Large magnetic entropy change in Nd₂In near the boiling temperature of natural gas

Natural gas is useful for the transition from traditional fossil fuels to renewable energies, and it plays an important role in current primary energy supply.¹ It is predicted that by 2040 the share of the natural gas in the primary energy will surpass oil and coal, and the demand for liquefied natural gas (LNG) will double, exceeding the volume of gas transported through pipelines.¹ Recently, interest in magnetocaloric gas liquefaction as an alternative to the traditional gas-compression expansion technology has increased due to its high efficiency, associated potential for the expanding market of LNG, and emerging large-scale utilization of liquid hydrogen.^1–6 Many material systems, such as RECo₂ (RE: rare-earth elements) and REAl₂, demonstrate large magnetocaloric effects near 112 K,^7,8 the boiling temperature of natural gas. For example, Dy_0.7Er_0.3Co₂ and TbAl₂ show a maximum magnetic entropy change of 11.9 at 108 and 7.4 J/kg K at 105 K under a magnetic field change of 2 T, respectively.^9,10

Recently, Eu₂In and light rare-earth based alloy Pr₂In have been reported to show giant magnetic entropy changes of 26 and 15 J/kg K, respectively, at about 58 K with almost no thermal hysteresis under a magnetic field change of 2 T,^11,12 whereas most of the giant magnetocaloric materials, such as NiMn-based Heusler alloys and La(Fe,Si)₁₃, showing first-order phase transition possess a non-negligible thermal hysteresis leading to a reduction in the cycling performance.^13–15 Although the authors pointed out that their volume changes during transition are about 0.1% and about 0.01%, respectively, both being negligibly small, the giant magnetic entropy changes of both alloys are reported to be caused by the first-order phase transition,^11,12 as supported by the measurements of magnetic and electric hyperfine interactions reported by Forker et al. for Pr₂In,¹⁶ and by the heat capacity measurement reported by Guillou et al.¹¹ and the Mössbauer study by Ryan et al.¹⁷ for Eu₂In. In this work, we studied the magnetocaloric properties of Nd₂In and found out that this compound exhibits a large magnetic entropy change of 7.42 J/kg K under a magnetic field change of 2 T at 109 K. The maximum adiabatic temperature change reaches 1.13 K and is fully reversible due to the negligible thermal hysteresis. Our analyses on the magnetic entropy change, and the heat capacity demonstrate that this magnetic transition is of first-order type. Further studies on the thermal expansion and the magnetostriction reveal an unreported two-stage phase transition in Nd₂In with a negligible volume change and a considerable negative spontaneous magnetostrictive strain.

The Nd₂In sample was prepared by arc melting using raw materials Nd (99.5%) and In (99.99%). To compensate Nd evaporation during arc melting, 0.5% Nd excess was added. To ensure homogeneity, the sample was re-melted five times. Magnetization measurements were carried out using a Physical Properties Measurement System (PPMS) from Quantum Design in magnetic fields up to 12 T. Heat capacity in different magnetic fields was measured in the same PPMS with the 2τ approach. X-ray diffraction data were collected on a powder diffractometer (Stadi P, Stoe & Cie GmbH) equipped with a Ge111-Monochromator using MoK_α1-radiation (λ = 0.70930 Å) in the Debye–Scherrer geometry. It was observed that the surface of the sample decomposes quickly in air or being exposed to water. Therefore, hermetically sealed capillaries filled with the ground powders for x-ray diffraction were prepared in an Ar-filled glovebox [p(O₂) < 0.1 ppm, MBraun] to avoid oxidation and decomposition. The data were evaluated by Rietveld refinement using the FullProf software.¹⁸ Backscatter electron (BSE) imaging was obtained by a Tescan Vega 3 scanning electron microscope (SEM). For direct measurements of the adiabatic temperature change, a piece of the sample was cut into a block of 6.20 × 3.35 × 1.33 mm³ and was connected to the thermocouple using silver-based glue. Our home-built measurement setup can apply a magnetic field up to 1.95 T.¹⁹ For the simultaneous measurements of magnetization, magnetostriction, electrical resistivity, and temperature of the sample, a purpose-built experimental setup has been designed and built within our group,²⁰ which enhances the measurement capacity of the vibrating sample magnetometer (VSM) option of the PPMS from Quantum Design.

Nd₂In crystallizes in the Ni₂In-type hexagonal structure (space group $P63/mmc$⁠). Figure 1(a) shows the x-ray diffraction pattern and the Rietveld refinement results. The main phase of the sample is confirmed to be Nd₂In with a fraction of 95%. The second phase with a fraction of 5% is confirmed to be Nd₃In crystallizing in the Cu₃Au-type cubic structure (space group $Pm3¯m$⁠). Figure 1(b) shows the BSE image of Nd₂In. No obvious secondary phase was observed, which confirms further that our Nd₂In sample is of sufficient quality.

The magnetocaloric effect of Nd₂In was never studied before. In Fig. 2(a), iso-field magnetization curves are plotted as a function of temperature in magnetic fields of 0.05, 1, and 2 T. From the M(T) curves, the transition temperature is observed to be around 108 K and almost no thermal hysteresis is present. There is no obvious sign of a second magnetic phase transition within the displayed temperature range, demonstrating that the small amount of ferromagnetic impurity phase Nd₃In with Curie temperature of 114 K (Ref. 21) only has a negligible influence on the magnetization measurement. Figure 2(b) shows the magnetic entropy change as a function of temperature under magnetic field changes up to 2 T with a step of 0.25 T. The magnetic entropy change was calculated by using the Maxwell equation from the measured data of the magnetization as a function of temperature under given magnetic field changes according to the protocol proposed in Ref. 22. Under a magnetic field change of 2 T, the maximum magnetic entropy change $ΔSmmax$ is 7.42 J/kg K at 109 K. For comparison, we plot the magnetic entropy changes of the RE₂In family and some other materials with transition temperatures near the boiling temperature of natural gas under a magnetic field change of 2 T in Fig. 2(c). Nd₂In shows a lower magnetic entropy change than the first-order magnetocaloric material Dy_0.7Er_0.3Co₂ (11.9 J/kg K) but compares with TbAl₂ (7.4 J/kg K).

Interestingly, the magnetic entropy changes of Eu₂In and the light rare-earth based alloys Pr₂In and Nd₂In are larger than or comparable to those of the other rare-earth based RE₂In alloys. For many other material systems, exactly the opposite is observed because the heavy rare-earth elements of Gd, Tb, Dy, Ho, and Er carry a larger magnetic moment than Eu and the light rare-earth elements of Pr and Nd. For example, the magnetic entropy change of DyAl₂ with a Curie temperature of 60 K is about twice as large as that of NdAl₂ with a Curie temperature of 77 K under magnetic field changes of 5 T.⁸ The same holds true for the magnetic entropy change of HoAl₂ being significantly larger than that of PrAl₂ with similar Curie temperature of about 32 K.⁸ The behavior that Pr₂In, Nd₂In, and Eu₂In do not follow this empirical rule is possibly related to the fact that the materials Eu₂In and Pr₂In undergo a first-order phase transition,^11,12 which might be also the case for Nd₂In, as reported by Forker et al.¹⁶ All other RE₂In alloys undergo a second-order phase transition, as reported by Refs. 23–25.

Figure 2(d) shows the direct measurements of the adiabatic temperature change obtained under a magnetic field change of 1.95 T using continuous protocol (approaching to each new target temperature occurs gradually, without overheating and overcooling the sample). The adiabatic temperature shows a maximum value of 1.13 K at 110.1 K. In the inset of Fig. 2(d), the adiabatic temperature changes of the first and second cycles were plotted as a function of magnetic fields up to 1.95 T at 110.1 K. Both cycles reach an almost identical maximum value of about 1.13 K, agreeing with the observations of the negligible thermal hysteresis. Our results on the magnetic and magnetocaloric properties of Nd₂In demonstrate that Nd₂In has a large magnetic entropy change and a moderate but fully reversible adiabatic temperature change.

Triggered by the question whether the abnormally large magnetic entropy change in Nd₂In is due to the first-order phase transition, we applied a quantitative criterion featuring on the exponent $n$ from the field dependence of the magnetic entropy change [$nT, H=dln|ΔSm| / dlnμ0H$] for determining the order of magnetic phase transitions.²⁷ This method was developed by Law et al., and the detailed descriptions are presented in Ref. 27. The main advantage of this method is that it is quantitative, which is suitable for studying the nature of the phase transition even being close to the critical point where one cannot easily decide on the order using other methods, such as differential scanning calorimeter (DSC) which relies on the experiences of a researcher.²⁷ We plot the value of the exponent $n$ in Fig. 3(a) as function of temperature in fields of 0.5, 1, 1.5, and 2 T. The maximum values of all curves are above 2, which is exactly the sign of a first-order phase transition.

Additionally, we carried out heat capacity measurements for Nd₂In. Figure 3(b) presents the results of the measurements in magnetic fields of 0, 2, 6, and 12 T. The inset shows the heat capacity in zero field down to the temperature of about 2 K. In addition to the magnetic phase transition at about 108 K, we noticed another phase transition at around 50 K, as indicated by the arrows in the inset. From the heat capacity curves $Cp(T, H)$⁠, it is apparent that the inflection point shifts with magnetic field at a considerable rate of about 1 K/T, which is a typical feature of a first-order phase transition.²⁸ Our analyses on the exponent $n$ and the heat capacity reveal that the magnetic phase transition of Nd₂In is of first-order type, and it is the reason why Nd₂In shows an abnormally large magnetic entropy change at around the boiling temperature of the natural gas. These conclusions are in good agreement with the measurements of magnetic and electric hyperfine interactions that support the first-order phase transition in Nd₂In.¹⁶ 

For better understanding of the magnetocaloric effects and the nature of the phase transition in RE₂In, we studied the electrical resistivity and the thermal expansion in different magnetic fields upon heating using our purpose-built simultaneous measurement system.²⁰ Figure 4(a) plots the magnetization as a function of temperature in magnetic fields up to 12 T. As seen in the M(T) curves, there is an additional phase transition around 50 K, agreeing with the indication of the heat capacity measurement. A similar phase transition was also reported by Biswas et al. in Pr₂In¹² and by Gamari-Seale et al. in Ho₂In.²⁹ It is suspected to be a spin reorientation transition or an electronic transition between two states with different moments of the rare-earth sites.¹² Further investigations into the nature of this transition are still proceeding. In this work, we focus on the magnetocaloric effects associated with the phase transition around 108 K. Figure 4(b) shows the electrical resistivity in magnetic fields up to 12 T. The electrical resistivity decreases with the decreasing temperature, and the curves shift toward lower values with the increasing magnetic fields.

Since first-order phase transition is often coupled with an obvious structural change, we did a study on the strain in different magnetic fields as well. Figure 4(c) presents the longitudinal strain in magnetic fields of 0.05 and 12 T with temperatures between 200 and 4 K. As the strain change during transition is too small to be observed clearly, the transition area as indicated by the blue circle is magnified and is plotted in Fig. 4(d), showing the longitudinal strain in temperatures between 120 and 100 K in magnetic fields of 0.05, 2, 4, 8, and 12 T. The inset in Fig. 4(d) demonstrates the volume change in magnetic fields of 0.05, 2, and 12 T. The volume change is calculated by summarizing the longitudinal strain and the other two lateral strains in the other two axial directions.³⁰ Although as mentioned above, the transition of Nd₂In is confirmed to be of the first-order type, we did not observe a large strain change during the transition. As a matter of fact, the strain changes are small, accounting for only 0.02%, whereas the first-order phase transition materials, such as the all-d-metal NiCoMnTi,³¹ reach 1% easily. In addition, the volume changes around 108 K are negligible, whereas the values are around 1% for (La,Ce)(Fe,Si)₁₃.³² However, we emphasize here that a small strain at the first-order phase transition was also reported by Kokorin et al. in Ni–Mn–Ga Heusler materials in zero magnetic field.³³ It is worth mentioning that when extrapolating the strain curve in the paramagnetic regime linearly down to low temperatures in Fig. 4(c) (Grüneisen parameter is out of consideration),^34,35 we found that the spontaneous magnetostrictive strain is negative and considerable, reaching a maximum absolute value of about 0.03%, which is comparable to that of Sm₂Co₇ single crystal along the c axis (0.036%).³⁶ Interestingly, as seen in Fig. 4(d), the strain around the transition temperature of 108 K increases and then decreases with magnetic fields. The strain curves in 2 and 4 T shift upward, whereas the strain curves of 8 and 12 T shift downward. These observations indicate that the phase transition undergoes two stages. Such observations were not reported in the RE₂In families. Although we have confirmed the magnetic phase transition to be of first-order type, our results of the abnormally large magnetic entropy change, the negligible hysteresis, and the small strain and volume changes during phase transition demonstrate that the phase transition of Nd₂In is complex and needs further research, especially on its electronic structure.

It should be emphasized that, in contrast to the other heavy rare-earth elements, the 4f electrons of Eu and the light rare-earth elements Nd, Pr, and Sm are less localized.³⁷ Hence, the interactions of the 4f electrons with the itinerant electrons are generally more complicated. First-principles calculation done by Mendive-Tapia et al. in 2020 shows that the free energy with respect to the ferromagnetic order of Eu₂In possesses two minima, suggesting a first-order phase transition.³⁸ They conclude that the positive feedback between the magnetism of itinerant electrons and the ferromagnetic ordering of 4f electron moments triggers a topological change to the Fermi surface, which might relate to the small structure change in Eu₂In during phase transition.³⁸ Though it is not clear whether the abnormal behavior of the longitudinal strain with magnetic field relates the first-order phase transition of Nd₂In, we highly suspect that there is a correlation in between.

To summarize, in this work, we reported the large magnetocaloric effect in Nd₂In. Under a magnetic field change of 2 T, the magnetic entropy change of Nd₂In reaches a value of 7.42 J/kg K with a negligible thermal hysteresis near the boiling temperature of natural gas, and a fully reversible adiabatic temperature change with a value of about 1.13 K under a magnetic field change of 1.95 T. Within the RE₂In family, the light rare-earth based Nd₂In shows a significantly larger magnetic entropy change than that of the heavy rare-earth based Gd₂In, Tb₂In, Dy₂In, and Ho₂In. The magnetic phase transition is confirmed to be of the first-order type, such as Eu₂In and the light rare-earth based material Pr₂In.¹⁶ The strain and volume changes around the transition are small compared to other materials, such as (La,Ce)(Fe,Si)₁₃³² and all-d-metal NiCoMnTi,³¹ but the spontaneous magnetostrictive strain is considerable. In addition, an abnormal two-stage strain behavior near the phase transition was observed.

Our work shows that these interesting properties are useful for the practical design of a magnetocaloric natural gas liquefaction system. However, to further understand the physics behind the unusual large magnetic entropy change of Nd₂In and its mechanism of phase transitions, much research is required.

The authors gratefully acknowledge the financial support from the Helmholtz Association via the Helmholtz-RSF Joint Research Group (Project No. HRSF-0045); the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL); the DFG through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat (EXC 2147, Project No. 39085490); and TU Darmstadt. This project was also supported by the DFG (Project No. 405553726-TRR 270, Germany).