Simulation of the magnetocaloric effect by means of theoretical models in Gd3Ni2 and Gd3CoNi systems

In this work, using a scaling data method based on the mean-field theory (MFT), magnetization isotherms M (H, T) and magnetic entropy change -ΔSM(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \Delta S_{M} (T) $$\end{document} curves have been simulated using MFT combined with the Bean–Rodbell model which verified the second-order phase transition for Gd3Ni2 and Gd3CoNi compounds. Under a low value of magnetic field H, M (T) and -ΔSM(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \Delta S_{M} (T) $$\end{document} curves have been successfully simulated and some magnetocaloric properties have been evaluated using the phenomenological model.


Introduction
The search for new materials with a large refrigerant capacity (RC) and high magnetocaloric effect (MCE) that can be applied in room and low-temperature magnetocaloric refrigerators is still one of the actual research topics advanced by the low-energy consumption and environmental impact of these devices [1,2]. Among them, we cited intermetallic systems composed of rare earth (R) and transition metal (M) such as 'Nd 2 Co 1.7 ', 'Gd3Ni 2 ' and 'Lu 2 Pd 5 '… [3][4][5]. Recently, Provino et al. [6] have reported the crystal structure, thermal stability, magnetic behavior and MCE of Gd 3 Ni 2 and Gd 3 CoNi compounds.
As shown in the work of Provino et al. for Gd 3 Ni 2 and Gd 3 CoNi compounds [6], the magnetic entropy change (− S M ) increases proportionally to H 2/3 (H is the applied magnetic field). This is expected from the mean-field theory (MFT).
In this way, a scaling data method based on MFT [7,8] and a phenomenological model proposed by Hamad [9] are exploited to study the MCE. First, following the scaling method [10] based on the MFT combined with the Bean-Rodbell model [11], we have estimated the exchange parameter (λ), the saturation magnetization (M 0 ), the total angular momentum (J) and the gyromagnetic factor (g) of Gd 3  depend powerfully on the interatomic distances. The critical temperature T C is given as follows: where ω v−v 0 v 0 with v and v 0 being, respectively, the sample volume with and without exchange interactions and T 0 is the transition temperature. The parameter β controls the spin-lattice coupling strength. The Gibbs free energy is given then as [8]: with σ (x) B J (x), S and K being, respectively, reduced magnetization, the magnetic entropy and the compressibility.
The above free energy is minimized dG dω 0 at: The following magnetic state equation can be obtained after minimizing G, with respect to σ , [11,12]: with where the parameter η checks the nature of the magnetic phase transition: a first order for η > 1 and a second order for η < 1 [15,16].

Phenomenological model
Based on the phenomenological model [9,17], the dependence of magnetization on temperature under a low magnetic field is represented as: with M i and M f being, respectively, the initial and final values of magnetization at ferromagnetic-paramagnetic transition, A is the magnetization sensitivity dM dT in the ferromagnetic region before transition, S c is the magnetization sensitivity dM dT at T C and C M i −M f 2 − BT C ; and the magnetic entropy change of a magnetic system under arising magnetic field from 0 to final value H max is given by: S M reaches its maximum at T T C . Such a maximum can be evaluated as: The RCP is given by: being the full width at half maximum in (− S M (T )) curve.  [6]. Measures have been taken near Curie temperature, T C 147 K for Gd 3 Ni 2 and T C 176 K for Gd 3 CoNi [6]. It is clear from Fig. 1a, b that the substitution of Ni in Gd 3 Ni 2 by Co in Gd 3 CoNi favors magnetization values. Based on the work of Pecharsky et al. [18], the saturation moment reaches 7.5 μ B per Gd for Gd 3 Al 2 under 5.5 T applied field and at 5.04 K temperature. However for Gd 3 Ni 2 , it has been found 7 μ B per Gd under 6.5 T applied field and at 5 K temperature [6]. As known, Ni is magnetic but Al is not. So, the substitution of Al by Ni contributes to decrease magnetization values.

Results and discussion
To determine the mean-field exchange (H exch ) for Gd 3 Ni 2 and Gd 3 CoNi compounds, we have used the scaling data method [7]. This method consists of plotting the evolution of H T vs. 1 T , taken at constant values of magnetization M (10 emu g −1 step) from 110 to 200 K, using Fig. 1a for Gd 3 Ni 2 , and from 125 to 250 K, using b, converging in one curve, are fitted by Eq. (2), using MATLAB software to determine M 0 , J and g. A good agreement between adjusted and theoretical parameters J and g is observed (Table 1). This agreement confirms the coupling state between sublattices suggested above.  The adjusted values λ, M 0 , J and g are injected in Eq. (1) to generate simulated magnetization isotherms M(T, H ), indicated by red lines in Fig. 5a, b for Gd 3 Ni 2 and Gd 3 CoNi, respectively. As shown in this figure, generated curves correlated well with the experimental ones (black symbols).
The use of the Bean-Rodbell model started by replacing x in Eq. (1) with Y (see Eq. 13) and choosing an average value of η and T 0 to see a correlation between generated and experimental M(T ) curves. The generated M(T ) curves (red lines) correlated with the experimental ones [6] (black symbols), as shown in Fig. 6a, b, after taking average values: η 0.1 and T 0 147 K for Gd 3 Ni 2 , whereas for Gd 3 CoNi, η 0.01 and T 0 176 K. Since η < 1 for these two compounds, the second-order transition is confirmed. Using the Mean-Field Simulation Suite software [19], Eq. (8) may be resolved. Such resolution allows us to establish the evolution of − S M (T ) curves (red lines), which are represented in Fig. 7a, b with experimental − S M (T ) curves (black symbols), using Eq. (7), for Gd 3 Ni 2 and Gd 3 CoNi, respectively. A significant shift between theoretical (red lines) and experimental (black symbols) − S M (T ) is observed in this figure, probably due to the roughness of the mean-field model to describe accurately well the magnetization of system having more than one magnetic system (Gd, Ni, Co). We should note that this model did not take into account the roles played by the Jahn-Teller effect and the charge ordering, which affect the MCE value even for the case of conventional ferromagnetic systems [20,21]. Nonetheless, this model has been useful to describe qualitatively − S M (T ), particularly at temperatures near to T C [22]. Such discrepancy between the theoretical calculations and the experimental data was also reported by Balli et al. [23].     Table 2. Simulated curves (red lines) of M versus T , obtained using Eq. (14), agree well with the experimental ones (black symbols) as shown in Fig. 9a, b for Gd 3 Ni 2 and Gd 3 CoNi, respectively. In comparison, the correlation between simulated and experimental M (T ) curves under low magnetic field obtained from the phenomenological model is better than the ones deduced after using the mean-field theory combined by the Bean-Rodbell model. This may suggest that the phenomenological model can complete the MFT under low magnetic applied fields. Generated − S M (T ) curves (red lines), in Fig. 10a, b for Gd 3 Ni 2 and Gd 3 CoNi, respectively, can be determined using Eq. (15). They are correlated with the experimental ones (black symbols) determined using Eq. (7). The − S max M , the δT FW H M and the RCP for Gd 3 Ni 2 and Gd 3 CoNi under 0.02 T are calculated using Eqs. (16) and (17). They are compared to those of other compounds in Table 3. We have noticed from Table 3 that these magnetic properties are comparable with those of (001)-oriented MnAs film [24] and La 0.8 Sr 0.05 Ca 0.15 MnO 3 [25]. While they are significantly larger than the ones of Ge 0.95 Mn 0.05 film [26] and GdCaBaCo 2 O 5.5 [27].

Conclusion
Based on the MFT, the scaling data method allows us to investigate the coupling between sublattices of Gd 3 Ni 2 and Gd 3 CoNi systems. Simulated M vs. H , M vs. T and − S M vs. T curves are achieved using the MFT combined with the Bean-Rodbell model. A good agreement between simulated and experimental M vs. H , M vs. T curves is observed. While, a significant shift has been noted between simulated and experimental − S M vs. T curves probably due to the inability of the MFT to describe accurately well the magnetization of system having more than one magnetic (Gd, Co and Ni). Under low value of magnetic field H , the phenomenological model was exploited to simulate successfully M and − S M versus T and to evaluate some magnetocaloric properties.
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