Magnetocaloric materials for hydrogen liquefaction

The expected energy transition to hydrogen gas as a greener energy vector has revived the interest in magnetic refrigeration at the cryogenic range, specifically between 20 and 80 K, with the vision to develop a new generation of hydrogen gas liquefiers. From the materials science point of view, the search for magnetocaloric materials containing mainly non-critical elements with a significant response in that temperature range, together with good cyclability and stability, is a challenging task. Given the increasing interest of the research community on this topic, we aim to establish a comprehensive catalog of the magnetocaloric compounds characterized so far, to be used as a starting point for further research. For this purpose, a systematic outlook of the state of the art is presented here, with the analysis and classification of more than 400 cryogenic magnetocaloric materials, divided into five large families according to their physicochemical properties. Moreover, we provide detailed information about their magnetocaloric properties, magnetic behavior, and transition characteristics together with criticality, which will facilitate the future search for optimal compounds.


INTRODUCTION
The current global energy landscape is highly dependent on carbon-based fuels, which have a significant impact on global warming that is already affecting our way of life.This motivates a pressing need to replace our actual energy carriers.2][3] Liquefying the gas is a crucial step in the final adoption of hydrogen as an energy vector because of its efficient transportation and storage. 1,2his will require liquefaction systems to increase energy efficiency, reduce losses, enhance safety, and enable up-scaling; these requirements, up to the moment, are not completely fulfilled by conventional systems based on Joule-Thomson expansion.
][5] The MCE is defined as the reversible temperature/entropy change produced in a magnetic material subjected to magnetic field variations in adiabatic/isothermal conditions.Since its discovery by P. Weiss and A. Picard in 1917 in nickel at 630 K and its subsequent physical description in 1921 by P. Weiss, 6,7 the capability of the MCE was rapidly exploited.The first breakthrough was in 1933 when W. F. Giauque and D. P. MacDougall 8 reached temperatures below 1 K, using Gd 2 (SO 4 ) 3 •8H 2 O, a paramagnetic gadolinium salt, that shows a strong temperature dependence of the magnetization with a large MCE close to zero kelvin.Since those early stages, the possibility of taking advantage of the MCE in every temperature range, depending on the working material, was evidenced, with a larger relevance in the vicinity of magnetic phase transitions where usually the magnetocaloric response is maximized.While paramagnetic salts offer the possibility of achieving ultralow temperatures, it is shown that the use of materials with near-room temperature phase transitions could be used for refrigerators. 9A paradigmatic example is elemental gadolinium, a ferromagnet with a Curie temperature of 292 K.The research focused on near-room temperature magnetic refrigeration was drastically enhanced at the end of the 20 th century by the discovery of materials with an extremely large (or giant) MCE response, [10][11][12][13][14][15] promoting the fabrication of many working prototypes of refrigerators all over the world. 5Nowadays, hundreds of magnetocaloric materials have been fully characterized, with transition temperatures ranging from one to several hundred kelvin.These include alloys, intermetallic compounds, oxides, ceramics, molecular solids, etc.. [16][17][18][19][20] More recently, an intermediate temperature range between 10 and 80 K is progressively gaining much attention due to its potential applications for gas liquefaction and storage.Given that liquid nitrogen can be produced below 77 K using conventional techniques based on the Hampson-Linde cycle, the challenge arises when trying to obtain other gases of lower boiling points, like hydrogen, in large quantities.Molecular hydrogen has a boiling point of 20.3 K and is one of the very few gases -along with helium and neon -that heats up during thermal expansion, which is a significant drawback for current industrial technology. 214][5] The core of these setups is a magnetocaloric material that acts as the working material and is controlled by the application and removal of magnetic field, which is typically generated by permanent or superconducting magnets.In general, the system absorbs heat from a cold load and releases it to the hot reservoir following a standard thermodynamic cycle, such as the ideal Carnot type illustrated in Figure 1, where the intensive variable is magnetic field instead of pressure.The heat transfer between the working material and the reservoirs occurs through a transfer medium made of gas, liquid or the magnetocaloric material itself.The idea of using magnetic refrigeration to produce liquid hydrogen dates to the 1990s, when the first prototypes appeared. 23Typically, they are designed to use helium as a heat transfer fluid and precooled liquid nitrogen as an input to the hot reservoir.Since compression-based refrigeration has very low efficiencies at cryogenic temperatures (usually less than 20 % of the ideal Carnot efficiency), 24 magnetic refrigeration has been established as a promising alternative to fulfill this industrial requirement.6][27][28][29][30][31] The most competitive ones are those based on two-stage active magnetic refrigeration schemes and have already demonstrated good performance. 26,27Moreover, the use of superconducting magnets is feasible given the operating temperature range of these devices.Thus, devices can operate under high magnetic fields of several tesla.Even so, this technology is still at an early stage of development and if it is to become a fully competitive option for the production of liquid hydrogen, the appropriate selection of magnetocaloric materials must be addressed.The present review article aims to provide a comprehensive reference on magnetocaloric materials suitable for hydrogen liquefaction.
Magnetocaloric working materials can be classified into those undergoing first (FOPT) or second order phase transitions (SOPT).Depending on the application, the magnetic transition should be optimized by modifying its Entropy T (K) 20  80 Liquid H 2 Vapor H 2

Magnetocaloric Carnot Cycle
Heat out Heat in Figure 1.An ideal magnetocaloric Carnot cycle operating between the temperatures of liquid nitrogen (hot reservoir) and liquid hydrogen (cold reservoir) This thermodynamic cycle, and many others, is capable of producing liquid hydrogen starting from liquid nitrogen temperatures.
order or tuning the response towards the desired temperature range.Typically, FOPT responses are larger in magnitude but, at the same time, they usually display thermal hysteresis that hinders the cyclability of the material and seriously affects its performance in devices. 32For cryogenic applications in the range of 20-80 K, the highest magnetocaloric responses are found in rare-earth (RE) based materials due to their typically low transition temperatures and large magnetic moment. 26,27,29,33However, the criticality of some elements might prevent its large-scale applications.The reduction of RE elements as well as other critical elements without reducing magnetocaloric response will push forward the industrial implementation of this technology.
This review paper analyzes more than 400 cryogenic magnetocaloric materials suitable for hydrogen liquefaction, paying special attention to composition, magnetocaloric properties, and magnetic phase transition.They are grouped into five different groups or families according to their physicochemical properties, as illustrated in Figure 2.This includes (1) binary and pseudobinary intermetallics; (2) ternary intermetallics with main group elements; (3) borides, nitrides, carbides and chalcogenide compounds (labeled as rare-earth non-metals in the figure); (4) rare-earth oxides; and (5) rare-earth amorphous / high-entropy alloys (HEA).Additionally, open topics pertinent to the material families and their performance are included in this compelling, and critically reviewed, compound materials catalog for gas liquefaction.This is complementary to recent reviews focused on specific families of magnetocaloric materials for cryogenics [34][35][36][37][38][39][40] and to a perspective paper on the current trends of magnetocaloric materials. 41

ΔS iso
The MCE is quantified by two complementary magnitudes: the isothermal entropy change ( ) and the adiabatic temperature change ( ) driven by a varying magnetic field H.Both and can be measured directly or calculated indirectly through other measured magnitudes, with the indirect calculation of being the most extended in the literature by employing the Maxwell relation: ΔT ad where is the magnetic permeability of vacuum.Similarly, for : where is the specific heat.Near the phase transition, if and the specific heat (c) is field independent, can be approximated to ΔT ad although it is based on strong approximations (specially for first-order transitions) and its proper use is not trivial. 42It has been shown that by employing an effective specific heat in martensitic transformation, it is possible to properly use it for first-order materials. 43To indirectly obtain , it is much more reliable to calculate it through the total entropy curves obtained from specific heat measurements at different fields: ΔT ad where the entropy at zero kelvin is neglected.Next, is obtained as

S (T, H) T (S, H)
ΔS iso where is inverted to .Similarly, can also be obtained from specific heat measurements as: Other magnitudes derived from and are routinely used to evaluate materials' performance, like the refrigerant capacity ( ), which accounts for the heat that can be exchanged between the hot and cold thermal reservoirs (at temperatures and , respectively). 44It is calculated as The reservoir temperatures depend on the refrigerator design, but they are usually approximated by the limits of the full width at half maximum .Classification of the cryogenic magnetocaloric materials analyzed in this review For each family of material, with its corresponding importance in the scientific literature, the maximum magnetocaloric response, the temperature range where these materials have a significant response, and the criticality range are shown.The calculation of criticality is explained in the last section of this review.

REVIEW
The Innovation Materials 1(3): 100045, December 13, 2023 3 ) in the curve.Thus, RC can be estimated by the product: RCP ΔT FWHM which is known as the relative cooling power ( ).The value should be considered with caution because it tends to overestimate the actual temperature span in materials with small magnetocaloric responses.It should be mentioned that more precise magnitudes or parameters have been developed.For instance, the Temperature Averaged Entropy Change (TEC), 45 corrects for deceptively large RC responses caused by shallow maxima.However, this magnitude is less discussed in the literature and, unfortunately, cannot be used for a general comparison.Similarly, the coefficient of performance (COP) offers a more precise tool than RC and RCP for evaluating the energy conversion of the materials as it also accounts for magnetization hysteresis during working conditions and allows comparison among different cooling technologies. 46However, this magnitude is also scarcely reported in publications.A recent overview of magnetocaloric measurement techniques, together with their application to the study of phase transitions, is presented in ref. 47

BINARY INTERMETALLIC COMPOUNDS
Binary compounds between a RE element and a metal, such as transition metal (Fe, Co, Ni, Mn, Cu, Ag, Rh, Pd, etc.), p-block metal (Al, Ga or In) or even a metalloid (Ge or Si) are included in this first group.These compounds can be found in a wide range of stoichiometric combinations, the most prevalent of which are RM, RM 2 , R 3 M, R 3 M 2 , RM 5 , etc.Even uncommon stoichiometries, such as R 5 M 4 , R 3 M 7 or R 7 M 12 , have a significant impact in this field and their basic magnetic properties have been reported in the last quarter of the previous century. 48The most relevant compounds with potential applications in cryogenic magnetic refrigeration will be discussed in the following.
Equiatomic compounds with generic formula RM have been extensively studied 40 and some of them display remarkable MCE.The most studied families are RZn, RNi and RGa.

ΔS iso
0][51][52] In addition, a spin reorientation (SR) transition is observed in some of them (R = Tb, Ho and Tm).In particular, HoZn has very broad curves due to two phase transitions, leading to a remarkable refrigerant capacity. 51S iso | ΔS iso RNi compounds have ferromagnetic behavior with possible low-temperature SR phase transitions at around 10 K. Non-collinear magnetism is observed in all compounds except for GdNi, which is a simple ferromagnet.They crystallize in FeB-or CrB-type crystal structures and their Curie temperatures vary between 8 and 71 K, being GdNi the one with the largest Curie temperature. 53Thus, the whole series has potential applications for hydrogen liquefaction.Most of them have been characterized for their MCE [54][55][56][57][58] except for the cases of NdNi and TmNi.Overall, from R = Gd to Er they present large values of between 15 and 30 J kg −1 K −1 for an applied field of 5 T. All of them also exhibit large values of RC, especially HoNi.This is due to the coexistence of two phase transitions that leads to a table-like MCE over a broad temperature range.Despite having an intermediate value of of 14.5 J kg −1 K −1 , the RCP value is estimated to be 780 J kg −1 for an applied field of 5 T. 56 Zheng et al. 59 have studied the pseudobinary Ho x Er 1-x Ni system for the whole range of x, producing samples with Curie temperatures between 11 and 36 K.The refrigerant capacity increases with increasing Ho content due to the table-like MCE of HoNi.
|ΔS iso | RGa gallides crystallize with the CrB-type structure and their magnetic properties are rather singular.3][64][65][66][67] One transition is associated with a ferromagnetic ordering, which can be either FOPT or SOPT, and the second is due to a SR transition.The former covers the range of 30-180 K with a decreasing trend from Gd to Er, while the latter always occurs at lower temperatures within the 15-40 K range.They present large values of , with TmGa ΔS iso being the one with the largest magnitude.Interestingly, HoGas exhibit two phase transitions occurring at 20 and 69 K, leading to a table-like MCE with a large refrigerant capacity. 62,66Moreover, the use of solid solutions combining two rare-earth elements is especially suitable for expanding magnetic refrigeration even further in these compounds.For instance, Tm 1−x Ho x Ga can be synthesized for all values of x, 62 leading to intermediate compounds exhibiting three phase transitions occurring within a relatively small temperature range.This is because they merge the spin reorientation transition of HoGa with the two successive phase transitions observed in TmGa.Another interesting strategy is to alloy Gd and Er. 68Since the transition temperatures for GdGa are much higher than those of ErGa, in the Gd x Er 1-x Ga system the peaks can be easily tuned by increasing the content of Gd, achieving compounds with extraordinary refrigerant capacity, as shown in Figure 3A.
Other examples of equiatomic binary compounds mainly crystallize in CsCl structure, such as TmCu, TmAg (both with very poor MCE), 72 GdCd 0.8 Ru 0.2 , 73 GdRh (both with better MCE responses), 74 NdSi, 75 NdPd, 76 ErAl 77 and HoAl 77 crystallize within orthorhombic DyAl-type structure, where the latter displays successive AFM to FM to PM transitions at 13 and 20 K, leading to remarkable values of 22.5 J kg −1 K −1 for 5 T.
The MCE of R 2 In ferromagnetic compounds reveals a few promising candidates for hydrogen liquefaction.Tb 2 In, Dy 2 In, Ho 2 In and Er 2 In crystallize in a Ni 2 In-type crystal structure and exhibit Curie temperatures of 165, 130, 82 and 42 K respectively.They possess remarkable MCE [78-81], especially Ho 2 In, whose additional SR transition at 32 K on top of the FM to PM transition leads to a large table-like MCE that expands over a broad temperature range. 78Surprisingly, Eu 2 In crystallizes in a different type of crystal structure (Co 2 Si-type), exhibits FOPT without apparent hysteresis, and its MCE magnitudes are the largest among the R 2 In series, with a maximum of 33 J kg −1 K −1 at 55 K for an applied field of 5 T (see Figure 3B). 69

ΔS iso ΔT ad
Many RM 2 compounds are known to crystallize within the cubic MgCu 2type crystal structure, or alternatively, the other two so-called Laves phases (i.e., hexagonal MgZn 2 or cubic MgNi 2 ).Over the last few years, their magnetic properties have been extensively studied, .Many of them, especially the RNi 2 , RCo 2 and RAl 2 series, have been fully characterized for their MCE.RCo 2 compounds could undergo a FOPT with a large MCE associated to it where among them only HoCo 2 and ErCo 2 show transition temperatures below 80 K (78 and 32 K respectively). 82,83Both compounds display very large values of and : 20 J kg −1 K −1 and 8.5 K for HoCo 2 84 and 32 J kg −1 K −1 and 7.2 K for ErCo 2 85,86 for an applied field of 5 T. Interestingly, the transition temperature can be finely tuned by alloying two different rare-earth elements 87 or shifted towards lower values after the partial substitution of Co by Ni 84,88 or other elements, as shown in Figure 3C.Solid solutions of RAl 2 compounds have been regarded as potential candidates for low temperature refrigeration since the early days of magnetocaloric research. 89,90This is because most of these compounds display FM to PM SOPTs between 13 and 80 K with a remarkable MCE. 91In particular, those compounds with R = Dy, Ho and Er show an additional SR transition below the Curie temperature which enhances their MCE response, 92,93 reaching the outstanding mark of 36.2J kg −1 K −1 and 11 K for and respectively for ErAl 2 . 92Very recently, Liu et al. 94 have explored the MCE of NdAl 2 and PrAl 2 together with some of their rare-earth solid solutions leading to remarkable figures.[97][98][99][100] For this reason, combinations of RM 2 Laves phases are regarded as strong candidates for hydrogen liquefaction applications and, indeed, are the preferential choice when building operating prototypes. 26,27,29,31However, it is important to note that, although the RM 2 series with M = Ni, Co and Al has attracted most of the attention in the materials community, 70,101 there are still other combinations that have received less attention despite having promising properties.This is the case with the RMn 2 series 102 or others that have not even been explored yet, like RMg 2 .][105] R 3 Co might be regarded as another promising family of compounds.They are mainly constituted by rare earth elements, which carry the magnetic moment, while Co acts as nonmagnetic element and, at the same time, stabi- lizes the AF ordering.They crystallize in Fe 3 C-type crystal structure and present complex magnetic behavior with transition temperatures between ~5 K (Tm 3 Co) 106 and ~130 K (Gd 3 Co). 107Er 3 Co, with arguably the best MCE response among the series, shows a very complex non-collinear magnetism and does not follow the general trend observed in the series. 108,109Most of the other compounds present two close successive phase transitions due to the presence of two types of magnetic orderings.The Tb 3 Co, 110 Dy 3 Co 111 and Ho 3 Co 112 compounds present transition temperatures that fall within the appropriate range between 20 and 80 K, with two peaks or a table-like character observed in their curves, which all give rise to high RC values.The transition temperatures of this class of compounds can be easily tuned by either incorporating a new transition metal or rare-earth elements by means of solid solutions.For instance, the Curie temperature of Gd 3 Co can be lowered after Ru incorporation in the Gd 3 Co x Ru 1-x . 113,114Alternatively, the solid solution between Er and Gd allows a great tunability of the transition temperature in Er 3-x Gd x Co. 115 There are more compounds with R 3 M stoichiometry and most of them have not been systematically studied for magnetocaloric applications.Even so, there are a few promising examples like Gd 3 Ru, whose Curie temperature is 54 K and displays a remarkable MCE with values up to 30 J kg −1 K −1 for 5 T. 116 Again, the solid solution between Gd and Er can be used to adjust the Curie temperature with remarkable precision. 117Gd 3 Rh and Tb 3 Rh also show good magnetocaloric properties but with transition temperatures above 100 K. 118,119 In addition, Ho 3 Ru and Ho 3 Rh have been investigated 120 in samples that contain other secondary phases that also contribute to the global MCE but with moderate performance.R 3 M 2 intermetallic compounds with Er 3 Ni 2 -type crystal structure are not so common and their MCE performance has not been studied systematically so far.R 3 Ni 2 compounds with R = Er and Ho behave as ferromagnets with an additional SR transition at low temperatures. 121The broad R 3 NiCo series, with solid solution of transition metals, have been analyzed by Herrero et al. 122 They present essentially the same features of R 3 Ni 2 compounds, same crystal structure and magnetic behavior without a clear SR transition.The Curie temperature monotonically decreases with increasing atomic number through the lanthanide series, from 97 to 7 K, following an almost exact geometric progression with 1/2 of common ratio.RC is quite large in these compounds, reaching up to 516 J kg −1 for Tb 3 NiCo.Similar features are observed in Er 3 Rh 2 and Ho 3 Rh 2 (despite having a different Y 3 Rh 2 -type crystal structure), while Nd 3 Rh 2 orders antiferromagnetically and shows poorer MCE performance. 123Another compound with this stoichiometry but with a distinctive tetragonal Zr 3 Al 2 -type crystal structure is Ho 3 Al 2 , which displays a spin reorientation transition at 31 K followed by a FM to PM phase transition at 40 K, leading to a remarkable RC value of 704 J kg −1 for an applied field of 5 T. 124 The series of RNi 5 is an interesting group of compounds due to its low content in rare-earth elements. 125They behave as simple ferromagnets with transition temperatures between 10 and 30 K, except PrNi 5 which is a param- ) )

REVIEW
The Innovation Materials 1(3): 100045, December 13, 2023 5 ΔT ad agnetic compound with inverse MCE.Among them, TbNi 5 and GdNi 5 seem more suitable for hydrogen liquefaction, with transition temperatures of 23 K and 32 K respectively 125,126 and remarkable .A very interesting feature of RNi 5 compounds is the possibility to produce ternary compounds of the type RNi 4 M or RNi 4 X with the incorporation of other metals or nonmagnetic main group elements, which has not yet been systematically studied.For instance, TbNi 5 shifts its transition temperature to larger values after partial replacement of Ni by Co or Fe, although its MCE worsens. 126Conversely, GdNi 4 Al and GdNi 4 Si display even better MCE than the parent GdNi 5 but have slightly lower Curie temperatures. 127 iso

ΔS iso
The R 5 (Si x Ge 1-x ) 4 pseudobinary alloys have attracted extraordinary attention since the discovery of giant MCE in Gd 5 Si 2 Ge 2 in 1997. 109][130] In brief, R 5 Si 4 and R 5 Ge 4 compounds usually crystallize in orthorhombic crystal structures of either Gd 5 Si 4 -or Sm 5 Ge 4 -type.However, for intermediate values of x~2, there is a monoclinic phase where the giant MCE could be observed.The FOPT is accompanied by a structural transformation that is responsible for the unexpected large magnetothermal response.The trends in magnetic ordering and transition temperature are difficult to rationalize, but the striking difference in transition temperature between the germanide and silicide counterparts can be attributed primarily to the formation of a different number of covalent bonds between the constituent slabs of the crystal structure in each phase (see Figure 3D).This fact was explained in the case of the Gd based alloy, 71 whose transition temperature gradually drops from 336 K for Gd 5 Si 4 to 32 K for Gd 5 Ge 4, with all the intermediate values for increasing Ge content.Analogous compounds containing Nd, Tb, Dy and Er have been studied in detail.Nd 5 Si 4 and Nd 5 Ge 4 display transition temperatures at 71 and 55 K respectively, but with moderate MCE responses. 131Moreover, it seems that a giant MCE is not observed in the whole Nd 5 (Si x Ge 1-x ) 4 . 131Tb 5 (Si x Ge 1-x ) 4 also shows remarkable MCE, but at temperatures above 100 K for the whole x range. 132Dy 5 (Si x Ge 1-x ) 4 is a particularly interesting system as its phase transitions expand over a temperature range from 130 K (Dy 5 Si 4 ) to 35 K (Dy 5 Ge 4 ), but its largest MCE is observed for Dy 5 Si 3 Ge at 65 K, with a larger than 30 J kg −1 K −1 . 133In the case of Er 5 (Si x Ge 1-x ) 4, the phase transitions take place between 14 and 30 K, showing a good yet moderate MCE. 134Other remarkable compounds include Gd 5 Rh 4 74 and especially Gd 5 Sn 4 , 135 with an excellent MCE response.However, it should be noted that pseudobinary R 5 M 4 phases display FOPT, usually with large thermal hysteresis.This means that its use in devices is difficult to implement due to the mechanical degradation of the materials upon operational cyclic conditions and because they usually show poor refrigerant capacities due to the sharp shape of the peaks, despite their large magnitude.The occupied f-orbitals of lanthanides allow the formation of intermetallic compounds with unusually large stoichiometric coefficients.One example of such type of compounds relevant to magnetocaloric applications is the R 12 Co 7 series.They crystallize following a complicated Ho 12 Co 7 -type monoclinic crystal structure containing four different types of coordination polyhedra for the RE sites in the unit cell.7][138] Conversely, the compounds with Ho and Er display a more complex behavior, with multiple phase transitions at lower temperatures. 139,140The possibility of using RE solid solutions to tune the MCE of these compounds had already been demonstrated in Gd 12-x Tb x Co 141 and Er 12-x Ho x Co 7 140 systems, the latter being very promising for hydrogen liquefaction.
3][144] Related compounds in the form of R 7 Pd 3 , such as Pr 7 Pd 3 145 and Nd 7 Pd 3 , 146 have also been considered as possible magnetocaloric materials.However, the high content of rare-earth in combination with Pd, an extremely expensive metal, suggests that such compounds are less economical for industrial applications, not even those with remarkable MCE like Nd 7 Pd 3 .

TERNARY INTERMETALLICS WITH MAIN GROUP ELEMENTS
In this second family of rare-earth compounds, we include those ternary combinations of a lanthanide element, a transition metal and a main group element in addition to Cd or Zn that can be considered in this group. 147Note that the combinations including non-metal elements of the second period, such as B, C, N or S, will be covered separately in the next section.Similar to binary rare earth compounds, the number of possible combinations in this second family is large.The most common stoichiometries for rare earth ternary compounds are RMX and R 2 M 2 X.The RMX compounds usually crystallize in a hexagonal ZrNiAl-type crystal structure though other structures could also exist. 148Their magnetocaloric properties have been investigated in many cases. 37On the other hand, the magnetocaloric R 2 M 2 X compounds mainly belong to tetragonal Mo 2 FeB 2 -type or W 2 CoB 2 -type crystal structures, along with some other less common structures.Many of these materials are known to display a remarkable MCE. 38In addition to these most common ones, there are many other stoichiometric possibilities that we will mention throughout this section.For clarity, the discussion thereafter in this section will focus on the various main group elements rather than the different stoichiometry.There are a few relevant cadmium, indium, or magnesium compounds among the ternary rare earth compounds with pertinent magnetocaloric properties, but the majority reported are aluminides, gallides, and silicides.
Aluminides represent a large group of this type of compounds whose magnetocaloric properties have been extensively studied.Magnetic equiatomic compounds, such as ferromagnetic elements (i.e., RNiAl, 149,150 RCoAl 151 and RFeAl 152 series) display good magnetocaloric properties.Those that contain Co and Fe behave as standard ferromagnets whereas those with Ni show antiferromagnetic ordering.The overall MCE performance follows the Co < Ni < Fe trend with transition temperatures in the range 10 -100 K. Their transition temperatures decrease with increasing atomic mass along the Gd-Er series.Surprisingly, compound series with non-magnetic atoms, such as Cu or Ag, are also promising materials in terms of MCE.4][155][156][157] A similar behavior is observed in the RAgAl series but with lower MCE. 158,159he remaining interesting aluminides for cryogenic magnetocaloric refrigeration are further combinations of Al with Ni or Co.The series R 2 Co 2 Al 164,165 and R 2 Ni 2 Al 165 share some similar features with their equiatomic counterparts, in the sense that Ni compounds display AF behavior while Co compounds behave as ferromagnets.However, their magnetothermal responses are weaker than their RMX counterparts, despite an additional SR transition at low temperatures.Interestingly, GdNiAl 2 and TbNiAl 2 display remarkable MCE with exceptionally high RCP at temperatures around 30 and 20 K respectively, 160 as shown in Figure 4A.Unfortunately, compounds with other lanthanides possess too low transition temperatures.In a similar manner, R 3 Ni 6 Al 2 compounds 166,167 show mainly ferromagnetic behavior and moderate MCE over a broad range of transition temperatures, from 105 K for Gd to 2.5 K for Er.Lastly, R 2 CoAl 3 compounds, 168 which are MgCu 2 -type Laves phases, show SOPTs and lower transition temperatures than their RCo 2 counterparts, with a significantly weaker MCE.

ΔS iso
Rare-earth gallides occur in a great variety of combinations and many of them have been fully characterized, showing moderate MCE magnitudes.The MCE of R 2 M 2 Ga compounds have been reported both for M = Co and Ni. 161,169,170All of them possess W 2 CoB 2 -type crystal structure.Co compounds behave as standard ferromagnets and exhibit moderate MCE (see Figure 4B), whereas Ni compounds display AF ordering that transforms to FM in the presence of magnetic fields, resulting in a poor MCE.Other representative compounds include the RNiGa 2 series 171 and equiatomic ternary HoAgGa. 172o MCE characterization of any other compound of the RAgGa series has been reported so far.Compounds with less common stoichiometries, such as R 2 CoGa 3 173 or R 6 Co 2 Ga, 174 have been studied.It is worth noting that despite the moderate magnetothermal response shown by rare-earth gallides, we can find some compounds with a large refrigerant capacity due to the broadness of their curves.This is the case for Gd 6 Co 2 Ga, whose Curie temperature is 78 K and its RCP value is estimated as high as 618 J kg −1 for 5 T. 174 There are many indium compounds containing rare-earth metals showing moderate magnetothermal responses.Although they are characterized by a rich variety of magnetic properties and phase transitions, none of them display a relevant MCE.Among them, we highlight the following as representative examples: RNiIn (see Figure 4C), 162  In (M = Cu, Ag and Au) display complex disordered crystal structures and a moderate magnetothermal response. 179The MCE of the R 2 Ni 2 In series has also been studied, but their transition temperatures are too low for their use in hydrogen liquefaction. 180

|ΔS iso |
ΔT ad A few magnesium and cadmium compounds have proven to be excellent candidates for magnetic refrigeration at low temperatures, e.g., equiatomic ternary RPtMg compounds, 181 or the more competitive R 4 MMg, [182][183][184] where M = Pt or Pd.Some of these compounds with Gd 4 RhIn-type crystal structure (i.e., Er 4 PdMg, Ho 4 PtMg or Er 4 PtMg) have FM to PM SOPT in the proximity of hydrogen liquefaction temperature with excellent magnetothermal response of around 6 J kg −1 K −1 and around 2 K for an applied field of 2 T.These values are often used as a good indicator for materials for cryogenic magnetic refrigeration.Isomorphous R 4 CoCd compounds also possess excellent MCE, comparable to magnesium compounds. 185,186In addition, the series R 2 Cu 2 Cd with Mo 2 FeB 2 -type crystal structure also shows good MCE properties, [187][188][189] especially Ho 2 Cu 2 Cd, whose Curie temperature is 30 K and is accompanied by a SR transition at 15 K, leading to large values around 10 and 20 J kg −1 K −1 for applied fields of 2 and 5 T, respectively. 189nfortunately for these combinations, Pt and Pd are extremely expensive, and Cd is often discouraged due to its high toxicity.
Silicides constitute an extensive family of interesting compounds.For instance, equiatomic ternary compounds, such as HoCoSi, 190 ErFeSi 163 or ErRuSi 191 are well-known ferromagnets with a remarkable MCE and appropriate Curie temperatures (see the curves of ErFeSi in Figure 4D).The main group of silicides are those with the general formula RM 2 Si 2 and ThCr 2 Si 2 -type crystal structure, with M = Mn, Cr, Ru, Fe, etc.Most of them present AF behavior with field induced AFM to FM FOPT and a moderate MCE.However, there are some notable exceptions, like ErCr 2 Si 2 , 192 which is a ferromagnet with a SOPT and impressive MCE values: maximum of 24.1 and 29.7 J kg −1 K −1 and of 8.4 and 17.4 K for 2 and 5 T, respectively.Unfortunately, its Curie temperature is only 4.5 K, which is too low to be considered as a competitive candidate for hydrogen liquefaction applications.Several rare-earth silicides containing Pd have been reported, such as those with the formula RPd 2 Si 193,194 or R 2 PdSi 3 , 195,196 very complex magnetic behavior and poor MCE performance.Note that the Gd 2 NiSi 3 compound 197 shares similar features with the aforementioned Pd-containing compounds.
Furthermore, while there are other compositions that contain other main group elements, none of them seem to exhibit a magnetothermal response good enough to be considered for magnetic refrigeration.For example, the study of magnetic properties of R 2 Ni 2 Sn compounds 198 finds them with extremely complex magnetic behavior and poor MCE magnitude.Other main group element compounds include EuAuZn, 199 EuAgZn, 200 EuPtZn, 200 EuAuGe, 201 PrMn 1.4 Fe 0.6 Ge 2 , 202 or EuFe 2 As 2 . 203

RARE-EARTH -NON-METAL COMPOUNDS
In this third group of compounds, we discuss rare earth compounds that combine lanthanides with nonmetallic main group elements.This section includes the limiting case of boron element, while those compounds containing oxygen will be discussed exclusively in the next section since oxides constitute a separate and broad group of compounds.Thus, this section focuses mainly on rare earth borides, carbides and nitrides with a few cases of sulfides.
The recent discovery of HoB 2 as a promising candidate for cryogenic magnetic refrigeration 204 has brought renewed attention to boron compounds.This is because HoB 2 possesses one of the largest (about 40 J kg −1 K −1 for 5 T) near the boiling point of hydrogen, with a of 12 K (see Figure 5A for other values of applied field).This feature was further confirmed by J. Li, et al., 205 pointing at slightly lower values of about 35 J kg −1 K −1 .Previously, the magnetocaloric properties of DyB 2 had been studied, with a lower but remarkable MCE. 206DyB 2 and HoB 2 behave as standard ferromagnets with Curie temperatures at 50 and 15 K respectively, and additional SR transitions at 20 and 12 K.The proximity between the FM to PM and SR phase transitions in HoB 2 might explain the large magnetothermal response observed in this compound.The pseudobinary system formed by the solid solution of Ho and Dy (Ho x Dy 1-x B 2 ) has been fully studied for the whole x range. 205,207The combination of Ho and Dy in the same compound decreases but leads to a table-like response, achieving very large RC values of over 600 J kg −1 . 205The Curie temperature can be tuned within the range of 15-50 K depending on the Ho and Dy ratio.More recently, the partial substitution of B by Al has been considered, 208 revealing that the MCE of original HoB 2 is barely affected (including the Curie point) in HoB 1.5 Al 0.5 .Given the striking price difference between Al and B, this could be a point worth of further exploration.Regarding other members of the RB 2 series, TbB 2 has a too high Curie temperature, i.e., 144 K, 209 while the MCE of ErB 2 or TmB 2 has not yet been investigated.
Other rare-earth borides include the ternary compounds RCo 2 B 2 or RCo 3 B 2 .The RCo 2 B 2 series is isomorphous with the RM 2 Si 2 silicides that has been discussed in previous section.They crystallize in ThCr 2 Si 2 -type structure and can display either ferromagnetic ordering associated to a SOPT or a metamagnetic FOPT with a field induced AFM to FM transition like GdCo 2 B 2 217 or TbCo 2 B 2 . 218As reported by L. Li, et al. [217][218][219][220] they display remarkable MCE properties and transition temperatures in the target range for hydrogen liquefaction.Especially, GdCo 2 B 2 presents large of 6.7 and 15.4 K for 2 T and 5 T, respectively, and also high RC in spite of its FOPT.On the other hand, RCo 3 B 2 compounds with CeCo 3 B 2 -type crystal structure behave as standard ferromagnets with suitable Curie temperatures for Gd, Tb and Dy compounds.][223][224] Rare-earth carbides containing transition metals show better overall MCE performance than equivalent borides.Among these carbides, we can highlight the RCoC, RCoC 2 and R 2 Cr 2 C 3 series for their excellent MCE.The RCoC ) for some promising ternary intermetallic materials (A) RNiAl 2 (R=Gd, Tb and Tm) for 5 T, 160 (B) R 2 Co 2 Ga (R=Dy, Ho, Er and Tm) up to 7 T, 161 (C) RNiIn (R=Gd, Tb, Dy, Ho and Er) for 5 T 162 and (D) ErFeSi for fields up to 5 T from magnetization (symbols) and specific heat (lines) measurements. 163(All figures reproduced with permission of AIP).

REVIEW
The Innovation Materials 1(3): 100045, December 13, 2023 7 |ΔS iso | compounds with R = Tb and Er 225 and R 2 Cr 2 C 3 with R = Dy, Ho and Er 226,227 possess good magnetocaloric properties but their transition temperatures tend to be lower than 25 K.They could undergo either FOPT or SOPT, exhibiting both ferromagnetic and antiferromagnetic ordering depending on the rare earth element, often with a field-induced phase transition from AF to FM order.On the other hand, the whole RCoC 2 series with R = Gd, Tb, Ho and Er 210,228,229 behaves as ferromagnetic materials with FM to PM SOPTs.They show low Curie temperatures in the range 11-28 K and large .For instance, GdCoC 2 , whose Curie temperature is 15 K, shows 16 and 28 J kg −1 K −1 for 2 and 5 T, respectively, 210 as shown in Figure 5B.Some compounds containing boron and carbon have also been studied for MCE, in particular, the quaternary RNiBC and RM 2 B 2 C compounds where M = Co and Ni.Those containing cobalt lead to ferromagnetic behavior and SOPTs, 230 while the presence of nickel stabilizes the AFM ordering associated to FOPTs. 231,232They exhibit characteristics that seem to be more similar to those of carbides than to those of their boride analogs.Equiatomic RNiBC compounds show ferromagnetic ordering and SOPTs, at least for GdNiBC and ErNiBC, 233 but their Curie temperatures are too low even for GdNiBC.
Surprisingly, with the notable exception of EuS, there has been very little research in recent years focused on the magnetocaloric properties of inorganic sulfides.It is a well-known magnetocaloric material 212,234,235 behaving as a ferromagnetic semiconductor with NaCl-type crystal structure and a Curie temperature of 18 K.It displays impressive MCE figures, namely of 22 and 37 J kg −1 K −1 for 2 and 5 T, respectively, leading to RCP values of 284 and 780 J kg −1 , respectively.The associated values are 7.5 and 10.4 K, respectively. 212The analogous EuSe is also a semiconductor that displays excellent MCE.However, its magnetic properties are different: it has a peculiar AF ordering after a FOPT at 4.6 K yet with very broad curves. 211The curves for 5 T of EuS and EuSe are presented in Figure 5C for comparison.
Typical inorganic sulfides containing rare earth, such as NaGdS 2 or LiGdS 2 , |ΔS iso | ΔT ad may have an excellent MCE 236 but their transition temperatures are below 10 K. Conversely, there are a few thiospinels with remarkable magnetic properties like CdCr 2 S 4 . 237This ferromagnetic compound has a Curie temperature of 87 K, slightly above our target temperature range, and show a of 7 J kg −1 K −1 and a of 2.6 K under an applied field of 5 T. Other related compounds like FeCr 2 S 4 or CoCr 2 S 4 possess much higher Curie temperatures. 238The partial replacement of Cd by Zn in CdCr 2 S 4 shifts the Curie temperatures to lower values but, at the same time, the MCE magnitude decreases. 239Notice that spinels are a large family of materials with the general formula of AB 2 X 4 that include oxides and other chalcogenide compounds with a unique cubic crystal structure.These compounds allow the accommodation of two different transition metals with both tetrahedral and octahedral coordination.This has led to substantial progress in the development of magnetic materials based on spinel compounds. 240In summary, further attempts are necessary to find new magnetic sulfides with transition temperatures between 10 and 80 K.We emphasize that even though many sulfides and other chalcogenides do not contain rare-earth elements, their potential as magnetocaloric compounds has not yet been thoroughly studied.
|ΔS iso | Rare earth nitrides represent another promising group of compounds with distinctive properties and very good MCE features.In particular, rare earth mononitrides with the formula of RN, where R = Gd, Tb, Dy, Ho, Er, have been investigated in depth, 213,214,241 along with many of their binary RE nitrides. 213,215,216,242,243All of them display NaCl-type crystal structure and a FM to PM SOPT with Curie temperatures in the region of 6 to 60 K, following the usual trend of decreasing Curie temperature along the series Gd < Tb < Dy < Ho < Er (see Figure 5D).They are dense materials in comparison with metallic materials and possess a high thermal conductivity.Rare earth nitrides are very stable from the chemical point of view and do not react with hydrogen gas, in contrast with most intermetallic compounds.For an applied field of 5 T, some of these nitrides, such as HoN or ErN, present very large

ΔS iso
][215][216] ΔT ad over 30 J K −1 kg −1 and around 10 K, 244 whose combined values make them emerge as the magnetocaloric materials with the best performance in the target temperature range (see Figure 5D).More recently, they have been synthesized as nanoparticles which leads to remarkable magnetocaloric properties. 245,246

RARE-EARTH OXIDES
Magnetic oxides are an extensive class of materials with a great variety of structures and properties.Regarding magnetic ordering, they usually present complex behaviors, including non-collinear magnetism, ferrimagnetism, or multiple phase transitions, often accompanied by structural transformations.Moreover, the presence of more than one magnetic atom in the structure could lead to a distinctive ordering for each species.For instance, there are many oxides that include a rare-earth element and a transition metal (Mn, Cr, Fe, etc.).In these cases, the interplay between 3d and 4f electrons usually plays a major role in the magnetic properties of the compounds, leading to unique features.

ΔS iso
From the point of view of the synthesis, oxides follow production methods which clearly differ from those of intermetallic compounds.As a result, the fabrication of relatively large single crystals is usually possible.This feature makes these single crystal potential candidates for rotating magnetocaloric devices in which the refrigeration driving force is created by rotating a strongly anisotropic sample inside a constant magnetic field rather than inducing phase transition on an isotropic material upon changing the modulus of the applied magnetic field.In the case of materials for rotating magnetocaloric applications, the difference between the values obtained along two well-defined crystallographic directions is the key parameter to maximize to obtain an appropriate material, and it is denoted as rotational .The larger the difference between two directions, the larger the refrigerant capacity will be when rotating the sample inside a magnetic field.
For the transition temperatures of oxides, they could range from a few K to more than 1000 K.Most of them undergo phase transitions at very low temperatures, below 10 K, but there are still many examples with higher transition temperatures, such as manganites, ferrites, spinels, etc.In this section, we focus on those with transition temperatures in our target range suitable for hydrogen liquefaction applications.Our discussion proceeds in increasing order of structural complexity, beginning with simple oxides with one metal in the anion (i.e., RCrO 4 , RTiO 3 , RVO 4 , etc.), and relatively simple structures, such as perovskite and zircon types.Finally, we discuss double oxides that include two metal atoms in the anion site and finalize with garnets and other complex oxides.
Several simple oxides with general stoichiometries of RMO 3 and RMO 4 have been investigated in the framework of magnetocaloric effect.RMO 3 compounds, such as RCrO 3 , RFeO 3 , RTiO 3 , RMnO 3 RVO 3, display perovskitetype structure whereas RMO 4 compounds, like RCrO 4 or RVO 4, crystallize within the tetragonal zircon-type symmetry.In addition, some spinel oxides could be included in this group.They are the simplest oxides showing a remarkable MCE.To the best of our knowledge, there is no other oxide simpler than these ternary ones with a remarkable MCE, except the notable EuO.This binary oxide crystallizes with a NaCl-type crystal structure and behaves as a ferromagnet with a Curie temperature of 69 K.For an applied field of 5 T, it shows of 17.5 J kg −1 K −1 , of 6.8 K and RCP of 665 J kg −1 . 247nvestigated chromite compounds are of the RCrO 3 type, with R = Sm, Dy, Ho, Er, Yb. [248][249][250] They show antiferromagnetic order and some of them display appropriate transition temperatures like HoCrO 3 or ErCrO 3 , whose transition temperatures are 20 and 10 K, respectively.In general, these compounds show a moderate MCE magnitude, except the aforementioned ErCrO 3 , with a remarkable value of -13 J kg −1 K −1 and an associated RC of 419 J kg −1 . 249

|ΔS
iso | 2][253] The values of rotational isothermal entropy changes between 10 and 20 J kg −1 K −1 for an applied field of 5 T can be achieved in these materials.Polycrystalline samples of GdFeO 3 have also been investigated, 254 where they exhibit very large values, over 30 J kg −1 K −1 for 5 T, but at around 5 K, which is too low for direct applications in hydrogen liquefaction refrigeration.
Manganites with the general formula of RMnO 3 are a well-known class of compounds for the magnetocaloric community since there are hundreds of them fully characterized. 260Although rare-earth free manganites usually display near-room temperature phase transitions, those with rare-earth elements may have much lower transition temperatures.The investigated RMnO 3 includes the magnetocaloric characterization of single crystals where R = Nd, 261 Dy, 262 Tb, 263 Ho 264 and Tm. 265Another important feature of manganites is that their transition temperatures can be easily tuned by element substitution or pressure.As an example, we can mention the La 0.65 Ca 0.35 Ti 1-x Mn x O 3 system in which the Curie temperature linearly varies from ~100 K for x=0 to ~40 K for x=0.4. 266Unfortunately, the MCE magnitude in rare-earth manganites is much weaker than the observed for other related oxides.
Rare-earth titanates (RTiO 3 ) present very interesting magnetocaloric features.They display a relatively simple ferromagnetic ordering compared with other similar oxides and transition temperatures in the range of 30-70 K in most cases.Y Su, et al. 267 systematically studied the MCE of RTiO 3 compounds with R = Dy, Ho, Er, Tm, and Yb, revealing very promising properties.The maximum values are larger than 10 J kg −1 K −1 accompanied by remarkable RCP values that could be larger than 400 J kg −1 .Similar values have been reported for GdTiO 3 single crystals. 268Conversely, a distinctive behavior has been observed for EuTiO 3 , which orders antiferromagnetically at only 5 K and shows huge and values of 40 J kg −1 K −1 and 16 K respectively. 269,270t is worth noting that the MCE of metavanadates (RVO 3 ) has not been studied with the exception of HoVO 3 , whose magnetic properties are clearly distinctive.It undergoes several phase transitions, either due to the Ho 3+ ion ordering or structural transitions, giving rise to a table-like MCE which spans over a broad temperature range. 271

ΔS iso
On the other hand, some rare-earth chromates (RCrO 4 ) have been studied for R = Gd, 255,272,273 Dy, 274 Ho 255,274 and Er. 272They crystallize in the tetragonal zircon-type structure and exhibit complex magnetic properties due to the strong competition between ferromagnetic and antiferromagnetic superexchange interactions of 3d and 4f spins.Overall, they behave as ferromagnets but with some anomalies and often with field-induced metamagnetic transitions.Nonetheless, they present remarkable over 20 J kg −1 K −1 for an applied field of 5 T, as shown in Figure 6A.Notably, other RCrO 4 compounds also exhibit transition temperatures between 10 and 25 K and complex magnetic properties, such as non-collinear magnetic ordering. 275,276Hence, they represent an interesting group of compounds for further investigation of their magnetocaloric properties.ΔS iso ΔT ad Rare-earth vanadates (RVO 4 ) are also a particularly interesting family of magnetic oxides.They present excellent magnetocaloric properties but at very low temperatures that are not completely suitable for hydrogen liquefaction purposes. 256,277This is because their antiferromagnetic order typically occurs below 5 K (see Figure 6B), but their curves are exceptionally broad, leading to a remarkable MCE even at temperatures around 20 K, at least for high fields. 277A similar behavior is observed for .In spite of this, there are still a few compounds worth highlighting.On the one hand, HoVO 4 displays the maximum MCE around 15 K.As it is assumed that its antiferromagnetic ordering occurs at very low temperatures (<1 K), this abnormal feature might be attributed to the crystal-field-split effect of Ho 3+ ions. 277On the other hand, compounds such as TbVO 4 or DyVO 4 show excellent tablelike MCE associated to the combination of the AF ordering at 14 K with a subsequent structural transformation from the original tetragonal symmetry to another orthorhombic phase, which takes place at 33 K 257,278 leading to a large conventional and rotating MCE (see Figure 6B).Because of their large relative cooling power and good performance over a broad temperature range, these compounds are particularly interesting.
Double oxides with more complex stoichiometries, such as RM 2 O 5 , R 2 M 2 O 5 , RM 2 O 6 , R 2 M 2 O 6 (typically with double perovskite structure) or R 2 M 2 O 7 , also deserve highlights from the magnetocaloric point of view.due to their strong anisotropy, finding remarkable rotating values over 10 J kg −1 K −1 .Moreover, TbMn 2 O 5 exhibits a giant rotating MCE when rotated around the b axis, achieving temperature changes up to 18.7 K. 280 Several cuprates with similar stoichiometries (i.e., R 2 Cu 2 O 5 and R 2 BaCuO 5 ) have been investigated but they usually display inverse MCE and their performance is

REVIEW
2][283] A larger MCE response is found after substituting Cu by Zn in Dy 2 BaZnO 5 or Ho 2 BaZnO 5 284 but their transition temperatures shift to very low values.
Polar aeschynite-type oxides with the general formula of RMWO 6 attract much attention nowadays due to their exotic magnetic properties and multiferroic character.However, the MCE magnitude in the RCrWO 6 series, which has been recently studied, is small, [285][286][287][288] often presenting inverse MCE.
Compounds with the general formula of R 2 M 2 O 6 usually crystallize in the socalled double perovskite structure.This structure is obviously closely related to the original perovskite structure of stoichiometry RMO 3 but due to its double nature, it can lead to several substructures that belong to different space groups.The MCE of many RE double perovskites has been investigated 36 but most of them display transition temperatures that are too low for hydrogen liquefaction.However, there are still a few examples with remarkable properties and suitable transition temperatures.The R 2 CoMnO 6 series has been studied with more detail for R = Gd, Tb, Dy, Ho and Er. 258,289,290espite moderate responses, they present multiple phase transitions, which enables the to span over a broad temperature region between 10 and 100 K, 289 an advantage for refrigeration applications.Gd 2 CoMnO 6 possesses the largest MCE among this series, achieving a maximum over 20 J kg −1 K −1 around 10 K for an applied field of 5 T.An inverse MCE is also observed around 40 K. Interestingly, the partial substitution of Gd by Sr suppresses the inverse MCE without significantly affecting the overall magnetothermal response. 290The MCE of Tb 2 CoMnO 6 single crystals has |ΔS iso | also been investigated 258 due to their large anisotropy arising from the non-collinear alignment of Mn 4+ /Co 2+ spins, which are perpendicular to those of Tb 3+ ions.Large rotational values between 7.5 and 20 J kg −1 K −1 can be obtained for an applied field of 5 T at temperatures between 2 and 15 K (see the curves in Figure 6C).It is worth noting that R 2 NiMnO 6 double perovskites show similar features, but their transition temperatures are lower than those of their Co counterparts. 291Other related compounds are R 2 FeCrO 6 with R = Er and Tm, which show moderate magnetothermal responses. 292

ΔS iso
Pyrochlores with general stoichiometry of R 2 M 2 O 7 and a distinctive cubic crystal symmetry are an important class of materials with diverse technological applications that might exhibit interesting magnetic behavior. 293owever, their magnetocaloric properties have not been explored in depth and only a few examples can be found in the literature.For instance, Dy 2 TiMnO 7 , Ho 2 TiMnO 7 and Er 2 Mn 2 O 7 behave as standard ferromagnets with SOPTs.The first two compounds display moderate magnetothermal responses with relatively large refrigerant capacities and Curie temperatures around 7.5 K. 294 On the other hand, Er 2 Mn 2 O 7 , whose Curie temperature is 34 K, presents very broad curves with a maximum absolute value as large as 16.1 J kg −1 K −1 for 5 T and a remarkable refrigerant capacity of 522 J kg −1 , 295 among the largest values of magnetocaloric materials with similar transition temperatures.It is likely that there are other pyrochlores with excellent MCE that have not been investigated.
If we go further in structural complexity, we will find magnetic oxides, such as spinels, complex borates derived or inspired from the mineral gaudefroyite, KEr(MoO 4 ) 2 or a large number of garnets.
|ΔS iso | Spinel compounds, which have been previously discussed, serve as another feasible magnetocaloric material free of RE elements.X. Luo and coworkers reported on MnV 2 O 4 , 296 a spinel showing ferrimagnetic ordering below 57 K and a maximum value as large as 24 J kg −1 K −1 for 4 T. Subsequently, several studies address various partial substitutions in Mn or V sites, [297][298][299] and reveal the complex nature of the magnetism in this compound.In detail, two successive phase transitions occur near to each other, attributed to the ferrimagnetic ordering and to the t 2g orbital ordering of V 3+ ions at a slightly lower temperature. 298audefroyite (Ca 4 Mn 3 O 3 (BO 3 ) 3 CO 3 ) contains chains of edge-linked MnO 6 octahedra on a Kagome lattice, which results in magnetic frustration between the chains and overall spin-liquid behavior below 9 K. Gaudefroyite itself displays a remarkable conventional MCE 300 but, more interestingly, it is possible to synthesize similar compounds, such as YCa 3 Mn 3 O 3 (BO 3 ) 4 or GdCa 3 Mn 3 O 3 (BO 3 ) 4, with better magnetocaloric performance and tunable transition temperature.300 It is likely that Y could be replaced with other rareearth.Additionally, related compounds with simplified structures inspired by gaudefroyite have been produced.This is the case for Pb 1-x Sr x MnBO 4 with x = ) -∆T iso,R (K)  255 (B) unit cell for RVO 4 oxides with R=Gd (upper panel) 256 and rotational response up to 5 T for R=Tb (lower panel), 257 (C) rotational effect for Tb 2 CoMnO 6 for 5 and 7 T (upper and lower panel, respectively) 258 and (D) rotational and (upper and lower panel, respectively) for KEr(MoO 4 ) 2 . 259(All figures are reproduced with permission of AIP and APS.) 0, 0.5 and 1, 301 which exhibit moderate magnetothermal responses over a broad tunable range of transition temperatures that can be adjusted from 15 to 30 K by varying the Pb/Sr ratio.Other similar borates were reported with better magnetocaloric responses but with much lower transition temperatures. 302On the other hand, complex oxides like KEr(MoO 4 ) 2 have proven to be excellent candidates for cryogenic magnetic refrigeration 259 due to their large of 14 J kg −1 K −1 around 10 K for 5 T but, more importantly, the strong magnetic anisotropy in their single crystal.This is because even for moderate fields of 2 T, a remarkable value of 10 J kg −1 K −1 is obtained with a simple rotation of the single crystal within the ab plane, as shown in Figure 6D.
Lastly, garnets are nesosilicates having the general formula of X 3 Y 2 (SiO 4 ) 3 , being X and Y divalent and trivalent metallic cations.In a more general context, silicon can be replaced by other elements, such as As, V, Fe, Al, etc. Their magnetocaloric properties are well-known from the early days of magnetocaloric research. 303Due to the large values found at 5 K in gadolinium gallium garnet (Gd 3 Ga 5 O 12 ), also known as GGG, it has been accepted as the standard magnetocaloric material in cryogenic magnetic refrigeration for a very long time. 304Other rare-earth gallium and aluminum garnets display a similar MCE, showing an overall antiferromagnetic ordering. 303Unfortunately, the few recent attempts to find garnets with higher transition temperatures lead to poor MCE. 305,306

RARE-EARTH AMORPHOUS AND HIGH ENTROPY ALLOYS
This last section is devoted to those materials with lack of long-range order and therefore, cannot be considered as crystalline.Very few amorphous alloys had been investigated for MCE, according to early review papers from as early as 2000.Most of the alloys were prepared by rapid solidification with rare-earth elements constituting majority of the alloy composition, typically at least more than 55 atomic %, balanced with transition metal elements.The latter procedure, from a different perspective, dilutes the lanthanide metal, resulting in the broadening of the MCE behavior with very low values though with large temperature spans.In 2006, two related reports, the MCE of Finemet-type alloy 307 and the discovery of universal curves for the field dependence of , 308 sparked the transition of rareearth-based amorphous alloys to transition metal-based compositions, marking a significant turning point for this class of materials.In addition to increased annual publication numbers (see the Web of Science survey presented in Figure 7A), amorphous magnetocaloric alloys show an exponential increase in citations after 2006.Recently, the focus of the work expanded to include amorphous high-entropy alloys (HEA), which use a revolutionary concept for material design, focusing on the central area of multiprincipal elements phase diagram (see Figure 7B).Instead of alloying elements to dilute one (or two) base constituent(s) as in case of conventional alloy development (located at the corners of the model ternary phase diagram), the HEA design concept utilizes a blend of several principal elements for a large configuration entropy of mixing (ΔS mix ) and thus are dubbed the "high-entropy alloy" name.Their mechanical properties have been found to outperform conventional alloys, thereby underscoring the potential of HEA design to drive the discovery of new materials as well as magnetocaloric materials with optimal mechanical stability. 309nitial HEA were restricted to five or more principal elements in equimolar concentrations with the aim of obtaining single-phase metallic solid solu-tions.In recent years, HEA have evolved to include both intermetallic and ceramic compounds, even with less than five principal elements and in nonequimolar compositions (for further information, readers are recommended to refer to refs. 310,312,313 mix Magnetocaloric HEA with amorphous structures are generally designed following the early HEA concept, namely five or more principal elements in equiatomic concentrations, except their atomic radius and enthalpies of mixing ( ) are largely dissimilar.The reason for this is to form the ΔS iso ΔS iso ΔS iso desired amorphous structure not just through rapid solidification.Hence, the blend of principal elements involves rare-earth and transition metal elements within the threshold predicted for formation of bulk metallic glass in the HEA space: atomic radius difference and . 314The common selections are Gd, Dy, Ho, Er, Tm and Tb for RE sites while Fe, Ni, Co, Cu and Al at the transition metal sites.Despite that 60% of the principal elements are RE elements (e.g., three out of five are RE elements in quinary HEA), the RE and transition metal (TM) blend has qualitative properties that are very similar to those of the amorphous conventional alloys mentioned earlier.As a result, amorphous magnetocaloric HEA exhibit broad magnetic ordering transitions and smeared-out MCE behavior (Figure 7C), which are not competitive to conventional magnetocaloric materials with high-performance (see Figure 7D).The same applies to the quaternary RE-TM HEA compositions.In general, they were reported showing spin-glass behavior followed by ferromagnetic-paramagnetic transition and undergo a SOPT.6][317][318][319][320] Among the magnetocaloric HEA reports in the cryogenic range, it is worth highlighting as-rolled FeCoCr x Ni HEA, a RE-free composition that could be tuned to lower transition temperatures in the cryogenic range while increasing upon heat treatment when x ≥ 1.05 (e.g., for x = 1.15, transition temperature decreased from ~41 to ~33 K and at the same time with a ~9% increase in upon heat treatment). 321ecent targeted property searches in the non-equiatomic HEA space have revealed that magnetocaloric HEA perform competitively or comparably to high-performing conventional magnetocaloric materials. 311,322Recent review on magnetocaloric HEA reveals that it is possible to search for improved MCE in the non-equiatomic space, 310 even though the current available reports are for temperatures above cryogenic applications.Nonetheless, these results provide insight into applying the HEA design concept for developing magnetocaloric materials with the best possible magnetocaloric and mechanical properties.This also applies to amorphous alloys (low lattice heat capacity) whose transitions are in the cryogenic temperature range.

CONCLUDING REMARKS AND OUTLOOK
In this review, the magnetothermal responses of over 400 cryogenic magnetocaloric materials have been systematically analyzed and categorized, focusing on hydrogen liquefaction applications.This is a crucial step to further guide research on this topic.We have classified the known magnetocaloric materials into different families mainly depending on their chemical composition, finding that each group contains promising materials with some advantages and drawbacks.We have also highlighted families of compounds that are significantly disregarded in the literature, despite their promising potential for magnetocaloric applications.
In addition to this classification, the potential for actual technological applications significantly depends on economic issues.In particular, criticality is a key factor towards the sustainability of any application that aims to be implemented at a large scale.As we have shown, there is extensive literature on the magnetocaloric response of materials, but criticality is usually disregarded in most of the papers.This concept, originally developed for metals, aims to discern among materials that are the most appropriate for a given industrial production process based on the evaluation of three key factors: supply risk, vulnerability to supply restriction, and environmental implications. 323To put the potential of all the presented materials in the proper context, we have employed the supply risk index (SRI) for the different constituents according to the European Commission. 324This index estimates the risk of disruptions in the supply considering factors such as the supply concentration, import reliance, governance performance measured by the World Governance Indicators, trade restrictions and agreements, existence, and criticality of substitutes.The greater the index, the more critical is the material.The evaluated materials typically range from values close to 0.3 up to 6.1, being considered critical if the index is above 1.For estimating the SRI of the alloy, we weighed the index of each constituent by its mass fraction in the composition.
Figure 8 presents, for each of the five major families considered in this ΔS iso ΔS iso review, the transition temperature, the magnetocaloric response ( ), which determine the location of the data in the figure, together with the SRI metrics, represented by the size of the sphere and its color.To make the materials selection easier, compounds with lower SRI (less critical) appear more to the front, while the larger SRI materials are positioned behind.In this way, less critical materials will be more visible and block the view of less desirable materials for the same and transition temperature ranges.We can see that most of the materials considered are from medium to high criticality, except for the two RE-free spinels.Apart from these cases, the remaining compounds are constituted by lanthanides, often accompanied by critical TM like Pt, Rh, Pd, etc.It is also possible to find other combinations with good MCE performance using less critical metals like Ni, Cu, etc.; however, many of those families exhibit medium criticality.It is therefore not necessary to choose highly critical materials for this specialized application.
Under this scenario, it might seem challenging to find new materials with exceptional magnetic properties that outperform those of the already known compounds, but it is not impossible.In this regard, the comprehensive information provided in this review may constitute a platform to search for new compounds.The recent discovery of the giant MCE in HoB 2 204 by machine learning demonstrates the value of computational screening methods for locating novel compounds with specific target features.The information provided in the supplementary material of this review can be used as a source of input data for such targeted search methodologies or as a reference for more conventional research studies.For example, most of characterization studies are devoted to Gd-Tm compounds while less attention has been paid to Eu compounds despite often showing excellent MCE properties. 69,211,235,247There are also other families with significant potential to become useful magnetocaloric materials in this temperature range but are currently understudied, like the case for RM 2 Laves phases with M different from M = Co, Ni and Al, spinels, complex oxides or RE borides.
Besides, there is still the challenge of assessing the actual performance of these materials for a particular target application, including the simultaneous use of several materials or composites constituted by multiple phases. 325The latter has been repeatedly suggested but only a few works performed proper characterization 326,327 for the target temperature range.Moreover, their properties have not been systematically studied for best material combinations.

Figure 2
Figure2.Classification of the cryogenic magnetocaloric materials analyzed in this review For each family of material, with its corresponding importance in the scientific literature, the maximum magnetocaloric response, the temperature range where these materials have a significant response, and the criticality range are shown.The calculation of criticality is explained in the last section of this review.

Figure 3 .
Figure 3. Magnetocaloric response for some promising binary intermetallic materials as well as some examples of representative crystal structures (A) for 5 T for Gd x Er 1-x Ga solid solutions (reproduced with permission of AIP), 68 (B) for 7 T for Eu 2 In together with its cell, 69 (C) and for 5 T (upper and lower panel, respectively) for several RM 2 phases 70 and (D) crystal structures of orthorhombic α-(left) and monoclinic β-Gd 5 (Si 2 Ge 2 ) (right) (reproduced with permission of APS).71 R 5 Ni 2 In 4 175 and R 2 Cu 2 In 176-178 with ZrNiAl-type, Lu 5 Ni 2 In 4 -type and Mo 2 FeB 2 -type crystal structures, respectively.EuM 5

Figure 6 .
Figure 6.Conventional and rotary magnetocaloric response for some promising oxides (A) Conventional up to 7 T for RCrO 4 (R=Gd and Ho),255 (B) unit cell for RVO 4 oxides with R=Gd (upper panel)256 and rotational response up to 5 T for R=Tb (lower panel),257 (C) rotational effect for Tb 2 CoMnO 6 for 5 and 7 T (upper and lower panel, respectively)258 and (D) rotational and (upper and lower panel, respectively) for KEr(MoO 4 ) 2 .259(All figures are reproduced with permission of AIP and APS.)

Figure 7 .
Figure 7. Magnetocaloric amorphous materials (A) Web of Science literature survey on amorphous magnetocaloric annual publications (left y-axis; bar graph) and citations (right y-axis; circles).An exponential increase fits the citation after 2006.Search terms include "amorphous" and "magnetocaloric" in the title field.(B) Contour ΔS mix plot of a model ternary alloy where the conventional alloy compositions are found at the corners.ΔS mix values increase to maximum (at the center of the plot) where the elements are in equimolar concentrations.(C) An example of broad MCE observed in amorphous magnetocaloric HEA.(D) MCE comparison reveals that the amorphous HEA are not competitive to high-performance conventional magnetocaloric materials and the HEA sought by targeted-property search (bluish region).Images B, C, D are reproduced licensed under an open access Creative Commons CC BY license from refs respectively.310,45,311