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*Crystals*
**2018**,
*8*(8),
329;
doi:10.3390/cryst8080329

Article

The Effect of Different Atomic Substitution at Mn Site on Magnetocaloric Effect in Ni50Mn35Co2Sn13 Alloy

^{1}

School of Materials Science and Engineering, University of Science and Technology of Beijing, Beijing 100083, China

^{2}

ChuanDong Magnetic Electronic Co., Ltd., FoShan 528513, GuangDong, China

^{3}

ChengXian Technology Co., Ltd., FoShan 528513, GuangDong, China

^{4}

Voggenreiter Technology (Beijing) Co., Ltd., 3006 Jinao International, No. 19 Madian East Rd., Haidian Dist., Beijing 100088, China

^{*}

Author to whom correspondence should be addressed.

Received: 23 June 2018 / Accepted: 15 August 2018 / Published: 18 August 2018

## Abstract

**:**

The effect of different atomic substitutions at Mn sites on the magnetic and magnetocaloric properties in Ni

_{50}Mn_{35}Co_{2}Sn_{13}alloy has been studied in detail. The substitution of Ni or Co for Mn atoms might lower the Mn content at Sn sites, which would reduce the d-d hybridization between Ni 3d e_{g}states and the 3d states of excess Mn atoms at Sn sites, thus leading to the decrease of martensitic transformation temperature T_{M}in Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{3}Sn_{13}alloys. On the other hand, the substitution of Sn for Mn atoms in Ni_{50}Mn_{34}Co_{2}Sn_{14}would enhance the p-d covalent hybridization between the main group element (Sn) and the transition metal element (Mn or Ni) due to the increase of Sn content, thus also reducing the T_{M}by stabilizing the parent phase. Due to the reduction of T_{M}, a magnetostructural martensitic transition from FM austenite to weak-magnetic martensite is realized in Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{2}Sn_{14}, resulting in a large magnetocaloric effect around room temperature. For a low field change of 3 T, the maximum ∆S_{M}reaches as high as 30.9 J/kg K for Ni_{50}Mn_{34}Co_{2}Sn_{14}. A linear dependence of ΔS_{M}upon μ_{0}H has been found in Ni_{50}Mn_{34}Co_{2}Sn_{14}, and the origin of this linear relationship has been discussed by numerical analysis of Maxwell’s relation.Keywords:

magnetocaloric effect; heusler alloys; magnetic propertiesPACS:

75.30.Sg; 81.30.Kf; 75.50.Cc## 1. Introduction

Over the past decades, Ni-Mn-Z (Z = Ga, In, Sn, and Sb) Heusler alloys have attracted significant attention due to its noteworthy multifunction properties, such as magnetic shape memory effect [1], magnetoresistance [2,3], exchange bias (EB) [4], and magnetocaloric effect (MCE) [5,6]. As one of the typical Ni-Mn-Z Heusler alloys, Ni-Mn-Sn alloy undergoes a martensitic transformation from ferromagnetic (FM) austenite to weak-magnetic martensite, which is accompanied with an abrupt change of magnetization ΔM [6]. This large ΔM across martensitic transformation results in a high difference of Zeeman energy E

_{zeeman}= μ_{0}HΔM, which drives a metamagnetic transition from the weak-magnetic martensite to FM austenite, thus leading to a large MCE [5,7]. Therefore, it is desirable to enhance the ΔM during martensitic transformation in order to obtain a large MCE.It has been reported that the stoichiometric Ni

_{2}MnSn alloy does not exhibit martensitic transformation while some Mn-rich Ni-Mn-Sn alloys show martensitic transformation from FM austenite to weak-magnetic martensite [8,9,10]. However, the excess Mn atoms would occupy the vacant Sn sites (4b positions), and are coupled antiferromagnetically (AFM) to the surrounding Mn atoms on the regular Mn site (4a positions) [11,12]. This fact suggests that excess Mn would lead to the weakness of ΔM during the martensitic transformation. The introduction of Co can act as a “FM activator” to induce the Mn moments to align in an FM order and enhance the magnetization of austenite phase, thus causing a larger ΔM as well as a large MCE [12,13]. Similar results have also been reported in other Heusler alloys [14,15], e.g., the substitution of Co for Ni modifies the magnetic structure of the austenite into FM as the preferred state and reduces the martensitic transformation temperature [15]. Furthermore, the martensitic transformation temperature (T_{M}) increases by substituting Mn with Co atoms, which is probably attributed to the rule of valence electron concentration [16]. Recently however, some studies have shown that the T_{M}does not increase monotonously by increasing the Co substitution for Mn atoms, suggesting that there is a disagreement of the rule of valence electron concentration [12,17]. The substitution of Mn by Ni atoms in Ni-Mn-Sn alloys increases the T_{M}remarkably while the MCE still remains nearly constant [18,19]. In addition, the substitution of Mn by Sn causes a reduction of T_{M}while the MCE remains nearly unchanged [20,21]. Consequently, different atomic substitutions at Mn sites have different effects on the martensitic transformation and the MCE. Unfortunately, to the best of our knowledge, a systematical study on different atomic substitutions at Mn sites in the Ni-Mn-Co-Sn system is still lacking. In the present work, we systematically study the effect of substituting Ni, Co, and Sn for Mn atoms for the magnetic and magnetocaloric properties in Ni_{50}Mn_{35}Co_{2}Sn_{13}alloy.## 2. Experimental

The Ni

_{50}Mn_{35}Co_{2}Sn_{13}(parent alloy), Ni_{51}Mn_{34}Co_{2}Sn_{13}(Ni for Mn), Ni_{50}Mn_{34}Co_{3}Sn_{13}(Co for Mn), and Ni_{50}Mn_{34}Co_{2}Sn_{14}(Sn for Mn) alloys were prepared by arc melting appropriate proportion of constituent components with a purity better than 99.9 wt.% under an argon atmosphere. The as-cast samples were wrapped by tantalum foil and annealed in a high-vacuum quartz tube at 1173 K for 96 h, followed by quenching in ice water. It is noted that the effect of different heat treatments on the magnetic and magnetocaloric properties has been studied intensively in NiMn-based Heusler alloys [22,23,24,25]. It is revealed that the MCE can be largely improved by optimizing the heat treatment, e.g., an optimized annealing method can reproduce the excellent functional properties of Ni-Co-Mn-Al films in ribbons [25]. Here, we chose the same heat treatment from Reference [17], which also studied Ni_{50}Mn_{34}Co_{2}Sn_{14}and presented giant MCE in this alloy. The final composition of the samples was determined by Energy Dispersive Spectrometry (EDS) using a JEOL-6060 Scanning Electron Microscope (SEM) from Akishima, Tokyo, Japan, and is shown in Table 1. It can be seen that the final composition is quite close to the nominal composition.The phase and crystal structure were investigated by using Rigaku D/max-2400 diffractometer with Cu Kα radiation from Tokyo, Japan. The differential scanning calorimetry (DSC) curves were measured using DSC 6220 with heating and cooling rates of 10 K/min. Magnetizations were measured as functions of temperature and the magnetic field using a cryogen-free cryocooler-based physical property measurement system (model VersaLab) from Quantum Design Inc., San Diego, CA, USA. In order to avoid the spurious magnetic entropy change (ΔS

_{M}), induced by the residual effect generated in standard process, the magnetization isotherms curves were measured in a loop process, in which the sample is cooled down to the weak-magnetic martensite and then warmed up to the target temperature before starting each M-H measurement. [26,27]. In this way, the phase transition is always crossed in the same sense. The M-μ_{0}H curves were corrected by taking into account the demagnetization effect, i.e., ${H}_{\mathrm{int}}={H}_{ext}-{N}_{d}M$.## 3. Results and Discussion

Figure 1 shows the powder X-ray diffraction (XRD) patterns at an ambient temperature for all the alloys. The Ni

_{50}Mn_{35}Co_{2}Sn_{13}parent alloy crystallizes into the 10 M modified orthorhombic martensitic structure at room temperature. In comparison, the XRD patterns reveal the matrix of austenitic phase with the Heusler L2_{1}cubic structure (Fm$\overline{3}$m space group) for the other substituted alloys. This result indicates that the T_{M}is above room temperature for the Ni_{50}Mn_{35}Co_{2}Sn_{13}parent alloy while it is reduced to below room temperature by the substitution of Mn. In addition, a small peak (denoted by “r”) is observed at the (2 2 0) Bragg peak of martensitic structure for Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{2}Sn_{14}, corresponding to the residual martensitic phase. Thus, it suggests that the T_{M}of these two alloys is close to room temperature [28].The DSC heat flow curves of Ni-Mn-Co-Sn alloys upon heating and cooling with a ramp rate of 10 K/min are displayed in Figure 2a. Well-defined exothermic and endothermic peaks, with distinct thermal hysteresis, indicate the first-order martensitic and reverse martensitic transformations upon cooling and heating, respectively [29]. It is clearly seen that the T
where $\frac{dQ}{dt}$ is the power of heat flow, $\frac{dT}{dt}$ is the heating or cooling rate, and T

_{M}of Ni_{50}Mn_{35}Co_{2}Sn_{13}parent alloy is above room temperature, the T_{M}of Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{2}Sn_{14}is just below room temperature, and the T_{M}of Ni_{50}Mn_{34}Co_{3}Sn_{13}is much lower than room temperature, respectively. This result is consistent with our analysis based on the XRD measurements. Based on the DSC curves, the entropy change (ΔS) associated to the structural transformation was calculated by the following equation:
$$\Delta S={\displaystyle {\int}_{{T}_{s}}^{{T}_{f}}\left[\left(\frac{dQ}{dt}\right){\left(\frac{dT}{dt}\right)}^{-1}\frac{1}{T}\right]}dT$$

_{s}and T_{f}are the starting and finishing temperatures of the structural transformation, respectively. Table 2 lists the ΔS values at the structural transition for Ni-Mn-Co-Sn alloys.The Figure 2b shows the martensitic transformation temperature (T

_{M}) and reverse martensitic transformation temperature (T_{A}) as a function of different atomic substitution and valence electron concentration e/a. The number of valence electrons for Ni, Mn, Co, and Sn atoms are 10 (3d^{8}4s^{2}), seven (3d^{5}4s^{2}), nine (3d^{7}4s^{2}), and four (5s^{2}5p^{2}), respectively. The e/a value of Ni-Mn-Co-Sn alloys is calculated by the following equation [30]:
$$e/a=\frac{10\times {\mathrm{Ni}}_{at.\%}+7\times {\mathrm{Mn}}_{at.\%}+9\times {\mathrm{Co}}_{at.\%}+4\times {\mathrm{Sn}}_{at.\%}}{{\mathrm{Ni}}_{at.\%}+{\mathrm{Mn}}_{at.\%}+{\mathrm{Co}}_{at.\%}+{\mathrm{Sn}}_{at.\%}}$$

Generally, the T

_{M}of NiMn-based Heusler alloys is related to the e/a and would increase with the increase of e/a [11,29,31]. However, it is found from Figure 2b that the structural transformation temperature does not monotonously increase with the enhancement of e/a. This non-monotonical dependence of T_{M}on e/a has also been reported in other NiMn-based Heusler alloys [32,33,34]. In Heusler alloys X_{2}YZ, there are four Wyckoff-positions, namely A (0, 0, 0), B (0.25, 0.25, 0.25), C (0.5, 0.5, 0.5), and D (0.75, 0.75, 0.75), respectively. Generally, the site preference of X and Y transition metal atoms is dependent upon the number of their valence electrons. The atom with more valence electrons prefers the A and C positions, while the atom with fewer valence electrons tends to occupy the B position, and the main group element Z always enters into the D site [35,36,37]. According to this rule, in the present case, Ni atoms with more valence electrons would occupy the A and C positions, while Mn atoms with the relatively fewest valence electrons would enter into the B position. Besides this, Sn, Co, and excess Mn atoms occupy the D site. This speculation about the atomic occupation needs to be confirmed by further experiments. Based on the study of the correlation between the electronic structure and martensitic phase transition of Ni-Mn-Sn by hard X-ray photoelectron spectroscopy and ab initio calculation, the d-d hybridization between Ni 3d e_{g}states and the 3d states of excess Mn atoms at Sn sites is believed to be the main driving force for the martensitic transformation [38,39]. Once the d-d hybridization between Ni and Mn atoms is established, any change in the Ni or Mn content would tend to weaken the hybridization and reduce T_{M}[38,39]. Here, the substitution of Ni or Co for Mn atoms might lower the Mn content at Sn sites, thus reducing the d-d hybridization between Ni 3d e_{g}states and the 3d states of excess Mn atoms at Sn sites—resulting in the decrease of T_{M}in Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{3}Sn_{13}alloys. On the other hand, the p-d covalent hybridization between the main group element (Sn) and the transition metal element (Mn or Ni) also plays an important role in stabilizing the parent phase [40,41], thus leading to the reduction of T_{M}by increasing the content of p-group elements [9]. In Ni_{50}Mn_{34}Co_{2}Sn_{14}, the increase of Sn content would enhance the p-d covalent hybridization and therefore reduce the T_{M}by stabilizing the parent phase.Figure 3a–d shows the temperature dependence of zero-field-cooling (ZFC) and field-cooling (FC) magnetization for all the alloys at 0.05 T and 3 T, respectively. For the Ni

_{50}Mn_{35}Co_{2}Sn_{13}parent alloy with the highest T_{M}(Figure 3a), the martensitic transformation nearly coincides with the paramagnetic (PM) to ferromagnetic (FM) magnetic transition of austenite, causing a small transition peak under 0.05 T. With the application of a high field of 3 T, the FM austenite can be induced by metamagnetic transition from both PM austenite and weak-magnetic martensite, which results in the decrease of T_{M}and the increase of the magnetic transition temperature of austenite (${T}_{C}^{A}$), thus causing the more prominent transition peak [42]. Additionally, the Ni_{50}Mn_{35}Co_{2}Sn_{13}parent alloy experiences a magnetic transition of martensite from a ferromagnetic to a weak-magnetic state at the = 190 K. With the T_{M}decreasing to below the ${T}_{C}^{A}$, the Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{2}Sn_{14}undergo a magnetostructural martensitic transition from FM austenite to weak-magnetic martensite with distinct thermal hysteresis (Figure 3b,d). Moreover, a large ΔM of 35 Am^{2}/kg can be obtained in Ni_{50}Mn_{34}Co_{2}Sn_{14}under 3 T through the magnetostructural transformation, which results in a large Zeeman energy difference between FM austenite and weak-magnetic martensite and implies a possibly high MCE according to the Clausius-Clapeyron relation $\Delta S=(\Delta M/\Delta T)\times \Delta {\mu}_{0}H$ [1]. For Ni_{50}Mn_{34}Co_{3}Sn_{13}alloy, as shown in Figure 3c, the T_{M}further reduces to below the ${T}_{C}^{M}$, and thus a martensitic transformation from FM austenite to FM martensite is obtained.The magnetization isotherms of Ni-Mn-Co-Sn alloys with increasing temperature upon field ascending and descending modes are presented in Figure 4a–d. The M-μ

_{0}H curves of Ni_{50}Mn_{35}Co_{2}Sn_{13}parent alloy increases almost linearly with increasing magnetic field, corresponding to the typical characteristic of PM/weak-magnetic state (Figure 4a). Meanwhile, M-μ_{0}H curves around T_{M}show a slight curvature with small magnetic hysteresis. This fact is attributed to the field-induced reverse martensitic transformation from weak-magnetic martensite to FM austenite, consistent with the result of thermomagnetic measurements in Figure 3a. Large magnetic hysteresis can be seen in the other substituted alloys, revealing the first-order martensitic transformation. As discussed above, the Ni_{50}Mn_{34}Co_{3}Sn_{13}alloy experiences a martensitic transformation in an FM state, which can be confirmed by the strong curvatures of M-μ_{0}H curves around the transition temperature T_{M}(Figure 4c). On the other hand, the Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{2}Sn_{14}undergo a magnetostructural martensitic transition from FM austenite to weak-magnetic martensite. Therefore, a dramatic field-induced metamagnetic transition from weak-magnetic martensite to FM austenite with more distinct magnetic hysteresis is observed in Figure 4b,d. For example, the maximum hysteresis loss of Ni_{50}Mn_{34}Co_{2}Sn_{14}reaches as high as 66 J/kg. This field-induced metamagnetic transition with remarkable hysteresis is attributed to the large Zeeman energy difference between the FM austenite and weak-magnetic martensite [43]. Meanwhile, it has to be pointed out that this large hysteresis loss during magnetization and demagnetization would lower the effective refrigerant capacity of the magnetic refrigerant, which is unfavorable for practical applications. Fortunately, the large hysteresis in Heusler alloys can be reduced effectively by fine-tuning the lattice parameters or using external bias stimuli such as hydrostatic pressure [5].Based on the magnetization isotherms, the ΔS

_{M}value can be calculated by using Maxwell relation [44]:
$$\Delta {S}_{M}={\mu}_{0}{{\displaystyle {\int}_{0}^{H}(\partial M/\partial T)}}_{H}dH$$

The validity of the Maxwell relation for first-order magnetic transition has been disputed in the past years since a giant spurious spike may be obtained by using the Maxwell relation for the first-order magnetic transition [45,46]. However, recently Amaral et al. [47,48] found that the breakdown of the Maxwell relation should not be interpreted as a consequence of the first-order magnetic transition, but a failure caused by not considering the non-equilibrium state of coexisting phases and the concomitant history dependence of the state. Furthermore, Caron et al. [26,49] pointed out that the spurious ΔS

_{M}spike can be avoided by measuring the isothermal magnetization in a loop process, and so the Maxwell relation is still feasible for the first-order magnetic transition. Consequently, the Maxwell relation is applicable in the present work since the magnetization isotherms were measured in a loop process. Figure 5a shows the temperature dependence of ∆S_{M}for Ni-Mn-Co-Sn alloys under different magnetic field changes of 1 T, 2 T, and 3 T, respectively. The Ni_{50}Mn_{35}Co_{2}Sn_{13}parent alloy shows a small ∆S_{M}value of 2.0 J/kg K for a field change of 3 T. On the other hand, large ∆S_{M}values can be obtained in the other substituted alloys, especially in the ones with magnetostructural martensitic transition. Ni_{50}Mn_{34}Co_{2}Sn_{14}exhibits the highest ∆S_{M}value in this series of alloys, e.g., the maximum ∆S_{M}is 30.9 J/kg K for a field change of 3 T. In comparison with the ΔS values at the structural transition listed in Table 2, it is seen that the ΔS_{M}(30.9 J/kg K) under a field change of 3 T for Ni_{50}Mn_{34}Co_{2}Sn_{14}is quite close the total entropy change ΔS of 31.6 J/kg K at the transition, suggesting that the 3 T is nearly the saturation magnetic field which leads to the completion of phase transformation from weak-magnetic martensite to FM austenite. Besides, the ΔS_{M}values for the rest of alloys are much lower than the ΔS values obtained from the calorimetric curves, indicating that the phase transformation in these alloys needs to be completed by a higher magnetic field.In order to investigate the magnetic field dependence of ΔS
where ΔS

_{M}, the maximum ΔS_{M}as a function of μ_{0}H for the Ni_{50}Mn_{34}Co_{2}Sn_{14}alloy is plotted as an example in Figure 5b. It is noted that the ΔS_{M}follows a linear relationship with the variation of the magnetic field when μ_{0}H > 0.2 T:
ΔS

_{M}= ΔS_{0}+ κ μ_{0}H_{0}is the intercept value when the field is zero, and κ is the slope factor which describes how strong the ΔS_{M}depends on μ_{0}H. The adjusted R-squared factor is 0.99922, indicating the excellent linear fitting. Similar linear relationships between ΔS_{M}and μ_{0}H have also been reported in other studies [50,51]. However, a slight deviation can be found in the low field range (inset of Figure 5b). The origin of this linear relationship and the deviation at low fields will be discussed in the following section.Since the magnetization isotherms were measured at discrete temperature intervals, the Maxwell relation can be numerically approximated to [52]:
where M(T

$$\begin{array}{ll}\Delta {S}_{M}(\frac{{T}_{1}+{T}_{2}}{2},H)& =\frac{{\mu}_{0}}{{T}_{2}-{T}_{1}}\left[{\displaystyle {\int}_{0}^{H}M({T}_{2},H)dH-{\displaystyle {\int}_{0}^{H}M({T}_{1},H)dH}}\right]\\ & \text{}={\mu}_{0}{\displaystyle \sum _{i}\frac{M({T}_{2},{H}_{i})-M({T}_{1},{H}_{i})}{{T}_{2}-{T}_{1}}\Delta {H}_{i}}\end{array}$$

_{1}, H_{i}) and M(T_{2}, H_{i}) are the magnetization values measured at temperatures T_{1}and T_{2}at a magnetic field H_{i}, respectively. Taking 278 K and 280 K as T_{1}and T_{2}for Ni_{50}Mn_{34}Co_{2}Sn_{14}alloy, $\frac{M({T}_{2},{H}_{i})-M({T}_{1},{H}_{i})}{{T}_{2}-{T}_{1}}=\frac{\Delta M}{2}$, where ΔM is the difference between M_{278 K}and M_{280K}at H_{i}upon field decreasing mode. Figure 6 shows the ΔM/2 between 278 K and 280 K as a function of the magnetic field for the Ni_{50}Mn_{34}Co_{2}Sn_{14}alloy. According to Equation (5), the ΔS_{M}value at 279 K is the integral area under the ΔM/2 vs. μ_{0}H curve. It is found that the ΔM/2 increases sharply at low fields, which is due to the dramatic change of magnetization as shown in Figure 4d. Then, the ΔM/2 reaches a maximum value and starts to decrease. The decrease of ΔM/2 becomes slow after the break point ΔM_{break}/2. Thus, the ΔS_{M}can be divided into two parts by ΔM_{break}/2. The first part ΔS_{M}_{1}is the integral area below the critical field ${\mu}_{0}{H}_{\Delta {M}_{break/2}}$, and it is a constant ΔS_{M}_{1max}when the field is higher than ${\mu}_{0}{H}_{\Delta {M}_{break/2}}$.When the field is higher than the critical field ${\mu}_{0}{H}_{\Delta {M}_{break/2}}$ of ΔM
where ΔM

_{break}/2, ΔS_{M}= ΔS_{M}_{1max}+ ΔS_{M}_{2}, where ΔS_{M}_{1max}is a constant as the integral area below ΔM_{break}/2 while ΔS_{M}_{2}is a variable as the integral area between the ${\mu}_{0}{H}_{\Delta {M}_{break/2}}$, and the final field is μ_{0}H. From Figure 6, the ΔS_{M}_{2}can be approximately considered to be a trapezoid, and so it could be estimated from
$$\begin{array}{ll}\Delta {S}_{M2}(T,H)& =\frac{1}{2}\times (\Delta {M}_{break}/2+\Delta M/2)\times ({\mu}_{0}H-{\mu}_{0}{H}_{\Delta {M}_{break/2}})\\ & \text{}=\Delta {M}_{ave}/2\times ({\mu}_{0}H-{\mu}_{0}{H}_{\Delta {M}_{break/2}})\\ & \text{}=(-\Delta {M}_{ave}/2\times {\mu}_{0}{H}_{\Delta {M}_{break/2}})+(\Delta {M}_{ave}/2\times {\mu}_{0}H)\end{array}$$

_{ave}/2 is the average value of $(\Delta {M}_{break}/2+\Delta M/2)$. Based on Equation (6), when field is higher than ${\mu}_{0}{H}_{\Delta {M}_{break/2}}$, the total ΔS_{M}can be obtained from
$$\begin{array}{ll}\Delta {S}_{M}& =\Delta {S}_{M1\mathrm{max}}+\Delta {S}_{M2}\\ & \text{}=(\Delta {S}_{M1\mathrm{max}}-\Delta {M}_{ave}/2\times {\mu}_{0}{H}_{\Delta {M}_{break/2}})+(\Delta {M}_{ave}/2\times {\mu}_{0}H)\end{array}$$

It is seen from Figure 6 that the ΔM

_{ave}/2 is nearly constant when μ_{0}H > ${\mu}_{0}{H}_{\Delta {M}_{break/2}}$. Therefore, by comparing Equations (4) and (7), the first bracket of Equation (7) can be considered as -ΔS_{0}in Equation (4), and the second bracket of Equation (7) equates with the κ μ_{0}H in Equation (4). Consequently, the above numerical analysis and discussion reveals the origin of the linear relationship between ΔS_{M}and μ_{0}H at high fields in Ni_{50}Mn_{34}Co_{2}Sn_{14}with first-order magnetostructural transition. On the other hand, this approximation does not hold when the field is lower than ${\mu}_{0}{H}_{\Delta {M}_{break/2}}$, thus leading to the deviation of the linear relationship at low fields. In addition to the ΔS_{M}peak value, it is also interesting to find that other ΔS_{M}values at different temperatures also follow the linear relation at high fields by performing the same numerical analysis. It has to be pointed out that the ΔS_{M}would not further increase by increasing μ_{0}H when it reaches saturation. Therefore, this linear relationship between ΔS_{M}and μ_{0}H only exists below the saturation magnetic field.## 4. Conclusions

In the present Ni-Mn-Co-Sn system, the martensitic transformation temperature T

_{M}reduces largely in the substituted alloys. The decrease of T_{M}is likely attributed to the reduction of d-d hybridization by substituting Mn with Ni or Co as well as the enhancement of p-d covalent hybridization by substituting Mn with Sn. The Ni_{51}Mn_{34}Co_{2}Sn_{13}and Ni_{50}Mn_{34}Co_{2}Sn_{14}exhibit a magnetostructural martensitic transition from FM austenite to weak-magnetic martensite, which results in a giant MCE around room temperature. Moreover, a linear relationship between ΔS_{M}and μ_{0}H is found in Ni_{50}Mn_{34}Co_{2}Sn_{14}, and the origin of this linear relationship is analyzed numerically based on the Maxwell relation.## Author Contributions

H.Z. (Hu Zhang) conceived and designed the experiments. C.X., Z.Q., D.H., and X.L. prepared the samples. C.X., Y.X., and H.Z. (HanNing Zhang) performed the measurements. H.Z. (Hu Zhang) and C.X. contributed to the data analysis and scientific interpretation. C.X. and H.Z. (Hu Zhang) drafted the article. K.L., Y.Z., and Y.L. made critical revisions to the article.

## Funding

This work was funded by the National Natural Science Foundation of China [Grant No.: 51671022 and 51571018]; the National Key Research and Development Program of China [Grant No.: 2017YFB0702704]; the Beijing Natural Science Foundation [No. 2162022]; and the Scientific and Technological Innovation Team Program of Foshan [2015IT100044].

## Conflicts of Interest

The authors declare no conflicts of interest.

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**Figure 1.**The powder X-ray diffraction (XRD) patterns at ambient temperature for Ni-Mn-Co-Sn alloys.

**Figure 2.**(

**a**) The differential scanning calorimetry (DSC) heat flow curves of Ni-Mn-Co-Sn alloys upon heating and cooling with a ramp rate of 10 K/min. (

**b**) The martensitic transformation temperature (T

_{M}) and reverse martensitic transformation temperature (T

_{A}) as a function of different atomic substitution and valence electron concentration e/a.

**Figure 3.**Temperature dependence of zero-field-cooling (ZFC) and field-cooling (FC) magnetization for Ni-Mn-Co-Sn alloys at 0.05 T and 3 T, respectively.

**Figure 4.**Magnetization isotherms of Ni-Mn-Co-Sn alloys with increasing temperatures upon field ascending and descending modes.

**Figure 5.**(

**a**) Temperature dependence of ∆S

_{M}for Ni-Mn-Co-Sn alloys under different magnetic field changes of 1 T, 2 T, and 3 T, respectively. (

**b**) The maximum ΔS

_{M}as a function of μ

_{0}H and the fitting line to ΔS

_{M}-μ

_{0}H curve for Ni

_{50}Mn

_{34}Co

_{2}Sn

_{14}alloy. The inset shows the ΔS

_{M}-μ

_{0}H curve and the fitting line at low fields.

**Figure 6.**The ΔM/2 between 278 K and 280 K as a function of magnetic field μ

_{0}H for Ni

_{50}Mn

_{34}Co

_{2}Sn

_{14}alloy.

**Table 1.**Comparison of nominal composition and final composition. The deviation is shown in the bracket.

Nominal Composition | Final Composition |
---|---|

Ni_{50}Mn_{35}Co_{2}Sn_{13} | Ni_{49.9(8)}Mn_{35.1(4)}Co_{1.9(1)}Sn_{13.1(4)} |

Ni_{51}Mn_{34}Co_{2}Sn_{13} | Ni_{50.9(11)}Mn_{34.1(7)}Co_{1.8(1)}Sn_{13.3(9)} |

Ni_{50}Mn_{34}Co_{3}Sn_{13} | Ni_{50.1(9)}Mn_{33.9(10)}Co_{2.9(8)}Sn_{13.2(7)} |

Ni_{50}Mn_{34}Co_{2}Sn_{14} | Ni_{50.0(10)}Mn_{34.0(8)}Co_{1.9(3)}Sn_{14.1(7)} |

**Table 2.**The ΔS values at the structural transition obtained from DSC curves for Ni-Mn-Co-Sn alloys.

Alloys | ΔS (J/kg K) |
---|---|

Ni_{50}Mn_{35}Co_{2}Sn_{13} | 42.3 |

Ni_{51}Mn_{34}Co_{2}Sn_{13} | 28.2 |

Ni_{50}Mn_{34}Co_{3}Sn_{13} | 14.9 |

Ni_{50}Mn_{34}Co_{2}Sn_{14} | 31.6 |

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