A New Formula for Thermal Conductivity Based on a Hierarchy of Hydrodynamical Models

Abstract

We present a hierarchy of macroscopic models which generalize Cattaneo equation and can be used to describe heat conduction in semiconductor materials. In particular, from these models we derive a new formula for the lattice thermal conductivity which is a able to reproduce the results achieved by means of the celebrated Callaway formula and to give some insights into the dynamical thermal conductivity of semiconductor materials. The models make use of a set of macroscopic state variables for the acoustic phonons, that are moments of their occupation numbers. The evolution equations for these variables are obtained starting from the Boltzmann–Peierls transport equations, and are closed by means of the maximum entropy principle. All the main interactions of phonons among themselves, with isotopes and boundaries are taken into account. Numerical results are shown for the cases of silicon and germanium.