Scaling description of the origin of the Urbach tail

Toyozawa's exciton self-trapping model for the Urbach tail is analyzed using scaling arguments. Following Halperin's analysis of the random impurity potential problem, we show that for the one-dimensional lattice the logarithm of the absorption coefficient scales as $T^{-\frac{2}{3}}$ or $E^{\frac{3}{2}}$. For the two- or three-dimensional lattice, spontaneous localization of excitons over a finite number of atomic sites with Gaussian potential fluctuations leads to Urbach behavior.