Energy levels of a hydrogenic impurity in GaAs/${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As multiple-quantum-well structures with narrow barriers in a magnetic field

A variational approach employing Gaussian-type trial wave functions and mixing of miniband states has been used to calculate the binding energies of the ground and first excited states of a hydrogenic donor in multiple-quantum-well structures consisting of a varying number of GaAs quantum wells separated by thin ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As barriers with periodicities of 89 Å (80-Å well and 9-Å barrier) and 49 Å (40-Å well and 9-Å barrier). The dependence of the binding energies on two different positions (on center and on edge of the central well) was investigated. It was found that the binding energies did not change in any significant way beyond 15 periods for either of the structures investigated at zero magnetic field. Calculations were also performed for superlattice structures with 15 periods in the presence of a magnetic field applied perpendicular to the interfaces. Results have been compared with recent experimental measurements for donor transition energies in GaAs/${\mathrm{Ga}}_{0.7}$${\mathrm{Al}}_{0.3}$As superlattices. Very good agreement is obtained.