Influence of Hysteresis on Magnetocaloric Performance at Cryogenic Temperatures: A Tb3Ni Case Study

RESEARCH ARTICLE www.afm-journal.de Influence of Hysteresis on Magnetocaloric Performance at Cryogenic Temperatures: A Tb3Ni Case Study Timo Niehoff,* Benedikt Beckmann, Konstantin Skokov, Aritz Herrero, Alberto Oleaga, Eduard Bykov, Catalina Salazar Mejía, Marc Straßheim, Oliver Gutfleisch, J. Wosnitza, and Tino Gottschall The magnetocaloric effect (MCE) offers a promising alternative for environmentally friendly cooling technologies, particularly at cryogenic temperatures. However, overestimating material capabilities can lead to misguided research efforts and hinder technological progress. Metamagnetic materials undergoing a transition from an antiferromagnetic to a ferromagnetic state are often predicted to exhibit a strong inverse MCE at cryogenic temperatures based on magnetization measurements. This assumption is critically assessed here using Tb3Ni as a case study. By employing a simple model and comparing results across various measurement techniques, it is demonstrated that the predicted inverse MCE does not exist. Specific-heat data reveal no evidence of this effect, while direct 𝚫Tad pulsed-magnetic-field measurements indicate significant heating caused by dissipative effects linked to hysteresis. Furthermore, total-entropy calculations derived from magnetization data violate the second law of thermodynamics, clearly ruling out the existence of an inverse MCE. These findings underscore the necessity of complementary experimental approaches and a precise understanding of the transitions to accurately characterize magnetocaloric materials and identify suitable candidates for cryogenic magnetic refrigeration. 1. Introduction In recent years, the magnetocaloric effect has become a strong contender in the development of environmentally friendly T. Niehoff, E. Bykov, C. Salazar Mejía, M. Straßheim, J. Wosnitza, T. Gottschall Dresden High Magnetic Field Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat Helmholtz-Zentrum Dresden-Rossendorf 01328 Dresden, Germany E-mail: t.niehoff@hzdr.de The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adfm.202505704 © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. DOI: 10.1002/adfm.202505704 cooling technologies at room temperature.[1–4]Increasing attention has also turned to its applications at cryogenic temperatures, where the MCE holds significant potential for hydrogen liquefaction[5–9]and helium-free cooling of cryostats,[10]offering a more efficient cooling alternative to existing methods. As a result, researchers are increasingly exploring new materials or revisiting known ones that exhibit a strong MCE at low temperatures. Particularly appealing are materials that combine a pronounced inverse MCE at very low temperatures with a conventional MCE at higher temperatures.[11,12]Such materials can cover the entire temperature range required for gas liquefaction, from liquid nitrogen at approximately 77 K to liquid hydrogen and helium down to 4 K. Previous studies have reported materials exhibiting this behavior either during a metamagnetic transition from an antiferromagnetic (AFM) to a fully polarized ferromagnetic (FM) phase[13] or during a transition from a paramagnetic (PM) to a FM phase. These studies predominantly rely on magnetization measurements to indirectly determine the MCE. The strong inverse MCE observed in such cases is often attributed to metamagnetic AFM-FM transitions,[13–42]linked to hysteresis,[43]explained through T. Niehoff, M. Straßheim, J. Wosnitza Institut für Festkörperund Materialphysik Technische Universität Dresden 01069 Dresden, Germany B. Beckmann, K. Skokov, O. Gutfleisch Institute of Materials Science Technical University of Darmstadt Peter-Grünberg-Str. 16, 64287 Darmstadt, Germany A. Herrero, A. Oleaga Departamento de Física Aplicada Escuela de Ingeniería de Bilbao Universidad del País Vasco UPV/EHU Bilbao 48013, Spain Adv. Funct. Mater. 2025,35, 2505704 2505704 (1 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH www.advancedsciencenews.com www.afm-journal.de phenomena such as domain-wall movement,[44]kinetic arrest,[45,46]charge-order transition,[47]and other mechanisms.[48,49]However, these studies did not validate their findings using complementary techniques, such as specific-heat measurements or direct determinations of the MCE in pulsed magnetic fields.[50–53] In this work, we demonstrate that the inverse MCE at low temperatures, as predicted by magnetization measurements using an incorrect application of the Maxwell relation, does not exist. Using a simple model, we reveal the origin of this apparent effect and provide a comprehensive comparison of results for Tb3Ni using various measurement techniques. This material, previously identified by some of the authors as an example for this behavior,[13]serves as a compelling case study. Magnetization data, combined with the Maxwell relation,[54] predict a strong inverse MCE for Tb3Ni. However, specific-heat data reveal no evidence of this effect. Direct pulsed-field measurements, instead, even show substantial irreversible heating, similar to ref. [55]. The strongest evidence for the absence of the inverse MCE comes from the calculated total entropy, as this would violate the second law of thermodynamics. Furthermore, we gain additional insight into the material and its hard magnetic properties through simultaneous measurements of magnetization, strain, and heat flow. These simultaneous measurements prove to be a powerful technique to understanding the properties of the crystal in detail. As mentioned, numerous papers have been published claiming an inverse magnetocaloric effect for various materials at low temperatures, which would hold great promise for cryogenic applications such as gas liquefaction. However, this effect is, in reality, absent. Such incorrect predictions could have severe consequences, as researchers would focus on the wrong materials. It would only become evident during practical application that these materials are not suitable. This parallels the colossal magnetocaloric effect, which was first predicted in 2004.[56]Several highly cited works followed,[57–61]predicting an extremely potent magnetocaloric effect that exceeded the theoretical limit. It was not until 2009 that it was revealed that this effect came from incorrect calculations.[54,62] Our findings, therefore, highlight the importance of employing complementary methods to accurately characterize magnetocaloric effects. 2. Results and Discussion 2.1. Phase Diagram To provide a clear overview and ensure better understanding of the presented measurements, Figure 1shows the H-T phase diagram of Tb3Ni for fields applied along the caxis. To construct the phase diagram, we identified the phase transitions through specific-heat measurements as well as by temperatureand fielddependent magnetization measurements performed using various protocols. The phase diagram aligns closely with a previously reported diagram.[63]However, we were unable to resolve the lock-in transition to the incommensurate phase with our mea020406080100 0 2 4 6 8 10 12 C(T) M(T) M(H) DP M(H) CSP Pulsed field μ 0 H[T] T[K] AFM IC SRO AFM FM Tb 3 Ni Hc Figure 1. H-T phase diagram of Tb3Ni for field applied along the caxis. The diagram highlights distinct magnetic phases, including antiferromagnetic (AFM), ferromagnetic (FM), paramagnetic-like short-range-ordered antiferromagnetic (SRO AFM), and incommensurate (IC) antiferromagnetic regions, as determined from specific heat Cand magnetization measurements Munder varying temperature Tand magnetic field Husing a discontinuous protocol (DP) and a continuous-sweep protocol (CSP). The crosshatched area represents a mixed FM+AFM phase, determined by the hysteresis shown in Figure 3b. The points within the mixed phase were identified by the additional step observed in the magnetization curve (for more details see ref. [63]). The anticipated lock-in to the incommensurate phase transition[63]is represented by a pink dashed line at 51 K. surements, which is expected to occur at 51 K. The anticipated transition is indicated as a dashed line in Figure 1. The following sections offer detailed discussion of the individual measurements and the associated phases. 2.2. Specific Heat Figure 2apresents the specific-heat data. Several features align well with the phase diagram previously reported by Gubkin et al.[63]The anomaly at Tf,around46KatH=0, was identified through neutron scattering[63]as a lock-in transition into the AFM state. This feature is visible during the temperature down sweep, but absent during the up sweep, and it shifts to lower temperatures as the magnetic field increases [left inset in Figure 2a]. Notably, this anomaly disappears entirely in 3 T. At zero field, the specific heat at TN1near 57 K displays a broad and rounded lambda-like anomaly, marking the transition from an incommensurate antiferromagnetic (IC AFM) phase to a short-range ordered antiferromagnetic phase (SRO AFM). As the field increases to 3 T, this transition is progressively suppressed and shifts to slightly lower temperatures. At 4 T, this anomaly is no longer observed. At 3 T and higher fields, a new 𝛿-peak-like anomaly emerges at TC(about 53 K at 3 T), indicating a first-order phase transition. With increasing field, this peak broadens and shifts to higher temperatures. Beyond 3 T, Tb3Ni has a transition from a lowtemperature FM to a PM state. At 3 T, the system undergoes a sequence of phase transitions with increasing temperatures. First from the FM phase to the IC AFM phase, followed by a transition Adv. Funct. Mater. 2025,35, 2505704 2505704 (2 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License www.advancedsciencenews.com www.afm-journal.de Figure 2. a) Specific heat of Tb3Ni as a function of temperature in magnetic fields up to 10 T, for zero-field-cooled (ZFC) condition. The anomalies at Tf =46 K and TN1=57 K in zero field correspond to transitions from the antiferromagnetic (AFM) to incommensurate antiferromagnetic (IC AFM) and short-range order antiferromagnetic (SRO AFM) phase, respectively. TC=53 K at 3 T corresponds to the ferromagnetic (FM) to paramagnetic (PM) transition, aligning well with the phase diagram by Gubkin et al.[63]The left inset highlights the field-induced shift of Tf. Additionally, the inset shows that upon cooling, this feature vanishes. The right inset displays a Schottky-like increase. The feature at TN2indicates a possible transition to a PM state. The dashed line indicates the Dulong-Petit value. b) Temperature-dependent magnetization of Tb3Ni for applied field up to 10 T along the easy caxis for field-cooled (FC) and ZFC conditions. The FC data reveal transitions corresponding to Tf,TN1,TN2,andTC, which align with specific-heat anomalies. The ZFC magnetization shows an additional transition at fields between 2.25 and 4 T and below 25K, disappearing at 6 T as the sample becomes ferromagnetic. This feature is absent in the specific-heat data. The dashed lines mark the temperatures that are examined later (in Figure 6). to the SRO AFM phase and finally a crossover to a PM phase as the temperature increases. Gubkin et al. mapped the phase diagram up to 80 K.[63]However, extending measurements to higher temperatures reveals additional features. Notably, at approximately 95 K, a weak anomaly appears at TN2in zero field, which persists up to 4 T without shifting in temperature. This feature could correspond to a transition from the SRO AFM phase to a PM state. Additionally, we observe a Schottky-like increase at temperatures below 2 K [right inset in Figure 2a]. Measuring the heat capacity at low temperatures is critical for accurately determining the entropy change and, with this, the magnetocaloric effect. Unaccounted low-temperature anomalies can lead to offsets in entropy calculations, resulting in substantial deviations from the correct values. Therefore, the heat capacity was extrapolated to 0 K using a polynomial fit in the range from 0.4 to 1 K to calculate the entropy. For better clarity, we summarize the transitions in the phase diagram shown in Figure 1. 2.3. Magnetization Figure 2b shows the temperature-dependent magnetization of Tb3Ni, measured in fields up to 10 T applied along c,forboth FC and ZFC conditions. Magnetization data up to 1.5 T was previously reported in Refs. [13,64]. The features observed in those lower fields, along with the anomalies found here in the FC magnetization, correspond well with the anomalies extracted from our specific-heat data (Figure 2a). These include the transitions at Tf,TN1,TN2,andTC, identified by maxima, minima, or slope changes of the curves. The ZFC magnetization data reveal additional complexity. At fields between 2.25 and 4 T, we observe a sharp increase in magnetization below 25 K, indicating an additional metastable phase. This feature shifts to lower temperatures as the applied field increases, and disappears at 6 T, with the sample entering a fieldinduced ferromagnetic state at least above 2 K. Interestingly, this transition from an antiferromagnetic to a ferromagnetic phase is not visible in the specific-heat measurements, neither in the ZFC nor in the FC data. To better understand this feature and the magnetocaloric properties, we conducted field-dependent magnetization measurements. The protocol used is crucial in ensuring reproducible results. Figure 3ashows an illustration of the discontinuous protocol we followed (upper panel), along with the corresponding field-dependent magnetization data (lower panel). This protocol is recommended in Refs. [54,65] to ensure that the entropy change calculated using the Maxwell relation is reliable and not overestimated due to mixed-phase states within a hysteresis loop of a first-order phase transition. Even when using the discontinuous protocol, we observe the metamagnetic transition, clearly marked by a sharp jump in the magnetization. Figure 3b shows the magnetization of Tb3Ni between 14 T and −14 T at different temperatures using continuous field sweeps. These data exhibit significantly different behavior compared to the magnetization measured using the discontinuous protocol and were reported previously by Gubkin et al.[63]Below 20 K, Tb3Ni turns into a hard magnet with a coercive field of up to 1.5 T and a remanence close to the saturation magnetization. During this process, the material transitions from a FM to a canted magnetic structure and then back to the FM state, as indicated by the magnetostriction data (see Figure 6). The magnetization values obtained using temperature sweeps (Figure 2b) and continuous-sweep protocols align well at high fields. However, the values measured using the discontinuous protocol are lower. This discrepancy likely arises from the fact that these measurements were conducted on different pieces of the sample. While Gubkin et al. recorded a magnetization of approximately 8.2μB/Tb(μBis the Bohr magneton) at 7 T, our measurements yield 7.37μB/ Tb (230.5Am 2kg−1) using the continuous-sweep protocol. This deviation may result from factors such as misalignment of the sample, or twinning of the crystal in our measurements. Additionally, we observe a weak hysteresis between 9 T and 12 T that diminishes for increasing Adv. Funct. Mater. 2025,35, 2505704 2505704 (3 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License www.advancedsciencenews.com www.afm-journal.de temperatures (see inset in Figure 3b). At 14 T, beyond the hysteresis, we obtained a magnetization of 8.06μB/ Tb (252 A m2kg−1). This is lower than the theoretical maximum of gJ =9μBper Terbium atom. This indicates that strong magnetocrystalline anisotropy prevents full spin alignment along the magnetic field until exceeding 12 T, at which point the complete spin-polarized phase is reached. The strong anisotropy can be seen in Figure S4 (Supporting Information). 2.4. Magnetocaloric Effect To quantify the magnetocaloric effect, we calculated the isothermal entropy change ΔSTfrom the measurements described above. Using the specific-heat data, we derived the fieldand temperature-dependent entropy S(T,H) according to the relation S(T, H)=∫T 0(C∕T)HdT. For field changes ΔH=Hf−Hi, the isothermal entropy change ΔST(T,ΔH) and the adiabatic temperature change ΔTad(T,ΔH)aregivenby: ΔST(T, ΔH)=S(T, Hf)−S(T, Hi)(1) ΔTad(T, ΔH)=T(S, Hf)−T(S, Hi)(2) Additionally, we computed the isothermal entropy change using both isofield and isothermal magnetization curves. In the latter case, we treated the ZFC and FC curves separately. For the calculations we used the Maxwell relation: ΔST(T, ΔH)=μ 0∫Hf Hi (𝜕M 𝜕T)HdH (3) Figure 4asummarizes these calculated curves. For clarity, we present only the entropy change ΔSTfor a field variation from 0 to 5 T. At temperatures above 40 K, all calculated results show excellent agreement, revealing a minimum value of ΔST =−17.5 Jkg−1K−1around 60 K. This minimum corresponds to the expected conventional magnetocaloric effect associated with the transition from a paramagnetic-like SRO AFM phase to the FM phase. In contrast, below 40 K, striking discrepancies emerge between the results obtained from specific-heat and FC magnetization data and those derived from ZFC and field-dependent magnetization data. The latter two results show a pronounced positive entropy change, indicating an inverse magnetocaloric effect. Notably, the calculation using the ZFC magnetization data exhibits a maximum near 6.3 K, with a remarkably large entropy change of ΔST=69 Jkg−1K−1. This effect originates from the anomaly observed in the ZFC and field-dependent magnetization data below 25 K and above 2.25 T. This feature is caused by the field-induced metamagnetic transition from the AFM to FM phase, which does not appear in the specific-heat and FC magnetization data. Contrary to the previous claim made by some of the authors in ref. [13] we argue that the calculated inverse magnetocaloric effect is not real and that no additional magnetocaloric effect is present. We show a compelling evidence that this effect cannot be real in Figure 4b, which displays the total entropy S−S0as a function of temperature. To determine this, we followed the standard approach of calculating the entropy diagram by deriving the zero-field entropy from specific-heat data and adding the entropy change to the result. For this, we used the 0 T curve shown in Figure 2a and calculated the entropy using: S(T, H =0) =∫T 0 CH=0 TdT (4) To obtain the total entropy, we added the ΔST(T,ΔH)values obtained from the ZFC magnetization data for different field changes to S(T,H=0). Typically, the expected ΔTad values are derived from such a diagram. However, for a field change larger than 3 T, the entropy decreases with increasing temperature above 6 K, directly violating the second law of thermodynamics. According to this fundamental principle, increasing the thermal energy should increase the number of accessible micro states, not reduce them. Figure 3. a) Magnetization data measured using the discontinuous protocol (DP) (illustrated in the upper panel), highlighting sharp jumps that mark metamagnetic transitions. b) Continuous field sweeps (CSP) (protocol sketched in the upper panel) of Tb3Ni from 14 T to −14 T and back, revealing large hysteresis below 20 K, where the compound is a hard magnet. A hysteresis feature near 11 T is highlighted in the inset. Adv. Funct. Mater. 2025,35, 2505704 2505704 (4 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License www.advancedsciencenews.com www.afm-journal.de Figure 4. a) Isothermal entropy change ΔSTcalculated from specific-heat, temperature-dependent ZFC, FC, and field-dependent magnetization data, measured using the discontinuous protocol (DP) for a field change from 0 to 5 T. b) Entropy diagram constructed from magnetization and zero-field specific-heat data. For field changes exceeding 3 T, the entropy curve decreases with rising temperature, contradicting the second law of thermodynamics. In conclusion, despite using the usual protocol to measure the magnetization, the inverse magnetocaloric effect, as derived from the Maxwell relation and the ZFC data, does not exist. In contrast, the entropy change obtained from the Maxwell relation and FC data shows good agreement with that derived from heatcapacity data. 2.5. Adiabatic Temperature Change To support our claim, we expanded the study by directly measuring the magnetocaloric effect. In these measurements, we determined the adiabatic temperature change at the maximum applied field, ΔTad, as a function of the initial sample temperature, TStart, for fields up to 50 T. We show the data as circles in Figure 5.For comparison, we included the adiabatic temperature changes indirectly determined from specific-heat data (that do not predict an Figure 5. Maximum adiabatic temperature change (ΔTad) as a function of initial sample temperature (TStart) in pulsed magnetic fields up to 50 T. Circles represent the measured ΔTad. Solid lines (pink and blue) indicate adiabatic temperature changes indirectly determined from specific-heat measurements for fields of 5 and 10 T. The grey solid line indicate the dissipative heating. The dashed lines depict the corrected pulsed-field data, matching the indirect heat-capacity results and thus also confirms that there is no inverse caloric effect. inverse magnetocaloric effect), plotted as solid lines, calculated using Equation (2) for fields of 5 and 10 T. At temperatures above 40 K, the direct and indirect measurements align well, but considerable deviations appear at lower temperatures. Here, the direct ΔTad measurements reveal a pronounced additional effect, similar to the discrepancies observed in the entropy changes shown in Figure 4a. The key difference is the reversed sign of the magnetocaloric effect. In the direct measurements, we observe heating, which contradicts the expected cooling, associated with an inverse magnetocaloric effect. We previously identified this irreversible heating in pulsedfield measurements of other materials as a dissipative effect associated with a first-order phase transition with hysteresis.[55]During the field-induced transition from the AFM to the FM phase, abrupt heating occurs when the phase boundary is crossed. We have quantified this effect and subtracted it from the pulsed-field data, yielding corrected results. These corrected pulsed-field data, represented by dashed lines, now closely match the indirect measurements across the entire temperature range, with an absolute temperature shift, which can be attributed to the different sample pieces and devices. The dissipative heating can be estimated using ΔTdiss =0.5⋅ qdiss ⋅C−1 eff , where the dissipative energy qdiss =μ0∮HdM corresponds to the area enclosed by the magnetization loop measured during the quasi-isothermal experiment. Ceff represents the effective heat capacity during the rapid temperature rise. Neglecting the temperature dependence can lead to huge uncertainties, particularly at low temperatures. Therefore, to improve the estimate, an effective temperature is first calculated as the average temperature between the start of the temperature rise Tdiss, start, and the final temperature reached after the dissipative heating, Tdiss, end. The effective average temperature is Teff =(Tdiss, end + Tdiss, start)/2. The effective zero-field heat capacity Ceff is then assigned based on this temperature. The estimated ΔTdiss is represented by a solid grey line in Figure 5. Interestingly, the dissipative effects are absent in the lowtemperature 5 T pulses at 4.8 and 9.5 K. At these low temperatures, higher fields are required to induce the first-order transition. This also demonstrates that the dissipative effects are real and cannot be attributed to eddy currents. Focusing only on the ΔTad above 40 K, we observe the strong potential of Tb3Ni for low-temperature magnetocaloric Adv. Funct. Mater. 2025,35, 2505704 2505704 (5 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License www.advancedsciencenews.com www.afm-journal.de Figure 6. Field dependence of adiabatically measured sample temperature Tad, magnetization M, strain, and heat flow ΔTflow ata)5K,b)15K,c)40K, and d) 60 K. The green curves show the simultaneously measured Tad and M. The FC magnetization, strain, and heat flow were measured simultaneously in quasi-static fields. Adv. Funct. Mater. 2025,35, 2505704 2505704 (6 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License www.advancedsciencenews.com www.afm-journal.de refrigeration. For 10 T pulses, the material exhibits a maximum reversible adiabatic temperature change of ΔTad =10.5 K at a starting temperature TStart of 66 K. As the field increases, the maximum shifts to higher temperatures, reaching a ΔTad of 33 K at TStart =163 K for 50 T pulses. These direct measurements are crucial for fully characterizing Tb3Ni, as they conclusively demonstrate that the inverse magnetocaloric effect suggested by the indirect calculation is not real. For an even deeper understanding, we present the field-dependent data of the individual pulses in Figure 6for various starting temperatures. These data illustrate the dissipative heating as well as the magnetocaloric effect. 2.6. Simultaneous Measurements Figure 6presents the simultaneous field-dependent measurements of FC magnetization, strain, and heat flow ΔTflow,aswell as ZFC magnetization, and pulsed-field data at various temperatures. To provide better insight, we indicated the corresponding temperatures as dashed lines in Figure 2b. We performed the strain and heat-flow measurements alongside the FC magnetization measurements shown in Figure 3b. These measurements followed the continuous sweep protocol, with the downsweep data represented as M FC down and the up-sweep data as MFCup. Additionally, we included magnetization measurements from the discontinuous protocol, labeled as M ZFC. The figure also includes the directly measured magnetocaloric effect in pulsed field, Tad, together with the simultaneously measured magnetization, labeled M pulsed. Figure 6a,b illustrate the direct transition from the AFM to the fully polarized FM phase at 5 and 15 K, respectively. This firstorder phase transition shows pronounced hysteresis, which we clearly observe in the magnetization, as well as in the adiabatic temperature changes during pulsed-field measurements. Dissipative heating, caused by the hysteresis, generates a sharp temperature jump as the transition occurs depicted in the top panel of Figure 6a,b. At 5 K, this jump occurs near 5.5 T, while at 15 K, it appears around 4.5 T. After the transition, both samples heat to approximately 25 K. This corresponds to the temperature, above which no hysteresis occurs. For this reason, no further dissipative heating can take place. The temperature reached through dissipative heating is consistently 25 K. Below this temperature, ΔTad, asshowninFigure5, only varies due to the different initial temperatures. During the down sweep, we observe a second small temperature increase around 2 T as the sample reenters the AFM phase. This results from the narrow hysteresis near 2 T at 25 K (see Figure 3b). The temperature increase is further reduced by the sample’s now larger heat capacity at 25 K. Beyond these dissipative effects, we detect no additional field-dependent temperature changes. The top panels of Figure 6c,d (40 and 60 K, respectively) demonstrate the typical behavior of a reversible magnetocaloric effect during a first-order phase transition. At the transition, the temperature Tad rises quickly and continues to increase as the sample is in the FM phase. During the down sweep, the temperature decreases congruently with the up-sweep. When the phase transition is reached again, with a hysteresis of approximately 1 T, the temperature returns to its initial value. These results confirm the presence of a reversible magnetocaloric effect at higher temperatures during both the IC-FM and SRO AFM-FM transitions, while only dissipative heating is present at lower temperatures during the AFM-FM transition. It is also noticeable, that the temperature jump in pulsedfield measurements appears about 2 T above the phase transition extracted from quasi-static measurements. This shift could result from dynamic effects during the rapid pulsed-field measurement. Highlighted in the phase diagram (Figure 1), this shift occurs only during the AFM-to-FM transition and disappears at higher temperatures, when the starting temperature lies within the IC or SRO AFM phase. Additionally, we observe that at 5 K Figure 6a), the continuous sweep protocol leads to a lower field to induce the FM transition compared to the discontinuous protocol. This difference vanishes at higher temperatures, where the sample retains its hard magnetic properties but no longer exhibits a mixed state, as evidenced by the intermediate step on the FC magnetization hysteresis data. This feature is evident in both strain and heatflow measurements. When the coercive field of 1 T is reached, the change in magnetization is accompanied by a temperature increase and a longitudinal elongation along the applied field of approximately 0.05 %, while the transverse strain only shows a small change of about −0.01 %. At 2 T, we observe an additional kink in the intermediate transition in both the strain and magnetization measurement, but not in the heat-flow data. At 3.5 T, as the sample fully transitions into the FM phase, the temperature rises again, and the strain returns to its initial state. Beyond this point, the strain increases steadily in the FM phase, which appears as well at higher temperatures, but with a stronger increase in strain. In the AFM phase, the strain remains constant, independent of temperature. At 15 K, the features at the AFM-toFM transition differs from Figure 6a because the sample is no longer a hard magnet, leading to distinct strain data. During the transition, the sample compresses along the field direction by approximately 0.05 %, similar to the behavior at 5 K, but additionally this compression is counteracted by expansion in the transverse direction of also about 0.05 %. A similar behavior appears at 40 K (Figure 6c). At 60 K (Figure 6d), the strain behavior changes again as Tb3Ni undergoes the IC-to-FM phase transition. In the IC phase, the sample contracts slightly with increasing field. As it transitions to the FM phase, both longitudinal and transverse strain increase, resulting in a volumetric expansion. 2.7. Mean-Field Model To explain why indirect calculations of the magnetocaloric effect from magnetization measurements may incorrectly predict an inverse magnetocaloric effect, and to demonstrate that this discrepancy is not unique to Tb3Ni, we analyze a simplified model system. This example highlights inconsistencies depending on the method used to determine the magnetocaloric effect. The model under consideration features antiferromagnetic interactions between nearest-neighbor atoms, competing with ferromagnetic couplings between next-nearest neighbors. For Adv. Funct. Mater. 2025,35, 2505704 2505704 (7 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License www.advancedsciencenews.com www.afm-journal.de Figure 7. a) Calculated normalized magnetization mas a function of reduced temperature tfor different reduced fields of h=0.5, 0.8, 0.9, 0.98, 1, 2, and 4 using Equation (5)with|J′/J|=1. The abrupt magnetization jumps of the AFM-FM transition are indicated by dashed lines. b) Entropy change Δst as a function of reduced temperature, comparing calculations based on the free energy with those derived indirectly using the Maxwell relation. reference, measurements on Tb3Ni have established that at zero field and low temperatures, antiferromagnetic coupling dominates.[13,63]Additionally, the system is represented by an Ising model, where spins can only align parallel or antiparallel to the caxis. This behavior mirrors the strong anisotropy observed in Tb3Ni between easy and hard axis[63](see Figure S4, Supporting Information). In contrast, a Heisenberg model, which allows spins to orient isotropically, would not reproduce the observed abrupt magnetization jump during the AFM-FM transition, and may even show spin canting.[66]The Hamiltonian for this simplified system is: =−J∑ ⟨i,j⟩ Si⋅Sj−J′∑ ⟨⟨i,k⟩⟩ Si⋅Sk−h∑ i Si(5) where Si=±1 represents an Ising spin at site i. The first term describes antiferromagnetic interactions between nearest neighbors (J<0). The second term accounts for ferromagnetic interactions between next-nearest neighbors (J′>0). The third term introduces an external magnetic field (h). In the past, research has primarily focused on ferromagnetic models of firstand second-order transitions,[67,68]with some studies also investigating antiferromagnetic systems.[69]Models that consider both antiferromagnetic and ferromagnetic couplings have mostly been analyzed using Heisenberg models.[70,71] Other approaches include multilayer systems with antiferromagnetic interplanar and ferromagnetic intraplanar couplings.[72] Ising models have also been investigated using Monte Carlo methods, but with strong ferromagnetic coupling |J′/J|>50[73] or in perovskite structures with |J′/J|=1.[74]However, neither Heisenberg nor the mentioned Ising models have been able to reproduce the abrupt increase in magnetization during the metamagnetic transition. A similar Ising model that is comparable to the one investigated here has been studied in ref. [75]. Even in this case, the abrupt metamagnetic transition was not examined in detail. We solve the model Equation (5) using the Bragg–Williams mean-field approximation, described in previous works.[66,76–78] For simplicity, we use reduced variables, with the temperature t=kBT/(|J+J′|), the Boltzmann constant kB, and the field h= μ0H/(|J+J′|). Further details can be found in the Supporting Information. For strong ferromagnetic coupling (|J′/J|>2), the system no longer shows an antiferromagnetic phase and for weak FM coupling (|J′/J|<0.2), the transition between AFM and FM states has no abrupt magnetization jump. For this study, we focus on a coupling ratio of |J′/J|=1, where the competition between the interactions produces the desired features. Additionally, we calculated the magnetization jump from the hysteresis of the model by minimizing the free energy. We present the normalized magnetization as a function of temperature in Figure 7a, with the magnetization jumps indicated by dashed lines. Despite the simplicity of the model, the magnetization closely resembles the temperature dependence observed for Tb3Ni (see Figure 2b). At low fields, the magnetization behaves as expected for an antiferromagnet up to h=0.8. As the field increases, the magnetization abruptly transitions to a fully polarized ferromagnetic state at lower temperatures. The size of the jump grows with field and occurs at progressively lower temperatures. Above h=1.0, the antiferromagnetic state no longer exists at the lowest temperatures, and the model behaves like a ferromagnet. The model also predicts strong hysteresis during the abrupt AFM-FM transition, which aligns well with the behavior observed in Tb3Ni (See Figures S1 and S2, Supporting Information). In particular, in experiment, we found a notable dissipative heating during pulsed-field measurements, that only occurs when the AFM-FM transition is crossed. However, the simple model does not account for the hard magnetic properties exhibited by Tb3Ni (which would require a more sophisticated theory). In Figure 7b, we present both the entropy change Δstdirectly calculated with the free energy f(t,h) obtained using Equation (5) (further details can be found in the Section S1, Supporting Information.), s(t, h)=− 𝛿f(t,h) 𝛿t, and Equation (1) as well as the indirectly determined value using the Maxwell relation: Δst(t, hf)=∫hc 0(𝜕m 𝜕t)hdh +𝛿s(t, hc) +∫hf hc (𝜕m 𝜕t)hdh (6) The first integral accounts for the region up to the AFMFM transition at the critical field hc, while the second integral Adv. Funct. Mater. 2025,35, 2505704 2505704 (8 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License www.advancedsciencenews.com www.afm-journal.de covers the region beyond the transition. The entropy jump 𝛿s(t,hc) at the transition can be estimated using the ClausiusClapeyron equation.[79] At low fields (h=0.5), the system exhibits both an inverse and a conventional magnetocaloric effect around the Néel temperature, resulting in positive and negative Δst, respectively. This behavior is expected for an AFM transition[71,74,80]and can also be observed at low fields in Tb3Ni, as shown in Figure S3 (Supporting Information). At higher fields (h=2), the calculations based on the free energy results in a conventional magnetocaloric effect, characterized by a negative entropy change, which is consistent with a ferromagnetic transition. However, when the entropy change at h=2 (after the magnetization jump), is calculated indirectly using the Maxwell relation Equation (6), a positive entropy change emerges at temperatures below t=1.3 predicting an inverse magnetocaloric effect. This additional effect is the result of the additional Clausius-Clapeyron term in Equation (6) and the discontinuities in the magnetization data. In real measurements, these are not magnetization jumps but rather differentiable functions. Therefore, the simple Maxwell relation is used in experiments, which would also lead to a non-existent inverse magnetocaloric contribution. At higher temperatures, the entropy change calculated by both methods is identical. At low fields (h=0.5), where no magnetization jump is observed, the entropy changes coincide across the entire temperature range for both calculation methods. The fact that a genuine inverse magnetocaloric effect occurs at low fields due to the antiferromagnetic transition, disappears at higher fields, and yet Maxwell’s relations predict an inverse effect for fields above the metamagnetic transition is probably the reason why this effect has often been misinterpreted in the literature, as it has been incorrectly attributed to the antiferromagnetic transition. In summary, the entropy change calculated from the free energy accurately predict the actual behavior, aligning with the magnetocaloric effect determined from heat-capacity measurements of Tb3Ni. In contrast, the inverse magnetocaloric effect suggested at high fields and low temperatures by the Maxwell relation is incorrect, both in this simplified model and in experiment, because of the first order transition. To further clarify the difference between the absence of the inverse magnetocaloric effect and the dissipative heating, Section 4 (Supporting Information) presents the calculated Δtand the dissipative heating. We emphasize that, while a more complex Hamiltonian, including additional couplings, would better capture the behavior of Tb3Ni, this simplified model can demonstrate the essential physics. Specifically, that the coexistence of competing AFM and FM interactions, combined with strong anisotropy, is sufficient to produce a magnetization jump accompanied by significant hysteresis. Notably, this does not result in an inverse MCE when calculated from the free energy, highlighting how the Maxwell relation can yield incorrect results 3. Conclusion In this work, we demonstrate that determining the MCE only based on magnetization measurements and the Maxwell relation, widely used in the literature due to its simplicity and experimental accessibility, can lead to drastic misinterpretations, especially in materials with large hysteresis. Despite using proper measurement protocols, many materials have been wrongly characterized, with overly optimistic predictions of promising effects that were not validated by complementary measurements. Using Tb3Ni as a case study, we confirm that the inverse caloric effect is not real. We achieve a complete understanding of the magnetocaloric behavior through a combination of experimental techniques, including magnetization, specific-heat, and direct pulsed-field ΔTad measurements. While the Maxwell relation predicts a pronounced inverse caloric effect, specific-heat measurements reveal no such effect, and pulsed-field ΔTad measurements reveal a strong conventional magnetocaloric effect. The apparent inverse caloric effect is attributed to an improper application of the Maxwell relation, and the strong heating observed in direct measurements is identified as a consequence of dissipative effects. Furthermore, we applied a simple model to illustrate the origin of these misinterpretations, demonstrating that this is not limited to Tb3Ni, but represents a fundamental phenomenon that can occur across a wide range of materials. Additionally, our findings highlight the value of combining simultaneous experimental techniques, such as magnetization, strain, and heat-flow measurements, to uncover the complex behavior of materials, providing deeper insights into their intrinsic properties. Finally, relying on magnetization measurements alone is insufficient for accurately characterizing the magnetocaloric effect. To fully understand the magnetocaloric effect of a material, the phase transitions and underlying physics of the system must be well understood. A comprehensive experimental approach, incorporating multiple techniques such as magnetization, heat capacity, and direct measurements of the magnetocaloric effect, for instance, in pulsed fields, is needed to ensure the reliability of reported results. Without such validation, the true potential of the magnetocaloric effect, particularly for emerging applications such as cryogenic cooling, may be misinterpreted and overstated, thereby hindering progress in this field of research. 4. Experimental Section Sample Preparation:In this work, the same Tb3Ni crystal was used as described in Refs. [13,63]. The crystal was synthesized by preparing a polycrystalline ingot of Tb3Ni through arc melting in a helium atmosphere, with terbium of 99.9 % purity and nickel of 99.99 % purity. Several singlecrystal samples were grown by remelting the ingot at temperatures slightly above the peritectic point in a resistance furnace, followed by slow cooling. From these, a single crystal measuring approximately 4 ×5×6m 3m was extracted. The crystal quality was assessed through X-ray diffraction using the back-reflection Laue method on various surfaces, confirming that no additional reflections from other grains were present. The resulting sample was shaped into a parallelepiped, with surfaces oriented perpendicularly to the crystallographic a,b,andcaxes. After the surfaces were polished, their orientation was once more verified through Laue diffraction. For magnetization and heat-capacity measurements, the sample was cut into several smaller pieces, each weighing approximately 4 mg. Specific Heat:The specific heat was measured in magnetic fields up to 10 T between 400 mK and 150 K. Below 2 K, the heat-pulse method was employed using a 3He cryostat (Oxford). For temperatures from 2 to 150 K, we Adv. Funct. Mater. 2025,35, 2505704 2505704 (9 of 11) © 2025 The Author(s). Advanced Functional Materials published by Wiley-VCH GmbH 16163028, 2025, 43, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/adfm.202505704 by Universidad Del Paã-S Vasco, Wiley Online Library on [10/11/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License