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Magnetic susceptibility and heat capacity of Ho0.5Nd0.5Fe3(BO3)4
Author's Accepted Manuscript
Magnetic susceptibility and heat capacity of Ho0.5Nd0.5Fe3(BO3)4 A.A. Demidov
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PII: DOI: Reference:
S0921-4526(14)00046-5 http://dx.doi.org/10.1016/j.physb.2014.01.024 PHYSB308128
To appear in:
Physica B
Received date: 11 December 2013 Revised date: 13 January 2014 Accepted date: 17 January 2014 Cite this article as: A.A. Demidov, Magnetic susceptibility and heat capacity of Ho0.5Nd0.5Fe3(BO3)4, Physica B, http://dx.doi.org/10.1016/j.physb.2014.01.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
Magnetic susceptibility and heat capacity of Ho0.5Nd0.5Fe3(BO3)4 A.A. Demidov1*
Bryansk State Technical University, 241035 Bryansk, Russia
ABSTRACT The magnetic properties of trigonal Ho0.5Nd0.5Fe3(BO3)4 substituted compound with the competitive Ho-Fe and Nd-Fe exchange interactions have been investigated theoretically. The spontaneous spin-reorientation transition near 9 K has been described. The increase in temperature of the spin-reorientation transition in Ho0.5Nd0.5Fe3(BO3)4 to 9 K in comparison with 5 K in HoFe3(BO3)4 was explained. The calculations were performed using a molecular-field approximation and a crystal-field model for the rare-earth subsystem. A good agreement between the experimental and calculated temperature dependences of magnetic susceptibility and heat capacity has been achieved.
PACS: 75.80.+q; 75.30.-m; 75.50.Ee Keywords: Phase transitions; Rare-earth ferroborates RFe3(BO3)4; Multiferroics
1. Introduction
Ho0.25Nd0.75Fe3(BO3)4 compound belongs to the family of trigonal rare-earth ferroborates RFe3(BO3)4, which exhibits a variety of phase transitions and multiferroic features [1-7]. The iron subsystem in these compounds becomes antiferromagnetically ordered at Neel temperatures TN 30-40 K. The rare-earth subsystem is magnetized by the f–d interaction and makes a significant contribution to the magnetic anisotropy and orientation of the magnetic moments. The magnetic moments of iron in pure HoFe3(BO3)4 are antiferromagnetically ordered at TN 38-39 K and lie in the basal plane ab when the temperature decreases to TSR 4.7-5 K [810], as in the case of the magnetic moments of Ho3+ ions. A spontaneous spin-reorientation * Corresponding author. Tel.: +7 4832 588227; fax: +7 4832 562939. E-mail address: [email protected] (A.A. Demidov).
2 transition in HoFe3(BO3)4 takes place at TSR 4.7-5 K; as a result, the magnetic moments of the Fe- and Ho-subsystems become oriented along the trigonal c axis [8-11]. For NdFe3(BO3)4 at T < TN 31 K all the magnetic moments are lying in the basal plane [2]. Thus, spontaneous and magnetic field–induced spin-reorientation transitions from axis c to plane ab can result from the competition of the contributions of Ho3+ and Nd3+ ions to the magnetic anisotropy of substituted ferroborate Ho1-xNdxFe3(BO3)4. In Ho1-xNdxFe3(BO3)4 (x = 0.5 [9], 0.75 [12]) compounds were observed spin-reorientation transitions produced by competition of various contribution of the Ho-, Nd- and Fe-subsystems in magnetic anisotropy. The substitution of Ho3+ ions by Nd3+ ions in Ho1-xNdxFe3(BO3)4 should in theory push the spin-reorientation from easy-plane to easy-axis state as found in HoFe3(BO3)4 at TSR | 5 K to even lower temperatures. Surprisingly it was found that in Ho0.5Nd0.5Fe3(BO3)4 this spinreorientation shifts up in temperature to 9 K [9] and in Ho0.25Nd0.75Fe3(BO3)4 temperature TSR is the same as that of pure HoFe3(BO3)4 [12]. This shows that the simple picture of a competition between the axial and the planar magnetic anisotropy in substituted ferroborate is not sufficient to explain the details of the magnetic structures adopted. In this paper the theoretical investigation of the magnetic properties of Ho0.5Nd0.5Fe3(BO3)4 has been performed and a possible explanation for increase TSR in Ho0.5Nd0.5Fe3(BO3)4 in comparison with HoFe3(BO3)4 was suggested.
2. Theory
The magnetic properties of Ho1-xNdxFe3(BO3)4 crystals are determined both by the magnetic subsystems and by the interaction between them. The iron subsystem in this compound can be considered as consisting of two antiferromagnetic sublattices. The rare-earth subsystem (magnetized due to the f–d interaction) can also be represented as a superposition of two sublattices. In the calculations we used a theoretical approach which has been successfully applied for description of the magnetic properties of the RFe3(BO3)4 (see e.g. [3-5,11,13-15]). This approach is based on a crystal-field model for the rare-earth ion and on the molecular-field approximation. Effective Hamiltonians describing the interaction of each R/Fe ion in the ith (i = 1, 2) sublattice of the corresponding subsystem in the applied magnetic field B can be written as i R = iCF g JR B J iR ª¬ B Rfd M iFe º¼ ,
i Fe
Ho Nd Nd º g S B Si ª¬ B M Fej 1 x Ho 1, 2, fd mi x fd mi ¼ , j
(1) jzi,
(2)
3 where iCF is the crystal-field Hamiltonian, g JR is the Lande factor, J iR is the operator of the angular moment of the R ion, gS = 2 is the g-value, Si is the operator of the spin moment of the iron ion, Rfd 0 and 0 are the molecular constants of the antiferromagnetic interactions (R– Fe and Fe–Fe, respectively). The magnetic moments of the ith iron M iFe and rare-earth miR sublattices per formula unit are determined by the relationships MiFe
3g S B Si , miR
g JR B J iR .
(3)
The crystal-field Hamiltonian CF can be expressed using irreducible tensor operators C qk as
CF = B 02 C 02 + B 04 C 04 + B 34 (C 4 3 – C 34 ) + B 60 60 +
B 63 (C 6 3
–
63 )
+
B 66 (C 66
+
(4)
C 66 ).
For Ho3+ and Nd3+ ions in Ho1-xNdxFe3(BO3)4, the crystal-field parameters Bqk are unknown and data on the splitting of the ground-state multiplet are unavailable. The computation of the values and orientations of the magnetic moments of the Fe- and Rsubsystems in solving the self-consistent problems using Hamiltonians (1) and (2) at the minimum of the corresponding thermodynamic potential makes it possible to calculate the stability regions of various magnetic phases, the phase-transition fields, magnetization curves, the susceptibility, and so on. In terms of the standard thermodynamic perturbation theory described in monograph [16] for f–d compounds, the corresponding thermodynamic potential can be written as
1 2 ª ¦ 1 x kBTln Zi Ho 2 i 1¬ 1 Fe xk BTln Zi Nd 1 x g JHo B J iHo Ho fd Mi 2 1 Nd Nd Nd Fe x g J B J i fd Mi 3k BTln Zi Fe 2 1 º Ho Ho Nd Nd 3g S B Si M Fe Ô ian » , j 1 x fd mi x fd mi 2 ¼
Ô T , B
(5)
where Z i R / Fe are the partition functions calculated with Hamiltonians (1) and (2) and Ô ian is the anisotropy energy for the ith sublattice of the Fe-subsystem. For a crystal of trigonal symmetry (see. e.g., [17]), this energy is
Ô ian
K 2Fe sin 2 -i K 4Fe sin 4 -i K 66Fesin 6-i cos6Mi ,
(6)
4 where anisotropy constant K 2Fe 0 stabilizes the easy-plane antiferromagnetic state; constant K 4Fe ! 0 stabilizes the easy-axis state; K 66Fe 0 is the anisotropy constant in the basal ab plane;
and -i and Mi are the polar and azimuth angles of magnetic moment vector M iFe of iron, respectively. The magnetization and magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4 are defined as M
1 2 ¦ MiFe 1 x miHo x miNd , k 2i1
kFe 1 x kHo xkNd , k
a , b, c . (7)
In the ordered phase, the initial magnetic susceptibility of the compound under consideration can be determined using the initial linear portions of the magnetization curves calculated for the corresponding directions of the external magnetic field. In the paramagnetic phase, where the interactions between rare-earth and iron subsystems can be ignored, the magnetic susceptibility of the rare-earth subsystem can be determined using the well-known Van Vleck formula for the energy spectrum and wave functions calculated on the basis of the crystalfield Hamiltonian (4). For the Fe-subsystem, the susceptibility can be described in terms of the Curie–Weiss law (with the corresponding paramagnetic Neel temperature 4 ). The contribution of the rare-earth subsystem to the magnetic part of the heat capacity of Ho1-xNdxFe3(BO3)4 compounds is calculated according to the conventional quantum-mechanical formula (per rare-earth ion, i.e., per formula unit)
C
The heat averages E 2 and E
1 x CHo xCNd , CR 2
kB
E2 E
k BÒ 2
2
.
(8)
are calculated from the spectrum of the rare-earth ion, which
is formed by the crystal-field and interactions with the iron subsystem and the external magnetic field.
4. Results and discussion In order to determine the parameters of the crystal-field Bqk , we have used the experimental data on the temperature dependences of the initial magnetic susceptibility a ,c T and heat capacity Cp/T(T) of Ho0.5Nd0.5Fe3(BO3)4 [9]. The initial values of the crystal-field parameters, from which the procedure of minimization of the corresponding target function was started, were selected from the values available for ferroborates: HoFe3(BO3)4 [11], NdFe3(BO3)4 [13] and
5 Nd1–xDyxFe3(BO3)4 (x = 0.05, 0.1, 0.15, 0.25) [5, 14, 15]. The best agreement is achieved for the following set of the crystal-field parameters Bqk , in cm 1 :
B02
612 , B04
1500 , B34
844 , (9)
B06
276 , B36
50 , B66 193 .
These parameters were determined in the calculations based on the ground multiplet; therefore, they can only be treated as effective parameters suitable for describing the thermodynamic properties of the compound. The set of parameters (9) corresponds to the energies of the lower levels of the ground multiplet of the Ho3+ and Nd3+ ions in Ho0.5Nd0.5Fe3(BO3)4 that are given in Table 1 for B = 0. These energies are given for T > TN, with allowance for the f–d interaction at T = 10 K > TSR (easy-plane state) and T = 8 K < TSR (initial state, see below). It can be seen that the inclusion of the f–d interaction at T < TN results in the removal of the degeneracy of the lower levels. At TSR, the energy levels shift with respect to each other: in the case of Ho, the shift of the lower energy levels increases the splitting from fd 3.5 to 11.2 cm–1; in the case of Nd, this shift causes a small narrowing from fd 10.1 to 7.2 cm–1. The calculated magnetic characteristics presented below in the figures were calculated for the parameters given in Table 2, which also gives the parameters of HoFe3(BO3)4 [11] and NdFe3(BO3)4 [13] for comparison. The parameter 2 enters into the Brillouin function, is responsible for the magnitude of the magnetic moment of iron at a given temperature and a given magnetic field, and determines the Neel temperature, because a three-dimensional order in the ferroborate structure is impossible without an exchange interaction between chains of the Fe3+ ions. In the calculations, we also use the uniaxial anisotropy constants of iron ( K2Fe
1.95 Ò Â and K 4Fe
Fe the basal plane ( K66
1.83 Ò Â at T = 4.2 K) and the anisotropy constant of iron in
1.35 102 Òë Â [13]).
To calculate the magnetic characteristics of Ho0.5Nd0.5Fe3(BO3)4 when an external field is directed along or perpendicular to trigonal axis c, we used the schemes of orientation of the magnetic moments of iron M iFe and a rare-earth miR subsystems shown in Fig. 1. The calculations according to the schemes in Figs. 1b and 1c were performed for a field directed along the trigonal axis (B||c). The scheme in Fig. 1d was used for the case of a magnetic field oriented in the basal plane (B٣c), and the scheme in Fig. 1a is shown for the case of B = 0 (cone of easy magnetization axes (CEMA)). The scheme in Fig. 1d shows the projections of the Fe ) and rare-earth ( miab magnetic moments of the iron ( Miab
Ho Nd ) subsystems onto x miab 1 x miab
6 the ab plane in domains with antiferromagnetism axes making angles Mi = 0 (L0) and Mi = 60q (L60) with the a axis. Both the iron subsystem ordered at T < TN and the rare-earth subsystem magnetized by the f–d interaction contribute to the initial magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4. Figure 2 shows the experimental and calculated temperature dependences of magnetic susceptibility a ,c T . As the temperature decreases, the experimental curve c T continues to increase,
which is characteristic of the easy-plane state of the magnetic subsystem. At TSR 9 K c T decreases sharply, which can be explained by the presence of a spin-reorientation transition from the easy-plane state into the initial low-temperature state, the character and specific features of which in Ho0.5Nd0.5Fe3(BO3)4 are not obvious. It is remarkable that TSR of Ho0.5Nd0.5Fe3(BO3)4 is almost twice as large as that of HoFe3(BO3)4. Thus, the substitution of Ho3+ ions with Nd3+ ions results an increase of the stability range of the initial low-temperature state. In [14, 15], we assumed the presence of a low-temperature magnetic state with the formation of a weakly noncollinear antiferromagnetic phase having the magnetic moments of iron deviating from axis c in order to explain the interesting steplike anomalies in the magnetization curves Mc(B) and the susceptibility curves c T of substituted ferroborates Nd1–xDyxFe3(BO3)4 (x = 0.05, 0.1, 0.15, 0.25). As a result, we were able to achieve agreement between the calculated and experimental data for the entire set of measured characteristics of Nd1–xDyxFe3(BO3)4 [14, 15]. Note that, when studying GdFe3(BO3)4 undergoing a spinreorientation transition, the authors of [18] concluded that the magnetic moments of iron deviate from axis c through high angles, which change at various temperatures and magnetic fields. Our extensive calculations of the magnetic phases that appear in Ho0.5Nd0.5Fe3(BO3)4 at various orientations of the magnetic moments of the Ho-, Nd- and Fe-subsystems suggest the presence of a low-temperature state that differs from both the easy-plane and the easy-axis states. An antiferromagnetic phase with the magnetic moments of iron deviating from axis c through an angle 48° (at T = 2 K) appears; as a result, a cone of easy magnetization axes forms at B = 0 (see Fig. 1a). This possible state can be caused by the competition between the contributions of the iron and rare-earth subsystems to the total magnetic anisotropy of Ho0.5Nd0.5Fe3(BO3)4. The magnetic anisotropy of the iron and neodymium subsystems stabilizes the easy-plane magnetic structure. At TSR 4.7 K HoFe3(BO3)4 undergoes a spin-reorientation transition, which results in an easy-axis magnetic structure at T < TSR. In Ho0.5Nd0.5Fe3(BO3)4 the spin-reorientation transition temperature increases noticeably to TSR 9 K. Thus we assume that the contribution of the Ho subsystem to Ho0.5Nd0.5Fe3(BO3)4 stabilizes an easy-axis magnetic structure at least to
7 temperatures 9 K. As a result, at certain temperatures and fields, the magnetic moments of iron can be oriented at angle to axis c. The calculations demonstrate that the antiferromagnetic phase with the magnetic moments of iron deviating from axis c through an angle 48° at T = 2 K (see Fig. 1a) can be used to explain and to quantitatively describe the anomaly detected experimentally in the c (T ) curve near 9 K. The sharp decrease in c (T ) at TSR = 9 K is caused by the change from the easy-plane (Fig. 1c) to the weakly noncollinear antiferromagnetic state with the magnetic moments of iron deviating from axis c (Fig. 1b). This spin-reorientation transition is caused by the different temperature dependences of the competing contributions of the iron and rare-earth subsystems to the total magnetic anisotropy of Ho0.5Nd0.5Fe3(BO3)4. It is seen that the resulting magnetization in the initial phase at c = 0.02 (Fig. 1b), M CEMA
1 Ho Nd M 1Fecos -1 M 2Fecos -2 1 x m1,2 c xm1,2 c , 2
and in the easy-plane state at > SR (Fig. 1c)
M EP
1 Fe Ho Nd M1,2 c 1 x m1,2 c xm1,2 c , 2
well describes the experimental curve c T . Thus, an increase in temperature of the spin reorientation transition in Ho0.5Nd0.5Fe3(BO3)4 is caused by the expansion of the stability range of the initial low-temperature state due to its status changes from easy-axis to CEMA state. Our calculations show that the difference between M CEMA and M EP in a temperature TSR for c = 0.02 is mainly caused by the change in the contributions of the Ho-subsystem to the magnetization of Ho0.5Nd0.5Fe3(BO3)4. As can be seen from the calculated temperature dependence of the angle of rotation -1 of vector M1Fe from trigonal axis c for c = 0.02 (see inset to Fig. 3), vector M1Fe tends to rotate into plane ab normal to the field direction as the temperature grows. As the temperature TSR 9 K is approached, the rate of increase of angle -1 increases and changes jump wise to almost 90° at TSR. When Ho0.5Nd0.5Fe3(BO3)4 is magnetized in the basal plane ab in fields lower than about 1.5 T, all three possible domains with antiferromagnetism axes located at an angle of 120° to each other contribute to the magnetization (see Fig. 1d). The magnetic susceptibility curve a T was calculated using the approach proposed in [13], where the magnetization processes
8 occurring in easy-plane NdFe3(BO3)4 were comprehensively studied with allowance for the possible existence of three types of domains. The total magnetization for field a = 0.1 (see. Fig. 1d) Ma
1 ª1 2 Ho Nd Fe Ho Nd º M 0Fe 1 x m1,2 M 60 1 x m1,2 a xm1,2 a a xm1,2 a » « 2 ¬3 3 ¼
describes
M 0Fe
M 1Fesin -1 M 2Fesin -2 is the contribution of iron to the magnetization of domain L0ab
Fe M 60
allowance
experimental
for
the
curve
a T
well
with
the
(10)
(see
projection
M 1Fesin -1 cos M1 M 2Fesin -2 cos M 2
is
onto the
Fig.
2).
In
ab,
plane
contribution
Eq.
of
iron
(10)
and to
the
magnetization of domain L0ab with allowance for the projection onto plane ab and axis a. After spin-reorientation Fe M 60
transition
at
>
SR
M 0Fe
M1Fe M 2Fe
and
M1Fecos M1 M 2Fecos M 2 .
The performed calculation for Ho0.5Nd0.5Fe3(BO3)4 ferroborate at temperatures T > 20 K (see Fig. 3) has predicted the behavior of the weakly anisotropic magnetic susceptibility curves a ,c T within the experimentally unexplored temperature range from 20 to 250 K.
The experimental data on the heat capacity of Ho0.5Nd0.5Fe3(BO3)4 at B = 0 [9] are plotted in Fig. 4 in the coordinates Cp/T(T). A sharp peak near 8 K on Cp/T(T) is due to the spontaneous spin-reorientation transition from the CEMA to easy-plane state. The calculated contribution from the R-subsystem to the heat capacity of Ho0.5Nd0.5Fe3(BO3)4 and the components of this contribution from the Ho- and Nd-subsystems are shown also in Fig. 4. The contribution from the R-subsystem was calculated for the CEMA state up to TSR and in the easy-plane state for T > TSR. The calculations showed that weakly pronounced Schottky anomaly on experimental curve Cp/T(T) near T | 3.5 K is due to the contribution from the Nd-subsystem and is associated with the redistribution of the populations of the levels of the ground doublet of the Nd3+ ion split by the f–d interaction. The Schottky anomaly on calculation curve for Ho-subsystem near T | 6.5 K (thin dashed curve) becomes less pronounced on the overall theoretical curve due to allowance for the contribution from the neodymium part of the rare-earth subsystem as well. The calculations have demonstrated that, if the spin-reorientation transition were not to occur in Ho0.5Nd0.5Fe3(BO3)4 compound with decreasing temperature, then the Schottky anomaly in the easy-plane state would be observed near 1.5 K.
9 5. Conclusions
The magnetic properties of substituted Ho0.5Nd0.5Fe3(BO3)4 ferroborate with competing Ho–Fe and Nd–Fe exchange interactions were studied theoretically and agreement between the calculated and experimental data was achieved. The parameters of the trigonal crystal-field for the rare-earth subsystem and the parameters of the Fe–Fe and R–Fe exchange interactions are determined (see Table 2). The proposed version of the magnetization processes in Ho0.5Nd0.5Fe3(BO3)4 in low magnetic fields with the formation of an antiferromagnetic state with the magnetic moments of iron deviating from axis c through angle ( 48° at T = 2 K and B = 0) made it possible to comprehensively analyze the behavior of the magnetic moments of the Rand Fe-subsystems. The spontaneous spin-reorientation transition between the cone of easy magnetization axes and easy-plane states has been described and suggested a possible explanation for the increase TSR in Ho0.5Nd0.5Fe3(BO3)4 in comparison with HoFe3(BO3)4. A correct calculation of magnetization processes of Ho0.5Nd0.5Fe3(BO3)4 in weak fields provides a description of the magnetic susceptibility curves a ,c T in agreement with the experimental data. A good agreement between the experimental heat capacity Cp/T(T) and calculated contribution from the rare-earth subsystem to the magnetic part of the heat capacity has been achieved.
Acknowledgements
This work was supported by Russian Foundation for Basic Research (Project No. 12-0231007 mol_a). References
[1]
A.K. Zvezdin, A.M. Kadomtseva, S.S. Krotov, et al., J. Magn. Magn. Mater. 300 (2006) 224.
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A.M. Kadomtseva, Yu.F. Popov, G.P. Vorob’ev, et al., Low Temp. Phys. 36 (2010) 511.
[3]
D.V. Volkov, E.A. Popova, N.P. Kolmakova, A.A. Demidov, et al., J. Magn. Magn. Mater. 316 (2007) e717.
[4]
E.A. Popova, D.V. Volkov, A.N. Vasiliev, A.A. Demidov, et al., Phys. Rev. B 75 (2007) 224413.
[5]
A.A. Demidov, I.A. Gudim, E.V. Eremin, Physica B 407 (2012) 393-397.
[6]
T.N. Stanislavchuk, E.P. Chukalina, M.N. Popova, et al., Phys. Lett. A 368 (2007) 408.
[7]
A.S. Krylov, S. N. Sofronova, I.A. Gudim, et al., Sol. St. Comm. 174 (2013) 26–29.
[8]
C. Ritter, A. Vorotynov, A. Pankrats, et al., J. Phys.: Condens. Matter 20 (2008) 365209.
10 [9]
R.P. Chaudhury, F. Yen, B. Lorenz, et al., Phys. Rev. B 80 (2009) 104424.
[10] A. Pankrats, G. Petrakovskii, A. Kartashev, et al., J. Phys.: Condens. Matter 21 (2009) 436001. [11] A.A. Demidov, D.V. Volkov, Phys. of the Sol. St., 53 (2011) 985. [12] R. P. Chaudhury, B. Lorenz, Y. Y. Sun, et al. J. Appl. Phys. 107, 09D913 (2010). [13] D.V. Volkov, A.A. Demidov, N.P. Kolmakova, JETP 104 (2007) 895. [14] A.A. Demidov, I.A. Gudim, E.V. Eremin, JETP 114 (2012) 259. [15] A.I. Begunov, A.A. Demidov, I.A. Gudim, E.V. Eremin, JETP 117 (2013) 862. [16] A.K. Zvezdin, V.M. Matveev, A.A. Mukhin, A.I. Popov, Rare-Earth Ions in MagneticallyOrdered Crystals, Nauka, Moscow, 1985 (in Russian). [17] I.E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 (1957). [18] S.A. Kharlamova, S.G. Ovchinnikov, A.D. Balaev, et al., JETP 101 (2005) 1098. Figure captions Fig. 1. Schemes of orientations of the magnetic moments of the iron M iFe and rare-earth miR
sublattices used in the calculation of the magnetic characteristics of Ho0.5Nd0.5Fe3(BO3)4 for different temperature ranges and different directions of the external magnetic field: (a) B = 0 (cone of easy magnetization axes), (b, c) B||c (the ab plane is perpendicular to the figure plane), and (d) BAc (the axis c is perpendicular to the figure plane). Fig. 2. Calculated (lines) and experimental (symbols) [9] temperature dependencies of the initial
magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4 along ( c ) and perpendicular ( a ) to the trigonal axis.
Fig. 3. Calculated (lines) up to 250 K and experimental (symbols) [9] temperature dependencies
of the initial magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4 along ( c ) and perpendicular ( a ) to the trigonal axis at = 0.1 . The inset shows calculated temperature dependence of angle of deviation -1 of magnetic moment M1Fe from axis c for B||c.
Fig. 4. Heat capacity of Ho0.5Nd0.5Fe3(BO3)4 (symbols) [9] and calculated contribution (lines) of
the R-subsystem (with components) to heat capacity Ho0.5Nd0.5Fe3(BO3)4.
11 Table 1 Energies of the eight lower levels of the ground multiplets of the Ho3+ and Nd3+ ions in
Ho0.5Nd0.5Fe3(BO3)4 at B = 0 that are split by the crystal-field (parameters (9)) in the paramagnetic and ordered (with allowance for the f–d interaction) temperature ranges
R
> TN
' = Ei – E1, -1 (i =1-8) 0, 0, 7.4, 18.6, 18.6, 24.8, 142, 154
Ho 10 > TSR 0, 3.5, 16.7, 23.5, 30, 37.3, 145, 160 8 < TSR 0, 11.2, 18, 24.6, 34.2, 42, 149, 157 > TN
0, 0, 50.8, 50.8, 72.2, 72.2, 218, 218
Nd 10 > TSR 0, 10.1, 51.6, 59.8, 77.7, 77.8, 223, 223 8 < TSR
0, 7.2, 52.3, 58.4, 70.8, 82.3, 218, 227
12 Table 2 Parameters of Ho0.5Nd0.5Fe3(BO3)4 and for comparison also parameters of HoFe3(BO3)4
[11] and NdFe3(BO3)4 [13] (the intrachain Fe-Fe exchange field Bdd1, the interchain Fe-Fe exchange field Bdd2, and the f–d exchange field Bfd are the low-temperature exchange fields corresponding to the molecular constants 1, 2, and Rfd , respectively; fd is the lowtemperature splitting of the ground state of an R ion due to the f–d interaction in the following states: a cone of easy magnetization axes (CEMA), easy-axis (EA) state, and easyplane (EP) state; TSR is the spin-reorientation transition temperature; TN is the Neel temperature; is the paramagnetic Neel temperature for the iron subsystem; 0 = |Mi(T = 0, B = 0)| = 15 PB is the magnetic moment of iron per formula unit).
Bfd = Rfd M0,
HoFe3(BO3)4 [11] 84 –5.6 27 –1.8 2.5
Rfd , T/PB
–0.16
Bdd1 = O1M0, O1, T/PB O1, T/PB Bdd2 = O2M0, O2, T/PB
fd
a4.7-5 [8-10] -1 o 0 (T < TSR)
Ho0.5Nd0.5Fe3(BO3)4 60 –4 27 –1.8 3.75 (Ho) 7.5 (Nd) –0.25 (Ho) –0.5 (Nd) a 11.2 Ho (CEMA) a 3.5 (EP) a 7.2 Nd (CEMA) a 10.1 (EP) a9 [9] -1 o 48 (T < TSR)
-1 o 90 (T > TSR)
-1 o 90 (T > TSR)
a38-39 [8-10] –210
a 32 [9] –210
PB g | O fd | M 0 , cm-1 a 7.7 (EA) a 3.4 (EP)
TSR, -1 , q(B = 0) N,
,
NdFe3(BO3)4 [13] 58 –3.87 27 –1.8 7.1 –0.47 8.8 (EP)
90 a 31 –130
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Magnetic susceptibility and heat capacity of Ho0.5Nd0.5Fe3(BO3)4 A.A. Demidov
www.elsevier.com/locate/physb
PII: DOI: Reference:
S0921-4526(14)00046-5 http://dx.doi.org/10.1016/j.physb.2014.01.024 PHYSB308128
To appear in:
Physica B
Received date: 11 December 2013 Revised date: 13 January 2014 Accepted date: 17 January 2014 Cite this article as: A.A. Demidov, Magnetic susceptibility and heat capacity of Ho0.5Nd0.5Fe3(BO3)4, Physica B, http://dx.doi.org/10.1016/j.physb.2014.01.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
Magnetic susceptibility and heat capacity of Ho0.5Nd0.5Fe3(BO3)4 A.A. Demidov1*
Bryansk State Technical University, 241035 Bryansk, Russia
ABSTRACT The magnetic properties of trigonal Ho0.5Nd0.5Fe3(BO3)4 substituted compound with the competitive Ho-Fe and Nd-Fe exchange interactions have been investigated theoretically. The spontaneous spin-reorientation transition near 9 K has been described. The increase in temperature of the spin-reorientation transition in Ho0.5Nd0.5Fe3(BO3)4 to 9 K in comparison with 5 K in HoFe3(BO3)4 was explained. The calculations were performed using a molecular-field approximation and a crystal-field model for the rare-earth subsystem. A good agreement between the experimental and calculated temperature dependences of magnetic susceptibility and heat capacity has been achieved.
PACS: 75.80.+q; 75.30.-m; 75.50.Ee Keywords: Phase transitions; Rare-earth ferroborates RFe3(BO3)4; Multiferroics
1. Introduction
Ho0.25Nd0.75Fe3(BO3)4 compound belongs to the family of trigonal rare-earth ferroborates RFe3(BO3)4, which exhibits a variety of phase transitions and multiferroic features [1-7]. The iron subsystem in these compounds becomes antiferromagnetically ordered at Neel temperatures TN 30-40 K. The rare-earth subsystem is magnetized by the f–d interaction and makes a significant contribution to the magnetic anisotropy and orientation of the magnetic moments. The magnetic moments of iron in pure HoFe3(BO3)4 are antiferromagnetically ordered at TN 38-39 K and lie in the basal plane ab when the temperature decreases to TSR 4.7-5 K [810], as in the case of the magnetic moments of Ho3+ ions. A spontaneous spin-reorientation * Corresponding author. Tel.: +7 4832 588227; fax: +7 4832 562939. E-mail address: [email protected] (A.A. Demidov).
2 transition in HoFe3(BO3)4 takes place at TSR 4.7-5 K; as a result, the magnetic moments of the Fe- and Ho-subsystems become oriented along the trigonal c axis [8-11]. For NdFe3(BO3)4 at T < TN 31 K all the magnetic moments are lying in the basal plane [2]. Thus, spontaneous and magnetic field–induced spin-reorientation transitions from axis c to plane ab can result from the competition of the contributions of Ho3+ and Nd3+ ions to the magnetic anisotropy of substituted ferroborate Ho1-xNdxFe3(BO3)4. In Ho1-xNdxFe3(BO3)4 (x = 0.5 [9], 0.75 [12]) compounds were observed spin-reorientation transitions produced by competition of various contribution of the Ho-, Nd- and Fe-subsystems in magnetic anisotropy. The substitution of Ho3+ ions by Nd3+ ions in Ho1-xNdxFe3(BO3)4 should in theory push the spin-reorientation from easy-plane to easy-axis state as found in HoFe3(BO3)4 at TSR | 5 K to even lower temperatures. Surprisingly it was found that in Ho0.5Nd0.5Fe3(BO3)4 this spinreorientation shifts up in temperature to 9 K [9] and in Ho0.25Nd0.75Fe3(BO3)4 temperature TSR is the same as that of pure HoFe3(BO3)4 [12]. This shows that the simple picture of a competition between the axial and the planar magnetic anisotropy in substituted ferroborate is not sufficient to explain the details of the magnetic structures adopted. In this paper the theoretical investigation of the magnetic properties of Ho0.5Nd0.5Fe3(BO3)4 has been performed and a possible explanation for increase TSR in Ho0.5Nd0.5Fe3(BO3)4 in comparison with HoFe3(BO3)4 was suggested.
2. Theory
The magnetic properties of Ho1-xNdxFe3(BO3)4 crystals are determined both by the magnetic subsystems and by the interaction between them. The iron subsystem in this compound can be considered as consisting of two antiferromagnetic sublattices. The rare-earth subsystem (magnetized due to the f–d interaction) can also be represented as a superposition of two sublattices. In the calculations we used a theoretical approach which has been successfully applied for description of the magnetic properties of the RFe3(BO3)4 (see e.g. [3-5,11,13-15]). This approach is based on a crystal-field model for the rare-earth ion and on the molecular-field approximation. Effective Hamiltonians describing the interaction of each R/Fe ion in the ith (i = 1, 2) sublattice of the corresponding subsystem in the applied magnetic field B can be written as i R = iCF g JR B J iR ª¬ B Rfd M iFe º¼ ,
i Fe
Ho Nd Nd º g S B Si ª¬ B M Fej 1 x Ho 1, 2, fd mi x fd mi ¼ , j
(1) jzi,
(2)
3 where iCF is the crystal-field Hamiltonian, g JR is the Lande factor, J iR is the operator of the angular moment of the R ion, gS = 2 is the g-value, Si is the operator of the spin moment of the iron ion, Rfd 0 and 0 are the molecular constants of the antiferromagnetic interactions (R– Fe and Fe–Fe, respectively). The magnetic moments of the ith iron M iFe and rare-earth miR sublattices per formula unit are determined by the relationships MiFe
3g S B Si , miR
g JR B J iR .
(3)
The crystal-field Hamiltonian CF can be expressed using irreducible tensor operators C qk as
CF = B 02 C 02 + B 04 C 04 + B 34 (C 4 3 – C 34 ) + B 60 60 +
B 63 (C 6 3
–
63 )
+
B 66 (C 66
+
(4)
C 66 ).
For Ho3+ and Nd3+ ions in Ho1-xNdxFe3(BO3)4, the crystal-field parameters Bqk are unknown and data on the splitting of the ground-state multiplet are unavailable. The computation of the values and orientations of the magnetic moments of the Fe- and Rsubsystems in solving the self-consistent problems using Hamiltonians (1) and (2) at the minimum of the corresponding thermodynamic potential makes it possible to calculate the stability regions of various magnetic phases, the phase-transition fields, magnetization curves, the susceptibility, and so on. In terms of the standard thermodynamic perturbation theory described in monograph [16] for f–d compounds, the corresponding thermodynamic potential can be written as
1 2 ª ¦ 1 x kBTln Zi Ho 2 i 1¬ 1 Fe xk BTln Zi Nd 1 x g JHo B J iHo Ho fd Mi 2 1 Nd Nd Nd Fe x g J B J i fd Mi 3k BTln Zi Fe 2 1 º Ho Ho Nd Nd 3g S B Si M Fe Ô ian » , j 1 x fd mi x fd mi 2 ¼
Ô T , B
(5)
where Z i R / Fe are the partition functions calculated with Hamiltonians (1) and (2) and Ô ian is the anisotropy energy for the ith sublattice of the Fe-subsystem. For a crystal of trigonal symmetry (see. e.g., [17]), this energy is
Ô ian
K 2Fe sin 2 -i K 4Fe sin 4 -i K 66Fesin 6-i cos6Mi ,
(6)
4 where anisotropy constant K 2Fe 0 stabilizes the easy-plane antiferromagnetic state; constant K 4Fe ! 0 stabilizes the easy-axis state; K 66Fe 0 is the anisotropy constant in the basal ab plane;
and -i and Mi are the polar and azimuth angles of magnetic moment vector M iFe of iron, respectively. The magnetization and magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4 are defined as M
1 2 ¦ MiFe 1 x miHo x miNd , k 2i1
kFe 1 x kHo xkNd , k
a , b, c . (7)
In the ordered phase, the initial magnetic susceptibility of the compound under consideration can be determined using the initial linear portions of the magnetization curves calculated for the corresponding directions of the external magnetic field. In the paramagnetic phase, where the interactions between rare-earth and iron subsystems can be ignored, the magnetic susceptibility of the rare-earth subsystem can be determined using the well-known Van Vleck formula for the energy spectrum and wave functions calculated on the basis of the crystalfield Hamiltonian (4). For the Fe-subsystem, the susceptibility can be described in terms of the Curie–Weiss law (with the corresponding paramagnetic Neel temperature 4 ). The contribution of the rare-earth subsystem to the magnetic part of the heat capacity of Ho1-xNdxFe3(BO3)4 compounds is calculated according to the conventional quantum-mechanical formula (per rare-earth ion, i.e., per formula unit)
C
The heat averages E 2 and E
1 x CHo xCNd , CR 2
kB
E2 E
k BÒ 2
2
.
(8)
are calculated from the spectrum of the rare-earth ion, which
is formed by the crystal-field and interactions with the iron subsystem and the external magnetic field.
4. Results and discussion In order to determine the parameters of the crystal-field Bqk , we have used the experimental data on the temperature dependences of the initial magnetic susceptibility a ,c T and heat capacity Cp/T(T) of Ho0.5Nd0.5Fe3(BO3)4 [9]. The initial values of the crystal-field parameters, from which the procedure of minimization of the corresponding target function was started, were selected from the values available for ferroborates: HoFe3(BO3)4 [11], NdFe3(BO3)4 [13] and
5 Nd1–xDyxFe3(BO3)4 (x = 0.05, 0.1, 0.15, 0.25) [5, 14, 15]. The best agreement is achieved for the following set of the crystal-field parameters Bqk , in cm 1 :
B02
612 , B04
1500 , B34
844 , (9)
B06
276 , B36
50 , B66 193 .
These parameters were determined in the calculations based on the ground multiplet; therefore, they can only be treated as effective parameters suitable for describing the thermodynamic properties of the compound. The set of parameters (9) corresponds to the energies of the lower levels of the ground multiplet of the Ho3+ and Nd3+ ions in Ho0.5Nd0.5Fe3(BO3)4 that are given in Table 1 for B = 0. These energies are given for T > TN, with allowance for the f–d interaction at T = 10 K > TSR (easy-plane state) and T = 8 K < TSR (initial state, see below). It can be seen that the inclusion of the f–d interaction at T < TN results in the removal of the degeneracy of the lower levels. At TSR, the energy levels shift with respect to each other: in the case of Ho, the shift of the lower energy levels increases the splitting from fd 3.5 to 11.2 cm–1; in the case of Nd, this shift causes a small narrowing from fd 10.1 to 7.2 cm–1. The calculated magnetic characteristics presented below in the figures were calculated for the parameters given in Table 2, which also gives the parameters of HoFe3(BO3)4 [11] and NdFe3(BO3)4 [13] for comparison. The parameter 2 enters into the Brillouin function, is responsible for the magnitude of the magnetic moment of iron at a given temperature and a given magnetic field, and determines the Neel temperature, because a three-dimensional order in the ferroborate structure is impossible without an exchange interaction between chains of the Fe3+ ions. In the calculations, we also use the uniaxial anisotropy constants of iron ( K2Fe
1.95 Ò Â and K 4Fe
Fe the basal plane ( K66
1.83 Ò Â at T = 4.2 K) and the anisotropy constant of iron in
1.35 102 Òë Â [13]).
To calculate the magnetic characteristics of Ho0.5Nd0.5Fe3(BO3)4 when an external field is directed along or perpendicular to trigonal axis c, we used the schemes of orientation of the magnetic moments of iron M iFe and a rare-earth miR subsystems shown in Fig. 1. The calculations according to the schemes in Figs. 1b and 1c were performed for a field directed along the trigonal axis (B||c). The scheme in Fig. 1d was used for the case of a magnetic field oriented in the basal plane (B٣c), and the scheme in Fig. 1a is shown for the case of B = 0 (cone of easy magnetization axes (CEMA)). The scheme in Fig. 1d shows the projections of the Fe ) and rare-earth ( miab magnetic moments of the iron ( Miab
Ho Nd ) subsystems onto x miab 1 x miab
6 the ab plane in domains with antiferromagnetism axes making angles Mi = 0 (L0) and Mi = 60q (L60) with the a axis. Both the iron subsystem ordered at T < TN and the rare-earth subsystem magnetized by the f–d interaction contribute to the initial magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4. Figure 2 shows the experimental and calculated temperature dependences of magnetic susceptibility a ,c T . As the temperature decreases, the experimental curve c T continues to increase,
which is characteristic of the easy-plane state of the magnetic subsystem. At TSR 9 K c T decreases sharply, which can be explained by the presence of a spin-reorientation transition from the easy-plane state into the initial low-temperature state, the character and specific features of which in Ho0.5Nd0.5Fe3(BO3)4 are not obvious. It is remarkable that TSR of Ho0.5Nd0.5Fe3(BO3)4 is almost twice as large as that of HoFe3(BO3)4. Thus, the substitution of Ho3+ ions with Nd3+ ions results an increase of the stability range of the initial low-temperature state. In [14, 15], we assumed the presence of a low-temperature magnetic state with the formation of a weakly noncollinear antiferromagnetic phase having the magnetic moments of iron deviating from axis c in order to explain the interesting steplike anomalies in the magnetization curves Mc(B) and the susceptibility curves c T of substituted ferroborates Nd1–xDyxFe3(BO3)4 (x = 0.05, 0.1, 0.15, 0.25). As a result, we were able to achieve agreement between the calculated and experimental data for the entire set of measured characteristics of Nd1–xDyxFe3(BO3)4 [14, 15]. Note that, when studying GdFe3(BO3)4 undergoing a spinreorientation transition, the authors of [18] concluded that the magnetic moments of iron deviate from axis c through high angles, which change at various temperatures and magnetic fields. Our extensive calculations of the magnetic phases that appear in Ho0.5Nd0.5Fe3(BO3)4 at various orientations of the magnetic moments of the Ho-, Nd- and Fe-subsystems suggest the presence of a low-temperature state that differs from both the easy-plane and the easy-axis states. An antiferromagnetic phase with the magnetic moments of iron deviating from axis c through an angle 48° (at T = 2 K) appears; as a result, a cone of easy magnetization axes forms at B = 0 (see Fig. 1a). This possible state can be caused by the competition between the contributions of the iron and rare-earth subsystems to the total magnetic anisotropy of Ho0.5Nd0.5Fe3(BO3)4. The magnetic anisotropy of the iron and neodymium subsystems stabilizes the easy-plane magnetic structure. At TSR 4.7 K HoFe3(BO3)4 undergoes a spin-reorientation transition, which results in an easy-axis magnetic structure at T < TSR. In Ho0.5Nd0.5Fe3(BO3)4 the spin-reorientation transition temperature increases noticeably to TSR 9 K. Thus we assume that the contribution of the Ho subsystem to Ho0.5Nd0.5Fe3(BO3)4 stabilizes an easy-axis magnetic structure at least to
7 temperatures 9 K. As a result, at certain temperatures and fields, the magnetic moments of iron can be oriented at angle to axis c. The calculations demonstrate that the antiferromagnetic phase with the magnetic moments of iron deviating from axis c through an angle 48° at T = 2 K (see Fig. 1a) can be used to explain and to quantitatively describe the anomaly detected experimentally in the c (T ) curve near 9 K. The sharp decrease in c (T ) at TSR = 9 K is caused by the change from the easy-plane (Fig. 1c) to the weakly noncollinear antiferromagnetic state with the magnetic moments of iron deviating from axis c (Fig. 1b). This spin-reorientation transition is caused by the different temperature dependences of the competing contributions of the iron and rare-earth subsystems to the total magnetic anisotropy of Ho0.5Nd0.5Fe3(BO3)4. It is seen that the resulting magnetization in the initial phase at c = 0.02 (Fig. 1b), M CEMA
1 Ho Nd M 1Fecos -1 M 2Fecos -2 1 x m1,2 c xm1,2 c , 2
and in the easy-plane state at > SR (Fig. 1c)
M EP
1 Fe Ho Nd M1,2 c 1 x m1,2 c xm1,2 c , 2
well describes the experimental curve c T . Thus, an increase in temperature of the spin reorientation transition in Ho0.5Nd0.5Fe3(BO3)4 is caused by the expansion of the stability range of the initial low-temperature state due to its status changes from easy-axis to CEMA state. Our calculations show that the difference between M CEMA and M EP in a temperature TSR for c = 0.02 is mainly caused by the change in the contributions of the Ho-subsystem to the magnetization of Ho0.5Nd0.5Fe3(BO3)4. As can be seen from the calculated temperature dependence of the angle of rotation -1 of vector M1Fe from trigonal axis c for c = 0.02 (see inset to Fig. 3), vector M1Fe tends to rotate into plane ab normal to the field direction as the temperature grows. As the temperature TSR 9 K is approached, the rate of increase of angle -1 increases and changes jump wise to almost 90° at TSR. When Ho0.5Nd0.5Fe3(BO3)4 is magnetized in the basal plane ab in fields lower than about 1.5 T, all three possible domains with antiferromagnetism axes located at an angle of 120° to each other contribute to the magnetization (see Fig. 1d). The magnetic susceptibility curve a T was calculated using the approach proposed in [13], where the magnetization processes
8 occurring in easy-plane NdFe3(BO3)4 were comprehensively studied with allowance for the possible existence of three types of domains. The total magnetization for field a = 0.1 (see. Fig. 1d) Ma
1 ª1 2 Ho Nd Fe Ho Nd º M 0Fe 1 x m1,2 M 60 1 x m1,2 a xm1,2 a a xm1,2 a » « 2 ¬3 3 ¼
describes
M 0Fe
M 1Fesin -1 M 2Fesin -2 is the contribution of iron to the magnetization of domain L0ab
Fe M 60
allowance
experimental
for
the
curve
a T
well
with
the
(10)
(see
projection
M 1Fesin -1 cos M1 M 2Fesin -2 cos M 2
is
onto the
Fig.
2).
In
ab,
plane
contribution
Eq.
of
iron
(10)
and to
the
magnetization of domain L0ab with allowance for the projection onto plane ab and axis a. After spin-reorientation Fe M 60
transition
at
>
SR
M 0Fe
M1Fe M 2Fe
and
M1Fecos M1 M 2Fecos M 2 .
The performed calculation for Ho0.5Nd0.5Fe3(BO3)4 ferroborate at temperatures T > 20 K (see Fig. 3) has predicted the behavior of the weakly anisotropic magnetic susceptibility curves a ,c T within the experimentally unexplored temperature range from 20 to 250 K.
The experimental data on the heat capacity of Ho0.5Nd0.5Fe3(BO3)4 at B = 0 [9] are plotted in Fig. 4 in the coordinates Cp/T(T). A sharp peak near 8 K on Cp/T(T) is due to the spontaneous spin-reorientation transition from the CEMA to easy-plane state. The calculated contribution from the R-subsystem to the heat capacity of Ho0.5Nd0.5Fe3(BO3)4 and the components of this contribution from the Ho- and Nd-subsystems are shown also in Fig. 4. The contribution from the R-subsystem was calculated for the CEMA state up to TSR and in the easy-plane state for T > TSR. The calculations showed that weakly pronounced Schottky anomaly on experimental curve Cp/T(T) near T | 3.5 K is due to the contribution from the Nd-subsystem and is associated with the redistribution of the populations of the levels of the ground doublet of the Nd3+ ion split by the f–d interaction. The Schottky anomaly on calculation curve for Ho-subsystem near T | 6.5 K (thin dashed curve) becomes less pronounced on the overall theoretical curve due to allowance for the contribution from the neodymium part of the rare-earth subsystem as well. The calculations have demonstrated that, if the spin-reorientation transition were not to occur in Ho0.5Nd0.5Fe3(BO3)4 compound with decreasing temperature, then the Schottky anomaly in the easy-plane state would be observed near 1.5 K.
9 5. Conclusions
The magnetic properties of substituted Ho0.5Nd0.5Fe3(BO3)4 ferroborate with competing Ho–Fe and Nd–Fe exchange interactions were studied theoretically and agreement between the calculated and experimental data was achieved. The parameters of the trigonal crystal-field for the rare-earth subsystem and the parameters of the Fe–Fe and R–Fe exchange interactions are determined (see Table 2). The proposed version of the magnetization processes in Ho0.5Nd0.5Fe3(BO3)4 in low magnetic fields with the formation of an antiferromagnetic state with the magnetic moments of iron deviating from axis c through angle ( 48° at T = 2 K and B = 0) made it possible to comprehensively analyze the behavior of the magnetic moments of the Rand Fe-subsystems. The spontaneous spin-reorientation transition between the cone of easy magnetization axes and easy-plane states has been described and suggested a possible explanation for the increase TSR in Ho0.5Nd0.5Fe3(BO3)4 in comparison with HoFe3(BO3)4. A correct calculation of magnetization processes of Ho0.5Nd0.5Fe3(BO3)4 in weak fields provides a description of the magnetic susceptibility curves a ,c T in agreement with the experimental data. A good agreement between the experimental heat capacity Cp/T(T) and calculated contribution from the rare-earth subsystem to the magnetic part of the heat capacity has been achieved.
Acknowledgements
This work was supported by Russian Foundation for Basic Research (Project No. 12-0231007 mol_a). References
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E.A. Popova, D.V. Volkov, A.N. Vasiliev, A.A. Demidov, et al., Phys. Rev. B 75 (2007) 224413.
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A.A. Demidov, I.A. Gudim, E.V. Eremin, Physica B 407 (2012) 393-397.
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T.N. Stanislavchuk, E.P. Chukalina, M.N. Popova, et al., Phys. Lett. A 368 (2007) 408.
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A.S. Krylov, S. N. Sofronova, I.A. Gudim, et al., Sol. St. Comm. 174 (2013) 26–29.
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C. Ritter, A. Vorotynov, A. Pankrats, et al., J. Phys.: Condens. Matter 20 (2008) 365209.
10 [9]
R.P. Chaudhury, F. Yen, B. Lorenz, et al., Phys. Rev. B 80 (2009) 104424.
[10] A. Pankrats, G. Petrakovskii, A. Kartashev, et al., J. Phys.: Condens. Matter 21 (2009) 436001. [11] A.A. Demidov, D.V. Volkov, Phys. of the Sol. St., 53 (2011) 985. [12] R. P. Chaudhury, B. Lorenz, Y. Y. Sun, et al. J. Appl. Phys. 107, 09D913 (2010). [13] D.V. Volkov, A.A. Demidov, N.P. Kolmakova, JETP 104 (2007) 895. [14] A.A. Demidov, I.A. Gudim, E.V. Eremin, JETP 114 (2012) 259. [15] A.I. Begunov, A.A. Demidov, I.A. Gudim, E.V. Eremin, JETP 117 (2013) 862. [16] A.K. Zvezdin, V.M. Matveev, A.A. Mukhin, A.I. Popov, Rare-Earth Ions in MagneticallyOrdered Crystals, Nauka, Moscow, 1985 (in Russian). [17] I.E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 (1957). [18] S.A. Kharlamova, S.G. Ovchinnikov, A.D. Balaev, et al., JETP 101 (2005) 1098. Figure captions Fig. 1. Schemes of orientations of the magnetic moments of the iron M iFe and rare-earth miR
sublattices used in the calculation of the magnetic characteristics of Ho0.5Nd0.5Fe3(BO3)4 for different temperature ranges and different directions of the external magnetic field: (a) B = 0 (cone of easy magnetization axes), (b, c) B||c (the ab plane is perpendicular to the figure plane), and (d) BAc (the axis c is perpendicular to the figure plane). Fig. 2. Calculated (lines) and experimental (symbols) [9] temperature dependencies of the initial
magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4 along ( c ) and perpendicular ( a ) to the trigonal axis.
Fig. 3. Calculated (lines) up to 250 K and experimental (symbols) [9] temperature dependencies
of the initial magnetic susceptibility of Ho0.5Nd0.5Fe3(BO3)4 along ( c ) and perpendicular ( a ) to the trigonal axis at = 0.1 . The inset shows calculated temperature dependence of angle of deviation -1 of magnetic moment M1Fe from axis c for B||c.
Fig. 4. Heat capacity of Ho0.5Nd0.5Fe3(BO3)4 (symbols) [9] and calculated contribution (lines) of
the R-subsystem (with components) to heat capacity Ho0.5Nd0.5Fe3(BO3)4.
11 Table 1 Energies of the eight lower levels of the ground multiplets of the Ho3+ and Nd3+ ions in
Ho0.5Nd0.5Fe3(BO3)4 at B = 0 that are split by the crystal-field (parameters (9)) in the paramagnetic and ordered (with allowance for the f–d interaction) temperature ranges
R
> TN
' = Ei – E1, -1 (i =1-8) 0, 0, 7.4, 18.6, 18.6, 24.8, 142, 154
Ho 10 > TSR 0, 3.5, 16.7, 23.5, 30, 37.3, 145, 160 8 < TSR 0, 11.2, 18, 24.6, 34.2, 42, 149, 157 > TN
0, 0, 50.8, 50.8, 72.2, 72.2, 218, 218
Nd 10 > TSR 0, 10.1, 51.6, 59.8, 77.7, 77.8, 223, 223 8 < TSR
0, 7.2, 52.3, 58.4, 70.8, 82.3, 218, 227
12 Table 2 Parameters of Ho0.5Nd0.5Fe3(BO3)4 and for comparison also parameters of HoFe3(BO3)4
[11] and NdFe3(BO3)4 [13] (the intrachain Fe-Fe exchange field Bdd1, the interchain Fe-Fe exchange field Bdd2, and the f–d exchange field Bfd are the low-temperature exchange fields corresponding to the molecular constants 1, 2, and Rfd , respectively; fd is the lowtemperature splitting of the ground state of an R ion due to the f–d interaction in the following states: a cone of easy magnetization axes (CEMA), easy-axis (EA) state, and easyplane (EP) state; TSR is the spin-reorientation transition temperature; TN is the Neel temperature; is the paramagnetic Neel temperature for the iron subsystem; 0 = |Mi(T = 0, B = 0)| = 15 PB is the magnetic moment of iron per formula unit).
Bfd = Rfd M0,
HoFe3(BO3)4 [11] 84 –5.6 27 –1.8 2.5
Rfd , T/PB
–0.16
Bdd1 = O1M0, O1, T/PB O1, T/PB Bdd2 = O2M0, O2, T/PB
fd
a4.7-5 [8-10] -1 o 0 (T < TSR)
Ho0.5Nd0.5Fe3(BO3)4 60 –4 27 –1.8 3.75 (Ho) 7.5 (Nd) –0.25 (Ho) –0.5 (Nd) a 11.2 Ho (CEMA) a 3.5 (EP) a 7.2 Nd (CEMA) a 10.1 (EP) a9 [9] -1 o 48 (T < TSR)
-1 o 90 (T > TSR)
-1 o 90 (T > TSR)
a38-39 [8-10] –210
a 32 [9] –210
PB g | O fd | M 0 , cm-1 a 7.7 (EA) a 3.4 (EP)
TSR, -1 , q(B = 0) N,
,
NdFe3(BO3)4 [13] 58 –3.87 27 –1.8 7.1 –0.47 8.8 (EP)
90 a 31 –130
Fig. 1
Fig. 2
Fig. 3
Fig. 4