Analytical description of phonon heat flow with diffusive boundary scattering

Abstract

The paper presents an analytical solution to the problem of heat flow due to the ballistic motion of phonons, which are diffusely reflected from the side walls of a two-dimensional dielectric material. An explicit analytical expression is obtained for the temperature profile in the approximation of linear dependence; explicit equations are derived for the heat flux and the thermal conductivity coefficients of the sample with arbitrary ratios of the width W and the length L. The analytical expressions are in very good agreement with numerical solutions of the integral equations, which describe these quantities and were derived in our previous study [J. Appl. Phys. 129, 085105 (2021)]. The obtained analytical relations can be used for subsequent studies in the case of mixed specular and diffuse reflection, as well as in the study of phonon systems of specific materials.