Asymptotic corrections to the potential of impurity ions in semiconductors with spatially variable dielectric constants

In a recent paper, the problem of the screening of a (point) donor ion in Si and Ge has been reexamined by introducing into the theory the spatial dielectric functions of Azuma and Shindo and Okuro and Azuma, respectively. Poisson's equation, with the neglect of a small term, has been solved approximately by making use of a variational principle. The result was an impurity-ion potential that consists of a linear combination of two exponentially screened Coulomb potentials (with two different screening lengths) which is scaled by the static dielectric constant of the medium. The present paper shows that, at distances larger than Dingle's screening length $R_0$ the term neglected in Poisson's equation would add to this impurity-ion potential two types of correction. One type consists of terms which are proportional to Dingle's potential, but with proportionality constants so small that the correction terms are rendered utterly negligible. The other type consists of terms each one of which is proportional to functions of the form $\left(\frac{1}{r^m}\right)\exp (-q_{i}r)$, where $m=2,3,4,\dots$ and the parameters $q_i$ are related to the constants appearing in the spatial dielectric functions of Si and Ge. The finding that the correction terms to the impurity-ion potential fall off faster than $\frac{1}{r}$ does, in the author's opinion, establish the asymptotic validity of the previous variational theory.