A theoretical analysis of the Monte Carlo (MC) method for both semiclassical and quantum device simulation is presented. A link between physically-based MC methods for semiclassical transport calculations and the numerical MC method for solving integrals and integral equations is established. The integral representations of the transient and the stationary Boltzmann equations are presented as well as the respective conjugate equations. The structure of the terms of the Neumann series and their evaluation by MC integration is discussed. Using this formal approach the standard MC algorithms and a variety of new algorithms is derived, such as the backward and the weighted algorithms, and algorithms for linear small-signal analysis. Applying this theoretical framework to the Wigner-Boltzmann equation enables the development of particle models for quantum transport problems.

1 Introduction

The MC method is now well established for studying semiconductor devices and exploring material properties. The method simulates the motion of charge carriers in the six-dimensional phase space formed by position and momentum. Subjected to the action of an external force field, the point-like carriers follow trajectories governed by Newton's law and the carrier's dispersion relation. These drift processes are interrupted by scattering events, which are assumed local in space and instantaneous in time. The duration of a drift process, the type of scattering mechanism and the state after scattering are selected randomly from given...