Excitons in extremely shallow quantum wells

Excitons in extremely shallow semiconductor quantum wells are considered in the limit when both the conduction band and the valence band confining potentials are small compared to the binding energy of a three-dimensional (3D) exciton. Under these circumstances it is found that the quantization of the center-of-mass motion can make a sizable contribution to energies of excitonic optical transitions. A simple effective Hamiltonian is derived for describing this situation, with a potential that confines the motion of the exciton center of mass. The shape of the potential is approximated either by a parabolic profile (when quantum wells are narrow compared with the 3D exciton Bohr radius), or by a rectangular potential (for wide quantum wells), and the resultant eigenvalue problem is solved accordingly. The results are compared to experimental data obtained in magnetooptical studies of ${\mathrm{Z}\mathrm{n}\mathrm{S}\mathrm{e}/\mathrm{Z}\mathrm{n}}_{1-x}{\mathrm{Mn}}_x\mathrm{Se}$ spin superlattices, giving excellent quantitative agreement.