Simulation of semiconductor devices using a Galerkin/spherical harmonic expansion approach to solving the coupled Poisson-Boltzmann system

This paper describes a Galerkin/spherical harmonics approach for solving the coupled Poisson-Boltzmann system of equations for the electron distribution function and the electric potential, which can then be used to calculate other parameters of interest such as current flow and electron temperature. The Galerkin approach described here has some pragmatic advantages in space-dependent problems over more commonly used term-matching techniques for arbitrary order spherical harmonic expansions in momentum space, but the method requires a careful treatment of the boundary conditions and upwinded discretization methods. Results are presented for nonuniformly doped one-dimensional devices using up to third order spherical harmonics to show the importance of including higher order harmonics to accurately calculate the distribution function in high field regions.