Self-consistent Monte Carlo Solution of Wigner and Poisson Equations Using an Efficient Multigrid Approach

Abstract

An accurate self-consistent solution of the coupled Wigner and Poisson equations is of high importance in the analysis of semiconductor devices. The proposed solver has two main components: a Wigner equation solver which treats the Wigner potential as a generating mechanism and is responsible for the generation and annihilation of signed particles used in the Monte Carlo method, and a Poisson equation solver which uses an efficient multigrid approach to take the electron distribution into account, and update the value of the potential in each time step. Results for the electron distribution, the electrostatic potential, and the electrostatic force calculated as the gradient of the potential energy are presented for a Cartesian xy-region which is not charged by any external doping or other sources of fixed charge in the beginning of the simulation. However, wavepackets representing electrons are constantly injected from one edge every femtosecond. Comparing the electron distribution in two cases, namely, obtained without taking the Poisson equation into account and with self-consistently solving the Poisson equation with the Wigner equation, demonstrates the repulsion of the injected wavepackets in the latter using the vector visualization of the force.