Crystal field model simulations of magnetic response of pairs, triplets and quartets of Mn$^{3+}$ ions in GaN

Abstract

A ferromagnetic coupling between localized Mn spins was predicted in a series of \textit{ab initio} and tight binding calculations and experimentally verified for the dilute magnetic semiconductor Ga$_{1-x}$Mn$_x$N. In the limit of small Mn concentrations, $x \lesssim 0.01$, the paramagnetic properties of this material were successfully described using a single ion crystal field model approach. In order to obtain the description of magnetization in (Ga,Mn)N in the presence of interacting magnetic centers, we extend the previous model of a single substitutional Mn$^{3+}$ ion in GaN by considering pairs, triplets and quartets of Mn$^{3+}$ ions coupled by a ferromagnetic superexchange interaction. Using this approach we investigate how the magnetic properties, particularly the magnitude of the uniaxial anisotropy field, change as the number of magnetic Mn$^{3+}$ ions in a given cluster increases from 1 to 4. Our simulations are then exploited in explaining experimental magnetic properties of Ga$_{1-x}$Mn$_x$N with $x \cong 0.03$, where the presence of small magnetic clusters gains in significance. As a result the approximate lower and upper limits for the values of exchange couplings between Mn$^{3+}$ ions in GaN, being in nearest neighbors $J_{\mathrm{nn}}$ and next nearest neighbors $J_{\mathrm{nnn}}$ positions, respectively, are established.