Magnetic evolution of spinel Mn_1−xZn_xCr₂O₄ single crystals

Abstract

Mn_1−xZn_xCr₂O₄ (0 ≤ x ≤ 1) single crystals have been grown using the chemical vapor transport (CVT) method. The crystallographic, magnetic, and thermal transport properties of the single crystals were investigated by room-temperature X-ray diffraction, magnetization M(T) and specific heat C_P(T) measurements. Mn_1−xZn_xCr₂O₄ crystals show a cubic structure, the lattice constant a decreases with the increasing content x of the doped Zn²⁺ ions and follows the Vegard law. Based on the magnetization and heat capacity measurements, the magnetic evolution of Mn_1−xZn_xCr₂O₄ crystals has been discussed. For 0 ≤ x ≤ 0.3, the magnetic ground state is the coexistence of the long-range ferrimagnetic order (LFIM) and the spiral ferrimagnetic one (SFIM), which is similar to that of the parent MnCr₂O₄. When x changes from 0.3 to 0.8, the SFIM is progressively suppressed and spin glass-like behavior is observed. When x is above 0.8, an antiferromagnetic (AFM) order presents. At the same time, the magnetic specific heat (C_mag.) was also investigated and the results are coincident with the magnetic measurements. The possible reasons based on the disorder effect and the reduced molecular field effect induced by the substitution of Mn²⁺ ions by nonmagnetic Zn²⁺ ones in Mn_1−xZn_xCr₂O₄ crystals have been discussed.

---

1 Introduction

The chalcogenide spinel compounds, named as ACr₂X₄ (A = 3d transition metals, Cd and Hg, X = O, S, Se), have attracted special interest in the past ten years because a variety of important physical effects have been found in these compounds, such as colossal magnetocapacitance (MC), multiferroicity, spin frustration and so on.^1–9 Because of the strong coupling among charge, spin and lattice degrees of freedom, ACr₂X₄ presents not only an interesting phenomena but also complicated magnetic structures.

Among ACr₂X₄ compounds, cubic spinel ACr₂O₄ oxides have special characters. The unfilled 3d³ shells of Cr³⁺ ions form isotropic S = 3/2 degree of freedom on a lattice of corner-sharing tetrahedron. When the tetrahedral A-site (A = Mg, Zn, Cd and Hg) is occupied by non-magnetic ions, the main magnetic interaction is the strong J_CrCr antiferromagnetic (AFM) direct exchange between the nearest-neighbor ions.^5–7 And these compounds show strongly geometrical frustration.^4,8,10–14 On the other hand, when the tetrahedral A-site is occupied by a magnetic ion, such as A = Mn, Co, Fe, and Ni, it helps to overcome the frustration of the pyrochlore lattice by the J_ACr coupling between the A site and the Cr³⁺ S = 3/2 spins.^5,15 For MnCr₂O₄, previous studies have calculated or evaluated from experimental data, J_MnCr, J_CrCr and J_MnMn and obtained that J_CrCr is larger than J_MnCr or approximately equal to J_MnCr.^7,16,18,24 In this case, the system presents nearly degenerated ground states and it develops complex low temperature magnetic order. Among the spinel ACr₂O₄ compounds, MnCr₂O₄ and ZnCr₂O₄ are two typical ones. In MnCr₂O₄, the long-range ferrimagnetic (LFIM) temperature T_C is observed around 41–51 K, which are dependent on the polycrystalline samples and single crystals. In addition, the sample exists a characteristic temperature T_S. As T < T_S, the long-range FIM and the short-range spiral FIM (SFIM) coexists. Between T_C and T_S, the long-range FIM with an easy axis parallel to the 〈110〉 direction occurs.^7,9,17–19 Very recently, the multiferroicity has also been reported in MnCr₂O₄ below the T_S.⁹ However, ZnCr₂O₄ shows strikingly different characters, such as strongly geometrical frustration (the frustration factor f ≈ 31) and high Curie–Weiss temperature θ = 390–400 K. The AFM with the spin-Jahn–Teller distortion, which favors a relief of the geometrical frustration, appears around T_N = 12 K with the character of a first-order phase transition.^8,20,21 From above reported works, it seems to mean that the molecular field of the A sites can be effectively tuned and has the important effect on the ground state of the spinel oxides. Because the emergent phenomena present in spinel MnCr₂O₄ and ZnCr₂O₄ compounds, the magnetic evaluation of Mn_1−xZn_xCr₂O₄ oxides are really deserved to be investigated. Although few work has been done on the Zn²⁺ ions doped MnCr₂O₄ compounds, the comprehensive study is still missing and the evolution of magnetic ground state is not very clear.^22,23 In order to further understand magnetic evolution of the ground state, herein, we investigate the effect of non-magnetic Zn²⁺ ions doping at the magnetic A sites of Mn_1−xZn_xCr₂O₄ single crystals. The magnetic phase diagram of Mn_1−xZn_xCr₂O₄ single crystals is obtained. We also discussed the magnetic evolution based on the disorder effect and the reduced molecular field one induced by the substitution of Mn²⁺ ions by nonmagnetic Zn²⁺ ones in Mn_1−xZn_xCr₂O₄ crystals.

2 Experimental results

Mn_1−xZn_xCr₂O₄ single crystals were grown by the chemical vapor transport (CVT) method, with CrCl₃ powders as the transport agent. Firstly, polycrystalline Mn_1−xZn_xCr₂O₄ were made by the solid state reaction. The stoichiometric amounts of Cr₂O₃ (99%, Alfa Aesar), ZnO (99.9%, Alfa Aesar) and MnO (99%, Alfa Aesar) powders were mixed in air and sintered at 1300 °C for 20 h for several times. Powder X-ray diffraction (XRD) at room temperature revealed a single-phase without any detected impurities. Secondly, polycrystalline Mn_1−xZn_xCr₂O₄ mixed with the CrCl₃ powders were ground again and sealed into several quartz tubes. All were done in the Ar-filled glove-box. All sealed quartz tubes were put in a two-zone tube furnace. The hot side is about 1060 °C and the cold side is about 1015 °C, and dwelled for 10 days, then slowly cooled down to room temperature with a rate of 15 °C h⁻¹. The crystals are octahedral with shining surfaces and the size is about 1.2 × 1.2 × 1.2 mm³. Heat capacity was measured using the Quantum Design physical properties measurement system (PPMS-9T) and magnetic properties were performed by the magnetic property measurement system (MPMS-XL5).

3 Results and discussion

Fig. 1(a) shows the XRD pattern of Mn_0.8Zn_0.2Cr₂O₄ powder obtained by crushing the single crystals, which is selected as a typical sample. It also presents the structural Rietveld refinement profiles of the XRD data by the Highscore software. The refinements of the XRD data indicate that the crystals are single-phase since no extra peaks were observed. It is found that the crystals belong to the normal spinel structure and the space group is Fd3̄m (space group no. is 227). X-ray diffraction patterns of Mn_1−xZn_xCr₂O₄ powder are presented in Fig. 1(b). It also presents a single-phase for all samples. The crystal structure of these samples does not change with the Zn doping content x at room temperature, the lattice parameter a decreases from 0.844 nm for x = 0 to 0.833 nm for x = 1.0, which confirms that the partial Mn²⁺ sites are occupied by the Zn²⁺ ions. The reduced lattice parameter is related to the fact that the Zn²⁺ ion radius (0.06 nm) is smaller than that of Mn²⁺ one (0.066 nm). The lattice constant a for all samples follows the Vegard law. On the other hand, these Bragg peaks of XRD pattern move towards higher angles with increasing x, which is consistent with the refined lattice parameter decreases with increasing x. Fig. 1(d) shows the enlargements of XRD pattern near the peak (511), which is selected as a typical sample.

Fig. 1 (a) The refined powder XRD patterns at room temperature for the Mn_0.8Zn_0.2Cr₂O₄ powders crushed by single crystals. The black dots are the experimental data and the red line is the fitting result. The solid line (green line) at the bottom corresponds to the difference between experimental and calculated intensities. The blue bars are the Bragg positions. The inset shows the typical crystal of Mn_0.8Zn_0.2Cr₂O₄, the scale bar is 1 mm; (b) powder XRD patterns for Mn_1−xZn_xCr₂O₄ powders crushed by the single crystals; (c) the content x of doped Zn²⁺ ions dependence of the fitted lattice parameter a; (d) the doping amount x of Zn²⁺ ions dependence of the position of Bragg peak (511).

In order to investigate the macroscopic magnetic properties of Mn_1−xZn_xCr₂O₄ single crystals, we carried out the measurement of the magnetization M(T) as the function of temperature. Fig. 2(a)–(f) show the M(T) under the zero field-cooled (ZFC), field-cooled (FCC) and field-warming (FCW) modes with the applied magnetic field parallel to the 〈111〉 direction for Mn_1−xZn_xCr₂O₄ single crystals, respectively. Fig. 2(a) and (b) show a FIM behavior for x ≤ 0.2. A collapse of the cusp is observed for the compounds with higher x in Fig. 2(c), which indicates the magnetic ground state is changed by the Zn²⁺ ion doping, namely, the SFIM may be suppressed and the magnetic ground state changes into the spin glass state in the spinel oxides.^9,24 For x ≥ 0.6, compared with the lower doped content x, the value of the magnetization M is much smaller and the ZFC and FCC curves are obviously irreversibility in x = 0.6 and 0.8 (as shown in Fig. 2(d) and (e)). It may mean that both AFM and FIM orders are perturbed, then destroyed and eventually the spin-glass-like ground state presents.²⁴ In Fig. 2(f), accompanied by the sharp drop of magnetization M(T), the AFM order occurs at T_N = 12 K, which is in agreement with the reported data. It is related to a structural phase transition from cubic Fd3̄m phase to the tetragonal I4₁/amd one at T_N for ZnCr₂O₄.^6,25 In order to further investigate the nature of the magnetic structure of Mn_1−xZn_xCr₂O₄ single crystals, the magnetic field dependence of the magnetization (M(H)) for all crystals at T = 5 K are shown in Fig. 3. Except for the parent MnCr₂O₄, it shows that the coercivity H_C increases with the increasing x for x ≤ 0.6. The saturated magnetization M_S is nearly decreasing with increasing content x.

Fig. 2 (a–f) The temperature dependence of magnetization M(T) and specific heat C_P(T) for Mn_1−xZn_xCr₂O₄ (0 ≤ x ≤ 1) single crystals. Black, red and blue lines are the magnetization measured under the ZFC, FCC and FCW modes, respectively. T_C are defined as the temperature of the minimum slope of the M(T) data under the ZFC modes. Except for MnCr₂O₄, T_S are defined as the temperature of the maximum slope of the M(T) data under the ZFC modes. T_S of MnCr₂O₄ is obtained from the peak of the heat capacity C_p/T. The glass freezing temperature T_g is defined as the temperature of maximum magnetization in M–T curves. The olive line represents the specific heat data at H = 0 T. Insets of (a) and (c) present the enlarged view of the specific heat data around T_C or T_S.

Fig. 3 The magnetic field dependence of magnetization M(H) at T = 5 K for Mn_1−xZn_xCr₂O₄ single crystals. The applied magnetic field is along the [111] direction. The left inset presents the enlarged view of the magnetic hysteresis loop for the doping level x > 0.6. The right inset shows the content x of Zn²⁺ ions dependence of the coercivity H_C and the saturation magnetization M_S for Mn_1−xZn_xCr₂O₄ single crystals at T = 5 K.

Now we focus on the nature of Zn doped MnCr₂O₄ single crystals, we did the analysis on the temperature dependent inverse susceptibility χ⁻¹(T). Firstly, we pay attention to the parent compound MnCr₂O₄, the FIM order T_C and SFIM one T_S are 52.7 K and 24.4 K obtained from the peak of the heat capacity C_p/T, respectively. From the mean-field theory, for a AFM system and FIM one, the temperature dependent inverse susceptibility above T_C can be described by the Curie–Weiss law (eqn (1)) and the hyperbolic behavior characteristic of ferrimagnets (eqn (2)):^7,26

where C is the Curie constant, θ is the Weiss temperature. In eqn (2), the first term is the hyperbole high-T asymptote that has a CW form and the second term is the hyperbole low-T asymptote. All the fitting results are summarized in Fig. 4(b). The substitution of Mn²⁺ ions by Zn²⁺ ions progressively perturbs the FIM structure, leading to a reduction of the effective magnetic moment μ_eff () and magnetic fraction factor f (f = θ/T_N) for increasing x. The tetragonal position occupied by the non-magnetic Zn²⁺ ions is responsible for the above results, which are consistent with the ones obtained from the above M(T) and M(H) measurements.

Fig. 4 (a) The inverse susceptibility dependence of the temperature for Mn_1−xZn_xCr₂O₄ single crystals. The solid lines are the fitting results according to eqn (1) and (2); (b) the obtained effective moment μ_eff, Weiss temperature θ, the magnetic order temperature T (including T_C and T_N) and the frustration factor f dependence of the doping level x of the Zn²⁺ ions.

To study the thermal property, we performed the detailed analysis on the temperature dependent specific heat C_P/T of Mn_1−xZn_xCr₂O₄ single crystals, as shown in Fig. 5(a). For ZnCr₂O₄, which yields a sharp specific heat anomaly with the character of the first-phase transition at T_N = 12.2 K. With the decrease of x, this heat capacity anomaly is clearly suppressed. For 0.6 ≤ x ≤ 0.8, the heat capacity peak disappears and the specific heat presents a monotonous smooth curve, which shows a spin glass behavior.²⁵ It agrees well with the magnetic results. When x ≤ 0.4, just one specific heat peak is observed, as present in the inset of Fig. 5(a), which is related to the LFIM. When x continues to decrease, two specific heat peaks are observed for 0 ≤ x ≤ 0.2, and which are corresponding to the LFIM and SFIM orders. As x ≤ 0.2, it is obvious that the magnetic structure of Mn_1−xZn_xCr₂O₄ samples in the FIM regions is in good agreement with MnCr₂O₄. In addition, as shown in Fig. 5(b), the low-temperature magnetic specific heat presents linear variation in Mn_1−xZn_xCr₂O₄. The linear variation suggests a constant density of states of the low-temperature magnetic excitations, which is claimed to be a common feature of spin glasses.^27,28 For 0.4 ≤ x ≤ 0.8, a broad magnetic specific heat anomaly is observed for spin glasses, indicating that short-range-order contributions extend up to very high temperatures. Fig. 5(b) shows the temperature dependence of the magnetic heat capacity C_mag. for Mn_1−xZn_xCr₂O₄. Since all the samples show insulating behavior, we can ignore the electronic contribution to the heat capacity, the C_mag. can be calculated by the following equations:²⁶

C_mag.(T) = C_p(T) − nC_VDebye(T) (4)

where n = 7 is the number of atoms per formula unit, R is the molar gas constant and Θ_D is the Debye temperature. The sum of Debye functions accounts for the lattice contribution to specific heat. We can get the C_mag. by eqn (4). We also can obtain the magnetic entropy S_mag. from the C_mag., which is calculated by integral of the C_mag./T versus T:²⁶

Fig. 5 (a) C_P/T curves of Mn_1−xZn_xCr₂O₄ single crystals. The inset presents partial enlarged detail for 0 ≤ x ≤ 0.4. The red and blue arrows are corresponding to T_C and T_S, respectively; (b) shows the linear dependence of magnetic specific heat C_mag. at low temperature. The straight line guides the linear dependence.

The T dependence of S_mag. is shown in Fig. 6. S_mag. is monotonously decreasing with the increasing content x except for ZnCr₂O₄. In addition, the change of Debye temperature behaves as firstly decreasing and then increasing with the increasing content x except for ZnCr₂O₄, the x = 0.4 sample has a minimum value Θ_D = 270 K. The abnormality of ZnCr₂O₄ may be attributed to the character of the first-phase transition at T_N = 12.2 K.

Fig. 6 The temperature dependence of the magnetic entropy S_mag. of Mn_1−xZn_xCr₂O₄ single crystals. The inset shows the doping content x dependence of the Debye temperature Θ_D.

Based on our obtained results, we summarize the magnetic phase diagram of Mn_1−xZn_xCr₂O₄ single crystals as plotted in Fig. 7. Now, let us try to understand the magnetic evolution in Mn_1−xZn_xCr₂O₄ single crystals. As we know, the substitution of Mn²⁺ (S = 5/2) by Zn²⁺ (S = 0) ions usually can induce following effects: the shrinkage of lattice, the disorder of A sites and the decreased content of magnetic Mn²⁺ ions. As shown in Fig. 2, for x ≤ 0.4, it can be seen that all the samples undergo a transition from PM to FIM. The ZFC and FCC curves are obviously irreversibility in 0.01 T, which can be attributed to magnetic frustration or a transition into a spin-glass phase.^29–31 We note that the magnetization in 0.01 T decreases and the transition temperature T_C is lower than that of MnCr₂O₄ with the increase of x. Meanwhile, the FIM order presents an easy axis along the [11̄0] direction in MnCr₂O₄ below T_C, which is identical with the axial component of Mn²⁺ ion.^7,9,18,32 Although the shrinkage of lattice could enhance the exchange interaction between Cr–Mn ions via oxygen 2p orbits, the decreased content of Mn ions is the major factor, which is responsible for the above observed phenomenon.³³ In MnCr₂O₄, the axial component of the tetrahedral A (Mn²⁺) site is antiparallel to that of the octahedral B (Cr³⁺) site.

Fig. 7 The magnetic evolution of Mn_1−xZn_xCr₂O₄ single crystals. PM, SFIM, LFIM, SG and AFM are corresponding to the paramagnetism, spiral ferrimagnetism, long-range ferrimagnetism, spin glass and antiferromagnetism, respectively.

According to the previous experimental evaluations and theoretical calculations, J_CrCr is larger than J_MnCr or approximately equal to J_MnCr.^7,16,18,24 However, as it is shown in Fig. 3, the abnormality of MnCr₂O₄ may be attributed to the cation distribution of A and B sites.²⁹ This is because Mn²⁺ site is slightly occupied by Cr³⁺ ions, which leads to the enhanced coercivity and the reduced magnetization.³⁴ With the substitution of magnetic ions in A site by the Zn²⁺ (has preferentially A-site occupancy), the cation distribution of A and B sites will be consistent with that of normal spinel structure. For x ≥ 0.05, the replacement of Mn²⁺ with non-magnetic Zn²⁺ leads to the weakening of the A–O–B super exchange interaction. This would further disturb the magnetic couplings and lead to a reduction of the magnetization. The substitution of Zn²⁺ ion for Mn²⁺ or Co²⁺ ions has relatively similar physical properties. For example, Brent C Melot et al.²⁴ has reported magnetic phase evolution in the spinel compounds Co_1−xZn_xCr₂O₄, which is similar to our results obtained in Mn_1−xZn_xCr₂O₄. At the same time, the structure of Co_1−xZn_xCr₂O₄ at low temperature (T = 5 K) is still modeled by the cubic space group Fd3̄m for x ≤ 0.9.²⁵ For x ≥ 0.6, the magnetic coupling interactions become more complicated. The Mn–O–Cr superexchange interaction partially breaks the spin degeneracy of the ground state and the coupling between the spin and lattice degrees of freedom becomes weaker than that of ZnCr₂O₄ in Mn_1−xZn_xCr₂O₄. It is reasonable that the Mn–O–Cr superexchange interaction could disrupt the coherency of Cr–Cr exchange coupling paths, and then inhibit the spin-Jahn–Teller distortion in Mn_0.2Zn_0.8Cr₂O₄ and Mn_0.6Zn_0.4Cr₂O₄.²⁵ However, more detail structural experiments at low temperature, including magnetic structure determined using neutron scattering method, are needed in future.

4 Conclusion

From the room-temperature X-ray diffraction, magnetization M(T) and specific heat C_P(T) measurements, the crystallographic, magnetic, and thermal transport properties of the Mn_1−xZn_xCr₂O₄ single crystals were obtained. The lattice constant a decreases with the increasing content x of the doped Zn²⁺ ions and follows the Vegard law. The magnetic evolution of Mn_1−xZn_xCr₂O₄ crystals based on the magnetization and specific heat measurements has been discussed. For 0 ≤ x ≤ 0.3, the magnetic ground state is the coexistence of LFIM and SFIM, which is similar to that of the parent MnCr₂O₄. x changes from 0.3 to 0.8, the SFIM is progressively suppressed and spin glass-like behavior is observed. While x is above 0.8, an AFM order presents. The magnetic specific heat (C_mag.) was also calculated and the results are coincident with the magnetic measurements. The possible reasons based on the disorder effect and the reduced molecular field effect induced by the substitution of Mn²⁺ ions by nonmagnetic Zn²⁺ ones in Mn_1−xZn_xCr₂O₄ crystals have been discussed.

Acknowledgements

This work was supported by the Joint Funds of the National Natural Science Foundation of China and the Chinese Academy of Sciences' Large-Scale Scientific Facility under contracts U1432139, U1532152, the National Nature Science Foundation of China under contracts 51171177, 11404339, and the Nature Science Foundation of Anhui Province under contract 1508085ME103.

References

1. J. Hemberger, P. Lunkenheimer, R. Fichtl, H.-A. Krug von Nidda, V. Tsurkan and A. Loidl, Nature, 2005, 434, 364–367.
2. Y. J. Choi, J. Okamoto, D. J. Huang, K. S. Chao, H. J. Lin, C. T. Chen, M. van Veenendaal, T. A. Kaplan and S.-W. Cheong, Phys. Rev. Lett., 2009, 102, 067601.
3. K. E. Sickafus, J. M. Wills and N. W. Grimes, J. Am. Ceram. Soc., 1999, 82(12), 3279–3292.
4. C. Lacroix, P. Mendels and F. Mila, Introduction to Frustrated Magnetism, in Solid-State Sciences, Springer Series, 2011.
5. S. Bordács, D. Varjas, I. Kézsmárki, G. Mihály, L. Baldassarre, A. Abouelsayed, C. A. Kuntscher, K. Ohgushi and Y. Tokura, Phys. Rev. Lett., 2009, 103, 077205.
6. M. C. Kemei, P. T. Barton, S. L. Moffitt, M. W. Gaultois, J. A. Kurzman, R. Seshadri, M. R. Suchomel and Y.-I. Kim, J. Phys.: Condens. Matter, 2013, 25, 326001.
7. E. Winkler, S. Blanco Canosa, F. Rivadulla, M. A. López-Quintela, J. Rivas, A. Caneiro, M. T. Causa and M. Tovar, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 104418.
8. S.-H. Lee, C. Broholm, W. Ratcliff, G. Gasparovic, Q. Huang, T. H. Kim and S.-W. Cheong, Nature, 2002, 418, 856–858.
9. K. Dey, S. Majumdar and S. Giri, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 184424.
10. M. T. Rovers, P. P. Kyriakou, H. A. Dabkowska and G. M. Luke, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 66, 174434.
11. I. Kagomiya, H. Sawa, K. Siratori, K. Kohn, M. Toki, Y. Hata and E. Kita, Ferroelectrics, 2002, 268, 327–332.
12. J.-H. Chung, M. Matsuda, S.-H. Lee, K. Kakurai, H. Ueda, T. J. Sato, H. Takagi, K. P. Hong and S. Park, Phys. Rev. Lett., 2005, 95, 247204.
13. R. Valdes Aguilar, A. B. Sushkov, Y. J. Choi, S.-W. Cheong and H. D. Drew, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 092412.
14. H. Ueda, H. Mitamura, T. Goto and Y. Ueda, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 094415.
15. D. Tobia, J. Milano, M. Teresa Causa and E. L. Winkler, J. Phys.: Condens. Matter, 2015, 27, 016003.
16. S. Blanco-Canosa, F. Rivadulla, V. Pardo, D. Baldomir, J.-S. Zhou, M. Garca-Hernndez, M. A. Lpez-Quintela, J. Rivas and J. B. Goodenough, Phys. Rev. Lett., 2007, 99, 187201.
17. J. M. Hastings and L. M. Corliss, Phys. Rev., 1962, 126, 556–565.
18. K. Tomiyasu, J. Fukunaga and H. Suzuki, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 70, 214434.
19. D. Y. Yoon, S. Lee, Y. S. Oh and K. H. Kim, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094448.
20. A. Miyata, H. Ueda, Y. Ueda, H. Sawabe and S. Takeyama, Phys. Rev. Lett., 2011, 107, 207203.
21. V. N. Glazkov, A. M. Farutin, V. Tsurkan, H.-A. Krug von Nidda and A. Loidl, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 024431.
22. K. Le Dang, M. C. Mery and P. Veillet, J. Magn. Magn. Mater., 1984, 43, 161–165.
23. F. Leccabue, B. E. Watts, D. Fiorani, A. M. Testa, J. Alvarez, V. Sagredo and G. Bocelli, J. Mater. Sci., 1993, 28, 3945–3950.
24. B. C. Melot, J. E. Drewes, R. Seshadri, E. M. Stoudenmire and A. P. Ramirez, J. Phys.: Condens. Matter, 2009, 21, 216007.
25. M. C. Kemei, S. L. Moffitt and L. E. Darago, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 174410.
26. C. Kittle, Introduction to Solid State Physics, Wiley, New York, 4th edn, 1966.
27. N. P. Raju, E. Gmelin and R. K. Kremer, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 5405–5411.
28. D. Meschede, F. Steglich, W. Felsch, H. Maletta and W. Zinn, Phys. Rev. Lett., 1980, 44, 102–105.
29. D. Y. Yoon, S. Lee, Y. S. Oh and K. H. Kim, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094448.
30. K. Binder and A. P. Young, Rev. Mod. Phys., 1986, 58, 801–976.
31. Z. Qu, Z. Yang, S. Tan and Y. Zhang, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 71, 184430.
32. R. Plumier, J. Appl. Phys., 1968, 39, 635–636.
33. X. Luo, Y. P. Sun, W. J. Lu, X. B. Zhu, Z. R. Yang and W. H. Song, Appl. Phys. Lett., 2010, 96, 062506.
34. A. Franco Jr and F. C. e. Silva, J. Appl. Phys., 2013, 113, 17B513.

---