An enhanced Fourier law derivable from the Boltzmann transport equation and a sample application in determining the mean-free path of nondiffusive phonon modes

An enhanced Fourier law that we term the unified nondiffusive-diffusive (UND) phonon transport model is proposed in order to account for the effect of low-frequency phonon modes of long mean-free path that propagate concomitantly to the dominant high-frequency modes. The theory is based on spherical harmonic expansions of the phonon distribution functions, wherein the high-frequency mode distribution function is truncated at the first order in the expansion, while the low-frequency mode distribution function, which is farther out of thermal equilibrium, is truncated at the second order. As an illustrative application, the predictions of the proposed model are compared with data from a recent experiment that utilized the transient gratings method to investigate the deviation of thermal transport in a silicon membrane from the predictions of the Fourier law. The good fit of the experimental effective thermal conductivity (ETC) with the analytical solution derived in this work yields quantitative information about the mean-free path of the dominant low-frequency heat-transfer mode in silicon.