Numerical aspects of a Godunov-type stabilization scheme for the Boltzmann transport equation

Abstract

We discuss the numerical aspects of the Boltzmann transport equation (BE) for electrons in semiconductor devices, which is stabilized by Godunov’s scheme. The k-space is discretized with a grid based on the total energy to suppress spurious diffusion in the stationary case. Band structures of arbitrary shape can be handled. In the stationary case, the discrete BE yields always nonnegative distribution functions and the corresponding system matrix has only eigenvalues with positive real parts (diagonally dominant matrix) resulting in an excellent numerical stability. In the transient case, this property yields an upper limit for the time step ensuring the stability of the CPU-efficient forward Euler scheme and a positive distribution function. Similar to the Monte-Carlo (MC) method, the discrete BE can be solved in time together with the Poisson equation (PE), where the time steps for the PE are split into shorter time steps for the BE, which can be performed at minor additional computational cost. Thus, similar to the MC method, the transient approach is matrix-free and the solution of memory and CPU intensive large systems of linear equations is avoided. The numerical properties of the approach are demonstrated for a silicon nanowire NMOSFET, for which the stationary I–V characteristics, small-signal admittance parameters and the switching behavior are simulated with and without strong scattering. The spurious damping introduced by Godunov’s (upwind) scheme is found to be negligible in the technically relevant frequency range. The inherent asymmetry of the upwind scheme results in an error for very strong scattering that can be alleviated by a finer grid in transport direction.