Numerical Experiments on the Absorption Lineshape of the Exciton under Lattice Vibrations. III. The Urbach Rule

We analyze the low-energy tail of the exciton absorption spectra calculated numerically on the standard model of exciton-phonon interaction. The empirical Urbach rule is established in its exponential decay for several orders of magnitude with a decay constant proportional to inverse temperature T -1 in two and three dimesnions and to T -2/3 in one dimension. A posteriori, the exponential tail is unambiguously ascribed to the "momentarily localized" excitons by considering the behaviour of the average oscillator strength per state. We introduce the steepness index s which relates the steepness coefficient σ and the exciton-phonon coupling constant g through σ= s / g . Our calculated values of 1.24 for square and 1.50 for simple cubic lattices enable us to correlate experimental observations on Urbach behaviour and self-trapping consistently.