Band tails, path integrals, instantons, polarons, and all that

This paper reviews the explanations recently developed by the authors and their collaborators of how disorder leads to exponential band tails and to Urbach tails in optical absorption. It starts with the simplest single-potential-well models which, despite their simplicity, are remarkably successful in accounting for the experimental facts. It then identifies the weaknesses, hidden or explicit, in these models and shows, step by step, how they can be corrected by increasing the sophistication of the procedures used. Exact results are finally achieved through use of field-theoretic techniques, and appropriately formulated single-potential-well models are shown to reproduce these quite accurately. It is also shown that the probability distribution of the random potential must be close to Gaussian, with an autocorrelation function which cuts off fairly rapidly with distance for there to be a well-defined, broad energy range in which there are exponential band tails in the density of states and Urbach tails in the optical absorption.