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Magnetic Magnetoelectric and Magnetoelastic Properties of new multiferroic material NdFe3(BO3)4

Complex experimental and theoretical study of the magnetic, magnetoelectric, and magnetoelastic properties of neodymium iron borate NdFe3(BO3)4 along various crystallographic directions have been carried out in strong pulsed magnetic fields up to 230 kOe in a temperature range of 4.2-50 K. It has been found that neodymium iron borate, as well as gadolinium iron borate, is a multiferroic. It has much larger (above 3 10^(-4) C/m^2) electric polarization controlled by the magnetic field and giant quadratic magnetoelectric effect. The exchange field between the rare-earth and iron subsystems (~50 kOe) has been determined for the first time from experimental data. The theoretical analysis based on the magnetic symmetry and quantum properties of the Nd ion in the crystal provides an explanation of an unusual behavior of the magnetoelectric and magnetoelastic properties of neodymium iron borate in strong magnetic fields and correlation observed between them.


Introduction
In the family of rare-earth iron borates RFe 3 (BO 3 ) 4 , gadolinium iron borate GdFe 3 (BO 3 ) 4 is the most thoroughly studied (the ground state of the Gd 3+ ion is 8 S 7/2 ) [1][2][3][4]. It exhibits the structural phase transition at 156 K, antiferromagnetic ordering of the spins of Fe 3+ at T N = 38 K, and spin-reorientation transition at T R ~ 10 K. Magnetic field-induced phase transitions recently found in it and the effects of the magnetic field control over the electric polarization that accompany these phase transitions indicate multiferroic properties of this compound [5]. It was of interest to investigate the effect of the ground state and the ionic radius of the rare-earth ion in iron borates on their properties and phase transitions. In this end, complex investigation of various properties of a single crystal of neodymium iron borate NdFe 3 (BO 3 ) 4 ( T N ~ 32 K) were carried out. The Nd 3+ ion has the ion radius larger than the Gd 3+ ion and is a Kramers ion with the 4 I 9/2 ground state. Neodymium iron borate NdFe 3 (BO 3 ) 4 , as well as gadolinium iron borate GdFe 3 (BO 3 ) 4 has the rhombohedral structure with the space group R32 and the effect of the magnetic-field control over the electric polarization is possible in this compound as well as in GdFe 3 (BO 3 ) 4 [5]. In order to compare the properties of GdFe 3 (BO 3 ) 4 and NdFe 3 (BO 3 ) 4 , the magnetic, magnetoelectric, and magnetoelastic properties of single crystals of neodymium iron borate NdFe 3 (BO 3 ) 4 in strong pulsed magnetic fields were investigated.

Experimental results
The magnetization, electric polarization, and magnetostriction of the NdFe 3 (BO 3 ) 4 single crystal were measured in strong magnetic fields up to 230 kOe for H||c and H ⊥ c for temperatures 4.2-50 K. Figure 1 shows the magnetization curves along the a-and caxes of the NdFe 3 (BO 3 ) 4 crystal for temperatures of 4.5, 15, 20, and 30 K (the a-axis is one of the three twofold axes perpendicular to the threefold c-axis).
According to the measurements, when the field was applied along the c-axis, the magnetization was almost independent of temperature T . The magnetization curves along the a-and c-axes of the crystal coincided with each other and exhibited temperature dependence (see Fig. 1). The total magnetization of NdFe 3 (BO 3 ) 4 consists of the slightly T -dependent contribution from the antiferromagnetic iron sublattice and the strongly T -dependent contribution from the rare-earth sublattice of the Nd 3+ ions: When the magnetic field was oriented along the a-axis of the crystal, longitudinal magnetostriction strains appeared at low temperatures 4.2-25 K and these deformations exhibited dome-like magnetic field dependence (see Fig. 2). In particular, at a temperature of 4.5K in a magnetic field of H crit~1 0kOe, a positive jump was observed in the longitudinal magnetostriction (~10 -5 ). As temperature increased, this jump decreased and vanished at 25K. When the magnetic field increased, longitudinal magnetostriction changed its sign and negative magnetostriction observed above 28 K was a quadratic function of the magnetic field. As shown below, a field of 50 kOe for which magnetostriction changes sign corresponds in the theoretical model to the exchange field generated by the antiferromagnetic iron subsystem on rare-earth ions. The stepwise appearance of large positive longitudinal electric polarization (~400 µC/m 2 at 4.5 K) was also found in the same temperature range 4.2-25 K for fields H crit~1 0kOe parallel to the a-axis. When the field increased, this polarization decreased and changed its sign for the same field 50kOe as magnetostriction. However, in contrast to magnetostriction, electric polarization was saturated at a value of about -320 µC/m 2 for fields above ~ 80 kOe (see Fig. 3). It is worth noting that the saturation electric polarization P a (H a ) in NdFe 3 (BO 3 ) 4 is 30 times larger than the value   orientation of the iron-ion spins in the ab plane. In the absence of the field, three equivalent directions corresponding to three twofold axes exist in the ab plane. The field applied in the plane reorients the magnetic moments of the iron sublattices in a field of H ~ 10 kOe in the direction perpendicular to the field. The fact that the longitudinal magnetostriction jumps for the magnetic field orientation along the b axis of the crystal (i.e., perpendicularly to one of the twofold axes) are an order of magnitude smaller [λ b (H b ) ~ 10 -6 ] than the polarization jumps for H || a possibly indicates that spins are predominantly oriented along the twofold axes. In contrast to the longitudinal electric polarization P a (H b ), the transverse electric polarization P a (H a ) in a strong magnetic field was not saturated and reached a large value of 2mC/m 2 in a field of 200 kOe at 4.5K (see Fig. 4).

Magnetic symmetry, theory, and discussion
To discuss the experimental results, we use the model of the magnetic structure of NdFe 3 (BO 3 ) 4 that was previously considered in [2,5] for gadolinium iron borate but supplement it with additional concepts concerning the subsystem of rare-earth ions.
The essence of this model is as follows. The subsystem of Fe 3+ ions is described by means of two magnetic sublattices antiferromagnetically coupled through the exchange field. Let , be the magnetic moments of the iron sublattices; According to the symmetry of the crystalline structure of iron borates, the basal plane contains three easy magnetization axes coinciding with the twofold axes. Therefore, the crystal under equilibrium conditions is divided into three types of domains, the vector L oriented along the corresponding twofold axis in each of them. This circumstance should be taken into account when considering the magnetization processes in relatively weak fields (up to 10 kOe, as follows from experimental magnetoelectric and magnetostriction dependences).
For further analysis, it is natural to separately consider two magnetic field ranges: (i) relatively weak fields up to 10 kOe and (ii) strong fields above 10 kOe. In the former range, the process of magnetic and domain structure rearrangement in the sample takes place so that the vector L becomes uniform over the sample and directed perpendicularly to the external magnetic field. The latter remagnetization process is determined by both the rare-earth subsystem and the spin flip of the iron sublattice and it extends to the field range of about 10 6 Oe; for this reason, for fields up to (1-2) ×10 5 Oe, one may take

M H
here, ⊥ χ is the transverse susceptibility of the iron sub-lattice (interaction with the rare-earth subsystem almost does not affect the spin-flip of the iron sublattices in neodymium iron borate).
Let us consider the system of Nd 3+ ions. As usual, we disregard the Nd-Nd exchange interaction (its energy 1 K); i.e., the Nd ion subsystem is treated as paramagnetic and is subjected to the exchange field created by iron ions. Another important action on Nd ions is provided by the crystal field that splits the main multiplet of these ions into five Kramers doublets.
The main multiplet of Nd 3+ is 4 I 9/2 with S = 3/2, L = 6, and J= 9/2. According to the spectroscopic experiment reported in [6], the main doublet of the 4 Here, g α is the effective g factor of the main doublet of the Nd 3+ ion; the subscript α = ⊥ and c specifies the orientation of the magnetic moment, ⊥ (small effect of magnetic anisotropy in the plane on g is disregarded here) and c refer to the basal plane and c axis, respectively; H H , where is the exchange field that is created by the iron sublattices and acts on rare-earth ions, the signs ± correspond to two antiferromagnetic sublattices, and the superscript i = 1, 2 specifies the rare-earth subsystems; and is the van ⊥ as shown below, whereas the contribution from the rare-earth subsystem for magnetization along the c axis is equal to zero (i.e., ), as indicated by the absence of the temperature dependence of (see Fig. 1).
In view of the above remarks, the total magnetization (per molecule) of the rare-earth ion system that is obtained by summing Eq. (3) with respect to i is equal to where summation is performed over two paramagnetic subsystems of the rare-earth sublattice.
Since the electric polarization in neodymium iron borate is larger than that in GdFe 3 (BO 3 ) 4 3)]. Table 1. Transformation properties of the basic physical tensors in the group 32 G ; R is the 120°-rotation matrix; P i , L i , and Mi are the components of the electric polarization, antiferromagnetism, and magnetization, respectively; and u ij are the magnetostriction tensor components We first consider the case H = (H x , 0, 0). Suggesting that domains with the vectors L oriented along the three easy axes exist in the sample for H x = 0, it is easy to verify that the polarization components averaged over these domains (it is assumed that they have the same total volume) <Pi> vanish, where < > is the averaging symbol. As the field increases, the domains are shifted and the iron spins are reoriented to the y axis (as it usually occurs in antiferromagnets due to the anisotropy of susceptibility) so that a uniform state with L = (0, L y , 0) and with corresponding maximum of P x is realized in the sample for H x ≥ 10 kOe. A jump in the P x component at H ≈ 10 kOe is remarkable in Fig. 3 where P x (H || x) is given by Eq. (6). Formula (6) is qualitatively correct for the field dependence of the electric polarization P x (H || y) for fields from 10 kOe to 70-80 kOe with the same value H exch ≈ 50 kOe. However, we point to certain systematic overestimation of experimental curves by Eq. (7). This discrepancy is discussed below.
As seen in Fig. 4, the behavior of the curve P x (H || y) for H > 80 kOe differs significantly from the behavior of the curve P x (H || x): in contrast to the latter, the former exhibits a strong, almost quadratic increase up to H > 200 kOe. This difference should be specially analyzed.
Formulas (5)- (7), which are used for above analysis of the field dependence P i (H), do not take into account that the crystal symmetry changes from class 32 to class 2 [at P = (P x , 0, 0)] when electric polarization appears in the sample. In particular, the magnetic susceptibility of rare-earth ions in class 2 is not a diagonal tensor as in class 32. Additional components χ 23 and χ 32 appear in the tensor χ ik (in the coordinate system, where the x axis is parallel to the twofold axis). This means that the z component m iz = χ 32 H y of the magnetization of Nd ions appears for H = (0, H, 0), which leads to the appearance of an additional term in P x (H || y): This circumstance explains the increase in polarization in strong fields, as well as the above systematic overestimation of the experimental curves by Eq. (7) in moderate fields (the comparatively small disregarded contribution from the iron sublattice, which is proportional to , affects in the same direction).   (2)].
Thus, the investigations reported above indicate that neodymium iron borate, as well as gadolinium iron borate, is a multiferroic, but it has much larger electric polarization (above 300 µC/m 2 ) controlled by the magnetic field and giant quadratic magnetoelectric effect (on the order of 10 -17 s/A). On the basis of the theoretical model describing the magnetic properties of the neodymium ion in the crystal, the experimental magnetization curves of neodymium iron borate are explained and the f-d exchange field (50 kOe), as well as the g factor for the neodymium ion, is estimated for the directions in the basal plane  3 and the principal axis g ⊥ g c ~ 0. The extraordinary behavior of magnetoelectric and magnetoelastic properties and correlation observed between them are explained in the framework of the symmetry approach. The difference between the field dependences for the longitudinal P a (H a ) and transverse P a (H b ) electric polarizations are also explained.