Exchange coupling and Mn valency in GaN doped with Mn and co-doped with Mg

We study 1 isolated or 2 neighboring Mn impurities, as well as 1 Mn and 1, 2, or 3 Mg ions as substitutional impurities for Ga in zinc-blende and wurtzite supercells of GaN by means of density functional calculations. Taking into account the electron correlation in the local spin density approximation with explicit correction of the Hubbard term (the LSDA+U method) and full lattice relaxation we determine the nearest neighbor exchange J for a pair of Mn impurities. We find J to be ferromagnetic and in between 9 and 17 meV in the Hamiltonian $\hat{H}=-2 J \hat{\vec{S}}_1\cdot \hat{\vec{S}}_2$ for the different lattices. That J is only weakly influenced by the U parameter (varying between 2 and 8 eV). From a detailed analysis of the magnetization density distribution we get hints for a ferromagnetic super-exchange mechanism. Also the Mn valence was found to be 3+ in the absence of co-doping with Mg. Co-doping with Mg leads to a valence change to 4+ for 1 Mg and to 5+ for 2 Mg, independently whether the Mn and Mg ions are close or far from each other. We show that the valence change can already be concluded from a careful analysis of the density of states of GaN doped with Mn without any Mg.