Ballistic-Diffusive Heat Conduction in Thin Films by Phonon Monte Carlo Method: Gray Medium Approximation Versus Phonon Dispersion

Abstract

The gray medium approximation treating all phonons with an averaged and representative mean-free-path (MFP) is an often used method in analyzing ballistic-diffusive heat conduction at nanoscale. However, whether there exists a reasonable value of the average MFP which effectively represents the entire spectrum of modal MFPs remains unclear. In this paper, phonon Monte Carlo (MC) method is employed to study the effects of the gray medium approximation on ballistic-diffusive heat conduction in silicon films by comparing with dispersion MC simulations. Four typical ways for calculating the average MFP with gray medium approximation are investigated. Three of them are based on the weighted average of the modal MFPs, and the remaining one is based on the weighted average of the reciprocals of the modal MFPs. The first three methods are found to be good at predicting effective thermal conductivity and heat flux distribution, but have difficulties in temperature profile, while the last one performs better for temperature profile than effective thermal conductivity and heat flux distribution. Therefore, none of the average MFPs can accurately characterize all the features of ballistic-diffusive heat conduction for the gray medium approximation. Phonon dispersion has to be considered for the accurate thermal analyses and modeling of ballistic-diffusive heat transport. Our work could be helpful for further understanding of phonon dispersion and more careful use of the gray medium approximation.