Thermal-response Functions and the Peierls-Boltzmann Equation for Second Sound and Phonon Hydrodynamics in Graphene

Abstract

We connect expressions for phonon phase-space distribution functions to microscopic physics of the evolution of heat waves. The role of interference effects that arise as a result of a periodic heating source typically encountered in transient thermal grating (TTG) experiments is then explored. The distribution functions are evaluated as solutions to the Peierls-Boltzmann equation (PBE) in the relaxation-time approximation (RTA). Starting from the PBE, we next develop thermal response functions. The response functions are computed using data from density-functional theory (DFT) calculations. Using this approach, it is shown how solutions to the PBE can be related to the propagation of second phonons as elementary excitations, and within this perspective the necessary conditions for the propagation and observation of second sound is elucidated. The approach developed therefore shows how PBE theory for phonon hydrodynamics and second sound can be modified to properly describe interference and phonon decoherence effects that are likely important at shorter length scales. Finally, we then discuss how many-body theory can be extended to include coupled scattering channels and hence provide a quantitative theory beyond the RTA.