Numerical Simulation of Quantum Waveguides

Abstract. This chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano–scale semiconductor devices. We discuss transparent boundary conditions (TBCs) for the time–dependent Schrödinger equation on a two dimensional domain. First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank– Nicolson–type finite difference scheme and a compact nine–point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary–valued problem is unconditionally stable and completely reflection–free at the boundary. Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.