A general solution to the Schr\"odinger-Poission equation for charged hard wall: Application to potential profile of an AlN/GaN barrier structure

TL;DR

This paper introduces a universal numerical and analytical approach to solving the Schrödinger-Poisson equation for charged hard walls, with applications to AlN/GaN barriers, improving understanding of potential profiles in doped semiconductor structures.

Contribution

It develops a general, system-independent formulation and approximations for the Schrödinger-Poisson equation, applicable to charged hard walls and semiconductor barriers.

Findings

The solution relates band bending to charge density at interfaces.

Analytical forms approximate the potential and wave functions effectively.

The approach provides good estimates for potential profiles in doped structures.

Abstract

A general, system-independent formulation of the parabolic Schr\"odinger-Poisson equation is presented for a charged hard wall in the limit of complete screening by the ground state. It is solved numerically using iteration and asymptotic-boundary conditions. The solution gives a simple relation between the band bending and charge density at an interface. I further develop approximative analytical forms for the potential and wave function, based on properties of the exact solution. Specific tests of the validity of the assumptions leading to the general solution are made. The assumption of complete screening by the ground state is found be a limitation; however, the general solution still provides a fair approximate account of the potential when the bulk is doped. The general solution is further used in a simple model for the potential profile of an AlN/GaN barrier, and gives an approximation which compares well with the solution of the full Schr\"odinger-Poisson equation.