Cite this Article
Liu Pengfei, Peng Jie, Xue Mianqi, Wang Bosen. Magnetocaloric effect and critical behavior of the Mn-rich itinerant material Mn3GaC with enhanced ferromagnetic interaction. Chinese Physics B, 2020, 29(4): 047503  
Permissions
Magnetocaloric effect and critical behavior of the Mn-rich itinerant material Mn3GaC with enhanced ferromagnetic interaction
Liu Pengfei1, 2, Peng Jie1, Xue Mianqi2, Wang Bosen3, †
Hunan Key Laboratory for Micro-Nano Energy Materials and Devices and School of Physics and Optoelectronics, Xiangtan University, Xiangtan 411105, China
Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

 

† Corresponding author. E-mail: bswang@iphy.ac.cn

Project supported by National Key Research and Development Program of China (Grant Nos. 2018YFA0305700 and 2018YFA0305800), One Hundred-Talent Program in Institute of Physics (Grant No. Y7K5031X61), Youth Promotion Association of CAS (Grant No. 2018010).

Abstract

We revisit the reversible magnetocaloric effect of itinerant ferromagnet Mn3GaC near the ferromagnetic to paramagnetic phase transition by adopting the experimental and theoretical methods and critical behavior of Mn-rich Mn3GaC with an enhanced FM interaction. Landau theory model cannot account for temperature dependent magnetic entropy change which is estimated from thermal magnetic methods only considering magnetoelastic coupling and the electron–electron interaction, apart from molecular mean-field model. Critical behavior is studied by adopting the modified Arrott plot, Kouvel–Fisher plot, and critical isotherm analysis. With these critical exponents, experimental data below and above Tc collapse into two universal branches, fulfilling the single scaling equation m = f±(h), where m and h are renormalized magnetization and field. Critical exponents are confirmed by Widom scaling law and just between mean-field model and three-dimensional Heisenberg model, as the evidence for the existence of long-range ferromagnetic interaction. With increasing the Mn content, Tc increases monotonously and critical exponents increases accordingly. The exchange distance changes from J(r) ∼ r–4.68 for x = 0 to J(r) ∼ r–4.71 for x = 0.08, respectively, which suggests the competition of the Mn–Mn direct interaction and the itinerant Mn–C–Mn hybridization. The possible mechanism is proposed.

PACS: ;75.20.En;;75.30.Sg;;75.60.Ej;
Keyword:;magnetocaloric effect;;critical behavior;;Mn3GaC;
1. Introduction

Itinerant electronic materials are excellent candidates to explore functional materials. Among them, Mn-based itinerant materials are such examples that have attracted more attention due to the important findings including giant magnetoresistance (GMR),[1,2] magnetocaloric effect (MCE),[3–6] negative thermal expansion (NTE),[7–13] and magnetostriction (MS),[14–16] etc. Mn3GaC exhibits various magnetic/structural phase transitions and has been widely investigated.[1–5,17–19] With decreasing temperature, it transits from paramagnetic (PM) to ferromagnetic (FM) phases at 250 K, to an intermediate canted ferromagnetic (IFM) phase at 158 K, and to an antiferromagnetic ground state with discontinuous expansion in lattice parameter near 155 K.[17,19] Accompanied by magnetic/structural phase transitions, GMR can reach nearly 80% at 5 T and larger “trapeziform”-like negative magnetic entropy changes exists.[1–5] Meanwhile, giant isotropic MS (1700 ppm) is found near the AFM–IFM phase transition.[15] In addition to important experimental findings, some basic scientific issues remain controversial on these FM materials after many years of exploration.

The first controversial point is Mn–Mn interaction of this system although different models have been proposed, such as the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, the direct-coupling model, and the mean field models. No systematic and in-depth theoretical calculations were performed to reveal the internal mechanism of magnetic interactions although similar tries were carried out in other itinerant FM materials. Another point is why FM–PM transition is so sensitive to Mn-doping only considering lattice changes. In other words, its Fermi surface structure nesting and energy band filling effect can play a key role. For Mn-rich Mn3GaC, a reversible MCE with large temperature spans at Tc is reported,[20] while the origin of quickly enhanced Tc with Mn-doping and the studies of MCE based on various theoretical models are lacking. Previous investigations have proved that critical exponent analysis in the vicinity of magnetic transition is a powerful to single out the relevant microscopic interaction.[21–27] According to the reports,[17,20] Mn3GaC is an itinerant electronic material and electron-electron interaction is important to carrier transporting and mainly comes from Mn-3d band electrons. Its ground state can be well-described by Fermi-liquid behavior. Meanwhile, electron-phonon interaction may be important in contributing magnetic entropy change near phase transitions. In this respect, more theoretical and experimental investigations are required to deepen our understanding of magnetic properties of this system.[21,22] In this work, we present a detailed study on the MCE near the FM–PM transition based on Landau theory of the transition model and the molecular mean-field model and on the specific heat. We find that the simple Landau theory model can not account for the temperature dependence of magnetic entropy in Mn3GaC if only considering magnetoelastic coupling and electron-electron interaction. The scaling plots confirm that critical exponents are reliable and all the critical exponents are just between the mean-field model and the three-dimensional (3D) Heisenberg model, as the evidence for the existence of long-range FM interaction. The exchange distance changes from J(r) ∼ r–4.68 for x = 0 to r–4.71 for x = 0.08, respectively, which suggests the competition of Mn–Mn direct interaction (180°) and the itinerant Mn–C–Mn (90°) hybridization.

2. Experimental details

Polycrystalline samples Mn-rich Mn3GaC was prepared as reported previously,[1,2] and structural parameters were collected by x-ray diffraction (Cu Kα, λ = 0.15406 nm). Magnetic properties were measured on a Quantum Design superconducting quantum interference device magnetometer (1.8 K ≤ T ≤ 400 K, 0 ≤ H ≤ 50 kOe). The sample for magnetic measurements can be considered as ellipsoids and the applied field is parallel to its longest semiaxis. Thus, a uniform field exists throughout the sample and sharp demagnetizing fields could be reduced. Thermopower coefficient was measured by using standard four-probe technique in the Quantum Design physical property measurement system (1.8 K ≤ T ≤ 400 K, 0 ≤ H ≤ 90 kOe). The Rietveld refinement of XRD pattern indicates that all the samples are single-phase (SG: ) with lower Rp and Rwp. Refined lattice constant of Mn3GaC is about 0.3897(3) nm, close to the reported value (0.3896 nm).[1,3,17] With increasing the substitution of Ga by the excess Mn, the lattice parameter decreases linearly.

3. Results and discussion
3.1. Direct-current magnetization

Figure 1 shows the temperature dependence of M(T) curves for Mn3 + xGa1 − xC under the zero-field-cooled (ZFC) process. In Fig. 1(a), M(T) is measured at different fields for Mn3GaC under the ZFC process and the temperature dependence of dM/dT is used to search the corresponding transition temperature. The inset describes magnetic structures of the FM phase. We find that its peak temperature shifts to higher temperature with increasing the field from 0.1 kOe to 10 kOe, which indicates enhanced FM interaction at higher fields.[1,17] Inverse susceptibility is plotted and well-described by the modified Curie–Weiss law: χ(T) = C/(T − θ) + χ0(1 + AT2), where the first term C/(T − θ) represents the Curie–Weiss contribution, the second term χ0(1 + AT2) is the Pauli PM contribution from enhanced exchanges with the χ0AT2 being the modified term, and A reflects electronic energy state shape at the Fermi level. In an external field, a small amount of energy is transmitted near the Fermi surface owing to different energies between spin-up and spin-down conduction electrons: χ0 = 3.56(8) emu·K/mol, A = –2.99(4) × 10–6 emu/(mol·K2). Effective moment is given by μeff = 2.83(C/η)0.5 μB = 3.086 μB, where η is the number of magnetic atoms in a unit cell. Similarly, both the M(T) and dM/dT of Mn3 + xGa1 − xC are shown in Fig. 1(b). We find that, with the increase of Mn-doping content, the peak height decreases and the peak width increases, and it Tc shifts to high temperature. The derivatives are found to be 250 K, 281.5 K, 296.5 K, and 323.5 K for x = 0, 0.06, 0.07, and 0.08, respectively. The enhancement of the Tc can be attributed to lattice contraction, which leads to an enhanced FM exchange interactions. However, this enhancement cannot be explained if ignoring band filling as a function of Mn-doping according to the compressibility of Mn3GaC under pressure. In this work, we discuss the MCE by experimental/theoretical methods and the critical behaviors of Mn-rich Mn3GaC based on the thermo-dynamical theory.[20] Different theoretical models are adopted to quantify entropy change of this system, such as the Landau theory model and the mean-field model.[28–32]

Fig. 1. (a) Temperature dependent M(T) in zero-field-cooled (ZFC) process. Inset: magnetic structures and dM/dT. Inverse susceptibility on the right of Fig. 2(a); (b) M(T) for Mn3 +xGa1 − xC at 1 T.
3.2. The MCE — Landau theory model

In Fig. 2(a), M(H) data are plotted for Mn3GaC (192–288 K) Arrott plots derived from M(H) covering a broad temperature. The Gibbs free energy G is expressed as

where A and B are the temperature-dependent parameters containing the magneto-elastic coupling and the electron interaction. From the equilibrium condition ∂G/∂M = 0, the magnetic equation of state is

The H/M vs. M2 plot has positive slopes, indicating second-order transition by Banerjee criterion.[33] From the differential Gibbs free energy, magnetic entropy is

The temperature dependences of A and B are obtained by linear fitting (Fig. 2(b)). It is found that A varies from negative to positive with increasing the temperature, and the temperature is corresponding to A = 0 and is basically consistent with the Tc. As shown in Fig. 2(c), B keeps positive and is the feature of ferromagnic materials.[28,34] As we know, A can describe the electron interaction part of Gibbs free energy, and B represents the magneto-elastic part, which plays an important role in determining entropy changes −ΔSM.[29,35] From Eq. (3), −ΔSM is calculated with different ΔH.

Fig. 2. (a) Isothermal magnetization curves M(H); (b) Arrott plots derived from M(H). Inset: temperature-dependent thermopower coefficient. (c) A (on the left) and B (on the right), where A and B are magneto-elastic coupling part and electron interaction part of the Gibbs free energy, respectively. (d) The calculated and experimental ΔSM under ΔH = 20 kOe and 45 kOe, respectively.

Figure 2(d) displays the calculated and experimental −ΔSM. We find that −ΔSM from Eq. (3) is about four times larger than the experimental value and the peak is 35 K lower. Theoretical/experimental results are of great discrepancy, which is mainly originated from the degradation linearity in the H/M vs. M2 plot below Tc. This is related to the sign change of thermopower coefficient S(T) as indicated in the previous investigations.[28,36] In inset of Fig. 2(b), temperature dependence of S(T) for Mn3GaC has two characteristics: a slope change of S(T) exists at 250 K closely associated with the FM–PM transition; its sign changes from negative to positive around 215 K, matching with the peak of calculated −ΔSM. Lastly, we may conclude that larger differences between the experimental and theoretical entropy changes are attributed to some ignored and critical factors such as the phonon-electron interaction.

3.3. The MCE — mean-field scaling method

Another approach was adopted to apply the mean-field scenario to isothermal curves and calculate entropy changes.[31,32] It does not depend on the numerical integration of Maxwell relation and is a complementary approach to Landau theory.[29,31] Magnetic equation of state is expressed as

where f is the mean-field state function, Hexch is the exchange field, and λ is the exchange parameter. Generally, Hexch could be obtained by a series of odd powers of M. In this work, Hexch is taken as Hexch = λ1 M + λ3M3, where λ3 describes the magneto-volume effects as the Bean and Rodbell formulation.[37] The inverse f–1(M) function is

For a constant M, the Hexch, f–1(M), and f–1(M)/M are constant if Hexch only depends on M. Accordingly, the plots of H/T vs. 1/T and H/MT vs. 1/T should be linear. According to M(H) in Fig. 2(a), numerical interpolations on H (in 1 emu/g step) are carried out. Figure 3(a) presents the plots of H/T vs. 1/T for isometric M up to 50 emu/g (in 5 emu/g step), Hexch is obtained by linear fitting of H/T vs. 1/T. Obviously, the lines with different M are parallel, indicating that the value of λ is weakly dependent on M and T. The Hexch is fitted according to Hexch = λ1M + λ3M3 and the fittings give λ3 ≈ 0, indicating that the Hexch linearly depends on M.[38] Figure 3(b) illustrates the plot of M vs. (H + Hexch)/T based on M(H). Obviously, almost all the data lie in the same curve which is corresponding to the mean-field state function, indicating reliable fittings. From an odd polynomial fitting of M, the function f is obtained in Fig. 3(b). During this fitting, only those points with higher H are used. In Fig. 3(c), the calculated and experimental M(H) data are compared. Obviously, the calculated results match well with the experimental data except for the magnetic domain regime at the lower magnetic field. From Eq. (5) and using the classical thermo-dynamical theory and Maxwell’s relation,[39] −ΔSM induced by the variation from H1 to H2 is given as follows:[31]

Generally, the first part of the integral is larger than the second one and the contribution from the second part is smaller. Thus the integral is transformed into an integration of f function. In Fig. 3(d), −ΔSM is calculated for Mn3GaC by Eq. (6) and compared with numerical integration of a Maxwell relation. It is found that both the results are in good agreement with each other. Therefore, it is concluded that the mean-field theory can well account for the entropy changes of this system.[40,41]

Fig. 3. (a) H/T vs. 1/T with constant M. Inset: the plots of H/MT vs. 1/T. (b) Scaled data in M vs. (H + Hexch) and polynomial fit. Inset: the exchange field Hexch vs. M with the formula Hexch = λ1 M+ λ3M3. (c) The experimental and the calculated curves. (d) ΔSM with ΔH = 20 kOe, 30 kOe, and 45 kOe for comparison.
3.4. The MCE — heat capacity measurement

Another method used to estimate entropy change is based on heat capacity.[42,43] In inset of Fig. 4, the Cp vs. T is plotted under H = 0 Oe and 20 kOe. A broad peak can be found nearly at 250 K, corresponding to the FM–PM transition. Meanwhile, the peak broadens with increasing the magnetic field. All these characteristics are consistent with the above results. Magnetic entropy change can be from the second thermodynamics law.[42,43] As shown in Fig. 4, temperature-dependent entropy change −ΔSM curves for ΔH = 20 kOe with three different methods are compared. It is found that the value of the −ΔSM exhibits the temperature dependence similar to those of magnetic measurements and the calculations based on the mean-field model except for some differences in details.

Fig. 4. ΔSM near the FM–PM phase transition. Inset: heat capacity under H = 0 and 20 kOe.
3.5. Direct-current magnetization-critical behavior

According to the scaling hypothesis, a second-order transition near Tc is characterized by critical exponents β (spontaneous magnetization), γ (initial magnetic susceptibility), and δ (critical magnetization isotherm) as follows:[44,45]

ε = (T−Tc)/Tc is the reduced temperature, M0, h/M0, and D are the critical amplitudes. Figure 5(a) shows M(H) with Tc ± 22 K. The FM behavior near Tc is investigated by Arrott plots which cover 228–272 K with positive intercepts.[43,46] According to the mean-field theory, M2 vs. H/M should be parallel lines and the line at T = Tc passes through the origin. In this work, the curves of Arrott plots are flexural, β and γ could not be obtained from M(H) directly. From the Arrott plots in Fig. 5(b), the curves above T > Tc can be extended smoothly into the H/M axis, yielding the 1/χ0(T). The polynomial fitting and extrapolation for T < Tc give the reliable MS(T, 0). For this extrapolation, M2 vs. H/M is fitted with a fourth-order polynomial from 4 kOe to 45 kOe, then extrapolated to H = 0 to obtain MS(T, 0).[21,22,27]

Fig. 5. (a) Isothermal M(H) curves from 228 K to 272 K (in 2 K step). (b) Arrott plots of H/M vs. M2 derived from M(H) curves.

MS(T, 0) vs. T and 1/χ0(T) vs. T are given in Fig. 6(a). They denote the power law fitting of MS(T, 0) vs. T and 1/χ0(T) vs. T according to Eqs. (7) and (8), respectively. For x = 0, β = 0.462 ± 0.02 with Tc = 247.29 ± 0.11 K and γ = 1.096 ± 0.01 with Tc = 247.91 ± 0.17 K. For x = 0.06, β = 0.556 ± 0.03 with Tc = 284.52 ± 0.04 K and γ = 1.068 ± 0.02 with Tc = 284.87 ± 0.20 K. For x = 0.07, β = 0.578 ± 0.01 with Tc = 296.74 ± 0.13 K and γ = 1.099 ± 0.04 with Tc = 297.16 ± 0.20 K. For x = 0.08, β = 0.657 ± 0.01 with Tc = 328.26 ± 0.15 K and γ = 1.122 ± 0.03 with Tc = 328.70 ± 0.26 K. Tc shifts to high temperature with the increase of x. MS(T, 0) and 1/χ0(T) decreases, and reaches the maximum at x = 0. Here β keeps increasing as x increases but γ starts to decrease and then increases.

Fig. 6. Temperature dependence of the spontaneous magnetization MS(T, 0) and the inverse initial susceptibility 1/χ0(T) for Mn3 + xGa1 − xC.
Fig. 7. The Kouvel–Fisher plots for the MS(T, 0) and 1/χ0(T).

The exponents are obtained by the Kouvel–Fisher (KF) method,[26,47]

MS(T)[dMS(T)/dT]–1 and yield the straight lines with slopes of 1/β and 1/γ, respectively, and the intercept on the T axis is equal to the value of Tc. For x = 0, β = 0.460 ± 0.004 with Tc = 247.56 ± 0.09 K and γ = 1.103 ± 0.005 with Tc = 247.69 ± 0.07 K. For x = 0.06, β = 0.561 ± 0.003 with Tc = 284.54 ± 0.06 K and γ = 1.061 ± 0.05 with Tc = 284.94 ± 0.06 K. For x = 0.07, β = 0.579 ± 0.001 with Tc = 296.74 ± 0.02 K and γ = 1.091 ± 0.007 with Tc = 297.27 ± 0.08 K. For x = 0.08, β = 0.663 ± 0.001 with Tc = 328.60 ± 0.05 K and γ = 1.128 ± 0.01 with Tc = 328.65 ± 0.10 K. The β keeps increasing as x increases but γ starts to decrease and then increases at x = 0.07. The value of MS(T)[d MS(T)/dT]–1 reaches the maximum nearly 55 K at x = 0.06 while with the maximum 22.5 K at x = 0.0. As shown in Fig. 8, the critical isotherms M vs. H can be correlated by Eq. (9). The reliable fittings are 3.133 ± 0.05 K, 2.950 ± 0.06 K, 2.988 ± 0.05 K, 2.893 ± 0.05 K, respectively. The exponents derived from the scaling analysis are related by the Widom relations.[48,49] We have δ = 1 + γ/β from the KF method, δ is close to the value from critical isotherms at Tc, implying that β and γ are reliable.

Fig. 8. Critical isotherms on log–log scale for Mn3 + xGa1 − xC.

In the critical region, magnetic equation of state is given by[50]

where f+ for T > Tc and f− for T < Tc are regular analytical functions. Equation (12) implies that the plot of M|ε|–β and H|ε|–(β + γ) produces two universal curves: one for temperatures below Tc and the other for temperatures above Tc. All the data points lie in two curves, one for T < Tc and the other for T > Tc, indicating that exponents and Tc are reliable. In Fig. 9, the plots are shown in log–log scale. We can find that the M|ε|–β increases with the Mn-doping content and the field increases sharply at lower magnetic fields, and flattens out at higher magnetic fields. Obviously, the scaling is reliable, β, γ and δ are close to those of the mean-field model, indicating the appearance of long-range FM interaction in this system. We also find that both the values of β and γ increase with increasing the Mn-doping and are slightly larger than those of the theoretical model. For clarify, the effective exponents βeff and γeff are compared in Fig. 10.[50] Here we define the normalized ε = (T − Tc)/Tc to be the reduced temperature. It is noticed that the value of βeff increases from 0.460 to 0.663 monotonously with the increase of the Mn-doping, while the γeff value decreases from 1.103 for x = 0 to 1.061 for x = 0.06 firstly and increases again at x = 0.07.

Fig. 9. Scaling plots for Mn3 + xGa1 − xC below and above Tc using β and γ determined by the Kouvel–Fisher method. The inset shows the plots on a log–log scale.
Fig. 10. The dependence of the critical exponent βeff and γeff on ε for Mn3 + xGa1 − xC, where ε = (T−Tc)/Tc is the reduced temperature.

By comparison, we can find that critical exponents of this system are apart from those of theoretical models, e.g., 3D Heisenberg and/or 3D Ising models, and mean-field theory.[44,51,52] However, both the values of β and γ from the magnetic measurements are just between the predicted mean-field model and the 3D Heisenberg model, and somewhat close to the mean-field model. Usually, the 3D Heisenberg model and the Ising model represent short-range FM interaction and the mean-field model is corresponding to long-range FM coupling.[44,51] Thus, it may be thought that FM interaction is dominant and mainly comes from long-range Mn–C–Mn. Generally, critical exponents depend on lattice dimension, order parameter and the range of interaction (short range, long range or infinite).[53] For magnets, the universality class of transition depends on the exchange interaction J(r) = 1/rd + σ, where d represents the spatial dimensions and σ is range of interaction and a positive constant.[53] For 3D materials (d = 3), there is a relation J(r) = r–(3 + σ). When σ = 2, the Heisenberg model is valid for a 3D isotropic ferromagnet, where J(r) decreases faster than r–5. When σ = 3/2, the mean-field model is satisfied, which indicates that J(r) decreases slower than r–4.5. In this work, σ = 1.681 ± 0.005 for x = 0, σ = 1.634 ± 0.002 for x = 0.06, σ = 1.671 ± 0.006 for x = 0.07, σ = 1.714 ± 0.004 for x = 0.08, implying that J(r) changes from r–4.68, r–4.63, r–4.67, r–4.71 for x = 0, 0.06, 0.07, 0.08, respectively. It indicates that J(r) of Mn3 + xGa1 − x C is close to the mean-field model (r–4.5) and trends to the Heisenberg model (r–5) with increasing the Mn-doping. This means that Mn–Mn FM interaction is dominant and the enhanced Mn–C–Mn hybridization derives short-range FM interaction with increasing the Mn doping. Some short-range FM interactions become stronger. The detailed mechanism requires more investigations.

4. Conclusions

MCE near FM–PM transition in Mn3GaC has been investigated. Molecular mean-field model can well describe −ΔSM, while the Landau theory model does not. In addition, critical behavior is investigated by magnetization measurements using three different techniques. Critical exponents of Mn-rich Mn3GaC are between the mean-field model and the Heisenberg model. Mn–Mn FM interaction is dominant and the enhanced Mn–C–Mn hybridization derives short-range FM interaction with increasing the Mn doping.

Reference
[1] Kamishima K Goto T Nakagawa H et al. 2000 Phys. Rev. 63 024426
[2] Wang B S Tong P Sun Y P Li L J Tang W Lu W J Zhu X B Yang Z R Song W H 2009 Appl. Phys. Lett. 95 222509
[3] Tohei T Wada H Kanomata T 2003 J. Appl. Phys. 94 1800
[4] Yu M H Lewis L H Moodenbaugh A R 2003 J. Appl. Phys. 93 10128
[5] Tohei T Wada H Kanomata T 2004 J. Magn. Magn. Mater. 272 E585
[6] Wang B S Tong P Sun Y P Luo X Zhu X B Zhang S B Zhu X D Yang Z R Dai J M Song W H 2009 Europhys. Lett. 85 47004
[7] Takenaka K Asano K Misawa M Takagi H 2008 Appl. Phys. Lett. 92 011927
[8] Tong P Louca D Kingdom G Liobet A Lin J C Sun Y P 2013 Appl. Phys. Lett. 102 041908
[9] Iikubo S Kodama K Takenaka K et al. 2008 Phys. Rev. Lett. 101 205901
[10] Takenaka K Takagi H 2009 Appl. Phys. Lett. 94 131904
[11] Huang J R Li L F Cai F S Xu X D Qian L H 2008 Appl. Phys. Lett. 93 081902
[12] Ding L Wang C Sun Y Colin C V Chu L H 2015 J. Appl. Phys. 117 213915
[13] Guo X G Lin J C Tong P Wang M Wu Y Yang C Song B Lin S Song W H Sun Y P 2015 Appl. Phys. Lett. 107 202406
[14] Shimizu T Shibayama T Asano Takenaka K 2012 J. Appl. Phys. 111 07A903
[15] Guo X G Tong P Lin J C Yang C Zhang K Lin S Song W H Sun Y P 2017 Appl. Phys. Lett. 110 062405
[16] Asano K Koyama K Takenaka K 2008 Appl. Phys. Lett. 92 161909
[17] Kim W S Chi E O et al. 2001 Solid State Commun. 119 507
[18] Shim J H Kwon S K Min B I 2002 Phys. Rev. 66 020406(R)
[19] Fruchart D Bertaut E F Sayetat F et al. 1970 Solid State Commun. 8 91
[20] Wang B S Tong P Sun Y P Zhu X B Luo X Li G Song W H Yang Z Dai R J M 2009 J. Appl. Phys. 105 083907
[21] Cabassi R Bolzoni F Gauzzi A Licci F 2006 Phys. Rev. 74 184425
[22] Yang J Lee Y P Li Y 2007 Phys. Rev. 76 054442
[23] Lin J C Tong P Cui D P Yang C Yang J Lin S Wang B S Tong W Zhang L Zou Y M Sun Y P 2015 Scr. Rep. 5 7933
[24] Cui D P Wang B S Tong P Lin J C Lin S Sun Y P 2015 J. Magn. Magn. Mater. 382 93
[25] Zhang L Wang B S Sun Y P Tong P Fan J Y Zhang C J Pi L Zhang Y H 2012 Phys. Rev. 85 104419
[26] Babu P D Kaul S N 1997 J. Phys.: Condens. Matter 9 7189
[27] Pramanik A K Banerjee A 2009 Phys. Rev. 79 214426
[28] Yang J Lee Y P Li Y 2007 J. Appl. Phys. 102 033913
[29] Amaral V S Amaral J S 2004 J. Magn. Magn. Mater. 272�?76 2104
[30] Lu W J Luo X Hao C Y Song W H Sun Y P 2008 J. Appl. Phys. 104 113908
[31] Amaral J S Silva N J O Amaral V S 2007 Appl. Phys. Lett. 91 172503
[32] Amaral J S Amaral V S 2009 Appl. Phys. Lett. 94 042506
[33] Banerjee S K 1964 Phys. Lett. 12 16
[34] Landau L D Lifshitz E M 1980 Statistical Physica 3 Oxford Pergamon
[35] Amaral J S Reis M S et al. 2005 J. Magn. Magn. Mater. 290�?91 686
[36] Yang J Sun Y P Song W H Lee Y P 2006 J. Appl. Phys. 100 123701
[37] Bean C P Rodbell D S 1962 Phys. Rev. 126 104
[38] Menyuk N Dwight K Reed T B 1971 Phys. Rev. 3 1689
[39] Gschneidner K A Pecharsky V K 2000 Annu. Rev. Mater. Sci. 30 387
[40] Yang F Y Chien C L Li X W Xiao G Gupta A 2001 Phys. Rev. 63 092403
[41] Yanagihara H Cheong W et al. 2002 Phys. Rev. 65 092411
[42] Gschneidner K A Jr. Pecharsky V K Tsokol A O 2005 Rep. Prog. Phys. 68 1479
[43] Tishin A M Spichkin Y I 2003 The Magnetocaloric Effect and its Applications Bristol Institute of Physics Publishing
[44] Stanley H E 1972 Introduction to Phase Transitions and Critical Phenomena Oxford Clarendon Press
[45] Fisher M E 1967 Rep. Prog. Phys. 30 615
[46] Mira J Rivas J et al. 1999 Phys. Rev. 60 2998
[47] Kouvel J S Fisher M E 1964 Phys. Rev. 136 A1626
[48] Widom B 1964 J. Chem. Phys. 41 1633
[49] Widom B 1965 J. Chem. Phys. 43 3898
[50] Stanley H E 1999 Rev. Mod. Phys. 71 S358
[51] Huang K 1987 Statistical Mechanics 2 New York Wiley
[52] Le J C Zinn-Justin J 1980 Phys. Rev. 21 3976
[53] Fisher M E Ma S K Nickel B G 1972 Phys. Rev. Lett. 29 917