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Magnetothermal Properties and Magnetocaloric Effect in R 3 Co 11 B 4

We present a study on the magnetic properties and magnetocaloric effect (MCE) in R 3 Co 11 B 4 , where R = Pr, Nd, Tb, Dy, and Ho. The two-sublattice model is used for calculating magnetization, magnetic heat capacity, isothermal entropy change ∆S m , and adiabatic temperature change ∆T ad , for different magnetic field changes ∆H = 1.5, 3, and 5 T and at temperatures up to 600 K. Direct and inverse MCE are shown to take place in the ferrimagnetic compounds with R = Tb, Dy, and Ho. The maximum isothermal magnetic entropy change and maximum adiabatic temperature change have been calculated for ferromagnetic Nd 3 Co 11 B 4 compound to be 1.85 J/K mol and 6.5 K at T c = 432 K, for a field change ∆H = 5 T. The relative cooling power (RCP) is in the 44–161 J/mol range for the same field change. Also, the type of phase transition is investigated in the light of Arrott plots, universal curves, and the features of the temperature and field dependences of the magnetization, heat capacity, entropy, and the magnetocaloric properties. Those features confirm that the transition at the Curie temperature of these compounds is of the second order.


Introduction
The magnetocaloric effect has been discussed before for rare earth inter-metallic compounds [1][2][3]. A large variety of functional materials useful for technological applications such as magnetic refrigeration has been reported. Magnetic refrigeration is hoped to be a good, more efficient, and environmentally safer alternative to traditional refrigeration [4][5][6]. Many studies have been done to investigate the room-temperature magnetocaloric effect (MCE) materials such as La 1-x Li x MnO 3 [7], Gd 5 Si 2 Ge 2 [8], NiMnIn [9], and NiMnSb [10]. Also, the MEC was studied on other materials such as RCo 2 [11], RCuAl [12], RMn 2 Si 2 , and HoCoSi [13]. The direct and inverse MCE have been observed in Heusler alloys [14] and ferrimagnetic materials such as RFe 2 [15,16] and DySb [17]. The magnetic properties of R 3 Co 11 B 4 have been reported by Tetean et al. [18] and Xiang-Mu et al. [19]. In this work, we present a theoretical calculation of the magnetothermal properties and the magnetocaloric effect in R 3 Co 11 B 4 compounds. In particular, we report on the isothermal change in entropy, the adiabatic change in temperature, and the relative cooling power in fields up to 5 T and at temperatures up to and above the Curie temperature of the compounds under study.

Model and Analysis
The exchange field of two sublattices systems can be expressed as follows [20,21]: where H is the external applied magnetic field, and M R (T) and M Co (T) are the magnetic moments for rare earth and cobalt, respectively, at temperature T. The factor d = N A ρ μ B /A converts the moment per formula unit of R 3 Co 11 B 4 from μ B to A/m, where ρ is the density of the compound in kg/m 3 , N A is Avogadro's number, and A is a formula unit weight of compound in kg per mole. The molecular field coefficients n RR , n CoCo , and n RCo are dimensionless.
M Co (0) and M R (0) are the magnetic moments of cobalt and rare earth at T = 0 K, respectively, and M R (0) = g R J R , where g and J have their usual meaning. M co (0) is, however, obtained from the experimental low temperature data. B J (y) is the Brillouin function: and y = μH kT The total magnetization can be calculated from: The magnetic entropy change is calculated from the thermodynamic Maxwell relation: The magnetic entropy change is calculated also by using trapezoidal rule, as reported, for example, by Tishin et al. [22].
where M(H) is the isothermal magnetization. A universal curve [23] is the relation between △S m /△S peak m and ϴ, where ϴ is defined as follows: One may choose the reference temperature T r such that [23]: The total heat capacity C tot includes three contributions: the magnetic heat capacity C m , the electronic heat capacity C e , and the lattice heat capacity C l [24,25]: First, we calculate the magnetic energy for the system in order to calculate the magnetic contribution to heat capacity:  The magnetic contribution to heat capacity is calculated from the temperature-derivative of magnetic energy: Second, the electronic contribution to heat capacity is proportional to temperature and may become a dominant term at very low temperatures; the electronic heat capacity is given by [26]: where γ e is the electronic heat-capacity coefficient and N (E f ) is the density-of-states at Fermi energy. Third, the lattice contribution to heat capacity is expressed in the Debye model, as follows: where x = ϴ D /T and ϴ D is Debye temperature.
The adiabatic temperature change [24] can be calculated from the following: According to Arrot-Belov-Kouvel (ABK) [27,28], the Arrott plots M 2 vs. H/M and M 2 vs. ∆S m in the ferromagnetic region at different temperatures, close to T c , can be T dH used to estimate the spontaneous magnetization and the Curie temperature. From Landau-Ginsburg theory [29], Gibb's free energy is expressed as follows: The magnetic entropy change obtained from Gibb's free energy: From the equilibrium condition at T c , F M = 0 the magnetic equation of state is: where A(T) and B(T) are Landau's coefficients.
The relative cooling power (RCP) [30] is defined as follows: where ∆S max (T) is the maximum magnetic entropy change and δT FWHM is the full width at half maximum of the magnetic entropy change curve.

Magnetization
In this study, we calculated the temperature dependences of the magnetization of the rare-earth, Co, and R 3 Co 11 B 4 system, where R = Pr, Nd, Tb, Dy, and Ho. Ferromagnetic coupling is present in compounds with R = Nd and Pr, whereas compounds with R = Tb, Dy, and Ho show ferrimagnetic coupling with compensation points. Figure 1(a) exhibits the magnetization of the two sublattices of Nd and Co, and the total magnetization of the compound. Figure 1(b) shows the same but for the ferrimagnetic compound with R = Dy where a compensation point is present. We may remark that the magnetic moments, at 0 K, for the Nd and Dy atoms are as follows: M Nd (0) = g Nd × J Nd = (8/11) × (7/2) = 2.54 μ B /atom and M Dy (0) = g Dy × J Dy = (4/3) × (15/2) = 10 μ B / atom. It is known that the nature of the magnetic moment in R-atoms is localized whereas that of the 3d-transition elements (Co in the present study) is relatively itinerant. Therefore, the magnetic moment of the cobalt sublattice is determined from the experimental data of the total magnetization and from the localized moments of the R-elements, as shown above. In the Nd compound, the cobalt moment per atom is M Co (0) = 4.4/11 = 0.4 μ B , but it is different in the Dy compound, i.e., M Co (0) = 13.36/11 = 1.214 μ B [19]. We may recall that, in the elemental cobalt, the magnetic moment of a Co atom, at 0 K, is 1.7 μ B ; however, it might be drastically different in some R-Co compounds; e.g., in RCo 2 compounds, the Co moment vanishes [31]. For the elemental Dy, however, M Dy (0) = 10 μ B , this same value of M Dy (0), in the Dy 3 Co 11 B 4 compound, led to the observed experimental total magnetization of the compound. Results of total magnetization and Curie temperatures, for the R 3 Co 11 B 4 system, are shown in Table 1. In this table, the percentage difference between our calculated magnetic moment, at very low temperatures, and the experimentally determined moments [18] is displayed. This difference is only ≤ 0.6%. The corresponding difference in the Curie temperature data is ≤ 7%. We may also refer to the value of the magnetic moment of the Nd 3 Co 11 B 4 compound, as evaluated by the band calculation [32], i.e., 15.87 µ B /f.u which is about 30% off the experimental value 12.2 µ B /f.u reported in the same reference. It is worth mentioning that, in our mean-field calculation of M (T, H), we picked up initial magnetic moment values, at temperatures (with a step of 1 K) up to Curie temperature and in a specific field, and checked, in an iteration process, whether they are possible simultaneous solutions to the well-known Brillouin functions. This iteration process was performed with an accuracy of 10 −7 .

Total Heat Capacity
Basically, the calculation of the adiabatic change in temperature depends on both of the temperature change of the magnetization and total heat capacity (Eq. 16). There are three contributions to the total heat capacity for R 3 Co 11 B 4 ,  as we mentioned before in Eq. (11). First, the magnetic heat capacity has been calculated from Eq. (13), and is shown, for different magnetic fields: 0, 1.5, 3, and 5 T, for Nd 3 Co 11 B 4 and Ho 3 Co 11 B 4 in Fig. 2a and b, respectively. Second, electronic heat capacity is calculated from the coefficient γ e , which is obtained from the materials project [33], A Szajek [34], and A Kowalczyk et al. [32] as shown in

The Isothermal Entropy Change
The isothermal change in entropy has been calculated by two methods: by Maxwell's relation (Eq. 7) and by the trapezoidal method (Eq. 8). Figure 3(a) shows the isothermal magnetic entropy change for R = Nd. Only direct MCE is present in this (and in the R = Pr) ferromagnetic system. For the ferrimagnetic compound with R = Dy, the isothermal magnetic entropy, for different magnetic field changes, is shown in Fig. 3(b) and it exhibits both direct and inverse MCE; i.e., two peaks are present: the first one at the Curie temperature, and the second one at a temperature below the compensation temperature. Comparing data of △S m , using Maxwell relation and the trapezoidal method, showed a reasonable agreement between the two methods, in particular for small field changes ( Table 2). In the trapezoidal method, which casts the Maxwell integral into a summation, we chose small field changes in order for the summation to be as accurate as possible. For the sake of comparison with bench-mark materials and other R 3 Co 11 B 4 compounds, e.g., (Gd x Y 1-x ) 3 Co 11 B 4 , we compare our results of ΔS m , which is in the range 0.7 to 2.5 J/ mol K at field 5 T, with that of Gd metal, i.e., 1.48 J/mol K at H = 5 T as reported by Wang et al. [36] and with the work of Burzo et al. on (Gd x Y 1-x ) 3 Co 11 B 4 [37], who reported a ΔS m about 2.4 J/ mol K at ΔH = 3 T.

Adiabatic Temperature Change
In this part, we report on the adiabatic temperature change.
Because of a weak dependence of the total heat capacity on the applied magnetic field, around T c for the studied compounds, for example, for R = Dy (Fig. 4), the term T/C tot is taken out of the integral in Eq. 16. Figure 5(a and b) shows   Table 3. The maximum value for ΔT is 6.5 K, for an applied magnetic field 5 T, in case of Nd 3 Co 11 B 4 ; i.e., the temperature is decreasing by a rate of 1.3 K/T.

The Relative Cooling Power
The relative cooling power is a figure-of-merit in the field of magnetic refrigeration and is calculated, for different field changes by Eq. (21). The RCP increases with increasing the applied magnetic field, and the results are summarized in   Figure 8 shows the universal curve of Dy 3 Co 11 B 4 in applied magnetic fields of 1.5, 3, and 5 T. The normalized magnetic entropy change curves fairly collapse onto a unique curve, which confirms that the phase transition in the R 3 Co 11 B 4 system is of the second order. Figure 9 shows the field dependence of the magnetic entropy change for R 3 Co 11 B 4 system. According to the MFT, the relation ΔS m vs. (H/T C ) 2/3 is another criterion for existence of a second-order magnetic phase transition [38,39].

Conclusion
We calculated the magnetothermal and MC properties, △S m and △T ad for R 3 Co 11 B 4 system using the meanfield theory. Magnetization calculations showed that compounds with R = Tb, Dy, and Ho are ferrimagnetic compounds, whereas those with R = Pr and Nd are ferromagnetic. △S m is calculated by two methods: Maxwell's relation and the trapezoidal method. The two methods give fairly similar results. The highest ordinary MCE △S m , △T ad , and RCP are 2.5 J/mol K, 6.6 K, and 161 J/mol for R = Nd for a magnetic field change of 5 T. The temperature and field dependencies of the magnetic properties, △S m , △T ad , Arrott plots, and the universal curves showed that the phase transition in these compounds is of the second order. The mean-field analysis is suitable for studying the magnetothermal properties and magnetocaloric effect of the R 3 Co 11 B 4 system. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.