Thermal transport in semiconductors studied by Monte Carlo simulations combined with the Green-Kubo formalism

In this paper, we demonstrate that it is possible to combine two computational approaches, usually used in very different contexts, namely, the Green-Kubo formalism and statistical approaches, to solve the Boltzmann transport equation to calculate the thermal conductivity of semiconductors. In this framework, first the phonon transport is solved by the Monte Carlo method according to the Debye-Callaway or Holland formalisms, then the flux autocorrelation is calculated. This new approach has been implemented to calculate the properties of Si, Ge, GaN, and C (diamond) in an extended temperature range, from 50 to 600 K. The simulation results are compared with those given by experimentation, with very good agreement.

Figure 1

Five phonon trajectories in bulk Si at $T=300$ K after 100 000 time steps ($\delta t=1$ ps) without boundary scattering (bulk case). The initial particle's locations are within the green box ($V=1\times 1\times 1\mu \mathrm{m}$).

Figure 2

Analysis of the particle mfp and phonon state during successive time steps of the simulation.

Figure 3

Top: Phonon mfp distribution for Si at 300 K. Bottom: Phonon mfp as a function of polarization, frequency, and group velocity. Five particles were followed during 100 000 time steps of 1 ps. Phonon lifetimes according to Holland's formalism [11].

Figure 4

Top: Heat flux autocorrelation for $x$, $y$, and $z$ components for Si at 300 K in the bulk. Bottom: Thermal conductivity components $k_{xx}$, $k_{yy}$, and $k_{zz}$. Five hundred particles were followed during 100 000 time steps of 1 ps. Phonon lifetimes according to Holland's formalism [11].

Figure 5

Temperature fluctuations in the domain vs simulation time according to the number of sampled particles. $N_p=50$, 100, and 500.

Figure 6

Computed TCs vs their experimental counterparts for Si [15]. Inset: $k$ vs $T$.

Figure 7

Computed TCs vs their experimental counterparts for Ge [15]. Inset: $k$ vs $T$.

Figure 8

Computed TCs vs their experimental counterparts for GaN [16]. Inset: $k$ vs $T$.

Figure 9

Computed TCs vs their experimental counterparts for C diamond [17]. Inset: $k$ vs $T$.