A self-consistent solution of Schrödinger–Poisson equations using a nonuniform mesh

A self-consistent, one-dimensional solution of the Schrödinger and Poisson equations is obtained using the finite-difference method with a nonuniform mesh size. The use of the proper matrix transformation allows preservation of the symmetry of the discretized Schrödinger equation, even with the use of a nonuniform mesh size, therefore reducing the computation time. This method is very efficient in finding eigenstates extending over relatively large spatial areas without loss of accuracy. For confirmation of the accuracy of this method, a comparison is made with the exactly calculated eigenstates of GaAs/AlGaAs rectangular wells. An example of the solution of the conduction band and the electron density distribution of a single-heterostructure GaAs/AlGaAs is also presented.