Magnetocaloric Properties in Gd3Ni2 and Gd3CoNi Systems

*Mohamed Hsini and Souhir Bouzidi*

### **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<sup>1</sup> K<sup>1</sup> , 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.

**Keywords:** magnetic energy, magnetic entropy change, magnetization, phase transition

### **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 [1–5]. 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, Δ*SM*. FromΔ*SM*, 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 Nd2Co1.7 [8–10]. 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 [13–16]. As shown in the work of Provino et al. [12], for Gd3Ni2 and Gd3CoNi compounds, the applied magnetic field, H, dependence on peaks of �Δ*SM* obtained near the Curie temperature, *TC*, increases proportionally (�Δ*SM*�*H*<sup>2</sup> 3). This behavior is matching with the mean-field theory (MFT). Moreover, the MFT establishes relations between �Δ*SM* 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– 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, *Ms* and �Δ*SM*. First, we used the �Δ*SM* values deduced from isothermal magnetization measurements to sort out the *Ms* using the MFT. Results are then compared with those determined from the Arrott plots extrapolation (<sup>H</sup> <sup>M</sup> vs. M2 ). Second, the Landau theory was applied to estimate the Gibbs free energy, *G* and�Δ*SM* near *TC*. Generated�Δ*SM* 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]:

$$G(T,M) = G\_0 + \frac{1}{2}A(T)M^2 + \frac{1}{4}B(T)M^4 + \frac{1}{6}C(T)M^6 - MH \tag{1}$$

where the coefficients *A T*ð Þ, *B T*ð Þ and *C T*ð Þ 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, <sup>∂</sup><sup>G</sup> <sup>∂</sup><sup>M</sup> ¼ 0, the magnetic equation of the state is obtained as:

$$\frac{\text{H}}{\text{M}} = \text{A(T)} + \text{B(T)}\text{M}^2 + \text{C(T)}\text{M}^4 \tag{2}$$

The magnetic entropy is obtained as:

$$-\Delta \mathbb{S}\_{\mathcal{M}}(T, M) = \left(\frac{\partial G(H, T)}{\partial T}\right)\_{H} = \frac{1}{2} A' M^2 + \frac{1}{4} B' M^4 + \frac{1}{6} C M^6 \tag{3}$$

where *<sup>A</sup>*<sup>0</sup> <sup>¼</sup> *<sup>∂</sup><sup>A</sup> <sup>∂</sup>T*, *<sup>B</sup>*<sup>0</sup> <sup>¼</sup> *<sup>∂</sup><sup>B</sup> <sup>∂</sup><sup>T</sup>* and *<sup>C</sup>*<sup>0</sup> <sup>¼</sup> *<sup>∂</sup><sup>C</sup> ∂T*. *Magnetocaloric Properties in Gd3Ni2 and Gd3CoNi Systems DOI: http://dx.doi.org/10.5772/intechopen.102065*

According to the renormalization group approach to scaling, Dong et al. [21] have reported that the zero-field spontaneous magnetization, *Ms*, Consequently, �Δ*SM* should not be null. Then, Eq. (3) can be rewritten as:

$$-\Delta \mathbf{S}\_{\mathbf{M}}(T, \mathbf{M}) = \frac{1}{2} \mathbf{A}' \left( \mathbf{M}^2 - \mathbf{M}\_s^2 \right) + \frac{1}{4} \mathbf{B}' \left( \mathbf{M}^4 - \mathbf{M}\_s^4 \right) + \frac{1}{6} \mathbf{C}' \left( \mathbf{M}^6 - \mathbf{M}\_s^6 \right) \tag{4}$$

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

$$S(\sigma) = -Nk\_B \left[ \ln \left( 2J + 1 \right) - \ln \left[ \frac{\sinh \left( \frac{2J + 1}{2l} B\_l^{-1}(\sigma) \right)}{\sinh \left( \frac{1}{2l} B\_l^{-1}(\sigma) \right)} \right] + B\_l^{-1}(\sigma)\sigma \right] \tag{5}$$

where *N* is the number of magnetic moments, *kB* is the Boltzmann constant, *J* is the angular spin value, <sup>σ</sup> is the reduced magnetization (*<sup>σ</sup>* <sup>¼</sup> *<sup>M</sup> M*<sup>0</sup> , *M*<sup>0</sup> ¼ *NJgμ<sup>B</sup>* : saturation magnetization) and *BJ* 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�Δ*SM* is proportional to *<sup>σ</sup>*<sup>2</sup> *<sup>M</sup>*<sup>2</sup> � � :

$$-\Delta \mathbf{S}\_{\mathcal{M}}(\sigma) = \frac{\mathfrak{J}}{2\mathfrak{J}+\mathbf{1}} N k\_B \sigma^2(\mathcal{M}^2) + \mathcal{O}\left(\sigma^4\right) \tag{6}$$

Below *TC*, the ferromagnetic materials acquire *Mspont*, as a result, the *σ* = 0 state is never reached. Then, the contribution of the reduced spontaneous magnetization *<sup>σ</sup>spont* <sup>¼</sup> *Mspont <sup>M</sup>*<sup>0</sup> should be added. Consequently, if we consider only the first term of Eq. (2), the magnetic entropy change may be written as:

$$-\Delta \mathbf{S}\_{\mathcal{M}}(\sigma) = \frac{3}{2} \frac{J}{(J+1)M\_0} Nk\_B \left(\mathbf{M}^2 - \mathbf{M}\_s^{-2}\right) \tag{7}$$

### **3. Results and discussions**

**Figure 1** presents the isothermal �Δ*SM* vs. *<sup>M</sup>*<sup>2</sup> plots, in the ferromagnetic region (*T* <*TC*). Curves (black symbols) present horizontal drift from the origin, corresponding to the value of *Ms* 2 ð Þ *T* .

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.

Linear fits are applied on the isothermal *-*Δ*SM* vs. *M*<sup>2</sup> plots inside the ferromagnetic region to sort out *Ms*. The same stuff is following to obtain *Ms* from the Arrott plots: <sup>H</sup> M vs. M2 in **Figure 2** for the Gd3Ni2 and Gd3CoNi systems.

**Figure 3** shows practically the same curve *Ms* vs. *<sup>T</sup>* from �Δ*SM* vs. *<sup>M</sup>*<sup>2</sup> (red symbols) and from Arrott plots (black symbols).

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.

**Figure 1.** *Linear fits of* �Δ*SM* vs. *<sup>M</sup>*<sup>2</sup> *plots for the Gd3Ni2 and Gd3CoNi alloys*.

**Figure 2.** *Linear fits of <sup>H</sup> <sup>M</sup> vs. M*<sup>2</sup> *plots for the Gd3Ni2 and Gd3CoNi alloys.*

Based on the scaling hypothesis, the critical exponent, *β,* the reduced temperature, *<sup>ε</sup>* <sup>¼</sup> *<sup>T</sup>*�*TC TC* , and the saturation magnetization, *M*0, as [22]:

$$M\_{\text{spont}}(T) \propto M\_0 (-\varepsilon)^{\beta}. \tag{8}$$

By changing Eq. (8) to log–log scale, the value of *β* corresponds to the slope of the curve ln ð Þ *Ms* vs. ln ð Þ �*ε* in **Figure 4**.

*Magnetocaloric Properties in Gd3Ni2 and Gd3CoNi Systems DOI: http://dx.doi.org/10.5772/intechopen.102065*

**Figure 3.** *Ms vs. T from* �Δ*SM vs. M*<sup>2</sup> *(red symbols) and from Arrott plots (black symbols)* for the Gd3Ni2 and Gd3CoNi samples.

**Figure 4.** *Linear fit of ln M*ð Þ*<sup>s</sup> vs. ln* ð Þ �*ε 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 T*ð Þ, *B T*ð Þ, shown in **Figure 6**, and *C T*ð Þ shown in **Figure 7** for the Gd3Ni2 and Gd3CoNi compounds.

The *A T*ð Þ curve is positive and would get a minimum value at the vicinity of *TC*. On the other hand, the magnetic phase transition order is governed by the sign of *B T*ð Þ

**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.*

at the transition: a SO occurs when *B T*ð Þ≥0 while a first-order transition happens if *B T*ð Þ<0. In this work, the positive sign of *B T*ð Þ *<sup>C</sup>* indicates a SO magnetic phase transition for the Gd3Ni2 and Gd3CoNi compounds. Besides, *C(T)* is positive at *TC* but in other cases, it is negative or positive. After sorting *A T*ð Þ*,B T*ð Þ*,* and *C T*ð Þ, Gibb's free energy change, Δ*G* ¼ *G* � *G*<sup>0</sup> can be estimated. The temperature dependence of Δ*G* under *H* ¼ 1 to 5 T is plotted in **Figure 7**.

As shown in **Figure 7**, Δ*G T*ð Þ 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 ð Þ �ΔSM is calculated using Eq. (4), under magnetic field varying from 1 to 5 T.

#### **Figure 7.** *Temperature dependence of* Δ*G* ¼ *G* � *G*<sup>0</sup> *under H* ¼ 1 *to 5* T *for the Gd3Ni2 and Gd3CoNi alloys.*

**Figure 8** shows a good agreement between the Landau plots (red lines) and the experimental plots of ð Þ �ΔSM vs. *T* for the Gd3Ni2 and Gd3CoNi alloys.

For the Gd3Ni2 and Gd3CoNi compounds, the peak of �ΔSM is achieved near their *TC*. Under 1 T applied magnetic field, the entropic peak is about 2.1 and 2.3 J.Kg�<sup>1</sup> K�<sup>1</sup> or under 5 T, it increases to be 8 and 8.3 J.Kg�<sup>1</sup> K�<sup>1</sup> for Gd3Ni2 and Gd3CoNi, respectively. These alloys exhibit relatively large MCE at intermediate temperatures. The �ΔSM 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.

#### **Figure 8.**

*Comparison between experimental (black symbols) and simulated (red lines)*�ΔSMð Þ *T using Landau theory under H* ¼ 1 *to 5* T *for the* Gd3Ni2 *and* Gd3CoNi *compounds.*

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 �ΔSM vs*:T* curves (between *<sup>T</sup>*<sup>1</sup> and *<sup>T</sup>*2) as *RC* <sup>¼</sup> <sup>Ð</sup> *<sup>T</sup>*2ð Þ *hot <sup>T</sup>*1ð Þ *cold* j j <sup>Δ</sup>SM *dT*, where T1 and T2 represent the temperatures of the hot and cold reservoir, respectively. This returns to assume T1 and T2 the low and hot temperatures, respectively, at full width at half maximum (FWHM) of �ΔSM. For the two compounds, the RC values are of the order of about 540 JKg�<sup>1</sup> under 5 T magnetic field. This high RC selects the intermetallic Gd3Ni2 and Gd3CoNi compounds as a good magnetic refrigerator.
