Features of phonon transport in silicon rods and thin plates in the boundary scattering regime. The effect of phonon focusing at low temperatures

The effect of phonon focusing on the phonon relaxation characteristics in the boundary scattering regime is studied in cubic crystals. The phonon relaxation times due to diffusive scattering at boundaries are calculated for rod-shaped samples and thin plates of finite lengths having uniform rectangular cross-sections. It is shown that relaxation times can be represented by piecewise smooth functions for different intervals of angles defined by the relation between components of the phonon group velocity and geometrical parameters of samples. Thermal conductivity and phonon free paths have been calculated for silicon samples with different orientation of the temperature gradient and side faces in the anisotropic continuum model. Anisotropies of thermal conductivity and phonon free paths for silicon change strongly with the width-to-thickness (aspect) ratio in the range from 1 to 1000, demonstrating steep dependence on the orientation of temperature gradient and the wide face of thin plates with high values of aspect ratio.