Phonon-Broadened Optical Spectra: Urbach's Rule

Starting from Frölich's Hamiltonian for conduction-electron-optical-phonon interactions, an approximate Green's function $G(p, t)$ is derived. The spectral density derived from this Green's function is shown to possess an exponential tail at low energies, $\frac{E}{{\omega }_0}\ll 0$. This demonstrates that Urbach's rule can be derived from Frölich's Hamiltonian. The approximate Green's function $G(p, t)$ is shown to be related to the intermediate-coupling models for polarons.