Transient Simulation of k⋅p-Schrödinger Systems Using Discrete Transparent Boundary Conditions

Abstract

This chapter deals with the derivation and analysis of discrete transparent boundary conditions (TBCs) for transient systems of Schrodinger-type equations in one space dimension. These systems occur i.e. in the physics of layered semiconductor devices as the so called k⋅p-Schrodinger equations, which are a well established tool for band structure calculations.The new TBCs are constructed directly for the chosen finite difference scheme, in order to ensure the stability of the underlying scheme and to completely avoid any numerical reflections. The discrete TBCs are constructed using the solution of the exterior problem with Laplace and \(\mathcal{Z}\)-transformation, respectively.These discrete TBCs can easily be obtained by an inverse \(\mathcal{Z}\)-transformation based on FFT, but these exact discrete TBCs are non-local in time and thus very costly. Hence, as a remedy, we present approximate discrete TBCs, that allow a fast calculation of the boundary terms using a sum-of-exponentials approach.