Impurity-band density of states in heavily doped semiconductors: A variational calculation

The impurity-band density of states ${\rho }_1\mathrm{}\left(E\right)\mathrm{}$ for electrons in the field of dense, random attractive impurities is calculated in the tail region using an analytic expression derived previously by one of us (V.S.). The analytic expression contains a single free parameter that fixes the curvature of the trial harmonic well which models the Gaussian random potential seen by the electrons. This parameter is determined here by implementing the rigorous variational principle derived by Lloyd and Best. The resulting density of states is significantly larger than that obtained by Halperin and Lax, although the Halperin and Lax values can be reproduced if the free parameter is determined as in their theory by minimizing the exponent in ${\rho }_1\mathrm{}\left(E\right)\mathrm{}$. The present ${\rho }_1\mathrm{}\left(E\right)\mathrm{}$ crosses the Kane value at higher $E$. A simple interpolation scheme is proposed to determine $\rho \mathrm{}\left(E\right)\mathrm{}$ for all energies $E$; use the present tail $\rho \mathrm{}\left(E\right)\mathrm{}$ from low $E$ up to where it crosses the Kane value and thereafter use ${\rho }^{\mathrm{kane}}\mathrm{}\left(E\right)\mathrm{}$.