Theory of the screening of impurity ions in semiconductors with spatially-variable dielectric constants

Abstract

In this paper, Dingle's theory of the screening of impurity ions in semiconductors is generalized. This is accomplished by inserting in the theory of screening the spatial dielectric function of the medium. Poisson's equation for the potential of an impurity ion follows from one of Maxwell's equations. With the analytical form for the spatial dielectric function for Si and Ge, obtained for these materials by Azuma and Shindo and by Okuro and Azuma, Poisson's equation assumes a specific form. The asymptotic form of this differential equation, the region where the theory is expected to be valid, is solved approximately by an equivalent variational principle. The end result of the present theory is an impurity-ion potential which is represented by a linear combination of two exponentially screened Coulomb potentials scaled by the static dielectric constant of the medium. Each of the screening lengths appearing in the screened Coulomb potentials is related to Dingle's screening length. Numerical data, as functions of charge carrier concentration, are given both for the screening lengths, and for the coefficients involved in the linear combination of the two screened Coloumb potentials. It is concluded that the form obtained in the present paper for the potential of an impurity ion leads to modifications in theories of ionized-impurity scattering in semiconductors.