Indirect exchange interaction in dilute magnetic semiconductors

We consider the indirect exchange interaction between magnetic ions embedded in direct-gap semiconductors with $s$ and $p$-band edges located at the center of the Brillouin zone. Based on the band structure calculated by the isotropic $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ method, we show that the indirect exchange interaction consists of two competing components, one ferromagnetic and one antiferromagnetic, mediated by electrons occupying different valence bands. For interspin distance equal to next-nearest-neighbor cation separation and beyond, the antiferromagnetic part coming from the higher valence bands usually dominates the ferromagnetic one. For two magnetic ions occupying nearest-neighbor cation sites, the reverse may be true. Although our quantitative results are based on $\overrightarrow{\mathrm{k}}·\overrightarrow{\mathrm{p}}$ theory, our qualitative conclusion about the existence of two competing components is in fact model independent; it results from wave-function symmetry.