Abstract

The gray medium approximation treating all phonons with an averaged and representative mean-free-path (MFP) is an often used method in analyzing ballistic-diffusive heat conduction at nanoscale. However, whether there exists a reasonable value of the average MFP which effectively represents the entire spectrum of modal MFPs remains unclear. In this paper, phonon Monte Carlo (MC) method is employed to study the effects of the gray medium approximation on ballistic-diffusive heat conduction in silicon films by comparing with dispersion MC simulations. Four typical ways for calculating the average MFP with gray medium approximation are investigated. Three of them are based on the weighted average of the modal MFPs, and the remaining one is based on the weighted average of the reciprocals of the modal MFPs. The first three methods are found to be good at predicting effective thermal conductivity and heat flux distribution, but have difficulties in temperature profile, while the last one performs better for temperature profile than effective thermal conductivity and heat flux distribution. Therefore, none of the average MFPs can accurately characterize all the features of ballistic-diffusive heat conduction for the gray medium approximation. Phonon dispersion has to be considered for the accurate thermal analyses and modeling of ballistic-diffusive heat transport. Our work could be helpful for further understanding of phonon dispersion and more careful use of the gray medium approximation.

References

References
1.
Allu
,
P.
, and
Mazumder
,
S.
,
2016
, “
Hybrid Ballistic-Diffusive Solution to the Frequency-Dependent Phonon Boltzmann Transport Equation
,”
Int. J. Heat Mass Transf.
,
100
, pp.
165
177
.10.1016/j.ijheatmasstransfer.2016.04.049
2.
Hanks
,
D. F.
,
Lu
,
Z.
,
Sircar
,
J.
,
Salamon
,
T. R.
,
Antao
,
D. S.
,
Bagnall
,
K. R.
,
Barabadi
,
B.
, and
Wang
,
E. N.
,
2018
, “
Nanoporous Membrane Device for Ultra High Heat Flux Thermal Management
,”
Microsyst. Nanoeng.
,
4
(
1
), pp.
1
10
.10.1038/s41378-018-0004-7
3.
Moore
,
A. L.
, and
Shi
,
L.
,
2014
, “
Emerging Challenges and Materials for Thermal Management of Electronics
,”
Mater. Today
,
17
(
4
), pp.
163
174
.10.1016/j.mattod.2014.04.003
4.
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
.10.1063/1.119402
5.
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
6.
Liu
,
W.
, and
Asheghi
,
M.
,
2004
, “
Phonon-Boundary Scattering in Ultrathin Single-Crystal Silicon Layers
,”
Appl. Phys. Lett.
,
84
(
19
), pp.
3819
3821
.10.1063/1.1741039
7.
Song
,
D.
, and
Chen
,
G.
,
2004
, “
Thermal Conductivity of Periodic Microporous Silicon Films
,”
Appl. Phys. Lett.
,
84
(
5
), pp.
687
689
.10.1063/1.1642753
8.
Hopkins
,
P. E.
,
Reinke
,
C. M.
,
Su
,
M. F.
,
Olsson
,
R. H.
,
Shaner
,
E. A.
,
Leseman
,
Z. C.
,
Serrano
,
J. R.
,
Phinney
,
L. M.
, and
El-Kady
,
I.
,
2011
, “
Reduction in the Thermal Conductivity of Single Crystalline Silicon by Phononic Crystal Patterning
,”
Nano Lett.
,
11
(
1
), pp.
107
112
.10.1021/nl102918q
9.
Maznev
,
A. A.
,
Cuffe
,
J.
,
Eliason
,
J. K.
,
Minnich
,
A. J.
,
Kehoe
,
T.
,
Torres
,
C. M. S.
,
Chen
,
G.
,
Nelson
,
K. A.
, and
Johnson
,
J. A.
,
2013
, “
Direct Measurement of Room-Temperature Nondiffusive Thermal Transport Over Micron Distances in a Silicon Membrane
,”
Phys. Rev. Lett.
,
110
(
2
), p.
025901
.10.1103/PhysRevLett.110.025901
10.
Maire
,
J.
,
Anufriev
,
R.
,
Hori
,
T.
,
Shiomi
,
J.
,
Volz
,
S.
, and
Nomura
,
M.
,
2018
, “
Thermal Conductivity Reduction in Silicon Fishbone Nanowires
,”
Sci. Rep.
,
8
(
1
), p.
4452
.10.1038/s41598-018-22509-0
11.
Volz
,
S.
,
Shiomi
,
J.
,
Nomura
,
M.
, and
Miyazaki
,
K.
,
2016
, “
Heat Conduction in Nanostructured Materials
,”
J. Therm. Sci. Tech.
,
11
(
1
), p.
JTST0001
.10.1299/jtst.2016jtst0001
12.
Bao
,
H.
,
Chen
,
J.
,
Gu
,
X. K.
, and
Cao
,
B. Y.
,
2018
, “
A Review of Simulation Methods in Micro/Nanoscale Heat Conduction
,”
ES Energy Environ.
,
1
, pp.
16
55
.10.30919/esee8c149
13.
Chen
,
G.
,
1998
, “
Thermal Conductivity and Ballistic-Phonon Transport in the Cross-Plane Direction of Superlattices
,”
Phys. Rev. B
,
57
(
23
), pp.
14958
14973
.10.1103/PhysRevB.57.14958
14.
Chen
,
G.
,
2001
, “
Ballistic-Diffusive Heat-Conduction Equations
,”
Phys. Rev. Lett.
,
86
(
11
), pp.
2297
2300
.10.1103/PhysRevLett.86.2297
15.
Alvarez
,
F. X.
, and
Jou
,
D.
,
2010
, “
Boundary Conditions and Evolution of Ballistic Heat Transport
,”
ASME J. Heat Transfer
,
132
(
1
), p.
012404
.10.1115/1.3156785
16.
Li
,
B. W.
, and
Wang
,
J.
,
2003
, “
Anomalous Heat Conduction and Anomalous Diffusion in One-Dimensional Systems
,”
Phys. Rev. Lett.
,
91
(
4
), p.
044301
.10.1103/PhysRevLett.91.044301
17.
Sellan
,
D. P.
,
Turney
,
J. E.
,
McGaughey
,
A. J. H.
, and
Amon
,
C. H.
,
2010
, “
Cross-Plane Phonon Transport in Thin Films
,”
J. Appl. Phys.
,
108
(
11
), p.
113524
.10.1063/1.3517158
18.
Aoki
,
K.
, and
Kusnezov
,
D.
,
2001
, “
Fermi-Pasta-Ulam Beta Model: Boundary Jumps, Fourier's Law, and Scaling
,”
Phys. Rev. Lett.
,
86
(
18
), pp.
4029
4032
.10.1103/PhysRevLett.86.4029
19.
Hua
,
Y. C.
, and
Cao
,
B. Y.
,
2017
, “
Slip Boundary Conditions in Ballistic-Diffusive Heat Transport in Nanostructures
,”
Nanoscale Microscale Thermophys. Eng.
,
21
(
3
), pp.
159
176
.10.1080/15567265.2017.1344752
20.
Li
,
H. L.
,
Hua
,
Y. C.
, and
Cao
,
B. Y.
,
2018
, “
A Hybrid Phonon Monte Carlo-Diffusion Method for Ballistic-Diffusive Heat Conduction in Nano- and Micro- Structures
,”
Int. J. Heat Mass Transfer
,
127
, pp.
1014
1022
.10.1016/j.ijheatmasstransfer.2018.06.080
21.
Parrish
,
K. D.
,
Abel
,
J. R.
,
Jain
,
A.
,
Malen
,
J. A.
, and
McGaughey
,
A. J.
,
2017
, “
Phonon-Boundary Scattering in Nanoporous Silicon Films: Comparison of Monte Carlo Techniques
,”
J. Appl. Phys.
,
122
(
12
), p.
125101
.10.1063/1.4993601
22.
Liu
,
W.
,
Etessam-Yazdani
,
K.
,
Hussin
,
R.
, and
Asheghi
,
M.
,
2006
, “
Modeling and Data for Thermal Conductivity of Ultrathin Single-Crystal SOI Layers at High Temperature
,”
IEEE T. Electron Dev.
,
53
(
8
), pp.
1868
1876
.10.1109/TED.2006.877874
23.
Chung
,
J. D.
,
Mcgaughey
,
A. J. H.
, and
Kaviany
,
M.
,
2004
, “
Role of Phonon Dispersion in Lattice Thermal Conductivity Modeling
,”
ASME J. Heat Transfer
,
126
(
3
), pp.
376
380
.10.1115/1.1723469
24.
Baillis
,
D.
, and
Randrianalisoa
,
J.
,
2009
, “
Prediction of Thermal Conductivity of Nanostructures: Influence of Phonon Dispersion Approximation
,”
Int. J. Heat Mass Transfer
,
52
(
11–12
), pp.
2516
2527
.10.1016/j.ijheatmasstransfer.2009.01.017
25.
Jeong
,
C.
,
Datta
,
S.
, and
Lundstrom
,
M.
,
2011
, “
Full Dispersion Versus Debye Model Evaluation of Lattice Thermal Conductivity With a Landauer Approach
,”
J. Appl. Phys.
,
109
(
7
), p.
073718
.10.1063/1.3567111
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.
Lacroix
,
D.
,
Joulain
,
K.
, and
Lemonnier
,
D.
,
2005
, “
Monte Carlo Transient Phonon Transport in Silicon and Germanium at Nanoscales
,”
Phys. Rev. B
,
72
(
6
), p.
064305
.10.1103/PhysRevB.72.064305
28.
Jeng
,
M.
,
Yang
,
R.
,
Song
,
D.
, and
Chen
,
G.
,
2008
, “
Modeling the Thermal Conductivity and Phonon Transport in Nanoparticle Composites Using Monte Carlo Simulation
,”
ASME J. Heat Transfer
,
130
(
4
), p.
042410
.10.1115/1.2818765
29.
Hao
,
Q.
,
Chen
,
G.
, and
Jeng
,
M. S.
,
2009
, “
Frequency-Dependent Monte Carlo Simulations of Phonon Transport in Two-Dimensional Porous Silicon With Aligned Pores
,”
J. Appl. Phys.
,
106
(
11
), p.
114321
.10.1063/1.3266169
30.
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
), pp.
1555
1569
.10.1103/PhysRevB.84.205331
31.
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
32.
Hori
,
T.
,
Chen
,
G.
, and
Shiomi
,
J.
,
2014
, “
Thermal Conductivity of Bulk Nanostructured Lead Telluride
,”
Appl. Phys. Lett.
,
104
(
2
), p.
021915
.10.1063/1.4862323
33.
Zeng
,
L.
,
Collins
,
K. C.
,
Hu
,
Y.
,
Luckyanova
,
M. N.
,
Maznev
,
A. A.
,
Huberman
,
S.
,
Chiloyan
,
V.
,
Zhou
,
J.
,
Huang
,
X.
,
Nelson
,
K. A.
, and
Chen
,
G.
,
2015
, “
Measuring Phonon Mean Free Path Distributions by Probing Quasiballistic Phonon Transport in Grating Nanostructures
,”
Sci. Rep.
,
5
, p.
17131
.10.1038/srep17131
34.
Hu
,
Y.
,
Zeng
,
L.
,
Minnich
,
A. J.
,
Dresselhaus
,
M. S.
, and
Chen
,
G.
,
2015
, “
Spectral Mapping of Thermal Conductivity Through Nanoscale Ballistic Transport
,”
Nat. Nanotechnol.
,
10
(
8
), pp.
701
706
.10.1038/nnano.2015.109
35.
Jiang
,
P. Q.
,
Lindsay
,
L.
, and
Koh
,
Y. K.
,
2016
, “
Role of Low-Energy Phonons With Mean-Free-Paths >0.8 μm in Heat Conduction in Silicon
,”
J. Appl. Phys.
,
119
(
24
), p.
245705
.10.1063/1.4954674
36.
Majumdar
,
A.
,
1993
, “
Microscale Heat Conduction in Dielectric Thin Films
,”
ASME J. Heat Transfer
,
115
(
1
), pp.
7
16
.10.1115/1.2910673
37.
Alvarez
,
F. X.
, and
Jou
,
D.
,
2007
, “
Memory and Nonlocal Effects in Heat Transport: From Diffusive to Ballistic Regimes
,”
Appl. Phys. Lett.
,
90
(
8
), p.
083109
.10.1063/1.2645110
38.
Hua
,
Y. C.
, and
Cao
,
B. Y.
,
2016
, “
Ballistic-Diffusive Heat Conduction in Multiply-Constrained Nanostructures
,”
Int. J. Therm. Sci.
,
101
, pp.
126
132
.10.1016/j.ijthermalsci.2015.10.037
39.
Hua
,
Y. C.
, and
Cao
,
B. Y.
,
2016
, “
The Effective Thermal Conductivity of Ballistic-Diffusive Heat Conduction in Nanostructures With Internal Heat Source
,”
Int. J. Heat Mass Transf.
,
92
, pp.
995
1003
.10.1016/j.ijheatmasstransfer.2015.09.068
40.
Hua
,
Y. C.
, and
Cao
,
B. Y.
,
2017
, “
Cross-Plane Heat Conduction in Nanoporous Silicon Thin Films by Phonon Boltzmann Transport Equation and Monte Carlo Simulations
,”
Appl. Therm. Eng.
,
111
, pp.
1401
1408
.10.1016/j.applthermaleng.2016.05.157
41.
Li
,
H. L.
, and
Cao
,
B. Y.
,
2019
, “
Radial Ballistic-Diffusive Heat Conduction in Nanoscale
,”
Nanoscale Microscale Thermophys. Eng.
,
23
(
1
), pp.
10
24
.10.1080/15567265.2018.1520763
42.
Lepri
,
S.
,
Livi
,
R.
, and
Politi
,
A.
,
2003
, “
Thermal Conduction in Classical Low-Dimensional Lattices
,”
Phys. Rep.
,
377
(
1
), pp.
1
80
.10.1016/S0370-1573(02)00558-6
43.
Hua
,
Y. C.
, and
Cao
,
B. Y.
,
2014
, “
Phonon Ballistic-Diffusive Heat Conduction in Silicon Nanofilms by Monte Carlo Simulations
,”
Int. J. Heat Mass Transf.
,
78
, pp.
755
759
.10.1016/j.ijheatmasstransfer.2014.07.037
44.
Maassen
,
J.
, and
Lundstrom
,
M.
,
2015
, “
Steady-State Heat Transport: Ballistic-to-Diffusive With Fourier's Law
,”
J. Appl. Phys.
,
117
(
3
), p.
011305
.10.1063/1.4905590
45.
Sobolev
,
S. L.
,
2017
, “
Discrete Space-Time Model for Heat Conduction: Application to Size- Dependent Thermal Conductivity in Nano-Films
,”
Int. J. Heat Mass Transf.
,
108
, pp.
933
939
.10.1016/j.ijheatmasstransfer.2016.12.051
46.
Kaiser
,
J.
,
Feng
,
T.
,
Maassen
,
J.
,
Wang
,
X.
,
Ruan
,
X.
, and
Lundstrom
,
M.
,
2017
, “
Thermal Transport at the Nanoscale—A Fourier's Law vs. Phonon Boltzmann Equation Study
,”
J. Appl. Phys.
,
121
(
4
), p.
044302
.10.1063/1.4974872
47.
Chen
,
G.
,
2005
, “
Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons
,”
Molecules, Phonons, and Photons
,
Oxford University Press
,
New York
.
48.
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 Transf.
,
42
(
6
), pp.
478
491
.10.1007/s00231-005-0645-6
49.
Vallabhaneni
,
A. K.
,
Chen
,
L.
,
Gupta
,
M. P.
, and
Kumar
,
S.
,
2017
, “
Solving Nongray Boltzmann Transport Equation in Gallium Nitride
,”
ASME J. Heat Transfer
,
139
(
10
), p.
102701
.10.1115/1.4036616
50.
Cahill
,
D. G.
,
Braun
,
P. V.
,
Chen
,
G.
,
Clarke
,
D. R.
,
Fan
,
S.
,
Goodson
,
K. E.
,
Keblinski
,
P.
,
King
,
W. P.
,
Mahan
,
G. D.
,
Majumdar
,
A.
,
Maris
,
H. J.
,
Phillpot
,
S. R.
,
Pop
,
E.
, and
Shi
,
L.
,
2014
, “
Nanoscale Thermal Transport—II: 2003–2012
,”
Appl. Phys. Rev.
,
1
(
1
), p.
011305
.10.1063/1.4832615
51.
Sobolev
,
S. L.
,
2018
, “
Hyperbolic Heat Conduction, Effective Temperature, and Third Law for Nonequilibrium Systems With Heat Flux
,”
Phys. Rev. E
,
97
(
2
), p.
022122
.10.1103/PhysRevE.97.022122
52.
Peterson
,
R. B.
,
1994
, “
Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal
,”
ASME J. Heat Transfer
,
116
(
4
), pp.
815
822
.10.1115/1.2911452
53.
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
54.
Brockhouse
,
B. N.
,
1959
, “
Lattice Vibrations in Silicon and Germanium
,”
Phys. Rev. Lett.
,
2
(
6
), pp.
256
258
.10.1103/PhysRevLett.2.256
55.
Esfarjani
,
K.
,
Chen
,
G.
, and
Stokes
,
H. T.
,
2011
, “
Heat Transport in Silicon From First Principles Calculations
,”
Phys. Rev. B
,
84
(
8
), pp.
293
293
.10.1103/PhysRevB.84.085204
56.
Holland
,
M. G.
,
1963
, “
Analysis of Lattice Thermal Conductivity
,”
Phys. Rev.
,
132
(
6
), pp.
2461
2471
.10.1103/PhysRev.132.2461
57.
Kukita
,
K.
, and
Kamakura
,
Y.
,
2013
, “
Monte Carlo Simulation of Phonon Transport in Silicon Including a Realistic Dispersion Relation
,”
J. Appl. Phys.
,
114
(
15
), p.
154312
.10.1063/1.4826367
58.
Chen
,
G.
,
1997
, “
Size and Interface Effects on Thermal Conductivity of Superlattices and Periodic Thin-Film Structures
,”
ASME J. Heat Transfer
,
119
(
2
), pp.
220
229
.10.1115/1.2824212
59.
Pohl
,
R. O.
, and
Swartz
,
E. T.
,
1989
, “
Thermal Boundary Resistance
,”
Rev. Mod. Phys.
,
61
(
3
), pp.
605
668
.10.1103/RevModPhys.61.605
You do not currently have access to this content.