# Modeling quasi-ballistic transient thermal transport with spatially sinusoidal heating: a McKelvey-Shockley flux approach

## TL;DR

This paper extends the McKelvey-Shockley flux method to model transient ballistic phonon transport under sinusoidal heating, accurately capturing non-diffusive effects and aligning well with experimental and Boltzmann solutions.

## Contribution

The work introduces an extended McKelvey-Shockley flux approach that includes inelastic scattering and supports full phonon dispersion, providing a simple yet accurate model for ballistic to diffusive heat transport.

## Key findings

- The extended model agrees with experimental data.
- It accurately reproduces Boltzmann transport solutions.
- Inelastic scattering is crucial for the diffusive limit.

## Abstract

Ballistic phonon effects, arising on length scales comparable to the mean-free-path, result in non-diffusive heat flow and alter the thermal properties of materials. Simple theoretical models that accurately capture non-diffusive transport physics are valuable for experimental analysis, technology design, and providing physical insight. In this work, we utilize and extend the McKelvey-Shockley (McK-S) flux method, a simple and accurate framework, to investigate ballistic effects in transient phonon transport submitted to a spatially sinusoidal heating profile, simulating a transient thermal grating. We begin by extending a previous McK-S formulation to include inelastic scattering, then obtain an analytical solution in the single phonon energy case (gray approximation), and after show how this approach can readily support a full phonon dispersion and mean-free-path distribution. The results agree with experimental data and compare very well to solutions of the phonon Boltzmann transport equation in the diffusive and weakly quasi-ballistic transport regimes. We discuss the role of ballistic and non-equilibrium physics, and show that inelastic scattering is key to retrieving the heat equation solution in the diffusive limit. Overall the McK-S flux method, which takes the form of a diffusion-like equation, proves to be a simple and accurate framework that is applicable from the ballistic to diffusive transport regime.