Nature of magnetocrystalline anisotropy in the basal plane of iron borate

Basal (hexagonal) magnetocrystalline anisotropy of iron borate FeBO 3 , including crystal ﬁeld and dipole dipole contributions, is discussed in detail. Previously, the latter contribution had been ruled out on the basis of symmetry; indeed, considering the magnetic dipole dipole interaction in the approximation of point dipoles, only uniaxial magnetocrystalline anisotropy is accounted for. In the present work we con sider the dipole dipole interaction energy for extended dipoles and calculate the dipole dipole contribu tion to basal magnetocrystalline anisotropy for two different models of an extended dipole: a pair of ﬁctitious magnetic charges and a circular current loop. A comparison between theoretical expressions developed and experimental results obtained by antiferromagnetic resonance allows estimating the effective size of magnetic dipoles and calculating the dipole dipole contribution to basal magnetocrys talline anisotropy constants of iron borate.


Introduction
From the standpoint of magnetic properties, iron borate FeBO 3 is an easy plane antiferromagnet, possessing a weak in plane fer romagnetic moment. Iron borate has rhombohedral calcite struc ture of point group symmetry D 3d and the space group D 6 3d [1]. The effective basal (hexagonal) magnetocrystalline anisotropy energy for FeBO 3 can be expressed as follows [2]: where u is the azimuthal angle of the antiferromagnetic vector and e eff is the effective basal anisotropy constant, e eff e FeBO 3 þ 1 4 d FeBO 3 and e FeBO 3 being the basal magnetocrystalline anisotropy con stants and a eff being the effective uniaxial magnetocrystalline aniso tropy constant, In the latter equation a FeBO 3 ; D FeBO 3 and E FeBO 3 are, respectively, the uniaxial magnetocrystalline anisotropy, Dzyaloshinskii Moriya and exchange constants for FeBO 3 .
As far as for Fe 3+ (3d 5 electronic configuration) the orbital moment equals zero, the exchange energy in a good approximation is isotropic [3], so that the magnetocrystalline anisotropy energy of FeBO 3 includes only crystal field (cf) and dipole dipole (dip) terms: The crystal field contributions to the abovementioned constants have been calculated by Seleznev in the mean field approximation [4]. Indeed, the effective exchange field H E in iron borate is in order of 10 3 kOe, thus, Fe 3+ ions experience strong exchange coupling [5]. The crystal field contribution to the anisotropic part of the magne tocrystalline anisotropy energy can be calculated in perturbation theory using the spin Hamiltonian for isolated Fe 3+ ion in a dia magnetic crystal isomorphous to iron borate, e.g., gallium borate [6,7]. Thus, a cf ; d cf and e cf can be expressed through the parame ters of the spin Hamiltonian. The corresponding expressions will be given and the parameters will be specified in the next Section.
The dipole dipole contributions to these constants are usually calculated using the lattice sum method. The value of a dip 3:82 10 5 Jm 3 at 0 K for FeBO 3 has been obtained previously [8]. a FeBO 3 has been determined by antiferromagnetic resonance ⇑ Corresponding author.

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(AFMR) in a wide temperature range by Velikov et al. [9]; its value extrapolated to 0 K is a exp Earlier, the occurrence of the dipole dipole contribution to hexagonal basal anisotropy in iron borate had been ruled out on the grounds of symmetry. Indeed, the dipole dipole interaction energy is usually considered for ''point dipoles" having a negligible size, in which case only the uniaxial anisotropy is accounted for. Meanwhile, more sophisticated anisotropies, in particular, the hexagonal magnetocrystalline anisotropy can be described by higher than first order terms in the expansion of the dipole dipole interaction energy in a Taylor series in the small parameter dipole size/interdipole distance. Thus, taking into consideration ''extended dipoles", having non negligible size, opens the possibil ity to reasonably account for the dipole dipole contribution to the basal anisotropy constants and, subsequently, to estimate effective dipole dimensions in iron borate.
With this aim in mind, we have developed a theoretical descrip tion of two models of the extended magnetic dipole: (1) an assem bly of two fictitious ''magnetic charges" AEq a distance d apart and (2) an Ampérian current, i.e., a circular current loop of a radius R delimiting an area S pR 2 . Previously, we have obtained and dis cussed the expressions of vector potentials and magnetic fields produced by these two models [10]. The purpose of the present work has been to calculate the dipole dipole contribution to the basal magnetocrystalline anisotropy constants of iron borate and to evaluate the size of dipoles in FeBO 3 in the framework of the suggested models.

Crystal field contribution to magnetocrystalline anisotropy
In order to calculate the crystal field contribution we have to consider a Fe 3+ ion in a diamagnetic crystal isomorphous with FeBO 3 . The conventional spin Hamiltonian in this case is [11,12]: where g is close to the free electron g factor g e 2:0023, b is the Bohr magneton, H is the magnetizing field, S 5 2 is the electron spin of Fe 3+ , D is the second order axial fine structure constant, a and F are, respectively, the fourth order cubic and axial fine structure are extended Stevens operators, as defined in the textbook by Al'tshuler and Kozyrev [13]. The Ç signs refer to two non equivalent iron sites with local magnetic axes rotated through the angle Ça about the C 3 axis, see Fig. 1.
The spin Hamiltonian parameters of Fe 3+ in diamagnetic GaBO 3 have been determined previously by Lukin et al. [11] and recently specified by Seleznyova et al. [12].
Seleznev has obtained the expressions for a cf ; d cf and e cf using the mean field approximation [4]: where N is the number of Fe 3+ ions per unit volume, N 2:236 10 28 m 3 for FeBO 3 and tðxÞ and rðxÞ are the following functions: where x gbH E =2kT, H E is an effective exchange field, vide infra, k is the Boltzmann constant and T is the absolute temperature.
As one can see from Eq. (6), the crystal field gives no contribu tion to e FeBO 3 .

Model of a pair of fictitious magnetic charges
Here we consider the interaction energy E between two paral lel/antiparallel identical dipoles for different models. Fig. 2 shows a system of two interacting dipoles implemented as a pair of Fig. 1. Two non-equivalent sites of Fe 3+ . ðx; y; zÞ is a Cartesian coordinate system with xkC2; ykm; zkC3 where C2; m and C3 are, respectively, a two-fold axis, a symmetry plane and the three-fold axis [1]. The z-axis is perpendicular to the plane of the figure and points towards the reader. The full and empty circles represent ions located above and below this plane, respectively. ''magnetic charges" AEq spaced a distance d apart. For the dipole dipole energy one gets: where l 0 is the permeability of vacuum, m qd qde m is the mag netic moment, defined by analogy with electrostatics and directed along the unit vector e m ðsin#cosu; sin#sinu; cos#Þ; where e r is the unit vector in the direction of r, see Fig. 2 for the notation. The choice of the AE signs refers to parallel and antiparallel dipoles, respectively.
Introducing e d=r in Eq. (9) one gets: For our purpose we need only an approximate expression of E for e ( 1. An expansion in the Taylor series up to the fourth order yields: where P n are Legendre polynomials [14] of the scalar product e r e m , and the and + signs correspond to parallel and antiparallel dipoles, respectively.

Model of a circular current loop
Next, we calculate the interaction energy between two identical and parallel circular current loops (Ampérian currents) of the same radii R and areae S pR 2 , carrying a current I, see Fig. 3. By defini tion, the magnetic moment of a loop is m SIe m pR 2 Ie m .
For definiteness, we choose the loops centered at the space ori gin O p and at an arbitrary point O s as the primary and secondary loops, respectively.
The dipole dipole interaction energy in this model is straight related to the mutual inductance M of the loops: Here the sign in the right hand member occurs because the interaction energy between two coaxial (attracting) currents is negative while M in this case is positive. By definition, where U, the magnetic flux induced by the current in the primary loop and passing through the secondary loop, can be calculated as follows: Here l p and l s are perimeters of the primary and secondary loops, dA is a differential element of the vector potential at a point M s on the secondary loop, produced by the primary loop: M p M s is the distance between two points on the primary and sec ondary loops, and dl p and dl s are differential elements of the corre sponding loops. In order to evaluate the closed curve integrals in Eq. (14) we express M p M s , dl p and dl s as follows: jM p M s j 2 r 2 þ 2R 2 ½1 sinð/ sÞ þ2Rr x ½sin uðcos / þ sin sÞ þ cos u cos #ðsin / cos sÞ þ2Rr y ½sin u cos #ðsin / cos sÞ cos uðcos / þ sin sÞ 2Rr z sin #ðsin / cos sÞ ; dl p Rðcos u cos # sin s þ sin u cos sÞds Rð sin u cos # sin s þ cos u cos sÞds R sin # sin s ds  3 In Eqs. (16) to (18) s and / are polar angles of arbitrary points of the primary and secondary loops, respectively. Putting these expressions in Eq. (14), for the interaction energy, cf. Eq. (12), we get: where the integrations are over s and /.
Since, as in the previous case, we need only an approximate expression of E, we can first expand the integrand in Eq. (19) in a Taylor series in the small parameter e R=r and then integrate the result. In terms of the Legendre polynomials, up to the fourth order we get: where the and + signs correspond to parallel and antiparallel dipoles, respectively.

Calculation of the dipole-dipole interaction energy in FeBO 3
In order to calculate the dipole dipole contribution to the mag netocrystalline anisotropy constants for FeBO 3 , we have put for ward a computer code implementing the lattice sum method. We have chosen to do the summation in the volume of a rhombo hedron congruent to the primitive rhombohedron shown in Fig. 4.
The axes of the rhombohedral coordinate system x 0 ; y 0 ; z 0 coin cide with the edges of the rhombohedron, see Fig. 4. In transform ing the radius vector from the Cartesian to the rhombohedral system, we express the coordinates of iron sites through the edge length l of the rhombohedron: x 0 m l, y 0 n l and z 0 k l, where m; n; k are integers numbering the sites along the corresponding axes. The radius vector in the new coordinate system is r l  where the factor ð 1Þ mþnþk takes into account antiferromagnetic ordering and Eðm; n; kÞ is the dipole dipole interaction energy between ions at the origin (numbered 0; 0; 0) and at a site num bered m; n; k. Henceforth, the magnetic dipole moment at T 0 K will be expressed as m gbS ð23Þ where g; b and S have the same meanings as in Eq. (5). The dipole dipole contributions at 0 K, together with those of the crystal field, described in Section II, are listed in Table 1. One can see that the models of a pair of magnetic charges and a circular current loop result in substantially different expressions for the dipole dipole energies.
In order to get the dipole dipole contributions to the magne tocrystalline anisotropy constants at different temperatures, these contribution at 0 K should be multiplied by ðM T =M 0 Þ 2 , where M T is the sublattice magnetization at the temperature T. ðM T =M 0 Þ for FeBO 3 have been tabulated [4]. The temperature dependences of the crystal field contributions are given in Eq. (6).

Comparison with experiment
From the AFMR experiments the effective basal magnetocrys talline anisotropy constant e exp eff , cf. Eq. (2), can be determined [15]. The EMR studies have been carried out at 77 K with a laboratory developed spectrometer at microwave frequencies m from 15 to 36 GHz and magnetizing field H up to 10 kOe applied in the basal plane of the crystal. FeBO 3 crystal has been synthesized used to calculate the dipole-dipole energy.  by solution in the melt technique and had the shape of a thin hexagonal plate [6]. For H applied in the basal plane of the crystal, the low frequency AFMR mode for FeBO 3 is described by the following expression [15,16]: where c is the gyromagnetic ratio for g 2:0; H D and H E , H E H hex and H 2 D are, respectively, effective Dzyaloshinskii Moriya and exchange fields, anisotropic and isotropic energy gaps, H hex being the effective field of basal magnetocrystalline anisotropy, and u is the angle between H and the x axis. As far as different authors use different definitions of H D , H E and H hex , the definitions used in this work are given in Table 2  This value of H D confirms the previous finding [9], see Table 2. Using this value, we have determined the product H E H hex , see Eq.
(24), from AFMR measurements at a fixed m, carried out by rotating H in the basal plane of the crystal. Fig. 6 shows the angular dependence of the quantity One can see that this dependence can be accounted for by a superposition of hexagonal and uniaxial magnetocrystalline aniso tropies in the basal plane. The occurrence of uniaxial magnetocrys talline anisotropy in this case can be due to a slight deviation of H from the basal plane or caused by mechanical stresses [17]. There fore, the angular dependence of the resonance field has been fitted to by the following expression, cf. Eq. (24): From H E H hex and H E we get H hex 0:913 10 5 kOe; then, using the definition of H hex given in Table 2, we get e exp eff 0:936 Jm 3 . The H hex value corroborates that earlier reported by Doroshev et al. [15]; note that the negative sign of the latter value is due to a dif ferent definition of the angle u.  Finally, from the results described above we estimate the dipole size. In the following we assume that e eff e exp eff , see Eq. (2) for e eff . Substituting e exp eff in Eq. (2) and taking into account Eqs. (3) and (4), we get: In what follows, for a FeBO 3 we use the experimental value a exp FeBO 3 3:2 10 5 Jm 3 determined by AFMR at 77 K [4,9]. yielding two positive solutions: R 1 0:2189 and R 2 0:0797Å: In order to assess the plausibility of such values, they should be compared with the ionic radius R i of Fe 3+ ; indeed, we can reason ably infer that the effective size of a dipole should be of the same order of magnitude as the size of the physical object producing the corresponding dipole moment. For high spin Fe 3+ in sixfold oxygen coordination R i 0:645 Å [18]; therefore the R 1 value seems to be a more realistic estimate than R 2 , the latter value being an order of magnitude smaller than R i .
The dipole dipole contributions to the magnetocrystalline ani sotropy constants of FeBO 3 at 77 K calculated with the R 1 value are: e dip 12:2 Jm 3 and d dip 7:57 Á 10 3 Jm 3 : ð29Þ

Conclusions
Possible contributions to the basal magnetocrystalline aniso tropy of iron borate, namely, crystal field and dipole dipole contri butions have been considered in detail. The former contribution has been calculated using the spin Hamiltonian parameters for iso lated Fe 3+ ions in (diamagnetic) gallium borate. The latter contribu tion has been evaluated assuming that the ratio dipole size/ interdipole distance is non negligible, i.e., that we are dealing with extended dipoles. The dipole dipole interaction energy has been calculated for two extended dipole models, viz., a pair of magnetic charges and a circular current loop. The dipole dipole contribution has been calculated by the lattice sum method.
In order to determine the basal magnetocrystalline anisotropy constants of iron borate, we have carried out AFMR studies at 77 K. A comparison between the experimental and calculated val ues of this constant has shown that the model of a pair of magnetic charges fails at explaining the experimental results. In contrast, the model of circular current loop provides consistent evidence in sup port of the dipole dipole contribution to the basal magnetocrys talline anisotropy of iron borate and, incidentally, yields a more or less realistic estimate of the size of the magnetic dipole associ ated with Fe 3+ ion.
In spite of the fact, that the modeling considering extended dipoles, put forward in this work provides new insight in the nat ure of the basal magnetocrystalline anisotropy of iron borate, it is certainly oversimplified. More sophisticated (ab initio) calculations are necessary in order to get further insight in the spatial distribu tion of the magnetic field produced by paramagnetic ions at short and intermediate distances.