Boltzmann Solver for Phonon Transport

Boltzmann Transport Equation is solved numerically to model phonon transport in a subcontinuum domain in order to study heat transfer in thin film semiconductors. The phonon distribution function is modified to get an Energy equation from the Boltzmann Transport Equation. Gray form of the Energy equation is solved in the Relaxation time approximation to get the Phonon Energy Density distribution. The phonon group velocity and the relaxation times are obtained using other methods. Structured Finite Volume Method is used to discretize the Energy equation and a recursive solution procedure is used to solve it. Temperatures in the domain are obtained by assuming statistical equilibrium. The temperature profiles and heat fluxes for different acoustic thicknesses agree with theoretical radiation results by Heaslet and Warming. Silicon bulk thermal conductivity is reproduced under the acoustically thick limit. Boundary scattering and confinement effects are studied by working with specularity and confinement parameters. NOMENCLATURE f phonon distribution function g v phonon group velocity k r phonon wave vector eff τ effective relaxation time ω phonon frequency h modified Planck’s constant ' ' ω e phonon energy density ( ) ω D phonon density of states k Boltzmann constant Thermal conductivity s unit direction vector r position vector t time o e angular averaged phonon energy density C specific heat ref T reference temperature