Thomas-Fermi Approach to Impure Semiconductor Band Structure

Abstract

The density of states in highly impure semiconductors is studied using a semiclassical or Thomas-Fermi type approximation. The "local" density of states is assumed proportional to ${(E\ensuremath{-}\mathcal{U})}^{\frac{1}{2}}$, where $\mathcal{U}$ is the local potential. The problem then reduces to the calculation of the distribution function for the potential, which is found to be a Gaussian in the high-density limit. It is clear that this approach predicts tails on both band edges which are identical except for a multiplying factor of ${({m}^{*})}^{\frac{3}{2}}$, the density-of-states mass. The important contributions to the potential variation near the average potential are shown to arise from fluctuations in impurity clusters whose volume is of the order of the cube of the screening length. For energies far below the average potential the important cluster sizes are much smaller than the screening length cubed. Their size is determined by the kinetic energy of localization, an effect which is not accounted for by the ${(E\ensuremath{-}\mathcal{U})}^{\frac{1}{2}}$ assumption. The Thomas-Fermi method involves a number of approximations, all of which are valid in the limit of high density. The most serious approximation results from the improper treatment of the kinetic energy of localization which requires ${(n{{a}_{0}}^{*3})}^{\frac{1}{12}}\ensuremath{\gg}1$ for validity. Because of this requirement, the method is never highly accurate in any attainable concentration range. The effect of potential fluctuations on tunnel diode $I\ensuremath{-}V$ characteristics is also studied. The results agree satisfactorily with experimental studies of silicon junctions by Logan et al.