We develop a computational framework, based on the Boltzmann transport equation (BTE), with the ability to compute thermal transport in nanostructured materials of any geometry using, as the only input, the bulk cumulative thermal conductivity. The main advantage of our method is twofold. First, while the scattering times and dispersion curves are unknown for most materials, the phonon mean free path (MFP) distribution can be directly obtained by experiments. As a consequence, a wider range of materials can be simulated than with the frequency-dependent (FD) approach. Second, when the MFP distribution is available from theoretical models, our approach allows one to include easily the material dispersion in the calculations without discretizing the phonon frequencies for all polarizations thereby reducing considerably computational effort. Furthermore, after deriving the ballistic and diffusive limits of our model, we develop a multiscale method that couples phonon transport across different scales, enabling efficient simulations of materials with wide phonon MFP distributions length. After validating our model against the FD approach, we apply the method to porous silicon membranes and find good agreement with experiments on mesoscale pores. By enabling the investigation of thermal transport in unexplored nanostructured materials, our method has the potential to advance high-efficiency thermoelectric devices.

References

References
1.
Majumdar
,
A.
,
2004
, “
Thermoelectricity in Semiconductor Nanostructures
,”
Science
,
303
(
5659
), pp.
777
778
.10.1126/science.1093164
2.
Venkatasubramanian
,
R.
,
Siivola
,
E.
,
Colpitts
,
T.
, and
O' Quinn
,
B.
,
2001
, “
Thin-Film Thermoelectric Devices With High Room-Temperature Figures of Merit
,”
Nature
,
413
(
6856
), pp.
597
602
.10.1038/35098012
3.
Hochbaum
,
A. I.
,
Chen
,
R.
,
Delgado
,
R. D.
,
Liang
,
W.
,
Garnett
,
E. C.
,
Najarian
,
M.
,
Majumdar
,
A.
, and
Yang
,
P.
,
2008
, “
Enhanced Thermoelectric Performance of Rough Silicon Nanowires
,”
Nature
,
451
(
7175
), pp.
163
167
.10.1038/nature06381
4.
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
5.
Yu
,
J.-K.
,
Mitrovic
,
S.
,
Tham
,
D.
,
Varghese
,
J.
, and
Heath
,
J. R.
,
2010
, “
Reduction of Thermal Conductivity in Phononic Nanomesh Structures
,”
Nat. Nanotechnol.
,
5
(
10
), pp.
718
721
.10.1038/nnano.2010.149
6.
Tang
,
J.
,
Wang
,
H.-T.
,
Lee
,
D. H.
,
Fardy
,
M.
,
Huo
,
Z.
,
Russell
,
T. P.
, and
Yang
,
P.
,
2010
, “
Holey Silicon as an Efficient Thermoelectric Material
,”
Nano Lett.
,
10
(
10
), pp.
4279
4283
.10.1021/nl102931z
7.
Lee
,
J.-H.
,
Galli
,
G. A.
, and
Grossman
,
J. C.
,
2008
, “
Nano Si as an Efficient Thermoelectric Material
,”
Nano Lett.
,
8
(
11
), pp.
3750
3754
.10.1021/nl802045f
8.
Chen
,
G.
,
2005
,
Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons
,
Oxford University Press
,
New York
.
9.
Casimir
,
H.
,
1938
, “
Note on the Conduction of Heat in Crystals
,”
Physica
,
5
(
6
), pp.
495
500
.10.1016/S0031-8914(38)80162-2
10.
Majumdar
,
A.
,
1993
, “
Microscale Heat Conduction in Dielectric Thin Films
,”
ASME J. Heat Transfer
,
115
(
1
), pp.
7
16
.10.1115/1.2910673
11.
Chen
,
G.
,
1998
, “
Thermal Conductivity and Ballistic-Phonon Transport in the Cross-Plane Direction of Superlattices
,”
Phys. Rev. B
,
57
(
3
), pp.
14958
14973
.10.1103/PhysRevB.57.14958
12.
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
13.
Esfarjani
,
K.
,
Chen
,
G.
, and
Stokes
,
H. T.
,
2011
, “
Heat Transport in Silicon From First-Principles Calculations
,”
Phys. Rev. B
,
84
(
8
), p.
085204
.10.1103/PhysRevB.84.085204
14.
Minnich
,
A. J.
,
Chen
,
G.
,
Mansoor
,
S.
, and
Yilbas
,
B. S.
,
2011
, “
Quasiballistic Heat Transfer Studied Using the Frequency-Dependent Boltzmann Transport Equation
,”
Phys. Rev. B
,
84
(
23
), p.
235207
.10.1103/PhysRevB.84.235207
15.
Hsieh
,
T.-Y.
,
Lin
,
H.
,
Hsieh
,
T.-J.
, and
Huang
,
J.-C.
,
2012
, “
Thermal Conductivity Modeling of Periodic Silicon With Aligned Cylindrical Pores
,”
J. Appl. Phys.
,
111
(
12
), p.
124329
.10.1063/1.4730962
16.
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
17.
Minnich
,
A. J.
,
Johnson
,
J.
,
Schmidt
,
A.
,
Esfarjani
,
K.
,
Dresselhaus
,
M.
,
Nelson
,
K. A.
, and
Chen
,
G.
,
2011
, “
Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths
,”
Phys. Rev. Lett.
,
107
(
9
), p.
095901
.10.1103/PhysRevLett.107.095901
18.
Minnich
,
A. J.
,
2012
, “
Determining Phonon Mean Free Paths From Observations of Quasiballistic Thermal Transport
,”
Phys. Rev. Lett.
,
109
(
20
), p.
205901
.10.1103/PhysRevLett.109.205901
19.
Ziman
,
J. M.
,
2001
,
Electrons and Phonons: The Theory of Transport Phenomena in Solids
,
OUP
,
Oxford, UK
.
20.
Mingo
,
N.
,
Stewart
,
D.
,
Broido
,
D.
,
Lindsay
,
L.
, and
Li
,
W.
,
2014
, “
Ab Initio Thermal Transport
,”
Length-Scale Dependent Phonon Interactions
,
Springer
,
Berlin
, pp.
137
173
.10.1007/978-1-4614-8651-0_5
21.
Yang
,
F.
, and
Dames
,
C.
,
2013
, “
Mean Free Path Spectra as a Tool to Understand Thermal Conductivity in Bulk and Nanostructures
,”
Phys. Rev. B
,
87
(
3
), p.
035437
.10.1103/PhysRevB.87.035437
22.
Paraud
,
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
23.
Paraud
,
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
24.
Jeng
,
M.-S.
,
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
25.
Chen
,
G.
,
2001
, “
Ballistic-Diffusive Heat-Conduction Equations
,”
Phys. Rev. Lett.
,
86
(
11
), pp.
2297
2300
.10.1103/PhysRevLett.86.2297
26.
Romano
,
G.
, and
Di Carlo
,
A.
,
2011
, “
Multiscale Electrothermal Modeling of Nanostructured Devices
,”
IEEE Trans. Nanotechnol.
,
10
(
6
), pp.
1285
1292
.10.1109/TNANO.2011.2129574
27.
Giannozzi
,
P.
,
Baroni
,
S.
,
Bonini
,
N.
,
Calandra
,
M.
,
Car
,
R.
,
Cavazzoni
,
C.
,
Ceresoli
,
D.
,
Chiarotti
,
G. L.
,
Cococcioni
,
M.
,
Dabo
,
I.
,
Dal Corso
,
A.
,
De Gironcoli1
,
S.
,
Fabrisi
,
S.
,
Fratesi
,
G.
,
Gebauer
,
R.
,
Gerstmann
,
U.
,
Gougoussis
,
C.
,
Kokalj
,
A.
,
Lazzeri
,
M.
,
Martin-Samos
,
L.
,
Marzari
,
N.
,
Mauri
,
F.
,
Mazzarello
,
R.
,
Paolini
,
S.
,
Pasquarello
,
A.
,
Paulatto
,
L.
,
Sbraccia
,
C.
,
Scandolo
,
S.
,
Sclauzero
,
G.
,
Seitsonen
,
A. P.
,
Smogunov
,
A.
,
Umari
,
P.
, and
Wentzcovitch
,
R. M.
,
2009
, “
quantum espresso: A Modular and Open-Source Software Project for Quantum Simulations of Materials
,”
J. Phys.: Condens. Matter
,
21
(
39
), p.
395502
.10.1088/0953-8984/21/39/395502
28.
Slack
,
G. A.
, 1994,
Handbook of Thermoelectrics
, D. M. Rowe, ed., CRC Press, Boca Raton, FL, p. 407.
29.
Hashin
,
Z.
, and
Shtrikman
,
S.
,
1962
, “
A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials
,”
J. Appl. Phys.
,
33
(
10
), pp.
3125
3131
.10.1063/1.1728579
30.
Esfarjani
,
K.
,
Chen
,
G.
, and
Stokes
,
H. T.
,
2011
, “
Heat Transport in Silicon From First-Principles Calculations
,”
Phys. Rev. B
,
84
(
8
), p.
085204
.10.1103/PhysRevB.84.085204
31.
Ravichandran
,
N. K.
, and
Minnich
,
A. J.
,
2014
, “
Coherent and Incoherent Thermal Transport in Nanomeshes
,”
Phys. Rev. B
,
89
, p.
205432
.10.1103/PhysRevB.89.205432
32.
Davis
,
B. L.
, and
Hussein
,
M. I.
,
2014
, “
Nanophononic Metamaterial: Thermal Conductivity Reduction by Local Resonance
,”
Phys. Rev. Lett.
,
112
, p.
055505
.10.1103/PhysRevLett.112.055505
33.
Jain
,
A.
,
Yu
,
Y.-J.
, and
McGaughey
,
A. J.
,
2013
, “
Phonon Transport in Periodic Silicon Nanoporous Films With Feature Sizes Greater Than 100 nm
,”
Phys. Rev. B
,
87
(
19
), p.
195301
.10.1103/PhysRevB.87.195301
You do not currently have access to this content.