Variational Approach to Solving the Spectral Boltzmann Transport Equation in Transient Thermal Grating for Thin Films

TL;DR

This paper introduces a variational method to analytically solve the phonon Boltzmann transport equation in thin film transient thermal grating experiments, enabling better understanding of non-diffusive heat transfer in nanostructures.

Contribution

It develops a variational approach with an analytical solution for the BTE in TTG geometry, linking thermal decay and conductivity to material and geometry parameters.

Findings

Analytical expression for thermal decay rate matches Monte Carlo simulations.

Derived closed-form effective thermal conductivity depending on material and geometry.

Recovers known limits for thick films and large grating spacings.

Abstract

The phonon Boltzmann transport equation (BTE) is widely utilized to study non-diffusive thermal transport. We find a solution of the BTE in the thin film transient thermal grating (TTG) experimental geometry by using a recently developed variational approach with a trial solution supplied by the Fourier heat conduction equation. We obtain an analytical expression for the thermal decay rate that shows excellent agreement with Monte Carlo simulations. We also obtain a closed form expression for the effective thermal conductivity that demonstrates the full material property and heat transfer geometry dependence, and recovers the limits of the one-dimensional TTG expression for very thick films and the Fuchs-Sondheimer expression for very large grating spacings. The results demonstrate the utility of the variational technique for analyzing non-diffusive phonon-mediated heat transport for nanostructures in multi-dimensional transport geometries, and will assist the probing of the mean free path (MFP) distribution of materials via transient grating experiments.