Hybrid ballistic–diffusive solution to the frequency-dependent phonon Boltzmann Transport Equation

Abstract

The phonon Boltzmann Transport Equation (BTE) is appropriate for modeling heat conduction in semiconductor materials at the nanoscale. However, the BTE is difficult to solve on account of the directional and spectral nature of the phonon intensity, which necessitates angular and spectral discretization, and ultimately results in a large number (typically few hundreds) of four-dimensional partial differential equations. In the ballistic (large Knudsen number) regime, the phonon intensity is highly anisotropic, and therefore, angular resolution is desirable. However, in the diffusive (small Knudsen number) regime, the intensity is fairly isotropic, and hence, angular discretization is wasteful. In such scenarios, the method of spherical harmonics (P N approximation) may be effectively used to reduce the large number of directional BTEs to a few partial differential equations. Since the Knudsen number is frequency dependent, the decision to preserve or eliminate angular discretization may be made frequency by frequency based on whether the spectral Knudsen number is large or small. In this article, a hybrid method is proposed in which for some frequency intervals (bands), full angular discretization is used, while for others, the P 1 approximation is invoked to reduce the number of directional BTEs. The accuracy and efficiency of the hybrid method is tested by solving several steady state and transient nanoscale heat conduction problems in two and threedimensional geometries. Silicon is used as the candidate material. It is found that hybridization is effective in significantly improving the efficiency of solution of the BTE-sometimes by a factor of three-without significant penalty on the accuracy.

Prediction of thermal conductivity in dielectrics using fast, spectrally-resolved phonon transport simulations

“…It is clear that q 1 (with Λ = 3120 nm) carries the dominant portion of energy throughout the entire range of domain sizes, as it barely begins to approach the asymptotic limit at a length of 10 microns. However, it is overtaken early on by groups 2-5, Diffuse groups (7,8,9) always carry low amounts of heat and remain relatively flat independent of domain size.…”

“…The problem is that the flow of energy into the implied phonon bath in the SMRT is not reflected in the explicit phonon bath used to compute temperature in Eq. (9), and so we must include an additional source term to correct this.…”

“…Advances have been made in the simulation of spectrally-dependent phonon transport, providing important insights into the spectral contribution to heat transport, but to date the computational burden associated with BTE solutions have limited the size, dimensionality, and or physics of the systems considered. Allu et al [9] investigated frequency dependence of the BTE in silicon slabs and reported spectral thermal conductivity based on Knudsen number (the mean free path divided by the domain length). However, the trade-off of this approach was their reported high computational expense, and the discrete phonon groups had no coupling.…”

An implicit kinetic scheme for multiscale heat transfer problem accounting for phonon dispersion and polarization

“…Over the past decades, a lot of numerical methods are developed to solve the phonon BTE based on the non-gray model, such as the (control angle) discrete ordinate method (DOM, CADOM) [20,21], hybrid Ballistic-Diffusive or Fourier-BTE method [22,23], Monte Carlo (MC) method [24,25,26,27] and so on. The Monte Carlo method [26,28,27,29,30] is one of the most widely used statistics methods and has made great progress in thermal application.…”

Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference

“…Although it is a great challenge to solve the phonon BTE as the temperature difference is large, some numerical methods have been developed, such as the Monte Carlo method [24,25,26,27,28,6], discrete ordinate method (DOM) [29,30,31] and the hybrid Ballistic-Diffusive method [32]. The DOM is one of the most widely used deterministic methods, which discretizes the whole wave vector space into small pieces.…”