Electron density of states in a Gaussian random potential: Path-integral approach

The behavior of the electron density of states in a Gaussian random potential is studied in the limit of low energies using the Feynman path-integral method. Two different impurity potentials are considered: the Gaussian potential and the screened Coulomb potential. It is shown that the density of states deep in the tail, in three dimensions, can be expressed analytically in the form proposed by Halperin and Lax, $\rho \mathrm{}\left(E\right)=\left[\frac{A\left(E\right)\mathrm{}}{{\xi }^2}\right]\exp [-\frac{B\left(E\right)\mathrm{}}{2\xi }]$, where $\xi$ is proportional to the density of impurities and to the square of the strength of the impurity potential. For a Gaussian potential with autocorrelation length $L$, we find $A\left(E\right)={\left(\frac{E_L}{L}\right)}^{3}a\left(\nu \right)$ and $B\left(E\right)=E_L^{2}b\left(\nu \right)$, where $a\left(\nu \right)=\frac{{[{(1+16\mathrm{}\nu )}^{\frac{1}{2}}-1]}^{\frac{3}{2}}{[{(1+16\mathrm{}\nu )}^{\frac{1}{2}}+7]}^{\frac{9}{2}}}{2^{12}{2}^{\frac{1}{2}}{\pi }^2}$ and $b\left(\nu \right)=\frac{{[{(1+16\mathrm{}\nu )}^{\frac{1}{2}}-1]}^{\frac{1}{2}}{[{(1+16\mathrm{}\nu )}^{\frac{1}{2}}+7]}^{\frac{7}{2}}}{2^8}$, with $\nu$ being the energy below the mean potential $E_0$ in units of $E_L=\frac{{\hslash }^2}{2m{L}^2}$. For screened Coulomb potential with inverse screening length $Q$, we find $A\left(E\right)={\left(E_{Q}Q\right)}^{3}a(\nu ,z)$ and $B\left(E\right)=E_Q^{2}b(\nu ,z)$ where $a(\nu ,z)=\frac{{(\frac{3}{2}{z}^2+\nu )}^{\frac{3}{2}}}{8\pi 2^{\frac{1}{2}}{z}^6\exp \left(\frac{z^2}{2}\right)D_{-3\mathrm{}}^2\mathrm{}\left(z\right)\mathrm{}}$ and $b(\nu ,z)=\frac{{\pi }^{\frac{1}{2}}{(\frac{3}{2}{z}^2+\nu )}^2}{2^{\frac{3}{2}}\exp \left(\frac{z^2}{4}\right)D_{-3\mathrm{}}\mathrm{}\left(z\right)\mathrm{}}$, with $z$ satisfying the equation $D_{-3\mathrm{}}\mathrm{}\left(z\right)=\left(\frac{z^3}{2}\right)(\frac{3}{2}{z}^2+\nu )D_{-4\mathrm{}}\mathrm{}\left(z\right)\mathrm{}$, $\nu$ being the energy below the mean potential $E_0$ in units of $E_Q=\frac{{\hslash }^2{Q}^2}{2m}$ and $D_p\mathrm{}\left(z\right)\mathrm{}$ denoting the parabolic cylinder function. Numerical results and calculated curves are presented. A detailed comparison with the minimum counting method of Halperin and Lax is given.