Energy spectra of donors and acceptors in quantum-well structures: Effect of spatially dependent screening

The energy spectra of shallow donors and acceptors in GaAs-${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Al}}_{\mathrm{x}}$As quantum-well structures have been calculated. The binding energies of the impurities were obtained within a variational calculation in the effective-mass approximation. Calculations were performed for simple neutral and double singly ionized impurities as functions of the position of the impurity in a GaAs quantum well of infinite depth and for various slab thicknesses. The effect of the spatially dependent screening is modeled with a dielectric response of the form ${\ensuremath{\epsilon}}^{\mathrm{\ensuremath{-}}1}$(r)=${\ensuremath{\epsilon}}_{0}^{\mathrm{\ensuremath{-}}1}$+(1-${\ensuremath{\epsilon}}_{0}$ $^{\mathrm{\ensuremath{-}}1}$)${e}^{\mathrm{\ensuremath{-}}r/a}$, with a screening parameter a\ensuremath{\approxeq}1.1 a.u. characteristic of bulk GaAs. Results are compared with Bastard's theory, which is based on a constant-${\ensuremath{\epsilon}}_{0}$ screening, and it is found that spatially dependent screening effects are small for donors down to very thin slab thicknesses, but can be quite important for all acceptors in GaAs quantum wells over a large range of slab thicknesses. Calculated results with improved statistics are in quantitative agreement with experimental data on neutral donors and acceptors.