Ionic-charge modification at the surface of polar crystals

Abstract

A simple variational principle has been applied to obtain a bound on the change in ionic charge of a polar crystal between the surface and the interior. Central to the estimate is the assumption that the energy involves terms analytic in the ionic charge $q$ and that these may be treated as a sum of Madelung, intra-atomic, and interatomic (overlap and hybridization) contributions. The Madelung terms are calculated explicitly, the intra-atomic contributions are obtained from the following paper [Phys. Rev. B 27, 6428 (1983)], and the change in interatomic terms on going from the bulk to the surface is estimated by scaling with coordination number. The results do not replace detailed quantum-mechanical calculations but they are simple computationally, and they suggest trends in the competition between the Madelung potential, which encourages ion charging, and hybridization, which discourages ionic charging, both at the surface and in the interior of a crystal. Nonpolar surfaces are of principal concern but one polar surface is considered indicating that the charge disturbance penetrates much further in from the surface than it does in nonpolar cases where the effect is largely limited to the first layer of atoms.