Effective Electron-Hole Interactions in Polar Semiconductors

We study the problem of an electron and a hole interacting with each other and with longitudinal optical phonons. Our method consists of examining the poles of the t matrix for dressed-particle-hole scattering due to the Coulomb interaction and the exchange of phonons. This approach is carried out in the two limits: (i) EBω₀ and (ii) EBω₀, where EB is the binding energy of the exciton state formed and ω₀ is the optical phonon energy. In both cases, we have an effective-mass equation for the electron-hole pair with the same form of nonlocal potential: however, in case (i) the self-energies occurring are polaron self-energies, while in case (ii) the self-energies are eliminated. We find that the first corrections in both limits are more important for the self-energy than for the interaction potential. We make the ansatz that this is true for arbitrary values of EBω₀ so that the potential is left unaltered, but the self-energy scales with the parameter EBω₀. The calculated binding energies obtained from this procedure are in excellent agreement with the measured binding energy of excitons in a variety of ionic semiconductors. The effective nonlocal potential we obtain satisfies the physical requirements of going asymptotically to (ε₀r)^-1, where ε₀ is the static dielectric constant, for rpolaronradius and EBω₀1, and to (ε_∞r)^-1, where ε_∞ is the high-frequency dielectric constant for rpolaronradius, and EBω₀1. The first corrections go as r^-2. We discuss in detail the form of the potential and its nonlocality, etc., as the parameters EBω₀, ε_∞ε₀, and mₑmₕ (ratio of electron mass to hole mass) vary. We define EB^' as the energy to separate to infinity the electron and the hole without altering the self-energy they have in the bound state. For appreciable electron-phonon coupling strength, EB^' and EB differ considerably. The exciton radius and the oscillator strength is to be estimated from EB^'. For TlCl, the actual exciton radius is estimated to be about three times smaller than one might estimate from EB.