Goodenough-Kanamori rules applied to wurtzite crystals

Goodenough-Kanamori (GK) criteria have provided a significant contribution to our understanding of the importance of the symmetry and the electron orbital characteristics in the development of the magnetic superexchange coupling [antiferromagnetic (AFM) or ferromagnetic (FM)] applied primarily to systems with bond angles of 180 degrees and 90 degrees. In the present work, we quantify and apply the GK criteria to wurtzite systems. Our approach is based on calculations of (i) the spin electron densities of the anions which are first nearest neighbors (1nn) to the magnetic dopants and, (ii) the generalized exchange integrals which are derived by investigating the electronic properties of the systems under a magnetization density constraint. We demonstrate that the magnetization constraint can be used as a probe in investigating the magnetic properties of the materials under magnetization constraints. Our results indicate that the GK criteria applied to diluted magnetic semiconductors and transition metal oxides of wurtzite structures always lead to a FM coupling between two 1nn dopants of the same type. This is justified by ab initio calculations obtained for ZnO and GaN doped with 3d-transition metal dopants.