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Using spin-polarized density-functional theory with exchange and correlation potential, approximated by both the generalized gradient approximation (GGA) and the $\mathrm{GGA}+\mathrm{U}$ methods, we show that the coupling between a pair of Cr atoms substituted in GaN nanoholes is ferromagnetic. The interaction between the two Cr atoms, each carrying an average magnetic moment of around $2.5{\ensuremath{\mu}}_{\mathrm{B}}$, is short ranged and mediated by the neighboring N atom. As the concentration of Cr increases, clustering ensues and the coupling changes from ferromagnetic to ferrimagnetic resulting in a significant reduction in the average magnetic moment per Cr atom.

Ferromagnetic to ferrimagnetic crossover in Cr-doped GaN nanohole arrays