A Wigner Potential Decomposition in the Signed-Particle Monte Carlo Approach

Abstract

The description of the electron evolution, provided by the Wigner equation, involves a force-less Liouville operator, which is associated with particles moving over Newtonian trajectories, and a Wigner potential operator associated with generation of positive and negative particles. These concepts can be combined to develop stochastic algorithms for solving the Wigner equation, consolidated by the so-called signed particle approach. We investigate the option to split the Wigner potential into two parts and to approximate one of them by a classical force term. The purpose is two-fold: First, we search for ways to simplify the numerical complexity involved in the simulation of the Wigner equation. Second, such a term offers a way to a self-consistent coupling of the Wigner and the Poisson equations. The particles in the signed-particle approach experience a force through the classical component of the potential. A cellular automaton algorithm is used to update the discrete momentum of the accelerated particles, which is then utilized along with the Wigner-based generation/annihilation processes. The effect of the approximation on generic physical quantities such as current and density are investigated for different cut-off wavenumbers (wavelengths), and the results are promising for a self-consistent solution of the Wigner and Poisson equations.