Characteristics of quasi-ballistic heat conduction in a multiple materials system based on the solution of the Boltzmann transport equation

Abstract

Phonons will exhibit quasi-ballistic heat transport when the mean free path is longer than the dimension of the medium through which the phonons travel. Such a transport regime can yield information about the phonon mean free path. The primary focus of this study is on numerically solving the one-dimensional transient Boltzmann transport equation to investigate how the phonon mean free path can be determined in accordance with the characteristics of quasi-ballistic heat conduction in a multiple materials system. The Boltzmann transport equation is solved for heat conduction within an aluminum film with a double-layered structure based on the discrete ordinates method. The metal film is deposited onto a silicon substrate. Frequency-dependent boundary conditions are used to define the difference in phonon dispersion between the metal film and the substrate. The frequency-dependent Boltzmann transport equation is used to determine how the thermal conductivity changes with the modulation frequency when phonons travel quasi-ballistically. The distribution of the phonon mean free path is discussed, and comparisons are made between theoretical results at different modulation frequencies. The results indicated that the effective thermal conductivity of the multiple materials system does not depend upon the modulation frequency. Phonons with mean free paths longer than the penetration depth do not contribute to the effective thermal conductivity. The effect of quasi-ballistic heat conduction is noticeable in semiconductor alloys even at room temperature.