A new kinetic Monte Carlo method for modeling phonon transport in quantum dot superlattices is presented. The method uses phonon scattering phase functions and cross sections to describe collisions between phonons and quantum dots. The phase functions and cross sections are generated using molecular dynamics simulation, which is capable of including atomistic effects otherwise unavailable in Monte Carlo approaches. The method is demonstrated for a test case featuring a Si-Ge quantum dot superlattice, and the model is compared against published experiments. It is found that molecular dynamics-derived cross sections must be weighted by diffuse mismatch model-type weighting factors in order to satisfy detailed balance considerations. Additionally, it is found that thin alloy “base layer” films strongly reduce thermal conductivity in these systems and must be included in the modeling to obtain agreement with published experimental data.
Issue Section:
Research Papers
References
1.
Goldsmid
, H. J.
, 2009
, Introduction to Thermoelectricity
, R.
Hull
, R. M.
Osgood
, Jr., J.
Parisi
, and H.
Warlimont
, eds., Springer
, New York
.2.
Dresselhaus
, M. S.
, Chen
, G.
, Tang
, M. Y.
, Yang
, R.
, Lee
, H.
, Wang
, D.
, Ren
, Z.
, Fleurial
, J.-P.
, and Gogna
, P.
, 2007
, “New Directions for Low-Dimensional Thermoelectric Materials
,” Adv. Mater.
, 19
, pp. 1043
–1053
.10.1002/adma.2006005273.
Lee
, S.-M.
, Cahill
, D. G.
, and Venkatasubramanian
, R.
, 1997
, “Thermal Conductivity of Si-Ge Superlattices
,” Appl. Phys. Lett.
, 70
, p. 118755
.10.1063/1.1187554.
Chen
, G.
, 2006
, “Nanoscale Heat Transfer and Nanostructured Thermoelectrics
,” IEEE Trans. Compon. Packag. Technol.
, 29
(2
), pp. 238
–246
.10.1109/TCAPT.2006.8758955.
Balandin
, A.
, 2005
, “Nanophononics: Phonon Engineering in Nanostructures and Nanodevices
,” J. Nanosci. Nanotechnol.
, 5
, pp. 1
–8
.10.1166/jnn.2005.1756.
Liu
, J. L.
, Khitun
, A.
, Wang
, K. L.
, Borca-Tasciuc
, T.
, Liu
, W. L.
, Chen
, G.
, and Yu
, D. P.
, 2001
, “Growth of Ge Quantum Dot Superlattices for Thermoelectric Applications
,” J. Cryst. Growth
, 227–228
, pp. 1111
–1115
.10.1016/S0022-0248(01)00998-87.
Chen, G., 1996, “Nonlocal and Nonequilibrium Heat Conduction in the Vicinity of Nanoparticles,”
ASME J. Heat Transfer
, 118
, pp. 539–545.10.1115/1.28226658.
Yang
, R.
, and Chen
, G.
, 2004
, “Thermal Conductivity Modeling of Periodic Two-Dimensional Nanocomposites
,” Phys. Rev. B
, 69
, p. 195316
.10.1103/PhysRevB.69.1953169.
Ni
, C.
, and Murthy
, J. Y.
, 2012
, “Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times
,” ASME J. Heat Transfer
, 134
, p. 082401
.10.1115/1.400616910.
Chen
, G.
, 2002
, “Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to Macroscales
,” ASME J. Heat Transfer
, 124
, pp. 320
–328
.10.1115/1.144793811.
Nabovati
, A.
, Sellan
, D. P.
, and Amon
, C. H.
, 2011
, “On the Lattice Boltzmann Method for Phonon Transport
,” J. Comput. Phys.
, 230
, pp. 5864
–5876
.10.1016/j.jcp.2011.03.06112.
Zuckerman
, N.
, and Lukes
, J. R.
, 2008
, “Acoustic Phonon Scattering From Particles Embedded in an Anisotropic Medium: A Molecular Dynamics Study
,” Phys. Rev. B
, 77
, p. 094302
.10.1103/PhysRevB.77.09430213.
Khitun
, A.
, Balandin
, A.
, Liu
, J. L.
, and Wang
, K. L.
, 2001
, “The Effect of the Long-Range Order in a Quantum Dot Array on the In-Plane Lattice Thermal Conductivity
,” Superlattices and Microstruct.
, 30
, pp. 1
–8
.10.1006/spmi.2001.098114.
Zhang
, W.
, Fisher
, T. S.
, and Mingo
, N.
, 2007
, “The Atomistic Green’s Function Method: An Efficient Simulation Approach for Nanoscale Phonon Transport
,” Numer. Heat Transfer, Part B
, 51
, pp. 333
–349
.10.1080/1040779060114475515.
Klitsner
, T.
, VanCleve
, J. E.
, Fischer
, H. E.
, and Pohl
, R. O.
, 1988
, “Phonon Radiative Heat Transfer and Surface Scattering
,” Phys. Rev. B
, 38
, pp. 7576
–7594
.10.1103/PhysRevB.38.757616.
Peterson
, R. B.
, 1994
, “Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal
,” ASME J. Heat Transfer
, 116
, pp. 815
–822
.10.1115/1.291145217.
Jacoboni
, C.
, and Reggiani
, L.
, 1983
, “The Monte Carlo Method for the Solution of Charge Transport in Semiconductors With Applications to Covalent Materials
,” Rev. Mod. Phys.
, 55
, pp. 645
–705
.10.1103/RevModPhys.55.64518.
Pop
, E.
, Dutton
, R. W.
, and Goodson
, K. E.
, 2004
, “Analytic Band Monte Carlo Model for Electron Transport in Si Including Acoustic and Optical Phonon Dispersion
,” J. Appl. Phys.
, 96
, pp. 4998
–5005
.10.1063/1.178883819.
Farmer
, J. T.
, and Howell
, J. R.
, 1994
, “Monte Carlo Prediction of Radiative Heat Transfer in Inhomogeneous, Anisotropic, Nongray Media
,” AIAA J. Thermophys. Heat Transfer
, 8
, pp. 133
–139
.10.2514/3.51120.
Khitun
, A.
, Liu
, J.
, and Wang
, K. L.
, 2004
, “On the Modeling of Lattice Thermal Conductivity in Semiconductor Quantum Dot Superlattices
,” Appl. Phys. Lett.
, 84
(10
), pp. 1762
–1764
.10.1063/1.166831721.
Liu
, J. L.
, Khitun
, A.
, Wang
, K. L.
, Liu
, W. L.
, Chen
, G.
, Xie
, Q. H.
, and Thomas
, S. G.
, 2003
, “Cross-Plane Thermal Conductivity of Self-Assembled Ge Quantum Dot Superlattices
,” Phys. Rev. B
, 67
, p. 165333
.10.1103/PhysRevB.67.16533322.
Metropolis
, N.
, Rosenbluth
, A. W.
, Rosenbluth
, M. N.
, Teller
, A. H.
, and Teller
, E.
, 1953
, “Equation of State Calculations by Fast Computing Machines
,” J. Chem. Phys.
, 21
, pp. 1087
–1092
.10.1063/1.169911423.
Jeng
, M.-S.
, Yang
, R.
, Song
, D.
, and Chen
, G.
, 2008
, “Modeling the Thermal Conductivity and Phonon Transport in Nanoparticle Composites Using Monte Carlo Simulation
,” ASME J. Heat Transfer
, 130
, p. 042410
.10.1115/1.281876524.
Klemens
, P. G.
, 1951
, “The Thermal Conductivity of Dielectric Solids at Low Temperatures (Theoretical)
,” Proc. R. Soc., London, Ser. A
, 208
, pp. 108
–133
.10.1098/rspa.1951.014725.
Klemens
, P. G.
, 1955
, “The Scattering of Low-Frequency Lattice Waves by Static Imperfections
,” Proc. Phys. Soc., London
, A68
, pp. 1113
–1128
.10.1088/0370-1298/68/12/30326.
Broido
, D. A.
, Ward
, A.
, and Mingo
, N.
, 2005
, “Lattice Thermal Conductivity of Silicon From Empirical Interatomic Potentials
,” Phys. Rev. B
, 72
, p. 014308
.10.1103/PhysRevB.72.01430827.
Jacoboni
, C.
, and Reggiani
L.
, 1983
, “The Monte Carlo Method for the Solution of Charge Transport in Semiconductors With Applications to Covalent Materials
,” Rev. Mod. Phys.
, 55
, pp. 645
–705
.10.1103/RevModPhys.55.64528.
McGaughey
, A. J.
, and Kaviany
, M.
, 2004
, “Quantitative Validation of the Boltzmann Transport Equation Phonon Thermal Conductivity Model Under the Single-Mode Relaxation Time Approximation
,” Phys. Rev. B
, 69
, p. 094303
.10.1103/PhysRevB.69.09430329.
Chernatynskiy
, A.
, Turney
, J. E.
, McGaughey
, A. J. H.
, Amon
, C. H.
, and Phillpot
, S. R.
, 2011
, “Phonon-Mediated Thermal Conductivity in Ionic Solids by Lattice Dynamics-Based Methods
,” J. Am. Ceram. Soc.
, 94
, pp. 3523
–3531
.10.1111/j.1551-2916.2011.04743.x30.
Mazumder
, S.
, and Majumdar
, A.
, 2001
, “Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization
,” ASME J. Heat Transfer
, 123
, pp. 749
–759
.10.1115/1.137701831.
Chen
, Y.
, Li
, D.
, Lukes
, J. R.
, and Majumdar
, A.
, 2005
, “Monte Carlo Simulation of Silicon Nanowire Thermal Conductivity
,” ASME J. Heat Transfer
, 127
, pp. 1129
–1137
.10.1115/1.203511432.
Zuckerman
, N.
, 2011
, “Propagation and Scattering of Mechanical Vibrations in Semiconductor Materials
,” Ph.D. thesis, The University of Pennsylvania, Philadelphia, PA.33.
Holland
, M. G.
, 1963
, “Analysis of Lattice Thermal Conductivity
,” Phys. Rev.
, 132
(6
), pp. 2461
–2471
.10.1103/PhysRev.132.246134.
Majumdar
, A.
, 1993
, “Microscale Heat Conduction in Dielectric Thin Films
,” ASME J. Heat Transfer
, 115
(7
), pp. 7
–16
.10.1115/1.291067335.
Lacroix
, D.
, Joulain
, K.
, and Lemonnier
, D.
, 2005
, “Monte Carlo Transient Phonon Transport in Silicon and Germanium at Nanoscales
,” Phys. Rev. B
, 72
, p. 064305
.10.1103/PhysRevB.72.06430536.
Tian
, W.
, and Yang
, R.
, 2007
, “Thermal conductivity modeling of compacted nanowire composites
,” J. Appl. Phys.
, 101
, p. 054320
.10.1063/1.265377737.
Randrianalisoa
, J.
, and Baillis
, D.
, 2008
, “Monte Carlo Simulation of Steady-State Microscale Phonon Heat Transport
,” ASME J. Heat Transfer
, 130
, p. 072404
.10.1115/1.289792538.
Randrianalisoa
, J.
, and Baillis
, D.
, 2008
, “Monte Carlo Simulation of Cross-Plane Thermal Conductivity of Nanostructured Porous Silicon Films
,” J. Appl. Phys.
, 103
, p. 053502
.10.1063/1.284169739.
Moore
, A.
, Saha
, S.
, Prasher
, R.
, and Shi
, L.
, 2008
, “Phonon Backscattering and Thermal Conductivity Suppression in Sawtooth Nanowires
,” Appl. Phys. Lett.
, 94
, p. 083112
.10.1063/1.297004440.
Hao
, Q.
, Chen
, G.
, and Jeng
, M.
, 2009
, “Frequency-Dependent Monte Carlo Simulations of Phonon Transport in Two-Dimensional Porous Silicon With Aligned Pores
,” J. Appl. Phys.
, 106
, p. 114321
.10.1063/1.326616941.
Hao
, Q.
, Zhu
, G.
, Joshi
, G.
, Wang
, X.
, Minnich
, A.
, Ren
, Z.
, and Chen
, G.
, 2010
, “Theoretical Studies on the Thermoelectric Figure of Merit of Nanograined Bulk Silicon
,” Appl. Phys. Lett.
, 97
, p. 063109
.10.1063/1.347845942.
Henry
, A.
, and Chen
, G.
, 2008
, “Spectral Phonon Transport Properties of Silicon Based on Molecular Dynamics Simulations and Lattice Dynamics
,” J. Comput. Theor. Nanosci.
, 5
, pp. 1
–12
.10.1166/jctn.2008.001a43.
Mittal
, A.
, and Mazumder
, S.
, 2010
, “Monte Carlo Study of Phonon Heat Conduction in Silicon Thin Films Including Contributions of Optical Phonons
,” ASME J. Heat Transfer
, 132
, p. 052402
.10.1115/1.400044744.
Narumanchi
, S. J.
, Murthy
, J. Y.
, and Amon
, C. H.
, 2004
, “Submicron Heat Transport Model in Silicon Accounting for Phonon Dispersion and Polarization
,” ASME J. Heat Transfer
, 126
, pp. 946
–955
.10.1115/1.183336745.
Han
, Y.-J.
, and Klemens
, P. G.
, 1993
, “Anharmonic Thermal Resisitivity of Dielectric Crystals at Low Temperatures
,” Phys. Rev. B
, 48
(9
), pp. 6033
–6042
.10.1103/PhysRevB.48.603346.
Wang
, Z.
, Zhao
, R.-J.
, and Chen
, Y.-F.
, 2010
, “Monte Carlo Simulation of Phonon Transport in Variable Cross-Section Nanowires
,” Sci. China Technol. Sci.
, 53
, pp. 429
–434
.10.1007/s11431-009-0338-347.
Huang
, M.-J.
, Tsai
, T.-C.
, Liu
, L.-C.
, Jeng
, M.-S.
, and Yang
, C.-C.
, 2009
, “A Fast Monte-Carlo Solver for Phonon Transport in Nanostructured Semiconductors
,” Comput. Model. Eng. Sci.
, 42
(2
), pp. 107
–130
.10.3970/cmes.2009.042.10748.
Huang
, M.-J.
, Tsai
, T.-C.
, and Liu
, L.-C.
, 2010
, “A Study of Phonon Transport in Si/Ge Superlattice Thin Films Using a Fast MC Solver
,” J. Electron. Mater.
, 39
, pp. 1875
–1879
.10.1007/s11664-009-1066-y49.
Chen
, G.
, 1998
, “Thermal Conductivity and Ballistic-Phonon Transport in the Cross-Plane Direction of Superlattices
,” Phys. Rev. B
, 57
, pp. 14958
–14973
.10.1103/PhysRevB.57.1495850.
Masao
, Y.
, Okano
, M.
, and Matsumoto
, M.
, 2011
, “DSMC Scheme to Study Phonon Dynamics
,” J. Mech. Sci. Technol.
, 25
(1
), pp. 21
–26
.10.1007/s12206-010-1111-z51.
Péraud
, J.-P.
, and Hadjiconstantinou
, N. G.
, 2011
, “Efficient Simulation of Multidimensional Phonon Transport Using Energy-Based Variance-Reduced Monte Carlo Formulations
,” Phys. Rev. B
, 84
, p. 205331
.10.1103/PhysRevB.84.20533152.
Zuckerman
, N.
, and Lukes
, J. R.
, 2008
, “Monte Carlo Modeling of Phonon Transport Using Scattering Phase Functions
,” Proceedings of the 3rd ASME Energy Nanotechnology International Conference (ENIC2008)
, Jacksonville, FL, Aug. 10–14, Paper No. ENIC2008-53022.53.
Ashcroft
, N. W.
, and Mermin
, N. D.
, 1976
, Solid State Physics
, Thomson Learning
, London.54.
Brockhouse
, B. N.
, 1959
, “Lattice Vibrations in Silicon and Germanium
,” Phys. Rev. Lett.
, 2
(6
), pp. 256
–258
.10.1103/PhysRevLett.2.25655.
Goicochea
, J.
, Madrid
, M.
, and Amon
, C.
, 2010
, “Thermal Properties for Bulk Silicon Based on the Determination of Relaxation Times Using Molecular Dynamics
,” ASME J. Heat Transfer
, 132
, p. 012401
.10.1115/1.321185356.
Sharma
, P. C.
, and Rose
, M. F.
, 1988
, “Three-Phonon Scattering Processes and Their Role in Phonon Thermal Conductivity of Silicon
,” J. Solid State Chem.
, 73
, pp. 92
–97
.10.1016/0022-4596(88)90058-857.
Glassbrenner
, C. J.
, and Slack
, G. A.
, 1964
, “Thermal Conductivity of Silicon and Germanium from 3 K to the Melting Point
,” Phys. Rev.
, 134
, pp. A1058
–A1069
.10.1103/PhysRev.134.A105858.
Ward
, A.
, and Broido
, D. A.
, 2010
, “Intrinsic Phonon Relaxation Times From First-Principles Studies of the Thermal Conductivities of Si and Ge
,” Phys. Rev. B
, 81
, p. 085205
.10.1103/PhysRevB.81.08520559.
Kim
, W.
, and Majumdar
, A.
, 2006
, “Phonon Scattering Cross Section of Polydispersed Spherical Nanoparticles
,” J. Appl. Phys.
, 99
, p. 084306
.10.1063/1.218825160.
Stillinger
, F. H.
, and Weber
, T. A.
, 1985
, “Computer Simulation of Local Order in Condensed Phases of Silicon
,” Phys. Rev. B
, 31
, pp. 5262
–5271
.10.1103/PhysRevB.31.526261.
Ding
, K.
, and Andersen
, H. C.
, 1986
, “Molecular-Dynamics Simulation of Amorphous Germanium
,” Phys. Rev. B
, 34
, pp. 6987
–6991
.10.1103/PhysRevB.34.698762.
Vincenti
, W. G.
, and Kruger
, C. H.
, 1977
, Introduction to Physical Gas Dynamics
, Robert Krieger
, New York
.63.
Hall
, J. J.
, 1967
, “Electronic Effects in the Elastic Constants of n-Type Silicon
,” Phys. Rev.
, 161
, pp. 756
–761
.10.1103/PhysRev.161.75664.
Zuckerman
, N.
, and Lukes
, J. R.
, 2007
, “Atomistic Visualization of Ballistic Phonon Transport
,” Proceedings of the 2007 ASME-JSME Thermal Engineering Summer Heat Transfer Conference
, Vancouver, British Columbia, July 8–12, Paper No. HT2007-32674.65.
Przybilla
, J.
, Korn
, M.
, and Wegler
, U.
, 2006
, “Radiative Transfer of Elastic Waves Versus Finite Difference Simulations in Two-Dimensional Random Media
,” J. Geophys. Res.
, 111
, p. B04305
.10.1029/2005JB00395266.
Yamamoto
, M.
, and Sato
, H.
, 2010
, “Multiple Scattering and Mode Conversion Revealed by an Active Seismic Experiment at Asama Volcano, Japan
,” J. Geophys. Res.
, 115
, p. B07304
.10.1029/2009JB00710967.
Aki
, K.
, 1992
, “Scattering Conversions P to S Versus S to P
,” Bull. Seismol. Soc. Am.
, 82
(4
), pp. 1969
–1972
.68.
Ying
, C. F.
, and Truell
, R.
, 1956
, “Scattering of a Plane Longitudinal Wave by a Spherical Obstacle in an Isotropically Elastic Solid
,” J. Appl. Phys.
, 27
(9
), pp. 1086
–1097
.10.1063/1.172254569.
Swartz
, E. T.
, and Pohl
, R. O
, 1989
, “Thermal Boundary Resistance
,” Rev. Mod. Phys.
, 61
(3
), pp. 605
–668
.10.1103/RevModPhys.61.60570.
Korneev
, V. A.
, and Johnson
, L. R.
, 1993
, “Scattering of Elastic Waves by a Spherical Inclusion —I. Theory and Numerical Results
,” Geophys. J. Int.
, 115
, pp. 230
–250
.10.1111/j.1365-246X.1993.tb05601.x71.
Grundmann
, M.
, Ledentsov
, N. N.
, Kirstaedter
, N.
, Heinrichsdorff
, F.
, Krost
, A.
, Bimberg
, D.
, Kosogov
, A. O.
, Ruvimov
, S. S.
, Werner
, P.
, Ustinov
, M.
, Kop’ev
, P.S.
, and Alferov
, Zh. I.
, 1998
, “Semiconductor Quantum Dots for Application in Diode Lasers
,” Thin Solid Films
, 318
, pp. 83
–87
.10.1016/S0040-6090(97)01144-972.
Bauer
, G.
, Darhuber
, A. A.
, and Holy
, V.
, 1999
, “Self-Assembled Germanium-Dot Multilayers Embedded in Silicon
,” Cryst. Res. Technol.
, 34
, pp. 197
–209
.10.1002/(SICI)1521-4079(199902)34:2<197::AID-CRAT197>3.0.CO;2-A73.
Lipsanen
, H.
, Sopanen
, M.
, Tulkki
, J.
, Ahopelto
, J.
, Brasken
, M.
, and Lindberg
, M.
, 1999
, “Growth and Optical Properties of Strain-Induced Quantum Dots
,” Phys. Scr.
, T79
, pp. 20
–26
.10.1238/Physica.Topical.079a0002074.
Thanh
, L.
, and Yam
, V.
, 2003
, “Superlattices of Self-Assembled Ge/Si (001) Quantum Dots
,” Appl. Surf. Sci.
, 212
, pp. 296
–304
.10.1016/S0169-4332(03)00078-375.
Bao
, Y.
, Balandin
, A. A.
, Liu
, J. L.
, Liu
, J.
, and Xie
, Y. H.
, 2004
, “Experimental Investigation of Hall Mobility in Ge/Si Quantum Dot Superlattices
,” Appl. Phys. Lett.
, 84
, pp. 3355
–3357
.10.1063/1.171304976.
Lee
, M. L.
, and Venkatasubramanian
, R.
, 2008
, “Effect of Nanodot Areal Density and Period on Thermal Conductivity in SiGe/Si Nanodot Superlattices
,” Appl. Phys. Lett.
, 92
, p. 053112
.10.1063/1.284238877.
Khitun
, A.
, Liu
, J.
, and Wang
, K. L.
, 2004
, “On the Modeling of Lattice Thermal Conductivity in Semiconductor Quantum Dot Superlattices
,” Appl. Phys. Lett.
, 84
(10
), pp. 1762
–1764
.10.1063/1.166831778.
Liu
, J. L.
, Khitun
, A.
, Wang
, K. L.
, Liu
, W. L.
, Chen
, G.
, Xie
, Q. H.
, and Thomas
, S. G.
, 2003
, “Cross-Plane Thermal Conductivity of Self-Assembled Ge Quantum Dot Superlattices
,” Phys. Rev. B
, 67
, p. 165333
.10.1103/PhysRevB.67.16533379.
Alvarez-Quintana
, J.
, Alvarez
, X.
, Rodriguez-Viejo
, J.
, Jou
, D.
, Lacharmoise
, P. D.
, Bernardi
, A.
, Goñi
, A. R.
, and Alonso
, M. I.
, 2008
, “Cross-Plane Thermal Conductivity Reduction of Vertically Uncorrelated Ge/Si Quantum Dot Superlattices
,” Appl. Phys. Lett.
, 93
, p. 013112
.10.1063/1.295703880.
Pernot
, G.
, Stoffel
, M.
, Savic
, I.
, Pezzoli
, F.
, Chen
, P.
, Savelli
, G.
, Jacquot
, A.
, Schumann
, J.
, Denker
, U.
, Mönch
, I.
, Deneke
, C.
, Schmidt
, O. G.
, Rampnoux
, J. M.
, Wang
, S.
, Plissonnier
, M.
, Rastelli
, A.
, Dilhaire
, S.
, and Mingo
, N.
, 2010
, “Precise Control of Thermal Conductivity at the Nanoscale Through Individual Phonon-Scattering Barriers
,” Nature Mater.
, 9
, pp. 491
–495
.10.1038/nmat275281.
Bernardi
, A.
, Alonso
, M. I.
, Goñi
, A. R.
, Ossó
, J. O.
, and Garriga
, M.
, 2006
, “Density Control on Self-Assembling of Ge Islands Using Carbon-Alloyed Strained SiGe Layers
,” Appl. Phys. Lett.
, 89
, p. 101921
.10.1063/1.234931782.
Schelling
, P. K.
, and Phillpot
, S. R.
, 2003
, “Multiscale Simulation of Phonon Transport in Superlattices
,” J. Appl. Phys.
, 93
(9
), pp. 5377
–5387
.10.1063/1.156160183.
Crain’s Petrophysical Handbook, accessed
2008
, http://spec2000.net/01-index.htm84.
Balamane
, H.
, Halicioglu
, T.
, and Tiller
, W. A.
, 1992
, “Comparative Study of Silicon Empirical Interatomic Potentials
,” Phys. Rev. B
, 46
, pp. 2250
–2279
.10.1103/PhysRevB.46.225085.
Jian
, Z.
, Kaiming
, Z.
, and Xide
, X.
, 1990
, “Modification of Stillinger-Weber Potentials for Si and Ge
,” Phys. Rev. B
, 41
, pp. 12915
–12918
.10.1103/PhysRevB.41.12915Copyright © 2014 by ASME
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