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 $E_1$. The magnitudes and dependences of $E_1$ and the width of the tail on disorder agree with the experiment.