Shallow donor impurities in GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As quantum-well structures: Role of the dielectric-constant mismatch

We present a variational method to compute donor eigenstates in a GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Al}}_{\mathit{x}}$As quantum well. The effective-mass approximation is used, and the envelope function is expanded in the complete set of the states of the quantum well at ${\mathbf{k}}_{\mathrm{?}}$=0, including the continuum states. The convergence is good; the contribution of the continuum is very small, except for odd states in narrow wells, where there are no bound subband states of the required symmetry in the quantum well. The results obtained for all impurity positions and for various values of the well thickness show a much larger effect of the dielectric-constant mismatch than previously anticipated, such effect being larger for off-center impurity positions. The far-infrared absorption coefficient is computed for both polarizations of the radiation, with the spatial impurity distribution taken into account. Good agreement is found with available experimental data.