Copyright © 2007 American Scientiﬁc Publishers All rights reserved Printed in the United States of America

Journal of Computational and Theoretical Nanoscience Vol. 4, 252–256, 2007

Si-Nanotrees: Structure and Electronic Properties Madhu Menon1 , Ernst Richter2 ∗ , Ingyu Lee3 , and Padma Raghavan3 1

Department of Physics and Astronomy and Center for Computational Sciences, University of Kentucky, Lexington, KY 40506, USA 2 DaimlerChrysler AG, 89081 Ulm, Germany 3 Department of Computer Science and Engineering, Pennsylvania State University, 308 Pond Lab, University Park, PA 16802-6106, USA We investigate structure and electronic properties of branched silicon nanowires (or “nanotrees”) using large scale quantum mechanical simulations. Our calculations show that such structures are stable for underlying tetrahedral as well as cage-like cores making up the “stems” and “branches” of these nanotrees. Electronic structure analysis shows that the formation of “nanotrees” are accomDelivered by Ingenta to: panied by a narrowing of the HOMO–LUMO gap. This has important implications in their application Amanda Barnard in molecular devices due to the predicted enhancement in the conducting properties.

IP : 129.67.84.229

Keywords: Molecular Dynamics, Electronics. Tue,Nanostructures, 13 Mar 2007Silicon 07:37:44

RESEARCH ARTICLE

1. INTRODUCTION Nanowires are increasingly becoming popular from scientiﬁc and technological perspectives. These wires can be grown from a variety of different materials and can even attain lengths approaching tens of micrometers. The current interest in silicon nanowires (Si-NWs) is mainly derived from their tremendous technological potential in nanoscale devices.1–5 One of the major advantages of these nanowires is that they afford tailoring of their electronic properties through selective doping. This property makes them a prime candidate for use in electronic industries where the miniaturization trend needs to be sustained for the foreseeable future. Tetrahedral wire-like structures have been theoretically investigated by various groups.6–10 More recently, structural predictions for small diameters Si-NWs were made by Menon and Richter11 and Marsen and Sattler.12 If two nanowires, one semiconducting and the other metallic, are connected, a hetero-junction is formed that will act like a rectifying diode. Experimentalists, in fact, have succeeded in creating a nanowire diode by joining two differently doped silicon semiconductor nanowires.13 While such two terminal hetero-junctions have many uses, they still lack the versatility of the 3-terminal devices where the 3rd terminal could be used for controlling the switching mechanism, power gain, or other transisting applications that are needed in any extended molecular electronic circuit. Nanowires containing branching structures, therefore, appear to be more promising in ∗

Author to whom correspondence should be addressed.

252

J. Comput. Theor. Nanosci. 2007, Vol. 4, No. 2

technological applications. Very recently, experimentalists working with semiconductor nanowires have reported the synthesis of branched nanowires.14 Furthermore, they have demonstrated that these tree-like nanostructures (“nanotrees”) can be formed in a highly controlled way. The high-resolution transmission electron microscopy (TEM) analysis showed that the branching mechanism gives continuous crystalline (monolithic) structures throughout the tree-like structures. Such structures can lead to the isolation of “Y-” and “T-junctions” of nanowires with tremendous potential in use as three-terminal devices. Nanowires and nanotrees made of Si have the added advantage that they permit a natural extension of silicon technology to the nanoscale regime and allow a natural integration of nanoscale devices into traditional largerscale silicon electronic technology. Si-nanowires have been synthesized by laser ablation of metal-containing Si.15 16 The resulting Si-nanowires are reported to consist of pure Si and have fairly uniform diameter. Further experimental analysis using electron diffraction and high-resolution transmission electron microscopy (HRTEM) suggested the nanowires to consist of an outer layer and a Si crystalline core.15 16 Synthesis of nanowires of diameter as small as 1 nm has been reported recently.17 In the present work, we investigate the stability of branched nanowires (nanotrees) made of Si atoms using a quantum molecular dynamics method. While Si can exist in many crystalline phases, the most stable of them are all four-fold coordinated. They include the diamond and the clathrate phases.18 19 Recent theoretical works have shown the Si-nanowires with the crystalline core consisting of the diamond or the clathrate types to be the most stable.20 1546-198X/2007/4/252/005

doi:10.1166/jctn.2007.008

Menon et al.

Si-Nanotrees: Structure and Electronic Properties

We, therefore, focus our investigations on Si-nanotrees whose “stem” and “branches” consist of Si-nanowires with diamond or clathrate type of inner cores.

2. TECHNIQUE The theoretical method used in the present work is the generalized tight-binding molecular dynamics (GTBMD) scheme of Menon and Subbaswamy18 that allows for full relaxation of covalent systems with no symmetry constraints. This method has been found to be very reliable in obtaining very good agreement with experiment for structural and vibrational properties of Si from dimer all the way to the bulk.18 The details of the technique can be found in Ref. [18]. Here we give a brief summary. If the total energy U of a system is known, the force Fx associated with an atomic coordinate x is given by U x The total energy is given by the sum: Fx = −

U = Uel + Urep + U0

(1)

Uel =

n

En

and Urep is given by a repulsive pair potential rij Urep = i

where K is the non-orthogonality coefﬁcient. The diagonal elements of Hij , as in the orthogonal theory, are taken to be the valence s and p energies. The off-diagonal interatomic matrix elements are given in terms of the Hamiltonian matrix elements in orthogonal theory, Vij , by 1 Hij = Vij 1 + − S22 (10) K where,

√ Sss − 2 3Ssp − 3Spp S2 = (11) 4 is the nonorthogonality between sp3 hybrids.18 The quantities S , in turn, are determined from

Delivered by Ingenta to: 2V Amanda Barnard S = (12) K IP : 129.67.84.229 + (2) Tue, 13 Mar 2007 We 07:37:44 take a simple exponential distance dependence in the

where Uel is the sum of the one-electron eigenvalues En for the occupied states: occ

In the non-orthogonal scheme, the overlap matrix is calculated in the spirit of extended Hückel theory23 by assuming a proportionality between H and S:18 Hij 2 (9) Sij = K Hii + Hjj

(3)

(4)

j>i

U0 is constant that merely shifts the zero of energy. In the non-orthogonal tight-binding scheme the characteristic equation for obtaining the eigenvalues is given by H − En SCn = 0

(5)

V r = V d0 e− r−d0

(7)

where d0 is the sum of the covalent radii of the pair of interacting atoms and is an adjustable parameter. The scaling of the repulsive term (Eq. (4)) is also taken to be exponential: r = 0 e− r−d0 (8) where = 4.18 J. Comput. Theor. Nanosci. 4, 252–256, 2007

2

K r = K0 e r−d0

(13)

Making use of all these terms, Eq. (1) can now be used to obtain trajectories of the particles for performing molecular dynamics simulations.

3. PARALLELIZING GTBMD SCHEME For large scale simulations, GTBMD involve the following calculations at each time step: • Obtaining Hamiltonian and the overlap matrices as parameterized analytical functions of inter-atomic distances. • Solving a large eigenvalue problem. • Computing the forces using the eigenvalues and eigenvectors. • Moving the atoms to new positions. The computational costs are dominated by those for solving the eigenvalue problem. The matrix dimension is number of basis times the number of atoms in the simulation. For models of interest with several thousand atoms, the matrix dimension N is in the range 10,000 to 50,000. In each time-step of the simulation, the generalized eigenvalue problem has to be solved and a simulation typically involves nearly a thousand time-steps. An additional challenge is that the GTBMD requires at least N /2 eigenvalues as well as the corresponding eigenvectors at each time-step. The computational cost of one instance of the eigenvalue problem grows as O N 3 . Consequently, for matrix dimensions of interest, the computational costs are prohibitive. Furthermore, storage demands also grow prohibitively. Such simulations are, therefore, not feasible 253

RESEARCH ARTICLE

where H and S are the Hamiltonian and overlap matrices, respectively.18 The Hellmann-Feynman theorem for obtaining the electronic part of the force is given by,18 n − En S C Cn † H En x x (6) = n n† C SC x In the Slater-Koster scheme the Hamiltonian matrix elements are obtained from the parameters, V , in terms of the bond direction cosines l, m, n.18 21 22 We take the parameters V r to decrease exponentially with r:18

nonorthogonality coefﬁcient, K,

Si-Nanotrees: Structure and Electronic Properties

Menon et al.

coordination. They belong to two general category; tetrahedral and clathrate (cage-like). The tetrahedral nanowire is denoted by Si(Td). The clathrate nanowires are further divided into two categories. The ﬁrst has an inner core made of the Si clathrate structure consisting of a face centered cubic (fcc) lattice with a 34 atom basis.19 We denote this by Si(34). The other has as its inner core a Si clathrate structure consisting of a simple cubic (sc) lattice with a 46 atom basis, denoted by Si(46).19 All the structures shown in Figure 1 are fully relaxed without any symmetry constraints. The surface atoms of all these nanowires show reconstructions that lower the surface energy. We next investigate the stability of “nanotrees” with stems and branches consisting of Si(Td), Si(34), and Si(46) nanowires. The total number of Si atoms in each of these nanotrees range from 2500–3000. Each nanotree is constructed by attaching smaller diameter nanowire branches 4. APPLICATIONS TO Si-NANOTREES with cores that are identical to the stem in a such a way that all the tree structures have four-fold coordinated inner We next present applications of the formalismDelivered to branchedby Ingenta to: cores surrounded by outer surface of atoms with three-fold Si-nanowires or Si-“nanotrees.” It should be notedAmanda that the Barnard coordination. We then fully relax the resulting structures nanowires synthesized in experiments are hydrogen IP : cov129.67.84.229 without any symmetry constraints. The atoms in the juncered, while the nanowires used in our simulations Tue,are 13made Mar 2007 tion07:37:44 region are the most affected by the relaxation process of pristine Si. The Si-nanotrees and stems considered in as they move to relieve the strain. Portions of the relaxed the present work consists of several thousand Si atoms. structures are shown in Figure 2. The surface reconstrucA fully quantum mechanical simulations for such large tion, coupled with branching results in interesting juncsystems using the serial GTBMD method is computation regions. These regions appear smooth for Si(34) and tionally prohibitive. We, therefore, use the parallelized Si(46) “nanotrees,” while in the case of Si(Td) “nanotree” GTBMD method in Section 3. the junctions appear abrupt. This is understandable since We perform GTBMD simulations for the Si-nanowires the underlying cage-like arrangements of Si(34) and Si(46) at zero temperature. The nanowires are carved out from the structures allow seamless connection between stems and most stable crystalline forms of Si so that their interiors the branches. This is because the cage-like forms are more resemble bulk Si. The relaxation process results in surisotropic than the tetrahedral forms. face reconstruction common to bulk Si surfaces. We conThe branching also gives rise to interesting electronic sider three different types of Si-nanowires; portions of properties. The “stems” are essentially ﬁnite length which are shown in Figure 1. The largest diameter of Si-nanowires that have core atoms with four-fold these nanowires is ≈4 nm. The total number of Si atoms coordination, while the surface atoms have three-fold coorin each of these nanowires range from 2000–2500. All dinations and show reconstructions that minimize the surthese nanowires have a four-fold coordinated inner core face energies for these structures. Formation of branches surrounded by an outer surface of atoms with three-fold induces additional strains on the surface atoms in the

RESEARCH ARTICLE

on even the best serial computers; simulation times can take several weeks. We solve the generalized eigenvalue problem of the form Hv = Sv in parallel using SCALAPACK.24 We used a cluster with 80 compute nodes where each node has a dual 2.4 GHz AMD processor with 8G main memory. The re-orthogonalization required for the eigenvector computation constitutes the bulk of the costs. These eigenvalue computations typically have the following three main steps: (i) compute a tridiagonal matrix T using orthogonal transformations and an orthogonal factor, Q, (ii) compute all eigenvalues of T and, (iii) compute eigenvectors of T and then use Q to compute eigenvectors of the original matrix.

Fig. 1. Figure showing portions of three different types of Si-nanowires. The symmetry axis of the nanowires are along the vertical direction. The left ﬁgure shows a nanowire with an inner crystalline core consisting of tetrahedral structure, while the wire in the middle ﬁgure is made of a clathrate structure with 34 atom basis in a face-centered cubic unit cell. The right ﬁgure shows a wire made from another clathrate structure with 46 atoms basis in a simple-cubic unit cell.

254

J. Comput. Theor. Nanosci. 4, 252–256, 2007

Menon et al.

Si-Nanotrees: Structure and Electronic Properties

Fig. 2. Portions of Si-nanotrees with stems and branches made of; tetrahedral (top), clathrate structure with 34 atom fcc unit cell (lower left), and another clathrate structure with 46 atom sc unit cell (lower right). Delivered by Ingenta to:

Amanda Barnard

junction area resulting in the introduction of states the to enhance conductivity due to more conduction channels IP in : 129.67.84.229 being available. For small diameter pristine tetrahedral electronic band gaps. This is illustrated in Figure 3 where Tue, 13 Mar 2007 07:37:44 Si-nanowire, it is well-known that quantum conﬁnement we show the computed electronic densities of states (DOS) increases the gap. For large enough diameter, however, the for “stems” and “nanotrees” of Si(Td). The electronic gap is expected to approach that of the Si bulk. A sysstructure analysis is performed using a sp3 s ∗ tight-binding tematic study of gap dependence on diameter and growth model25 that correctly reproduces the band gap for bulk direction can be found in Ref. [26]. Si in the diamond and clathrate phases. The DOS for In summary, we have presented results for structure the “stems” show a highest occupied molecular orbital and stability studies using large scale quantum mechani(HOMO)—lowest unoccupied molecular orbital (LUMO) cal simulations of “nanotrees” made from Si and shown gap value of 0.57 eV which is smaller than that for the bulk that such structures are stable. Electronic structure analysis diamond phase of silicon (1.1 eV). This is because of the shows that formation of “nanotrees” are accompanied by presence of a signiﬁcant number of three-fold coordinated a narrowing of the HOMO–LUMO gap. This has imporunpassivated surface Si-atoms. Formation of “nanotrees” tant implications in their application in molecular devices causes the creation of states in the HOMO–LUMO gap, due to the predicted enhancement in the conducting propnarrowing the gap value to 0.16 eV. Furthermore, more erties. The present work is supported through grants states from the unoccupied levels are pulled in towards the by NSF (ITR-0221916), DOE (00-63857), and US-ARO Fermi level, EF (shown as dashed line). This is expected (W911NF-05-1-0372).

References

J. Comput. Theor. Nanosci. 4, 252–256, 2007

255

RESEARCH ARTICLE

Fig. 3. The densities of states (DOS) for “stem” and “nanotree” structures of tetrahedral Si, Si(Td). The Fermi level is shown as dashed line. The gap is seen to be narrower for the “nanotree” suggesting better potential for conducting applications.

1. A. M. Morales and. C. M. Lieber, Science 279, 208 (1998). 2. S.-W. Chung, J.-Y. Yu, and J. R. Heath, Appl. Phys. Lett. 76, 2068 (2000). 3. D. Appell, Nature 419, 553 (2002). 4. C. M. Lieber, Nano Lett. 2, 81 (2002). 5. G. Zheng, W. Lu, S. Jin, and C. M. Lieber, Adv. Mater. 16, 1890 (2004). 6. S. Watanabe, Y. A. Ono, T. Hashizume, Y. Wada, J. Yamauchi, and M. Tsukada, Phys. Rev. B 52, 10768 (1995). 7. J. Xia and K. W. Cheah, Phys. Rev. B 55, 15688 (1997). 8. A. M. Saitta, F. Buda, G. Fiumara, and P. V. Giaquinta, Phys. Rev. B 56, 1446 (1996). 9. A. J. Read, R. J. Needs, K. J. Nash, L. T. Canham, P. D. J. Calcott, and A. Qteish, Phys. Rev. Lett. 69, 2400 (1992). 10. T. Ohno, K. Shiraishi, and T. Ogawa, Phys. Rev. Lett. 69, 2400 (1992). 11. M. Menon and E. Richter, Phys. Rev. Lett. 83, 792 (1999). 12. B. Marsen and K. Sattler, Phys. Rev. B 60, 11593 (1999).

Si-Nanotrees: Structure and Electronic Properties 13. Y. Cui and C. M. Lieber, Science 291, 851 (2001). 14. K. A. Dick, K. Deppert, M. W. Larsson, T. Martensson, W. Seifert, L. R. Wallenberg, and L. Samuelson, Nature Mater. 3, 380 (2004). 15. Y. F. Zhang, Y. H. Tang, N. Wang, D. P. Yu, C. S. Lee, I. Bello, and S. T. Lee, Appl. Phys. Lett. 72, 1835 (1998). 16. A. M. Morales and C. M. Lieber, Science 279, 208 (1998). 17. D. D. D. Ma, C. S. Lee, F. C. K. Au, S. Y. Tong, and S. T. Lee, Science 299, 1874 (2003). 18. M. Menon and K. R. Subbaswamy, Phys. Rev. B 55, 9231 (1997). 19. M. Menon, E. Richter, and K. R. Subbaswamy, Phys. Rev. B 56, 12290 (1997).

Menon et al. 20. I. Ponomareva, M. Menon, D. Srivastava, and A. N. Andriotis, Phys. Rev. Lett. 95, 265502 (2005). 21. J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). 22. W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco (1980). 23. R. Hoffmann, J. Chem. Phys. 39, 1397 (1963). 24. L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, “ScaLAPACK Users’ Guide,” SIAM, Philadelphia, PA (1997). 25. P. Vogl, H. P. Hjalmarson, and J. D. Dow, J. Phys. Chem. Solids 44, 365 (1983). 26. I. Ponomareva, M. Menon, E. Richter, and A. N. Andriotis, Phys. Rev. B 74, 125311 (2006).

Received: 26 April 2006. Accepted: 3 August 2006.

RESEARCH ARTICLE

Delivered by Ingenta to: Amanda Barnard IP : 129.67.84.229 Tue, 13 Mar 2007 07:37:44

256

J. Comput. Theor. Nanosci. 4, 252–256, 2007

Journal of Computational and Theoretical Nanoscience Vol. 4, 252–256, 2007

Si-Nanotrees: Structure and Electronic Properties Madhu Menon1 , Ernst Richter2 ∗ , Ingyu Lee3 , and Padma Raghavan3 1

Department of Physics and Astronomy and Center for Computational Sciences, University of Kentucky, Lexington, KY 40506, USA 2 DaimlerChrysler AG, 89081 Ulm, Germany 3 Department of Computer Science and Engineering, Pennsylvania State University, 308 Pond Lab, University Park, PA 16802-6106, USA We investigate structure and electronic properties of branched silicon nanowires (or “nanotrees”) using large scale quantum mechanical simulations. Our calculations show that such structures are stable for underlying tetrahedral as well as cage-like cores making up the “stems” and “branches” of these nanotrees. Electronic structure analysis shows that the formation of “nanotrees” are accomDelivered by Ingenta to: panied by a narrowing of the HOMO–LUMO gap. This has important implications in their application Amanda Barnard in molecular devices due to the predicted enhancement in the conducting properties.

IP : 129.67.84.229

Keywords: Molecular Dynamics, Electronics. Tue,Nanostructures, 13 Mar 2007Silicon 07:37:44

RESEARCH ARTICLE

1. INTRODUCTION Nanowires are increasingly becoming popular from scientiﬁc and technological perspectives. These wires can be grown from a variety of different materials and can even attain lengths approaching tens of micrometers. The current interest in silicon nanowires (Si-NWs) is mainly derived from their tremendous technological potential in nanoscale devices.1–5 One of the major advantages of these nanowires is that they afford tailoring of their electronic properties through selective doping. This property makes them a prime candidate for use in electronic industries where the miniaturization trend needs to be sustained for the foreseeable future. Tetrahedral wire-like structures have been theoretically investigated by various groups.6–10 More recently, structural predictions for small diameters Si-NWs were made by Menon and Richter11 and Marsen and Sattler.12 If two nanowires, one semiconducting and the other metallic, are connected, a hetero-junction is formed that will act like a rectifying diode. Experimentalists, in fact, have succeeded in creating a nanowire diode by joining two differently doped silicon semiconductor nanowires.13 While such two terminal hetero-junctions have many uses, they still lack the versatility of the 3-terminal devices where the 3rd terminal could be used for controlling the switching mechanism, power gain, or other transisting applications that are needed in any extended molecular electronic circuit. Nanowires containing branching structures, therefore, appear to be more promising in ∗

Author to whom correspondence should be addressed.

252

J. Comput. Theor. Nanosci. 2007, Vol. 4, No. 2

technological applications. Very recently, experimentalists working with semiconductor nanowires have reported the synthesis of branched nanowires.14 Furthermore, they have demonstrated that these tree-like nanostructures (“nanotrees”) can be formed in a highly controlled way. The high-resolution transmission electron microscopy (TEM) analysis showed that the branching mechanism gives continuous crystalline (monolithic) structures throughout the tree-like structures. Such structures can lead to the isolation of “Y-” and “T-junctions” of nanowires with tremendous potential in use as three-terminal devices. Nanowires and nanotrees made of Si have the added advantage that they permit a natural extension of silicon technology to the nanoscale regime and allow a natural integration of nanoscale devices into traditional largerscale silicon electronic technology. Si-nanowires have been synthesized by laser ablation of metal-containing Si.15 16 The resulting Si-nanowires are reported to consist of pure Si and have fairly uniform diameter. Further experimental analysis using electron diffraction and high-resolution transmission electron microscopy (HRTEM) suggested the nanowires to consist of an outer layer and a Si crystalline core.15 16 Synthesis of nanowires of diameter as small as 1 nm has been reported recently.17 In the present work, we investigate the stability of branched nanowires (nanotrees) made of Si atoms using a quantum molecular dynamics method. While Si can exist in many crystalline phases, the most stable of them are all four-fold coordinated. They include the diamond and the clathrate phases.18 19 Recent theoretical works have shown the Si-nanowires with the crystalline core consisting of the diamond or the clathrate types to be the most stable.20 1546-198X/2007/4/252/005

doi:10.1166/jctn.2007.008

Menon et al.

Si-Nanotrees: Structure and Electronic Properties

We, therefore, focus our investigations on Si-nanotrees whose “stem” and “branches” consist of Si-nanowires with diamond or clathrate type of inner cores.

2. TECHNIQUE The theoretical method used in the present work is the generalized tight-binding molecular dynamics (GTBMD) scheme of Menon and Subbaswamy18 that allows for full relaxation of covalent systems with no symmetry constraints. This method has been found to be very reliable in obtaining very good agreement with experiment for structural and vibrational properties of Si from dimer all the way to the bulk.18 The details of the technique can be found in Ref. [18]. Here we give a brief summary. If the total energy U of a system is known, the force Fx associated with an atomic coordinate x is given by U x The total energy is given by the sum: Fx = −

U = Uel + Urep + U0

(1)

Uel =

n

En

and Urep is given by a repulsive pair potential rij Urep = i

where K is the non-orthogonality coefﬁcient. The diagonal elements of Hij , as in the orthogonal theory, are taken to be the valence s and p energies. The off-diagonal interatomic matrix elements are given in terms of the Hamiltonian matrix elements in orthogonal theory, Vij , by 1 Hij = Vij 1 + − S22 (10) K where,

√ Sss − 2 3Ssp − 3Spp S2 = (11) 4 is the nonorthogonality between sp3 hybrids.18 The quantities S , in turn, are determined from

Delivered by Ingenta to: 2V Amanda Barnard S = (12) K IP : 129.67.84.229 + (2) Tue, 13 Mar 2007 We 07:37:44 take a simple exponential distance dependence in the

where Uel is the sum of the one-electron eigenvalues En for the occupied states: occ

In the non-orthogonal scheme, the overlap matrix is calculated in the spirit of extended Hückel theory23 by assuming a proportionality between H and S:18 Hij 2 (9) Sij = K Hii + Hjj

(3)

(4)

j>i

U0 is constant that merely shifts the zero of energy. In the non-orthogonal tight-binding scheme the characteristic equation for obtaining the eigenvalues is given by H − En SCn = 0

(5)

V r = V d0 e− r−d0

(7)

where d0 is the sum of the covalent radii of the pair of interacting atoms and is an adjustable parameter. The scaling of the repulsive term (Eq. (4)) is also taken to be exponential: r = 0 e− r−d0 (8) where = 4.18 J. Comput. Theor. Nanosci. 4, 252–256, 2007

2

K r = K0 e r−d0

(13)

Making use of all these terms, Eq. (1) can now be used to obtain trajectories of the particles for performing molecular dynamics simulations.

3. PARALLELIZING GTBMD SCHEME For large scale simulations, GTBMD involve the following calculations at each time step: • Obtaining Hamiltonian and the overlap matrices as parameterized analytical functions of inter-atomic distances. • Solving a large eigenvalue problem. • Computing the forces using the eigenvalues and eigenvectors. • Moving the atoms to new positions. The computational costs are dominated by those for solving the eigenvalue problem. The matrix dimension is number of basis times the number of atoms in the simulation. For models of interest with several thousand atoms, the matrix dimension N is in the range 10,000 to 50,000. In each time-step of the simulation, the generalized eigenvalue problem has to be solved and a simulation typically involves nearly a thousand time-steps. An additional challenge is that the GTBMD requires at least N /2 eigenvalues as well as the corresponding eigenvectors at each time-step. The computational cost of one instance of the eigenvalue problem grows as O N 3 . Consequently, for matrix dimensions of interest, the computational costs are prohibitive. Furthermore, storage demands also grow prohibitively. Such simulations are, therefore, not feasible 253

RESEARCH ARTICLE

where H and S are the Hamiltonian and overlap matrices, respectively.18 The Hellmann-Feynman theorem for obtaining the electronic part of the force is given by,18 n − En S C Cn † H En x x (6) = n n† C SC x In the Slater-Koster scheme the Hamiltonian matrix elements are obtained from the parameters, V , in terms of the bond direction cosines l, m, n.18 21 22 We take the parameters V r to decrease exponentially with r:18

nonorthogonality coefﬁcient, K,

Si-Nanotrees: Structure and Electronic Properties

Menon et al.

coordination. They belong to two general category; tetrahedral and clathrate (cage-like). The tetrahedral nanowire is denoted by Si(Td). The clathrate nanowires are further divided into two categories. The ﬁrst has an inner core made of the Si clathrate structure consisting of a face centered cubic (fcc) lattice with a 34 atom basis.19 We denote this by Si(34). The other has as its inner core a Si clathrate structure consisting of a simple cubic (sc) lattice with a 46 atom basis, denoted by Si(46).19 All the structures shown in Figure 1 are fully relaxed without any symmetry constraints. The surface atoms of all these nanowires show reconstructions that lower the surface energy. We next investigate the stability of “nanotrees” with stems and branches consisting of Si(Td), Si(34), and Si(46) nanowires. The total number of Si atoms in each of these nanotrees range from 2500–3000. Each nanotree is constructed by attaching smaller diameter nanowire branches 4. APPLICATIONS TO Si-NANOTREES with cores that are identical to the stem in a such a way that all the tree structures have four-fold coordinated inner We next present applications of the formalismDelivered to branchedby Ingenta to: cores surrounded by outer surface of atoms with three-fold Si-nanowires or Si-“nanotrees.” It should be notedAmanda that the Barnard coordination. We then fully relax the resulting structures nanowires synthesized in experiments are hydrogen IP : cov129.67.84.229 without any symmetry constraints. The atoms in the juncered, while the nanowires used in our simulations Tue,are 13made Mar 2007 tion07:37:44 region are the most affected by the relaxation process of pristine Si. The Si-nanotrees and stems considered in as they move to relieve the strain. Portions of the relaxed the present work consists of several thousand Si atoms. structures are shown in Figure 2. The surface reconstrucA fully quantum mechanical simulations for such large tion, coupled with branching results in interesting juncsystems using the serial GTBMD method is computation regions. These regions appear smooth for Si(34) and tionally prohibitive. We, therefore, use the parallelized Si(46) “nanotrees,” while in the case of Si(Td) “nanotree” GTBMD method in Section 3. the junctions appear abrupt. This is understandable since We perform GTBMD simulations for the Si-nanowires the underlying cage-like arrangements of Si(34) and Si(46) at zero temperature. The nanowires are carved out from the structures allow seamless connection between stems and most stable crystalline forms of Si so that their interiors the branches. This is because the cage-like forms are more resemble bulk Si. The relaxation process results in surisotropic than the tetrahedral forms. face reconstruction common to bulk Si surfaces. We conThe branching also gives rise to interesting electronic sider three different types of Si-nanowires; portions of properties. The “stems” are essentially ﬁnite length which are shown in Figure 1. The largest diameter of Si-nanowires that have core atoms with four-fold these nanowires is ≈4 nm. The total number of Si atoms coordination, while the surface atoms have three-fold coorin each of these nanowires range from 2000–2500. All dinations and show reconstructions that minimize the surthese nanowires have a four-fold coordinated inner core face energies for these structures. Formation of branches surrounded by an outer surface of atoms with three-fold induces additional strains on the surface atoms in the

RESEARCH ARTICLE

on even the best serial computers; simulation times can take several weeks. We solve the generalized eigenvalue problem of the form Hv = Sv in parallel using SCALAPACK.24 We used a cluster with 80 compute nodes where each node has a dual 2.4 GHz AMD processor with 8G main memory. The re-orthogonalization required for the eigenvector computation constitutes the bulk of the costs. These eigenvalue computations typically have the following three main steps: (i) compute a tridiagonal matrix T using orthogonal transformations and an orthogonal factor, Q, (ii) compute all eigenvalues of T and, (iii) compute eigenvectors of T and then use Q to compute eigenvectors of the original matrix.

Fig. 1. Figure showing portions of three different types of Si-nanowires. The symmetry axis of the nanowires are along the vertical direction. The left ﬁgure shows a nanowire with an inner crystalline core consisting of tetrahedral structure, while the wire in the middle ﬁgure is made of a clathrate structure with 34 atom basis in a face-centered cubic unit cell. The right ﬁgure shows a wire made from another clathrate structure with 46 atoms basis in a simple-cubic unit cell.

254

J. Comput. Theor. Nanosci. 4, 252–256, 2007

Menon et al.

Si-Nanotrees: Structure and Electronic Properties

Fig. 2. Portions of Si-nanotrees with stems and branches made of; tetrahedral (top), clathrate structure with 34 atom fcc unit cell (lower left), and another clathrate structure with 46 atom sc unit cell (lower right). Delivered by Ingenta to:

Amanda Barnard

junction area resulting in the introduction of states the to enhance conductivity due to more conduction channels IP in : 129.67.84.229 being available. For small diameter pristine tetrahedral electronic band gaps. This is illustrated in Figure 3 where Tue, 13 Mar 2007 07:37:44 Si-nanowire, it is well-known that quantum conﬁnement we show the computed electronic densities of states (DOS) increases the gap. For large enough diameter, however, the for “stems” and “nanotrees” of Si(Td). The electronic gap is expected to approach that of the Si bulk. A sysstructure analysis is performed using a sp3 s ∗ tight-binding tematic study of gap dependence on diameter and growth model25 that correctly reproduces the band gap for bulk direction can be found in Ref. [26]. Si in the diamond and clathrate phases. The DOS for In summary, we have presented results for structure the “stems” show a highest occupied molecular orbital and stability studies using large scale quantum mechani(HOMO)—lowest unoccupied molecular orbital (LUMO) cal simulations of “nanotrees” made from Si and shown gap value of 0.57 eV which is smaller than that for the bulk that such structures are stable. Electronic structure analysis diamond phase of silicon (1.1 eV). This is because of the shows that formation of “nanotrees” are accompanied by presence of a signiﬁcant number of three-fold coordinated a narrowing of the HOMO–LUMO gap. This has imporunpassivated surface Si-atoms. Formation of “nanotrees” tant implications in their application in molecular devices causes the creation of states in the HOMO–LUMO gap, due to the predicted enhancement in the conducting propnarrowing the gap value to 0.16 eV. Furthermore, more erties. The present work is supported through grants states from the unoccupied levels are pulled in towards the by NSF (ITR-0221916), DOE (00-63857), and US-ARO Fermi level, EF (shown as dashed line). This is expected (W911NF-05-1-0372).

References

J. Comput. Theor. Nanosci. 4, 252–256, 2007

255

RESEARCH ARTICLE

Fig. 3. The densities of states (DOS) for “stem” and “nanotree” structures of tetrahedral Si, Si(Td). The Fermi level is shown as dashed line. The gap is seen to be narrower for the “nanotree” suggesting better potential for conducting applications.

1. A. M. Morales and. C. M. Lieber, Science 279, 208 (1998). 2. S.-W. Chung, J.-Y. Yu, and J. R. Heath, Appl. Phys. Lett. 76, 2068 (2000). 3. D. Appell, Nature 419, 553 (2002). 4. C. M. Lieber, Nano Lett. 2, 81 (2002). 5. G. Zheng, W. Lu, S. Jin, and C. M. Lieber, Adv. Mater. 16, 1890 (2004). 6. S. Watanabe, Y. A. Ono, T. Hashizume, Y. Wada, J. Yamauchi, and M. Tsukada, Phys. Rev. B 52, 10768 (1995). 7. J. Xia and K. W. Cheah, Phys. Rev. B 55, 15688 (1997). 8. A. M. Saitta, F. Buda, G. Fiumara, and P. V. Giaquinta, Phys. Rev. B 56, 1446 (1996). 9. A. J. Read, R. J. Needs, K. J. Nash, L. T. Canham, P. D. J. Calcott, and A. Qteish, Phys. Rev. Lett. 69, 2400 (1992). 10. T. Ohno, K. Shiraishi, and T. Ogawa, Phys. Rev. Lett. 69, 2400 (1992). 11. M. Menon and E. Richter, Phys. Rev. Lett. 83, 792 (1999). 12. B. Marsen and K. Sattler, Phys. Rev. B 60, 11593 (1999).

Si-Nanotrees: Structure and Electronic Properties 13. Y. Cui and C. M. Lieber, Science 291, 851 (2001). 14. K. A. Dick, K. Deppert, M. W. Larsson, T. Martensson, W. Seifert, L. R. Wallenberg, and L. Samuelson, Nature Mater. 3, 380 (2004). 15. Y. F. Zhang, Y. H. Tang, N. Wang, D. P. Yu, C. S. Lee, I. Bello, and S. T. Lee, Appl. Phys. Lett. 72, 1835 (1998). 16. A. M. Morales and C. M. Lieber, Science 279, 208 (1998). 17. D. D. D. Ma, C. S. Lee, F. C. K. Au, S. Y. Tong, and S. T. Lee, Science 299, 1874 (2003). 18. M. Menon and K. R. Subbaswamy, Phys. Rev. B 55, 9231 (1997). 19. M. Menon, E. Richter, and K. R. Subbaswamy, Phys. Rev. B 56, 12290 (1997).

Menon et al. 20. I. Ponomareva, M. Menon, D. Srivastava, and A. N. Andriotis, Phys. Rev. Lett. 95, 265502 (2005). 21. J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). 22. W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco (1980). 23. R. Hoffmann, J. Chem. Phys. 39, 1397 (1963). 24. L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, “ScaLAPACK Users’ Guide,” SIAM, Philadelphia, PA (1997). 25. P. Vogl, H. P. Hjalmarson, and J. D. Dow, J. Phys. Chem. Solids 44, 365 (1983). 26. I. Ponomareva, M. Menon, E. Richter, and A. N. Andriotis, Phys. Rev. B 74, 125311 (2006).

Received: 26 April 2006. Accepted: 3 August 2006.

RESEARCH ARTICLE

Delivered by Ingenta to: Amanda Barnard IP : 129.67.84.229 Tue, 13 Mar 2007 07:37:44

256

J. Comput. Theor. Nanosci. 4, 252–256, 2007