This paper presents a Monte Carlo simulation scheme to study the phonon transport and the thermal conductivity of nanocomposites. Special attention has been paid to the implementation of periodic boundary condition in Monte Carlo simulation. The scheme is applied to study the thermal conductivity of silicon germanium (Si–Ge) nanocomposites, which are of great interest for high-efficiency thermoelectric material development. The Monte Carlo simulation was first validated by successfully reproducing the results of (two-dimensional) nanowire composites using the deterministic solution of the phonon Boltzmann transport equation reported earlier and the experimental thermal conductivity of bulk germanium, and then the validated simulation method was used to study (three-dimensional) nanoparticle composites, where Si nanoparticles are embedded in Ge host. The size effects of phonon transport in nanoparticle composites were studied, and the results show that the thermal conductivity of nanoparticle composites can be lower than that of the minimum alloy value, which is of great interest to thermoelectric energy conversion. It was also found that randomly distributed nanopaticles in nanocomposites rendered the thermal conductivity values close to that of periodic aligned patterns. We show that interfacial area per unit volume is a useful parameter to correlate the size effect of thermal conductivity in nanocomposites. The key for the thermal conductivity reduction is to have a high interface density where nanoparticle composites can have a much higher interface density than the simple 1D stacks, such as superlattices. Thus, nanocomposites further benefit the enhancement of thermoelectric performance in terms of thermal conductivity reduction. The thermal conductivity values calculated by this work qualitatively agrees with a recent experimental measurement of Si–Ge nanocomposites.

1.
Chen
,
G.
,
Dresselhaus
,
M. S.
,
Dresselhaus
,
G.
,
Fleurial
,
J.-P.
, and
Caillat
,
T.
, 2003, “
Recent Developments in Thermoelectric Materials
,”
Int. Mater. Rev.
0950-6608,
48
, pp.
45
66
.
2.
Dresselhaus
,
M. G.
,
Chen
,
G.
,
Tang
,
M. Y.
,
Yang
,
R. G.
,
Lee
,
H.
,
Wang
,
D. Z.
,
Ren
,
Z. F.
,
Fleurial
,
J. P.
, and
Gogna
,
P.
, 2007, “
New Directions for Low-Dimensional Thermoelectric Materials
,”
Adv. Mater. (Weinheim, Ger.)
0935-9648,
19
, pp.
1043
1053
.
3.
Kim
,
W.
,
Zide
,
J.
,
Gossard
,
A.
,
Klenov
,
D.
,
Stemmer
,
S.
,
Shakouri
,
A.
, and
Majumdar
,
A.
, 2006, “
Thermal Conductivity Reduction and Thermoelectric Figure of Merit Increase by Embedding Nanoparticles in Crystalline Semiconductors
,”
Phys. Rev. Lett.
0031-9007,
96
, pp.
045901
.
4.
Harman
,
T. C.
,
Taylor
,
P. J.
,
Walsh
,
M. P.
, and
LaForge
,
B. E.
, 2002, “
Quantum Dot Superlattice Thermoelectric Materials and Devices
,”
Science
0036-8075,
297
, pp.
2229
2232
.
5.
Venkatasubramanian
,
R.
,
Silvona
,
E.
,
Colpitts
,
T.
, and
O’Quinn
,
B.
, 2001, “
Thin-Film Thermoelectric Devices With High Room-Temperature Figures of Merit
,”
Nature (London)
0028-0836,
413
, pp.
597
602
.
6.
Hsu
,
K. F.
,
Loo
,
S.
,
Guo
,
F.
,
Chen
,
W.
,
Dyck
,
J. S.
,
Uher
,
C.
,
Hogan
,
T.
,
Polychroniadis
,
E. K.
, and
Kanatzidis
,
M. G.
, 2004, “
Cubic AgPbmSbTe2+m:Bulk Thermoelectric Materials With High Figure of Merit
,”
Science
0036-8075,
303
, pp.
818
821
.
7.
Majumdar
,
A.
, 2004, “
Thermoelectricity in Semiconductor Nanostructures
,”
Science
0036-8075,
303
, pp.
777
778
.
8.
Tritt
,
T. M.
,
Semicond. Semimetals
0080-8784, Vols.
69–71
(special issues on Recent Developments on Thermoelectrics).
9.
Golsmid
,
H. J.
, 1964,
Thermoelectric Refrigeration
,
Plenum
,
New York
.
10.
Yang
,
R. G.
, and
Chen
,
G.
, 2005, “
Nanostructured Thermoelectric Materials: From Superlattices to Nanocomposites
,”
Materials Integration
,
18
, pp.
31
36
(feature issue on “Thermoelectric Materials R&D in the World”).
11.
Chen
,
G.
, 2001, “
Phonon Transport in Low-Dimensional Structures
,”
Semicond. Semimetals
0080-8784,
71
, pp.
203
259
.
12.
Yang
,
R. G.
, and
Chen
,
G.
, 2004, “
Thermal Conductivity Modeling of Periodic Two-Dimensional Nanocomposites
,”
Phys. Rev. B
0163-1829,
69
, p.
195316
.
13.
Zhao
,
X. B.
,
Ji
,
X. H.
,
Zhang
,
Y. H.
,
Zhu
,
T. J.
,
Tu
,
J. P.
, and
Zhang
,
X. B.
, 2005, “
Bismuth Telluride Nanotubes and the Effects on the Thermoelectric Properties of Nanotube-Containing Nanocomposites
,”
Appl. Phys. Lett.
0003-6951,
86
, p.
062111
.
14.
Kapitza
,
P. L.
, 1941, “
Heat Tracer and Superfluidity of Helium II
,”
J. Phys. (Moscow)
0368-3400,
4
, pp.
181
210
;
Swartz
,
E. T.
, and
Pohl
,
P. P.
, 1989, “
Thermal Boundary Resistance
,”
Rev. Mod. Phys.
0034-6861,
61
, pp.
605
668
.
15.
Hasselman
,
D. P. H.
, and
Johnson
,
L. F.
, 1987, “
Effective Thermal Conductivity of Composites With Interfacial Thermal Barrier Resistance
,”
J. Compos. Mater.
0021-9983,
21
, pp.
508
515
.
16.
Benvensite
,
Y.
, and
Miloh
,
T.
, 1991, “
On the Effective Thermal Conductivity of Coated Short-Fiber Composites
,”
J. Appl. Phys.
0021-8979,
69
, pp.
1337
1344
.
17.
Every
,
A. G.
,
Tzou
,
Y.
,
Hasselman
,
D. P. H.
, and
Raj
,
R.
, 1992, “
The Effect of Particle Size on the Thermal Conductivity of ZnS∕Diamond Composites
,”
Acta Metall. Mater.
0956-7151,
40
, pp.
123
129
.
18.
Nan
,
C.-W.
,
Birringer
,
R.
,
Clarke
,
D. R.
, and
Gleiter
,
H.
, 1997, “
Effective Thermal Conductivity Of Particulate Composites With Interfacial Thermal Resistance
,”
J. Appl. Phys.
0021-8979,
81
, pp.
6692
6699
.
19.
Felske
,
J. D.
, 2004, “
Effective Thermal Conductivity of Composite Spheres in a Continuous Medium With Contact Resistance
,”
Int. J. Heat Mass Transfer
0017-9310,
47
, pp.
3453
3461
.
20.
Lu
,
S. Y.
, and
Song
,
J. L.
, 1996, “
Effect of Interfacial Characteristics on Effective Conductivities of Composites Containing Randomly Distributed Aligned Long Fibers
,”
Chem. Eng. Sci.
0009-2509,
51
, pp.
4393
4404
.
21.
Graham
,
S.
, and
McDowell
,
D. L.
, 2003, “
Numerical Analysis of the Transverse Thermal Conductivity of Composites With Imperfect Interfaces
,”
ASME J. Heat Transfer
0022-1481,
125
, pp.
389
393
.
22.
Torquato
,
S.
, and
Rintoul
,
M. D.
, 1995, “
Effect of the Interface on the Properties of Composite Media
,”
Phys. Rev. Lett.
0031-9007,
75
, pp.
4067
4070
.
23.
Lazarenkova
,
O. L.
, and
Balandin
,
A. A.
, 2002, “
Electron and Phonon Energy Spectra in a Three-Dimensional Regimented Quantum Dot Superlattice
,”
Phys. Rev. B
0163-1829,
66
, p.
245319
.
24.
Yang
,
B.
, and
Chen
,
G.
, 2003, “
Partially Coherent Phonon Heat Conduction in Superlattices
,”
Phys. Rev. B
0163-1829,
67
, p.
195311
.
25.
Yang
,
R. G.
,
Chen
,
G.
,
Laroche
,
M.
, and
Taur
,
Y.
, 2005, “
Multidimensional Transient Heat Conduction at Nanoscale Using the Ballistic-Diffusive Equations and the Boltzmann Equation
,”
ASME J. Heat Transfer
0022-1481,
127
, pp.
298
306
.
26.
Yang
,
R. G.
,
Chen
,
G.
, and
Dresselhaus
,
M. S.
, 2005, “
Thermal Conductivity of Simple and Tubular Nanowire Composites in the Longitudinal Direction
,”
Phys. Rev. B
0163-1829,
72
, p.
125418
.
27.
.
Howell
,
J. R.
, 1998, “
The Monte Carlo Method in Radiative Heat Transfer
,”
ASME J. Heat Transfer
0022-1481,
120
, pp.
547
560
.
28.
Moglestue
,
C.
, 1982, “
Monte-Carlo Particle Modeling of Small Semiconductor Devices
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
30
, pp.
173
208
.
29.
Jacoboni
,
C.
, and
Reggiani
,
L.
, 1983, “
The Monte-Carlo Method for the Solution of Charge Transport in Semiconductors With Applications to Covalent Materials
,”
Rev. Mod. Phys.
0034-6861,
55
, pp.
645
705
.
30.
Fischetti
,
M. V.
, and
Laux
,
S. E.
, 1988, “
Monte Carlo Analysis of Electron Transport in Small Semiconductor Devices Including Band-Structure and Space-Charge Effects
,”
Phys. Rev. B
0163-1829,
38
, pp.
9721
9745
.
31.
.
Fischetti
,
M. V.
, and
Laux
,
S. E.
, 1993, “
Monte Carlo Study of Electron Transport in Silicon Inversion Layers
,”
Phys. Rev. B
0163-1829,
48
, pp.
2244
2274
.
32.
Lugli
,
P.
,
Bordone
,
P.
,
Reggiani
,
L.
,
Rieger
,
M.
,
Kocevar
,
P.
, and
Goodnick
,
S. M.
, 1989, “
Monte Carlo Studies of Nonequilibrium Phonon Effects in Polar Semiconductors and Quantum Wells. I. Laser Photoexcitation
,”
Phys. Rev. B
0163-1829,
39
, pp.
7852
7865
.
33.
Peterson
,
R. B.
, 1994, “
Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal
,”
ASME J. Heat Transfer
0022-1481,
116
, pp.
815
822
.
34.
.
Klitsner
,
T.
,
VanCleve
,
J. E.
,
Fischer
,
H. E.
, and
Pohl
,
R. O.
, 1988, “
Phonon Radiative Heat Transfer and Surface Scattering
,”
Phys. Rev. B
0163-1829,
38
, pp.
7576
7594
.
35.
Mazumder
,
S.
, and
Majumdar
,
A.
, 2001, “
Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization
,”
ASME Trans. J. Heat Transfer
0022-1481,
123
, pp.
749
759
.
36.
Song
,
D.
, 2003, “
Phonon Heat Conduction in Nano and Micro-Porous Thin Films
,” Ph.D. thesis, University of California at Los Angeles, Los Angeles, CA.
37.
Lacroix
,
D.
,
Joulain
,
K.
, and
Lemonnier
,
D.
, 2005, “
Monte Carlo Transient Phonon Transport in Silicon and Germanium at Nanoscales
,”
Phys. Rev. B
0163-1829,
72
, p.
064305
.
38.
Chen
,
Y. F.
,
Li
,
D. Y.
,
Lukes
,
J. R.
, and
Majumdar
,
A.
, 2005, “
Monte Carlo Simulation of Silicon Nanowire Thermal Conductivity
,”
ASME Trans. J. Heat Transfer
0022-1481,
127
, pp.
1129
1138
.
39.
Prasher
,
R.
, 2006, “
Transverse Thermal Conductivity of Porous Materials Made From Aligned Nano- and Microcylindrical Pores
,”
J. Appl. Phys.
0021-8979,
100
, p.
064302
.
40.
Prasher
,
R.
, 2006, “
Thermal Conductivity of Composites of Aligned Nanoscale and Microscale Wires and Pores
,”
J. Appl. Phys.
0021-8979,
100
, p.
034307
.
41.
Chen
,
G.
, 1998, “
Thermal Conductivity and Ballistic-Phonon Transport in the Cross-Plane Direction of Superlattices
,”
Phys. Rev. B
0163-1829,
57
, pp.
14958
14973
.
42.
Brockhouse
,
B. N.
, 1959, “
Lattice Vibrations in Silicon and Germanium
,”
Phys. Rev. Lett.
0031-9007,
2
, pp.
256
258
.
43.
Nilsson
,
G.
, and
Nelin
,
G.
, 1971, “
Phonon Dispersion Relations in Ge at 80k
,”
Phys. Rev. B
0556-2805,
3
, pp.
364
369
.
44.
The temperature-dependent thermal conductivity of bulk Si and Ge is available at http://www.ioffe.rssi. ru/SVA/NSM/Semicond/http://www.ioffe.rssi. ru/SVA/NSM/Semicond/
45.
Goodson
,
K. E.
, and
Ju
,
Y. S.
, 1999, “
Heat Conduction in Novel Electronic Films
,”
Annu. Rev. Mater. Sci.
0084-6600,
29
, pp.
261
293
.
46.
Thijssen
,
J.
, 2007,
Computational Physics
,
2nd ed.
Cambridge University Press
,
UK
.
47.
Swartz
,
E. T.
, and
Pohl
,
R. O.
, 1989, “
Thermal Boundary Resistance
,”
Rev. Mod. Phys.
0034-6861,
61
, pp.
605
668
.
48.
Dames
,
C.
, and
Chen
,
G.
, 2004, “
Theoretical Phonon Thermal Conductivity of Si∕Ge Superlattice Nanowires
,”
J. Appl. Phys.
0021-8979,
95
, pp.
682
693
.
49.
Lee
,
H.
, 2005, “
Experimental Study of Thermal Conductivity Reduction of Silicon-Germanium Nanocomposite for Thermoelectric Application
,” MS thesis, Massachusetts Institute of Technology, Boston, MA.
50.
Dresselhaus
,
M. S.
,
Chen
,
G.
,
Ren
,
Z. F.
,
Fleurial
,
J. P.
, and
Gogna
,
P.
, 2005, Second Annual Technical Report for NASA Contract No. NAS3-03108.
51.
Tian
,
W. X.
, and
Yang
,
R. G.
, 2007, “
Effect of Interface Scattering on Phonon Thermal Conductivity Percolation in Random Nanowire Composites
,”
Appl. Phys. Lett.
0003-6951,
90
, p.
263105
.
52.
Tian
,
W. X.
, and
Yang
,
R. G.
, 2007, “
Thermal Conductivity Modeling of Compacted Nanowire Composites
,”
J. Appl. Phys.
0021-8979,
101
, p.
054320
.
53.
Tian
,
W. X.
, and
Yang
,
R. G.
, “
Phonon Transport and Thermal Conductivity Percolation in Random Nanoparticle Composites
,”
Comput. Model. Eng. Sci.
1526-1492(CMES), in press.
54.
Minnich
,
A.
, and
Chen
,
G.
, 2007, “
Modified Effective Medium Formulation for the Thermal Conductivity of Nanocomposites
,”
Appl. Phys. Lett.
0003-6951,
91
, p.
073105
.
You do not currently have access to this content.