Introduction

The motivation for studying HEA for magnetocalorics

Data statistics consolidated by Enerdata © show a long-term trend of increasing electricity consumption [1]. This prompts for energy efficiency concerns in order to maintain sustainable living. The magnetocaloric effect (MCE), which describes the reversible temperature change of a magnetic material when subjected to an adiabatically varying magnetic field, forms the basis of magnetic refrigeration. Utilizing solid magnetic material as the regenerator bed, this cooling technology does not rely on ozone-depleting or hazardous gaseous refrigerants used in conventional vapor-compression refrigerators and at the same time offers a noiseless system (omitting the compressor). In addition, with an attractive energy efficiency in magnetic refrigeration (doubling that of conventional refrigerators), magnetocalorics would be very useful for specialty applications, such as the refrigeration in space or submarine settings where noiseless, space-saving and avoidance of compressed gases would be crucial. Additionally, it could address some of the global energy issues we face today in a more environmental-friendly way. However, to make this promise a reality, the ongoing search of appropriate magnetocaloric material with optimized performance including mechanical stability is crucial.

A new revolutionary material design concept has found new alloys with excellent mechanical properties, corrosion and radiation resistance [2]. It is widely known as the High Entropy Alloy (HEA) design concept and it is the current go-to technique when it comes to designing materials for structural / defense-related applications. The approach utilizes a combination of multiprincipal elements in large proportions to yield high configurational entropy of mixing (∆Smix) values. HEAs are originally sought for within single-phase solid solution metallic alloys but today the focus has expanded to multiphases and intermetallics / ceramics. This entails a large compositional freedom with wide window of opportunities for property exploration. Hence, applying the HEA design concept for developing magnetocaloric materials would create new combined capabilities of high performance and improved material service life.

Current state-of-the-art of high-entropy alloys in the field of magnetocalorics

High-entropy alloys (HEAs) have received a global interest since their discovery with outstanding mechanical properties, being a rapidly growing field of research. The number of their publications has tremendously surged: an increase of 32% in annual publications for 2021 vs. 2020; Web of Science (WoS) bibliographic search (up to May 2022) already shows a record of 732 publications for 2022, which is expected to increase by the year´s end. However, these endeavors are mainly focused on mechanical properties and structural applications while functional reports are scarce (5% out of the total number of publications are for functional high-entropy materials). As an example, we present the cumulative number of HEA research articles as shown in Fig. 1 and their different focus topics for 2021 (blue shaded regions) and 2022 (orange solid lines). The WoS survey shows that the structural reports (on the left of Fig. 1) clearly surpass those of the functional endeavors (on the right of Fig. 1), being magnetic and MCE of HEAs receiving most attention among the functional properties.

Figure 1
figure 1

The number of HEA research articles and their different focus (mechanical properties, microstructure, etc.) presented in log scale. Bibliographic search was restricted to article titles containing “high-entropy ∗ ” with the focus labels (surveyed in WoS up to May 2022).

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Fundamentals

The magnetocaloric effect

The magnetocaloric effect (MCE) refers to the reversible temperature change of a magnetic material when adiabatically magnetized or demagnetized. During this process, the magnetic field manipulates the degrees of freedom of the magnetic sub-lattice including their coupling to the degrees of freedom associated with the structure. Hence, MCE can be indirectly determined from the isothermal entropy change correlated with magnetization from the following equation as derived from the Maxwell relation:

$$\Delta S_{{{\text{isothermal}}}} = \mu _{0} \int_{{(H_{{{\text{initial}}}} )}}^{{(H_{{{\text{final}}}} )}} {\frac{{\partial M}}{{\partial T}}dH} ,$$
(1)

where ΔSisothermal refers to the isothermal entropy change, H is the magnetic field, M represents magnetization and T is temperature. Therefore, for materials exhibiting an abrupt change of magnetization with temperature influenced by magnetic field a considerable MCE will be observed. In other words, MCE would peak near the transition temperature of the magnetic material since ΔSisothermal is directly related to \(\partial M/\partial T\) as depicted in Fig. 2.

Figure 2
figure 2

Left y-axis: normalized magnetization of FOPT (blue) and SOPT (orange) magnetocaloric material. Right y-axis: Derivatives of magnetization vs temperature (dashed lines) which are directly related to ΔSisothermal according to Eq. 1. Hence, the peak values of MCE lie in the vicinity of the transition temperatures of the materials where the change of magnetization with respect to temperature plays a crucial role in MCE magnitude.

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In general, magnetocaloric materials are classified according to their order of phase transitions, namely first-order phase transition (FOPT) and second-order (or higher order) phase transition (SOPT). It has been reported that Eq. 1 can be directly applied to materials that undergo SOPT while discontinuous measurement protocols are essential for FOPT materials in order to use the same equation for determining MCE without the influence of prior history of the sample during each set of measurements. Ref. [3] contains details of the various discontinuous measurement protocols for performing magnetization measurements to determine for the MCE appropriate for both types of magnetocaloric materials. For reviews on magnetocalorics, one can refer to recent comprehensive reports on materials [4,5,6] and devices [7].

HEA

Rather than employing one or two main constituents in the traditional alloy development method, HEAs adopt a different design concept that blends multiple principal elements in large concentrations without any base element. In that way, they form multiprincipal elements alloys of high values of configurational entropy of mixing (ΔSmix), focusing on the central regions of multicomponent phase diagrams as illustrated in the ΔSmix contour plot in Fig. 3. For simplicity, a ternary alloy system is used for illustration, where it can be noticed that ΔSmix increases as the composition approaches the central equiatomic region, arriving at a maximum value. HEAs were originally defined comprising of five or more principal elements in equiatomic compositions being single-phase random solid solution metallic alloys. Today, the HEA design has expanded from equiatomic to non-equiatomic compositions, enabling a greater compositional space than that of the traditional alloying method and thus permits new (and more) viable element combinations and interesting properties to be discovered. Hence, the scope of HEA research now also includes materials of four principal elements, intermetallic and ceramic compounds, including microstructures of any number and type of phases [8]. Thus, alternative names to HEAs can be found for embracing this broader research scope: complex concentrated alloys (CCAs) and high-entropy materials (HEMs). In general, there are two widely accepted ways of defining HEAs [8, 9]:

Figure 3
figure 3

Contour plot of ΔSmix for a model ternary alloy. Conventional alloys based on 1 or 2 constituents are found at the corners and edges. As the alloy compositions modify toward large proportion of each element, ∆Smix values are also observed to increase (as we move toward the center of the plot). The largest ∆Smix is found at the center of the plot, where the elements are in equiatomic compositions.

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Composition-based definition

In early reports, HEAs had been defined with five or more principal elements in equiatomic compositions [10]. Today, with more knowledge about this new class of materials from their growing works, this restricted definition is expanded to a compositional definition: alloys that include multiple principal elements with concentration of each element being 5 to 35 atomic percent (at.%), which could also contain minor elements for property tuning purposes [8].

This composition-based definition for HEAs further expands the compositional space from equiatomic to non-equiatomic regions and is not restricted to requirement of single-phase solid solutions.

Entropy-based definition

For this definition, the equation to calculate for ΔSmix is

$$\Delta S_{{{\text{mix}}}} = R\sum x_{i} \ln x_{i}$$
(2)

where R is the gas constant and xi is the mole fraction of the ith element. The threshold for HEA design utilizes ΔSmix ≥ 1.5 R though other definitions or interpretations (i.e., 1.61 R) could be found in the literature. As an example, a quinary alloy formula of 5–35 at.% will yield ΔSmix = 1.36 R (35% A, 35% B, 20% C, 5% D and 5% E) while the ΔSmix value of 1.61 R is the maximum attainable for a quinary HEA in equiatomic compositions. This case would be considered a HEA by the composition-based definition but not by the entropy-based one. To find a compromise between both definitions, it is currently accepted that ΔSmix ≥ 1.5 R is the threshold for being considering a HEA. There have been several modifications and compromises between these definitions, being the most general the compositional one. For further details, the reader is referred to Ref. [8].

Typical MCE-HEA design strategies

The search for magnetocaloric HEAs was traditionally centered on the equiatomic compositional design and usually comprised of fully rare-earth (RE) elements or a mixture of RE and transition metal (TM) elements. For the former, they yield zero enthalpy of mixing values and small atomic radii difference, resulting in crystalline microstructures. On the other hand, RE-TM HEAs yield negative enthalpy of mixing values and large atomic radii difference, which frustrates crystallization and form high-entropy bulk metallic glasses (HE-BMGs). As microstructures could be further manipulated through thermal processing, there are more works on magnetocaloric HE-BMGs than RE-HEAs, with ref. [11] covering a comprehensive discussion in bringing together how the different reports (or compositions) are interlinked to one another. In addition to compositional search or effects of processing, many of the early results focus on the tunability of the transition temperatures and the exploitation of distributed exchange considering the d-orbital overlap [12,13,14,15]. Recently, magnetocaloric HEA reports evolve to the exploration of the non-equiatomic region, finding MCE that surpasses the previous limitations of RE-HEAs and RE-free HEAs. In the following sections, we will highlight the methods by which magnetocaloric HEAs are designed, their limitations, as well as how to approach the design in the non-equiatomic regions in an efficient manner.

Increasing number of principal elements

The most common approach for designing magnetocaloric HEAs is to increase the number of principal elements in equiatomic proportions toward the HEA definition. This was firstly reported by Yuan et al. [16]. They found that the peak isothermal entropy change, │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│, values (and transition temperatures, Ttransition) increase as they tune from ternary TbHoEr → quaternary GdTbHoEr → quinary GdTbDyHoEr. Following this approach, other magnetocaloric rare-earth (RE) HEA with increasing ∆Smix values were reported, with their data being summarized in Fig. 4. As presented in the figure legend, the works can be classified into two series: Dy-containing and Dy-free. For the former [17], though the │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ values (including their Ttransition) decrease from ̴10 ± 1 to 8.6 J kg−1 K−1 (for a field of 5 T) when increasing the number of principal elements to GdTbDyHoEr, such a MCE magnitude could still be considered a relatively good magnetocaloric material. On the other hand, the trend of Dy-free series is less straightforward. Overall, the MCE decreases when tuning from quaternary to quinary or senary HEA in equiatomic compositions. Among the quinary and senary HEAs, the addition of magnetic Pr to form GdTbHoErPr enables the highest │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ value, unlike the additions of non-magnetic Y and La. Furthermore, the comparison of these works (as shown in Fig. 4) reveals that the design strategy by increasing ∆Smix of magnetocaloric HEA is highly dependent on the nature of the selected elements (in this case, the number of 4f electrons of the RE element) for the alloy composition including their proportions (though the reported studies focus only on equiatomic amounts). In other words, it is not very likely to experimentally find remarkable MCE in these novel compositions in an efficient manner without an extensive trial and error approach.

Figure 4
figure 4

The magnetocaloric effect depends on the nature of the selected element when designing HEAs by increasing number of principal elements. Data taken from refs. [17,18,19].

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Extending toward the non-equiatomic high-entropy regions

HEAs started with an initial design concept of a single-phase solid solution comprising of more than five major elements. Very commonly, these multicomponent alloys present more than one phase when advanced microscopic investigations are performed, which drove many efforts in the direction of understanding or predicting the phase formation in this vast compositional space. As more HEA studies enable better understanding about this new class of materials, their earlier limitations to the abovementioned definitions have evolved to encompassing non-equiatomic multiphase alloys, including intermetallic and ceramics, forming the second-generation HEAs [2]. With the HEA concept evolution, it enables more novel materials to be designed and fabricated as well as to meet specific needs of the applications, which are very crucial in functional aspects. Many structural second-generation HEAs are found with remarkable properties that are formerly not found in first-generation HEAs and conventional alloys [20].

For magnetocalorics, the second-generation HEAs of non-equiatomic compositions has overcome the previous limitations of equiatomic magnetocaloric HEAs. For the RE-containing amorphous HEAs as presented in Fig. 5a, the equiatomic compositions (see the solid symbols) show transition temperatures (Ttransition) that tend to cluster at low temperatures (< 75 K) though they can show significant MCE. Taking the RE-containing amorphous HEA with the highest Ttransition (i.e., 73 K) whose composition is GdTbCoAl (marked by black arrow), its non-equiatomic Gd36Tb20Co20Al24 counterpart attains Ttransition = 81 K, which shows an increase of 11% [21]. With minor Fe additions to Gd36Tb20Co20Al24, Ttransition as high as 108 K is found, overcoming the previous low temperature limit of RE-containing amorphous HEAs. These compositions are marked by orange arrows in Fig. 5a. Unlike the RE-containing HEAs, magnetocaloric RE-free HEAs (see Fig. 5b) are not limited to low temperature range but have very modest performance (see the inset of Fig. 5b), especially for their equiatomic compositions (solid symbols). These│\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ values mainly saturate to less than 1 J kg−1 K−1 over a wide Ttransition range in the chart, while the non-equiatomic RE-free HEAs enable at least an order of magnitude enhancement (represented as the open circles and dash lines in Fig. 5b) [22, 23]. However, it has to be noted that such magnitudes are still not competitive to high-performing conventional magnetocaloric materials due to the distributed exchange interactions that occur from the disorder as pointed out in refs. [24, 25]. Despite this, extending beyond the equiatomic HEA region has indeed enlarged the compositional space of HEAs, where it lies a wide window of opportunities in obtaining appropriate compositions with good magnetocaloric properties. The key to finding the solution to this challenge of locating large MCE in the huge compositional space is to perform an effective search in this large region with the targeted goal in mind. This has been recently reported in refs. [26, 27] where a significant improvement in the MCE is found for RE-free HEAs, whose data are plotted as the greenish open circles and dash lines in Fig. 5b. The literature data collected for plotting Fig. 5 are further tabulated in Tables 1 and 2 for RE-containing and RE-free magnetocaloric HEAs, respectively (Ref. [11] provides a supplementary document with the various MCE parameters).

Figure 5
figure 5

Magnetocaloric performance of HEAs for (a) RE-amorphous (5 T) and RE-free HEA (2 T): equiatomic (solid symbols) versus non-equiatomic (open circles). Arrows in (a) indicate the evolution from equiatomic to non-equiatomic (see text). Inset in (b): A magnified subplot for equiatomic RE-HEAs with smaller scale limits.

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TABLE 1 Literature data collected for the performance comparison of equiatomic vs. non-equiatomic RE-containing amorphous magnetocaloric HEAs for 5 T.
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TABLE 2 Literature data collected for the performance comparison of equiatomic vs. non-equiatomic RE-free magnetocaloric HEAs for 2 T.
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A game changer in magnetocaloric HEAs

As the MCE is intrinsic to magnetic materials, it is very common to find researchers designing for large magnetization values in magnetocaloric materials, which is also the case for magnetocaloric HEAs. Thus, their compositional designs center on using ferromagnetic TM elements (such as Fe, Co, Ni, etc.) as well as RE elements (due to their large intrinsic magnetic moments). However, combining such elements, especially in large concentrations to design for multicomponent alloys in HEAs, be it all RE-HEAs or HEAs with combination of RE and ferromagnetic TM elements, most likely leads to a compensated magnitude in the overall magnetization value of the alloy. This could be due to the average performance from the various elements or to the coupling effects among the elements. Hence, the typical magnetocaloric HEAs often shows sub-par performance as well as a broadened MCE, especially for those shown in the abovementioned sections.

Although extending the compositional space for HEAs from equiatomic to non-equiatomic does alleviate magnetocaloric HEA limitations by increasing the opportunity window, this large degree of freedom in HEA search resembles of finding a needle in the haystack. It is challenging to perform search in the vast HEA space in a rational and efficient manner although there are many high-throughput tools developed today for predictions or synthesis experiments. They will need an appropriate starting composition to maximize the full potential of these advanced techniques.

Another method of improving MCE is to consider the fact that its large magnitude lies near the transition temperature, where an abrupt dM/dT in Eq. 1 (also presented in Fig. 2) will lead to a large MCE. Materials that show the most abrupt change are those which undergo first-order phase transitions (FOPT). The earlier-mentioned HEAs exhibit gradual dM/dT, undergoing second-order phase transitions (SOPT). Hence, the hypothesis for large MCE enhancement in HEAs is the existence of FOPT in the vast HEA space. This has been recently reported by Law et al., by using a targeted property search strategy in the HEA space for further performance optimization, achieving significant MCE values (│\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ = 13.1 J kg−1 K−1 for 2.5 T) in RE-free HEAs [27].

Targeted property search strategy

A comparison of the earlier-mentioned HEA design schemes versus the targeted property search strategy is presented as schematic ternary phase diagrams (for simplicity) in Fig. 6. The conventional alloys based on one or two constituents followed by dilution of minor additions are found at the corners of the phase diagram. The yellow center of the phase diagram indicates the equiatomic HEA composition. Thus, Fig. 6a indicates the first design strategy adopted for designing magnetocaloric HEAs, which is to increase the number of principal elements with equiatomic compositions. The magnetocaloric HEA design approach evolves from equiatomic to non-equiatomic compositions as depicted in Fig. 6b. It has to be highlighted that the extension endeavors have been made with equiatomic compositions as the starting point as indicated by the arrows. In this case, the challenge is to determine for the search direction that will lead to the desired property. Contrastingly, the targeted property search strategy that helps us find large MCE values in RE-free HEAs reverses the direction of the search arrows as shown in Fig. 6c. The starting composition is selected from conventional alloys already with the desired behavior intended, which exhibits a FOPT, and tuned it toward the HEA space, in combination with the following considerations:

Figure 6
figure 6

The various approaches taken for designing magnetocaloric HEAs: (a) increasing number of principal elements, (b) extending from equiatomic to non-equiatomic HEA compositions and (c) targeted property search strategy.

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(i) choosing doping element(s) with chemically compatibility for isostructural substitutions, and.

(ii) preserving the elemental stoichiometry (including the substitutions).

The Fe–Mn-Ni-Ge-Si HEA case

One of the cases where magnetocaloric materials undergo a FOPT and give a large MCE is associated with the presence of a magnetostructural transformation. With this aim, Law et al. tuned FOPT-MnNiSi toward HEA region with isostructural substitutions of Fe and Ge in appropriate concentrations, resulting in a magnetostructural transformation in RE-free quinary Fe22.2Mn22.3Ni22.2Ge16.65Si16.65 (ΔSmix = 1.60 R) as evidenced in their thermomagnetic data and temperature-dependent synchrotron X-ray diffraction results [26].

Further performance optimization was made by varying the Ge/Si ratio following this starting composition in Fe22.2Mn22.3Ni22.2GexSiy HEAs, while adhering to the second condition of the directed search approach [27]. Their magnetostructural transformation temperatures are further tuned to higher temperatures (from 143 to 203 K) in addition to an ~ 79% increase in │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ (refer to Ref. [27] for more details). This MCE enhancement not only finds Fe–Mn–Ni–Ge–Si HEAs with the largest magnitude of │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ among magnetocaloric HEAs, but it also closes the pre-existing gap of magnetocaloric HEAs versus high-performing conventional magnetocaloric materials. This is depicted in the performance matrix presented in Fig. 7 where the sizes of the circles represent the magnitude of │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ with a dashed line separating the two groups of materials: conventional versus HEAs. In addition, annealing of Fe–Mn–Ni–Ge–Si HEAs further optimize their performance, as recently presented by the same co-authors in conferences, finding MCE magnitudes that are as competitive as the highly regarded conventional materials, La(Fe,Si)13 and Fe2P [47].

Figure 7
figure 7

Magnetocaloric performance matrix for magnetocaloric HEAs (greenish text labels) vs high-performance conventional magnetocaloric materials (gray text labels) for 2 T. The gray dash line in the matrix is a guide to the eye for separating these two groups of samples. Size of the circles corresponds to │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│. Literature data collected from refs. [4, 12, 16, 18, 19, 21, 26,27,28, 36, 47,48,49,50,51,52,53].

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Methods of MCE analysis applicable to HEA systems

Being magnetocalorics an emergent topic for HEAs, many of their reports focus on studying the microstructural, phase transition temperatures and the ΔSisothermal of magnetocaloric HEAs. Due to many of their reports on MCE broadening behavior, in this section, we only select the appropriate developed tools among all others used for further analyzing magnetocaloric materials. They include the temperature averaged entropy change (TEC) and universal scaling. More recently, the MCE field dependence exponent n has been used for revealing the multiphases in magnetocaloric HEAs as well as for evaluating if the new alloys undergo a FOPT or SOPT.

Temperature averaged entropy change

The temperature averaged entropy change, TEC(ΔTlift) [54], relates to the maximum average of the ΔSisothermal over a specified temperature span and has been widely used as an alternate figure of merit of magnetocaloric materials for avoiding the artificially large refrigerant capacity of shallow peaks. As HEAs comprise of multiple principal elements in large concentrations, they typically show very broad MCE due to the dilution of the overall magnetization of the alloy. Although reports frequently highlight for their large refrigerant capacity values, this could be misleading unless the peak entropy change is also large, which is usually not the case. Thus, the TEC figure of merit plays an essential role in evaluating for the performance of magnetocaloric HEAs. It is calculated using the following equation:

$$\it {\text{TEC}}\left( {\Delta T_{{{\text{lift}}}} } \right) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\Delta T_{{{\text{lift}}}} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\Delta T_{{{\text{lift}}}} }$}}\mathop {\max }\limits_{{T_{{{\text{mid}}}} }} \mathop \smallint \limits_{{T_{{{\text{mid}}}} - \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \Delta T_{{{\text{lift}}}} }}^{{T_{{{\text{mid}}}} + \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \Delta T_{{{\text{lift}}}} }} \Delta S\left( T \right)_{\Delta H,T} {\text{d}}T,$$
(3)

where ΔTlift represents the specified range of temperatures (usually 3–10 K) and Tmid refers to the temperature corresponding to the center of the average. In cases that the whole magnetocaloric curve is not available, such as analyzing work from literature reports, a TEC approximation has been found in good agreement with calculated values [21]. This simplification of calculating for TEC(ΔTlift) is approximated as:

$$\it {\text{TEC}}\left( {\Delta T_{{{\text{lif}}t}} } \right) \approx \frac{{\Delta S_{{{\text{isothermal}}}} \left( {T_{{{\text{peak}}}} - \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \Delta T_{{{\text{lift}}}} } \right) + \Delta S_{{{\text{isothermal}}}} \left( {T_{{{\text{peak}}}} } \right) + \Delta S_{{{\text{isothermal}}}} \left( {T_{{{\text{peak}}}} + \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \Delta T_{{{\text{lift}}}} } \right)}}{3}$$
(4)

where Tpeak is the temperature corresponding to │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│.

Universal curves method

Scaling laws [55] have been widely applied to magnetocaloric materials, especially to evaluate for the order of phase transitions they undergo and to extrapolate MCE values at higher or lower magnetic fields for literature comparison. This method is based on the assumption that a SOPT scales. Thus, when the rescaled magnetocaloric curves collapse onto a universal curve, it indicates that it is a SOPT magnetocaloric material, which is also successfully applicable to the adiabatic temperature change [56].

Rescaling of ΔSisothermal(T) curves measured for different maximum magnetic fields is performed by following a phenomenological procedure [57]:

  1. (i)

    Normalizing the ΔSisothermal(T) curves to their respective │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ values (they are field dependent),

  2. (ii)

    Rescaling the temperature axis with reference temperature (Tr) by imposing that the reference points in the curve correspond to θ =  ± 1, where

    $$\theta = \left( {T - T_{{\text{C}}} } \right)/\left( {T_{{\text{r}}} - T_{{\text{C}}} } \right)$$
    (5)

and Tr is selected at the corresponding points where 0.5 – 0.7 times of \(\left| {\Delta S_{{{\text{isothermal}}}}^{{{\text{peak}}}} } \right|\). These rescaled curves (plots of normalized ΔSisothermal versus θ) have all their normalized ΔSisothermal data fall at the same θ point, being then said to collapse onto a universal curve, when the scaling laws are applicable to the material.

Universal scaling had been applied to families of Finemet, Nanoperm, HiTperm, and bulk amorphous alloys [58], which are soft magnetic amorphous alloys with SOPT character. The rescaled curves collapse onto a single universal curve, further confirming the order of the transition. Subsequently, the types of materials tested were significantly extended to rare-earths, crystalline phases, etc. [55].

It has to be noted that when the results do not collapse onto a single universal curve, it can imply that there are several magnetic phases with their SOPTs overlapping one another [59] or that the demagnetizing field of the samples is not negligible [60]. These effects could be resolved by rescaling the temperature axis with two Tr:

$$\theta = \left\{ {\begin{array}{*{20}c} { - \left( {T - T_{{\text{C}}} } \right) / \left( {T_{{{\text{r}},1}} - T_{{\text{C}}} } \right) ,\;T \leq T_{{\text{C}}} } \\ {\left( {T - T_{{\text{C}}} } \right) / \left( {T_{{{\text{r}},2}} - T_{{\text{C}}} } \right), \;T > T_{{\text{C}}} } \\ \end{array} } \right.$$
(6)

where Tr1 and Tr2 are selected at the corresponding points where 0.5 – 0.7 times of │\(\Delta {S}_{\text{isothermal}}^{\text{peak}}\)│ below and above Tc.

In addition, this universal scaling analysis can further uncover the presence of additional thermomagnetic phase transitions [61], indicating for a multiphase composite even when a single MCE peak is apparently observed. In the case of composites, deviations from the universal curve become more apparent when phase fractions or the MCE responses of the several coexisting phases are similar (for further details on magnetocaloric composite materials, readers can refer to ref. [5]). Recently, a phase deconvolution procedure based on universal scaling of the MCE has been successfully applied to a biphasic SOPTs composite for predicting the response of the pure constituent phases [62] and then further extended to deconvolute SOPT from FOPT in a Heuser alloy [63]. Refs. [3, 55] summarize further applications of the universal curves procedure. There have been many magnetocaloric HEA reports that include the universal scaling method to show that their new alloys undergo SOPT. However, as these new HEAs are frequently multiphase materials, we expect an increase of the publications applying the abovementioned phase deconvolution procedures to identify the response of each of the individual phases. In this way, it would be possible to determine to what extent additional synthesis efforts for obtaining a phase-pure alloy of a specific composition would enhance the performance of the material.

When the universal curve cannot be obtained even by using Eq. (6) and the sample is single phase, this can be used as a qualitative method that overcomes the contradiction between experimental calorimetric data of FOPT material and results from Banerjee´s criterion [64] as reported in ref. [65].

MCE field dependence exponent n

As the order of phase transition plays a main role in the magnitude of MCE, it is particularly important to determine the order of phase transitions in magnetocaloric materials, including magnetocaloric HEAs. Many authors rely on the quantitative Banerjee´s criterion [64], which identifies SOPT when the Arrott plots ((H/M) as a function of M2) show positive slopes and FOPT when there are negative slopes. Since the earlier-mentioned discrepancies with calorimetric data of FOPT magnetocaloric material (as reported in ref. [65]), an alternate quantitative criterion based on the MCE has been developed by Law et al. [66]. This new method is based on the magnetic field dependence of ΔSisothermal expressed as: ΔSisothermal α Hn, where exponent n is field and temperature dependent, as illustrated in Fig. 8. It can be locally calculated as:

$$n = \frac{{{\text{d}}\ln \left| {\Delta S_{{{\text{isothermal}}}} } \right|}}{{{\text{d}}\ln H}}$$
(7)
Figure 8
figure 8

Schematic representation of the (a) direct MCE and (b) its exponent n as a function of temperature. In panel (c), a sample showing both inverse and direct MCE is schematically illustrated with its exponent n presented in (d). The overshoot of n > 2 criterion for FOPT (shaded in yellowish color) is observed for cases of ferromagnetic (FM) → paramagnetic (PM) in panel (a) as well as from inverse MCE as shown in panel (c). Characteristic features of switching the signs of isothermal entropy change (pattern-filled region) in (d) should not be mistaken as the FOPT criterion.

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The fingerprint for FOPT identifies an overshoot of n > 2 near the transition temperature and its absence indicates for a SOPT. It has to be taken into account that in cases of a direct MCE switching into an inverse MCE, the n features will indicate for the characteristic switching signs instead of a FOPT. As an example, a schematical illustration of the MCE types, FOPT criterion (in yellowish shade) and the sign-switching features (pattern-filled) is presented. Figure 8(a) shows the direct MCE undergoing FOPT and SOPT and their exponent n in Fig. 8(b) yields an overshoot of n > 2 near the transition temperature for FOPT. For a sample exhibiting both inverse and direct MCE as shown in Fig. 8(c), its FOPT exponent n criterion (Fig. 8(d)) is observed near the transition temperature of inverse MCE while the sign-switching characteristic feature should be noted in the pattern-filled region. In the reports of the FOPT magnetocaloric HEAs, this exponent n criterion was applied for identifying the FOPT [26, 27].

In addition, exponent n as a function of temperature has been used for identifying the presence of multiphases in Gd–Tb–Co–Al–Fe HEAs [21, 67]. The authors found broadened widths in the minima of n(T) curves for their samples, which attribute to the observed enhancement in the refrigerant capacity values.

Temperature first-order reversal curve (TFORC)

In spite of a significantly larger magnetocaloric response, FOPTs come at a cost, namely the presence of thermal hysteresis, which is undesirable for magnetocaloric refrigeration. Recently, an extension of the first-order reversal curve analysis (FORC) to thermal hysteresis, called TFORC [68], has been developed. It can provide valuable information on the origin of hysteresis and fine details about the transformation kinetics that occurs in the material. In particular, the combination of modeling [69] and experimentation with well-known samples [70], allows the identification of unexpected processes that take place in the samples under study.

For performing these TFORC experiments, the temperature of the sample has to be increased to the point of complete transformation (saturation) then it must be swept back to a certain reversal temperature in the hysteretic region. Starting from that reversal temperature, the measurement of M(T) back to saturation is called first-order reversal curve. This procedure is repeated for a complete set of reversal temperatures and, by performing the cross-derivative of magnetization vs. temperature and reversal temperature, the distribution of individual square hysteresis loops that conforms the full process is obtained. This TFORC distribution would present characteristic features that can be identified from a catalogue, providing valuable knowledge about the processes that the material undergoes [3].

While TFORC has been studied on different magnetocaloric materials, it has not yet been studied on HEAs. The reason for this absence of studies is that until very recently, MCE-HEAs did not exhibit a FOPT, making TFORC unapplicable. Considering that HEAs with large MCE are now associated with FOPT, we anticipate a surge of these studies.

Outlook

The HEA design concept yields newly unexplored compositional space in multicomponent alloy phase diagrams that could provide breakthroughs in structural and defense applications via combinations of properties. Thus, this propels the interest in advancing the design idea to functional sectors, which can directly address the current limitations of functional materials. This is especially so for the limited mechanical stability faced by conventional magnetocaloric materials with high-performance for magnetic refrigeration.

While the efforts of exploring HEAs for magnetocalorics are surging, only the recent breakthrough with a targeted search strategy into the HEA compositional space found magnetocaloric properties that show HEAs´ potential for this functional application. The implementation of magnetostructural behavior in HEAs greatly boosts their MCE performance to be on-par with conventional high-performing magnetocaloric materials. While this design method has led to some clear and quantifiable parameters for designing competitive magnetocaloric HEA compositions, exploration and development of HEAs with high MCE performance along these lines will be required as the field evolves. Efficient property optimization can be then enabled with the help of high-throughput predictive thermodynamic models, and/or compositional substitutions and processing techniques. There are precedents of shape memory alloys exhibiting martensitic transformation that can have good mechanical integrity upon cycling as well as superplasticity. Therefore, the objective of discovering HEAs with first-order magnetostructural phase transition accompanied by mechanical integrity should be reachable.

With the know-how approach to designing competent magnetocaloric HEAs, combining them with optimal mechanical properties and economic viability, reliability of devices would be improved and service life would be extended. Thus, this leads to improving sustainability in various sectors. This spurs the search for competitive magnetocaloric HEAs, which is still at an ascent stage. Clear pathways of designing such HEAs and understanding their material and property analysis play important roles in this development.