Operator Splitting Methods for the Wigner-poisson Equation

1. I n t r o d u c t i o n . In this paper we shall discuss operator splitting methods to discretize the linear Wigner equation and the coupled Wigner-Poisson system. The Wigner formalism, which represents a phase-space description of quantum mechanics, has in recent years a t t racted considerable at tent ion of solid s tate physicists for including quantum effects in the simulation of ul tra-integrated semiconductor devices, like resonant tunneling diodes, e.g. ([7], [8], [4]). Also, the Wigner (-Poisson) equation has recently been the objective of a detailed mathematical analysis ([11] and references therein). The real-valued Wigner (quasi) distribution function w = w(x, v, t) describes the motion of an electron ensemble in the 2<i-dimensional position-velocity (a:, y)-phase space under the action of the electrostatic potential V. In the absence of collision and scattering, and in the effective-mass approximation, the time evolution is governed by the Wigner equation (q denotes the unit charge and m* the effective mass of the electron): wt + v • Vxw + £-G[V]w = 0, x,veR , d1,2 or 3, (1.1) with the pseudo-differential operator Q[V]w = iSV (x, \VV, t)w= (1.2) = (2^F JR* JR*, 8V(x,r,,t)w(x,v'tt)e« -'>"dv'dTi, 6V(x, iM) = * f [V (x + £rti, t)-V(x^ r , , t)] . In this kinetic framework the particle density n and the current density J are defined by n(x,t) = J w(x,v,t)dv and J(x,t) = —qfvw(x,v,t)dv. In order to account for electronelectron interactions in a simple mean-field approximation, (1.1) has to be coupled to the Poisson equation AV(x, t) = q/e [n(x, t) — D(x)], where e and D(x) denote the permitt ivity and the doping profile of the semiconductor, respectively. The Wigner function representation is equivalent to the conventional wave function formalism ([17], [10]), and any L-solution of the Wigner equation can be expanded into a series of pure states: