Experimental demonstration of generalized Fourier's Law for heat conduction at the nanoscale

Heat conduction under large temperature gradients that occur over scales of mean free paths of the energy carriers in solids is a topic of intense interest. Although the failure of Fourier's law is well understood, the appropriate replacement has been the topic of debate. A concise relation that links temperature gradient to heat flux based on a rigorous mathematical interpretation is not only necessary but crucial to advance our knowledge of nanoscale heat transport. Here, we derived a generalized Fourier's law based on Peierls-Boltzmann transport equation. This generalized Fourier's law contains two parts, nonlocality of thermal conductivity, which has been previously hypothesized, and nonlocality of external effects, i.e. volumetric heat generation, which has long been ignored in literatures. We demonstrated the validity of this generalized Fourier's law through comparisons with a series of time-domain thermoreflectance (TDTR) measurements. Furthermore, we showed that misinterpreting the generalized Fourier's law in the experimental observation of nanoscale heat conduction would lead to erroneous microscopic information of phonons. To map the macroscopic observations to intrinsic phonon properties, it is crucial to appropriately take account into the microscopic heat input.