A one-dimensional, self-consistent numerical solution of Schrödinger and Poisson equations

Publisher Website
Google Scholar
Abstract
A self-consistent, one-dimensional, numerical solution of Schrödinger and Poisson equations has been obtained. To solve Schrödinger equation, instead of the conventional finite difference approach, we start by dividing the space in intervals of constant potential energy, in which the solution type is well known. Next we match the wave functions and their first derivatives, divided by the effective mass on each side of the potential steps. This approach is very efficient on finding the eigenvalues in structures with large regions of almost constant potential energy such as quantum well structures or heterojunctions. Validation is presented by comparing the exact solution of Schrödinger equation for a triangular well with that obtained by our method. Poisson equation is solved considering the deep (DX) and shallow centers assuming a donor with one ground state and two excited states. Applications to isotype n-GaAs/AlxGa1−xAs graded heterojunctions show that the density of the two-dimensional electron gas (2DEG) is almost independent of graduality when this is smaller than about 200 Å and that DX centers may lower the 2DEG concentration by as much as 20%.