Multicaloric effects in Metamagnetic Heusler Ni-Mn-In under uniaxial stress and magnetic field

The world's growing hunger for artificial cold on the one hand, and the ever more stringent climate targets on the other, pose an enormous challenge to mankind. Novel, efficient and environmentally friendly refrigeration technologies based on solid-state refrigerants can offer a way out of the problems arising from climate-damaging substances used in conventional vapor-compressors. Multicaloric materials stand out because of their large temperature changes which can be induced by the application of different external stimuli such as a magnetic, electric, or a mechanical field. Despite the high potential for applications and the interesting physics of this group of materials, only few studies focus on their investigation by direct methods. In this paper, we report on the advanced characterization of all relevant physical quantities that determine the multicaloric effect of a Ni-Mn-In Heusler compound. We have used a purpose-designed calorimeter to determine the isothermal entropy and adiabatic temperature changes resulting from the combined action of magnetic field and uniaxial stress on this metamagnetic shape-memory alloy. From these results, we can conclude that the multicaloric response of this alloy by appropriate changes of uniaxial stress and magnetic field largely outperforms the caloric response of the alloy when subjected to only a single stimulus. We anticipate that our findings can be applied to other multicaloric materials, thus inspiring the development of refrigeration devices based on the multicaloric effect.


I. INTRODUCTION
Our society is facing an increasing demand for refrigeration 1 going hand in hand with a pronounced increase in the energy spent for various cooling applications. Furthermore, conventional refrigeration systems use fluids with a strong global warming potential 2 (their effect per unit mass is about thousand times larger than carbon dioxide). It is therefore urgent to develop technologies which are both energy efficient and respectful with the environment. Solid-state cooling, based on the giant caloric effects exhibited by a variety of materials undergoing ferroic phase transitions, are considered as the best alternatives to replace present refrigerators that use harmful fluids 3,4 . For the last two decades, a worldwide intensive research has been devoted to the study of magnetocaloric 5 , electrocaloric 6 , and mechanocaloric (which include elastocaloric and barocaloric) materials [7][8][9] , and very recently the study has been extended to multicaloric materials for which good prospects are envisaged [10][11][12] .
A broad variety of materials exhibiting large isothermal entropy and adiabatic temperature changes, induced by the application or removal of different fields (magnetic, electric, mechanical) have been discovered up to now [13][14][15][16][17] . This includes recent reports on plastic crystals 18,19 for which the pressure-induced entropy changes compare to the values given by standard commercial fluid refrigerants (although lower pressures are required for the latter). Elastocaloric materials are known to exhibit the largest values for the adiabatic temperature changes among all caloric materials, and very recently reversible values larger than 30 K have been reported (at readily accessible stresses) for the elastocaloric effect of shape memory alloys formed by all 3d elements 20 .
In spite of the significance of the achievements up to now, there are a series of bottlenecks, mostly related to the first-order character of the phase transitions that are at the origin of the giant caloric effects. On the one hand, the required fields to achieve giant values are in general still too large, and on the other hand, the hysteresis associated with the phase transitions reduces the caloric efficiency and compromises the reversibility of the caloric effect. It has been proposed that the combination of more than one external field may help in overcoming some of the mentioned limitations 10,11,21 . Materials with significant coupling between degrees of freedom have a cross-response to different external stimuli. In those materials, entropy and temperature changes can be driven by either a single stimulus (single caloric effect) or by multiple stimuli (multicaloric effect), which can be applied/removed either simultaneously or sequentially 11,22 . The study of multicaloric ef-fects and materials is a quite novel research field, but very interesting results have already been achieved [23][24][25] . It has been shown that a suitable combination of magnetic field and hydrostatic pressure enables a drastic reduction of the magnetic-field-effective hysteresis in materials with magnetostructural transformations 10,26,27 . In a recent work, we have undertaken a different strategy to show that in a metamagnetic Ni-Mn-In shape-memory alloy the combination of uniaxial stress and magnetic field enables designing a multicaloric cycle which now takes advantage of the thermal hysteresis associated with the martensitic transition (which is termed "hysteresis positive" approach) 12 . However, the study of multicaloric effects is still challenging. Although the thermodynamic framework is well established 22 , experimental studies are very scarce 21,25,26 , mostly because the obtention of the physical quantities describing these multicaloric effects typically requires the use of non-commercial advanced characterization systems 28 .
In this work, we have used a purpose-built calorimeter that works under the application of uniaxial stress and magnetic field, to study the multicaloric response in terms of isothermal entropy and adiabatic temperature changes of a prototypical metamagnetic shape memory alloy subjected to the combined application of magnetic field and uniaxial stress. We have selected a Ni-Mn-In alloy with a martensitic transition temperature close to the austenitic Curie temperature. The proximity between martensitic and magnetic transitions results in a pronounced coupling between magnetism and structure so that the application of magnetic field has a strong influence on the martensitic transition. Our results evidence the advantages of the multicaloric effect in comparison with the single caloric (magnetocaloric and elastocaloric) ones.

II. RESULTS
Thermomagnetization curves were recorded as a function of temperature for selected values of magnetic field (H). Our results ( Fig. 1(a)) conform to the reported behaviour for metamagnetic Ni-Mn-In Heusler alloys 29,30 . On cooling, magnetization increases due to the onset of the ferromagnetic order in the austenite (T c ∼ 303 K), and upon further cooling magnetization sharply decreases at the martensitic transition, as a consequence of the short range antiferromagnetic interactions in the martensitic phase of Ni-Mn-In alloys 31 .
Calorimetric measurements were conducted at 0.6 and 1 Kmin −1 for heating and cooling runs, respectively, for selected values of applied (constant) magnetic field in the range 0 -6 T, and applied (constant) uniaxial load in the range 0 -1 kN. Changes in the cross-section of the specimen are expected to be negligibly small and stresses (σ ) correspond to the ratio between applied load and the unstressed specimen's cross-section measured at room temperature. Examples of the measured calorimetric curves are shown in Fig. 1(b) for 0 and 4 T magnetic field, and 0 and 50 MPa uniaxial stress. On cooling, the exothermal peak corresponds to the forward martensitic transition while the endothermal peak on heating corresponds to the reverse martensitic transition. The transition strain increases with increasing uniaxial stress, as a result of the increase in the percentage of favourably oriented martensitic variants 32 . It is worth noticing the good correlation for the transition region from the two sets of measurements, which indicates that both latent heat release (absorption) and strain are proportional to the transformed fraction.
From both, calorimetric and strain data, it is apparent that the application of a magnetic field shifts the martensitic transition to lower temperatures while uniaxial stress shifts the transition to higher temperatures. By identifying the transition temperature from the peak position of the calorimetric curves on heating and cooling, it is possible to determine the phase diagram in the H − σ coordinate space, which is shown in Fig. 2(a). The red plane corresponds to the reverse martensitic transition which occurs either by increasing temperature, increasing magnetic field or decreasing stress (as indicated by arrows in the figure). Well above that plane the sample is in austenite 33 . The blue plane corresponds to the forward martensitic transition induced either by decreasing temperature, decreasing magnetic field or increasing stress (as indicated by arrows in the figure). Well below that plane the sample is in martensite 33  with increasing field, with a slope in the absence of applied stress that compares well with typical data for this kind of alloys 30 . Also, for all values of applied magnetic field, transition temperatures linearly increase with increasing stress. In this case, the slope of the transition lines in the absence of magnetic field is lower but comparable to the values reported for similar alloys 32 . The slope of these transition lines has been found to depend on the application of the secondary field, as illustrated in the inset of Fig. 2(b), which shows a decrease in |dT /dµ 0 H| with increasing stress and in the inset of Fig. 2(c) which shows an increase in dT /dσ with increasing magnetic field. There are very few studies on the caloric response of materials subjected to more than one external stimulus, and most of them refer to the combined effect of hydrostatic pressure and magnetic field 25,26,34 .
The effect of uniaxial stress and magnetic field was studied for the metamagnetic transition in Fe-Rh 27 . For that compound, it was found that the application of a secondary field does not affect the values for |dT /dµ 0 H| and dT /dσ .

A. Transition entropy change and entropy curves
The complexity of the experimental set-up allowing the application of uniaxial load makes our calorimeter less accurate than a conventional DSC. As a result, calorimetric signals are affected by a poorer base-line, particularly on cooling runs and at high magnetic fields. For this reason, we will restrict the following analysis to the heating protocol and magnetic fields up to 4 T.
The transition entropy change (∆S t ) can be computed by suitable integration of the base-line corrected calorimetric curves ( Fig. 3(a-e)) as For the range of applied stresses, we have not found any systematic dependence of ∆S t with applied stress, but conversely ∆S t decreases with increasing magnetic field, as illustrated in Fig. 3(f) which shows ∆S t as a function of magnetic field. For each applied field, ∆S t values obtained for different applied stresses are within the experimental error (indicated by the error bars). The observed decrease in ∆S t with increasing magnetic field is a consequence of the increase in the magnetic contribution to the total entropy, which opposes the phonon contribution being larger (in absolute value) than its magnetic counterpart, and to a good approximation, magnetic field independent 35,36 .
The entropy curves (S(T, σ , H)) referenced to a value at a specific temperature T 0 (chosen below the phase transition region) can be computed 17 by combining calorimetric curves (recorded at selected values of stress and magnetic field) with specific heat (C) data . In these computations it is common to take C to be independent of magnetic field and stress. While the stress independence of C is still a good approximation in our case, the proximity of the martensitic transition to the austenitic Curie temperature in our sample makes C to be dependent on the applied magnetic field. For this reason, we have measured the temperature dependence of C at selected magnetic fields. Results for the sample under study (Ni 50 Mn 35.5 In 14.5 ) are shown in Fig. 4(a), which have been measured using a bespoke universal calorimeter that operates up to a temperature of 310 K and under an external magnetic field 38 . From the figure, three different regions with distinct behaviour for C vs T are clearly identified. At low temperatures, below the martensitic transition there is no dependence of the specific heat of the martensite (C M ) with the magnetic field, and a linear temperature dependence can be assumed (black line in the inset of Fig. 4(a)). Within the transition region (around room temperature), the latent heat of the phase transition results in an apparent peak in the specific heat. This peak shifts to lower temperature under a 2 T magnetic field, which is in good agreement with the shift observed in DSC data (shown in Fig. 3). Above the martensitic transition, a small peak (centered at 303 K) associated with the Curie point of the austenite is clearly visible for the 0 T curve. This peak is smoothed when a 2 T magnetic field is applied. Because our bespoke calorimeter is limited at high temperatures, it is not possible to accurately determine the specific heat in the austenitic phase (C A ). For this reason, we have used a second (commercial) calorimeter (PPMS from Quantum Design) to measure C over a broader range of temperatures and magnetic fields. In order to separate the contributions from the latent heat and the Curie point we have used a second sample (Ni 50 Mn 34 In 16 ) for which the martensitic transition takes place at a temperature (180 K) well below its Curie point. It is not expected that small differences in composition may affect the value of C A in the paramagnetic austenitic phase.
Results from these measurements are summarized in Fig. 4(b). As shown in the inset of Fig.   4(b), above the Curie temperature, C A is almost independent of temperature but it shows a small dependence in magnetic field.
Using data for C M and C A , and the base-line corrected calorimetric curves (    due to the hysteresis of the transition, the application of both a magnetic field and mechanical stress will take the sample through a minor loop within the two-phase coexistence region (for which no experimental data are available). For these reasons, the analysis of single caloric and multicaloric effects will be restricted to values obtained from the application of the magnetic field and the removal of stress (they correspond to trajectories from below to above the red plane in Fig.   2(a)).
From S(T, σ , H), the isothermal entropy change (∆S) associated with a given caloric effect arising from the application of a magnetic field H, and the stress removal from a value σ is obtained as The previous expression provides ∆T as a function of entropy, but it is customary to plot adiabatic temperature changes as a function of temperature. Such a temperature dependence is easily obtained by plotting each ∆T data at the temperature given by the S (T, σ , H)   We have used the following function to fit the entropy curves with T 0 = 256 K.
The fitting function X(T, σ , H) is taken as: where B and T t are free parameters of the fit which depend on H and σ , and the best fit to experimental data is found for  Fig. 9(e)), and as a function of magnetic field and temperature ( Fig. 9 (f)).
It is worthwhile to remember that due to the hysteresis of the transition, these surfaces are only representative for increasing field and decreasing stress (as indicated by arrows in the axis of the figures). The cross-over behaviour for the magnetocaloric effect is evident within all the range of temperatures and stresses under study ( Fig. 9(e)). Furthermore, the marked decrease in entropy as magnetic field increases together with the weak dependence in stress is also apparent from Fig.   9(f). In relation to the magnetocaloric effect (Fig. 11), the influence of uniaxial stress is very weak (due to the low range of applied stresses). There is a slight shift of the region where the giant (inverse) magnetocaloric effect occurs towards higher temperatures under an applied uniaxial stress.
Furthermore, the overall magnetocaloric effect slightly shifts to higher magnetic fields when stress is applied. For instance, if we focus on the contour plot corresponding to ∆S ∼ 19 J kg −1 K −1 , it is observed that a magnetic field of 2.7 T is needed in the stress free case (Fig. 11(a)), while a magnetic field of 3.3 T is necessary to achieve this value when a 40 MPa stress is applied (Fig. 11(b)).
This shift towards higher magnetic field values is due to the fact that uniaxial stress stabilizes the martensitic phase. The maximum magnetic-field induced entropy change is ∆S = 23.1 kg −1 K −1 , corresponding to a magnetic field of 4 T and the absence of applied stress.  It is to be noted that in general the multicaloric response of a given thermodynamic system ∆S(T, 0 → σ , 0 → H) is not given by the sum of single caloric effects ∆S(T, 0 → σ , 0) and ∆S(T, 0, 0 → H), because there is also a contribution from a cross-coupling term 11,22 , which accounts for the cross-response of the material to the application of non-conjugated fields. In our case, there is no contribution from such a coupling term because we are computing the multicaloric effect arising from the removal of one field (stress) and application of another field (magnetic field). In that case, ∆S(T, σ → 0, 0 → H) can be obtained by the addition of ∆S(T, 0, 0 → H) and ∆S(T, σ → 0, 0, ).
The reversibility under field cycling is a relevant feature for a potential application of a given caloric (multicaloric) effect for refrigeration. A thorough analysis of the reversibility of the multicaloric effect in our Ni-Mn-In alloy has not been possible due to the poor quality of the thermo-grams recorded on cooling. However, the determined hysteresis in temperature, stress and magnetic field of the martensitic transition together with the stress and magnetic-field dependences of the transition temperatures enable us to make some estimates on the reversibility of the caloric effects. By considering a thermal hysteresis of ∼ 12 K, and taking representative values (see Figs.
2(b) and 2(c)) for the typical shift of the transition with field and stress of dT /dµ 0 H ∼ -2 KT −1 and dT /dσ ∼ 0.08 KMPa −1 , it is expected that the magnetocaloric effect is reversible for fields larger than 6 T, while the elastocaloric effect is expected to be reversible for stresses larger than 150 MPa. However, the effective hysteresis in the given external field can be drastically reduced by application of a secondary field 10,11 . While magnetic fields in the order of ∼ 6 T are unfeasible for practical applications, the application of stress can enhance the reversibility of the magnetocaloric effect. In our case, if we consider a magnetic field of 1 T, the magnetocaloric effect is expected to be reversible under the following sequence (which is schematized in the Supplementary Material,

IV. CONCLUSION
We have used a unique calorimeter working under magnetic field and uniaxial load to study the multicaloric response of Ni-Mn-In, a prototype metamagnetic shape-memory alloy. A numerical treatment of our calorimetric data has enabled us to obtain the entropy of the alloy over the whole temperature, magnetic-field, and uniaxial-stress phase-space. Based on this, we could compute single caloric (elastocaloric and magnetocaloric) and multicaloric effects for arbitrary combinations of magnetic field and stress. Our results show that the multicaloric response of Ni-Mn-In exceeds that of single caloric effects. In particular, a suitable combination of magnetic field and stress gives rise to isothermal entropy and adiabatic temperature changes larger than those achievable when only a single external stimulus is applied. Furthermore, the combination of two external stimuli enlarges the temperature window where the alloy exhibits a giant caloric response, which is also accompanied by an enhancement in the reversibility of the caloric response when the external stimuli are cyclically changed.
The advantages of the multicaloric effect in relation to single caloric effects found here may lead to clear improvements in the use of multicaloric materials for cooling applications in refrigeration devices designed to work at low values of the external fields. For instance, the combination of a magnetic field of 1 T, with a uniaxial stress of 40 MPa, yields an isothermal entropy change of 15 J kg −1 K −1 , which is more than double the maximum value achievable for the pure magnetocaloric effect.
It is expected that many of the trends found here for a metamagnetic shape-memory alloy may also be valid for other multiferroic materials with strong coupling between different degrees of freedom. Present results should inspire the development of refrigeration devices, which take advantage of the multicaloric response of multiferroic materials.

V. EXPERIMENTAL SECTION
A sample with nominal composition Ni 50 Mn 35.5 In 14.5 was prepared by arc melting. The ingot was turned upside down and remelted several times to ensure chemical homogeneity. The button was further treated using the suction-casting option of the arc melter. The specimen was subsequently annealed at 900 o C for 24 h, followed by water quenching. From the heat-treated rod, a block with dimensions 2 × 2 × 4.9 mm 3 was cut and polished. A smaller piece was cut for thermomagnetization measurements which were carried out using a Vibrating Sample Magnetometer.
Differential Scanning Calorimetry measurements were conducted by means of a bespoke differential scanning calorimeter, which enables simultaneous measurements of the length of the specimen. The system operates under magnetic fields up to 6 T, and uniaxial loads up to 1. Specific heat was measured for temperatures up to 310 K using a bespoke experimental system 38 , for fields up to 2 T; and for temperatures up to 380 K and fields up to 4 T, using a commecial PPMS (Quantum Design) calorimeter. Temperature changes resulting from the application and removal of magnetic fields under applied uniaxial compressive load were measured by means of a purpose-built experimental system. The specimen temperature was measured by a fine gauge K thermocouple (0.075 mm diameter) in contact with the surface of the sample. The magnetic field and uniaxial load ranges are 0 -2T and 0 -1kN, respectively. The set-up is described in detail in Ref. [27].

SUPPLEMENTARY MATERIAL
Supplementary Material shows a comparison between direct and indirect magnetocaloric adiabatic temperature changes. It also shows three dimensional plots for the isothermal entropy and adiabatic temperature changes for elastocaloric effects under magnetic field and magnetocaloric effects under uniaxial stress. It finally provides a scheme for a reversible multicaloric cycle. Direct measurements of the adiabatic temperature change have been made using an experimental system 39 which enables simultaneous application of uniaxial load and magnetic field, while the sample temperature is measured by a fine gauge K thermocouple (75µm diameter) in thermal contact with the surface of the sample. Results for a magnetic field ot 1.64 T in the absence of stress and for an applied stress (constant) of 40 MPa are shown in Figure S1 as solid symbols, and they are compared to data derived from entropy curves (lines). Direct ∆T data confirm the crossover form the inverse to the conventinal magnetocaloric effects. The temperature region where the inverse magnetocaloric effect derived from entropy curves takes place perfectly matches the region determined from direct measurements. For the inverse magnetocaloric effect, the differences between measured and computed ∆T values (∼ 0.7 K) are mostly due to the lack of absolute adiabaticity of the thermometric system, and are in the order of those expected for that particular device 40 . In contrast, larger differences are found for the conventional magnetocaloric effect. In that case, the discrepancy between measured and computed data must be ascribed to the fact that quasi-direct methods provide reliable ∆T values around a first-order phase transition, however, ∆T data at temperatures beyond the phase transtion can be affected by a considerable error.