Exciton Binding Energy in Extremely Shallow Quantum Wells

The usual approach to the problem of excitons in semiconductor quantum wells is to assume that both the electron or the hole are primarily localized in the potential well regions defined by the band offsets, i.e., that the quantum wells are deep. We re-examine the problem of the exciton in the presence of a very shallow square well potential due to the (small) conduction and valence band offsets in a semiconducting heterostructure. We show that the combined effects of the shallow well and the Coulomb interaction between the electron and the hole are equivalent to an effective potential acting on the center-of-mass of a three-dimensional exciton. We calculate the shape of such a potential and show it to be satisfactorily approximated by the potential of a parabolic well.