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. However, debate exists over the appropriate constitutive law that defines thermal conductivity in the nondiffusive regime. 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 provide evidence for the validity of this generalized Fourier's law through direct comparison with time-domain thermoreflectance measurements in the nondiffusive regime without adjustable parameters. Furthermore, we show that interpreting experimental data without the generalized Fourier's law can lead to inaccurate measurement of thermal transport properties.
