Size dependent thermal conductivity of Si nanosystems based on phonon gas dynamics

The silicon nanosystems have shown a great potential for high efficient energy conversion devices due to the strong size effect on thermal conductivity. An accurate and convenient prediction model for such size effect is highly desired. In this paper a macroscopic heat conduction model for nano-systems is presented based on the phonon gas dynamics, in which heat conduction is regarded as phonon gas flow in a porous medium. The resistant term in the momentum equation of the phonon gas flow consists of two parts. One is the Darcyʹs term, representing the volume resistance and another is the Brinkman term, representing the surface resistance. The latter is usually negligible compared with the former for the medium at the normal scale, while the relative importance of the Brinkman term increases and consequently, the thermal conductivity decreases with size reduction. The effective phonon gas viscosity is extracted from the experiments and found to be proportional to the system size in nanoscale based on the rarefied gas dynamics. In this way an explicit expression for the size dependent thermal conductivity of silicon nanosystems is obtained, which agrees well with the experimental results for both nano-wires and films.