Thermal Transport at the Nanoscale - A Fourier's Law vs. Phonon Boltzmann Equation Study

Steady-state thermal transport in nanostructures with dimensions comparable to the phonon mean-free-path is examined. Both the case of contacts at different temperatures with no internal heat generation and contacts at the same temperature with internal heat generation are considered, and Fourier Law results are compared to finite volume method solutions of the phonon Boltzmann equation in the gray approximation. When the boundary conditions are properly specified, results obtained using Fourier's Law without modifying the bulk thermal conductivity are in essentially exact quantitative agreement with the phonon Boltzmann equation in the ballistic and diffusive limits and in good agreement between these two limits. Fourier's Law results are nearly identical to those obtained from a widely-used ballistic-diffusive approach, but analytically much simpler. Although limited to steady-state conditions with spatial variations in one dimension and to a gray model of phonon transport, the results show that Fourier's Law can be used over a much wider range of length scales than is generally appreciated. For the structures examined, Fourier's Law provides simple, accurate, analytical solutions valid from the ballistic to diffusive limits.