Device modeling in the Wigner picture

Abstract

Basic computational aspects of the Wigner function approach for modeling and simulation of electronic transport are discussed, beginning with the coherent problem, followed by the dissipative problem including scattering events, which has been recently restated in terms of a scattering-induced Wigner function correction. These alternative formulations of the computational task are discussed along with a method for separation of the Wigner potential into classical (force) and quantum components.

Inflow boundary conditions and nonphysical solutions to the Wigner transport equation

“…The expression for the Wigner potential in Eq. ( 4) can be rewritten as [17] where Ṽ(k) is the spatial Fourier transform of V(z) with the convention In the case of an even potential,…”

“…The Wigner function formalism has also been used to model quantum transport in semiconductor nanostructures [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The appeal of the Wigner function formalism for describing quantum transport stems from an intuitive link with semiclassical transport, captured by the Boltzmann transport equation, and a relatively straightforward implementation of the common boundary condition scheme, often referred to as inflow or U-boundary conditions [15,22,33].…”

Implementation of the Wigner-Boltzmann transport equation within particle Monte Carlo simulation

“…is the Boltzmann collision term including scattering effects in the weak and fast scattering approximation [15,17,31]. The scattering probabilities per unit of time s i (k, k ) are calculated for each scattering process i in the 3D reciprocal space within the first order perturbation theory.…”

“…It should be noted that in some cases it is relevant to separate the potential V (x) may be separated into a slowly varying part V slow (x) and a rapidly varying part V rapid (x), or in other words into a classical component V cl (x) and a quantum mechanical component V qm (x), i.e. [31] V…”

“…The semi-classical component yields a Boltzmann-like evolution of the WF via the classical force term, while only the quantum mechanical component is included in the Wigner potential (4) [31]. The transport equation (2) becomes…”