# Experimental demonstration of a generalized Fourier's Law for non-diffusive thermal transport

## TL;DR

This paper derives and experimentally validates a generalized Fourier's law that accurately describes non-diffusive thermal transport from ballistic to diffusive regimes, highlighting the importance of nonlocal effects and inhomogeneous source terms.

## Contribution

The paper introduces a generalized Fourier's law derived from the Peierls-Boltzmann equation, accounting for nonlocal effects and inhomogeneous sources in thermal transport.

## Key findings

- The generalized Fourier's law matches TDTR measurements in the non-diffusive regime.
- Neglecting the generalized law leads to inaccurate thermal property measurements.
- The law applies across ballistic to diffusive regimes for various geometries.

## Abstract

Phonon heat conduction over length scales comparable to their mean free paths is a topic of considerable interest for basic science and thermal management technologies. Although the failure of Fourier's law beyond the diffusive regime is well understood, debate exists over the proper physical description of thermal transport in the ballistic to diffusive crossover. Here, we derive a generalized Fourier's law that links the heat flux and temperature fields, valid from ballistic to diffusive regimes and for general geometries, using the Peierls-Boltzmann transport equation within the relaxation time approximation. This generalized Fourier's law predicts that thermal conductivity not only becomes nonlocal at length scales smaller than phonon mean free paths, but also requires the inclusion of an inhomogeneous nonlocal source term that has been previously neglected. We demonstrate the validity of this generalized Fourier's law through direct comparison with time-domain thermoreflectance (TDTR) measurements in the nondiffusive regime without adjustable parameters. Furthermore, we show that interpreting experimental data without this generalized Fourier's law leads to inaccurate measurement of thermal transport properties.