Variational principles for solving nonlinear Poisson equations for the potential of impurity ions in semiconductors

Abstract

The present paper concerns itself with corrections to the potential of an impurity ion obtained by Dingle for charged impurities in semiconductors with spherical energy surfaces. In Dingle's theory, the Fermi-Dirac integral ${\mathcal{F}}_{\frac{1}{2}}$, appearing in the density of the screening charge, is expanded in powers of the potential. Dingle, retaining only terms up to and including the linear term in the potential, has obtained a linearized Poisson equation. The solution of this differential equation results in a screened Coulomb potential for the impurity ion which is scaled by the static dielectric constant of the medium. In the present paper, a variational principle is suggested that is equivalent to the Poisson equation to any given order in the impurity-ion potential. The analytical groundwork is carried through to include Poisson equations resulting from the retention of quadratic and cubic terms in the expansion of ${\mathcal{F}}_{\frac{1}{2}}$. Use of the variational principle permits the representation of the approximate solutions to the nonlinear Poisson equations as linear combinations of exponentially screened Coulomb potentials which are scaled by the static dielectric constant of the semiconductor. This feature has the advantage of permitting a straightforward modification of theories of ionized impurity scattering which are based on the use of a single screened Coulomb potential.