An acceleration technique for the solution of the phonon Boltzmann transport equation
Advanced nanofabrication techniques have enabled the creation of novel engineering devices. As a consequence of shrinking size, new physics must be explored due to subcontinuum behavior. Specifically, phonons have been identified as a major player in thermal management in semiconductor devices. Non-gray phonon transport solvers based on the Boltzmann transport equation (BTE) are frequently employed to simulate sub-micron thermal transport. Typical solution procedures using sequential solution schemes encounter numerical difficulties because of the large spread in scattering rates. For frequency bands with very low Knudsen numbers, strong coupling between the directional BTE’s results in slow convergence for sequential solution procedures. In this M. S. thesis, a hybrid Fourier-BTE model is presented which addresses this issue. By establishing a phonon group cutoff (say Knc=0.1 ), phonon bands with low Knudsen numbers are solved using a modified Fourier equation (MFE) which includes a scattering term as well as corrections to account for boundary temperature slip. Phonon bands with high Knudsen numbers are solved using a BTE solver. Once the governing equations are solved for each phonon group, their energies are then summed to find the total lattice energy and correspondingly, the lattice temperature. An iterative procedure combining the lattice temperature determination and the solutions to the modified Fourier and BTE equations is developed. The procedure is shown to work well across a range of Knudsen numbers.^
Jayathi Y. Murthy, Purdue University.