Numerical Solution of the Boltzmann Transport Equation using Spherical Harmonics Expansions

This thesis deals with the numerical solution of the Boltzmann transport equation using an expansion of the distribution function in spherical harmonics for the purpose of electronic device simulation. Both the mathematical and physical backgrounds are discussed, then the Boltzmann transport equation is projected onto spherical harmonics without posing unnecessary restrictions on the energy band structure. From entropy principles a stabilisation is found which serves as a Scharfetter-Gummel-like stabilisation for the discretisation. The finite volume method using a full Galerkin scheme is proposed for the discretisation of the projected equations, which has the advantage of ensuring current continuity by virtue of construction. To reduce computational costs and speed up the assembly of the system matrix, analytical formulae for the integral terms in the discretised equations are derived. Complexity analysis shows that higher order spherical harmonics expansions suffer from huge memory requirements, especially for two and three dimensional devices. A compressed matrix storage scheme is therefore introduced, which reduces the memory requirements for the storage of the system matrix especially for higher order spherical expansions by up to several orders of magnitude. Finally, simulation results for a n+nn+-diode prove the applicability of the full Galerkin method. Self-consistent solutions are obtained by coupling the system of projected equations with the Poisson equation. The resulting systems of linear equations turn out to be poorly conditioned, thus preconditioners are proposed and compared.