Binding energies of ground and excited states of shallow acceptors in GaAs/${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As quantum wells

We calculate binding energies of shallow acceptors in GaAs/${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As quantum wells (QW’s) for varying well widths. Variational calculations are performed in the framework of a multiband effective-mass theory, which accounts for the mixing between heavy and light holes. The Hamiltonian also takes into account the mismatch between the band parameters and the dielectric constants of well and barrier materials. The envelope function is expanded into a basis set consisting of products of two-dimensional hydrogeniclike functions and impurity-free QW eigenfunctions at ${\mathbf{k}}_{\mathrm{?}}$=0. QW eigenfunctions of the continuum are included till convergence of the acceptor energies is reached. The present method is suited for ground as well as excited acceptor states of ${\mathrm{\Gamma }}_6$ and ${\mathrm{\Gamma }}_7$ symmetry. We do not include central-cell effects, but give the dependence of the on-center density as a function of the well width in order to estimate these corrections. Comparison with recent experiments, which determine the 1s-2s energy separation, shows very good agreement.