Magnetic and magnetocaloric properties of Dy_1−xEr_xNi₂ solid solutions and their promise for hydrogen liquefaction

Jacek Ćwik *^a, Yurii Koshkid’ko ^a, Kiran Shinde *^b, Joonsik Park ^b, Nilson Antunes de Oliveira ^c, Michał Babij ^a and Agata Czernuszewicz ^d
aInstitute of Low Temperature and Structure Research, PAS, Okólna 2, Wrocław, 50-422, Poland. E-mail: j.cwik@intibs.pl
^bDepartment of Materials and Science and Engineering, Hanbat National University, Daejeon 34158, Republic of Korea. E-mail: yourkirans@gmail.com
^cInstituto de Física Armando Dias Tavares-Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro 20550-013, RJ, Brazil
^dAmes National Laboratory, U.S. Department of Energy, Iowa State University, Ames, IA 50011, USA

First published on 15th August 2024

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Introduction

The ability to cool on demand has had a profound impact on the modern world. Access to refrigeration and space conditioning has become an integral part of our daily lives. However, cooling has a much broader scope than what we typically think. One of the crucial areas where cooling is essential is gas liquefaction. This process involves reducing the temperature of a substance, often to cryogenic levels, i.e., below 123 K. Liquefied gas has a much higher density, making it easier to store and transport. Consequently, the ability to liquefy gases has become an essential aspect of many industrial and scientific processes.

Hydrogen, the most abundant element on earth, can be used to provide clean and renewable energy. Hydrogen typically does not exist freely in nature. It is generated from a variety of sources, such as fossil fuels, biomass, and water. Hydrogen fuel cells combine oxygen and hydrogen to produce electricity and heat, with water vapor as the sole by-product, making it an environmentally friendly and sustainable energy option. Hydrogen fuel can be used in a variety of applications in industries such as transportation, power generation, and portable devices. It can provide electricity to remote areas without power grids and can also be used in emergencies to power buildings. For practical applications, to facilitate storage and transportation, hydrogen gas needs to be transformed into its liquid state, which increases its energy density from 0.01 MJ L⁻¹ to 8.50 MJ L⁻¹.¹ This is achieved by precooling it to 80 K with liquid nitrogen bath and subsequently compressing and expanding the precooled gas until condensation at 20 K. Hydrogen liquefaction is, however, an energy-intensive and expensive process that is typically accomplished by compressing and expanding gas at an efficiency below 25%.²

Magnetic cooling is a promising alternative to traditional liquefaction technology. It can achieve liquefaction efficiencies above 50%,² and there are no greenhouse gas emissions related to refrigerant leaks. Magnetic cooling is based on the magnetocaloric effect (MCE), a magneto-thermodynamic phenomenon that manifests itself in the absorption or emission of heat by a solid refrigerant when exposed to a changing external magnetic field. The MCE is characterized by the magnetic entropy change, ΔS_M, in an isothermal process and the temperature change, ΔT_ad, in an adiabatic process. Typically, the values of magnetocaloric potentials, ΔS_M and ΔT_ad, are highest near the magnetic ordering temperature and gradually decrease to zero beyond the magnetic phase transition region.

The prospect of high efficiency and low environmental impact resulted in immense focus on the development of magnetic cooling systems.^2–4 Extensive research has been conducted to assess the impact of the work cycle,^5,6 component design,^7,8 and operating parameters⁹ on system performance. A lot of emphasis has also been placed on developing new magnetocaloric materials that can operate at cryogenic temperatures. To replace hydrogen liquefiers, magnetic coolers need to be able to cool down hydrogen from 80 K to 20 K. This requires the utilization of multistage systems. At each subsequent stage, different magnetocaloric materials with lower Curie temperature and operating temperature range need to be used. The employed magnetic refrigerants should provide continuous magnetocaloric response from 80 K to below 20 K.¹⁰ Alternatively, the same results can be achieved by using multilayered regenerators composed of magnetocaloric materials operating in the hydrogen liquefier temperature range.¹¹ Rare-earth nickel, RNi₂, intermetallic compounds based on heavy rare earth elements undergoing a second-order ferromagnetic to paramagnetic phase transition are of particular interest. This is because they might show a large reversible MCE in the temperature range of hydrogen liquefaction.¹² Their special properties are associated with highly localized magnetic moments originating from the incompletely filled 4f-electron shell of the R atoms. In these compounds, Ni atoms stay in the non-magnetic state.^13,14 Because both nickel and rare earth elements are perceived as critical metals with a high supply risk due to their importance in technological applications, it is essential to develop recovery and separation methods for these materials, especially if magnetic cooling technology is commercialized.¹⁵

In our previous studies, we found that the crystal lattice of the Dy_1−xEr_xNi₂ solid solutions, similar to Tb_1−xDy_xNi₂ (ref. 16) and Tb_1−xEr_xNi₂ (ref. 17) systems, can be adequately described by the C15-type (space group F3m) Laves-phase structure with the doubled lattice parameter and that the lattice parameter decreases with increasing Er content from a = 14.295(4) Å for x = 0.25 to a = 14.264(1) Å for x = 0.75.¹⁸ Replacing Dy with Er leads to expected magnetic dilution, weakening exchange interactions and decreasing the ordering temperature. The Curie temperatures fall linearly from 16.8 K for Dy_0.75Er_0.25Ni₂ to 10 K for Dy_0.25Er_0.75Ni₂, making these solid solutions useful for a lower temperature range of multistage magnetic hydrogen liquefiers. Based on heat capacity data, Dy_0.75Er_0.25Ni₂ exhibits the maximum magnetic entropy change of 12.3 J kg⁻¹ K⁻¹ around 17 K for a magnetic field change of 2 T. For Dy_0.5Er_0.5Ni₂ and Dy_0.25Er_0.75Ni₂ compositions, the maximum magnetic entropy change is slightly lower, reaching 11.6 and 11.1 J kg⁻¹ K⁻¹, respectively. The maximum values of ΔT_ad for the 2 T magnetic field change reach 4.1 K for both Dy_0.75Er_0.25Ni₂ and Dy_0.5Er_0.5Ni₂, and 3.9 K for Dy_0.25Er_0.75Ni₂.¹⁸

This work continues the investigation of Dy_1−xEr_xNi₂ solid solutions and is focused on deepening understanding of their magnetic and magnetocaloric properties in high magnetic fields. Isothermal magnetic entropy changes were calculated based on magnetic measurements, and direct measurements of adiabatic temperature changes were performed in magnetic fields of up to 14 T. The obtained experimental data were compared with theoretical calculations performed employing a model Hamiltonian that includes the crystalline electric field effects and exchange magnetic interaction. Since the MCE is strongly correlated to the phase transition, it is important to explore the magnetic interactions that cause it. The critical behavior around phase transition was investigated through isothermal magnetization measurements. Observed large magnetocaloric quantities and transition temperatures below hydrogen liquefaction temperature showed that Dy_1−xEr_xNi₂ solid solutions are strong candidates for low-temperature magnetic refrigerants.

Experimental details

Polycrystalline samples of Dy_1−xEr_xNi₂ (x = 0.25, 0.5, and 0.75) solid solutions were prepared as described in ref. 18. The samples used for magnetization measurements had a mass of ∼2 mg and were nearly spherical in shape, allowing for the demagnetizing field to be accounted for. Isothermal magnetization curves, M(μ₀H), were measured in magnetic fields up to 14 T at various temperatures below and above T_C with a step dT of 2 K. Measurements were performed using a vibration sample magnetometer equipped with a Bitter-type magnet and a step motor.¹⁹ For direct magnetocaloric effect measurements (ΔT_ad), a system described in ref. 20 was used. Experiments were performed on ∼50 mg samples in magnetic fields up to 14 T generated by a Bitter-type magnet. The temperature of the sample was measured using a differential thermocouple, and a Hall sensor placed in the sample holder was used to measure the magnetic field. A thermal screen around the sample helped to minimize heat losses to the environment.

Theoretical formulation

To describe the magnetic properties and the magnetocaloric effect in the doped compound Dy_1−xEr_xNi₂, a model Hamiltonian of local interacting magnetic moments with two magnetic sublattices was employed.²¹ The presence of the two body interactions and the chemical disorder at the rare earth sites complicate the theoretical description of this compound. To simplify the analysis and account for the two-body interactions and doping, we used an extended mean-field approximation.^18,22 The Hamiltonian suitable to describe the magnetic properties and the magnetocaloric effect in Dy_1−xEr_xNi₂ can be written as:

	(1)

where B is the applied magnetic field, stands for the exchange interaction energy between two Dy ions, represents the exchange interaction parameters between two Er ions, and are the exchange interaction parameters between two different rare earth ions. g^Er = 1.2 and g^Dy = 1.33 are the Lande factors as determined by Hund's rule. The term describes the crystalline electric field.^23–26 In cubic symmetry, it can be written as:

	(2)

where x_c and W are model parameters, Oⁿ_m are the Stevens’ operators,²³ and F₄ and F₆ are common factors that depend on the angular momentum of rare earth ions.²⁴ For Dy and Er, their values are F₄ = 60 and F₆ = 13860.

The free energy for the Hamiltonian (1) is F^α_M = −k_BTlnZ^α, where is the partition function and E^α_i (α = Dy,Er) are the energy eigenvalues of the Hamiltonian. The magnetic part of the entropy, determined via the usual thermodynamic relation S^α_M = −(∂F^α_M/∂T), is:

	(3)

In order to calculate the magnetocaloric quantities, besides the magnetic entropy, we also need to include the contributions from the conduction electrons and crystalline lattice. The entropy for the α-sublattice is S^α(T,μ₀,H,x) = S^α_M(T,μ₀,H,x) + S_lat(T) + S_el(T), where S_el (T) = γT is the contribution from the conduction electrons, γ is the Sommerfeld coefficient and the S_lat (T) represents the contribution from the crystalline lattice, which is given in the Debye approximation by:

	(4)

where R is the gas constant, and N_i is the number of atoms per unit formula. The total entropy of the doped compound Dy_1−xEr_xNi₂ as a function of temperature and impurity concentration is S(T,μ₀,H,x) = (1 − x)S^Dy(T,μ₀,H,x) + xS^Er(T,μ₀,H,x), where S^Dy(T,μ₀,H,x) and S^Er(T,μ₀,H,x) are the entropy contributions associated with Dy and Er sublattices, respectively. Once the total entropy as a function of temperature for two different values of the applied magnetic field is known, the magnetocaloric quantities, i.e., the isothermal entropy change ΔS(T,μ₀ΔH,x) = S(T,μ₀H₂,x) − S(T,μ₀H₁,x) where μ₀ΔH = μ₀H₂ − μ₀H₁ and the temperature change ΔT_ad(T,μ₀ΔH,x) = T₂ − T₁ under the adiabatic condition S(T,μ₀H₂,x) = S(T,μ₀H₁,x) can be determined.

Results and discussion

Magnetic measurements

All investigated Dy_1−xEr_xNi₂ (x = 0.25, 0.5, and 0.75) solid solutions are characterized by a ferromagnetic order and relatively low Curie temperatures below the temperature of liquid hydrogen. Fig. 1 shows the magnetization versus temperature dependence at an applied field of μ₀H = 0.5 T. The M(T) curves exhibit a single magnetic transition from a paramagnetic (PM) to a ferromagnetic (FM) state as the temperature decreases. The transition temperatures T_C, determined from the critical point in the derivative dM/dT curves, are found to be 16.2, 12.7, and 10.2 K for Dy_0.75Er_0.25Ni₂, Dy_0.5Er_0.5Ni₂, and Dy_0.25Er_0.75Ni₂, respectively (Fig. 1 inset). These results are comparable to those previously estimated from heat capacity and magnetic measurements.¹⁸

The magnetization curves measured at 4.2 K are presented in Fig. 2. The magnetization shows a rapid increase in low fields (<2 T) for all samples. However, at higher fields (>6 T), it varies linearly with the field, and saturation is not achieved even at the maximum field of 14 T. The saturation magnetization, M_S, values were determined from the M vs. 1/μ₀H curves extrapolated to 1/μ₀H = 0 (see Fig. 3 inset). The obtained M_S values decrease with the increase of Er content from 8.2μ_B per f.u. for Dy_0.75Er_0.25Ni₂ to 7.8μ_B per f.u. for Dy_0.25Er_0.75Ni₂. The results are ∼1.5μ_B below the expected theoretical values calculated assuming that nickel is non-magnetic and the magnetic moments originate from the rare-earth sublattice.

According to our previous studies on parent compounds,^27,28 the reduced M_S moment results from partial quenching of the orbital moment by the crystalline field, and this effect increases with a decrease in magnetic ordering temperatures. In the case of DyNi₂ with a 4f¹⁰ electron configuration, the measured M_S value is 0.2μ_B lower than the theoretical calculations.²⁷ In contrast, the ErNi₂ sample with a 4f¹² electron configuration and the lower magnetic ordering temperature T_C of 6.5 K, at 4.2 K and 14 T shows an M_S of 7.7μ_B per f.u. (ref. 28). This value is ∼12% higher than the value reported by Farrell et al.²⁹ and close to the one reported for the ribbon sample by Llamazares et al.³⁰ It can be assumed that the low M_S values in Dy_1−xEr_xNi₂ solid solutions are due to the Er sublattice. To confirm our assumption, neutron diffraction is necessary. The spontaneous magnetic moments, M_SP, also decrease with the increasing Er content as a consequence of the lower total angular momentum of Er atoms than Dy atoms. The calculated values are 7.3, 6.3, and 6.2μ_B per f.u. for Dy_0.75Er_0.25Ni₂, Dy_0.5Er_0.5Ni₂, and Dy_0.25Er_0.75Ni₂, respectively. The magnetic and structural parameters obtained for the investigated Dy_1−xEr_xNi₂ solid solutions, together with parent DyNi₂ and ErNi₂ compounds, are listed in Table 1.

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Magnetization isotherms were measured at selected temperatures near ordering temperatures at applied magnetic fields up to 14 T. The measurements were performed for all investigated solid solutions. Fig. 3 shows, as an example, the magnetization isotherms for Dy_0.5Er_0.5Ni₂ from 4.2 to 30 K with a 2 K step. We noticed that at low temperatures, well below T_C, the magnetization increases abruptly in the low-field range and then gradually rises linearly with increasing magnetic field. No saturation is observed even at the maximum external magnetic field of 14 T. Above T_C, M(μ₀H) shows a downward curvature and increases slowly, with no saturation-like behavior observed.

High-field magnetocaloric properties

To identify magnetocaloric properties, we used indirect and direct methods. The magnetic entropy change (ΔS_M) was calculated from magnetization data, and the adiabatic temperature change (ΔT_ad) was measured directly. The experimental results were compared with theoretical calculations.

Using the magnetization isotherms obtained close to the Curie temperature, the change in magnetic entropy was calculated for each Dy_1−xEr_xNi₂ solid solution by applying the integrated Maxwell's relation:

	(5)

In order to calculate the magnetocaloric properties using the previously presented model, a set of model parameters had to be selected. The exchange parameters were chosen to fit the experimental data for the magnetic ordering temperature. The values of these parameters were . The crystalline electric field parameters were x_c = −0.1, W = 0.019 meV for the intermetallic compound DyNi₂, and x_c = −0.54, W = 0.05 meV for ErNi₂. These values were obtained by fitting experimental data of specific heat capacity.^18,31,32 The Sommerfeld coefficient was chosen as γ = 20 mJ mol⁻¹ K⁻² for all investigated concentrations. These values fall in the range commonly used in the literature.^21,32

Fig. 4 compares experimental data and results of theoretical calculations of the isothermal entropy change resulting from the variation of the magnetic field, ranging from 1 to 14 T, for all solid solutions studied. The theoretical calculations are in reasonable agreement with the experimental data. For all the samples, −ΔS_M reaches relatively large quantities near the temperature of magnetic phase transitions. Increasing Er content leads to a decrease in the Curie temperature, resulting in a significant shift of the peak of −ΔS_M towards lower temperatures. At the magnetic field change of 14 T, the maximum experimental −ΔS_M values are 33.5 J kg⁻¹ K⁻¹ at 20 K, 28 J kg⁻¹ K⁻¹ at 16.9 K, and 28.9 J kg⁻¹ K⁻¹ at 13.0 K for Dy_0.75Er_0.25Ni₂, Dy_0.5Er_0.5Ni₂, and Dy_0.25Er_0.75Ni₂, respectively. A similar trend has been observed for a field change of 10 T, where −ΔS_M for Dy_0.75Er_0.25Ni₂ is 30.2 J kg⁻¹ K⁻¹ at 16.9 K, and for Dy_0.5Er_0.5Ni₂ and Dy_0.25Er_0.75Ni₂, −ΔS_M values are close to 26 J kg⁻¹ K⁻¹ at 15 and 13 K, respectively. It should also be noted that the values obtained at 10 T are lower than the ones obtained for the initial DyNi₂ and ErNi₂ compounds, where −ΔS_M reaches 34 (ref. 27) and 29.1 J kg⁻¹ K⁻¹,³⁰ respectively (see Table 2). The reason behind reduced magnetocaloric properties of Dy_1−xEr_xNi₂ solid solutions is likely due to the formation of a superstructure during crystallization. This effect was also observed in other solid solutions, such as Tb_1−xEr_xNi₂ (ref. 17) or Er_1−xHo_xNi₂.³³ The characteristic quantity, −ΔS_M/μ₀ΔH, was observed to decrease with the increase in the external magnetic field change. For magnetic field change rising from 2 to 14 T, −ΔS_M/μ₀ΔH decreases from 6.3 J kg⁻¹ K⁻¹ to 2.4 J kg⁻¹ K⁻¹ for Dy_0.75Er_0.25Ni₂, from 5.2 J kg⁻¹ K⁻¹ to 2.0 J kg⁻¹ K⁻¹ for Dy_0.5Er_0.5Ni₂, and from 6.1 J kg⁻¹ K⁻¹ to 2.1 J kg⁻¹ K⁻¹ for Dy_0.25Er_0.75Ni₂.

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For all studied samples, as the external magnetic field increases, the peak corresponding to the maximum entropy shifts toward a higher temperature range. This behavior is associated with the electronic entropy and electronic heat capacity, which exhibit strong nonlinear dependencies on both temperature and magnetic field in the low-temperature range, where the electronic heat capacity coefficient drastically changes under the influence of external magnetic field or due to the coexistence of magnetic phase transitions, as is the case with the solid solutions being investigated. Other Laves-phase solid solutions that are magnetically ordered at low temperatures also showed similar behavior.^17,33

Fig. 5 shows theoretical and directly measured ΔT_ad results. The ΔT_ad values measured for field applications are plotted against the initial sample temperature for magnetic fields changing from 0 up to 14 T for Dy_0.75Er_0.25Ni₂ and Dy_0.25Er_0.75Ni₂, and from 0 up to 10 T for Dy_0.5Er_0.5Ni₂. The experimentally observed ΔT_ad data are in good agreement with the theoretical calculations. As expected, the field application leads to an increase in the sample temperature. One can notice that the maximum adiabatic temperature changes for Dy_0.75Er_0.25Ni₂ and Dy_0.5Er_0.5Ni₂ are similar. For instance, with a magnetic field change of 10 T, ΔT_ad for both Dy_0.75Er_0.25Ni₂ and Dy_0.5Er_0.5Ni₂ reaches 10.8 K at temperatures of 16.7 and 13.5 K, respectively. The values for Dy_0.25Er_0.75Ni₂ are slightly lower, with a 10.5 K temperature change at 10 K. The values obtained at 1 and 2 T are similar to the ones previously calculated from heat capacity measurements and published in ref. 18.

Additionally, it was observed that for Dy_0.25Er_0.75Ni₂, the temperature at which ΔT_ad reaches its maximum value shifts toward higher temperatures as the magnetic field increases. The behavior is similar to that seen earlier for the parent ErNi₂ compound.²⁸ It is caused by a combination of factors such as low magnetic ordering temperature, the magnitude of the magnetic field, and the shape of the heat capacity curve above the Curie temperature. In the low-temperature region, the lattice heat capacity increases in proportion to T³, while at higher temperatures, this growth slows down. Such displacements were reported previously for a high magnetic field change mentioned in ref. 34 and 35. This behavior was not observed for the other two studied compounds. The characteristic quantity, ΔT_ad/μ₀ΔH, was observed to decrease with the increase in the external magnetic field change, similar to −ΔS_M/μ₀ΔH. For the magnetic field change rising from 2 to 14 T, ΔT_ad/μ₀ΔH decreases from 1.9 K T⁻¹ to 0.9 K T⁻¹ for Dy_0.75Er_0.25Ni₂, from 1.9 K T⁻¹ to 1.1 K T⁻¹ for Dy_0.5Er_0.5Ni₂, and from 1.5 K T⁻¹ to 0.9 K T⁻¹ for Dy_0.25Er_0.75Ni₂. All the experimental −ΔS_M and ΔT_ad results are presented in Table 2.

Critical behavior

The critical behavior of the FM–PM, second-order magnetic transition (SOMT), can be analyzed with a set of three critical exponents, β, γ, and δ,^36,37 associated with the magnetocaloric properties of materials. To gain a comprehensive understanding of the intrinsic magnetic interactions within Dy_1−xEr_xNi₂ (x = 0.25, 0.5, 0.75), different techniques were employed to study the critical behavior near the Curie temperature. According to the Arrott–Noakes equation,³⁸ the magnetization of a ferromagnetic material undergoing the second-order phase transition is expressed as,

	(6)

where A and B are temperature-dependent parameters, β and γ are the critical exponents, and ε is the reduced temperature calculated as,

	(7)

The critical exponents β, γ, and δ are related to the spontaneous magnetization (M_SP) below T_C, the inverse magnetic susceptibility (χ₀⁻¹) above T_C, and the critical magnetization isotherm M at T_C, respectively:^39,40

MSP(T) = M0(−ε)β, ε < 0, T < TC,	(8)

	(9)

M = RH1/δ,  at ε = 0, T = TC,	(10)

where the critical amplitudes are represented as constants M₀, h₀/M₀, and R.

Based on the Arrott–Noakes equation, if the values of the critical exponents β and γ are approached, the modified Arrott plots (MAP) of M^1/βversus (μ₀H/M)^1/γ will form a set of parallel lines at temperatures near T_C with a line at T = T_C passing through the coordinate origin. However, as evidenced in Fig. 6, the Arrott plots of M²versus μ₀H/M (β = 0.5 and γ = 1.0) for all investigated solid solutions do not exhibit the expected characteristics.⁴¹ Therefore, we can conclude that the mean-field model is not suitable for our materials. The reason behind such behavior is the existence of the short-range FM order in the studied compounds.

Fig. 7 shows the M^1/βversus (μ₀H/M)^1/γ plots for Dy_0.75Er_0.25Ni₂ at different temperatures, using three different theoretical models: (a) 3D-Heisenberg model (β = 0.365, γ = 1.386), (b) tricritical mean-field model (β = 0.265, γ = 1), and (c) 3D-ising model (β = 0.325, γ = 1.24).⁴² Different results for each approach as shown in Fig. 7 prove that identifying the most appropriate theoretical model for the studied solutions is a challenging task.

The relative slope (RS) method is frequently utilized to determine which model best describes the magnetic isotherms. This involves normalizing the slope of magnetic isotherms by the slope at T_C, as indicated by the following relation,

	(11)

where S(T) and S(T_C) are the slopes of isotherms around and at T_C, respectively. Fig. 8 compares the RS versus T plots for the Dy_0.75Er_0.25Ni₂ sample obtained using four different models. The most suitable model is expected to have the RS value closest to the unity.^43,44 The tricritical mean-field model meets this requirement, and therefore, we will exclusively examine the critical behavior of Dy_0.75Er_0.25Ni₂ under this model.

The spontaneous magnetization M_SP(T) and the inverse magnetic susceptibility χ₀⁻¹(T) were extracted from the modified Arrott plots for the tricritical mean-field model. The M_SP(T) and χ₀⁻¹(T) values as the intercepts with the M^1/β and (μ₀H/M)^1/γ axes, respectively, were calculated by linear extrapolation in a higher magnetic field region. The obtained values of M_S(T) and χ₀⁻¹(T) are plotted against temperature (T) as shown for the Dy_0.75Er_0.25Ni₂ sample in Fig. 9(a). By fitting eqn (8) and (9), the corresponding critical exponents are determined. The results indicate β = 0.21 with T_C = 16.3 K and γ = 0.76 with T_C = 16.1 K.

The accuracy and reliability of the β, γ, and T_C values can also be verified independently using the Kouvel–Fisher (K–F) method.^45,46 The Kouvel–Fisher plot does not require any prior knowledge of T_C. Instead, it is derived from the intercept of the fitted straight lines on the temperature axis, and the critical exponents can be directly determined from the following Kouvel–Fisher relations,

	(12)

	(13)

Based on these equations, a straight line with slopes 1/β and 1/γ, respectively, should be obtained by plotting M_SP(T)[dM_SP(T)/dT]⁻¹ and χ⁻¹₀(T)[dχ₀⁻¹(T)/dT]⁻¹ against T. Fig. 9(b) shows the Kouvel–Fischer plot of the Dy_0.75Er_0.25Ni₂ sample. The linear fitting provides the estimated exponents and T_C as β = 0.21, T_C = 16.4 K and γ = 0.93, T_C = 16.5 K.

In the critical region, the critical exponents should follow the standard critical state and scaling equation, which is given as:⁴⁷

	(14)

where f₊ and f₋ are magnetic state functions for the M(μ₀H) data at the temperatures above and below T_C, respectively. Moreover, the re-normalized magnetization and field can be expressed as m = M|ε|^−β and h = μ₀H|ε|^−(β+γ), respectively.^48,49 The m vs. h curves near the T_C should yield two independent branches, one below T_C and the other above T_C, as depicted in Fig. 10(a). Fig. 10(b) shows the m²versus h/m curves for T < T_C and T > T_C. It is evident that all curves fall on two independent branches: T < T_C and T > T_C. The critical exponent δ can be estimated by employing the Widom scaling relation,⁵⁰

	(15)

Utilizing the critical exponents β and γ obtained through the MAP and K–F methods in eqn (15), the values of critical exponent δ were determined as 4.61 and 5.42, respectively. Table 3 compares the critical exponents for Dy_0.75Er_0.25Ni₂ obtained using various theoretical models. The discrepancies between experimentally determined critical exponents and theoretical values can be attributed to several factors, such as the finite size effect, sample purity and quality, and the critical region. The finite size effects in real experimental systems, which are not present in theoretical models assuming infinite size, can cause deviations in critical exponents. The sample preparation method produces impurities and defects in the materials that introduce additional interactions not considered in theoretical models. The critical region, where scaling laws and critical exponents are valid, can be very narrow. Experiments may not always probe sufficiently close to the critical point due to practical limitations, leading to observed deviations from the theoretical critical behavior. The characteristic exponents of Dy_0.75Er_0.25Ni₂ coincide well with the tricritical mean-field model. The findings suggest the presence of short-range FM interactions in this compound.^51–55 It should be noted that a recent study conducted by Fan et al.⁵⁴ analyzed the critical behavior of ErNi₂ compounds, and their findings indicate that the critical behaviors of these compounds also align most closely with the tricritical mean-field model with final critical exponents determined as β = 0.164(2) and γ = 0.821(9).

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In the case of SOMT materials, there is an additional critical exponent, n, which can be specified by the magnetic field dependence of −ΔS^max_M, which follows a scaling law,⁵⁶

ΔSmaxM ≈ a(μ0H)n,	(16)

where n is a scaling exponent for the field dependence of the peak in the isothermal magnetic entropy change as shown in Fig. 11. The value of n was extracted by fitting with eqn (16), resulting in n = 0.62 for Dy_0.75Er_0.25Ni₂. Alternatively, n can be calculated using the following equation because it is field- and temperature-dependent, as shown in the inset of Fig. 11:

	(17)

According to mean field theory (MFT), the exponent n(T,μ₀H) for the long-range ferromagnet tends to achieve the lowest value of 0.67 at Curie temperature. However, at T ≪ T_C and T ≫ T_C, it tends to be 1 and 2, respectively.^57,58 For Dy_0.75Er_0.25Ni₂, n initially slightly decreases as temperature increases, reaching a minimum around T_C. Further temperature increase causes a rapid rise of n. The observed value of n near T_C is 0.62 at μ₀ΔH = 3 T, which is lower than the value for the mean-field model (0.67) and indicates the occurrence of short-range magnetic ordering. In the Fig. 11 inset, there is no overshoot of n = 2 near the transition temperature for Dy_0.75Er_0.25Ni₂, confirming that this alloy follows SOMT behavior.

Conclusions

Solid solutions of Dy_1−xEr_xNi₂ with x = 0.25, 0.50, and 0.75 were prepared using the arc-melting method and their magnetic and magnetocaloric properties in a wide magnetic field range, up to 14 T, have been studied theoretically and experimentally. The magnetic measurements confirmed that the substitution of Dy atoms for Er atoms noticeably modifies the magnetic behavior of the Dy_1−xEr_xNi₂ solutions. The presence of Er atoms in the rare-earth sublattice results in common magnetic dilution, which weakens exchange interactions and reduces the ordering temperature. The T_C decreases from 16.2 K for Dy_0.75Er_0.25Ni₂ to 10.2 K for Dy_0.25Er_0.75Ni₂. The saturation magnetic moments, estimated from the magnetization curves at a temperature of 4.2 K, exhibit a decreasing trend with an increase in Er content and are approximately 1.5μ_B lower than theoretically anticipated values. The reduced moment is attributed to partial quenching of the orbital moment by the crystalline field, and this effect increases as the magnetic ordering temperatures decrease.

The critical behavior of the Dy_0.75Er_0.25Ni₂ solid solution under magnetic field variations of up to 3 T was studied using various techniques. The study was validated by using the scaling theory and Widom scaling relation. The results show that the critical exponents are close to the theoretical prediction of the tricritical mean-field model. These findings suggest that the Dy_0.75Er_0.25Ni₂ solid solution exhibits short-range ferromagnetic interactions.

The experimentally obtained magnetocaloric properties, both the isothermal magnetic entropy change and the adiabatic temperature change, demonstrate good agreement with the theoretical results calculated using the model Hamiltonian. The highest entropy change was observed for Dy_0.75Er_0.25Ni₂. At the maximum applied magnetic field change of 14 T, the maximum entropy change was of 33.5 J kg⁻¹ K⁻¹ at 20 K. Dy_0.5Er_0.5Ni₂ and Dy_0.25Er_0.75Ni₂ −ΔS_M reached slightly lower values of 28 J kg⁻¹ K⁻¹ at 16.9 and 28.9 at 13.0 K, respectively. At the 2 T field change, the maximum −ΔS_M values were between 10.4 and 12.6 J kg⁻¹ K⁻¹, depending on the composition. The maximum observed adiabatic temperature changes upon a 10 T magnetic field change were 10.8 K, 10.8 K, and 10.5 K for Dy_0.75Er_0.25Ni₂, Dy_0.5Er_0.5Ni₂, and Dy_0.25Er_0.75Ni₂, respectively, measured at 16.3, 12.7, and 10.2 K. At a field of 2 T, the maximum ΔT_ad values were between 3 and 3.8 K, depending on the composition.

The observed magnetocaloric properties are high and reach their maximum within a narrow temperature range below 20 K. This leads us to believe that Dy_1−xEr_xNi₂ (x = 0.25, 0.5, 0.75) solid solutions may be a viable option for refrigerants in magnetic cryocoolers suitable for hydrogen liquefaction.

Data availability

The data are available from the corresponding author upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The work was supported by the National Science Center, Poland through the OPUS Program under Grant No. 2019/33/B/ST5/01853. N. A. de Oliveira acknowledges the financial support from CNPq and FAPERJ.

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