The thermal conductivity of silicon thin films is predicted in the directions parallel and perpendicular to the film surfaces (in-plane and out-of-plane, respectively) using equilibrium molecular dynamics, the Green-Kubo relation, and the Stillinger-Weber interatomic potential. Three different boundary conditions are considered along the film surfaces: frozen atoms, surface potential, and free boundaries. Film thicknesses range from and temperatures from . The relation between the bulk phonon mean free path and the film thickness spans from the ballistic regime at to the diffusive, bulk-like regime at . When the film is thin enough, the in-plane and out-of-plane thermal conductivity differ from each other and decrease with decreasing film thickness, as a consequence of the scattering of phonons with the film boundaries. The in-plane thermal conductivity follows the trend observed experimentally at . In the ballistic limit, in accordance with the kinetic and phonon radiative transfer theories, the predicted out-of-plane thermal conductivity varies linearly with the film thickness, and is temperature-independent for temperatures near or above the Debye’s temperature.
Issue Section:
Micro/Nanoscale Heat Transfer
Keywords:
thermal conductivity,
silicon,
elemental semiconductors,
semiconductor thin films,
molecular dynamics method,
Green's function methods,
potential energy functions,
surface potential,
phonons,
Debye temperature,
ballistic transport,
equilibrium molecular dynamics,
silicon thin films,
in-plane,
out-of-plane thermal conductivity,
ballistic phonon transport
1.
International Technology Roadmap for Semiconductors
, 2005, http://public.itrs.nethttp://public.itrs.net.2.
Cahill
, D. G.
, Ford
, W. K.
, Goodson
, K. E.
, Mahan
, G. D.
, Majumdar
, A.
, Maris
, H. J.
, Merlin
, R.
, and Phillpot
, S. R.
, 2003, “Nanoscale Thermal Transport
,” J. Appl. Phys.
0021-8979, 93
(2
), pp. 793
–818
.3.
Li
, D.
, Huxtable
, S. T.
, Abramson
, A. R.
, and Majumdar
, A.
, 2005, “Thermal Transport in Nanostructured Solid-State Cooling Devices
,” ASME J. Heat Transfer
0022-1481, 127
, pp. 108
–114
.4.
Yang
, R.
, Chen
, G.
, Laroche
, M.
, and Taur
, Y.
, 2005, “Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation
,” ASME J. Heat Transfer
0022-1481, 127
, pp. 298
–306
.5.
Srivastava
, G. P.
, 1990, The Physics of Phonons
, A. Hilger
, New York
, pp. 175
–178
.6.
Hone
, J.
, Llaguno
, M. C.
, Biercuk
, M. J.
, Johnson
, A. T.
, Batlogg
, B.
, Benes
, Z.
, and Fischer
, J. E.
, 2002, “Thermal Properties of Carbon Nanotubes and Nanotube-Based Materials
,” Appl. Phys. A
0947-8396, A74
(3
), pp. 339
–343
.7.
Dresselhaus
, M. S.
, Dresselhaus
, G.
, Charlier
, J. C.
, and Hernandez
, E.
, 2004, “Electronic, Thermal and Mechanical Properties of Carbon Nanotubes
,” Philos. Trans. R. Soc. London, Ser. A
0962-8428, 362
(1823
), pp. 2065
–2098
.8.
Zhong
, H.
, and Lukes
, J. R.
, “Thermal Conductivity of Single-Wall Carbon Nanotubes
,” Proc. ASME International Mechanical Engineering Congress and Exposition
, Anaheim, CA.
, IMECE2004-61665.9.
Volz
, S. G.
, and Chen
, G.
, 1999, “Molecular Dynamics Simulation of Thermal Conductivity of Silicon Nanowires
,” Appl. Phys. Lett.
0003-6951, 75
(14
), pp. 2056
–2058
.10.
Li
, D.
, Wu
, Y.
, Kim
, P.
, Shi
, L.
, Yang
, P.
, and Majumdar
, A.
, 2003, “Thermal Conductivity of Individual Silicon Nanowires
,” Appl. Phys. Lett.
0003-6951, 83
(14
), pp. 2934
–2936
.11.
Li
, D.
, Wu
, Y.
, Fan
, R.
, Yang
, P.
, and Majumdar
, A.
, 2003, “Thermal Conductivity of Si∕SiGe Superlattice Nanowires
,” Appl. Phys. Lett.
0003-6951, 83
(15
), pp. 3186
–3188
.12.
Ju
, Y. S.
, and Goodson
, K. E.
, 1999, “Phonon Scattering in Silicon Thin Films with Thickness of Order 100nm
,” Appl. Phys. Lett.
0003-6951, 74
(20
), pp. 3005
–3007
.13.
Liu
, W.
, and Asheghi
, M.
, 2004, “Phonon-Boundary Scattering in Ultrathin Single-Crystal Silicon Layers
,” Appl. Phys. Lett.
0003-6951, 84
(19
), pp. 3819
–3821
.14.
Liu
, W.
, and Asheghi
, M.
, “Thermal Conductivity in Ultra-Thin Pure and Doped Single Crystal Silicon Layers at High Temperatures
,” Proc. ASME Summer Heat Transfer Conference
, San Francisco, CA.
, HT2005-72540.15.
Volz
, S.
, Saulnier
, J. B.
, Chen
, G.
, and Beauchamp
, P.
, 2000, “Computation of Thermal Conductivity of Si∕Ge Superlattice by Molecular Dynamics Technique
,” Microelectron. J.
0026-2692, 31
(9–10
), pp. 815
–819
.16.
Abramson
, A. R.
, Tien
, C.-L.
, and Majumdar
, A.
, 2002, “Interface and Strain Effects on the Thermal Conductivity of Heterostructures: A Molecular Dynamics Study
,” ASME J. Heat Transfer
0022-1481, 124
(5
), pp. 963
–970
.17.
Daly
, B. C.
, and Maris
, H. J.
, 2002, “Calculation of the Thermal Conductivity of Superlattices by Molecular Dynamics Simulation
,” Physica B
0921-4526, 316–317
, pp. 247
–249
.18.
Asheghi
, M.
, Touzelbaev
, M. N.
, Goodson
, K. E.
, Leung
, Y. K.
, and Wong
, S. S.
, 1998, “Temperature-Dependent Thermal Conductivity of Single-Crystal Silicon Layers in SOI Substrates
,” ASME J. Heat Transfer
0022-1481, 120
, pp. 30
–36
.19.
Liu
, W.
, and Asheghi
, M.
, “Thermal Conductivity of Ultra Thin Single Crystal Silicon Layers, Part 1—Experimental Measurements at Room and Cryogenic Temperatures
,” Proc. ASME International Mechanical Engineering Congress and Exposition
, Anaheim, CA.
, IMECE2004-62105.20.
Majumdar
, A.
, Fushinobu
, K.
, and Hijikata
, K.
, 1995, “Effect of Gate Voltage on Hot-Electron and Hot-Phonon Interaction and Transport in a Submicrometer Transistor
,” J. Appl. Phys.
0021-8979, 77
(12
), pp. 6686
–6694
.21.
Fushinobu
, K.
, Majumdar
, A.
, and Hijikata
, K.
, 1995, “Heat Generation and Transport in Submicron Semiconductor Devices
,” ASME J. Heat Transfer
0022-1481, 117
, pp. 25
–31
.22.
Lai
, J.
, and Majumdar
, A.
, 1996, “Concurrent Thermal and Electrical Modeling of Sub-Micrometer Silicon Devices
,” J. Appl. Phys.
0021-8979, 79
(9
), pp. 7353
–7361
.23.
Narumanchi
, S. V. J.
, Murthy
, J. Y.
, and Amon
, C. H.
, 2003, “Simulation of Unsteady Small Heat Source Effects in Sub-Micron Heat Conduction
,” ASME J. Heat Transfer
0022-1481, 125
(5
), pp. 896
–903
.24.
Narumanchi
, S. V. 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
0022-1481, 126
, pp. 946
–955
.25.
Amon
, C. H.
, Ghai
, S. S.
, Kim
, W. T.
, and Jhon
, M. S.
, 2006, “Modeling of Nanoscale Transport Phenomena: Application to Information Technology
,” Physica A
0378-4371, 362
, pp. 36
–41
.26.
Escobar
, R. A.
, Smith
, B.
, and Amon
, C. H.
, 2006, “Lattice Boltzmann Modeling of Subcontinuum Energy Transport in Crystalline and Amorphous Microelectronic Devices
,” ASME J. Electron. Packag.
1043-7398, 128
(2
), pp. 115
–124
.27.
Escobar
, R. A.
, Ghai
, S. S.
, Jhon
, M. S.
, and Amon
, C. H.
, 2006, “Multi-Length and Time Scale Thermal Transport Using the Lattice Boltzmann Method With Applications to Electronics Cooling
,” Int. J. Heat Mass Transfer
0017-9310, 49
, pp. 97
–107
.28.
Ghai
, S. S.
, Kim
, W. T.
, Escobar
, R.
, Amon
, C. H.
, and Jhon
, M. S.
, 2005, “A Novel Heat Transfer Model and Its Application to Information Storage Systems
,” J. Appl. Phys.
0021-8979, 97
, pp. 10P703
-1–3
.29.
Narumanchi
, S. V. J.
, Murthy
, J. Y.
, and Amon
, C. H.
, 2005, “Comparison of Different Phonon Transport Models for Predicting Heat Conduction in Silicon-on-Insulator Transistors
,” ASME J. Heat Transfer
0022-1481, 127
(7
), pp. 713
–723
.30.
Narumanchi
, S. V. J.
, Murthy
, J. Y.
, and Amon
, C. H.
, 2006, “Boltzmann Transport Equation-Based Thermal Modeling Approaches for Hotspots in Microelectronics
,” Heat Mass Transfer
0947-7411, 42
(6
), pp. 478
–491
.31.
Klitsner
, T.
, VanCleve
, J. E.
, Fischer
, H. E.
, and Pohl
, R. O.
, 1988, “Phonon Radiative Heat Transfer and Surface Scattering
,” Phys. Rev. B
0163-1829, 38
, pp. 7576
–7594
.32.
Peterson
, R. B.
, 1994, “Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal
,” ASME J. Heat Transfer
0022-1481, 116
, pp. 815
–822
.33.
Mazumder
, S.
, and Majumdar
, A.
, 2001, “Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization
,” ASME J. Heat Transfer
0022-1481, 123
, pp. 749
–759
.34.
Chen
, Y.
, Li
, D.
, Lukes
, J.
, and Majumdar
, A.
, 2005, “Monte Carlo Simulation of Silicon Nanowire Thermal Conductivity
,” ASME J. Heat Transfer
0022-1481, 127
, pp. 1129
–1137
.35.
Li
, J.
, Porter
, L. J.
, and Yip
, S.
, 1998, “Atomistic Modeling of Finite-Temperature Properties of Crystalline β-SiC. II. Thermal Conductivity and Effects of Point Defects
,” J. Nucl. Mater.
0022-3115, 255
, pp. 139
–152
.36.
Lukes
, J. R.
, Li
, D. Y.
, Liang
, X.-G.
, and Tien
, C.-L.
, 2000, “Molecular Dynamics Study of Solid Thin-Film Thermal Conductivity
,” ASME J. Heat Transfer
0022-1481, 122
, pp. 536
–543
.37.
Volz
, S. G.
, and Chen
, G.
, 2000, “Molecular-Dynamics Simulation of Thermal Conductivity of Silicon Crystals
,” Phys. Rev. B
0163-1829, 61
(4
), pp. 2651
–2656
.38.
Schelling
, P. K.
, Phillpot
, S. R.
, and Keblinski
, P.
, 2002, “Comparison of Atomic-Level Simulation Methods for Computing Thermal Conductivity
,” Phys. Rev. B
0163-1829, 65
, pp. 144306
.39.
Gomes
, C. J.
, Madrid
, M.
, and Amon
, C. H.
, “Thin Film In-Plane Thermal Conductivity Dependence on Molecular Dynamics Surface Boundary Conditions
,” Proc. ASME International Mechanical Engineering Congress and Exposition
, Anaheim, CA
, IMECE2004-62264.40.
McGaughey
, A. J. H.
, and Kaviany
, M.
, 2004, “Thermal Conductivity Decomposition and Analysis Using Molecular Dynamics Simulations. Part II. Complex Silica Structures
,” Int. J. Heat Mass Transfer
0017-9310, 47
, pp. 1799
–1816
.41.
Che
, J.
, Cagin
, T.
, Deng
, W.
, and Goddard
, W. A.
, 2000, “Thermal Conductivity of Diamond and Related Materials From Molecular Dynamics Simulations
,” J. Chem. Phys.
0021-9606, 113
(16
), pp. 6888
–6900
.42.
Volz
, S. G.
, and Chen
, G.
, 1999, “Lattice Dynamic Simulation of Silicon Thermal Conductivity
,” Physica B
0921-4526, 263–264
, pp. 709
–712
.43.
Gomes
, C. J.
, Madrid
, M.
, and Amon
, C. H.
, “Parallel Molecular Dynamics Code Validation Through Bulk Silicon Thermal Conductivity Calculations
,” Proc. ASME International Mechanical Engineering Congress and Exposition
, Washington, DC.
, IMECE2003-42352.44.
Che
, J.
, Cagin
, T.
, and Goddard
, W. A.
, III, 2000, “Thermal Conductivity of Carbon Nanotubes
,” Nanotechnology
0957-4484, 11
, pp. 65
–69
.45.
Lee
, Y. H.
, Biswas
, R.
, Soukoulis
, C. M.
, Wang
, C. Z.
, Chan
, C. T.
, and Ho
, K. M.
, 1991, “Molecular-Dynamics Simulation of Thermal Conductivity in Amorphous Silicon
,” Phys. Rev. B
0163-1829, 43
(8
), pp. 6573
–6580
.46.
Ding
, K.
, and Andersen
, H. C.
, 1986, “Molecular-Dynamics Simulation of Amorphous Germanium
,” Phys. Rev. B
0163-1829, 34
(10
), pp. 6987
–6991
.47.
Feng
, X.-L.
, Li
, D.
, and Guo
, Z.-Y.
, 2003, “Molecular Dynamics Simulation of Thermal Conductivity of Nanoscale Thin Silicon Films
,” Microscale Thermophys. Eng.
1089-3954, 7
, pp. 153
–161
.48.
Weakliem
, P. C.
, and Carter
, E. A.
, 1992, “Constant Temperature Molecular Dynamics Simulations of Si(100) and Ge(100): Equilibrium Structure and Short-Time Behavior
,” J. Chem. Phys.
0021-9606, 96
(4
), pp. 3240
–3250
.49.
Yu
, Q.
, and Clancy
, P.
, 1994, “Molecular Dynamics Simulation of the Surface Reconstruction and Strain Relief in Si1−xGex∕Si(100) Heterostructures
,” Modell. Simul. Mater. Sci. Eng.
0965-0393, 2
, pp. 829
–844
.50.
Chantrenne
, P.
, and Barrat
, J.-L.
, 2004, “Finite Size Effects in Determination of Thermal Conductivities: Comparing Molecular Dynamics Results With Simple Models
,” ASME J. Heat Transfer
0022-1481, 126
, pp. 577
–585
.51.
Stillinger
, F. H.
, and Weber
, T. A.
, 1985, “Computer Simulation of Local Order in Condensed Phases of Silicon
,” Phys. Rev. B
0163-1829, 31
(8
), pp. 5262
–5271
.52.
Broughton
, J. Q.
, and Li
, X. P.
, 1987, “Phase Diagram of Silicon by Molecular Dynamics
,” Phys. Rev. B
0163-1829, 35
(17
), pp. 9120
–9127
.53.
Cook
, S. J.
, and Clancy
, P.
, 1993, “Comparison of Semi-Empirical Potential Functions for Silicon and Germanium
,” Phys. Rev. B
0163-1829, 47
(13
), pp. 7686
–7699
.54.
Karimi
, M.
, Yates
, H.
, Ray
, J. R.
, Kaplan
, T.
, and Mostoller
, M.
, 1998, “Elastic Constants of Silicon Using Monte Carlo Simulations
,” Phys. Rev. B
0163-1829, 58
(10
), pp. 6019
–6025
.55.
Kallman
, J. S.
, Hoover
, W. G.
, Hoover
, C. G.
, Groot
, A. J. D.
, Lee
, S. M.
, and Wooten
, F.
, 1993, “Molecular Dynamics of Silicon Indentation
,” Phys. Rev. B
0163-1829, 47
(13
), pp. 7705
–7709
.56.
Allen
, M. P.
, and Tildesley
, D. J.
, 1987, Computer Simulation of Liquids
, Clarendon Press
, Oxford, UK
, pp. 61
–62
.57.
Frenkel
, D.
, and Smit
, B.
, 2001, Understanding Molecular Simulation: From Algorithms to Applications
, Academic
, San Diego, CA.
, p. 90
.58.
McQuarrie
, D. A.
, 1976, Statistical Mechanics
, University Science Books
, Sausalito, CA.
, p. 521
.59.
Hardy
, R. J.
, 1963, “Energy-Flux Operator for a Lattice
,” Phys. Rev.
0031-899X, 132
(1
), pp. 168
–177
.60.
Reif
, F.
, Fundamentals of Statistical and Thermal Physics
, McGraw-Hill
, New York
, pp. 480
–483
.61.
Flubacher
, P.
, Leadbetter
, A. J.
, and Morrison
, J. A.
, 1959, “Heat Capacity of Pure Silicon and Germanium and Properties of Their Vibrational Frequency Spectra
,” Philos. Mag.
0031-8086, 4
(39
), pp. 273
–294
.62.
Desai
, P. D.
, 1986, “Thermodynamic Properties of Iron and Silicon
,” J. Phys. Chem. Ref. Data
0047-2689, 15
(3
), pp. 967
–983
.63.
Porter
, L. J.
, Justo
, J. F.
, and Yip
, S.
, 1997, “The Importance of Gruneisen Parameters in Developing Interatomic Potentials
,” J. Appl. Phys.
0021-8979, 82
, pp. 5378
–5381
.64.
Ashcroft
, N. W.
, and Mermin
, N. D.
, 1976, Solid State Physics
, Holt, Rinehart and Winston
, New York
, p. 461
.65.
Aono
, M.
, Hou
, Y.
, Oshima
, C.
, and Ishizawa
, Y.
, 1982, “Low-Energy Ion Scattering From the Si(001) Surface
,” Phys. Rev. Lett.
0031-9007, 49
, pp. 567
–570
.66.
Yang
, W. S. F.
, Jona
, F.
, and Marcus
, P. M.
, 1983, “Atomic Structure of Si{001}2×1
,” Phys. Rev. B
0163-1829, 28
(4
), pp. 2049
–2059
.67.
Ho
, C. Y.
, Powell
, R. W.
, and Liley
, P. E.
, 1972, “Thermal Conductivity of the Elements
,” J. Phys. Chem. Ref. Data
0047-2689, 1
(2
), pp. 279
–421
.68.
Majumdar
, A.
, 1993, “Microscale Heat Conduction in Dielectric Thin Films
,” ASME J. Heat Transfer
0022-1481, 115
, pp. 7
–16.
Copyright © 2006
by American Society of Mechanical Engineers
You do not currently have access to this content.
