Energy levels of Wannier excitons in $\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}-{\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ quantum-well structures

Energy levels of Wannier excitons in a quantum-well structure consisting of a single slab of GaAs sandwiched between two semi-infinite layers of ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ are calculated with the use of a variational approach. Owing to lowering of symmetry along the axis of growth of this quantum-well structure and the presence of energy-band discontinuities at the interfaces, the degeneracy of the valence band of GaAs is removed, leading to two exciton systems, namely, the heavy-hole exciton and the light-hole exciton. The values of the binding energies of the ground state and of a few lowlying excited states of these two exciton systems are calculated as a function of the size of the GaAs quantum well for several values of the heights of the potential barriers and their behavior is discussed. The results thus obtained are also compared with the available experimental data. The reliability of the various approximations made in this calculation is discussed.