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) \ensuremath{\rho}(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)\ensuremath{\equiv}〈V(x)V(0)〉 of the form (i) ${V}_{\mathrm{rms}{}^{2}\mathrm{exp}[\mathrm{\ensuremath{-}}(\mathit{\ensuremath{\Vert}}\mathit{x}\mathit{\ensuremath{\Vert}}/\mathit{L}{)}^{m}]}$ for 1\ensuremath{\le}ml\ensuremath{\infty}. For short-range disorder characterized by two correlation lengths ${L}_{1}$ and ${L}_{2}$, we consider the model (ii) B(x)=${\mathit{V}}_{\mathrm{rms}}^{2}$[\ensuremath{\alpha} exp(-${\mathit{x}}^{2}$/${\mathit{L}}_{1}^{2}$)+(1-\ensuremath{\alpha})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.