Dzyaloshinskii-Moriya interaction constant in iron-gallium borate single crystals

The Dzyaloshinskii-Moriya interaction constant for iron-gallium borate FexGa1-xBO3 single crystals has been calculated on the basis of considering antisymmetric exchange in a pair of nearest-neighbouring magnetic ions. Both numerical and analytical calculations predict a square-law dependence of the Dzyaloshinskii-Moriya constant on the contents of magnetic ions, x, in the crystals.


Introduction
Recently, in the Crystal Growth Laboratory at the Crimean Federal University, high quality single crystals of iron-gallium borates, FexGa1-xBO3, with 01 x  , have been synthesised [1]. All these crystals are isostructural to iron borate [1]; meanwhile, their physical properties are very different, depending on the contents of magnetic ions, x.
Iron borate, FeBO3, 1 x  , is an easy-plane weakly ferromagnetic antiferromagnet, with the Néel temperature of 348 K [2]. It possesses a number of outstanding physical characteristics and unusual its combination, for instance, it is magnetically-ordered, albeit possessing a transparency window in the visible spectral range [2]. From the viewpoint of applied research, FeBO3 is an extremely "high-tech" material with a wide possibilities of practical applications, in particular, in the field of modern synchrotron technologies [3].
The crystals with 1 x  are, in fact, diamagnetically diluted iron borates; they all remain transparent for visible light; meanwhile, their magnetic properties depend on the degree of dilution. The possibility of modifying iron contents greatly increases their interest for fundamental research in magnetism; indeed, it allows to elucidate detailed mechanisms of transformation of the magnetic structure under the transition from magnetically ordered state to non-magnetic one [4][5][6][7]. On the other hand, the use of mixed iron-gallium borate crystals, instead of pure iron borate, is expected to significantly optimize the characteristics of technical devices, e.g., in the field of synchrotron technologies [8].
Obviously, in this context, theoretical and experimental studies of fine magnetic characteristics of iron-gallium borates become of paramount importance. We have already reported the results of electron magnetic resonance (EMR) [4], nuclear magnetic resonance [6] and theoretical studies [7,9] of these crystals; and the aim of the present work is to carry out numerical calculations of the Dzyaloshinskii-Moriya interaction constant for FexGa1-xBO3 crystals. This interaction determines the weak ferromagnetism of these crystals and of some other trigonal antiferromagnetics [10,11].

Crystal and magnetic structures of iron-gallium borate crystals
The crystal structure of FeBO3 is described by the space symmetry group D3d 6 . The corresponding rhombohedral unit cell is shown in Figure 1(a) [2]. Iron borate possesses two magnetic sublattices; the magnetic moments of ferric ions Fe1 3+ and Fe2 3+ belonging to these sublattices, 12 and μμ , ( ),   see Figure 1(a) are almost antiparallel and lie in the basal plane of the crystal perpendicular to the threefold axis C3 [2]. The sublattice magnetizations are defined as: where n = 2.236•10 28 m -3 is the iron concentration, A slight break of antiparallelism between 12 and μμ (the tilt angle ca. 55 [12]) produces a weak ferromagnetic moment M = M1 + M2, apart from a strong antiferromagnetic moment L = M1 − M2 , see Figure 1(b). In further calculations, we use the reduced ferromagnetic m and antiferromagnetic l vectors:  In pure iron borate, the nearest environment of an iron ion consists of Z = 6 irons located at a distance of 3,601 Å [2] in the planes above and below the given ion. In diamagnetically diluted FexGa1-xBO3 single crystals, a part of iron ions is randomly substituted by diamagnetic gallium ions; therefore, the number of nearest magnetic neighbours of Fe 3+ ion decreases, see Figure 1(c).

Dzyaloshinskii-Moriya constant in diamagnetically diluted crystals
Considering only the nearest magnetic neighbours, the density of the Dzyaloshinskii-Moriya energy for diamagnetically diluted crystals can be expressed as follows: where   12 ,, μ μ u is the scalar triple product of the magnetic moments of iron ions belonging to the two sublattices and u is the C3 axis unit vector, dex is the constant of antisymmetric exchange for a pair of nearest magnetic ions, Zi is the number of nearest magnetic neighbours for the i th ion and the summation is over all magnetic ions, N, in the volume V. Taking in account Equation (1) we get: is the concentration of iron ions in diluted crystals. Expressing in Equation (4) the sublattice magnetizations through the reduced magnetic vectors, see Equation (2), we get: is the sublattice magnetization for diluted crystals. Transforming the triple scalar product, Equation (5) can be rewritten as: where mix DM D is the Dzyaloshinskii-Moriya interaction constant for diluted crystals: Following the approach described by Seleznyova et al. [13], we assume that dex for the pair of nearest magnetic ions does not depend on the degree of diamagnetic dilution. With this assumption, we get: where DDM = 1.05×10 7 Jm -3 is the Dzyaloshinskii-Moriya constant for iron borate [14]. Thus, In order to numerically calculate mix DM D , a computer code implementing the summation in Equation (10) has been developed. The random substitution of a part of the magnetic iron ions by the diamagnetic ones has been modelled using the Monte Carlo technique [15]. Figure 2 shows the result of this calculation. As one can see, with decreasing x mix DM D decreases following a square low. This behaviour can be explained as follows. Taking into account that for FexGa1-xBO3, the mean number of nearest magnetic neighbours of a given iron ion is:  (11) and mix  n nx , using Equations (9) and (6), one can rewrite Equation (10) as: thus confirming the numerical calculation, see Figure 2. for diluted iron-gallium borates FexGa1-xBO3 is obtained under the assumption that dex does not change with diamagnetic dilution. In order to model diamagnetically diluted crystal lattice and implement a summation over the nearest neighbours of iron ions, a computer code is put forward. Both numerical and analytical calculations show that the mix DM D vs. x dependence follows the square-law. The results of the present work will provide the basis for detailed interpretation of experimental results on magnetometry and EMR studies of FexGa1-xBO3 crystals which are actually in progress.