Donor energies in a quantum well: A perturbation approach

Abstract

The binding energies of some low-lying donor states in a quantum well are calculated by a perturbation-variational approach introduced herein. It is significant that a comparison of the ground-state binding energies in a ${\mathrm{Ga}}_{0.47}{\mathrm{In}}_{0.53}{\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{0.48}{\mathrm{In}}_{0.52}\mathrm{As}$ quantum well with those computed by a variational method shows good quantitative agreement for varying well sizes and donor positions. The perturbation-variational method, which is shown in detail to depend on only a single parameter, is fast computationally even when the difference in dielectric constants of well and barrier materials is taken into account. The method also gives an analytic expression for the energy correction due to different electron effective masses. An analysis of ${2p}_{0}$ states as a function of donor position ${z}_{i}$ reveals a peak in the binding energy when the impurity is located about midway between the well center and well edge. The various perturbation energies due to magnetic field, effective mass, and dielectric mismatch are also studied as a function of ${z}_{i}.$ Application of degenerate perturbation theory to various ${p}_{\ifmmode\pm\else\textpm\fi{}1}$ states yields reasonable agreement with experiment for the transition energies between donor states in a 150 \AA{} ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ quantum well over a range of magnetic fields.