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Review

Viable Materials with a Giant Magnetocaloric Effect

by
Nikolai A. Zarkevich
1,* and
Vladimir I. Zverev
2
1
Ames Laboratory, U.S. Department of Energy, Ames, IA 50011, USA
2
Physics Department, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia
*
Author to whom correspondence should be addressed.
Crystals 2020, 10(9), 815; https://doi.org/10.3390/cryst10090815
Submission received: 3 September 2020 / Accepted: 12 September 2020 / Published: 15 September 2020
(This article belongs to the Special Issue Magnetocalorics)
Browse Figures
Versions Notes

Abstract

:
This review of the current state of magnetocalorics is focused on materials exhibiting a giant magnetocaloric response near room temperature. To be economically viable for industrial applications and mass production, materials should have desired useful properties at a reasonable cost and should be safe for humans and the environment during manufacturing, handling, operational use, and after disposal. The discovery of novel materials is followed by a gradual improvement of properties by compositional adjustment and thermal or mechanical treatment. Consequently, with time, good materials become inferior to the best. There are several known classes of inexpensive materials with a giant magnetocaloric effect, and the search continues.

Graphical Abstract

1. Introduction

A change of temperature T can be converted to electricity [1]. A significant fraction of consumed electricity is used for heat pumping, which results in a targeted cooling or heating [2]. Magnetocaloric materials have a growing role in facilitating these energy transformations [3,4,5].
Under adiabatic condition, when there is no heat exchange between a material and the environment, a change of internal entropy ΔS of a material by an external stimulus is compensated by a change in its temperature ΔT. This phenomenon is called the caloric effect, which is typically the largest at a phase transition. Examples of phase transitions include liquid-gas (evaporation), solid-liquid (melting), and solid-solid transformations—the last ones include structural, magnetic, and magnetostructural transitions in solids. The magnetocaloric effect (MCE) consists in a reversible change of temperature of a magnetic solid upon exposure to a varying external magnetic field [5,6,7,8,9,10,11,12].
Liquid-gas phase transitions are used in vapor-compression refrigeration [13]. A solid-liquid phase transition in isentropically compressed Helium-3 is used for Pomeranchuk cooling to sub-Kelvin temperatures [14,15]. Solid-solid phase transitions in magnetocaloric materials can be used for energy transformations: for cooling and heat pumping [10,16,17] at various T, including cryogenic and room temperature (RT), as well as for generating electricity from a changing temperature [1,18]. There is a hope that use of caloric materials will lead to a more cost-effective, energy-efficient, and environmentally friendly alternative to the traditional vapor-compression refrigeration [3].
A magnitude of the magnetocaloric effect (in terms of an isentropic temperature change ΔTS or an isothermal entropy change ΔST) depends on a material and an applied magnetic field H. Typically, magnetocaloric effect is maximal at the phase transition between two states with different net magnetization and different total entropy, which combines electronic (including magnetic) and lattice (phonon) contributions [19]. There is no strict definition of a giant magnetocaloric effect, which is giant relative to that in known materials. The number of known materials (composed from a finite set of chemical elements, see Figure 1) and the largest known MCE increase with time, see Figure 2.
Magnetocaloric materials are magnetic: they have constituting atoms with non-zero atomic magnetic moments Ma. In a nonmagnetic (NM) state, atomic magnetic moments are zero and there is no net magnetization. A ferromagnetic (FM) state is characterized by a magnetic long-range order (LRO) of the parallelly aligned atomic magnetic moments Ma and a net magnetization M ≠ 0. In a ferrimagnetic (FiM) state with LRO, antiparallel alignment of atomic magnetic moments with different amplitudes also results in a nonzero net magnetization. Zero net magnetization is obtained from an antiparallel alignment of nonzero atomic magnetic moments with the same amplitudes (but opposite directions) in an antiferromagnetic (AFM) state with LRO. A paramagnetic (PM) state has a zero net magnetization due to orientationally disordered directions of nonzero atomic magnetic moments, which may have short range order (SRO), but no LRO. Sometimes, a PM state is approximated by a homogeneously random uncorrelated distribution of disordered local moments (DLM) without SRO and without LRO. Orientations of atomic moments dynamically change with time in a PM state, which differs from a spin glass with a static magnetic structure.
Dynamic orientational disorder of nonzero atomic magnetic moments results in a nonzero magnetic entropy, which is large in a PM state. In contrast, magnetic entropy is negligible in the LRO states with ordered magnetic alignment (FM, FiM, AFM, and NM).
The MCE is characterized by the isentropic temperature change Δ T S , which increases with the isothermal change of the total entropy Δ S T . A large MCE is typically observed in magnetic or magnetostructural phase transitions between a highly magnetic (FM or FiM) and a weakly magnetic (PM or AFM) states. The change of magnetic entropy is large in a transition between the PM state and any state with magnetic LRO (FM, FiM, or AFM). However, magnetic contribution to the total entropy competes with electronic and lattice contributions. In theory, the total entropy change Δ S T is expected to be maximal in a magnetostructural phase transition between FM and PM states in a material with large atomic magnetic moments, if magnetic, electronic, and lattice entropy changes sum in Δ S T with the same signs and sufficient amplitudes. In practice, if any particular contribution to Δ S T is large, then the MCE can be large even if the other contributions are small. For example, in spite of a negligible difference in magnetic entropy between the FM and AFM states (both have LRO and a small magnetic entropy), the difference in the total entropy can be large due to a dominant contribution of electronic entropy [20], which is proportional to the total electronic density of states (DOS) at the Fermi energy EF [19].
History of a giant magnetocaloric effect near room temperature probably starts from the theoretical prediction [20] of a giant change of electronic entropy in the isostructural metamagnetic phase transition between the FM and AFM states in FeRh, followed by direct experimental observation [21] of a drop in temperature in a quenched near-stoichiometric Fe49Rh51 sample, later confirmed by other measurements [22,23,24]. The prediction [20] was based on the earlier measurements [25,26] of magnetic properties and unpaired spin densities in the Fe-Rh alloys.
In spite of an implementation of the caloric cooling using the second-order phase transition in metallic gadolinium [27], followed by the observation of a large MCE in expensive FeRh alloy containing precious rhodium, a broad interest remained dormant until the discovery of a giant MCE in Gd5Si2Ge2 [28], see Figure 2b,d. After an extensive search [29], other materials with a giant MCE were found. Most of them contained critical, rare, or toxic elements, see Figure 2c. However, to be viable for industry, magnetocaloric materials should be made from inexpensive, abundant, and safe elements [30,31].
The cost of a material combines prices of its constituents, manufacturing, shipping, and handling, and can be further increased by taxation and human greed. With caution, estimated prices of chemical elements can be used to evaluate the lower bound on the cost of a materials. The market price is expected to form as a balance between supply and demand. Chemical composition is not the only factor determining the price. For example, cheap coal and expensive diamonds are composed predominantly by carbon, a chemical element which is quite common, see Figure 1.
Cheap and abundant materials are typically made from abundant elements. In general, more abundant elements are mined in larger quantities. Global production of chemical elements correlates with their abundance in the upper crust, see Figure 2a. For the profitable production of a material, its market price should be higher than the sum of the prices of constituents, cost of manufacturing, processing, shipping and handling, and added expenses due to a monopoly in mining or distribution, as well as political risks (the last are significant for the rare earth). Poisonous or toxic materials are more expensive to handle during the whole life cycle of materials and products.
Magnetic materials contain atoms or ions with nonzero atomic magnetic moments Ma. Magnetocaloric materials with a large MCE typically contain a large atomic fraction of magnetic ele-ments, such as f-metals (Gd, Tb, Dy, Ho, Er, Tm, Eu, Sm) or d-metals (Mn, Fe, Co, Ni, Cr) with a partially filled f or d band, see Figure 1. Magnetization in the FM state and magnetic entropy in the PM state correlate with the amplitudes of atomic moments Ma, which are large near half-filling of f or d band (e.g., in Gd or Mn in a neutral state). Electronic entropy correlates with electronic DOS at EF, which is large near ¼ or ¾ filling of f or d band.
Thus, elemental abundance, toxicity, safety, electronic structure, and magnetism impose constraints on composition of magnetocaloric materials, viable for industry. For example, most of the f-metals (except for the relatively cheap La and Ce) are rare critical materials, and their use in mass production is undesired. Due to toxicity, undesired are As and Sb.
A material is economically viable if it is available (people can make it), affordable (can be obtained for a reasonable price), machinable (can be processed using existing equipment), non-toxic, ecological, and (most importantly) useful. Historically, the first discovered materials with a large MCE contained expensive or rare elements. Although scientifically interesting [32,33], they were not economically viable. It was a challenge to discover magnetocaloric materials made of abundant and non-toxic elements [30,31]. For example, in the CaloriCool approach [3,29] over 104 (ten thousand) magnetic phase transformations were rapidly screened and less than 102 (hundred) systems were pre-selected, out of which <10 were labeled as the most promising, but only two materials were patented [34,35]. The preferable choice of compositions was based on elemental abundance, shown in Figure 1. Disregarded were compositions containing critical or toxic chemical elements, as well as gas-emitting hydrides. Recent discoveries [30,31] of a giant MCE in materials without critical or toxic elements enable production and use of such materials in large quantities.

2. Materials with a Giant MCE

How large should be a “giant” MCE? One can claim a giant effect if it competes with or exceeds that in known materials. In 1997, the MCE in Gd5Si2Ge2 [28] was giant, because it exceeded MCE in the known at that time [36] rare metallic Gd [27,37] and precious FeRh [21], see Figure 2d.
There are thousands of magnetocaloric publications, including hundreds of those mentioning a giant caloric effect, see Figure 2b. However, most considered materials are not economically viable, because they contain precious metals (Rh, Pd) [21,22,24], rare earth [12,38,39,40,41], expensive germanium [42,43] or gallium [28,44], toxic elements [45,46,47,48,49,50,51], or hydrogen [52,53,54,55], which is released in a gas phase, see Figure 2c. Before 2005, a large MCE (exceeding that in metallic Gd) was claimed in precious FeRh [21,22], expensive Gd5Si2Ge2 [28,56,57,58,59,60,61,62,63,64,65,66,67,68] and related materials (GdGe4 [69,70], GdSn4 [71,72], Tb5Si2Ge2 [73]), Heusler alloys [74] containing Ga, partially hydrated La(Fe,Si)13H1.5−x [52,53], toxic MnAs1−xSbx [45,46,47,75], MnFeP0.45As0.55 [48,49], Mn1.82V0.18Sb [50], and other materials [76,77,78,79].
Table
Table 1. Composition (with critical or undesired elements, if any), caloric response ΔST (J kg−1 K−1) and ΔTS (K) at 2 Tesla magnetic field change, phase transition Tc (K), and hysteresis width ΔTh (K).
Table 1. Composition (with critical or undesired elements, if any), caloric response ΔST (J kg−1 K−1) and ΔTS (K) at 2 Tesla magnetic field change, phase transition Tc (K), and hysteresis width ΔTh (K).
CompositionΔSTTc (K)ΔTS (K)ΔTh (K)CriticalRef.Year
Ni37.5Co12.5Mn35Ti1527290 -[30]2019
Ni36.3Co13.7Mn35Ti1518260 -[30]2019
Ni35Co15Mn35Ti155180 -[30]2019
Mn0.5Fe0.5NiSi0.95Al0.0515.5316.5 -[31]2019
Mn0.5Fe0.5NiSi0.945Al0.05519.5287 -[31]2019
Mn0.5Fe0.5NiSi0.94Al0.0620.5268 -[31]2019
Mn0.5Fe0.5NiSi0.935Al0.06522243 -[31]2019
Mn0.5Fe0.5NiSi0.93Al0.0722208 -[31]2019
Mn0.5Fe0.5NiSi0.94Al0.06B0.0522283 25 1-[31]2020
Ni44.5Mn37Fe6.7Sn11.85.6293 -[80]2010
Ni44.2Mn39.3Fe4.9Sn11.66.4284 -[80]2010
Ni44.4Mn44.1Sn11.510.6260 -[80]2010
Ni50.2Mn35.0In14.818301 1.23.6In[81]2016
Ni49.6Mn35.6In14.87.5268 1.67.9In[81]2016
Ni45.7Mn36.6Co4.2In13.510.52893.010In[82]2015
Ni55.4Mn20Ga24.6403131.58Ga[74]2005
MnFeP0.45As0.5514.5300 1 1As 4[50]2002
MnAs38315 As 4[45]2001
La0.7Ca0.3MnO35.72602.3-0-[83]2019
La(Fe,Co,Al)13-0–400---[84]1992
LaFe11.4Si1.614 1209 -[85]2000
LaFe11.4Si1.614.3208 -[86]2001
LaFe11.4Si1.614.2199 -[87]2009
LaFe11.7Si1.32518842-[88]2005
LaFe11.2Co0.7Si1.112.52742.42-[88]2005
LaFe11.12Co0.711Al1.174.6279 -[85]2000
La(Fe0.96 Co0.04)11.9 Si1.116.4243 -[89]2005
La(Fe0.88 Co0.12)13 H1.5193236.81 1H 5[52]2003
La(Fe0.89 Co0.11)13 H1.3242916.91 1H 5[52]2003
La(Fe0.9Co0.1)13 H1.1282877.11 1H 5[52]2003
Fe49Rh51 3 313−12.98.5Rh 6[21]1990
Fe49Rh5113 1317 −9.2 210 1Rh 6[24] 2016
Fe49Rh5113.9 Rh 6[23]2020
Gd5Si2Ge218.42624.95Gd,Ge[28]1997
Gd5Si2Ge2152775 1 Gd,Ge[63]2003
Gd5Si2Ge2 3272707 Gd,Ge[63]2003
Gd5Si44.2346 Gd[90]2001
Gd6 12934 1−0Gd[27]1976
Gd5.62945 1−0Gd[38]1998
Gd5.22924.7−0Gd[91]2019
1 Approximate value. 2 ΔTS reduces from 9.2 K to 6.2 K during cycling in field of 1.9 Tesla. 3 Heat treated. 4 Arsenic is toxic. 5 Hydrides emit H2 gas. 6 Rhodium is precious.
Known materials made of abundant and non-toxic elements include those with body-centered cubic (bcc) A2, B2 [30], or Heusler structures [80,92,93,94,95,96,97,98,99,100,101,102,103] (with or without a possible tetragonal distortion), Mn0.5Fe0.5NiSi1−xAlx [31] with TiNiSi-type orthorhombic to Ni2In-type hexagonal phase transition, LaFe13−xSix [87,104,105] with the NaZn13-type 1:13 phase (Figure 3), a subset of magnetic shape memory alloys (SMA) containing Ni and Mn, manganites based on La0.7Ca0.3MnO3, etc.
The MCE depends on crystal structure. A subset of crystal structures of giant magnetocaloric materials is shown in Figure 3. Importantly, materials with a close-packed structure, such as the face centered cubic (fcc) or hexagonal close-packed (hcp), are less likely to show a giant MCE, because atomic magnetic moments Ma become smaller at a smaller volume per atom. A suppression of MCE by anisotropy was reported for metallic Gd [106]. In general, materials with larger Ma are more responsive to a changing external magnetic field. Continued increase of diversity of considered structures is an important task of materials discovery.
The MCE depends not only on composition and structure, but also on atomic ordering, which is controlled by thermal history of a sample. For example, a quenched FeRh sample showed a larger MCE than an annealed one [21]. A dependence of MCE on atomic LRO was observed in B2 Ni-Co-Mn-Ti alloys [30] and other materials [107]. Figure 3a,b shows binary B2 and quaternary Heusler structures decorating the same bcc lattice; these structures differ by atomic ordering. Typically, experimental samples with the Heusler structure are not fully ordered. Figure 3d,e shows the low-T orthorhombic and the high-T hexagonal structures of Mn0.5Fe0.5NiSi1−xAlx [31], which has atomic disorder on (Mn,Fe) and (Si,Al) sublattices. The MCE is a function of many parameters, which include composition, structure, atomic ordering, temperature, and applied external fields. Due to the multi-dimensional parametric space, direct comparison of MCE in different materials can be tricky [108].
There is a significant progress in discovery and understanding of materials with a giant magnetocaloric effect [19,24,30,31,39,43,75,79,81,82,85,92,93,94,95,96,97,98,99,100,107,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294], see Figure 2. Many among the considered materials are scientifically interesting [295], but not economically viable. A few contain precious Rh (e.g., FeRh [24,222,278,279] and Tb3Rh [270]). Many contain critical rare earth, such as Eu in Eu2In [140], EuTiO3 [114,135,156,195] doped with Al [113], Cr [196], Mn [134], Co [151], Ni [119], Ba [157,206], Nb [110,170,195] and Eu in other compounds [133,232,240,251]; Gd in GdAlO3 [158], GdFeO3 [138,162], GdCrO3 [183], GdCrO4-ErCrO4 [202], GdScO3 [117,124], Gd2CoMnO6 [155], Gd2BaNiO5 [152], GdCrTiO5 [143], GdCo2B2 [271], GdCoC2 [171], RuSr2GdCu2O8 [208], etc.; Tb in TbFeO3 [176], TbMn2O5 [179], Tb2CoMnO6 [132], Tb5Ge2−xSi2−xMn2x [255], Tb4Gd1Si2.035Ge1.935Mn0.03 [197], etc.; Dy in DyAl2 [275], DySb [263], DyNi2B2C [250], Dy0.9Tm0.1Ni2B2C superconductor [272], DyNiSi [181], DyCuSi [266], DyFeO3 [199], DyCrO4 [230], DyVO4 [209], etc.; Ho in thin Ho films [248], HoMn2O5 [127,221], Ho3Al2 [237], HoCuAl [224], Ho12Co7 [223], HoCoGe [166], HoCoSi [235], HoCuSi [268], HoGa [267], Ho2PdSi3 [194], Ho5Pd2 [187], amorphous HoErGdCuNi [112], etc.; Er in ErNiBC [243], ErCr2Si2 [244], ErMn2Si2 [242], ErRu2Si2 [286], ErRuSi [234], Er2Mn2O7 [164], ErK(MoO4)2 [188], amorphous ErxCo1−x [220], etc.; Tm in TmZn [198] and metallic glasses [233]; rare earth in clathrates [277,296], etc. Manganites [258] include La0.7Ca0.3MnO3 [227,262,264,285], La0.48Ca0.52MnO3 [174], Na-deficient La0.8Na0.2−xMnO3 [185], La0.67Sr0.16Ca0.17MnO3 [207], La0.8−xKxBa0.05Sr0.15MnO3 [297], as well as containing critical rare earth La0.7−xPrxCa0.3MnO3 [160], GdMnO3 [186], Gd2NiMnO6 and Gd2CoMnO6 [193], TbMnO3 [165,252], DyMnO3 [247], PrPb manganites [294], etc. Materials containing highly toxic arsenic include MnAs [123,131,210,238,245,246,256,265,273,276,282,289] and MnFe(P,As) [48,49,239,298]. However, interest is shifting towards less toxic materials without As, such as (Mn,Fe)2(P,Si) [125,126,150] doped with Ge [109,205] or B [184,201,215]. Examples with toxic antimony include Mn1.9Co0.1Sb [211], Mn2−xCrxSb [236], NiMn0.9Sb0.1 [254], Ni0.5−xCoxMn0.38Sb0.12 [92], etc. Hydrides emit H2 gas [52,53,54,55]. A giant MCE was found in cobalt hydroxides Co(OH)2 [212,259,280]. Ni-Mn-Ga alloys [169,213,253] contain expensive Ga. Mn-Fe-Ge [287], Mn-Ni-Ge [121] and Mn-Co-Ge [116,129,168,173,178,182] contain critical Ge [231]; these alloys can be doped with In [141], Si [159] and Fe [111,120,142]; Ge was successfully substituted by (Si,Al) [31]. Magnetic shape memory alloys [148,204] include Ni-Mn-In [167,175,192,261,283] doped with Co [136,172,241] and Si [122]; Ni-Mn-Sn [115,226]; NiMn-based B2 (Ni-Co)(Mn-Ti) [30], Ni-Co-Mn-Al films [189,190] and Heusler Ni40Co10Mn36Al14 [100]; Ni-Co-Mn-Sn alloys, such as Ni40Co10Mn41Sn9 [146], Ni40Co10Mn40Sn10 [214], Ni42.8Mn40.3Co5.7Sn11.2 [219], Ni40Co10Mn40Sn9Al1 [144], and Ni42−xTixCo9Mn39Sn10 [153]. A large MCE was found in amorphous Fe78−xCrxSi4Nb5B12Cu1 [281], Fe73.5−xCrxSi13.5B9Nb3Cu1 [293], and metallic glasses [274]. Significant interest was devoted to the potentially viable LaFe13−xSix [137,200,284,288,290,292], doped with Co [249] and other additives [35], or partially hydrated [52,53,54,55]. For completeness, we mention Mn3CuN1−xCx [147] and molecular magnets [128].
One can see trends towards increasing MCE, replacing toxic elements, and eliminating expensive and critical elements, leading to a cost reduction of the best-in-class materials [30,31,296]. Discussed applications of magnetocaloric materials include cooling and heat pumping [2,3,4,5,6,7,8,299,300,301], as well as energy generation [1]. MCE can be used for the thermo-magneto-electric energy transformations, such as conversion of energy into a temperature change [3,4,5,6], or conversion of a changing temperature [1] into a changing magnetic field, which generates voltage in a coil [18].
One can use thermodynamic estimators [19] for systematic screening of caloric materials [29]. It is convenient to store properties in a database [302]. Properties of the best-in-class materials are compared in Figure 2c. An historic timeline is presented in Figure 2d. Table 1 augments data from the reviews [83,303]. Next, we proceed to consideration of the properties and relations among them.

3. Properties

3.1. Thermodynamic Relations

The caloric effect results in heat generation or absorption, when an external field is applied, and the subsystem changes its entropy. In the adiabatic process, which happens without transfer of heat or mass between a thermodynamic system and its surrounding, this leads to an isentropic change in temperature by ΔTS. At adiabatic switching-off of a magnetic field there is a demagnetization of a ferromagnetic material, i.e., destruction of a magnetic order that leads to increase of magnetic entropy. This, in turn, leads to an adjustment in the lattice temperature by ΔTS, because the process of magnetic order destruction (demagnetization) in the subsystem of magnetic moments requires energy, which is supplied by a crystal lattice. Thus, during adiabatic magnetization and demagnetization of the substance, there is a reversible process of heat transfer from the magnetic subsystem to the lattice and vice versa. Thus, MCE is the result of an entropy change in magnetic subsystem under the influence of an applied external magnetic field. The total entropy S of a magnetic material, in which magnetization is formed due to localized magnetic moments, can be represented as the sum of the lattice SL and electronic (including magnetic) parts S e = S e l + S M —the last includes magnetic contribution SM. At constant pressure P, all three components are functions of temperature T [5],
S ( T , H ) P = [ S L ( T , H ) + S e l ( T , H ) + S M ( T , H ) ] P .
Among them, magnetic entropy is highly dependent on the magnetic field H, while “pure” electronic Sel and lattice SL contributions are usually almost independent on the field H near room T.
Under adiabatic conditions (in an isentropic process) the total entropy is not changed, therefore
Δ S M ( T , H ) P = − [ Δ S e l ( T , H ) + Δ S L ( T , H ) ] P .
The separation of the lattice vibrational (phonon) entropy SL is possible if one neglects the electron-phonon interactions, which have a noticeable effect on the spectrum of electronic excitations. Using the Debye temperature θD, the lattice entropy can be calculated from Debye’s interpolation:
S L = − 3 R [ l n ( 1 − e − θ D / T ) + 12 ( T θ D ) 3   ∫ 0 θ D / T x 3   d x e x − 1 ] .
If the change of the electronic contribution Sel in the magnetic field H is insignificant, then ∆SM is linked mainly to the change of the lattice entropy ∆SL. However, in itinerant magnets containing 3d metals a separation of the electronic and magnetic entropy into purely magnetic and electronic parts is difficult, because d-electrons forming local magnetic moments contribute to conductivity, and their contribution to the electronic density of states n(EF) at the Fermi energy EF is comparable with those from p and s-electrons. The electronic (including magnetic) entropy can be estimated by the Sommerfeld’s expansion:
S e ≈ ( π 3 ) k B 2 T   n ( E F )
The value of the electronic contribution to entropy can be calculated from experimental values of electronic heat capacity:
S e l = γ T ,
where γ is the electronic heat capacity factor.
Let’s consider the total entropy S of a magnetic material at constant pressure P as a function of temperature T and magnetic field H. The exact differential of the total entropy is
d S ( T , H ) P = ( ∂ S ∂ T ) H , P d T + ( ∂ S ∂ H ) T , P d H .
The Maxwell relation [304]
− ∂ 2 F ∂ H   ∂ T = ( ∂ S ∂ H ) T , P = ( ∂ M ∂ T ) H , P ,
where M is the magnetization of the system and F is the free energy, allows to estimate the entropy change ∆ST at constant T by the thermal integration:
Δ S T = ∫ H 1 H 2 ( ∂ M ( T , H ) ∂ T ) H , P d H .
The partial derivatives ∂ M ∂ T can be extracted from the processed experimental M(H,T) data.
Let us consider various contributions to MCE. In the case of a ferromagnetic materials with two sublattices (e.g., TbxGd1−x alloys at x < 0.8), MCE includes contributions due to true magnetization (paraprocess), changes in the magnetic anisotropy energy, changes in the interlattice interaction between Tb and Gd sublattices, magnetostrictive deformations, domain boundary displacements, and irreversible processes. Each contribution depends on temperature, value and direction of the external field. However, it is possible to neglect several contributions, if they are small. The ones due to displacement and other irreversible processes are not experimentally observed. Estimates show that the contribution of magnetostrictive deformations is small (especially near the Curie temperature). Contributions due to paraprocess, anisotropy, and sublattice interaction play a significant role in TbxGd1−x alloys. In the case of a ferromagnetic material, the work done to increase the magnetization of the domain M D = M S + M I (where MI is the change of the domain magnetization due to the paraprocess, MS is spontaneous magnetization), equals to d W S = − H d M D .
From the second law of the thermodynamics, d Q = T d S , and from definition of the heat capacity C = ( d Q d T ) at constant external field, C H = ( d Q d T ) H = T ( d S d T ) H , using Equations (6) and (7), one can find:
T d S = C H d T + T ( ∂ M ∂ T ) H d H .
In the adiabatic process d S = 0 , hence the isentropic temperature change [305] due to the magnetocaloric effect caused by a change of external magnetic field is
Δ T S = T 2 − T 1 = − ∫ H 1 H 2 T C H ( T , H ) × ( ∂ M ( T , H ) ∂ T ) H d H .
Here, the integration is from H1 to H2; H1 is the initial field at which the initial sample’s temperature is T1, H2 is the final field at which the final sample’s temperature is T2, and C H is the heat capacity at constant external fields. In general, the MCE due to the paraprocess in fields lower than those of technical saturation is extremely small. As a rule, a significant MCE is observed in magnetocaloric materials at field values that exceed the technical saturation field for a given material, and therefore in most practical cases the contribution due to the paraprocess is crucial. The absolute value of the magnetization derivative has an extremum at the phase transition temperature. Thus, the maximal MCE is observed near the phase transition temperature. This fact is associated with the correlations among the magnetothermal and other properties of magnetocaloric materials.

3.2. Correlated Physical Properties

Magnetothermal phenomena include the MCE, magnetic contributions to the heat capacity and entropy, as well as special parameters introduced to characterize magnetic nanoparticles in the method of magneto-liquid hyperthermia, such as specific absorption rate (SAR) and intrinsic loss parameter (ILP) [306,307]. These thermodynamic parameters quantify the key properties of a magnetic material: magnetization, heat capacity, magnetic susceptibility, etc. Knowledge of these dependencies allows to characterize a material and consider its practical applications.
From Equation (10), a field change Δ H = H 2 − H 1 causes an isentropic temperature change Δ T S that depends on both T C ( T , H ) P , H and ( ∂ M ( T , H ) ∂ T ) H . The MCE is large when ( ∂ M ( T , H ) ∂ T ) H is large and C ( T , H ) is small at the same T. The extremums of both ( ∂ M ( T , H ) ∂ T ) H and C ( T , H ) do not coincide in a general case. At an ideal first-order phase transition the derivative ( ∂ M ∂ T ) H doesn’t exist, and the listed equations are not fulfilled as they are derived from Landau theory [308] for the second-order phase transition. The mutual influence of these properties was considered in [309], with a theoretical estimate of the maximal possible magnetocaloric effect for bulk ferromagnetic materials with a second-order phase transition near the room temperature. It was demonstrated that the maximum MCE cannot exceed 18 K/T in the ideal case, which is not achievable in practice. The estimates for realistic materials provide the maximum values from 8 to 10 K/T.
From the equation for heat capacity at constant pressure C p ( T , H ) = T ( ∂ S   ( T , H ) ∂ T ) H , P and Equation (1),
d S   ( T , H ) P = C L ( T ) P T d T + C e l ( T ) P T d T + C M ( T ) P T d T + ( ∂ S M ( T , H ) ∂ H ) T , P d H ,
where CL (T) is the lattice contribution to the heat capacity, Cel (T) is the electronic (without magnetic) contribution, CM (T,H) is the heat capacity of the magnetic subsystem. Thus, one can say that the heat capacity of a magnetic material is the sum of electronic, lattice, and magnetic heat capacities. The electronic and lattice heat capacity in most materials have a negligibly weak dependence on external magnetic field, while the magnetic contribution strongly depends on the applied magnetic field and therefore determines the MCE value.
As shown in [11,310], the exact temperature at which MCE reaches its maximum in simple ferromagnetic materials should be above the temperature of the heat capacity maximum at zero magnetic field, due to the following equation:
Δ T S ( T , H ) m a x = − T 1 Δ C ( T , H ) ,
where
Δ C ( T , H ) = C ( T 1 , H ) − C ( T 2 , H ) C ( T 1 , H ) .
Here, T1 and T2 are the initial and the final temperature of the sample, respectively.
In the case of low magnetic fields and relatively high temperatures (e.g., near room temperature), Δ T S ( T 1 , H ) m a x T 1 can be neglected. Thus, Equation (13) at Tmax(H), where MCE reaches its extremum value, can be approximated:
C ( T m a x , H ) ≅ C ( T 2 , H )
Magnetic materials can have several characteristic temperatures associated with magnetic transformations [11,310]. Nevertheless, the MCE sign is directly related to the sign of dHcr/dT derivative, where Hcr is the critical field. For example, in Dy with a negative MCE in low fields the phase boundary shifts towards lower temperatures, see Figure 2 in [311].
Thus, a change in the sample temperature due to MCE cannot bring the sample to a new phase. In particular, Gd does not undergo the FM-PM transition when heated only due to the magnetocaloric effect; pure field transition in Gd is not observed experimentally, see Figure 1 in [311]. Accordingly, it can be assumed that the phase transformation shifts away from the transition temperature in zero field faster than MCE increases when the magnetic field is applied, that is, MCE cannot exceed the value of dTc/dH, where Tc is the transition temperature.
For many magnetocaloric materials, the influence of the magnetic field on the MCE maximum value in high fields is known from literature. However, the behavior of anomalies of the magnetocaloric effect, magnetization, magnetic entropy change and heat capacity (in particular, their relative locations and field shifts relative to each other) in low fields still needs further investigated. For example, the heat capacity maximum typically shifts towards higher temperatures in strong magnetic fields [38]. At the same time, it is unclear why the temperature at which the heat capacity reaches its maximum in 2 T (Figure 7 of [38]) is obviously lower than the corresponding temperature for the zero-field dependence (see Figure 4).
In [312], the shift of the heat capacity maximum is investigated by means of Landau theory, and it is shown that in the weak fields below 2.4 Tesla the maximum shifts towards lower temperatures, while in stronger magnetic fields above 2.4 T it shifts in the opposite direction.
For simple ferromagnetic materials and for the known magnetic field change it is possible to determine a specific temperature Θ(H), in the vicinity of which the MCE maximum should be observed [310]. This temperature is higher than the temperature at which the heat capacity maximum is reached in zero magnetic field. If the magnetic field decreases in the adiabatic process, the MCE maximum is close to the characteristic temperature Θ(H), at which the magnetic field has no effect on heat capacity. In general, when magnetic order differs from the LRO (FM or AFM) state, or when more than one magnetic phase transition is observed, the MCE dependence on temperature becomes more complex (additional maxima and minima appear); its functional form depends on how many new extrema have appeared and where the characteristic temperatures Θ(H) are now located.
Figure 4 summarizes the displacements of the maximum in the magnetic heat capacity of gadolinium, terbium and holmium [313,314], where T/Tpk is the ratio of the measured temperature to the temperature at which the maximum of the magnetic heat capacity in a given magnetic field is observed. The experimental data is compared with the theoretical line, taken from Figure 9 in [312]. There is a critical field value, at which the maximum begins to shift towards higher temperatures. Thus, the field-induced non-monotonic behavior of the maximum in the magnetic heat capacity is commonly expected for the rare earth metals, although there are exceptions (e.g., Tb).
Physical properties are correlated, and there is a frequent combination of outstanding magnetothermal properties, such as the caloric effect and a property of another origin. Materials like NiTi show both shape memory and caloric effect at the phase transition [315,316]. This provides an inspiration to search for a giant MCE in magnetic shape memory alloys. FeRh alloys reveal a large MCE, a significant spontaneous volume magnetostriction [317] and magnetoresistance [318], and a remarkable decrease in electrical resistivity [319] in the vicinity of the AFM-FM transition. Similar correlations are observed in Gd5(SixGe1−x)4 alloys [320]. In addition, magnetic transformations are often accompanied by an abnormal thermal expansion [139,321,322]. The caloric effect also scales with the total entropy change. Many catalysts [323] have a large electronic density of states at the Fermi energy, and thus a large electronic entropy. Electronic entropy increases with electronic DOS at EF (see Equation (4)), affecting the caloric response.

3.2.1. Multicaloric Effect

A striking example of correlated physical properties in one material is the multicaloric effect. Caloric materials have a wide range of properties, among which there are the caloric effects associated with a reversible change of entropy (under isothermal conditions) or temperature (under adiabatic conditions) due to a variation of external fields (magnetic H, electric E, mechanical stress σ, or hydrostatic pressure P). The corresponding individual caloric effects caused by a variation of a single physical field are called magneto-, electro-, elasto-, and baro-caloric [324,325,326], respectively.
A phase transition can be driven by more than one external stimulus. If more than one physical field is capable to produce a caloric response, then such effect is called multicaloric [327]. The corresponding entropy change due to a simultaneous or sequential application of several fields is
d S = ( ∂ S ∂ H ) T , P , E d H + ( ∂ S ∂ E ) T , P , H d E + ( ∂ S ∂ P ) T , E , H d P + ( ∂ S ∂ T ) P , E , H d T ,
and consequently
d S = − T C P , E , H [ ( ∂ M ∂ T ) P , E , H d H + ( ∂ p ∂ T ) P , E , H d E − ( ∂ V ∂ T ) P , E , H d P ] .
Here, M is magnetization, p is polarization, V is volume, H is magnetic field, E is electric field, P is pressure, and T is temperature.
In particular, magnetic transitions can be governed by a changing external magnetic field, while a transition between two phases with different densities (and different volume per formula unit) is affected by pressure. Also, a variable magnetic field can not only cause MCE, but also induce an electrical field, which in turn will change the polarization p and generate an electro-caloric effect. At constant entropy S, the equation for this combined effect in a variable magnetic field is [327]
d T S = − T C ( T , H ) P , E , H [ α m ε 0 χ e ( ∂ p ∂ T ) P , E , H + ( ∂ M ∂ T ) P , E , H ] d H .
Here, αm is the direct magnetoelectric interaction ratio in d E = α m ( ε 0 χ e ) − 1 d H ; ε0 is vacuum permeability; χe is electrical susceptibility.
Finding the right material for a magnetocaloric refrigerator is only a part of the task, since there is a need to generate large variable magnetic fields (with amplitudes up to several Tesla). Without water-cooled solenoids or superconducting magnets, which diminish the efficiency of the magnetocaloric cooling method, a variable magnetic field can be created by a mechanical motion of permanent magnets relative to the active caloric material. Motion produces noise and friction, resulting in wear and tear of the moving parts.
There was a suggestion [328] to avoiding such problems by using artificial magnetoelectric materials and multiferroics. According to [328], the magnetic properties of the material can be controlled using the electrical voltage applied to the piezoelectric layer rigidly bonded to the magnetic material. A multicaloric device exploits the synergistic effect that occurs in an interconnected system and amplifies the resulting caloric effect.
In general, transition temperature Tc of a magnetostructural [329] or magneto-volume phase transition depends on external magnetic field, stress, and strain. Response of a caloric material to one physical field does not disallow response to others. Thus, many caloric materials are multicaloric [330]. Multicaloric materials and effects are extensively studied nowadays [331,332].

3.2.2. Hysteresis

Most of the materials with a giant MCE undergo a first-order phase transition, whose latent heat contributes to the amplitude of the caloric effect in these materials. Hysteresis is a common feature of the first-order phase transitions [333]. A large thermomagnetic hysteresis results in an energy loss, released as a parasitic heat, which limits the usability of materials for caloric cooling [3]. Hysteresis is increased by nucleation barriers, activation energy, and strain due to a lattice misfit [334]. For structural transformations in solids, there is a connection between the hysteresis width and the middle eigenvalue λ2 of the transformation stretch matrix [333,334]. Hysteresis is a rather well studied phenomenon: its causes and the ways to overcome negative consequences are described in the review [335]. A negligible hysteresis was observed at the first-order magnetic transitions in Eu2In [140] and Pr2In [336] line compounds. However, not only the hysteresis, but also aging due to cycling in a changing magnetic field affects the amplitude ΔT of the magnetocaloric response [24,337].

3.2.3. Aging

The phenomenon of a reduction of MCE with cycling is called aging. Let us consider aging in the FeRh alloys that manifest a negative magnetocaloric effect with a maximum near room temperature [21,22,23,24], see Figure 5a,b. Typical experimental curves ΔT(H) are presented in Figure 3 in [337]. In addition to the field hysteresis (e.g., ~1.2 Tesla in Figure 5b), there is an irreversible trajectory: the sample temperature does not return to the initial value after one full cycle of the changing magnetic field. This aging effect is observed at all temperatures in the range of 300–340 K, and its value depends on T, see Figure 4 in [337]. The maximum of this effect is shifted by ~2 K compared to the peak of ΔT(T) curve. The value of ΔT in FeRh is large in the first cycle, but becomes smaller in the subsequent cycles, see Figure 5b. In the Fe50.4Rh49.6 sample, ΔT changes from ΔT1 = −7.5 K in the first cycle to ΔT2 = −3.4 K in the second one [337]. Aging is observed in many magnetocaloric materials that experience the first-order phase transition during work cycles [335].
From a practical point of view, together with hysteresis, aging lowers cooling efficiency in magnetic refrigerators using multistage magnetization/demagnetization cycles. Thus, the elucidation of the origin of irreversibility during the first cooling cycle and a drop of MCE during subsequent cycles may lead to discovery of new materials that retain their magnetocaloric properties during cycling.

3.3. Materials Life Cycle and Recyclability

Currently, most of materials are moved irreversibly from the natural resources to landfills (Figure 6). This consumption of natural resources is not sustainable and results in accumulation of waste. Material discovery alters waste compositions, while wasteless technologies are still needed.
The explosive growth of materials research results not only in the rapid increase of the number of useful materials, but also in a substantial broadening of the spectrum of their practical applications. In magnetocalorics, along with the traditional technologies for obtaining ultra-low temperatures and magnetic cooling, the biomedical applications come to the fore: the method of magneto-liquid hyperthermia treatment of malignant neoplasms and a targeted drug delivery [300,338]. However, products have their lifetime, at the end of which they need to be disposed of [339].
Life cycle of caloric materials includes extraction of elements, materials synthesis, manufacturing and assembly into a device, packaging and transportation, followed by storage, use, and disposal together with the device. After disposal, the disassembly of a device into individual components and their recycling is preferred for precious and high-cost materials. However, the cost of the magnetocaloric materials lowers with time; this makes them more suitable for mass production, but less attractive for recycling, which requires disassembly of disposed items. Most of the magnetocaloric materials are synthesized from high-purity ingredients, because a phase transition temperature and hysteresis width are highly sensitive to impurities. Reuse of old low-purity materials for synthesis of new high-purity ones is diminished due to high cost of purification.
There are established recycling methods for structural and scrap metals, magnets, and batteries. There was a suggestion of a separate recycling for materials containing critical elements, such as metallic Gd and rare earth compounds (Gd5Si2Ge). However, those materials are being replaced by others, composed by more abundant elements. A separate recycling of magnetocaloric materials is problematic due to a rapid change of compositions of the best-in-class materials, purity constraints, and a relatively low market share. Due to a diversity of magnetocaloric materials, there is no single recycling technique that fits them all. For example, precious FeRh and toxic MnAs require different handling methods and should not be mixed during recycling. Non-toxic caloric materials (such as Ni37.5Co12.5Mn35Ti15) contain metals and can be recycled together with metals.

4. Summary

The correlation between elemental abundance and mass production provides constraints on chemical compositions of materials, viable for industry. Cost is an important economic factor. Toxicity affects the cost of handling during the whole life cycle of materials and products. Properties of materials depend on composition, structure, atomic ordering, heat treatment, applied fields, and external conditions. For magnetocalorics, important properties include the MCE, hysteresis, aging, recyclability, toxicity, and cost. We considered the caloric effect and its correlations with other physical properties. A magnetocaloric effect is useful for energy transformations, with applications in refrigeration, heat pumping, and energy generation. There is a vast literature on magnetocalorics. We focused on the economically viable materials with a giant MCE near room temperature. We pointed at the best-in-class materials and constructed a timeline (see Figure 2). Diversity among the known magnetocaloric materials increases with time. Recent observation [30,31] of a giant MCE in materials composed by abundant and non-toxic chemical elements can become a game changer in commercial applications of magnetocalorics.

Author Contributions

N.A.Z. prepared the original draft and wrote Section 1, Section 2 and Section 4. V.I.Z. wrote most of Section 3. Both authors have agreed to publish. All authors have read and agreed to the published version of the manuscript.

Funding

Basic research at the Ames Laboratory is funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Caloric materials discovery is supported by the U.S. DOE, Advanced Manufacturing Office of the Office of Energy Efficiency and Renewable Energy through CaloriCool™-the Caloric Materials Consortium established as a part of the U.S. DOE Energy Materials Network. Ames Laboratory is operated for the U.S. DOE by Iowa State University under contract DE-AC02–07CH11358.

Acknowledgments

Anis Biswas, Yaroslav Mudryk, Andrey Smirnov, Duane D. Johnson, and Vitalij K. Pecharsky are cordially acknowledged for discussions.

Conflicts of Interest

The authors declare no conflict of interest. Opinion of the authors does not represent opinions of any governmental organizations.

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Figure 1. Periodic Table with abundances of chemical elements in solids of the Earth’s upper crust. Marked are the most magnetic elements (atoms or ions in neutral or partially oxidized states).
Figure 1. Periodic Table with abundances of chemical elements in solids of the Earth’s upper crust. Marked are the most magnetic elements (atoms or ions in neutral or partially oxidized states).
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Figure 2. (a) Correlation between global annual production and abundance of chemical elements in the upper crust of the Earth. (b) The number of publications per year on magnetocalorics, a giant magnetocaloric effect and a multicaloric effect in the subject (title, abstract, and keywords) in the Web of Science core collection. (c) The isothermal entropy change ΔST (J kg−1 K−1) at 2 Tesla (20 kOe) magnetic field change versus phase transition temperature Tc (K). (d) Timeline for the selected classes of magnetocaloric materials: ΔST at 2 Tesla near room T versus time. Viable materials made from abundant non-toxic elements are represented by black symbols. Colored are critical or toxic materials; hydrides (blue) emit H2 gas.
Figure 2. (a) Correlation between global annual production and abundance of chemical elements in the upper crust of the Earth. (b) The number of publications per year on magnetocalorics, a giant magnetocaloric effect and a multicaloric effect in the subject (title, abstract, and keywords) in the Web of Science core collection. (c) The isothermal entropy change ΔST (J kg−1 K−1) at 2 Tesla (20 kOe) magnetic field change versus phase transition temperature Tc (K). (d) Timeline for the selected classes of magnetocaloric materials: ΔST at 2 Tesla near room T versus time. Viable materials made from abundant non-toxic elements are represented by black symbols. Colored are critical or toxic materials; hydrides (blue) emit H2 gas.
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Figure 3. Crystal structures: (a) cubic CsCl-type binary B2 (with nearest-neighbor bonds); (b) LiMgPdSn-type quaternary Heusler (red lines show 4-atom primitive unit cell); (c) cubic NaZn13-type La(Fe,Si)13 (La is black, Fe1 is red, Fe2 sites occupied by Fe+Si are orange); (d) TiNiSi-type orthorhombic ternary structure; and (e) Ni2In-type hexagonal structure in orthorhombic supercell (Ni1 is red, Ni2 is blue, In is yellow). Images are prepared using VESTA software (https://jp-minerals.org/vesta).
Figure 3. Crystal structures: (a) cubic CsCl-type binary B2 (with nearest-neighbor bonds); (b) LiMgPdSn-type quaternary Heusler (red lines show 4-atom primitive unit cell); (c) cubic NaZn13-type La(Fe,Si)13 (La is black, Fe1 is red, Fe2 sites occupied by Fe+Si are orange); (d) TiNiSi-type orthorhombic ternary structure; and (e) Ni2In-type hexagonal structure in orthorhombic supercell (Ni1 is red, Ni2 is blue, In is yellow). Images are prepared using VESTA software (https://jp-minerals.org/vesta).
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Figure 4. Position of the maximum in the magnetic heat capacity of gadolinium, terbium and holmium versus the applied magnetic field (Tesla). Theory [312] is compared to experimental data for Gd [38], Tb [313], and Ho [314].
Figure 4. Position of the maximum in the magnetic heat capacity of gadolinium, terbium and holmium versus the applied magnetic field (Tesla). Theory [312] is compared to experimental data for Gd [38], Tb [313], and Ho [314].
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Figure 5. (a) Theoretical and experimental ΔT(T) at 1.8 Tesla for the first magnetic field cycle (H = 0→+1.8 T→0→−1.8 T→0) versus initial temperature T, and (b) experimental ΔT(H) at 324 K for three cycles of magnetization/demagnetization in the Fe50.4Rh49.6 sample. Data is from [337].
Figure 5. (a) Theoretical and experimental ΔT(T) at 1.8 Tesla for the first magnetic field cycle (H = 0→+1.8 T→0→−1.8 T→0) versus initial temperature T, and (b) experimental ΔT(H) at 324 K for three cycles of magnetization/demagnetization in the Fe50.4Rh49.6 sample. Data is from [337].
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Figure 6. How mankind moves materials. Natural resources are shrinking; demand and consumption are increasing; landfills are growing; water and atmosphere are polluted. Images: Baikal lake; coal mining in Kuznetsk basin (Kuzbass); Ekibastuz GRES-2 power plant with 420-m chimney; Moscow city; a landfill. Images are from the open sources.
Figure 6. How mankind moves materials. Natural resources are shrinking; demand and consumption are increasing; landfills are growing; water and atmosphere are polluted. Images: Baikal lake; coal mining in Kuznetsk basin (Kuzbass); Ekibastuz GRES-2 power plant with 420-m chimney; Moscow city; a landfill. Images are from the open sources.
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Zarkevich, N.A.; Zverev, V.I. Viable Materials with a Giant Magnetocaloric Effect. Crystals 2020, 10, 815. https://doi.org/10.3390/cryst10090815

AMA Style

Zarkevich NA, Zverev VI. Viable Materials with a Giant Magnetocaloric Effect. Crystals. 2020; 10(9):815. https://doi.org/10.3390/cryst10090815

Chicago/Turabian Style

Zarkevich, Nikolai A., and Vladimir I. Zverev. 2020. "Viable Materials with a Giant Magnetocaloric Effect" Crystals 10, no. 9: 815. https://doi.org/10.3390/cryst10090815

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