Approximate treatment of the Thomas-Fermi equation for the potential due to a negative ion in a semiconductor

Abstract

The nonlinear Thomas-Fermi equation, due to Resta, for the potential $V(r)$ associated with an impurity ion of charge $Z<0$ is approximated and solved. The present approximation differs from the linearization of Resta in that, here, the region within the Coulomb hole ($r<{R}_{C}$) is treated exactly. The result is a simple analytic expression for $V(r)$. The corresponding spatial dielectric function $\overline{\ensuremath{\epsilon}}(r)$ exhibits both the correct asymmetry in comparison with the positive $Z$ case as well as the correct dependence upon $Z$ for $Z=\ensuremath{-}1, \ensuremath{-}2, \ensuremath{-}3, \ensuremath{-}4$.