Binding energies of Wannier excitons in GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Al}}_{\mathrm{x}}$As quantum-well structures in a magnetic field

Binding energies of the ground state of both the heavy-hole and light-hole excitons in a GaAs quantum well sandwiched between two semi-infinite layers of ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Al}}_{\mathrm{x}}$As, are calculated as a function of the size of the well in the presence of an arbitrary magnetic field. A variational approach is followed where the trial wave functions are expressed in terms of Gaussian basis sets. The applied magnetic field is assumed to be parallel to the axis of growth and the binding energies are calculated for a finite value of the height of the potential barrier. As expected, for a given value of the magnetic field, the binding energies are found to be larger than their values in a zero magnetic field. The contribution to the binding energy due to the magnetic field, at a given field, increases slowly as the well size is reduced. A comparison with the available experimental data is made.