Rare-earth tuned magnetism and magnetocaloric effects in double perovskites R₂NiMnO₆

Magnetic materials have undoubtedly played an important role in advancing the technology to its commonly used current form [1]. What underlies these technologies are the key concepts or phenomena in magnetism that have been harnessed to our advantage via a careful design. Some of the technologically useful phenomena displayed by magnetic materials are, giant magnetoresistance, magnetocapacitance and magnetocaloric effect (MCE) [2–6]. At the level of fundamental physics, a common ingredient seems to be a delicate interplay of spin, charge, and lattice degrees of freedom that leads to various magnetic phase transitions, and the sensitivity of these transitions to external fields or pressure manifests in the form of a technologically useful effect [7–9].

Oxides of transition metals are perhaps the most well known materials that are equally interesting from fundamental physics as well as application points of view [10–12]. Double perovskites (DP) with formula R₂BB'O₆, where R is rare earth and B, B' are the transition metal (TM) ions, belong to this interesting class of materials [13–16]. The presence of two TM ions leads to multiplet of possibilities in terms of tuning the properties in a desired manner [17–19]. The coupling between Rare earth and TM network further enriches the low temperature magnetic properties [7].

MCE remains a topic of immense interest as it opens pathway toward realization of clean and energy efficient magnetic refrigeration technology with clear advantage over the conventional vapor-compression techniques that use potentially harmful chlorofluorocarbon gases [20]. Recent studies show a giant and reversible MCE in various magnetic materials such as MnFeP_0.45As_0.55 [21] (magnetic entropy change −ΔS_M = 18.0 J kg⁻¹ K⁻¹ for ΔH = 0–5 T), (MnNiSi)_0.56(FeNiGe)_0.44 (−ΔS_M = 70 J kg⁻¹ K⁻¹ for ΔH = 0–5 T) [22], nano-crystalline films of EuTiO₃ (−ΔS_M = 24 J kg⁻¹ K⁻¹ for ΔH = 0–2 T) [23], GdCrTeO₆ (−ΔS_M ≈ 42 J kg⁻¹ K⁻¹ for ΔH = 0–7 T) [24], Gd₅(Si_x Ge_1−x ) [25], Ni–Mn–In [26] (adiabatic temperature change ΔT_ad = 6.2 K), Gd₂NiMnO₆ [27] (−ΔS_M = 35.5 J kg⁻¹ K⁻¹ for ΔH = 0–7 T). The materials which show MCE at low temperatures can be advantageous for cryogenic magnetic cooling to obtain a sub-kelvin temperature as an alternative option of He3/He4 liquid whose prices are constantly increasing. Observation of a significant MCE in DPs opens up possibilities for using the high degree of flexibility available in these oxides to enhance the effect further. We have recently studied the magnetic and magneto-caloric properties of Nd₂NiMnO₆ and proposed that an interplay of the two magnetic sub-lattices can be used as a control knob to tune the MCE properties of magnetic materials with multiple magnetic sublattices and specifically the DPs [8]. It would be instructive to test this proposal by varying the moment size on one of the magnetic sublattices to see how the magnetic and MCE properties evolve. This motivates our present study.

In this work, we present detailed experimental investigations of magnetization and magnetocaloric behavior from 2 K to 300 K and in the field range 1 < H < 8 T, on R₂NiMnO₆ (R = Sm, Gd, Pr, Nd, Tb, and Dy). The data for Nd₂NiMnO₆ is taken from reference [8] for comparison with the other DPs. All studied DPs go through a paramagnetic to ferromagnetic phase transition due to the ordering of Ni²⁺ and Mn⁴⁺ magnetic ions. The choice of R, in addition to affecting the value of T_C, also influences the magnetization and consequently the MCE behavior as inferred from the nature of low-temperature anomalies for different R. We propose a simple explanation for this behavior in terms of a Heisenberg model that takes into account the coupling of the spin on R with the spin on Ni–Mn network, as well as the local spin–orbit coupling on R sites. We support our experimental findings via Monte Carlo simulations and mean-field analysis on the two-sublattice Heisenberg model. The remainder of the paper is organized as follows. In section 2, we present the magnetization measurements for all studied DPs. The low temperature anomalies in magnetization as well as in magnetic entropy change are presented and discussed here. In section 3, we present a two-sublattice Heisenberg model for magnetism in DPs and describe the mean-field approximation and Monte-Carlo simulation method to study the model. Further, we discuss the results obtained by these theoretical methods in this section. We summarize and conclude in section 4.

Although several previous studies have been devoted to the study of the magnetic and MCE properties of DPs. However, we believe that a detailed study of the evolution of these properties across the whole series, a tracking of these properties as a function of lattice volume and rare-earth moment size, and a theoretical estimation of the temperature dependence of these properties using a simple Heisenberg Hamiltonian which takes into account the magnetic coupling of both sub-lattices with each other, have not been attempted before. In this sense our work presents new results.

Polycrystalline samples of R₂NiMnO₆ where R = Pr, Nd, Sm, Gd, Tb, and Dy were synthesized using a solid state reaction method as described previously in references [28, 29]. It is known that synthesis conditions are important in controlling the oxygen stoichiometry and anti-site disorders. Off-stoichiometry or anti-site disorder have been shown to effect the magnetic properties significantly, leading to a much reduced magnetization and to glassy magnetic features at low temperatures and fields [8, 30–32]. The materials used in this study do not show any glassy magnetic behaviors and the magnetization is similar to the highest values reported for various DPs [30, 31, 33]. The phase purity of all DPs was examined by powder x-ray diffraction using a Rigaku Ultima IV Diffractometer. These x-ray diffraction results confirmed that all materials were single phase and crystallized in the expected crystal structure. Figure 1 shows the temperature dependent magnetization M(T) for all synthesized DPs at different fields as indicated in the plots.

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All the DPs show a rapid increase in magnetization (M) at temperatures T_C which is associated with the ferromagnetic ordering of Ni²⁺–Mn⁴⁺ sub-lattice. For higher field this upturn in magnetization moves to higher temperature which is a common feature of a ferromagnetic transition. At lower temperatures, a second anomaly is observed for all the DPs at T₂. The values of T_C and T₂ for all DPs which were studied, are given in table 1, which are consistent with the previous reports [29, 30, 34]. The nature of the low temperature anomaly depends on the R ion. For R = Nd, and Sm, there is a downturn in M at T₂ when measured in low magnetic fields. This downturn in M can be suppressed on the application of a magnetic field and changed to an upturn at larger magnetic fields as previously reported [8]. For R = Pr this downturn is not observed, possibly because the lowest field of measurement, H = 1 T is already strong enough to suppress it. This downturn in magnetization however is not a signature of long range ordering of R ions as demonstrated for Nd₂NiMnO₆ for which a low temperature heat capacity study did not show any anomaly [8] and microscopic probes like XMCD did not show any ordered moment on Nd³⁺ ions in Nd₂NiMnO₆ [33]. On the other hand, for R = Gd, Tb, and Dy there is an increase in M at T₂ at low fields which is enhanced in larger fields. Taken at face value the above observations seem to indicate that there is antiferromagnetic coupling between the R and Ni–Mn sublattice for R = Pr, Nd, and Sm, while this coupling is ferromagnetic for R = Gd, Tb, and Dy [35–37]. However, recent DFT calculations of the exchange constants in Nd₂NiMnO₆ have shown that the magnetic exchange between the Nd spin and the Ni–Mn sublattice is ferromagnetic [33]. The downturn in the magnetization can then be understood by realizing that for less than half-filled 4f shells, the orbital moment of the R is oppositely aligned to it's spin moment and is larger in magnitude than the spin moment. Therefore, although the spin is coupled ferromagnetically to the Ni–Mn sublattice, the orbital moment is opposite to the Ni–Mn sublattice resulting in a downturn in the magnetization for R = Pr, Nd, and Sm. At large enough magnetic fields, it becomes energetically favorable for the net effective magnetic moment to align with the field, which leads to a switching of the magnetization anomaly at T₂ from a downturn to an upturn as field is increased. For R = Gd, Tb, and Dy the orbital moment is in the same direction as the spin moment and so the ferromagnetic coupling between the spin of R and the Ni–Mn sublattice leads to an upturn at T₂.

Figure 2 shows the isothermal magnetization M(H) of all R₂NiMnO₆ at different temperatures as indicated in plots. All of the DPs were found to show a negligible hysteresis in magnetization down to the lowest measured temperature 2 K, indicating soft ferromagnetic behavior which is a desirable feature for the magnetic refrigeration techniques. At 300 K, in the paramagnetic state, M(H) for all the DPs varies linearly. Below T_C but above T₂, M(H) shows tendency to saturate but with a weak linear increase at large H as expected in the ferromagnetic state of Ni–Mn, while the R ions are still paramagnetic. Below T₂ the M(H) curves are complicated by the different behavior for DPs with less than (R = Pr, Nd, Sm) and more than or equal to (R = Gd, Tb, Dy) half filled 4f-shells. For R = Pr, Nd, and Sm, the M(H) at low fields is lower than for T > T₂ as seen in the downturn in the M(T) below T₂ due to the orbital moment being anti-aligned with the Ni–Mn sublattice. At higher fields for which the downturn in M(T) changes to an upturn due to field induced aligning of the orbital moment with Ni–Mn sublattice, the M(H) increases and shows a tendency to saturate. However, a weak linear increase in M(H) with field remains down to T = −2 K, suggesting that the full magnetic moment of R has still not aligned with the field. The values of observed saturation magnetization (M'_S ) at T = 2 K, H = 9 T, and the theoretically expected saturation magnetization (M_S ) if all (R, Ni, and Mn) magnetic moments have saturated are also given in table 1. The value of observed saturated magnetization for all R₂NiMnO₆ is lower than the theoretically predicted saturation magnetization except for Gd₂NiMnO₆ probably because Gd has no orbital moment (L).

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These field and temperature dependent magnetic responses also manifest themselves in the magnetic entropy change. The isotherms of magnetization M(H) were collected over a broad temperature range from 2 K to 300 K at temperature intervals of 2 K. Using these M(H) data, the change in magnetic entropy (ΔS_M ) can be calculated from Maxwell's thermodynamic relation [2]:

$$\begin{equation}{\Delta}{S}_{M}(H,T)={\int }_{0}^{H}{\left(\frac{\mathrm{d}M}{dT}\right)}_{{H}^{\prime }}\mathrm{d}{H}^{\prime }.\end{equation} \tag{ 1 }$$

Figure 3 show the magnetic entropy change (−ΔS_M ) versus temperature at different applied magnetic fields up to 8 T for all the R₂NiMnO₆ compounds. All the compounds show two prominent features. The first one at the ferromagnetic phase transition at T_C and second one at lower temperature. The value of −ΔS_M is positive for all the DPs across T_C and it's magnitude increases with field, which is consistent with ferromagnetic ordering of the Ni–Mn sublattice at T_C as indicated in the magnetization versus T and H. When the temperature is lowered, the behavior of −ΔS_M like M(T), is of two kinds. For R = Pr, Nd, and Sm, the −ΔS_M monotonically decreases below T_C and even becomes negative for R = Nd and Sm at low fields. On increasing the applied magnetic field, the negative −ΔS_M is suppressed and a positive peak or anomaly appears at low temperatures and high fields. This can be understood interms of the anti-alignment of the total effective moment of the R ions with the Ni–Mn sublattice below T₂ at low fields. So the magnetic entropy is high. At higher fields, the R moment switches and aligns with the Ni–Mn sublattice and also the applied magnetic field, resulting in a low magnetic entropy like in the case of the ferromagnetic transition at T_C. Gd₂NiMnO₆, Tb₂NiMnO₆, and Dy₂NiMnO₆ show a positive anomaly in −ΔS_M below T₂ since the effective moment of R is already tending to align with the Ni–Mn sublattice. This alignment simply increases as higher magnetic fields are applied resulting in an enhancement in the magnitude of −ΔS_M . For R = Gd, Tb, and Dy the value of −ΔS_M is very large at low temperatures, particularly for R = Gd, which make them potentially useful for low temperature magnetic refrigeration.

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Another quantity which is used to quantify the usefulness of MCE materials is called RCP which is defined as:

$$\begin{equation}\mathrm{R}\mathrm{C}\mathrm{P}=-{\Delta}{S}_{M}(T,H)\times \delta {T}_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}},\end{equation} \tag{ 2 }$$

where δT_FWHM is the full width at half maxima of −ΔS_M for a specific value of applied magnetic field. Values of −ΔS_M (at T_C), δT_FWHM, RCP, and −ΔS'_M (at low temperature below T₂) were calculated for all DPs for ΔH = 8 T and are given in table 2.

Figure 4 shows the evolution of T_C and −ΔS_M at T_C with the ionic radius of R³⁺ in R₂NiMnO₆. We find that the ferromagnetic phase transition temperature T_C increases linearly and −ΔS_M at T_C decreases monotonically with the ionic size of R³⁺ for all R₂NiMnO₆. The decrease in T_C in going from Pr to Dy is understood to arise from a decrease in the Ni–O–Mn bond angle which leads to a decrease of the super-exchange strength [39]. Additionally from table 2 we observe that the paramagnetic background of rare earth ions spreads the magnetic entropy over a large temperature range across T_C.

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3.1. Heisenberg model for R₂NiMnO₆

The experimental data discussed above presents intriguing magnetic behavior for Ni–Mn DPs. What is particularly interesting is the dramatically different magnetic response for different R at low temperatures. In order to comprehend the experimental observations, we propose a simple Heisenberg model on a body-centered cubic (BCC) lattice that takes into account spin degrees of freedom on B sites and both spin and orbital degrees of freedom on R site. The model is specified by the Hamiltonian,

$$\begin{align}\hfill \mathcal{H}& =-{J}_{1}\sum\limits _{\langle ij\enspace \rangle }\;{\mathbf{S}}_{i}^{\text{Mn}}\cdot {\mathbf{S}}_{j}^{\text{Ni}}-{J}_{2}\sum\limits _{\langle ij\enspace \rangle }\;{\mathbf{S}}_{i}^{\text{R}}\cdot {\mathbf{S}}_{j}^{\text{Mn}}\hfill \\ \hfill & \quad -{J}_{2}^{\prime }\sum\limits _{\langle ij\enspace \rangle }\;{\mathbf{S}}_{i}^{\text{R}}\cdot {\mathbf{S}}_{j}^{\text{Ni}}+\lambda \sum\limits _{i\in \mathrm{A}}\;{\mathbf{L}}_{\text{R}}^{i}\cdot {\mathbf{S}}_{i}^{\text{R}}\hfill \\ \hfill & \quad -{g}_{s}H\sum\limits _{i\in \mathrm{A},\mathrm{B}}{S}_{i,\mathrm{z}}-{g}_{L}H\sum\limits _{i\in \mathrm{A}}{L}_{i,\mathrm{z}}.\hfill \end{align} \tag{ 3 }$$

In the above, S_i with the appropriate superscript denotes the spin of the relevant Ni, Mn or R ions, and ${\mathbf{L}}_{i}^{R}$ denotes the orbital angular momentum on the rare earth ion. The model explicitly considers two coupled sublattices A and B. Spins on R³⁺ ions on sublattice A are coupled to those on sublattice B of Mn⁴⁺ and Ni²⁺. In the model, the first three terms have the summation over nearest neighbors (nn) as indicated by angular brackets. i ∈ A (B) represents lattice sites on A (B) sublattices. J₁ is the ferromagnetic coupling between Mn⁴⁺ and Ni²⁺ and ${J}_{2}({J}_{2}^{\prime })$ is the ferromagnetic coupling of R³⁺ to Mn⁴⁺ (Ni²⁺). The strength of spin–orbit coupling in rare earth ions is represented by λ. H symbolizes the strength of uniform external field which couples to the z components of the spin and orbital magnetic moments. We use g_s = 2 and g_L = 1 as the spin and orbital g-factors. In the BCC lattice, Mn⁴⁺ $(S=\frac{3}{2})$ couples to 6 Ni²⁺ (S = 1) and vice versa. Ni²⁺ couples to 8 R³⁺ and similarly coordination number of Mn⁴⁺ with R³⁺ is 8. J₁ = 1 sets the energy scale for the model. For simplicity we assume ${J}_{2}^{\prime }={J}_{2}$, and different R ions are parameterized by different choices of J₂ and λ, with the choice J₁ ≫ J₂, λ inspired by the experimental data.

3.2. Methodology

Mean-field approximation. The approximation proceeds by rewriting the model as a sum of three effective single-spin Hamiltonians as,

$$\begin{align}\hfill {\mathcal{H}}_{\mathrm{M}\mathrm{F}}& =-{g}_{s}\sum\limits _{\text{Mn}}{\mathbf{S}}^{\text{Mn}}\cdot {\mathbf{H}}_{\mathrm{e}\mathrm{ff}}^{\text{Mn}}-{g}_{s}\sum\limits _{\text{Ni}}{\mathbf{S}}^{\text{Ni}}\cdot {\mathbf{H}}_{\mathrm{e}\mathrm{ff}}^{\text{Ni}}\hfill \\ \hfill & \quad -{g}_{s}\sum\limits _{R}{\mathbf{S}}^{R}\cdot {\mathbf{H}}_{\mathrm{e}\mathrm{ff}}^{R}-{g}_{L}\sum\limits _{R}{\mathbf{L}}^{R}\cdot {\mathbf{H}}_{\mathrm{e}\mathrm{ff}}^{L}.\hfill \end{align} \tag{ 4 }$$

The effective magnetic field or the molecular field for the spins on different inequivalent sites is given by,

$$\begin{align}\hfill {\mathbf{H}}_{\mathrm{e}\mathrm{ff}}^{\text{Mn}(\mathrm{N}\mathrm{i})}& =\mathbf{H}+6\frac{{J}_{1}\langle {\mathbf{S}}^{\text{Ni}(\text{Mn})}\rangle }{{g}_{s}}+8\frac{{J}_{2}\langle {\mathbf{S}}^{R}\rangle }{{g}_{s}}\hfill \\ \hfill & =\mathbf{H}+6\frac{{J}_{1}{\mathbf{M}}^{\text{Ni}(\mathrm{M}\mathrm{n})}}{{g}_{s}^{2}}+8\frac{{J}_{2}{\mathbf{M}}_{s}^{R}}{{g}_{s}^{2}}.\hfill \end{align} \tag{ 5 }$$

In the above, we replace each spin operator S^Ni(Mn) by its average value $\langle {\mathbf{S}}_{i}^{\text{Ni}(\mathrm{M}\mathrm{n})}\rangle$ in the mean-field spirit, and the sublattice-resolved magnetization is then given by, M^Ni(Mn) = g_s ⟨S^Ni(Mn)⟩. The subscript in M_s is necessary for the R sites in order to differentiate between spin and orbital contributions. Similarly we find,

$$\begin{equation}{\mathbf{H}}_{\mathrm{e}\mathrm{ff}}^{R}=\mathbf{H}+4\frac{{J}_{2}}{{g}_{s}^{2}}[{\mathbf{M}}^{\text{Ni}}+{\mathbf{M}}^{\text{Mn}}]-\lambda \frac{{\mathbf{M}}_{L}^{R}}{{g}_{s}{g}_{L}}.\end{equation} \tag{ 6 }$$

In the above equation, we used ${M}_{L}^{R}={g}_{L}\langle {L}_{R}\rangle$. Also,

$$\begin{equation}{\mathbf{H}}_{\mathrm{e}\mathrm{ff}}^{L}=\mathbf{H}-\lambda \frac{{\mathbf{M}}_{s}^{R}}{{g}_{s}{g}_{L}}.\end{equation} \tag{ 7 }$$

Following the textbook procedure of writing statistical average ⟨S⟩ in terms of the partition function, we get magnetization on different sublattices as [38],

$$\begin{equation*}{M}_{s}^{i}={g}_{s}{S}^{i}{B}_{{S}^{i}}(x),\quad \forall \enspace i\in \left\{\mathrm{M}\mathrm{n},\mathrm{N}\mathrm{i},R\right\}\end{equation*}$$

$$\begin{equation}{M}_{L}^{R}={g}_{L}{L}^{R}{B}_{{L}^{R}}(y),\end{equation} \tag{ 8 }$$

with $x=\frac{{g}_{s}{S}^{i}{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{{S}^{i}}}{T}$ and $y=\frac{{g}_{L}{L}^{R}{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{{L}^{R}}}{T}$, where Sⁱ and L^R take the maximum value of the respective spin and orbital angular momenta and the Brillouin function is given by,

$$\begin{equation}{B}_{p}(x)=\frac{2p+1}{2p}\enspace \mathrm{coth}\left(\frac{2p+1}{2p}x\right)-\frac{1}{2p}\enspace \mathrm{coth}\left(\frac{x}{2p}\right).\end{equation} \tag{ 9 }$$

This leads to a system of coupled equations which are then solved self-consistently. The total magnetization is calculated as, $M={M}^{\mathrm{M}\mathrm{n}}+{M}^{\mathrm{N}\mathrm{i}}+{M}_{s}^{R}+{M}_{L}^{R}$. We also determine the change in entropy (ΔS_M ) as a function of temperature via,

$$\begin{equation}{\Delta}{S}_{M}({T}_{i},H)=\sum\limits _{j=1}^{p}\frac{M({T}_{i+1},{H}_{j})-M({T}_{i-1},{H}_{j})}{{T}_{i+1}-{T}_{i-1}}({H}_{j+1}-{H}_{j}).\end{equation} \tag{ 10 }$$

ΔS_M (T_i , H) is thus the change in magnetic entropy at a certain discrete value T_i of temperature. H_j arises from uniform discretization of interval [0, H] such that, H₁ = 0 and H_p = H. The above equation is the discrete version of continum equation, equation (1).

Classical Monte Carlo method. Since the mean-field approximations discussed above completely ignores spatial correlations, we use Monte-Carlo simulations to ensure that the low-temperature magnetic behavior is correctly captured. We employ the standard Markov chain Monte Carlo method with the Metropolis algorithm [40]. Typical simulated lattice size is 2 × 16³ sites, with periodic boundary conditions. For thermal equilibration and averaging of quantities, we use $\sim 1{0}^{5}$ Monte Carlo steps each. Alongside, the change in entropy (ΔS_M ) as a function of temperature (equation (10)), the z component of total magnetization is calculated as,

$$\begin{equation}{M}_{z}=\frac{1}{N}\left\langle {g}_{s}\sum\limits _{i\in \mathrm{A},\mathrm{B}}{S}_{z}^{i}\right\rangle +\frac{1}{N}\left\langle {g}_{L}\sum\limits _{i\in \mathrm{A}}{L}_{z}^{i}\right\rangle ,\end{equation} \tag{ 11 }$$

where N is total number of spins and the angular bracket denotes the thermal average over Monte Carlo generated equilibrium configurations.

3.3. Results and discussions

We begin with the results obtained from the classical Monte Carlo simulations using Hamiltonian equation (3). First, we discuss the variation of magnetization (M_z ) with temperature at various values of magnetic field strength (H). In figure 5 we find that for all choices of R, there is a smooth rise in M_z as the temperature is lowered. This indicates the presence of a second-order phase transition from paramagnetic to ferromagnetic state below a critical temperature (T_C ≈ 4). The transition is due to ferromagnetic coupling between the magnetic moments of Ni–Mn network in the paramagnetic background of rare earth moments [33]. The interesting behavior is noted at low temperature where M_z shows an upturn or a downturn depending on the choice of R. The experimental data clearly indicates an upturn for R = Gd, Tb, Dy, and a weak downturn for R = Sm, Pr, Nd. Since Gd does not support any orbital contribution to magnetism, the upturn in magnetization for Gd confirms that the coupling J₂ is ferromagnetic. The sign of the coupling between spin and orbital moments on R ions, however, depends on the filling of f-shells. Indeed, in accordance with the Hund's rule, the orbital and spin moments will antialign (align) for less (more) than half filled band. If this argument borrowed from atomic physics holds then we expect that a simple change in sign of λ should capture the experimental behavior of magnetization across the entire rare earth series. Indeed, using the magnitude of ferromagnetic J₂ and λ as model parameters, our Monte Carlo results describe the experimental data very well. For R = Sm, Pr, Nd, at lower values of H, we notice a downturn in magnetization much below T_C (see figures 5(a), (c) and (d)). The downturn at low temperatures indicates an antiferromagnetic correlation of rare earth moments with Ni–Mn moments. However, the origin of this distinct behavior lies in the spin–orbit coupling. Once the applied field becomes stronger than the antiferromagnetic tendency, magnetization begins to show an upturn. The simulations for R = Gd, Tb, Dy are also consistent with the experiments, and display an upturn (see figures 5(b), (e) and (f)).

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We have seen how the Monte Carlo data resembles the experimental data for all choices of R by simply tuning the values of λ and J₂. We clarify further the underlying key idea by even simpler mean-field calculations described in the methods section. In order to emphasize the essential features, we set J₁ = 1 and J₂ = 0.1 for the mean field calculations, and use L^R = 6 corresponding to R = Nd. The low-temperature behavior of magnetization obtained within mean-field calculations (see figures 6(a) and (b)) confirms that the Hund's rule controlling the antialignment or alignment of spin and orbital moments on the rare-earth ion can explain the qualitatively distinct behavior for different R. Also from figure 6(c), we find that for all values of λ, R spin moments begin to align ferromagnetically with the Ni–Mn sublattice as the temperature is lowered starting at T_C. On the other hand, the orbital moments on R remain disordered until the temperature scale becomes lower than the energy scale of the spin orbit coupling λ. For a positive sign of λ, the orbital and spin moments on R antialign (see figure 6(d)). Similar mean-field results are obtained by using the L_R values corresponding to Sm and Pr (not shown). As mentioned earlier, the AFM spin–orbit interaction on R ions can be justified on the basis of Hund's rule. The R⁺³ ions with R = Sm, Pr, Nd have 4f subshells which are less than half-filled. According to Hund's rule, if the subshell is less than half-filled, then the stable configuration has the minimum total angular momentum (J) value. The minimum J is possible only if the orbital moment of rare-earth ion aligns oppositely to the spin magnetic moment.

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The low temperature downturn in M_z , for instance in Nd₂NiMnO₆ can be understood as follows. The orbital moment of Nd⁺³, L = 6, is much larger than the corresponding spin value. As the orbital moment and the spin moment of Nd⁺³ are oppositely aligned, the total moment is in the direction of orbital moment. Additionally, the Nd spin moments on A sublattice is ferromagnetically coupled to Ni–Mn spin moments on B sublattice. Therefore, the total magnetic moment of the Nd sublattice is antialigned to the total magnetic moment of the Ni–Mn sublattice leading to a decrease in magnetization at low temperatures. With increasing magnetic field the Hund's rule energy is compensated by the external field leading to an upturn in M_z . Similar physics persists in Sm₂NiMnO₆ and Pr₂NiMnO₆ DP. On the other hand, in DP with R = Tb, Dy we find that ferromagnetic coupling between the rare earth orbital moment and the spin moment captures the upturn in M_z in the low-temperature regime. The alignment of spin and orbital contributions to magnetization in case of Tb and Dy is again attributed to Hund's rules. As the 4f ions have more than half-filled f orbitals so the minimum energy state must have maximum J. The maximum J is obtained when the orbital moment is parallel to the spin magnetic moment of rare-earth ion. The upturn can thus be related to the large total magnetic moment of rare-earth ion on A sublattice coupling ferromagnetically to B sublattice. R = Gd represents a special case where the orbital contribution does not exist, and therefore it also serves as a checkpoint for the choice of sign of J₂ used in our model. Our mean-field results on Gd₂NiMnO₆ are consistent with a previous mean-field study on this material [39].

From the above discussion, we conclude that the low temperature behavior in DPs with R = Sm, Nd, Gd, Tb, Dy observed in our experiments is qualitatively captured via a minimal model that explicitly considered the orbital degree of freedom on R sites. The case of R = Pr presents a slight disagreement as we do not find a downturn in M_z at low temperatures in our low-field data. One possibility for this could be a smaller spin–orbit coupling in R = Pr that can be easily overcome even by a small magnetic field.

Further, we present results of change in entropy as a function of temperature obtained via our Monte-Carlo simulations. The results are shown in figure 7. Given that ΔS_M is a quantity derived from M_z (H, T), it is not surprising that except for R = Pr the results match well with the data reported in our experiments. We obtain −ΔS_M > 0 near the ferromagnetic transition of Ni–Mn sublattice for all choices of R. The anti-alignment (alignment) of the total magnetic moment of Nd sublattice to the magnetic moment of Ni–Mn sublattice at low temperatures leads to −ΔS_M < (>)0 as seen in figures 7(a)–(c) (figure 7(d)–(f)) [37]. In case of antialignment, external magnetic field can enforce an alignment and hence ΔS_M also shows a sign reversal at sufficiently large magnetic field (see, for example, figure 7(b)). As a result, the IMCE at low temperatures changes to conventional MCE upon increasing the magnetic field strength.

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From figure 7 it can be seen that for R = Gd, Tb, Dy there exist two peaks with −ΔS_M > 0, one at T_C and other at low temperatures. Upon increasing magnetic field, both the peaks display characteristic features associated with convensional MCE, such as the broadening of temperature range and increase in the peak height. These observations are in accordance with the experimental findings.

In conclusion, we have presented a detailed experimental and theoretical study of the magnetization and MCE of rare-earth based DP R₂NiMnO₆ (R = Pr, Nd, Sm, Gd, Tb, and Dy) as a function of temperature (2 ⩽ T ⩽ 300 K) and magnetic field (1 ⩽ H ⩽ 8 T). In contrast to Y₂NiMnO₆, which shows a single anomaly at the ferromagnetic transition temperature T_C caused by the ordering of Ni²⁺ and Mn⁴⁺ magnetic ions, all the R₂NiMnO₆ show anomalies in magnetization and MCE at T_C, where the Ni–Mn sublattice orders ferromagnetically, as well as additional features in the magnetization and MCE at low temperature (T₂) because of the coupling of R³⁺ ions and ordered Ni–Mn sublattice. Pr₂NiMnO₆, Nd₂NiMnO₆ and Sm₂NiMnO₆ show a reduction in magnetization below T₂ which occurs inspite of the ferromagnetic coupling between the R spins and the Ni/Mn spins. This can be understood as the orbital moment is anti-aligned to the spin for the first half of the R elements. This decrease in magnetization below T₂ can be overcome by a large field strong enough to polarize the full effective moment of the R³⁺ ions. This leads to a negative value of (−ΔS_M ) in MCE measurements at low temperature. Just like the magnetization, the reduction of the MCE can be reversed on the application of a large enough magnetic field. Materials from the second half of the R series, R₂NiMnO₆ (R = Gd, Tb, and Dy) show an upturn in the magnetization below T₂ because the orbital and spin moments of these R ions are aligned with the Ni–Mn sublattice. The large rare earth moments of these heavier R ions leads to a huge and positive value of (–ΔS_M ) at low temperature, which may potentially be useful in magnetic refrigeration. Additionally, the paramagnetic background of rare-earth ions spreads out the magnetic entropy over a larger temperature range around T_C which broadens the MCE profile at T_C resulting in a large RCP. The large RCPs make these materials potentially attractive for MCE based refrigeration techniques. For the microscopic understanding of the magnetic behavior, we investigated a phenomenological two-sublattice Heisenberg model that takes into account both spin and orbital degrees of freedom. To study the model we use mean-field theory and classical Monte Carlo simulations. Both these methods reproduce the main features of the experimental observation for magnetization and for magnetic entropy change. We find that the magnetic behavior at low temperatures in the DP containing f block elements is controlled by spin–orbit coupling. The more than or less than half-filled 4f orbitals decides the nature of spin–orbit coupling to be ferromagnetic or antiferromagnetic. The weak ferromagnetic coupling J₂ between the rare earth ion sublattice and Ni–Mn sublattice and antiferromagnetic spin–orbit coupling (λ) in DPs with R = Nd, Pr and Sm is responsible for the downturn in magnetization at low temperature. On the other hand, the weak ferromagnetic coupling J₂ and ferromagnetic λ in DPs with R = Gd, Tb and Dy explains the upturn in magnetization at low temperatures.

We acknowledge the support of the x-ray facility at IISER Mohali for powder XRD measurements. KP thanks DST, Government of India, for the award of Inspire faculty fellowship.

The data that support the findings of this study are available upon reasonable request from the authors.