Structural, magnetic and magnetocaloric properties in distorted RE₂NiTiO₆ double perovskite compounds

The development of novel magnetic materials with promising functional characters is of great significance for the advancement of modern technology and society. Among them, the magnetocaloric effect (MCE) based magnetic refrigeration (MR) technology promises to replace the conventional vapor cycle refrigeration, but also has a wide range for potential applications from room temperature to ultra-low temperature [1–6]. The MCE is a magnetic field induced thermodynamic phenomenon, which can be manifested as the heat generated or absorbed by the materials when the external magnetic field is changed. As a result, a large number of theoretical and experimental researches have been carried out on a variety of MR materials. Up to now, a series of MR materials with potential application near ambient temperature have been identified, such as Gd–Si–Ge [6], La–Fe–Si/Al [7], Mn–Fe–P–As/Ge/Si [8, 9], and Ni–Mn–X (X= Ga, In and Sn) [10, 11], etc. In comparison, rare earth (RE)-based compounds and oxides are regarded as the promising cryogenic MR materials for MR technology on account of their unique advantages and application prospects in the liquefaction of extremely low-temperature resources including helium (He), hydrogen (H₂) and nitrogen (N₂) [12–16].

In recent years, some of the RE-based oxides were fabricated and systematically checked, which are found to exhibit many intriguing physical properties in terms of magnetodielectric effect, magnetoresistive effect, luminescence characteristic and large/giant MCE effects, etc. For examples, Zhao et al have implemented the phase transition from antiferromagnetic-paraelectric to ferromagnetic–ferroelectric order in the (EuTiO₃)_0.5:(MgO)_0.5 composite through the enhancement of spin-phonon coupling by negative pressure [17]. Arh et al have found that strong spin–orbit coupling (SOC) plays a key role in stabilizing spin liquid derived from magnetic anisotropy in the Ising triangular-lattice antiferromagnet NdTa₇O₁₉ oxide [18]. Blasco et al reported that each Fe²⁺ ion in the NaREFeWO₆ (RE = Pr and Sm) oxides has strong antiferromagnetic coupling with the three nearest RE³⁺ ions and follows a unique magnetic order of 'up-up-down-down' sequence running along the [1 0 0] direction [19]. Moreover, Zeng et al have investigated the MCE performances in the weberite-type oxides Gd₃ MO₇ (M= Nb, Sb and Ta), and found that they are promising alternatives for MR techniques [20]. Fkhar et al have found the employment composite or spray drying as the effective strategy to improve magnetic properties in the La_0.45Nd_0.25Sr_0.3MnO₃/CuO compound [21, 22]. Koskelo et al have reported a series of fcc oxides A₂GdSbO₆ (A = Ca, Sr and Ba) with small superexchange interactions, the maximum magnetic entropy change ($- \Delta S_{\text{M}}^{\max }$) are 20% higher than that of the standard MR material Gd₃Ga₅O₁₂ using in extremely low-temperature [23]. Very recently, Xu et al have used the combination of experiment and density functional theory (DFT) calculation to reveal electronic, magnetic and MCE properties of the isostructural RE₂BaZnO₅ compounds, in which Dy₂BaZnO₅ and Ho₂BaZnO₅ compounds exhibit excellent magnetocaloric performances at liquid He temperature [24].

As an important branch of RE-based oxides, the RE₂ TMTM'O₆ (where TM and TM' are transition metal elements) double perovskite (DP) compounds derived from the ABO₃ perovskite oxides have become the research focus in recent years. Because their intrinsic properties can be regulated by the variability of ions at RE, TM and TM' sites, they are endowed with outstanding optical, electrical, catalytic, magnetic and MCE properties [25–28]. In particular, the RE₂ TMTM'O₆ compounds have attracted extensive attention in exploring the exotic magnetic properties caused by strong SOC, structural variation diversity and anisotropic exchange interaction [29–31]. The competition of ferromagnetic and antiferromagnetic couplings among RE³⁺, Ni²⁺ and Ir⁴⁺ ions in the RE₂NiIrO₆ (RE = La, Pr and Nd) compounds can not only change the magnetic moment arrangement of Ni²⁺ and Ir⁴⁺ ions, but also increase the corresponding magnetically ordered temperatures with the decrease of the size of RE³⁺ [32]. The spontaneous magnetization of the RE₂LiFeO₆ (RE = Sm and Eu) compounds with abnormally high valence Fe⁵⁺ ions is closely related to the Dzyaloshinskii Moriya interaction, and the magnetization increases with increasing local geometric spin frustration between the nearest Fe⁵⁺ ions in the [1 1 1] direction [33]. In terms of MCE performances, the RE₂ TMTM'O₆ compounds commonly have lower hysteresis and higher chemical stability than rare earth (RE)-transition (TM) intermetallic compounds and alloys, as well as higher resistivity that facilitates reduced eddy current losses. Up to now, the excellent MCE performances in the field of extremely low-temperature have also been researched in the RE₂CuMnO₆ [34], RE₂ZnMnO₆ [35] and RE₂FeAlO₆ compounds [36]. It is evident from the above results that the exploration and investigation of the RE₂ TMTM'O₆ compounds with promising MCE performances is of great interest in the field of extremely low-temperatures MR, meanwhile, understanding its magnetic exchange interaction can also provide valuable theoretical information for the discipline of condensed matter physics.

Based on this background, we found that the RE₂NiTiO₆ compounds containing light RE ions have unique properties in the areas of optics, electricity and magnetism through the results of the relevant literatures [37–39]. However, there are few reports on the RE₂NiTiO₆ compounds containing heavy RE ions to date, especially in term of the magnetic and MCE properties. Therefore, we have studied the crystal structures, magnetic properties, electronic structures as well as MCE of the RE₂NiTiO₆ (RE = Gd, Tb and Ho) DP compounds by combining experiments with DFT calculations. As a consequence, we found that the anisotropic distortion in the monoclinic crystal structure of the RE₂NiTiO₆ compounds increase with decreasing RE³⁺ ion radius, resulting in the reduction of crystal symmetry. Furthermore, considerable reversible MCE performances have been observed in Gd₂NiTiO₆ compound, which provides a crucial clue for the exploration of MR materials suitable for extremely low-temperature.

The RE₂NiTiO₆ (RE = Gd, Tb and Ho) polycrystalline compounds were prepared by citric acid-assisted sol-gel route. The precursor materials Gd(NO₃)₃ (⩾99.99%), Tb(NO₃)₃ (⩾99.99%), Dy(NO₃)₃ (⩾99.99%), Ni(NO₃)₂ (⩾99.99%) and Ti(SO₄)₂ (⩾99.99%) are purchased from the Shanghai Macklin Biochemical Co., Ltd. Firstly, stoichiometric amounts high purity raw materials of Gd(NO₃)₃, Tb(NO₃)₃ Dy(NO₃)₃, Ni(NO₃)₂ and Ti(SO₄)₂ were stirred in distilled water until they were completely dissolved. Subsequently, anhydrous citric acid was added to the precursor mixture solution. As a complex adhesive, anhydrous citric acid can help the good dispersion of various elements in the precursor gel. The obtained precursor mixture solutions were kept under constant magnetic stirring at 353 K for 12 h until viscous gels were formed. The gels were then fired at 1073 K for 2 h in an air atmosphere to remove organic residues and produce fluffy black powders. Finally, the black powders obtained after grinding were cold-pressed into pellets with the pressure of 30 MPa, and then annealed at 1473 K for 24 h. The phase characterization of the polycrystalline RE₂NiTiO₆ (RE = Gd, Tb and Ho) compounds were identified by the Rigaku x-ray diffraction technique (SmartLab-9 KW diffractometer). The surface morphology and chemical composition were investigated by using the field emission scanning electron microscope (FESEM, JEOL-JSM5800) and the attached energy dispersive X-ray spectroscopy (EDS). DC-susceptibility measurements of the RE₂NiTiO₆ (RE = Gd, Tb and Ho) compounds were conducted by the magnetic properties measurement system (MPMS, QD).

The structural optimizations, electronic and magnetic properties calculations were determined based on the projector augmented-wave (PAW) method within the DFT calculation by using the Vienna ab initio simulation package code [40, 41]. The spin-polarized generalized gradient approximation (GGA) was used to describe the exchange-correlation effects, and combined with both the GGA + U (Hubbard potential) and SOC to optimize the on-site Coulomb repulsion of localized RE-f electrons and determine the magnetic anisotropy, respectively [42, 43]. The value of U_eff with 3.4 eV has been set for Ni²⁺ ion according to the relation U_eff = U−J, where U and J are Coulomb and Hund's exchange parameters, respectively. The PAW method was used to resolve the electron–core interaction and the valence electron contributions for Gd, Tb, Ho, Ni, Ti and O were generated as [5s²5p⁶4f⁷5d¹6s²], [5s²5p⁶4f⁹5d⁰ 6s²], [5s²5p⁶4f¹⁰5d⁰ 6s²], [3p⁶3d⁸4s²], [3p⁶3d²4s²] and [2s²2p⁴], respectively [44, 45]. The kinetic energy cutoff was set at 600 eV to expansion the electronic wave functions. To optimize the structure, the conjugate gradient algorithm was used to completely relax the lattice constant and atomic position when the Hellman–Feynman forces converge to less than 0.01 eV·Å⁻¹. The Brillouin zone was sampled using k-point mesh of 9 × 9 × 7 under the Monkhorst-Pack scheme, which can improve the accuracy of the density of states (DOS) and SOC calculations.

Figures 1(a)–(c) illustrate the obtained XRD patterns at room temperature and the Rietveld refinement results by aid of the FULLPROF software of the RE₂NiTiO₆ (RE = Gd, Tb and Ho) compounds. The present RE₂NiTiO₆ compounds are crystalized in the monoclinic structure with B-site ordered DP structure (P2₁/n space group, No. 14). In the Tb₂NiTiO₆ and Ho₂NiTiO₆ compounds, small amounts of Tb₄O₇ and Ho₂Ti₂O₇ impurity phases were detected around 29° and 30.5° (the symbols ''and '$\diamondsuit$'), and the corresponding weight ratios were determined to be 1.83 and 3.25 wt%, respectively [46, 47]. The Rietveld indices were calculated, and the values of the refinement factors R_p, R_wp, R_exp and χ² are all lower than 10% (as listed in table 1), which verifies the reliability of refinements. A schematic presentation of the crystal structure of the Gd₂NiTiO₆ compound along the b axis is illustrated in figures 2(a)–(d) with the environments of metal atom in RE₂NiTiO₆ compounds. The formation of the Gd₂NiTiO₆ compound superstructure makes the alternating distribution of the Ni located on the 2c and Ti on the 2d Wyckoff positions, which are surrounded by six O (4e) atoms to establish NiO₆ and TiO₆ octahedrons, respectively. Each TiO₆ octahedron and the adjacent NiO₆ octahedron share an O atom in the form of alternating z-shape chains along the a-axis. In parallel, Gd atoms are located on the 4e Wyckoff position, which not only occupy the cavities formed by co-point NiO₆ and TiO₆ octahedrons, but also superpose alternately with Ni/Ti atomic layers along the c-axis. In order to further distinguish the changes in crystal structure caused by different RE³⁺ ions, the refined lattice constants (a, b and c), the unit cell volume (V) and the average-octahedral-tilt angle ($\varphi$) are illustrated in figures 3(a)–(c). Generally, the a, b, c and V decrease monotonically with decreasing RE³⁺ ions theoretical radii, obeying the principle of the lanthanide contraction. Moreover, the average-octahedral-tilt angle $\varphi = (180 - \eta )/2$ can reflect the degree of anisotropic distortion of the unit cell in monoclinic compounds. The $\eta$ is defined as the evolution of the average tilting angle ∠Ni–O–Ti and the corresponding values are 156.14°, 153.58° and 149.79° for RE = Gd, Tb and Ho. The values of $\eta$ suggest that the monoclinic distortion of the present RE₂NiTiO₆ compounds will increase monotonically with the decrease of the radii of RE³⁺ ions and result in the reduction of crystal symmetry. The FESEM micrographs for the present RE₂NiTiO₆ compounds together with the corresponding EDS mappings of each element are presented in figures 4(a)–(c). The RE₂NiTiO₆ (RE = Gd, Tb and Ho) compounds are composed of uniformly distributed irregular particles with average particle sizes about 1.82, 1.86 and 1.87 μm. This microstructure with irregular particles and high density confirms the polycrystalline nature of the RE₂NiTiO₆ compounds. The randomly selected regions of EDS results analysis verified that the atomic ratios of the present RE₂NiTiO₆ series compounds are close to the standard 2:1:1:6. The ratios of each element are 20.16, 10.99, 10.68, 58.17 at %, 21.13, 9.22, 10.91, 58.74 at % and 21.19, 11.05, 11.09, 56.67 at % for Gd₂NiTiO₆, Tb₂NiTiO₆ and Ho₂NiTiO₆ compounds, respectively.

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The temperature dependent magnetization, the M(T) curves for the RE₂NiTiO₆ compounds are illustrated in figures 5(a)–(c), which were carried out in zero-field-cooled (ZFC)/field-cooled (FC) modes under the magnetic flux densities (${\mu _0}H$) of 200, 500 and 1000 Oe. All the M(T) curves of present RE₂NiTiO₆ compounds increase with the increase ${\mu _0}H$ in the whole temperature range. Another feature observed in each M(T) curve under different ${\mu _0}H$ is that ZFC and FC modes have good consistency, which confirms these compounds have negligible thermal hysteresis. From a technical point of view, the perfect reversibility of magnetic phase transitions (MPTs) makes the present RE₂NiTiO₆ series compounds have potential application prospects of extremely low-temperature MR technology. Insets are the derivative curves of the corresponding M(T) curves for the RE₂NiTiO₆ compounds. It can be observed that the minimum values of all the RE₂NiTiO₆ compounds shift toward higher temperatures with increasing ${\mu _0}H$, implying that their dominant MPTs are of the FM type. When ${\mu _0}H$ increases to 1000 Oe, the magnetic ordering temperatures (T_M) are determined to be 4.3, 4.5 and 3.9 K, which are obtained from the minimum values of the temperature axes for the Gd₂NiTiO₆, Tb₂NiTiO₆ and Ho₂NiTiO₆ compounds, respectively.

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The magnetization and inverse magnetic susceptibility (${\chi ^{ - 1}} = {{{\mu _0}H} \mathord{\left/ \right. } M}$) of present RE₂NiTiO₆ compounds are plotted as a function of temperature in the presence of magnetic flux densities ${\mu _0}H$ = 10 kOe, as illustrated in figures 6(a)–(c). In the high-temperature part (50 ⩽ T ⩽ 300 K), all the χ⁻¹(T) curves of present RE₂NiTiO₆ compounds are almost linear, which satisfies the modified Curie-Weiss law: $\chi (T) = C/(T - {\theta _{\text{p}}})$, in which $C = N{({\mu _{\text{B}}}{\mu _{{\text{eff}}}})^2}/3{k_B}$ denotes the Curie constant, ${\theta _{\text{P}}}$ and ${\mu _{{\text{eff}}}}$ are named as paramagnetic Curie temperature and effective magnetic moment, respectively. The values of ${\mu _{{\text{eff}}}}$ are calculated as 11.11 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$ for Gd₂NiTiO₆, 13.42 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$ for Tb₂NiTiO₆, as well as 14.89 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$ for Ho₂NiTiO₆, which are significantly larger than the values of theoretical magnetic moments provided by free Gd³⁺ (7.94 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$), Tb³⁺ (9.72 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$) and Ho³⁺ (10.64 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$) ions, respectively. This result implies the magnetic moment contribution of Ni²⁺ should be considered. In this regard, the values of ${\mu _{{\text{eff}}}}$ are compared with the theoretical values calculated by the formula ${\mu _{{\text{theor}}}} = {\left[ {2\mu {{(R{E^{3 + }})}^2} + \mu {{({\text{N}}{{\text{i}}^{2 + }})}^2}} \right]^{{1 \mathord{\left/ \right. } 2}}}$, and the gained quite close values establish the existence of the spin-only magnetic moments of Ni²⁺ (2 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$).

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To further understand the magnetic properties and electronic structures of the RE₂NiTiO₆ compounds in the ground states, the total and partial DOS have been evaluated by the GGA and GGA + U methods in the framework of ab initio calculation, as illustrated in figures 7(a)–(f). It can be seen that the Gd₂NiTiO₆ and Ho₂NiTiO₆ compounds behave as semiconductors in the GGA+U method, with the band gaps of 1.83 and 0.25 eV, respectively. The Tb₂NiTiO₆ compound is semi-metallic in both GGA and GGA+U methods. Moreover, the magnetic properties of all compounds are mainly provided by RE³⁺ and Ni²⁺ ions. The Gd₂NiTiO₆ and Ho₂NiTiO₆ compounds have FM ground states, while the Tb₂NiTiO₆ compound has an AFM ground state contributing from the antiparallel arrangement of the Ni²⁺ ion. The atomic magnetic moments (${\mu _{{\text{atom}}}}$) of the present RE₂NiTiO₆ series compounds have been determined by spin-polarized calculation according to the obtained magnetic ground states. The $\mu _{{\text{calc}}}^S$ values of each RE³⁺ and Ni²⁺ ion are determined to be 6.83 and 1.65 ${\mu _{\text{B}}}$/atom for the Gd₂NiTiO₆ compound, 5.81 and 1.66 ${\mu _{\text{B}}}$/atom for the Tb₂NiTiO₆ compound, as well as 3.75 and 1.65 ${\mu _{\text{B}}}$/atom for the Ho₂NiTiO₆ compound, respectively. Moreover, the magnetic anisotropy energies ($\Delta {E_{{\text{MAE}}}}$) of the present RE₂NiTiO₆ series compounds have been further investigated based on the SOC method. The $\Delta {E_{{\text{MAE}}}}$ is defined as the difference of energy in various directions, which can be formulated by the equation: $\Delta {E_{{\text{MAE}}}} = {E_{{\text{max}}}} - {E_{{\text{min}}}}$. Where ${E_{\max }}$ and ${E_{\min }}$ are expressed as the maximum and minimum energies of the quantum axes along the [0 0 1], [0 1 0] and [1 0 0] directions, respectively. The acquired results of the present RE₂NiTiO₆ series compounds are presented as bar charts in figure 8, where the red and orange blocks are corresponding to the GGA and GGA+U methods, respectively. Obviously, the values of $\Delta {E_{{\text{MAE}}}}$ calculated by both methods exhibit gradual increase, which indicates that the magnetic moments of the present RE₂NiTiO₆ series compounds decrease with the increase of $\Delta {E_{{\text{MAE}}}}$. In addition, the values of $\Delta {E_{{\text{MAE}}}}$ for the Gd₂NiTiO₆ compound in both methods are close to zero, demonstrating that the $\Delta {E_{{\text{MAE}}}}$ of Ni²⁺ ions in all RE₂NiTiO₆ compounds are extremely small. Hence, it can conclude that the $\Delta {E_{{\text{MAE}}}}$ of the Tb₂NiTiO₆ and Ho₂NiTiO₆ compounds are mainly caused by Tb³⁺ and Ho³⁺ ions.

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A series of $M({\mu _0}H)$ curves for present RE₂NiTiO₆ compounds are recorded at 2–35 K with $\Delta {\mu _0}H$ up to 70 kOe, as presented in figures 9(a)–(c). Generally, the $M({\mu _0}H)$ curves of all compounds at 2 K tend to be saturated with the increasing $\Delta {\mu _0}H$. The saturation magnetization (${M_{{\text{sat}}}}$) values are determined to be 14.67, 9.87 and 10.21 ${{{\mu _{\text{B}}}} \mathord{\left/ \right. } {{\text{f.u.}}}}$ for the Gd₂NiTiO₆, Tb₂NiTiO₆ and Ho₂NiTiO₆ compounds, respectively. In the case of Gd₂NiTiO₆ compound, the value of ${M_{{\text{sat}}}}$ approximates to the sum of two isolated Gd³⁺ ions and one Ni²⁺ ion. In contrast, the values of ${M_{{\text{sat}}}}$ deviate from the individual magnetic ions for the Tb₂NiTiO₆ and Ho₂NiTiO₆ compounds, which may be due to the non-negligible magnetic anisotropy [48, 49]. The temperature dependent magnetic entropy change ($\Delta {S_{\text{M}}}$) curve provides crucial information about the present RE₂NiTiO₆ series compounds, which can not only verify whether they are suitable for the application of MR technology, but also further judge their MPTs type. The $\Delta {S_{\text{M}}}$ is given by the following formula based on thermodynamic theory [50]:

$$\begin{equation}{\left( {\frac{{\partial S}}{{\partial {\mu _0}H}}} \right)_T} = {\left( {\frac{{\partial M}}{{\partial T}}} \right)_{{\mu _0}H}} \Rightarrow \Delta {S_{\textrm M}}\left( {T,\Delta {\mu _0}H} \right) = \mathop \int ^{{\mu _0}{H_{\max }}}_0 {\left( {\frac{{\partial M}}{{\partial T}}} \right)_{{\mu _0}H}}{\text{d}}({\mu _0}H).\end{equation} \tag{ 1 }$$

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From the well measured $M({\mu _0}H)$ curve, the $\Delta {S_{\text{M}}}$ can be approximately evaluated by the following expression:

$$\begin{equation}\left| {\Delta {S_{\textrm M}}\left( {T,{\mu _0}H} \right)} \right| = \sum\limits_i {\frac{{{M_{i + 1}}({T_{i + 1}},{\mu _0}{H_i}) - {M_i}({T_i},{\mu _0}{H_i})}}{{{T_{i + 1}} - {T_i}}}} \Delta {\mu _0}{H_i}.\end{equation} \tag{ 2 }$$

Here, M_i and M_{i+ 1} can be obtained by the magnetization at the corresponding T_i and T_{i+ 1} when a certain ${\mu _0}H$ is fixed. According to equation (2), the resulting $- \Delta {S_{\text{M}}}(T)$ curves for the RE₂NiTiO₆ compounds collected at several typical are presented in figures 10 (a), (c) and (e). The maximal values ($- \Delta S_{\text{M}}^{\max }$) of the present RE₂NiTiO₆ series compounds around T_M increase with the increase of $\Delta {\mu _0}H$ and gradually shift to high temperatures, which is consistent with the MR materials with typical second-ordering MPT characteristics. The values of $- \Delta S_{\text{M}}^{\max }$ around T_M with $\Delta {\mu _0}H$ of 0–20 and 0–50 kOe were calculated to be 6.92 and 31.28 J·kg⁻¹·K⁻¹ for the Gd₂NiTiO₆ compound, 5.86 and 13.08 J·kg⁻¹·K⁻¹ for the Tb₂NiTiO₆ compound as well as 6.08 and 11.98 J·kg⁻¹·K⁻¹ for the Ho₂NiTiO₆ compound, respectively. In view of the strong correlation between the MCE and MPT of magnetic solid in the actual MR cycle, we can analyze the $- \Delta {S_{\text{M}}}(T)$ curves to determine the types of first/second-ordering (FO/SO) MPT. The scaling relations have become a universal model for determining MR material with the nature of SO-MPT, which is expressed as assembling the $\Delta {S_{\text{M}}}(T)$ curves into a single master curve [51, 52]. The insets of figures 10 (a), (c) and (e) illustrate the rescaled temperature $\theta$ ($\theta = \left\{ \begin{gathered} - (T - {T_C})/({T_{r1}} - {T_C});T \leqslant {T_C} \\ (T - {T_C})/({T_{r2}} - {T_C});T > {T_C} \\ \end{gathered} \right.$) dependent $\Delta S^{^{\prime} }$ ($\Delta S^{^{\prime} } = \Delta S_\text{M}^{}(T)/\Delta S_\text{M}^{\max }$) curves by conducting the normalizations around T_M for the present RE₂NiTiO₆ series compounds, where ${T_{r1}}$ (above T_M) and ${T_{r2}}$ (below T_M) are two reference points corresponding to $\Delta {S_{\text{M}}}({T_{r1,2}}) = 0.6 \times \Delta S_{\textrm M}^{\max }$. It can be found from the normalization $\Delta S^{^{\prime} }(\theta )$ curves that the right part can converge to one single curve when the present RE₂NiTiO₆ compounds take T_M as the symmetry point, which indicates that they have the characteristics of SO-MPTs around T_M. Since the temperatures of the present RE₂NiTiO₆ compounds below T_M are close to the absolute zero, the left part of the normalization $\Delta S^{^{\prime} }(\theta )$ curves do not in accordance with the scaling relations, so that it is difficult to confirm the type of MPT. Accordingly, a novel criterion for the relationship between MCE and the exponent n proposed by Law et al [53, 54] was used for verification research, which can be expressed as following formula:

$$\begin{equation}n\left( {T,{\mu _0}H} \right) = \frac{{d\ln \left| {\Delta {S_{\textrm M}}} \right|}}{{d\ln {\mu _0}H}}.\end{equation} \tag{ 3 }$$

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The n(T) curves converted from the $- \Delta {S_{\text{M}}}(T)$ curves for the present RE₂NiTiO₆ series compounds collected at $\Delta {\mu _0}H$ up to 0–70 kOe are illustrated in figure 10(b), (d) and (f). Scrutiny over the MPTs type of present RE₂NiTiO₆ compounds reveals that the exponents of n are less than 2 around T_M with the further increase of $\Delta {\mu _0}H$, indicating the nature of SO-MPT. In addition, two or one points in n value are larger than 2 below T_M can be found in Gd₂NiTiO₆ and Tb₂NiTiO₆ compounds, which is probability owing to the instability of their magnetic ground states close to absolute zero.

In addition, the parameters of temperature averaged $\Delta {S_{\text{M}}}(TEC)$, relative cooling power (RCP) and refrigerant capacity (RC) obtained from the $- \Delta {S_{\text{M}}}(T)$ curve can also be used as variety effective ways to evaluate the MCE performance, which can be calculated by the following formulas:

$$\begin{equation}{TEC}(\Delta {T_{{\text{lift}}}}) = \frac{{\text{1}}}{{\Delta {T_{{\text{lift}}}}}}\mathop {\max }\limits_{{T_{{\text{mid}}}}} \left\{ {\mathop \int \limits_{T{\text{mid}} - \frac{1}{2}\Delta {T_{{\text{lift}}}}}^{{T_{{\text{mid}}}} + \frac{1}{2}\Delta {T_{{\text{lift}}}}} \Delta S{{(T)}_{\Delta {\mu _0}H,T}}{\text{d}}T} \right\},\end{equation} \tag{ 4 }$$

$$\begin{equation}{RCP} = \left| { - \Delta S_\text{M}^{\max }} \right| \times \delta {T_{{\text{FWHM}}}},\end{equation} \tag{ 5 }$$

$$\begin{equation}{RC} = \mathop \int \limits_{{T_{{\text{cold}}}}}^{{T_{{\text{hot}}}}} \left| { - \Delta {S_\text{M}}(T)} \right|{\text{d}}T.\end{equation} \tag{ 6 }$$

Griffith et al defined the temperature averaged entropy change (TEC) as the average value of $\Delta {S_{\text{M}}}(T)$ curve in a selected temperature range ($\Delta {T_{{\text{lift}}}}$) under a fixed $\Delta {\mu _0}H$, and ${T_{{\text{mid}}}}$ is expressed as the center of the $\Delta {T_{{\text{lift}}}}$ [55]. According to equation (4), the resulting values of TEC (5 K) were calculated to be 5.88 and 27.38 J·kg⁻¹·K⁻¹ for the Gd₂NiTiO₆ compound, 5.24 and 12.71 J·kg⁻¹·K⁻¹ for the Tb₂NiTiO₆ compound, as well as 5.62 and 11.72 J·kg⁻¹·K⁻¹ for the Ho₂NiTiO₆ compound with the $\Delta {\mu _0}H$ of 0–20 and 0–50 kOe, respectively. It can be observed that the values of TEC (5 K) deviate greatly from the corresponding $- \Delta S_{\text{M}}^{\max }$ with the same $\Delta {\mu _0}H$, which is particularly prominent in the Gd₂NiTiO₆ compound. Such behavior is related to the narrow temperature span of the $\Delta {S_{\text{M}}}(T)$ curve for Gd₂NiTiO₆ compound. The RCP and RC are commonly used as the coefficients of MCE performance, which can reliably predict the amounts of energy transferred from T_cold to T_hot in the MR cycle. The T_cold and T_hot are expressed as the two sides at ${{\left| { - \Delta S_{\text{M}}^{\max }} \right|} \mathord{\left/ \right. } 2}$ values of $- \Delta S_{\text{M}}^{\max }(T)$ curves collected at several types of $\Delta {\mu _0}H$, and the value of $\delta {T_{{\text{FWHM}}}}$ is equal to T_hot –T_cold. According to equations (5) and (6), the calculated values of RCP (RC) collected at $\Delta {\mu _0}H$ up to 0–20 and 0–50 kOe were 45.44 (34.79) and 242.11 (185.24) J·kg⁻¹ for the Gd₂NiTiO₆ compound, 49.15 (38.21) and 213.41 (168.36) J·kg⁻¹ for the Tb₂NiTiO₆ compound, as well as 57.33 (45.27) and 221.73 (174.56) J·kg⁻¹ for the Ho₂NiTiO₆ compound, respectively. Moreover, the calculated values of $- \Delta S_{\text{M}}^{\max }$, RC and RCP under the $\Delta {\mu _0}H$ of 0–50 kOe for the present RE₂NiTiO₆ series compounds together with some reported promising RE-based MR materials with T_M below 10 K are summarized in table 2. It is evident that the present RE₂NiTiO₆ series compounds can be used as excellent MR materials in the extremely low-temperature region. In particular, the maximal value ($- \Delta S_{\text{M}}^{\max }$ = 42.64 J·kg⁻¹·K⁻¹, $\Delta {\mu _0}H$ = 0–70 kOe) of the Gd₂NiTiO₆ compound is quite impressive when compared to the benchmark of commercial application, Gd₃Ga₅O₁₂ ($- \Delta S_{\text{M}}^{\max }$ = 38.4 J·kg⁻¹·K⁻¹, $\Delta {\mu _0}H$ = 0–70 kOe).

Table 2. The values of $- \Delta S_{\text{M}}^{\max }$, TEC (5 K), RCP and RC with the $\Delta {\mu _0}H$ of 0–50 kOe for present RE₂NiTiO₆ compounds and some reported promising RE-based MR materials.

Material	TM (K)	$- \Delta S_{\text{M}}^{\max }$ (J·kg−1·K−1)	TEC (5 K) (J·kg−1·K−1)	RCP (J·kg−1)	RC (J·kg−1)	References
Gd2NiTiO6	4.3	31.28	27.38	242.11	185.24	present
Tb2NiTiO6	4.5	13.08	12.71	213.41	168.36	present
Ho2NiTiO6	3.9	11.98	11.72	221.73	174.56	present
Gd2ZnMnO6	6.4	15.1	—	226.2	176.8	[35]
Gd2FeAlO6	3.5	19.2	—	—	136.1	[36]
Sr2GdNbO6	2.0	26.07	19.02	195.5	148.2	[5]
GdFe2Si2	8.6	23.25	21.01	276.56	205.67	[56]
TbScO3	3.1	23.71	—	—	—	[57]
GdCrTiO5	5	25.1	—	—	—	[58]
Er2FeC4	6	21.62	—	—	280.3	[59]
ErRu2Si2	5.5	17.6	—	—	—	[60]
HoCuSi	7	33.1	—	—	385	[61]

To conclude, we have successfully fabricated the polycrystalline RE₂NiTiO₆ (RE = Gd, Tb and Ho) DP compounds and their crystallographic structures, electronic structures, magnetic properties and MCE were investigated in combination with experiment and theory. All the RE₂NiTiO₆ compounds are confirmed to crystallize in monoclinic structure with P2₁/n space group by XRD refinements. The ground states of all compounds are FM ordering around T_M, and the characteristics of SO-MPTs are further confirmed by the novel criterion of n(T) curves and the normalization $\Delta S^{^{\prime} }(\theta )$ curves. The MCEs are determined by an indirect approach of experimental data, which yields the $- \Delta S_{\text{M}}^{\max }$, TEC (5 K), RCP and RC values with $\Delta {\mu _0}H$ of 0–70 kOe as high as 42.64 J·kg⁻¹·K⁻¹, 38.84 J·kg⁻¹·K⁻¹, 419.36 J·kg⁻¹ and 321.66 J·kg⁻¹ for the Gd₂NiTiO₆ compound, 15.12 J·kg⁻¹·K⁻¹, 14.85 J·kg⁻¹·K⁻¹, 330.38 J·kg⁻¹ and 259.81 J·kg⁻¹ for the Tb₂NiTiO₆ compound, as well as 13.85 J·kg⁻¹·K⁻¹, 13.63 J·kg⁻¹·K⁻¹, 346.18 J·kg⁻¹ and 269.15 J·kg⁻¹ for the Ho₂NiTiO₆ compound, respectively. Compared with the reported RE-based MR materials, the superior MCE performances of the Gd₂NiTiO₆ compound make it an excellent candidate as MR material suitable for extremely low-temperature.

The present work was supported by the National Natural Science Foundation of China (Grant No. 52171174). The authors acknowledge the Supercomputing Center of Hangzhou Dianzi University for providing computing resources.

All data that support the findings of this study are included within the article (and any supplementary files).

There are no conflicts to declare.