Temperature in a Peierls-Boltzmann Treatment of Nonlocal Phonon Heat Transport

TL;DR

This paper develops a theoretical framework for understanding nonlocal phonon heat transport in insulators, solving the Peierls-Boltzmann equation and analyzing the transition from diffusive to ballistic regimes.

Contribution

It introduces a formal solution for nonlocal heat transport using the PBE and RTA, and defines a nonlocal thermal susceptibility, providing new insights into temperature measurement at microscopic scales.

Findings

Demonstrates diffusive to ballistic crossover in heat transport

Highlights fundamental issues in defining temperature in RTA

Provides analytical solutions for periodic heating scenarios

Abstract

In nonmagnetic insulators, phonons are the carriers of heat. If heat enters in a region and temperature is measured at a point within phonon mean free paths of the heated region, ballistic propagation causes a nonlocal relation between local temperature and heat insertion. This paper focusses on the solution of the exact Peierls-Boltzmann equation (PBE), the relaxation time approximation (RTA), and the definition of local temperature needed in both cases. The concept of a non-local "thermal susceptibility" (analogous to charge susceptibility) is defined. A formal solution is obtained for heating with a single Fourier component $P(\vec{r},t)=P_0 \exp(i\vec{k}\cdot\vec{r}-i\omega t)$, where $P$ is the local rate of heating). The results are illustrated by Debye model calculations in RTA for a three-dimensional periodic system where heat is added and removed with $P(\vec{r},t)=P(x)$ from isolated evenly spaced segments with period $L$ in $x$. The ratio $L/\ell_{\rm min}$ is varied from 6 to $\infty$, where $\ell_{\rm min}$ is the minimum mean free path. The Debye phonons are assumed to scatter anharmonically with mean free paths varying as $\ell_{\rm min}(q_D/q)^2$ where $q_D$ is the Debye wavevector. The results illustrate the expected local (diffusive) response for $\ell_{\rm min}\ll L$, and a diffusive to ballistic crossover as $\ell_{\rm min}$ increases toward the scale $L$. The results also illustrate the confusing problem of temperature definition. This confusion is not present in the exact treatment but is fundamental in RTA.