Investigation of the inverse magnetocaloric effect with the fraction method

In this study, we examine Ni49Nb1Mn36In14 (nom. at.%) magnetic shape memory alloy (MSMA) to illustrate the inverse magnetocaloric effect (MCE) using the fraction method. The magnetic entropy change, ΔSmag , was calculated with both, the fraction method and the thermomagnetic Maxwell relation. Our results demonstrate that there exists a large magnetization difference between field-cooling and field-heating histories in Ni49Nb1Mn36In14 (nom. at.%) MSMA, which can be attributed to the pinning of lattice entropy and magnetic entropy, as it is well-known that the temperature and applied magnetic fields have an opposing effect on the total entropy change. In addition, we describe the inverse MCE and the contradictory roles on the total entropy change between the two stimuli via the fraction method.


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Introduction
Caloric effects refer to the isothermal entropy change (∆S) or the adiabatic temperature change (∆T ad ) exhibited by a material under the application or removal of an external field. It is well-known that these effects are enhanced at temperatures close to magnetic and structural phase transition temperatures [1]. First-order phase transitions, where the latent heat of a structural transition provides a contribution to the fieldinduced entropy change has been shown to exhibit an unprecedentedly large ∆S [2][3][4]. Two studies, in particular, exhibited significant ∆S when a magnetic field, H, was applied to a magnetocaloric material, i.e. (1) Gd 5 Si 2 Ge 2 [5] and (2) Ni-Mn-In shape memory alloy [6]. Both materials exhibited a meta-magnetic phase transition across structural transition and a giant inverse magnetocaloric effect (MCE). We thus focus our attention to the giant inverse MCE in NiMnIn alloys with tailored compositions that can achieve near room-temperature transitions.
The total entropy change, ∆S T , of an alloy that exhibits a magnetostructural transition is usually denoted by [7], In equation (1), ∆S lat is the entropy change of the lattice and ∆S el and ∆S mag are electronic and magnetic contributions, respectively. It was experimentally demonstrated that the lattice contribution plays a dominant role in ∆S T , while the electronic contribution was negligibly small [8]. Several studies have attempted to explain the nature of the inverse MCE in meta-magnetic SMAs, such as NiMnIn alloys [1,[7][8][9][10]. In these works, a contradictory role was found between ∆S mag and ∆S lat . Specifically, a large change in magnetization was measured during the application of H, which implied a larger ∆S mag on demagnetization, however it was shown that ∆S lat simultaneously decreased. These two contributions to the total entropy change were shown to compete during a magnetic field-driven structural transition and has subsequently been coined 'the dilemma of inverse magnetocaloric materials'.
Interestingly, in NiMnIn magnetic shape memory alloys (MSMAs), it was shown that the magnetic field-induced decrease in martensitic transformation temperatures was closely related to the temperature difference between the hightemperature austenite phase's Curie point (T C ) and the average martensitic transformation temperature (T M ) [11], where higher values of (T C -T M ) usually corresponded to a larger magnetic field-induced transformation temperature shift. Generally, alloying some other elements, especially Nb [12,13] or Ti [11,[14][15][16] in substitution for Ni, can effectively decrease the T M and tune the T C in Ni-Mn-X based alloys. Likewise, several investigations have been performed on the effect of alloying elements such as Pd [17] and Fe [18] on phase transformation characteristics and magnetocaloric performance of Ni-Co-Mn-X (X = In, Sn, Sb, Ga) alloys. The substitution of Ni with Nb or Ti was found to be capable of adjusting the T M , but with a detrimental effect of also decreasing the latent heat of the structural transition [12,14]. In order to investigate the effect of Nb addition on the transition temperatures, Ni 50 Mn 36 In 14 (nom. at. %) was selected as the base MSMA composition. This alloy was found to exhibit a structural transition with martensitic transition temperatures of martensite start (M s ), martensite finish (M f ), austenite start (A s ), and austenite finish (A f ) at 342 K, 319 K, 341 K, and 358 K, respectively. Moreover, the austenite Curie temperature was expected to be 292 K [15,16,19]. The latent heat accompanying the magnetostructural transition was reportedly 50.0 J kg −1 K −1 [15,16,19]. Therefore, we added a small amount (∼1 at.%) of Nb in place of Ni to decrease the transition temperatures with the aim of overlapping T M with the T C in the austenite phase. The purpose of the present investigation was to explore the relationship between ∆S lat and ∆S mag and the inverse MCE Ni 49 NbMn 36 In 14 (at %) MSMA. We employ the (T C -T M ) quantity to rationalize our results and demonstrate the dilemma of the inverse MCE by means of the fractional method, which is applicable to several other MSMA systems.

Experimental
Ni 49 Nb 1 Mn 36 In 14 (nom. at.%) MSMA alloy was prepared by conventional arc melting and then annealed at 1173 K for 24 h in protective argon atmosphere. The compositions of the alloys were determined using energy-dispersive x-ray spectroscopy (EDX) within four different grains after homogenization. Differential scanning calorimetry (DSC) was performed in a temperature range between 200 K and 425 K with 10 K min −1 for both heating and cooling. For temperature and magnetic field-dependent magnetization measurements, a physical property measurement system (Quantum Design), equipped with a vibrating sample magnetometer, was employed. The temperature and magnetic field sweep rates were at 2 K min −1 and 0.2 mT s −1 , respectively. Sample mass was determined with 0.01 mg accuracy before the magnetic measurements and placed into plastic capsules and measured with a contamination-free half-cylinder brass sample holder. Contamination and the reproducibility of the measurements were regularly controlled as described in [20].
Isothermal magnetization data were processed using the traditional Maxwell relation. Additionally, we performed analyses on isofield thermomagnetic curves via the fraction method to calculate the isothermal field-induced entropy changes. The fractional method used in this study was based on the Clausius-Clapeyron (CC) equation for first-order transition and equilibrium thermodynamics to estimate the practical MCE [11,14,[21][22][23][24].

Results and discussion
According to our EDX measurements, the compositions (in at.%) of the samples were determined to be Ni 48.89±1.20 Nb 1.10±0.10 Mn 36.10±0.80 In 13.89±0.73 , where they were within 1 at.% of the nominal target (see figure 1 in supp. mat.). Figure 1 shows the DSC curve of the alloy. The exothermic peak (on cooling) and endothermic peak (on heating) corresponds to the latent heat of the magnetostructural transition. M s , M f , A s , and A f temperatures are determined as 285 K,  278 K, 285 K, 297 K and Curie temperature, was determined as 290 K. The structural transition occurred with a thermal hysteresis of approximately 15 K and the ∆S T was found to be nearly 40.0 J kg −1 K −1 for cooling and 36.6 J kg −1 K −1 on heating. As expected, martensitic transition temperatures and ∆S T decreased with the substitution of Ni with Nb, from the base alloy, whereby the structural transition temperatures, T M , and magnetic transition temperatures T C , nearly overlapped. In prior works, the close proximity of T C and M s was related to very sharp magnetostructural transitions [25][26][27][28].
Figures 2(a)-(e) show the temperature dependence of magnetization under several constant applied magnetic fields. The characteristic temperatures for the martensitic transition under 7 T are indicated by arrows. For all applied field levels, the field heated (FH) data in all M(T) measurements do not retrace the field cooled (FC) data, but instead exhibit some hysteresis, which is also a signature of a first-order structural transition. The transformation temperatures under 0.01 T are comparable to those from DSC. The Ni 49 Nb 1 Mn 36 In 14 (at. %) shows successive magnetic transitions on cooling namely, such as from the paramagnetic (PM) austenite phase to a ferromagnetic (FM) austenite phase at T A C ≈ 290 K with further under cooling the MSMA transforms from the FM austenite phase to the low magnetic martensite state. As shown in figure 2(a), the zero-FC (ZFC) and FC curves bifurcate at temperatures below T G , which indicates the presence of magnetic heterogeneities or mixed exchange interactions in the martensite phase. The martensite in Heusler alloys has been shown to possess a variety of magnetic configurations such as PM/anti-FM (AFM) or spin glass clusters when cooled to certain temperatures (i.e. the blocking temperature) [29][30][31]. Interestingly, we observed fluctuations in the thermogram on heating (figure 1) above A f . In addition, we observed a shift to higher temperatures in magnetic phase transition of the second order (T A ′ C ) of in the M(T) curves. This behavior was attributed to the existence of a non-collinear FM phase [32]. We believe these are due to magnetic heterogeneities or mixed exchange interactions in the martensite phase that persist to austenite on heating. A similar shoulder-like peak was observed in DSC data for Ni 50 Mn 35 In 13.9 B 1.1 [32], which it was attributed to the co-existence of two magnetic phases near T C . Nevertheless, it is known that even though the structural transition temperatures are, in general, very sensitive to composition in the Ni-Mn-In alloys, the T C of the austenite is frequently composition independent, a feature that also holds for the present case. Similar T C temperatures were reported for close compositions such as 304 K for Ni 50 Mn 34 In 16 [33,34], 289 K for Ni 50 Mn 36 In 14 [16], and 310 K Ni 50 Mn 35 In 15 [35,36].
As shown in figure 2(a), there exists a magnetic transition for the martensite phase at 100 K. This transition resembles the Néel temperature in ferrimagnetic materials. Furthermore, the presence of the peak in the ZFC curve is might be due the to spin-glass behavior with a glass transition temperature (T G ≈ 110 K) or from AFM to ferrimagnetic transition [36]. In figures 2(b)-(e) we observed that 7 T was insufficient to saturate the magnetization in either phase and the minimum value of magnetization was decreased with increasing the magnetic field, which is further evidence of mixed magnetic ordering in the martensite phase [36]. Therefore, as proposed by Bennett et al, we believe that there may exist both, ferrimagnetic and FM ordering in the martensite phase [36]. Figures 2(b)-(e) presents a considerable change in magnetization (∆M) during the martensitic transition. It was apparent from this data that there was a significant difference between the ∆M of cooling and heating directions, where ∆M on cooling was greater than on heating. Specifically, ∆M during cooling (forward) and heating (reverse) was determined as 67.5 emu g −1 and 48.5 emu g −1 , respectively, for 5 T. As the applied magnetic field increased the martensitic transitions temperatures shifted to lower temperatures due to the stabilization of the FM austenitic phase. The characteristic temperature of M s and M f , linearly decreased with the applied magnetic field at a rate of −2.02 K T −1 and −1.90 K T −1 shown in figure 2(f) and those of A s and A f were measured to be −2.07 K T −1 and −0.55 K T −1 , respectively. Another interesting feature was that the transformation range (M s -M f ) while cooling was 5 K for all applied magnetic field levels. Nevertheless, while heating the transition range (A f -A s ) was slightly increased from 14 K under 0 T to 24 K under 7 T. Also, the thermal hysteresis increased from 29.5 K to 34 K with applied 7 T magnetic field. As such, the applied magnetic field increased, and the difference between FC and FH transformation temperature ranges also increased. This change was an indication of the difficulty of phase front motion between martensite and austenite phases with higher applied fields. While heating, the change of characteristic temperatures of A f and A s with the applied magnetic field were distinct from each other, this distinction might be also due to the difficulty of phase front motion between martensite and austenite phases. In figure 2(b), another significant feature was a fluctuation observed below the A f temperature in the FH in the presence of a 1 T magnetic field. As applied magnetic field increased, this fluctuation shifted to lower temperatures, shown as triangle in figures 2(b)-(d). The fluctuation temperature decreased with the applied magnetic field at a rate of −1.22 K T −1 . This fluctuation might be related with magnetic heterogeneities or mixed exchange interactions when transforming from martensite to austenite.
The magnetic entropy change of magnetocaloric materials with first-order magnetostructural transformation can be determined from the isofield magnetization data or isothermal magnetization data. We used two of different techniques to calculate the magnetic entropy change in the Ni 49 Nb 1 Mn 36 In 14 alloy to obtain a reliable evaluation of the MCE. First, we estimated the magnetic entropy change ∆S mag with the transformation fraction method, since the transformed fraction of the material directly reflects the yielded MCE [11,14,21,23,24]. The fraction method in equation (2) was based on the CC slope and is used to compute the magnetic field induced entropy change ∆S mag [11,14,21,23,24]. Figure 3(a) shows the FH and FC-M(T) curves from 200 K to 350 K under the application of 5 T magnetic field. Red/blue lines refer to FH/FC magnetization data, respectively, and cyan/magenta lines, on the same graph, correspond to fraction lines, f(T) calculated with equation (2): where M A (T) and M M (T) represent the magnetization of the low-temperature and high-temperature phases, respectively. The total magnetization was assumed to be related to the phase volume fraction, thereby the austenite or martensite phase fraction, f(T) could be computed with the M(T) data [37][38][39].
Because the temperature and magnetic field are known to be thermodynamic forces which trigger the structural martensite phase change, the phase fraction was calculated quantitatively using thermomagnetization data acquired under various magnetic fields [37]. Figure 3(b) shows the fraction data for 0.1 mT and 5 T applied fields and the fraction difference upon heating and cooling. Furthermore, ∆S mag with the phase fraction induced by the application of a magnetic field, namely the MCE [11,14,21,23,24], was calculated by equation (3), where ∆M was the difference between 100 Oe and 5 T in magnetization between the two phases at the transition temperature, dT/dH was the rate of change of the transition temperature during cooling/heating direction with the magnetic field ( figure 2(f)). The negative values in the ∆f FH curve (yellow) of figure 3(b) above 300 K were due to magnetic interactions that existed in the material. The magnetic field induced entropy change calculated with equation (4) which was plotted in figure 4(b) for 5 T as a function of temperature for FH and FC, which is labeled as CC equation using isofield magnetization measurements (CCIF)-FH and CCIF-FC, respectively. The maximum value of CCIF-FC was around 34 J kg −1 K −1 , whereas for CCIF-FH was around 13 J kg −1 K −1 .
To calculate ∆S mag via the Maxwell relation, we measured magnetic field-dependent magnetization curves, and the measurement was done by following the procedure reported in [38]. Figure 4(a) shows M(H) curves at several temperatures, whereby the samples showed a metamagnetic transition and above 305 K sample shows FM behavior. The maximum magnetic hysteresis was measured to be 3.0 T at 290 K with a critical magnetic field, i.e. the field needed to initiate the martensite to austenite transformation, of 2.7 T. The ∆S mag can be computed across T C using the Maxwell relation (denoted as Maxwell relation with a numerical form using isothermal magne-tization measurements (MRIT) in figure 4(b)) using the relationship [11,40,41], where, T k is an isothermal test temperature, ∆T k = (T k+1 − T k ), and T k = (T k+1 + T k )/2. The maximum value of ∆S mag (MRIT) was estimated to be 11.5 J kg −1 K −1 , which is in good agreement with ∆S mag calculated by the CCIF-FH method. However, ∆S mag estimated from MRIT was quite different from that estimated with CCIF-FC. The inapplicability of MRIT was by addressed Niemann et al [42], where they ascribed field-induced martensite reorientation to spurious peaks in the inverse MCE. In line with [42] we also observed a sudden increase in magnetization at the critical field, H cr , see figure 4(a), however, we did not observe spurious peaks. Thus, we concluded that we had reliable MRIT results with no martensite reorientation. According to the work in [38], the martensite phase in our base alloy (Ni 50 Mn 35 In 15 at.%) is ferrimagnetic at low temperatures and transforms into a ferromagnet near room temperature. As the applied magnetic field is increased and it reaches a critical value (H cr ), the field-induced ferri-to-FM transition begins and eventually saturates as the applied magnetic field increased to 5 T. However, in the present study, the magnetization had not saturated at 5 T at 305 K. It was also stated in [36] that while decreasing the applied magnetic field, a significant fraction of the sample was preserved in the FM state (presumably austenite) and did not transform to the ferrimagnetic state, which was also observed in the present case. The hysteresis loss (HL) was determined by integrating the areas between magnetization and demagnetization branches of the M(H) isothermal magnetization data (shaded area in figure 4(a)). The average HL (AHL), calculated by averaging the integral area under the temperature range of the full width at half maximum of the hysteresis peak AHL  Ni 49 Mn 37 In 14 [47]. Moreover, ∆M increased in Ni 49 Mn 37 In 14 by adding 1 at.% Co in place of Ni [47]. Conversely, ∆M, was found to decrease in Ni 50 Mn 36 Sn 14−x Bi x , as the Bi content increased [48] while the temperature interval between T C and T M subsequently decreased. Eventually, the smallest ∆M was observed for the highest Bi ratio in [48]. Similar observations were also made for Ni 50 Mn (50−x) In x (x = 15.5 and 16), where (T A C -T M ) and ∆M were determined to be 13 K and 61 K, 17 emu g −1 and 35 emu g −1 [17], respectively. In this study, the (T A C -T M ) temperature interval was observed as −7 K. It was stated that T M > T C MCE exhibits low entropy values due to the weak magnetism [1,17], which is valid in this study. Therefore, the temperature interval (T A C -T M ) appears to affect the magnitude of ∆M and dT/dH, and thereby ∆S mag . Together, these results provide important insights into the proximity of T A C and T M for alloy design. On account of the calculated values of ∆S mag for the current study, we observed similar values for CCIF-FH and MRIT as in line with other studies [11,14,20,[23][24][25][26], however, there exists a discrepancy between the calculated values of ∆S mag for and CCIF-FC and MRIT. This is mainly because a smaller amount of the sample is allowed to convert to martensite at 5 T during the FH process with respect to FC process which is shown in figure 3(b). Namely, based on equation (2), this discrepancy could be attributed to the large ∆M difference and higher dT/dH difference between cooling and heating. The main reason for this discrepancy might be related to the dilemma of the inverse MCE while heating as it is already known that there exists a contradictory role of lattice entropy and magnetic entropy [7,27,28]. Previous studies have reported that applied magnetic field stabilizes FM austenite, and thereby the martensitic transition temperatures decrease. Additionally, cooling the sample simultaneously with the application of a magnetic field ensures that a greater fraction of the sample is under the phase transition. Consequently, the change of phase fraction induced by the driving forces (magnetic field and temperature) for the phase transition which is ∆f (T, ∆H) directly reflects the phase transformation completion of the FC process is higher than the FH process as shown in figure 3(b). This corresponds to the CCIF-FC case, in our present study. While heating, the temperature driven transitions force the sample into the austenite phase (reverse), but the application of magnetic field forces the sample into a martensitic phase, consequently a lesser fraction of the sample undergoes a martensitic transition. Considering that the ∆S T of 40 J kg −1 K −1 was measured from DSC data for Ni 49 Nb 1 Mn 36 In 14 (at %) MSMA, the lattice entropy and magnetic entropy obviously counteract one another.

Conclusion
In this paper, we utilized the fraction method to explain the contradictory roles of the applied temperature and magnetic field on the MCE in Ni 49 Nb 1 Mn 36 In 14 (at %) MSMA. A small negative contribution to ∆S mag was measured in the austenite phase when T M and T C were in close proximity. The following conclusions can be drawn from the present study. Ni 49 Nb 1 Mn 36 In 14 (at %) has a mixed magnetic structure. The difference in magnetization (∆M) of FC and FH curves under high magnetic fields might be due to the pinning effect of the lattice entropy and magnetic entropy of each other while having contradictory roles. In addition, the dilemma of inverse magnetocaloric materials, which is demonstrated by the fraction method while heating, is due to contradictory role the temperature and magnetic field. Finally, one could have maximum MCE whether the temperature and magnetic field drive the sample in the same direction, but this is contrary to the nature of materials which show inverse magnetocaloric properties.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).