In this article, two models for phonon transmission across semiconductor interfaces are investigated and demonstrated in the context of large-scale spatially three-dimensional calculations of the phonon Boltzmann transport equation (BTE). These include two modified forms of the classical diffuse mismatch model (DMM): one, in which dispersion is accounted for and another, in which energy transfer between longitudinal acoustic (LA) and transverse acoustic (TA) phonons is disallowed. As opposed to the vast majority of the previous studies in which the interface is treated in isolation, and the thermal boundary conductance is calculated using closed-form analytical formulations, the present study also considers the interplay between the interface and intrinsic (volumetric) scattering of phonons. This is accomplished by incorporating the interface models into a parallel solver for the full seven-dimensional BTE for phonons. A verification study is conducted in which the thermal boundary resistance of a silicon/germanium interface is compared against the previously reported results of molecular dynamics (MD) calculations. The BTE solutions overpredicted the interfacial resistance, and the reasons for this discrepancy are discussed. It is found that due to the interplay between intrinsic and interface scattering, the interfacial thermal resistance across a Si(hot)/Ge(cold) bilayer is different from that of a Si(cold)/Ge(hot) bilayer. Finally, the phonon BTE is solved for a nanoscale three-dimensional heterostructure, comprised of multiple blocks of silicon and germanium, and the time evolution of the temperature distribution is predicted and compared against predictions using the Fourier law of heat conduction.

References

References
1.
Ju
,
Y. S.
, and
Goodson
,
K. E.
,
1999
, ‘‘
Phonon Scattering in Silicon Films With Thickness of Order 100 nm
,’’
Appl. Phys. Lett.
,
74
(
20
), pp.
3005
3007
.10.1063/1.123994
2.
Tien
,
C. L.
,
Majumdar
,
A.
, and
Gerner
,
F. M.
, eds.,
1998
,
Microscale Energy Transport
,
Taylor and Francis
,
Bristol, PA
.
3.
Zhang
,
Z.
,
2007
,
Nano/Microscale Heat Transfer
,
McGraw-Hill
,
New York
.
4.
Swartz
,
E. T.
, and
Pohl
,
R. O.
,
1989
, “
Thermal Boundary Resistance
,”
Rev. Mod. Phys.
,
61
(
3
), pp.
605
668
.10.1103/RevModPhys.61.605
5.
Zhao
,
H.
, and
Freund
,
J. B.
,
2009
, “
Phonon Scattering at a Rough Interface Between Two fcc Lattices
,”
J. Appl. Phys.
,
105
(
1
), p.
013515
.10.1063/1.3054383
6.
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
.10.1063/1.1524305
7.
Vincenti
,
W. G.
, and
Kruger
,
C. H.
,
1977
,
Introduction to Physical Gas Dynamics
,
Kreiger Publishing
,
Malabar, FL
.
8.
Dames
,
C.
, and
Chen
,
G.
,
2004
, “
Theoretical Phonon Thermal Conductivity of Si/Ge Superlattice Nanowires
,”
J. Appl. Phys.
,
95
(
2
), pp.
682
693
.10.1063/1.1631734
9.
Hopkins
,
P. E.
,
2009
, “
Multiple Phonon Processes Contributing to Inelastic Scattering During Thermal Boundary Conductance at Solid Interfaces
,”
J. Appl. Phys.
,
106
(
1
), p.
013528
.10.1063/1.3169515
10.
Hopkins
,
P. E
,
Duda
,
J. C.
, and
Norris
,
P. M.
,
2011
, “
Anharmonic Phonon Interactions as Interfaces and Contributions to Thermal Boundary Conductance
,”
ASME J. Heat Transfer
,
133
(
6
), p.
062401
.10.1115/1.4003549
11.
Duda
,
J. C.
,
Norris
,
P. M.
, and
Hopkins
,
P. E.
,
2011
, “
On the Linear Temperature Dependence of Phonon Thermal Boundary Conductance in the Classical Limit
,”
ASME J. Heat Transfer
,
133
(
7
), p.
074501
.10.1115/1.4003575
12.
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
,
48
(
22
), pp.
16373
16387
.10.1103/PhysRevB.48.16373
13.
Lyeo
,
H.-K.
, and
Cahill
,
D. G.
,
2006
, “
Thermal Conductance of Interfaces Between Highly Dissimilar Materials
,”
Phys. Rev. B
,
73
(
14
), p.
144301
.10.1103/PhysRevB.73.144301
14.
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
,
50
(
19–20
), pp.
3977
3989
.10.1016/j.ijheatmasstransfer.2007.01.040
15.
Landry
,
E. S.
, and
McGaughey
,
A. J. H.
,
2009
, “
Thermal Boundary Resistance Predictions From Molecular Dynamics Simulations and Theoretical Calculations
,”
Phys. Rev. B
,
80
(
16
), p.
165304
.10.1103/PhysRevB.80.165304
16.
Duda
,
J. C.
,
Beechem
,
T. E.
,
Smoyer
,
J. L.
,
Norris
,
P. M.
, and
Hopkins
,
P. E.
,
2010
, “
Role of Dispersion on Phononic Thermal Boundary Conductance
,”
J. Appl. Phys.
,
108
(
7
), p.
073515
.10.1063/1.3483943
17.
Reddy
,
P.
,
Castelino
,
K.
, and
Majumdar
,
A.
,
2005
, “
Diffuse Mismatch Model of Thermal Boundary Conductance Using Exact Phonon Dispersion
,”
Appl. Phys. Lett.
,
87
(
21
), p.
211908
.10.1063/1.2133890
18.
Singh
,
D.
,
Murthy
,
J. Y.
, and
Fisher
,
T. S.
,
2011
, “
Effect of Phonon Dispersion on Thermal Conduction Across Si/Ge Interfaces
,”
ASME J. Heat Transfer
,
133
(
12
), p.
122401
.10.1115/1.4004429
19.
Little
,
W. A.
,
1959
, “
The Transport of Heat between Dissimilar Solids at Low Temperatures
,”
Can. J. Phys.
,
37
(
3
), pp.
334
349
.10.1139/p59-037
20.
Modest
,
M. F.
,
2013
,
Radiative Heat Transfer
,
3rd ed.
,
Academic
, New York.
21.
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
,
123
(
1
), pp.
105
112
.10.1115/1.1338138
22.
Kazan
,
M.
,
2011
, “
Interpolation Between the Acoustic Mismatch Model and the Diffuse Mismatch Model for the Interface Thermal Conductance: Application to InN/GaN Superlattice
,”
ASME J. Heat Transfer
,
133
(
11
), p.
112401
.10.1115/1.4004341
23.
Ni
,
C.
, and
Murthy
,
J. Y.
,
2009
, “
Parallel Computation of the Phonon Boltzmann Transport Equation
,”
Numer. Heat Transfer, Part B
,
55
(
6
), pp.
435
456
.10.1080/10407780902864771
24.
Murthy
,
J. Y.
,
Narumanchi
,
S. V. J.
,
Pascual-Gutierrez
,
J. A.
,
Wang
,
T.
,
Ni
,
C.
, and
Mathur
,
S. R.
,
2005
, “
Review of Multi-Scale Simulation in Sub-Micron Heat Transfer
,”
Int. J. Multiscale Comput. Eng.
,
3
(
1
), pp.
5
32
10.1615/IntJMultCompEng.v3.i1.20.
25.
Raithby
,
G. D.
, and
Chui
,
E. H.
,
1990
, “
A Finite-Volume Method for Predicting a Radiant Heat Transfer in Enclosures With Participating Media
,”
ASME J. Heat Transfer
,
112
(
2
), pp.
415
423
.10.1115/1.2910394
26.
Chai
,
J. C.
,
Lee
,
H. S.
, and
Patankar
,
S. V.
,
1994
, “
Finite-Volume Method for Radiative Heat Transfer
,”
J. Thermophys. Heat Transfer
,
8
(
3
), pp.
419
425
.10.2514/3.559
27.
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
.10.1115/1.1518495
28.
Ali
,
S. A.
,
Kollu
,
G.
,
Mazumder
,
S.
,
Sadayappan
,
P.
, and
Mittal
,
A.
,
2014
, “
Large-Scale Parallel Computation of the Phonon Boltzmann Transport Equation
,”
Int. J. Therm. Sci.
,
86
, pp.
341
351
.10.1016/j.ijthermalsci.2014.07.019
29.
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
.10.1115/1.2824212
30.
Yang
,
R.
, and
Chen
,
G.
,
2004
, “
Thermal Conductivity Modeling of Periodic Two-Dimensional Nanocomposities
,”
Phys. Rev. B
,
69
(
19
), p.
195316
.10.1103/PhysRevB.69.195316
31.
Majumdar
,
A.
,
1993
, “
Microscale Heat Transfer in Dielectric Thin Films
,”
ASME J. Heat Transfer
,
115
(
1
), pp.
7
16
.10.1115/1.2910673
32.
Whitaker
,
S.
,
1983
,
Fundamental Principles of Heat Transfer
,
Krieger Publishing
,
Malabar, FL
.
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
,
123
(
4
), pp.
749
759
.10.1115/1.1377018
34.
Brockhouse
,
B. N.
,
1959
, “
Lattice Vibrations in Silicon and Germanium
,”
Phys. Rev. Lett.
,
2
(
6
), pp.
256
258
.10.1103/PhysRevLett.2.256
35.
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
(
5
), p.
052402
.10.1115/1.4000447
36.
Chai
,
J. C.
,
Lee
,
H. S.
, and
Patankar
,
S. V.
,
1993
, “
Ray Effect and False Scattering in the Discrete Ordinates Method
,”
Numer. Heat Transfer, Part B
,
24
(
4
), pp.
373
389
.10.1080/10407799308955899
37.
Mittal
,
A.
, and
Mazumder
,
S.
,
2011
, “
Generalized Ballistic-Diffusive Formulation and Hybrid SN-PN Solution of the Boltzmann Transport Equation for Phonons for Non-Equilibrium Heat Conduction
,”
ASME J. Heat Transfer
,
133
(
9
), p.
092402
.10.1115/1.4003961
38.
Mittal
,
A.
, and
Mazumder
,
S.
,
2011
, “
Hybrid Discrete Ordinates—Spherical Harmonics Solution to the Boltzmann Transport Equation for Phonons for Non-Equilibrium Heat Conduction
,”
J. Comput. Phys.
,
230
(
18
), pp.
6977
7001
.10.1016/j.jcp.2011.05.024
39.
Saad
,
Y.
,
2003
,
Iterative Methods for Sparse Linear Systems
,
2nd ed.
,
SIAM
,
Philadelphia, PA
.10.1137/1.9780898718003
40.
Srinivasan
,
S.
, and
Miller
,
R. S.
,
2004
, “
Parallel Computation of the Boltzmann Transport Equation for Microscale Heat Transfer in Multilayered Thin Films
,”
Numer. Heat Transfer, Part B
,
36
, pp.
31
58
.
41.
Lacroix
,
D.
,
Joulain
,
K.
, and
Lemonnier
,
D.
,
2005
, “
Monte Carlo Transient Phonon Transport in Silicon and Germanium at Nanoscale
,”
Phys. Rev. B
,
72
(
6
), p.
064305
.10.1103/PhysRevB.72.064305
42.
Holland
,
M. G.
,
1963
, “
Analysis of Lattice Thermal Conductivity
,”
Phys. Rev.
,
132
(
6
), pp.
2461
2471
.10.1103/PhysRev.132.2461
43.
Alan
,
J.
McGaughey, private communication.
44.
Sellan
,
D. P.
,
Landry
,
E. S.
,
Turney
,
J. E.
,
McGaughey
,
A. J. H.
, and
Amon
,
C. H.
,
2010
, “
Size Effects in Molecular Dynamics Thermal Conductivity Predictions
,”
Phys. Rev. B
,
81
(
21
), p.
214305
.10.1103/PhysRevB.81.214305
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