Hydrogenic impurity states in quantum-well wires: Shape effects

Abstract

The binding energies for the bound states of a hydrogenic impurity placed in a quantum-well wire are calculated with the use of variational solutions to the effective-mass equation. The quantum-well wire is a quasi-one-dimensional region of GaAs. The electrons are confined to this region by an infinite barrier. The calculations are performed to determine the effect that the shape of the wire's cross section has on the binding energies. Results for quantum-well wires with rectangular cross section are determined and compared with results previously obtained for cylindrical quantum-well wires. The results for cylindrical wires with diameter w and for square wires with width w are nearly identical. Even closer agreement occurs between the results obtained for cylindrical and square wires with equal cross-sectional area. When the wire becomes nonsquare by expanding one side while keeping the other side fixed, the binding energies drop rapidly to the values appropriate for two-dimensional wells of finite thickness. When the wire becomes nonsquare by expanding one side and contracting the other side to maintain a constant cross-sectional area, the binding energies change very slowly. The binding energies approach the values expected for narrow two-dimensional wells only after the cross section has deviated substantially from being square. The results indicate that the binding energies for impurities in wires with comparable shapes are most closely correlated to the cross-sectional area.