Exponential tails in the density of states

The exponential tails in the density of states in amorphous tetrahedral films, chalcogenide glasses, and ionic crystals are associated with local states bound in single potential "wells". The exponential distribution is due (a) to the Gaussian distribution of the depths of the single potential "wells", and (b) to an approximately linear relation between the binding energy and the square of the depth of the potential well. This simple picture accounts for the qualitative features of the Urbach tails and it is consistent with quantitative data in a-Si:H, As₂Se₃, and in ionic crystals. Various levels of refinement of this simple picture have been worked out by us and our collaborators. These refinements provide quantitative improvement in the Urbach region, and extend the validity of our treatment to the Halperin-Lax and Tauc regions as well as the extremely deep states.