Sequential numerical solution methods are commonly used for solving the phonon Boltzmann transport equation (BTE) because of simplicity of implementation and low storage requirements. However, they exhibit poor convergence for low Knudsen numbers. This is because sequential solution procedures couple the phonon BTEs in physical space efficiently but the coupling is inefficient in wave vector (K) space. As the Knudsen number decreases, coupling in K space becomes dominant and convergence rates fall. Since materials like silicon have K-resolved Knudsen numbers that span two to five orders of magnitude at room temperature, diffuse-limit solutions are not feasible for all K vectors. Consequently, nongray solutions of the BTE experience extremely slow convergence. In this paper, we develop a coupled-ordinates method for numerically solving the phonon BTE in the relaxation time approximation. Here, interequation coupling is treated implicitly through a point-coupled direct solution of the K-resolved BTEs at each control volume. This implicit solution is used as a relaxation sweep in a geometric multigrid method which promotes coupling in physical space. The solution procedure is benchmarked against a traditional sequential solution procedure for thermal transport in silicon. Significant acceleration in computational time, between 10 and 300 times, over the sequential procedure is found for heat conduction problems.

References

References
1.
International Technology Roadmap for Semiconductors (ITRS),
2002
, http://www.itrs.net/Links/2002Update/2002Update.pdf
2.
Plumbridge
,
W. J.
,
Matela
,
R. J.
, and
Westwater
,
A.
,
2004
,
Structural Integrity and Reliability in Electronics
,
Kluwer
,
Dordecht, The Netherlands
.
3.
Rowlette
,
J. A.
, and
Goodson
,
K. E.
,
2008
, “
Fully Coupled Nonequilibrium Electron-Phonon Transport in Nanometer-Scale Silicon FETs
,”
IEEE Trans. Electron Devices
,
55
(
1
), pp.
220
232
.10.1109/TED.2007.911043
4.
International Technology Roadmap for Semiconductors,
2009
, http://www.itrs.net/Links/2009ITRS/Home2009.htm
5.
Yu
,
J.
,
Xiong
,
J.
,
Cheng
,
B.
, and
Liu
,
S.
,
2005
, “
Fabrication and Characterization of Ag–TiO2 Multiphase Nanocomposite Thin Films With Enhanced Photocatalytic Activity
,”
Appl. Catal., B
,
60
(
3–4
), pp.
211
221
.10.1016/j.apcatb.2005.03.009
6.
Shahil
,
K. M. F.
, and
Balandin
,
A.
,
2012
, “
Graphene-Multilayer Graphene Nanocomposites as Highly Efficient Thermal Interface Materials
,”
Nano Lett.
,
12
(
2
), pp.
861
867
.10.1021/nl203906r
7.
Li
,
D.
,
Huxtable
,
S. T.
,
Abramson
,
A. R.
, and
Majumdar
,
A.
,
2005
, “
Thermal Transport in Nanostructured Solid-State Cooling Devices
,”
ASME J. Heat Transfer
,
127
(
1
), pp.
108
114
.10.1115/1.1839588
8.
Zebarjadi
,
M.
,
Joshi
,
G.
,
Zhu
,
G.
,
Yu
,
B.
,
Minnich
,
A.
,
Lan
,
Y.
,
Wang
,
X.
,
Dresselhaus
,
M.
,
Ren
,
Z.
, and
Chen
,
G.
,
2011
, “
Power Factor Enhancement by Modulation Doping in Bulk Nanocomposites
,”
Nano Lett.
,
11
(
6
), pp.
2225
2230
.10.1021/nl201206d
9.
Granstrom
,
M.
,
Petritsch
,
K.
,
Arias
,
A. C.
,
Lux
,
A.
,
Andersson
,
M. R.
, and
Friend
,
R. H.
,
1998
, “
Laminated Fabrication of Polymeric Photovoltaic Diodes
,”
Nature
,
395
(
6699
), pp.
257
260
.10.1038/26183
10.
Goodey
,
A. P.
,
Eichfeld
,
S. M.
,
Lew
,
K.-K.
,
Redwing
,
J. M.
, and
Mallouk
,
T. E.
,
2007
, “
Silicon Nanowire Array Photoelectrochemical Cells
,”
J. Am. Chem. Soc.
,
129
(
41
), pp.
12344
12345
.10.1021/ja073125d
11.
Bidkar
,
R.
,
Tung
,
R. C.
,
Alexeenko
,
A.
,
Sumali
,
H.
, and
Raman
,
A.
,
2009
, “
Unified Theory of Gas Damping of Flexible Microcantilevers at Low Ambient Pressures
,”
Appl. Phys. Lett.
,
94
(
16
), p.
163117
.10.1063/1.3122933
12.
Mahapatro
,
A.
,
Chee
,
J.
, and
Peroulis
,
D.
,
2009
, “
Fully Electronic Method for Quantifying the Post-release Gapheight Uncertainty of Capacitive RF MEMS Switches
,”
International Microwave Symposium Digest
, Boston, MA, June 7–12, pp.
1645
1648
.
13.
Singh
,
D.
,
Guo
,
X.
,
Alexeenko
,
A.
,
Murthy
,
J. Y.
, and
Fisher
,
T. S.
,
2009
, “
Modeling of Subcontinuum Thermal Transport Across Semiconductor-Gas Interfaces
,”
J. Appl. Phys.
,
106
(
2
), p.
024314
.10.1063/1.3181059
14.
Majumdar
,
A.
,
1993
, “
Microscale Heat Conduction in Dielectric Thin Films
,”
ASME J. Heat Transfer
,
115
(
1
), pp.
7
16
.10.1115/1.2910673
15.
Kittel
,
C.
,
1996
,
Introduction to Solid State Physics
,
Wiley
,
New York
.
16.
Sun
,
L.
, and
Murthy
,
J. Y.
,
2006
, “
Domain Size Effects in Molecular Dynamics Simulation of Phonon Transport in Silicon
,”
Appl. Phys. Lett.
,
89
(
17
), p.
171919
.10.1063/1.2364062
17.
McGaughey
,
A. J. H.
, and
Kaviany
,
M.
,
2006
, “
Phonon Transport in Molecular Dynamics Simulations: Formulation and Thermal Conductivity Prediction
,”
Advances in Heat Transfer
,
Academic Press
,
New York
, Vol.
39
, pp.
169
255
.10.1016/S0065-2717(06)39002-8
18.
Narumanchi
,
S. V. J.
,
Murthy
,
J. Y.
, and
Amon
,
C. H.
,
2005
, “
Boltzmann Transport Equation-Based Thermal Modeling Approaches for Hotspots in Microelectronics
,”
Heat Mass Transfer
,
42
(
6
), pp.
478
491
.10.1007/s00231-005-0645-6
19.
Bansal
,
A.
,
Meterelliyoz
,
M.
,
Singh
,
S.
,
Murthy
,
J.
, and
Roy
,
K.
,
2006
, “
Compact Thermal Models for Estimation of Temperature-Dependent Power/Performance in finFET Technology
,”
Asia and South Pacific Conference on Design Automation
, Yokohama, Japan, Jan. 24–27, pp.
237
242
.
20.
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
,
126
(
6
), pp.
946
955
.10.1115/1.1833367
21.
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
22.
Pascual-Gutiérrez
,
J.
,
Murthy
,
J. Y.
, and
Viskanta
,
R.
,
2009
, “
Thermal Conductivity and Phonon Transport Properties of Silicon Using Perturbation Theory and the Environment-Dependent Interatomic Potential
,”
J. Appl. Phys.
,
106
(
6
), p.
063532
.10.1063/1.3195080
23.
Henry
,
A. S.
, and
Chen
,
G.
,
2008
, “
Spectral Phonon Transport Properties of Silicon Based on Molecular Dynamics Simulations and Lattice Dynamics
,”
J. Comput. Theor. Nanosci.
,
5
(
2
), pp.
1
12
.10.1166/jctn.2008.001a
24.
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
,
127
(
7
), pp.
713
723
.10.1115/1.1924571
25.
Loy
,
J. M.
,
Murthy
,
J. Y.
, and
Singh
,
D.
,
2013
, “
A Fast Hybrid Fourier–Boltzmann Transport Equation Solver for Nongray Phonon Transport
,”
ASME J. Heat Transfer
,
135
(
1
), p.
011008
.10.1115/1.4007654
26.
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
27.
Péraud
,
J.-P. M.
, and
Hadjiconstantinou
,
N. G.
,
2011
, “
Efficient Simulation of Multidimensional Phonon Transport Using Energy-Based Variance-Reduced Monte Carlo Formulations
,”
Phys. Rev. B
,
84
(
20
), p.
205331
.10.1103/PhysRevB.84.205331
28.
Péraud
,
J.-P. M.
, and
Hadjiconstantinou
,
N. G.
,
2012
, “
An Alternative Approach to Efficient Simulation of Micro/Nanoscale Phonon Transport
,”
Appl. Phys. Lett.
,
101
(
15
), p.
153114
.10.1063/1.4757607
29.
Raithby
,
G. D.
, and
Chui
,
E. H.
,
1990
, “
A Finite-Volome 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
30.
Chui
,
E. H.
, and
Raithby
,
G. D.
,
1992
, “
Implicit Solution Scheme to Improve Convergence Rate of Radiative Transfer Problems
,”
Numer. Heat Transfer, Part B
,
22
(
3
), pp.
251
272
.10.1080/10407799208944983
31.
Fiveland
,
W. A.
, and
Jessee
,
J.
,
1996
, “
Acceleration Schemes for the Discrete Ordinates Method
,”
J. Thermophys. Heat Transfer
,
10
(
3
), pp.
445
451
.10.2514/3.809
32.
Hassanzadeh
,
P.
,
Raithby
,
G. D.
, and
Chui
,
E. H.
,
2008
, “
Efficient Calculation of Radiation Heat Transfer in Participating Media
,”
J. Thermophys. Heat Transfer
,
22
(
2
), pp.
129
139
.10.2514/1.33271
33.
Mathur
,
S. R.
, and
Murthy
,
J. Y.
,
2009
, “An Acceleration Technique for the Computation of Participating Radiative Heat Transfer,” IMECE, Lake Buena Vista, FL, Nov. 13–19, pp.
709
717
.
34.
Mazumder
,
S.
,
2005
, “
A New Numerical Procedure for Coupling Radiation in Participating Media With Other Modes of Heat Transfer
,”
ASME J. Heat Transfer
,
127
(
9
), pp.
1037
1045
.10.1115/1.1929780
35.
Mathur
,
S. R.
, and
Murthy
,
J. Y.
,
1999
, “
Coupled Ordinates Method for Multigrid Acceleration of Radiation Calculations
,”
J. Thermophys. Heat Transfer
,
13
(
4
), pp.
467
473
.10.2514/2.6485
36.
Holland
,
M. G.
,
1963
, “
Analysis of Lattice Thermal Conductivity
,”
Phys. Rev.
,
132
(
6
), pp.
2461
2471
.10.1103/PhysRev.132.2461
37.
Chai
,
J. C.
,
Lee
,
H. S.
, and
Patankar
,
S. V.
,
1994
, “
Finite Volume Method for Radiation Heat Transfer
,”
J. Thermophy. Heat Transfer
,
8
(
3
), pp.
419
425
.10.2514/3.559
38.
Patankar
,
S. V.
,
1980
,
Numerical Heat Transfer and Fluid Flow
,
Taylor & Francis
,
New York
.
39.
Murthy
,
J. Y.
, and
Mathur
,
S. R.
,
2003
, “
An Improved Computational Procedure for Sub-Micron Heat Conduction
,”
ASME J. Heat Transfer
,
125
(
5
), pp.
904
910
.10.1115/1.1603775
40.
Brandt
,
A.
, and
Livne
,
O. E.
,
2011
,
Multigrid Techniques: 1984 Guide With Applications to Fluid Dynamics
,
SIAM
,
Philadelphia, PA
.10.1137/1.9781611970753
41.
Mathur
,
S. R.
, and
Murthy
,
J. Y.
,
1997
, “
A Pressure-Based Method for Unstructured Meshes
,”
Numer. Heat Transfer, Part B
,
31
(
2
), pp.
195
215
.10.1080/10407799708915105
42.
Heaslet
,
M. A.
, and
Warming
,
R. F.
,
1965
, “
Radiative Transport and Wall Temperature Slip in an Absorbing Planar Medium
,”
Int. J. Heat Mass Transfer
,
8
(
7
), pp.
979
994
.10.1016/0017-9310(65)90083-9
43.
Modest
,
M. F.
,
1993
,
Radiative Heat Transfer
,
McGraw-Hill
,
New York
.
44.
Gironcoli
,
S. De
,
1992
, “
Phonons in Si-Ge Systems: An Ab Initio Interatomic-Force-Constant Approach
,”
Phys. Rev. B
,
46
(
4
), pp.
2412
2419
.10.1103/PhysRevB.46.2412
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
.10.1103/PhysRevB.56.8542
46.
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
.10.1021/nl034721i
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