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Long range superexchange - an exchange interaction through empty bands

We derive a generalization of the RKKY interaction to semiconductors using perturbation theory on a non-degenerated two-impurity Anderson model. In metals the interaction is mediated by excitations of free carriers over the Fermi-energy. In semiconductors, where no carriers are present, the only possible excitations are those of the localized impurity electrons (or holes) themselves. Thus a possible interaction is closely related to superexchange. We find an oscillating anti-ferromagnetic spin-spin coupling due to impurity electron (hole) excitations. By treating the coupling through empty bands (superexchange) along the same route as carrier mediated interactions (RKKY) it is easy to compare these two kinds of spin-spin coupling. The interaction derived here is of special interest for diluted magnetic semiconductors.