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

Abstract

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}m<\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.We find that for reasonable choices of the correlation lengths and rms potential fluctuation that the entire experimentally observable part of the DOS in three dimensions lies in the crossover regime between the shallow energy Halperin-Lax tail \ensuremath{\rho}(E)\ensuremath{\sim}\ensuremath{\Vert}E${\mathrm{\ensuremath{\Vert}}}^{3/2}$exp(-const\ifmmode\times\else\texttimes\fi{}\ensuremath{\Vert}E${\mathrm{\ensuremath{\Vert}}}^{1/2}$) and the deep Gaussian tail \ensuremath{\rho}(E)\ensuremath{\sim}\ensuremath{\Vert}E${\mathrm{\ensuremath{\Vert}}}^{3}$exp(-\ensuremath{\Vert}E${\mathrm{\ensuremath{\Vert}}}^{2}$/${2\mathrm{V}}_{\mathrm{rms}}^{2}$) where the energy E is measured relative to the shifted continuum band edge. For systems with rapidly decaying short-range correlations (m\ensuremath{\ge}2) the crossover regime exhibits a linear exponential (Urbach) tail which easily spans five decades in the DOS. The extent of linearity is highly sensitive to the range of the correlation function B(x). The screened-Coulomb impurity model (m=1) requires a screening length considerably smaller than the interatomic spacing to give an Urbach tail. These results are obtained numerically by saddle-point (instanton) evaluation of a replica-functional-integral representation of the one-electron propagator. The instanton method provides an asymptotic expansion for the band-tail DOS, which is nearly exact for all energies below the shifted continuum edge. Comparison is made to the Feynman path-integral method and to a simple physical argument which yields to a high degree of accuracy the results of the instanton method. Our results provide a basis for understanding the extent, precision, and universality of Urbach tails in disordered materials.