Analytical solution for phonon transport across thin films

The numerical solution of the Boltzmann equation for the thin film applications requires extensive computational power. An analytical solution to the phonon transport equation is fruitful in order to reduce the computational effort and cost. In the present study, an analytical solution for the phonon radiative transport equation in thin film is carried out. The analytical treatment of the problem reduces the two identical radiative transport equations to Fredholm integral equation of the second kind. The resulting phonon intensity data are presented in terms of the dimensionless temperature across the gray thin films of silicon and diamond. The findings are compared with their counterparts predicted from the numerical simulations. The study is extended to include the effect of the film thickness on the dimensionless temperature distribution. It is found that the analytical solution for temperature agrees well with the numerical predictions. Reducing the film thickness increases the temperature jump at the film edges, which is more pronounced for the diamond film.