Modified Ballistic-diffusive Equation for Phonon Transport in Micro- and Nano-Devices

Boltzmann transport equation (BTE) has been proven to be an accurate tool for modelling phonon transport, especially in cases when the size of the device is comparable to phonon mean free path. Though very simple in form, BTE is very difficult to solve because it involves variables in both real and momentum spaces, as well as time. Simplified equation and efficient simulation methods are desirable for modelling realistic problems. The Ballistic-Diffusive equation (BDE), derived by Chen, is one such an attempt. Although it is a much better approximation to BTE than Fourier law and Cattaneo equation, there exists an inconsistence, which leads to inaccurate results particularly the temperature profile near the boundary. In this work, a set of modified ballistic-diffusive equations (MBDE) is derived based on the key idea that for the modelling of the diffuse part, the P1-approximation should not be applied on the entire solid angle, but rather on each section of the solid angle. As such, the boundary conditions can be applied naturally, and the inconsistence between the P1-approximation and boundary conditions inherent in the original BDE is resolved. Several benchmark problems are employed to validate the accuracy of the new method (MBDE) and results have shown that the new method is a much more accurate and efficient approximation of BTE than the original BDE.