Gap-modulated doping effects on indirect exchange interaction between magnetic impurities in graphene

A dilute distribution of magnetic impurities is assumed to be present in doped graphene. We calculate the interaction energy between two magnetic impurities which are coupled via the indirect-exchange or Ruderman-Kittel-Kasuva-Yosida (RKKY) interaction by the doped conduction electrons. The current model is a half-filled $AB$-lattice structure. Our calculations are based on the retarded lattice Green's function formalism in momentum-energy space which is employed in linear response theory to determine the magnetic susceptibility in coordinate space. Analytic results are obtained for gapped graphene when the magnetic impurities are placed on the $A$ and $B$ sublattice sites of the structure. This interaction, which is important in determining spin ordering, has been found to be significantly different for $AA$ and $BB$ exchange energies in doped graphene due to the existence of an energy gap, and is attributed to a consequence of the local fields not being equal on the $A$ and $B$ sublattices. For doped graphene, the oscillations of all three RKKY interactions from ferromagnetic to antiferromagnetic with increasing Fermi energy is significantly modified by the energy gap both in magnitude and phase. Additionally, the $AB$ exchange energy may be modified by the presence of a gap for undoped graphene but not for doped graphene due to the dominance of doped conduction electrons.