Supplementary material for Heat conduction across a solid-solid interface:
Understanding nanoscale interfacial effects on thermal resistance
Ganesh Balasubramanian and Ishwar K. Puri*
Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, USA
*
Corresponding author
Virginia Tech, Department of Engineering Science and Mechanics,
223A Norris Hall, Blacksburg, VA 24061, USA
Tel.: +1-540-231-3243, Fax: +1-540-231-4574, Email: ikpuri@vt.edu
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Using either a solely continuum approach or instead only molecular simulations to describe
the overall influence of the thermal resistance across an interface can be problematic. Continuum
relations do not adequately describe the interfacial heat transfer at the nanoscale.1 However, the
alternative of conducting molecular simulations for a mesoscale system containing thousands of
atoms can be computationally expensive and an inefficient use of resources. We circumvent this
difficulty by incorporating the effect of Rk into a modified continuum relation for steady-state
heat conduction.
Dependence of Ge thermal conductivity on length
Since the heat transfer changes from ballistic transport at lengthscales smaller than the
characteristic phonon mean free path to diffusive transport for larger systems,2 the material
thermal conductivity, a required parameter in continuum relations, also changes.3, 4 The literature
discusses the variation of Si thermal conductivity with length3, 4, but a similar study for Ge is
absent.
We use equilibrium MD to simulate Ge cubes of various edge dimensions L to describe the
dependence of its thermal conductivity kGe on length. Periodic boundary conditions in all
directions are employed to eliminate phonon-boundary scattering. Tersoff potentials model the
atomic interactions.5, 6 The value of kGe is obtained from equilibrium MD simulations using the
Green-Kubo method based on the fluctuation-dissipation theorem.3, 7, 8 The heat current vector S
in such systems oscillates about zero at equilibrium and the thermal conductivity is correlated
with the time required for these fluctuations to dissipate. Therefore, for a cubic isotropic material

kGe  (1/ 3kBVT 2 )  S (t ).S (0) dt , where kB denotes the Boltzmann constant, V and T the
0
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system volume and absolute temperature, t the time, and
S (t ).S (0) is the heat current
autocorrelation function (HCAF). We determine kGe from converged values of the HCAF.
Each cube is initialized at 300 K and 1 atm. Using 0.002 ps timesteps, it is (1) simulated for
0.2 ns under constant temperature and pressure (NPT ensemble) to relax the internal stresses in
the lattice, (2) followed by a constant volume and temperature (NVT) equilibration for further
0.2 ns. We employ the Nosé-Hoover thermostat and barostat, each with a coupling time of 0.1 ps
to maintain the temperature and pressure, respectively. Finally, the system is allowed to evolve
freely with constant volume and energy (NVE) over the next 2 ns, while the heat current values
are sampled over every 0.02 ps and the time correlation between them is calculated. kGe is
calculated by averaging the thermal conductivities kxx, kyy and kzz along the three orthogonal (x-yz) directions of the cubic lattice recorded every timestep over that last 200 ps. To accurately
determine the HCAF integral, simulation times (2 ns) that are much longer than phonon
relaxation time of ~100 ps are employed.3
The variation of 1/kGe with 1/L is presented in Fig. S-1 for L in the range from 2.5 nm – 10
nm. As the system length increases so does the thermal conductivity with a linear correlation
between 1/L and 1/kGe. At very small lengthscales, phonons do not have sufficient space for their
natural evolution and decay. This enhances the energy loss due to boundary scattering so that
transport is predominantly ballistic. As the system size increases, boundary scattering
diminishes, leading to lower energy losses and higher thermal conductivity. Here, thermal
transport becomes diffusive. Extrapolating the predictions for L  ∞, we obtain the bulk
thermal conductivity for Ge kGe-Tersoff = 111 Wm-1K-1. While this value agrees with the simulation
literature,9 it differs from experimental measurements because (1) empirical potential models are
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used to mimic the interatomic interactions, (2) a pure defect-free Ge is simulated in comparison
to isotopically impure Ge typically used in experiments and (3) the simulations being classical,
do not account for variations in phonon frequencies.
Variation of thermal resistance with interfacial contact area
Effects of varying interfacial contact areas on thermal resistance Rk are considered using
nonequilibrium MD simulations. Superlattices are simulated with Lx = 10 nm (= 2 Lx-Si = 2 Lx-Ge)
while Ly = Lz are varied between 1 – 10 nm. Values for Rk presented in Table I show that beyond
2.5 nm, the lateral dimensions of the superlattice have negligible influence on the magnitude of
Rk. Hence, this resistance is dependent only on the dimensions along the direction of heat
conduction, not on those perpendicular to it.
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References
1.
S. Murad and I. K. Puri, Applied Physics Letters 95 (5), 051907-1 (2009).
2.
G. Chen, International Journal of Thermal Sciences 39 (4), 471 (2000).
3.
D. P. Sellan, E. S. Landry, J. E. Turney, A. J. H. McGaughey and C. H. Amon, Physical
Review B 81 (21), 214305-1 (2010).
4.
Y. W. Yang, X. J. Liu and J. P. Yang, Molecular Simulation 34 (1), 51 (2008).
5.
J. Tersoff, Physical Review B 37 (12), 6991 (1988).
6.
J. Tersoff, Physical Review B 39 (8), 5566 (1989).
7.
P. K. Schelling, S. R. Phillpot and P. Keblinski, Physical Review B 65 (14), 144306-1
(2002).
8.
A. Guajardo-Cuellar, D. B. Go and M. Sen, Journal of Chemical Physics 132 (10),
104111-1 (2010).
9.
J. J. Dong, O. F. Sankey and C. W. Myles, Physical Review Letters 86 (11), 2361 (2001).
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List of Figures:
Fig. S-1: Length L dependence of the Ge thermal conductivity kGe at 300 K obtained from
equilibrium MD simulations. The continuous line is a linear fit to the data. It is extrapolated to
1/L = 0 to determine the bulk Ge thermal conductivity kGe = 111 Wm-1K-1. The error bars depict
the data scatter in the form of standard errors for the thermal conductivities that are recorded
every 0.002 ps over that last 200 ps of each simulation.
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Table I: Interfacial thermal resistance Rk values obtained from MD simulations of 10 nm × y nm
× z nm Si-Ge superlattices with different cross-sections are listed. Beyond 2.5 nm lateral
dimension, Rk remains constant, implying that dimensions orthogonal to direction of heat transfer
do not exert a notable influence on the thermal transport through the interface.
Cross section
1.0 × 1.0
2.5 × 2.5
5.0 × 5.0
7.5 × 7.5
10.0 × 10.0
(y nm × z nm)
Rk (m2KW-1)
4.02 × 10-8 2.85 × 10-9 2.65 × 10-9 2.72 × 10-9
2.75 × 10-9
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Figure S-1
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