## Structural analysis

The XRD patterns recorded for the Ho1xErxNi2 solid solutions at room temperature were analyzed by the Rietveld method and are depicted in Fig. 1. Through the substitution of erbium for holmium in Ho0.5Er0.5Ni2 and Ho0.25Er0.75Ni2, the ordering of R vacancies preserved in the structure of HoNi2 phase takes place, and the 2*a* cubic superstructure (space group *F*-43*m*) forms, indicated by indexed peaks marked with S in Figs. 1(a), 1(b). For the Ho0.75Er0.25Ni2 stoichiometric composition, in contrast to Ho0.5Er0.5Ni2 and Ho0.25Er0.75Ni2, this effect is not so evident, reflections of the superstructure do not appear in the X-ray diffraction pattern, and the structure can be described by the space group *Fd*-3*m*.

According to Delsante et al. [5], formation of the regular C15 structure (space group *Fd*-3*m*) is expected for the RNi2 compounds with the enthalpy of formation Δf*H*o at 300 K of less than 40 kJ/mol. The enthalpies of formation Δf*H*o of HoNi2 and ErNi2 equal to − 48 and − 50 kJ/mol, respectively, suggest the emergence of the regular C15 structure in these compounds, which was indeed confirmed in our earlier work [10]. However, in the case of Ho0.5Er0.5Ni2 and Ho0.25Er0.75Ni2 solid solutions, this rule is not confirmed. Additional vacancies are induced and are responsible for the formation of the superstructure. Vacancies arise as structural defects resulting from differences in the atomic radii of elements comprising a solid solution. Owing to the difference in the atomic radii, Ho - Ni and Er - Ni bonds in the solid solutions differ in length; this fact has a direct impact on the formation of vacancies. Similar results were obtained for the Tb distribution in Tb1xHoxNi2 solid solutions [12] and are in line with the data obtained for the other ternary Laves-phase solid solutions, e.g., Tb1xDyxNi2 [13] studied previously.

According to the data given in Fig. 1, small amounts of Ho2O3 and Er2O3 impurity phases are present in the Ho0.5Er0.5Ni2 and Ho0.25Er0.75Ni2 samples, the total content of which is not more than 3 wt. %. For Ho0.75Er0.25Ni2, the lattice parameter is equal to 7.1462 Å. For the two consecutive substitutions, the lattice parameter decreases as the Er content increases to *x* = 0.75. This is due to the fact that, in accordance with the lanthanide contraction, the radius of Er atoms is lower than that of Ho. It should be noted that the parent compounds, similarly to the Ho0.75Er0.25Ni2 compound, solidify with the formation of the cubic C15 crystal structure.

The typical SEM image and EDX studies of the characteristic microstructure of the polished section as representative of Ho0.25Er0.75Ni2 are shown in Fig. 2. The EDX analysis performed for large areas of samples showed that their chemical compositions are consistent with the nominal ones. Similar results were also obtained for the other samples.

## Evaluation Of Magnetocaloric Effect By Indirect Method

Figures 3(a)-(c) show the temperature dependences of the total heat capacity, *C*tot(*T*), of the Ho1xErxNi2 solid solutions in zero magnetic field. Filled symbols correspond to the experimental data, and open symbols correspond to the magnetic part of heat capacity, *C*mag(*T*), obtained after subtraction of the electron and phonon contribution, *C*el+ph(*T*), which was estimated by Debye function (solid lines in Figs. 3(a)-(c)) according to the equation:

$${C}_{el+ph}\left(T\right)=\gamma T+9NR{\left(\frac{T}{{\varTheta }_{D}}\right)}^{3}{\int }_{0}^{{\varTheta }_{D}/T}\frac{{x}^{4}{e}^{x}}{{\left({e}^{x}-1\right)}^{2}}dx$$

1

where the first term represents the electronic heat capacity; the second term corresponds to the phonon Debye contribution; *N* = 3 is the number of atoms per formula unit; *R* is the universal gas constant; and \(x\equiv \hslash \omega /{k}_{B}T\). In the temperature range 2–100 K, the best fit for all the studied samples, could be obtained by fixing the parameter γ = 3.8 mJ/molK2, while the Debye temperature of the Ho1 − xErxNi2 system, similarly to that of the Dy1 − xErxNi2 system [14], increases as the Er content increases. In the absence of magnetic field, the temperature dependence of the heat capacity shows a peak corresponding to magnetic phase transition typical of ferromagnetic compounds. The Curie temperatures *T*C of the Ho0.75Er0.25Ni2 (Fig. 3(a)), Ho0.5Er0.5Ni2 (Fig. 3(b)), and Ho0.25Er0.75Ni2 (Fig. 3(c)) compounds are 12.0, 9.7, and 7.7 K, respectively.

Insets in Fig. 3(a)-(c) show the heat capacity, as a function of temperature, measured in zero, 1- and 2-T magnetic fields. The feature observed for all of the studied compositions is the broadening of the *C*tot(*T*) peak and reduction of its height, which takes place with the increasing applied magnetic field.

The magnetic part of the entropy *S*mag(*T*) was calculated by integrating the dependence *C*mag(*T*)/*T* for each composition (Fig. 3(d)-(f)). The fact that the dependence of entropy exhibits a strong tendency to saturation, but the entropy does not approach the theoretical maximum value *S*max = *R*ln (2*J* + 1) (where *J* is the total angular momentum of a rare earth ion) at the Curie temperatures can be explained by peculiarities in the ground-state level splitting by the crystal electric field (CEF) when several CEF levels are separated from others by a substantial energy gap [15]. Similar behavior was observed for other pseudo-binary Laves-phase compounds [14, 16, 17]. According to the theoretical calculations, the maximum magnetic entropy should equal to 23.2–23.4 J/molK. In the case of the tested solid solutions, the maximum value of *S*mag for Ho0.75Er0.25Ni2 and Ho0.5Er0.5Ni2 is 21.4 J/molK at 100 K and is 22.3 J/molK for Ho0.25Er0.75Ni2. This means that almost the total magnetic entropy associated with the magnetic process is utilized.

The temperature behaviour of the magnetic entropy in 1- and 2-T magnetic fields shows that the applied magnetic field leads to the decrease in *S*mag near *T*C. In particular, the maximum value of *S*mag for Ho0.75Er0.25Ni2 near *T*C decreases from 15.1 to 10.5 J/molK in the applied magnetic field. The temperature dependences of the isothermal magnetic entropy change Δ*S*mag(*T*) calculated using the heat capacity data according to the procedure reported in [14] and caused by 1- and 2-T magnetic field change, are shown in insets in Figs. 3(d)-(f). For a magnetic field change of 0–2 T, the experimental maximum − Δ*S*mag in the case of the Ho0.75Er0.25Ni2 compound reaches the highest value of 4.6 J/mol K (16.3 J/kg K) near 12.1 K and, as the Er content increases, becomes lower and equals to 3.9 J/mol K (13.7 J/kg K) for the Ho0.25Er0.75Ni2 sample near 8 K. Figures 4(a)-(c) show dependences of the adiabatic temperature change, Δ*T*ad, for Ho1 − xErxNi2 with x = 0.25, 0.5, and 0.75, which were derived from the heat capacity data obtained in 1- and 2-T magnetic fields. As is seen, the increase in the applied magnetic field leads to an increase in the adiabatic temperature change near *T*C. Both at 1- and 2-T magnetic field changes, the highest magnetocaloric effect was observed for Ho0.75Er0.25Ni2. The maximum Δ*T*ad for Ho0.75Er0.25Ni2 reaches 2.8 K (4.9 K) at 12.0 K, and, with increasing Er content, the maximum peak value of Δ*T*ad decreases to 2.2 K (3.9 K) for Ho0.25Er0.75Ni2 at 7.7 K for a magnetic field change of 1 (2) T. Table 1 summarizes the data on the experimental isothermal magnetic entropy change Δ*S*mag(*T*) and adiabatic temperature change Δ*T*ad(*T*) for low external magnetic field changes, which were estimated by the indirect method using the heat capacity data.

Table 1

Magnetocaloric properties for the selected binary RNi2 compounds and for the investigated Ho1xErxNi2 solid solutions estimated from heat capacity measurements for magnetic field changes of 1 and 2 T. *T*C is the magnetic phase transition temperature; -Δ*S*mag is the maximum magnetic entropy change; Δ*T*ad is the maximum adiabatic temperature change; *RC* is the refrigerant capacity; *RCP* is the relative cooling power; and *TEC* is the temperature averaged entropy change.

Compound | *T*C (K) | -Δ*S*mag (J/kgK) | Δ*T*ad (K) | *RC/RCP* (J/kg) | *TEC*(3)/ *TEC*(10) (J/kgK) |

| | 0–1 T | 0–2 T | 0–1 T | 0–2 T | 0–1 T | 0–2 T | 0–1 T | 0–2 T |

TbNi2 [23] | 37.1 | 3.4 | 6.5 | 1.4 | 2.4 | 34/44 | 94/122 | 3.2/2.9 | 6.4/6.0 |

DyNi2 [23] | 21.8 | 6.3 | 11.1 | 2.4 | 3.8 | 48/63 | 117/154 | 5.9/4.8 | 11.1/9.1 |

HoNi2 [10] | 13.5 | 8.8 | 14.6 | 2.7 | 4.6 | 55/72 | 128/171 | 8.0/6.2 | 14.9/11.4 |

Ho0.75Er0.25Ni2 | 12.0 | 10.0 | 16.2 | 2.8 | 4.9 | 49/65 | 117/155 | 8.9/4.9 | 15.1/9.1 |

Ho0.5Er0.5Ni2 | 9.7 | 10.3 | 15.5 | 2.7 | 4.5 | 52/72 | 123/163 | 9.0/6.8 | 15.1/11.8 |

Ho0.25Er0.75Ni2 | 7.7 | 8.8 | 13.7 | 2.2 | 3.9 | 45/58 | 102/133 | 8.0/5.6 | 12.6/10.4 |

ErNi2 [10] | 6.2 | 8.6 | 13.0 | 2.2 | 3.8 | 43/55 | 96/122 | 7.8/5.3 | 12.1/9.3 |

Composite 1 | - | 6.7 | - | - | - | 57/67 | - | 6.6/5.6 | - |

Composite 2 | - | - | 12.0 | - | - | - | 121/150 | - | 11.9/10.4 |

To compare the refrigeration properties of Ho1 − xErxNi2 with those of the other previously investigated RNi2 compounds, the refrigerant capacities (*RC*), relative cooling power (*RCP*) and temperature averaged entropy change (*TEC*) were estimated. The first parameter is a measure of the amount of heat that can be transferred between the cold and hot sinks in one ideal refrigeration cycle and were estimated by integrating the Δ*S*mag(*T*) curve over the full width at half maximum [18, 19]. It should be noted that, as the magnetic entropy change decreases owing to the Er doping in Ho1 − xErxNi2, the *RC* also reduces, but it is still high, namely, ~ 45 J/kg and ~ 102 J/kg for a field change of 1 and 2 T, respectively.

The second parameter is defined as ǀΔ*S*magǀ(max) × δ*T*FWHM, where δ*T*FWHM denotes the full width temperature span of ǀΔ*S*magǀ vs. *T* curve at its half maximum [20]. As the Er content increases, the *RCP* values decrease from 65 J/kg for Ho0.75Er0.25Ni2 to 58 J/kg for Ho0.25Er0.75Ni2 at the 1-T magnetic field change and from 155 J/kg for Ho0.75Er0.25Ni2 to 133 J/kg for Ho0.25Er0.75Ni2 at the 2-T magnetic field change. It should be noted that, in the case of the Ho0.5Er0.5Ni2 solid solution, there are slight deviations for both the obtained *RC* and *RCP* values from the expected ones.

The third parameter, the temperature averaged entropy change (*TEC*), was introduced by Griffith et al. [21] and the magnitude is calculated by the following formula:

$$TEC\left(\varDelta {T}_{\text{l}\text{i}\text{f}\text{t}}\right)=\frac{1}{{\varDelta T}_{\text{l}\text{i}\text{f}\text{t}}}\underset{{T}_{\text{mid}}}{\text{max}}\left\{\underset{{T}_{\text{m}\text{i}\text{d}}-\frac{{\varDelta T}_{\text{l}\text{i}\text{f}\text{t}}}{2}}{\overset{{T}_{\text{m}\text{i}\text{d}}+\frac{{\varDelta T}_{\text{l}\text{i}\text{f}\text{t}}}{2}}{\int }}\left|{\varDelta S}_{\text{M}}\right|{\left(T\right)}_{{\mu }_{0}\varDelta H,T}dT\right\}$$

2

where Δ*T*lift is the desired lift of temperature and *T*mid is the temperature of the center of the *TEC* and is determined by maximizing the *TEC* value. Accordingly, two different ∆*T*lift values of 3 and 10 K are chosen to calculate *TEC* for the Ho1 − xErxNi2 solid solutions under study. The resulted values of *TEC* (3 K) and *TEC* (10 K) at µ0∆*H* = 1 T oscillate between 8.0-9.4 and 4.9–6.8 J/kgK and, at µ0∆*H* = 2 T, oscillate between 12.6–15.1 and 9.1–11.8 J/kgK, respectively.

The obtained values are of a high level and are comparable to those obtained for other promising low temperature magnetocaloric materials, such as TbNi2 [22, 23] DyNi2 [23, 24], ErNi2, HoNi2 [10], Dy1 − xErxNi2 [14], Tb1 − xHoxNi2 [12], TmCoAl [25], ErRu2Si2 [26], or HoNi2B2C [27].

Due to the fact that the ideal Ericsson cycle employs a constant value of Δ*S*mag in the temperature range of refrigeration, which is necessary for improving regeneration processes, composite materials were considered. It is expected that a composite material formed by at least two magnetic Ho1xErxNi2 compounds differing in the Er concentration could exhibit a “table-like” behavior of MCE in a wider temperature range. In this context, according to a procedure proposed in Refs. [11, 28, 29], numerical simulations were done to construct a composite material formed by Ho1 − xErxNi2 compounds. The isothermal magnetic entropy change of a magnetic composite ǀΔ*S*magǀcomp based on *N* kinds of magnetic materials is equal to the sum of their magnetic entropy changes ǀΔ*S*magǀj weighted by a molar ratio *y*j. In our case, for a magnetic field change of 0–1 T (composite 1), optimal molar ratios are y1 = 0.599 for Ho0.25Er0.75Ni2, y2 = 0.046 for Ho0.5Er0.5Ni2, and y3 = 0.355 for Ho0.75Er0.25Ni2, while, in the case of a magnetic field change of 0–2 T (composite 2), two compounds are sufficient with y1 = 0.706 for Ho0.25Er0.75Ni2 and y2 = 0.294 for Ho0.75Er0.25Ni2.

Figure 5 shows the calculated isothermal magnetic entropy changes for the composite based on Ho1 − xErxNi2 compounds, which are obtained for magnetic field changes of 1 and 2 T. It should be noted that, both in 1- and 2-T magnetic field changes, the maximum magnetic entropy change of the composite material exhibits an almost constant value of ǀΔ*S*magǀcomp that is around 6.7 J/kgK for µ0Δ*H* = 1 T and 12 J/kgK for µ0Δ*H* = 2 T. For both composites, calculated ǀΔ*S*magǀcomp remains almost unchanged in a temperature range of 8 to 12 K. These results suggest that, in order to design the appropriate composition of a refrigerant, it is necessary to evaluate the corresponding optimal molar ratios using the value of external magnetic field change at which the refrigerator should operate. To compare the magnetocaloric performance of the proposed composites with that of their constituents, the values of *RC, RCP*, and *TEC* have been calculated. The magnitudes computed by the methods described earlier for both composites are of a high level and are comparable to those of the individual solid-solution constituents; the value of *RC*(*RCP*) for composite 1 (µ0Δ*H* = 1 T) is equal to 57(67) J/kg and, for composite 2 (µ0Δ*H* = 2 T), it is 122(150) J/kg. The *TEC*(3) values obtained for both composites are comparable to their maximum isothermal magnetic entropy change values, which result directly from the scope of ∆*T*lift values. In the case of *TEC*(10), the values are slightly smaller in comparison with *TEC*(3); however, they are still of a high level and comparable to those of the solid solution constituents (see Table 1).

## Evaluation Of The Magnetocaloric Effect With Direct Measurements

The adiabatic temperature change Δ*T*ad caused by the magnetic field change µ0Δ*H*, i.e., the magnetocaloric effect, has been additionally determined by direct temperature measurements in the range of magnetic fields up to 14 T. Figures 6(a) and (b) show experimental Δ*T*ad vs. the initial temperature, as obtained in the magnetizing process, for Ho0.75Er0.25Ni2 and Ho0.5Er0.5Ni2, respectively. The initial field was zero in all cases. As expected, the increase of the applied magnetic field leads to an increase in Δ*T*ad. The maximum value of Δ*T*ad at µ0Δ*H* = 14 T reaches 16.4 K at *T*C for Ho0.75Er0.25Ni2, and 15.1 K at *T*C for Ho0.5Er0.5Ni2. The maxima of Δ*T*ad obtained at 1- and 2-T magnetic field changes by both direct and indirect methods have been detected at the same temperature and the determined values are in good agreement.

Table 2

Experimental data characterizing the adiabatic temperature change, Δ*T*ad, due to MCE caused by the magnetic field change, µ0Δ*H*, for the selected binary RNi2 intermetallic compounds and for Ho0.75Er0.25Ni2 and Ho0.5Er0.5Ni2 solid solutions. *A* is the coefficient from equation Δ*T*ad = *A*(µ0Δ*H*)2/3. The data were obtained by direct measurements of Δ*T*ad during the field change, µ0Δ*H*, achieved by using the extraction method in a Bitter magnet. The values of Δ*T*ad marked with **‡** symbol are estimated by the extrapolation of the Δ*T*ad = *A*(µ0Δ*H*)2/3 relation.

Compound | *T*C (K) | Δ*T*ad (K) | *A* (K/T2/3) |

| | 0–1 T | 0–2 T | 0–5 T | 0–8 T | 0–10 T | 0–14 T | |

TbNi2 [23] | 37.1 | 1.5**‡** | 2.4 | 4.6 | 6.2 | 6.9 | 8.4**‡** | 1.45 |

DyNi2 [23] | 21.8 | 2.3 | 3.6 | 7.1 | 9.2 | 10.6 | 13.4**‡** | 2.31 |

HoNi2 [10] | 13.5 | 2.8**‡** | 4.2 | 8.7 | 11.4 | 12.9 | 16.3**‡** | 2.8 |

Ho0.75Er0.25Ni2 | 12.0 | 2.9 | 5.1 | 9.6 | 12.7 | 13.9 | 16.3 | 2.9 |

Ho0.5Er0.5Ni2 | 9.7 | 2.6 | 4.5 | 8.3 | 11.2 | 12.9 | 15.1 | 2.6 |

ErNi2 [10] | 6.2 | 2.1**‡** | 3.5 | 6.2 | 8.8 | 9.8 | 12.2**‡** | 2.1 |

Directly measured maximum Δ*T*ad as a function of the final magnetic field is plotted in Figs. 6(c)-(d). For both Ho0.75Er0.25Ni2 and Ho0.5Er0.5Ni2 solid solutions, Δ*T*ad grows nonlinearly with increasing µ0Δ*H*. Characteristic quantity Δ*T*ad/µ0Δ*H* decreases from 2.8 K/T at 1 T to 1.2 K/T at 14 T for Ho0.75Er0.25Ni2, and from 2.7 K/T at 1 T to 1.1 K/T at 14 T for Ho0.5Er0.5Ni2.

To check the applicability of the thermodynamic Landau theory for the description of our experimental results, the adiabatic temperature change Δ*T*ad was plotted as a function of (µ0Δ*H*)2/3, as shown in insets in Figs. 6(c)-(d). The linear behavior of the dependences for both investigated compounds near their Curie temperature demonstrates a good agreement between the experimental results and thermodynamic Landau theory.

By plotting the maximum Δ*T*ad value versus (µ0Δ*H*)2/3 and using an equation: Δ*T*ad = *A*(µ0Δ*H*)2/3, where *A* is a characteristic parameter of magnetocaloric materials, one can obtain information about the magnetocaloric properties of investigated samples [30]. By fitting the experimental data, we find *A* = 2.9 K/T2/3 for Ho0.75Er0.25Ni2 and *A* = 2.6 K/T2/3 for Ho0.5Er0.5Ni2. These values are comparable with those obtained for the parent compounds and other binary Laves-phase compounds and are also comparable with the values of the most efficient magnetic refrigerants, such as Gd (*A* = 3.83 K/T2/3) and LaFe11.2Si1.8 (*A* = 2.16 K/T2/3) [30]. The data obtained by direct measurements are gathered in Table 2.