Low-lying excited states of a hydrogenic donor and resonant states in a quantum well

Some of the low-lying excited states of a hydrogenic donor in a quantum well formed by GaAs and ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Al}}_{\mathrm{x}}$As are calculated for various well widths and different compositions x. The nonparabolicity of the subband is included by using an energy-dependent effective mass. Limiting values of the binding energy as the well width goes to zero are presented. Results are also discussed in the light of some recent work on the 2$p_0$-like state. The position of a resonant state relative to a localized donor state and the width of the resonant state are calculated as a function of well width. The results are compared with experiments and earlier theoretical estimates.