Toward a Unified Theory of Urbach's Rule and Exponential Absorption Edges

Exponential absorption edges $\alpha =A{e}^{g(\hslash \omega -\hslash {\omega }_0)}$ have been observed in both ionic (Urbach's rule: $g=\frac{\sigma }{k_B{T}^{*}}$ and covalent materials. Arguments are given to show that a unified theory of exponetial absorption edges must (i) rely on electric microfields as the cause, (ii) include exciton effects and the final-state interaction between the electron and the hole, and (iii) ascribe Urbach's rule to the relative, internal motion of the exciton. An approximate calculation has been made in which the nonuniform microfields are replaced by a statistical distribution of uniform microfields; this calculation is a generalization to physically relevant intermediate-strength fields of previous strong- and weak-field theories of Redfield and Dexter. In contrast with the other microfield models, which obtain the exponential spectral shape by averaging over microfield distributions, the present theory obtains a quantitatively exponential edge as an inherent feature. The temperature dependences of the edges in various materials follow qualitatively from the nature of the microfield sources. The specific temperature dependence of Urbach's rule in ionic crystals is obtained from this model, with supplementary arguments to account for nonuniformity of the fields.