On the Kinetic Theory of Thermal Transport in Crystals

Abstract

The phonon Boltzmann transport equation is a reliable model for describing heat transfer in dielectric crystals and, coupled with first-principles methods, has shown to provide an accurate description of heat transport. Early solutions, based on simplifying assumptions, such as the single-mode relaxation-time approximation, have fostered the description of thermal transport as a kinetic theory of the phonon gas. In this chapter, we summarize some recent advances in the field of semiclassical transport. In particular, we show how the exact solutions of the linearized Boltzmann transport equations can be framed in terms of relaxons, collective excitations of phonon populations defined as the eigenvectors of the scattering operator. We show how an exact kinetic theory of heat transport can be constructed using relaxons, rather than phonons, as the relevant heat carriers. We showcase this change of perspective with a first-principles study of the transport properties of graphene. We discuss the relaxons’ mean free paths, velocities, relaxation times, and their contributions to thermal conductivity. We contrast these results with those, approximate, obtained in the single-mode relaxation time approximation, and show how in this case the approximation fails qualitatively and quantitatively. Consequently, also Matthiessen’s rule becomes invalid. Then, we discuss how surface or edge scattering can be properly recast in terms of friction, giving rise to heat profiles that resemble the hydrodynamic flow of liquids. Finally, we show how the Laplace transform of the linearized Boltzmann equation allows the definition of collective excitations, termed transport waves. These excitations are characterized by dispersion relations, and, in the context of lattice transport, they correspond to second-sound modes at finite wavelength, leading to relaxons in the long-wavelength limit.