Continued reduction in characteristic dimensions in nanosystems has given rise to increasing importance of material interfaces on the overall system performance. With regard to thermal transport, this increases the need for a better fundamental understanding of the processes affecting interfacial thermal transport, as characterized by the thermal boundary conductance. When thermal boundary conductance is driven by phononic scattering events, accurate predictions of interfacial transport must account for anharmonic phononic coupling as this affects the thermal transmission. In this paper, a new model for phononic thermal boundary conductance is developed that takes into account anharmonic coupling, or inelastic scattering events, at the interface between two materials. Previous models for thermal boundary conductance are first reviewed, including the diffuse mismatch model, which only considers elastic phonon scattering events, and earlier attempts to account for inelastic phonon scattering, namely, the maximum transmission model and the higher harmonic inelastic model. A new model is derived, the anharmonic inelastic model, which provides a more physical consideration of the effects of inelastic scattering on thermal boundary conductance. This is accomplished by considering specific ranges of phonon frequency interactions and phonon number density conservation. Thus, this model considers the contributions of anharmonic, inelastically scattered phonons to thermal boundary conductance. This new anharmonic inelastic model shows improved agreement between the thermal boundary conductance predictions and experimental data at the Pb/diamond and Au/diamond interfaces due to its ability to account for the temperature dependent changing phonon population in diamond, which can couple anharmonically with multiple phonons in Pb and Au. We conclude by discussing phonon scattering selection rules at interfaces and the probability of occurrence of these higher order anharmonic interfacial phonon processes quantified in this work.
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.
0021-8979, 93
, pp. 793
–818
.2.
Chang
, C. W.
, Okawa
, D.
, Garcia
, H.
, Majumdar
, A.
, and Zettl
, A.
, 2008, “Breakdown of Fourier’s Law in Nanotube Thermal Conductors
,” Phys. Rev. Lett.
0031-9007, 101
, p. 075903
.3.
Prasher
, R. S.
, 2008, “Thermal Boundary Resistance and Thermal Conductivity of Multiwalled Carbon Nanotubes
,” Phys. Rev. B
0556-2805, 77
, p. 075424
.4.
Lee
, S. -M.
, Cahill
, D. G.
, and Venkatasubramanian
, R.
, 1997, “Thermal Conductivity of Si–Ge Superlattices
,” Appl. Phys. Lett.
0003-6951, 70
, pp. 2957
–2959
.5.
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
, pp. 3186
–3188
.6.
Gundrum
, B. C.
, Cahill
, D. G.
, and Averback
, R. S.
, 2005, “Thermal Conductance of Metal-Metal Interfaces
,” Phys. Rev. B
0556-2805, 72
, p. 245426
.7.
Hopkins
, P. E.
, Stevens
, R. J.
, and Norris
, P. M.
, 2008, “Influence of Inelastic Scattering at Metal-Dielectric Interfaces
,” ASME J. Heat Transfer
0022-1481, 130
, p. 022401
.8.
Lyeo
, H. -K.
, and Cahill
, D. G.
, 2006, “Thermal Conductance of Interfaces Between Highly Dissimilar Materials
,” Phys. Rev. B
0556-2805, 73
, p. 144301
.9.
Kapitza
, P. L.
, 1941, “The Study of Heat Transfer in Helium II
,” Zh. Eksp. Teor. Fiz. Pis'ma Red.
, 11
, pp. 1
–31
.10.
Little
, W. A.
, 1959, “The Transport of Heat Between Dissimilar Solids at Low Temperatures
,” Can. J. Phys.
0008-4204, 37
, pp. 334
–349
.11.
Swartz
, E. T.
, and Pohl
, R. O.
, 1989, “Thermal Boundary Resistance
,” Rev. Mod. Phys.
0034-6861, 61
, pp. 605
–668
.12.
Landry
, E.
, and Mcgaughey
, A. J. H.
, 2009, “Thermal Boundary Resistance Predictions From Molecular Dynamics Simulations and Theoretical Calculations
,” Phys. Rev. B
0556-2805, 80
, p. 165304
.13.
Swartz
, E. T.
, and Pohl
, R. O.
, 1987, “Thermal Resistances at Interfaces
,” Appl. Phys. Lett.
0003-6951, 51
, pp. 2200
–2202
.14.
Costescu
, R. M.
, Wall
, M. A.
, and Cahill
, D. G.
, 2003, “Thermal Conductance of Epitaxial Interfaces
,” Phys. Rev. B
0556-2805, 67
, p. 054302
.15.
Koh
, Y. K.
, and Cahill
, D. G.
, 2007, “Frequency Dependence of the Thermal Conductivity of Semiconductor Alloys
,” Phys. Rev. B
0556-2805, 76
, p. 075207
.16.
Hopkins
, P. E.
, Norris
, P. M.
, Stevens
, R. J.
, Beechem
, T.
, and Graham
, S.
, 2008, “Influence of Interfacial Mixing on Thermal Boundary Conductance Across a Chromium/Silicon Interface
,” ASME J. Heat Transfer
0022-1481, 130
, p. 062402
.17.
Stevens
, R. J.
, Smith
, A. N.
, and Norris
, P. M.
, 2005, “Measurement of Thermal Boundary Conductance of a Series of Metal-Dielectric Interfaces by the Transient Thermoreflectance Technique
,” ASME J. Heat Transfer
0022-1481, 127
, pp. 315
–322
.18.
Reddy
, P.
, Castelino
, K.
, and Majumdar
, A.
, 2005, “Diffuse Mismatch Model of Thermal Boundary Conductance Using Exact Phonon Dispersion
,” Appl. Phys. Lett.
0003-6951, 87
, p. 211908
.19.
Phelan
, P. E.
, 1998, “Application of Diffuse Mismatch Theory to the Prediction of Thermal Boundary Resistance in Thin-Film High-Tc Superconductors
,” ASME J. Heat Transfer
0022-1481, 120
, pp. 37
–43
.20.
Majumdar
, A.
, and Reddy
, P.
, 2004, “Role of Electron-Phonon Coupling in Thermal Conductance of Metal-Nonmetal Interfaces
,” Appl. Phys. Lett.
0003-6951, 84
, pp. 4768
–4770
.21.
Beechem
, T. E.
, Graham
, S.
, Hopkins
, P. E.
, and Norris
, P. M.
, 2007, “The Role of Interface Disorder on Thermal Boundary Conductance Using a Virtual Crystal Approach
,” Appl. Phys. Lett.
0003-6951, 90
, p. 054104
.22.
Prasher
, R. S.
, and Phelan
, P. E.
, 2001, “A Scattering-Mediated Acoustic Mismatch Model for the Prediction of Thermal Boundary Resistance
,” ASME J. Heat Transfer
0022-1481, 123
, pp. 105
–112
.23.
Beechem
, T.
, and Hopkins
, P. E.
, 2009, “Predictions of Thermal Boundary Conductance for Systems of Disordered Solids and Interfaces
,” J. Appl. Phys.
0021-8979, 106
, p. 124301
.24.
Stoner
, R. J.
, and Maris
, H. J.
, 1993, “Kapitza Conductance and Heat Flow Between Solids at Temperatures From 50 to 300 K
,” Phys. Rev. B
0556-2805, 48
, pp. 16373
–16387
.25.
Huberman
, M. L.
, and Overhauser
, A. W.
, 1994, “Electronic Kapitza Conductance at a Diamond-Pb Interface
,” Phys. Rev. B
0556-2805, 50
, pp. 2865
–2873
.26.
Sergeev
, A. V.
, 1998, “Electronic Kapitza Conductance Due To Inelastic Electron-Boundary Scattering
,” Phys. Rev. B
0556-2805, 58
, p. R10199
.27.
Sergeev
, A. V.
, 1999, “Inelastic Electron-Boundary Scattering in Thin Films
,” Physica B
0921-4526, 263–264
, pp. 217
–219
.28.
Chen
, Y.
, Li
, D.
, Yang
, J.
, Wu
, Y.
, Lukes
, J.
, and Majumdar
, A.
, 2004, “Molecular Dynamics Study of the Lattice Thermal Conductivity of Kr/Ar Superlattice Nanowires
,” Physica B
0921-4526, 349
, pp. 270
–280
.29.
Stevens
, R. J.
, Zhigilei
, L. V.
, and Norris
, P. M.
, 2007, “Effects of Temperature and Disorder on Thermal Boundary Conductance at Solid-Solid Interfaces: Nonequilibrium Molecular Dynamics Simulations
,” Int. J. Heat Mass Transfer
0017-9310, 50
, pp. 3977
–3989
.30.
Kittel
, C.
, 1996, Introduction to Solid State Physics
, Wiley
, New York
.31.
Hopkins
, P. E.
, Salaway
, R. N.
, Stevens
, R. J.
, and Norris
, P. M.
, 2007, “Temperature Dependent Thermal Boundary Conductance at Al/Al2O3 and Pt/Al2O3 Interfaces
,” Int. J. Thermophys.
0195-928X, 28
, pp. 947
–957
.32.
Snyder
, N. S.
, 1970, “Heat Transport Through Helium II: Kapitza Conductance
,” Cryogenics
0011-2275, 10
, pp. 89
–95
.33.
Klemens
, P. G.
, 1966, “Anharmonic Decay of Optical Phonons
,” Phys. Rev.
0031-899X, 148
, pp. 845
–848
.34.
Norris
, P. M.
, and Hopkins
, P. E.
, 2009, “Examining Interfacial Diffuse Phonon Scattering Through Transient Thermoreflectance Measurements of Thermal Boundary Conductance
,” ASME J. Heat Transfer
0022-1481, 131
, p. 043207
.35.
Chen
, G.
, 1998, “Thermal Conductivity and Ballistic-Phonon Transport in the Cross-Plane Direction of Superlattices
,” Phys. Rev. B
0556-2805, 57
, pp. 14958
–14973
.36.
Dames
, C.
, and Chen
, G.
, 2004, “Theoretical Phonon Thermal Conductivity of Si/Ge Superlattice Nanowires
,” J. Appl. Phys.
0021-8979, 95
, pp. 682
–693
.37.
Hopkins
, P. E.
, and Norris
, P. M.
, 2007, “Effects of Joint Vibrational States on Thermal Boundary Conductance
,” Nanoscale Microscale Thermophys. Eng.
1556-7265, 11
, pp. 247
–257
.38.
Kosevich
, Y. A.
, 1995, “Fluctuation Subharmonic and Multiharmonic Phonon Transmission and Kapitza Conductance Between Crystals With Very Different Vibrational Spectra
,” Phys. Rev. B
0556-2805, 52
, pp. 1017
–1024
.39.
Hopkins
, P. E.
, and Norris
, P. M.
, 2009, “Relative Contributions of Inelastic and Elastic Diffuse Phonon Scattering to Thermal Boundary Conductance Across Solid Interfaces
,” ASME J. Heat Transfer
0022-1481, 131
, p. 022402
.40.
Hopkins
, P. E.
, 2009, “Multiple Phonon Processes Contributing to Inelastic Scattering During Thermal Boundary Conductance at Solid Interfaces
,” J. Appl. Phys.
0021-8979, 106
, p. 013528
.41.
Chen
, G.
, 2005, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons
, Oxford University Press
, New York
.42.
Zhang
, Z.
, 2007, Nano/Microscale Heat Transfer
, McGraw-Hill
, New York
.43.
Gray
, D. E.
, 1972, American Institute of Physics Handbook
, McGraw-Hill
, New York
.44.
Touzelbaev
, M. N.
, and Goodson
, K. E.
, 1997, “Impact of Nucleation Density on Thermal Resistance Near Diamond-Substrate Boundaries
,” J. Thermophys. Heat Transfer
0887-8722, 11
, pp. 506
–512
.45.
Duda
, J. C.
, Beechem
, T.
, Smoyer
, J. L.
, Norris
, P. M.
, and Hopkins
, P. E.
, 2010, “The Role of Dispersion on Phononic Thermal Boundary Conductance
,” J. Appl. Phys.
0021-8979, 108
, p. 073515
.46.
Chen
, Z.
, Jang
, W.
, Bao
, W.
, Lau
, C. N.
, and Dames
, C.
, 2009, “Thermal Contact Resistance Between Graphene and Silicon Dioxide
,” Appl. Phys. Lett.
0003-6951, 95
, p. 161910
.47.
Duda
, J. C.
, Hopkins
, P. E.
, Smoyer
, J. L.
, Bauer
, M. L.
, English
, T. S.
, Saltonstall
, C. B.
, and Norris
, P. M.
, 2010, “On the Assumption of Detailed Balance in Prediction of Diffusive Transmission Probability During Interfacial Transport
,” Nanoscale Microscale Thermophys. Eng.
1556-7265, 14
, pp. 21
–33
.48.
Vincenti
, W. G.
, and Kruger
, C. H.
, 2002, Introduction to Physical Gas Dynamics
, Krieger
, Malabar, FL
.49.
Majumdar
, A.
, 1993, “Microscale Heat Conduction in Dielectric Thin Films
,” ASME J. Heat Transfer
0022-1481, 115
, pp. 7
–16
.50.
Chen
, G.
, 1997, “Size and Interface Effects on Thermal Conductivity of Superlattices and Periodic Thin-Film Structures
,” ASME J. Heat Transfer
0022-1481, 119
, pp. 220
–229
.51.
Yang
, R.
, and Chen
, G.
, 2004, “Thermal Conductivity Modeling of Periodic Two-Dimensional Nanocomposites
,” Phys. Rev. B
0556-2805, 69
, p. 195316
.52.
Brockhouse
, B. N.
, Arase
, T.
, Caglioti
, G.
, Rao
, K. R.
, and Woods
, A. D. B.
, 1962, “Crystal Dynamics of Lead. I. Dispersion Curves at 100 K
,” Phys. Rev.
0031-899X, 128
, pp. 1099
–1111
.53.
Weber
, W.
, 1977, “Atomic Bond Charge Model for the Phonons in Diamond, Si, Ge, and α-Sn
,” Phys. Rev. B
0556-2805, 15
, pp. 4789
–4803
.54.
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
0556-2805, 79
, p. 224305
.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States Government purposes.
You do not currently have access to this content.
