Urbach's Rule in an Electron-Phonon Model

Abstract

We have investigated the optical absorption in the region below the direct edge in a model electron-phonon semiconductor. The absorption coefficient has been shown to have the form $A(\ensuremath{\omega})={A}_{0}\mathrm{exp}[\ensuremath{-}\frac{\ensuremath{\sigma}({E}_{g}\ensuremath{-}\ensuremath{\omega})}{\mathrm{KT}}]$, and the temperature dependence of the coefficient has been found to agree, at least qualitatively, with the observed dependence. (This behavior of the absorption coefficient is associated with the name of F. Urbach.) To obtain these results we have used the method of electron Green's functions, and the electron self-energy has been determined in an approximation which involves the electron-phonon interaction to all orders.