Polarons with Spatially Dependent Mass in a Finite Parabolic Quantum Well

We study the energy levels of an electron (or hole) polaron in a parabolic quantum well structure, including the spatial dependence of the effective mass. We also consider the two-mode behaviour of longitudinal optical phonon modes of the ternary mixed crystals in the structure, in the calculation of the effect of the electron-phonon interaction. We calculate the ground state, the first excited state and the transition energy of an electron (or hole) in the GaAs/Al_xGa_1-xAs parabolic quantum well structure. The numerical results show that the electron-phonon interaction obviously affects the energy levels of the electron (hole), which are in agreement with experimental results.