Strain Engineering in Ni-Co-Mn-Sn Magnetic Shape Memory Alloys: Influence on the Magnetic Properties and Martensitic Transformation

Ni-Mn-Sn ferromagnetic shape memory alloys, which can be stimulated by an external magnetic field, exhibit a fast response and have aroused wide attention. However, the fixed and restricted working temperature range has become a challenge in practical application. Here, we introduced strain engineering, which is an effective strategy to dynamically tune the broad working temperature region of Ni-Co-Mn-Sn alloys. The influence of biaxial strain on the working temperature range of Ni-Co-Mn-Sn alloy was systematically investigated by the ab initio calculation. These calculation results show a wide working temperature range (200 K) in Ni14Co2Mn13Sn3 FSMAs can be achieved with a slight strain from 1.5% to −1.5%, and this wide working temperature range makes Ni14Co2Mn13Sn3 meet the application requirements for both low-temperature and high-temperature (151–356 K) simultaneously. Moreover, strain engineering is demonstrated to be an effective method of tuning martensitic transformation. The strain can enhance the stability of the Ni14Co2Mn13Sn3 martensitic phase. In addition, the effects of strain on the magnetic properties and the martensitic transformation are explained by the electronic structure in Ni14Co2Mn13Sn3 FSMAs.


Introduction
Ni-Mn-Sn ferromagnetic shape memory alloys (FSMAs) with various related magnetic effects, including excellent magnetocaloric effect (MCE) [1,2], extraordinary magnetic shape memory effect (MSME) [3][4][5], and magnetoresistance effect (MR) [6,7]. These multifunctional properties are attributable to the coupling between the magnetic and structural transitions by the magnetic field [8][9][10], i.e., magnetic field-induced martensitic transformation (MFIMT). It is the magnetic field-driven shape memory behavior that makes ferromagnetic shape memory alloys different from conventional shape memory alloys, which must be actuated through temperature. The application of magnetic fields is fast and easy, and its fast response to magnetic fields makes this alloy more widely used than conventional shape memory alloys. Moreover, Ni-Mn-Sn FSMAs are cheap, non-toxic, and have simple fabrication processes, which highlights their advantages over conventional shape memory alloys. Although Ni-Mn-Sn FAMAs have so many excellent properties, the relatively narrow working temperature range is still a key drawback in extensive practical application [11,12]. Previous studies have pointed out that the narrow working temperature 3 × 6 × 6 is used for two phases. The cut-off energy is 500 eV. The L2 1 austenite structure (Fm3m) of Ni 16 Mn 13 Sn 3 with three inequivalent Wyckoff positions (4a, 4b, 8c) is shown in Figure 1a. The Sb and Mn atoms occupy 4a (0, 0, 0) and 4b (0.50, 0.50, 0.50) positions respectively and Ni atoms occupy the 8c ((0.25, 0.25, 0.25) and (0.75, 0.75, 0.75)) sites. In addition, the calculation method used in the transformation process from austenite to martensite is tetragonal distortion. That is, on the premise of keeping the cell volume unchanged, the optimized austenite structure is subjected to lattice distortion with different tetragonal distortion rates c/a so as to obtain the most stable martensite structure. As seen in Figure 1b, we substituted Co atoms for Ni atoms directly in our study, and Mn Sn is the designation for the excess Mn atoms at the deficient Sn atoms. The Mn atoms that remain at their sites are called Mn Mn in the Ni 16−x Co x Mn 13 Sn 3 (x = 0, 1, 2) FSMAs. For both the austenitic and martensitic phases, we calculated two situations: the magnetic properties of Ni 16−x Co x Mn 13 Sn 3 (x = 0, 1, 2) alloys are FM states and AFM states. The FM configuration was set by magnetic moments of all Mn atoms (Mn Sn and Mn Mn ) parallel to each other. AFM configuration was decided by magnetic moments of Mn Sn , which are opposite in the direction of the magnetic moments of the Mn Mn . According to the VASP user manual [39], the calculation of spin polarization requires the parameter ISPIN = 2, while the setting of FM and AFM is determined by the parameter MAGMOM. Therefore, the spin polarization of both ferromagnets and antiferromagnets can be achieved by VASP. In first-principles calculations, we simulate biaxial strain by changing lattice vectors directly. That is, fixing the lattice constant in the c-axis while relaxing the lattice constants in the a-axis and b-axis. It is worth mentioning that 0% represents no deformation, positive deformation represents stretching, and negative deformation represents compression. works were performed on the basis of the density functional theory (DFT). As the ex change-correlation functional, we used the Perdew-Burke-Ernzerhof (PBE) method an the generalized gradient approximation (GGA) [38]. For Ni-Mn-Sn FSMAs, a k-mesh of × 6 × 6 is used for two phases. The cut-off energy is 500 eV. The L21 austenite structur ( 3 ) of Ni16Mn13Sn3 with three inequivalent Wyckoff positions (4a, 4b, 8c) is shown i Figure 1a. The Sb and Mn atoms occupy 4a (0, 0, 0) and 4b (0.50, 0.50, 0.50) positions re spectively and Ni atoms occupy the 8c ((0.25, 0.25, 0.25) and (0.75, 0.75, 0.75)) sites. I addition, the calculation method used in the transformation process from austenite t martensite is tetragonal distortion. That is, on the premise of keeping the cell volume un changed, the optimized austenite structure is subjected to lattice distortion with differen tetragonal distortion rates c/a so as to obtain the most stable martensite structure. As see in Figure 1b, we substituted Co atoms for Ni atoms directly in our study, and MnSn is th designation for the excess Mn atoms at the deficient Sn atoms. The Mn atoms that remai at their sites are called MnMn in the Ni16−xCoxMn13Sn3 (x = 0, 1, 2) FSMAs. For both th austenitic and martensitic phases, we calculated two situations: the magnetic propertie of Ni16−xCoxMn13Sn3 (x = 0, 1, 2) alloys are FM states and AFM states. The FM configuratio was set by magnetic moments of all Mn atoms (MnSn and MnMn) parallel to each othe AFM configuration was decided by magnetic moments of MnSn, which are opposite in th direction of the magnetic moments of the MnMn. According to the VASP user manual [39 the calculation of spin polarization requires the parameter ISPIN = 2, while the setting o FM and AFM is determined by the parameter MAGMOM. Therefore, the spin polarizatio of both ferromagnets and antiferromagnets can be achieved by VASP. In first-principle calculations, we simulate biaxial strain by changing lattice vectors directly. That is, fixin the lattice constant in the c-axis while relaxing the lattice constants in the a-axis and b axis. It is worth mentioning that 0% represents no deformation, positive deformation rep resents stretching, and negative deformation represents compression.

Results and Discussions
First, we investigated the two phasic structures, martensitic transition and magnet properties of the Ni16Mn13Sn3. Table 1 shows the results of our calculations for the mag netic properties and equilibrium lattice parameters of the Ni16Mn13Sn3 alloys. For th Ni16Mn13Sn3 austenitic phase, the AFM state of the alloy is low energy, and the equilibrium lattice parameter is 5.94 Å. The magnetic ground state and lattice constant are in goo agreement with other theoretical values [40]. In the Ni16Mn13Sn3 martensitic phase, the FM

Results and Discussion
First, we investigated the two phasic structures, martensitic transition and magnetic properties of the Ni 16 Mn 13 Sn 3 . Table 1 shows the results of our calculations for the magnetic properties and equilibrium lattice parameters of the Ni 16 Mn 13 Sn 3 alloys. For the Ni 16 Mn 13 Sn 3 austenitic phase, the AFM state of the alloy is low energy, and the equilibrium lattice parameter is 5.94 Å. The magnetic ground state and lattice constant are in good agreement with other theoretical values [40]. In the Ni 16 Mn 13 Sn 3 martensitic phase, the FM state has higher energy and is more unstable than the AFM state at c/a~1.35. That is, austenite and martensite phase are AFM states under 0% strain. This is also consistent with the theoretical results [3]. According to the energy corresponding to 0% strain in Figure 2, the energy of AFM austenite is higher than that of AFM martensite, so MPT will occur, which is a prerequisite for shape memory alloys. This is also verified experimentally [41]. The above results confirm the correctness of our calculation, so we can apply biaxial strain based on it, and then we calculated the total energies E of the Ni 16 Mn 13 Sn 3 austenitic phase and martensitic phase with strain (−1.5~1.5%), as shown in Figure 2a,b respectively, to reveal the effect of strain on the phase structures, MPT, and magnetic properties. Figure 2a indicates that the energy of the AFM state is lower than that in the FM state for the austenite phase, and the total energies E of Ni 16 Mn 13 Sn 3 austenitic phase firstly decrease with strain from 1.5% to 0% and then increase with strain from 0% to −1.5%. For the Ni 16 Mn 13 Sn 3 martensitic phase, the total energy E of both FM and AFM increases with strain from 1.5% to −1.5%, and the AFM state energies are lower than the FM state energies. Therefore, we can conclude that austenite and martensite of Ni 16 Mn 13 Sn 3 alloy are still in AFM state under the application of biaxial strain; that is, the biaxial strain will not affect the magnetic ground state of the system. In addition, the value of ∆E A-M and ∆E P-F in Ni 16 Mn 13 Sn 3 alloys under strain (−1.5~1.5%) are shown in Figure 3 to show the impact of strain on T M and T C . It is obvious that the ∆E A-M increases with strain, while the ∆E P-F decrease with strain under strain from 1.5% to −1.5%. This shows that T M and T C increase with the increase in ∆E A-M and ∆E P-F , respectively. According to the results, applying strain can tune the working temperature of Ni 16 Mn 13 Sn 3 alloys. Based on the above results, it can be concluded that the stability of austenite will be reduced no matter whether biaxial compressive strain or biaxial tensile strain is applied. However, the stability of martensite increases with compressive strain and decreases with tensile strain. Moreover, the biaxial strain does not affect the occurrence of MPT and the most stable magnetic configuration of each phase.    In order to tune the working temperature of alloys, alloys must have stable MFIMT. The ΔM is larger, and the MFIMT is more stable [41,42]. Therefore, we calculated the ΔM of Ni16Mn13Sn3 alloys with strain in Table 1. Table 1 further accurately shows that the ΔM of Ni16Mn13Sn3 is very small, and the strain has a weak effect on the value of ΔM (0.06-0.01 μB). The low ΔM cannot satisfy the stable MFIMT. Therefore, the biaxial strain alone cannot satisfy the stable MFIMT, which is a necessary condition for an adjusted wide working temperature region. Fortunately, the Co element enhances ferromagnetism in the In order to tune the working temperature of alloys, alloys must have stable MFIMT. The ∆M is larger, and the MFIMT is more stable [41,42]. Therefore, we calculated the ∆M of Ni 16 Mn 13 Sn 3 alloys with strain in Table 1. Table 1 further accurately shows that the ∆M of Ni 16 Mn 13 Sn 3 is very small, and the strain has a weak effect on the value of ∆M (0.06-0.01 µB). The low ∆M cannot satisfy the stable MFIMT. Therefore, the biaxial strain alone cannot satisfy the stable MFIMT, which is a necessary condition for an adjusted wide working temperature region. Fortunately, the Co element enhances ferromagnetism in the austenitic phase and T C of Ni 16 Mn 13 Sn 3 alloys. Thus, the strain may be an efficacious strategy to adjust the wide working temperature of Ni-Co-Mn-Sn.
The impact of Co doping on the physical properties of the Ni-Co-Mn-Sn system, particularly on the operating temperature, must also be taken into account. We first evaluate the equilibrium lattice parameters and magnetic properties of the Ni 16−x Co x Mn 13 Sn 3 (x = 1, 2) and present them in Table 1 to show the change in phase structures, MPT, and magnetic properties of the Ni 16 Mn 13 Sn 3 with Co doping. The findings demonstrate that as Co increases, the equilibrium lattice parameters of the Ni 16−x Co x Mn 13 Sn 3 (x = 1, 2) austenitic phases increase from 5.91 Å to 5.93 Å. The change of Ni 16−x Co x Mn 13 Sn 3 (x = 1, 2) lattice constant can be attributed to the fact that the atomic radius of the substitution elements is slightly larger than that of the substituted element, and the lattice parameters are close to the value of the experiment (5.987 Å) and theory (5.973 Å) [43,44]. The origin of the experimental error is the slight difference between the actual compound and the nominal compound, and then the DFT calculation is carried out at T = 0 K, while the equilibrium lattice constant measured by XRD is carried out at room temperature. The theoretical error may be caused by the error between different calculation software. The austenitic Ni 14 Co 2 Mn 13 Sn 3 phase is the FM state, whereas the martensitic phase is the AFM state, and there is a large ∆M (5.48 µB) between these two phases as shown in Table 1. This is also consistent with the experimental facts (6.68 µB) [45]. In short, the Ni 14 Co 2 Mn 13 Sn 3 alloys meet the stable MFIMT, and the strain method may be an efficient strategy to tune the wide working temperature of Ni 14 Co 2 Mn 13 Sn 3 alloys.
Subsequently, we calculated the total energies of Ni 14 Co 2 Mn 13 Sn 3 alloys with strain (−1.5~1.5%) and shown in Figure 4 to show the impact of the magnetic properties and working temperature on this alloy. It is obvious that the total energies of austenitic phases firstly decrease with strain (−1.5~0%) and then increase with strain (0-1.5%), and FM states are the most stable magnetic configuration for austenitic phases. The total energies of martensitic phases increase with strain (−1.5~1.5%), as shown in Figure 4b. Moreover, the most stable magnetic configuration of martensitic phases is AFM states. Combined with (a) and (b) of Figure 4, it can be seen that the energy of AFM martensite is always less than that of FM austenite under the action of biaxial strain. It shows that the alloy will undergo magnetic structure coupling transformation, which further confirms that there is a large magnetic moment difference in the system. To show the change of T M and T C of Ni 14 Co 2 Mn 13 Sn 3 alloys with strain, we listed the ∆E A-M and ∆E P-F in Table 2. The value of ∆E A-M increase with strain from 1.5% to −1.5%, while the value of ∆E P-F decrease with strain from 1.5% to −1.5%. The results show the strain can increase T M and decrease T C . In consideration of the wide working temperature of Ni 14 Co 2 Mn 13 Sn 3 alloys, one of the conditions is that T C must be higher than T M . Therefore, we need to evaluate the value of T M and T C . Generally, the T M and T C increase linearly with ∆E A-M and ∆E P-F in Ni-Mn-based Heusler alloys. To further explore the relationship between T M and ∆E A-M , as depicted in Figure 5, we made the T M and ∆E A-M fitting curves [1,10,34,35,[46][47][48][49][50][51].
According to the Heisenberg model and Stoner theory [52], the relationship of T M and ∆E A-M is represented by ∆E P-F = −k B TcM/M 0 , where M is the magnetic moment at T = 0 K, and M 0 is the equilibrium magnetic moment at T = 0 K [53]. Based on it, we calculated the T M and T C of Ni 14 Co 2 Mn 13 Sn 3 alloys with strain in Figure 6. It shows that the T M and T C increase with strain, and the T M is lower than T C with strain (−1.5~1.5%), which indicates that the alloy has been fully qualified to dynamically adjust the wide working temperature range. In addition, the changing trend of T M is consistent with the experiment in other Ni-Mn based [54]; that is, the working temperature moves to a high temperature under the compressive strain. Then, with strain from 0% to −1.5%, the Ni 14 Co 2 Mn 13 Sn 3 FSMAs show a tunable wide working temperature (from 168 K to 330 K), which meets the application in different temperatures. The operating temperature range of Ni 14 Co 2 Mn 13 Sn 3 is 160 K; the wide range may be overestimated compared to experimental values. In short, the strain method is an effective way to tune effectively by using the strain approach for Ni 14 Co 2 Mn 13 Sn 3 alloys. Table 2. Calculated energy difference ∆E A-M in meV/atom between the austenite and martensite phases, ∆E P-F in meV/atom between the paramagnetic and ferromagnetic state, martensite transition temperature T M and Curie temperature T C in Ni 16−x Co x Mn 13 Sn 3 (x = 0, 1, 2) alloys with strain (−1.5~1.5%).

Alloys
Strain %    Ni-Mn-Ga [50][51][52] Ni-Mn-In [47,48] /Sb [49] Ni-Mn-Sn [1,10,34,35,49] T M (K)    Figures 7-9 respectively to further illuminate the physical mechanism of the MPT and magnetic properties [55,56]. In addition, the phase stability is strongly dependent upon the TDOS at the Fermi level plays (E F ) [57,58]. Usually, the low TDOS indicates the stable phase in Ni 14 Co 2 Mn 13 Sn 3 FSMAs. Figure 7a shows that the TDOS at E F of strain (−1.5%, 1.5%) is similar in Ni 14 Co 2 Mn 13 Sn 3 austenitic phases. The strain has a weak effect on austenitic phase stability. Moreover, for martensitic phases, the TDOS of −1.5% strain is lower than the TDOS of 1.5% strain at E F , as shown in Figure 7b. The −1.5% strain decreases the instability of the Ni 14 Co 2 Mn 13 Sn 3 martensitic phase. Then, the instability of the martensitic phase decreases, and the stability of austenitic phases show a few changes. In addition, austenite is a peak at E F , while martensite is a pseudopotential valley. It shows that the stability of austenite is lower than that of martensite, which leads to the MPT according to the Jahn teller splitting effect. The results show that strain can tune the T M of Ni 14 Co 2 Mn 13 Sn 3 FSMAs.     For the magnetic properties of Ni 14 Co 2 Mn 13 Sn 3 austenitic phases, as can be seen in Figure 8b, the PDOS of Mn Mn and Mn Sn are similar and mostly up-spin states under the E F . With applying strain, for Figure 8a,c, the PDOS of Mn Mn and Mn Sn has almost no change. This is indicated that the most stable magnetic configuration of austenitic phases is always FM, and this configuration is unaffected by strain. For martensite, in Figure 9b, the Mn Mn is in up-states while the Mn Sn is in down-states. With applying strain, the distribution of the PDOS of Mn Mn and Mn Sn is still different, the Mn Sn is down-state, but the Mn Mn is up-state, as shown in Figure 9a,c. The most stable magnetic configuration of martensite phases is AFM. In addition, the difference in the distribution of austenite and martensite also explains the existence of large ∆M. In conclusion, the strain cannot change the magnetic properties of Ni 14 Co 2 Mn 13 Sn 3 FSMAs. The Ni 14 Co 2 Mn 13 Sn 3 FSMAs meet the condition of the stable MFIMT.

Conclusions
To summarize, to achieve the tunable broad working temperature region of Ni-Mn-Sn alloys, we have systematically studied the influence of strain on the structures, MPT, and magnetic properties Ni 16−x Co x Mn 13 Sn 3 (x = 0, 1, 2) by first-principles calculation. The value of ∆E A-M increases with the strain from 1.5% to −1.5%, bringing about T M enhancement. According to the results, the strain method can reveal the ability of the tunable wide working temperature range in Ni 14 Co 2 Mn 13 Sn 3 FSMAs. Particularly, with a slight strain (0~1.5%), Ni 14 Co 2 Mn 13 Sn 3 with a large working temperature region of~160 K and the working temperature (168-330 K) of Ni 14 Co 2 Mn 13 Sn 3 satisfy the application from low-temperature to high-temperature. This work predicts an adjusted broad working temperature region of Ni-Mn-Sn alloys, which shows a great application range of Ni-Mn-Sn FSMAs. Therefore, the strain method provides the reference for designing the tunable wide working temperature FSMAs and makes it possible for the wide application of FSMAs.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors declare no conflict of interest.