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

A ferromagnetic coupling between localized Mn spins was predicted in a series of ab initio and tight binding works and experimentally verified for the dilute magnetic semiconductor Ga$_{1-x}$Mn$_x$N. In the limit of small Mn concentration, x $\lesssim$ 0.03, the paramagnetic properties of this material were successfully described using single ion crystal field model approach. However, there is still a need to investigate the effect of magnetic coupling on magnetic properties of dilute magnetic semiconductors using approach that goes beyond the classical approximation, e.g. Landau-Lifshitz-Gilbert one. Therefore, in order to obtain the magnetization M(T,B) of (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 uniaxial anisotropy field change as the number of magnetic Mn$^{3+}$ ions in a given cluster increases from 1 to 4.

A ferromagnetic coupling between localized Mn spins was predicted in a series of ab initio and tight binding works and experimentally verified for the dilute magnetic semiconductor Ga1−xMnxN. In the limit of small Mn concentration, x 0.03, the paramagnetic properties of this material were successfully described using single ion crystal field model approach. However, there is still a need to investigate the effect of magnetic coupling on magnetic properties of dilute magnetic semiconductors using approach that goes beyond the classical approximation, e.g. Landau-Lifshitz-Gilbert one. Therefore, in order to obtain the magnetization M (T, B) of (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 uniaxial anisotropy field change as the number of magnetic Mn 3+ ions in a given cluster increases from 1 to 4.

I. INTRODUCTION
Magnetism in reduced dimensions, such as in magnetic nanostructures and magnetic clusters, has received a great research interest in the recent years due to its unexpected features and potential applications in highdensity storage [1], nanoelectronics and quantum computations [2]. A major advantage with respect to analogous bulk based materials originates from an additional degrees of freedom of nanoparticles to tune the magnetic properties by modifications of their size, shape, number of magnetic ions and/or coupling with the substrate [3].
For example, recently it has been shown that single cobalt atoms deposited onto platinum (111) surface pose a very large magnetic anisotropy energy (MAE) of about 9 meV [4]. The single-ion MAE depends on the arrangements of atoms around the magnetic ion through the spinorbit interaction and crystal field induced anisotropy of quantum orbital angular momentum (L). In the bulk materials the magnitude of L is usually quenched or strongly diminished by electron delocalization, ligand fields and hybridization effects, what result in small values of MAE of the order of 0.01 meV/atom [5]. However, it is possible to enhance the MAE by using low-coordination geometries, such as atoms deposited on the surface, 1D atomic chains, magnetic clusters or molecular complexes. The values of MAE of the order of 1 ÷ 10 meV per atom have * sztenkiel@ifpan.edu.pl been routinely reported in such systems [4,6,7]. Experiments on small particles of iron, cobalt, and nickel revealed the strong dependence of per-atom magnetic moments on the cluster size [8]. The ferromagnetism was present even for the clusters composed of about 30 atoms, with atom-like magnetization. These magnetic moments per one atom decreased with the number of ions in a given particle, approaching the bulk limit for about 500 atoms. Even in the material investigated here, the magnetic anisotropy strongly depends on the Mn ion concentration x, due to the dependence of lattice parameters c and a of Ga 1−x Mn x N on x. The magnetic anisotropy is high in very diluted case [9] and then decreases with x. [10]. High MAE reduces the magnitude of the thermal fluctuations in superparamagnetic nanostructures and thus determines the potential applicability of these small-scale systems in high-density recording and magnetic memory operations. It is thus very relevant to investigate magnetic anisotropy properties of systems with reduced symmetry and/or coordination of magnetic aggregates.
In this paper we numerically study how the MAE evolves from single isolated magnetic Mn 3+ impurity in  [14,15] for II-VI dilute magnetic semiconductors doped with Cr, and then successfully applied for other DMSs [9,[16][17][18][19][20][21][22][23]. Recently it was shown that CFM simulations can explain magnetic [9,21,22], magnetooptic [19] and even magnetoelectric [10] properties in dilute Ga 1−x Mn x N, with x ≤ 0.03. Therefore it is a natural way to extend aforementioned model of a single substitutional Mn 3+ ion in GaN by considering pairs, triplets and quartets of Mn 3+ ions coupled by ferromagnetic superexchange interaction [22,24,25]. Due representing the partition function of the j-th center, where E j k , ϕ j k are the k-th eigenenergy and the eigenfunction of the Mn 3+ ion being in j-th J-T center, respectively. In this work, we consider singles, pairs, triplets and quartets of Mn 3+ ions coupled by a ferromagnetic superexchange interaction H exch (1, 2) = JŜ 1Ŝ2 . The exact value of nearest neighbor superexchange coupling J is not known. The magnitudes of J obtained from first-principles methods [28] or tight binding approximations [29] are rather high J > 10 meV. On the other hand, in our recent LLG simulations (not published yet) that described reasonably well ferromagnetic properties of Ga 1−x Mn x N with x = 6% we use J=1.4 meV.

SINGLE
Therefore we assume J=2 meV here. Now, the rel-   In order to quantify the strength of magnetic anisotropy we use two different approaches. Firstly, we plot in Fig. 4 and N =2 (pair). Secondly, we calculate MAE in a standard way. The MAE is the energy needed to rotate the magnetization from its easy axis into the hard one and it can be obtained from the following formula: where B a is the uniaxial anisotropy field (see also Fig. 4(b)). The results of this procedure are summarized in Tab. II. We observe a rather weak dependence of MAE per atom on cluster size, but still the MAE increases with N . It seems that the magnetic anisotropy at small magnetic fields increases with N (see Fig. 2