Theory of ferromagnetism driven by superexchange in dilute magnetic semiconductors

Magnetic properties of Ga$_{1-x}$Mn$_x$N are studied theoretically by employing a tight binding approach to determine exchange integrals $J_{ij}$ characterizing the coupling between Mn spin pairs located at distances $R_{ij}$ up to the 16th cation coordination sphere in zinc-blende GaN. It is shown that for a set of experimentally determined input parameters there are no itinerant carriers and the coupling between localized Mn$^{3+}$ spins in GaN proceeds via superexchange that is ferromagnetic for all explored $R_{ij}$ values. Extensive Monte Carlo simulations serve to evaluate the magnitudes of Curie temperature $T_\mathrm{C}$ by the cumulant crossing method. The theoretical values of $T_\mathrm{C}(x)$ are in quantitative agreement with the experimental data that are available for Ga$_{1-x}$Mn$_x$N with randomly distributed Mn$^{3+}$ ions with the concentrations $0.01 \leq x \leq 0.1$.