b .N 75.80.+q 75.30.�m 75.50.Ee for sed or le a tion tin & 2008 Elsevier B.V. All rights reserved. e3(BO3 c effec e iron ically w e rar e intr N changing the susceptibility of the Fe subsystem from wJ to w?. This ange ed in ature 3+ data for magnetostriction in the compounds RFe3(BO3)4 with ARTICLE IN PRESS Contents lists availabl .el ica B Physica B 404 (2009) 213–216 � reorientation is accompanied by the orientation of RE magnetic moments along the magnetic field direction. At T ¼ 4.2K the magnetization jump occurs to the value �9mB at Bsf ¼ 3.5 T in R ¼ Pr [4] and Tb [9] in the magnetic field along the trigonal axis. 2. Theoretical background Consideration is performed on the basis of the theoretical approach presented in our work [8], which is devoted to Corresponding author. Tel.: +74832575741; fax: +74832562939. E-mail address:
[email protected] (N.P. Kolmakova). 0921-45 doi:10.1 in the Fe subsystem, the Fe magnetic moments change their direction from the trigonal axis to basal plane by jump, thus dependences of multipole moments of Nd ions. This work is concerned with theoretical consideration of recent plane. In the RFe3(BO3)4 with R ¼ Pr, Tb, Dy at the magnetic field along the trigonal axis for ToT , a spin–flop transition occurs this compound. There is a jump in the vicinity of 1 T and a ch of sign in higher fields (�6T). Observed features were explain our work [8] in terms of magnetic field and temper Fe subsystem tends to orientate the magnetic moments in the basal plane. The resulting magnetic structure depends on the RE subsystem, the magnetic anisotropy of which is formed by the crystal field. In the compounds with R ¼ Pr, Tb, Dy, the magnetic moments of both subsystems are directed along the trigonal axis c. For R ¼ Nd and Er, all the magnetic moments lie in the basal In NdFe3(BO3)4 magnetization curves are more complicated due to the existence of domain structure of this trigonal easy- plane antiferromagnet. However, a spin–flop transition occurs at the magnetic field in the basal plane �1T in one of the three possible domains [5] (see also [6]). In Ref. [7] peculiarities of longitudinal magnetostriction along the a-axis were observed in Keywords: Magnetostriction Rare-earth ferroborates RFe3(BO3)4 Quadrupole moments 1. Introduction The trigonal 4f–3d crystals RF demonstrate various magnetoelasti on the rare-earth element R. Th compounds orders antiferromagnet ture in the interval 30–40K. Th magnetized due to f–d coupling. Th 26/$ - see front matter & 2008 Elsevier B.V. A 016/j.physb.2008.10.025 )4 (space group R32) ts depending strongly subsystem in these ith the Neel tempera- e-earth subsystem is insic anisotropy of the TbFe3(BO3)4 [1] and �7mB at Bsf ¼ 2.8 T in DyFe3(BO3)4 [2] (see also Ref. [3]). Recent measurements of magnetic character- istics of PrFe3(BO3)4 [4] showed that in this compound the magnetization jump to �1mB takes place at Bsf ¼ 4.5 T for T ¼ 4.2K. The transition field grows and the magnetization jump decreases with increasing temperature in these compounds. Such behaviour is typical for a uniaxial antiferromagnet magnetized along the easy axis. Magnetostriction in the rare-earth ferro A.A. Demidov a, N.P. Kolmakova a,�, D.V. Volkov b, A a Bryansk State Technical University, 241035 Bryansk, Russia b M.V. Lomonosov Moscow State University, 119992 Moscow, Russia a r t i c l e i n f o Article history: Received 25 June 2008 Received in revised form 4 September 2008 Accepted 15 October 2008 PACS: a b s t r a c t Recent experimental data R ¼ Pr and Tb are discus calculated in the framew approximation. Quadrupo longitudinal magnetostric are deduced when accoun transition. journal homepage: www Phys ll rights reserved. orates RFe3(BO3)4, R ¼ Pr and Tb . Vasiliev b magnetostriction in the rare-earth (RE) ferroborates RFe3(BO3)4 with from a theoretical point of view. Multipole moments of RE ions are k of a crystal-field model for the RE ion and the molecular-field pproximation is shown to be sufficient for interpretation of data for at the magnetic field along the trigonal axis. Parameters of PrFe3(BO3)4 g for the experimental magnetization curves that manifest a spin–flop sevier.com/locate/physb e at ScienceDirect Ref. [8]) should be fulfilled. Contribution of the Fe subsystem to magnetostriction is neglected because it is known (see e.g. To account for the field dependence of longitudinal magnetos- have calculated the magnetization curves of PrFe3(BO3)4 at BJc for several temperatures and have compared them with the experi- mental ones (Fig. 2 in Ref. [4]) with the aim to find the remaining parameters. Agreement between calculated and experimental magnetization curves turned out to be quite satisfactory. Then with all these parameters we have calculated the multipole moments of the Pr3+ ion in an effort to account for the field dependences of longitudinal magnetostriction along the c-axis also measured in Ref. [4] at temperatures ToTN (Fig. 3 in Ref. [4]). It turned out that it is not possible to describe even the main features of these curves, either in quadrupole approximation (see Eqs. (1) and (2)) or regarding the multipole moments of fourth and sixth orders. For example, an increase of magnetostriction value in the flop phase at B4Bsf cannot be described. Jumps of relevant multipole moments at spin–flop transition turned out to be many times (�10) greater than those in TbFe3(BO3)4. ARTICLE IN PRESS sica Ref. [10]) that magnetoelastic properties of f–d compounds are usually governed mainly by the RE subsystem. Calculations of all multipole moments of RE ions and their comparison with the experimental data [4,9] for longitudinal magnetostriction along the trigonal axis lc in RFe3(BO3)4, R ¼ Pr, Tb, show that quadru- pole approximation is sufficient. In this approximation, the behaviour of magnetostriction for BoBsf is determined only by the change of moment Q20 ¼ aJ/O02S (aJ is the Stevens factor, O02 is the equivalent operator) lcðBÞ ¼ A1 Qþ20ðBÞ þ Q�20ðBÞ 2 � Q20ðB ¼ 0Þ � � ; BoBsf (1) Q20 7 (B) is the quadrupole moment of the RE ion with magnetic moment (+) along the magnetic field and (�) opposite to the field direction. At B4Bsf the trigonal symmetry is broken because the Fe magnetic moments are oriented at an angle to the basal plane. So, low-symmetrical quadrupole moments of the type of aJ/O21S, aJ/O22S, etc., which are absent in the collinear phase by symmetry, should be taken into account. If we form a generalized quadrupole moment Q, for magnetostriction we can write the following expression: lcðBÞ ¼ A2 DQ ¼ A2½Q ðBÞ � Q20ðB ¼ 0Þ�; B4Bsf (2) In Eqs. (1) and (2) the quantities A1 and A2 are relevant combinations of magnetoelastic coefficients and elastic constants, which have a very cumbersome form (see Ref. [8]). 3. Magnetostriction in TbFe3(BO3)4 Magnetostriction along three crystallographic directions in TbFe3(BO3)4 was measured in Ref. [9] at the magnetic field along the trigonal axis for T ¼ 10K. All the curves l(B) show a jump at the spin–flop transition field, which is equal to �4T according to magnetic measurements [1] and to �4.4 T according to magnetos- triction data [9] at this temperature. The jump value of �10�5 is much the same for lc and lb; for la it is �0.6�10�5. la and lb are positive and grow with increasing field at B4Bsf, lc is negative and practically does not change at these fields. So, the jump values in TbFe3(BO3)4 and NdFe3(BO3)4 [7] are approximately equal. Comparison of jump values for the relevant multipole moments for Nd3+ and Tb3+ ions in ferroborates, calculated in our work [8], permits us to account for this relation in the following way. In NdFe3(BO3)4, the necessary multipole moments are 2–3 times magnetoelastic effects in trigonal 4f–3d crystals. Expressions for deformation tensor components listed in Ref. [8] permit writing the expressions for magnetostriction for any geometry of experi- ment. RE contribution to magnetostriction is determined by changes of multipole moments of the RE ion at the magnetic field in the chosen direction. Magnetoelastic Hamiltonian for the RE subsystem is written out in Ref. [8] in the multipole approximation. Multipole moments of the RE ion is calculated on the basis of Hamiltonians also given in Ref. [8]. The RE Hamiltonian includes the crystal-field Hamiltonian of trigonal symmetry, Zeeman term and f–d coupling, which depends on values and directions of Fe magnetic moments. They are found when solving a self-consistent problem because the Hamiltonian of the Fe subsystem includes the f–d coupling term in addition to the Fe–Fe exchange interaction; in doing so, the condition of minimum of corresponding thermodynamic potential (Eq.(5) in A.A. Demidov et al. / Phy214 larger than those in TbFe3(BO3)4. Considering that a spin–flop transition in NdFe3(BO3)4 occurs only in one of the three possible domains, only this domain contributes to the jump. triction lc(B) at BJc in PrFe3(BO3)4 measured in Ref. [4], the parameters of the compound should be available. They are, firstly, the crystal-field parameters, the Fe–Fe and Pr–Fe exchange parameters, and the Fe anisotropy constant. Using the well- known Van–Vleck formula for paramagnetic susceptibility for the Pr subsystem and the Curie–Weiss law for the Fe subsystem, we have found the crystal-field parameters of trigonal symmetry from experimental data [4] for wc,?(T) at T4TN; the paramagnetic Neel temperature was also varied. These parameters provide the energy of the first excited state E1E38 cm �1, as it was found in Ref. [4], since we have included this value in the target function. Description of wc,?(T) was as good as in Ref. [4] (see Fig. 1 in Ref. [4]). With the above-mentioned crystal-field parameters we All multipole moments of the Tb3+ ionwere calculated with the parameters of TbFe3(BO3)4 determined from the analysis of its magnetic properties in our work [1]. Analysis of this numerical material and comparison with experimental field dependence of longitudinal magnetostriction [9] presented in Fig. 1 show that its interpretation is possible in quadrupole approximation. The corresponding calculated curve is also displayed in Fig. 1. Coefficient A2 ¼ ATb ¼ �2.5�10�3 in Eq.(2) is deduced from the comparison of jump values of magnetostriction and of the quadrupole moment Q. It is seen that the calculated change of quadrupole moment describes a very small value of magnetos- triction in the collinear phase and its weak change with increasing field in the flop phase. 4. Magnetic characteristics and magnetostriction of PrFe3(BO3)4 Fig. 1. Field dependencies of longitudinal magnetostriction (experimental data digitized from Ref. [9]) and appropriate quadrupole moment of the Tb3+ ion in TbFe3(BO3)4. B 404 (2009) 213–216 According to the data of the same authors, the longitudinal magnetostriction in TbFe3(BO3)4 [9] is over two times more than the magnetostriction in PrFe3(BO3)4. Explanation of this situation would be possible only if one supposes that the corresponding coefficients (coefficient A2 in quadrupole approximation, see Eq. (2)) are 10 times less in praseodymium compound. Consider- ing that magnetic structures, magnetic field direction and type of phase transition are the same in these compounds, this assump- tion for ions belonging to the rare–earth series seems to us little reasonable. So we have tried to describe all these data (magnetic susceptibility, magnetization and magnetostriction curves along the trigonal axis) for PrFe3(BO3)4 all together with one set of parameters of the compound; the value of E1E38 cm �1 found in Ref. [4] was rejected. Our calculations have shown that the curves wc,?(T) at T4TN can be described in a very similar manner with different values of increasing of magnetostriction of PrFe3(BO3)4 in the flop phase is described (see Fig. 3). 5. Conclusions Our calculations have shown that field dependences of long- itudinal magnetostriction at the magnetic field along the trigonal axis in PrFe3(BO3)4 [4] and in TbFe3(BO3)4 [9] are governed by the rare-earth contribution, which is sufficient to describe in quadru- pole approximation. Field dependences of the quadrupole mo- ment of the RE ion in the flop phase account for a small change of magnetostriction with increasing field in TbFe3(BO3)4 and an increase of its value in PrFe3(BO3)4; a jump of quadrupole moment at the phase transition in TbFe3(BO3)4 is several times greater than that in PrFe3(BO3)4, this reflects the relation of magnetostriction values in these compounds. Both ions, Pr3+ and Tb3+, in ARTICLE IN PRESS digitized from Ref. [4]) and appropriate quadrupole moment of the Pr3+ ion in A.A. Demidov et al. / Physica E1 (including �180 cm�1, which results from the crystal-field parameters for the isostructural compound NdAl3(BO3)4 [11]). Crystal-field parameters for other RE ferroborates give rise to the splitting of two lower-lying singlets E1 for the Pr 3+ ion in PrFe3(BO3)4 of the order of 60–80 cm �1. So, we have considered three available sets of the crystal-field parameters for another light neighbouring in the RE series ion Nd3+ in NdFe3(BO3)4 [12,6,13] (cited also in [6]) and have calculated the dependences wc,?(T) for T4TN. All three sets of crystal-field parameters give rise to description of wc,?(T) of reasonable accuracy. The best agreement is achieved for the crystal-field parameters [13] and is much the same as in Ref. [4], the paramagnetic Neel temperature equals �123K. For the crystal-field Hamiltonian written in the irreducible tensor operators [14] they are (in cm�1): B20 ¼ 604; B40 ¼ �1203; B43 ¼ 701, B60 ¼ 466; B63 ¼ 135; B66 ¼ 416. (3) With these crystal-field parameters we have calculated the magnetization curves at BJc for several temperatures and from their comparison with the experimental ones we have chosen the remaining parameters of the compound, which give rise to the best possible agreement. We used this approach to deduce the parameters for description of magnetization curves in RFe3(BO3)4 with R ¼ Nd, Tb, Dy [1,3,6]. All the experimental material is considered simultaneously, and a sensitivity of a definite feature to a given parameter is analysed. For example, a slope of magneti- zation curve in the flop phase is determined primarily by the value of intrachain Fe–Fe exchange interaction, characterized by the exchange field Bdd, because bending of iron magnetic moments occurs against the intrachain exchange. We note that in case of strong magnetic ions Tb and Dy the influence of Fe anisotropy on the spin–flop transition field Bsf was found to be negligibly small [1,3]. In praseodymium compound, description of magnetization curves with spin–flop transitions without regard to anisotropy in Fe subsystem is impossible. Fig. 2. Magnetization curves of PrFe3(BO3)4. Symbols are experimental data digitized from Ref. [4]; lines are calculations. Experimental (digitized from Ref. [4]) and calculated magnetiza- tion curves of PrFe3(BO3)4 at BJc for several temperatures are displayed in Fig. 2. The parameters used in calculations are as follows: the crystal-field parameters (3), the intrachain Fe–Fe exchange field Bdd ¼ 43T, the f–d exchange field Bf�d ¼ 11T, the Fe anisotropy field BA Fe ¼ 0.07 T, the Fe–Fe exchange field, which includes the interchain interaction and is responsible for the value of Fe magnetic moment at a given T and B (see Ref. [1]), Bdd2 ¼ 25T. Temperature dependence of the Fe anisotropy con- stant was found from experimental dependence Bsf(T). Inset in Fig. 2 shows that this temperature dependence is steeper than the dependence according to Akulov–Zener’s law, which is usually not valid for antiferromagnets. It is seen in Fig. 2 that the agreement between experimental and calculated magnetization curves is reasonable. With all these parameters, we have calculated field depen- dences of multipole moments of the Pr3+ ion in PrFe3(BO3)4 for collinear and flop phases at T ¼ 4.2K. Analysis of numerical material and comparison with experimental data for longitudinal magnetostriction along the c-axis lc(B) (see Fig. 3) show that for interpretation of lc(B), the quadrupole approximation is sufficient, just as in the case of TbFe3(BO3)4. As for TbFe3(BO3)4, the coefficient A2 in Eq. (2) is determined from comparison of jumps of magnetostriction and quadrupole moments Q. It equals APr ¼ �3.7�10�3. APr differs from ATb only one and a half times which seems to be quite reasonable. We note that in NdFe3(BO3)4 the value of analogous coefficient can be found from the data of Refs. [7,6]. Depending on the approximation, it comprises from �1.7�10�3 to �3�10�3. We call attention to the fact that PrFe3(BO3)4. Fig. 3. Field dependences of longitudinal magnetostriction (experimental data B 404 (2009) 213–216 215 ferroborates of trigonal symmetry possess a singlet ground state. This state in Tb3+ is formed due to f-d coupling, which splits a quasi-doublet formed by the crystal field. The f–d exchange field in TbFe3(BO3)4 equals 3.8 T. Tb 3+ ion is an Ising ion in trigonal symmetry (gcE17.8, g?E0.2) [1]. Weak in magnetic aspect, the Pr3+ ion is much less anisotropic and possesses a singlet ground state due to splitting of the ground multiplet by the crystal field. For interpretation of magnetostriction in PrFe3(BO3)4, from its magnetic characteristics (temperature dependences of magnetic susceptibility and magnetization curves at B||c) we have found the parameters that are required for calculations of multipole moments of the Pr3+ ion. The magnetization curves along the trigonal axis are described reasonably with these parameters. Acknowledgement We thank A.A. Mukhin for his proposal to discuss magnetos- triction in the compounds under consideration. This study was supported, in part, by ISTC, Project 3501. References [1] E.A. Popova, D.V. Volkov, A.N. Vasiliev, A.A. Demidov, N.P. Kolmakova, et al., Phys. Rev. B 75 (2007) 224413. [2] E.A. Popova, N. Tristan, A.N. Vasiliev, et al., Eur. Phys. J. B 62 (2008) 123. [3] D.V. Volkov, A.A. Demidov, N.P. Kolmakova, JETP 106 (2008) 724. [4] A.M. Kadomtseva, Yu.F. Popov, G.P. Vorobev, et al., JETP Lett. 87 (2008) 39. [5] E.A. Popova, N. Tristan, C. Hess, et al., JETP 105 (2007) 105. [6] D.V. Volkov, A.A. Demidov, N.P. Kolmakova, JETP 104 (2007) 895. [7] A.K. Zvezdin, G.P. Vorob’ev, A.M. Kadomtseva, et al., JETP Lett. 83 (2006) 500. [8] A.A. Demidov, N.P. Kolmakova, L.V. Takunov, D.V. Volkov, Physica B 398 (2007) 78. [9] A.M. Kadomtseva, Yu.F. Popov, G.P. Vorob’ev, et al., in: Proceedings of the International Meeting Multiferroics, 2007, 188pp. (in Russian). [10] A.K. Zvezdin, V.M. Matveev, A.A. Mukhin, A.I. Popov, Rare-earth Ions in Magnetically-Ordered Crystals, Nauka, Moscow, 1985 (in Russian). [11] C. Cascales, C. Zaldo, U. Caldino, et al., J. Phys.: Condens. Matter 13 (2001) 8071. [12] M.N. Popova, E.P. Chukalina, T.N. Stanislavchuk, et al., Phys. Rev. B 75 (2007) 224435. [13] M.N. Popova, E. Antic-Fidancev, Private communication. [14] B.J. Wybourne, Spectroscopic Properties of Rare Earths, Wiley, USA, 1965. ARTICLE IN PRESS A.A. Demidov et al. / Physica B 404 (2009) 213–216216 Magnetostriction in the rare-earth ferroborates RFe3(BO3)4, RequalPr and Tb Introduction Theoretical background Magnetostriction in TbFe3(BO3)4 Magnetic characteristics and magnetostriction of PrFe3(BO3)4 Conclusions Acknowledgement References