Longitudinal-optical-phonon effects on the exciton binding energy in a semiconductor quantum well

The logitudinal-optical-phonon effect on the exciton binding energies in a quantum well consisting of a single slab of GaAs sandwiched between two semi-infinite slabs of ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As are studied. By using the Lee-Low-Pine unitary transformation, the Hamiltonian can be separated into two parts which contain the phonon variables and exciton variables, respectively, providing that the virtual phonon-electron and virtual phonon-hole interactions are neglected. A trial wave function, which is able to reproduce the correct exciton binding energy in a quantum well, is obtained by using a perturbative variational technique. The trial wave function consists of a product of the envelope function in the z direction (perpendicular to the layers) for electron and hole and a purely two-dimensional exciton wave function. The dependence of the ground-state binding energy, which includes the effect of electron-phonon interaction on the well width, is investigated. It is found that the correction due to polaron effects on the exciton binding energy is quite significant for a well width of several hundred angstroms and the effects of either surface phonons or bulk phonons on the binding energy of the heavy-hole exciton is always larger than that of the light-hole exciton. Our results are compared with some previous results, and satisfactory agreements are obtained.