Magnetocaloric Properties in Gd₃Ni₂ and Gd₃CoNi Systems

Abstract

Intermetallic Gd3Ni2 and Gd3CoNi undergo second-order ferromagnetic paramagnetic phase transition at the Curie temperature, TC. They exhibit a large magnetocaloric effect (MCE). This MCE is manifested with a high entropic peak of 8 and 8.3 J.Kg−1 K−1, at the vicinity TC under 5 T magnetic applied field for Gd3Ni2 and Gd3CoNi, respectively. With their boosted MCE and large refrigerant capacity, Gd3Ni2 and Gd3CoNi compounds can be a candidate as a magnetocaloric refrigerator which is still one of the current research projects recommended by the low energy consumption and low environmental impact of these devices. Based on the Landau theory, Gibb’s free energy leads to determine temperature-dependent parameters which correspond to the electron condensation energy and magnetoelastic coupling and the magnetic entropy change which is a very crucial parameter to evaluate the MCE of a given magnetic system.

1. Introduction

The study of magnetic materials having boosted magnetocaloric effect (MCE) and large refrigerant capacity applied in low- and room-temperature magnetocaloric refrigerators is one of the current research projects recommended by the low energy consumption and the safe environmental impact of these materials [1, 2, 3, 4, 5]. The MCE is observed when magnetic systems are subjected to an external magnetic field. For a ferromagnet, in an adiabatic process, the MCE presents itself as follows: when an external magnetic field is applied to the ferromagnet the temperature increases and decreases when this magnetic field is removed. From this, a famous quantity can characterize the MCE which is the magnetic entropy change, $-∆S_M$. From$-∆S_M$, one may evaluate the refrigerant capacity of the material, which expresses the exchanged heat in a thermodynamic cycle of magnetic refrigerators. The optimization and development of magnetic refrigerator devices depend on a solid thermodynamic description of the magnetic material, and its properties throughout the steps of the cooling cycles [6, 7]. Among these magnetic systems, intermetallic alloys formed by rare earth (R) and transition metal (M) such as Gd₃Ni₂, Lu₂Pd₅ and Nd₂Co_1.7 [8, 9, 10]. The projected applications of these materials include magnetic refrigeration, magnetic memory, spintronics and magnetic sensors, etc. Rare earth intermetallic alloys have been attractive for researchers because of their richness in their role in a variety of applications and fundamental physics. Due to their highly localized unpaired 4f electrons, the rare-earth ions in these solid systems can retain their atomic moments. As a result, a very large magnetic moments range can be established. Also, the rare-earth atoms in these samples are heavy, the spin-orbit interaction dominates to be responsible for strong magneto crystalline anisotropy [11]. Recently, Provino et al. [12] reported the crystal structure, thermal stability, magnetic behavior and MCE of Gd₃Ni₂ and Gd₃CoNi compounds.

Generally, SO ferromagnetic-paramagnetic (FM-PM) phase transitions are one of the vital issues related to the functionalities and fundamental physics of magnetic systems. The Landau theory for phase transitions was used to describe the MCE in Gd₃Ni₂ and Gd₃CoNi systems with magnetoelastic and magnetoelectronic couplings [13, 14, 15, 16]. As shown in the work of Provino et al. [12], for Gd₃Ni₂ and Gd₃CoNi compounds, the applied magnetic field, H, dependence on peaks of $-∆S_M$ obtained near the Curie temperature, $T_C$, increases proportionally ($-∆S_M$∼$H^{\frac{2}{3}}$). This behavior is matching with the mean-field theory (MFT). Moreover, the MFT establishes relations between $-∆S_M$ and magnetization, $M$ [17]. In addition, the theory of critical phenomena indicates the existence of universal magnetocaloric behavior in materials undergoing SO FM-PM [18, 19, 20]. However, the critical exponents can set the behavior magnetic phase transitions.

Since Gd₃Ni₂ and Gd₃CoNi magnetic materials can be described by the MFT, we chose to study, in this paper, the MCE of these samples using both the Landau model and MFT. These two approaches provide side by side the estimation of both spontaneous magnetization, $M_s$ and $-∆S_M$. First, we used the $-∆S_M$ values deduced from isothermal magnetization measurements to sort out the $M_s$ using the MFT. Results are then compared with those determined from the Arrott plots extrapolation ($\frac{\mathrm{H}}{\mathrm{M}}$ vs. ${\mathrm{M}}^2$). Second, the Landau theory was applied to estimate the Gibbs free energy, $G$ and$-∆S_M$ near $T_C$. Generated$-∆S_M$ results were compared with the ones estimated using the classical Maxwell relation.

2. Theory

Based on the Landau theory, the Gibb’s free energy reads as [15]:

$$G\left(T,M\right)=G_0+\frac{1}{2}A\left(T\right)M^2+\frac{1}{4}B\left(T\right)M^4+\frac{1}{6}C\left(T\right)M^6-\mathit{MH}$$E1

where the coefficients $A\left(T\right)$, $B\left(T\right)$ and $C\left(T\right)$ are temperature-dependent parameters that correspond to the electron condensation energy and magnetoelastic coupling, $M$ is the magnetization and $H$ is the magnetic applied field. At the equilibrium condition, $\frac{\mathrm{\partial G}}{\mathrm{\partial M}}=0$, the magnetic equation of the state is obtained as:

$$\frac{\mathrm{H}}{\mathrm{M}}=\mathrm{A}\left(\mathrm{T}\right)+\mathrm{B}\left(\mathrm{T}\right){\mathrm{M}}^2+\mathrm{C}\left(\mathrm{T}\right){\mathrm{M}}^4$$E2

The magnetic entropy is obtained as:

$$-∆S_M\left(T,M\right)={\left(\frac{\partial G\left(H,T\right)}{\partial T}\right)}_H=\frac{1}{2}{A}^{\prime }{M}^2+\frac{1}{4}{B}^{\prime }{M}^4+\frac{1}{6}{C}^{\prime }{M}^6$$E3

where $A^{\prime }=\frac{\partial A}{\partial T}$, $B^{\prime }=\frac{\partial B}{\partial T}$ and $C^{\prime }=\frac{\partial C}{\partial T}$.

According to the renormalization group approach to scaling, Dong et al. [21]

have reported that the zero-field spontaneous magnetization, $M_s$, Consequently, −$∆S_M$ should not be null. Then, Eq. (3) can be rewritten as:

$$-∆S_M\left(T,M\right)=\frac{1}{2}{A}^{\prime }\left(M^2-{M_s}^2\right)+\frac{1}{4}{B}^{\prime }\left(M^4-{M_s}^4\right)+\frac{1}{6}{C}^{\prime }\left(M^6-{M_s}^6\right)$$E4

To estimate the zero-field spontaneous magnetization, $M_s$, we have a look on the expression of the magnetic entropy from the mean-field theory [9]:

$$S\left(\sigma \right)=-N{k}_B\left[\mathit{ln}\left(2J+1\right)-\mathit{ln}\left[\frac{\mathit{sinh}\left(\frac{2J+1}{2J}{B}_J^{-1}\left(\sigma \right)\right)}{\mathit{sinh}\left(\frac{1}{2J}{B}_J^{-1}\left(\sigma \right)\right)}\right]+B_J^{-1}\left(\sigma \right)\sigma \right]$$E5

where N is the number of magnetic moments, k_B is the Boltzmann constant, J is the angular spin value, $\mathrm{\sigma }$ is the reduced magnetization ($\sigma =\frac{M}{M_0}$, $M_0=\mathit{NJg}{\mu }_B$: saturation magnetization) and B_J is the Brillouin function for a given J value. For small $M$ values, Eq. (5) can be performed using a power expansion, and the magnetic entropy change$-∆S_M$ is proportional to ${\sigma }^2\left(M^2\right)$:

$$-∆S_M\left(\sigma \right)=\frac{3J}{2J+1}N{k}_B{\sigma }^2\left(M^2\right)+O\left({\sigma }^4\right)$$E6

Below $T_C$, the ferromagnetic materials acquire $M_{\mathit{\text{spont}}}$, as a result, the σ = 0 state is never reached. Then, the contribution of the reduced spontaneous magnetization ${\sigma }_{\mathit{\text{spont}}}=\frac{M_{\mathit{\text{spont}}}}{M_0}$ should be added. Consequently, if we consider only the first term of Eq. (2), the magnetic entropy change may be written as:

$$-∆S_M\left(\sigma \right)=\frac{3}{2}\frac{J}{\left(J+1\right)M_0}N{k}_B\left(M^2-{M_s}^2\right)$$E7

3. Results and discussions

Figure 1 presents the isothermal $-∆S_M$ vs. $M^2$ plots, in the ferromagnetic region ($T<T_C$). Curves (black symbols) present horizontal drift from the origin, corresponding to the value of ${M_s}^2\left(T\right)$.

Figure 1.

Linear fits of $-∆S_M$ vs. $M^2$ plots for the Gd₃Ni₂ and Gd₃CoNi alloys.

As shown in Figures 1 and 2, all curves at different temperatures obey the same regularity and a series of linear dependence with an approximately constant slope occurs. This indicates that it is possible to analyze the current experimental results with the mean-field theory.

Figure 2.

Linear fits of $\frac{H}{M}$ vs. $M^2$ plots for the Gd₃Ni₂ and Gd₃CoNi alloys.

Linear fits are applied on the isothermal -$∆S_M$ vs. $M^2$ plots inside the ferromagnetic region to sort out $M_s$. The same stuff is following to obtain $M_s$ from the Arrott plots: $\frac{\mathrm{H}}{\mathrm{M}}$ vs. ${\mathrm{M}}^2$ in Figure 2 for the Gd₃Ni₂ and Gd₃CoNi systems.

Figure 3 shows practically the same curve $M_s$ vs. $T$ from $-∆S_M$ vs. $M^2$ (red symbols) and from Arrott plots (black symbols).

Figure 3.

$M_s$ vs. $T$ from $-∆S_M$ vs. $M^2$ (red symbols) and from Arrott plots (black symbols) for the Gd₃Ni₂ and Gd₃CoNi samples.

As seen in Figure 3, as the temperature decreases, the spontaneous magnetization becomes larger, suggesting that the systems are approaching a spin ordering state and a strong localization of moments is formed.

Based on the scaling hypothesis, the critical exponent, β, the reduced temperature, $\epsilon =\frac{T-T_C}{T_C}$, and the saturation magnetization, $M_0$, as [22]:

$$M_{\mathit{\text{spont}}}\left(T\right)\propto M_0{\left(-\epsilon \right)}^{\beta }.$$E8

By changing Eq. (8) to log–log scale, the value of β corresponds to the slope of the curve $ln\left(M_s\right)$ vs. $ln\left(-\epsilon \right)$ in Figure 4.

Figure 4.

Linear fit of $\mathit{ln}\left(M_s\right)$ vs. $\mathit{ln}\left(-\epsilon \right)$for the Gd₃Ni₂ and Gd₃CoNi samples.

The value of the exponent β, is found to be 0.49 with Gd₃Ni₂ and 0.47 with Gd₃CoNi. The β values are consistent with the standard mean-field model (β = 0.5 [22]).

In the next, Fitting the Arrott plots in Figure 5 gives the parameters $A\left(T\right)$, $B\left(T\right)$, shown in Figure 6, and $C\left(T\right)$ shown in Figure 7 for the Gd₃Ni₂ and Gd₃CoNi compounds.

Figure 5.

Temperature dependence of Landau coefficients $A$, $B$ for the Gd₃Ni₂ and Gd₃CoNi samples.

Figure 6.

Temperature dependence of Landau coefficient $C$ for the Gd₃Ni₂ and Gd₃CoNi samples.

Figure 7.

Temperature dependence of $∆G=G-G_0$ under $H=1$ to 5 T for the Gd₃Ni₂ and Gd₃CoNi alloys.

The $A\left(T\right)$ curve is positive and would get a minimum value at the vicinity of $T_C$. On the other hand, the magnetic phase transition order is governed by the sign of $B\left(T\right)$ at the transition: a SO occurs when $B\left(T\right)\ge 0$ while a first-order transition happens if $B\left(T\right)<0$. In this work, the positive sign of $B\left(T_C\right)$ indicates a SO magnetic phase transition for the Gd₃Ni₂ and Gd₃CoNi compounds. Besides, C(T) is positive at $T_C$ but in other cases, it is negative or positive. After sorting $A\left(T\right)$, $B\left(T\right)$, and $C\left(T\right)$, Gibb’s free energy change, $∆G=G-G_0$ can be estimated. The temperature dependence of $∆G$ under $H=1$ to 5 T is plotted in Figure 7.

As shown in Figure 7, $∆G\left(T\right)$ changes quickly from the high absolute values to the low ones while going from the FM to PM region. The first principle of thermodynamics indicates that there is conservation of energy and in this case, if the internal energy of the system varies, it is because there is an exchange of energy with the external environment either in the form of work or in the form of heat.

The temperature dependence of $\left(-∆{\mathrm{S}}_{\mathrm{M}}\right)$ is calculated using Eq. (4), under magnetic field varying from 1 to 5 T.

Figure 8 shows a good agreement between the Landau plots (red lines) and the experimental plots of $\left(-∆{\mathrm{S}}_{\mathrm{M}}\right)$ vs. T for the Gd₃Ni₂ and Gd₃CoNi alloys.

Figure 8.

Comparison between experimental (black symbols) and simulated (red lines)$-∆{\mathrm{S}}_{\mathrm{M}}\left(T\right)$ using Landau theory under $H=1$ to 5 T for the Gd₃Ni₂ and Gd₃CoNi compounds.

For the Gd₃Ni₂ and Gd₃CoNi compounds, the peak of $-∆{\mathrm{S}}_{\mathrm{M}}$ is achieved near their $T_C$. Under 1 T applied magnetic field, the entropic peak is about 2.1 and 2.3 J.Kg⁻¹ K⁻¹ or under 5 T, it increases to be 8 and 8.3 J.Kg⁻¹ K⁻¹ for Gd₃Ni₂ and Gd₃CoNi, respectively. These alloys exhibit relatively large MCE at intermediate temperatures. The $-∆{\mathrm{S}}_{\mathrm{M}}$ is not the only parameter to quantify the potential of a magnetic refrigerant: the cooling power or the refrigerant capacity, RC, is another important quantity. The RC quantifies the efficiency of a magnetic system in terms of the energy transfer between the cold and the hot reservoir in a perfect thermodynamic refrigeration cycle. It estimates the transferred heat between the hot and the cold ends; so, for practical applications, a boost RC over a wide temperature range coupled with hight MCE is desirable. The RC values can be calculated by integrating the area of the $-∆{\mathrm{S}}_{\mathrm{M}}$ vs$.T$ curves (between $T_1$ and $T_2$) as $\mathit{RC}={\int }_{T_1\left(\mathit{\text{cold}}\right)}^{T_2\left(\mathit{hot}\right)}\left|∆{\mathrm{S}}_{\mathrm{M}}\right|\mathit{dT}$, where ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ represent the temperatures of the hot and cold reservoir, respectively. This returns to assume ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ the low and hot temperatures, respectively, at full width at half maximum (FWHM) of $-∆{\mathrm{S}}_{\mathrm{M}}$. For the two compounds, the RC values are of the order of about 540 JKg⁻¹ under 5 T magnetic field. This high RC selects the intermetallic Gd₃Ni₂ and Gd₃CoNi compounds as a good magnetic refrigerator.

4. Conclusion

In this work, we used the derivative of the Gibbs free energy to estimate the magnetic entropy change and the mean-field theory to sort out the spontaneous magnetization from the dependence of magnetic entropy change on magnetization, $\left(-∆{\mathrm{S}}_{\mathrm{M}}\right)$ vs. M² in Gd₃Ni₂ and Gd₃CoNi systems. The obtained spontaneous magnetization values are in good agreement with those found from the extrapolation from the Arrott plots (H/M vs. M²). An excellent agreement has been found between the $\left(-∆{\mathrm{S}}_{\mathrm{M}}\right)$ values estimated by Landau theory and those obtained using the classical Maxwell relation. The intermetallic Gd₃Ni₂ and Gd₃CoNi exhibit a large MCE manifested with hight entropic peak at the vicinity of $T_C$ and boost refrigerant capacity which make these alloys as a good candidate for magnetic refrigerator.

Conflict of interest

The authors declare no conflict of interest.