Double perovskite of the general formula A′A″B′B″X₆; where the (A′A″)-sites are occupied by rare or alkaline earth ions, (B′B″)-sites by transition metal ions and (X)-sites by oxygen or halide ions, have been known for several decades.^[1,2] The structure, magnetic, and dielectric properties of the double perovskite oxides, R₂NiMnO₆ (R = Pr, Nd, Sm, Gd, Tb, Dy, Ho, and Y) are investigated in Ref. [3]. La₂NiMnO₆, exhibiting a ferromagnetic Curie temperature T_C of 280 K, has received considerable attention recently,^[4–7] not only because of the possibility of combining multiple electronic properties (ferromagnetism, magnetoresistance, magneto capacitance, and semiconductivity) in this material, but also because of the expectation that a fundamental understanding of the Ni²⁺–O–Mn⁴⁺ electronic interaction would provide new guidelines for designing the multiple property materials. The structural, electronic, and magnetic properties of the double perovskite Pb₂FeReO₆ have been studied by using the first principles.^[8] The magnetic ordering was explained as the superexchange interaction between Ni²⁺(d⁸, S = 1) and Mn⁴⁺ (d³, S = 3/2) ions.^[7,9] Compared with other ferromagnetic semiconductors and insulators which only exhibit magnetic ordering at very low temperatures, La₂NiMnO₆ demonstrates ferromagnetic ordering near room temperature, as high as 280 K.^[4,10] The magnetic hysteresis displacement, thermal inversion of the magnetization, hysteresis loops jumps, and crossing branches of hysteresis loops at low magnetic fields are reviewed.^[11] The electronic and magnetic structures for the ferromagnetic and antiferromagnetic states of La₂MnNiO₆ and the site-projected density of states and electronic dispersion curves are analyzed in Ref. [12]. La₂NiMnO₆ is a ferromagnetic insulator with a positive superexchange interaction between Ni and Mn cations.^{[5,6,13–15]} The influences of bismuth on the magnetic and electrical properties of La₂MnNiO₆ are studied in Ref. [16]. The Monte Carlo data by finite-size scaling, and the location of a line of critical points of the Ising universality class are analyzed.^[17] The magneto-caloric effect of a Gd₅₅Co₂₅Al₁₈Sn₂ bulk metallic glass is investigated.^[18] The crystal field and external magnetic field-split ground state of Dy³⁺ in Dy₃Al₅O₁₂ are calculated based on the quantum theory.^[19] In this paper, theoretical work on the dependence of the magnetization on temperature for La₂MnNiO₆ compound is simulated to predict magnetocaloric properties such as magnetic entropy change, relative cooling power, and heat capacity change. The thermal magnetization, dM/dT, specific heat, and magnetic entropy are given for a fixed magnetic parameter. The transition temperature T_C values of La₂NiMnO₆ double perovskite are deduced for different values of external magnetic field. The thermal total magnetization, specific heat, and magnetic entropy are established with the external magnetic field h. We also give the adiabatic temperatures with different values of external magnetic field. Finally, the field dependence of relative cooling power (RCP) is obtained.

The Hamiltonian of an Ising model with the La₂NiMnO₆ double perovskite includes nearest neighbor interactions and external magnetic field and is given as

The La₂NiMnO₆ double perovskite as given in Fig. 1 is assumed to reside in the unit cell and the system consists of the total number of spins N = N_Mn + N_Ni, with N_Mn = N_Ni = 1024.

Fig. 1.

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	Fig. 1. Double perovskite La2NiMnO6 structure.

We use a standard sampling method to simulate the Hamiltonian given by Eq. (1). Cyclic boundary conditions on the lattice are imposed and the configurations are generated by sequentially traversing the lattice and making single-spin flip attempts. The flips are accepted or rejected according to a heat-bath algorithm under the Metropolis approximation. Our data are generated with 10⁵ Monte Carlo steps per spin, discarding the first 10⁴ Monte Carlo simulations. Starting from different initial conditions, we perform the average of each parameter and estimate the Monte Carlo simulations, averaging over many initial conditions. Our program calculates the following parameters:

In a previous work, we have studied the magnetocaloric effect in NdMnO₃ perovskite by using the Monte Carlo simulations.^[20] La₂NiMnO₆ is a double perovskite structure and is so named because the unit cell is twice that of perovskite. They do not have the same structures. In NdMnO₃, we consider the first nearest neighbor but in the La₂NiMnO₆ system we consider the first and the second exchange interactions for each atom.

Shown in Fig. 2(a) are the thermal total magnetizations of La₂NiMnO₆ and thermal magnetization of Ni and Mn. The dM/dT, specific heat, and magnetic entropy of La₂NiMnO₆ are given in Figs. 2(b)–2(d), respectively under h = 0. The ferromagnetic (FM) transition temperature is calculated from the minimum position of the (dM/dT) versus temperature curve. The observed ferromagnetism could be explained within the framework of FM interaction between the Mn and Ni ions.^[6] The transition temperature obtained is T_C = 275 K. The transition from a paramagnetic (PM) state to an FM state around T_C ≈ 275 K for the parent phase as reported in Refs. [6] and [16]. The near room temperature interaction and superexchange interaction between Ni⁺² and Mn⁺⁴ are shown to be due to the ferromagnetism of LNMO. This behavior is observed in the previous work.^[21] The double perovskite La₂NiMnO₆ is ferromagnetic at transition temperature T_C = 280 K, which has been predicted according to the Goodenough–Kanamori rules and confirmed by the experimental data.^[15,22–24]

Fig. 2.

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	Fig. 2. Thermal magnetizations (a), dM/dT (b), the specific heat (c), and magnetic entropy (d) of La2NiMnO6 for h = 0.

The magnetic entropy increases with increasing temperature until the transition temperature T_C, which corresponds to the maximum value of the magnetic entropy. The thermal total magnetization and dM/dT of La₂NiMnO₆ are shown in Figs. 3(a) and 3(b), respectively for external magnetic field h = 1, 4, 7, and 10 T. The total magnetization increases with the external magnetic field increasing. On the other hand, Blasse argued that ferromagnetism in La₂NiMnO₆ is entirely due to Mn⁴⁺–O–Ni²⁺ superexchange interactions,^[16] which was later concluded^[25] from the measurement of ⁵⁵Mn nuclear magnetic resonance line width. The curves of magnetic entropy versus the temperature of La₂NiMnO₆ are given in Fig. 4 for different values of external magnetic field h = 1, 4, 7, and 10 T. The magnetic entropy increases with increasing temperature until the transition temperature T_C and becomes constant for different values of external magnetic field.

Fig. 3.

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	Fig. 3. Thermal total magnetizations (a) and dM/dT curves (b) of La2NiMnO6 for external magnetic field h = 1, 4, 7, and 10 T.

Fig. 4.

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	Fig. 4. Curves of magnetic entropy versus the temperature of La2NiMnO6 for different values of external magnetic field h = 1, 4, 7, and 10 T.

We show in Fig. 5(a) the temperature dependences of the magnetic entropy for different values of external magnetic field h = 1, 4, 7, and 10 T. The value of ΔS_m is negative in the entire temperature range and is extended over a wide range of temperature around the Curie temperature, which is useful for above room temperature magnetic refrigeration. The values of −ΔS_m reach 8.44, 15.45, 20.65, and 25 J/(kg·K) at T_C ≈ 275 K when the applied magnetic fields are 1, 4, 7, and 10 T, respectively. The paramagnetic–ferromagnetic transition is extremely sharp, indicating its second-order phase character as given in Fig. 5(b). One can see that all the ΔS_m(T,h) data points are collapsed into a universal curve in the whole temperature range as given in Ref. [26]. As is well known, the magnetic order is a second order phase transition in the La₂NiMnO₆. Therefore, the corresponding lattice and electronic entropy change will be much smaller. These also rule out the possibility that the magnetic entropy change is partially compensated for by the lattice and electronic contributions. This behavior is observed in previous work.^[15]

Fig. 5.

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	Fig. 5. Temperature dependences of the magnetic entropy change (a) and master curve behaviors of the curve each as a function of temperature (b) for La2NiMnO6 for different values of external magnetic field h = 1, 4, 7, and 10 T.

A large change in magnetic entropy due to adiabatic magnetization is observed, which argues well for potential applications as magnetic refrigerant around liquid nitrogen temperature. Luo et al. suggested that the large magnetic entropy change in perovskite manganites could originate from the spin–lattice coupling in the magnetic ordering process.^[27] We have also presented in Fig. 6, the temperature dependences of the adiabatic temperature in La₂NiMnO₆ for different values of external magnetic field h = 1, 4, 7, and 10 T. A large magnetic entropy change is observed and the changes of magnetic entropy and adiabatic temperature at moderate magnetic fields are quite significant. A maximum adiabatic temperature change about 275 K is obtained to be close to the transition temperature with applied field h = 1, 4, 7, and 10 T. These results indicate that our samples are very good substances for magnetic refrigeration applications. To explain a large magnetic entropy change in perovskite manganite, the Zener double exchange model has been strongly recommended.^[28,29] A maximum entropy, −ΔS_M = 20.54 J · kg⁻¹ · K⁻¹ and adiabatic temperature change 4.84 K are comparable to these values: 35.5 J · kg⁻¹ · K⁻¹ (∼ 24 J · kg⁻¹ · K⁻¹, and 10.5 K (6.5 K) is observed in Gd₂NiMnO₆ (Gd₂CoMnO₆) for a magnetic field change of 7 T at low temperatures.^[30]

Fig. 6.

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	Fig. 6. Temperature dependences of the adiabatic temperature change for La2NiMnO6 for different values of external magnetic field h = 1, 4, 7, and 10 T.

Figure 7 shows the field dependence of relative cooling power (RCP) on external magnetic field for La₂NiMnO₆. It can be seen from this figure that RCP increases monotonically as the field increases, and reaches the value of 905 J·kg for h = 10 T. The RCP is another important factor for assessing the usefulness of a magnetic refrigerant material.^[31] The RCP depends not only on the magnitude of ΔS_m, but also on the temperature dependence of ΔS_m (e.g., the full width at half maximum of the ).^[32] The values obtained around T_C = 275 K increase with applied field increasing. All the parameters related to the magnetocaloric effect, such as the magnetic entropy change, relative cooling power, and refrigerant capacity depend strongly on the magnitude of the applied magnetic field h and exchange interaction between Mn and Ni ions.

Fig. 7.

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	Fig. 7. Field dependence of relative cooling power (RCP) for La2NiMnO6.

Finally, we show in Fig. 8 the hysteresis loops of La₂NiMnO₆ for different temperatures T = 260, 275, and 285 K. The coercive field decreases with temperature increasing. The same behavior has been observed experimentally.^[33] The system exhibits the superparamagnetic behavior for T ⩾ T_C. However, although the large abrupt change in magnetization causes correspondingly a giant magnetic entropy change, this appears at the expense of thermal and magnetic hysteresis, which should be avoided in order to be able to apply these materials in refrigerators appliances. The coercive field in this double perovskite is sample-dependent, which might indicate that the role of structural defects such as antiphase boundaries^[34] acting as domain pinning centres can be very important in such highly anisotropic compounds. Antiphase boundaries in double perovskite can occur when there appears a lattice shift equal to half a lattice parameter along a crystallite plane. This gives rise to a plane formed by antiferromagnetic Ni–O–Ni bonds or Mn–O–Mn bonds. The extremely large magnetic entropy change may be attributed to a large number of weakly interacting spins. These indicate that the material is a promising magnetocaloric effect (MCE) candidate for low temperature application, and possibly could make ultra-low temperatures easily achievable for most laboratories and for space application as well.

Fig. 8.

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	Fig. 8. Hysteresis loops of La2NiMnO6 for different temperatures T = 260, 275, and 285 K.

The magnetocaloric effect and the critical behavior of the double perovskitesLa₂NiMnO₆ are investigated by Monte Carlo simulations. The maximum of the magnetocaloric effect corresponds to transition temperature T_C. The La₂NiMnO₆ shows the second order PM–FM phase transition around 12 with a large magnetic entropy change over a wide range of temperature, which is comparable to the magnetic entropy changes of other manganites. The changes in magnetic entropy and the adiabatic temperature are also significant at moderate magnetic fields. Therefore, a great deal of attention has been focused on finding new materials with a large MCE and small magnetic hysteresis.^[35,36] These results are shown in our material. The exchange interaction between Mn–Ni atoms is the main mechanism that controls the magnetic and transport properties of manganite and are chosen the positive values as given in Refs. [5], [6], [13], and [15]. Finally, in order to study the nature of the paramagnetic–ferromagnetic phase transition, a master curve behavior for the temperature dependence of predicted for different maximum fields is proposed and the hysteresis loop is obtained for the temperatures superior and inferior to those of the transition temperature.