Effective Electron-Hole Interaction in Polar Semiconductors

The effective-mass equation for an exciton in a polar semiconductor is derived, including the corrections to the effective electron-hole interaction originating in electron (hole)-optical-phonon interaction. It is assumed that the electron (hole)-phonon coupling is weak (the Fr\"ohlich constant αe,h1) and that the binding energy of the exciton is small [β=(EBω₀)1/21]. The electron and hole self-energies and the irreducible vertex part are calculated to include terms up to the order αβ⁴ω₀. The homogeneous Bethe-Salpeter equation is reduced to a hydrogenlike equation containing additional terms of the forms pe,h⁴ and δ(→r). They are of the order αβ⁴ω₀ and correspond to the nonparabolicity of the dispersion laws and to the effect of the phonon-field fluctuations, respectively. Contrary to other theories, no term of the order αβ³ω₀ behaving asymptotically as 1r² is present in the effective Hamiltonian.