Exponential Band Tails in Random Systems

We present a simple derivation of the exponential band tails universally observed in three-dimensional disordered materials. The physical picture employed is that of Halperin and Lax in which states are localized by long-wavelength potential fluctuations. When the effect of the small-scale fluctuations of the potential are included as well, via the scaling arguments of Thouless, there results an exponential dependence of the density of states on energy below an energy E1. The magnitudes and dependences of E1 and the width of the tail on disorder agree with the experiment.