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}
^{a}Institute of Low Temperature and Structure Research, PAS, Okólna 2, Wrocław, 50-422, Poland. E-mail: j.cwik@intibs.pl
^{b}Department of Materials and Science and Engineering, Hanbat National University, Daejeon 34158, Republic of Korea. E-mail: yourkirans@gmail.com
^{c}Instituto de Física Armando Dias Tavares-Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro 20550-013, RJ, Brazil
^{d}Ames National Laboratory, U.S. Department of Energy, Iowa State University, Ames, IA 50011, USA

Received
31st May 2024
, Accepted 23rd July 2024

First published on 15th August 2024

Magnetic cooling is perceived as an enticing option for the liquefaction of hydrogen. Compared to the currently employed method, it offers significantly enhanced efficiency. However, further research is needed to identify refrigerants suitable for low-temperature applications. Rare-earth nickel, RNi_{2}, intermetallic compounds based on heavy rare earth elements have garnered considerable attention owing to their unique properties linked to highly localized magnetic moments that emanate from the incompletely filled 4f-electron shell of the R atoms. In this work, the impact of simultaneous substitution within the rare earth sublattice on the magnetic and magnetocaloric properties of Dy_{1−x}Er_{x}Ni_{2} (x = 0.25, 0.5, 0.75) Laves phase solid solutions has been studied and the critical behavior around ferromagnetic–paramagnetic phase transition was analyzed. The samples were synthesized by arc melting and tested in a wide magnetic field range of up to 14 T. Both direct and indirect methods were used to characterize magnetocaloric properties. Experimental data were compared with theoretical results obtained in the frame of the microscopic model that considers exchange magnetic interaction and crystal electric field anisotropy. At temperatures below 20 K, all samples studied are ferromagnets. As the erbium content increases, Curie temperatures and magnetic entropy changes decrease while temperature changes remain stable. With an increasing magnetic field, the peak value of magnetic entropy change shifts to higher temperatures. At a magnetic field change of 14 T, the largest observed magnetic entropy and temperature changes are equal to 32 J kg^{−1} K^{−1} and 13 K, respectively, demonstrating that these solid solutions have high potential for low-temperature applications.

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 parameters^{9} 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, RNi_{2}, 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−x}Er_{x}Ni_{2} solid solutions, similar to Tb_{1−x}Dy_{x}Ni_{2} (ref. 16) and Tb_{1−x}Er_{x}Ni_{2} (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 _{0.75}Er_{0.25}Ni_{2}** to 10 K for

This work continues the investigation of Dy_{1−x}Er_{x}Ni_{2} 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−x}Er_{x}Ni_{2} solid solutions are strong candidates for low-temperature magnetic refrigerants.

(1) |

(2) |

The free energy for the Hamiltonian (1) is F^{α}_{M} = −k_{B}TlnZ^{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,μ_{0},H,x) = S^{α}_{M}(T,μ_{0},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) |

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/μ_{0}H curves extrapolated to 1/μ_{0}H = 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 _{0.75}Er_{0.25}Ni_{2}** to 7.8μ

Fig. 2 Magnetization of Dy_{1−x}Er_{x}Ni_{2} (x = 0.25, 0.5 and 0.75) solid solutions and binary DyNi, _{2}ErNi measured at 4.2 K. The inset shows the high-field region of M vs. μ_{2}_{0}H plots. |

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 **DyNi _{2}** with a 4f

Compound | Space group |
T
_{C} (K) |
M
_{S} (μ_{B} per f.u.) |
M
_{SP} (μ_{B} per f.u.) |
Ref. | |
---|---|---|---|---|---|---|

Theor. | Expt. | |||||

DyNi
_{2} |
Fdm | 21.8 | 10.0 | 9.8 | 8.9 | 27 |

Dy
_{0.75}Er_{0.25}Ni_{2} |
F3m | 16.2 | 9.75 | 8.2 | 7.3 | Present |

Dy
_{0.5}Er_{0.5}Ni_{2} |
F3m | 12.7 | 9.5 | 8.0 | 6.3 | Present |

Dy
_{0.25}Er_{0.75}Ni_{2} |
F3m | 10.2 | 9.25 | 7.8 | 6.2 | Present |

ErNi
_{2} |
Fdm | 6.2 | 9.0 | 7.7 | 6.1 | 28 |

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 _{0.5}Er_{0.5}Ni_{2}** from 4.2 to 30 K with a 2 K step. We noticed that at low temperatures, well below T

Using the magnetization isotherms obtained close to the Curie temperature, the change in magnetic entropy was calculated for each Dy_{1−x}Er_{x}Ni_{2} 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 **DyNi _{2}**, and x

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 _{0.75}Er_{0.25}Ni_{2}**,

Compound |
μ
_{0}ΔH (T) |
ΔT_{ad} |
−ΔS_{M} |
Ref. | ||||
---|---|---|---|---|---|---|---|---|

T
_{max} (K) |
ΔT_{ad} (K) |
ΔT_{ad}/μ_{0}ΔH (K T^{−1}) |
−ΔS_{M} (J kg^{−1} K^{−1}) |
−ΔS_{M}/μ_{0}ΔH (J kg^{−1} K^{−1} T^{−1}) |
||||

DyNi
_{2} |
2 | 21.7 | 3.6 | 1.8 | 19.3 | 11.1 | 5.6 | 27 |

5 | 20.8 | 7.1 | 1.4 | 19.1 | 23.0 | 4.6 | 27 | |

10 | 21.3 | 10.6 | 1.1 | 19.1 | 34.0 | 3.4 | 27 | |

Dy
_{0.75}Er_{0.25}Ni_{2} |
2 | 16.3 | 3.8 | 1.9 | 16.9 | 12.6 | 6.3 | Present |

5 | 16.3 | 7.3 | 1.5 | 16.9 | 22.0 | 4.4 | Present | |

10 | 16.3 | 10.8 | 1.1 | 16.9 | 30.2 | 3.0 | Present | |

14 | 16.3 | 12.6 | 0.9 | 20.0 | 33.5 | 2.4 | Present | |

Dy
_{0.5}Er_{0.5}Ni_{2} |
2 | 12.7 | 3.8 | 1.9 | 12.9 | 10.4 | 5.2 | Present |

5 | 12.7 | 7.3 | 1.5 | 12.9 | 19.4 | 3.9 | Present | |

10 | 12.7 | 10.8 | 1.1 | 15.0 | 26.0 | 2.6 | Present | |

14 | 16.9 | 28.0 | 2.0 | Present | ||||

Dy
_{0.25}Er_{0.75}Ni_{2} |
2 | 9.1 | 3.0 | 1.5 | 9.2 | 12.1 | 6.1 | Present |

5 | 10.1 | 6.4 | 1.3 | 9.2 | 19.5 | 3.9 | Present | |

10 | 10.2 | 10.5 | 1.1 | 13.0 | 25.5 | 2.6 | Present | |

14 | 10.3 | 12.2 | 0.9 | 13.0 | 28.9 | 2.1 | Present | |

ErNi
_{2} |
2 | 6.5 | 3.4 | 1.7 | 6.2 | 15.1 | 7.6 | 30 |

5 | 7.8 | 6.2 | 1.2 | 6.2 | 24.3 | 4.9 | 30 | |

10 | 10.0 | 9.8 | 1.0 | 6.2 | 29.1 | 2.9 | 30 |

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 _{0.75}Er_{0.25}Ni_{2}** and

Additionally, it was observed that for **Dy _{0.25}Er_{0.75}Ni_{2}**, the temperature at which ΔT

(6) |

(7) |

The critical exponents β, γ, and δ are related to the spontaneous magnetization (M_{SP}) below T_{C}, the inverse magnetic susceptibility (χ_{0}^{−1}) above T_{C}, and the critical magnetization isotherm M at T_{C}, respectively:^{39,40}

M_{SP}(T) = M_{0}(−ε)^{β}, ε < 0, T < T_{C}, | (8) |

(9) |

M = RH^{1/δ}, at ε = 0, T = T_{C}, | (10) |

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/β}versus (μ_{0}H/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 M^{2}versus μ_{0}H/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/β}versus (μ_{0}H/M)^{1/γ} plots for **Dy _{0.75}Er_{0.25}Ni_{2}** 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).

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) |

Fig. 8 Relative slope (RS) defined as RS = S(T)/S(T_{C}) of the Dy solid solution as a function of temperature._{0.75}Er_{0.25}Ni_{2} |

The spontaneous magnetization M_{SP}(T) and the inverse magnetic susceptibility χ_{0}^{−1}(T) were extracted from the modified Arrott plots for the tricritical mean-field model. The M_{SP}(T) and χ_{0}^{−1}(T) values as the intercepts with the M^{1/β} and (μ_{0}H/M)^{1/γ} axes, respectively, were calculated by linear extrapolation in a higher magnetic field region. The obtained values of M_{S}(T) and χ_{0}^{−1}(T) are plotted against temperature (T) as shown for the **Dy _{0.75}Er_{0.25}Ni_{2}** 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

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^{−1}_{0}(T)[dχ_{0}^{−1}(T)/dT]^{−1} against T. Fig. 9(b) shows the Kouvel–Fischer plot of the **Dy _{0.75}Er_{0.25}Ni_{2}** sample. The linear fitting provides the estimated exponents and T

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

(14) |

(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 _{0.75}Er_{0.25}Ni_{2}** 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

Sample/model | Method |
T
_{C} (K) |
β | γ | δ |
---|---|---|---|---|---|

Dy
_{0.75}Er_{0.25}Ni_{2} |
MAP | 16.3 | 0.21 | 0.76 | 4.61 |

KF | 16.4 | 0.21 | 0.93 | 5.42 | |

WSR | — | — | — | 5.7 | |

Mean-field model | Theory | — | 0.5 | 1.0 | 3.0 |

3D-Heisenberg model | Theory | — | 0.365 | 0.386 | 4.8 |

3D-Ising model | Theory | — | 0.325 | 1.24 | 4.82 |

Tricritical mean-field model | Theory | — | 0.265 | 1.0 | 5.0 |

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}

ΔS^{max}_{M} ≈ a(μ_{0}H)^{n}, | (16) |

(17) |

According to mean field theory (MFT), the exponent n(T,μ_{0}H) 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 _{0.75}Er_{0.25}Ni_{2}**, n initially slightly decreases as temperature increases, reaching a minimum around T

The critical behavior of the **Dy _{0.75}Er_{0.25}Ni_{2}** 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

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 _{0.75}Er_{0.25}Ni_{2}**. At the maximum applied magnetic field change of 14 T, the maximum entropy change was of 33.5 J kg

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−x}Er_{x}Ni_{2} (x = 0.25, 0.5, 0.75) solid solutions may be a viable option for refrigerants in magnetic cryocoolers suitable for hydrogen liquefaction.

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