Green's Functions of the Boltzmann Transport Equation with the Full Scattering Matrix for Phonon Nanoscale Transport beyond the Relaxation Time Approximation

TL;DR

This paper develops a comprehensive Green's function solution for the phonon Boltzmann transport equation that incorporates the full scattering matrix, enabling accurate analysis of nanoscale heat transport beyond the relaxation time approximation, especially in high Debye temperature materials.

Contribution

It introduces a full scattering matrix formalism and Green's function solution for the BTE, surpassing the limitations of the RTA in modeling nanoscale thermal phenomena.

Findings

Full scattering matrix formalism captures complex phonon interactions.

Numerical results demonstrate differences from RTA in graphene.

Method describes phenomena like heat dissipation and second sound.

Abstract

The phonon Boltzmann transport equation (BTE) has been widely utilized to study thermal transport in solids. While for a number of materials the exact solution to the BTE has been obtained for a uniform heat flow, problems arising in micro/nanoscale heat transport have been analyzed within the relaxation time approximation (RTA). Since the RTA breaks down at temperatures low compared to the Debye temperature, this approximation prevents the study of an important class of high Debye temperature materials such as diamond, graphite, graphene and some other 2D materials. We present a full scattering matrix formalism that goes beyond the RTA approximation and obtain a Green's function solution for the linearized BTE, which leads to an explicit expression for the phonon distribution and temperature field produced by an arbitrary spatio-temporal distribution of heat sources in an unbounded medium. The presented formalism is capable of describing a wide range of phenomena, from heat dissipation by nanoscale hot spots to the propagation of second sound waves. We provide numerical results for graphene for a spatially sinusoidal heating profile and discuss the importance of using the full scattering matrix compared to the RTA.