Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

TL;DR

This paper investigates the nonlocal effects in phonon thermal conductivity during the ballistic to diffusive crossover using the Peierls-Boltzmann transport theory, proposing improved methods for bulk thermal conductivity estimation.

Contribution

It extends Callaway's correction method to the non-local regime, analyzing spatial temperature variations and phonon wavevector dependencies in thermal transport.

Findings

Nonlocal effects cause nonlinear temperature profiles.

Improved extrapolation methods for bulk thermal conductivity.

Generalization of Callaway's correction to non-local phonon transport.

Abstract

Simulations (e.g. Zhou et al., Phys. Rev. B 79, 115201 (2009)) show nonlocal effects of the ballistic/diffusive crossover. The local temperature has nonlinear spatial variation not contained in the local Fourier law $\vec{j}(\vec{r})=-\kappa\vec{\nabla}T(\vec{r})$. The heat current $\vec{j}(\vec{r})$ depends not just on the local temperature gradient $\vec{\nabla}T(\vec{r})$, but also on temperatures at points $\vec{r}^{ \ \prime}$ within phonon mean free paths, which can be micrometers long. This paper uses the Peierls-Boltzmann transport theory in non-local form to analyze the spatial variation $\Delta T(\vec{r})$. The relaxation-time approximation (RTA) is used because full solution is very challenging. Improved methods of extrapolation to obtain the bulk thermal conductivity $\kappa$ are proposed. Callaway invented an approximate method of correcting RTA for the $\vec{q}$ (phonon wavevector or crystal momentum) conservation of N (normal as opposed to Umklapp) anharmonic collisions This method is generalized to the non-local case where $\kappa(\vec{k})$ depends on wavevector of the current $\vec{j}(\vec{k})$ and temperature gradient $i\vec{k}\Delta T(\vec{k})$.