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Steady-state thermal transport in nanostructures with dimensions comparableto the phonon mean-free-path is examined. Both the case of contacts atdifferent temperatures with no internal heat generation and contacts at thesame temperature with internal heat generation are considered. Fourier's Lawresults are compared to finite volume method solutions of the phonon Boltzmannequation in the gray approximation. When the boundary conditions are properlyspecified, results obtained using Fourier's Law without modifying the bulkthermal conductivity are in essentially exact quantitative agreement with thephonon Boltzmann equation in the ballistic and diffusive limits. The errorsbetween these two limits are examined in this paper. For the four casesexamined, the error in the apparent thermal conductivity as deduced from acorrect application of Fourier's Law is less than 6%. We also find that theFourier's Law results presented here are nearly identical to those obtainedfrom a widely-used ballistic-diffusive approach, but analytically much simpler.Although limited to steady-state conditions with spatial variations in onedimension and to a gray model of phonon transport, the results show thatFourier's Law can be used for linear transport from the diffusive to theballistic limit. The results also contribute to an understanding of how heattransport at the nanoscale can be understood in terms of the conceptualframework that has been established for electron transport at the nanoscale.

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