We report finite-volume simulations of the phonon Boltzmann transport equation (BTE) for heat conduction across the heterogeneous interfaces in SiGe superlattices. The diffuse mismatch model incorporating phonon dispersion and polarization is implemented over a wide range of Knudsen numbers. The results indicate that the thermal conductivity of a Si/Ge superlattice is much lower than that of the constitutive bulk materials for superlattice periods in the submicron regime. We report results for effective thermal conductivity of various material volume fractions and superlattice periods. Details of the nonequilibrium energy exchange between optical and acoustic phonons that originate from the mismatch of phonon spectra in silicon and germanium are delineated for the first time. Conditions are identified for which this effect can produce significantly more thermal resistance than that due to boundary scattering of phonons.
Issue Section:
Micro/Nanoscale Heat Transfer
Keywords:
Boltzmann equation,
elemental semiconductors,
finite volume methods,
germanium,
heat conduction,
phonon dispersion relations,
semiconductor superlattices,
silicon,
thermal conductivity,
superlattices,
diffuse mismatch model,
ballistic-diffusive transition,
phonon scattering,
interfacial thermal resistance
References
1.
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.
, 93
(2
), pp. 793
–818
.2.
Bo
, X. -Z. -Z.
, Lee
, C. Y.
, Strano
, M. S.
, Goldfinger
, M.
, Nuckolls
, C.
, and Blanchet
, G. B.
, 2005 “Carbon Nanotubes-Semiconductor Networks for Organic Electronics: The Pickup Stick Transistor,”
Appl. Phys. Lett.
, 86
(18
), 182102
.3.
Cola
, B. A.
, Xu
, J.
, Cheng
, C.
, Hu
, H.
, Xu
, X.
, and Fisher
, T. S.
, 2007, “Photoacoustic Characterization of Carbon Nanotube Array Thermal Interfaces,”
J. Appl. Phys.
, 101
(5
), 054313, pp. 1
–9
.4.
Dresselhaus
, M. S.
, Lin
, M. S.
, Koga
, T.
, Cronin
, S. B.
, Rabin
, O.
, Black
, M. R.
, and Dresselhaus
, G.
, 2001, Semiconductors and Semimetals: Recent Trends in Thermoelectric Materials Research III
, T. M.
Tritt
, ed., Academic Press
, San Diego, CA
, Vol. 71
, pp. 1
–121
.5.
Rowe
, D. M.
, 2005, Thermoelectrics—Macro to Nano
, CRC Press
, Boca Raton, FL
.6.
Li
, D.
, Wu
, Y.
, Kim
, P.
, Shi
, L.
, Yang
, P.
, and Majumdar
, A.
, 2003, “Thermal Conductivity of Individual Silicon Nanowires,”
Appl. Phys. Lett.
, 83
(14
), pp. 2934
–2936
.7.
Kim
, P.
, Shi
, L.
, Majumdar
, A.
, and McEuen
, P. L.
, 2001, “Thermal Transport Measurements of Individual Multiwalled Nanotubes,”
Phys. Rev. Lett.
, 87
(21
), 215502
.8.
Asheghi
, M.
, Leung
, Y. K.
, Wong
, S. S.
, and Goodson
, K. E.
, 1997, “Phonon-Boundary Scattering in Thin Silicon Layers,”
Appl. Phys. Lett.
, 71
(13
), pp. 1798
–1800
. 9.
Lü
, X.
, Shen
, W. Z.
, and Chu
, J. H.
, 2002, “Size Effect on the Thermal Conductivity of Nanowires,”
J. Appl. Phys.
91
(3
), pp. 1542
1552.10.
Dingle
, R. B.
, 1949, “The Electrical Conductivity of Thin Wires,”
Proceedings of the Royal Society (London)
, A 201
, pp. 545
–560
.11.
Sondheimer
, E. H.
, 1952, “The Mean Free Path of Electrons in Metals,”
Adv. Phys.
1
(1
), pp. 1
–42
.12.
Mingo
, N.
, Yang
, L.
, Li
, D.
, and Majumdar
, A.
, 2003, “Predicting the Thermal Conductivity of Si and Ge Nanowires,”
Nano Lett.
, 3
(12
), pp. 1713
–1716
.13.
Zhang
, W.
, Mingo
, N.
, and Fisher
, T. S.
, 2007, “Simulation of Phonon Transport Across a Nonpolar Nanowire Junction Using an Atomistic Green’s Function Method,”
Phys. Rev. B
, 76
, 195429
.14.
Zhang
, W.
, Mingo
, N.
, and Fisher
, T. S.
, 2007, “Simulation of Interfacial Phonon Transport in Si–Ge Heterostructures Using an Atomistic Green’s Function Method,”
ASME J. Heat Transfer
, 129
, pp. 483
–491
.15.
Bulusu
, A.
, and Walker
, D. G.
, 2007, “Quantum Modeling of Thermoelectric Performance of Strained Si/Ge/Si Superlattices Using the Nonequilibrium Green’s Function Method,”
J. Appl. Phys.
, 102
(7
), 073713
.16.
Simkin
, M. V.
, and Mahan
, G. D.
, 2000, “Minimum Thermal Conductivity of Superlattices
,” Phys. Rev. Lett.
, 84
(5
), pp. 927
–930.
17.
Venkatasubramanian
, R.
, 2000, “Lattice Thermal Conductivity Reduction and Phonon Localization Like Behavior in Superlattice Structures
,” Phys. Rev. B
, 61
(4
), pp. 3091
–3097.
18.
Chen
, Y.
, Li
, D.
, Lukes
, J. R.
, Ni
, Z.
, and Chen
, M.
, 2005, “Minimum Superlattice Thermal Conductivity From Molecular Dynamics,”
Phys. Rev. B
, 72
(17
), 174302
.19.
Landry
, E. S.
, and McGaughey
, A. J. H.
, 2009, “Effect of Interfacial Species Mixing on Phonon Transport in Semiconductor Superlattices
,” Phys. Rev. B
, 79
, 075316.
20.
Little
, W. A.
, 1959, “The Transport of Heat Between Dissimilar Solids at Low Temperatures,”
Can. J. Phys.
, 37
, pp. 334
–349
.21.
Swartz
, E. T.
, and Pohl
, R. O.
, 1989, “Thermal Boundary Resistance
,” Rev. Mod. Phys.
, 61
(3
), pp. 605
–668.
22.
Schelling
, P. K.
, Phillpot
, S. R.
, and Keblinski
, P.
, 2004, “Kapitza Conductance and Phonon Scattering at Grain Boundaries by Simulation
,” J. Appl. Phys.
, 927 95
(11
), pp. 6082
–6091.
23.
Schelling
, P. K.
, Phillpot
, S. R.
, and Keblinski
, P.
, 2002, “Phonon Wave-Packet Dynamics at Semiconductor Interfaces by Molecular-Dynamics Simulation
,” Appl. Phys. Lett.
, 80
(14
), pp. 2484
–2486.
24.
Sun
, L.
, and Murthy
, J. Y.
, 2008, “Molecular Dynamics Simulation of Phonon Scattering at Rough Semiconductor Surfaces,”
Proceedings of ITHERM 2008
, Orlando, FL
.25.
Zhao
, H.
, and Freund
, J. B.
, 2009, “Phonon Scattering at a Rough Interface Between Two fcc Lattices,”
J. Appl. Phys.
, 105
(1
), 013515
.26.
Ren
, S. Y.
, and Dow
, J. D.
, 1982, “Thermal Conductivity of Superlattices
,” Phys. Rev. B
, 25
(6
), pp. 3750
–3755.
27.
Broido
, D. A.
, and Reinecke
, T. L.
, 2004, “Lattice Thermal Conductivity of Superlattice Structures,”
Phys. Rev. B
, 70
(8
), 081310
.28.
Ward
, A.
, and Broido
, D. A.
, 2008, “Intrinsic Lattice Thermal Conductivity of Si/Ge and GaAs/AlAs Superlattices,”
Phys. Rev. B
, 77
(24
), 245328
.29.
Cahill
, D. G.
, 1997, “Heat Transport in Dielectric Thin Films and at Solid–Solid Interfaces
,” Microscale Thermophys. Eng.
, 1
, pp. 85
–109.
30.
Lee
, S. M.
, Cahill
, D. G.
, and Venkatasubramanian
, R.
, 1997, “Thermal Conductivity of Si–Ge Superlattices
,” Appl. Phys. Lett.
, 70
(22
), pp. 2957
–2959.
31.
Chen
, G.
, 1998, “Thermal Conductivity and Ballistic Phonon Transport in the Cross-Plane Direction of Superlattices
,” Phys. Rev. B
, 57
(23
), pp. 14958
–14973.
32.
Borca-Tasciuc
, T.
, Liu
, W. L.
, Liu
, J. L.
, Zeng
, T.
, Song
, D. W.
, Moore
, C. D.
, Chen
, G.
, Wang
, K. L.
, Goorsky
, M. S.
, Radetic
, T.
, Gronsky
, R.
, Sun
, X.
, and Dresselhauss
, M. S.
, 2000, “Thermal Conductivity of Symmetrically Strained Si/Ge Superlattices
,” Superlattices Microstruct.
, 28
, pp. 199
–206.
33.
Huxtable
, S. T.
, Abramson
, A.
, Majumdar
, A.
, Tien
, C. L.
, Zeng
, G.
, LaBounty
, C.
, Shakouri
, A.
, Bowers
, J. E.
, and Croke
, E.
, 2002, “Thermal Conductivity of Si/SiGe and SiGe/SiGe Superlattices
,” Appl. Phys. Lett.
, 80
(10
), pp. 1737
–1739.
34.
Li
, D.
, Wu
, Y.
, Fan
, R.
, Yang
, P.
, and Majumdar
, A.
, 2003, “Thermal Conductivity of Si/SiGe Superlattice Nanowires
,” Appl. Phys. Lett.
, 83
(15
), pp. 3186
–3188.
35.
Dames
, C.
, and Chen
, G.
, 2004, “Theoretical Phonon Thermal Conductivity of Si–Ge Superlattice Nanowires
,” J. Appl. Phys.
, 95
(2
), pp. 682
–693.
36.
Chen
, G.
, and Neagu
, M.
, 1997, “Thermal Conductivity and Heat Transfer in Superlattices
,” Appl. Phys. Lett.
, 71
(19
), pp. 2761
–2763.
37.
Chen
, G.
, 1997, “Size and Interface Effects on the Thermal Conductivity of Superlattices and Periodic Thin Film Structures
,” ASME J. Heat Transfer
, 119
(2
), pp. 220
–229.
38.
Yang
, R.
, and Chen
, G.
, 2004, “Thermal Conductivity Modeling of Periodic Two-Dimensional Nanocomposities,”
Phys. Rev. B
, 69
(19
), 195316, pp. 1
–10
.39.
Pascual-Gutiérrez
, J. A.
, Murthy
, J. Y.
, and Viskanta
, R. V.
, 2007, “Limits of Size Confinement in Silicon Thin Films and Wires,”
J. Appl. Phys.
, 102
(3
), 034315
.40.
Murthy
, J. Y.
, Narumanchi
, S. V. J.
, Pascual-Gutierrez
, J. A.
, Wang
, T.
, Ni
, C.
, and Mathur
, S. R.
, 2005, “Review of Multiscale Simulation in Submicron Heat Transfer
,” Int. J. Multiscale Comp. Eng.
, 3
(1
), pp. 5
–32.
41.
Joshi
, A. A.
, and Majumdar
, A.
, 1993, “Transient Ballistic and Diffusive Phonon Heat Transport in Thin Films
,” J. Appl. Phys.
, 74
(1
), pp. 31
–39.
42.
Ni
, C.
, 2009, “Phonon Transport Models for Heat Conduction in Sub-micron Geometries with Application to Microelectronics,”
Ph.D. thesis, Purdue University, West Lafayette, IN.43.
McGaughey
, A. J. H.
, 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
, 094303
.44.
Reddy
, P.
, Castelino
, K.
, and Majumdar
, A.
, 2005, “Diffuse Mismatch Model of Thermal Boundary Conductance Using Exact Phonon Dispersion,”
Appl. Phys. Lett.
, 87
(21
), 211908
.45.
Bazant
, M. Z.
, Kaxiras
, E.
, and Justo
, J. F.
, 1997, “Environment Dependent Interatomic Potential for Bulk Silicon,”
Phys. Rev. B
, 56
(14
), pp. 8542
–8552
.46.
Harrison
, W. A.
, 1989, Electronic Structure and the Properties of Solids
, Dover
, Mineola, NY
.47.
Klemens
, P. G.
, 1958, Solid State Physics
, Vol. 7
, F.
Seitz
and D.
Turnbull
, eds., Academic
, New York
, p.
1
.48.
Glassbrenner
, C. J.
, and Slack
, G. A.
, 1964, “Thermal Conductivity of Silicon and Germanium From 3 K to the Melting Point
,” Phys. Rev.
, 134
(4A
), pp. A1058
–A1069.
49.
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. 141
–152.
50.
Turney
, J. E.
, McGaughey
, A. J. H.
, and Amon
, C. H.
, 2009, “Assessing the Applicability of Quantum Corrections to Classical Thermal Conductivity Predictions,”
Phys. Rev. B
, 79
, 224305
.51.
Pascual-Gutiérrez
, J. A.
, Murthy
, J. Y.
, and Viskanta
, R. V.
, 2009, “Thermal Conductivity and Phonon Transport Properties of Silicon Using Perturbation Theory and the Environment-Dependent Interatomic Potential,”
J. Appl. Phys.
, 106
, 063532
.52.
Murthy
, J. Y.
, and Mathur
, S. R.
, 2002, “Computation of Sub-Micron Thermal Transport Using an Unstructured Finite Volume Method,”
ASME J. Heat Transfer
, 124
(6
), pp. 1176
–1181
.53.
Roe
, P. L.
, 1986, “Characteristic-Based Schemes for the Euler Equations
,” Annu. Rev. Fluid Mech.
, 18
, pp. 337
–365.
54.
Majumdar
, A.
, and Reddy
, P.
, 2004, “Role of Electron–Phonon Coupling in Thermal Conductance of Metal–Nonmetal Interfaces
,” Appl. Phys. Lett.
, 84
, pp. 4768
–4770.
55.
Sungtaek Ju
, Y.
, Hung
, M. T.
, and Usui
, T.
, 2006, “Nanoscale Heat Conduction Across Metal-Dielectric Interfaces
,” J. Heat Transfer
, 128
(9
), pp. 919
–925.
Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
