Density of states for an electron in a correlated Gaussian random potential: Theory of the Urbach tail

A detailed study of the density of states (DOS) ρ(E) in the tail for an electron in a spatially correlated Gaussian random potential V(x) is presented. For disordered solids characterized by short-range order extending a distance L, of the order of the interatomic spacing, we consider autocorrelation functions B(x)≡〈V(x)V(0)〉 of the form (i) $V_{\mathrm{rms}{}^2\exp \left[\mathrm{-}\right(\mathit{\Vert }\mathit{x}\mathit{\Vert }/\mathit{L}{)}^m]}$ for 1≤m<∞. For short-range disorder characterized by two correlation lengths $L_1$ and $L_2$, we consider the model (ii) B(x)=${\mathit{V}}_{\mathrm{rms}}^2$[α exp(-${\mathit{x}}^2$/${\mathit{L}}_1^2$)+(1-α)exp(-${\mathit{x}}^2$/${\mathit{L}}_2^2$)]. Finally, we consider the case of longer-range correlations (iii) B(x)=${\mathit{V}}_{\mathrm{rms}}^2$[1+(x/L${)}^2$]-${\mathit{m}}_1$/2, which may be relevant to system with topological disorder or disordered polar materials in which the randomness may be modeled by frozen-in longitudinal-optical phonons.