Effect of diffuse phonon boundary scattering on heat flow

Abstract

We propose a model for the thermal conductivity of 2D-samples when the mean free paths due to the phonon–phonon interactions exceed the sample dimensions. The physical mechanisms that ensure the stationary heat flux and the stationary nonequilibrium temperature distribution are examined. A recursive equation is derived to quantify the contribution of phonon scattering to the net heat flux in the pure elastic isotropic diffusive regime, as a function of the number of scattering at the boundaries. As the length to width ratio L/W increases above unity, our model shows an increase in the phonon mean free path compared to that predicted by the Casimir model in which the lateral walls are assumed to be black bodies. The present model also reveals that the multiple phonon interactions with the lateral walls lead to a stationary, but non-linear temperature distribution. The dependence of the heat flux and the thermal conductivity on the sample dimensions is then shown to be non-monotonic. This infers that the location of the thermometers on the sample influences experimental measurements of the thermal conductivity.