Exciton states in type-I and type-II GaAs/${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Al}}_{\mathrm{x}}$As superlattices

We develop an expression for the exciton envelope function in a superlattice in terms of localized ‘‘exciton Wannier functions’’. Through an expansion of the Wannier functions in the basis of ‘‘two-well’’ exciton states, we present the exciton minibands in the type-I GaAs/${\mathrm{Ga}}_{0.7}$${\mathrm{Al}}_{0.3}$As superlattice and the ‘‘confinement-induced’’ type-II GaAs/AlAs superlattice. We also calculate the absorption coefficient for the type-I system. We find that our binding energies are in good agreement with experimental results down to layer thicknesses of 30 Å.