Solution of the Space-dependent Wigner Equation Using a Particle Model

Abstract

The Wigner equation is well suited for numerical modeling of quantum electronic devices. In this work, the stationary, position-dependent Wigner equation is considered. Carrier scattering is described semi-classically by the Boltzmann collision operator. The development of Monte Carlo algorithms is complicated by the fact that, as opposed to the semiclassical case, the integral kernel is no longer positive semi-definite. Particle models are presented which interpret the potential operator as a generation term of numerical particles of positive and negative statistical weight. The problem arising from the avalanche of numerical particles is thereby solved for the steady state. When constructing the algorithms particular emphasis has been put on the conservation laws implied by the Wigner equation. If particles of opposite sign are generated pairwise, charge is conserved exactly. If the free-flight time is reduced such that only one particle is generated each time, then the sign of the particle weight is selected randomly, and charge is conserved only on average.