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) $E_B\ll {\omega }_0$ and (ii) $E_B\gg {\omega }_0$, where $E_B$ is the binding energy of the exciton state formed and ${\omega }_0$ 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 $\frac{E_B}{{\omega }_0}$ so that the potential is left unaltered, but the self-energy scales with the parameter $\frac{E_B}{{\omega }_0}$. 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 ${\left({\epsilon }_0\mathcal{r}\right)}^{-1\mathrm{}}$, where ${\epsilon }_0$ is the static dielectric constant, for $\mathcal{r}\gg \mathrm{polaron}\mathrm{radius}$ and $\frac{E_B}{{\omega }_0}\ll 1$, and to ${\left({\epsilon }_{\infty }\mathcal{r}\right)}^{-1\mathrm{}}$, where ${\epsilon }_{\infty }$ is the high-frequency dielectric constant for $\mathcal{r}\ll \mathrm{polaron}\mathrm{radius}$, and $\frac{E_B}{{\omega }_0}\gg 1$. The first corrections go as ${\mathcal{r}}^{-2\mathrm{}}$. We discuss in detail the form of the potential and its nonlocality, etc., as the parameters $\frac{E_B}{{\omega }_0}$, $\frac{{\epsilon }_{\infty }}{{\epsilon }_0}$, and $\frac{m_e}{m_h}$ (ratio of electron mass to hole mass) vary. We define $E_B^{\prime }$ 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, $E_B^{\prime }$ and $E_B$ differ considerably. The exciton radius and the oscillator strength is to be estimated from $E_B^{\prime }$. For TlCl, the actual exciton radius is estimated to be about three times smaller than one might estimate from $E_B$.