k-space formulation of Γ-X mixing for excitons in a thin GaAs/AlAs quantum well

Using a nonvariational method in k space, we study the effects of Γ-X valley mixing on the structure of excitons in an asymmetric ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As/AlAs/GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As quantum well. The formulation of excitons in this system involves two complications due to Γ-X mixing: the nonparabolic subband structures in the xy plane, and the Coulomb interaction between different electronic subbands. These effects lead to a set of two coupled integral equations. The eigenvalues and functions thus obtained are then used to compute the optical absorption spectra for various structures near the Γ-X crossing.