Calculated shallow-donor-level binding energies in GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As quantum wells

We numerically solve Schrödinger’s equation for a single electron located in a GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As quantum well with a shallow Coulombic donor impurity either in the well or in the barrier, as a function of well width, well depth, and impurity position. We calculate the binding energy of the ground state and of the first two ‘‘s-like’’ excited states and also the binding energy of the lowest ‘‘p-like’’ excited state. We compare our exact numerical results for the ground-state binding energy with earlier variational calculations and show that they are nearly coincident with these calculations. We calculate the electronic charge density as a function of impurity position and show that the asymmetry in the charge density peaks and then decreases as the impurity is taken from the center of the well to the edge and then into the barrier.