Numerical Experiments on the Absorption Lineshape of the Exciton under Lattice Vibrations. IV. The Indirect Edge

Abstract

Employing the standard model of exciton-phonon interaction we compute the lineshape of the indirect exciton absorption edge. Around the absorption edge the features of the density of states are reproduced; for the tail at lower energy the empirical Urbach rule is established in its exponential decay, with a decay constant proportional to inverse temperature in three and two dimensions. The analysis of the average oscillator strength per state furnishes proof that the Urbach tail is caused by “momentarily localized” exciton states appearing below the exciton band due to thermal lattice vibrations. We derive steepness indices of 0.85 and 0.80 for the three- and two-dimensional indirect absorption tails to correlate the occurrence of exciton self-trapping and the steepness of the Urbach tail. Excellent agreement with the optical study on mixed silver halide crystals is obtained.