A variational approach to the study of strongly bound impurity states in crystals

Abstract

Working within the Wannier function representation Koster and Slater have shown how to solve the problem of a localized perturbation in a simple cubic lattice having a single electronic energy band which is describable in terms of a Wannier function with only nearest-neighbour interactions. Here it is shown how to set up a simple variational method which leads to results in fair agreement with those of Koster and Slater for bound impurity states and at the same time is readily extended to single-band models of other crystal structures described by Wannier functions which may have other than nearest-neighbour interactions. Some results are given for the body-centred cubic lattice with a single band and only nearest-neighbour interactions. On making the assumption that it is reasonable to separate the band index and lattice site dependence of the expansion coefficients the method is extended to multi-band models of crystals. In particular, it is shown how to locate the energy of a localized impurity state produced by a strongly repulsive perturbation in a diamond-like crystal having eight bands for which the corresponding Wannier functions have only nearest-neighbour interactions. It is indicated that the results may have relevance to the discussion of bound states around a vacancy in diamond. Finally, it is expected that the method will be useful in the study of localized vibrational and spin states in perturbed crystals.