A variational principle for the potential of impurity ions in semiconductors with spatially variable dielectric constants

Abstract

Dingle's theory of the screening of (point) impurity ions in semiconductors (with standard energy surfaces) is generalized to semiconductors with spatially variable dielectric constants. With the analytical approximations to the spatial dielectric functions of Si and Ge, obtained for these materials by Azuma and Shindo and by Okuro and Azuma, Poisson's equation for the potential of an impurity ion assumes a specific form. The asymptotic form of this differential equation is solved approximately by an equivalent variational principle. The result is a linear combination of two exponentially screened Coulomb potentials (with different screening lengths) scaled by the static dielectric constant of the medium.