the magnetic and topological con®nement of this form can be combined with electric con®nement to create even more exotic, 1D and 2D structures. Calvo [1546] analyzed the problem of a 2DEG constrained to a channel by adjustable 1D gate potential V…x†, which changes smoothly across the channel and tends asymptotically to V for x ! 1, which was parametrized as V ˆ V0 …1  14 D†2 . The applied magnetic ®eld was assumed to attain its maximum, B0 , at x ˆ 0 and to decay symmetrically to zero for jxj > d. More closely, he studied a smooth magnetic barrier geometry of different shape and found that the discrete and continuum energy states of such a system overlap. Foden et al. [1547] reported results on the band structure and the conductance of axially symmetric curved non-interacting 2DEG, topologically equivalent to a Corbino disk, in the presence of a nonhomogeneous magnetic ®eld arising as a result of an applied axial magnetic ®eld. They found that the dispersion relation for a sphere is asymmetric as electrons moving in the same sense as the ®eld have a higher velocity than in the absence of a ®eld, whereas those moving in the opposite sense either slow down or are con®ned by the ®eld and have the sign of their velocity reversed. Furthermore, they added that for a system in which the normal component of the magnetic ®eld has the same sign everywhere, the quantum conductance monotonically approaches the semiclassical integer values; if there are regions of different sign, the quantum conductance can be greater than the semiclassical one as a result of electron tunneling. You et al. [1549] analyzed the 2DEG consisting of magnetic barriers produced by

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deposited ferromagnetic stripes on top of the heterostructures. It was shown that the electron tunneling through multiple-barrier magnetic structures exhibits complicated resonant features; but, due to the averaging of the transmission over half the Fermi surface, the conductance shows much simpler resonant structure. As for the single- or double-barrier structures, their calculations show that the conductance preserves the main features of electron tunneling. Krakovsky P [1551] analyzed the energy band structure of a 2DEG in a periodic magnetic ®eld: ~ B ˆ Bm a nˆ0;1;... ‰d…y ‡ c ‡ an† ÿ d…y ‡ an†Š^z, with homogeneous ®eld B0 ˆ 0; this corresponds to the extreme case of the other limit, jB0 j  jBm j. It was shown that the band structure exhibits a pronounced asymmetry in the lateral (x) direction and consists of the two types of bands. The states in broad bands have a ®nite mobility in the longitudinal (y) direction, while the mobility of the electrons occupying states in narrow bands is in®nitesimally small. A generic Fermi surface contains both types of states. It was argued that, at low enough electron density, only the narrow bands will be occupied resulting in vanishing of the longitudinal conductance, and leading to the directional insulating regime. Gerhardts [1552] extended the classical approach of Beenakker [1476] to 2D superlattices de®ned by spatially periodic electrostatic and magnetic ®elds of arbitrary shape but equal periods. The experimentally relevant situation of modulation ®elds produced by periodic arrays of magnetized stripes or dots on the surface of a heterostructure containing a 2DEG was analyzed. It was shown that the magnetic modulation ®elds of different symmetries, tuned by the orientation of the magnetization, superimposed on the built-in electrostatic modulation lead to characteristic interference effects on the Weiss oscillations of the magnetoresistance. Ji and Sprung [1553] addressed the following question: what are the transport properties of electrons in a quantum wire, a ®nite section of which is subjected to a non-homogeneous magnetic ®eld, which is periodic in one direction? They focused on the effects of magnetic ®eld pro®le, magnetic ®eld strength, and the thickness of the magnetic barriers. It was shown that the conductance exhibits dips below the mode thresholds, and the quantum transmission along the wire is enhanced by magnetic ®elds with periodic pro®les. The effect of variation of ®eld strength, the number of periods, and their length had revealed some systematic trends in the resulting conductance of the channel. Mackinnon and coworkers [1555] embarked on the dispersion relations of a 2DEG subjected to a perpendicular magnetic ®eld varying linearly with the position along one direction. They calculated the electronic and current densities, which revealed rich structure as a function of the position along the longitudinal direction. The Fermi energy EF required for this purpose was determined by minimizing the total energy with the constraint that the number of electrons is constant. Their numerical results clearly reveal, in contrast to the assumption in Ref. [1533], that EF is not independent of the magnetic ®eld. They speculate that the fact that electronic density is not constant could imply some interesting phenomena associated with spin-polarized currents. Li et al. [1556] extended theory of Peeters and Vasilopoulos [1536] and that of Xue and Xiao [1534] to include higher-order correction in perturbation scheme for studying the energy spectrum and transport properties of 2DEG under (relatively general form of) 1D periodic magnetic ®eld. They found that at low temperatures there are two kinds of oscillations: the Weiss oscillations which stem from the oscillatory bandwidth of the modulationbroadened LLs at the EF , and the SdH oscillations which stem from the oscillatory DOS at EF . They also noticed Weiss oscillations in ryy and attributed it solely to their inclusion of second-order perturbation term in the theory. A surprising result of this investigation is that while the Hall resistance rxy displays quantized plateaus, electron transport across the magnetic barriers can be almost dissipationless. They admit that the contribution of higher-order correction in modulation is appreciable

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only at high ®elds, but is negligible at low ®elds. In analogy with their work on electric modulation [1503], where they reported that the response function is not only broadened by the additional electric modulation but contained a series of van Hove singularities at the band edges, Stewart and Zhang [1557] have shown that such a behavior is also reciprocated in the magnetic case, with the obvious difference re¯ected due to the larger amplitude of bandwidth in the magnetic case. Results on the collective excitations reveal that the principal magnetoplasmon oscillates aperiodically with respect to wave vector along the modulation, between oc and 2oc , such that the oscillation minima are now shifted above oc . The ground-state energy of a 2D, spinless, charged particle in a 2D periodic magnetic ®eld is studied by Rom [1558], using an exact quantum wave-packet-propagation method. The periodic magnetic-®eld modulation was shown to remove the degeneracy of the lowest LL, and produce a cluster of states both hoc, the lowest LL for homogeneous magnetic ®eld. The magnetic modulation was above and below 12  de®ned by ~ B ˆ fB0 ‡ Bm ‰ cos…Kx† ‡ cos…Ky†Šg^z, with period a taken as a ˆ …lf †1=2 lc , such that lf ¯ux units thread through each magnetic unit cell. It was demonstrated that the lowest LL of a spinless charged particle is split into energy subbands by a periodic modulation like all the higher LLs, with a simple dependence on lf . The width of the (split) energy bands depends on the amplitude of the periodic modulation. Addition of spin energy term to the Hamiltonian of the charged particle in the periodic magnetic ®eld was shown to leave the lowest LL unaffected by the magnetic modulation, while all other levels are split into bands, in accordance with Dubrovin and Novikov [1512]. Gerhardts and coworkers [1559] studied the density response of a 2DEG in a 1D magnetic cosine modulation added to a homogeneous quantizing magnetic ®eld, and compared it with the response to an electric cosine modulation. They also included self-consistently the built-in potentials which were shown to reduce the density ¯uctuation. It was demonstrated that the electrostatic effects on the spectrum depend on the temperature and on the ratio of the cyclotron radius Rc and the length scale adr of the density modulation. They found that adr cannot only be equal to the modulation period a, but also much smaller. For Rc  adr, the spectrum in the vicinity of m (the chemical potential) remains essentially the same as in the non-interacting system, while for Rc  adr it may be drastically altered by the Hartree potential. In this work [1559], the reader is introduced to a very nice discussion differentiating the electric and magnetic modulations both in the classical (ho0  kB T) and quantum (ho0  kB T) regimes. 2.3.3. Hofstadter's butter¯y spectrum The study of the physical properties of metals and semiconductors in external magnetic ®elds has been among the most fruitful techniques for gaining insight into their electronic structure. In many cases, the experiments give us information about the electronic energy levels in a magnetic ®eld. Examples are the diamagnetic susceptibility, de Haas±van Alphen effect, SdH effect, CR, magnetooptic effects, and the quantum Hall effects in 2D systems, to mention a few. It is therefore not surprising that the theory of motion of Bloch electrons in a uniform magnetic ®eld has received a good deal of attention, during almost the past six decades. Following the classic work of Landau [1562] on the quantum theory of free electrons in a (uniform) magnetic ®eld, the ®rst analysis of Bloch electrons in a magnetic ®eld was carried through by Peierls [1563]. This latter work was based on the tight-binding approximation and hence its results are known to have only qualitative validity. Since then a great many contributions to this problem have been made, some of them dealing with the single energy levels, others emphasizing the free energy of the entire system. However, because of the great mathematical

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complexity of the problem, many early authors [108±112] found it necessary to use one or the other uncontrolled approximation, so that the reliability of their results was often questioned in the literature [112]. The 1950s had seen two interesting developments. One was an expression due to Onsager [109], which relates the levels in a magnetic ®eld to simple geometrical properties of the energy bands and applies to bands of arbitrary shape. This expression has been extremely useful in analyzing de Hass± van Alphen experiments. However, it was derived by means of a semiclassical argument whose range of validity has, as the author himself admitted, not been entirely clear. The other concerned the extent to which magnetic levels, highly degenerate in the absence of a periodic potential, are spread into bands [110,111]. That such broadening occurs was beyond any doubt, but the width of the bands was an issue which was not settled at that time, and there was substantial disagreement among different authors [1564±1569,1787,1788]. Much of the early work is based on the so-called single-band Hamiltonian: k†, where ~ k ˆ …~ p ‡ e~ A=c†, Em …~ k† is the energy band in question, and ~ A is the vector potential H  Em …~ giving rise to the uniform magnetic ®eld. Of particular relevance to the subject matter of this section are two papers published in the sixties: one by Azbel [1566], who studied the energy spectrum of a charged particle with a periodic dispersion law in a magnetic ®eld and showed (within an ill-famously impenetrable mathematical analysis) that each LL splits into sublevels, each of which consists of other sublevels, etc., in accordance with the expansion of the reduced magnetic ®eld into a continued fraction; and the other by Langbein [1568], who treated the energy spectrum for lattice electrons in a magnetic ®eld through the extension of either the tight-binding or nearly free-electron method and demonstrated that the secular problem resulting from both methods are identical under certain conditions. The latter author performed a Bloch-type transformation reducing these essentially 2D secular problems to 1D and proved how, for rational magnetic ¯uxes F ˆ 2pN=M per unit cell of the real lattice, one is left with a ®nite secular determinant of order M (tight-binding method) or N (nearly free-electron method). These works proved to the hallmarks for the later development of the subject on the self-similarity of the diamagnetic band structure for Bloch electrons in a magnetic ®eld. In 1970s, Wannier and coworkers [1570±1572] had been engaged in analyzing the energy spectrum for Bloch electrons, which results when an electron is acted upon simultaneously by a periodic lattice potential and an externally applied homogeneous magnetic ®eld. They were interested in the high-®eld regime where the ®eld contributes more than a minor correction to the Bloch band structure. Customarily, as it then was, the Landau spectrum was obtained from Peierls±Onsager effective Hamiltonian (POEH) hypothesis, which yields a certain difference equation. Despite its reasonable success, the validity of this hypothesis was not rigorously established in those days, and it was not clear at all in what function space this effective Hamiltonian operates. An alternative formulation due to Obermair and Wannier [1572] writes the electron wave function as a linear combination of freeelectron Landau functions. For high ®elds and low quantum numbers, the spacing hoc of the free electron levels is large compared to their broadening and splitting by the lattice potential. Therefore, in the ®rst-order perturbation, one may neglect the coupling between these levels. One then obtains a difference equation for coef®cients of Landau functions of equal energy and different wave vector. The resulting equation is formally identical to that obtained in the Peierls±Onsager treatment, with certain symbols taking on a different meaning, however. The use of this alternative approach was reasoned on account of the fact that the function space in which it operates is well understood: the space of Landau functions. In a series of papers [1570±1572], the authors embarked on the rationality and irrationality of the magnetic ®elds re¯ected in and characterizing the deduced energy spectrum.

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The problem of Bloch electrons in magnetic ®elds is a very peculiar problem, because it is one of the very few places in physics where the difference between rational and irrational numbers makes itself felt [1567]. Common sense tells us that there can be no physical effect stemming from the irrationality of some parameter, because an arbitrary small change in that parameter would make it rational Ð and this would create some physical effect with the property of being everywhere discontinuous, which is unreasonable. The only alternative, then, is to show that a theory which apparently distinguishes between rational and irrational values of some parameter does so only in a mathematical sense, and yields physical observables which are nevertheless continuous. It was essentially the purpose of a historic paper by Hofstadter [1573,1807] to present a philosophical strategy which effects such a reconciliation of ``rational'' and ``irrational'' magnetic ®elds. The methodology was illustrated in a maximally simple model of physical situation, but the ideas which arose have been later seen to be applicable to more realistic models of the physical situation [1574±1608,1789±1793]. We take this opportunity to borrow an essentially important part of this work [1573,1807], which is paramount in order for understanding his so-called butter¯y spectrum, and which is a rare live example that advocates the coexistence of science and beauty. Hofstadter's sample model involved a 2D square lattice of spacing (or period) a, immersed in a uniform magnetic ®eld B perpendicular to it. He restricted his consideration to what happens to a single Bloch band when the ®eld is applied. This is one strong simplifying feature of the model; the next is that he postulates the following tight-binding form for the Bloch energy function: W…~ k† ˆ 2E0 ‰ cos…kx a† ‡ cos…ky a†Š. Perhaps the most dif®cult step to justify on physical grounds is the following one, which he referred to as ``Peierls substitution'': replace h~ k in the above function by the operator …~ p ÿ e~ A=c†, to create an operator out of W…~ k†, which he treated as an effective single-band Hamiltonian. Work to justify this had already been done (see, e.g., Ref. [1565]). When this substitution is made, the effective Hamiltonian is seen to contain translation operators exp…apx =h† and exp…apy =h†. Depending on the gauge chosen, there are, in addition, certain phase factors dependent on the magnetic ®eld strength, which multiply the translation operators. If the Landau gauge Ð ~ A ˆ B…0; x; 0† Ð is chosen, then only the translations along y are multiplied by phases. From now on, we assume this gauge. Then when effective Hamiltonian is introduced into a time-independent SchroÈdinger equation with 2D wave function, the following eigenvalue equation results:       ieBax ieBax c…x; y ‡ a† ‡ exp c…x; y ÿ a† ˆ Ec…x; y†: E0 c…x‡a; y†‡ c…x ÿ a; y†‡exp ÿ hc  hc (2.15) Note how the wave function at (x, y) is linked to its four nearest neighbors in the lattice. It is convenient to make the following substitutions: x ˆ ma, y ˆ na, and E=E0 ˆ E. It is furthermore reasonable to assume plane-wave behavior in the y direction, since the coef®cients in the above equation only involve x. Therefore, we write c…ma; na† ˆ exp…inn†g…m†:

(2.16)

Finally, we introduce a parameter about which all the fuss is made: aˆ

Ba2 : …hc=e†

(2.17)

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Thus a equals the ratio of the magnetic ¯ux through a unit cell and the ¯ux quantum, and is therefore a dimensionless parameter. Felix Bloch is known to have suggested Hofstadter that this a can be interpreted as the ratio of two characteristic periods of this problem: one is the period of the motion of an electron in a state (with crystal momentum 2p h=a) given by a2 m =2ph; the other is the reciprocal of  the cyclotron frequency oc ˆ eB=m c. A value of a ˆ 1 implies an enormous magnetic ®eld (on the Ê ). Despite this, order of a billion gauss), if the period a is typical of real crystals (on the order of 2 A Hofstadter became interested in the results for such values of a. With all these substitutions, Eq. (2.15) can be cast into a 1D difference equation: g…m ‡ 1† ‡ g…m ÿ 1† ‡ 2 cos…2pma ÿ n†g…m† ˆ Eg…m†:

(2.18)

This equation is sometimes called ``Harper equation'', and had been studied by a number of authors [1566±1572], but it was Hofstadter who unveiled its charismatic characteristics. Another way of writing Eq. (2.18) is:      g…m ‡ 1† E ÿ 2 cos…2pma ÿ n† ÿ1 g…m† ˆ : (2.19) g…m† 1 0 g…m ÿ 1† $

$

The 2  2 matrix is referred to as ``A …m†''. When a product of m successive A matrices is multiplied with the two vector hg…1†; g…0†i, the result is the two vector hg…m ‡ 1†; g…m†i. The physical condition which must be imposed on the wave function g…m† is boundedness, for all m. This translates into a $ $ are periodic in m (which may condition on the products of successive A matrices. Now if the A matrices $ very well be, since m enters only under a cosine), then long products of A matrices consist essentially in $ $ repetitions of one Block of A matrices, whose length is the period in m. Let us assume that the A matrices are indeed periodic in m, with period q. This is a requirement on a, namely that there should exist an integer p such that 2pa…m ‡ q† ÿ n ˆ 2pam ÿ n ‡ 2pp:

(2.20)

Algebra reveals the fact that this condition on a is precisely that of rationality [1567]: a ˆ p=q. We now proceed, making full use of this somewhat$bizarre ansatz. (Presently,$we will consider the case when a is irrational.) The product of q successive A$matrices will be called ``Q''. The condition of physicality is now$transformed from the g's to the matrix Q. It can be shown without trouble that the correct condition on Q is that its two eigenvalues be of unit magnitude. That condition can then be shown to be equivalent to requiring its trace to be less than or equal to 2, in absolute value. Hence, a concise test for the boundedness of the g's is the following: $

jTr Q…E; n†j  2:

(2.21)

Trace conditions of this type had already been found by other authors [1567]. This one was discovered by Obermair and extensively used by Hofstadter. Next, it can be shown that the $ only way that n affects $ the value of Tr Q is additively, i.e., as n changes, the shape of the graph of Tr Q, plotted against E, is unaltered Ð it merely moves up and down. (A proof of essentially this fact can be found in Ref. $ $ [1567].) Therefore, Tr Q…E; n† ˆ Tr Q…E† ‡ 2f …n†, where f …n† is a periodic function of unit amplitude, $ $ and Q…E† is de®ned as Q…E; 1=2q†. We are interested in all values E which, for some n, yield bounded g's (such values are called ``eigenvalues'' of the difference equation). Therefore, we want to form the union of all eigenvalues E, as n varies. Since 2f …n† ranges between ‡2 and ÿ2, the condition on the trace can

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be written as follows: $

jTr Q…E†j  4: $

(2.22)

The trace of Q is always a polynomial of degree q; hence one might expect the above condition to be satis®ed in roughly q distinct regions of the E axis Ð one region centered on each root. This is indeed the case, and is the basis for a very striking (and at ®rst disturbing) fact about this problem: when a ˆ p=q, the Bloch band always breaks up into precisely q distinct energy bands. Since small variations in the magnitude of a can produce enormous ¯uctuations in the value of the denominator q, one is apparently faced with an unacceptable physical prediction. However, nature is ingenious enough to ®nd a way out of this apparent anomaly. Before we go into the resolution, however, let us mention certain facts about the spectrum belonging to any value of a. Most can be proven trivially: (i) spectrum (a) and spectrum (a ‡ N) are identical, (ii) spectrum (a) and spectrum (ÿa) are identical, (iii) E belongs to spectrum (a) if and only if ÿE belongs to spectrum (a), (iv) if E belongs to spectrum (a) for any a, then ÿ4  E  ‡4. The last property is a little subtler than the previous three; it can be proven in different ways. One proof was published by Rauh (see, e.g., Ref. [13] in Ref. [1573]). From properties (i) and (iv), it follows that a graph of the spectrum need only include values of E between ‡4 and ÿ4, and values of a in any unit interval. Hofstadter looked at interval (0, 1). Furthermore, as a consequence of these properties, the graph inside the above-de®ned rectangular region must have two axes of re¯ection, namely the horizontal line E ˆ 0, and the vertical line a ˆ 12 (see, e.g., Fig. (1) in [1573] for the originality). Such a spectrum possesses some very unusual characteristics. The large gaps form a very striking pattern somewhat resembling a ``butter¯y'' Ð and hence the name ``butter¯y spectrum''; perhaps equally striking are the delicacy and beauty of the ®negrained structure. These are due to a very intricate scheme, by which bands cluster into groups, which themselves may cluster into larger groups, and so on. The exact rules of formation of these hierarchically organized clustering patterns were detailed originally by Hofstadter himself. His description of clustering patterns is based on three statements, each of which describes some aspects of the structure of the graph. All of these statements are based on extremely close examination of the numerical data, and are taken as ``empirically proven'' theorems of mathematics. For further details, we refer the reader to the original work by Hofstadter [1573,1807]. Hofstadter had also suggested the possibility of looking for the features of the butter¯y spectrum predicted by his model experimentally. At ®rst glance, as it was originally commented, the idea seems totally out of range of possibility, since a value of a ˆ 1 in a crystal with the rather generous lattice  spacing of a ˆ 2 A demands a magnetic ®eld of roughly 109 G. It had been suggested, however, that one could manufacture a synthetic 2D lattice of considerably greater spacing than that which characterizes real crystals. In particular, one could make a tight-binding model so that the electronic energy bands are approximately given by his simple sum of two cosines. Moreover Ð and this was the   crux of the idea Ð one can choose the lattice spacing; thus, e.g., with a spacing of 200 A instead of 2 A , a magnetic ®eld of 100 kG gives a value of a equal to one. All that remains to be done is to apply a uniform magnetic ®eld perpendicular to the plane of the 2D gas, and to measure the transitions when the sample is irradiated with EM radiation of various wavelengths. With this proposal he did not mean that the idea was easy; but such an intriguing spectrum was expected a deserving experimental test. Lately fabricated 2D periodic lattices of dots (and antidots), which seem to serve the purpose for the sample model proposed by Hofstadter, have been occasionally put to such a test [1458,1471], but a successful observation of the butter¯y spectrum continues to remain a dream.

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The late 1970s had seen further clari®cation of the features associated with the butter¯y spectrum, in particular, by Wannier and coworkers [1574±1577,1789,1790]. Subsequent interest in the problem of 2D Bloch electrons has arisen in many different physical contexts. Every time it emerges with a new face to describe another physical application, and brings a new excitement. The cornerstone of the problem is that the equations of motion can be reduced to 1D ones, which appear in many different problems, ranging from the quantum Hall effects [1581±1583,1586,1587,1589] to chaotic dynamics [1590,1596,1792] to quasi-periodic systems [1593]. Much of the rich and interesting features of the energy spectrum are characterized, as above mentioned, by the parameter a ˆ p=q, where p and q are integers which are prime to each other. When a is a rational number (i.e., the magnetic ¯ux is commensurate with the periodic potential), it consists of q subbands, each subband carries an integral Hall conductance. As a is changed continuously, p and q change wildly. If the Fermi level lies in an energy gap such that r bands are completely full (and separated from each other by gaps), and all others are empty, one has the Diophantine relation: r ˆ qsr ‡ ptr (where sr and tr are integers), in connection with the quantum Hall effect [1594]. The full conductance is given by ÿtr e2 =h [1581,1589]. When a is irrational (i.e., the magnetic ¯ux is incommensurate with the periodic potential), the spectrum is known to have an extremely rich structure like Cantor set and to exhibit a mutifractal behavior [1593,1604], usually termed as butter¯y spectrum. Hofstadter butter¯y is obtained in two complementary, but mathematically equivalent, limits [1568]: ®rst, in a strong potential modulation and a weak magnetic ®eld, and second, a weak potential modulation and a strong magnetic ®eld. Both approaches deliver the so-called Harper equation [1568,1573,1577,1581,1583,1585,1589,1591,1595,1599,1601,1604,1606±1608,1790,1793,1807], determining the energy spectrum. The energy spectrum has usually been investigated in the intermediate region in the absence of collisional broadening. The coupling of the LLs by the periodic potential modulation strongly reduces the original high symmetry of the Hofstadter spectrum, but retains a very complicated subband structure [1596]. It has been possible to transform the equation of motion, the SchroÈdinger equation, into a vectorial form reminiscent of the Harper equation, but exact under the most general conditions and exhibiting chaos in the classical limit [1596]. Two of the key factors responsible for stimulating ever increasing research interest in the butter¯y spectrum, after Hofstadter [1573,1807], have been (i) the duality (between the momentum and spatial coordinates) due to Aubry and Andre [1578], who showed the existence of a transition between the localized and extended states C when a is an irrational number, (ii) and a rigorous justi®cation of the POEH for Bloch electrons in a homogeneous magnetic ®eld due to Nenciu [1588]. The latter (POEH) had also seen some partial support from the ®rst-principles results [1579,1580], but only for non-degenerate bands. Recent theoretical developments of the problem of Bloch electrons in a magnetic ®eld pertaining to the study of energy spectrum have added quite interesting dimensions to the already existing excitement. These include, e.g., a new approach based on the Bethe ansatz equations of Wiegmann and Zabrodin [1597,1598,1604], the effects of e±e interactions on the energy spectrum and the thermodynamic DOS [1600], the effects of spatially varying magnetic ®elds [1601,1602], the FIR absorption of a 2DEG in a short-period strongly modulated square lattice, showing the underlying Hofstadter subband structure due to either intra or inter-band magnetoplasmons by tuning the density or the ®lling factor within the lowest Landau band [1603], the Hofstadter spectrum obtained for the sample models having ¯at (dispersionless) single-particle band(s) originally proposed for itinerant spin ferromagnetism [1605], the two interacting butter¯y spectrum for two interacting particles in the Harper model, leading to explore the interaction-induced localization effect [1606], the study of the

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butter¯y spectrum in a 2D lattice with hexagonal symmetry [1607] Ð thereby extending the earlier work by Claro and Wannier [1576], and the effects of electron correlations on butter¯y spectrum, showing that the structure of the butter¯y for the system of correlated electrons is modi®ed only in the band-gaps and bandwidths, but not in the characteristics of self-similarity and the homeomorphism of the Cantor set [1608]. Finally, it is worth mentioning that in spite of the strong theoretical support (see, e.g., Refs. [1583,1592,1600]) providing, in particular, the rigorous proof of the statements [1583] used by Hofstadter to characterize the recursive structure of the butter¯y spectrum, the observation of unique spectrum continues to be a challenging experimental problem requiring extremely high quality samples and demanding ®nesse. However, it is repeatedly believed [1592] that transport measurements on 2D quantum-dot arrays can, at least in principle, be a viable approach, and may some day bring Hofstadter's dream to reality. 3. Methodologies for inhomogeneous plasmas Many physical theories describe observables in terms of elementary collective excitations [637±639]. In order to evaluate these quantities within the approximations involved in the appropriate theories, it is usually necessary to determine the dispersion relations for the collective excitations in a system in question. Here we are concerned with the collective (plasmon) excitations in the conventional (classical) systems as well as in the non-conventional (quantum) systems, reviewed in the preceding section. Collective plasmon excitations in both kinds of systems have been studied using both classical and quantum methodologies. The purpose of this section is to present a brief critique of these formal techniques, which are all well established since the late 1960s. By this we do not mean to go through the mathematical details and to review where and under what circumstances any of such methodologies has been successful and where it has failed to give a reasonable answer. This is because none of these schemes gives better than a reasonable answer in different (physical) circumstances and different (physical) systems. And such a remark remains often valid even if there is a good agreement between experimentally measured and theoretically calculated quantities. A theorist, no matter what his ®eld of specialization, is repeatedly faced with a need to construct a model theory which can adequately describe the physics of the system at hand, and which should be simple enough for de®nite conclusions to be drawn. Elementary collective excitations are no exception to this rule. Ideally one should set up and solve a complete SchroÈdinger equation for a given condensed matter system. However, it is impractical. As such, one is faced with the necessity of developing a simpli®ed model, which will enable one to proceed with the calculations and explain the existing experimental results and predict the new ones. The available experimental results then serve as a check on the degree of adequacy of the model theory. The number of such (macroscopic and microscopic) theoretical methods for studying the collective excitations (in conventional and non-conventional systems), developed to date, is so large that it is neither possible nor desirable to review all of them in great detail. We do not intend to specify and criticize the model theories described below with respect to their validity regarding the local or non-local effects, the external (electric and/or magnetic) ®eld effects, the size effects, the diverse interaction effects, and the dimensionality of the systems; neither we are interested to expand on their quantitative comparison. In addition, this is not the central theme of the present review. So, we would like to recall some of the principal formal techniques used to study the

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plasmon excitations in the systems reviewed in Section 2, and present our collection of observations (gathered from the existing literature) regarding their success and failure in describing the collective excitations in the conventional systems for which they were originally proposed; and refer the reader to the relevant original references for details. 3.1. Macroscopic model theories Macroscopic model theories widely used for studying plasmon excitations in both conventional and non-conventional systems include the HDM, the TMM, and, more recently, the IRT. The idea of essentially using hydrodynamics to study the electron gas originated with Bloch [1609], and slightly later work by Jensen [1610]. The HDM presents a simplest way to include spatial dispersion in the plasma dielectric function E…q; o†. The essential idea of hydrodynamics is that when a system is disturbed, its relaxation to equilibrium is so fast that ``local equilibrium'' is always maintained and the relaxation to global equilibrium can therefore be described entirely via macroscopic variables. In the simplest version of the HDM one includes spatial dispersion (or non-locality) within the metal, e.g., in the approximation El …q; o† ˆ 1 ÿ

o2p o…o ‡ in† ÿ bq2

;

Et …o† ˆ El …0; o†

(3.1)

with phenomenological parameters n (the collisional frequency) and b. The subscript l …t† refers to the longitudinal (transverse). Historically, the HDM was ®rst derived from a hydrodynamic equation of motion for the degenerate electrons with dispersive term (/ b) arising from the Fermi pressure (see, e.g., Refs. [1611,1612]). A great advantage of the HDM over (nonlocal) microscopic theories is, as mentioned before (see Section 2.2.1), the analytical simplicity. The HDM, however, sacri®ces the inclusion of the electron±hole (single-particle) excitation spectrum and intersubband excitations at the expense of this analytical simplicity. The relevant question, whether this analytical simplicity is not purchased at too high a price in terms of the ability to describe the physics of surface dielectric response is addressed by Feibelman [248], in addition to discussing the quantitative comparison of theoretical approaches such as the RPA, HDM, and PPA employed to study the surface EM waves. The HDM, as applied to the degenerate free-electron gas, is known to suffer from serious drawback, namely, it is valid only for frequencies o  n and o  n. Quite recently, it has, however, been generalized to arbitrary frequencies [1613]. As already discussed (in respective sections in Section 2), the HDM has been employed for studying, e.g., plasmon excitations in 2D [92,479,642], 1D [959], and 0D [1291] systems; the EMPs in ®nite-size 2D systems [1398±1400]; and plasmon excitations in electrically modulated 2DEG [1485]. The TMM, sometimes also known as the algebraic Bethe ansatz, frequently used for studying EM properties of multilayered system, as well as for investigating tunneling phenomena in multi-well systems in quantum mechanics, has been implicitly employed since the time of Lord Rayleigh [1614]. The speci®c theoretical framework of TMM, simplifying Rayleigh's treatments of wave propagation in a stack of thin homogeneous ®lms, was developed in a classic paper by AbeleÁs [1615]. Since then the TMM has become the most favorite macroscopic tool for studying any wave phenomenon in a multilayer (periodic or non-periodic) system in the local approximation. All the properties of the $ multilayer system are contained in the transfer matrix T , whose general characteristics we discuss

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149 $

below. The most fundamental result is that, in the absence of damping, T is unimodular, i.e., $ $ of unimodular matrices. An immediate detjT j ˆ 1. This follows from the fact that T is a product $ consequence is that the eigenvalues l1 and l2 of (say, 2  2) T satisfy l1 l2 ˆ 1. This is only possible provided that (i) l1 and l2 lie on the unit circle, and are complex conjugates of each other, or (ii) l1 and l2 both lie on the real axis. It has been known that in the frequency region in which (i) holds, Bloch waves propagate through in®nitely (periodic) $superlattice system, while (ii) corresponds to a stop band. The above-mentioned characteristics of T underlie the proof of the Chebyshev identity for the $ matrix T N   N  AUNÿ1 ÿ UNÿ2 A B BUNÿ1 ˆ ; (3.2) CUNÿ1 DUNÿ1 ÿ UNÿ2 C D where UN ˆ sin ‰…N ‡ 1†KLŠ= sin ‰KLŠ and KL ˆ cosÿ1 ‰12 …A ‡ D†Š (L being the period of the system), corresponding to$a transfer of the ®eld amplitudes across N unit cells. The proof relies on the fact that $ $N the matrix (say) S that diagonalizes T also diagonalizes T . A convenient account is given by Yeh et al. [706]. Rytov [1616] is known to have considered ®rst the case of EM wave propagation in a layered medium, and showed that the system behaves as if it were homogeneous but anisotropic (uniaxial crystal) Ð with the optic axis along the normal to the layer planes. Due to tremendous research interest in the properties of quantum wells and superlattices, over recent past, a large majority of researchers have made use of the TMM to study the phonons, plasmons, magnons, magnetoplasmons, etc. in such systems (see, e.g., Refs. [706±726,745±759,1752,1753]). The scheme of using the TMM for in®nite (perfectly periodic), semi-in®nite, and ®nite superlattice systems was briefed in Section 2.2.2, and the details can be found in the original articles (see, e.g., Refs. [749,757]). The Maxwell equations with appropriate boundary conditions worked out for any multilayered system yields numerically identical results to those obtained from the TMM. But the analytical results in the two methodologies look (almost always) formally dissimilar; and the intent of transforming corresponding results from one methodology to the other is known to be a ``hard nut to crack''. This remains true irrespective of the waves in question. The IRT is probably the youngest macroscopic theory, which was introduced for composite systems by Dobrzynski in 1986 [799]; it proceeds through the matrix formulation. The response functions (or Green functions) in the IRT are evaluated as functions of bulk response functions of each subsystem and of the interface response operators. These operators are shown to be the linear superposition of the responses to a cleavage operator of the corresponding ideal free surfaces of all subsystems and of the responses to the coupling operator of all interfaces. The resultant response functions can then be made use of to derive, literally speaking, any physical property of the system at hand. They play a crucial role in the theories of light scattering (both Raman and Brillouin), as well as in a variety of other physical phenomena [1617]. The IRT has seen a great deal of success for studying, e.g., the bulk and surface plasmon±polaritons, both with and without an applied magnetic ®eld in different con®gurations, for N… 2†-layered system. Quite recently, it (the IRT) has also been generalized to account for diverse 2D systems, both in the absence [730] and in the presence of an applied magnetic ®eld [731±733]. The details of the formal procedure behind the development of the IRT have been reported by Dobrzynski in a number of excellent review papers and we refer the reader to them [1618,1794±1796]. The limitation of the IRT, just as of any other macroscopic theory, is that it does not account for the quantum-size effects. However, the general framework of the IRT is independent of any particular

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model. For instance, one can always incorporate the spatial dispersion by using the dielectric tensor elements evaluated through the HDM, the damping effects by allowing imaginary parts in the dielectric tensor elements, the coupling to the optical phonons by considering frequency dependence of the background dielectric constant, and, of course, the retardation effects. The numerical work in the framework of IRT shares some interesting aspects with other macroscopic theories. For example, treating frequency (wave vector) as a complex variable can help determine the lifetime (propagation length) of the collective excitations in question. The computation of the inverse penetration depth is useful for determining the limitation of certain experimental probes. The advantage of working with the IRT, over other macroscopic theories, is the simplicity for deriving the required results in an elegantly compact form. The most general, and almost ®nal, result of the IRT is, e.g., the complete response function of the composite system (of N-layered media): $

$

$

$ÿ1

$

g …D; D† ˆ G…D; D† ÿ G…D; M†G …M; M†G…M; D† $

$ÿ1

$

$ÿ1

$

‡ G…D; M†G …M; M†gp …M; M†G …M; M†G…M; D†;

(3.3)

where the functionals …M; M† and …D; D† refer, respectively, to the interface space and the full space of the composite system, considered to be made up of independent slabs each limited by two black box surfaces (BBSs). By BBS we mean an entirely opaque surface through which EM ®elds cannot propagate. This is achieved by stressing that c (the speed of light) and E (the dielectric function) vanish $ for z  0, while the conducting medium resides in the upper half space (z > 0). In Eq. (3.3), gp …M; M† $ and G…a; b† (with a; b  M; D) are, respectively, the response function in the interface space and the bulk Green function in the respective space. The required propagating or localized modes in the system in question are determined through condition: $ÿ1

detjg

…M; M†j ˆ 0:

(3.4)

Such results have been derived and tested for diverse planar multilayered structures, which correspond to (N ÿ 1)-interface systems (N  2). It is noteworthy that the framework of the IRT is also comparable to any microscopic theory for a planar composite system comprised of non-interacting free electron gas, where quantum-size effects are ignored. 3.2. Microscopic model theories Every topic in condensed matter physics involves, at some level of investigation, the problem of e±e interactions. It is, however, fortunate, in view of the complexity of this question, that simple independent-particle models, in which interactions are included only through self-consistent ®eld, work rather satisfactorily. Collective excitations in homogeneous or inhomogeneous systems are an important hallmark of many-particle systems. They depend for their existence on the particle±particle interactions and are delocalized excitations of the entire system. The basic difference, from the point of view of the formulation, between the homogeneous and the inhomogeneous systems is that the wave vector component along the direction of inhomogeneity is not conserved and hence a simpli®cation of the basic equations by a Fourier transformation with respect to the corresponding coordinates is not allowed, thereby considerably complicating the formal methodologies. In the search of the collective excitations, one is bound to resort to the formal derivation of the non-local, dynamic dielectric function

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E…r; r 0 ; o†. It is also quite often necessary to know not only the dielectric function, but also its inverse Eÿ1 …r; r0 ; o†. The dielectric function and its inverse are related to each other through an integral equation: Z 00 00 0 00 0 (3.5) Eÿ1 ij …r; r ; o†Ejk …r ; r ; o† dr ˆ dik d…r ÿ r †: The formal determination of the exact IDFs for quasi-n dimensional electron systems (n  2) has recently been reported by Kushwaha and Garcia±Moliner [705]. The obtention of the dielectric function for the purpose of investigating the collective excitations in inhomogeneous systems has been achieved by employing the same formal techniques, which had already been used for the homogeneous systems. In order for deriving the dielectric function for a system at hand, it is ®rst necessary to derive the appropriate density±density response function (or current±current response function if one is interested to take into account the retardation effects). This has been done by a number of alternative equivalent techniques, such as, e.g., the HFA [1619,1797], the RPA ([119,126,133,136]), the equation of motion method (EMM) [139,185], the SCFA [158], the diagrammatic perturbation theory (DPT) [100,149,637± 639,1620,1621], DFT [190±192,1622,1623], and the Kubo correlation functions (KCFs) [171,172]; all of which are well established since the late 1960s. All these techniques have been used from time to time for studying the elementary excitation spectra in the conventional as well as non-conventional systems. The microscopic theories mentioned above are capable to describe both macroscopic and microscopic phenomena. They are, however, subject to one of the following limitations: (i) they are exact for only a limited range of particle density or interaction, one that rarely corresponds to physical reality. (ii) The extent to which they provide an approximate account of a real physical system can rarely be estimated with the desired precision. Microscopic theories in the ®rst category may be thought of as solutions to model problems, i.e., solutions based on well-de®ned approximations which can be justi®ably valid for a given class of particle interactions and densities. For electronic systems there exist two such models: the high-density electron gas and low-density electron solid. The measure of particle density is represented by a dimensionless parameter rs ˆ r0 =a0, where r0 is the interelectron separation h2 =me e2 is the Bohr radius; rs speci®cally refers to the ratio of the average potential energy and a0 ˆ  and the kinetic energy. The behavior of the electron liquid in the high-density limit (rs  1) is simple because here the Coulomb interaction represents a relatively small perturbation. In the low-density limit (rs  1) the electron behavior is dominated by the Coulomb interaction. As Wigner [448,1718,1719] ®rst remarked, in this strong-coupling limit the electron ``liquid'' will not be a liquid at all, but rather a solid; the electrons form a stable lattice in the sea of uniform positive charge. The region of actual metallic densities (1:89rs 95:6) is essentially an intermediate regime, and the kinetic energy and potential energy are roughly comparable. As such, it is far more dif®cult to treat. In the usual treatment of the dielectric response the low-density electron solid is not considered, primarily because its behavior is not relevant to the physical situation concerned with the collective excitations. The microscopic theories can be regarded, to a certain extent, as independent and general schemes to deal with the quantum many-particle problem. That is not to say that they solve the many-body problem the way one would long for; none of these approaches does that Ð if one miraculously did, physics would be much less interesting and challenging. But they do furnish a method of attack and a workable ®rst approximation. What we intend to say is that all of these proposed formal techniques suffer from one or the other drawback in a given circumstance. Known is the fact that the dielectric function E…q; o†

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of the interacting system can be determined only approximately by using the various simple descriptions of a many-particle system. This is what we would like to point out in what follows. It is vital to stress that the following remarks about the adequacy of the formal techniques are quite general as have been noticed in the course of their applications to the homogeneous systems, and we do not specify their advantages or disadvantages with respect to different physical conditions or the size and dimensionality of the mesoscopic systems. That would, in fact, be a somewhat long and tortuous path one must follow and may undesirably lead us to extensively expand on and digress from the main theme of this review. To begin with, we consider the HFA, which was the ®rst attempt to calculate the many-particle properties and thus represents a useful starting point. The HFA is known to stand for a great improvement over the Hartree approximation, which is known to give the same results as the Sommerfeld free-electron gas model. The ®rst disadvantage of the Hartree approximation is that the cohesive energy (the energy necessary to dissociate the solid into separated atoms) comes out to be completely wrong. This is due to the fact that the interaction between the electrons are completely neglected; there is no repulsion among the electrons, and two electrons can occupy precisely the same site. A second disadvantage is that the method does not take into account the Pauli exclusion principle, i.e., the wave function is not antisymmetric with respect to the interchange of any two electrons. Within the HFA, while the Pauli principle is appropriately taken into account, only the exchange part of the Coulomb interaction is considered. The Pauli principle dictates that two electrons of the parallel spin cannot be at the same site. This is what is exactly re¯ected from the calculation of the pair correlation function g…r† ˆ 12 ‰g"" …r† ‡ g"# …r†Š, which gives the probability that when one particle is observed at point r0 , another particle will be found in a characteristic volume V0 ˆ O=…N ÿ 1† at a distance r away from the point r0 . It is then obvious that if there is no correlation between the positions of two different particles, g…r† becomes unity. Within the HFA, the pair correlation function turns out to be gHF …r† ˆ 1 ÿ 92 ‰… sin…r† ÿ r cos…r††=r 3 Š2 , where r is measured in the units of kFÿ1 . It is thus evident that gHF …r† ˆ 12, because g"# …r† ˆ 1 for any r. A decrease in gHF …r† with r ! 0 is exclusively associated with taking into account of the Pauli principle, which forbids the presence of electrons with parallel spins at the same point and which states that g"" …r† ˆ 0. The restriction imposed by the Pauli principle extends also to a certain region around the electron, where one is not likely to ®nd another electron with the same spin. The effect is said to consist of an ``exchange hole'' around the electron (the term ``Fermi hole'' is also used). Because one deals, within the HFA, with a repulsive interaction, any tendency of the electrons to stay away from one another will give rise to a reduction in the ground-state energy of the system. We may regard the exchange energy as arising from the spin-induced correlations in the electron gas; the fact that the electrons possess a charge, and interact via Coulomb interaction, plays no role in determining gHF …r†. In fact, the electrons with antiparallel spins also avoid one another at small distances because of a direct Coulomb repulsion, as a result of which g"# …r† < 1. This effect is known to consist in that the electron is surrounded by a ``correlation hole'' (the term ``Coulomb hole'' is also used). At small distance (rs ), however, the effect of Coulomb hole is weak as compared to the Fermi hole, which is why the above expression for gHF …r† is asymptotically exact in the limit rs ! 0. Whenever a more sophisticated theory is sought after, the pressing need has been the concern for ®nding the correlation energy de®ned by the difference in the total energy between the HFA and any such better theory. However, it is well recognized in the literature that the many-particle theory of determining the reliable ground-state energy, particularly in the extended systems, is generally too dif®cult to

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implement beyond the HFA. There have been many efforts devoted to improve the theoretical framework of the HFA, but none has been conceptually so sound as the GWA introduced by Hedin and Lundquist [424,1702]. GWA is a schematic notation to denote a ®rst-order expansion of the electron self-energy (sometimes also called the mass operator) in terms of the Green function G and the screened e±e interaction W. The GWA is derived from many-particle perturbation theory using the Green function method, which is the most suitable approach for studying excited-state properties of the extended systems. The form of the self-energy operator S in the GWA is the same as in the HFA, but the Coulomb interaction is dynamically screened, remedying the most serious de®ciency of the HFA. The self-energy in the GWA is therefore non-local and energy dependent, and contains the effects of exchange and correlations. It is, in general, complex with the imaginary part describing the damping of the quasi-particles. It has been known that calculation of S is unfortunately very dif®cult even for the electron gas and one must resort to certain approximations. The GWA is physically sound because it provides a fruitful approximation to the self-energy and results in a reasonably correct answer in some limiting cases that allows applications to a large class of systems including electron gas, metals, and semiconductors. A quantity which provides a substantial amount of information on an interacting ÿ1 many-particle system is the electron self-energy S…q; o† ˆ Gÿ1 0 …q; o† ÿ G …q; o†, where G0 …q; o† is the bare non-interacting Green function without the exchange-correlation interactions. The self-energy is roughly the correction to the non-interacting electron single-particle energy due to the interactions. Once G…q; o† or S…q; o† Ð related to each other through the Dyson equation: G ˆ G0 ‡ G0 SG Ð is known, the single-particle properties such as the spectral density function, electron distribution function, and band-gap renormalization, etc., can be computed. For the critical review on the GWA, the reader is referred to a recent extensive review article by Aryasetiawan and Gunnarsson [1624]. It is worthwhile mentioning that GWA has also been extensively used for studying many-body effects in the semiconductor quantum wires by Das Sarma and coworkers [947]. The next most non-trivial methodology introduced by BoÈhm and Pines (see, e.g., Ref. [119]) in the early ®fties for calculating the non-local, dynamic dielectric function E…q; o† is the so-called RPA. Before going any further, let us make it clear what we mean by the RPA. The approximation acquired its name on the basis of the physical argument that in the derivation of the equation of motion for rk (the Fourier transform of the density ¯uctuation), under suitable circumstances, a sum of exponential terms with randomly P varying phases could be neglected as compared to r0 …ˆ N†. Thus the approximation: rkÿl ˆ i exp‰ÿi…k ÿ l†  ri Š  Ndk;l was frequently made when rkÿl appeared as a multiplicative factor in the basic equations of theory. More precisely, one can say that in the RPA one determines a given property of the system which depends upon a particular momentum transfer k, by keeping only that component of the Coulomb interaction which involves the momentum transfer k. In other words, in making the RPA one neglects the coupling between different momentum transfers for either the e±e or electron±plasmon interactions. The development of the RPA for the electron gas offers a useful object lesson to the theoretical physicist. First, it illustrates the splendid variety of ways that can be developed for saying the same thing. Second, it suggests the usefulness of learning more than one ``language'' of theoretical physics, and of attempting the reconciliation of seemingly different, but obviously related, results. The RPA goes under wide variety of names Ð RPA, independent-pair approximation, SCFA [158], time-dependent HFA, etc.; in fact, there are as many names as there are ways of deriving the answer. The way in which the RPA differs from the HFA can be seen through the derivation of the IDF. It is thus found that in the RPA (HFA) the electrons respond to effective (external) ®eld. In both cases, the response is given by the free-electron polarizability.

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It is worth mentioning that though the RPA is a high-density approximation, its reasonable success in studying collective excitations in the low-density, low-dimensional systems (e.g., quantum wells, quantum wires, and quantum dots, etc.) has made it to stand for the most enviable approach ever employed in many-particle systems. At this point one may ask: why does the RPA work at any density? It turns out that in the low-density, quantum systems usually fabricated out of low-effective mass systems (e.g., GaAs), the dimensionless parameter rs ˆ r0 =a0 is small (since a0 is inversely proportional to the effective mass, m ˆ 0:067m0 in GaAs). It has, however, been known that the RPA runs into dif®culties for large momentum transfers. As Nozieres and Pines [127] and Hubbard [143] have argued, there is no distinction in the RPA between the contribution to the correlation effects arising from electrons of parallel spin and antiparallel spin. On the other hand, physically, one would expect that electrons of parallel spin simply do not feel the short-range part of the interaction, inasmuch as they are kept apart by the Pauli principle. Mathematically, such an effect appears because for large momentum transfers the ``exchange'' parts of the perturbation-theoretic expansion (which occur only for electrons of parallel spin, and which are neglected in the RPA) cancel that one-half of the ``direct'' interaction that may be attributed to electrons of parallel spin. The result: only electrons of antiparallel spin interact via the large momentum transfer part of the Coulomb interaction; and for electrons of antiparallel spin no exchange processes are possible. For small momentum transfers, for which the RPA is valid, the exchange terms are unimportant too [637]. It is only when one goes toward metallic densities that the large-momentum-transfer contributions become appreciable; where they are important one must go beyond the RPA to calculate the system properties. The RPA dielectric function is only approximate and, in general, becomes a poor approximation as electron correlations increase. Essentially the same shortcoming of the RPA appears when it is attempted to calculate other quantities. Therefore, efforts have been made to improve the RPA framework. We discuss in what follows some of these efforts dedicated to improve the RPA results in one-component plasma in 3D homogeneous systems. In an attempt to improve the RPA by taking into account of the short-range exchanges, Hubbard [143] has suggested that the RPA dielectric function E…q; o† ˆ 1 ÿ V…q†w0 …q; o†

(3.6)

be replaced by EH …q; o† ˆ 1 ÿ

V…q†w0 …q; o† ; 1 ‡ V…q†G…q†w0 …q; o†

(3.7)

where G…q† ˆ 12 ‰q2 =…q2 ‡ kF2 †Š; the factor 12 comes from the fact that only electrons of parallel spin can have exchange interaction, and the q-dependent part is called the Thomas±Fermi dielectric function. The appearance of the Fermi wave vector kF indicates that, at low-temperatures, momentum exchange due to interaction takes place only near the Fermi surface. In these equations w0 …q; o† is the Lindhard (single-particle) polarization function. Eq. (3.7) represents the static local ®eld correction and is an approximation which becomes poor for large q. The curious formula due to Hubbard was regarded as an improvement to the RPA in many properties. Yet its true worth was unappreciated, because it could not be compared with the better dielectric function which only became available later. As Mahan remarked [639], the best dielectric functions today are written in precisely the form of Eq. (3.7) with G…q† which is only slightly different from the simple form proposed by Hubbard. Thus his result, which was somewhat of a ``stab in the dark'', is now regarded as being well ahead of its time. The factor G…q†

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was introduced, in a sense, to account for the effects of the exchange hole around the electron, but clearly neglects the effects of correlation hole. This implies that when one electron is participating in the dielectric screening, others are less likely to be found nearby. This should have some effect upon the nature of dielectric screening [637±639]. The function G…q† is usually taken to be frequency independent. It is, however, worthwhile to introduce a certain function G…q; o† which also has frequency dependence, and hence stands for the dynamic local ®eld correction. This is because the exchange hole is presumably frequency dependent [1625,1798,1799]. Historically, a number of authors [1626±1628] have attempted to account empirically for correlation hole by modifying G…q† to the form: G…q† ˆ 12 ‰q2 =…q2 ‡ xkF2 †Š, where x is a parameter. In particular, Geldart and Vosko [1627] stressed the importance of satisfying the compressibility sum rule, which would require x ˆ 2, and it was argued that a proper theory must take account of the correlation hole in addition to the exchange hole considered by Hubbard. Another most important (from a physical point of view) technique for improving the RPA is the one proposed by Singwi and collaborators [441], previously referred to as STLS. Their dielectric function has apparently the same form as Hubbard's, but with a different choice of G…q†. Singwi is known to have collaborated with a variety of authors to develop improvements in the way of choosing G…q†. Thus one could, humorously, remark that G…q† is time-dependent because of its improvements over the years. The original proposal of STLS is very attractive because it explicitly incorporates not only the exchange hole but also the correlation hole surrounding the electron. In order to ®nd the correction to the local ®eld in the STLS scheme, it is suf®cient to consider the standard relation between the induced density nind and the exchange-correlation potential Vxc : Vxc …q; o† ˆ ÿV…q†G…q; o†nind …q; o†. In their scheme the required G…q† can only be evaluated with the knowledge of the static structure factor S…q†. This, in turn, is obtained from a knowledge of E…q; o†. But E…q; o† depends on G…q†. Thus the expressions for S…q†; G…q†, and E…q; o† form a triad and have to be solved self-consistently on the computer; and this must be done for every value of rs , since the results depend on the density. The simpli®ed expression for GSTLS turns out to be Z 1 d~ k ~ q ~ k GSTLS …q† ˆ ÿ ‰S…j~ q ÿ~ kj† ÿ 1Š: (3.8) n0 …2p†3 k2 Thus, the local-®eld correction in this scheme does not depend on the frequency. This is quite natural, especially as we recall that the local-®eld effect accounted for in the basic equations are timeindependent. Nevertheless, the STLS approximation has the advantage Ð it is self-consistent. The local-®eld correction in the STLS scheme is constructed so that it satis®es the Shaw±Kimball rule [1629,1800] G…q†jq!1 ˆ 1 ÿ g…0†:

(3.9)

According to Eq. (3.9), the static correction to the local ®eld remains to be ®nite and its value lies somewhere in the middle between G…q; 1† and G…q; 0†. Thus the function G…q†, which satis®es Eq. (3.9), may be regarded as a frequency-averaged local-®eld correction. Therefore, it cannot serve as a good approximation to G…q; o† at o ˆ 0. The GSTLS …q†, however, provides a correct behavior of the static structure factor S…q† in the large q limit and leads to a relatively correct description of the pair correlation function g…r† at small inter-electron distances Ð g…r† may be obtained from the knowledge of S…q† [639]. The function gSTLS …r† does not become negative practically over the entire region of the

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metallic densities of the electron gas. This is in contrast to the gRPA …r† and gH …r† which are seen to have negative values at small values of r. This is viewed as a great de®ciency of the RPA and Hubbard approximation, since g…r† is Ð by de®nition Ð strictly a positive function. An early method of calculating local-®eld corrections in the calculation of non-local, dynamic dielectric function was given by Adler [1630] and Wiser [1631]. They clari®ed the relation between the local-®eld correction and the matrix character of the dynamic response (in the homogeneous systems). A simultaneous use of the RPA (see also Refs. [1632,1633,1801] for the later development of the RPA), the Hubbard approximation, and the STLS approximation was made by Jonson [485] in his study of the plasmon dispersion in the quasi-2D system of inversion layers (see Section 2.2.1). As one moves away from the long-wavelength limit, two techniques offer themselves for the evaluation of the many-body effects in the microscopic framework: the DPT [1621] and the equationof-motion method [185]. Both techniques have been widely applied to the problem of electron ¯uids, and either presents special advantages; and enable one to calculate the dielectric function to any degree of accuracy. The diagrammatic approach has a precise theoretical meaning in the context of a perturbation treatment of the interactions, but one has to sum an in®nite set of diagrams even in the simplest approximation; and both the choice and the evaluation of the relevant set of diagrams present dif®culties as one goes beyond the RPA. Here comes in the aid from Wick's theorem which forms the basis for the diagrammatic representation. The advantage of the diagrammatic technique consists mainly in that it enables one to perform the summation of the whole classes of diagrams on the basis of simple topological considerations. Note that the diagrammatic technique of many-body perturbation theory makes use of the causal Green function rather than the retarded Green function. Here the correct value of E…q; o† is obtained only for o > 0. The correct value for E…q; o† in the region o < 0 can, however, be obtained with the aid of the relation E…q; ÿo† ˆ E …ÿq; o†, which follows from the condition of the realness of E…r; r 0 ; t†. In many-body perturbation theory the summation of a certain set of diagrams is also equivalent to the solution of a certain integral equation. The given system of integral equations has a purely formal meaning, since each of the equations expresses only one unknown function via another unknown function, which makes the diagnosis more complex. The equation-of-motion method presents one of the most ef®cient methods of many-body theory based on the solution of equations for the double-time Green functions [139,185]. However, in this method the calculation of terms of higher order in the interaction does not require the integration over ``implicit'' frequencies. An approximate solution of the hierarchy of equations of motion can be given by decoupling schemes that compactly account for some effects of the interaction to in®nite order, but one may ®nd a justi®cation on intuitive grounds and in the veri®cation of sum rules and limiting behaviors rather than in a precise theoretical development. What one needs to know is how to calculate the commutators of the ®eld operators in the second-quantization representation. Moreover, the derivation of many physically reasonable approximations to E…q; o†, otherwise obtained usually as a result of very cumbersome computations, appears to be simpler and compact when use is made of this technique. The practical importance of the method, however, consists in that at a certain stage of calculations one usually introduces an additional approximate equation, which relates the Green function of the last rank taken into account to the Green function of the lower ranks on the basis of physical considerations. Consequently, the system of equations is found to be closed and its solution presents no serious dif®culties. In other words, unless special circumstances occur, the hierarchy of equations must be terminated in a possibly arbitrary manner by an ansatz that expresses the

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…n ‡ 1†-particle function as a combination of n-particle functions. Such an ansatz is frequently referred to as a decoupling procedure. For extensive details, the reader is referred to the classic references on the subject [638,639,1634±1636]. All the quantum methodologies for the many-particle systems described hitherto were originally proposed before 1960s, although they have been continuously improved upon in the subsequent years. The density-functional theory (DFT), introduced by Hohenberg, Kohn, and Sham (hereinafter referred to as HKS) [190±192], provided a practical, new approach to electronic structure, applicable to large-N systems. The DFT is couched in terms of the electron density n…r† (or for magnetized systems, spin-density ns …r†; s ˆ 1) instead of the many-electron wave function C. It leads to the Kohn±Sham (KS) self-consistent equations, similar to the Hartree equations, in which, however, exchange and correlation effects are included (in principle, exactly) by the addition of the exchange-correlation potential of Vxc …r†. The theory allows parameter-free calculations of densities, ground-state energies, as well as related quantities such as lattice structures and constants; elastic coef®cients; work functions, surface energies, and atom±surface interaction energies; phonon dispersion relations; plasmon dispersion relations; magnetic moments, etc. Accuracies typically range from 1 to 2% depending upon the context; geometries emerge very accurately, typically 1%. The formal DFT is, in principle, exact, but in practice requires an approximation for the exchange-correlation energy Exc. The simplest, the LDA, rests on the accurate Monte Carlo calculations of a uniform, interacting electron gas. The scaling of the computation time with N is a relatively very favorable N a , where 1  a  3, so that the calculations for ®nite systems with N  100±1000 have been quite feasible. The DFT has now-a-days become the method of choice for calculating electron densities and ground-state energies of most condensed-matter systems. The fundamental status of the DFT rests on the two HKS theorems, which characterize the basic ground-state properties of a non-uniform interacting many-fermion system. The ^ g i, where H ^ is the full many-body Hamiltonian and jg ®rst is that the ground-state energy E…ˆ hjg jHjj represents the ground-state wave function) is uniquely given by the density n…r†, i.e., E ˆ E‰nŠ Ð perhaps more basic, the jjg i is given uniquely by n…r† from which E ˆ E‰nŠ. The second is that if we write Z (3.10) E‰nŠ  Ev ˆ F‰nŠ ‡ d3 rV…r†n…r†; then the absolute minimum of Ev (the variational principle), when the external potential is ®xed, is given by this density n…r†. These two theorems are known to comprise the so-called DFT. What HKS have actually shown is that E ˆ EN ‰nŠ, i.e., it is a functional of n…r† as well as a function of the total particle number N. Incidentally, since in the thermodynamic limit E should depend on the intensive variables, then EN ‰nŠ ˆ E‰n0 ; nŠ, where n0 can be taken as the average density of n…r†. As it has been shown, this additional dependence on n0 , which was almost overlooked when approximate structures of E‰nŠ were suggested, is at the heart of the band-gap discontinuity. KS went a step further and rigorously mapped E‰nŠ to the self-consistent single-particle equations, becoming known as the KS equations. Very brie¯y, KS split the internal energy F‰nŠ into F‰nŠ ˆ EKE ‰nŠ ‡ EHE ‰nŠ ‡ EXC ‰nŠ;

(3.11)

where EKE ; EHE , and EXC are the kinetic energy, the Hartree energy, and the exchange-correlation energy, respectively, of the non-interacting non-uniform fermions with the same density n…r†. They then

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write EKE in terms of auxiliary functions ji …r† as  Z N  X h2   3 2 EKE ‰nŠ ˆ d r ÿ  ji …r†r ji …r† 2m iˆ1

(3.12)

with n…r† ˆ

N X iˆ1

ji …r†ji …r†

(3.13)

and minimize Eq. (3.10). The complete proof of the basic HKS theorems have been given by several authors (see, e.g., Ref. [1623]). These theorems provide a general method for calculating ground-state properties. An important result follows from the fact that if the ground state is non-degenerate, min jmin ng ˆ jg ; if the ground state is degenerate, jng is equal to one of the ground-state wave functions, and others can also be obtained [1623]. The ground-state charge-density then determines the groundstate wave function(s), from which all ground-state properties can be calculated. These properties are therefore functionals of the density Ð the cornerstone of the DFT. This is the formal justi®cation for working with the density instead of the wave function. A variational principle of the problem allows one to derive a set of single-particle SchroÈdinger equations of the form   Z 0 h2 2 2 3 0 n…r † (3.14) dr ‡ VXC …r† ji …r† ˆ Ei …r†ji …r†; ÿ  r ‡ V…r† ‡ e jr ÿ r0 j 2m VXC …r† ˆ

dEXC ‰n…r†Š : dn…r†

(3.15)

Eqs. (3.13)±(3.15) have to be solved in a self-consistent manner. The formulation is exact, but the problem is that EXC and VXC are not known in general. The LDA consists in assuming that, locally, the relation between EXC and n…r† is the same as that of the free-electron gas of identical density. That is to say that VXC …r† depends in a simple manner only on the charge-density at the point of interest, VXC …r† ˆ VXC ‰n…r†Š. The merit of the DFT in the LDA is that it is nearly as simple to carry through as the Hartree theory; it gives in most cases a good semi-quantitative account of the exchange and correlation effects. In practice, it is found [1622] that typically LDA results are better than those of HFA, even though the latter are computationally very much demanding and, for real solids, almost prohibitive. The reason is that the DFT includes, fairly accurately, both exchange and correlation, while the HFA includes exchange exactly but neglects correlations completely. It is worth mentioning that some authors have practiced the separation of the exchange-correlation energy by writing, e.g., Z (3.16) EXC ‰nŠ ˆ EX ‰nŠ ‡ n…r†EC ‰n…r†Š dr and have used the relation corresponding to Eq. (3.15) only for the correlation part of the potential. This has been rigorously discouraged (see, e.g., Ref. [1637,1802]) arguing that it is not proper to separate exchange and correlation energies in the LDA, since a regular power series in density gradient does not exist for these energies separately, but only for their sum. Eq. (3.14) is a single-particle equation, so that the N-particle problem has been formally reduced to N single-particle problems. All many-body effects are included in the exchange-correlation potential, VXC , which is local (LDA), in contrast to Hartree±Fock potential. In the original form, the HKS theorems involve a very signi®cant restriction to spinless fermions, which do not exist, and the LDA is a

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ground-state theory and is formally not applicable to excited-state properties. Subsequent years have, however, seen several generalizations of the simplest form of DFT to include, e.g., ®nite temperature [1638], spin [1639], multi-component systems such as electron-hole droplets in semiconductors [1640], and, to some extent, excited states [1641]. The DFT has also been used to investigate the dependence of energy level structure and the effective mass on the 2D electron density in Si-inversion layers forming a quasi-2D EG [440] in order to interpret the infrared absorption measurements [402]. Quite recently, Beck has investigated the plasmon dispersion in the superlattice structures of 2DEG layers within the framework of TDLDA [826]. 3.3. The strategy of working within the RPA The history of collective excitations such as CDEs, SDEs, as well as SPEs is, in view of many, the most remarkable chapter in the 20th century history of condensed matter physics. To the understanding of these collective excitations, one is bound to develop a theoretical framework for deriving the nonlocal, dynamic dielectric function, and/or conveniently the IDF for a system in question. There are different means of doing this (see the preceding section). However, we prefer starting with Kubo's correlation functions and use the ``language'' of second quantization to arrive at the desired results. This section concerns with the derivation of the dielectric function and its inverse for a quasi-ndimensional (n  2; 1; 0) electron system (Q-nDES). For this purpose, and for the sake of selfconsistency, we would ®rst like to derive the density±density response function (DDRF) for a 3D system, which would then be specialized corresponding to the system of quasi-n dimensionality. To start with, we calculate the linear response (with the assumption that the external potential is weak enough) and write the condition of self-consistency as r; t† ˆ Vext …~ r; t† ‡ Vind …~ r; t†; Vtot …~

(3.17)

where the subscripts ``tot'', ``ext'', and ``ind'' stand for the total, external, and induced, respectively. r; t† may be assumed to arise from some Here ~ r refers to the 3D position vector and t to the time. Vext …~ r; t† is due to the induced external point charge distribution, r…~ r; t† ˆ ÿed…~ r ÿ~ r…t††, say, and Vind …~ charge density in the system and is expressed as Z Vind …~ r; t† ˆ d3 r 0 Vee …~ r;~ r 0 †nind …~ r 0 ; t†; (3.18) r;~ r 0 †, the binary Coulombic interaction energy between the two electrons occupying the where Vee …~ spatial sites ~ r and ~ r 0 , is expressed in terms of the electrostatic Green function such that r;~ r 0† ˆ Vee …~

e2 e2 1 G…~ r;~ r 0† ˆ ; E0 E0 j~ r ÿ~ r 0j

(3.19)

where E0 is the background dielectric constant. We express the induced charge density nind …~ r; t† in terms of Kubo's correlation function [172] Z Z r; t† ˆ d3~ rt;~ r 0 t0 †Vtot …~ r 0 ; t0 †; (3.20) r 0 dt0 w0 …~ nind …~ Z Z 3 0 dt0 w…~ r nind …~ r; t† ˆ d ~ rt;~ r 0 t0 †Vext …~ r 0 ; t0 †; (3.21)

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where w and w0 are the total and single-particle DDRF, respectively, and are related to each other through an integral (Dyson) equation Z Z 0 0 0 3 00 d3~ r;~ r †‡ d~ r;~ r 00 †Vee …~ r 00 ;~ r 000 †w…~ r 000 ;~ r 0 †; (3.22) w…~ r;~ r † ˆ w …~ r r 000 w0 …~ where both sides are Fourier transformed with respect to time and the o-dependence is suppressed for the sake of brevity. Since we are interested in the RPA scheme which has proved to be as much successful in the systems of reduced dimensionality as in the conventional solids, we will be working throughout in terms of w0 , Eq. (3.20), which is de®ned by i rt;~ r 0 t0 † ˆ ÿ y…t ÿ t0 †hf0 j‰n0 …~ r; t†; n0 …~ r 0 ; t0 †Šjf0 i; w0 …~ h  where the electron density operator n0 …~ r; t† for spin-free fermions is given by X  r; t† ˆ c‡ r†fj …~ r†: n0 …~ i …t†cj …t†fi …~

(3.23)

(3.24)

i;j

Here c‡ i …ci † is the creation (annihilation) operator corresponding to the single-particle wave function r† of the unperturbed system; these operators are all in the interaction picture. jf0 i is the timefi …~ independent Heisenberg ground state of the unperturbed system. In the interaction picture (in the absence of the inter-particle interactions), the time-dependent operators c‡ i …t† and ci …t† are expressed in …0† and c …0†: terms of the time-independent SchroÈdinger operators c‡ i i   E t ÿiEi t i ‡ ci …t† ˆ ci …0† exp ; c‡ ; (3.25) i …t† ˆ ci …0† exp h  h hoi is the energy eigenvalue of the eigenstate fi . The time dependence may then be where Ei ˆ  factored out of the commutator in Eq. (3.23). It is then a simple matter to prove that the expectation value of the commutator in Eq. (3.23) is given by Z Z ‡ ‡ 0  ‡ ‡ ‡ hf0 j‰ci cj ; ck cl Šjf0 i ˆ dt f0 ci cj ck cl f0 ÿ dt0 f0 c‡ (3.26) k cl ci cj f0 : Note that the prime on the volume element dt0 indicates that the multiple integral includes the sum over the two values of each of the spin variables. The orthogonality of f0 imposes the condition that the ®rst integral vanishes until and unless ‡ c‡ i cj ck cl f0 ˆ f0

(3.27)

and similarly, the second integral vanishes until and unless ‡ c‡ k cl ci cj f0 ˆ f0 :

(3.28)

With the aid of the simple arguments of second quantization, the orthonormal properties of f0 , and anticommutation relations for fermions, we deduce that the commutator in Eq. (3.26) becomes: ‡ hf0 j‰c‡ i cj ; ck cl Šjf0 i ˆ ‰ f0 …Ei † ÿ f0 …Ej †Šdil dkj ;

(3.29)

where f0 …Ei † is the Fermi±Dirac distribution function. It is clearly unnecessary to specify that i 6ˆ j, since the square bracket on the right-hand side of Eq. (3.29) vanishes when i ˆ j. Now, Eq. (3.23), with

M.S. Kushwaha / Surface Science Reports 41 (2001) 1±416

the aid of Eqs. (3.24), (3.25) and (3.29), assumes the form:   iX i…Ei ÿ Ej †…t ÿ t0 †  0 0 0 w …~ rt;~ r t†ˆÿ ‰ f0 …Ei † ÿ f0 …Ej †Š exp r†fj …~ r†fi …~ r 0 †fj …~ r 0 †: fi …~ h i; j  h Fourier transforming with respect to time leaves us with X f0 …Ei † ÿ f0 …Ej † w0 …~ r;~ r 0 ; o† ˆ f …~ r†fj …~ r†fi …~ r 0 †fj …~ r 0 †; ‡ i E ÿ E ‡ h  o j i;j i

161

(3.30)

(3.31)

where o‡ ˆ o ‡ ig0 , with g0 being a vanishingly small positive constant (an appropriate reference is [1104]). In the usual sense, g0 ! 0 serves to switch the perturbation adiabatically on in the remote past. Note that our time dependence is de®ned as  exp…ÿiot† throughout. Eq. (3.31), for single-particle DDRF, is an important result to be used in the calculation of the dielectric functions for the respective Q-nDES. A familiar relation which associates the total perturbing potential (energy) in the system to an applied external potential (energy) is given by Z r; o† ˆ d3~ r;~ r 0 ; o†Vtot …~ r 0 ; o†; (3.32) r 0 E…~ Vext …~ where E…~ r;~ r 0 ; o† is the non-local, dynamic dielectric function of the conventional system. A simple and straightforward calculation, using Eqs. (3.17), (3.18), (3.20) and (3.32), yields Z 0 0 r ÿ~ r † ÿ d3~ r;~ r 00 †w0 …~ r 00 ;~ r 0 ; o†; (3.33) r 00 Vee …~ E…~ r;~ r ; o† ˆ d…~ where Vee and w0 are as speci®ed by Eqs. (3.19) and (3.31). Appropriate Fourier transformation of Eq. (3.33) with respect to 3D spatial coordinates yields X f0 …Ek † ÿ f0 …Ek‡q † 1 ; (3.34) E…q; o† ˆ 1 ÿ Vee …q† O Ek ÿ Ek‡q ‡ ho‡ k where the 3D Fourier transform of the screened Coulombic interactions Vee …q† ˆ 4pe2 =E0 q2 . This result was ®rst obtained by Lindhard [156] with the SCFA and later by Nozieres and Pines [122±127] within the RPA. In what follows, we will use Eq. (3.33) to derive the dielectric function and its inverse, in the general situations, for the Q-nDES. Let us ®rst start with a quasi-two dimensional electron systems (Q-2DESs) con®ned in the x±y plane wherein the translational invariance prevails. This means that we can Fourier transform all the quantities only with respect to the planar coordinates. Further, we write the single-particle wave r† and the respective energy eigenvalues Ei …~ kk † as function fi …~ 1 r† ˆ p exp…i~ rk †fn …z†; (3.35) fi …~ kk ~ A kk † ˆ Ei …~

h2 kk2

‡ En ; (3.36) 2m where ~ kk and ~ rk are the 2D vectors, A the normalization area, and the subscript n refers to the electric subband index due only to the size quantization along the spatial dimension z of the electron gas

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system. Fourier transforming Eq. (3.33) with respect to ~ rk  …x; y† and making use of Eq. (3.35) yields   Z X E…qk ; o; z; z0 † ˆ d…z ÿ z0 † ÿ Pnn0 …qk ; o† dz00 Vee …qk ; z; z00 †fn …z00 †fn0 …z00 † fn …z0 †fn0 …z0 †; nn0

(3.37) where Pnn0 and Vee are given by 1 X f0 …En;kk † ÿ f0 …En0 ;kk ‡qk † Pnn0 …qk ; o† ˆ ; ho‡ A k En;kk ÿ En0 ;kk ‡qk ‡ 

(3.38)

mid

Vee …qk ; z; z0 † ˆ

2pe2 exp…ÿqk jz ÿ z0 j†; E 0 qk

(3.39)

are, respectively, the polarizability function and the 2D Fourier transform of the Coulombic interactions. Now, we determine the IDF Eÿ1 …qk ; o; z; z0 † such that the integral equation Z dz00 Eÿ1 …z; z00 †E…z00 ; z0 † ˆ d…z ÿ z0 † (3.40) is satis®ed. Just as in this equation, we will henceforth suppress the (qk ; o) dependence for the sake of brevity. Let us ®rst cast Eq. (3.37) in the form X Lm …z†Pm Sm …z0 †: (3.41) E…z; z0 † ˆ d…z ÿ z0 † ÿ m

The composite index m‰  …n; n0 †Š ˆ ms ; ma ; with subscript s (a) referring to the symmetric (antisymmetric) wave function depending upon whether n ‡ n0 ˆ even (odd). This is a quite a general scheme and singles out only the symmetric structures from the asymmetric ones. The symbols Lm …z† and Sm …z0 † are de®ned by Z 0 (3.42) Lm …z†  Lnn …z† ˆ dz0 Vee …z; z0 †Znn0 …z0 †; Sm …z†  Snn0 …z† ˆ Znn0 …z† ˆ fn …z†fn0 …z†:

(3.43)

Clearly, Lm and Sm stand for the long-range and the short-range parts of the response function, respectively. Abiding by our motivation (and keeping in mind the space being covered) prevents us from expanding on the reasons and usefulness of labeling the pair of subband indices …nn0 † by a single composite index m. It is noteworthy that the range of summation over m in Eq. (3.41) is an interval …a; b†, which can well be taken to be …ÿ1; ‡1† for the sake of generality. We presume Eÿ1 …z; z0 † to be given by, say X An …z†Hn Bn …z0 †: (3.44) Eÿ1 …z; z0 † ˆ d…z ÿ z0 † ‡ n

Then substituting E…z; z0 † and Eÿ1 …z; z0 † from Eqs. (3.41) and (3.44) in Eq. (3.40) leaves us with X X Lm …z†Pm Sm …z0 † ‡ An …z†Hn ‰Pm Sm …z0 †amn ÿ Bn …z0 †dmn Š ˆ 0; (3.45) m

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where

Z

amn ˆ

dzLm …z†Bn …z†:

(3.46)

Multiplying Eq. (3.45) by Lm …z0 † and integrating over z0 gives X X Lm …z†Pm bgm ‡ An …z†Hn ‰Pm bgm amn ÿ agn …z0 †dmn Š ˆ 0; m

where

(3.47)

m;n

Z

bgm ˆ

163

dz Lg …z†Sm …z†:

(3.48)

Now let us de®ne Bm …z† ˆ lm Sm …z†;

(3.49)

where li is a non-zero integer, so that amn ˆ ln bmn :

(3.50)

As such, Eq. (3.47) assumes the form

! X X bgm Lm …z†Pm ‡ An …z†Hn ln ‰Pm bmn ÿ dmn Š ˆ 0: m

(3.51)

n

Either the ®rst or the second factor is zero. Since bgm 6ˆ 0, we are left with X Lm …z†Pm ˆ An …z†Hn ln ‰dmn ÿ Pm bmn Š:

(3.52)

If Lmn is assumed to be an inverse of …dmn ÿ Pm bmn † such that X …dmn ÿ Pm bmn †Lng ˆ dmg ;

(3.53)

n

n

then multiplying Eq. (3.52) by Lmg and summing over m yields 1 X  Ag …z† ˆ L …z†Pm Lmg : Hg lg m m Substituting An …z† and Bn …z†, from Eqs. (3.54) and (3.49), in Eq. (3.44) leaves us with X Lm …z†Pm Lmn Sn …z0 †: Eÿ1 …z; z0 † ˆ d…z ÿ z0 † ‡

(3.54)

(3.55)

m;n

That this is the correct inverse of E…z; z0 †, Eq. (3.41), can be justi®ed by substituting Eqs. (3.41) and (3.55) in the left-hand side of Eq. (3.40) to get exactly d…z ÿ z0 †, the right-hand side of Eq. (3.40). Next, is Eÿ1 …z; z0 † in Eq. (3.55) the unique inverse of E…z; z0 † in Eq. (3.41)? In order to be sure let us answer this question in the negative and suppose that kÿ1 …z; z0 † 6ˆ Eÿ1 …z; z0 † is another inverse of E…z; z0 †. Then there

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must be a matrix L0mn satisfying an identity such as the one given in Eq. (3.53), i.e. X …dmn ÿ Pm bmn †L0ng ˆ dmg :

(3.56)

Subtracting Eq. (3.56) from Eq. (3.53) piecewise leaves us with X …dmn ÿ Pm bmn †…Lng ÿ L0ng † ˆ 0:

(3.57)

n

n

Since the ®rst term is non-zero Ð otherwise, the identity in Eq. (3.49) or (3.53) makes no sense Ð the second term equated to zero is the proper solution of Eq. (3.57); i.e., Lng ˆ L0ng . This leads us to infer that kÿ1 …z; z0 † ˆ Eÿ1 …z; z0 †. In case that n ˆ 1, Eq. (3.53) gives L ˆ …1 ÿ Pb†ÿ1 and Eq. (3.55) assumes the form 1 L …z†PS…z0 †: Eÿ1 …z; z0 † ˆ d…z ÿ z0 † ‡ (3.58) 1 ÿ Pb Note that Eq. (3.58) is valid only for intrasubband excitations, where only the lowest subband is occupied. Now, let us turn to the quasi-one dimensional electron system (Q-1DES). We assume that the electron gas is free along the spatial dimension x and is laterally quantized in the y±z plane. We write the singleparticle wave function and the corresponding energy eigenvalue such that 1 r† ˆ p exp…ikx†fa …~ r? †; fi …~ L

(3.59)

2 k 2 h ‡ Ea ; (3.60) 2m where k  kx ;~ r?  …y; z†, L is the normalization length, and a  …m; n† is a composite index for the transverse motion Ð m and n stand for quantum numbers which characterize the nodes in the envelope wave functions in the y and z directions, respectively. It should be pointed that we do not impose any simplifying assumptions regarding the relative magnitude of the size quantization in the y±z plane. We prefer to keep the formal analysis quite general and hence work in terms of the composite index a. The limiting cases, e.g., for zero width along the z direction of the Q-1DES can be obtained from the general expressions at every stage of the analytical diagnosis. Fourier transforming Eq. (3.33) with respect to x and making use of Eqs. (3.59) and (3.60) leaves us with Ei …k† ˆ

r 0? † ˆ d…~ r?ÿ~ r 0? † E…q; o;~ r? ;~ Z  X  00 2 00 00 00 0 d~ r? ;~ r ? †fa …~ r ? †fa0 …~ r ? † fa …~ r 0? †fa0 …~ r 0? †; (3.61) ÿ Paa …q; o† r ? Vee …q;~ aa0

where the polarizability function Paa0 and the 1D Fourier transform of the Coulombic interactions Vee are given by 1 X f0 …Ea;k † ÿ f0 …Ea0 ;k‡q † ; (3.62) Paa0 …q; o† ˆ L k Ea;k ÿ Ea0 ;k‡q ‡  ho‡ Vee …q;~ r? ;~ r 0? † ˆ

2e2 K0 …qj~ r? ÿ~ r 0? j†; E0

(3.63)

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where K0 …x† is the modi®ed Bessel function of the second kind. The dielectric function for a Q-1DES r 0? † and its inverse Eÿ1 …q; o;~ r? ;~ r 0? † should satisfy an integral with a ®nite width and height E…q; o;~ r? ;~ equation Z r 00? Eÿ1 …~ r? ;~ r 00? †E…~ r 00? ;~ r 0? † ˆ d…~ r? ÿ~ r 0? †: (3.64) d2~ Let us rewrite Eq. (3.61), suppressing …q; o† dependence such that X r 0? † ˆ d…~ r? ÿ~ r 0? † ÿ Lm …~ r? †Pm Sm …~ r 0? †; E…~ r? ;~

(3.65)

r? † and short-range Sm …~ r? † parts are de®ned by where m  …a; a0 † and the long-range Lm …~ Z 0 r? Lm …~ r? †  Laa0 …~ r? † ˆ d2~ Vee …~ r? ;~ r 0? †Zaa0 …~ r 0? †;

(3.66)

m

Sm …~ r? †  Saa0 …~ r? † ˆ Zaa0 …~ r? † ˆ fa …~ r? †fa0 …~ r? †: Following exactly analogous to the case of Q-2DES yields X r? ;~ r 0? † ˆ d…~ r? ÿ~ r 0? † ‡ Lm …~ r? †Pm Lmn Sn …~ r 0? †: Eÿ1 …~

(3.67)

(3.68)

m;n

Note that what one has to be careful about is that here all the integral equations, corresponding to, e.g., Eqs. (3.46) and (3.48), are double integrals. One can immediately check that E…~ r? ;~ r 0? † and ÿ1 0 r? ;~ r ? †, given by Eqs. (3.65) and (3.68), respectively, do satisfy the integral equation (3.64). E …~ r? ;~ r 0? † is found to to be the unique inverse of E…~ r? ;~ r 0? †, just as in the previous case (of Moreover, Eÿ1 …~ Q-2DES). It is interesting to note that Eqs. (3.65) and (3.68) can easily be approximated for the situation where quantum con®nement along z direction is much stronger (or when the width along z direction is taken to be zero) than that along the y direction. This is done by writing the envelope r? † ˆ fm …y†‰d…z†Š1=2 and expressing E…~ r? ;~ r 0? † ˆ E…y; y0 †d…z†d…z0 †, e.g., here m will wave functions fa …~ refer to the index of energetically separated 1D subbands and one is always in the lowest subband of the z-motion. Finally, we consider a quasi-zero dimensional electron systems (Q-0DES) where the electrons are quantum-con®ned in all three spatial dimensions and the electron states are discrete. Such structures offer a dispersionless electronic system with an electron-energy spectrum that can be modulated either by varying the gate voltage or by applying an external magnetic ®eld. Since the translational invariance does not exist in any of the three spatial dimensions, there is no chance of thinking of making Fourier transforms. Any way, we write down the single-particle wave function and energy eigenvalue for such a quantum dot symbolically as r†  fa …~ r†; fi …~

Ei  Ea ;

(3.69)

where a  …l; m; n† is a composite index for the collective motion Ð l; m, and n refer to the quantum r† in the x; y, and z directions, numbers which characterize the nodes in the envelope wave functions fa …~ respectively. For the sake of generality, we will be working with the composite index a, just as in the

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preceding case of Q-1DES. Eq. (3.33) with the aid of Eq. (3.69) yields Z  X  00 0 0 3 00 00 00 E…~ r;~ r ; o† ˆ d…~ r Vee …~ r ÿ~ r †ÿ Paa0 …o† d~ r;~ r †fa …~ r †fa0 …~ r † fa …~ r 0 †fa0 …~ r 0 †;

(3.70)

aa0

where Paa0 …o† ˆ

f0 …Ea † ÿ f0 …Ea0 † ; ho‡ Ea ÿ Ea0 ‡ 

Vee …~ r;~ r 0† ˆ

e2 1 : r ÿ~ r 0j E0 j~

(3.71) (3.72)

r;~ r 0 ; o† have to satisfy an integral equation The dielectric function E…~ r;~ r0 ; o† and its inverse Eÿ1 …~ Z r;~ r 00 †E…~ r 00 ;~ r 0 † ˆ d…~ r ÿ~ r 0 †: (3.73) d3~ r 00 Eÿ1 …~ We write Eq. (3.70), suppressing o-dependence, as follows: X E…~ r;~ r 0 † ˆ d…~ r ÿ~ r 0† ÿ Lm …~ r†Pm Sm …~ r 0 †;

(3.74)

m

r† and Sm …~ r† representing the long-range and short-range parts of the response where m  …a; a0 † and Lm …~ functions, respectively, are given by Z 0 r 0 Vee …~ r†  Laa …~ r† ˆ d3~ r;~ r 0 †Zaa0 …~ r 0 †; (3.75) Lm …~ r†  Saa0 …~ r† ˆ Zaa0 …~ r† ˆ fa …~ r†fa0 …~ r†: Sm …~

(3.76)

Following exactly the same methodology as in the previous two cases (of Q-1DES and Q-2DES) leaves us with the inverse of E…~ r;~ r 0 †, Eq. (3.74), given by X r;~ r 0 † ˆ d…~ r ÿ~ r 0† ‡ Lm …~ r†Pm Lmn Sn …~ r 0 †: (3.77) Eÿ1 …~ m;n

0

ÿ1

0

r;~ r †, given by Eqs. (3.74) and (3.77), respectively, satisfy Eq. (3.73) serves to That E…~ r;~ r † and E …~ ÿ1 0 r;~ r † is the correct inverse of E…~ r;~ r 0 †. In addition, one can prove, with the same prove that E …~ r;~ r 0 † is the unique inverse of E…~ r;~ r 0 †. Note convention as adopted in the previous two cases, that Eÿ1 …~ that the general results in Eqs. (3.74) and (3.77) can easily be speci®ed for the geometry where the quantum con®nement along z direction is much stronger than in the x±y plane. This would lead the respective quantities correspond to the quantum dot structure fabricated out of 2DES. A simpler alternative of investigating the collective (intra and intersubband) excitations is searching the poles of the IDF of the respective system. Eqs. (3.55), (3.68) and (3.77) for the quasi-2D, quasi-1D, and quasi-0D systems, respectively, reveal at once that these poles are given by jMj  jI ÿ Pbj ˆ 0;

(3.78)

where I is a unit matrix. Note that M is, in the practical sense, a ®nite N  N matrix, where N refers to the maximum number at which an 1  1 matrix is truncated, i.e., the sums over m and n run from 1 to N. Remember, the condition M ˆ 0 is exactly equivalent to the statement that the self-sustaining

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collective modes of the system are given by the zeros of E…. . .† or by the poles of Eÿ1 …. . .†. An extensive formal diagnosis has led us to ®nd even the simpler (than Eq. (3.78)) condition for studying the collective excitations in these systems [1642]. We remark that the analytical results obtained in Eqs. (3.55), (3.68) and (3.77) for the respective systems are quite general and know no bounds with respect to the subband occupancy. These results will ®nd useful applications in the studies of EELS for quasi-n DES. It so happens that Im Eÿ1 …. . .† is central to the theory of EELS. Since the ®rst term in the above mentioned equations is always real, the second term is a direct measure of the important contribution to the EELS in the respective systems. It is noteworthy that this methodology for ®nding the IDFs does remain valid even when the systems are subject to an external magnetic ®eld [1642]. 4. Plasmons on a semiconductor surface This section and the following ones will be devoted to the brief discussion of the illustrative examples on plasmons and magnetoplasmons in diverse planar structures, until and unless stated otherwise. For this purpose, we need to make a few remarks which will remain valid for all these sections, in particular with respect to the geometry under consideration. All planar geometries will be considered to be made up of the structures with the interfaces between two (or more) different material media lying in the x±y plane(s) and the normal to the interfaces along z-axis (in the Cartesian coordinate system). The direction of (wave) propagation will be taken in the y±z plane, with qx (the wave vector component along x-axis) taken to be zero, with no loss of generality. This implies that y±z plane is the sagittal plane Ð the plane de®ned by the wave vector and the normal to the surface/interface (see Fig. 10). Moreover, all the analytical results cited or referred to in the original papers will be based on the temporal and spatial dependence of the form  exp‰i…~ q ~ r ÿ ot†Š. The magnetic-®eld orientation will be speci®ed whenever it is required for. The EM boundary conditions, with or without an applied magnetic ®eld, comprise of the continuity of the tangential components of the EM ®elds and the normal component of the displacement vector. ~ in the Maxwell We will always be concerned with the non-magnetic materials, which means ~ BH equations. This implies that the boundary conditions can be speci®ed as the continuity of Ex ; Ey ; Bx ; By ,

Fig. 10. Schematics of the surface/interface geometry considered in Section 4. The half-space z < 0 (z > 0) refers to the semiconducting (insulating) medium characterized by frequency dependent (independent) dielectric constant.

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and Dz . However, it turns out that in no case there are ®ve independent boundary conditions. This is because in the present con®guration, e.g., the continuity of Dz  the continuity of Bx . That this is so can easily be proved through a simple mathematical manipulation of the Maxwell equations. We will consider throughout a semiconductor which is n- or p-type and in which the free charge carriers occupy an energy band which is simple parabolic. What is usually desired is seeking appropriate solutions to the wave-®eld equation: ~r ~ ~ E ˆ 0; r E ÿ q20~E  ~

(4.1)

where q0 ˆ o=c is the vacuum wave vector and the components of the dielectric tensor ~E for a conductor in the presence of an external magnetic ®eld are de®ned as " # o2p (4.2) …o2 dij ÿ oci ocj ‡ ioock dijk † ; Eij ˆ EL dij ÿ 2 2 o …o ÿ o2c † where the background dielectric constant EL can further be de®ned as EL ˆ E1

o2 ÿ o2L o2 ÿ o2T

(4.3)

for a polar semiconductor (where oL and oT are the zone-center longitudinal and transverse optical phonon frequencies, respectively); otherwise it is a constant (frequency independent) for a non-polar semiconductor. Using EL , given by Eq. (4.3), allows coupling of the EM waves to the optical phonons. The symbol oc ˆ eB=m c and op ˆ …4pn0 e2 =m EL †1=2 are the cyclotron frequency and the (screened) plasma frequency, respectively. The symbol dij is the Kronecker delta, and dijk is the third rank antisymmetric tensor de®ned by 8 < ‡1 if ijk is an even permutation; dijk ˆ ÿ1 if ijk is an odd permutation; (4.4) : 0 otherwise: In writing Eq. (4.2) we neglect the effects of damping, spatial dispersion, and the interaction of the plasmons and the optical phonons. The last effect is true if the frequency dependence of the background dielectric constant is ignored. Employing a local dielectric tensor ~E…o† puts an upper bound on the magnitude of the propagation vector ~ q, because the local theory is not valid for arbitrary q…ˆ j~ qj†. Therefore, the short wavelength limit q  q0 discussed in some sections later must be understood to be limited to values for which the local theory is still valid. Eq. (4.1) can, after some algebraic manipulation, be cast in the matrix form: 32 3 2 q20 Exy ÿ @ x @ y q20 Exz ÿ @ x @ z q20 Exx ‡ @ 2y ‡ @ 2z Ex 7 6 2 (4.5) q20 Eyy ‡ @ 2z ‡ @ 2x q20 Eyz ÿ @ y @ z 54 Ey 5 ˆ 0; 4 q0 Eyx ÿ @ y @ x 2 2 2 2 2 E z q0 Ezx ÿ @ z @ x q0 Ezy ÿ @ z @ y q0 Exz ÿ @ x ‡ @ y where @ j ˆ @=@ j ˆ iqj , j  x; y; z. It should be pointed out that writing @ j ˆ iqj is all right for j  x; y; but such a substitution can be dangerous and possibly cause a serious error for j  z (where z-axis is the normal to the surface/interface). This is because when one seeks the solution to Eq. (4.1) of the form ~ Eˆ~ E0 exp‰azŠ exp‰i…qx x ‡ qy y ÿ ot†Š;

(4.6)

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where a ˆ iqz and the sign ‡…ÿ† in the ®rst exponential term corresponds to the half-space z < 0…z > 0†, the @ z includes the derivative of the ®rst exponential term, which means @ z could turn into either ‡a or ÿa, as the case may be. It is therefore not advisable to replace @ z by iqz outrightly. In the geometrical con®guration considered here @ x ˆ 0, and we will, for the sake of simplicity, write q  qy . It is noteworthy that cubic symmetry and inherent isotropy of the semiconducting medium considered throughout also leads to further simpli®cation of Eq. (4.5). For example, if the external magnetic ®eld ~ B kz-axis, then Exx ˆ Eyy ; Eyx ˆ ÿExy , and Ezx ˆ Exz ˆ Ezy ˆ Eyz ˆ 0. This means Eq. (4.5) will involve only three independent components of the dielectric tensor in the problem. These are B-dependent Exx , and Exy , and B-independent Ezz . The inclusion of the phenomenological damping factor in the local dielectric function is a simple matter (see, e.g., Ref. [304]). It should be pointed that major part of this section is based on the classic work on surface magnetoplasmons by Wallis and coworkers [268±271]. 4.1. Zero magnetic ®eld In order for providing a basis for comparison, we ®rst consider the dispersion relation for zero magnetic ®eld. In the absence of an applied magnetic ®eld, Eq. (4.5) with Eij ˆ E…0† for i ˆ j (i 6ˆ j), simpli®es considerably and yields the decay constant a for the semiconducting medium (z < 0) de®ned by a2 ˆ ÿq2z ˆ q2 ÿ q20 E; where E  E…o† for the semiconducting medium is de®ned by ! 1 ÿ o2p E…o† ˆ EL o2

(4.7)

(4.8)

by neglecting the damping and spatial dispersion. The omission of spatial dispersion (or non-local effects) amounts to assuming (implicitly) a d-function electron distribution or, hydrodynamically, to the neglect of pressure of the electron gas. This type of omission, or assumption, is often called the ``cold plasma approximation'' because it is classically equivalently for setting the thermal velocity equal to zero. A cold plasma is characterized by the local form of the dielectric function given by Eq. (4.8). It is only a function of frequency and is independent of the propagation vector. This is quite a good approximation provided that the phase velocity of the wave in the system is much larger than the hydrodynamic speed or thermal velocity. Further, we consider an insulating medium (z > 0) characterized by a frequency-independent dielectric constant E0 with decay constant a0 ; the latter is de®ned analogous to a in Eq. (4.7) with E replaced by E0. In the absence of an applied magnetic ®eld, the p-polarized (TM) modes characterized by the non-vanishing ®eld components Ey ; Ez , and Bx , and the s-polarized (TE) modes characterized by non-vanishing ®eld components By ; Bz , and Ex are entirely decoupled. We will be interested in the TM modes where the electric ®eld of the surface plasmon is con®ned to the sagittal plane. Matching the appropriate EM boundary conditions and using the condition of non-trivial solutions leaves us with E0 a ‡ E…o†a0 ˆ 0:

(4.9)

This is the desired dispersion relation for the (p-polarized) surface plasmons. Both decay constants a

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Fig. 11. Dispersion curve for surface plasmon±polaritons propagating at InSb±air interface in the absence of an applied magnetic ®eld. (Redone after Wallis, Ref. [304].)

and a0 must be real and positive for the bona®de surface plasmons. Eq. (4.9) can be simpli®ed to the form   E…o†E0 1=2 : (4.10) q ˆ q0 E…o† ‡ E0 If E…o† < 0 (which is usually the case for the ``surface-wave active'' medium), then E…o† ‡ E0 must be a negative quantity, i.e., E…o† < ÿE0 . A plot of the dispersion relation for InSb±air interface is given, in terms of the dimensionless variables in Fig. 11. The dashed line demarked as a0 ˆ 0 is the light line in the dielectric medium (z > 0). The bona®de surface plasmon±polariton starts from the zero and propagates towards the right of the light line to become asymptotic to the frequency characterized by   E0 ÿ1=2 ; (4.11) E…o† ‡ E0 ˆ 0 ) o ˆ op 1 ‡ EL which is well established through the numerical results of the dispersion relation cited in Eq. (4.9). 4.2. Non-zero magnetic ®eld Now we impose a magnetostatic ®eld ~ B and consider several speci®c con®gurations with respect to the orientation of ~ B relative to the surface/interface and the propagation vector. These are the Voigt

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con®guration, perpendicular con®guration, and Faraday con®guration. An obvious complexity arising because of the applied magnetic ®eld is the inseparable coupling of the TE and TM modes, with the exception in the Voigt con®guration, where the TE and TM modes decouple even in the presence of the magnetic ®eld, just as in its absence. 4.2.1. Voigt con®guration In this con®guration the applied magnetic ®eld ~ B kx-axis and the wave propagates along the y-axis. The result is that Eyy ˆ Ezz ; Ezy ˆ ÿEyz , and Exy ˆ Eyx ˆ Exz ˆ Ezx ˆ 0. As such Eq. (4.5) assumes the following form: 32 3 2 0 0 q20 Exx ÿ q2 ‡ @ 2z Ex 7 6 0 q20 Eyy ‡ @ 2z q20 Eyz ÿ iq@ z 54 Ey 5 ˆ 0: (4.12) 4 2 2 2 E z 0 ÿq0 Eyz ÿ iq@ z q0 Eyy ÿ q It is thus evident that the electric vector con®ned to the sagittal plane represents the TM modes, and these are the only ones that exhibit magnetic ®eld dependence. The non-trivial solution of Eq. (4.12) speci®es the decay constant a2 ˆ ÿq2z ˆ q2 ÿ q20 Ev ;

(4.13)

where the Voigt dielectric function Ev is de®ned as Ev ˆ Eyy ‡

E2yz : Eyy

(4.14)

The dispersion relation for magnetoplasmons in this Voigt geometry is given by aa0 Eyy ‡ E0 k2 ÿ iqa0 Eyz ˆ 0; 2

2

(4.15)

q20 Eyy †.

where k ˆ …q ÿ A simple algebraic manipulation can lead one to prove that Eq. (4.15) is exactly identical to the dispersion relation derived by Wallis (see, e.g., Eq. (38) in Ref. [304]). One can note at a glance that this dispersion relation is non-reciprocal with respect to the direction of propagation, i.e., the positive and negative values of the wave vector q are not equivalent. In the nonretardation limit (NRL) (c ! 1), Eq. (4.15) reduces to E0 ‡ Eyy ÿ iEyz sgn…q† ˆ 0: Substituting Eij from Eq. (4.2) in this equation gives, after some algebra " #1=2 o2p o2c 1 ‡ os ˆ ‡ oc sgn…q†: 2 4 …1 ‡ E0 =EL †

(4.16)

(4.17)

In the limit of small oc =op , this equation reduces to Eq. (40) of Wallis [304]. The computed dispersion relation for InSb±air interface is shown, both for q > 0 and < 0 in Fig. 12 for oc =op ˆ 0:5. Let us ®rst discuss the case q > 0. An interesting feature is immediately apparent Ð the dispersion curve consists of two parts with a gap between them. The lower curve starts from the origin, rises just to the right of the light line (a0 ˆ 0), bends over, and terminates when it intersects the dispersion curve for the bulk magnetoplasmons speci®ed by a ˆ 0. The upper branch starts on the line de®ned by Eyy ˆ 0, rises, and then approaches the asymptotic frequency for the non-retarded magnetoplasmons de®ned by

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Fig. 12. Dispersion curves for surface magnetoplasmon±polaritons (solid curves) at the InSb±air interface in the Voigt geometry, for Oc ˆ 0:5. Dimensionless variables x ˆ o=op and z ˆ cq=op . BMP waves (dashed curves) are demarked as a ˆ 0. The dashed line designated as a0 ˆ 0 is the light line in the vacuum. Upper (lower) horizontal dashed line refers to the asymptotic limit for the propagation vector q > 0 (q < 0). (Redone after Wallis, Ref. [304].)

E0 ‡ Eyy ÿ iEyz ˆ 0. Note that this branch stops a little before it reaches the light line. The reduced wave vector at which the upper branch starts is speci®ed by the equation  2 cq 1 ‡ O2c 2 ˆ ; (4.18) zs  op 1 ÿ …E0 =EL †2 …1 ‡ O2c †=O2c where Oc ˆ oc =op . In order for zs to be ®nite and positive, EL and Oc must satisfy the following inequality: EL …1 ‡ O2c †1=2 oH > ˆ ; E0 Oc oc

(4.19)

where oH ˆ …o2c ‡ o2p †1=2 is the hybrid cyclotron±plasmon frequency. The inequality in Eq. (4.19) speci®es a critical value of the magnetic ®eld below which the upper branch will cease to exist. For InSb (EL ˆ 15:68) and E0 ˆ 1, this is given by Oc ' 0:064. So, to have a surface branch above the line Eyy ˆ 0, the magnetic ®eld should be such that Oc  0:064. That a gap can exist in the dispersion curve is evident from a glance at Fig. 13, where the dimensionless Voigt dielectric function Ev =EL is plotted against the reduced frequency x ˆ o=op. There is a region below the line Eyy ˆ 0 where Ev is large and positive, implying that a2 < 0 for ®nite propagation vector and where no surface magnetoplasmon is allowed. If the non-retarded surface magnetoplasmon lies above the line Eyy ˆ 0 (or o ˆ oH ), as it does

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Fig. 13. Reduced Voigt dielectric function (Ev =EL ) versus reduced frequency x ˆ o=op for Oc ˆ 0:5.

for InSb±air interface with Oc ˆ 0:5, then a surface wave exists both above and below this line, and a gap must persist just below this line. For a value of Er …ˆ EL =E0 ) just below the right-hand side of Eq. (4.19), the upper branch lies below the line Eyy ˆ 0 and to the right of the lower bulk dispersion curve. As Er decreases, the gap rapidly diminishes, and only a single branch remains. Turning now to the case of q < 0 reveals that the situation is qualitatively very different from that where q > 0. We now have a complete lower branch starting from the origin out to the asymptotic value speci®ed by E0 ‡ Eyy ‡ iEyz ˆ 0. In addition, however, we have an upper branch which starts at the light line where Eyy ˆ 1, rises to the right of the light line, and stops where it meets the upper bulk magnetoplasmon curve, a ˆ 0. The upper branch for q < 0 is seen to exist whenever Er > 1 and Oc > 0. It is noteworthy that, for both q > 0 and < 0, there are certain frequencies at which both a bulk and a surface wave can propagate Ð a situation which does not exist in the absence of an applied magnetic ®eld. 4.2.2. Perpendicular con®guration We specify this con®guration by considering the applied magnetic ®eld ~ B kz-axis and to the propagation vector q, itself parallel to the y-axis. This implies that Exx ˆ Eyy ; Eyx ˆ ÿExy , and Exz ˆ Ezx ˆ Eyz ˆ Ezy ˆ 0, due to the symmetry requirements. As a result, Eq. (4.5) may be written for the present geometry, as 2 6 4

q20 Exx ÿ q2 ‡ @ 2z ÿq20 Exy 0

q20 Exy

q20 Exx ‡ @ 2z ÿiq@ z

0

32

Ex

3

76 7 ÿiq@ z 54 Ey 5 ˆ 0: Ez q20 Ezz ÿ q2

(4.20)

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The condition of non-trivial solutions imposed on Eq. (4.20) leaves us with the following relation: a2 ˆ ÿq2z ˆ

1 f‰q2 …Exx ‡ Ezz † ÿ 2q20 Exx Ezz Š  ‰q4 …Exx ÿ Ezz †2 ‡ 4k32 q20 E2xy Ezz Š1=2 g; 2Ezz

(4.21)

where k32 ˆ …q2 ÿ q20 Ezz †. The message Eq. (4.20) conveys is that the TE and TM modes are strongly coupled and can be decoupled under no circumstances, except in the NRL; the coupled modes are usually called the transverse EM modes. In general, both solutions in Eq. (4.21) must be utilized in order to satisfy the boundary conditions. It is worth mentioning that we are interested in the situation where q is real when the absorption is neglected. Then a0 (the decay constant in the insulating medium in the upper half space z > 0) is either real or pure imaginary, just as in the Voigt geometry. In the presence of damping the condition Re a0 > 0 must be imposed and Re a may be imposed with no loss of generality. The magnetoplasma modes with real q and a0 may be further classi®ed according to the nature of a . Depending upon the spectral range in the o±q plane the following possibilities may arise: (i) a‡ and aÿ are both real and positive, (ii) a‡ and aÿ are both pure imaginary, (iii) a‡ is real and aÿ is purely imaginary, or vice versa, and (iv) a‡ and aÿ are the complex conjugates of each other. We have classi®ed the magnetoplasma modes corresponding to the aforementioned possibilities as the surface modes, bulk (or waveguide) modes, hybrid surface±bulk modes (the term pseudo-surface modes is also used), and generalized (or complex) modes. The dispersion relation for the magnetoplasma modes obtained in this perpendicular con®guration is given by [304] k32 E0 2 ‰k ‡ a‡ aÿ ‡ a0 …a‡ ‡ aÿ †Š ‡ a‡ aÿ …a‡ ‡ aÿ † ‡ a0 …a2‡ ‡ a‡ aÿ ‡ a2ÿ † ÿ a0 k12 ˆ 0; a0 Ezz 1

(4.22)

where k12 ˆ …q2 ÿ q20 Exx †. Eqs. (4.21) and (4.22) can be solved for the dispersion curves using a highspeed computer. Typical results for the InSb±air interface with Oc ˆ 0:5, are shown in Fig. 14. Since q is perpendicular to ~ B and the direction of ~ q is immaterial, q appears only as even powers in Eqs. (4.21) and (4.22), and the wave propagation for positive and negative q is reciprocal. The dispersion curve starts on the light line at o ˆ oc , rises just to the right of the light line, ¯attens out, and then approaches an asymptotic value for large q. For o < oc, helicons propagate, but there are no surface waves [304]. The asymptotic frequency of the surface magnetoplasmons is speci®ed by  1=2 Exx ˆ ÿE0 : (4.23) Ezz Ezz Since E0 is taken as a positive quantity, it follows that Ezz must be negative, and …Exx =Ezz †1=2 > 0. Thus the correct asymptotic behavior demands Exx < 0 and Ezz < 0. In view of this, the asymptotic solutions predicted by Eq. (4.23) must lie in the frequency window speci®ed by oc  o  op. For the magnetoplasma model speci®ed by Eq. (4.2), Eq. (4.23) may be solved to obtain  2 o 1 2 ˆ (4.24) f‰…E2L ÿ E20 †O2c ‡ 2E2L Š  ‰…E2L ÿ E20 †2 O4c ‡ 4E20 E2L Š1=2 g: Os  2 2 op 2…EL ÿ E0 † It can easily be proved that Eq. (4.24) for E0 ˆ 1, is exactly the same as Eq. (29) of Wallis [304]. The asymptotic decay constant, from Eq. (4.21), is given by a ˆ q…Exx =Ezz †1=2. One also notes in Fig. 14 that there is another surface branch (the generalized modes) which lies in a frequency range above op . This

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Fig. 14. Dispersion curves for surface magnetoplasmon±polaritons (solid curves) at InSb±air interface in the perpendicular geometry for Oc ˆ 0:5. The BMP waves (dashed curves) are labeled as a ˆ 0. The dashed line demarked as a0 ˆ 0 is the light line in the vacuum. (Redone after Wallis, Ref. [304].)

branch never reaches the light line, and has no asymptotic limit. In fact, it starts at the frequency where Ezz ˆ 1 and terminates when it intersects the bulk magnetoplasmon dispersion curve speci®ed by a ˆ 0. It is noteworthy that this branch does not exist if EL ˆ 1 (metallic plasma), since then Ezz ˆ 1 only at o ! 1. For the present values of parameters Oc ˆ 0:5, pseudo-surface modes exist in the region indicated in Fig. 14. The criterion for the onset of pseudo-surface waves is that one of the decay constants (a ) be zero, i.e., the surface wave dispersion curve intersects the bulk magnetoplasmon dispersion curve. Then from Eqs. (4.21) and (4.22) one can show that the onset frequencies are speci®ed by the following equation:  1=2 1=2 Ev …Exx ‡ Ezz † ÿ 2Exx ‡E0 …Ev ÿ Exx † ˆ 0; (4.25) Exx …Ev ÿ E0 † ‡ …Ev ÿ E0 † Ezz where Ev ˆ Exx ‡ E2xy =Exx is the Voigt dielectric function. A corresponding expression with some apparent typos, was derived by Wallis (see Eq. (31) in Ref. [304]); the computation was, we believe, performed with the correct expression. It was demonstrated that no pseudo-surface waves exist for Oc  0:32. Above the latter value, pseudo-surface waves exist over a range of frequencies whose width increases with increasing oc . Below plasma frequency, the lower branch of the pseudo-surface wave region decreases until it reaches oc . The frequency of the upper boundary increases until it becomes equal to op . The pseudo-surface wave in the region above op speci®es the right-hand side end-point of the upper branch in Fig. 14. This upper branch exists for InSb±air interface only for Oc  0:37 (see, e.g., Fig. 4 in Ref. [304]).

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4.2.3. Faraday con®guration This con®guration is speci®ed by the applied magnetic ®eld oriented parallel to the direction of propagation, which is taken to be along the y-axis. In the present con®guration (~ B ky-axis) the dielectric tensor (~E) is speci®ed by the symmetry requirements such that Exx ˆ Ezz ; Ezx ˆ ÿExz , and Exy ˆ Eyx ˆ Eyz ˆ Ezy ˆ 0. As such, Eq. (4.5) may be written as follows: 2 2 32 3 Ex 0 q20 Exz q0 Exx ÿ q2 ‡ @ 2z 6 7 6 7 2 (4.26) 4 0 q20 Eyy ‡ @ z ÿiq@ z 54 Ey 5 ˆ 0: Ez ÿq20 Exz ÿiq@ z q20 Exx ÿ q2 This equation subjected to the condition of non-trivial solutions leaves us with the following relation: 1 f‰k2 …Exx ‡ Eyy † ÿ q20 E2xz Š  ‰…k12 …Exx ÿ Eyy † ÿ q20 E2xz †2 ÿ 4q2 q20 E2xz Eyy Š1=2 g: (4.27) a2 ˆ ÿq2z ˆ 2Exx 1 Just as in the perpendicular geometry, both solutions must be utilized to satisfy the boundary conditions. The rest of the discussion regarding the coupling of TE and TM modes and the classi®cation of the magnetoplasma modes related with the perpendicular con®guration is still valid. The dispersion relation for the magnetoplasma modes obtained in this con®guration is given by [304] a0 Exx Eyy ‰a0 …a‡ ‡ aÿ † ‡ a2‡ ‡ a2ÿ ‡ a‡ aÿ Š ‡ a‡ aÿ E0 Exx …a0 ‡ a‡ ‡ aÿ † ‡ a0 k12 Eyy …E0 ÿ Eyy † ˆ 0: (4.28) Note that just as in the perpendicular geometry, q appears only as even powers, both in Eqs. (4.27) and (4.28); implying that reciprocity holds. Numerical results for the magnetoplasmons for the InSb±air interface with Oc ˆ 0:5 are illustrated in Fig. 15. The dispersion curve starts at the origin, rises just to the right of the light line, and then ¯attens out to an asymptotic value given by  1=2 Eyy ˆ ÿE0 : (4.29) Exx Exx Since E0 is taken as a positive quantity, it follows that Exx must be negative, and …Eyy =Exx †1=2 > 0. Thus the correct asymptotic behavior is given by: Exx < 0 and Eyy < 0. Therefore the asymptotic solutions predicted by Eq. (4.29) must lie in the frequency window speci®ed by oc  o  op, just as in the perpendicular geometry. The non-retarded magnetoplasmon frequency obtained by solving Eq. (4.29) is the same as that speci®ed by Eq. (4.24). The asymptotic decay constant, from Eq. (4.27), is given by a ˆ q…Eyy =Exx †1=2 . One should note in Fig. 15 the manner in which the surface magnetoplasmon dispersion curve is repelled by the bulk magnetoplasmon dispersion curve a ˆ 0. No pseudo-surface or generalized modes exist in the present case. Also, no upper branch analogous to that in Fig. 14 exists. However, as Oc is increased to 0.9, one gets a pseudo-surface wave region, as was shown by Wallis [304], but no upper branch. An equation specifying the end points of the pseudo-surface region can be obtained in the same fashion as that employed for the perpendicular geometry. The result is ‰Exx …Exx  iExz ÿ E0 †Š1=2 ‰iExz …Exx ‡ Eyy † ÿ E2xz Š1=2 ‡ iExz …iExz  Exx  E0 † ˆ 0:

(4.30)

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Fig. 15. Dispersion curves for surface magnetoplasmon±polaritons (solid curves) at the InSb±air interface in the Faraday geometry for Oc ˆ 0:5. The rest is the same as in Fig. 14. (Redone after Wallis, Ref. [304].)

Replacing iExz by Exz and substituting E0 ˆ 1 reproduces Eq. (48) of Wallis [304]. It was noted by Wallis that, for Oc ˆ 0:9, there is a region of reduced wave vector, from z ˆ 1:8 to 4.4, where the decay constants are complex conjugates of each other, and the generalized surface wave prevails. The end points of a region of generalized surface modes are characterized by a‡ ˆ aÿ. This implies that the discriminant of Eq. (4.27) must be zero, one obtains aq4 ÿ bq2 ‡ c ˆ 0;

(4.31)

where a ˆ …Exx ÿ Eyy †2 ;

b ˆ 2q20 ‰Exx …Exx ÿ Eyy †2 ÿ E2xz …Exx ‡ Eyy †Š;

c ˆ q40 ‰Exx …Exx ÿ Eyy † ‡ E2xz Š2 ; (4.32)

instead of Eq. (49) of Wallis [304], which is erroneous. Eq. (4.31) and the corresponding decay constant from Eq. (4.27) determine the end points of the generalized surface wave region. It should be remarked that the so-called ``degenerate'' surface waves reported by Rao and Uberoi [288] are characterized by a‡ ˆ aÿ . These waves, however, do not satisfy the proper boundary conditions and are spurious [194]. For rigorous numerical details on the surface magnetoplasma modes in isotropic as well as anisotropic systems, including the effects of damping and coupling to the optical phonons, the reader is referred to an excellent review paper by Wallis [304].

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It is worth mentioning that many of the analytical results discussed in this section would prove to be extremely useful in reviewing and understanding similar results on plasmons and magnetoplasmons in relatively more sophisticated (periodic as well as non-periodic) composite planar systems to be discussed in the following sections. 5. Plasmons in thin semiconducting ®lms This section is devoted to discuss some exact results on the plasmons and magnetoplasmons in the two-interface (thin-®lm) geometries. We will consider the situation both with and without an external magnetic ®eld in different con®gurations. In the absence of an applied magnetic ®eld, we discuss the illustrative examples of plasmon±polaritons that propagate in the supported (when the bounding media are unidentical) and unsupported (when the bounding media are identical) thin ®lms. In the presence of an applied magnetic ®eld Ð considering the three principal con®gurations: the Voigt, the perpendicular, and the Faraday geometries Ð we discuss, for the sake of simplicity, only the results on magnetoplasma modes in the thin ®lms identically bounded by the dielectric media. Most of the interesting approximate situations, in all these con®gurations, have been considered by Kushwaha and Halevi [346±350]. We would therefore con®ne our attention to the exact results including the effect of retardation, but neglecting the absorption. For the sake of consistency, we would retain the ``coordinate geometry'' just as described in Section 4. 5.1. Zero magnetic ®eld Consider, for the sake of generality, a conducting ®lm unidentically bounded by two dielectric media (see Fig. 16). The two bounding media, I and III, are characterized by the frequency independent dielectric constants E1 and E3 . The conducting medium (II) of ®nite thickness (0  z  d) is characterized by the local, dynamic dielectric function E2 …o†. We will consider only the p-polarization

Fig. 16. The schematics of the ®lm geometry considered in Section 5. Regions ÿ1  z  0, 0  z  d, and d  z  ‡1 B†], and dielectric (E3 ) media, respectively. The y-axis is considered to be the refer to the dielectric (E1 ), semiconductor [~E…o; ~ direction of propagation (q  qy ). In general, the ®lm is unidentically bounded by two different dielectrics Ð asymmetric con®guration (also termed as supported ®lm); when E1 ˆ E3 , the geometry is called a symmetric con®guration (also termed as unsupported ®lm).

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(TM) modes. The Maxwell equations with appropriate boundary conditions now give the dispersion relation       E1 E2 E2 E3 E1 E2 E2 E3 ‡ ‡ ÿ ÿ ‡ exp…ÿ2a2 d† ˆ 0; (5.1) a1 a2 a2 a3 a1 a2 a2 a3 where the decay constants ai (i  1; 2; 3) are de®ned by ai ˆ …q2 ÿ q20 Ei †1=2

(5.2)

and the local, dynamic dielectric function for the semiconducting ®lm is given just as in Eq. (4.8). In general, Eq. (5.1) is a transcendental function and is not so easy to solve. Our approach has been to develop suitable approximations. We obtain reliable results for the real frequencies and real wave vectors, both for the symmetric and asymmetric con®gurations. Such illustrative examples are the issues to be discussed brie¯y in what follows. 5.1.1. Supported ®lms The supported ®lm structure corresponds to the situation where E1 6ˆ E3 . We use the following parameters: E1 ˆ 2:25; EL ˆ 15:70, and E3 ˆ 1:0. These material parameters specify our layered structure (Fig. 16) as made up of glass substrate (region I), an InSb ®lm (region II), and air (region III). The numerical results in terms of dimensionless variables are shown in Fig. 17. There are two modes

Fig. 17. The surface plasmon±polaritons (solid curves) in the thin InSb ®lm in the asymmetric con®guration in zero magnetic ®eld. The material parameters are as listed in the ®gure. The two dashed lines demarked as a1 ˆ 0 and a3 ˆ 0 refer to the light lines in medium ®rst (ÿ1  z  0) and third (d  z  ‡1). The horizontal dashed lines stand for the asymptotic limits of the two decoupled modes in the NRL (c ! 0). Note that the part of the upper polariton mode lying between the two light lines is a radiative mode because a1 is purely imaginary in this region of o±q plane; the rest part of this mode and the whole of the lower mode are the true, surface modes characterized by both a1 and a3 being real and positive. As regards a2 (for the ®lm), it is always real and positive for x  1, since E…o† < 0 there.

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which both start from the origin, rise just to the right of the respective light lines, and ¯atten out at the asymptotic frequencies speci®ed by E2 …o† ˆ ÿEi ; i  1; 3. While the lower branch stands for the pure surface plasmon±polariton, the upper branch has different meaning before and after crossing the light line in medium I. The latter branch represents the pure surface mode only towards the right of the light line in medium I, because both a1 and a3 are real and positive there. The part of this curve between the two light lines is, in fact, characterized by real and positive a3 , but positive imaginary a1. This portion of the upper mode is usually referred to as the radiative mode (in medium I); it is only localized and decays exponentially in medium III, but radiated at the corresponding frequencies in medium I. 5.1.2. Unsupported ®lms Unsupported ®lm structure realized in the present work corresponds to the material parameters: E1 ˆ 1:0 ˆ E3 , and EL ˆ 15:7 (InSb ®lm). The numerical results for the unsupported InSb ®lm in terms of dimensionless variables are depicted in Fig. 18. As one notes, both surface modes start from the origin, rise just to the right of the light line, and become asymptotic to the frequency speci®ed by E2 …o† ˆ ÿ1. In this case, both of these modes are characterized by real and positive a1 ˆ a3 , and hence are pure surface polaritons characterized by the EM ®elds localized at and decaying exponentially away from both interfaces. Such non-vanishing EM ®eld components for TM modes under consideration are Ey , Ez , and Bx . That is to say that the electric vector lies in the sagittal (y±z) plane Ð the plane de®ned by the normal to the interfaces and the direction of propagation.

Fig. 18. The same as in Fig. 17, but for the symmetric con®guration with E1 ˆ E3 ˆ 1. Note that, unlike in the asymmetric con®guration, both modes represent the bona®de surface plasmon±polaritons in the whole o±q plane towards the right of the light line.

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5.2. Non-zero magnetic ®eld The effect of a magnetostatic ®eld (~ B) is known to cause various qualitative changes in the behavior characteristics of the EM modes [1643]. Such effects in the thin-®lm geometries, both supported and unsupported structures, were extensively investigated by Kushwaha and Halevi (KH) [346±350] in all the three principal con®gurations. They derived exact dispersion relations for magnetoplasmons in the Voigt [347], perpendicular [348], and Faraday [346] geometries. These exact results are independent of any particular model, and leave options to include, e.g., the effects of damping, coupling to the optical phonons, and spatial dispersion through hydrodynamical model. Instead of feeding the expressions for the exact results into the machine, KH embarked on investigating numerous approximate situations. These include, e.g., the non-retarded limits, thin-®lm approximation, surface phonon±polaritons modi®ed by magnetized overlayer, magnetized ®lms symmetrically bounded by dielectric media, splitting of the magnetoplasmon modes due to the transition layer, etc. Such approximate analytical and numerical results provided realizable physical situations of magnetoplasmon propagation in thin-®lm geometries. However, the exact numerical results remained an unaccomplished object of that project. Here we will present some exact results on the magnetoplasmons in the three con®gurations with respect to the orientation of the external magnetic ®eld. We will consider all the three (Voigt, perpendicular, and Faraday) geometries just as speci®ed in Section 4. 5.2.1. Voigt con®guration The exact analytical results for the magnetoplasmon dispersion in the supported semiconducting ®lms in the Voigt geometry (~ Bjjx-axis) reads [347]   Eyz 2 E1 E3 ‡ iq …a3 E1 ÿ a1 E3 † tanh…bd† ‡ b…a3 E1 ‡ a1 E3 † ˆ 0; (5.3) a1 a3 Ev ‡ k Eyy Eyy where k2 ˆ …q2 ÿ q20 Eyy †. The decay constants ai (i  1; 3) in the bounding media are de®ned just as in Eq. (5.2). The decay constant in the ®lm medium (region II) b is de®ned by, from Eq. (4.13) b2 ˆ ÿq2z ˆ …q2 ÿ q20 Ev †;

(5.4)

where Ev is the Voigt dielectric function speci®ed by Eq. (4.14). Eq. (5.3) is the dispersion relation for the (TM) magnetoplasma waves in the Voigt con®guration. For B ˆ 0, Eq. (5.3) can be shown with a little algebra, to reproduce the plasmon dispersion relation given by Eq. (5.1). One notes immediately from Eq. (5.3) that the dispersion relation for supported ®lm is non-reciprocal, i.e., the positive and negative values of the propagation vector q are not equivalent. However, for the symmetric con®guration (unsupported ®lm with E1 ˆ E3 ) the reciprocity holds. We are interested in the propagating solutions of Eq. (4.3), i.e., q must be real in the absence of damping. Then a1 and a3 are either real or pure imaginary. The latter case is of certain interest in the waveguide theory (``substrate modes'' and ``air modes'') [1644]. Here, we will limit our attention to the ®eld solutions which decay exponentially away from both interfaces of the ®lm. Such solutions are characterized by both a1 and a3 being real and positive. The magnetoplasma modes with real q, a1 , and a3 may be further classi®ed according to the nature of b, given by Eq. (5.4). Depending upon the spectral range in the o±q plane following possibilities may arise: (i) b is real and positive (we may always choose the positive root of b). This corresponds to the surface modes decaying exponentially away from both interfaces, inside as well as outside the ®lm, and (ii) b is pure imaginary (one may

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Fig. 19. Magnetoplasmon dispersion in thin unsupported (E1 ˆ E3 ) InSb ®lm in the Voigt geometry (~ B k ^x;~ q k ^y). The material parameters are: E1 ˆ 1 ˆ E3 , EL ˆ 15:7, oc =op ˆ 0:5, and (normalized ®lm-thickness) op d=c ˆ 0:1. Note that a1 (ˆ a3 ) is always real and positive towards (here invisible) light line in the bounding media; however, b (in the ®lm) can be real or pure imaginary. The solid curves in blue refer to the pure surface modes with both a1 and b being real and positive, while those in pink stand for the waveguide modes with a1 real and b pure imaginary. The latter kinds of modes are characterized by the oscillatory ®eld dependence inside the ®lm. The two dashed curves demarked as b ˆ 0 refer to the BMP modes. The two face-to-face arrows indicate the existence of an excitation gap of width Do ' o2p =‰2…EL ÿ 1†oH ] (see text). Fig. 20. Magnetoplasmon dispersion in a thin, supported (E1 6ˆ E3 ) InSb ®lm in the Voigt geometry (~ B k ^x; ~ q k ^y). The material parameters are: E1 ˆ 11:7, EL ˆ 15:7, E3 ˆ 1:0, oc =op ˆ 0:5, and op d=c ˆ 0:1. This corresponds to an InSb ®lm on Si substrate. The left (right) panel refers to q < 0 (q > 0). The dashed lines (a3 ˆ 0) starting from zero refer to the light lines in vacuum. The dashed curves labeled b ˆ 0 have been referred to (in the text) as the bulk boundaries separating the bulk and surface wave regions. The straight (horizontal) lines demarked as S stand for the asymptotic frequencies speci®ed by Eqs. (5.6) and (5.7). The blue (pink) curves stand for the pure surface (waveguide) magnetopolaritons.

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choose Im b > 0). These are waveguide (or bulk) modes with an oscillatory ®eld dependence inside the ®lm. This classi®cation of the modes into surface and bulk modes is valid if and only if the dissipation is neglected (i.e., the carrier collision frequency n ˆ 0). With ®nite absorption (n 6ˆ 0) q, a1 , and a3 are, of course, all complex quantities. We will con®ne here only to the case of n ˆ 0, where Eyy and Exx are real and Eyz is pure imaginary. We ®rst discuss the numerical results for the magnetoplasma waves in the symmetric con®guration (E1 ˆ E3 ) where the reciprocity holds. The material parameters used in the computation are: E1 ˆ E3 ˆ 1:0, EL ˆ 15:7, and oc =op ˆ 0:5, and the normalized ®lm-thickness op d=c ˆ 0:1. These parameters correspond to the unsupported InSb ®lm. The numerical results in terms of dimensionless variables are illustrated in Fig. 19. There are three sharp boundaries that are seen to divide the bulk and surface wave regions. In the lowest region, below the lower bulk solutions b ˆ 0, there are two surface modes (shown in blue) both of which start from the origin and rise just to the right of the (invisible) light line. The lower surface mode is seen to become asymptotic to the frequency osÿ given by " #1=2 o2p EL o2c 1 os ˆ  oc ; (5.5) ‡ 2 4 EL ‡ 1 whereas the upper one terminates when it intersects the bulk boundary speci®ed by b ˆ 0 (the lower dashed curve). In the second lowest region, between the lower bulk boundary and the hybrid cyclotron± plasmon frequency oH ˆ …o2c ‡ o2p †1=2, the bulk solutions (shown in pink) are found to exist. Because of the singularity in the Voigt dielectric function Ev near oH, there is a pile-up of these bulk solutions just below oH. In the third lowest region, between oH and the upper bulk boundary speci®ed b ˆ 0 (upper dashed curve), two new surface branches appear; the lower beginning at o ˆ oH and the upper one at o ˆ ‰o2H ‡ o2p =…EL ÿ 1†Š1=2 . The lower one of these becomes asymptotic to a frequency os‡, de®ned by Eq. (5.5), while the upper one terminates where it intersects the upper bulk boundary b ˆ 0. In the fourth lowest region, above the upper bulk boundary, there is a bulk solution rising just to the right of the (invisible) light line. The numerical results in Fig. 19 bear a coherent correspondence with those in Fig. 2 of De Wames and Hall [341]; this is true in spite of the fact that these authors used a set of parameters very different from ours. The absence of a number of bulk wave solutions, shown in Fig. 2 of Ref. [341], seems to be related to our choice of very thin ®lm [347]. We may, however, conclude that an unsupported ®lm in the Voigt con®guration supports at least four bona®de surface magnetoplasmons. The second and the fourth surface modes were shown to change their character to bulk modes in the thick ®lm employed by De Wames and Hall [341]. Next, we discuss an illustrative example of magnetoplasmons in an asymmetric (E1 ˆ E3 ) con®guration. The numerical results with the material parameters E1 ˆ 11:70, E3 ˆ 1:0, EL ˆ 15:70, oc =op ˆ 0:5, and op d=c ˆ 0:1, are depicted in Fig. 20. The left (right) panel corresponds to the ÿq and ‡q directions. In the right panel, the surface mode (shown in blue) starts from the origin, rises Fig. 21. Magnetoplasmon dispersion in an unsupported InSb ®lm in the perpendicular geometry (~ B k ^z; ~ q k ^y). The material parameters are the same as in Fig. 19. All the solutions are restricted towards the right of the light line in vacuum. The solid curves in blue, green, red, and black refer to the surface, pseudo-surface, waveguide, and generalized surface modes, respectively (see text). The dashed curves in black have been referred to as the bulk boundaries making distinction among different classes of magnetoplasmon waves. Fig. 22. The same as in Fig. 21 but for the magnetic-®eld orientation in the Faraday geometry (~ qk~ B k ^y-axis).

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towards the right of the light line, and ¯attens out to the asymptotic frequency designated as Sÿ . Above the lower bulk boundary b ˆ 0 (dashed curve), there is a pile-up of the bulk solutions (shown in pink) just below oH, just as in the symmetric case (Fig. 19). Above oH , there is a new surface mode (in blue) that starts from the light line, propagates to the right of the light line, and is seen to become asymptotic to the frequency designated as S‡ . Turning to the left panel, we ®nd a surface mode, analogous to the one in the right panel, which becomes asymptotic the frequency designated as S‡ . Above the lower bulk boundary b ˆ 0 and below oH, there is a bulk wave solution that propagates with negative group velocity until a1 ˆ 0 in the o±q plane (i.e., the light line in the substrate), and then changes sign of the group velocity from negative to positive. This mode changes its character from bulk to surface (blue portion) until it meets the light line in medium III. In other words, the second bona®de surface mode starts from the light line (in medium III) with negative group velocity, and crosses the line o ˆ oH , where it changes the character from surface to bulk. In the close vicinity of o  oH , there is again a pile-up of the bulk solutions, just as in the right panel. However, we do not ®nd a second surface mode, analogous to the one above oH in the right panel, that propagates with positive group velocity. This leads us to infer that the non-reciprocity can lead to a drastic distinction not only in the nature but also in the number of certain modes in the positive and negative directions of propagation. The asymptotic limits demarked as S‡ and Sÿ are speci®ed by the following expressions: " #1=2 o2p EL o2c 1 ‡ sgn…q†oc ; (5.6) ‡ os‡ ˆ 2 4 EL ‡ E3 " #1=2 o2p EL o2c 1 osÿ ˆ ÿ sgn…q†oc : (5.7) ‡ 2 4 EL ‡ E1 This means that there are two asymptotic solutions (corresponding to q ! 1) for each of the two interfaces in the NRL. It is noted that the two asymptotic modes corresponding to a given interface are separated by joc j. Eqs. (5.6) and (5.7), for E1 ˆ E3 ˆ 1, reduces to Eq. (5.5) for the symmetric con®guration. As regards the sign-change of the group velocity (Vg ) of certain modes (see above), it was suggested by KH [347] that there exists a critical value of the magnetic ®eld (o c ) at which Vg of a certain mode in a given direction of propagation changes sign from negative to positive. What is most interesting in the Voigt geometry is that the dynamical Hall effect plays a signi®cant role even in the NRL Ð unlike in the perpendicular and Faraday geometries, which are otherwise known to be considerably sophisticated due, in particular, to the allowance of strong coupling of TE and TM modes. 5.2.2. Perpendicular con®guration We consider an asymmetric con®guration where a semiconducting ®lm is bounded by two dissimilar dielectric media characterized by dielectric constants E1 and E3 . The non-trivial solutions of Eq. (4.20) are speci®ed by b2 ˆ ÿq2z ˆ

1 f‰q2 …Exx ‡ Ezz † ÿ 2q20 Exx Ezz Š  ‰q4 …Exx ÿ Ezz †2 ‡ 4l2 q20 E2xy Ezz Š1=2 g; 2Ezz

(5.8)

where l2 ˆ …q2 ÿ q20 Ezz †. Making use of the Maxwell equations and the appropriate EM boundary conditions, we obtain the dispersion relation for the magnetoplasma waves in the supported (E1 6ˆ E3 )

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semiconducting ®lms given, after rigorous mathematics, by [348] A2‡ ‰…b2ÿ ‡ a1 a3 †Tÿ ‡ bÿ …a1 ‡ a3 †Š‰…b2‡ a1 a3 E2zz ‡ l4 E1 E3 †T‡ ‡ b‡ …a1 E3 ‡ a3 E1 †l2 Ezz Š ‡ A2ÿ ‰…b2‡ ‡ a1 a3 †T‡ ‡ b‡ …a1 ‡ a3 †Š‰…b2ÿ a1 a3 E2zz ‡ l4 E1 E3 †Tÿ ‡ bÿ …a1 E3 ‡ a3 E1 †l2 Ezz Š ÿ A‡ Aÿ f‰…b2ÿ a1 Ezz ‡ l2 a3 E1 †Tÿ ‡ bÿ …a1 a3 Ezz ‡ l2 E1 †Š  ‰…b2‡ a3 Ezz ‡ l2 a1 E3 †T‡ ‡ b‡ …a3 a1 Ezz ‡ l2 E3 †Š ‡ ‰…b2ÿ a3 Ezz ‡ l2 a1 E3 †Tÿ ‡ bÿ …a3 a1 Ezz ‡ l2 E3 †Š‰…b2‡ a1 Ezz ‡ l2 a3 E1 †T‡ ‡ b‡ …a1 a3 Ezz ‡ l2 E1 †Šg ‡ 2A‡ Aÿ b‡ bÿ …l2 E1 ÿ a21 Ezz †…l2 E3 ÿ a23 Ezz †…1 ÿ T‡2 †1=2 …1 ÿ Tÿ2 †1=2 ˆ 0;

(5.9)

where A ˆ k2 ÿ b2 ;

k2 ˆ q2 ÿ q20 Exx ;

T ˆ tanh…b d†;

(5.10)

where d is the thickness of the ®lm. It is worth mentioning that we are interested in the situation where q is real when absorption is neglected. Then the decay constants in the bounding media, a1 and a3 , are either real or pure imaginary. Here we con®ne our attention to the solutions for which the EM ®elds decay exponentially away from both interfaces. Such solutions are characterized by both a1 and a3 being real and positive. In the presence of damping the conditions Re ai > 0 (i  1, 3) must be imposed and Re b > 0 may be imposed with no loss of generality. The magnetoplasma modes with real q, a1 , and a3 may be further classi®ed according to the nature of b ; depending upon the spectral region in the o±q plane (see Section 4.2.2). In general, the three components of the electric ®eld in the three media are all non-vanishing. Hence the electric ®eld is elliptically polarized in a tilted plane (neither parallel nor perpendicular to the interfaces). Similarly, the magnetic ®eld of the wave is polarized in another tilted plane. The dispersion relation, Eq. (5.9), is even in the propagation constant q, i.e., o…ÿq† ˆ o…‡q† for a given solution. For such reciprocal propagation, we may limit the discussion, with no loss of generality, to the case q > 0. In fact, we present here the exact numerical results for the symmetric con®guration (E1 ˆ E3 ) for real o and real q. For many approximate analytical and numerical results derived from Eq. (5.9), we refer the reader to the original paper [348]. The exact numerical results for the magnetoplasma waves in an unsupported InSb ®lm are illustrated in Fig. 21. All the solutions are restricted towards the right of the light line, i.e., where a ˆ a1 ˆ a3 is real and positive. Furthermore, we have searched the solutions with the following four sets of conditions imposed one by one. The curves in blue correspond to the solutions with b being real and positive, implying that these are the true surface magnetoplasmon±polaritons. The curves in red correspond to the solutions with b being pure imaginary, which means the bulk waves for which the EM ®elds are oscillatory inside the ®lm. The curves in green correspond to the solutions where b‡ is real positive and bÿ is pure imaginary, or vice versa; these are the so-called pseudo-surface waves. The solid curve in black corresponds to the solutions with b‡ and bÿ being complex conjugates of each other, implying that this is the generalized surface mode. The two dashed curves (in black) demarked as b ˆ 0 represent the bulk magnetoplasmons, and make a clear distinction among different magnetoplasma-wave regions. For instance, between o ˆ oc and the lower bulk boundary (b ˆ 0), only true surface waves survive until considerably large propagation vector q; for o  oc, there is no surface wave. In between the lower bulk boundary (b ˆ 0) and the plasma frequency op, only the bulk and the pseudo-surface waves prevail. Just above o ˆ op , there is a generalized surface mode. A new

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true surface mode starts at the intersection of the light line (in the bounding media) and the upper bulk boundary (b ˆ 0); this mode propagates just below the upper bulk boundary up to a considerably large value of q. Above the upper bulk boundary (b ˆ 0), there is no true surface mode. One should note the manner in which certain true surface modes (in blue) and pseudo-surface modes (in green) change their character to become pseudo-surface modes and bulk modes (in red), respectively. The asymptotic limits attained by the true surface modes are well speci®ed by, e.g., Eq. (4.24) in the NRL. We conclude that there are at least four true surface magnetoplasmon waves that may propagate in an unsupported semiconducting ®lm in the presence of an applied magnetic ®eld in the perpendicular con®guration. It should, however, be remarked that we still believe the nature, the number, and the propagation range of all the four kinds of magnetoplasma waves to be subject to the material parameters, the intensity of the applied magnetic ®eld, and the ®lm-thickness. 5.2.3. Faraday con®guration We consider a semiconducting ®lm (medium II) of ®nite thickness characterized by the dielectric tensor ~E…o; ~ B† which is assumed to be independent of the wave vector (local effects). Two media, I and III, characterized by the dielectric constants E1 and E3 , respectively, bound the ®lm. The magnetostatic ®eld ~ B is assumed to be parallel to the propagation vector taken to be parallel to the y-axis, the Faraday con®guration. Then the non-trivial solutions of Eq. (4.26) are speci®ed by b2 ˆ ÿq2z ˆ

1 f‰k2 …Exx ‡ Eyy † ÿ q20 E2xz Š  ‰…k2 …Exx ÿ Eyy † ÿ q20 E2xz †2 ÿ 4q2 q20 E2xz Eyy Š1=2 g; 2Exx

(5.11)

where k2 ˆ …q2 ÿ q20 Exx †. Making use of the Maxwell equations and the appropriate EM boundary conditions, we obtain the dispersion relation for the magnetoplasma waves in the supported semiconducting ®lms given, after laborious mathematics, by [346] A2‡ ‰…a1 a3 ‡ b2ÿ †Tÿ ‡ …a1 ‡ a3 †bÿ Š‰…a1 a3 E2yy ‡ E1 E3 b2‡ †T‡ ‡ Eyy …a3 E1 ‡ a1 E3 †b‡ Š ‡ A2ÿ ‰…a1 a3 ‡ b2‡ †T‡ ‡ …a1 ‡ a3 †b‡ Š‰…a1 a3 E2yy ‡ E1 E3 b2ÿ †Tÿ ‡ Eyy …a3 E1 ‡ a1 E3 †bÿ Š ÿ A‡ Aÿ f‰…a1 a3 Eyy ‡ E1 b2ÿ †Tÿ ‡ …a1 Eyy ‡ a3 E1 †bÿ Š‰…a3 a1 Eyy ‡ E3 b2‡ †T‡ ‡ …a3 Eyy ‡ a1 E3 †b‡ Š ‡ ‰…a3 a1 Eyy ‡ E3 b2ÿ †Tÿ ‡ …a3 Eyy ‡ a1 E3 †bÿ Š‰…a1 a3 Eyy ‡ E1 b2‡ †T‡ ‡ …a1 Eyy ‡ a3 E1 †b‡ Šg ‡ 2A‡ Aÿ b‡ bÿ a1 a3 …Eyy ÿ E1 †…Eyy ÿ E3 †…1 ÿ T‡2 †1=2 …1 ÿ Tÿ2 †1=2 ˆ 0;

(5.12)

where A ˆ

k2 ÿ b2 ; b2 ‡ q20 Eyy

k2 ˆ q2 ÿ q20 Exx ;

T ˆ tanh…b d†:

(5.13)

A careful inspection reveals a close analogy between Eqs. (5.9) and (5.12). This leads us to infer that one of these two dispersion relations should be obtainable from the other. It is, in fact, found that rede®ning A in Eq. (5.13) such that it is given by Eq. (5.10) and replacing ai Eyy =Ei in Eq. (5.12) by l2 Ei =ai Ezz yields Eq. (5.9). In the Faraday geometry, the electric-®eld vector for the general value of the propagation vector ~ q, traces out an ellipse which contains the normal to the interfaces and is inclined to the direction of propagation. It was demonstrated [346] that the exact dispersion relation, Eq. (5.12), in the NRL is devoid of the off-diagonal element of the dielectric tensor Exz, just as in the perpendicular geometry [348]. This should mean that, in the NRL, the transverse (Hall) ®eld is negligible. Thus there

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is no dynamical Hall effect and we do not expect helicon modes for q  q0. This, however, does not preclude long-range propagation under suitable conditions. The rest of the remarks made in the paragraph following Eq. (5.10), regarding classi®cation of the modes, absorption, and reciprocal propagation, are still valid. Just as in the perpendicular con®guration, we present here the exact numerical results for symmetric con®guration (E1 ˆ E3 ) for real o and real q. For extensive approximate analytical and numerical results derived from Eq. (5.12), we refer the reader to the original paper [346]. The exact numerical results for the magnetoplasma waves in an unsupported InSb ®lm are depicted in Fig. 22. All the solutions are restricted towards the right of the light line in the bounding media, i.e., a ˆ a1 ˆ a3 is real and positive. Just as in the perpendicular con®guration, we have searched the solutions with the following four sets of conditions one by one. The curves in blue correspond to the solutions with b being real and positive, implying that these are the true surface magnetoplasma modes. The curves in red correspond to the solutions with b being pure imaginary, which means the bulk waves for which EM ®elds are oscillatory inside the ®lm. The curves in green correspond to the solutions where b‡ is real positive and bÿ is pure imaginary, or vice versa; these are the so-called pseudo-surface waves. The (solid) curves in black correspond the solutions with b‡ and bÿ being complex conjugates of each other, implying that these are the generalized surface modes. The two dashed curves in black demarked as b ˆ 0 represent the bulk magnetoplasmon±polaritons, and make a clear distinction among the different magnetoplasma wave regions. The important difference between the perpendicular and Faraday con®gurations is that the bona®de (generalized) surface modes start from the origin in the latter case, instead from o ˆ oc in the former one. Two generalized surface modes propagating in the close vicinity of the light line and indiscernible from each other change their character from generalized to true surface modes at the reduced wave vector z ' 0:38. The upper one of the latter modes is a doubly degenerate mode and changes its character (at z ' 0:81) to pseudo-surface mode at the intersection with the lower bulk boundary (b ˆ 0). The lower true surface mode is seen to be repelled by the bulk magnetoplasmon dispersion curve and changes its character (at z ' 1:66) to bulk (or waveguide) modes, which again becomes a true surface mode (after at z ' 3:97). Above the lower bulk boundary (b ˆ 0), we do not ®nd any other true or generalized surface modes. In the region above the lower bulk boundary, only pseudo-surface and waveguide modes survive; the uppermost (apparently degenerate) pseudo-surface wave changes its character, at the intersection with the upper bulk boundary (b ˆ 0), to waveguide modes, just as the lower two pseudo-surface modes change their character to the bulk modes and retain their character at large wave vector. The asymptotic limits attained by the magnetoplasma modes (in the vicinity of op and oH ) in the NRL are well speci®ed by Eq. (4.24). We conclude that there are at least two clearly discernible true surface modes, which both start from the light line (in the air). Again, we remark that the nature, number, and the propagation range of all the four kinds of modes should be understood to be subject to the material parameters, the intensity of the magnetic ®eld, and the ®lm thickness. Finally, it should be pointed out that the magnitude of the gap above the hybrid cyclotron±plasmon frequency oH, in all the three symmetric con®gurations (Figs. 19, 21 and 22), is roughly speci®ed by [346±348] ÿ  op D o=op ˆ : (5.14) 2…EL ÿ 1†oH The existence of this gap in the magnetoplasmon spectrum is attributed to the presence of the thin ®lm. In the absence of an applied magnetic ®eld oH ˆ op , and the gap becomes D…o=op † ˆ ‰2…EL ÿ 1†Šÿ1 .

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We add that the expression (5.14) was derived within the thin-®lm approximation (q0 d  1) for the surface phonon±polaritons modi®ed by the magnetized overlayer [346±348]. 5.2.4. Resonance splittings Here we consider the thin-®lm con®guration Fig. 16 such that the media I±III are assumed to be B†, and E3 …ˆ 1†, respectively. The substrate (region I) is assumed to be characterized by E1 …o†, ~E…o; ~ undoped and its dispersion is entirely due to phonons: E1 …o† ˆ E1 …o2 ÿ o2L †=…o2 ÿ o2T †, where E1 is the high-frequency dielectric constant. The thin ®lm is strongly doped and in it we neglect the phonons. This approximation is justi®ed provided that o2  o2Lf , where oLf is the longitudinal optical phonon frequency of semiconducting ®lm (which, however, should not necessarily be a polar semiconductor). With this understanding, we have computed the splitting de®ned by D…o† ˆ o‡ …q‡ † ÿ oÿ …q‡ †:

(5.15)

This is just the vertical distance between the two branches evaluated at the initial point (o‡ ˆ cq‡ ) of the upper branch. The splitting is, in general, just the repulsion in the two branches at the resonance frequencies, and is a consequence of the resonance between the thin-®lm magnetoplasmons and the substrate phonons. It is worth mentioning that, in the course of the computation of such splittings, we have discarded the solutions with q < q0 , the reason being that these would give an imaginary decay constant a3 , which would thereby result in radiative polariton modes in medium III. We choose the following parameters in our computation: EL ˆ 15:70, E1 ˆ 9:65, oT ˆ 12 THz, oL ˆ 21:6 THz, and op d=c ˆ 2p  10ÿ2 . This corresponds to a strongly doped InSb ®lm on an undoped MgO substrate. In Fig. 23 we compare the computed splittings D…o=op †, as a function of the reduced cyclotron frequency oc =op, in the Voigt, perpendicular, and Faraday geometries [348,349]. The curves designated as V, F, and Pi refer to the Voigt, Faraday, and perpendicular con®gurations, respectively. It is evident that D…o† in the Voigt (for both positive (V‡ ) and negative (Vÿ ) directions of propagation) and the Faraday geometries depends strongly on the intensity of the applied magnetic ®eld. The situation in the perpendicular geometry is rather different. For oc < oT, where one ®nds only one B. This splitting at o ' op (see Fig. 2 in Ref. [348]), D…o† is a very slowly varying function of ~ behavior may be understood from the fact that in the vicinity of the resonance o ' op , the magnetic ®eld dependence is negligible (see, e.g., Eq. (31) in Ref. [348]), resulting in the ®eld-independent expression. For oc > oT (with both oc and op now being inside the surface phonon±polariton range) there are two resonances (see, e.g., Fig. 3 in Ref. [348]). The curves labeled as Pp and Pc correspond to the splittings at op and oc , respectively. In the special case of oc ˆ op , there is only one resonance; the corresponding value of D…o† has been marked by a thick dot. This causes a discontinuity in curves Pp and Pc at oc ˆ op . For oc > op, the curves Pp and Pc exchange roles. It is interesting to note that the two curves exhibit opposite behaviors: when Pp (oc ) increases the Pc (oc ) decreases, and vice versa. Generally speaking, Fig. 23 shows that the magnitude of the splitting depends in a qualitative way on the direction and the intensity of the applied magnetic ®eld. 6. Plasmons in double inversion layers The purpose of this section is to present results of theoretical calculations of the dispersion relations for magnetoplasmon excitations associated with the double inversion layers (DILs) in the

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Fig. 23. The resonance splitting Do, as de®ned by Eq. (5.15), as a function of normalized cyclotron frequency. The material parameters correspond to a strongly doped InSb ®lm on an undoped MgO substrate (see the text). We compare results for the Voigt (V ), the Faraday (F), and the perpendicular (Pp or Pc ) geometries. The splittings arise as a result of a resonance between the magnetoplasmons of the ®lm (oc , op , oH ) and the phonon±polaritons of the substrate. Notice the strong dependence of Do on both the direction and the intensity of the applied magnetic ®eld.

semiconductor±insulator±semiconductor (SIS) heterostructure. We will do so in two different geometries: (i) where the thickness of the 2DEG and 2DHG is zero and the system is subjected to a magnetic ®eld in the perpendicular con®guration, (ii) where the thickness of the 2DEG and 2DHG is non-zero and the system is subjected to a magnetic ®eld in the Voigt con®guration. The theoretical framework (as will be discussed brie¯y below) employed to investigate the magnetoplasmon excitations is different in the two cases; both approaches allow to account for the retardation effects, damping, and coupling to optical phonons, however. It should be mentioned that both methodologies belong to the local approximation, and thus spatial dispersion (non-local effects) will be neglected. Consideration of the non-local effects is known to excite cyclotron-resonance harmonics in a 2D system [503]. The strength of such non-local effects is governed by the parameter …qvF =oc †2, where q is the plasmon wave vector and vF is the Fermi velocity. The coupling of magnetoplasma oscillations and cyclotron-resonance harmonics splits the magnetoplasma resonance line near the frequencies o ˆ noc with n ˆ 2; 3; 4; . . . [1645].

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Fig. 24. Schematics of the DILs system considered in Section 6.1, with 2D layers of zero thicknesses. Region B is an insulator ®lm (of thickness dB ˆ 2d) asymmetrically bounded by a p-type semiconductor (semi-in®nite region A) and an ntype semiconductor (semi-in®nite region C). The 2DEG (2DHG) is assumed to exist at the A±B (B±C) interface in the x±y plane and characterized by the 2D polarizability tensor ~ wA (~ wC ). The magnetic ®eld is considered to be oriented perpendicular to the interfaces (~ B k ^z). The material parameters correspond to a Si-MOSFET-like structure, where region A and region C are differently doped and region B is an undoped SiO2 : EA ˆ 11:7 ˆ EC , EB ˆ 3:7.

6.1. DILs with zero thicknesses We consider an insulator ®lm (region B) asymmetrically bounded by p-type semiconductor (region A) and n-type semiconductor (region C), as shown in Fig. 24. The 2DEG and 2DHG are assumed to exist at A±B and B±C interfaces, respectively. The insulator and semiconductors are characterized by the background dielectric constants EB , EA , and EC ; subscripts refer to the respective media. An external magnetic ®eld is taken to be perpendicular to the interfaces and to the direction of propagation. The inversion layers at the two interfaces have the polarizability tensors ~wA and ~wC . This means that an electric ®eld of the form  E…z† exp‰i…qy ÿ wt†Š evaluated at the planes z ˆ d will cause a polarization of the form ~ P…~ r; t† ˆ ‰Px …q; o†; Py …q; o†Šd…z  d†;

(6.1)

~ such that where ~ P…q; o† ˆ ~ w…q; o†  ~ E. The polarizability tensor ~w is related to the conductivity tensor s ~ ˆ ÿio~ s w. It should be pointed out that the 2D behavior of the plasmons does not require that the motion of the charge carriers in the inversion layers be purely 2D. It only requires that the wavelength of the plasmons (lp ) be large compared to the width of the charge sheet (di ). In the present work, we consider the inversion layers to be ideally 2D (i.e., lp  di ). In this case, both quantum and classical considerations yield identical results. We intend to treat the general properties of the 2D plasmons followed by the Maxwell equations for the corresponding oscillations of the EM ®elds. Elimination of the magnetic ®eld variable from Maxwell's curl ®eld equations yields the following wave-®eld equation in terms of the macroscopic electric ®eld (~ E): 4pi ~ ~  …r ~ ~ q0 J ÿ q20 E~ E ˆ 0; r E† ÿ c

(6.2)

where E is the dielectric constant appropriate to the medium in which the wave equation is being

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applied, and ~ J…~ r; t† ˆ ~ J…q; o† exp‰i…qy ÿ ot†Šd…z  d†

(6.3)

is the surface current density associated with the respective inversion layers. More precisely, ~ J…q; o† ˆ ÿio~ w…q; o†  ~ E. In the presence of the surface current, the standard EM boundary conditions take the form E2 †  ^ n ˆ 0; …~ E1 ÿ ~

~1 ÿ H ~2 †  ^ …H nˆ

4p ~ J; c

(6.4)

where ^ n is the normal unit vector directed from mediums 1 to 2. Imposing these standard boundary conditions at z ˆ  d leads to the following dispersion relation [512]: ‰…A1 C1 ‡ B21 † tanh…y† ÿ …A1 ‡ C1 †B1 Š‰…A2 C2 ‡ B22 † tanh…y† ‡ …A2 ‡ C2 †B2 Š ‡ Ax Ay …B1 ÿ C1 tanh…y††…B2 ‡ C2 tanh…y†† ‡ Cx Cy …B1 ÿ A1 tanh…y††…B2 ‡ A2 tanh…y†† ‡ B1 B2 …Ax Cy ‡ Ay Cx †…1 ÿ tanh2 …y†† ‡ Ax Ay Cx Cy tanh2 …y† ˆ 0; where y ˆ aB dB with dB ˆ 2d, and rest of the symbols are de®ned such that aj Ej ; j2 ˆ w jyy ‡ ; jx ˆ w jxy ; jy ˆ w jyx ; j1 ˆ w jxx ÿ 4paj 4pq20 aB EB B1 ˆ ; B2 ˆ ; a2j ˆ q2 ÿ q20 Ej ; j  A; C: 2 4paB 4pq0

(6.5)

(6.6)

Here the symbol j (as a principal term or in the suf®x) refers to the speci®c quantities in the respective regions (Fig. 24). The symbol aj ( j  A, B, C) stands for the decay constants for the ®elds in the respective media. In the presence of a magnetic ®eld the Cartesian elements of the polarizability tensor (~ w) are de®ned as follows: w jxx ˆ w jyy ˆ ÿ

nj e2 ; mj …o2 ÿ o2cj †

w jxy ˆ ÿw jyx ˆ i

nj e2 ocj ; mj o…o2 ÿ o2cj †

(6.7)

where nj , mj , e, and ocj are the number of charge carriers per unit area, effective mass, electron charge, and the cyclotron frequency, respectively; j  A, C. In writing Eq. (6.7), we have retained only the nonoscillatory part of ~ w which is insensitive to the temperature and thus excludes consideration, e.g., of Shubnikov±de Haas oscillations. Eq. (6.5) is an implicit general dispersion relation for the coupled magnetoplasmon excitations (CMEs) associated with the 2DEG and 2DHG in the DILs in the SIS heterostructure at hand. It contains all the known results as the limiting cases. Let us now study a few special cases by subjecting Eq. (6.5) to some speci®c limits. The case d ! 1: In this limit Eq. (6.5) simpli®es to the following form: ‰…A1 ÿ B1 †…A2 ‡ B2 † ÿ Ax Ay Š‰…C1 ÿ B1 †…C2 ‡ B2 † ÿ Cx Cy Š ˆ 0:

(6.8)

Either the ®rst or the second factor is zero. Equating the ®rst factor to zero gives, after substitutions     aA ‡ aB 1 EA EB A A A ‡ (6.9) wyy ‡ ÿ wA wxx ÿ xy wyx ˆ 0 4p aA aB 4pq20

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and the second factor equated to zero yields, after substitutions     aC ‡ aB 1 EC EB C wCxx ÿ w ÿ wCxy wCyx ˆ 0: ‡ ‡ yy 4p aC aB 4pq20

(6.10)

Eqs. (6.9) and (6.10) represent the dispersion relations for the 2D magnetoplasmons of a plane of electrons (holes) localized at A±B (C±B) interface and embedded in two dielectrics A and B (C and B). Eqs. (6.9) and (6.10) are exact analogs of Eq. (58a) of Chiu and Quinn [478]. The case ~ B ˆ 0: The limit of zero magnetic ®eld implies to an isotropic case (wixy ˆ 0 ˆ wiyx ; i  A, C). As a result, Eq. (6.5) reduces to the form ‰…A1 C1 ‡ B21 † tanh…y† ÿ …A1 ‡ C1 †B1 Š‰…A2 C2 ‡ B22 † tanh…y† ‡ …A2 ‡ C2 †B2 Š ˆ 0:

(6.11)

Either the ®rst or the second factor is zero. It can readily be shown, after substitutions, that the vanishing of the ®rst factor yields the TE (s-polarization) CME and that of the second factor gives the TM (p-polarization) CME. In the presence of an external magnetic ®eld, the TM and TE waves remain coupled due, in particular, to the Hall currents in the 2D layers. The second factor equated to zero yields the following dispersion relation for the plasmon dispersion in the double-inversion-layer system with wi  wixx ˆ wiyy     2 EB EC EB EA EB 2 csch …y† ˆ 4pwC ‡ ‡ coth…y† 4pwA ‡ ‡ coth…y† : (6.12) aB aC aB aA aB Eq. (6.12) is exactly the same as Eq. (13) of Kushwaha [505]. If d ! 1, Eq. (6.12) reproduces the two independent modes speci®ed by EA EB 4pwA ‡ ‡ ˆ 0; (6.13) aA aB EC EB 4pwC ‡ ‡ ˆ 0: (6.14) aC aB Thus Eqs. (6.13) and (6.14) represent the dispersion relations for the decoupled plasmons of the 2DEG and 2DHG, respectively. We ®nd that Eq. (6.12) can be further simpli®ed to reproduce several interesting results derived by different authors [88,92,454,477,481]; the same is true for the magnetic®eld dependent equations. For instance, in the NRL (c ! 1), Eq. (6.9) simpli®es to 4pqwA yy ‡ Es ˆ 0;

(6.15)

where Es ˆ …EA ‡ EB †. Substituting the expression for wA yy, from Eq. (6.7), Eq. (6.15) assumes the form   1=2 4pne e2 q : (6.16) o ˆ o2ce ‡ Es me This substantiates the fact that in the NRL there are no bona®de solutions below the cyclotron frequency oc. In the absence of an external magnetic ®eld, this equation becomes  1=2 4pne e2 q1=2 (6.17) oˆ Es me as expected for the 2D plasmon dispersion. Eq. (6.17) can also be readily obtained from Eq. (6.13) in the NRL.

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Next, we turn to discuss the exact numerical results, both with and without an applied magnetic ®eld, obtained using the general dispersion relation, Eq. (6.5). It is noteworthy that the strategy of our numerical computation is based on the fact that we search the zeros of the real part of the transcendental function, Eq. (6.5), irrespective of whether or not its imaginary part is zero. This sometimes (depending upon the material parameters) also includes some unphysical solutions. For instance, this is known that in the presence of an applied magnetic ®eld in the perpendicular con®guration there is no bona®de solution for a 2D magnetoplasmon±polariton in the NRL below the cyclotron frequency [1646]. And yet, the above strategy can allow low-frequency solutions in the gap below oc. Such unphysical solutions correspond to, in our terminology, the imaginary ai (i  A, C), implying the EM ®elds growing exponentially away from the 2D planes. These unphysical solutions can be discarded by searching the zeros of the real part of the transcendental function and accepting only the solutions for which the imaginary part of the function is zero. If the 2D layers are embedded in different materials (EA 6ˆ EB 6ˆ EC ), the latter scheme of performing computation in the search of genuine solutions can be quite time consuming. We consider a Si-MOSFET-like structure, where region A and region C are differently doped and region B is an undoped SiO2 . This is to say that EA ˆ EC ˆ 11:7 and EB ˆ 3:7. Our numerical results do (sometimes) include the low-frequency unphysical solutions, because we adopted the former, rather than the latter, strategy of searching zeros of, in general, a complex transcendental function, Eq. (6.5). 6.1.1. Zero magnetic ®eld In spite of the fact that we consider 2DEG and 2DHG of zero thickness, we introduce (mathematically) an effective thickness for the purpose of normalization. As such, we present the numerical results in terms of the dimensionless frequency x ˆ o=op and the wave vector z ˆ cq=op, where ope ˆ 4pne e2 =EA me di is the effective 3D plasma frequency, where di is the hypothetical thickness of the inversion layers, taken to be equal for the 2DEG and 2DHG. First, we consider the case d ! 1, implying the decoupled modes of excitations of the 2DEG and the 2DHG. The numerical results for this case are illustrated in Fig. 25. The symbol ds  DB stands for the thickness of the spacer between the 2DEG and 2DHG, and deff ˆ ope di =c is the normalized effective thickness of the inversion layers. The two modes demarked as 2DEG and 2DHG correspond to the 2D plasmons of the electron and hole layers, respectively. The mode for the 2DHG is lower in frequency because the effective mass of the holes (mh ) is heavier than that of the electrons (me ). Both of them start from zero and propagate to the right of the light line in media A (  C). The numerical results for the coupled modes of excitations are shown in Fig. 26. There are three sets of modes corresponding to three values of the effective thicknesses. Intuitively the upper (lower) mode of each pair of CME corresponds to 2DEG (2DHG), for the reason stated above. In principle, such a distinction of the modes within the CME is not allowed, particularly when the spacer thickness is reasonably small but ®nite (r ˆ ds =di ˆ 100). The variation in deff clearly reveals that the frequencies of the CME increase as the thickness of the 2D layers increases. This is understood in view of the fact that as the dimensions of the 2D layers approach those of the 3D ®lms, the system can allow the 3D (or quasi-3D) plasmons with higher energies. Finally, the effect of variation of spacer thickness on the CME is depicted in Fig. 27. There are ®ve sets of modes corresponding to ®ve values of r, for a given deff . It is observed that all sets of CME start from the origin and propagate to the right of the light line, implying that all the decay constants ai 's are real and positive. The coupled plasmon modes corresponding, intuitively, to the 2DEG show a

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Fig. 25. Dispersion of the decoupled (dB ˆ 2d ! 1) modes of excitation in the DIL system in the zero magnetic ®eld. The parameters are: ds  dB ˆ 2d ! 1, deff ˆ ope di =c, di being the mathematically introduced thickness of the inversion layers and taken to be equal for 2DEG and 2DHG. The subscripts 1, 2, and 3 on E and a correspond to regions A, B, and C, respectively.

negligibly small variation with r over the whole range of propagation vector q. The plasma modes due to 2DHG, on the other hand, display a considerable change in frequency with the variation in the spacer thickness. Overall, it is observed that the frequency corresponding both to the 2DEG and 2DHG increases with increasing spacer thickness. It is noteworthy that the dispersion curves due to the 2DHG for thinner spacer are almost straight lines, implying a constant group velocity. These results are in close correspondence with those of Kushwaha (see, e.g., Ref. [505]). 6.1.2. Non-zero magnetic ®eld In the presence of the magnetic ®eld, the plasmon dispersion for the decoupled modes (ds ! 1) is shown in Fig. 28. It is found that both magnetoplasma polariton modes start at and propagate towards the right of the light line in the bounding media. Evidently, both modes have bona®de solutions even below the respective cyclotron frequencies, in contrast to the non-retardation dispersion which starts at above the cyclotron frequencies. This is in coherent correspondence with the earlier work [478,512,1726], where it was shown that the magnetoplasmon±polariton excitations arise at frequencies below the cyclotron frequency when retardation effects are taken into account. However, it should be pointed out that there does exist a low-frequency gap extending up to zero, i.e., the bona®de solutions do not start right from the origin. Such details were missed previously [512,1726], but are in accord with the later work [1646]. The low-frequency gap is noticed only when one searches the zeros of the tran scendental function with its imaginary part restricted to zero, but not when the latter is left free.

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Fig. 26. Dispersion of the couples modes of excitation in the DIL system in the zero magnetic ®eld. The solid curves in blue, red, and green correspond to deff ˆ 0:01, 0.10, and 1.0, respectively, for a given normalized spacer thickness r ˆ ds =di ˆ 100. The dashed lines in black demarked as a2 ˆ 0 (a1 ˆ a3 ˆ 0) is the light line in the spacer region B (bounding media A  C). Fig. 27. Dispersion of the coupled modes of excitation in the DIL system in the zero magnetic ®eld. The set of modes in blue, red, green, pink, and black correspond to the value of r ˆ 10, 20, 50, 102 , and 103 , respectively. It is only the modes corresponding (intuitively) to the 2DHG layer that shows well discernible change with r; the mode corresponding (intuitively) to the 2DEG remains practically unaffected by the variation in r.

Again, we are able to distinguish the upper (lower) mode belonging to the 2DEG (2DHG) because of the reasons stated above. Fig. 29 depicts the effects of variation in the effective thickness (deff ) of the inversion layers. We have shown three sets of coupled magnetoplasma modes, for a normalized cyclotron frequency oce =ope ˆ 0:5 and r ˆ 100. It is clearly noticeable that the starting frequencies, for both modes of each set, are lower for the higher effective thickness. The deviation of the modes from the light line is seen to be higher for the lower effective thickness. For low enough thickness of the inversion layers, the magnetoplasma modes start at almost the respective cyclotron frequencies and propagate with negligible dispersion, roughly in accordance with Eq. (6.16). Fig. 30 illustrates other intricacy of the magnetopolariton dispersion in the DILs, where we show the effect of the variation in the spacer thickness (r) for a given deff ˆ 10ÿ3 . In this case, we only restrict the decay constant aB to be real and positive. The variation of r produces, for these values of parameters, practically indiscernible change in the frequencies of the bona®de magnetopolariton modes; although a close inspection reveals that the frequency of both modes increases with increasing r. Note that the bona®de polariton modes are only the two ¯attened ones that appear towards the right of the light line in the bounding media, for each value of r. However, if one forgets this criterion for a moment, the picture of mode dispersion for a given r is describable as follows. Both coupled modes of excitation are seen to start from the frequency lower than the hole±cyclotron frequency, observe two splittings at the two cyclotron frequencies, and ¯atten out in the proximity of the cyclotron frequencies with a short-wavelength dispersion speci®ed approximately by Eq. (6.16). It is seen that the magnitude of the resonance splitting at the two cyclotron frequencies increases with increasing r (the normalized

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Fig. 28. Dispersion of the decoupled (ds ! 1) modes of excitation in the DIL system in the presence of a magnetic ®eld for normalized (electron) cyclotron frequency oce =ope ˆ 0:5. The retardation effect allows the bona®de solution even below the lower (the hole) cyclotron frequency och ˆ 12 oce, unlike in the non-retardation case where the genuine solutions occur only at o  oci . Noticeable low-frequency gap persists in the spectrum (see the text).

spacer thickness); of course, one should expect a critical value of this parameter r where the coupled modes of excitation will decouple. Moreover, such a repulsion is found to decrease with increasing deff (keeping r ®xed), provided that y  2 () tanh…y† ! 1) is maintained. A simple approximate result (for aA ˆ aC ˆ 0) then yields o ˆ oci ÿ opi Pj , where Pj ˆ 12 ‰Ej =…Ej ÿ EB †1=2 Šdeff (here i  e (h) ) j  A (C)). For Pj  1 (which is usually the case), o ! oci (i ˆ e, h), this justi®es the resonance splitting at the two cyclotron frequencies. An implicit requirement in such approximate result is that Ej > EB . However, if one thinks of the observability of the bona®de modes, then only the two ¯attened modes come to mind, because these are the only ones that are the true polaritons characterized by real and positive decay constants. Note that such a behavior as depicted in Fig. 30 was also noticed in the context of a single 2DEG screened by perfectly conducting sheets at a distance [511]. 6.1.3. Effect of optical phonons in the spacer layer We now consider the effect of optical phonons in an undoped spacer layer on the coupled modes of excitation in the double-inversion-layer system. We use roughly hypothetical parameters to demonstrate the effect. The numerical results for the zero magnetic ®eld are shown in terms of dimensionless variables in Fig. 31. It is clearly seen that the lower pair of modes starting from the origin propagate just towards the right of the light line (in the bounding media) and become asymptotic to oT (the transverse optical phonon frequency at the zone center). A new pair of coupled modes of excitation start from the light line at oT , and grows almost monotonically with the propagation vector.

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Fig. 29. Dispersion of the coupled modes of excitation in the DIL system in the presence of a magnetic ®eld, for oce =ope ˆ 0:5 and r ˆ 102 . The pair of modes in blue, pink, and green correspond to the values of deff ˆ 0:01, 0.10, and 1.0, respectively. The dashed line in black is the light line in the bounding media. Notice that as deff decreases, both modes of each pair tend to propagate with lower frequencies at large wave vectors, and exhibit progressively much less dispersion. Fig. 30. Dispersion of the coupled modes of excitation in the DIL system in the presence of a magnetic ®eld, for oce =ope ˆ 0:5, and deff ˆ 10ÿ3 . The modes in blue, pink, and green correspond to r ˆ 10, 102 , and 103 , respectively. Notice the resonance splitting of the bona®de magnetopolariton modes with their corresponding radiative modes (between the two light lines) at the two cyclotron frequencies. The large parts of the modes in blue and pink remain indiscernible from those in green; this is particularly so for the ¯attened parts in the proximity of the cyclotron frequencies. This is attributed to the virtually vanishing thicknesses of the 2D layers; a close inspection reveals that the above-mentioned repulsion, as well as the frequency of the pair of magnetoplasma modes, increases with increasing r.

In the presence of an external magnetic ®eld, we study the evolution of the CME with the spacer thickness (r) for a given effective thickness (deff ) of the inversion layers. The numerical results are illustrated in Fig. 32(a)±(d), for the normalized (electron) cyclotron frequency oce =ope ˆ 0:5. For r ˆ 103 , there are two modes which start from the light line (in the bounding media) just above oT with the lower one becoming asymptotic to the hole±cyclotron frequency (och ), the third mode that starts from the light line just above och propagates almost parallel to the second mode (from the bottom) and becomes asymptotic to some characteristic frequency, and the fourth mode starts from the light line just above the electron±cyclotron frequency (oce ) and shows nearly ¯at dispersion slightly above oce (see Fig. 32(a)). As r decreases to r ˆ 5  102 (Fig. 32(b)), the ®rst and second (from bottom) modes show a repulsive character at z ' 1:22; similar repulsion is observed between the second and third modes at z ' 0:95 in the vicinity of och . However, the uppermost mode remains intact. A further decrease in r to r ˆ 102 (Fig. 32(c)) reveals that the lowest mode decreases in frequency over almost the whole range of propagation vector. The second and the third modes observe a splitting at o ˆ och , just to the right of the light line; the fourth one now seems to start slightly from the right of the light line with positive group velocity, but maintains its character at higher z corresponding to that in Fig. 32(a) and (b). Finally, we reduce the r such that r ˆ 10 (see Fig. 32(d)). The lowest mode still goes further down to become almost linear with q. The second and third modes now show a discernible splitting at o ˆ och just to the right of the light line; a similar, but stronger, splitting is now seen between the third and

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Fig. 31. Effect of the optical phonons in the spacer (region B) on the dispersion of the coupled modes of excitation in the DIL system in the zero magnetic ®eld. The parameters satisfying the well-known LST relation Ð E2 …0†=E2 …1† ˆ o2LO =o2TO Ð are quite hypothetical, and were chosen just to have a feel of the resulting interaction. Notice how the resonance at oTO (the transverse optical phonon frequency) splits the spectrum into two separate propagation windows for the 2D plasmon spectrum in the system. The rest of the parameters are the same as in the previous ®gures in this section.

fourth modes at o ˆ oce ; the fourth mode is expected to become asymptotic to oL . The two splittings observed in Fig. 32(d) are attributed to the resonance between the spacer phonons and the interacting 2DEG and 2DHG magnetoplasmons for a reasonably small (but ®nite) spacer thickness. Their number ``two'' is a consequence of the fact that both och and oce lie between oT and oL (i.e., oT < och < oce < oL ). If this condition is violated such that, e.g., och < oT < oce < oL , then we observe only one splitting at o ˆ oce , as shown in Fig. 33. This is because the magnetoplasmons in 2DHG now fall beyond the possibility of having resonance with the spacer phonons. Such splittings as seen in Figs. 32(c) and (d), or 33 are disallowed in Fig. 32(a) and (b), possibly because of the thicker spacer that prevents the 2DEG and 2DHG magnetoplasmons to interact with each other. However, a single weak splitting felt in Fig. 32(a) and (b) also signals the likelihood of resonance between the spacer phonons and the 2DHG magnetoplasmons. 6.2. DILs with ®nite thicknesses A practically feasible and physically interesting system would be where one has 2DEG and 2DHG layers of ®nite thicknesses. Such a system of DILs, both with and without an applied magnetic ®eld in the Voigt con®guration, was investigated by Kushwaha and Djafari±Rouhani [730,731], within the framework of Green-function (or response function) theory. The schematic geometrical structure is as shown in Fig. 34, where de and dh (the thicknesses of the 2DEG and 2DHG layers) were ®nally limited

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Fig. 32. Effect of the optical phonons in the spacer on the magnetoplasmon dispersion in the DIL system in the presence of a magnetic ®eld, for oce =ope ˆ 0:5, and deff ˆ 10ÿ3 . The results in (a)±(d) reveal the evolution of the resonance splitting (highlighted by circles) at the two cyclotron frequencies with decreasing spacer thickness r, for the situation when oTO < och < oce < oLO .

such that di ! 0, i  e, h. The simple, but physically sound, approximation that was seen to work excellently for many 2D multilayered structures is that (see, e.g., Eq. (4.1) in Ref. [731]) Eiv di ! 4pwiv ) a2 di ! ÿ4pq20 wiv ;

(6.18)

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Fig. 33. The same as in Fig. 32, but for a ®xed deff ˆ 10ÿ3 and r ˆ 10; and in the case when och < oTO < oce < oLO . A strong splitting (highlighted by a circle) is observed only at oce , as expected.

where wiv ˆ wiyy ‡ …wiyz †2 =wiyy is the magnetic-®eld dependent 2D Voigt polarizability function, which is related to the conductivity (siv ) such that siv ˆ ÿiowiv . Eq. (6.18), which is applicable to any electron or hole layer of thickness di (such that di ! 0) and characterized by a Voigt dielectric function Eiv , is known to play a crucial role in obtaining the desired results on plasmons and magnetoplasmons in

Fig. 34. Schematics of the DIL system: 2DEG (shaded region) and a 2DHG (blank region) separated by a spacer (medium II) of thickness d2 . The symbols de and dh refer to the thicknesses of 2DEG and 2DHG layers, respectively. Regions I and III are semi-in®nite. The magnetic ®eld is assumed to be in the Voigt con®guration: ~ B0 k ^x and ~ q k ^y, where y±z is the sagittal plane. We embark on the situation where de , dh ! 0. Eventually, we consider the whole system to be made up of a single (doped or undoped) semiconductor: GaAs with background dielectric constant EL ˆ 12:8.

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diverse periodic and non-periodic systems [730,731]. The ®nal expression for the magnetoplasmon dispersion relation in the DIL system subjected to an external magnetic ®eld in the Voigt con®guration is given by [731]       E1 E2 E3 E2 E2 E2 e h ‡ coth…y† Ev ‡ ‡ coth…y† ÿ Ev ‡ csch2 …y† ˆ 0; a1 de a2 de a3 dh a2 dh a2 de a2 dh

(6.19)

where y ˆ a2 d2 . The subscript i (  1, 2, 3) corresponds to the quantities in the respective medium (regions I±III) in Fig. 34, and subscript or superscript e (h) stands for the 2DEG (2DHG). The symbol di is the thickness of the inversion layers taken to be equal for both layers (i.e., de ˆ dh ˆ di ). In the limit of a zero magnetic ®eld, Eq. (6.19) reduces (within the aforesaid approximation) to Eq. (6.12), as expected. Eq. (6.19) represents the magnetic-®eld dependent TM modes in the Voigt geometry. This time we will consider the system to be made up of only a single (doped or undoped) semiconductor. A careful diagnosis of the energy scales usually explored in the 2D systems let us choose GaAs (with background dielectric constant EL ˆ 12:8) as the most suitable candidate for the 2D (electron and/or hole) layers, as well as for the dielectric materials (wherein the 2D layers are embedded) where one could employ the intrinsic GaAs. 6.2.1. Zero magnetic ®eld First we discuss an illustrative example without an applied magnetic ®eld, for a ®xed normalized spacer thickness r ˆ d2 =di ˆ 100. The numerical results are presented for three different values of the normalized thickness of the inversion layers (deff ˆ ope di =c ˆ 0:001, 0.007, and 0.05) in terms of the dimensionless variables z ˆ cq=ope and x ˆ o=ope , where ope ˆ 4pne e2 =EL me di , in Fig. 35. The interesting aspect of the numerical results depicted in Fig. 35 that emerges while comparing with Fig. 26 is that there is no coherent correspondence between the two. This is attributed to the fact that now we have a realistic system (with 2DEG and 2DHG of ®nite thickness) made up of a single semiconductor, instead of two, implying that the problem now contains a single decay constant ‡ and one is not bound to control two different decay constants in order to obtain the bona®de solutions for CME. In spite of this, the general conclusions drawn from Fig. 26 still remain valid. This is to say that the frequencies of both the modes (of each pair) increase with increasing deff . 6.2.2. Non-zero magnetic ®eld For the better understanding of the numerical results in the presence of the magnetic ®eld, we plot the Voigt dielectric function for electrons and holes in Fig. 36. In order to understand the importance of this ®gure, we recall Eq. (6.19) that has been fully used in the computation. A careful look at this equation leads us to infer that its left-hand side remains always real and positive (i.e., every term retains its own sign) in all frequency regimes (for ai real and positive). (This criterion can fail where one is towards the left of the light line, where the terms with ai 's become pure imaginary.) This then implies that, for the magnetoplasmon±polariton modes we are interested in, one needs Ev to be negative for the purpose of ®nding zeros of this transcendental function. Fig. 36 dictates that there are two frequency windows (for each Eiv ) where Ev remains negative. These are de®ned by 0  x  Ol and Oj  x  Ou , where Oj ( j  e, h) refers to the hybrid (electron or hole) cyclotron±plasmon frequency, and Oi (i  l, u) to the (lower or upper) zeros of Ev . For the parameters used in the present computation, these frequency windows are speci®ed by 0  x  0:5931 (0  x  0:7808) and 0:75  x  0:8431

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Fig. 35. Dispersion of the coupled modes of excitation in the DIL system in the zero magnetic ®eld, for normalized spacer thickness r ˆ ds =di ˆ 100. The dotted, dashed, and solid curves correspond to the effective thickness of the inversion layers deff ˆ 0:001, 0.007, and 0.05, respectively. The variation of deff leads us to draw virtually similar conclusions as in Fig. 26, although there is no coherent correspondence between the two ®gures. Dashed line marked as light-line is the light line in GaAs.

(1:118  x  1:281) for holes (electrons). In addition, Fig. 36 also dictates a complete gap in the excitation spectrum within the frequency range de®ned by Oi  x  Oj ; with i ˆ l and j  e, h. This is true irrespective of whether the system is made up of one or both types of charge carriers. In the latter case, the subscript i (j)ˆu (e) for holes (electrons), presuming that holes are heavier than electrons. We have seen that all the dispersion relations for the magnetoplasmon±polaritons in different structures subjected to an external magnetic ®eld in the Voigt geometry abide by the rules dictated by Fig. 36 (see, e.g., Refs. [730,731]). The numerical results for the magnetoplasmon±polaritons based on Eq. (6.19) are illustrated in Fig. 37, for a ®xed r ˆ 100. The interesting aspect concerned with this system in this Voigt geometry is that it enables us to understand how the pair of CME Ð with ®nite spacer Ð split when there are two hybrid cyclotron±plasmon frequencies to play an independent role in the problem. The lower panel exhibits three pairs of CME propagating towards the right of the light line. The lowest (dotted curves), middle (dashed curves), and the upper (solid curves) pairs correspond to deff ˆ0.001, 0.007, and 0.05, respectively. The distinction of the lower (upper) mode of each pair as the magnetoplasmons of 2DEG (2DHG), and the variation of their frequencies with di , have the same reason as explained above with respect to Fig. 26 The lower (upper) mode of each pair becomes asymptotic to Ohl ˆ 0:5931 (Oel ˆ 0:7808). Each of the middle and upper panel depicts only one mode of excitation for each value of d. All the modes in the middle (upper) panel will become asymptotic to Ohu ˆ 0:8431 (Oeu ˆ 1:281).

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Fig. 36. The Voigt dielectric function for electrons (solid lines) and holes (dashed lines) versus the reduced frequency x, for Oce ˆ oce =ope ˆ 0:5. The y-axis displays Eiv =EL ; i  e, h.

The splitting of the CME in the middle and upper panel is attributed to the fact that the Voigt dielectric function, due to the simultaneous presence of holes and electrons, blows up at different frequencies, given by Oh and Oe , respectively. The reader should not be surprised to note that the split modes in the middle (upper) panel possess the dominant contribution due solely to 2DHG (2DEG) magnetoplasmons. That this is so cannot, however, easily be seen from the analytical expression in Eq. (6.19). The matter of fact is that the coupling term (the last term on the left-hand side) in Eq. (6.19) becomes negligibly small above the lower (i.e., the hole) hybrid frequency, and the bona®de solutions (i.e., the zeros of the function) within the middle (upper) panel are given by the second (®rst) square bracket equated to zero. As such, we are led to infer that the concept of the CME prevails only within the lower panel. The coupling strength of the CME Ð the thinner (but ®nite) the spacer layer the stronger the coupling effect Ð is more pronounced at the longer wavelengths. The corresponding effect at shorter wavelengths is weaker because both the lower and the upper modes of each pair of CME (in the lower panel) will, after all, approach the same well-de®ned asymptotic limits. Finally, we remark that there is a complete gap within the excitation spectrum in the frequency range speci®ed by Ohu  x  Oe . 7. Plasmons in binary compositional superlattices The 1980s have seen a great deal of interest in the properties of superlattices which typically consist of alternating layers of two materials produced by MBE. Since the thicknesses of the layers can be

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Fig. 37. Dispersion of the coupled modes of excitation in the DIL system in the presence of a magnetic ®eld in the Voigt con®guration, for Oce ˆ oce =ope ˆ 0:5. The dotted, dashed, and solid curves correspond to deff ˆ 0:001, 0.007, and 0.05, respectively. The middle (upper) panel displays the split modes of the pair of coupled modes of excitation above the hole (electron) hybrid cyclotron±plasmon frequency Oh (Oe ). The dash-dotted line refers to the light line in the spacers (GaAs).

controlled at a precise number of lattice spacings, one can produce a periodic structure which has an additional periodicity superimposed on the normal atomic periodicity of a crystal Ð hence the name superlattice. Superlattices involving semiconductors as well as metals have been fabricated. To the understanding of electronic and optical properties of superlattices, the knowledge of the elementary collective excitations in the systems is of fundamental importance. The set of such excitations is characterized by a Bloch vector (q? ) normal to the interfaces as well as a wave vector (qjj ) parallel to the interfaces. If d is the period (the width of the unit cell) of the superlattice, the boundaries of the bulk plasmon bands exist for q? ˆ 0 or p. The truncation of the superlattice yields surface plasmon± polariton modes located at the surface but still affected by the layered structure of the bulk constituents. These polariton modes appear in the gaps between the bulk bands, and above and below the bands. The elementary excitations in superlattice systems have been studied by many research groups using macroscopic as well as microscopic theories (see Section 2.2.2), both with and without an applied magnetic ®eld. There are usually two situations. (i) The case where the layer widths are much smaller than the electron mean free path and the electrons are quantized into a series of miniband energy levels Ð usually referred to as the quantum limit. In this case the conduction electrons correspond to those of a quasi-2DEG and their dynamical behavior cannot be characterized by the macroscopic (local) dielectric functions; a theoretical treatment is followed usually by a self-consistent ®eld or random phase approximation. (ii) The case where the layer widths are much larger than the electron mean free

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Fig. 38. Schematic representation of a periodic semiconductor heterostructure consisting of two different types of semiconductor slabs A and B of widths dA and dB . The period of the system d ˆ dA ‡ dB ; n designates the number of the unit cell. With or without an applied magnetic ®eld, the direction of propagation is chosen such that ~ q k ^y.

path so that the quantum size effects become negligible and the constituent layers can be treated as bulk characterized by the macroscopic (local) dielectric functions Ð usually referred to as the classical limit. In this case the theoretical treatment has been followed by using either a hydrodynamical model or the Maxwell equations with proper EM boundary conditions. While the former situation usually ignores the retardation effects, the latter allows their embodiment. In the case where only the lowest subband is occupied, the theoretical approach in the latter situation has signi®cant advantages owing to its analytical simplicity, and also retains the essential features of the more complicated treatment in the former situation. In addition, such a model theory yields analytical results which are found to be readily conclusive. In this section our aim is to review the study of the collective (bulk and surface) excitations of metal± dielectric, semiconductor±dielectric, and semiconductor±semiconductor binary superlattices carried out within the framework of the macroscopic theories, both with and without an applied magnetic ®elds (Fig. 38). Such a macroscopic approach can safely be termed as the TMM, even though the analytical results in the original papers to be reviewed here [708,719,723,1752] were not always written in the formalism adequate to TMM. It should be pointed out that we would be succinct in the mathematical formulation and recall only brief expressions, the lengthy expressions will be referred to the original papers. However, some approximate diagnoses without much rigor will desirably be given at the appropriate places. To start with, we recall the general expression for the bulk collective excitations in this binary (semiconductor±semiconductor) superlattices in the absence of any magnetic ®eld given by Kushwaha [720] …1 ‡ n21 † sinh…aA dA † sinh…aB dB † ‡ 2n1 ‰ cosh…aA dA † cosh…aB dB † ÿ cos…kd†Š ˆ 0; where k (  qz ) is the Bloch vector and other symbols are as de®ned below: ! 1 ÿ o2pi EA aB ; ai ˆ …q2 ÿ q20 Ei †1=2 ; i  A; B: ; Ei …o† ˆ ELi n1 ˆ aA EB o2

(7.1)

(7.2)

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Note that in writing the second equation in (7.2) we have ignored the damping effects. This is a general situation where both layers A and B in Fig. 38 are considered to be semiconductors. The special case where, say, layer A is a metal (or semiconductor) and layer B is a dielectric can easily be obtained from Eq. (7.1). In the former case, e.g., we simply take ELA ˆ 1 and EB as a frequency independent dielectric constant. In the latter case, we would only consider EB to be a frequency independent dielectric constant. In general, Eq. (7.1) is a transcendental function which has to be solved in a manner as stated in the preceding sections. It is, however, worth attempting to diagnose the asymptotic limits attained by the bulk bands. For this purpose, we impose the NRL …c ! 1 or q  q0 † on Eq. (7.1) to obtain EA ‡ EB ˆ 0:

(7.3)

This can be explicitly written in terms of the dimensionless frequency x (ˆ o=opA ) as follows: " #1=2 o2pB 1 ELA ‡ ELB 2 : (7.4) xˆ opA …ELA ‡ ELB †1=2 In the case that the layer A is a metal and B is a dielectric, the following substitution has to be made: ELA ˆ 1 and opB ˆ 0. On the other hand, if the layer A is a semiconductor and B is a dielectric only the substitution opB ˆ 0 is required. It is noteworthy that in this (latter) case Eq. (7.4) with opA ! op ; ELB ! E0 , and ELA ! EL simpli®es to Eq. (4.11) and hence represents the asymptotic frequency that corresponds to the surface plasmon frequency in the NRL. We will see that both bulk bands in the binary superlattices approach the asymptotic limits speci®ed by Eq. (7.4). Consider now that the perfect periodic system (Fig. 38) is truncated at z ˆ 0 such that the region ÿ1  z  0 is occupied by a third medium (``C'') whose characteristic dielectric function EC and decay constant aC are de®ned analogously. The last layer assumed to be in touch with this semi-in®nite region is considered to be the layer A. The general dispersion relation for surface excitations of such a semi-in®nite superlattice is given by [720] …1 ÿ n21 n22 † cosh…aA dA † sinh…aB dB † ‡ n1 …1 ÿ n22 † sinh…aA dA † cosh…aB dB † ‡ n2 …1 ÿ n21 † sinh…aA dA † sinh…aB dB † ˆ 0;

(7.5)

where n2 ˆ

EC aA : aC EA

(7.6)

In the NRL, Eq. (7.5) assumes the form: …EA ‡ EB †…EB ÿ EC †…EC ‡ EA † ˆ 0:

(7.7)

Either of these three factors equated to zero speci®es a certain asymptotic frequency approached by the surface mode(s) of the truncated superlattice. Equating the ®rst factor to zero is identical with Eq. (7.3) and reproduces exactly the same asymptotic limit as the one attained by the bulk bands, Eq. (7.4). Equating the second factor to zero gives " #1=2 o2pB o2pC 1 ELB 2 ÿ ELC 2 : (7.8) xˆ opA opA …ELB ÿ ELC †1=2

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In the case that layer B is a dielectric …opB ˆ 0† and the medium C is also dielectric …opC ˆ 0†, Eq. (7.8) yields no bona®de solution. In the case where layer B is a semiconductor but the medium C is a dielectric, Eq. (7.8) simpli®es to   ELC ÿ1=2 opB : (7.9) xˆ 1ÿ ELB opA When medium C is vacuum …ELC ˆ 1† and layer B is metallic …ELB ˆ 1†, Eq. (7.9) makes no sense. Even when the layer B is semiconductor and medium C is vacuum ELB  1, Eq. (7.9) yields o ! opB , which means EB ! 0. This latter condition will, in that case, violate the correct asymptotic limits attained by the bulk bands and by one of the surface modes, speci®ed by Eq. (7.4). It may thus be concluded that the second factor in Eq. (7.7) does not vanish. Equating the third factor to zero yields " #1=2 o2pC 1 ELA ‡ ELC 2 : (7.10) xˆ opA …ELA ‡ ELC †1=2 In the case that medium C is a dielectric, Eq. (7.10) simpli®es to   ELC ÿ1=2 xˆ 1‡ : ELA

(7.11)

This gives a bona®de asymptotic limit attained by one of the surface modes, irrespective of whether the layer A is a metal or a semiconductor. Note that Eq. (7.11) is identical to Eq. (4.11) and represents a surface plasmon mode that corresponds to a single semiconductor±dielectric interface in the semiin®nite geometry. We stress that the foregoing analysis is devoid of the absorption in the material media, as well as coupling to the optical phonons in case any of the material media is a polar semiconductor. In what follows, we would discuss several results on the elementary collective excitations in different types of superlattice systems, both with and without an applied magnetic ®eld in different con®gurations. 7.1. Plasmon polaritons in metal±dielectric superlattices Almost 15 years ago, Camley and Mills [708,1752] investigated the collective (bulk and surface) excitations in a binary superlattice system in the absence of an applied magnetic ®eld and excluding retardation effects. They practically studied the case where one (metallic) layer contains free charge carriers, while the other is described by a frequency-independent dielectric constant. They also explored the energy-loss spectrum of electrons backscattered from such a stack of ®lms. Through the explicit calculations of the energy-loss functions, they argued that EELS may be used to study the collective excitations of such arrays. The purpose of this section is to discuss some of their interesting results on the collective excitations and energy-loss spectrum. 7.1.1. Collective excitations For the sake of consistency, we comment that their [708,1752] parameterization is such that: ELA ˆ 1; EB and EC are frequency independent dielectric constants (opB ˆ opC ˆ 0), layer-thickness dA…B† ˆ d1…2† , period d  L; opA  op (ˆ15 meV) and they plot the absolute frequencies o (in meV). Moreover, since they worked in the NRL throughout, (our) ai ! q  (their) k; and (our) k  (their) q.

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Fig. 39. Dispersion of the collective excitations Ð bulk bands (hatched areas) and surface polaritons (solid curves) Ð in a binary metal±dielectric superlattice made up of Al ®lms (op ˆ 15 eV). The normalized Bloch vector 0  q  p speci®es the bulk band boundaries. The thickness ratio is d1 =d2 ˆ 2 with 1…2†  A…B). (After Camley and Mills, Ref. [708].)

Fig. 39 illustrates both the bulk and surface excitations, for the case EB ˆ EC ˆ 1 and d1 ˆ 2d2 , as a function of the dimensionless propagation vector kd1. (The parameters correspond to a model semiin®nite structure of aluminum ®lms separated by a dielectric spacer B with a dielectric constant EB and the semi-in®nite medium C with a dielectric constant EC ˆ 1.) We see that the bulk excitations fall into two bands separated by a gap wherein lies the only existing surface mode. Note that as one scans through the frequency spectrum of bulk modes with a ®nite propagation vector ~ k parallel to the interfaces, the modes tend to crowd together, forming a high DOS near the boundary lines with qL ˆ p. The reason that there is no other surface mode in the spectrum lies in the choice of the parameters: EB ˆ EC ˆ 1. In this case the otherwise expected two surface modes, speci®ed by Eqs. (7.4) and (7.11), are identical. Fig. 40(a) and (b) depicts the excitation spectra, for two values of the parameter kd1, as a function of the ratio d1 =d2 ; with EB ˆ EC ˆ 1. The o‡ and oÿ bulk bands are seen to be always separated by a gap, except for the situation d1 ˆ d2 , where their lower bounds (qL ˆ 0) just touch. There lies a surface mode within this gap, but it exists only when d1 > d2 ; this is a physically realistic situation as discussed in Refs. [708,1752]. A simple, but relatively more general, situation is shown in Fig. 41, which illustrates the excitation spectrum for the situation EB (ˆ3 for Al2 O3 † 6ˆ EC …ˆ 1†, and d1 > d2 . One notes that, for large values of kd1 , both o‡ and oÿ bulk bands are down-shifted in frequency by virtue of the screening provided by the oxide layer B. The upper surface mode for large values of kd1 corresponds, in accordance with Eq. (7.11), to the surface plasmon mode appropriate to the Al±vacuum interface, o ' 10:6 meV. Thus,

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Fig. 40. Dispersion of the collective excitations as a function of thickness ratio d1 =d2 for a given wave vector kd1 ˆ 1:0 (a) and 0.1 (b) for EB ˆ EC ˆ 1:0. Notice that the surface mode is absent for d1 < d2. The system is the same as in Fig. 39. (After Camley and Mills, Ref. [708].)

in contrast to the previous example (with EB ˆ EC ˆ 1), a new surface mode emerges from the upper edge of the o‡ bulk band, at kd1 ' 1:5, for this choice of d2 . The other surface mode, appearing at lower values of kd1 in the gap, merges with the lower edge of the o‡ bulk band at about the same value of kd1 at which the upper surface mode emerges.

Fig. 41. The same as in Fig. 39, but for EB ˆ 3:0 and EC ˆ 1:0. Notice the existence of two surface branches, in contrast to one in Fig. 39. (After Camley and Mills, Ref. [708].)

210

Fig. 42.

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The same as in Fig. 41, but for d1 =d2 ˆ 12. Again the material A is Al. (After Camley and Mills, Ref. [708].)

Fig. 42 demonstrates the existence of two surface modes, for EB …ˆ 3† > EC …ˆ 1†, even when d1 < d2 . The upper surface mode behaves in a manner similar to the corresponding mode in Fig. 41, except for emerging at a considerably lower value of kd1 . The surface mode lying in the gap is not seen to merge with any of the two bulk bands, rather it survives even in the limit kd1 ! 1, always trapped between the two bulk bands. 7.1.2. Electron energy loss functions In the quest of how the collective excitations of superlattice structures can be probed, Camley and Mills [708,1752] developed a simple, but reasonably suitable, theory of energy-loss spectroscopy. They argued that if one is concerned with metallic superlattices, possibly with insulating spacers between adjacent metal ®lms, then small-angle inelastic scattering may serve as a suitable experimental probe since it had already been used to study the surface excitations on the metal surfaces [1647]. Furthermore, since the losses of interest lie in range of several electron volts, very-high-resolution techniques are not required. In the regime of small-angle de¯ections, where surface±plasmon contributions dominate the loss mechanism for backscattering off the surface, the electron interacts with excitations in the substrate through the ¯uctuating electric ®elds in the vacuum outside the surface. The theory of the loss cross-section, under such circumstances, was discussed many years ago for scattering off of the surface of a semi-in®nite material [1648], or for a material upon which an optically active layer is present [230]. The summary of such theoretical treatments, along with numerous applications of the resulting formulas, can be found in a book by Ibach and Mills [1649]. Camley and Mills [708,1752] extended the theory to the superlattice structures. It was argued that they could have done so by following the quantal analysis given earlier but they followed the simple classical trajectory procedure (CTP) introduced by Schaich [1650], which reproduces one limiting form of the results provided by the full analysis. Note that this limiting form follows from a full treatment only after certain simplifying assumptions are introduced [1649]; their validity is questionable when the energy loss (of the electron) is in the range of several electron volts, but the CTP yields a formula which is believed to contain the essential features expected in the observed data.

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The CTP proceeds such that the incoming electron polarizes the substrate, viewed as a dielectric medium. The induced polarization produces an electric ®eld which does work on the electron as it approaches the crystal, and then exits after re¯ection off the surface. One calculates the total work performed by the induced ®eld to obtain the total energy loss the electron suffers, and an appropriate decomposition of this expression gives the energy distribution of those electrons which suffer an inelastic scattering. Summarizing this understanding, we write the expression for the energy lost by the electron in the form: Z 1 do  hoP…o†; (7.12) Wˆ ÿ1

where P…o† is the probability (per unit frequency) that the electron has lost the energy ho. Noting that the response function [708,1752] R…Qk ; o† ˆ 1 ÿ where



EA ‡ EB Fˆ EA ÿ EB

1 ; 1 ‡ EA …o†…1 ÿ F†=…1 ‡ F†

  exp…ÿQk d1 † ÿ exp…ÿbL† exp…Qk d2 † exp…Qk d1 † ÿ exp…ÿbL† exp…Qk d2 †

(7.13)

(7.14)

with b ˆ ÿiq and k  Qk , is an odd function of frequency, the probability function reads [708, 1752] Z Z 2p 4e2 v2? 1 2 dQk Qk Im‰R…Qk ; o†Š dy ; (7.15) P…o† ˆ 2 2 hp  ‰…v? Qk † ‡ …o ÿ Qk vk cos y†2 Š2 0 where vjj …v? † is the parallel (perpendicular) velocity component of incoming electron with prescribed rjj the projection of ~ r onto a plane parallel to the surface. The integral on y path ~ r ˆ~ rjj …t† ‡ z…t†^z, with ~ may be evaluated in the closed form [708]. Let yi be the angle of incidence of the electron beam measured relative to the normal to the surface. Then vjj ˆ v0 sin yi and v? ˆ v0 cos yi , where v0 is the speed of the incoming electron. In the calculations to be discussed in what follows, the aluminum ®lm was modeled by the choice [708]: EA …o† ˆ 1 ÿ

o2p o…o ‡ ig†

;

(7.16)

where r is the phenomenological relaxation rate. The model parameters used in the computation are: EB ˆ EC ˆ 1; op ˆ 15 eV; g ˆ 0:2 eV; yi ˆ 45 , and the incident-electron kinetic energy E0 ˆ 200 eV.  Fig. 43 illustrates the loss spectrum for two cases: The ®rst has d1 ˆ 20 and d2 ˆ 15 A , while the  second has d1 ˆ 15 and d2 ˆ 20 A . One clearly notices a prominent peak at 10.6 eV for the former case, while only a hole exists for the latter, near the surface±plasmon frequency (10.6 eV) corresponding to Al±vacuum interface. The broad feature which rises dramatically with decreasing energy loss has its origin in scattering off of the continuum of bulk excitations which, as one notes from Fig. 39, extend down to very low frequency when Qjj d1 is small. This feature should be evident in data as a broadening of the quasi-elastic peak, which varies as the thickness of the Al ®lm is changed. Such a

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Fig. 43. Electron-energy loss cross-section for near-specular scattering of 200 eV electrons off of a semi-in®nite superlattice   of very thin Al ®lms interspersed with vacuum. Two cases are considered: d1 ˆ 15, d2 ˆ 20 A; and d1 ˆ 20, d2 ˆ 15 A. (After Camley and Mills, Ref. [708].)

broadening of the quasi-elastic peak was also noticed in later studies of thin ®lms of Ag deposited on GaAs [1651]; there the whole feature was seen to observe a down-shift to much lower energies by virtue of the smaller bulk-plasmon frequency of Ag and by softening of the low-frequency collective modes of Ag produced by the screening of the electric ®elds by the GaAs substrate. Note that with increasing Al-®lm thickness, the intensity of the quasi-elastic background decreases for ®lms thicker than a few monolayers, just as in the earlier example [1651]. It is becoming well known that as one scatters electrons off a semi-in®nite sample of Al, the loss mechanism considered here leads to a loss peak at the frequency of the surface plasmon at Al±vacuum interface, 10.6 eV for the model considered here. It is interesting to inquire how such a spectrum evolves from that given in Fig. 43 if d1 is increased, but the ratio d2 =d1 held ®xed at a value greater than unity. The point is that in the limit d1 ! 1, we must observe a loss peak at 10.6 eV as the only prominent feature in the spectrum, but the earlier discussion shows we never have a surface mode at any ®nite value of d1 if d1 < d2 , and EB ˆ EC …ˆ 1†. This point is explored in Fig. 44, where we demonstrate the loss spectrum for several values of d1 with the ratio d1 =d2 held ®xed and d1 increasing  to rather large values. For d1 ˆ 120 A, we see a shoulder near 2.5 eV produced by scattering off the oÿ branch of bulk excitations. As d1 increases, this shoulder evolves into a prominent peak, and moves to progressively higher energy, but always below 10.6 eV. Similarly, scattering from o‡ branch produces a loss peak above 10:6 eV which softens as d1 decreases, always remaining above 10.6 eV. As d1 increases, the two peaks (below and above 10.6 eV) coalesce, and the gap between them shows up. What is left in the limit d1 ! 1 with d2 =d1 …> 1† ®xed is a single feature centered at 10.6 eV with a width controlled by the damping factor g. Note that as d1 increases, the low-energy tail in the loss

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Fig. 44. Electron-energy loss cross-section for near-specular scattering of 200 eV electrons from a semi-in®nite superlattice of very thin Al ®lms with vacuum in between. In all cases, thickness ratio d2 =d1 ˆ 43 is held ®xed. (After Camley and Mills, Ref. [708].)

spectrum decreases dramatically, so that very little broadening of the quasi-elastic peak is expected for scattering off thick Al ®lms. With a comparison of the dramatic differences between the energy-loss spectra presented in Figs. 43 and 44, one may appreciate that even when the Al ®lms is quite thick, the loss spectrum is in¯uenced by the whole structure and not just by the properties of its outermost constituent. This point was further reinforced through extensive computation of the loss spectra even for the case d2 < d1 in Refs. [708,1752]. 7.2. Plasmon polaritons in semiconductor±dielectric superlattices In this section, we will discuss the collective (bulk and surface) excitations of a binary superlattice system (Fig. 38) whose one layer is a semiconductor and the other a dielectric, both with and without an applied magnetic ®eld in the Voigt con®guration. The underlying physical aspects are such that the retardation is included, but the damping, spatial dispersion and quantum-size effects are neglected. In other words, we have the same kinds of system as discussed in the preceding section where the material layers are characterized by macroscopic dielectric function in the local approximation, except for the inclusion of the retardation effects and an applied magnetic ®eld (in qjj^y). The semiconducting layer (GaAs) is assumed to contain the Voigt con®guration; ~ B0 jj^x; ~ conduction electrons with effective mass m ˆ 0:07m0 , where m0 is the free-electron mass, and to have a background dielectric constant EL ˆ 13:1; the dielectric (intrinsic Ga1ÿx Alx As) layer has the background dielectric constant EL ˆ 12:4. We will introduce a dimensionless layer thickness d ˆ op d=c, frequency x ˆ o=op, and propagation vector z ˆ cq=op. The cyclotron frequency

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involving applied magnetic ®eld is assumed to be oc ˆ 0:25op , for all calculations. The medium (C) that ®lls the space ÿ1  z < 0 is assumed to be vacuum (i.e., EC ˆ 1). We examine situations where either GaAs or Ga1ÿx Alx As constitutes the surface layer (A) and situations where GaAs layer thickness is twice that of the Ga1ÿx Alx As layer thickness or vice versa. We now discuss the results for various cases considered. 7.2.1. Zero magnetic ®eld We ®rst present the dispersion relation for plasmon±polaritons in the absence of an external magnetic ®eld, when the surface layer A is Ga1ÿx Alx As and B is GaAs, and dA =dB ˆ 0:5. All the numerical results will be plotted strictly towards the right of the light line in the layer Ga1ÿx Alx As, irrespective of whether or not this is the surface layer. The numerical results in terms of the dimensionless variables are shown in Fig. 45. There is a single, doubly degenerate surface polariton mode that lies in the gap between the bulk bands. This mode starts from the light line and becomes asymptotic to a frequency speci®ed by Eq. (7.4), by interchanging suf®x A and B, i.e., x ' 0:7167. It is noteworthy that this is the only surface mode in this case, since the other surface mode substantiated (in the NRL) by Eq. (7.11) is not allowed. This is because the third factor (EA ‡ EC ) in Eq. (7.7) does not vanish in this case. Next, we turn to the case where Ga1ÿx Alx As is still the surface layer, but the GaAs layers are half as thick as the Ga1ÿx Alx As layers (dA =dB ˆ 2:0). Such numerical results are plotted in Fig. 46. In this case it is seen that the gap between the bulk bands is divided into two regions, due to the touching point at the reduced wave vector z ' 2:912. This touching point, in the o±q plane, is explicable by the Brewster effect, which occurs when the so-called ``Brewster decay constant'' aBR ˆ …q2 ÿ q20 EB sin…yBR ††1=2 ˆ 0, where the Brewster angle yBR ˆ tanÿ1 …EA =EB †. The corresponding solutions are shown by the dashed curve in green. It has been noted that in the case when the bulk bands touch in this manner, the bona®de surface modes between the bulk bands exist either towards the left or the right of this touching point. In the present situation the doubly degenerate surface mode exists towards the left of this touching point (shown by the solid curve in pink) and terminates at the light line. The case when GaAs is the surface layer (A), and dA =dB ˆ 0:5, is illustrated in Fig. 47. We ®nd that the bulk bands exhibit similar behavior as seen in Fig. 46, but the surface modes are quite different. There are two doubly degenerate surface-polariton branches: One branch lies in the gap between the bulk bands, starting at the touching point of the band edges and extending out to large values of the propagation vector; it does not merge with any of the bulk bands and approaches the same asymptotic limit as the one attained by the bulk bands. The second surface mode lies above the upper bulk band; it starts from the light line and extends out to large propagation vector. The asymptotic frequency attained by this higher surface mode is speci®ed, for the parameters used, by Eq. (7.11) and is given by x ' 0:9636. Finally, we present the results when GaAs is still the surface layer, but dA =dB ˆ 2. The numerical results are depicted in Fig. 48. We ®nd that there is no surface branch lying in the gap between the bulk bands, instead there is a one that lies above the upper bulk band, starting at the light line and extending out to large wave vector. It thus seems that as the ratio of the active-medium-layer thickness to inactivemedium-layer thickness increases, the touching point of the bulk bands moves towards the larger wave vector, carrying the surfaces mode in the gap with it. This seems to be the reason of the absence of the surface mode in the gap for the case where the active±inactive layer thickness ratio is greater than one.

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Fig. 45. Dispersion of collective excitations for bulk (hatched regions in blue) and surface (solid curves in pink) in zero magnetic ®eld. GaAs layers are twice as thick as the Ga1ÿx Alx As layers, and Ga1ÿx Alx As is the surface layer. The dashed line in black is the light line in the Ga1ÿx Alx As. Background dielectric constant EL ˆ 13:1 (12.4) for GaAs …Ga1ÿx Alx As†. (Redone after Wallis and Quinn, Ref. [719].) Fig. 46. The same as in Fig. 45, but when GaAs layers are half as thick as the Ga1ÿx Alx As layers, and Ga1ÿx Alx As is the surface layer. The dashed curve in green represents the Brewster dispersion curve aBR ˆ ‰q2 ÿ q20 EB sin…yBR †Š1=2 ˆ 0, where the Brewster angle yBR ˆ tanÿ1 …EA =EB †. (Redone after Wallis and Quinn, Ref. [719].) Fig. 47. The same as in Fig. 45, but when Ga1ÿx Alx As layers are twice as thick as the GaAs layers, and GaAs is the surface layer. The dashed curve in green is the Brewster dispersion curve. (Redone after Wallis and Quinn, Ref. [719].) Fig. 48. The same as in Fig. 45, but when Ga1ÿx Alx As layers are half as thick as the GaAs layers, and GaAs is the surface layer. (Redone after Wallis and Quinn, Ref. [719].)

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7.2.2. Non-zero magnetic ®eld qjj^y), the dispersion In the presence of an applied magnetic ®eld in the Voigt con®guration (~ B0 jj^x and ~ relation for the magnetoplasmons in the in®nite (perfectly periodic) superlattice reads [719] 2…TA‡ ÿ TAÿ †…TB‡ ÿ TBÿ † exp…ÿaA dA † exp…ÿaB dB † cos…Q†

‡ …TA‡ TAÿ ‡ TB‡ TBÿ †…1 ÿ exp…ÿ2aA dA ††…1 ÿ exp…ÿ2aB dB †† ˆ …TA‡ ÿ TAÿ exp…ÿ2aA dA ††…TB‡ ÿ TBÿ exp…ÿ2aB dB ††

‡ …TAÿ ÿ TA‡ exp…ÿ2aA dA ††…TBÿ ÿ TB‡ exp…ÿ2aB dB ††;

(7.17)

where Q ˆ kd is the dimensionless Bloch vector, … j† Tj ˆ E…yyj† S j ‡ iEyz ;

S j ˆ

…qE…yyj†  iaj E…yzj† † … j† …ÿiqEyz



… j† aj Eyy †

;

a2j ˆ q2 ÿ q20 E…v j† ;

(7.18)

and E…v j†

ˆ

E…yyj†

‡

…E…yzj† †2 … j†

Eyy

(7.19)

is the Voigt dielectric function, with j  A, B. The quantities aj is the decay constant for the ®eld amplitudes in the presence of an external magnetic ®eld. All the symbols in Eqs. (7.18) and (7.19) are written according to our geometrical con®guration, for the sake of consistency. For general purposes, we have assumed that both layers A and B contain free charge carriers. For truncated superlattice in the presence of an external magnetic ®eld in the Voigt geometry, the dispersion relation for the surface-magnetoplasmon±polaritons has the form [727]: …TC ‡ TBÿ †‰…TC ‡ TAÿ †…TB‡ ÿ TA‡ † ÿ …TC ‡ TA‡ †…TB‡ ÿ TAÿ †exp…ÿ2aA dA †Šexp…ÿ2aB dB † ˆ …TC ‡ TB‡ †‰…TC ‡ TAÿ †…TBÿ ÿ TA‡ † ÿ …TC ‡ TA‡ †…TBÿ ÿ TAÿ †exp…ÿ2aA dA †Š;

(7.20)

where TC ˆ q…EC =aC †. This relation is equivalent to, but simpler than, the dispersion relation given by Wallis and coworkers [718]. It is evident that Eq. (7.20) should exhibit non-reciprocal behavior, i.e., o…ÿq† 6ˆ o…‡q†. In order for understanding the asymptotic (non-retardation) limits, it is worth commenting that such limits can quite well be understood through Eqs. (5.6) and (5.7). This remark is valid both for the bulk bands and the surface magnetoplasma modes. More precisely, Eq. (7.17), in the NRL, simpli®es to …TA‡ ÿ TBÿ †…TAÿ ÿ TB‡ † ˆ 0:

(7.21)

Similarly, Eq. (7.20), in the NRL, assumes the following form: …TA‡ ÿ TBÿ †…TBÿ ‡ TC †…TAÿ ‡ TC † ˆ 0:

(7.22)

A careful substitution of the quantities involved reproduces, from Eqs. (7.21) and (7.22), equations analogous to Eqs. (5.6) and (5.7). This then leads one to infer that bulk bands, in the NRL, are separated in magnitude by joc j (the cyclotron frequency); the same is true for the pair of surface modes in the presence of an external magnetic ®eld. We now present the results for the various (analogous to the zero-®eld) cases considered.

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We ®rst present the results for the case where surface layer A is Ga1ÿx Alx As and B is GaAs, and dA ˆ 0:5, dB ˆ 1:0. The numerical results for oc ˆ 0:25op and for both directions of propagation are illustrated in terms of dimensionless variables in Fig. 49. What is evident is that, in the presence of a magnetic ®eld, the gap between the bulk bands widens, particularly at large wave vectors, and the surface mode (shown in pink) splits into two branches Ð one for the ‡q, and the other for ÿq. For the surface mode along ÿq, the external magnetic ®eld, the direction of propagation, and the normal to the surface form a right-hand triple, whereas for the surface along ‡q, they form a left-hand triple. This can be viewed involving opposite directions of propagation of the surface polariton, and the fact that they are inequivalent is an indication of the non-reciprocity produced by the applied magnetic ®eld in this Voigt con®guration. The positive and negative surface branches asymptotically approach the upper (x ' 0:8525) and lower (x ' 0:6025) bulk bands, respectively. Next, we turn to the case where Ga1ÿx Alx As is still the surface layer, but its thickness is double of the thickness of GaAs layer; dA ˆ 1:0, dB ˆ 0:5.The numerical results are illustrated in Fig. 50, for oc ˆ 0:25op . An outright consequence of an external magnetic ®eld is that the gap widens, and the touching point of the bulk bands disappears. The surface branch (in the absence of a magnetic ®eld) now (in the presence of a magnetic ®eld) splits into two, just as in Fig. 49. These branches, along both directions of propagation, extend from the light line (aA ˆ 0) toward large wave vectors until they merge in the bulk continuum at z ' 3:0. There are no surface polaritons in the NRL in either the presence or absence of an applied magnetic ®eld. The case when the surface layer A is GaAs and B is Ga1ÿx Alx As is depicted in Fig. 51, for dA ˆ 0:5 and dB ˆ 1:0. Application of an external magnetic ®eld causes the removal of the touching point of the bulk bands (see Fig. 47) and each surface branch splits into two branches. The positive and negative surface branches in the gap emerge from the lower (at z ' 3:15) and upper (at z ' 2:42) bulk bands, respectively, and extend out within the gap to considerably large wave vector, though propagating in the close vicinity of the respective bulk bands. The upper surface branch along ‡q…ÿq† has the positive (negative) group velocity, and their asymptotic limits are de®ned by x ' 0:8525…q < 0† and x ' 1:1025…q > 0†. Both of these (upper) branches start at the light line and extend out to the large wave vectors on both sides. Finally, we present the results when the GaAs is still the surface layer, but dA ˆ 1:0; dB ˆ 0:5. The numerical results are illustrated in Fig. 52. Comparing Fig. 52 with Fig. 48 reveals that the application of an external magnetic ®eld results in widening the gap between the bulk bands, and that both edges of the upper bulk band now show a positive group velocity, instead of negative (in the absence of an applied ®eld). Moreover, the surface branch now splits into two, the positive branch has higher frequency than the negative branch, and they both exhibit positive group velocity. Their asymptotic limits are just as de®ned above, and they both start from the light line and extend out to large wave vectors. It should be pointed out that this whole section (i.e., Section 7.2 rests heavily on the original paper by Wallis and Quinn [719]. 7.3. Plasmon polaritons in semiconductor±semiconductor superlattices This section is devoted to discuss the collective (bulk and surface) excitations of a binary superlattice system (Fig. 38) whose both layers are semiconductors, both with and without an applied magnetic ®eld qjj^y). While the retardation effects are taken into account, the in the perpendicular con®guration (~ B0 jj^z;~ damping effects, spatial dispersion, and quantum-size effects will be neglected. Theoretical treatment in

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Fig. 49. Dispersion of the collective excitations for bulk (hatched regions in blue) and surface (solid curves in pink) in the non-zero magnetic ®eld, for oce =ope ˆ 0:25. Left (right) panel corresponds to q < 0 (q > 0). The GaAs layers are twice as thick as the Ga1ÿx Alx As layers, and Ga1ÿx Alx As is the surface layer. Dashed lines demarked as L‡ and Lÿ are the light lines in Ga1ÿx Alx As. Non-reciprocity in the propagation of surface modes in the q < 0 and > 0 directions is noteworthy. (Redone after Wallis and Quinn, Ref. [719].) Fig. 50. The same as in Fig. 49, but when GaAs layers are half as thick as the Ga1ÿx Alx As layers, and Ga1ÿx Alx As is the surface layer. (Redone after Wallis and Quinn, Ref. [719].) Fig. 51. The same as in Fig. 49, but when Ga1ÿx Alx As layers are twice as thick as GaAs layers, and GaAs is the surface layer. (Redone after Wallis and Quinn, Ref. [719].) Fig. 52. The same as in Fig. 49, but when Ga1ÿx Alx As layers are half as thick as GaAs layers, and GaAs is the surface layer. (Redone after Wallis and Quinn, Ref. [719].)

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this section is based on the local approximation in the framework of the TMM, where material layers are characterized by macroscopic dielectric function, which becomes a tensor in the presence of an applied magnetic ®eld, with non-vanishing non-diagonal components, implying the strong coupling of the TE and TM modes. The perpendicular geometry, unlike the Voigt geometry (in the preceding section), makes the analytical results quite cumbersome, and often hampers the simple diagnosis thereof for studying, e.g., the asymptotic (non-retardation) limits. However, we have succeeded, after laborious algebraic manipulation, to substantiate the asymptotic limits approached both by bulk bands and surface modes in the presence of an external magnetic ®eld in the perpendicular con®guration. Some relevant remarks pertaining to the difference in the numerical results from those reported in the original papers [721±723] will be made at the appropriate place. The numerical results on the collective (bulk and surface) excitations in the binary semiconducting superlattices, both with and without an applied magnetic ®eld, have been obtained in three different situations with respect to the relative thickness of layer A and layer B: (i) dA > dB , (ii) dA ˆ dB , and (iii) dA < dB . Here we discuss the numerical results on the plasmon±polaritons of a binary superlattice whose both layers are semiconductors having free charge carriers. The normalization of the quantities involved is done with respect to the plasmon frequency of layer A …opA †, which is always treated to be the surface layer. Therefore our dimensionless quantities, in the absence of a magnetic ®eld, are the reduced wave vector z ˆ cq=opA, reduced frequency x ˆ o=opA, OpB ˆ opB =opA ˆ 0:5 (always), dA ˆ opA dA =c; dB ˆ opA dB =c. In the presence of an applied magnetic ®eld, the additional parameters involved in the theoretical treatment are the cyclotron frequencies of the two layers, which are de®ned such that OcA ˆ ocA =opA ˆ 0:25 and OcB ˆ ocB =opA ˆ OcA throughout. In the case of a truncated (semi-in®nite) superlattice, the truncating medium is characterized by the dielectric constant EC ˆ 1 (vacuum). The background dielectric constants of layers A and B are taken to be equal (ELA ˆ ELB ˆ EL ˆ 13:13), i.e., the semiconducting properties of the two layers differ only due to the doping concentrations. This implies that our numerical results are appropriate for the binary superlattices made up of GaAs as the single host material with differing doping concentration in layers A and B. We now discuss our numerical results for various cases considered. 7.3.1. Zero magnetic ®eld We ®rst present the results for plasmon±polaritons for the case where dA ˆ 1:0 and dB ˆ 0:5. The numerical results in terms of the dimensionless variables are illustrated in Fig. 53. The hatched regions (with blue boundaries) and thick curves in pink refer to the bulk bands and surface polaritons, respectively. The dashed line demarked as aC ˆ 0 is the light line in the truncating medium. Note that both surface branches start from the light line and extend out to large wave vector. Both surface modes touch the edges of the upper bulk band (at z ' 0:9) to change the sign of their group velocities. The asymptotic limits attained by the bulk bands and the gap-surface mode is the same and is speci®ed by Eq. (7.4). The upper surface polariton approaches the asymptotic limit speci®ed by Eq. (7.11). The asymptotic limit attained by the upper surface mode is thus identical to the surface plasmon±polariton corresponding to the layer A±vacuum interface, and is thus unaltered by the plasma frequency of layer B as well as by the stack of the layers. Next case we consider is when the thickness of layer A is equal to that of layer B; speci®cally, dA ˆ dB ˆ 1:0. The other parameters are the same as in Fig. 53. The numerical results are shown in Fig. 54. We notice that the gap between the bulk bands reduces as compared to that in Fig. 53, and both surface modes seem to start practically from the edges of the upper bulk band almost at their

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intersection with the light line. The gap surface mode, although apparently seems to be trapped between the two bulk bands, remains a bona®de mode up to a large wave vector. In addition, the band-width of the upper bulk band decreases with increasing layer thickness of the rarer medium (the layer B). This could well be understood in view of the fact that the large gaps between the denser ®lms reduces the coupling strength between the surface excitations on each such ®lm, which in turn reduces the bandwidth of the bulk excitations, a result similar to the reduction in band-width for electronic states when the atoms are moved farther apart. Our earlier remark on the asymptotic limits attained both by the bulk bands and the surface polaritons still remains valid. Finally, we present the results for the case where the thickness of layer A is smaller than that of layer B; speci®cally, dA ˆ 1:0 and dB ˆ 2:0. The numerical results are depicted in Fig. 55. The second dashed line demarked as aB ˆ 0 is the light line in the dispersive material medium B (or A). Note that both gap and the bandwidth of the upper bulk band are further compressed for the reasons stated above. In addition, the bulk bands seem to exhibit a touching point at the reduced wave vector z ' 2:55; however, the gap surface mode, though trapped between the two bulk bands, remains a well-de®ned excitation. The upper surface mode, above the upper bulk band, now really starts from the light line in layer A (or B); towards the left of this light line this surface mode is a guided wave rather than a true surface mode. This reminds us of our earlier remark (see Section 7.1) that the upper surface mode requires dA  dB for its proper existence. We have shown these guided modes between the two light lines just for the sake of completeness; they could be washed out, however. A little carelessness in the computation could lead one interpret them as part of the bona®de surface mode, which actually starts from at the light line in medium A (or B) in the present situation. 7.3.2. Non-zero magnetic ®eld In the presence of an applied magnetic ®eld in the perpendicular con®guration (~ B0 jj^z;~ qjj^y) the dispersion relation for the magnetoplasmons in the in®nite (perfectly periodic) superlattice, in the framework of TMM, reads [723] " # ! 4 4 X X 2 Tii cos…Q† ‡ …Tii Tjj ÿ Tij Tji † ÿ 2 ˆ 0; (7.23) 4 cos …Q† ÿ 2 iˆ1

i6ˆjˆ1

where Tij are the matrix elements of the transfer matrix [723]. For ®xed q (the propagation vector), Eq. (7.23) yields the frequency bands when o is plotted versus the dimensionless Bloch vector Q ˆ kd, where k  qz and d is the superlattice period. For truncated superlattice in the presence of an applied magnetic ®eld in the perpendicular con®guration, the dispersion relation for the magnetoplasmon±polaritons has the form [723] …a1 b2 ÿ a2 b1 †…b1 c2 ÿ b2 c1 † ÿ …c1 a2 ÿ c2 a1 †2 ˆ 0;

(7.24)

where ai ; bi and ci , in terms of the matrix elements of the transfer matrix, are as de®ned in Ref. [723]. Eq. (7.24) constitutes the implicit dispersion relation for the magnetoplasmon±polaritons in the truncated binary superlattice system whose both layers A and B are assumed, for the sake of generality, to be different semiconductors. We point out that both Eqs. (7.23) and (7.24) involve q as its even powers implying that reciprocity holds, i.e., the propagation along ‡q and ÿq are equivalent. It is worth pointing out that Eqs. (7.23) and (7.24) are, respectively, implicitly the same as Eqs. (21) and (34) in Ref. [721]. However, it seems to be a ``hard nut to crack'' to transform Eqs. (21) and (34) in

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Ref. [721] to the present Eqs. (7.23) and (7.24), respectively, or vice versa. Note that the proper classi®cation of the magnetoplasmon±polaritons here requires relatively more stringent physical conditions than in the case where a single layer is a semiconductor, instead of both of them. This is because in the case when both layers A and B are semiconductors, there are two magnetic-®elddependent decay constants in each of them; speci®cally, aA and aB (see, e.g., Eq. (4.21) in Section 4). We are interested in the situation where q is real and the absorption is neglected. We will con®ne our attention to the solutions for which the EM ®elds are localized at and exponentially decay away from the interfaces. The magnetoplasma modes with real q and aC (the decay constant in the truncating medium) may, however, be further classi®ed according to the nature of ai , i  A, B. Depending upon the spectral range in the o±q plane the following possibilities may arise: (i) ai‡ and aiÿ are both real and positive, (ii) ai‡ and aiÿ are both pure imaginary, (iii) ai‡ is real and aiÿ is pure imaginary, or vice versa, and (iv) ai‡ and aiÿ are complex conjugates of each other. Just as before, we can term the magnetoplasma modes corresponding to the above-mentioned scheme as surface polariton modes, guided modes, pseudo-surface modes, and generalized modes. This classi®cation can help us understand and specify the behavior of the magnetoplasma modes in the spectral region with respect to both of the constituent layers (i  A, B). The question, however, is: why should we, in general, expect that aA and aB will have their identical nature in order for the above classi®cation to be valid? The answer is de®nitely dubious, because it may and may not occur. In view of this, we have simply searched the zeros of the real parts of the transcendental functions, Eqs. (7.23) and (7.24), irrespective of whether or not their imaginary parts are zero. We have found that this scheme has provided us quite a reasonable answer, since we have been able to substantiate all the magnetoplasma modes, both bulk bands and surface modes, through the analytical diagnosis of both of the dispersion relations in the NRL (c ! 1), as discussed below. In the NRL, a careful analytical diagnosis of Eq. (7.23) leads to obtain  A 1=2  B 1=2 A Exx B Exx ‡Ezz B ˆ 0: (7.25) Ezz A Ezz Ezz Now because of our convention that q > 0 and ai > 0, we must choose the positive root B (Eixx =Eizz †1=2 > 0. Eq. (7.25) thus makes a sense only when EA zz and Ezz have opposite signs. We decide to B A take Ezz > 0, so that Ezz < 0, which implies a well-de®ned propagation window speci®ed by EA xx < 0, A B B Ezz < 0 ) ocA  o < opA , and Exx > 0, Ezz > 0 ) o  oHB . Eq. (7.25) can be explicitly written as 1 x ˆ p ‰1 ‡ O2cA ‡ O2pB Š1=2 : (7.26) 2 In writing this equation we have used our convention: ELA ˆ ELB and OcA ˆ OcB . We will see later that Eq. (7.26) correctly substantiates the asymptotic limits attained by the magnetoplasma bulk bands in all the numerical examples. Similarly, imposing NRL on Eq. (7.24) results, after careful laborious analytical diagnosis, into three independent physical conditions, namely A 1=2 ‡ EC ˆ 0; …EA xx Ezz †  A 1=2  B 1=2 A Exx B Exx ‡Ezz B ˆ 0; Ezz A Ezz Ezz

(7.27) (7.28)

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Fig. 53. Dispersion of the collective bulk (hatched regions in blue) and surface (solid curves in pink) excitations in binary semiconductor±semiconductor superlattice in zero magnetic ®eld. The parameters used are: ELA ˆ ELB ˆ 13:13, EC ˆ 1:0, opB ˆ 0:5opA , dA ˆ 1:0, and dB ˆ 0:5. The dashed line in black is the light line in vacuum (the truncating medium C in the region ÿ1  z  0). (After Kushwaha, Ref. [723].) Fig. 54. The same as in Fig. 53, but for dA ˆ dB ˆ 1:0. Notice the reduction in the gap-width as compared to that in Fig. 53. (After Kushwaha, Ref. [723].) Fig. 55. The same as in Fig. 53, but for dA ˆ 1:0, and dB ˆ 2:0. Notice some radiative modes in pink towards the left of the bulk boundary aB ˆ 0; their existence is attributed to the fact that dA < dB . We also call attention to the touching point of the bulk band at z ' 2:55. (After Kushwaha, Ref. [723].) Fig. 56. Dispersion of the collective magnetoplasmon excitations for bulk (hatched regions in blue) and surface (solid curves in pink) in binary semiconductor±semiconductor superlattice in the presence of a magnetic ®eld in the perpendicular geometry (~ B k ^z axis). The parameters used are: ELA ˆ ELB ˆ 13:13, EC ˆ 1:0, opB ˆ 0:5opA , ocA =opA ˆ 0:25, ocB ˆ ocA , dA ˆ 1:0, and dB ˆ 0:5. The dashed line in black labeled as aC ˆ 0 is the light line in vacuum. The dashed curves in black demarked as ai ˆ 0, i  A, B, have been referred to as the bulk boundaries. There are four surface modes designated as S1 ; S2 ; S3 , and S4 ; S1 and S4 represent the bunch of surface modes which gradually widen out to attain a fork-like shape at large wave vector in the proximity of oHA and opA . (Redone after Kushwaha, Ref. [723].)

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Fig. 57. The same as in Fig. 56, but for dA ˆ dB ˆ 1:0. Notice the widening of the lowest (S1 ) and highest (S4 ) bunch of surface modes at large wave vector, and reduction in the gap-width as compared to Fig. 56. (Redone after Kushwaha, Ref. [723].) Fig. 58. The same as in Fig. 56, but for dA ˆ 1:0 and dB ˆ 2:0. The reduction in the gap-width and widening of the lowest (S1 ) and highest (S4 ) bunch of surface modes at large q are more pronounced than in Fig. 57. A sudden anomalous dip in the S3 (at z ' 2:5) and in the top of S4 (at z ' 2:75) is noteworthy. The touching point of the bulk bands at z ' 2:55, just as in Fig. 55, is also noticeable. (Redone after Kushwaha, Ref. [723].)

and EA zz



EA xx EA zz

1=2

ÿEBzz



EBxx EBzz

1=2 ˆ 0;

(7.29)

where all the quantities have their usual meanings. Solving Eq. (7.27) yields 8 #1=2 9    "   < = 2 2 2 2 1 EC EC EC 2 4 x2 ˆ 2 ‡ O : 1 ÿ 1 ÿ ‡4  O cA cA ; 2…1 ÿ E2C =E2L † : E2L E2L E2L

(7.30)

This equation is exactly identical to Eq. (4.24) and hence it represents, in the NRL, the surface magnetoplasmon±polariton appropriate for the interface between media A and C. Thinking of the valid solutions of Eqs. (7.28) and (7.29) leads us to argue in the following way. Note that both of these equations, as such, produce, within the present convention (ELA ˆ ELB ˆ EL , ocB ˆ ocA ), an equation identical to Eq. (7.26). This means that there is a surface magnetoplasma mode that approaches the same asymptotic limit as the bulk bands. Furthermore, these Eqs. (7.28) and (7.29), are derived from the two prefactors (of two equations) where one of them is, literally speaking, (P1 ‡ Q1 )ˆ 0, and the other is …P1 ÿ Q1 † ˆ 0 [723]. These (later) equations are compatible if and only if P1 ˆ Q1 ˆ

A B B B 1=2 …EA xx ÿ Ezz †…Exx ÿ Ezz †…Exx † B A B 1=2 EA xy Exy Ezz …Ezz †

A B B A 1=2 …EA xx ÿ Ezz †…Exx ÿ Ezz †…Exx † B B A 1=2 EA xy Exy Ezz …Ezz †

ˆ 0;

(7.31)

ˆ 0:

(7.32)

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If the solutions o ˆ oci , and o ˆ opi (i  A, B) are ruled out, and none of the two parentheses in the numerator vanishes (unless j~ B0 j ˆ 0), then the reasonable answer, or so it looks like, is that EBxx ˆ 0 and EA xx ˆ 0. The former gives o ˆ oHB , (the hybrid cyclotron±plasmon frequency for the material medium B) and the latter yields o ˆ oHA (the hybrid cyclotron±plasmon frequency for the material medium A). This leads us to infer that something strange must happen at the two hybrid frequencies (oHB and oHA ). That this is so will be commented on a little later while discussing the illustrative examples in what follows. The material parameters used in the numerical work are as listed above. Since the existence of surface magnetoplasma modes in the case under consideration depends on the thickness ratio dA =dB , we have investigated three representative values of this ratio which are typical of the regimes dA > dB ; dA ˆ dB , and dA < dB , analogous to the zero-®eld case. The ®rst illustrative example in terms of dimensionless variables depicted in Fig. 56 corresponds to dA ˆ 1:0 and dB ˆ 0:5. The two bulk bands indicated by hatched regions are shown to start from the light line (demarked as aC ˆ 0) and approach the asymptotic limit (x ' 0:8101), speci®ed by Eq. (7.26), at large wave vector; the lower bulk band starts at the frequency slightly higher than opB . The lowest surface mode starts below the lower bulk band at x ' ocA , propagates just to the right of the light line, ¯attens out at x ' 0:5, and widens out after crossing the bulk boundary (aB ˆ 0) to attain a fork-like shape at higher wave vector (demarked as S1 ). The extremes of this fork lie between oPB and oHB at large wave vector. The second surface mode (designated as S2 ) starts at the light line just above the lower bulk band, propagates in the gap, and becomes trapped between the two bulk bands to approach the same asymptotic limit as that attained by bulk bands. The third surface mode (demarked as S3 ) also starts from the light line in the gap, crosses the upper bulk band, intersects twice the upper bulk boundary (aA ˆ 0), and becomes asymptotic to a frequency (x ' 0:9769) speci®ed by Eq. (7.30). In the NRL, this mode (S3 ) represents the surface magnetoplasmon±polariton propagating at the interface between the mediums A and C. The uppermost bunch of surface modes starts from the light line above the upper bulk boundary (aA ˆ 0) and gradually widens out to attain a fork-like shape at higher wave vector in the vicinity of oHA above opA , analogous to the lowest bunch of surface modes (S1 ). The uppermost dashed (horizontal) line represents some sophisticated asymptotic frequency (x ' 1:0405), which we could not substantiate through analytical diagnoses. Comparing Fig. 56 with Fig. 5 in the original paper [723] leads one to notice several similarities as well as dissimilarities, regarding the surface modes. The two forks in the vicinity of oHB and oHA are almost similar; and S3 mode in the present Fig. 56 and S21 mode in Fig. 5 in Ref. [723] are identical. However, the gap mode in the original paper was missing; similarly, the starting of the lower fork and the lower bulk boundary (aB ˆ 0) were misunderstood. Given a complicated transcendental function, represented by Eq. (7.24) here, and the different numerical schemes adopted to search the surface modes, such differences are not unexpected. In the original paper [723], the numerical results were based on the search of an absolute minimum of this complex (and complicated) transcendental function at a given wave vector q in a chosen frequency window. In the present computation, on the other hand, we searched zeros of the real part of the function, irrespective of whether or not its imaginary part was zero. It is quite likely that any better designed scheme to search reliable solutions of a complex transcendental function (like this) may allow a desired classi®cation of the surface modes, and economize the computational time as well. Any such yearned for software was beyond our reach, and we feel that we have reported considerably well-reasoned solutions because these have been substantiated, at least in the NRL, through the sophisticated analytical diagnoses.

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Next, we discuss an illustrative example for equal thickness of layers A and B; speci®cally, dA ˆ dB ˆ 1:0. The numerical results for this case are depicted in terms of dimensionless variables in Fig. 57. A few differences are noticeable with increasing thickness of the rarer medium (layer B). The gap between the bulk bands reduces, the starting of the upper bulk band is pushed towards lower frequencies, and its upper edge now exhibits a positive group velocity, unlike the negative one in Fig. 56. In addition, the bulk bands trapping the gap-surface mode close together at lower wave vector. The lower fork is seen to have a tendency to start gradually widening from at the lower wave vector, split at the higher wave vector, and become broader and broader with q; but is always seen to be surfaced by the lower bulk boundary (aB ˆ 0). The third surface mode (S3 ) now does not intersect the upper bulk boundary (aA ˆ 0). The rest of the discussion related with Fig. 56 is still valid. The last case we investigate is that where the thickness of the rarer medium B is larger than that of the denser medium A. The numerical results for this case are illustrated by the example dA ˆ 1:0 and dB ˆ 2:0 in Fig. 58. The result is a further reduction in the size of the gap between the bulk bands, and pushing down of the upper band towards lower frequencies. In addition, the bulk bands seem to touch at the reduced wave vector z ' 2:25 before (slightly) opening up again, analogous to their zero-®eld counterparts (Fig. 55). The lower fork of the surface modes now ®lls the whole space in the o±q plane between the plasmon frequency of layer B (oPB ) and the lower bulk boundary (aB ˆ 0). The third surface mode (S3 ) again intersects the upper bulk boundary (aA ˆ 0) and exhibits a strange dip at z ' x…EL †1=2 , and so does the upper boundary of the upper fork. However, the remarkable change in the behavior characteristics of the magnetoplasma modes with increasing thickness of the rarer medium (layer B) leaves the rest of the discussion related to Fig. 56 practically unaltered. 8. Plasmons in periodic multi-heterostructures The motivation in the present section is to study the collective excitations in the periodic multiheterostructures (PMHs) consisting of a unit cell made up of three constituents leading to a new concept of man-made polytype heterostructures (ABCABC . . . ; ABACABAC . . . ; ACBCAC . . ., etc.) which can never be achieved with dual-constituent system. Such triple-constituent multi-heterostructures, which appeared to offer not only a novel class of materials for scienti®c investigations but also a new breed of semiconductor devices for high-speed applications, were proposed by Esaki et al. [1652] in the early 1980s. The PMH in the present treatment is considered to be comprised of three layers of dissimilar, isotropic semiconductors: A, B, C; where A ˆ AlSb, B ˆ GaSb, and C ˆ InAs, characterized by the background dielectric constants ELA …ˆ 9:86†, ELB …ˆ 14:44†, and ELC …ˆ 11:83†. Here AlSb is a relatively wide indirect-gap semiconductor (1.6 eV) as compared to the direct-gap semiconductors, GaSb (0.68 eV), and InAs (0.36 eV). Their relative band-edge energies are such that the energy gap of AlSb extends over those of GaSb and InAs, making possible for AlSb layers to serve as potential barriers for electrons as well as holes in the multi-heterostructures and superlattices. It should be pointed out that these three semiconductors, while signi®cantly differing in their band    structures, are closely matched in lattice constants: 6:136 A (AlSb), 6:095 A (GaSb), and 6:058 A (InAs). This compatibility, crucial for heteroepitaxy, represents a rather unique situation among III±V semiconductors. We are interested in the situation where the layer-thickness (widths of the potential wells and barriers) in the resultant structure are much larger than the de Broglie wavelength so that quantum-size

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Fig. 59. Schematics of a PMH consisting of three dissimilar types of semiconductors (or metals), leading to a new concept of man-made polytype heterostructures: ABC ABC ABC . . .. The growth direction is taken to be along z-axis, the period of the system is d ˆ dA ‡ dB ‡ dC , and the wave propagates along y-axis (~ q k ^y).

effects are unimportant; i.e., the semiconductor layers can safely be characterized by macroscopic dielectric functions in the local approximation. The periodicity of the structure lies in the direction of the heterostructure growth (taken to be the ^z-axis), with period d ˆ dA ‡ dB ‡ dC , where dA , dB , and dC are widths of the layers A, B, and C, respectively (Fig. 59). The frequency (o) of the modes depends upon both ~ q…jj^y† and k…  qz †, the wave vector components parallel and perpendicular to the layers. The general dispersion relations derived both for in®nite (perfectly periodic) and truncated (semi-in®nite) superlattices, employing Maxwell's curl-®eld equations with appropriate EM boundary conditions along with the Bloch theorem, are found to be functions of the dielectric properties and size of the semiconducting layers. We have performed the computation including the retardation effects, while damping effects and spatial dispersion are neglected. In this section, we have also not incorporated, for the sake of simplicity, any external magnetic ®eld. For generality, we have considered all three layers to be doped semiconductors containing free charge carriers. The general dispersion relation for plasmon±polaritons in the ternary semiconductor PMH is given by [720] 1 1 …1 ‡ n21 † sinh…aA dA † sinh…aB dB † cosh…aC dC † ‡ …1 ‡ n22 † cosh…aA dA † sinh…aB dB † sinh…aC dC † n1 n2 1 ‡ …1 ‡ n23 † sinh…aA dA † cosh…aB dB † sinh…aC dC † n3 (8.1) ÿ 2‰ cos…kd† ÿ cosh…aA dA † cosh…aB dB † cosh…aC dC †Š ˆ 0; where n1 ˆ

EA aB ; aA EB

n2 ˆ

EB aC ; aB EC

n3 ˆ

EC aA : aC EA

(8.2)

Eq. (8.1), in the limit dC ! 0, reduces to Eq. (7.1) which represents the bulk excitations for in®nite, binary semiconducting superlattices. In the NRL …c ! 1†, Eq. 8 can be shown, after some algebraic steps, to simplify to the form …EA ‡ EB †…EB ‡ EC †…EC ‡ EA † ˆ 0:

(8.3)

M.S. Kushwaha / Surface Science Reports 41 (2001) 1±416

Either ®rst, second, or third factor is zero. Equating the ®rst factor to zero yields "   #1=2 opB 2 1 ELA ‡ ELB ; xˆ opA …ELA ‡ ELB †1=2 where x ˆ o=opA . Similarly, equating the second and third factors to zero yields " #1=2 o2pB o2pC 1 xˆ ELB 2 ‡ ELC 2 ; opA opA …ELB ‡ ELC †1=2 and xˆ

1 …ELA ‡ ELC †1=2

" ELA ‡ ELC

o2pC

o2pA

227

(8.4)

(8.5)

#1=2 :

(8.6)

In what follows, we will see that all the three bulk bands of in®nite PMH abide by the asymptotic limits speci®ed by Eqs. (8.4) ± (8.6). Consider now that the perfectly periodic system (Fig. 59) is truncated at the interface between an Atype slab and a semi-in®nite medium (in the region ÿ1  z < 0) characterized by the dielectric function ED ; this fourth medium is also considered to be dispersive, for the sake of generality. The standard procedure yields the following dispersion relation for p-polarized (TM) modes in the truncated (semi-in®nite) superlattice system [720]: ‰…1 ÿ n21 n24 † cosh…aA dA † sinh…aB dB † ‡ n1 …1 ÿ n24 † sinh…aA dA † cosh…aB dB † ‡ n4 …1 ÿ n21 † sinh…aA dA † sinh…aB dB †Š cosh…aC dC † n1 ‡ ‰n2 …n23 ÿ n24 † cosh…aA dA † cosh…aB dB † ‡ n3 …n22 ÿ n24 † sinh…aA dA † sinh…aB dB † n2 n3 ÿ n2 n4 …1ÿn23 † sinh…aA dA † cosh…aB dB † ÿ n3 n4 …1 ÿ n22 † cosh…aA dA † sinh…aB dB †Š sinh…aC dC † ˆ 0 (8.7) where

ED aA : aD EA Eq. (8.7) subjected to the NRL (c ! 1) simpli®es to the following form: n4 ˆ

(8.8)

…EB ‡ EC †…EC ‡ EA †…EA ‡ ED †…EB ÿ ED † ˆ 0:

(8.9)

Equating the ®rst factor to zero reproduces Eq. (8.5), and equating the second factor to zero yields Eq. (8.6). We equate the third factor to zero to obtain " #1=2 o2pD 1 ELA ‡ ELD 2 : (8.10) xˆ opA …ELA ‡ ELD †1=2 Equating the fourth factor to zero yields " #1=2 o2pB o2pD 1 ELB 2 ÿ ELD 2 : xˆ opA opA …ELB ÿ ELD †1=2

(8.11)

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It has been noticed, in what follows, that Eqs. (8.5), (8.6), (8.10) and (8.11) substantiate the numerical results for surface excitations in the NRL. It is worthwhile commenting that in the case that truncating medium is a dielectric (opD ˆ 0) and ELB ˆ ELD , Eq. (8.11) gives an absurd, unphysical answer. This can happen, e.g., in the case that we consider an all metallic multi-heterostructure (ELA ˆ ELB ˆ ELC ˆ 1) and when the truncating material medium is taken to be vacuum (ELD ˆ 1). We have also considered such metallic systems and noted that there the total number of surface excitations is three instead of four (in the case of semiconductor multi-heterostructures). It is noteworthy that in all the numerical examples, we have assumed the truncating medium to be vacuum (i.e., opD ˆ 0, and ELD ˆ 1). The numerical results will be plotted in terms of dimensionless wave vector z ˆ cq=opA, and frequency x ˆ o=opA. Other relevant parameters employed throughout are: opB ˆ 0:5opA , and opC ˆ 0:25opA . 8.1. Semiconducting multi-heterostructures We consider the semi-in®nite multi-heterostructure where all the three layers in the unit cell are semiconductors. The normalized thickness of layer A, which is always considered to be the surface layer, is de®ned as d ˆ opA dA =c ˆ 0:3 throughout. The ®rst illustrative example depicted in Fig. 60 corresponds to the relative thicknesses dB =dA ˆ 0:8 and dC =dA ˆ 0:6. The dashed line demarked as aD ˆ 0 is the light line in the vacuum (the truncating medium D), and the one designated as aB ˆ 0 is the light line in medium B (GaSb with highest background dielectric constant). The three shaded

Fig. 60. Dispersion of the collective excitations Ð bulk bands (hatched regions) and surface modes (solid curves) Ð for ternary semiconductor heterostructure: A  AlSb, B  GaSb, C  InAs. The parameters used are: ELA ˆ 9:86; ELB ˆ 14:14; ELC ˆ 11:83; ELD ˆ 1:0 (vacuum), d ˆ opA dA =c ˆ 0:3; dB =dA ˆ 0:8; dC =dA ˆ 0:6; opB =opA ˆ 0:5; opC =opA ˆ 0:25. The dashed line labeled as aD ˆ 0 is the light line in the vacuum. The dashed line demarked as aB ˆ 0 is also referred to as the ``light line'' in the dispersive medium B.

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regions are the bulk plasmon bands shown towards the right of the light line aD ˆ 0; their boundaries correspond to kd ˆ 0 and p, just as in the binary superlattices. The asymptotic limits of the lowest, second-lowest, and third-lowest (topmost) bulk-bands, counting from the bottom, are well speci®ed by Eqs. (8.4)± (8.6), respectively. Note that the upper edge of the topmost bulk band shows a negative group velocity, starting from the light line aD ˆ 0 until considerably large wave vector. The solid curves are the surface plasmon±polariton modes. Note that the lowest surface mode starts from the origin, propagates just towards the right of the light line (understood to be the light line in medium D unless stated otherwise), repelled from the lower edge of the lowest bulk band, and becomes asymptotic to the frequency speci®ed by Eq. (8.5). The second-lowest surface mode starts from the light line slightly above the upper edge of the lowest bulk band, propagates in the proximity of the light line until reaching the lower edge of the second-lowest bulk band, repelled from its lower edge, and becomes asymptotic to a frequency speci®ed by Eq. (8.11). The third-surface mode starts above the upper edge of the second-lowest bulk band, propagates towards the right of the light line in medium B, ¯attens below the lower edge of the topmost bulk band, and then travels (after z ' 3:8) in the (indiscernibly) close vicinity of the upper edge of the second lowest bulk band to become asymptotic to the frequency speci®ed by Eq. (8.6). The fourth (topmost) surface mode starts from the light line aB ˆ 0 above the upper edge of the uppermost bulk band (at z ' 3:13, x ' 0:9623) with negative group velocity to become asymptotic to a frequency speci®ed by Eq. (8.10). The next case we consider is when dB ˆ dC ˆ dA ; the rest of the parameters are the same as in the previous case. The numerical results in terms of the dimensionless variables are illustrated in Fig. 61.

Fig. 61. The same as in Fig. 60, but for dB ˆ dC ˆ dA. Notice the discernible change in the widths of the bulk bands and the propagation characteristics of, particularly, the third lowest (counting from the bottom) surface mode, as compared to those in Fig. 60.

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Fig. 62. The same as in Fig. 60, but for dB =dA ˆ 1:2 and dC =dA ˆ 1:6. The change in the band-widths and propagation characteristics of surface modes in this case of dA < dB < dC is noticeable, as compared to that of dA > dB > dC (Fig. 60) and dA ˆ dB ˆ dC (Fig. 61).

Evidently, the noteworthy consequences of increasing the thickness of the rarer media (B and C) are the pushing down the topmost bulk plasmon band to the lower frequencies, sudden increase in the width of the topmost bulk band, and its sharp decrease with increasing wave vector. In addition, the second surface mode touches the upper edge of the second bulk band at lower wave vector and then propagates discernibly along its upper edge. One can also note that maximum widths of the second as well as third bulk bands occur at lower wave vectors as compared to those in the previous case (Fig. 60). The lowest and the second-lowest bulk bands start at approximately opC and opB , respectively, just as in the previous case. The rest of the discussion related with Fig. 60 still remains valid. Finally, we investigate the case where thicknesses of the rarer media (layers B and C) are larger than the thickness of the denser medium (surface layer A). This is exempli®ed speci®cally for dB ˆ 1:2dA and dC ˆ 1:6dA . The numerical results are shown in Fig. 62. We notice that the topmost bulk band is further pushed down to lower frequencies, and widths of all the three bulk bands reduce with increasing thicknesses of the rarer media; the starting of the lowest and second-lowest bulk bands is seen to retain the same position as in the previous case. Furthermore, the second surface mode now touches the upper edge of the second-lowest bulk band at still lower wave vector and then makes itself considerably distinguishable, but to approach the same asymptotic limit as the one attained by the second bulk band. It is noteworthy that all the surface modes illustrated in Figs. 60±62 are bona®de plasmon±polaritons, except for the third-lowest surface mode which is not a true surface mode before diverging from the light line demarked as aB ˆ 0. This is because the third decay constant aC remains purely imaginary in this region of o±q plane, which means that the corresponding EM ®elds do not decay exponentially into the layer C.

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It is worthwhile mentioning that the theoretical results depicted in this paper reveal that both bulk and surface excitations in ternary superlattices are richer than those in the binary superlattices. These excitations in such ternary systems as studied here also reveal complete and pseudogaps between the bulk bands just as are becoming the target issues in the photonic and phononic crystals.4 Within these gaps, only surface modes, which are a consequence of truncating the perfect periodicity of the system, propagate. This then leads us to suspect that ternary superlattices are advantageous over their binary counterparts in, e.g., their use as ®lters for allowing free propagation at some frequencies while forbidding at others. This is attributed to the fact that, while in binary systems the gap between the two bulk bands is closed at large wave vectors, the same is not true for these ternary systems, as we have seen above. This is understandable, at least mathematically. In binary superlattices, the gap between the two bulk bands is almost closed at the asymptotic frequency speci®ed (in the present notations) by EA ‡ EB ˆ 0. However, the ternary systems can allow complete or pseudogaps even at very large wave vectors, since there are three different asymptotic frequencies (see Eqs. (8.4) ± (8.6)) which could be tailored so as to be widely separated from each other. Other factor that seems to be uncommon in binary and ternary superlattices is the location of the surface modes. In the binary superlattices (including those which are made up of quasi-2D layers and treated so as to consider the quantum-size effects), it is usually known that if the ratio (of the background dielectric constants of the surface layer and the truncating medium) ELA =ELD > 1…< 1† the corresponding surface mode appears above (below) the upper (lower) bulk plasmon band. In this connection, it is also noteworthy that this statement is, by all means, justi®ed if and only if opB < opA . The situation takes a different turn if opB > opA . In the later case, the aforesaid surface mode starts from the origin to appear below the lower bulk band (throughout), even though ELA =ELD > 1; and its asymptotic frequency remains independent of opB . This leads us to infer that the location of this bona®de surface mode (with respect to the bulk bands) is not independent of the relative carrier concentration (or, in general, the ratio of the relative plasmon frequency) of the constituent layers in the unit cell. The situation in the ternary superlattices is still more interesting. We ®nd that there are, in general (with ELA 6ˆ ELB 6ˆ ELC 6ˆ ELD ), four surface modes; two of them appear inside the two gaps and the other two show up outside the extremes of the bulk bands, one above the uppermost bulk band and the other below the lowest bulk band. We stress that all the four surface modes depicted in Figs. 60±62 are the bona®de modes which should be detectable in the suitable experimental techniques such as the Raman scattering, EELS, and the usual ATR. We discuss in what follows some illustrative examples in the case that such ternary systems (as discussed in this section) are made up of metallic layers and the truncating medium is still vacuum. 8.2. Metallic multi-heterostructures In the case of ternary metallic superlattices we consider the normalized thickness of layer A to be given by d ˆ opA dA =c ˆ 1:0; the other parameters are the same as used in the case of semiconductor system (in the preceding section), except that here ELA ˆ ELB ˆ ELC ˆ 1:0. The ®rst case we consider is when the layer thicknesses of the rarer media B and C are smaller than the thickness of the denser medium A; speci®cally, dB ˆ 0:8dA and dC ˆ 0:6dA . The numerical results in terms of the dimensionless variables are illustrated in Fig. 63. All the bulk and surface excitations are restricted 4

For an extensive recent review on electronic, phononic, photonic, and vibrational band structures, see Ref. [1653].

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Fig. 63. Dispersion of the collective excitations Ð bulk bands (hatched regions) and surface modes (solid curves) Ð for ternary metallic heterostructure. The parameters used are: ELA ˆ ELB ˆ ELC ˆ ELD ˆ 1:0, d ˆ opA dA =c ˆ 1:0, dB =dA ˆ 0:8, dC =dA ˆ 0:6, opB =opA ˆ 0:5, opC =opA ˆ 0:25. The dashed line labeled as aD ˆ 0 is the light line in the truncating medium taken to be the vacuum. Notice that there is no surface mode appearing above the uppermost bulk band, which could be substantiated in the NRL by Eq. (8.11). This is because Eq. (8.11) in the present situation (ELB ˆ 1 ˆ ELD ) yields x ! 1 ) o ! 1. This is in contrast to the semiconducting ternary heterostructures (see Figs. 60±62 where the fourth polariton mode appearing above the uppermost bulk band is a bona®de surface mode.

towards the right of the light line in the semi-in®nite (truncating) medium D (with ELD ˆ 1 and opD ˆ 0). The shaded regions are the bulk plasmon bands of an in®nite superlattice and solid curves are the surface plasmon±polariton modes of semi-in®nite superlattice. The lower edges of the lowest and second-lowest bulk bands start at above x ' 0:25 and x ' 0:5, respectively; this is in accordance with the fact that opB ˆ 0:5opA and opC ˆ 0:25opA . Both boundaries of each of the bulk bands start with positive group velocity, except for the upper boundary of the topmost bulk band which starts with a negative group velocity. The lowest, second-lowest, and third-lowest (uppermost) bulk bands approach the asymptotic limits speci®ed by Eqs. (8.4) ± (8.6), respectively. The lowest surface mode starts from the origin, propagates just towards the right of the light line and then below the lower edge of the lowest bulk band, and becomes asymptotic to the frequency speci®ed by Eq. (8.5). The second surface mode starts from at the light line above the upper edge of the lowest bulk band, propagates towards the right of the light line in the ®rst gap monotonously, and becomes asymptotic to the frequency speci®ed by Eq. (8.10) at higher wave vector. The third surface mode starts from the light line just above the upper edge of the second bulk band, rises in the close vicinity of its upper edge until z ' 1:4, and then propagates with a negative group velocity to become asymptotic to the frequency speci®ed by Eq. (8.6). There is no other (fourth) surface mode, which can be substantiated, in the NRL, by Eq. (8.11). This is because Eq. (8.11) in the present situation (ELD ˆ 1 ˆ ELB ) yields x ! 1 ) o ! 1.

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Fig. 64. The same as in Fig. 63, but for dB ˆ dC ˆ dA. Notice the change in the band-widths and the propagation characteristics of the surface modes as compared to the previous case of dA > dB > dC (Fig. 63).

The next case we consider is when dB ˆ dC ˆ dA ; the rest of the parameters are the same as in the previous case. The numerical results are depicted in Fig. 64. The consequence of increasing thicknesses of the rarer media is re¯ected in various qualitative differences as compared to Fig. 63. The widths of all the three bulk bands reduce due to the pushing down of the upper edges to lower frequencies. The lowest surface mode that still starts from the origin propagates in the closer vicinity of the lower edge of the lowest bulk band. There is a wider separation between the lower edge of the second bulk band and the second surface mode at lower wave vectors. The third surface mode starts from the light line above the upper edge of the second bulk band, propagates without touching its upper edge, attains maximum at lower wave vector (z ' 1:2), and then propagates with a negative group velocity until it becomes asymptotic to the frequency speci®ed by Eq. (8.6). The rest of the discussion related with Fig. 63 still remains valid. Finally, we investigate the case where thicknesses of the rarer media are larger than the thickness of the denser medium. This is illustrated in Fig. 65; speci®cally, for the case where dB ˆ 1:2dA and dC ˆ 1:6dA . One can note that the widths of bulk bands reduce still further, the separation between the second surface mode and the lower edge of the second bulk band widens, and the third surface mode now starts from the light line at much higher frequency in the gap between the second and the third bulk bands. The rest of the discussion, regarding the propagation of and asymptotic limits attained by the bulk and surface excitations, related with Fig. 63 is still valid. In simple metals and degenerate semiconductors, the presence of the surface destroys the translational invariance along the normal to the surface, and the surface plasmons in these systems can

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Fig. 65. The same as in Fig. 63, but for dB =dA ˆ 1:2 and dC =dA ˆ 1:6. The change in the band-widths and propagation characteristics of surface modes is noticeable. Compare this case of dA < dB < dC with that of dA > dB > dC (Fig. 63) and dA ˆ dB ˆ dC (Fig. 64).

always excite an electron±hole pair conserving both energy and parallel component of the wave vector and are thus subject to the Landau damping. This can be attributed to the fact that the electronic energy spectrum in a 3D model is a continuous function of the normal component of the wave vector. In semiconducting superlattices, on the other hand, the quantization (if taken into account) of the energy levels by the superlattice potential makes it impossible to conserve both the energy and the parallel wave vector in the creation of an electron±hole pair by an elementary excitation lying outside the single-particle continuum (SPC). The collective surface excitations (in superlattices) lying outside the SPC cannot, therefore, decay into a single electron±hole pair and hence do not suffer from the Landau damping. In the present macroscopic approach, we do not come across, in the o±q plane, the low-lying SPC, which is responsible for the Landau damping of the surface plasmons. In other words, we are in the range of o±q where the condition o <  hq…kF ‡ 12 q†=m is effectively not valid. However, the surface excitations discussed here do suffer from Landau damping due speci®cally to their 3D character, even if their branch does not lie inside the SPC. 9. Plasmons in doping (n±i±p±i) superlattices This section deals mainly with the collective excitations in the doping superlattices, i.e., semiconductor superlattices in a homogeneous bulk (e.g., GaAs) which is only modulated by

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periodically alternating n- and p-doping, possibly separated by undoped (intrinsic, i-) regions (n±i±p±i crystal). It is known that the superlattice potential (in the doping superlattices), which is exclusively space-charge induced band-edge modulation in contrast to the band-gap variation in the case of familiar compositional superlattices, leads to an indirect energy gap in real space. From microscopic point of view, the EMA provides a very satisfactory description. The space-charge potential varies slowly over distances on the order of the bulk lattice constant and the concerned energies of the bands are suf®ciently close to the band edges. Therefore the neglect of corrections due to nonparabolicity of the bands is justi®ed. The most important consequences are a spatial separation between electron and hole states, extremely enhanced recombination lifetimes, and the metastability of nonequilibrium free-carrier distributions which provides the unique chance of tuning the carrier concentration over a wide range by photoexcitation or by current injection. Tuning of the carrier concentration is, in turn, associated with a strong modulation of the effective energy gap of the subband structure and of the absorption coef®cient in a given n±i±p±i crystal. Because of the tunability of the carrier concentration (and hence of the subband structure) almost all electronic and optical properties of n±i±p±i systems are no longer ®xed material parameters but rather tunable quantities over a wide range for a given sample. Doping superlattices are of particular interest for the study of elementary collective excitations in dynamically 2D, but spatially 3D, many-body systems for at least two reasons. First, due to the tunability of the carrier concentration and the subband structure it is possible to investigate many-body effects as a function of carrier concentration within a single sample. Second, because of the spatial separation between electrons and holes these systems represent an ideal model environment for the observation of the non-Landau-damped acoustic plasmons [743]. These two features are seldom exhibited by compositional superlattices. Moreover, the relation between the carrier concentration and the quasi-Fermi level difference provides a convenient tool for obtaining the carrier concentration from luminescence or photo-voltage measurements [537]. It is worth mentioning that in spite of relatively less (as compared to the compositional superlattices) attention drawn by doping superlattices, the existing experimental work on GaAs n±i±p±i systems has already con®rmed a large number of theoretically predicted exotic properties [532,1730±1734]. The major part of the work on the electronic and optical properties of n±i±p±i superlattices, theoretical as well as experimental, has been pursued by DoÈhler and coworkers [532,1730±1734]. The present section is aimed at presenting some investigation of the collective (bulk and surface) excitations in the n±i±p±i superlattices in the classical limit. The classical limit used here refers to the situation in which the layers' thicknesses are large enough so that the quantum-size effects become negligible and the constituent layers can be described by macroscopic dielectric tensor (in the local approximation). Almost all the analytical results, both with and without an applied magnetic ®eld, employed to compute the dispersion relations were obtained in the framework of a TMM. We are interested to discuss mainly the results on the (bulk and surface) plasmon±polaritons in the absence [749] as well as in the presence of an external magnetic ®eld in the Voigt [754] and perpendicular [751] con®gurations. As stated before, we would only recall the short analytical results from and refer the lengthy expressions (or substitutions) to the original papers. It is noteworthy that while the whole formalism, particularly in Refs. [749,751], is quite general, where the unit cell of the superlattice is assumed to be comprised of four different semiconducting layers (all characterized by frequencydependent dielectric functions), the computation is speci®ed for the n±i±p±i superlattices. The schematics of the n±i±p±i superlattice considered in the present investigation shown in Fig. 66.

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Fig. 66. Schematics of the n±i±p±i superlattice geometry considered in Section 9. Material layers A, B, C, and D are ndoped, intrinsic, p-doped, and intrinsic, respectively, with respective thickness di ; i  A, B, C, D. The integer n refers to the number of unit cell. The semi-in®nite medium E (ÿ1  z  0) is an insulator.

9.1. Zero magnetic ®eld In the absence of an external magnetic ®eld, the general dispersion relation for collective (bulk) excitations of the four-layer in®nite superlattice, in the framework of TMM, reads [749] $

cos…Q† ÿ 12 Tr T ˆ 0;

(9.1)

where Q ˆ kd, with k  qz (the normal component of the wave vector along the growth axis) and the ‡ dD , is the dimensionless Bloch vector. The symbol Tr stands for the trace of period d ˆ dA ‡ dB ‡ dC $ the 2  2 transfer matrix T (see, e.g., Ref. [749]). Eq. (9.1) has been examined by subjecting it to the special limits, viz., dB ˆ dD ˆ 0. It can readily be seen that Eq. (9.1), for these limits, reproduces exactly the dispersion relation for the collective (bulk) excitations in a binary semiconducting superlattice (see, e.g., Eq. (7.1) in this review). In the NRL (c ! 1), Eq. (9.1) simpli®es to the form [749] …EA ‡ EB †…EB ‡ EC †…EC ‡ ED †…ED ‡ EA † ˆ 0;

(9.2)

where Ej ( j  A±D) is, in general, a frequency-dependent, local dielectric function. Let us specify the

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foregoing analytical results applicable to the n±i±p±i superlattice made up of a single host material with background dielectric constant EL ˆ ELA ˆ ELB ˆ ELC ˆ ELD . For this purpose, we assume that layers A and C are n- and p-doped, respectively, and B and D are the intrinsic semiconductors. Consequently, the plasma frequency opA  ope, opC  oph , and opB ˆ 0 ˆ opD . As such EB ˆ ED and Eq. (9.2) yields two possible, independent solutions (9.3) EA ‡ EB ˆ 0; EC ‡ EB ˆ 0:

(9.4)

Since we will be interested in the numerical results in terms of the dimensionless frequency x ˆ o=ope (and dimensionless wave vector z ˆ cq=ope ), Eqs. (9.3) and (9.4) are simpli®ed to give 1 x ˆ p ; (9.5) 2 1 oph : (9.6) x ˆ p 2 ope We will see in what follows that Eqs. (9.5) and (9.6), for the parameters used in the computation, reproduce correctly the asymptotic frequencies approached by the upper and the lower pairs of bulk bands for an in®nite n±i±p±i superlattice, at large propagation vector. Next, we consider the semi-in®nite superlattice system where the superstructure is truncated at z ˆ 0 such that the surface layer A remains in the immediate contact with the insulator medium E, characterized by a frequency-independent dielectric constant EE , in the region ÿ1  z < 0. We seek solutions to Maxwell's equations in which EM ®elds are localized at and exponentially decay away from each interface of the superlattice as well as at the interface between the surface layer (A) and the semi-in®nite insulator (E). The general dispersion relation for the surface plasmon±polaritons in such a semi-in®nite, four-layer superstructure reads [749] b…T11 ‡ bT12 † ˆ T21 ‡ bT22 ;

$

(9.7)

where Tij 's are the elements of the 2  2 transfer matrix T and the symbol b stands for the substitution [749], nA ÿ nE ; (9.8) bˆ nA ‡ nE

where nj ˆ Ej =aj and aj ˆ …q2 ÿ q20 Ej †1=2 is the decay constant appropriate to the medium j characterized by the frequency-dependent dielectric function Ej . Eq. (9.7), in the spatial limits dB ˆ dD ˆ 0 reproduces exactly Eq. (7.5) that represents the dispersion relation for the collective (surface) excitations in a binary semiconducting superlattice. Just as the bulk bands, the surface plasmon±polaritons in the semi-in®nite superlattice become asymptotic to certain characteristic frequencies in the non-retardation limit. Let us now focus on the analytical diagnoses of the exact dispersion relation Eq. (9.7), in order to understand the asymptotic limits attained by the surface excitations. We ®nd that Eq. (9.7) in the NRL assumes the form [749] …EA ‡ EB †…EB ‡ EC †…EC ‡ ED †…EA ‡ EE †…ED ÿ EE † ˆ 0:

(9.9)

Let us recall that material layer B  D (intrinsic semiconductor) and the truncating (semi-in®nite) insulator medium E 6ˆ B  D. In view of this, we arrive at the conclusion that the ®rst factor equated to zero yields, Eq. (9.5). Similarly, the second (or third) factor equated to zero yields Eq. (9.6). Equating

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fourth factor to zero simpli®es to   EE ÿ1=2 xˆ 1‡ ; EL

(9.10)

where EL is the background dielectric constant of the host Ð doped or undoped Ð semiconductor medium. The ®fth factor, in Eq. (9.9), does not vanish since ED 6ˆ EE . We now present some illustrative numerical examples for the case where thickness of the p-doped medium (layer C) is varied, and refer the reader for extensive numerical results to the original paper [749]. We will consider GaAs (with is made of. The plasma frequency EL ˆ 13:13) as the single host semiconductor the n±i±p±i superlattice p of layer C is assumed to be such that OpC ˆ opC =opA ˆ 1= 2, throughout. The ®rst case we consider is demonstrated by specifying the normalized thickness dA ˆ opA dA =c ˆ 0:5, dB ˆ opA dB =c ˆ 1:0, dC ˆ opA dC =c ˆ 0:5, and dD ˆ opA dD =c ˆ 1:0. The numerical results in terms of dimensionless variables x and z are illustrated in Fig. 67. The hatched regions refer to the bulk plasmons bands and the solid curves (in pink) to the surface plasmon±polaritons. One can notice that the lower pair of bulk bands starts from the origin, rises towards the right of the light line in medium B (or D), and becomes asymptotic to the frequency (x ˆ 0:5) speci®ed by Eq. (9.6). The other two bulk bands of the upper pair shown to start from the light line (aE ˆ 0) in the semi-in®nite medium (taken to be the vacuum) approach the asymptotic limit (x ˆ 0:7071) speci®ed by Eq. (9.5), at large wave vector. The inner edges of the two bulk bands of the upper pair are seen to touch at the reduced wave vector z ' 1:8. The lowest surface-polariton mode starts from the origin, rises in the gap between the bulk bands of the lower pair, and merges with the upper edge of the lowest bulk band at z ' 3:5. The second surface mode also starts from the origin, rises just towards the right of the ®rst light line, repelled from the bottom of the third bulk band, then propagates with negative group velocity until z ' 0:6, and rises again to merge with the upper edge of the second lowest bulk band at z ' 3:5. The third surface mode starts from the ®rst light line just at the upper edge of the third lowest bulk band, rises in the gap between the upper two bulk bands, repelled from the lower edge of the topmost bulk band to further propagate with negative group velocity, then rises again to cross the touching point (of the bulk bands), and propagates within the gap to become asymptotic to the frequency speci®ed by Eq. (9.5). The fourth surface mode, above the uppermost bulk band, approaches the asymptotic limit (x ' 0:9639) speci®ed by Eq. (9.10). We stress that all the solutions, for bulk and surface modes, above the uppermost bulk band and between the two light lines are mathematical, correspond to the radiative modes, and irrelevant; they are shown exactly as they come out of the machine without any restriction. The only restriction in our computer program was that no solutions are desired towards the left of the ®rst light line (in the vacuum). However, it deserves mention that the surface mode towards the right of the second light line are the only bona®de surface polaritons because all the decay constants are real and positive in this region of o±q plane. The next case we present is when thickness of p-doped layer (C) is increased to dC ˆ 0:75, keeping the rest of the parameters in Fig. 67 ®xed. The numerical results are depicted in Fig. 68. The noteworthy differences (as compared to Fig. 67) are that the gap between the lower pair of bulk bands reduces, the upper pair of bulk bands is pushed towards lower frequencies, and the touching point between the upper pair of bulk bands tends to move towards the lower wave vector. Moreover, the lowest and the second lowest surface modes move towards higher frequencies, while the third lowest surface mode occur at lower frequencies at small wave vectors. The rest of the discussion related with Fig. 67 still remains valid.

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Fig. 67. Dispersion of the collective excitations Ð bulk bands (hatched regions) of the in®nite and surface polaritons (solid curves in pink) Ð of the truncated p n±i±p±i superlattice in the zero magnetic ®eld. The parameters used are: EL ˆ 13:13, EE ˆ 1:0, OpC ˆ opC =opA ˆ 1= 2, dA ˆ dC ˆ 0:5, and dB ˆ dD ˆ 1:0. This implies that the surface layer A is half as thick as the intrinsic layer B and justi®es, to a certain extent, the occurrence of the radiative (both bulk and surface) modes above the uppermost bulk band, in between the two light lines demarked as aE ˆ 0 and aB ˆ aD ˆ 0. (After Kushwaha, Ref. [749].) Fig. 68. The same as in Fig. 67, but when dC ˆ 0:75. Notice the touching point between the upper pair of bulk bands moves towards the lower wave vector, as compared to Fig. 67. (After Kushwaha, Ref. [749].) Fig. 69. The same as in Fig. 67, but when dC ˆ 1:0. It is noteworthy that a further increment in the thickness of the p-doped layer results in the disappearance of touching point. (After Kushwaha, Ref. [749].)

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The last case we investigate is when the p-doped layer is assigned a thickness of dC ˆ 1:0, while rest of the parameters are the same as in Fig. 67. The numerical results are illustrated in Fig. 69. One immediately notices that there persists only a small gap between the lower pair of bulk bands at large wave vectors; within this gap the lowest surface mode is discernible only in a smaller range of wave vector (2:2  z  3:5). The touching point between the upper pair of bulk bands disappears altogether, and the widths of both bulk bands of the upper pair are seen to reduce. The third-lowest surface mode appears to start from the ®rst light line just above the upper edge of the third-lowest bulk band, and remains trapped within the gap without merging into either of these bulk bands up to a large wave vector. Our illustrative examples are subject to the case dA < dB ; dC ; dD . We have seen that when all layer thicknesses are equal (speci®cally, for dA ˆ dB ˆ dC ˆ dD ˆ 0:5 or 1.0) the touching point between the upper pair of bulk bands persists; the same is true for dA > dB ; dC ; dD (see, e.g., Ref. [749]). 9.2. Non-zero magnetic ®eld This section is aimed at uncovering the effects of an external magnetic ®eld on the plasmon dispersion in n±i±p±i superstructures. This is done for two different con®gurations: the Voigt [754] and the perpendicular con®guration [751]. Many studies in the past have been attempted at investigating the effects of various symmetry lowering on the collective mode spectra in different periodic as well as non-periodic structures. In addition to the lowering of symmetry caused by physical perturbations in the layering of superlattice, one may also lower the symmetry by applying an external magnetic ®eld. Studies involving magnetoplasmon±polaritons in n±i±p±i superlattices have considered the situation where the applied magnetic ®eld is assumed to be in the perpendicular con®guration [751]. In such a case, the direction of propagation remains independent of the ®eld, and is thus the situation of higher symmetry than the consideration of the magnetic ®eld in the Voigt con®guration [754]. In the latter con®guration, the surface magnetoplasmons can show a non-reciprocal behavior in frequency with respect to the direction of propagation: o…ÿq† 6ˆ o…‡q†. In what follows, we will review some published results on the magnetoplasmon±polaritons in the Voigt and perpendicular con®gurations. 9.2.1. Voigt con®guration Ð non-reciprocal propagation The effect of an external magnetic ®eld in the Voigt geometry on the plasmon±polaritons in the ®nite n±i±p±i superlattices was considered by Johnson and Camley [754]. They were motivated by the analogous study of the effects of applied magnetic ®elds in antiferromagnetic ®lms and in superlattices. It was shown that the effect of an external magnetic ®eld on thin-®lm antiferromagnets depends upon the symmetry of the sublattices. For a structure consisting of alternate layers of oppositely directed spins, labeled A and B, it turns out that for an equal number of layers A and B (ABABAB, for instance) the magnon dispersion is non-reciprocal. On the other hand, for an unequal number of layers A and B (ABABABA, for instance) the propagation remains reciprocal. Similar results were reported for superlattices composed of ferromagnetic ®lms, where the magnetization of alternating ®lms are oppositely directed. In order for understanding such effects, the authors [754] presented the following symmetry arguments. For the propagation to be reciprocal, there must exist a set of symmetry operations which will take the propagation vector ‡q to ÿq while leaving the structure in its original con®guration. If no such set of operations can be found, then the propagation need not be reciprocal. Therefore, the

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symmetry of the structure becomes a critical deciding feature. In the even case of ABABAB, with the ~ q k ^y and ~ B0 k ^x, there is no symmetry operation which takes ‡q to ÿq and leaves the structure and ®eld direction unchanged. In contrast, the odd case of ABABABA can be taken into itself and take ‡q into ÿq by a simple rotation of 180 about an axis parallel to x centered at the mid-point of the central ®lm. Thus this structure has reciprocal propagation. They argued that, in addition to non-reciprocity induced in the dispersion relations, the symmetry-lowering effects of the magnetic ®eld can also serve to localize certain spin waves. If one examines the amplitude of the excitations for surface magnons as a function of depth, two distinct cases emerge, depending upon the symmetry of the structure. For the even case, ABABAB, there is a re¯ection symmetry about the mid plane Ð even though at ®rst glance this does not appear to be the case. In this geometry, when spins are re¯ected about a mirror plane parallel to the axis of the spins, not only does the position go from ‡z to ÿz, but the directions of the spins are reversed as well. Because of this, the excitations which had a de®nite parity in the absence of an applied ®eld retain a de®nite parity with an applied ®eld. For the odd case, ABABABA, there is no mid-plane symmetry, and as a result the surface modes which had de®nite parity without a ®eld can be signi®cantly localized by an applied ®eld. The localization, either to the top surface or the bottom surface, depends upon the direction of propagation. With the same token, one might expect that a superlattice composed of alternating ®lms of materials containing charge carriers of opposite sign would exhibit a similar character, since the sense of rotation of electrons about a magnetic ®eld is opposite to that of holes. This motivated Camley [754] to consider ®nite n±i±p±i superlattices, subjected to an applied magnetic ®eld in the Voigt geometry, in order for studying the plasmon propagation and their localization. He did not derive an explicit dispersion relation; however, a theoretical development was presented valid for any number of layers (and their thicknesses) including magnetic ®eld. Two types of n±i±p±i geometries were considered: (i) an ``ideal'' GaAs superlattice, which is bounded both on top and bottom by vacuum, and (ii) a GaAs superlattice which rests on a semi-in®nite intrinsic GaAs substrate. It was assumed further that the doping concentrations of the conducting layers is the same for both n- and p-doped layers and that these active layers are characterized by the local Voigt dielectric function. The assumed doping level is n ˆ 1018 cmÿ3 , which gives a screened plasma frequency op ˆ 0:04075 eV. The thickness dn of the doped ®lms …n ˆ 1; 3; 5; . . .† was considered to be double of that of the undoped ones (n ˆ 2; 4; 6; . . .). In what follows, we present some numerical examples which illustrate the symmetry-lowering effects of the magnetic ®eld leading to non-reciprocal propagation as well as non-reciprocal localization of magnetoplasmons [754]. The ®rst set of results we discuss are for the ideal geometry with four layers of doped GaAs (two each of n- and p-doped) separated by three insulating ®lms. This corresponds to the even-spin geometry, and as a result the dispersion relations are non-reciprocal while the potentials retain a de®nite parity. Fig. 70 displays the dispersion curves, for positive and negative k…ˆ q†, for the four-layer geometry with a ®eld of 1 kG. The ordinate (abscissa) represents the absolute frequency in eV (dimensionless propagation vector kd1 ). We note the non-reciprocity, particularly for the surface modes, the lowest- and highest-frequency bulk modes. The uppermost modes are surface modes and show the highest degree of non-reciprocity. Fig. 71 depicts a typical bulk-mode potential in the four-layer geometry; the lowest picture gives the zero-®eld results, and the upper pictures are with a non-zero ®eld (of 1 kG). While the upper pictures illustrate the expected retention of de®nite parity under an applied ®eld for the four-layer geometry, the application of the ®eld does dramatically alter the scalar potential

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Fig. 70. Dispersion curves for a four-layer ideal n±i±p±i structure symmetrically surrounded by vacuum, with an applied magnetic ®eld (of 1 kG) in the Voigt geometry. The material parameters used are: EL ˆ 13:13, Eout ˆ 1:0, ne ˆ nh ˆ 1018 cmÿ3 ) (the screened plasma frequency) op ˆ 0:04075 eV. Notice the non-reciprocal behavior produced by the ®eld in the Voigt geometry. The upper two modes on both sides (‡k and ÿk) are too close to be resolved on this scale, as are the second highest mode on the ‡k side. (After Johnson and Camley, Ref. [754].)

as a function of depth for ‡k. Thus even though there is a de®nite parity in all cases, there is still a clear non-reciprocity in that the scalar potential is clearly different for ‡k and ÿk. Next, we turn attention to the case of a GaAs n±i±p±i superlattice resting on an intrinsic GaAs substrate. In this case, the existence of the substrate is itself a sign of the lowering symmetry of the structure, and therefore we expect the results to be slightly less simple than for the ideal geometry. For example, the above-mentioned rotation-symmetry operations no longer apply; if we rotate the structure 180 about the x-axis (k ! ÿk), this brings the substrate above the superlattice, and there is no way to return the structure to its original con®guration. Therefore, the dispersion relation need not be reciprocal for any (even or odd) layer geometry. Moreover, the presence of the substrate changes the symmetry of the boundary conditions Ð the top surface is in contact with vacuum, while the bottom surface is in contact with an insulator with higher background dielectric constant Ð and hence the potentials need not display a de®nite parity. In order for the computation, a ®ve-layer geometry was considered. Fig. 72 shows the dispersion curves for both directions of propagation with an applied ®eld (of 1 kG). Note the non-reciprocity caused by the joint effect of the ®eld and the substrate. Fig. 73 depicts the behavior of a typical bulk mode as a function of the applied ®eld. The bottom panel is the mode for zero ®eld; the absence of a de®nite parity here is a consequence of the substrate. The middle picture (kd1 ˆ ÿ1:5), although not a clear signal of localization at either the top or bottom surface, shows a distinct difference from the zero-®eld case. The top picture (kd1 ˆ ‡1:5), however, demonstrates a strong localization of the potential to the upper surface.

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Fig. 71. Evolution of the electrostatic potential of a bulk mode for the four-layer ideal structure of Fig. 70. The bottom panel represents the mode in the zero ®eld (o ˆ 0:03633 eV), while the upper two panels represent the mode with an applied ®eld (o ˆ 0:03657 eV for ‡k and o ˆ 0:03753 eV for ÿk). Note that the mode retains its de®nite parity regardless of the ®eld or propagation direction. (After Johnson and Camley, Ref. [754].)

In summary, the n±i±p±i structure responds to an applied magnetic ®eld in the Voigt geometry in a way analogous to antiferromagnetic thin ®lms and magnetic superlattices. Depending upon the symmetry of the structure, the applied ®eld can cause (i) signi®cant non-reciprocity in the dispersion relation, and/or (ii) strong localization of the excitations themselves. The presence of substrate serves to further lower the symmetry, and the combination of the applied ®eld and the substrate has a dramatic effect on the plasmon propagation. 9.2.2. Perpendicular con®guration Ð collective excitations This section is aimed at investigating the effect of an applied magnetic ®eld oriented along the superlattice axis (the perpendicular geometry) on the collective (bulk and surface) excitations in semiin®nite n±i±p±i superlattices. The superlattice geometry considered for this purpose is depicted in Fig. 66. Material layers A and C have frequency and magnetic-®eld dependent dielectric tensors E A and E C , respectively; whereas the intrinsic layers B and D are assumed to have frequency independent dielectric constants EB and ED , respectively. The thicknesses of the layers are di with i  A, B, C, D. The period of the superlattice is de®ned as d ˆ dA ‡ dB ‡ dC ‡ dD . The magnetoplasma modes are assumed to propagate along the ^y axis with wave vector q and frequency o. We are interested in the collective excitations characterized by the EM ®elds localized at and exponentially decaying away from

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Fig. 72. Dispersion curves for a ®ve-layer n±i±p±i structure resting on an intrinsic GaAs substrate in the applied ®eld of 1 kG. The substrate lowers the symmetry and therefore the dispersion becomes non-reciprocal. It was noted that the second highest mode on ÿk side is really an unresolved pair of modes. (After Johnson and Camley, Ref. [754].)

the interfaces. A static magnetic ®eld is applied along the superlattice axis (taken to be the z-axis) while the interfaces lie in the x±y plane. The mathematical formalism for deriving the dispersion relations, both for bulk and surface excitations, is based on the TMM. We would recall short formal results from and refer the lengthy expressions (and relevant substitutions) to the original paper [751]. In the presence of a perpendicular magnetic ®eld, the general dispersion relation for the collective (bulk) excitations in a n±i±p±i in®nite superlattice, in the framework of TMM, reads [751] " # ! 4 4 X X 2 Tii cos…Q† ‡ …Tii Tjj ÿ Tij Tji † ÿ 2 ˆ 0; (9.11) 4 cos …Q† ÿ 2 iˆ1

i6ˆjˆ1

where Q ˆ kd is the dimensionless Bloch vector and Tij are the matrix elements of the 4  4 transfer matrix [751]. For a given propagation wave vector q, Eq. (9.11) yields the frequency (o) bands when o is plotted against Bloch vector Q. Eq. (9.11) is a quadratic equation in cos…Q† which means that there are, in principle, two solutions for each value of Q (ˆ p and 0). In other words, Eq. (9.11) is a very important analytical form of the dispersion relation which explicitly reveals that it is quite likely that the bulk bands display broken degeneracy at some wave vectors in the o±q plane in the presence of a magnetic ®eld. This kind of broken degeneracy was, however, not seen in the binary (semiconducting or metallic) superlattices [716±723], where the layers have been characterized by the macroscopic dielectric functions in the local approximation. In the n±i±p±i superlattices, on the other hand, it is

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Fig. 73. Evolution of the electrostatic potential for the ®ve-layer-substrate geometry of Fig. 72. The bottom panel is for zero ®eld (o ˆ 0:03656 eV), the absence of de®nite parity is caused by the substrate. Here only the ‡k direction (o ˆ 0:03640 eV) is strongly localized by the combination of ®eld and substrate (top panel). The ÿk case has a frequency of o ˆ 0:03740 eV. (After Johnson and Camley, Ref. [754].)

found that the bulk bands exhibit a dramatic broken degeneracy between the two cyclotron frequencies of the doped layers by splitting into a large number of bulk bands (see in what follows). This leads us to reaf®rm the prediction [537] that n±i±p±i doping superlattices display relatively richer and unusual properties as compared to their compositional counterparts. In order for studying the magnetoplasmon±polaritons in the n±i±p±i superlattice system, we consider the geometry when the superstructure is truncated at, say z ˆ 0, such that the medium E in the region ÿ1  z < 0 is replaced by an insulator characterized by a frequency independent dielectric constant EE . The general dispersion relations for the collective (surface) magnetoplasmons in the truncated n±i±p±i superlattice reads [751] …a1 b2 ÿ a2 b1 †…b1 c2 ÿ b2 c1 † ÿ …c1 a2 ÿ c2 a1 †2 ˆ 0;

(9.12)

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where ai , bi , and ci , in terms of the matrix elements of the transfer matrix T are as de®ned in Ref. [751]. Eq. (9.12) constitutes an implicit dispersion relation for the magnetoplasma polaritons in the semiin®nite n±i±p±i superlattice system, which is a result of an appropriate scheme that seeks solutions to the Maxwell equations in which the EM ®elds are localized at and decay exponentially away from each interface of the system. It is worthwhile mentioning that the mathematical complexity of the general dispersion relations, both for bulk and surface collective excitations, prevents us from making analytical diagnoses to understand, for instance, the asymptotic behavior of the magnetoplasma modes in the NRL (c ! 1). However, we shall justify the bona®de regions of propagation on the basis of some approximate diagnosis. The material parameters employed in the present computation are EL …ˆ ELA ˆ ELB ˆ ELC ˆ ELD † ˆ p 13:13, EE ˆ 1:0, ne ˆ nh , mh ˆ 2me ) opC ˆ opA = 2, ocA ˆ 0:5 opA ) ocC ˆ 0:25 ocA . Here ni and mi are the carrier concentration and effective mass, respectively; i  e, h. The subscripts e (h) refers to the electrons (holes) residing in layer A (C). The symbol opi …oci † stands for the plasmon (cyclotron) frequency of the doped layer. Before discussing the illustrative numerical examples, it is instructive to consider the possibilities of how the propagating plasma modes, consisting of bulk and surface excitations, can be classi®ed. We are interested in the situation where the propagation vector q is real and absorption is neglected. As pointed out earlier, we shall con®ne our attention to the solutions for which the EM ®elds are localized at and decay exponentially away from the interfaces. Such solutions are characterized by ai (i  A, C) and aj ( j  B, D, E) being real and positive. However, depending upon the spectral range the following possibilities may arise: (i) ai‡ and aiÿ are both real and positive, (ii) ai‡ and aiÿ are both pure imaginary, (iii) ai‡ is real and aiÿ is pure imaginary, or vice versa and (iv) ai‡ and aiÿ are complex conjugates of each other. In all these cases, aj ( j  B, D, E) may attain either real or pure imaginary values. We shall term the magnetoplasma modes corresponding to the above-mentioned scheme the surface polariton modes, bulk (or waveguide) modes, pseudo-surface modes, and generalized complex modes, respectively. This classi®cation can (and should) help us understand the behavior of magnetoplasma modes in the speci®c spectral range. It should be remembered, however, that the foregoing nomenclature of the modes requires the same nature of the decay constants ai (i  A, C) in both doped layers. Such regions are unfortunately quite rare in the spectral range of interest, even though the magnetoplasma modes may turn out to be bona®de. Interestingly, this happens where the decay constants aA are real and aC are complex conjugates of each other, or vice versa; their occurrence is possible in a superlattice system whose unit cell consists of two (or more) different semiconducting (or differently doped) layers. Note that q in both dispersion relations, Eqs. (9.11) and (9.12), appears as even powers; this implies that reciprocity holds. Let us now turn to what we referred to above as the approximate diagnoses. In the NRL (c ! 1; q  q0 ), the seven decay constants Ð ai (i  A, C) and aj ( j  B, D, E) Ð assume the following forms [751]:  ai‡ ˆ

Eixx Eizz

1=2

q;

i  A; C;

aiÿ ˆ aj ˆ q;

j  B; D; E:

(9.13)

Since we treat the propagation vector q as a real positive quantity, the true surface modes require ai‡ to be a real and positive quantity. This in turn requires that Eixx and Eizz must have the same sign. In view of this, the asymptotic solutions predicted by the aforesaid conditions should lie in the frequency windows

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speci®ed by ocC  o  opC

and=or

o  oHC ;

ocA  o  opA

and=or o  oHA ;

(9.14)

where oHi ˆ …o2pi ‡ o2ci †1=2 is the hybrid cyclotron±plasmon frequency of the ith doped layer. It is found that the bona®de windows are those that lie between the respective cyclotron and plasmon frequencies. However, the true surface modes require both aA‡ and aC‡ to be real and positive in the same region of o±q plane. The latter condition again reduces the width of the propagation window allowed for the true surface modes. This happens to be the frequency range speci®ed by ocA  o  opC :

(9.15)

The surface modes in the frequency interval ocC  o  opA are characterized as follows. The decay constant aE is real and positive towards the right of the ®rst light line (in vacuum) and aj ( j  B, D) is pure imaginary between the two light lines and real and positive towards the right of the second light line (in the intrinsic layers). The magnetic-®eld dependent decay constants ai (i  A, C) assume usually different nature in the different frequency intervals. For instance, aAÿ is pure imaginary in the interval ocC  o  ocA and the rest of them are real and positive. All ai are real and positive in the frequency range ocA  o  opC . aCÿ is pure imaginary in the interval opC  o  oHC , and the rest of them are real and positive. In the frequency range oHC  o  opA , aC‡ is the complex conjugate of aCÿ , and aA are real and positive. The foregoing analysis clearly reveals that classifying the magnetoplasma modes according to the above-mentioned scheme is really a ``hard nut to crack'', and may lead us really go astray. At this point, it should be stressed that the aforesaid analysis partially assumes our speci®c set of parameters which satisfy the inequalities ocC < ocA < opC < opA ; the other inequalities, e.g., ocC < opC < ocA < opA could also be worth attempting. For this reason, what we have attempted is the search of zeros of the real parts of the transcendental functions, Eqs. (9.11) and (9.12), irrespective of whether or not their imaginary parts are zero. We have found that this scheme provides us quite a reasonable answer. We will discuss in what follows our neat illustrative numerical examples obtained without imposing absolutely any restriction on the nature of decay constants, which do contain numerous super¯uous and irrelevant radiative, both bulk and surface, modes in between the two light lines; we could have eliminated, at least, some of them, but we did not do so for the sake of completeness. We ®rst solve Eq. (9.11) for a given reduced wave vector z ˆ 1:5. The Bloch bands in terms of the dimensionless frequency x ˆ o=opA and Bloch vector kd are plotted in Fig. 74. All the material parameters used for this purpose were the same as listed in Fig. 75. We have shown the results with only two restrictions: no modes were desired (i) towards the left of the ®rst light line (in the semi-in®nite medium, taken to be the vacuum) and (ii) below the lowest cyclotron frequency ocC (in the doped layer C). The frequency interval was restricted to ocC  o  opC . The effect of multilayer system in the broadening of a single-®lm plasmons into bands along the superlattice axis is evident. In addition, one can notice that there is an accumulation of Bloch bands in the close vicinity of the cyclotron frequency of the n-doped layer, ocA , as well as in the vicinity of the plasmon frequency of the p-doped layer opC. This is exactly what is observed in Fig. 75, where we have plotted the complete excitation spectrum in terms of the dimensionless frequency x ˆ o=opA and wave vector z ˆ cq=opA. The hatched regions are the bulk magnetoplasmon bands whose boundaries correspond to kd ˆ 0 and p. The solid curves in red are the surface magnetoplasma modes. The dashed line marked as aE ˆ 0 (aB ˆ aD ˆ 0) is the light line in the vacuum (in the intrinsic layer B or D). The horizontal dashed line refers to the cyclotron

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Fig. 74. Bloch bands for an in®nite n±i±p±i superlattice in the perpendicular magnetic ®eld, for the normalized propagation vector z ˆ cq=ope ˆ 1:5. The other parameters used in the computation are the same as listed in Fig. 75. Notice the pile-up of the bands at o ˆ ocA and at o ˆ opC . (After Kushwaha, Ref. [751].)

Fig. 75. Dispersion of the collective excitations Ð bulk bands (hatched regions) and surface modes (solid curves in red) Ð for an n±i±p±i superlattice in the perpendicular magnetic ®eld. The dashed lines in black are the light lines in vacuum and intrinsic GaAs. Notice a sophisticated, but interestingly rich behavior of the surface excitations carried through on the top of each of the multiple bulk bands, particularly, between the two cyclotron frequencies ocC and ocA . A pile-up of the bulk as well as surface modes in the vicinity of ocA and opC is noticeable. The existence of annoyingly complex behavior of the large number of radiative modes between the two light lines is partially attributed to the relative thickness of the surface layer A, dA =dB ˆ 12. This is entirely neat picture of the collective excitations obtained without imposing absolutely any restriction on the nature of the decay constants. The parameters used are as listed in the picture. (After Kushwaha, Ref. [751].) Fig. 76. Scale-up of the excitation spectrum between the two cyclotron frequencies ocC and ocA in Fig. 75.

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frequency in the doped layer C. The material parameters used for this computation are as listed in the ®gure. As one can notice, we have restricted the spectral region to frequency range ocC  o  opA . This is because the frequency interval o  ocC corresponds, generally, to the pseudo-surface wave region; and nothing interesting or relevant turns out in this region. Note that the frequency region explored in Fig. 75 can be divided into three regions: (i) ocC  o  ocA , (ii) ocA  o  opC , and (iii) opC  o  opA . This is, as we realized, important to do the computation in three parts corresponding to these three regions, since the machine feels handicapped to handle the whole region (ocC  o  opA ) in a single stroke, and this is because the transcendental functions, both Eqs. (9.11) and (9.12), blow up at ocA as well as at opC [751]. The numerical results in the presence of an applied magnetic ®eld in Fig. 75 should be compared with those in the absence of an applied magnetic ®eld in Fig. 67. Our understanding of such a comparison is as discussed below. In the ®rst regime (ocC  o  ocA ), the two lowest bulk bands start from o ˆ ocC , and the next two (clearly visible) start from the ®rst light line (in vacuum). All these four bulk bands exhibit a number of touching points, repulsions, and crossing (of their boundaries) with each other in the wave vector range, particularly towards the right of the second light line (in the intrinsic layers B or D) before displaying the Zeeman-like splittings at large wave vectors. Each of these bulk bands carries over its upper boundary a surface mode; each of which is seen to propagate in the gap between the two nearest neighboring bulk bands. In the vicinity of o ˆ ocA , there is a pile-up of the bulk bands as well as of surface modes. This can be quite clearly seen in Fig. 76 which is a scaled-up plot of Fig. 75 between the two cyclotron frequencies (i.e., ocC  o  ocA ). These four clearly visible bulk bands in the frequency range ocC  o  ocA were interpreted in the original paper [751] as if the lower pair of bulk bands (in the absence of a magnetic ®eld) splits into an even pair of bulk bands (in the presence of a magnetic ®eld). The ®fth bulk band starts from at o ˆ ocA , and the sixth one starts from the ®rst light line; inside the gap between these two bulk bands propagates a surface mode, which is a bona®de polariton mode only towards the right of the second light line (where, at least, three of the ®eld independent decay constants are real and positive). In the close vicinity of the next characteristic frequencies o ˆ opC (the plasmon frequency associated with layer C), there is again a pile-up of both bulk and surface modes; in fact, it is like a fork which tends to widen out with the increasing wave vector. Above the sixth bulk band, we see a single surface mode which becomes asymptotic to a certain characteristic frequency below the plasmon frequency opA. The only ambiguity in Fig. 75 is the appearance of the two oddlooking surface modes, which start from o ˆ ocC at different wave vectors and both (seem to) cross the lowest bulk band and the (lowest bona®de) surface mode it carries over its head, at large wave vector. Moreover, we do not want to expand on the strange radiative, both bulk and surface, modes which occur in between the two light lines. It is noteworthy that the upper pair of bulk bands, above ocA , does not display any touching point, unlike the zero-®eld case (Fig. 67); the presence of the ®eld is, however, re¯ected in the variation of the widths of these bulk bands. A word of warning on the comparison of Fig. 75 with the corresponding Fig. 7 in the original paper [751] is in order. The apparent difference lies due, in fact, only to the different strategies of ®nding reliable solutions of cumbersome and, in general, complex transcendental functions, Eqs. (9.11) and (9.12). In the original paper, all the solutions, both for bulk and surface excitations, were based on the scheme of looking for the absolute minimum of the functions, whereas the present computation is established on the scheme of looking for the zeros of the real part of the function irrespective of whether or not its imaginary part is zero. Given the complexity of the nature of these transcendental functions, such apparent minor differences, particularly with respect to the surface modes in the

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frequency range ocA  o  opC (the second regime), is not unexpected. What is overall worth commenting is the fact that the n±i±p±i superstructure exhibits a richer excitation spectrum with respect to both the bulk and the surface modes; and this is true both for the zero and non-zero magnetic ®eld. It should also be pointed out that the variation in the relative thicknesses of the doped and undoped layers does reveal the difference in the frequency of both bulk and surface excitations, as well as the width of the bulk bands; the overall picture of the excitation spectrum, as a result of such variations, is still describable almost analogously to Fig. 75. Therefore, we withdrew from including a large number of such ®gures, particularly for the reasons of space. 10. Finite-size effects in superlattices The purpose of this section is to investigate the theory of plasmon±polaritons in the ®nite superlattices, i.e., when an in®nite (perfectly periodic) superlattice is truncated somewhere both on the top and bottom surfaces perpendicular to the superlattice axis. Such a ®nite superstructure bears a practical relevance, since such are the structures which are actually probed experimentally in the laboratory. A theoretical treatment of a ®nite superlattice poses many interesting questions. For instance, (i) What is the effect of ®niteness on the widths of the bulk bands (in an otherwise in®nite superlattice)? (ii) What is the size effect on the quantization of the bulk bands? (iii) Is there any frequency shift of the surface polaritons due to the variation in the size of the superstructure? (iv) Is there any unique feature of the ®nite superstructure which does not exist in the semi-in®nite superstructure? A few of these questions were addressed by Camley and coworkers [753] in the case of a binary (metal±insulator) superlattice in the NRL. Subsequently, Kushwaha [757] investigated a ®nite n±i±p±i superlattice with the same purpose. Here we intend to review the theoretical results obtained in the aforesaid works. The theoretical treatment of the plasmon±polaritons in both of these works is based directly [757] or indirectly [753] on the TMM. We consider a ®nite four-layer superstructure as depicted in Fig. 77. Material layers A±D are assumed, for the sake of generality, to have frequency-dependent dielectric functions (in the local approximation) EA ; EB ; EC ; and ED , respectively. The growth direction will be taken to be the z-axis which is usually termed as the superlattice axis. The period of the superstructure is de®ned as d ˆ dA ‡ dB ‡ dC ‡ dD . The plasma modes are assumed to propagate along the y direction parallel to the interface, with wave vector q and frequency o. We replace the parts of the superlattice lying in the regions ÿ1  z  0 and pd  z  ‡1 by the dielectric materials E and F, respectively; p being the total number of unit cells. Thus the ®nite superstructure lies in the region 0  z  pd. We are interested in the collective excitations characterized by the EM ®elds localized at and decaying exponentially away from each interface as well as from the ends of the superstructure. Although the whole formalism is quite general where the unit cell of the superstructure is assumed to be made up of four different semiconducting layers, all characterized by frequency-dependent dielectric functions, we later specify our analytical results corresponding to the n±i±p±i superstructure bounded by two unidentical dielectrics. For generality, we consider even the bounding media E and F to be characterized by frequency-dependent dielectric functions. Since the periodicity in the z direction is now destroyed (due to the truncation of the superlattice), we can no longer use the Bloch theorem which relates the ®eld amplitude in one layer to that in the other through the envelope function exp…iknd†; n, numbers the unit cells. Instead, we have the envelope functions exp…ÿbnd† and exp…ÿb…p ÿ n†d†, which correspond to the localization of the plasmon±

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Fig. 77. Schematics of the ®nite n±i±p±i superlattice geometry, reducible to a ®nite binary compositional superlattice, considered in Section 10. Material layers A, B, C, and D are n-doped, intrinsic, p-doped, and intrinsic (GaAs) semiconductors, respectively, with respective thickness di ; i  A±D. The bounding media E (ÿ1  z  0) and F (pd  z  ‡1) are taken to be unidentical insulators, for the sake of generality. The integer n numbers the unit cell, and p is the total number of unit cells composing the ®nite superstructure.

polaritons at the top and bottom surfaces of the superstructure, respectively. For bulk plasmons in the superlattice one replaces b by ÿik and ‡ik, respectively. An elegantly compact scheme of working in the framework of TMM yields a desired expression [757] e2pl ˆ

‰nA …1 ÿ k† ‡ nF …1 ‡ k†Š‰nA …1 ÿ k0 † ÿ nE …1 ‡ k0 †Š ; ‰nA …1 ÿ k0 † ‡ nF …1 ‡ k0 †Š‰nA …1 ÿ k† ÿ nE …1 ‡ k†Š

(10.1)

where l ˆ bd, nj ˆ Ej =aj , Ej ˆ ELj …1 ÿ o2pj =o2 †, and aj ˆ …q2 ÿ q20 Ej †1=2 ; j  A±F. The symbols k and k0 are de®ned as follows: kˆ

eÿbd ÿ T11 ; T12

k0 ˆ

e‡bd ÿ T11 ; T12

(10.2) $

where Tij (i; j  1, 2) are the elements of the matrix elements of the transfer matrix T [757]. Eq. (10.1) is the implicit general dispersion for the plasmon±polaritons in the ®nite, four-layered, semiconducting

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superstructure; each layer and the semi-in®nite media (E and F) being characterized by the background dielectric constant ELj . Within the special limits, viz., dC ˆ dD ˆ 0, and dB ! 1, Eq. (10.1) reproduces exactly the proper results for a ®nite binary superlattice (generalized to include the retardation in Eq. (45) of Camley and coworkers [753]) and for a ®lm A bounded by two dissimilar dielectrics E and B, respectively. In the special limit p ! 1 (semi-in®nite case), Eq. (10.1) can be analyzed to draw two conclusions, depending upon the sign of l. For l > 0 (l < 0), the dispersion relation reduces to the denominator (numerator) equated to zero. Thus the signs of l, for p ! 1, correspond to the plasmon±polaritons localized either at the bottom or at the top surface of the superstructure. Furthermore, depending upon the signs of k and k0 , and the frequency range, one can decide which of the two factors (equated to zero) in each of the above-mentioned two cases will give the correct dispersion relation for the concerned plasmon±polaritons. However, it is noteworthy that the foregoing remark is only analytical (rather mathematical) and the fact is the other way round. It is beyond dispute that l (the normalized inverse penetration depth) is almost a real and positive quantity for the pure surface polaritons, regardless of the structure at hand. Therefore, the possibility of imagining l < 0 must be ruled out. It should be pointed out that the aforesaid criterion (of l > 0) is unfailingly valid only for semi-in®nite superlattice; for a ®nite superlattice, both l > 0 and < 0 are valid. The parameter b implicitly involved in Eq. (10.1) is, in general, a complex quantity whose real part is positive and represents the inverse penetration depth of the plasmon±polariton modes. The value of b is determined by the requirement that the bulk mode condition be satis®ed for the same value of frequency and wave vector. This leads to the following relationship: $ 1 (10.3) b ˆ coshÿ1 ‰12 Tr T Š: d A direct inspection of Eq. (10.3) makes it apparent that b is real everywhere in the o±q plane except inside the bulk plasmon bands where it is purely imaginary and coincides with k, and in the gap region between the bulk bands where it acquires an imaginary part equal to p=d. As such, the dispersion relation for the plasmon±polaritons can be obtained with the use of Eq. (10.3) in Eq. (10.1). Now we discuss some illustrative numerical examples for plasmon±polaritons in the ®nite, binary compositional and n±i±p±i doping, superstructures. 10.1. Compositional superlattices The illustrative examples for the ®nite, binary compositional superlattices investigated by Camley and coworkers [753] are based on the corresponding dispersion relation in the NRL. Conforming our geometry (in Fig. 77) to theirs requires that dC ˆ dD ˆ 0 and our semi-in®nite medium E(F)  their C (D). Such a dispersion relation, but still generalized to include the effects of retardation, obtained from Eq. (10.1) reads ‰n2A …1 ÿ k†…1 ÿ k0 † ÿ nC nD …1 ‡ k†…1 ‡ k0 † ÿ nA …nC ÿ nD †…1 ÿ kk0 †Š tanh…pl† ÿ nA …nC ‡ nD †…k ÿ k0 † ˆ 0:

(10.4)

Eq. (10.4) subjected to the NRL (c ! 1; q  q0 ) assumes the following form: ‰E2A …1 ÿ k†…1 ÿ k0 † ÿ EC ED …1 ‡ k†…1 ‡ k0 † ÿ EA …EC ÿ ED †…1 ÿ kk0 †Š tanh…pl† ÿ EA …EC ‡ ED †…k ÿ k0 † ˆ 0:

(10.5)

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This is the dispersion relation derived and used for the computation of plasmon±polaritons in the ®nite, binary superlattices by Camley and coworkers [753]. In the short wavelength limit, Eq. (10.5) further simpli®es to the form (10.6) …EA ‡ EB †…EA ‡ EC †…EA ÿ ED † ˆ 0: This equation serves to provide the asymptotic limits of the plasmon±polaritons. We consider two situations: when the bounding media (C and D) are identical and when they are unidentical. First of all let us specify, in accordance with the numerical work of Camley and coworkers [753], that the bounding media C and D are insulators, i.e., opC ˆ 0 ˆ opD , and hence EC ˆ ELC and ED ˆ ELD . If medium layer B is also insulator, then opB ˆ 0, and so EB ˆ ELB . Moreover, if layer A is a metal, then ELA ˆ 1. The illustrative examples to be discussed here are presented in terms of the dimensionless frequency x ˆ o=opA and wave vector z ˆ kdA, unlike in Ref. [753]. In view of the foregoing speci®cation of the structural geometry studied in Ref. [753], equating ®rst factor in Eq. (10.6) to zero yields EA ‡ EB ˆ 0

)

x ˆ …1 ‡ ELB †ÿ1=2 :

(10.7)

Similarly, equating the second factor to zero gives EA ‡ EC ˆ 0

)

x ˆ …1 ‡ ELC †ÿ1=2 :

(10.8)

Clearly, then, the third factor does not vanish; otherwise Eq. (10.7) or (10.8) can never be satis®ed. In the case that the medium layer B is the same as the semi-in®nite medium C, Eqs. (10.7) and (10.8) are identical, and there exists only one asymptotic limit; otherwise they yield two different asymptotic limits. Now we discuss a few illustrative numerical examples of plasmon±polariton propagation in a ®nite, binary (metal±dielectric) superlattice. The ®rst case we consider is that of a ®nite superlattice made up of ®ve unit cells (i.e., p ˆ 5) and the normalized width of the layers dA =dB ˆ 2:0. The background dielectric constants ELB ˆ ELC ˆ ELD ˆ 1:0. The numerical results of the dispersion relation, Eq. (10.5), are depicted in Fig. 78. The curves in blue are the bulk bands separated by a gap within which the two surface modes (in pink), which start with opposite group velocities at z ' 0:27, propagate. The surface modes are characterized by the real l, considering both positive and negative signs. The surface modes become degenerate at z ' 2:0. Interesting effect that emerges as a result of the ®nite-size is that the bulk band continuum of states found in the in®nite or semi-in®nite superlattice breaks up into closely spaced discrete states in the ®nite superlattice. One further difference between the semi-in®nite and the ®nite cases considered here is that the surface modes actually tend to merge continuously into the bulk bands. It should be pointed out that the gap between the two (sets of discrete) bulk bands exists only for the case dA 6ˆ dB ; in the case that dA ˆ dB , the gap diminishes and no surface mode is allowed to propagate. The asymptotic limits attained both by the bulk bands and the surface modes is well speci®ed by Eq. (10.7) or (10.8). The effect of retardation on the plasmon±polaritons propagating in a ®nite superlattice, with the same parameters as employed in Fig. 78, is demonstrated in Fig. 79. We plot the dispersion curves based on Eq. (10.4) only towards the right of the light line (dashed line demarked as aB ˆ aC ˆ aD ˆ 0). As one can notice, the retardation effect allows both surface modes to start from the origin; these surface modes turn out to be apparently degenerate and indiscernible, particularly at lower and higher propagation vectors. In addition, the width of the (discrete) bulk bands is seen to reduce throughout. A word of warning regarding the computation that should enable one to accept only the physical solutions and discard the unphysical ones is in order. Let us ®rst mention that if there are p unit cells,

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Fig. 78. Collective excitation spectrum for a binary superlattice made up of ®ve unit cells: A  Al, B  vacuum, EC  EE ˆ 1, and ED  EF ˆ 1. The dimensionless variables are: x ˆ o=opA , and z ˆ kd1 . Note the discrete allowed modes (in blue) and splitting of the surface modes (in pink), as discussed in the text. (Redone after Johnson et al., Ref. [753].) Fig. 79. The same as in Fig. 78, but including the effect of retardation. Notice the effect of retardation on the propagation characteristics of both the discrete allowed modes (in blue) and the surface modes (in pink). The dashed line in black is the light line in vacuum.

with a total of n  p active layers, then the total number of bona®de solutions must unfailingly come out to be 2n. This is because only the (two) interfaces of an active layer (in each unit cell) will give rise to two bona®de modes. However, if one searches zeros of a transcendental function like the one in Eq. (10.4) or (10.5) a bit carelessly, then one may be surprised to obtain 4n modes, instead of 2n. One searches zeros of a function usually, or so I guess, by checking to see if the function has changed the sign, i.e., if the previous value of the function has a different sign than the current value as one scans through different values of o for a given q, or vice versa. Obviously, if the function crosses the axis you have a change of sign and this gives you a good solution. However, if there is a pole in the function at some particular frequency so that one value goes to ‡1 and the next one goes to ÿ1, then this also apparently looks a solution; and this happens quite often. In such a situation you might get exactly twice as many roots as expected. This state of affair is clearly shown in Fig. 80, where we plot the real part of the function versus the reduced frequency x, for a given wave vector z ˆ 1:0, and p ˆ 5. The bona®de solutions are marked with circles. The next case we consider is with p ˆ 10; the rest of the parameters are the same as in Fig. 78. The numerical results in terms of the dimensionless variables are plotted in Fig. 81. There are several noteworthy differences as compared to Fig. 78. The widths of the bulk bands increase, and their maximum width occurs at relatively lower wave vectors. Interestingly, both surface modes start and merge together at relatively lower wave vectors, as compared to Fig. 78. Camley and coworkers [753] had noted that the boundaries of the bulk bands as well as the frequency of the surface modes agree very nearly with those found for the semi-in®nite superlattice with the same parameters only for p  20. One signi®cant exception to this occurs in the region for z  0:3 (see Figs. 78 and 81), where the two surface modes start appearing. In this region, the wavelength of the plasmon is larger than the

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255

Fig. 80. A scheme outlining the strategy of choosing the physical solutions and discarding the unphysical ones. We plot the real part of the function versus the reduced frequency x, for a given reduced wave vector z ˆ 1:0 and p ˆ 5. The valid solutions are marked by circles (see the text for details).

thickness of a ®lm and can also be larger than the whole thickness of the superlattice. As a result, the surface plasmon at the top surface of the superlattice can interact with the surface plasmon at the bottom surface of the superlattice. This interaction produces the splitting of the surface plasmon frequency which is visible in both Figs. 78 and 81. A similar situation prevails in the case of a single ®lm. As the thickness is reduced, the surface plasmon propagating at the top and the bottom surfaces interact; the net result is that there are two surface±plasmon branches, instead of one for a very thick ®lm. We now turn to a more realistic case where material layer B is an insulating spacer, and the layered ®lms are resting on an insulating substrate. We consider a structure where EB ˆ 3:0, EC ˆ 1:0, and ED ˆ 2:5. These parameters are appropriate for the case where material B is Al2 O3 and the substrate is SiO2 . In addition, we have chosen dA =dB ˆ 2:0 and p ˆ 10, just as before. The numerical results in terms of dimensionless variables are presented in Fig. 82. There are several features which are uncommon in Figs. 81 and 82. (i) There is a surface mode for large z which starts from the top of the upper bulk band at z ' 1:44 and becomes asymptotic to a frequency (x ' 0:7071) speci®ed by Eq. (10.8); this frequency corresponds to that of a surface mode propagating at the interface of layer A and vacuum in a semi-in®nite geometry. (ii) Comparing the frequency of the bulk and surface modes in Fig. 82 with those in Fig. 81 leads us to conclude that they are generally down-shifted. (iii) There is second surface mode at lower frequencies that is analogous to the lower split mode in Fig. 81, and which approaches the same asymptotic limit (x ' 0:5) as the one attained by the bulk bands, speci®ed by Eq. (10.7). (iv) The most striking feature in Fig. 82 is a short segment of surface mode, which

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emerges from the lowest discrete mode of the upper bulk band and exists as a surface mode in the wave vector range 0:34  z  1:16, and then becomes a bulk mode again. In this wave vector range the dispersion curve is almost ¯at. Fig. 83 illustrates the case where the thickness ratio is inverted, i.e., dA =dB ˆ 0:5 rather than 2 as in Fig. 82. We notice that the frequencies are further down-shifted as compared to Fig. 82. This is because the insulating spacers (layers B) further isolate the active (metallic) ®lms (of layers A). Moreover, the bulk modes are considerably compressed in comparison to the preceding ®gure. This results due to the fact that the large gaps between active layers reduce the coupling strength between the surface excitations on each metal ®lm, which in turn reduces the band-width of the bulk excitations. The upper surface mode now starts from the top of the upper bulk band at z ' 0:37, and becomes asymptotic to the frequency x ' 0:7071, speci®ed by Eq. (10.8). The lower surface mode now starts from the top of the lower bulk band at z ' 0:48, and approaches, along with the bulk bands, an asymptotic limit speci®ed by Eq. (10.7). It is also interesting to note that the surface mode seen in Fig. 82, which existed only in a short range of z, has disappeared. A reader should be warned that our de®nition of k and k0 in the framework of TMM, in Eq. (10.2), are apparently different from that of Camley and coworkers (see Eqs. (42) and (43) in Ref. [753]). However, they are implicitly identical, because both de®nitions give exactly the same results. At this point, we stress again that seeking a direct (analytical) correspondence between the theoretical results derived in the framework of a TMM and in the form of the boundary conditions does not always make much sense, and, generally, turns out to be a ``hard nut to crack''. 10.2. Doping (n±i±p±i) superlattices This section will be devoted to discuss some illustrative numerical examples for plasmon±polaritons propagating in a ®nite n±i±p±i superlattice. For this purpose, as we stated before, we need to specify our four-layered structure (Fig. 77) as well as the general dispersion relation, Eq. (10.1), corresponding to the n±i±p±i geometry. For doing so, we assume that the layers A and C are, respectively, n- and p-doped, and B and D are the intrinsic semiconductors. Consequently, the plasma frequency opA ˆ ope, opC ˆ oph , and opB ˆ 0 ˆ opD . Also, the bounding media E and F will be considered to be unifentical dielectrics with opE ˆ 0 ˆ opF . Moreover, the n±i±p±i superlattice can be fabricated with a single host semiconductor, since there are no restrictions on the choice of materials due to the requirements of lattice matching. As such, we have made an extensive numerical computation for the n±i±p±i superstructure made up of GaAs as a single host semiconductor. However, we do consider some exceptional cases which are seen to give interesting numerical results on the plasmon±polariton propagation in ®nite, four-layered geometry. We ®rst focus on the analytical diagnoses of the exact dispersion relation, Eq. (10.1) in the NRL (c ! 1) in order to understand the asymptotic limits attained by the bulk bands and the surface modes. The latter are characterized, as usual, by the real, positive or negative, l (or b). In the NRL, Eq. (10.1) simpli®es to the form [757]: …EA ‡ EB †…EB ‡ EC †…EC ‡ ED †…ED ‡ EA †…EA ‡ EE †…EA ÿ EF †…1 ÿ e2pl † ˆ 0:

(10.9)

The possibility of vanishing of the last factor in this equation is ruled out for the simple reasons (p 6ˆ 0 and l 6ˆ 0). The ®rst, second, third, and fourth factors independently equated to zero yield the

M.S. Kushwaha / Surface Science Reports 41 (2001) 1±416

following:   ELB ÿ1=2 xˆ 1‡ ; ELA   ELB ÿ1=2 opC xˆ 1‡ ; ELC opA   ELD ÿ1=2 opC ; xˆ 1‡ ELC opA   ELD ÿ1=2 : xˆ 1‡ ELA

257

(10.10) (10.11) (10.12) (10.13)

Note that in the case when material media B and D are the same, Eqs. (10.10) and (10.13) give identical results. The same is true with respect to Eqs. (10.11) and (10.12). Now, ®fth factor equated to zero yields   ELE ÿ1=2 xˆ 1‡ : (10.14) ELA The vanishing of the sixth factor in Eq. (10.9) is ruled out; otherwise the conditions EA ‡ EB ˆ 0 leading to Eq. (10.10), …ED ‡ EA † ˆ 0 leading to Eq. (10.13), and …EA ‡ EE † ˆ 0 leading to Eq. (10.14) can never be satis®ed. In the general case that the material media A, B, C, D, E, and F are all taken to be different, we should have ®ve distinct asymptotic frequencies. However, our numerical computation is restricted to at least two situations: the bounding media E and F are identical and taken to be the vacuum, and the host semiconductor for the layer A and layer C is the same (i.e., ELA ˆ ELC ) but with different electron and hole concentrations. We now present the illustrative examples for several cases considered. The ®rst case we have investigated is that of a n±i±p±i superstructure made up of a single host semiconductor (GaAs) with ELA ˆ ELB ˆ ELC ˆ ELD ˆ 13:13 and the bounding media are both taken to be the i.e., EE ˆ EF ˆ 1:0. The plasma frequency of layer C is de®ned such that opC ˆ pvacuum,  opA = 2, and the thicknesses of the doped layers are half of those of the intrinsic layers; speci®cally dA ˆ opA dA =c ˆ 0:5, dB ˆ opA dB =c ˆ 1:0, dC ˆ opA dC =c ˆ 0:5, and dD ˆ opA dD =c ˆ 1:0. The number of the unit cells p ˆ 10. The numerical results in terms of the dimensionless frequency x ˆ o=opA and the propagation vector z ˆ cq=opA are plotted in Fig. 84. The lower pair of bulk bands starts from the origin, propagates towards the far right of the light line in vacuum, and becomes asymptotic to the frequency speci®ed by Eq. (10.11) or (10.12), at large wave vector. The upper pair of bulk bands shown to start from the light line observes a crossing point at z ' 1:78, and approaches an asymptotic limit speci®ed by Eq. (10.10) or (10.13). There are two surface modes which both start from the origin, rise just to the right of the light line, deviate from the light line with opposite group velocities, become apparently degenerate at z ' 3 and then propagate in the close vicinity of the top of the second lowest bulk band. For these values of the parameters, we do not ®nd any other surface mode that would approach the asymptotic limit speci®ed by Eq. (10.14). In other words, the surface modes are seen to propagate only within the gap between the two pairs of bulk bands. Fig. 85 illustrates the real (in black) and imaginary (in red) parts of the transcendental function, Eq. (10.1), for the parameters used in Fig. 84, for a given z ˆ 1:0. Our numerical results in Fig. 84 are

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Fig. 81. The same as in Fig. 78, but for (the number of unit cells) p ˆ 10. (Redone after Johnson et al., Ref. [753].) Fig. 82. The same as in Fig. 78, but for a more realistic case when EB ˆ 3:0 (Al2 O3 ), EC ˆ 1, and ED ˆ 2:5 (SiO2 ). Notice the change in the propagation characteristics, as well as in the number, of the surface modes (in pink). (Redone after Johnson et al., Ref. [753].) Fig. 83. The same as in Fig. 82, but for the case when dA =dB ˆ 0:5, i.e., the surface layer (on the side of the Al2 O3 substrate) is half as thick as the insulating layer B. Notice that the second-lowest surface mode propagating in a small range of z in Fig. 82 has disappeared. (Redone after Johnson et al., Ref. [753].) Fig. 84. Collective excitation spectrum of a ®nite n±i±p±i superstructure (Fig. 77), for p (the number of unit cells)ˆ10. Discrete allowed modes are shown in blue and the surface modes in pink. The dashed line in black is the light line in vacuum (the surrounding media E and F limiting the size of the superstructure). The material parameters are as listed in the ®gure. (After Kushwaha, Ref. [757].)

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Fig. 85. A plot of the real (in black) and imaginary (in red) parts of the transcendental function, Eq. (10.1), versus the reduced frequency x ˆ o=opA, for a given reduced wave vector z ˆ cq=opA ˆ 1:0. The rest of the parameters are the same as used in Fig. 84. See the text for the relevant discussion on this ®gure. Fig. 86. The same as in Fig. 84, but when layers B and D, as well as the outer media E and F, are all treated as vacuum (i.e., EB ˆ ED ˆ EE ˆ EF ˆ 1). Notice the drastic change in the propagation characteristics of both discrete allowed modes and the surface modes (in pink). The touching point between the upper pair of bands of discrete modes (in Fig. 84) has shifted to the lower z. Fig. 87. The same as in Fig. 86, but when EB ˆ EE ˆ EF ˆ 1:0 and ED ˆ 2:5 (SiO2 ). Notice the difference in the behavior characteristics of the discrete and surface modes, and the fact that the touching point has disappeared altogether. Fig. 88. The same as in Fig. 86, but when EB ˆ 2:5 and ED ˆ EE ˆ EF ˆ 1:0. A comparative look at the results in this ®gure and those in Fig. 86 is worth attention.

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based on the search of zeros of the real part of the function, irrespective of whether or not its imaginary part is zero. One can notice that Fig. 85 justi®es width of all the four bulk bands as well as the frequency of the lower surface mode (encircled solution) at z ˆ 1:0 in Fig. 84. However, the upper surface mode (in Fig. 84), though justi®ed as a bona®de solution characterized by real l (or b), is not fully justi®able, since it corresponds to the next zero (in Fig. 85) of the real part of the function which is apparently a solution due to the pole in the function. Such a situation that may occur quite often in some cases renders one in dilemma about the choice of the valid, physical solutions and even question the methodology. The present scheme of ®nding roots of the function rules out the existence of the lowest and uppermost surface modes in the corresponding ®gure in the original paper (see Fig. 2 in [757]), where the author searched the solutions by looking for the absolute minimum of the transcendental function. The next case we consider is when medium A and medium C are still the same n- and p-doped GaAs, but the layers B and D, as well as the outer media E and F, are vacuum. The relative thicknesses of the layers are the same as in the preceding ®gure. The dispersion curves in terms of the dimensionless variables are plotted in Fig. 86. The noteworthy features visible in Fig. 86 are the following. There is a pair of surface modes that starts from the origin, with both modes rising towards the right of the light line, in the gap between the two pairs of bulk bands. The lower one of them merges with the second lowest bulk band at z ' 2:0, whereas the upper one stops propagating at the plasmon frequency of layer C, opC . Two new surface modes start from the light line at the upper edge of the third lowest bulk band, propagate such as to cross the touching point (at z ' 1:0) between the upper pair of the bulk bands, and then interchange their characteristics; the lower one propagates in the close vicinity of the top of the third lowest bulk band, while the upper one rises to be trapped in the gap between the upper pair of bulk bands, but without merging with any of the two bulk bands. There are two asymptotic frequencies in the problem for these values of parameters; the ®rst speci®ed by Eq. (10.11) or (10.12), and the second given by Eq. (10.10) or (10.13) or (10.14). The lowest pair of bulk bands and the lowest surface mode approach the ®rst asymptotic frequency x ' 0:6816, and the upper pair of bulk bands and the pair of the gap-surface modes attain the second asymptotic limit x ' 0:9639. The upper pair of surface modes, in this sense, corresponds, in the large z limit, to the frequency of the surface plasmon propagating at the GaAs±vacuum interface in the semi-in®nite geometry. We now turn to the situation where material layers B and D are unidentical insulators; speci®cally, EB ˆ 1:0 and ED ˆ 2:5, and the rest of the parameters remain the same as in Fig. 86. The illustrative numerical results for the dispersion relation are shown in Fig. 87. Comparing results in Fig. 87 with those in Fig. 86 reveals several noteworthy features. The gap between the lower, as well as the upper, pair of bulk bands increases; resulting into a considerable reduction in the gap between the two pairs of bulk bands. The second lowest surface mode now propagates with the negative group velocity after reaching opC , unless it merges with the lowest surface mode (at z ' 1:65); at z  2:0, this doubly degenerate mode is seen to have the same characteristics as it did in Fig. 86. Although the touching point in-between the upper pair of bulk bands has disappeared altogether, the pair of the gap-surface modes is seen to suffer from two crossings, not just one. After the second crossing, the upper gap surface mode touches the lower boundary of the uppermost bulk band in the wave vector range 1:1  z  1:3, and then propagates at a slightly lower frequency in the vicinity of the uppermost bulk band; the lower gap-surface mode propagates at still lower frequency. Now there are four independent asymptotic frequencies in the problem: x1 ' 0:6481, x2 ˆ 0:6816, x3 ' 0:9165, and x4 ˆ 0:9639 speci®ed, respectively, by Eqs. (10.12), (10.11), (10.13) and (10.10) or (10.14). In the short wavelength

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limit, the lowest bulk band approaches x1 , the second bulk band as well as the lower (®nally degenerate) pair of surface modes approach x2 , the third bulk band reaches x3 , and the upper pair of surface modes and the topmost bulk band approach x4 . Fig. 88 illustrates the case where the material medium in layers B and D is interchanged, as compared to Fig. 87; speci®cally, EB ˆ 2:5 and ED ˆ 1:0. The rest of the parameters are the same as in the preceding case. A comparison of Figs. 87 and 88 indicates that the difference, particularly with respect to the surface modes, can be quite signi®cant. One sees a larger separation between the lowest surface mode and the second lowest bulk band before the former merges into the latter just as in Fig. 86, the second lowest surface mode stops propagating after arriving at opC ; just as in Fig. 86. The crossing point between the upper pair of surface modes, within the gap between the upper pair of bulk bands, now moves to the higher wave vector (z ' 1:53); after which both of these surface modes propagate in the close vicinity of the third lowest bulk band. However, there still remain the same four asymptotic limits as stated in the description of Fig. 87, except that now x1, x2 , x3 , and x4 are speci®ed, respectively, by Eqs. (10.11), (10.12), (10.10) and (10.13) or (10.14). The third lowest bulk band that starts from the light line at opC has almost no width at the starting point as compared to Fig. 84. This is also true for the corresponding bulk band in Fig. 87. It is also noteworthy that the frequencies of bulk and surface modes both in Figs. 87 and 88 are generally down-shifted as compared to those in Fig. 86, particularly at lower wave vectors. This is quite likely caused by the screening of the doped layers by the intervening insulating layers B or D (with higher dielectric constants). Also, the discrete modes making up the third lowest bulk band, both in Figs. 87 and 88, are seen to be relatively more compressed in comparison to Fig. 86. As pointed out before, this is usually seen to occur due to the larger insulating gaps between the active layers. Here, on the contrary, such gaps are kept ®xed. It then seems that such a compression of modes is also attributable to the insulating spacers with larger dielectric constants. The interesting cases worth investigating, and which should give deeper insight into the ®nite-size effects on the plasmon±polariton propagation in the ®nite superstructure, are those where one studies the effects of variation in the relative thicknesses of the constituent layers in the unit cell. The effects of externally applied magnetic ®elds in different con®gurations, which have not, to our knowledge, yet been investigated, are also worth attempting. Other intricate parameters which make the problem more realistic are the damping effects, and the coupling to the optical phonons, particularly when the host material is a polar semiconductor. 11. Quantum structures Ð intrasubband excitations The synthesis of semiconductor heterostructures with speci®ed band gaps has led to the discovery of totally unexpected 2D behavior of charge carriers. Because of their applications to novel and potentially useful devices, the physics of semiconductor heterostructures and superlattices has attracted a great deal of attention, both experimental and theoretical, in the past decade [1654]. Theoretically, the reduced degrees of freedom allow detailed and often exact calculations. Experimentally, new and unexpected phenomena have been observed. To the understanding of electronic and optical properties of superlattice systems, the studies of elementary collective excitations are of paramount importance. Studies of such elementary excitations were initiated and stimulated by a classic paper by Das Sarma and Quinn [658,1748]. Since then numerous workers have devoted to the understanding of different types of collective excitations in various kinds of such 2D systems (see Section 2.2.2 for detailed

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account). This section is exclusively concerned with both type-I (typi®ed by GaAs±Ga1ÿx Alx As systems) and type-II (typi®ed by InAs±GaSb systems) superlattices which have been the most widely studied systems in the past. The simplest model of type-I superlattice which we will be concerned with is a low-temperature periodic system of 2DEG. Similarly, it is suf®cient, for our purpose, to consider type-II superlattice as a periodic arrangement of 2D-con®ned alternating electron and hole layers. A 2D electron system, such as can be realized in the aforesaid semiconductor superlattices, consists of charge carriers which are mobile in the plane along the interfaces and con®ned in the perpendicular direction (i.e., along the superlattice axis, taken to be the z-direction). This con®nement leads to the formation of discrete subbands. Application of an external magnetic ®eld along the superlattice axis splits each subband into Landau levels and leads therefore to a completely quantized system. This case where the magnetic ®eld is oriented in the same direction as the con®ning electric ®eld is a very special one because only then cantle Hamiltonian be separated into an electric part leading to the subbands and a magnetic part leading to the Landau levels. For any other orientation, this separation is not possible anymore and the band structure becomes much more complicated. The effect of an applied magnetic ®eld in the perpendicular con®guration on the collective excitations in both types-I and type-II superlattices was studied fairly completely by Das Sarma and Quinn [658,1748], Tselis and Quinn [673,674], and Kushwaha [726], within the framework of both macroscopic and microscopic approaches. The present section is devoted to examine theoretical collective excitations, both bulk and surface, which occur in these systems. We con®ne mainly to the intrasubband modes with in the approximation that each layer (of the periodic system) is strictly 2D with no overlap between adjacent layers. This approximation is made just for the sake of convenience, allowing us to derive most of the desired results analytically. It is expected to be reasonably good for low electron densities when only the lowest 2D subband in each quantum well is occupied by electrons. Intrasubband collective modes, with or without an applied magnetic ®eld, can be investigated by using either macroscopic or microscopic approaches, with exactly the same degree of accuracy. The analytical results stated here are derived in the framework of simple EM theory, where the con®nement of the charge carriers is governed by the Dirac-delta function. We are interested in the situation where the widths of the potential wells or barriers in the superlattice structure are much larger than the de Broglie wavelength, so that the quantum-size effects remain unimportant. We will consider the applied magnetic ®eld in two different con®gurations: perpendicular geometry and Voigt geometry. The former geometry is dealt with indirectly in the framework of TMM, while the latter is worked out in the framework of Green-function (or response-function) theory. The zero-®eld results will also be discussed for the sake of comparison. The analytical as well as numerical results in Section 11.1 (Section 11.2) correspond to the situation where the thickness of the 2D layers is zero (non-zero). The schematics of the geometries will be described brie¯y in the respective sections. We consider both perfectly periodic (in®nite) and truncated (semi-in®nite) 2D systems; the latter systems are useful in exciting surface plasmons and magnetoplasmons Ð the modes characterized by the EM ®elds localized at and decaying exponentially away from the 2D planes. 11.1. Periodic systems of 2D layers with zero thicknesses We consider a periodic array of 2D layers: alternate layers contain electrons (with a real density ne and effective mass me ) and holes (with a real density nh and effective mass mh ). The superlattice

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Fig. 89. Schematics of type-II superlattice system with 2DEG and 2DHG layers of zero thicknesses. Alternative layers contain electrons (with areal density ne and effective mass me ) and holes (with areal density nh and effective mass mh ). The periodic system consists of a series of equally spaced, parallel 2D planes centered at z ˆ ld, where l is the layer index and d is spacing between the electron and hole layers. Thus the periodicity of the system lies along the superlattice axis, taken to be the z-axis of the Cartesian coordinate system. The even (l ˆ 0; 2; 4; . . .) and odd (l ˆ 1; 3; 5; . . .) number of layers are occupied by the 2DEG and 2DHG layers, respectively. The motion of the charge carriers is free in the x±y plane and the superlattice potential con®nes them to the planes z ˆ ld through the Dirac-delta function. The tunneling of the carriers between the consecutive layers is completely forbidden. The wave propagates along the y-axis (i.e., ~ q k ^y) and the magnetic ®eld is oriented along the growth axis (i.e., ~ B0 k ^z).

structure (see Fig. 89) consists of equally spaced, parallel 2D planes centered at z ˆ  ld, l ˆ 0; 1; 2; 3; . . ., where l is the layer index and d the spacing between electron and hole layers. Thus the periodicity of the system lies along the superlattice axis (taken to be the z-direction of the Cartesian coordinate system).The even (l ˆ 0; 2; 4; . . .) and odd (l ˆ 1; 3; 5; . . .) numbered layers are assumed to be occupied by 2DEG and 2DHG, respectively. The external magnetic ®eld (~ B0 ) is oriented along the superlattice axis. Further, we assume that the motion of the charge carriers is free in the 2D x± y plane, and the superlattice potential con®nes the charge carriers strictly to the planes z ˆ  ld. The tunneling of the carriers between the consecutive layers is completely forbidden. This model system of periodic 2D multilayers located at z ˆ ld is assumed to be embedded in a background characterized by the dielectric constant Es . The truncated superlattice systems, useful for exciting surface plasmons and magnetoplasmons, is described such that the in®nite periodic system is truncated at an interface, say z ˆ 0, so that the region ÿ1  z  0 is occupied by an insulating medium characterized by a dielectric constant E0 . The standard EM boundary conditions used for accomplishing the relevant task in this section are exactly analogous to those given in Eq. (6.4), accompanied by the Bloch theorem for in®nitely periodic system, or its surface analog for the truncated systems. It should be pointed out that our general dispersion relations, both with and without an applied magnetic ®eld, do account for the retardation effects, but we ignore the damping effects.

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11.1.1. Zero magnetic ®eld 11.1.1.1. In®nite type-II superlattice. In the absence of an applied magnetic field, the general dispersion relation for collective (bulk) excitations in an infinite type-II superlattice systems reads [724]:      2pa 2pa 2pa 2 1‡ w Se w Se ÿ we wh S2h ˆ 0; (11.1) 1‡ Es e Es h Es where the structure factors Se and Sh are de®ned by Se ˆ

sinh…2ad† ; cosh…2ad† ÿ cos…2kd†

Sh ˆ

2 sinh…ad† cos…kd† : cosh…2ad† ÿ cos…2kd†

(11.2)

Note that k…  qz † is the Bloch vector and 2d is the period of the type-II superlattice system. The decay constant a ˆ …q2 ÿ q20 Es †1=2 , and the 2D polarizability function wi ˆ ÿni e2 =mi o2 . Here q is the magnitude of the propagation vector ~ q k ^y axis, and o the (angular) frequency of the wave. Eq. (11.1), in the NRL (c ! 1; q  q0 ), can be written in the form 8 9 "   <  2 #1=2 = qd S h o2 ˆ Se …o2pe ‡ o2ph †  …o2pe ÿ o2ph †2 ‡ 4o2pe o2ph ; (11.3) : ; 2Es Se where opi ˆ ‰4pni e2 =…2dmi †Š1=2 is the effective 3D plasma frequency for the electron (i  e) and hole (i  h) system alone, with 3D electron and hole densities given by (ni =2d). Eq. (11.3), for kd ˆ p=2 ) Se ˆ tanh…qd† and Sh ˆ 0, yields  1=2  1=2 qd qd o‡ ˆ ope tanh…qd† ; oÿ ˆ oph tanh…qd† : (11.4) Es Es These solutions represent the inner band edges of the two bulk bands for all values of q in the NRL. Let us have a look at the collective (bulk) modes implied by Eq. (11.3) in the weak coupling (qd  1) and strong coupling (qd  1) limits. In the weak coupling limit, Se ' 1 and Sh ' 0, and hence one obtains, from Eq. (11.3), two independent solutions  1=2  1=2 2pne e2 2pnh e2 1=2 o‡ ˆ q ; oÿ ˆ q1=2 : (11.5) me Es mh Es Both of these solutions lead one to infer that each layer supports its own 2D electron or hole plasmon. In the strong coupling limit, we have two cases to consider: k 6ˆ 0, and k ˆ 0. For the case k 6ˆ 0 and qd  1, we ®nd from Eq. (11.2), Se '

2qd ; 1 ÿ cos…2kd†

Sh '

2qd cos…kd† : 1 ÿ cos…2kd†

(11.6)

Consequently, Eq. (11.3) assumes the following form: o2 '

1 q2 d2 f…o2pe ‡ o2ph †  ‰…o2pe ÿ o2ph †2 ‡ 4o2pe o2ph cos2 …kd†Š1=2 g: Es 1 ÿ cos…2kd†

(11.7)

It thus seems that both of these branches speci®ed by Eq. (11.7) are the acoustic modes, with o / q.

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For k ˆ 0 and qd  1, we get from Eq. (11.2) Se ' 1=qd ' Sh . As a result of which Eq. (11.3) yields " #1=2 …o2pe ‡ o2ph † o‡ ˆ and oÿ ˆ 0: (11.8) Es The ®rst, ®nite, valid mode is that of two independent simultaneously excited 3D electron and hole plasmons, in which the oscillating charge densities are in phase from one supercell to the next, and are out of phase within the supercell. Note that the diagnosis of the collective modes made by Tselis and Quinn [674] in this limit seems to be erroneous. For instance, the mode represented by oÿ was shown to be ®nite, whereas it turns out to be exactly zero. Second, there is possibly a printing error in the de®nitions of the effective 3D (electron and hole) plasmon frequencies. 11.1.1.2. Truncated type-II superlattice. Next, we consider the truncated type-II superlattice system. The general dispersion relation for collective (surface) excitations in semi-infinite type-II superlattice reads [724]:      4pa 2pa 2pa 2 w coth…2ad† 1 ‡ w coth…ad† ÿ we wh csch2 …ad† 1‡ Es e Es h Es    4pa 2pa w w tanh…ad† ; (11.9) ˆl l‡ 1‡ Es e Es h where l ˆ E0 a=Es a0 . Note that Eq. (11.9) represents the surface plasmon±polaritons in the situation where the 2DEG is the surface layer. As will be shown in what follows, type-II superlattices admit, in general, two branches of surface plasmons whose features and location in the o±q plane crucially depend upon the parameters Er …ˆ Es =E0 † and or …ˆ oph =ope †. A novel feature of the type-II superlattice is the existence of a surface plasmon branch lying within the gap between the two bulk plasmon bands. In order to understand this characteristic surface plasmon we analyze Eq. (11.9) in the NRL (c ! 1 ) a ! q), for small qd, to obtain o…q† ˆ f2‰Es …1 ‡ o2r †Šÿ1 g1=2 oph qd ‡ O…q2 d 2 †:

(11.10)

This substantiates that the gap-surface plasmon is an acoustic mode with a characteristic frequency o / q. This acoustic mode never merges into any of the two bulk plasmon bands, and hence it is a long-lived surface excitation mode. Eq. (11.9) is complemented by the expression for b…ˆ ÿik† which is a complex quantity whose real part is positive de®nite and represents the inverse penetration depth. The value of b is determinated by the requirement that the bulk-mode condition be satis®ed for the same values of o and q. This leads to the following relationship: ! " #  2 2 2 o ‡ o o o 1 pe ph pe ph y sinh…2y† ÿ coshÿ1 cosh …2y† ÿ y2 ‰1 ÿ cosh…2y†Š (11.11) bˆ 2d Es o2 Es o2 from Eq. (11.1), where y ˆ ad. A direct inspection of Eq. (11.11) reveals that b is real everywhere in the o±q plane except inside the bulk plasmon bands where it is purely imaginary and coincides with k. It is noteworthy that our surface±plasmon dispersion relation, Eq. (11.9), is independent of b.

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11.1.1.3. In®nite type-I superlattice. The model structure we use for this system is essentially the same as that considered for the type-II superlattice system, except that every other layer contains electrons instead of holes. Clearly, then, the period of the type-I superlattice will be d, instead of 2d. Substituting wh ˆ 0 and replacing 2d by d in Eq. (11.1) yields 2pa 1‡ w Se ˆ 0; (11.12) Es e where the structure factor Se is now de®ned by sinh…ad† : Se ˆ cosh…ad† ÿ cos…kd†

(11.13)

Eq. (11.12) is the dispersion relation for the collective (bulk) excitations in an in®nite type-I superlattice [724]. In the NRL (c ! 1 ) a ! q), Eq. (11.12) can be written as   2pne e2 2 o ˆ (11.14) qSe : me Es Let us consider the weak (qd  1) and strong (qd  1) coupling limits of Eq. (11.14). In the weak coupling limit, Se ' 1, and therefore Eq. (11.14) yields  1=2 2pne e2 q1=2 : (11.15) oˆ me Es Thus the type-I superlattice supports a 2D plasmon in the weak coupling limit. In the strong coupling limit, we again have two cases to consider: k ˆ 0 and k 6ˆ 0. For the case k 6ˆ 0, and qd  1, we ®nd Se ' qd=‰1 ÿ cos…kd†Š, and thus Eq. (11.14) becomes o2 ˆ

2pne e2 q2 d : me Es 1 ÿ cos…kd†

(11.16)

For the case k ˆ 0, Se ' 2=qd, and Eq. (11.14) assumes the form o2 ˆ

4pne e2 o2pe ˆ : me Es Es

(11.17)

As such, in the strong coupling limit, for k 6ˆ 0, the mode is an acoustic plasmon (o / q), whereas for, k ˆ 0, it is a 3D plasmon, with an effective 3D electron density ne ˆ ne =d. 11.1.1.4. Truncated type-I superlattice. Making similar substitutions as mentioned above (i.e., wh ˆ 0 and replacing 2d by d) in Eq. (11.9) yields   4pa 4pa 1‡ w coth…ad† ˆ l l ‡ w : (11.18) Es e Es e This is the dispersion relation for collective (surface) excitations in the truncated type-I superlattice [724]. In the NRL (a ! q ) l ! l0 ˆ E0 =Es ), Eq. (11.18) simpli®es to " #1=2 ope …coth…qd† ÿ l † 0 : (11.19) o ˆ p …qd†1=2 Es …1 ÿ l20 †

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This leads us to infer that in order for this to be a bona®de mode, both numerator and denominator in the square bracket must have the same signs. It has been observed that for lÿ1 0 > 1…< 1†, the surface mode occurs above (below) the bulk plasmon continuum [669,724]. Just as in the preceding case of type-II superlattices, Eq. (11.18) is complemented by the expression for b…ˆ ÿik†, a complex quantity which represents the inverse penetration depth of the surface modes. The expression of b for type-I superlattice can be directly written from Eq. (11.11), with oph ˆ 0 and 2d ! d. This leads to the relationship !  " # 2 o 1 ad pe b ˆ coshÿ1 cosh…ad† ÿ sinh…ad† : (11.20) d 2 Es o2 Eq. (11.20) can also be obtained from Eq. (11.12). Note that the dispersion relation for surface plasmons, Eq. (11.18), is independent of the inverse penetration depth b. 11.1.1.5. Numerical examples. Next, we present some illustrative numerical examples on the collective, bulk and surface, excitations in both type-I and type-II superlattice systems. For this purpose we use the following values of the parameters: Type-I: Es ˆ 13:0, ne ˆ 7:34  1011 cmÿ2 , me ˆ 0:0665m0 ,  d ˆ 600 A , E0 ˆ 1:0; Type-II: Es ˆ 13:0, ne ˆ nh ˆ 4:2  1011 cmÿ2 , me ˆ 2mh ˆ 0:0665m0 , d ˆ  550 A, E0 ˆ 1:0. The numerical examples will be presented in terms of the dimensionless frequency x ˆ …o=ope †  ope d=c† ˆ od=c and dimensionless wave vector z ˆ qd. Fig. 90 illustrates the plot of x

Fig. 90. Bloch bands for type-I superlattice system plotted as reduced frequency x versus reduced Bloch vector kd, for three values of reduced wave vector z ˆ 0:1, 0.5, and 1.0, in the zero magnetic ®eld. The parameters used are: Es ˆ 13:0, E0 ˆ 1:0,  ne ˆ 7:34  1011 cmÿ2 , me ˆ 0:0665 m0 , (period) d ˆ 600 A. (After Kushwaha, Ref. [726].)

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Fig. 91. Collective excitation spectrum for type-I superlattice system in the zero magnetic ®eld. We plot reduced frequency x versus reduced wave vector z. The hatched region is the bulk plasmon band of an in®nite superlattice and the solid curve is the surface plasmon±polariton for a semi-in®nite superlattice. The dashed line is the light line in the vacuum. The parameters used are as listed in the ®gure. (After Kushwaha, Ref. [726].)

versus kd (the dimensionless Bloch vector) for a given parallel (propagation) vector z for type-I superlattice. The effect of the multilayer system in the broadening of a single 2DEG layer plasmon into a band along the superlattice axis is evident. The band-width is determined by the properties of the superlattice as discussed by Bloss and Brody [654], and is adjustable simply by choosing the appropriate parameters characterizing the system. The complete dispersion relations of bulk and surface plasmons for type-I superlattice are plotted in Fig. 91. The shaded region shows the bulk plasmon band continuum whose boundaries correspond to k ˆ 0 and k ˆ p=d. For small values of z, corresponding to the strong coupling limit, the mode for p k ˆ 0 starts at an effective 3D plasma frequency o ˆ ope = Es which means x ' 0:004244, whereas that for k ˆ p=d starts with linear dispersion at x ˆ 0. The solid curve above the upper edge of the bulk plasmon band is the surface plasmon±polariton whose existence is in analogy with the criterion as discussed by Giuliani and Quinn [669] and by Kushwaha [725,1753]. The intersection of the polariton mode with the band edge occurs at a critical value of q ˆ q given by 1 1 ‡ l : (11.21) a ˆ ln d 1 ÿ l Eq. (11.21), in the NRL (c ! 1), reduces to Eq. (3) of Giuliani and Quinn [669]. For q < q, the polariton mode does not exist because there b is purely imaginary. It has been found that the value of q decreases when the retardation effect is included. This leads us to believe that the ATR should prove to

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Fig. 92. Bloch bands for type-II superlattice system plotted as reduced frequency x versus reduced Bloch vector kd, for three values of reduced wave vector z ˆ 0:1, 0.5, and 1.0, in the zero magnetic ®eld. The parameters used are: Es ˆ 13:0, E0 ˆ 1:0,  ne ˆ nh ˆ 4:2  1011 cmÿ2 , me ˆ 2mh ˆ 0:0665 m0 , (period) d ˆ 550 A. (After Kushwaha, Ref. [726].)

be an appropriate experimental probe for observing such a surface mode in the truncated superlattice system. In the NRL, the situation takes a different turn and ATR has been argued to be an unlikely method for the observation of these modes, particularly in the long wavelength limit [669, 724]. We now turn to the illustrative examples for bulk and surface excitations of the type-II superlattice systems. Fig. 92 shows the plot of x versus kd for a given propagation vector z. The broadening of the single 2DEG and 2DHG layer plasmons into the bands due to the multilayers is evident. In analogy with the two masses per unit cell in the phonon dynamics, the Brillouin zone is halved by the reduced periodicity producing one optical-like and the other acoustic-like plasmon branch. This results in the opening up of a gap at the zone boundaries. The band-widths and the magnitude of the resulting gaps are both adjustable and much depend on the choice of the material parameters characterizing the superlattice system at hand. The full dispersion relations for the collective (bulk and surface) excitations of type-II superlattice systems are illustrated in Fig. 93. The two shaded regions constitute the bulk plasmon bands whose extreme edges correspond to the Bloch vector k ˆ 0 and k ˆ p=2d; 2d now being the superlattice period. At the zone center, the upper and the lower modes are seen to start, respectively, at o ˆ ‰…1 ‡ o2r †=Es Š1=2 ope ) x ' 0:003765 for k ˆ 0 and at o ˆ 0 for k ˆ p=2d; or ˆ oph =ope . The variation of k from 0 to p=2d results into a separation between the two bulk bands provided that the ratio or is different from unity. At large wave vector, the asymptotic limits attained by both bulk bands

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Fig. 93. Collective excitation spectrum for type-II superlattice system in the zero magnetic ®eld. We plot reduced frequency x versus reduced wave vector z. The hatched regions are bulk plasmon bands of the in®nite superlattice and the solid curves are the surface polariton modes of a semi-in®nite superlattice. The dashed line demarked as a ˆ 0 is the light line in vacuum. The parameters used are as listed in the ®gure. (After Kushwaha, Ref. [726].)

are speci®ed by Eq. (11.3). The two solid curves Ð one appearing above the upper edge of the upper bulk band and another in the gap between the two bulk bands Ð are the surface plasmon±polariton modes. Similarly to the corresponding case of type-I superlattices, the features and location of the surface modes depend upon the parameters Er …ˆ Es =E0 † and or [669,724]. The upper plasmon±polariton emerges from the edge of the upper bulk band at very small wave vector and merges into it large wave vector. It has been noticed [670] that this surface mode, which directly corresponds to the one appearing in type-I superlattices, can only exist either above the upper edge of the upper bulk band or below the lower edge of the lower bulk band. In both cases, b turns out to be purely real. Moreover, except for extremely limiting situations, these modes never exist for very small values of z. At some ®nite value of q, they intersect the bulk continuum. The critical value of q where this occurs also corresponds to the value of b ! 0 as the surface mode turns into a bulk mode. The value of q , with retardation, is found to be such that 1 …1 ‡ l†…1 ‡ lo2r † : (11.22) ln aˆ 2d …1 ÿ l†…1 ÿ lo2r † This equation, in the NRL, reduces to Eq. (7) of Qin et al. [670]. Again, we ®nd that the inclusion of the retardation effects reduces the value of q , and hence the limitation of ATR to observe this mode is

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ruled out. The lower surface mode which appears within the gap between the two bulk bands never merges into either of the two bands. This is an acoustic surface plasmon which, in the strong coupling limit (ad  1), can be readily shown to be speci®ed by Eq. (11.10). A novel feature of this acoustic plasmon is that it is known to be free from Landau damping. That implies that the collective mode has a long lifetime and hence represents a well-de®ned excitation of the system. The extensive investigations on the collective excitations in the superlattice systems have, however, shown that this remarkable property (of being free from Landau damping) is characteristic feature of the surface plasmons of all types of superlattices and is attributed to the quantization of electronic motion along the superlattice axis. It should be remarked that there is a readily understandable close correspondence between the analytical and the numerical results for both kinds of superlattices. What is remarkable is that our dispersion relations for surface plasmons, in both types of superlattices, Eqs. (11.9) and (11.18), are found to be independent of the parameter b (ˆ ÿik), which represents the inverse penetration depth. This makes our analytical results physically more meaningful for the diagnoses of the surface modes. We stress that our semiclassical theory for the collective excitations in both types of superlattices is, to our knowledge, the ®rst of its kind to introduce the surface plasmons (in superlattices) which are explicitly independent of the inverse penetration depth. This new feature is a physically signi®cant peculiarity of surface plasmons in superlattices, in addition to their being free from Landau damping. Therefore, we believe that, in high-mobility superlattice systems, these surface modes are considerably long-lived excitations and potentially useful for the surface wave devices. The surface modes studied in the present work should be observable by means of well-established techniques such as ATR, resonant Raman and Brillouin scattering, and EELS. However, relatively large value of q leaves ATR an unlikely method of observation. In the following section, we generalize the present treatment to the case where the model superlattice systems are subject to an applied magnetic ®eld oriented along the superlattice axis. 11.1.2. Non-zero magnetic ®eld in the perpendicular geometry In this section, we consider the same in®nite (periodic) and truncated (semi-in®nite) 2D superlattices of both types subjected to an external magnetic ®eld in the perpendicular con®guration (~ B0 k ^z-axis, ~ q k ^y-axis). Just as in the preceding section, we will neglect the miniband structure of the model superlattices, consider only the ground subband, and hence con®ne our interest to the intrasubband magnetoplasmons. This is a good approximation for these low-density, high-mobility systems at low temperatures. Moreover, such a situation can be dealt with both macroscopic and microscopic approaches. Note that the simple EM theory used here provides us with the exact analogs of the analytical results for the collective (bulk and surface) excitations derived by Quinn and coworker [673,674], using somewhat more sophisticated (SCFA) method. Furthermore, the present theory allows us to include the effects of retardation and phenomenological damping, in addition to the coupling to the optical phonons, in a simpler and systematic way. We examine the collective excitation spectra of 2D magnetoplasmons in both types of superlattices in the situation where the con®nement of the charge carriers to the 2D planes is governed by the Dirac-delta function. The details of the theoretical treatment were reported by the author [726] to which we will refer quite frequently in what follows. We will recall only shorter analytical results and refer for lengthy expressions (and substitutions) to the original paper [726], where extensive analytical diagnoses of the exact analytical results were presented.

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11.1.2.1. In®nite type-II superlattice. In the presence of a perpendicular magnetic field, the dispersion relation derived for the collective (bulk) excitations in the type-II superlattice reads [726]:   E 2  Es s e h 2 2 e h wyy wyy …Se ÿ Sh † ‡ …w ‡ wyy †Se ‡ 2pa yy 2pa "  2 # a a  wexx whxx …S2e ÿ S2h † ÿ …we ‡ whxx †Se ‡ 2pq20 xx 2pq20   h Es i e 2 a e 2 2 2 ÿ wyy …Se ÿ Sh † ‡ Se …whxy whyx † Se wxx …Se ÿ Sh † ÿ 2pa 2pq20   h Es i h 2 a h 2 2 2 Se wxx …Se ÿ Sh † ÿ ÿ wyy …Se ÿ Sh † ‡ Se …wexy weyx † 2pa 2pq20 Es (11.23) ‡ wexy weyx whxy whyx …S2e ÿ S2h †2 ÿ 2 2 wexy whxy S2h ˆ 0; 2p q0 where the structure factors Se and Sh are the same as de®ned in Eq. (11.2). The polarizability tensor elements wijj0 (i  e; h and j; j0  x; y) in the presence of an external magnetic ®eld are de®ned by wiyy ˆ wixx ˆ ÿ

ni e2 1 ; mi …o2 ÿ o2ci †

wixy ˆ ÿwiyx ˆ i

ni e2 oci ; mi o…o2 ÿ o2ci †

(11.24)

where mi is the effective mass of the electron (i  e) or the hole (i  h), and oci …ˆ eB0 =mi c† is the electron or hole±cyclotron frequency. It can be readily veri®ed that, in the special limits of zero magnetic ®eld, Eq. (11.23) reproduces the well-known results of TM modes (Eq. (11.1)) and TE modes. The former (latter) modes are obtained by equating the ®rst (second) square bracket in the ®rst term of Eq. (11.23) to zero, in the absence of a magnetic ®eld. The other limiting cases, for example in the weak (ad  1) and the strong (ad  1) coupling limits, will be referred to while discussing the illustrative numerical results a little further on. Here, we would like to recall the analytical diagnosis in the limit of strong coupling (ad  1 ) Se ' Sh ' 2ad=‰1 ÿ cos…2kd†Š† and strong magnetic ®eld …o  oci †, with kd  1. In that case, Eq. (11.23) simpli®es to the form "  #1=2 K0 K 0 2 …ck†2 oˆ ‡ ; (11.25)  2K 2K K where the symbols K and K 0 stand, respectively, for " # " # 2 2 o2pe o2ph o o pe ph ‡ ÿ ; and K 0 ˆ ; Kˆ o2ce o2ch oce och and opi ˆ ‰4pni e2 =mi …2d†Š1=2 is the effective 3D plasma frequency of the ith component. Note that K0 ˆ

o2pe oce

ÿ

o2ph och

ˆ

2p…ne ÿ nh †ec : B0 d

There are two cases to be considered: compensated (ne ˆ nh ) and uncompensated (ne 6ˆ nh ) materials. In the case of compensated materials K 0 ˆ 0, and Eq. (11.25) yields o ˆ v A k;

(11.26)

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which ispthe  well-known dispersion relation of an Alfven wave [18±22] where the Alfven velocity v A ˆ c= K . Alfven waves occur in compensated two-component plasmas and consist of out-of-phase motion of the two components (electrons and holes, for example). In the case of uncompensated materials, we consider the long-wavelength limit, with the second term under the radical in Eq. (11.25) assumed to be much smaller than the ®rst. We ®nd the existence of two modes: one is the optical mode with K 0 …ck†2 oˆ ‡ K K0 and the other is a helicon with …ck†2 : K0 Interestingly, both modes have the same group velocity. It should be pointed out that these modes also occur in 3D plasmas with the only difference that classically K has Es added to it. It is also noteworthy that in the foregoing analysis we have completely ignored to effects of carrier collisions. The effects of collisions on the damping of the transverse modes, such as helicons, is known to be very important in 3D plasma. Our neglect of collisions, however, can be justi®ed in view of the fact that in the superlattice systems the charge carriers are spatially separated from the parent donor impurities. Moreover, the experiments are usually done at very low temperatures. As stated before, helicon-like modes were ®rst observed by Maan et al. [549] through transmission resonances in an InAs±GaSb superlattice. The mode frequency scaled as a function of wave vector and magnetic ®eld correctly. The electron mass they derived signi®cantly exceeded the effective mass of the conduction electrons in InAs; in the lack of suf®cient and convincing experimental efforts this result remains unclari®ed. It is worth pointing out that the sample used by Maan et al. [549] was actually a type-I, since the GaSb layers were very heavily doped. No measurements, of transverse modes have, to our knowledge, been made in type-II superlattices which seem to be potential candidates to allow both Alfven and helicon waves made to propagate along the superlattice axis. In the NRL (c ! 1 ) q0 ! 0), Eq. (11.23) assumes the following form:      2pq e 2pq h 2pq 2 e h 2 1‡ w Se w Se ÿ wyy wyy Sh ˆ 0: (11.27) 1‡ Es yy Es yy Es oˆ

This can be simpli®ed and cast in the form o2 ˆ 12 ‰…o2ce ‡ o2ch † ‡ …O2pe ‡ O2ph †Se Š  12 f‰…o2ce ÿ o2ch † ‡ …O2pe ÿ O2ph †Se Š2 ‡ 4O2pe O2ph S2h g1=2 ; (11.28) where Opi is the well-known 2D plasmon frequency de®ned by O2pi ˆ

o2pi Es

…qd† ˆ

2pni e2 q: mi Es

Let us now have a look at the weak (qd  1) and strong (qd  1) coupling limits invoked upon Eq. (11.28). In the case of a weak coupling, Se  1 and Sh ˆ 0, and Eq. (11.28) yields o2‡ ˆ o2ce ‡ O2pe ;

o2ÿ ˆ o2ch ‡ O2ph :

(11.29)

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This implies that, in this limit, each layer supports its own electron (or hole) 2D magnetoplasmons, as expected. Note that the same conclusions can easily be arrived at even without imposing the NRL. In the strong coupling limit, there are two cases to be considered: k ˆ 0 and k 6ˆ 0. For k ˆ 0, we have Se ' 1=qd ' Sh , and Eq. (11.28) yields two branches ( )1=2   2 1 1 1 1 4 …o2ce ÿ o2ch † ‡ …o2pe ÿ o2ph † ‡ 2 o2pe o2ph ; o2 ˆ …o2ce ‡ o2ch † ‡ …o2pe ‡ o2ph †  2 Es 2 Es Es (11.30) which are the coupled 3D magnetoplasma modes of a two-component plasma system. For k 6ˆ 0, on the other hand, Eq. (11.28) yields " # 2 1 2 2 …o2pe ‡ o2ph †…qd† 2 2 o ‡ och ‡ o ˆ 2 ce Es ‰1 ÿ cos…2kd†Š 8" 91=2 # 2 2 4 2 2 2 2 2 = …o ÿ o †…qd† o o …qd† cos …kd† 1< 2 2 16 pe ph pe ph  oce ÿ o2ch ‡ ‡ 2 : (11.31) 2: Es ‰1 ÿ cos…2kd†Š Es ‰1 ÿ cos…2kd†Š2 ; This represents the coupled electron±hole acoustic magnetoplasmons. In the case that kd ˆ …n ‡ 12†p, these coupled modes become decoupled by the screening, and reduce to o2‡ ˆ o2ce ‡

1 2 o …qd†2 ; Es pe

o2ÿ ˆ o2ch ‡

1 2 o …qd†2 : Es ph

(11.32)

These are independent electron and hole acoustic magnetoplasmons. Other relevant analytical diagnoses, for example, at the zone center, will be recalled later while discussing the illustrative numerical examples. Next, we turn to the dispersion relation for magnetoplasma polaritons in semiin®nite type-II superlattices. 11.1.2.2. Truncated type-II superlattices. Here we consider a practically more realizable situation, where the otherwise perfectly periodic superlattice is truncated at an interface z ˆ 0 such that the region ÿ1  z  0 is assumed to be occupied by an insulating material characterized by the dielectric constant E0 . To be succinct, we immediately write down the dispersion relation for magnetoplasma modes in 2D semi-infinite superlattice [726]: …a1 b2 ÿ a2 b1 †…b1 g2 ÿ b2 g1 † ÿ …g1 a2 ÿ g2 a1 †2 ˆ 0;

(11.33)

where the symbols aj ; bj and gj are as de®ned in the original paper [726]. In the special limit of zero magnetic ®eld, Eq. (11.33) reduces to the form ‰…d1 cosh…ad† ‡ d3 sinh…ad††…cosh…ad† ÿ l sinh…ad†† ÿ d1 Š  ‰…d2 cosh…ad† ‡ d4 sinh…ad††…cosh…ad† ÿ m sinh…ad†† ÿ d2 Š ˆ 0:

(11.34)

With appropriate substitutions for dj, l, and m [726], the ®rst factor and second factor independently equated to zero yield the dispersion relations for TM modes (Eq. (11.9)) and TE modes, respectively. Furthermore, in the limit d ! 1, Eq. (11.33) can be easily shown to reproduce two independent

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expressions:     1 E0 Es …a0 ‡ a† e e wyy ‡ ‡ wxx ÿ ÿ wexy weyx ˆ 0; 2 4p a0 a 4pq0     Es a whyy ‡ whxx ÿ ÿ whxy whyx ˆ 0: 2pa 2pq20

275

(11.35) (11.36)

Evidently, the former is the dispersion relation for the 2DEG at the interface l ˆ 0, and the latter is that of the 2DHG at the interface demarked as l ˆ 1 in Fig. 89, as expected. Other relevant diagnoses of Eq. (11.33) in, for example, the NRL will be referred to a little later while discussing the numerical examples. 11.1.2.3. In®nite type-I superlattices. The generalization of the analytical results obtained in the case of type-II superlattice to those of type-I is quite straightforward, as we have seen in the zero-field case in the preceding section. The keystone to obtain directly the dispersion relations for type-I superlattice is to substitute whij ˆ 0 and replace 2d by d in the corresponding result of the type-II superlattice; this is true both for infinite and semi-infinite superlattices. Such a substitution in Eq. (11.23) leaves us with [726]:   h Es i e a e wyy Se ‡ (11.37) w Se ÿ ÿ …wexy Se †…weyx Se † ˆ 0; 2pa xx 2pq20 where the structure factor Se is the same as de®ned in Eq. (11.13). This is the dispersion relation for the collective (bulk) excitations of 2D magnetoplasma in type-I superlattice. It is a simple matter to check that, in the special limit of the zero magnetic ®eld, Eq. (11.37) reproduces the well-known results of TM modes (Eq. (11.12)) and TE modes. The former (latter) modes are obtained by equating the ®rst (second) square bracket in the ®rst term of Eq. (11.37) to zero, in the absence of a magnetic ®eld. Let us now have a look at Eq. (11.37) subject to various special limits, viz., weak coupling (ad  1) limit, strong coupling (ad  1) limit, and NRL (c ! 1). In the weak coupling limit (Se ' 1), Eq. (11.37) assumes the form of Eq. (11.36) with superscript h replaced by e. This means that in the weak coupling limit, type-I superlattice supports its 2D magnetoplasmons. That this is so becomes more evident if Eq. (11.37) is analyzed in the NRL (see a little further on in what follows). In the strong coupling limit, with Se ' ad=‰1 ÿ cos…kd†Š, there are two cases to be considered: k 6ˆ 0, and k ˆ 0. The case k 6ˆ 0. First, we consider the situation where the propagation vector q ˆ 0. In this case, Eq. (11.37) simpli®es to the form oˆ

oce sin2 …kd=2†

‰…1=4†…ope d=c†2 ‡ sin2 …kd=2†Š

;

(11.38)

where ope ˆ …4pne e2 =me d†1=2 is the effective 3D plasma frequency. Note that for q ˆ 0, o does not depend on the background dielectric constant Es . This can be understood physically in the following way [673]. When an EM wave propagates in an electron gas parallel to an externally imposed magnetic ®eld, its plane of polarization rotates, with rotation angle proportional to the path length in the plasma. (This is just the Faraday rotation.) In our case, the plane of polarization changes only when the wave goes through an electron layer, and remains unaffected by the background medium in which no electronic transitions are allowed to occur. Hence, o turns out to be independent of Es . If q is slightly

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different from zero, however, so that the wave propagates slightly off the z-axis, this is no longer true, and o does depend on Es , in the manner shown below. The behavior of the frequency as a function of k is interesting. For kd  1, Eq. (11.38) simpli®es to the form o ' oce

…ck†2 ‰o2pe ‡ …ck†2 Š

(11.39)

which is the dispersion relation of a helicon wave propagating along the superlattice axis. As kd increases, the frequency increases to the cyclotron frequency. For kd ˆ p=2, the frequency dips slightly to 1 o ˆ oce : (11.40) ‰1 ‡ …1=2†…ope d=c†2 Š This is the frequency attained by the wave for kd ˆ …n ‡ 12†p. For kd ˆ p, the frequency again dips, by a smaller amount, to 1 : (11.41) o ˆ oce ‰1 ‡ …1=4†…ope d=c†2 Š This repeats at kd ˆ …2n ‡ 1†p. The behavior characteristic of the wave demonstrated in Eqs. (11.40) and (11.41) is, to our knowledge, not well established, and may well be spurious. Let us now turn to the situation where q 6ˆ 0. Assuming that kd  1 and o2pe  …ck†2 , we obtain, from Eq. (11.37), the wave frequency "  # 1 cq 2 ; (11.42) o ˆ ocp 1 ‡ 2Es ocp where ocp ˆ oce …ck=ope †2 . It should be pointed out that although our analytical diagnosis imply the existence of a mode with the dispersion relation of a helicon, the precise nature of this mode is not that of a helicon. This has been already discussed and cross-questioned by some authors (see, for example, Ref. [1655]). For this case (k 6ˆ 0), considering NRL (c ! 1) simpli®es the dispersion relation, Eq. (11.37), to the form (with qd  1† o2 ˆ o2ce ‡

o2pe

…qd†2 ; 2Es ‰1 ÿ cos…kd†Š

(11.43)

and establishes the existence of the acoustic magnetoplasmons propagating in type-I superlattices. The case k ˆ 0. In this case, we also impose, for the sake of simplicity, the NRL. In the strong coupling limit, with Se ' 2=qd, Eq. (11.37) can be simpli®ed to obtain o2 ˆ o2ce ‡

1 2 o : Es pe

(11.44)

As such, we note that in the strong coupling limit the k ˆ 0 mode is a 3D bulk magnetoplasmon. In the weak coupling limit, accompanied by the NRL, Eq. (11.37) simpli®es to assume the following from: 2pq e w Se ‡ 1 ˆ 0; Es yy

(11.45)

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where Se , in the NRL, is now simpli®ed such that Se ' 1, for all values of kd. Then Eq. (11.45) yields o2 ˆ o2ce ‡

o2pe …qd†; 2Es

(11.46)

which means that in the weak coupling limit (qd  1), the type-I superlattice in the presence of a perpendicular magnetic ®eld supports 2D magnetoplasmon with o / q1=2 . 11.1.2.4. Truncated type-I superlattice. The dispersion relation for the collective (surface) excitations in type-I superlattice subjected to an external magnetic field oriented along the superlattice axis can be immediately written down by substituting whij ˆ 0 and replacing 2d by d in Eq. (11.33). The result, formally, is [726]: …a01 b02 ÿ a02 b01 †…b01 g02 ÿ b02 g01 † ÿ …g01 a02 ÿ g02 a01 †2 ˆ 0;

(11.47)

where a0j , b0j and g0j are exactly the same as de®ned in Ref. [726]. In the special limit of ~ B0 ˆ 0, Eq. (11.45) simpli®es to attain, formally, the form of Eq. (11.34) with all dj replaced by d0j. This then is a simple matter to con®rm that Eq. (11.47), in the absence of a magnetic ®eld, reproduces proper results for the TM (Eq. (11.18)) and TE modes. Furthermore, in the limit of d ! 1, Eq. (11.47) reduces exactly to Eq. (11.35), which represents the dispersion relation for the magnetoplasmons of a 2DEG embedded at the interface between two different insulators characterized by the dielectric constants Es and E0 . Other relevant diagnoses of Eq. (11.47) in, for example, the NRL will be recalled at the time of discussing illustrative numerical results. 11.1.2.5. Numerical examples. The case of non-zero magnetic fields is illustrated by two specific examples: Oce ˆ oce =ope ˆ 0:5 and 0.9; the rest of the material parameters are the same as employed for the zero-field case (see Section 11.1.1). The numerical results will be presented in terms of the dimensionless variables: frequency x ˆ od=c, and propagation vector z ˆ qd. The illustrative examples of the collective magnetoplasma excitations for type-I superlattices are shown for Oce ˆ 0:5 and 0.9, respectively, in Figs. 94 and 95. One immediately notes that the application of the magnetic field results in pushing the whole spectrum up above the cyclotron frequency o ˆ oce. The shaded region is the bulk magnetoplasmon band whose lower edge, for k ˆ p=d, starts at q ˆ 0 with o ' oce . This can be seen by solving Eq. (11.37) in the strong coupling limit …ad  1† to obtain o ˆ oce ‰1 ‡ 14 P2 Šÿ1 ' oce ;

(11.48)

where P…ˆ ope d=c†  1. The upper edge of the bulk band, for k ˆ 0, starts at q ˆ 0 with the frequency speci®ed by 8 #1=2 9   " 2 < = 1 2 2 4 ÿ 2 ‡ O2ce ‡ ; (11.49) W2 ˆ O2ce ‡ ; 2: Es Es Es where W ˆ o=ope and Oce ˆ oce =ope . In the limit of zero magnetic ®eld (i.e., Oce ˆ 0), Eq. (11.49) p reproduces the starting point of the k ˆ 0 edge with o ˆ ope = Es (Fig. 91). It should be pointed out that the mode frequencies speci®ed by Eqs. (11.48) and (11.49) are ideal, theoretical values at q ˆ 0. In practice, feeding q ˆ 0 renders with computational dif®culties, and hence one is bound to start at an

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Fig. 94. Collective excitation spectrum for type-I superlattice system in the non-zero magnetic ®eld in the perpendicular con®guration. We take the normalized cyclotron frequency Oce ˆ 0:5; the rest of the parameters are the same as in Fig. 91. The hatched region is the bulk plasmon band of an in®nite superlattice, and the solid curve is the surface magnetopolariton mode of a semi-in®nite superlattice. The dashed line labeled a ˆ 0 is the light line in vacuum. Notice that switching on the magnetic ®eld results in pushing the whole spectrum above the cyclotron frequency. (After Kushwaha, Ref. [726].)

in®nitesimally small, but ®nite, value of q. This results in somewhat different values of x than dictated by these equations. The solid curve above the upper edge of the bulk band is a magnetoplasmon±polariton mode that starts at a frequency almost comparable to that at which the upper edge of the bulk band emerges at q ˆ 0. This surface polariton becomes a bona®de mode only after crossing the light line Ð a dashed line demarked as a ˆ 0 Ð where both a0 and a are real and positive. The numerical results reveal that the non-zero magnetic ®eld enhances (as compared to the zero-®eld case) the critical value of the wave vector q, where the surface mode intersects the upper edge of the bulk band and then emerges out. In the long wavelength range, the polariton mode propagates in the close vicinity of the upper edge of the bulk band, but it never traverses it. In the NRL, for z  1, Eq. (11.47) assumes a simple form W 2 ˆ O2ce ‡

1 …qd†: Es ‡ E0

(11.50)

Thus in this (equivalently weak coupling) limit, the surface mode is, in fact, a 2D magnetoplasmon supported by the 2DEG layer immersed in the two different dielectrics characterized by the dielectric constants Es and E0 . In the present system this corresponds to the z ˆ 0 interface, where the periodicity of the superlattice is truncated. We notice that the polariton modes in Figs. 94 and 95 are almost exactly reproducible from Eq. (11.50), for z  3:0.

M.S. Kushwaha / Surface Science Reports 41 (2001) 1±416

Fig. 95.

279

The same as in Fig. 94, but for Oce ˆ 0:9. (After Kushwaha, Ref. [726].)

The effect of the ®eld intensity on the excitation spectrum can be seen by comparing the results depicted in Figs. 94 and 95. As regards the propagation characteristics, one immediately notes that the width of the bulk magnetoplasmon band decreases as the magnetic ®eld increases. Moreover, the larger the magnetic ®eld, the smaller the (spatial) separation of the surface excitations from the upper edge of the bulk band. We now turn to the magnetic ®eld effects on the collective excitation spectra in type-II superlattices. The illustrative numerical results for Oce ˆ 0:5 and 0.9 are plotted in Figs. 96 and 97, respectively. The lower edges of the two bulk magnetoplasmon bands start at the respective cyclotron frequencies in the problem. This can be seen through analytical diagnosis of Eq. (11.23), for k ˆ p=2d, in the extreme long wavelength limit. The result is fW 2 ‰…1 ‡ P2 † ‡ …1 ‡ P2 o2r † ÿ 1Š ÿ Oce Och ‡ W‰Och …1 ‡ P2 † ÿ Oce …1 ‡ P2 o2r †Šg fW 2 ‰…1 ‡ P2 † ‡ …1 ‡ P2 o2r † ÿ 1Š ÿ Oce Och ÿ W‰Och …1 ‡ P2 † ÿ Oce …1 ‡ P2 o2r †Šg ˆ 0; (11.51) where Oci ˆ oci =opi , with i  e; h, and the rest of the symbols are the same as already de®ned in the text. Now, since or is of the order of one and P  1, Eq. (11.51) reduces to …W ‡ Oce †…W ‡ Och †…W ÿ Oce †…W ÿ Och † ˆ 0:

(11.52)

Either the third or fourth factor is zero. As such Eq. (11.52) yields two modes: o ˆ oce and o ˆ och . The upper edges of the two bulk bands have the same story as the corresponding modes in the type-I

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Fig. 96. Collective excitation spectrum for type-II superlattice system in the non-zero magnetic ®eld in the perpendicular con®guration. We take the normalized cyclotron frequency Oce ˆ 0:5; the rest of the parameters are the same as in Fig. 93. The hatched regions are the bulk magnetoplasmon bands of an in®nite superlattice and solid curves are the surface magnetopolariton modes of a semi-in®nite superlattice. The dashed line labeled a ˆ 0 is the light line in vacuum. Notice that the application of the magnetic ®eld results in pushing up the whole spectrum above the lower cyclotron frequency Oce in the problem. (After Kushwaha, Ref. [726].)

superlattice (Figs. 94 and 95). Their zone center (q ˆ 0) frequencies are speci®ed by    2 1 2 W ÿ …Oce ÿ Och †W ÿ Oce Och ‡ …1 ‡ Or † Es    1 ˆ 0:  W 2 ‡ …Oce ÿ Och †W ÿ Oce Och ‡ …1 ‡ O2r † Es

(11.53)

Here the signs of Och are already decided to be positive, and O2r ˆ och =oce . Equating the ®rst factor to zero yields " # 1=2 1 12 Wˆ 9O2ce ‡ ÿOce : (11.54) 2 Es Similarly, equating the second factor to zero gives " #  1 12 1=2 2 Wˆ 9Oce ‡ ‡Oce : 2 Es

(11.55)

Again, it is noteworthy that Eqs. (11.54) and (11.55) represent the ideal, theoretical starting frequencies at the zone center (q ˆ 0) Ð a situation, in practice, extremely dif®cult to achieve.

M.S. Kushwaha / Surface Science Reports 41 (2001) 1±416

Fig. 97.

281

The same as in Fig. 96, but for Oce ˆ 0:9. (After Kushwaha, Ref. [726].)

Unlike the type-I superlattice, the magnetic ®eld effects on the collective excitations in type-II superlattice are drastically different as compared to its zero-®eld case. In the presence of an external magnetic ®eld, we ®nd three magnetoplasmon±polariton modes which are designated as S1 ; S2 , and S3 (see Fig. 96). The lowest surface mode (S1 ) starts at x ' 0:00444 above the upper edge of the lower bulk band with a negative group velocity Vg, extends (after crossing the light line) in the extreme vicinity of the upper edge of the lower bulk band (0:1  z  0:3), and then emerges out with increasing frequency towards large wave vectors within the gap between the two bulk bands. The second lowest mode (S2 ) starts at x ' 0:00844 with a positive group velocity, changes the sign of Vg , after crossing the light line, and extends to intersect (at z ' 0:12) the upper edge of the upper bulk band to propagate with a negative group velocity until it reaches z ' 2:02. At large wave vector (z  2:3), it (S2 ) again assumes a positive group velocity (i.e., it propagates with frequencies increasing with the wave vectors). Thus we conclude that this is the surface mode that changes the sign of its group velocity twice in the o±q space. The third (the uppermost) surface mode (S3 ) starts at x ' 0:00854 with a negative group velocity at the lowest possible wave vector, and after intersecting the light line it propagates to merge throughout with the upper edge of the upper bulk band. This leads us to infer that, in the presence of an external perpendicular magnetic ®eld, there are, practically, two magnetoplasmon±polariton modes (S1 and S2 ), and both of them largely propagate within the gap between the two bulk bands. It is found that S1 and S2 modes also merge together at large wave vectors, but they never merge with either of the bulk bands. It is not dif®cult to prove that the merger of S1 and S2 does occur at large z. In order to substantiate this merger, we solved Eq. (11.33) in the NRL where it reduces to the following form: …1 ÿ l0 †…1 ‡ 12 C1 †…1 ‡ l0 ‡ D1 † ˆ 0;

(11.56)

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where l0 ˆ E0 =Es , C1 ˆ 4pqwhyy =Es , and D1 ˆ 4pqweyy =Es , in the NRL [726]. The ®rst factor does not vanish for the obvious reasons (because Es 6ˆ E0 ). The second factor equated to zero gives W 2 ˆ O2ch ‡

2 …qd† Es

(11.57)

and the third factor equated to zero yields W 2 ˆ O2ce ‡

2 …qd†: Es ‡ E0

(11.58)

Eq. (11.57) represents a 2D magnetoplasmon, and can be seen to reproduce exactly the frequency of the upper shrinked bulk band with which the S3 mode is merged. On the other hand, Eq. (11.58) substantiates the merger of the two surface modes (S1 and S2 ) at large wave vectors (z ' 5:3). We compare the results on the magnetoplasmon excitations plotted in Figs. 96 and 97, in order to examine the effects of variation in the ®eld intensity. We ®nd that the band-widths decrease with increasing magnetic ®eld. The Zeeman splitting of the bulk bands increases with increasing magnetic ®eld. There are still three magnetoplasmon±polaritons designated as S1 ; S2 and S3 ; they start, respectively, at x ' 0:00731, 0:01439, and 0:01458. Their propagation characteristics are describable just as in the previous case (Fig. 96), except that S1 and S2 now merge at relatively large z. It is, generally, seen that smaller the magnetic ®eld, the lower the z, where the true surface modes S1 and S2 merge together. It is important to note that, in the presence of the magnetic ®eld, the search of a critical wave vector q where the surface modes intersect the edges of the bulk bands is thwarted by the mathematical complexity. In addition, we remind the reader the resonance splittings of the upper (k ˆ 0) edges of the bulk bands in the close vicinity of the light line, as was pointed out in the original paper [726]. We could not afford to dedicate such an extensive numerical diagnoses to discuss such resonance splittings here, and hence refer the reader to the original work for such ®ndings [726]. We wish to point out that inclusion of the effects of carrier collisions and the coupling to the optical phonons, which both are of practical importance, should yield a richer excitation spectrum and give deeper insight into the subject. The application of an external electric ®eld, normal to the superlattice axis, which causes relative drift in the current carriers and thereby produces an instability that leads to ampli®cation is also worth attention. We hope to embark on such extensions of the present theory in the near future. As regards the present numerical results, particularly for type-II superlattices in the magnetic ®eld, we suspect some intricate phenomena with respect to the propagation characteristics of the surface modes at larger wave vectors. 11.2. Periodic systems of 2D layers with ®nite thicknesses In this section, we consider a 4-layer superlattice (Fig. 98) with respective thicknesses as de ; d1 ; dh , and d2 with the period D ˆ de ‡ d1 ‡ dh ‡ d2 . Out of these four layers in the unit cell, we will ®nally take the limit di …i  e; h† ! 0; and the medium I  II. The resultant periodic structure will then represent a type-II superlattice in which alternating 2DEG and 2DHG are embedded in a dielectric medium I  II. However, we begin with still more general situation where each layer of the unit cell will be considered as a semiconducting medium. All the interfaces lie in the x±y plane, i.e., z-axis is the superlattice axis which observes the periodicity D. We consider the Voigt con®guration, such that the

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Fig. 98. Schematics of type-II superlattice system with 2DEG and 2DHG layers of non-zero thicknesses. The symbol di …i  1; 2; e; h, with 1, 2, e, and h referring to the I, II, shaded and blank layers) stands for the thickness of the respective ~ (see layer. The encircled numbers refer to the four states (i.e., four interfaces) in the unit cell belonging to the reduced space M text). D ˆ de ‡ d1 ‡ dh ‡ d2 is the period of the superstructure. The superlattice-growth direction is along z-axis. The wave propagates along y-axis, and the external magnetic ®eld ~ B0 is assumed to be oriented parallel to the x-axis (the Voigt con®guration).

wave propagates along y-axis and an external magnetic ®eld is oriented parallel to the x-axis, just as in the earlier structures considered in this work. To start with, each layer of width di is labeled by the layer index i…  1; 2; e; h† within the unit cell designated by an index n. We replace the z-coordinate by two variables (m; z) such that m  n; i; ÿ1  n  ‡1. The equivalent notations are m  …n; i†  i ‡ Nn , where N is the number of layers within the unit cell. Also, there are two different ways to label one and the same interface:  …n; 1; 1†  …n ÿ 1; N; 1† if i ˆ 1; …m; 1†  …n; i; 1†  …n; i ÿ 1; 1† if i 6ˆ 1: This is quite a general criterion until the complete mathematical machinery is developed, i.e., before taking any special geometric limits whatsoever. We work in the framework of the IRT [799,1759] as brie¯y described in Section 3.1. The extensive theoretical framework of the general methodology, both with [731±733] and without [730] an external magnetic ®eld in the Voigt [731], perpendicular [732], and Faraday [733] con®gurations, have been well established. We will quote here only the necessary expressions of the dispersion relations or the magnetoplasmons in the in®nite and the semi-in®nite superlattices, both type-I and type-II, and refer for the details of the mathematical tools to Refs. [730± 733]. This section is motivated to expose the unique characteristics of the Voigt geometry with respect to the magnetoplasmon propagation in periodic and truncated superlattices made up of 2DEG and/or

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2DHG layers of small, but ®nite, thicknesses. We have already pointed out such uniqueness of the Voigt geometry before in the earlier sections (see, for example, Sections 4.2.1, 5.2.1, 6.2.2 and 9.2.1). It is worthwhile to stress that the Voigt geometry allows the simple polarization (TM) of the magnetoplasma waves, as compared to other (perpendicular and Faraday) geometries that involve strong, and generally inseparable, coupling of TM and TE modes, just as in the absence of an applied magnetic ®eld. This makes the life, with respect to the analytical diagnoses of and check on the exact results, relatively easier. We now proceed to recall the relevant results from the original papers [730,731] in the respective geometries. In®nite type-II superlattice. In the presence of an external magnetic ®eld in the Voigt con®guration, the general dispersion relation, including the ®eld dependence and the effects of retardation, for the type-II superlattices, was derived, in the limit of de ; dh ! 0, in Eq. (7.22) in Ref. [731]. This equation is subjected to the special limits of medium I  II (Fig. 98) taken to be identical insulators characterized by the frequency-independent dielectric constant Es , which implies that E1 ˆ E2 ˆ Es , a1 ˆ a2 ˆ a, d1 ˆ d2 ˆ ds ; y1 ˆ y2 ˆ y ) T1 ˆ T2 ˆ T ˆ coth…y†; S1 ˆ S2 ˆ S ˆ csch…y†; and D ˆ 2d. Then Eq. (7.22) in Ref. [731] simpli®es to      2pa e 2pa h 2pa 2 e h 2 wv Se wv Se ÿ wv wv Sh ˆ 0; (11.59) 1‡ 1‡ Es Es Es where the structure factors Se and Sh are just as de®ned in Eq. (11.2), and wiv ˆ wiyy ‡ …wiyz †2 =wiyy is the ®eld-dependent 2D Voigt polarizability function. The crux of the theoretical framework, and a consequence of taking the limit di …i  e; h† ! 0, is an excellent approximation [730,731]: 4pwiv ! Eiv di ;

(11.60)

where Eiv ˆ Eiyy ‡ …Eiyz †2 =Eiyy is the local, Voigt dielectric function. Eq. (11.59) represents the magnetoplasmon dispersion in an in®nite, type-II superlattice subjected to an applied magnetic ®eld in the Voigt geometry. Note that in the special limit of zero-magnetic ®eld, Eq. (11.59) simpli®es exactly to Eq. (11.1). Truncated type-II superlattice. Here we consider an in®nite type-II superlattice (Fig. 98) truncated at an interface z ˆ 0, such that the region ÿ1  z  0 is ®lled with a semi-in®nite medium, say III, characterized by a frequency-independent dielectric constant E0 and a decay constant a0 . The general dispersion relation, without any geometric restriction whatsoever, for the semi-in®nite type-II superlattice in the presence of an external magnetic ®eld was derived in Eq. (8.16) in Ref. [731]. Within the special limits as described above, the said equation reduces to [731]:      4pa e 2pa h 2pa 2 e h w coth…2ad† 1 ‡ w coth…ad† ÿ wv wv csch2 …ad† 1‡ Es v Es v Es    4pa e 2pa h w 1‡ w tanh…ad† (11.61) ˆl l‡ Es v Es v with l ˆ E0 a=a0 Es . Note that Eq. (11.61) in the special limit of zero magnetic ®eld reduces exactly to Eq. (11.9). Again, we will be working within an approximation stated in Eq. (11.60), in order for keeping the thicknesses of the 2DEG and 2DHG small, but ®nite. In®nite type-I superlattice. In order for deriving the dispersion relation for the collective (bulk) excitations for type-I superlattices, we adopt the same scheme as stated in Section 11.1 (i.e., substitute

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whv ˆ 0 and replace the period 2d by d). Making such substitutions in Eq. (11.59) leaves us with 1‡

2pa e w Se ˆ 0; Es v

(11.62)

where Se is now given by Eq. (11.13). In the special limit of zero magnetic ®eld, Eq. (11.62) reproduces Eq. (11.12). Use of the approximation stated in Eq. (11.60) is understood. Truncated type-I superlattice. Substituting whv ˆ 0 and replacing 2d by d in Eq. (11.61) yields   4pa e 4pa e w coth…ad† ˆ l l ‡ w : (11.63) 1‡ Es v Es v This equation, together with the approximation stated in Eq. (11.60), represents the dispersion relation for the collective magnetoplasmon excitations in truncated type-I superlattices made up of 2DEG layers of small, but ®nite, thicknesses. In the special limit of zero magnetic ®eld, Eq. (11.63) reproduces Eq. (11.18). Before the description of the illustrative numerical examples, it is instructive to shed some light on the strategy of the geometric and material parameters employed in the present computation. It is convenient to introduce the dimensionless wave vector z ˆ cq=ope, and the dimensionless frequency x ˆ o=ope , where opi …i  e; h† is the usual 3D screened plasmon frequency. This in turn implies the introduction of some other dimensionless parameters which lead us to understand readily the propagation characteristics of the magnetoplasmons in different situations at hand. These are the electron±cyclotron frequency Oce ˆ oce =ope, the hole±cyclotron frequency Och ˆ ÿa oce =ope, hybrid electron±cyclotron±plasmon frequency Oe ˆ …1 ‡ O2ce †1=2, and the hybrid hole±cyclotron±plasmon frequency Oh ˆ …a ‡ O2ch †1=2, where oce ˆ eB0 =me c is the usual electron cyclotron frequency, and a ˆ …me =mh †. In addition, we de®ne two dimensionless thicknesses: dr ˆ ds =di , and deff ˆ ope di =c, where di and ds represent the widths of the 2D (electron or hole) layers and the spacers (of medium I or II), respectively. It is also useful to recall, at this point, the behavior of the Voigt dielectric function Ev as discussed in Section 6.2. We state the material parameters used in the computation. A careful diagnosis of the energy scales usually explored in the 2D systems let us choose GaAs as the most suitable material for the 2D (electron and/or hole) layers, as well as for the dielectric materials (e.g., media I, II, and III)  where one could employ the pintrinsic GaAs. As such Es ˆ 12:8; E0 ˆ 1; me ˆ 0:067m0 ; a ˆ 0:5; Oce ˆ 0:5; or ˆ oph =ope ˆ a; deff ˆ 0:001; and dr ˆ 100 (type-II), 200 (type-I); m0 being the freeelectron mass. This choice of the material parameters implies that we are interested in the situation where doping concentration ne ˆ nh . For the sake of comparison, we will present the numerical results both with and without an applied magnetic ®eld. It is found that in the presence of an applied magnetic ®eld all the dispersion relations for the magnetoplasmons in both types of superlattices abide by the rules dictated by Fig. 36. We now discuss the numerical results for collective (bulk and surface) excitations for both type-I and type-II superlattices for the material parameters stated above. 11.2.1. Zero magnetic ®eld The collective excitations for type-I superlattice in the absence of an applied magnetic ®eld are illustrated in terms of the dimensionless variables in Fig. 99. The hatched region is the bulk plasmon band and the solid curve (in blue) is the surface plasmon±polariton. The upper (k ˆ 0) edge of the bulk band can be shown to start at the zone center with a frequency speci®ed by, from Eq. (11.62) with

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Fig. 99. Collective excitation spectrum for type-I superlattice in the zero magnetic ®eld. The parameters used are: Es ˆ 12:8, deff ˆ 0:001, dr ˆ 200, me ˆ 0:067m0 . The hatched region is the bulk plasmon band and the solid curve is the surface plasmon±polariton mode of a semi-in®nite superlattice. The dashed line labeled as a ˆ 0 is the light line in intrinsic GaAs spacers. Notice that there is no coherent correspondence between Figs. 99 and 91 due, in fact, to the different normalization scheme. (After Kushwaha and Djafari±Rouhani, Ref. [730].)

B0 ˆ 0, x ˆ …1 ‡ dr †ÿ1=2 :

(11.64)

This implies, for dr ˆ 200, x ' 0:07053. In the NRL, the surface mode is observed to obey the following trajectory:    1 E0 ÿ1=2 1‡ ; (11.65) xˆ 1‡ qdi Es where qdi ˆ deff z stands for the dimensionless wave vector. We have found that the excitation (bulk as well as surface) spectrum is pushed up in frequency with decreasing ratio dr , while keeping the period ®xed. This implies that the frequency of the excitation spectrum increases with increasing thicknesses of the 2DEG layers. An explanation to such a behavior characteristic on the thickness dependence of the mode frequencies was given in Section 6 with respect to the coupled modes of excitation in the double-inversion-layer systems. A similar check on the effects of variation of the parameter deff is worth exploring. We have also observed that the band-width also increases as the ratio dr decreases. The collective excitation spectrum for type-II superlattice in the absence of an external magnetic ®eld is depicted in Fig. 100. The shaded regions are the bulk bands, and the solid curves (in blue and red) are

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Fig. 100. Collective excitation spectrum for type-II superlattice in the zero magnetic ®eld. The parameters used are: p Es ˆ 12:8, deff ˆ 0:001, dr ˆ 100, me ˆ 0:5mh ˆ 0:067m0 , a ˆ me =mh ˆ 0:5, or ˆ aoph =ope . The hatched regions are the bulk plasmon bands and the solid curves in blue (in red) are the surface plasmon±polariton modes of a semi-in®nite superlattice truncated at the 2DEG (2DHG) layer. The dashed line labeled as a ˆ 0 is the light line in intrinsic GaAs spacers. Again, there is no coherent correspondence between Figs. 100 and 93. (After Kushwaha and Djafari±Rouhani, Ref. [730].)

the surface±plasmon polaritons. The surface modes marked with e (h) correspond to the truncation of the otherwise periodic superlattice at the 2DEG (2DHG) layer. The upper (k ˆ 0) edge of the upper bulk band is found to start at the zone-center (q ˆ 0) with a frequency speci®ed by, from Eq. (11.59) with B0 ˆ 0,  1=2 1 …1 ‡ o2r † x ˆ p : (11.66) 2 …1 ‡ dr † This implies, for dr ˆ 100, x ' 0:08617. In the case that the periodic superlattice is truncated at the 2DHG, the surface modes (in red) in the present case behave similarly to the corresponding modes in Fig. 93, where the truncation was at a 2DEG layer. The reason being that now we take me =mh ˆ 0:5, whereas there (in Fig. 93) the effective-mass ratio was taken to be me =mh ˆ 2:0, implying that the roles of the 2DEG and 2DHG layers have virtually interchanged. In the case that the superlattice is truncated at the 2DEG layer, the gap-surface mode intends to merge with the upper edge of the lower bulk band at large wave vector. In the NRL, the surface modes, Eq. (11.61) with B0 ˆ 0, satisfy the following relation:   (11.67) …1 ÿ l0 †‰x2 …1 ‡ l0 † ‡ qdi …x2 ÿ 1†Š x2 ‡ 12 qdi …x2 ÿ o2r † ˆ 0: The ®rst factor does not vanish for simple reasons (because E0 6ˆ Es ). The second factor equated to zero yields  1=2 qdi (11.68) xˆ 1 ‡ l0 ‡ qdi

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and the third factor equated to zero gives  1=2 qdi xˆ or : 2 ‡ qdi

(11.69)

Eqs. (11.68) and (11.69) reproduce reasonably well the frequencies of the surface modes at large wave vectors. It is noteworthy that these equations correspond to the surface modes in the case the superlattice is truncated at the electron layer. On the other hand, when the truncation is considered at the hole layer, we obtain similar results with the difference that the right-hand side of Eq. (11.68) is multiplied by or and that of Eq. (11.69) is divided by or. The rest of the propagation characteristics of both bulk and surface plasmons are describable just as before. We would like to stress that the surface plasmons are true, bona®de surface modes only towards the right of the light line demarked as a ˆ 0; since only there the decay constants a0 and a are both real and positive. This remark is true for both kinds of the superlattices. Other comments with respect to the effects of variation of the parameters deff and dr made in relation with Fig. 99 are still worth attention. Next, we discuss the charismatic effects of the magnetic ®eld on the collective excitation spectrum of both types of superlattices in the Voigt geometry. 11.2.2. Non-zero magnetic ®eld in the Voigt geometry Before the description of the illustrative numerical examples in the presence of an external magnetic ®eld, it is very important to recall the behavior characteristics of the Voigt dielectric function that dictates the bona®de propagation windows for the system made up of 2DEG and/or 2DHG, as stated in Section 6.2.2. We plot the collective magnetoplasmon excitation spectrum for type-I superlattices in terms of the dimensionless variables in Fig. 101. The shaded regions are the bulk bands and the solid curves (in blue) stand for the collective surface excitations. The surface magnetoplasmon±polariton emerges from the upper edge of the bulk band and rises towards the right of the light line; there are, in fact, two light lines: dotted (dashed) line corresponds to vacuum (GaAs) medium. We will usually refer to the light line in the spacer (GaAs).The spatial separation of the surface mode from the bulk band, over the large wave vector range, ensures that it is free from Landau damping. At large wave vector, both degenerate bulk band and the surface mode will become asymptotic to Oel ˆ 0:7808. The picture above the hybrid frequency Oe is as depicted in the upper panel. The surface mode still emerges from the upper edge of the bulk band towards the right of the light line in the spacer. The lower edge of the bulk band starts at Oe slightly towards the left of the light line in the spacer, while the upper one still starts from the origin. Both the bulk band and the surface mode in this panel are expected to become asymptotic to Oeu ˆ 1:2808. This implies that there is a frequency window which allows neither any bulk band nor the surface mode to propagate. In this case, this gap region is de®ned by Oel …ˆ 0:7808†  x  Oe (ˆ1.1180). Such a gap in the excitation spectrum is usually peculiar to the Voigt geometry and prevails in neither the perpendicular geometry nor the Faraday geometry. We now turn to the type-II superlattice system represented by the dispersion relations for bulk and surface excitations, respectively, in Eqs. (11.59) and (11.61). The corresponding complete excitation spectrum is illustrated in Fig. 102. The two shaded regions constitute the bulk magnetoplasmon bands whose boundaries correspond to k ˆ 0 and p=2d, where 2d is the superlattice period. The variation of k from 0 to p=2d results in the spatial separation between the two bulk bands, provided the ratio or ˆ oph =ope is different from unity. The two solid curves designated with letter e (h) Ð one above and the other below the upper bulk band Ð are the surface magnetoplasmon±polaritons corresponding

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Fig. 101. Collective excitation spectrum for type-I superlattice in the non-zero magnetic ®eld in the Voigt con®guration. We take the normalized cyclotron frequency Oce ˆ 0:5; the rest of the parameters are the same as in Fig. 99. The hatched regions are the bulk magnetoplasmon bands of an in®nite superlattice, and solid curves are the surface magnetopolariton modes of a semi-in®nite superlattice. The upper panel that starts at the hybrid cyclotron±plasmon frequency Oe is an exclusive right of the Voigt geometry. The dotted (dashed) line is the light line in vacuum (intrinsic GaAs spacers). (After Kushwaha and Djafari± Rouhani, Ref. [731].)

Fig. 102. Collective excitation spectrum for type-II superlattice system in the non-zero magnetic ®eld in the Voigt con®guration. We take the normalized cyclotron frequency Oce ˆ 0:5; the rest of the parameters are the same as in Fig. 100. The hatched regions are the bulk magnetoplasmon bands of an in®nite superlattice, and solid curves in blue (red) are the surface magnetopolariton modes of a semi-in®nite superlattice truncated at the 2DEG (2DHG) layer. The dotted (dashed) line is the light line in vacuum (intrinsic GaAs spacers). The lower (upper) inset corresponds to the split spectrum above the hybrid frequency Oh (Oe ), and refers to the dominant contribution due to 2DHG (2DEG) layers. (After Kushwaha and Djafari± Rouhani, Ref. [731].)

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to the truncation of the otherwise periodic superlattice at the 2DEG (2DHG) layer. The surface modes propagating in the gap between the two bulk bands are the acoustic surface magnetoplasmons which remain well-de®ned modes until the in-plane wave vector q ˆ 0. The surface modes above the upper bulk band start to emerge from its upper edge at a critical value of the wave vector q towards the right of the light line in the spacer (GaAs). One can notice that except for the lowest surface mode (demarked as e) all the rest are seen unintending to merge with any of the two bulk bands until a very large wave vector q. This guarantees these modes to be free from Landau damping and hence long-lived excitations. Let us now look into the o±q space just above the hole (electron) hybrid frequency as demonstrated by the lower (upper) inset in Fig. 102. In analogy with the remarks made in the discussion of Fig. 37 (see Section 6), we stress that the predominant contribution to formation of both bulk bands and the surface modes within the lower (upper) inset comes from the 2DHG (2DEG) layers. To be speci®c, both of the propagation windows Ð one above Oh (the lower inset) and the other above Oe (the upper inset) Ð support only one bulk band and one surface mode, unlike the main panel where two bulk bands and two surface modes are allowed. Only the surface mode corresponding to the truncation at the hole layer is clearly visible in the lower inset, while the one corresponding to the truncation at the electron layer is entirely merged with the upper edge of the bulk band and hence becomes indiscernible. The reverse is the case with the upper inset. It should be pointed out that the visible surface mode, in both of these insets, emerges from the respective bulk band towards the right of the light line in the spacer media. There is a complete gap in the excitation spectrum lying within the frequency range speci®ed by Ohu  x  Oe , as dictated by Fig. 36 in Section 6. Such a charismatic splitting of the excitation spectrum into three different frequency regimes and creation of a complete gap in the spectrum are the exclusive right reserved by the Voigt con®guration. For methodological and analytical details, the reader is referred to the original papers [730±733]. 11.2.3. Coupling to the optical phonons The purpose in this section is to examine the effect of electron±phonon coupling on the plasmon dispersion in both type-I and type-II superlattice systems. We consider the kinds of systems studied in the preceding section, where the thickness of the 2DEG and/or 2DHG layers is small but ®nite. In addition, we are interested, for the sake of simplicity, in the zero-®eld case. The introduction of heterostructures and superlattices synthesized mostly from the materials of III±V group has made the electron±phonon interaction an essential ingredient in understanding the physical properties of these socalled 2D systems. In the systems in which the background material is polarizable, as is the case with the weakly polar III±V semiconductors, it is important and (sometimes) necessary to allow the frequency dependence of the background dielectric constant Es . A suitable way is to have Es de®ned as [725,1753]: Es …o† ˆ E1

o2 ÿ o2L ; o2 ÿ o2T

(11.70)

where E1 is the high-frequency dielectric constant, and oL and oT are, respectively, the longitudinal and transverse optical-phonon frequencies at the zone center (q ˆ 0). This, in other words, allows the coupling of 2D plasmons to the optical (in fact LO) phonons in the superlattice systems. We are interested in replacing Es in the dispersion relations of type-II superlattice, Eqs. (11.59) and (11.61), and of type-I superlattice, Eqs. (11.62) and (11.63), by the Es …o† de®ned in Eq. (11.70). In order to properly

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Fig. 103. Collective excitation spectrum for type-I superlattice in the zero magnetic ®eld, but when coupling to the optical phonons in GaAs spacers is incorporated. We used the normalized phonon frequencies: OL ˆ 0:0344, and OT ˆ 0:0317; the rest of the parameters are the same as in Fig. 99. The hatched regions are the bulk plasmon bands of an in®nite superlattice, and the solid curves are the surface polariton modes of a semi-in®nite superlattice. The dotted (dashed) line is the light line in vacuum (intrinsic GaAs spacers excluding the frequency dependence). Notice the splitting at and occurrence of an equivalent excitation spectrum above OL in the upper panel.

account for the frequency-dependence of Es , we work throughout in terms of the unscreened plasma frequency ope. We also preserve the retardation effects in the problem. This makes the analytical diagnosis a little tougher. Coupling of the 2D (bulk and surface) plasmons to optical phonons in both types of superlattices in the NRL was considered by the author [725,1753]. Here we consider similar systems except for the difference that the 2DEG and 2DHG layers are considered to have small but ®nite widths. The illustrative numerical examples both for type-I and type-II superlattices are discussed in what follows. The material parameters are exactly the same as used in Figs. 99 and 100, except for the additional parameters: E0 ˆ 12:8, E1 ˆ 10:9, and the normalized (with respect to the unscreened plasma frequency ope ) longitudinal and transverse optical phonon frequencies Ð Type-I: OL ˆ 0:0344, OT ˆ 0:0317; Type-II: OL ˆ 0:0394, OT ˆ 0:0364. Fig. 103 illustrates the collective (bulk and surface) excitation spectrum for the type-I superlattice. Since Es …o†, for o < oT, remains a real and positive constant, the excitation spectrum below oT bears a close resemblance with that where coupling to the optical phonons is neglected (see Fig. 99), except for the fact that both the bulk band and the surface mode are seen to be suppressed in frequency, at higher wave vectors. Note that the upper (k ˆ 0) edge of the bulk band in the lower panel starts at exactly the same frequency as in Fig. 99. Both the bulk band and the surface mode in the lower panel seem to become asymptotic to oT . Es …o† also remains a positive constant above oL , and one again ®nds a corresponding bulk band and a surface mode. The lower edge of the bulk band in the upper panel starts at the resonant frequency o ˆ oL. The surface mode, in the upper panel, starts at the upper edge of the bulk band towards the right of the light line in the vacuum, shows a small hump in the vicinity of the light line in the spacer, merges with the bulk

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Fig. 104. Collective excitation spectrum for type-II superlattice in the zero magnetic ®eld, but when coupling to the optical phonons in the GaAs spacers is incorporated. We used the normalized phonon frequencies: OL ˆ 0:0394, and OT ˆ 0:0364; the rest of the parameters are the same as in Fig. 100. The hatched regions are the bulk plasmon bands of an in®nite superlattice, and the solid curves in blue (pink) are the surface polariton modes of a semi-in®nite superlattice truncated at 2DEG (2DHG) layer. The dotted (dashed) line is the light line in vacuum (intrinsic GaAs spacers excluding the frequency dependence). Notice the splitting at and occurrence of an equivalent excitation spectrum above OL in the upper panel.

band, and then emerges at higher wave vector than the surface mode in the lower panel. The bulk band in the upper panel is a result of the bulk longitudinal optical phonon mode becoming dispersive and broadening due to coupling to the intrasubband plasmons. In the frequency range speci®ed by oT  o < oL , Es …o† attains a negative value giving rise to a complete gap in the excitation spectrum. Fig. 104 illustrates the excitation spectrum for the type-II superlattice in the case where coupling to the optical phonons is allowed. The upper edge of the upper bulk band in the lower panel still starts at exactly the same frequency as in Fig. 100, where coupling to optical phonons is neglected. We have shown two types of surface modes: when the superlattice is truncated at the electron layer (in green) and when it is truncated at the hole layer (in pink). The physical mechanism regarding the existence of bulk bands and surface modes in the upper panel is the same as stated above with respect to Fig. 103. The propagation characteristic of both the bulk bands and the surface modes in the lower panel are describable just as in Fig. 100. In the upper panel, above oL , the bona®de surface modes are seen to emerge at larger wave vectors than their counterparts in the lower panel. The starting of the gap-surface (acoustic) modes in the upper panel is quite noteworthy; the gap mode due to the truncation at the electron (hole) layer emerges from the lower (upper) edge of the upper (lower) bulk band. The existence of humps and dips shown by the upper surface modes in the upper panel, towards the right of the light line in the vacuum until they emerge out from the bulk band, is quite an unusual phenomenon. The excitation spectrum is seen to observe a complete gap in the frequency range between oT and oL . To conclude with, what is interesting is that the coupling of plasmons to optical phonons makes the collective excitation spectrum richer than when such a coupling is neglected. This is true with respect to both the bulk and the surface excitations. Existence of a complete gap in the spectrum is quite useful and its magnitude can be tailored, depending upon the choice of the material media. We have dealt with

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the low-temperature properties where real phonons are nearly absent and where the coupling effect on the transport is rather small. However, in the real life, electron±phonon coupling is an important scattering mechanism limiting the mobility of the carriers at high temperatures. 12. Quantum structures Ð intra and intersubband excitations By now, we have been concerned with the theoretical results on plasmons and magnetoplasmons in the systems which, though apparently give an impression to be non-conventional, are penetrable reasonably successfully through a variety of classical or semiclassical approaches. This section is, on the other hand, devoted to review the collective excitations in the systems which can only be approached through the methodologies developed in the framework of quantum mechanics; and there are numerous of them which, inspite of their certain limitations, have provided us not only the expected results but also enabled a reasonable comparison with the existing experiments. We refer to the manmade 2D, 1D, and 0D systems Ð also broadly known as quantum wells, quantum wires, and quantum dots (see, for examples, Sections 2.2.2 ± 2.2.4). We are, however, very well convinced that the RPA introduced originally in the context of conventional (3D) systems by Bohm and Pines has been surprisingly quite successful in the studies of the collective Ð both intra and intersubband Ð excitations, both with and without an applied magnetic ®eld, in these systems of reduced dimensionality. Just as before, we will recall only short important, relevant mathematical results at the appropriate places, and refer the reader for the theoretical details to the original papers. Also, we would like to focus, for the sake of simplicity, only on the theoretical results obtained in the absence of any applied magnetic ®eld; even though the magnetic-®eld effects will no doubt appear somewhere in some different context. Collective excitations, particularly in these mesoscopic and nanosystems, are important hallmark of many-body effects, which are already becoming known to increase with the decreasing dimensionality. 12.1. Quantum wells The subband structure in semiconductor heterostructures, arising from ®nite thickness of the quasitwo-dimensional electron gas (Q-2DEG), gives rise to interesting quantum-size effects that are completely missed by a purely 2D treatment. The intersubband transitions correspond to charge-density oscillations perpendicular to the plane and there is a collective mode associated with them Ð the socalled intersubband plasmon. Also, the dispersion of the intrasubband plasmon, whose energy goes as p q in the long-wavelength limit in 2D, itself is affected by the ®nite thickness and by coupling to the higher subbands. These phenomena have been studied, both theoretically and experimentally, in great detail. All the calculations are, generally, carried out in the framework of the RPA, which has proven to be suf®ciently adequate for the quasi-2D systems in the past. Experimentally, most of the observations have been made through the Raman scattering measurements of the plasmons and magnetoplasmons (see Section 2.2.2). Some calculations of Raman intensities have, however, indicated possible dif®culties in the ability to resolve surface modes from the bulk spectrum [661±667]. This has brought into light an alternative experimental probe Ð the electron-energy-loss spectroscopy (EELS). In EELS, with electron energy in the appropriate range of 1±100 eV, the surface is illuminated with waves that have a wavelength comparable to the inter-particle separations. This is in contrast to the optical

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spectroscopies, which, in all cases, employ radiation with wavelength long compared to the interparticle separation. Through the electron energy loss studies in the impact scattering regime, one has the possibility, in principle, of obtaining detailed structural information from the loss cross-sections, which is not quite possible with optical methods. Some experimental works have, however, recommended the combination of Raman scattering and electron scattering for the precise characterization of the plasma modes in the system. We have, therefore, focused on and discussed, in what follows, the theoretical results based on both Ð the Raman intensities and energy-loss spectra. 12.1.1. Collective CDEs Self-sustaining collective modes of a system are given by the zeros of the dielectric function E…† or by the poles of the IDF Eÿ1 …†; see, for example, Section 3.3. The collective modes are well-de®ned excitations when the real part of E…† has zero in a region where the imaginary part of E…† vanishes; the latter describes the dissipative aspects of the excitations, while the former describes the reactive part. In all the theoretical results to be discussed below, we shall ignore the retardation effects. The problem of damping or ampli®cation [1655±1658] of the excitations is not a simple one, and we shall not shed any light on it here. Theoretical illustrative results on collective excitations discussed below are taken from the work by Jan and Das Sarma [666], and are based on a number of simplifying assumptions. For instance, the semiconductor quantum well is modeled by an in®nite square-well potential of width L, thus ignoring the ®nite-depth and band-bending effects. This can be remedied, to a certain extent, simply by treating the subband separation as an adjustable parameter. There they also ignore (i) coupling to LO phonons, because the concern is with the plasmon energies that are quite small compared with LO phonon energy; (ii) excitonic effects, which are supposed to be small enough in GaAs±Ga1ÿx Alx As; (iii) exchange-correlation effects; (iv) image-charge effects due to dielectric mismatch at the surface and the interfaces. They argue that all these approximations can be systematically relaxed and that their net effect on their ®nal results was expected to be qv Fl ; qv Fh ) and the lowfrequency region (qv Fl > o > qv Fh ). Since r > 1 and d ˆ 1, the condition for the existence of the acoustic plasmon mode in the low-frequency region is given by v Fh < v AP < v Fl. Due to the energy gap

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Fig. 126. Collective and SPE spectrum for a two-component, Q-1D quantum plasma. The parameters used are: d  kFl =kFh ˆ nl =nh ˆ 1:0, r  mh =ml ˆ 3:0, and (the dimensionless electron-gas parameter) rsl ˆ 1:0. Only the lowest subbands are assumed to be occupied by both types of (light and heavy) charge carriers. The shaded regions are the singleparticle transitions, and solid (dashed) curve is the ordinary (acoustic) plasmon. (After Tanatar, Ref. [981].)

between the two single-particle continua, the acoustic plasmon remains Landau-undamped up to a rather large wave vector. This undamped, 1D acoustic plasmon is fundamentally different from the corresponding mode in the higher dimensional systems, since the presence of the low-energy Landau damping region in 2D or 3D gives rise to the complete damping of the acoustic mode. Tanatar [981] stressed that for d ˆ 1 and rsl ˆ 1, the acoustic plasmon branch would be present for all r > 1. He further argues that as the coupling strength is increased, the existence of the acoustic plasmon is possible for some rmin ; for instance, at d ˆ 1 and rsl ˆ 3, rmin ˆ 1:7 Ð increase in d is reported to have an effect of decreasing rmin [981]. Theoretical results on the existence of this 1D acoustic plasmon [951,981] show that this mode should be readily observable in resonant RS experiments performed on photo-excited electron±hole plasma in narrow GaAs±Ga1ÿx Alx As quantum wires [1666]; in contrast to it higher dimensional counterpart where the inevitable Landau damping has prevented an unambiguous observation of a quantum acoustic plasmon in degenerate electron±hole plasma [1667]. Fig. 127 illustrates the effects of the local-®eld corrections in the HA on the excitation spectrum of a two-component, Q-1D quantum plasma, for r ˆ 5, d ˆ 1, and rsl ˆ 1. The dashed and solid curves stand for the dispersion of optical and acoustic plasmon with and without the local-®eld corrections, respectively. Note that the local-®eld effects in the HA reduce the plasma frequencies of both optical and acoustic plasmons as compared to the neat RPA results, just as in the higher-dimensional cases. One can also note that, for these parameters, the critical wave vector qc at which the acoustic plasmon enters the SPC decreases. Moreover, rmin is said to decrease upon the inclusion of the local-®eld effects [981], whereas the damping is increased. Knowing the fact that the optical mode exhausts the f-sum rule in the long-wavelength limit, the local-®eld correction has little effect on the optical mode; this is in contrast to the acoustic mode which is very strongly affected by the local-®eld effects [951]. For more extensive details of the plasmon propagation in two-component, Q-1D systems, the reader is referred to the recent work by Das Sarma [951], who studied the problem on a wider scale.

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Fig. 127. Effect of local-®eld corrections within the Hubbard-like approximation on the collective excitation spectrum for a two-component Q-1D quantum plasma (Fig. 126). The model parameters are: d ˆ 1, r ˆ 5, and rsl ˆ 1. Dashed and solid curves show the dispersion of the ordinary and acoustic plasmons with and without the local-®eld corrections, respectively. (After Tanatar, Ref. [981].)

12.2.3. Inelastic light scattering Ð theory and experiment 12.2.3.1. Theory. This section is devoted to the discussion of the theoretical development of the Raman (or inelastic light) scattering from the collective excitations in Q-1DEG. A simple theory of RS in single and multiple quantum wires was reported by Li and Das Sarma [944], and we will heavily rely on their work in this section. A major part of this work [944] was devoted to calculate the elementary excitation spectra in single wire, double-layered quantum wires, and multiple-wire systems. They calculated the excitation energies which correspond to the positions of the peaks in suitable spectroscopic experiments. By that time, no experimental data from RS experiments on the quantum wire systems were available; this unabled them to check their results. However, their choice of the material and, particularly, geometric parameters for the calculation of the excitation spectrum and the Raman intensities was not quite consistent; this was, possibly, the reason that they did not stress on substantiating the correspondence between the excitation energies and the Raman peaks. However, it is, by all means, worthwhile to recall their theoretical development of the (later reported) RS experiment [905]. The light-absorption or RS experiments are known to measure not only the position of the peaks, but also the line shape of the peaks, which, of course, contains some additional information of the system being probed. As such, to have a complete comparison with the experiments, one should also calculate the spectral weights (line shapes). Theoretically, the calculation of the spectral weight is more involved than the determination of the plasmon energies, because the position of the peak is determined by the pole of the DDCF and does not involve the full knowledge of the DDCF, whereas the spectral weight is proportional to the imaginary part of the dynamical polarizability function for RS experiment and is proportional to the real part of the conductivity for light-absorption experiments. In RS experiments, one measures the imaginary part of the DDCF directly for ®nite wave vector ~ q. Since the Coulomb interaction is spin independent, i.e., it does not ¯ip spins, the DDCF calculated without screening (single bare bubble) D0ij corresponds to the SDE, whereas the DDCF calculated with screening using the

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RPA Dijkl corresponds to the charge-density (or collective) excitations [944]. In the infrared±lightabsorption experiments, one is always in the long-wavelength limit, since the speed of light is always much larger than the Fermi velocity in 1DES, which is de®ned by the slope of the single-particle dispersion relation. In order to increase the range of wave vectors that can be probed, one can use the grating coupler of period d to allow jumps in the probed wave vectors by 2pn=d; n being an integer. The time-ordered DDCF is de®ned in the usual manner (with h ˆ 1) D…x; y; t; x0 ; y0 ; t0 † ˆ ÿihtn…x; y; t†n…x0 ; y0 ; t0 †i;

(12.52)

where we have assumed that the 1DES is con®ned in a zero-thickness x±y plane and the z coordinate drops out. The quantity n…x; y; t† is the electron density operator in the Heisenberg representation and t is the time-ordering operator. D…x; y; t; x0 ; y0 ; t0 † depends on x and x0 only through the difference x ÿ x0 due to the translational invariance in the x direction, and one can thus Fourier transform it in the variables x ÿ x0 and t ÿ t0 to get D…q; o; y; y0 †. One can write D…q; o; y; y0 † in terms of the wave functions in the y direction (of the quantum con®nement) as X Dijkl …q; o†fi …y†fj …y†fk …y0 †fl …y0 †: (12.53) D…q; o; y; y0 † ˆ ijkl

Within the standard RPA, Dijkl is given by, suppressing (q; o)-dependence X Dijkl ˆ D0ij dik djl ‡ D0ij Vijmn Dmn;kl ;

(12.54)

m;n

where Vijmn is the matrix element of the Coulomb interaction given by (12.50), and D0ij is the DDCF in the absence of Coulomb interactions X D0ij …q; o†fi …y†fj …y†fi …y0 †fj …y0 †: (12.55) D0 …q; o; y; y0 † ˆ ij

To calculate the RS intensity, Li and Das Sarma [944] used the Matsubara's ®nite-temperature formula [638,639] to determine D0ij : 2 X fF …Ei …p†† ÿ fF …Ej …p ‡ q†† D0ij …q; ion † ˆ ; (12.56) L p Ei …p† ÿ Ej …p ‡ q† ‡ ion where fF …Ei …p†† is the Fermi distribution function and on ˆ 2pn=b with b ˆ 1=kB T, is the complex frequency; n is even (odd) for bosons (fermions); p  px , q  qx . In the calculation of the Raman intensities, one usually replaces the complex frequency ion by o ‡ ig. But this simple replacement is known to violate the conservation of the number of particles. Instead, one should use the correct form of D0 as suggested by Mermin [1104], which is D0 !

D0 …q; o ‡ ig†…1 ‡ ig=o† : ‰1 ‡ i…g=o†D0 …q; o ‡ ig†=D0 …q; 0†Š

(12.57)

However, it was remarked that making this correction does not lead to any signi®cant difference in the results obtained through the simple replacement of ion by o ‡ ig. Here g refers to the phenomenological parameter characterizing the broadening of energy levels due to the impurities and imperfections inevitably surviving in the system. Note that in the interpretation of the peak position or spectral weight one does need the perpendicular component of the wave vector (qy ). This is

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accounted for by, e.g., integrating D…q; o; y; y0 † over y and y0 to write Z Z X D…q; o; qy † ˆ dy dy0 exp…ÿiqy …y ÿ y0 ††D…q; o; y; y0 † ˆ Dijkl …q; o†Bijkl …qy †

(12.58)

i;j;k;l

with

Z Bijkl …qy † ˆ

Z dy

dy0 fi …y†fj …y† exp…ÿiqy …y ÿ y0 ††fk …y0 † fl …y0 †:

(12.59)

It should be remarked, however, that such an integration of the correlation function is not equivalent to seeking its Fourier transform, which is evidently not the case (due to the quantization of the electronic motion along the y direction). Li and Das Sarma [944] con®ned to work within a two-subband model for a square-well con®nement, with a well-width a. Thus the symmetry of the potential well allowed them to obtain numerous intermediate analytical results explicitly. This was the case not only for the single-wire system, but also for the multiple-wire (superlattice) system. Next, we discuss brie¯y some illustrative numerical examples gathered from Ref. [944], on the RS intensity of CDEs of GaAs± Ga1ÿx Alx As 1DES. The RS intensity is proportional to the imaginary part of the DDCF, which was seen, from their analytical results (see, e.g., Eq. (5.18) in Ref. [944]), to be a clear-cut sum of two parts: intra and intersubband plasmon excitations. D…q; o; qy † ˆ D1111 B1111 ‡ …D1212 ‡ D1221 ‡ D2112 ‡ D2121 †B1212 :

(12.60)

Fig. 128 illustrates the RS intensity of the CDEs for a 1DES with Ne1D ˆ 0:885  105 cmÿ1 , a ˆ 39 nm, g ˆ  h2 =…2m a2 †, q ˆ 1=a, and T ˆ 0:10E21 =kB for three different values of qy ˆ 1=a [Fig. 128(a)], 2/a [Fig. 128(b)], and 4/a [Fig. 128(c)]. Here the symbol E21 ˆ E2 ÿ E1 stands for the subband separation. The peak at the lower energy refers to the intrasubband plasmon excitation, whereas the peak at the higher energy represents the intersubband plasmon excitation. The spectral weight of the intersubband excitation is seen to increase with increasing qy . Fig. 129 depicts the RS intensity of the CDE at ®xed qy ˆ 4=a for the same parameters as stated h2 =…2m a2 † [Fig. 129(b)]. above in Fig. 128, except for g ˆ 5:0 h2 =…2m a2 † [Fig. 129(a)] and g ˆ 10:0 As it is expected, both intra and intersubband plasmon peaks are seen to broaden with increasing g. The half-widths of the peaks are determined by g and the temperature T. Next, we consider the case of a 1D quantum-wire superlattice, and restrict to the case of an isolated (no tunneling) 1D periodic system with period d. As regards the SDEs, the RS intensity of the 1D superlattice is nothing but the sum of the contributions of each individual quantum wires, since it is calculated in the absence of the Coulomb interaction, and is therefore enhanced in a simple way by a factor of N (the number of quantum wires in the superlattice). For the CDEs, the RS intensity of the 1D superlattice can be calculated by replacing the Coulomb interaction matrix elements of a single wire (Eq. (12.49)) by those of a 1D quantum-wire superlattice [944] S…q; y ÿ y0 † ˆ

X exp‰ÿi…qy ‡ 2ps=d†…y ÿ y0 †Š s

‰q2 ‡ …qy ‡ 2ps=d†2 Š1=2

;

(12.61)

where the summation is over all integers s. It has been noted that the matrix elements of Coulomb interaction in the 1D isolated superlattices can be calculated explicitly for the square-well potential, considerably simplifying the computation of the spectral weight (see, e.g., Eq. (5.21) in Ref. [944]).

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Fig. 128. The calculated RS intensity of the CDEs in Q-1DES for three different values of the wave vector qy ˆ 1=a, 2=a, h2 =2m a2 , and 4=a in (a)±(c), respectively. The other parameters are: NW ˆ 0:888  105 cmÿ1 , a ˆ 390 nm, qx ˆ 1=a, g ˆ  and T ˆ 0:1E21 =kB . Such calculations embody the second subband completely. Notice that the spectral weight of the intersubband plasmon excitation increases with increasing qy . (After Li and Das Sarma, Ref. [944].)

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Fig. 129. The calculated RS intensity of the CDEs in Q-1DES at a ®xed qy ˆ 4=a for the same parameters as in Fig. 128, except for g ˆ 5:0  h2 =2m a2 (a) and 10:0  h2 =2m a2 (b). (After Li and Das Sarma, Ref. [944].)

Fig. 130 shows the theoretical results for the spectral weights of the CDEs in the 1D quantum-wire superlattices (with no tunneling). The material parameters used for this purpose are: Ne1D ˆ h2 =…2m a2 † ˆ 0:377 meV, d (the superlattice period) ˆ 2a, and 0:872  106 cmÿ1 ; a ˆ 39 nm, g ˆ  T ˆ 4:4 K, for qy ˆ 4=a (Fig. 130(a)) and 2=a (Fig. 130(b)). Just as before, the left (right) peak corresponds to the intrasubband (intersubband) plasmon excitation. Similar to the single-quantum-wire case, the spectral weight of the intersubband plasmon excitation increases as qy increases. In Fig. 130(b) the small peak at the extreme left is the intrasubband plasmon excitation of the second subband. Finally, Fig. 131 presents the calculated spectral weights of the CDEs of the 1D quantum-wire superlattice with the same parameters as in the preceding case of Fig. 130, except for g ˆ 0:754 meV (Fig. 131(a)), and 1.885 meV (Fig. 131(b)). Analogous to the single-wire case, the Raman peaks broaden with increasing g. In addition, the height of the peaks is also lowered as g increases, just as in Fig. 129. 12.2.3.2. Experiment. The first successful RS experiments determining the wave vector dispersions of the single-particle and collective, both intra and intersubband, excitations of 1DEG in GaAs quantum wires were performed by Pinczuk and coworkers [905]. MD multiple-QWWs were fabricated from a Ê -wide single GaAs±Ga1ÿx Alx As quantum well using electron-beam lithography followed by low250 A energy ion bombardment [897]. The electron density and mobility of their parent (2DEG) sample were

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Fig. 130. The calculated RS intensity of the CDEs in Q-1D isolated wire superlattice for two different values of qy ˆ 4=a (a) and 2=a (b). The parameters used are: NW ˆ 0:872  106 cmÿ1 , a ˆ 39 nm, qx ˆ 1=a, g ˆ 0:377 meV, T ˆ 4:4 K and (period) d ˆ 2a. The peak on the left (right) side is the intrasubband (intersubband) plasmon excitation. (After Li and Das Sarma, Ref. [944].)

n ˆ 3:2  1011 cmÿ2 and m ˆ 1:1  106 cm2 =V s, respectively. The 1D pattern consisted of  70 nmwide lines with a period of d ˆ 200 nm. The changes in the Fermi energy had been monitored by means of PL and PLE measurements [897±906]. The energy spacing E01 between two lowest 1D subbands was determined from light-scattering spectra of intersubband excitations as discussed below. At power densities of  1 W=cm2 , they obtained a Fermi energy EF ˆ 5:8  0:5 meV and intersubband spacing E01 ˆ 5:2  0:5 meV. The electron gas was thus in almost a 1D quantum limit, with only a slight occupation of the first excited subband. Resonant RS measurements were performed at 1.7 K using a tunable dye laser in the energy range of interband optical transitions from higher-lying valence to conduction subband states. Spectra were measured in a conventional backscattering geometry for different angles of incidence by rotating the sample around the y-axis. In this scattering geometry the wave vector component along the wires (x direction) is given by q ˆ …4p=l† sin y, where l is the wavelength of the incident light. Accessible wave vectors were in the range (0.5±1.5) 105 cmÿ1 (roughly one-tenth of the Fermi wave vector). Incident

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Fig. 131. The calculated RS intensity of the CDEs in Q-1D isolated wire superlattice for a ®xed value of qy ˆ 1=a. The parameters used are the same as in Fig. 130, except for g ˆ 0:754 meV (a) and 1.885 meV (b). (After Li and Das Sarma, Ref. [944].)

and scattered light were linearly polarized parallel (H) or perpendicular (V) to the wires. Both polarized (HH) and depolarized (HV) Raman spectra showed that peak energies increase with increasing q (see Figs. 1 and 2 in Ref. [905]), and the strong dispersive behavior indicated that these are due to intrasubband excitations of the electron gas. The higher-energy peaks observed only in HH polarization were assigned to the 1D plasmon of the wires. The lower-energy bands were interpreted as predominantly intrasubband SPEs. The broader spectra in HV polarization were understood to be a superposition of SPE and SDE. Note that for intrasubband SPE, RS experiments measure the imaginary part of the dynamical polarizability function w, which is the same, in a broad sense, as the DDCF. In 1D, this function is a symmetric peak centered at  hqv F ; v F being the Fermi velocity. From this analysis, they 5 ÿ1 estimated n0 ˆ …6:5  0:4†  10 cm , EF ˆ 5:8  0:6 meV, and a slight population of the second subband with Fermi energy  0:6 meV and density n1 ˆ …2:1  0:4†  105 cmÿ1. In a crossed (HV) polarization, the maximum peak position of measured spectra were seen at lower energies because there is a contribution due to intrasubband SDE. The calculated intrasubband peaks coincide with the structure on the high-energy side of the depolarized spectra. The energy shift of SDE with respect to the

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Fig. 132. Collective and SPE spectrum of a Q-1DEG in the quantum limit. Solid dots represent the intrasubband collective CDEs and SDEs. Open circles display the position of the peak at  hqvF of intrasubband SPEs. Squares correspond to data of 1D intersubband CDEs measured in VV polarization. The shaded areas indicate SPE regime given by Im…q; o† 6ˆ 0. Inset: comparison of the 1D intrasubband plasmon frequencies with those of a 2DEG with EF ˆ 3:8 meV as a function of q1=2 . (After GonÄi et al., Ref. [905].)

SPE band is a measure of the exchange Coulomb interaction in the ground state of the electron gas. It was remarked that exchange effects in 1D appear to be comparable to those reported in 2D case [905]. Fig. 132 summarizes the measured wave vector dependence of the Raman peaks. The 1D intrasubband plasmon, which corresponds to the charge-density oscillations along the wires, exhibits an almost linear dispersion that extrapolates to a ®nite positive frequency at q ' 0. The inset of Fig. 132 shows that the 1D system has markedly different behavior from the expected o / q1=2 dependence measured for 2D plasmon. The difference between the measured dispersions of 1D and 2D plasmons are larger than experimental uncertainty, as shown by the error bars. In order to give a quantitative interpretation, Pinczuk and coworkers [905] considered the case of two occupied 1D subbands with linear electron densities n0 and n1 , and allowed the coupling between the intrasubband collective excitations of both subbands. They choose parabolic potential well due to con®nement in the y direction, assumed the electron gas to be of zero thickness in the z direction, and did a good estimate of the excitation energies within the RPA. The curves through the data points correspond to such a ®t with electron density n…ˆ n0 ‡ n1 † ˆ …8:6  0:4†  105 cmÿ1. In these experiments, Coulomb coupling between wires results in a minor effect because 1 < qd < 3. The wire width a, de®ned as FWHM of the ground-state harmonic oscillator wave function, was found to be 33 nm from experimental value of the

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intersubband spacing. The overall q dependence is well accounted for within the RPA; a small discrepancy  10% was attributed to, possibly, an under estimation of the electron density and the effect of the mode coupling. The 1D intersubband CDE observed in the VV polarization (see also Fig. 4 in Ref. [905]) also display remarkable behavior of being almost dispersionless and exhibit a surprising q dependence of the bandwidths. The experimental observation is that at q ˆ 0 there is a broad band at  5:2 meV that partially overlaps a weak luminescence band. At larger q, the bands are much sharper but appear at roughly the same energy. These observations are understandable through the consideration of Landau damping effects. Since the SPE in the 1DES can be excited only with the wave vectors parallel to the Fermi wave vectors, the slight occupation of the ®rst excited subband opens up a gap in the intersubband SPE (hatched regions starting at  5:2 meV and characterized by Im w 6ˆ 0). At ®nite q; w has a minimum at hqvF . Undamped 1D intersubband CDE exist with energy E01. For q  0, the E01 and maxima at E01   two maxima in the loss function (w) merge into a single peak at E01 , and the excitations become strongly damped. The theoretical 1D intersubband CDE unaccounting for the depolarization shift occur above the uppermost intersubband SPE, just as in Fig. 125. Also worth reminding here is the fact that for the low-electron-density samples probed in Ref. [905], the con®ning potential is nearly parabolic and the subband spacing depends weakly on the density [1078,1773]. In a system where the background potential is quadratic, the resonance frequencies of the electron gas are independent of the electron±electron interaction Ð the GKT [1082,1774]. Thus the energy of the intersubband CDE is very close to E01 . It should be remarked the resonant RS experiments reported in Ref. [905] stimulated a tremendous research interest in the later, theoretical as well as experimental, development of the studies on the elementary electronic excitations in the quasi-1DES. 12.2.4. Inelastic electron scattering from a quantum wire The purpose of this section is to present the ®rst systematic theoretical investigation on the fastparticle energy loss to a quasi-1DEG. This has been carried out in the framework of a dielectric response theory (DRT) [229,1649], which is still the best available description of the EELS. The DRT has been successfully used to study the multi-subband superlattice systems [693,741,1755] and provided an excellent interpretation of the HREELS experiments [600,608] on GaAs±Ga1ÿx Alx As superlattices for all the electron impact energies (from 4 to 35 meV). DRT proceeds in two steps. The ®rst step consists in evaluating the work done by the polarization ®eld of the sample on the electron (responsible for the polarization) along its semiclassical trajectory in which an electron is regarded as an external time-dependent potential that causes transitions in the target. This ®rst, classical step is complemented by a suitable quantal description of the multiple excitations emitted or absorbed by the electron. The essential result of the second step is accomplished by con®ning ourselves within the RPA, so that the exchange-correlation effects are neglected. It is important to specify ®rst the geometry of the system to be dealt with. We start with an electron gas in a narrow quantum well with interfaces parallel to the x±y plane and the well-width smaller than any other length scale in the problem. This corresponds to a realizable experimental situation in which the con®nement in the z direction is much stronger than that in the lateral x±y plane. In other words, the electronic motion along z direction drops out of consideration. We consider an effective con®ning potential (which is a sum of bare and Hartree potential) along the y direction to be parabolic one; this is quite a reasonable approximation for low-electron concentration in GaAs quantum wires [1078,1773]. Thus fabricated quantum wire represents a Q-1DEG with free-electron motion along the x direction,

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and the subband structure along the y direction. For this parabolic con®nement V…y† ˆ 12 m o20 y2 ;

(12.62)

where m is the electron effective mass and o0 the characteristic frequency of the harmonic potential, the single-particle eigenstates and eigenenergies are given by 1 cnk …x; y; z† ˆ p eikx fn …y†‰d…z†Š1=2 ; Lx   1 h2 k2 Enk ˆ n ‡  ho0 ‡ ; 2 2m

(12.63) (12.64)

where n is the subband index, fn …y† the Hermite function, and Lx refers to the size of quantum wire along the x direction of free motion. Such a system is also characterized by the transverse con®nement h=m o0 Š1=2 . Now we specify the strategy of working practically with such a model length l0 ˆ ‰ quantum wire as de®ned above. We are interested to explore a simple situation of a two-subband model with only the lowest one occupied. We calculate self-consistently the Fermi energy EF, for a given particle density Ne1D and con®ning potential strength ho0 , through 2 X  Ne1D ˆ ‰2m …EF ÿ En †Š1=2 y…EF ÿ En †; (12.65) p h n ho0 . For  ho0 ˆ 4:6 meV and Ne1D ˆ 5:72  105 cmÿ1 , Eq. (12.65) yields where En ˆ …n ‡ 12† EF ˆ 4:59 meV; this corresponds to the Fermi level lying just around the bottom of the ®rst excited subband. The effective con®nement p width of the parabolic potential well, estimated by the extent of the Hermite function (weff ˆ 2 2n ‡ 1 l0 ), comes out to be 31.45 nm. The actual weff which we estimated from a ®t to the hard wall potential is weff ˆ 29:56 nm. The difference indicates that the actual con®nement potential is somewhere between square-well and parabolic, just as it was predicted by Stern and coworkers [1078,1773] on the basis of self-consistent solution of the 2D SchroÈdinger± Poisson solver. The EELS refers, in a broad sense, to every kind of electron spectroscopy wherein inelastic electron scattering is used to study excitations of surface or thin solid ®lms. We have studied the fast-particle energy loss phenomenon in three different geometries with respect to the motion of a coherent electron beam: the fast-particle (i) moving parallel to, (ii) being specularly re¯ected from, and (iii) shooting through the Q-1DEG. It should be pointed out that the IDF Eÿ1 …q; o; y; y0 † of the Q-1DEG is central to the description of energy loss phenomena we are interested in. The exact IDF for quasi-n-dimensional (n  2) electronic systems were derived by Kushwaha and Garcia-Moliner [705] for multiple subband occupancy within the RPA. The IDF for Q-1DEG is given by (see Section 3.3) X Lm …y†Pm Lmn Sn …y0 † (12.66) Eÿ1 …y; y0 † ˆ d…y ÿ y0 † ‡ m;n

suppressing the …q; o† dependence for the sake of brevity. The symbols in Eq. (12.66) have the same meaning as speci®ed in Section 3.3. The elementary electronic collective excitations are determined by the poles of Eÿ1 …y; y0 † [990]. In Fig. 133, we summarize the wave vector dependence of the excitation spectrum of a Q-1DEG for a two-subband model. Although illustrated together, intra and intersubband, both collective and single-

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Fig. 133. Collective and SPE spectrum for quasi-1DEG within a two-subband model, in the framework of full RPA. We use an exact expression for IDF and a parabolic potential to characterize the lateral con®nement. The parameters used are: ho0 ˆ 4:6 meV, Ne1D ˆ 5:72  105 cmÿ1 , EF ˆ 4:59 meV, weff ˆ 29:56 nm, and E0 ˆ 12:8 (GaAs). The horizontally  (vertically) hatched region refers to the intrasubband (intersubband) SPEs associated with the lowest occupied (®rst excited) subband at absolute zero. The bold lower (upper) curve represents the intrasubband (intersubband) collective excitations (see text for an explanation of the depolarization shift). (After Kushwaha and Zielinski [990].)

particle (hatched regions), excitations are decoupled modes. This is because, for a symmetric potential well (as is the case here), Vijmn (the matrix elements of the Fourier transformed Coulomb interaction, Eq. (12.50)) is strictly zero for arbitrary momentum transfer q if the sum i ‡ j ‡ m ‡ n is an odd number. It is noteworthy that the intersubband collective excitation frequency (o ) at q ˆ 0 is almost twice the respective SPE. This shift of the intersubband resonance ho to energies signi®cantly above the subband spacing ( ho0 ) is attributed to the many-body effects, just as in 2D systems [85]. We assume that the depolarization effects are dominant and thus have ho ˆ h…o20 ‡ o2d †1=2 , where od is the depolarization frequency. In the lack of a desired model an upper bound on depolarization shift can be put classically [1668] to yield o2d ˆ 8pe2 Ne1D =…~Em w2eff ). With our input parameters, this requires a constant ~E ˆ 3:55E0 , which signals the importance of the screening effects in a quantum wire; E0 ˆ 12:8 for GaAs. Knowing the fact that in an experimental situation the quantization is dominant, this classical idea gives only a physical feel and is expected to grossly overestimate the depolarization effects. Turning to the theory of EELS, we ®rst discuss the rate of loss of energy W 0 due to a fast-particle moving parallel to the Q-1DEG at a distance y0 determined through [990±992] Z  Z e2 0 0 0 ÿ1 0 W ˆ Im (12.67) dq dy …qv x †K0 …qjy ÿ y0 j†E …q; o ˆ qv x ; y0 ; y † ; p where v x is the particle velocity along the axis of the quantum wire. We have performed the numerical computation for the rate of energy loss per unit energy. Doing so allows us to get rid of the integral over

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Fig. 134. The rate of electron energy loss per unit energy for a fast-particle moving parallel to the quasi-1DEG. The fastparticle velocity and its distance from the quantum wire are as given in the picture. In this geometry, only the fast-particle velocities greater than the Fermi velocity make sense. The odd appearance of the top of ®rst two sets of loss peaks is an artifact of constraining the y-axis to this value (a bad graphic software!); otherwise, these loss peaks are extremely high. (After Kushwaha and Zielinski [990].)

q and hence a very substantial saving in computational time is achieved, without loss of the relevant information about the loss peaks. The numerical results for a given particle velocity are illustrated for three values of y0 in Fig. 134. One can immediately notice that the positions of the loss peaks (in energy) do not vary as a function of the distance y0 . However, it has been seen that the larger the distance between the fast particle and the Q-1DEG, the smaller the rate of energy loss, just as it is expected intuitively. The ®rst, second, and third loss peaks occurring at 7.16, 14.05, and 16.35 meV, corresponding, respectively, to q=qF ˆ 0:5; 0:98, and 1.14, explain exactly the intrasubband collective mode in the excitation spectrum (see Fig. 133). Similarly, by repeating the computation for v x ˆ 2:05v F , we have observed three loss peaks below 25 meV Ð the lowest peak explains the intrasubband collective mode, whereas the two higher ones interpret the intersubband collective mode. However, for v x ˆ 2:30v F, we have observed only one broad (at 8.37 meV) and the other d-like (at 11.62 meV) peak below 25 meV Ð the lower peak was seen to explain the upper edge of the intersubband SPEs whereas the upper (d-like) peak interprets the intersubband collective mode. The analogous extensive computation performed for other values of the particle velocity leads us to infer that the dominant contribution to the loss peaks comes from the collective modes (i.e., the plasmons). In this geometry, it is exclusively noteworthy that only the fast-particle velocities greater than or equal to the Fermi velocity make any sense.

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Next, we study the case where the fast particle is assumed to move along a prescribed path ~ r ~ r…t† ˆ v x t^x ‡ v y t sgn…t†^y. This refers to the situation where the perpendicular component of velocity (v y ) changes the sign after (at t ˆ 0) the particle impinges (at y ˆ ÿy0 ) and is specularly re¯ected from the surface of the Q-1DEG. After rigorous mathematics, we deduce the expression for the probability function [990±992], Z  Z e2 a 0 ÿiat 0 ÿ1 0 dt e F…q; o; v x ; v y ; y ‡ y0 †E …q; o; …ÿy0 ; ‡v y t†; y † ; (12.68) Im dy P…q; o† ˆ 2 2p  ho where the function F is de®ned by   p Z 1 p …1 ‡ c2 =q2 † ÿ3=2 F…q; o; v x ; v y ; y† ˆ exp…ÿ…q2 y2 =2†x† dx x exp ÿ x 2qv y 0   2   q yx ‡ 2ic ÿicy p ‡ c:c: : erfc  e 2q x

(12.69)

Here c ˆ a=v y ˆ …o ÿ qv x †=v y , and the second term in square brackets is the complex conjugate (c.c.) of the ®rst. The function P…q; o† has the interpretation that P…q; o† dq do is the probability that the probe particle is inelastically scattered into the range of energy losses between ho and (ho ‡ do) and into the range of momentum losses between  hq and (hq ‡ dq). P…q; o† completely speci®es the actual kinematics of an electron at the detector. The numerical results for a given propagation vector (q ˆ 0:5qF ) and a ®nite y0 ˆ 10 nm are depicted in Fig. 135. The sharp (d-like) peaks at 7.11 and 11.14 meV explain exactly the intrasubband and intersubband collective modes in the excitation spectrum (see Fig. 133), whereas the broad peaks

Fig. 135. The probability function P…q; o† for a fast-particle specularly re¯ected from the quasi-1DEG. The input fastparticle velocities, the propagation vector, and the non-zero distance (y0 ) used in the computation are as given in the picture. The d-like peaks explain exactly the collective modes in the excitation spectrum (Fig. 133). (After Kushwaha and Zielinski [990].)

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describe approximately the boundaries of the SPEs. We repeated our computation for q ˆ 0:2qF and q ˆ qF to observe similar behavior. It should be pointed out that, for any value of the particle velocity, the number of loss peaks observed in the EELS is less than or equal to six, just as it is expected within a two-subband model at hand. The two d-like peaks in the EELS spectrum occur due to the non-vanishing Dirac-delta functions associated with the imaginary parts of the polarizability functions which in turn correspond to the existence of the collective modes in the excitation spectrum. This remark is valid for all the three geometries considered in this work. It is noteworthy that the positions of the d-like peaks (in energy) do not vary with the variation in the fast-particle velocity. This abides by the fact that the propagation vector q is kept constant. The earlier remark made with respect to plasmons as the main energy loss mechanism still remains valid. Finally, we turn to an illustrative example of a fast-particle shooting through a Q-1DEG. In this case, we treat the particle velocity ~ v ˆ constant (fast particle shoots through) and derive the expression for the probability function given by [990±992], with v x ! 0, Z    Z e2 1 io…y0 ÿ y00 † ÿ1 0 00 00 0 q Im dy exp P…q; o† ˆ dy E …q; o; y ; y † ; vy 2p hv y o2 ‡ …qv y †2

(12.70)

where the symbols have their usual meanings. The numerical results for q ˆ 0:5qF and for three values of the fast-particle velocity are shown in Fig. 136. Apart from some broad peaks that describe approximately the boundaries of the SPEs, we observe two d-like peaks at 7.11 and 11.12 meV that explain exactly the intra and intersubband collective modes (see Fig. 133). Again, the positions of the sharp loss-peaks remain intact, even though the fast-particle velocity varies. Performing the

Fig. 136. The probability function P…q; o† for a fast-particle shooting through the quasi-1DEG. The input fast-particle velocities and the propagation vector are as given in the picture. The delta-like peaks explain exactly the collective modes in the excitation spectrum (Fig. 133). (After Kushwaha and Zielinski [990].)

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computation for q ˆ 0:1qF, 0:25qF , and qF led us to draw similar conclusions. Just like in other two geometries, we stress that the dominant contribution to the loss peaks comes from the intrasubband and intersubband collective excitations. The rest of the discussions related to previous two geometries still remains valid. In spite of the fact that the main physics regarding the loss mechanism is consistent in all three geometrics considered here, we feel that this geometry of fast-particle shooting through the Q1DEG yields, in general, sharper structures in the EELS spectrum that allows sometimes better interpretation of even the SPEs in the long-wavelength limit. An interesting and well-de®ned (at longwavelengths) feature observed in this geometry is that the widths of the d-like loss-peaks are seen to decrease with increasing particle velocity. To sum up, we have presented theoretical results on the EELS in an isolated model quantum wire characterized by harmonic potential. The predicted loss spectra are expected to be con®rmed by the so far unattempted HREELS (see the following paragraph), irrespective of whether such experiments are performed on an isolated wire or on array of wires. This is because the Coulomb coupling between neighboring wires (in the state-of-the-art high quality multiwire systems) is very weak and one is allowed to interpret the experiments performed, for sensitivity reasons, on multiwires in terms of excitations in an isolated wire. A word on the capability of EELS for detection of these low-energy excitations in quantum wires is in order. As far as we know, no effort so far has been made to use EELS on quantum wires. The ®rst and the foremost obstacle, or so it looks like, is the thought of energy resolution of EELS concerned with the low-energy excitations in quantum wires that scares. It is true that the essential problem with this technique, which has proved to be the most versatile and sensitive tool in surface vibration spectroscopy, has always been the resolution, which was signi®cantly less than for the competing optical techniques, such as FIR and Raman spectroscopies. In the latter techniques the resolution is typically about 0.25 meV, whereas in EELS a resolution of 5 meV was considered to be a good result until recently. With the increasing complexity of the problems, it became desirable to have a sensitive method with a better resolution. The technology of spectrometers is now based on science and excellent, easy to operate instruments capable of resolution down to 0.3 meV (theoretical limit) and 0.5 meV (experimentally achieved limit) have been built [1669]. In view of this, we believe that HREELS could prove to be a potential alternative of already employed optical techniques. We would like to draw attention to the fact that the ®rst two geometries represent practically realizable situations. As regards the actual experiments, we believe that the (non-zero) parameter y0 (see above) implicitly takes care of the expected multiple scattering and absorption of the coherent electron beam in the host material cladding the quantum wire. The third geometry seems to remain only of fundamental interest, unless some experimental arrangement is suggested such that the fast-particle can be made to shoot through the wire embedded in the host material. This is because the substrate materials lapping the Q-1DEG would hardly allow the fast-particle to shoot through the whole system. However, this dif®culty could possibly be surmounted in a system of quantum wires micromachined from a freely suspended 2DEG [1670]. To the best of our knowledge, this work [990] predicts the ®rst theoretical fast-particle energy loss spectra in a model quantum wire in the framework of a dielectric response theory within the full RPA. For this purpose, we made use of an exact analytical expression for the IDF, which knows no bound with respect to the subband occupancy and/or an applied magnetic ®eld [705]. We designed the Q1DEG with a parabolic potential well to characterize the lateral quantum con®nement and worked within a two-subband model. Our main conclusion is that the dominant contribution to the energy loss

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peaks comes from the intra and intersubband collective excitations. We hope this work will stimulate more theoretical work and encourage the experiments to evidence the capability of HREELS on the quantum-wire structures. Further details of the theory Ð incorporating explicitly the absorption in the substrate, considering larger number of occupied and unoccupied subbands, accounting for the effects of an applied magnetic ®eld, including the coupling to the optical phonons, and extending to the multiwire periodic systems Ð are in progress and the results will be reported shortly [991,992]. 12.3. Quantum dots The wizardry of modern semiconductor technology now makes it possible to fabricate heterostructures within which carriers are three-dimensionally (3D) con®ned to an ultrasmall region of space by a potential barrier. When the dimensions of the con®ning potential are smaller than the electron wavelength, electrons in these structures can display astounding behavior. In these so-called quantum structures, carrier energy levels are quantized and their energy depends crucially on the dimensions and magnitude of the con®ning potential. We refer speci®cally to 3D-con®ned quantum dots, quite often referred to as ``arti®cial atoms'', which have attracted world-wide interest in the recent past. Electronic transport through these systems is governed by the mechanism of CB and singleelectron tunneling (see, for extensive details, Section 2.2.4). To understand quantum dots it is helpful to know how to make them. Just as with quantum wires, one of the most common way is to start with an original 2DEG de®ning the x±y plane; lateral con®nement along both x and y directions yields the desired quantum dot structures, depending upon the strength of the con®ning potentials. In the experiment, one often applies a magnetic ®eld oriented (usually) perpendicular to the original 2DEG. This then allows one to exploit the interplay between electric and magnetic con®nements very effectively in order to characterize lengths, energies, and interactions in the quantum dots. If the quantum dot structure is a 2D array, one is also permitted to study the collective excitations in the system, which, in fact, refers to the band structure of a periodic 2D array of quantum dots, both with and without an allowance for tunneling between the neighboring dots. Our principal interest in this section is twofold: ®rst, to discuss the magnetic-®eld effects on the single-particle energy levels of parabolically con®ned single quantum dots; and second, to present some illustrative examples on the collective excitations in a 2D array of quantum dots. Extensive details of the electronic, optical, and transport studies, both experimental and theoretical, have already been discussed in Section 2.2.4. 12.3.1. Magneto-optical spectrum in single quantum dots In order to understand the behavior of the quantum dot containing more than one electron, the ®rst step is to understand in some detail the quantum spectrum of a single electron in the presence of a con®ning potential and the applied magnetic ®eld. We start with strictly 2D system in the x±y plane; B with a circularly symmetric con®ning potential v…~ r† ˆ 12 m o0 r 2 and an applied magnetic ®eld ~ 2 2 2 oriented along z direction; r ˆ x ‡ y for the moment and the motion along z direction is of no interest. Neglecting the effect of spin dynamics, the single-particle Hamiltonian for an electron with effective mass m and charge ÿe (e > 0) can be written as Hˆ

1 ~ e ~2 1  2 2 P ‡ A ‡ m o0 r : 2m c 2

(12.71)

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We are interested to write the SchroÈdinger equation in the polar coordinates (r; y; z). The result is     h2 @ 2 1 @  1 @2 i @ 1  2 2 2 ÿ hoc (12.72) ÿ  ‡ ‡ m …oc ‡ 4o0 †r c…r; y† ˆ Ec…r; y†; ‡ 2 @y 8 2m @r 2 r @r r 2 @y2 where oc is the cyclotron frequency, and o0 is the characteristic frequency of the con®ning potential; the rest of the symbols have their usual meanings. So, the basic aim is to determine the single-particle eigenfunction c…r; y† and the eigenenergy E in Eq. (12.72). This was done more than seven decades ago by Fock [1124], and later by Darwin [1125]. It is very curious to note that the same problem Ð but for the zero con®ning potential Ð was studied two years after Fock's work by Landau [1562], leading to the terms like LLs, Landau degeneracy, Landau diamagnetism, and what not. 12.3.1.1. Single-particle spectrum. A systematic, rigorous but straightforward, mathematical calculation leads to derive the eigenfunction cn;l …r; y† of the SchroÈdinger equation. The result is ‰…n ‡ a†!Š1=2 ÿz=2 a=2 e z F…b; g; z† eily ; cn;l …r; y† ˆ p 1=2 2pa!…n!†

(12.73)

gF…b; g; z† ÿ …g ÿ b†F…b; g ‡ 1; z† ÿ bF…b ‡ 1; g ‡ 1; z† ˆ 0;

(12.74)

p where a ˆ jlj; b ˆ ÿn; g ˆ a ‡ 1, and z ˆ r2 =2l2e , with the effective magnetic length le ˆ h=m O, and O ˆ …o2c ‡ 4o20 †1=2 . Here n…ˆ 0; 1; 2; . . .† and l…ˆ 0; 1; 2; . . .† are the radial and angular quantum numbers, corresponding to the fact that the electron moves in two dimensions. The symbol F…b; g; z† stands for the so-called con¯uent hypergeometric function, which satis®es the identity [1671] and is related to the Laguerre polynomial Lan …z† through   n‡a a Ln …z† ˆ F…ÿn; a ‡ 1; z†: n

(12.75)

The corresponding single-particle eigenenergies En;l are given by hO ‡ 12 l hoc En;l ˆ 12 …2n ‡ jlj ‡ 1†

(12.76)

depend on both quantum numbers n and l. We have already discussed the simple results obtained through the analytical diagnoses of Eq. (12.76) for the two limiting cases of o0 ˆ 0 and oc ˆ 0 (see Section 2.2.4). Besides having the quantum numbers n and l, electrons also have a spin quantum number s, which can take two values  12. An electron with s ˆ ‡ 12 has a spin which points along the zaxis perpendicular to the plane of the quantum dot (spin up); an electron with s ˆ ÿ 12 has a spin which points against it (spin down). In zero magnetic ®eld, electrons with either spin state have precisely the same energy. It is very useful to examine the energy levels determined by Eq. (12.76), both with and without an applied magnetic ®eld. This is illustrated in Fig. 137, where we plot the energy levels as a function of l. The upper (lower) panel depicts the results of zero (nonzero) magnetic ®eld; the dashed line refers to the Fermi energy for the total number of electrons N ˆ 30 (80) in the upper (lower) panel. The lines in the ®gure connect states that have the same value of quantum number l. The bottom ``V'' is for states with n ˆ 0. Each state can hold two electrons, one with electron spin up and the other with spin down. Each of the circle can hold two electrons; the ®lled circles represent the ®lled states in a quantum dot

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Fig. 137. Top panel: energies of the different quantum levels as a function of angular quantum number l in zero magnetic ®eld. The different ``V's'' correspond to different values of radial quantum number n. Bottom panel: the same as in the top panel, but in the presence of an applied magnetic ®eld. Essentially two things have happened due to the ®eld (see the text): the ``V's'' have rotated to the left, and the energy separation between the ``V's'' has increased.

containing 30 electrons. At low temperatures, non-interacting electrons will simply fall like marble into the lowest available states. Note that for the case of zero ®eld (upper panel), the symmetry of the ``V's'' allow typically several states for the same value of energy. This implies that the states are degenerate. The situation takes a different turn in the presence of a magnetic ®eld (see the lower panel in Fig. 137). Essentially two things have happened due to the ®eld. The ``V's'' have rotated to the left, and the energy separation between the ``V's'' has increased. If we again think of electrons as marbles ®lling slots, the marbles then fall from states on the right arm of the ``V's'' to the left arms. Moreover, because the spacing between ``V's'' increases with increasing ®eld, the marbles fall from the upper to the lower ``V's''. In a magnetic ®eld, we refer to each of the ``V's'' as LLs. At very strong ®elds, all of the marbles are in the left arm of the lowest ``V''. In this case, all of the electrons are in the ``lowest LL''. The peak positions in the SECS or the GTS (gated transport spectroscopy) experiments simply track the energy of the highest energy electron in the quantum dot [1119]. This electron may be in a different quantum level depending on the magnetic-®eld strength, and each of these levels has a different evolution in a magnetic ®eld. Therefore, the peak position is expected to zig-zag as the highest energy electron moves from state to state. In the absence of spin splitting, each of the energy levels depicted in Fig. 137 holds two electrons of precisely the same energy. In this case, one would expect the two electrons to undergo identical level shifts (zig-zags) as the magnetic-®eld strength is varied. In reality, each level in the lower panel of Fig. 137 is split into two different energies (two circles displaced vertically, one for spin up and the other for spin down) by the magnetic ®eld. It is reported that at

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Fig. 138. Single-particle energy levels in a parabolic quantum dot as a function of magnetic ®eld. The energy levels are indicated by their quantum numbers (n; l). The con®nement energy is  ho0 ˆ 4 meV. (Redone after Chakraborty, Ref. [1111].)

B ˆ 2 T the amount of this splitting is only about 0.05 meV [1119]. This value is rather smaller than the energy differences between orbits in a quantum dot which are known to be typically around 1±2 meV. Next, we plot the full single-particle energy spectrum as a function of the magnetic ®eld strength in Fig. 138. The electron energy levels are assigned a composite FDL index mFD  …n; l†. The con®nement energy parameter used for this purpose was  ho0 ˆ 4 meV. As noted before (see Section 2.2.4), without the con®ning potential the energies of the negative l would be independent of l, but in its presence they increase with l. For very large magnetic ®eld (oc  o0 ) and for n ˆ 0; l < 0, we get the ideal 2D limit hoc , while for n ˆ 1; l  0; Enl ˆ 32 hoc, etc. States with l > 0 have been for the lowest LL, i.e., En;l ˆ 12  clearly seen to have much higher energies than for l < 0. Note that at zero magnetic ®eld and for n ˆ 0; l  0, the electron energy levels are successively separated by the con®nement energy ho0 (i.e., ho0 ), as expected. As discussed in Section 2.2.4, the transport measurements on E0;l ÿ E0;lÿ1 ˆ  quantum dots, speci®cally by the MIT group (see, e.g., Ref. [1119]), have been able to map these electron energy levels from the oscillations of the conductance as a function of magnetic ®eld. It would not be out of place to mention that apparently similar, but classical, energy level spectrum was noticed by Robnik [1672] in the search of a perimetric correction to the Landau diamagnetism for a ®nite-size, free-electron gas. 12.3.1.2. Fermi energy. Because each quantum level takes two electrons with opposite spin, the Fermi energy EF of a system of N non-interacting electrons at absolute zero temperature is equal to the energy of the …12 N†th level. As such, we can define the self-consistent determination of the Fermi energy in a

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349

Fig. 139. Fermi energy of the N-particle quantum dot as a function of the magnetic ®eld for the harmonic potential with con®nement energy  ho0 ˆ 3 meV. The dashed lines are the LL energies of an ideal 2D system.

quantum dot containing N electrons through the following expression: X N ˆ 2 y…EF ÿ En;l †;

(12.77)

n;l

where the summation is over both quantum numbers n and l, and En;l is given by Eq. (12.76). Since the electronic spectrum is quite complex, the Fermi energy cannot be expected to be a smooth function of the magnetic ®eld. Fig. 139 illustrates the Fermi energy as a function of the magnetic ®eld for quantum dot systems with different number of electrons, for the con®ning potential characterized by ho0 ˆ 3 meV. The dashed lines represent the LL energies of an ideal 2D system. As it is expected, the  convergence to the lowest LL energy is much faster for quantum dots with a smaller number of electrons. The reason being that, for lower quantum number l, the effect of the con®ning potential is much smaller. For suf®ciently strong magnetic ®elds (such that oc  o0 ) Enl ' …N ‡ 12†hoc , with N ˆ ‰n ‡ 12 …jlj ‡ l†Š†, the Fermi energy approaches 12 hoc , i.e., the energy of the lowest LL. 12.3.1.3. Optical transitions Ð selection rules. The transition probability …n; l†„…n0 ; l0 † is proportional to the square of the interaction energy [1126,1777]. Due to the interaction between the applied electric field of the EM radiation and the electric dipole moments of the electrons, the transitions between the electronic state cn;l and cn0 ;l0 are governed by the transition amplitude A…nl; n0 l0 † ˆ hcnl jr eiy jcn0 l0 i;

(12.78)

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where the electronic state cn;l is as given by Eq. (12.73). The associated oscillator strength is de®ned by 2m …En0 l0 ÿ Enl †jA…nl; n0 l0 †j2 : (12.79) h2  Making use of the single-particle eigenfunction, Eq. (12.73), and performing integration over y yields, from Eq. (12.78)  1=2  1=2 Z p n! n0 ! 0 0 0 0 dl0 ;l1 dz eÿz z…a‡a ‡1†=2 Lan …z†Lan0 …z†: (12.80) A…nl; n l † ˆ 2le 0 0 …n ‡ a†! …n ‡ a †! 0 0

fnln l ˆ

This clearly reveals that only transitions with Dl ˆ l0 ÿ l ˆ 1 are allowed. This gives us a rigorous selection rule on the possible values of the quantum number l. The integral in Eq. (12.80) can still be solved analytically. The ®nal result is p p p (12.81) A…nl; nl0 † ˆ 2le ‰ n ‡ a ‡ 1 dn0 ;n ÿ …1 ÿ dl;0 † n ‡ 1 dn0 ;n‡1 Šdl0 ;l1 ; which leads to the selection rule on the possible n ! n0 such that Dn ˆ 0; 1. The corresponding transition energies are given by DE ˆ 12  hO  12  h oc ;

(12.82)

where the ‡ …ÿ† sign corresponds to left (right) circular polarization. This implies that only two transition energies are possible. Note that with the increasing magnetic ®eld (o2c  o20 ), DE‡ approaches the cyclotron energy  hoc, while DEÿ decreases with the magnetic ®eld. The allowed transitions as a function of the magnetic ®eld are plotted in Fig. 140, for several values of the con®nement energies  ho0 . The solid, dashed, and dotted curves correspond to the values of ho0 ˆ 1:5, 2.5 and 3.5 meV, respectively. The dash-dotted line starting from zero magnetic ®eld  corresponds to  ho0 ˆ 0:0; in this case the transition is purely CR, and the slope of this line gives the ho0 6ˆ 0, both transition energies are possible. At B ˆ 0; DE ˆ ho0 electron effective mass m . When  is the only allowed transition. As B ! 1, the upper branch approaches towards hoc , and the lower branch towards zero, indicating, respectively, the transitions between two successive LLs and the transition within the same LL. Several magneto-optical experiments as discussed in Section 2.2.4 have, in fact, observed such optical transitions con®rming the existence of both modes of dispersion in the presence of the non-zero con®nement potential. 12.3.1.4. Straight-line representation of single-particle spectrum. Let us recall Eq. (12.76) to introduce hO, and l ˆ oc =O. Eq. (12.76) then transforms into dimensionless quantities E ˆ En;l = E ˆ …n ‡ 12 jlj ‡ 12† ‡ 12 ll:

(12.83)

This form of eigenenergies allows us to discuss the gradual transition from pure spatial quantization (o0 > 0; oc ˆ 0; l ˆ 0) to pure magnetic quantization (o0 ˆ 0; oc > 0; l ˆ 1). In this representation the single-particle energy spectrum consists of a set of straight lines, as illustrated in Fig. 141. Each state cn;l corresponds to one line with a slope given by 12 l. For l  0, the quantum number n determines the intersection with the vertical axis l ˆ 1. At l ˆ 0, the degeneracy is equal to E, while at l ˆ 1, we obtain the well-known macroscopic degeneracy of the LLs. Fig. 141 is limited to states of angular momentum l  10; otherwise there would be an in®nite number of lines (for an in®nite sample area) of increasing slope arising from each of the levels at l ˆ 1. Note that only (n; l ˆ 0) states are independent

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351

Fig. 140. Allowed magneto-optical transitions as a function of the magnetic ®eld for the harmonic potential, with three different values of con®nement energies. For reference, the dash-dotted line is the cyclotron energy  hoc.

Fig. 141. Straight-line representation of the single-particle energy spectrum according to Eq. (12.83), for l  10. Notice that following the state (n ˆ 0; l ˆ 1), crossing with states of (n ˆ 0; l < 0) occur at l ˆ …l ÿ 1†=…l ‡ 1†. (See also Bockelmann, Ref. [1239].)

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of l and represent the straight lines with zero slope. Following the state (n ˆ 0; l ˆ 1), crossings with states of (n ˆ 0; l < 0) occur at l ˆ …l ÿ 1†=…l ‡ 1† [1239]. Almost all the theoretical results on the quantum dots discussed in this section are representative of the single-particle behavior. As the dots contain more than one electron in most of the cases, the question of dealing with the effect of e±e interaction still remains open. The relevant work focusing on the many-particle physics in quantum dot systems with diverse theoretical machineries has been extensively discussed in Section 2.2.4. We next take up the issue of 2D arrays of quantum dots, where a few efforts have been made to study the collective excitations proposed to be observable through optical (Raman) spectroscopy. 12.3.2. Collective excitations in 2D arrays of quantum dots This section is devoted to discuss some explicit theoretical results on the collective excitations in 2D arrays of quantum dots. Several authors have embarked on investigating the collective excitation spectra in quantum dots, both with and without allowance for tunneling, in the recent past [1200,1209,1235,1280,1287], as extensively discussed in Section 2.2.4. There are two factors common in these theoretical studies: ®rst all these have focused on the quantum dots forming 2D square/ rectangular lattice, and second all were dealt with in the framework of the RPA. The uncommon factor in these studies has been the way in which the tunneling between the neighboring dots was accounted for (see the detailed discussion in Section 2.2.4). We are mainly interested here to discuss the theoretical results on the collective excitation spectrum in 2D array of quantum dots obtained by Que and Kirczenow [1200], Que [1235], and Kim and Ulloa [1287]. First theoretical work on collective excitations in a 2D array of quantum dots was done by Que and Kirczenow [1200]. They considered the square array of the dots to be in the x±y plane, and z-axis to be along the columns. Since the electron con®nement in the z direction was assumed to be much stronger than in the lateral plane, it was only necessary to consider the lowest energy level in the z motion. In the x and y directions, quantum dots were assumed to be periodically spaced and wave functions of electron states for different dots were disallowed to overlap. Thus, in x and y directions, tight-binding wave functions X X eikx nd fi …x ÿ nd†; ciky …y† ˆ eiky nd fj …y ÿ nd† (12.84) cikx …x† ˆ n

n

proved to be perfectly suitable for describing the system. In these equations d is the period of the square lattice, and i and j label the states in the x and y directions. The energy of the Lth level E…L† depends both on i and j. If the con®ning potential of a dot is parabolic, then one can write L ˆ i ‡ j. The ®rst excited energy level has L ˆ 1, and is doubly degenerate, corresponding to states (i ˆ 0; j ˆ 1) and (i ˆ 1; j ˆ 0). This degeneracy exists as long as the quantum dot has xy symmetry, irrespective of the detailed form of the con®ning potential. If the quantum dots do not have xy symmetry then the labels L and L0 should be replaced by ij, and i0 j0 , respectively. The authors [1200] hypothesized that although the quantum dots are electrically isolated from each other in the sense that electrons cannot transfer from one quantum dot to another, the long-range Coulomb force couples the quantum dots and this coupling can lead to collective excitations in the system. This is, in fact, analogous to the situations in multiwire superlattices and multilayer superlattices. Within the RPA, their ®nal formal result (see Eq. (24) in [1200]) in terms of the polarizability function PLL0 …o† Ð which is non-zero only when L 6ˆ L0 (in their notation L ˆ 0…1† corresponds to the ground (®rst-excited) level) Ð predicts two non-degenerate

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353

Fig. 142. Collective excitation spectrum for a square lattice of quantum dots. Each quantum dot is assumed to have two   energy levels. Parameters used are: E10 ˆ 25 meV, m ˆ 0:041m0 , d ˆ 1000 A, z0 ˆ 504 A, 12 …ns0 ÿ ns1 † ˆ 2  1010 cmÿ2 , and Eb ˆ 6:5. L and T represent the longitudinal and transverse modes, respectively. (After Que and Kirczenow, Ref. [1200].)

collective modes, except for some special cases. At ®rst, it seems surprising that although the ®rst excited single-particle energy level is doubly degenerate, the collective excitations associated with electron excitations from the ground state to the ®rst excited state have two non-degenerate modes. It was, however, realized that the lifting of the degeneracy is due to the fact that the system can support both longitudinal and transverse collective modes, and excitation energies for these two modes are different except for some special wave vectors. Fig. 142 illustrates the dispersions of collective excitations for a square lattice of quantum dots considered in Ref. [1200]. In producing these results, the authors assumed that the con®ning potential of the quantum dots takes the parabolic form V…x; y† ˆ …12 m †…E10 =h†2 …x2 ‡ y2 †, and that the z-component wave function is the ``particle in a box''-like, i.e., xz ˆ …2=z0 †1=2 sin…pz=z0 †. It is further remarked [1200] that these assumptions are appropriate for the systems studied by Reed et al. [1128]. The material parameters used in Fig. 142 are [1128]: E10 ˆ 25 meV (the average dielectric constant of air and GaAs) Eb ˆ 6:5, m ˆ 0:041m0 , d ˆ 100 nm, z0 ˆ 5 nm and 12 …ns0 ÿ ns1 † ˆ 2  1010 cmÿ2 . G, X, and M are the high symmetry points in the reciprocal space and correspond to …0; 0†, (p=d; 0†, and (p=d; p=d), respectively. Along the directions of GX and MG, we can clearly identify the longitudinal and transverse modes. These collective modes can have energies signi®cantly higher than the energylevel splittings in individual quantum dots. RS experiments should be able to detect the collective modes depicted in Fig. 142. However, the dispersions of the longitudinal and transverse modes and their splittings are very small, making the experimental resolutions of the two modes dif®cult. However, it was clearly demonstrated that while a single quantum dot containing only one electron can only have SPEs, an array of such quantum dots can also support collective excitations. In other words, a system made up of the building blocks with a dispersionless single-particle energy spectrum can have a dispersive collective energy spectrum with a lifted degeneracy. Que [1235], in an effort to extend their previous work [1200], took an important step by predicting the existence of a tunneling plasmon in 2D array of quantum dots, and emphasized the role played by the dimensionalities. For this purpose, he considered a rectangular array of quantum dots, with small enough spacing between dots in the x direction so that electrons can tunnel (weakly) between the dots, but no tunneling was allowed in the y direction. Thus there are three dimensionalities in the problem: 0D dot dimension, 1D tunneling motion, and 2D array dimension. It is fascinating to understand the roles played by each of these dimensionalities in the optical spectra of such a system. Que remarked

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that one can achieve 1D tunneling by using either elliptic shaped dots, or isotropic dots as long as the lattice constants dx and dy (in the x and y directions) are properly chosen. It is already known from his earlier work on the semiconductor superlattices [804], that the presence of tunneling can lead to a new branch of plasmon in a forbidden energy range of a non-tunneling superlattice system. This new branch of plasmon has since been dubbed as ``tunneling plasmon'' due to its dependence on tunneling. Que carried out a detailed analysis of the tunneling plasmon in the 2D array of quantum dots as described above to show that although the dots are 0D and the tunneling is 1D, the tunneling plasmon, in the longwavelength limit, exhibits the characteristic 2D behavior, i.e., a square-root dependence on the wave vector. The single-particle miniband energies in the system considered were expressed by [1235] Ea …k† ˆ Ea ÿ 2tax cos…kx dx †;

(12.85)

where tax is the hopping integral for the miniband labeled by a, and the rest of the symbols have their usual meanings. With the assumption of a weak tunneling, the tight-binding wave functions offer a good description of the electronic states. He also assumed that the individual dots have sx and sy re¯ection plane symmetries, so that it is possible to classify collective excitations in the system in terms of symmetry properties. It should be remarked that Que [1235] took into account the effect of tunneling only through the energy dispersion, but not in the wave function overlap between the dots. If only the lowest miniband is occupied, and if we neglect the coupling between the tunneling plasmon and interminiband modes, then the general (formal) condition, jEab;a0 b0 j ˆ 0, for determining the collective modes simpli®es to the form: (1 ÿ SP† ˆ 0, with S representing the structure factor and P the polarization function [1235]. In the long-wavelength limit S / qÿ1 , while P / q2x ; thus for a tunneling p plasmon, with qy ˆ 0, the plasmon energy displays a square-root dispersion: o / qx . The physical reason for this behavior is clearly the long-range Coulomb coupling among the quantum dots of the 2D array. This result signals that it can be very dangerous to interpret the optical spectra of quantum dot arrays simply in terms of the single-dot models, unlike the case of quantum wire systems. It is noteworthy that the square-root dispersion discussed above is valid only for the long-wavelength limit, which, in the system considered by Que, is de®ned by, say, qx  min…dxÿ1 ; dyÿ1 †. Away from this limit, the plasmon dispersion can behave quite differently. Fig. 143 illustrates the complete dispersion of the tunneling plasmon. For a wide range of wave vector along GX direction, for instance, the dispersion appears linear. The linear dispersion is, however, a common occurrence in periodic systems; it has been observed in multilayer superlattices [98] as well as in multiwire superlattices [905]. One can also notice an extremely anisotropic spectrum with ¯at

Fig. 143. Collective excitation spectrum of the tunneling plasmon in a rectangular array of quantum dots, with lowest miniband half-®lled. The parameters used are: dx ˆ 200 nm, dy ˆ 3dx , m ˆ 0:067m0 , tx ˆ 0:1 meV, Eb ˆ 6:9, D ˆ 5 meV, where D is the energy spacing in a parabolic model potential. (After Que, Ref. [1235].)

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355

dispersion along the XM direction. This ¯at dispersion along XM does not necessarily imply that the Coulomb coupling in the y direction is weak; it is simply due to the lack of tunneling in the y direction. The same reason holds for the disappearance of the tunneling plasmon along the GX0 direction. Que also remarked that if one uses a square lattice of anisotropic dots with tunneling in the x direction only, the spectrum is still characterized by a ¯at dispersion along XM, but a considerable dispersion along GX, MG, and X0 M directions. In his concluding remarks, Que pointed out that there is a tendency among some workers to neglect the Coulomb coupling between quantum dots, although the interdot Coulomb energy e2 =Er for typical parameters is not small. (For E (averaged between GaAs and air)ˆ6.9, and r ˆ 400 nm, the Coulomb energy is  1 meV, comparable to typical energy-level spacings in GaAs dots.) His analysis of the tunneling plasmon reveals that in the long-wavelength limit the tunneling plasmon displays a 2D behavior, indicating that interdot Coulomb coupling can make not only a qualitative, but also a quantitative difference. In a 2D quantum dot array with 1D tunneling, the tunneling plasmon was also shown to have the unusual property, namely, the oscillatory dependence of its energy on the electron density, as discussed in Section 2.2.4. Finally, we refer to the work by Kim and Ulloa [1287], where the tunneling in both (x and y) directions of a 2D array of quantum dots was allowed through energy dispersion as well as in the wave function overlap between the quantum dots. The energy of the ath state was expressed by the usual nearest-neighbor tight-binding energy bands Ea ˆ Eij ÿ Wxij cos…kx dx † ÿ Wyij cos…ky dy †;

(12.86)

where W ij are the half-band-widths in the respective directions. The phenomenological parameter g was introduced in the polarizability function Paa0 …o† in order to account for the effects of collisional broadening in the system; this also helps to regularize the behavior of Paa0 …o† near the poles. While their analytical results are valid for the 2D rectangular lattice, they con®ned their numerical work to a square lattice (dx ˆ dy ˆ d). Each dot was described by three energy levels (the higher two being degenerate) yielding three tunneling bands. Just as in the work by Que and Kirczenow [1200], they also assumed the Wannier functions to be given by those of a local parabolic potential characterized by the h. This Gaussian representation simpli®es the calculations considerably harmonic frequency o0 ˆ E10 = and was expected to give a proper description of the problem. Moreover, only the lowest band was considered to be partially ®lled. Fig. 144 illustrates the collective excitation spectrum in the non-tunneling limit for a 2D square array, along the principal symmetry directions for typical structure parameters. Note that no self-consistent correction in the ground state [1219] was included in producing Fig. 144. Absence of this correction as well as the interdot coupling considered in the system causes a small depolarization shift at the G point. This shift is known to vanish for a single parabolic dot case, due to the GKT [1127,1210]. Note the drastic changes in the dispersion of the collective modes as the lattice constant d is varied. These changes are qualitatively explained by the enhanced (diminishing) Coulomb ®elds at smaller (larger) lattice constants. Introducing tunneling in the system, as was done in Ref. [1287], produces a band of extra modes at low energies due to the tunneling degree of freedom. In order to shed light on the characteristics of this extra mode, as well as the expected interband modes, they performed calculations of the relative oscillator strength of the modes, which is proportional to the imaginary part of DDCF w ˆ w0 ‡ w0 Vw. The numerical results of the typical calculations for ÿIm w on o±q plane are illustrated in Fig. 145.

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Fig. 144. Collective excitation spectrum for a square array of quantum dots in the non-tunneling limit. Parameters used are:   ho0 ˆ 25 meV, m ˆ 0:014m0 , z0 ˆ 50 A, Eb ˆ 6:5, and ``dot-size'' ˆ … h=m o0 †1=2 ˆ 148 A. The dash-dotted, solid, E10 ˆ   and dashed curves correspond, respectively, to the period d ˆ 1000, 500, and 250 A. G, X, and M are the high symmetry points in the irreducible part of the ®rst Brilloiun zone. (After Kim and Ulloa, Ref. [1287].)

These plots show the full response function peaks corresponding to the collective modes of this tunneling-allowed quantum dot array. By changing the band-widths from zero to Wy00 ˆ Wy00 ˆ 0:1E10 and Wx10 ˆ Wy10 ˆ 0:15E10 (see Fig. 145(a)), it was noticed that the main interband-mode curves acquire strong slopes along the GX and XM directions. For even larger band-widths, Wx00 ˆ Wy00 ˆ 0:2E10 and Wx10 …ˆ Wx01 † ˆ Wy10 …ˆ Wy01 † ˆ 0:3E10 , (see Fig. 145(b)), these slopes are noticed to increase further. This strong q dependence is argued [1287] to be due to the single-particle level dispersion, as one would expect a weaker effective Coulomb interaction (and then smaller slope) as the carriers are allowed to spread laterally [1209]. Notice that the tunneling in the system makes the interband collective modes become more like a broad-band. This broadening was argued to be brought about by the admixture of the interdot tunneling with the interband single-particle-like transitions. Note that the main plasmon peak rides at frequencies higher than all the possible single-particle transitions and, in fact, carries most of the oscillator strength. Noticeable in Fig. 145 is the band of low-frequency modes in the range o  0:5o0 . The characteristic frequencies of these modes increases weakly as a function of q; one would expect that for p larger tunneling this variation would become  q dependence [1235]. This is qualitatively observed in Fig. 145(b), where the larger tunneling enhances the oscillator strength of this low-energy tunneling plasmon. 13. Magnetic-®eld effects in some variants of quantum structures The historical survey of electronic, optical, and transport properties in the systems of reduced dimensionality (Section 2) leads one to believe that there is an ever increasing interest in the investigation of ultrasmall, laterally microstructured, originally 2DEG. Because of the reduced dimensionality and size, quantum con®nement and novel physical phenomena have been predicted and observed. One ultimate goal is the realization of arti®cial atoms in the quantum dot structures, which

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Fig. 145. Plot of ÿIm w…q; o† on o±q plane; notice the logarithmic scale on the vertical axis. Attention is drawn to both intraband and interband set of modes with non-vanishing oscillator strength for low and high energy, respectively. Parameters  used are as in Fig. 144; d ˆ 500 A, Z ˆ 0:02E10 , and the band-widths are: (a) top panel, Wx00 ˆ Wy00 ˆ 0:1E10 and Wx10 ˆ 10 Wy ˆ 0:15E10 ; (b) bottom panel, Wx00 ˆ Wy00 ˆ 0:2E10 and Wx10 ˆ Wy10 ˆ 0:3E10 . (After Kim and Ulloa, Ref. [1287].)

contain only a small number of electrons on discrete energy levels. A reversed structure with respect to dots is that of antidots where ``holes'' are ``punched'' into 2DEG. There are already numerous investigations on the electronic and transport properties of both quantum dots and antidots (see Section 2.2.4), in particular, in the search of EMPs (Section 2.3.1), commensurability phenomena in

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electrically and magnetically modulated 2DEG (Section 2.3.2), and Hofstadter's butter¯y spectrum (Section 2.3.3). This section is devoted to discuss some illustrative examples on such phenomena observed in diverse physically accessible situations. 13.1. EMPs in ®nite-size geometries The 2DEG on liquid helium has played an important role in clarifying the collective excitations (plasmons) of 2DEG in general. The ®rst experimental veri®cation of the gapless plasmon dispersion in 2D was made in this system [96], and later, an unexpected EMP excitation was discovered [1361,1362]. However, it is now realized that the ®rst observation of an EMP was actually made for small 2DEG (radius R  1:4 mm) in GaAs±Ga1ÿx Alx As heterojunction [1360]. Subsequent investigations on the EMP, both theoretical and experimental, have already made an exciting history, regarding the role played by the boundaries of the small systems in understanding their transport properties associated with, e.g., the quantum Hall effects (Section 2.3.1). The EMP is the 2D analog of a surface plasmon in 3D systems [1398±1400], with the peculiarity that it propagates around the edge of the sample at a frequency which is much smaller than the cyclotron frequency and which decreases with increasing magnetic ®eld. It has been suggested [1339] that the EMP response of the 2DEG may be an effective probe of edge effects in the quantum Hall effects especially if the plasmon-propagation direction becomes known. Investigation of plasmon excitations in bounded 2DES ranging from classical disks to strongly con®ned quantum dots has mostly been restricted to the simple case of circular disks (see Section 2.3.1). This geometry together with the approximation of a parabolic con®nement potential leads to frequencies of the allowed dipole excitations of the form: o ˆ …o2p ‡ 14 o2c †1=2  12 oc ;

(13.1)

where oc is the usual cyclotron frequency and op the plasma frequency of the con®ned system (at zero magnetic ®eld). For a classical disk approximated by a thin oblate spheroid with radius R, the latter is de®ned as o2p ˆ …3p e2 N=4E m R3 †. In a perpendicular magnetic ®eld the plasma resonance splits into two branches with upper branch approximating oc , whereas the lower one eventually decreases in frequency as Bÿ1 . This mode is connected with the presence of the boundary (or ®nite-size) of the system and hence has been termed the EMP. Note that physical situation discussed above and Eq. (13.1) is a special case of a general situation of circular disks in a rectangular array, where the magnetic-®eld dispersion is given by [1373] o2 ˆ 12 …o2x ‡ o2y ‡ o2c †  ‰14 …o2x ‡ o2y ‡ o2c †2 ÿ o2x o2y Š1=2

(13.2)

in a Coulomb-coupled, but electrically insulated, array of electron disks, where the non-interacting plasma frequency op can be interpolated from the measured ox and oy as the weighted mean, o2p ˆ …xx o2y ÿ xy o2x †=…xx ÿ xy †, where xi is the component of ``lattice tensor'' [1373]. Eq. (13.1) is a special case of Eq. (13.2) for o2p ˆ o2x ˆ o2y. In the general situation, Eq. (13.2), Kotthaus and coworkers [1373] have observed a gap in the magnetic-®eld dispersion at B ˆ 0, and a strong polarization dependence of the intensity at lower B. Both features manifest local-®eld corrections in the disk lattice. If the lattice is anisotropic, the Lorentz ®eld associated with the plasma excitation depends on the orientation of the polarization with respect to the axes of the disk lattice, so that two fundamental plasma frequencies occur at B ˆ 0. Their splitting is a measure of the coupling strength. This is

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359

Fig. 146. Resonance positions of the rectangular disk array; the parameters are as given in the text. The gap at zero magnetic ®eld is a measure of the intra-disk Coulomb interaction. The solid lines re¯ect the magnetic-®eld dispersion of the dipole model, Eq. (13.2), with B ˆ 0 resonance frequencies and m ˆ 0:068m0 as ®t parameters. Open symbols stand for higher harmonics. (After Dahl, Kotthaus, Nickel, and Schlapp, Ref. [1373].)

qualitatively different from the response of non-interacting disks or disks in a square array, which exhibit only a single isotropic fundamental plasma mode at B ˆ 0. It is the lowering of the symmetry of the rectangular lattice that enabled the experiment to unambiguously demonstrate and quantify the interdisk interaction. Fig. 146 displays the measured resonance positions [1373] together with a ®t of the magnetic-®eld dispersion according to Eq. (13.2). For experimental values at B ˆ 0; ox =2p ˆ 95:6 GHz, and oy =2p ˆ 112:3 GHz. If the disks are modeled as thin oblate spheroids [1360], one obtains ox =2p ˆ 105 GHz and oy =2p ˆ 119 GHz, using m ˆ 0:068m0 , E ˆ …EGaAs ‡ Eair †=2 ˆ 6:8, and ns ˆ 2:5 1011 cmÿ2 . At higher magnetic ®elds, the resonance positions of the lower branch deviate substantially from the theoretical dispersion. Since the model in Ref. [1373] essentially assumes a parabolic con®ning potential, this difference is, in some sense, a measure of anharmonicity. Here a strictly 2D calculation that predicts ¯atter dispersion leads to a better agreement with the experiment [1373]. A further indication of anharmonic con®nement is the appearance of higher modes (open symbols). Such modes have most comprehensively been studied on macroscopic electron disks on the surface of liquid helium [1361,1362]. Here only resonances with larger radial mode index have been observed, modes with azimuthal index other than 1 do not respond to an external dipole ®eld. The measurements made in [1373] lead to conclude that no spontaneous polarization can occur in arrays of circular disks, even for ®nite wave vector or different lattice type, since maxjxi j depends only relatively weakly on these parameters. It was thus shown how plasma oscillations in disk arrays could be determined by disk and lattice geometry. The experiment was concluded with the remark that, with circular disks in a rectangular array, one is allowed to separate intradisk and interdisk Coulomb interaction and thus to

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measure the latter. Its strength is given by the splitting of the plasma resonance at zero magnetic ®eld and is well described bydipole±dipole interaction. Theoretical studies on the EMPs on the bounded 2DES have mostly been restricted to the simply connected geometries such as, e.g., dots or disks (see Section 2.3.1). We recall here a corresponding investigation in the ring geometry where the existence of a new EMP, due to the propagation of density ¯uctuations on the inner boundary of the sample, was proposed by Proetto [1407]. He considered a classical 2DEG con®ned in a ring of inner (outer) radius a…b†. The ring contains a static positive and uniform background with areal charge-density en0 (e > 0) and a compressible electron ¯uid with areal charge-density ÿe…n0 ‡ nind †, where nind is the self-induced density (nind  n0 ). The system is subjected to a perpendicular magnetic ®eld ~ B k z direction and surrounded by dielectric material with dielectric constant E1 (E2 ) for z > 0 (z < 0). Furthermore, the system was considered to be grounded by two metallic electrodes above (z ˆ ‡h) and below (z ˆ ÿh) the 2DEG. He adopted the hydrodynamic approach to study the magnetoplasma excitations of such an electron ¯uid con®ned to a ring. The theoretical framework is based on the fact that rotational invariance around the z-axis and the translational invariance in time allow all the unknown quantities to have an angular and time dependence of the form exp‰i…ly ÿ ot†Š, where l is an integer and o is the angular frequency of the normal mode. In the fully screened limit jhj ! 0, the self-induced density and potential are related by a local relation [1407]. His ®nal formal results (see Eqs. (9) and (10) in Ref. [1407]) are not seen to be invariant with a change of ‡l and ÿl, if oc 6ˆ 0; this means that while the solutions corresponding to the normal modes with ‡l and ÿl are degenerate if oc ˆ 0; switching on the magnetic ®eld B gives rise to a splitting between the two modes. In real samples, however, any deviation from the cylindrical geometry will give a coupling between the ‡l and ÿl modes, and consequently a splitting even at B ˆ 0. Another important concern is the number of the EMPs accommodating in the ring in a strong magnetic ®eld regime oc =O0  1 for a given l, where O0 ˆ ‰4p e2 n0 h=…m a2 …E1 ‡ E2 ††Š1=2 ˆ cp =a, with cp as the screened plasma velocity. By de®nition the existence of the EMP implies o < oc ; in view of this, the asymptotic expansion of the modi®ed Bessel function (in Eq. (10) in Ref. [1407]) yields two EMPs:     oc 1 ; (13.3)  1 ! lO0 ‡ O l > 0 : o‡ oc O0     oc a 1 ; (13.4) l < 0 : oÿ  1 ! ÿl O0 ‡ O b oc O0 which could be cast in a more transparent form: o‡  ‡lcp =a, and oÿ  ÿlcp =b. This makes it crystal clear that the ‡l (ÿl) mode o‡ (oÿ ) corresponds to a density perturbation that circulates around the inner (outer) edge in the counterclockwise (clockwise) sense with the screened plasma velocity cp. The former (latter) mode is the analog of the antidot (dot) EMP; it is the latter that has been relatively extensively discussed in the literature, and its main feature is that its frequency decreases as the magnetic ®eld increases, reaching asymptotically the limiting value speci®ed by Eq. (13.4). Note that no bona®de solution is found for l ˆ 0, which is consistent with the fact that for these radial modes the angular component of the velocity is zero when B ˆ 0 [1407]. Turning to the illustrative examples, Fig. 147(a) displays the antidot EMP frequencies o‡ (dashed curves) and the dot EMP frequencies oÿ (solid curves) for several values of l, and the aspect ratio b=a ˆ 2; the dotted line corresponds to the cyclotron frequencies oc . Note that the plots stand for only lowest radial quantum number (no nodes in the density in the radial direction) that gives rise to the

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Fig. 147. (a) EMP frequencies as a function of magnetic ®eld for a ring with aspect ratio b=a ˆ 2. The full line (l < 0) corresponds to the dot EMP, the dashed line (l > 0) to the antidot EMP, and the dotted line is the cyclotron frequency. (b) The same as in (a), but for b=a ˆ 5. (After Proetto, Ref. [1407].)

EMP; otherwise there are in®nite number of solutions corresponding to different values of radial quantum number. As stated above, the degeneracy between the l modes is lifted by the magnetic ®eld, and we obtain a dot EMP whose frequency decreases with magnetic ®eld, and an antidot EMP whose frequency increases with magnetic ®eld, going asymptotically to the limiting value speci®ed by Eq. (13.3). As expected, for a given value of jlj and B, in the strong-®eld regime where the EMPs are well established, the dot EMP is closer to its limiting value ÿlO0 …a=b† than the antidot EMP to its corresponding value lO0 ; this is related to the fact that for a given value of jlj, ®nite-size effects are more important for the dot EMP than for the antidot EMP. To make this point clearer, we have shown in Fig. 147(b) the corresponding results, but for larger aspect ratio b=a ˆ 5. It is clearly noticeable that the dot EMP crosses the line o ˆ oc (which gives a rough estimate of the threshold magnetic ®eld at which

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the EMP begins to develop) much before the antidot EMP (compare, for instance, the jlj ˆ 5 case). It is also noteworthy that for l  1, the threshold magnetic ®eld at which the antidot EMP begins to develop becomes independent of b and is, in fact, given by …l…l ÿ 1††1=2 O0 [1407]. Theoretical prediction of this new, antidot EMP [1407] was soon followed by the experimental detection of this mode for a ring-shaped 2DEG in etched GaAs±Ga1ÿx Alx As heterojunctions [1378] and on the surface of liquid helium [1386]. The agreement between the experiment and theory was, however, only qualitative, since the frequency of the observed antidot EMP increases for small B, but decreases for large B, while theory predicted saturation to a constant frequency at large B. Besides, some of the observed BMP modes, whose squared frequency increases linearly with o2c , were seen to start with a negative dispersion, which again was missing in the calculation. These discrepancies between theory and experiment were soon realized [1408] to arise from the simplifying assumption made in Ref. [1407] that the rings were fully screened by metallic gates Ð the so-called LCA. Theoretical results obtained after relaxing the LCA [1408] were found to be in excellent agreement with the experimental results on etched GaAs±Ga1ÿx Alx As heterojunction [1378]. Such theoretical results, with smaller aspect ratios are shown in Fig. 148; b ˆ 25 mm, a ˆ 6 mm (Fig. 148(a)), and b ˆ 26 mm, a ˆ 15 mm (Fig. 148(b)). As before, full (dashed) curves correspond to l ˆ ‡1 (ÿ1) magnetoplasmons; the discrete points are the experimental data from Ref. [1378]. It is noteworthy that the remarkable agreement between theory and experiment is seen for the values of a; b, and n0 quoted in the experiment, and consequently there was no adjustable parameter in the theory. Both experimental and theoretical results are easily understood considering, e.g., this rather wide ring as a perturbed dot with the same outer radius (see Fig. 148(a)). A new feature of this ring is that the o1‡ mode shows, at low magnetic ®elds, a negative magnetodispersion, followed by a regime where the frequency tends asymptotically towards oc , as corresponds to the BMP. This negative dispersion is a consequence of the softening of the o1‡ mode, which as a result of the large size of the central hole starts at smaller magnetic ®elds. Fig. 148(b) depicts the results, both theoretical and experimental, for a rather narrow ring. Due to the small aspect ratio b=a ˆ 1:7, the rich set of resonances is much more attenuated; e.g., the o0ÿ and o0‡ modes are quite close, being almost degenerate at low and high magnetic ®elds. While the validity of the earlier assumption (of considering the wide rings as perturbed disks) is not quite obvious for this narrow ring, the qualitative dispersion of BMP and EMP is similar to the previous case. Equivalently, the zero-®eld excitations for this narrow ring can also approximately be described as 1D magnetoplasmons with wave vector q ˆ 2=d, where d is the mean diameter of the ring [1378]. In summary, the magnetoplasma excitations of a 2DEG con®ned to a ring geometry exhibit intricate features which are determined by the interplay between the geometry and the magnetic ®eld. The spectrum decomposes into a set of high-frequency BMP modes and two low-frequency EMPs which can unambiguously be connected with the two boundaries of this ring. It should be remarked that we have recently embarked [1673] on a systematic investigation of the EMPs in the cylindrical and ring geometries in the framework of a Green function (or response function) approach ®rst introduced by Dobrzynski [799,1618,1759,1794±1796] for the strati®ed structures, unlike the usual hydrodynamic scheme adopted by almost all the authors to date. We study the collective excitations (plasmons and magnetoplasmons) in the presence of an axial magnetic ®eld. New collective excitations, whose frequencies are found to be functions of the magnetic ®elds as well as the aspect ratios, are predicted. We consider both low- and high-®eld regimes, and compare our results with the available theoretical (obtained within the different schemes) and experimental results. The extensive numerical computation (which is still awaited for the ®nal conclusions to be drawn) leads

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Fig. 148. (a) Resonance frequencies for a ring with b ˆ 25 mm, and a ˆ 6 mm. Full (dashed) lines correspond to l ˆ ‡1…ÿ1† EMPs. The discrete points are experimental data from Ref. [1378]. (b) The same as in (a), but for b ˆ 26 mm, and a ˆ 15 mm. Only the lowest four low-lying doublets are shown, corresponding to an increasing number of radial nodes. (After Reboredo and Proetto, Ref. [1408].)

us to speculate that the EMP spectrum in such ®nite geometries ismore intricate than it has been believed so far. It is argued that the EMPs in such structures can be probed by the inter-edge magnetoplasma spectroscopy [1383,1392]. 13.2. Commensurability oscillations in periodically modulated structures The magnetotransport in the 2DEG subjected to periodic electric and magnetic modulation has attracted tremendous experimental and theoretical attention during the past decade (see Section 2.3.2

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for extensive discussion on the status of the theories and experiments emerging on the subject). Even though some theoretical predictions on the possibility of observing novel physical phenomena in 2DEG under periodic potential had already been made [1430,1432], the subject gained an intense momentum after the experimental discovery of the so-called Weiss oscillations in 1989 [1433]. Novel oscillations of the magnetoresistivity tensor rmn were observed at low magnetic ®elds B, distinctly different in period and temperature dependence from the usual SdH oscillations at higher B. These novel oscillations re¯ect the interesting comensurability between two length scales: the cyclotron radius at the Fermi level Rc ˆ …kF l2c † and the period of the imposed potential modulation (see, for instance, Eq. (2.9)). The purpose of this section is to discuss some explicit theoretical results which are at the heart of understanding the commensurability problem associated with the Weiss oscillations. The key feature connected with the novel Weiss oscillations is that the modulation potential lifts the degeneracy of the LLs and gives them a width ( jVn j) of ®nite extent. Since Vn is a function of Laguerre polynomial Ln , which is an oscillatory function of its index n, width of the Landau bands oscillates. Formally, this is due to the properties of the Laguerre polynomials. Physically, it re¯ects the fact that with increasing n, the spatial extent of the eigenfunction increases ( 2Rn ˆ 2lc …2n ‡ 1†1=2 ), and the latter effectively senses an average of the periodic modulation potential (of period a) over an interval of width 2Rn . This can most easily be understood within the ®rst-order perturbation theory with respect to V0 , which yields the energy spectrum speci®ed by Eq. (2.11). Let us ®rst see how the LLs for an unmodulated 2DEG turn into the Landau bands in the presence of a small but ®nite modulation potential. For this purpose, we cast Eq. (2.11) in the form: hoc ‡ E exp…ÿ 12 u†Ln …u†; En ˆ …n ‡ 12†

(13.5)

where E ˆ V0 cos…Kx0 † and u ˆ 12 K 2 l2 ; the rest of the symbols are the same as de®ned before (see Section 2.3.2). Fig. 149 illustrates the ranges of allowed energy values as dark areas for the ®rst ten bands (n ˆ 0; 1; 2; . . . ; 9), for jEj  0:2 and a…ˆ 2p=K† ˆ 200 nm. The ``¯at-band'' energies (i.e., the energies of the cyclotron orbits with radii Rc satisfying the commensurability condition, Eq. (2.9)) speci®ed by Em ˆ 18 m o2c a2 …m ‡ 34†2

(13.6)

are indicated by the dashed parabolas for m ˆ 1; 2, and 3. They intersect the Landau fan at the zeros of cos ‰2…nu†1=2 ÿ p=4Š, at the ¯at-band condition. Such a broadening of the LLs into Landau bands as depicted in Fig. 149 is attributed solely to the ®niteness of the periodic modulation potential. Next, we plot the energy bands as a function of Kx0 in Fig. 150, for V0 ˆ 1:5 meV, a ˆ 100 nm, and B ˆ 0:5 T. The ¯at-band energies Em are indicated for m ˆ 1; 2; . . . ; 4 as dashed horizontal lines. The origin of the band-width oscillations is that the wave functions, having a spatial extent of approximately 2Rn , sense effectively the average value of the periodic potential over an interval of length 2Rn . This gives rise to oscillations in the band-width, which in turn also leads to oscillations in the DOS. The DOS D…E†, with t ˆ Kx0 , takes the form: 1 Z p hoc X  dt d…E ÿ En;t †; (13.7) D…E† ˆ D0 2p nˆ0 0 where D0 ˆ m =p h2 is the DOS of a 2DEG at zero magnetic ®eld. The early magnetocapacitance measurements by Weiss et al. [1436] have already established directly the oscillatory broadening of the

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Fig. 149. Sketch of the modulation-broadened Landau fan diagram, calculated from Eq. (13.5) with jEj  0:2 meV and a ˆ 2p=K ˆ 200 nm. Allowed energy regions in the Landau bands with n ˆ 0; 1; 2; . . . ; 9 are darkened. The approximate ¯at band energies Em , calculated from the asymptotic form, Eq. (13.6), are shown for m ˆ 1; 2; 3 as dashed lines. (Redone after Pfannkuche and Gerhardts, Ref. [1493].)

DOS solely due to a 1D modulation potential. The band-width (of the Landau bands) becomes minimum for the nth LL, if the condition (2.9) is satis®ed for cyclotron radius corresponding to this level, Rc ˆ Rn ˆ lc …2n ‡ 1†1=2 . This minimum band-width is known [1481] to be accompanied by a maximum height of the DOS peak contributed by this LL. In practical systems, there will always be some broadening due to the presence of inevitable impurities or imperfections. If this is accounted for, then there will be a need to compare the relative effects of the two broadenings: the modulation-induced broadening and the collision-induced broadening. It has been noted that if the modulation-induced band-width becomes larger than the collision-induced one, a double-peak structure is resolved [1481]. On the contrary, if the collisioninduced broadening is so large that the DOS peaks start to overlap, in addition to the peak heights also the minimum values of the DOS between the peaks show an oscillatory modulation owing to the bandwidth oscillations. The amplitude oscillation of the DOS related to this band-width oscillation was what was observed in the magnetocapacitance experiments [1436]. Another evidence of the quantum origin of the phenomenon is the fact that at lower temperatures the Weiss oscillations occur as an amplitude modulation of the SdH oscillations [1455,1458]. As further consequences of these band-width oscillations, Weiss type oscillations have been predicted for the collective magnetoplasma excitations [1478,1484,1785] and for thermomagnetic transport coef®cients [1483]. The Weiss oscillations have also been observed in magnetoresistance experiments on samples with microstructured gates, where the 1D modulation strength could be tuned by the gate voltage (see, e.g., Refs. [1435,1444]). It has been noticed that if the modulation amplitude becomes too large, the Weiss oscillations disappear [1444].

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Fig. 150. Theoretical energy spectrum versus (normalized) center coordinate x0 for B ˆ 0:5 T, V0 ˆ 1:5 meV, a ˆ 100 nm, and material parameters of GaAs. Solid lines: exact; dashed lines: ¯at-band approximation of Eq. (13.5). (Redone after Gerhardts et al., Ref. [1434].)

The notion of the oscillatory behavior of any physical property in a 2DEG subject to a periodic potential modulation, in one or both directions, and a uniform perpendicular magnetic ®eld is based on the oscillatory Landau band-widths. This concept is the most important that lies at the heart of the Weiss oscillations in the magnetoresistance tensor rmn, or other Weiss-type oscillations predicted or observed in the similar situation. A direct calculation of the band-widths as a function of the magnetic ®eld is therefore highly desirable. The width of these broadened (due to potential modulation) LLs is given by W ˆ 2jVn j ˆ 2V0 exp…ÿ 12 u†jLn …u†j

(13.8)

with u ˆ 12 K 2 l2c , and is plotted in Fig. 151 for n ˆ nF, where nF ˆ EF =hoc is the LL index at the Fermi energy. Because nF is taken as an integer, the band-width (the solid curve) exhibits a step each time a new LL moves through the Fermi level. For the low magnetic ®elds, which are usually of interest for the system under consideration, we may estimate the band index at the Fermi level as hoc ˆ 12 kF2 l2c , so that the cyclotron radius RnF ˆ kF l2c results. Moreover, in order to make nF ‡ 12 ˆ EF = an estimate of the minima and maxima of the band-width, we take the limit of large band-index n for the Laguerre polynomial to write [1671] W ' …p2 nu†ÿ1=4 cos…2…nu†1=2 ÿ 14† ‡ O…1=n3=4 †:

(13.9)

This we have already analyzed (see Section 2.3.2) to state that the commensurability condition, Eq. (2.9), assumes the form 2Rc =a ˆ …j ‡ f†, where f ˆ 14 (34) for the maximum (minimum) band-

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Fig. 151. Theoretical band-width of the LLs at the Fermi energy as a function of magnetic ®eld. Solid curve: exact, using Eq. (13.8); dashed curve: large-n approximation, using Eq. (13.9). Note that our approximate results (the dashed curve) are different from the corresponding ones in Ref. [1491]. The parameters used are as listed in the ®gure. (Redone after Peeters and Vasilopoulos, Ref. [1491].)

width; the minimum band-width condition is also often referred to as the ¯at-band condition. The  physical parameters used in Fig. 151 are: V0 ˆ 1:0 meV, a ˆ 3500 A , and ns ˆ 3  1011 cmÿ2 . The magnetic ®eld, from the commensurability condition, can be expressed as B ˆ 2…hc=ae†…2pns †1=2 = … j ‡ f†. We are then led to express the position of the extreme in terms of B (in units of Tesla) as B…T† ˆ 0:5167=… j ‡ f†, where j in the foregoing discussion is an integer including zero. This results into the magnetic ®eld values (i) for maximum band-width: B…T† ˆ 2:067; 0:413; 0:229; 0:159; 0:121; 0:098; 0:082; 0:071; 0:062; 0:055; . . .; and (ii) for minimum band-width: B…T† ˆ 0:689; 0:295; 0:188; 0:138; 0:109; 0:089; 0:076; 0:066; 0:059; 0:053; . . . . These values are seen to agree well with the position of the maxima and minima in the band-width depicted in Fig. 151. The asymptotic expression given by Eq. (13.9) is also numerically calculated and plotted in Fig. 151 by the dashed curves, and is seen to be surprisingly very close to the exact result (the solid curves) based on Eq. (13.8). It is noteworthy that a good convincing picture of the above resonance condition can be presented by giving a classical description in real space. This was nicely done by Peeters and Vasilopoulos [1491]; they derived an asymptotic expression for the increment in the energy of the cyclotron motion due to the periodic potential. Their asymptotic expression was also seen to reproduce similar condition of maximum and minimum band-width as stated above, and turned out to approximate the exact results very well. Fig. 151 presents a crystal-clear evidence that the band-width of the modulation-broadened LLs is an oscillatory function of the magnetic ®eld. This oscillation of the Landau band-width is the origin of the observed Weiss oscillations of the resistivity tensor, which are periodic in Bÿ1 . Such Weiss-type

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oscillations have lately been found in numerous electrical, thermal, and transport properties, including collective magnetoplasma modes, in electrically modulated structures, where the band-width forms an explicit or implicit part of the problem (see, e.g., Refs. [1440±1475] and [1476±1511,1785]). Here we were aimed at giving a ¯avor to the subject that has added an interesting dimension to the already ongoing excitement, both fundamental and practical, in the systems of reduced dimensionality. Next, we sketch brie¯y the ®nding of Horing and coworkers [1478] that the intra-Landau-band magnetoplasma spectrum, speci®ed by X 8m oc e2 ~2 ˆ sin2 …px00 =a† jVn j…1 ÿ D2n †1=2 y…1 ÿ Dn †; (13.10) o 3 E0 p h qk n q where qk ˆ q2x ‡ q2y , x00 ˆ ÿqy l2c , Dn ˆ j…EF ÿ En †=Vn j, and y…x† is the Heaviside step function, exhibits novel commensurability oscillations in Bÿ1 for periodically density-modulated 2DEG, and that the origin of such oscillations is the same as in the Weiss oscillations of magnetoresistance. The magnetoplasma spectrum based on Eq. (13.10) is shown graphically in Fig. 152, using the parameters of the original experiments on the Weiss oscillations [1433±1435]. The modulation-induced oscillations are clearly evident, superposed on the sharp SdH oscillations. Also shown in the ®gure (inset) are two modes which are the exact solutions of 1ˆ

o2p …o2 ÿ o2c †

‡

~2 o ; o2

(13.11)

where op ˆ 2pn0 e2 q=E0 m is the ordinary 2D plasma frequency, including the coupling between the inter-Landau-band and the intra-Landau-band modes. It is seen that the former has superimposed on it the SdH oscillations, while the latter appears with further reduced amplitude, both resulting from the coupling. The two modes are nevertheless well resolved, separated by a ®nite gap.

Fig. 152. Intra-Landau-band frequency as a function of the inverse magnetic ®eld. Parameters used are: qx ˆ 0, qy ˆ 0:01kF , n0 ˆ 3:16  1011 cmÿ2 , a ˆ 382 nm, V0 ˆ 1:0 meV, and material parameters for GaAS. The arrows marked `1' and `2' correspond to the values of j ˆ 1 and j ˆ 2, respectively, in Eq. (2.9), with phase shifts of f ˆ ÿ0:22 for the former and f ˆ ÿ0:21 for the latter. Inset: the two roots of Eq. (13.11). (After Cui, Fessatidis, and Horing, Ref. [1478].)

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A closer analytical diagnosis of Eq. (13.10) reveals the origins of the two types of oscillations. With oc > jVn j, y-function vanishes for all but the highest occupied Landau band, corresponding to the h band index, say, N. It is argued [1478] that the sum over n is trivial and the plasma frequency is simply ~  jVn j1=2 …1 ÿ D2N †1=4 y…1 ÿ DN †. The formal function responsible for the sharp SdH-type given as  ho oscillations is the function y…1 ÿ DN †, which jumps periodically from zero (when the Fermi level is above the highest occupied Landau band) to unity (when the Fermi level is contained within the highest hkF2 . The periodic modulation of the amplitude occupied Landau band), with a period DSdH …c=B† ˆ e= of SdH-type oscillations is largely a consequence of the oscillatory nature of the function jVn j1=2 (see above), which is known to exhibit commensurability oscillations governed by Eq. (2.9). As such, it is not surprising that the period of the amplitude modulation in Fig. 152 is exactly the same as that predicted with Eq. (2.9), which is D…c=B† ˆ ea=2 hkF , using Rc ˆ kF l2c at the Fermi level. We recall brie¯y the supplementary remark made by Zhang [1484] on the work by Horing and ~ vanishes with coworkers [1478]. According to Ref. [1478], the intra-Landau-band frequency o ~ Zhang pointed modulation potential as jVn j1=2 and there are ®nite gaps between the Landau bands in o. out that while the former is correct within the perturbation scheme, the latter is qualitatively wrong. He ~ are not meaningful; stating that if one treats the Fermi energy selfadded that the gaps appearing in o consistently, the spectrum should be gapless regardless of how small the modulation potential is. ~ is non-zero everywhere (he states Zhang's analysis is as follows. For  hoc < 2jVn j, it is clear that o ~ that it can be shown that even for  hoc  2jVn j, o only contains discrete zeros but has no ®nite gaps). He further stressed that in the absence of other scatterers and when hoc > 2jVn j, the DOS contains ®nite gaps between Landau bands, and only the highest occupied Landau band (with band-index, say, ~ Moreover, he argues that B-dependent Fermi energy can only lie within the Landau N) contributes to o. ~ band except for integer ®lling factors. Then calculating the factor of …1 ÿ D2n †1=2 led him to state that o is ®nite and only has discrete zeros when the ®lling factor is an integer or VN  0; for hoc < 2jVn j, the ~ can be lifted. His plot of o ~ versus Bÿ1 is illustrated in Fig. 153 for two values of V0 (i) zeros of o V0 ˆ 0:1 meV and (ii) V0 ˆ 1:0 meV. The case (i) corresponds to hoc > 2jVn j for the whole regime of ~ has periodic zeros at integer ®lling factors. The case (ii) magnetic ®eld (1 < 1=B…Tÿ1 † < 4). The o

Fig. 153. The same as in Fig. 152. The arrows indicate the zeros of VN . The top curve: V0 ˆ 0:1 meV this curve is shifted by 0.35 meV. Bottom curve: V0 ˆ 1:0 meV. (After Zhang, Ref. [1484,1785].)

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~ only has zeros in the regime where hoc > 2jVn j. For 2 < corresponds to the case when o hoc < 2jVn j, and the Landau bands are overlapped near the Fermi energy. In this case 1=B…Tÿ1 † < 3:5,  ~ The zeros of o ~ are lifted; the minima of o ~ in this regime more Landau bands will contribute to o. simply re¯ect the degree of overlap between the Landau bands. In view of this, Zhang concludes that ~ the ®nite gaps between the Landau bands and the sharp cut in the low edge of the Landau band in the o spectrum presented in Ref. [1478] should not exist if the Fermi energy is treated properly. It should be pointed out that the foregoing remarks of Zhang were counteracted by Horing and coworkers [1484] agreeing only partially with him (see the brief discussion in Ref. [1484,1785]). The Weiss-type oscillations have also been predicted and observed in the 2DEG subject to the periodic magnetic-®eld modulation [1512±1561,1786]. The novel oscillations in this case, although have a similar origin as those found for a periodic potential modulation, are out of phase and have a much larger amplitude (see Section 2.3.2 for further extensive discussion). 13.3. Hofstadter's butter¯y spectrum The effects of a magnetic ®eld on Bloch electrons in the quantum regime have been extensive since the early works of Landau and Peierls, and proved to be a fascinating problem with rich physics. An indepth understanding has led to deep insights in the physics of electrons in the conventional as well as non-conventional condensed systems. Magnetic quantization is responsible for Fock±Darwin spectrum, Landau diamagnetism, de Haas±van Alphen effect, SdH effect, CR, and more recently, the quantum Hall effect in 2D systems, etc. It is becoming known that geometric, topological, and algebraic structures hidden in it are playing an interesting fundamental role in the problem of 2D electrons in a magnetic ®eld. In the problem of 2D Bloch electrons in a magnetic ®eld, one can use the ¯ux per plaquette, f, to characterize the system. When f is rational, the single-particle SchroÈdinger equation can be reduced to a 1D difference equation, the so-called Harper model, which appears in many different contexts ranging from the quantum Hall effects to quasi-periodic systems (see Section 2.3.3). When f is irrational, i.e., the magnetic ¯ux is incommensurate with the periodic potential, the spectrum is known to have extremely rich structure like the Cantor set, and to exhibit a multifractal behavior, with very unusual clustering of energy levels, which is popularly known as Hofstadter's butter¯y spectrum, after Douglas Hofstadter, who discovered it in a classic paper in 1976 [1573,1807]. The butter¯y spectrum turns out to be very sensitive to the smallest variation in the magnetic ®eld and, until today, there seems to be, to the best of my knowledge, no experimental evidence in its full support. This seems to be true although there have appeared a few pieces to be collected, which have merely a chance to prove to be ``a drop in the ocean'' (see Section 2.3.3 for extensive literature survey and the basic notions of a butter¯y). In this section, we are aimed at giving a few analytical details which are of immediate concern for a novice who would like to understand how practically a butter¯y emerges within a simple Harper model as described in a compact form in Section 2.3.3. It is not pretended that the analytical results given below are original with the author. The literature is a live example that several, in fact, many authors have entirely understood the strategy of getting a butter¯y spectrum. This could be taken for granted, even though there still remain many features behind this intricate spectrum which are generally accepted, but unproved. The most important of these is the nesting of the energy spectrum which is the very key to its understanding. A number of such issues have, however, been addressed by Wannier from time to time (see, e.g., Refs. [1570±1577,1789,1790,1807]).

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We start with the Harper equation, Eq. (2.18). Note that Eq. (2.18) is periodic in m such that m ! m ‡ q leaves this equation unaltered. Hence we can write by Floquet's theorem: g…m ‡ q† ˆ exp…im†g…m†;

g…m ÿ 1† ˆ exp…ÿim†g…m ‡ q ÿ 1†:

(13.12)

This equation does three things: (i) it closes the system of equations represented by Eq. (2.18), reducing it to q equations only, (ii) upon restricting Floquet's factor to the unit circle, it selects for us the bounded solutions, and (iii) it de®nes the parameter m. Since m is a phase, all values between 0 and 2p have equal weight. Another parameter is the phase n of the cosine in Eq. (2.18); it is also a phase and hence has constant weight between 0 and 2p. Only for the immediate concern, we do make the following substitution: b…m† ˆ ‰2 cos…n ÿ 2pma† ÿ EŠ:

(13.13)

Making use of Eqs. (13.12) and (13.13) in Eq. (2.18) allows us to construct the following set of secular equations: b…m†g…m† ‡ g…m ‡ 1† ‡    ‡ eÿim g…m ‡ q ÿ 1† ˆ 0;

m ! m ‡ 0;

g…m† ‡ b…m ‡ 1†g…m ‡ 1† ‡ g…m ‡ 2† ‡    ˆ 0;

m ! m ‡ 1;

0 ‡ g…m ‡ 1† ‡ b…m ‡ 2†g…m ‡ 2† ‡ g…m ‡ 3† ‡    ˆ 0;

m ! m ‡ 2;

0 ‡ 0 ‡ g…m ‡ 2† ‡ b…m ‡ 3†g…m ‡ 3† ‡ g…m ‡ 4† ‡    ˆ 0; .. .

m ! m ‡ 3;

   ‡ g…m ‡ q ÿ 3† ‡ b…m ‡ q ÿ 2†g…m ‡ q ÿ 2† ‡ g…m ‡ q ÿ 1† ˆ 0;

m ! m ‡ q ÿ 2;

im

e g…m† ‡    ‡ g…m ‡ q ÿ 2† ‡ b…m ‡ q ÿ 1† g…m ‡ q ÿ 1† ˆ 0;

m ! m ‡ q ÿ 1:

Note that the dots in these equations refer to the terms which are explicitly zeros. This set of secular equations can be cast in the following form of a tridiagonal matrix, with O$ as the null matrix: 3 32 2 g…m† b…m ˆ 0† 1      eÿim 76 g…m ‡ 1† 7 6 1 b…m ˆ 1† 1      7 76 6 7 76 6 76 g…m ‡ 2† 7 6  1 b…m ˆ 2† 1     7 76 6 7 76 6          7 $ 76 6 7ˆ O: 6 7 6 . . . . . . . . . 7 76 6 .. .. .. .. .. .. .. .. .. 7 76 6 7 76 6 7 76 6          7 76 6 7 76 6 54 g…m ‡ q ÿ 2† 5 4      1 b…m ˆ ÿ2† 1 eim











1

b…m ˆ ÿ1†

g…m‡qÿ1† (13.14)

The non-trivial solutions of this equation demand that the determinant of the coef®cients vanish. The resulting …q  q† determinant has the property that in all its terms the off-diagonal elements occur in products of pairs symmetric with respect to the main diagonal. There are only two exceptions to this, namely, two constant terms consisting of extreme off-diagonal elements. Their contribution is …ÿ1†qÿ1 2 cos…m†, while the highest power in E equals …ÿE†q . m does not appear in any other term. The

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resultant determinant therefore has the form [1575] Pq …E† ˆ 2 cos…m† ‡ 2 cos…qn†:

(13.15)

Here Pq …E† is a polynomial of degree q in E having the coef®cient of its highest power Eq ˆ 1; the other powers have coef®cients depending on p and q, but not on either m or n. Eq. (13.15) indicates that for each band of allowed values of E, Pq varies between ÿ4 and ‡4 (see also Section 2.3.3). Note that the symmetry of the square lattice considered here simpli®es the life considerably, since the energy eigenvalues E appear only in the principal diagonal and one does not really need to diagonalize$the Hermitian matrix in Eq. (13.14). The matrix in Eq. (13.14) is nothing but the one denoted by Q in Section 2.3.3. There are q real eigenvalues corresponding to the splitting of the single Bloch energy band into q subbands (a ˆ p=q). Fig. 154 illustrates the scaled energy eigenvalues E (on the vertical axis) as a function of a (on the horizontal axis), i.e., the famous butter¯y spectrum. We have con®ned our calculations to the rational values of a: 1  p  50 and p ‡ 1  q  50. As a consequence of the properties (see Section 2.3.3), the spectrum in the interval 0  a  1 has two axes of re¯ection, namely, the horizontal line E ˆ 0, and the vertical line a ˆ 0:5. Talking more about the so-well documented, 24 years old original butter¯y would surely sound less interesting. And then there are now so many of them Ð and younger ones Ð there. In physics, it must stop somewhere.

Fig. 154. The famous Hofstadter's original butter¯y: the scaled energy eigenvalues E (on the vertical axis) versus the ¯ux ratio a (on the horizontal axis), for rational values of a ˆ p=q. (Redone after Hofstadter, Ref. [1573,1807].)

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14. Epilogue Quite often in life we ®nd that our goals are incompatible: we have to renounce some of them and we feel frustrated. For instance, Mr. John wants to be a friend of both Mr. White and Mr. Black. Unfortunately, they hate each other, not unlike the scientists working in the same ®elds. It is then rather dif®cult for Mr. John to be a good friend of both of them (a very frustrating situation!). Something similar happens with anyone writing a review article or a book and giving more importance to some issues and less to the others. Knowing the fact that no one is expert in all subjects, the authors take the criticism for granted. During the course of writing this review we received considerable amount of suggestions from colleagues. Some said it should stress on the instructional material in the area while others emphasized the utility of current research. We tried to strike a compromise. The methodological strategies, analytical diagnoses, ®gures, and long list of references are included to obtain quickly the essential information in the area. All efforts have been made to cover basic ideas about the surface plasmon theory with applications to real, rather than model, materials. This review is written keeping both a novice and an expert in mind; and could also be considered as a resource review, which is certainly not meant to be exhaustive, complete, and utterly up-to-date. In this sense it does conform to a theorem by Dyson which tells us that a publication coming out at time t and taking into account research up to time t ÿ t0 will be outdated at time t ‡ t0 . So if you intend to prepare a seminar talk about some part of this review, you should ®nd out what has happened in research during around the publication of this review. As has been once indicated long ago by J.M. Ziman, a distinguished British theorist, the dif®culty of doing good physics is so great that a high proportion (say 90%) of the papers that get published do not prove of permanent value. How does this process of puri®cation and recrystallization of the scienti®c literature actually occur? By what means are readers, familiar and unfamiliar ones with the ®eld over many years, guided to the choice of the truly signi®cant works, containing the currently agreed knowledge of the topic in question? This is the function of the review article and, possibly, the monograph; the latter is, however, thought to cover a much wider ®eld than the former. The task of bringing together the results of primary research papers and presenting some uni®ed account of progress in the solution of some particular problem is of much more importance and responsibility than is given credit for in the reward system of the scienti®c profession. It is recognized to be something of an art to do well, by attempting to cover all (!) published literature on some particular problem over some particular period, indicating those points on which there is agreement, intending to resolve contradictions of observation or theory, and setting out the elements of con¯icting, controversial interpretations, or explanations. Such a review is, however, usually intended for experts rather than for beginners Ð and calls, of course, for considerable tact, generosity of spirit, and intellectual ®rmness; it must signify the ability to write such a review without losing either the respect or the friendship of one's scienti®c colleagues! Our intent here has, however, been ``keeping up with what is going on in the physics of plasmons and magnetoplasmons in the composite systems''. The present review has focused on the historical development of the studies on the surface plasmons starting with the early predictions by Ritchie in 1957, and covering the latest available status of the theory and observation. During the past four decades, research in the area of surface excitations has embarked on not only the conventional systems, such as surfaces/interfaces and thin ®lms, but also the unconventional ones, covering, e.g., the geometries with diminishing dimensions. Tremendous advances have been made in making the theoretical frameworks, the crystal-growth technology, and the

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observational techniques more and more ef®cient and transparent. Theoretical methodologies include both classical and quantal schemes appropriate for the system in question; observational tools include the sophisticated spectroscopies which involve electrons, photons, atoms, and ions. In the course of investigation, efforts have been made to explore the effects of the geometries, dimensionalities, periodicities, external probes, commensurabilities, etc. in different (classical or quantal) regimes on the physical phenomena of interest. This is seen to be done more so for understanding, in particular, the electronic, optical, and transport properties of the systems of interest due, in fact, to the world-wide drive for the ever-better devices. To such an understanding, the knowledge of collective (plasmon) excitations, as reviewed here, is of fundamental importance. We would like to summarize here what has been reviewed in the previous sections. This includes the plasmons and magnetoplasmons in the conventional systems of semiconductor surfaces/interfaces and thin ®lms; unconventional systems of diminishing dimensions such as quantum wells, wires, and dots; and some variants of the quantum structures such as ®nite-size 2DEG and periodically modulated structures. The systematic account of the development of the theories and experiments on the plasmons and magnetoplasmons in such diverse geometries follows by the critical review of the methodologies, both classical and quantal, being in current use for investigating the elementary collective excitations. These include, e.g., the transfer-matrix method which is usually based on the Maxwell equations accompanied by the appropriate EM boundary conditions, the HDM, the IRT, the HFA, the RPA, the diagrammatic technique, the equation-of-motion method, and the DFT; and several improvements thereof. Although physics is known to be deeply committed to the ``mathematical method of thought'', physics is not `merely' mathematics. The choice between mathematical and verbal exposition is therefore, to some extent, a matter of taste. An equation may be intensely powerful and exact, but the symbols are cryptic, and do not speak so directly to the comprehension. It takes long practice and deep concentration to `read' a page of algebra and grasp its signi®cance. To the extent, therefore, that the literature of physics consists of mathematical manipulations and full-length formulae, it becomes esoteric and intelligible only after deliberate and laborious deciphering. In view of this, our approach in this review of the theoretical schemes has been based on the thoughtful compromise between the mathematical and the verbal exposition of the limitations, advantages, and disadvantages of using one methodology over the other, keeping the equations to a minimum. Sections 4±13 contain numerous relevant, interesting illustrative examples of the numerical results on the dispersion characteristics of plasmons and magnetoplasmons in the respective systems. There we tried to give some simple analytical diagnoses of the exact results in order for the understanding of certain characteristic frequencies in the o±q plane. Since the ®eld has considerably matured over the years, it has huge literature and some, if not many, important works might have been inadvertently omitted. I have been rather much careful in citing all the original plus recent papers on the subject, instead of framing the citations on the philosophy of giving only the recent papers that would allow easy search of publication chain back in time. This is simply because not all the authors always intend to list the original works in the subject, sometimes for the reasons of space and sometimes for the reasons unknown to the readers. The summary of an ordinary or a review article is commonly believed to be an extremely condensed description (say within a few percent of the whole space) of what has been done in the rest of the article. More closely, it demands the author of the fundamental and technological importance of the issues involved. One of the many motivations behind the surface±plasmon investigation is that its observation serves as a diagnostic tool for characterizing the structure at hand. The surface±plasmon

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detection in the EELS of fast-particle passing through thin ®lms, in optical re¯ection, in the Auger electron spectroscopy, in tunneling of electrons through thin insulating layers, and in LEED has proved useful for surface diagnostics. Plasmon studies give information on the plasma densities, plasmon dispersion, penetration depth, and lifetimes resulting, e.g., from Landau damping, carrier collisions, and interactions of plasmons with each other and with acoustical and optical phonons. In addition, radiation from plasmons has been detected from surfaces, and plasmons have been indirectly observed through their effects on electron tunneling and on photoemission spectra. Surface plasmons in the conventional composite structures have been generally observed through a variety of experimental tools which include, e.g., ATR and EELS (see Section 2.1). Turning to the systems of diminishing dimensions, the plasmon excitations, both collective and single-particle, have been observed mostly through the optical spectroscopies such as FIR and/or Raman scattering. Just as in the conventional systems, the plasmon observation in these man-made quantum structures (quantum wells, quantum wires, and quantum dots) is of signi®cant importance in characterizing, e.g., the dimensional crossover, the periodicities, the ®eld-effects, the size-effects, the tunneling, the commensurability (between different length scales) oscillations in periodically modulated structures, several thermal and transport properties of the respective systems, compressible and incompressible states associated with the quantum Hall effects, etc. The geometric signi®cance is, of course, paramount in these quantized structures, particularly when the system is exposed to an external magnetic ®eld at low temperatures. Even though the main theme of this review has been the studies of plasmons and magnetoplasmons, a considerable attention has been focused on the exotic electronic, optical, and transport properties predicted and observed in these quantum structures. This has compelled us to give a due importance to discuss the issues related with the effects of temperature, disorder, electric ®eld, magnetic ®eld, dimensionality, periodicity, electric and magnetic modulations, and dimensional resonances, e.g., in both classical and quantum regimes (see Sections 2.2 and 2.3). The important question is the quest for the fabrication and investigation of such systems of reduced dimensionality and size. Theoretically the reduced degrees of freedom allow detailed and often exact calculations. Practically, new and unexpected phenomena have been observed. These include the observation of, e.g., the Bloch oscillations leading to the NDR effects that make feasible the designing of the high-frequency generators out of quantum wells/superlattices, the conductance quantization in quantum wires and point contacts whose characteristic (transverse) dimensions approach the electronic wavelengths, and single-electron tunneling in the 3D-con®ned Coulomb island (that usually contains a relatively large number (70±1000) of electrons) where single-electron charging is dominant. There have been exciting curiosities about the behavior of quantum dots (a 3D-con®ned ultrasmall region of electron gas that usually contains a fewer electrons, and where the size-quantization and single-electron charging can be of the same order) which is so unusual that it is natural to ask whether they will be useful for applications within the ®eld of electronics. Most suggestions in the literature have been made in the fabrication of ultrasmall logic gates, with conventional (i.e., classical) computation in mind [1674]. However, the most interesting novel application of coupled-dot systems as quantal logic gates in the newly emerging ®eld of ``quantum computation'' [1675] has recently been proposed [1676,1677]. Quantum computation is aimed at exploiting the speci®c properties of quantum mechanics to accomplish the computation. Penrose [1677] suggests that quantum computation may occur within the brain. Microtubles in each neuron contain arrays of coupled tublins which appear

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similar to coupled dots. Coupled dots are also expected to be useful for shedding light on some fundamental issues of quantum mechanics [1118]. One example is the Fermi two-atom causality problem which, in the language of the dots, relates to two adjacent dots with negligible inter-dot tunneling. If at time t ˆ 0 dot `1' is in an excited state but dot `2' is in the ground state, one can question the time t taken for dot `2' to become excited. Originally, Fermi had calculated that t  s=c, where s is the inter-dot separation and c is the speed of light. However, this result has recently been questioned [1678]. Even beyond the practical applications of quantum devices and the new intellectual territory they offer experimental physicist, quantum dots are exciting to researchers. The ability to manipulate matter on the atomic scale and create unique materials and devices with tailor-made properties has universal appeal. It marks a triumph of human ingenuity over the natural rules dictating the formation of materials. All of the strange and notorious phenomena predicted and/or observed in the daughter structures like quantum wires, quantum dots, Coulomb islands, and some newly proposed but still unfabricated ones such as quantum rings, quantum balls, quantum snakes, etc. (see Fig. 9), have their parentage in the 2DES at low temperatures exposed to external probes such as magnetic ®elds, periodic (electric and magnetic) modulations, etc. So, before diving into the mysterious world of fundamental and device physics emerging from n-dimensional (n  2) many-particle physics Ð the lower the dimension, the stronger the many-body effects Ð one needs to get to the genuine appreciation for the sophisticated fabrication technologies that make the journey possible. Recently emerging interesting aspects associated with the 2DES are the discoveries of the EMPs in the ®nite-size quantum dots, antidots, and cylindrical and ring geometries; the Weiss oscillations in the magnetoresistance tensor due to the periodic (electric and/or magnetic) modulation of the 2DEG; and the characteristic subband splitting of the LLs in periodically modulated 2DEG signaling the hope for the indirect observation of the famous Hofstadter's butter¯y. Advancement of the research on electronic systems has been predominantly toward more con®nement Ð from quantum well (two degrees of freedom) to quantum wire (one degree of freedom) to quantum dots (zero degree of freedom). Surprisingly, no comparable effort has, to our knowledge, yet been undertaken in the opposite direction, i.e., in the study of high-quality 3D systems. Although there has been extensive research in high-electron density 3D systems such as metals, our understanding of low-density 3D systems such as semiconductor structures is rather poor. There are several interesting but as yet unveri®ed predictions for low-density 3D electron systems. For instance, in the limit of zero temperature and in suf®ciently strong magnetic ®elds, a low-density electron gas is predicted to undergo a transition to a spin-density wave state [1679], where e±e interaction causes spin ordering in the ground state, producing highly anisotropic electrical properties. In still stronger magnetic ®elds or at lower electron densities, the electron gas is predicted to form a 3D Wigner crystal. One would expect a UD semiconductor (GaAs doped with Si comes to mind) to be a good candidate for this study. But the electron gas in case ¯oats in a sea of ®xed discrete background charges (of ionized donors). Electron scattering from the donor atoms inhibits observation of many interesting effects expected for such an ideal 3DEG. While the miniaturization in the quest of ever-smaller and ever-faster devices continues to provide exciting avenues for the fundamental physics and practical devices, some ``pessimism'' has started ¯oating. Such pessimism is, however, surfacing for not so long a list of reasons. There are several physical limitations that may slow the trend towards miniaturization before the quantum range is reached. One is the presence of the potential ¯uctuations and the statistical variations in characteristics

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from device to device that are expected to be increasingly signi®cant as devices get smaller. Another is the reduced current-carrying capacity of small devices. On the other hand, certain novel techniques, such as the ability to in¯uence the properties of a pair of contacts from a remote location, as has been demonstrated in the ballistic regime, may lead to new class of devices and applications (appropriate references on the foregoing remarks can be found in Ref. [844]). The technological promise that emerges from the research on these systems is the discovery of new routes to the designing of quantumbased electronic and optical devices and the discovery of new media for device structures in the submillimeter and far-infrared regimes. Already known practical applications include the high-mobility ®eld-effect transistors (HMFET), quantum lasers, Bloch oscillators, and the proposals for creating new devices such as quantum bolometers and ef®cient photon detectors made out of freely suspended 2DEG structures [1670]. It is hardly possible to cover, and do justice with, all the directions of this immensely growing ®eld of low-dimensional systems in a short survey. I have tried in this review to give a ¯avor of the many facets of the basic properties of the semiconductor quantized structures relevant to the fundamental physics through, I suppose, a balanced account of theoretical and experimental work; even though the major pretext has almost everywhere been the ``plasmons and magnetoplasmons''. We are witnessing in this ®eld a rare occurrence, where technological advances driven by the quest for the ever-better electronic and optical devices have yielded new physical systems which have in turn led to major new advances in the fundamental condensed matter physics. The importance of the ®eld can be well judged from the relative space devoted to the subject in the most relevant conference proceedings of, e.g., the biennial International Conference on Electronic Properties of Two (but better we now say n, n  2)-dimensional Systems (see, e.g., Refs. [87,1654]). Those attending these conferences know the spectacular excitement in and impact of these n-dimensional systems, witnessed by the very crowded and vivid atmosphere in the numerous simultaneously running specialized sessions. Actually, this exponential development has caused a major embarrassment to this reviewer: the number of signi®cant research papers being published in the ®eld is still increasing faster than his ability to grasp them all. I therefore apologize for any omissions of relevant material and refer the reader to the proceedings of various conferences in the ®eld, past or coming. I also apologize for obscurities and de®ciencies in scholarship and encourage the readers to tactfully bring them to my attention. Nevertheless, I hope you would ®nd these pages helpful. In the near end of this endeavor, I yearn to recall a few words on the human side of this noble job of scienti®c research, where the scientists have usually had a popular image of disinterested truth-seekers. There are, to be sure, minor deceptions in virtually all scienti®c affairs, just as they are in all other walks of human life. But to the extent that the literature of physics continues to ignore, until recently, the due credit and priority to a classic work done in the late 1920s is extremely disgusting and disappointing. I speci®cally refer to the problem of a single ideally 2D electron in a circular dot con®ned by a parabolic potential in the presence of an external magnetic ®eld solved more than seven decades ago by Fock [1124] and subsequently by Darwin [1125]. In this connection, it is interesting (!) to note that the same problem, but for zero con®nement potential, was studied 2 years after Fock's work, and almost contemporary to Darwin's work, by Landau [1562], leading to the now-fashionable terms such as LLs, Landau degeneracy, Landau diamagnetism, and what not. Does not it sound unethical on the part of the physics community to devoid Fock (and also possibly Darwin) of his (or their) due credit and priority? Would not it look justi®able and nicer, in the name of professional ethics, to replace `Landau' by `Fock±Darwin', or at least by `Fock', in the above-mentioned terms for future,

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rectifying the unimaginable damage already done to those who are no longer amidst us to claim? This is, quite likely, one of the many such unethical and undesirable examples that arise out of our false image of being disinterested seekers of truth, gathering facts with mind cleansed of prejudices and preconceptions. This is, as David Goodstein properly calls, ``the myth of the noble scientist''. We would like to end our odyssey by drawing attention to the fact that coincidences can and do happen, and an incorrect theory may agree with the experiment, just as a correct theory may disagree with the experiment, by mere coincidence. Therefore, agreement with experiment does not always prove that a theory is correct, as the following recent example illustrates. The example we intend to cite comes from the experimental ®nding [1044±1049] of the quenching of Hall effect or suppression of the Hall resistance of a realistic quantum wire in a weak magnetic ®eld. This phenomenon, as modestly discussed in Section 2.2.3, drew immediate attention of several theoretical research groups, but had resisted a complete and consistent explanation. We speci®cally refer to one of the earliest interpretation by Beenakker and van Houten [1051], who argued that the quenching of the Hall effect should always take place whenever the extent of the edge states in the lowest subband lt exceeds the width of the wire W. Brie¯y, they suggested the edge-state suppression as the quenching mechanism, which holds that quenching should occur when W  Dmin , the minimum transverse extension of an edge state. These conditions led them to derive a threshold ®eld Bth  2h=…ekF W 3 †; B < Bth was therefore concluded to be a necessary and suf®cient condition for the quenching of the Hall resistance (RH ) in narrow channels. Subsequent exact microscopic calculations [1052±1058,1771] assuming idealized weakly, as well as strongly, coupled Hall probes were seen to be outrightly con¯icting with edge-state-suppression theory. These theories have reasonably proved that quenching of the Hall effect is due to a property of the contact geometry, but it is not intrinsic to the quasi-1D limit. Generic quenching, as experimentally observed, has been shown [1056] to occur only due to the widening of the wires near the junction region. The resulting collimation of the electrons injected into the junction suppresses the Hall resistance, since electrons in these modes contribute a little to RH . The impact of these microscopic theories coming from different (and distant) research groups was that the authors of Ref. [1051] now no longer support their edge-state-suppression theory (see, e.g., their confession in Ref. [1057]). This example of interpreting, rather misinterpreting, the experimental data based on the unsound mathematical expressions reminds us of the immortal words that the English physicist Edmund Stoner (in 1950) used to criticize a different theory: ``It is not unfair to say that obscurities in the presentation do not seem to arise wholly from the inherent complexities of the problem''; and we conclude by quoting him: ``It is important to bear in mind that the validity of a mathematical (or physicomathematical) argument in itself cannot be con®rmed merely by an agreement with experiment of approximate relations which may have been derived; and that the value of a theory as interpretative of observable phenomena cannot be properly assessed until the essential details in the argument from the premises to the conclusions have been clearly presented''. The ideas I stand for are not mine, I borrowed them from Socrates. I swiped them from Chester®eld. I stole them from Jesus. If you don't like their rules, whose would you use? Ð Dale Carnegie

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Acknowledgements It is a pleasure for me to thank my collaborators, P. Halevi, B. Djafari-Rouhani, F. Garcia-Moliner, G. Gumbs, P. Zielinski, and T. Park, together with whom I have learned a great deal about this subject and who are co-authors of a number of research papers referenced in this review. I should gratefully acknowledge numerous very useful communications with Professors R.F. Wallis, B. Djafari-Rouhani, R.E. Camley and S.E. Ulloa. I am indebted to Professor L. Dobrzynski for his constant support and encouragement during the course of this review. I should specially thank Professor Maurice Glicksman for providing his own excellent review article and letting me know the source for availing some long sought-after references on the SSP. I wish to thank Dr. Daniela Pfannkuche for paying her attention to my queries on her years-old work with Professor R.R. Gerhardts. The continued hardships (and adversities!) concerned with both the software and the hardware faced during the course of this review would have delayed the outcome even more without the generous, occasional help extended by G. Martinez. My appreciation goes to J. Arriaga for allowing me to use his Dec-Workstation for many months. Raul Brito deserves a vote of thanks for his timely attention on some key points related with the software. I feel thankful to Mr. H. Hernandez for being so very helpful in bringing numerous articles photocopied from the National University of Mexico in the weekends during many months. This manuscript was written using a LaTeX editing system on a PC-Pentium-R computer, thanks to the unfailing availability of Ms. Monica Hernandez. I would also like to thank my better half, Rosy Pastrana, for spending innumerable painful hours late in the nights for so many months, waiting for me to open the doors, and serving me meals at many odd hours. Finally, I should like to thank Professor L. Dobrzynski (Editor at USTL) and Professor H. Weinberg (General Editor) for extending their kind invitation to contribute to this subject and for their extraordinary patience. This work was partially supported by CONACyT Grant No. 28110-E.

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