Investigation of the structural competing and atomic ordering in Heusler compounds Fe2NiSi and Ni2FeSi under strain condition

The structural competing and atomic ordering of the full Heusler compounds Fe2NiSi and Ni2FeSi under uniform and tetragonal strains have been systematically studied by the first-principles calculation. Both Fe2NiSi and Ni2FeSi have the XA structure in cubic phase and they show metallic band structures and large magnetic moments (greater than 3μB) at equilibrium condition. Tetragonal distortion can further decrease the total energy, leading to the possible phase transformation. Furthermore, different atom reordering behaviours have been observed: for Fe2NiSi, atoms reorder from cubic XA-type to tetragonal L10-type; for Ni2FeSi, there is only structural transformation without atom reordering. The total magnetic moments of Fe2NiSi and Ni2FeSi are mainly contributed by Fe atoms, and Si atom can strongly suppress the moments of Fe atoms when it is present in the nearest neighbours of Fe atoms. With the applied strain, the distance between Fe and Si atoms play an important role for the magnetic moment variation of Fe atom. Moreover, the metallic band nature is maintained for Fe2NiSi and Ni2FeSi under both uniform and tetragonal strains. This study provides a detailed theoretical analysis about the full Heusler compounds Fe2NiSi and Ni2FeSi under strain conditions.

The structural competing and atomic ordering of the full Heusler compounds Fe 2 NiSi and Ni 2 FeSi under uniform and tetragonal strains have been systematically studied by the first-principles calculation. Both Fe 2 NiSi and Ni 2 FeSi have the XA structure in cubic phase and they show metallic band structures and large magnetic moments (greater than 3μ B ) at equilibrium condition. Tetragonal distortion can further decrease the total energy, leading to the possible phase transformation. Furthermore, different atom reordering behaviours have been observed: for Fe 2 NiSi, atoms reorder from cubic XA-type to tetragonal L1 0 -type; for Ni 2 FeSi, there is only structural transformation without atom reordering. The total magnetic moments of Fe 2 NiSi and Ni 2 FeSi are mainly contributed by Fe atoms, and Si atom can strongly suppress the moments of Fe atoms when it is present in the nearest neighbours of Fe atoms. With the applied strain, the distance between Fe and Si atoms play an important role for the magnetic moment variation of Fe atom. Moreover, the metallic band nature is maintained for Fe 2 NiSi and Ni 2 FeSi under both uniform and tetragonal strains. This study provides a detailed theoretical analysis about the full Heusler compounds Fe 2 NiSi and Ni 2 FeSi under strain conditions. coordinates as 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). X and Y are transition metal elements and they enter A, B and C sites; Z is from main group elements and it enters D site. Different arrangements of transition metal elements X and Y at A, B and C Wyckoff positions generally result into two different structures: the Cu 2 MnAl-type and the Hg 2 CuTi-type. The former one is also known as L2 1 -type structure (Fm-3m space group, No. 225), where the two X atoms occupy A and C positions, and the latter one is known as XA-type structure (F-43m space group, No. 216), where the two X atoms occupy A and B positions. We select Fe 2 NiSi as an example and show the two cubic crystal structures in figure 1a,b. For L2 1 -type structure of Fe 2 NiSi, the four Wyckoff sites A, B, C and D are occupied by Fe1, Ni, Fe2 and Si atoms, respectively. Whereas, for the XA-type structure, the four Wyckoff sites A, B, C and D are occupied by Fe1, Fe2, Ni and Si atoms, respectively.
In general, for full Heusler alloys, site preferences of transition metal elements X and Y are determined by the number of their valence electrons [75,76]: the element with more valence electrons prefers the A and C sites yet the element with less valence electrons prefers the B site. This rule has been widely used for explaining the atom ordering in Heusler alloy and even applied for new Heusler design. However, there are also some studies showing the contrary results which violate this rule [77,78]. It should be stressed that the atom ordering in Heusler alloys strongly influences their properties. In order to determine the stable state of cubic structure Fe 2 NiSi and Ni 2 FeSi, we computed their total energy with both L2 1 -type and XA-type structures under different lattice constants. Besides, two magnetic states, ferromagnetic (FM) and non-magnetic (NM), have also been considered in each structure type. The results are shown in figure 2a,b for Fe 2 NiSi and Ni 2 FeSi, respectively. It is seen from figure 2a,b that the non-magnetic states of both L2 1 -type and XA-type structures have higher total energy than their ferromagnetic counterparts for both Fe 2 NiSi and Ni 2 FeSi Heusler alloys, meaning that the ferromagnetic state is more energetically stable. Moreover, the total energy curves of ferromagnetic states would converge to the non-magnetic ones at small lattice constants and this is due to the abatement of magnetic moment with decreasing lattice, which will be discussed later. The antiferromagnetic state is not considered simply because the experimentally synthesized Fe 2 NiSi compounds show ferromagnetic configuration with large magnetic moment [65].
More importantly, it is found the XA-type ferromagnetic structures have the lowest total energy for Fe 2 NiSi and Ni 2 FeSi, which implies the XA-type structure is the stable state. For Fe 2 NiSi, its XA structure obeys the general site preference rule: two Fe atoms have less valence electrons than Ni atom and they occupy the A and B sites. While for Ni 2 FeSi, it does not follow this rule and two Ni atoms with more valence electrons enter the A and B sites forming the XA structure instead of L2 1 structure. In addition, some studies show that the pure metal structure of X element from the Heusler alloy X 2 YZ has influence on its atomic ordering [79,80], i.e. when X metal prefers a FCC or HCP structure, the X 2 YZ Heusler crystalizes in the L2 1 structure; while X metal with BCC structure would lead to the XA structure. This rule is also coincident with Fe 2 NiSi, because pure Fe metal has BCC structure. But for Ni 2 FeSi, pure Ni metal prefers FCC structure and this would predict the L2 1 structure, contrary to the stable XA structure. Since both L2 1 and XA structures have been experimentally demonstrated in Heusler alloys, we still keep these two structures for the following consideration.
The lattice constants are determined by minimizing the total energy for each state, and their values are listed in table 1. The results from our calculations are in good agreement with other experimental measurements and theoretical calculations [37,[65][66][67][68]. In order to further check the structure stability, we computed the total energy difference between the XA structure and the L2 1 structure at ferromagnetic state under different lattice constants, and the results are plotted in figure 2c,d. For

Electronic and magnetic properties
In this section, we calculate the electronic and magnetic properties of Fe 2 NiSi and Ni 2 FeSi compounds at the determined equilibrium lattice constants, via calculating the energy band structure, the densities of states, the electronic spin densities distribution and the magnetic moments. In order to check the atomic site preference in these two Heusler alloys and also investigate the reason for the contradictory behaviour observed for Ni 2 FeSi, we include both L2 1 and XA structures. The electronic band structure and the corresponding density of state (DOS) for Fe 2 NiSi and Ni 2 FeSi in these two structures under their own equilibrium lattice are plotted in figure 3. It can be seen that the electronic bands exhibit an overlap with the Fermi energy level in both spin-up and spin-down directions for both Fe 2 NiSi and Ni 2 FeSi in either L2 1 or XA structures, which indicates their metallic nature. The effect of Hubbard U   on the electronic band structure is also considered for the transition metal elements Fe and Ni, and results show that the metallic feature is still preserved even with some variation of the band structures. Usually, the DOS spectrum plays an important role for the structural stability of the intermetallic compounds [52].
In particular, a lower DOS at the Fermi energy level can induce a more stable structure. Thus, we compare the DOS of Fe 2 NiSi and Ni 2 FeSi at the Fermi energy level in L2 1 and XA structures. It is found that the DOS values of Fe 2 NiSi in XA structure at the Fermi energy level is 0.96 states/eV and 2.65 states/eV in spin-up and spin-down channels, which are lower than 1.19 states/eV and 3.10 states/eV in L2 1 structure. For Ni 2 FeSi, the DOS values at Fermi energy are 1.02 states/eV and 2.31 states/eV in XA structure and 0.91 states/eV and 2.90 states/eV for L2 1 structure in spin-up and spin-down channels, respectively, indicating the total DOS in XA structure is smaller than that in L2 1 structure. Consequently, we think the DOS values near the Fermi level have a dominant role of determining the stable XA structure in both Fe 2 NiSi and Ni 2 FeSi. It is known that Fe-based ternary Heusler compounds often exhibit magnetism, which is very important for the application in spintronics and magnetoelectronics. So, we move on to examine the magnetic properties of Fe 2 NiSi and Ni 2 FeSi in different structures, and the calculated total and atomresolved magnetic moments at equilibrium lattice are listed in table 1. The total magnetic moment is mainly contributed by the transition metal elements Fe and Ni, and also they are ferromagnetically aligned because of the same positive sign. It can also be found that the different atom site ordering leads to a larger variation in both total and atom-resolved magnetic moments.
To further elucidate the magnetic variation under different atomic orderings, we calculated the electronic spin density distribution of Fe 2 NiSi and Ni 2 FeSi in the (110) plane at the equilibrium lattice, and the results are displayed in figure 4. This electronic spin density is defined as the electronic density difference between the spin-up and spin-down directions and, thus, it can give information about the magnetic origination and distribution. Note that the same colourmap scale is applied for different structures so as to provide a better visual comparison. We can immediately see the bright colour around Fe atoms in all different structures, and this implies that Fe atoms have large spin density difference and strong magnetism. This can be simply understood because Fe atom often shows ferromagnetic properties in most Fe-based compounds. The spin density difference is smaller around Ni atoms and becomes even indistinguishable with background for Si atoms. For Fe 2 NiSi in L2 1 structure, the colour around Fe atoms from two positions are all the same, leading to the same Afterwards, we also study the effect of uniform strain on the magnetic moments by varying the lattice constant around the equilibrium condition, and the results are reported in figure 5. It can be seen that the   For the L2 1 structure in both Fe 2 NiSi and Ni 2 FeSi, the magnetic moments in A and C sites always overlap with each other (Fe1 and Fe2, Ni1 and Ni2) throughout the whole strain variation because they have the same surrounding environment as explained above. The variation of the total magnetic moment is mainly from the Fe atoms and there is almost no change for Si atoms for all cases. While for Ni atoms, different changing trends are observed: a small decrease in L2 1 structure and a slightly bigger increase in XA structure for Fe 2 NiSi; no changes in L2 1 structure and a small increase in XA structure for Ni 2 FeSi. With lattice decrease, the distance between atoms gets smaller so that the interaction of the valence electrons from different atoms is enhanced and they become more delocalized. The larger variation with strain found from Fe 2 NiSi in L2 1 structure and Ni 2 FeSi in XA structure is all related with Fe atom and this is because of the surrounding Si atom as the nearest neighbours of Fe atom. With strain increase at positive side, the distance between Fe and Si atoms is larger so that Fe atom partially regain its spin moments, and vice versa. In particular, for Fe 2 NiSi in XA structure, the magnetic moment variation of Fe1 is a little larger than Fe2 and this is probably caused by the different neighbouring distance of Si atoms: Si atom is nearest to Fe1 while it is secondary nearest to Fe2. The electronic band structures for both Fe 2 NiSi and Ni 2 FeSi under uniform strain have also been calculated and they all show metallic overlap with the Fermi energy in both spin-up and spin-down directions, indicating the metallic behaviour under studied strain condition.

Tetragonal transformation
According to the theoretical calculation and experimental study [52,69], both Fe-based and Ni-based Heusler compounds may undergo a tetragonal distortion and transform from cubic phase into tetragonal phase. This phase transformation is still related with the crystal structure arrangement and, thus, we further investigate the tetragonal distortion in terms of c/a ratio variation in both L2 1 and XA structures under ferromagnetic state. Note, when the c/a ratio is varied, the unit cell volume is fixed at the equilibrium condition. The corresponding structures for Fe 2 NiSi in tetragonal L1 0 -type and tetragonal XA-type are shown in figure 1c,d, respectively. In order to understand the phase stability, we first calculate the total energy at different c/a ratios, and the results are presented in figures 6 and 7 for Fe 2 NiSi and Ni 2 FeSi, respectively. It is clearly seen that the tetragonal distortion can reduce the total energy for both Fe 2 NiSi and Ni 2 FeSi in either L2 1 or XA structure, which may lead to the phase transformation. As experimentally proved, Fe 2 NiSi prepared by arc-melting crystallizes in cubic structure [66] and this is due to the L2 1 B disorder effect. Also, obtained elastic modulus constants from previous study [68] confirmed the mechanical stability for Fe 2 NiSi in cubic structure. Thus, the possible tetragonal phase transformation is not caused by mechanical properties. Another comprehensive theoretical calculation from the energetic point of view shows that Fe 2 NiSi has tetragonal L1 0 ground state and attributes the reason to the density of states at the Fermi energy level [52]. This transformation is very interesting because not only the tetragonal structural distortion but also the atom site reordering occurs. From figure 6 at 100% volume, we can see that Fe 2 NiSi at cubic condition (c/a = 1) has stable XA structure because of the low total energy, as discussed before, and, with the variation of c/a value, its total energy decreases much more strongly in tetragonal L1 0 structure than in tetragonal XA structure to the minimum total energy when c/a ratio is equal to 1.5, meaning the stable cubic XA structure could change into tetragonal L1 0 structure with tetragonal distortion. This finding is consistent with previous study [52]. By varying the unit cell volume to 95% and 105%, an opposite effect is observed for the stable tetragonal distortion: the tetragonal XA structure has the lowest total energy at 95% volume, meaning that only tetragonal structural distortion is present without atom site reordering; while for 105% volume, the atom site ordering and tetragonal structural distortion have been enhanced because of the larger total energy difference. For Ni 2 FeSi as shown in figure 7, the tetragonal distortion can still decrease the total energy and may lead to the phase transformation. Through the 95% to 105% volume variation, the tetragonal XA structure always has the minimum energy value, indicating the presence of only the tetragonal structural distortion and no atom site reordering occurs, which is different from the case of Fe 2 NiSi. However, by observing the energy difference trend at different volume percentages, the minimum total energies under tetragonal L1 0 and XA structures are getting close to each other at smaller volume, meaning the similar stability of these two structures. In combination, it is found the tetragonal structural distortion can decrease the total energy of cubic Fe 2 NiSi and Ni 2 FeSi in either L2 1 or XA structure and possibly lead to phase transformation. The origin of these tetragonal ground states of Fe 2 NiSi and Ni 2 FeSi has been theoretically explained by the decreasing effect of the density of states at the Fermi energy level [52]. At the same time, atom site reordering may occur in this process dependent on the cell volume variation.
In addition, to further evaluate the stability of the tetragonal phase, we define the tetragonal transformation energy (ΔE T ) as the difference of the total energy at cubic equilibrium state and the minimum total energy at tetragonal distortion. Its variation under different volume percentages for both Fe 2 NiSi and Ni 2 FeSi is displayed in figure 8. The positive sign of all the values shows the stable tetragonal phase under currently studied volume range. With volume increase, the ΔE T values all decrease except Fe 2 NiSi in tetragonal L1 0 structure, meaning the structural stability of the tetragonal phase is becoming weaker at larger volume. It is also found that the values of ΔE T in tetragonal XA structure are much smaller than in tetragonal L1 0 structure, especially at larger volume side, which indicates the stronger stability in tetragonal L1 0 phase. It should be pointed out that the values of c/a ratio at the minimum total energy under tetragonal phase for Fe 2 NiSi and Ni 2 FeSi in different for Ni 2 FeSi in tetragonal XA and L1 0 structures, respectively. In order to check the dynamic stability of the tetragonal phase, the phonon dispersion spectra for Fe 2 NiSi and Ni 2 FeSi are also calculated and are displayed in figure 9. It is clearly seen that the phonon curve in the tetragonal phases for the two compounds exhibits no imaginary frequency and, therefore, it indicates the dynamic stability of the tetragonally distorted structure. Besides, the electronic band structures calculated at the minimum total energy point of tetragonal distortion for both Fe 2 NiSi and Ni 2 FeSi have been calculated, and they all have metallic overlap with the Fermi energy in both spin directions. As shown in §3.1 and also demonstrated in both experiment and theory, cubic structure Fe 2 NiSi and Ni 2 FeSi Heusler compounds are ferromagnets with considerably large magnetic moments. Magnetic variation under tetragonal distortion is also of high interest and great importance because it is directly related with the magnetic shape memory effect and the development of spin-transfer torque magnetic random access memory. The calculated total and atom-resolved magnetic moments for Fe 2 NiSi and Ni 2 FeSi in both tetragonal L1 0 and tetragonal XA structures under different c/a ratios are shown in figures 10 and 11, respectively. We can see that the variation of the total magnetic moment is still mainly from the Fe atoms, the same as in the cubic structure. Consequently, we focus our discussion on the magnetic moment of Fe atoms. For the tetragonal L1 0 structure in both Fe 2 NiSi and Ni 2 FeSi, the two atoms in sites A and C always have the same moment values and thus overlap throughout the whole c/a variation because of the same surrounding environment even in tetragonal structure.    Figure 10. The calculated total and atomic spin magnetic moments of Fe 2 NiSi under two ferromagnetic crystal structures of L1 0 -type (a) and XA-type (b) as functions of c/a ratio. Atomic sites are referred to the crystal structure in figure 1. The distance between Fe atom and its nearest Si atom in the two corresponding structures are shown in (c,d ).  royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191007 In order to detail the moment variation of Fe atom, we need to go back to the crystal structure. As an example for Fe 2 NiSi in tetragonal L1 0 structure, the two Fe atoms in sites A and C have the same surrounding atoms of four Si atoms and four Ni atoms forming two tetrahedrons, see figure 1c. By varying the c/a ratio, the distance between Fe atom and its nearest Si atom is calculated and shown in figure 10c. It is found that this distance value is increased when the c/a ratio is changed from 1 on both sides. As discussed in §3.2, this distance plays an important role for the magnetic moment of Fe atom and its increase would lead to the moment gain of Fe atom, as found in figure 1a. For Fe 2 NiSi in tetragonal XA structure, the two Fe atoms have different environments so that we need to calculate the nearest Fe-Si distance for each of them, and the results are shown in figure 10d. The magnetic moment variation of the two atoms can be still related with the Fe-Si distance change in the same way. For Ni 2 FeSi in both tetragonal L1 0 and tetragonal XA structures, the partial moment variation of Fe atom can still be understood with the Fe-Si distance in the same way.

Conclusion
In the current work, we systematically studied the structural configuration of the full Heusler compounds Fe 2 NiSi and Ni 2 FeSi by employing the first-principles calculations based on density functional theory, in terms of the structural, electronic and magnetic properties. Besides, the effects of uniform and tetragonal strains have been also considered and discussed. Results show that both Fe 2 NiSi and Ni 2 FeSi prefer XA-type atomic ordering rather than L2 1 -type in cubic phase due to the lower total energy. The obtained lattice constants agree with previous experimental and theoretical studies. Both Fe 2 NiSi and Ni 2 FeSi show metallic band structures and large magnetic moments (greater than 3μ B ) at equilibrium condition. Under tetragonal distortion, total energy can be further decreased, leading to the possible phase transformation, but different atom site reordering behaviours have been observed: for Fe 2 NiSi, tetragonal L1 0 structure has smaller total energy than tetragonal XA structure at 100% unit cell volume, which implies that atoms reorder from cubic XA-type to tetragonal L1 0 -type; for Ni 2 FeSi, the XA structure always has the lowest total energy, which means there is only structural transformation without atom reordering. This atom reordering behaviour under tetragonal phase transformation is very interesting and can open up a variation of studies for material property under distortion conditions. The total magnetic moments of Fe 2 NiSi and Ni 2 FeSi are mainly contributed by Fe atoms, and Si atom can strongly suppress the moments of Fe atoms when Si atom is present in the nearest neighbours of Fe atoms. With strain applied, the distance between Fe and Si atoms plays an important role for the magnetic moment variation of Fe atom. Moreover, the metallic band structure is maintained for Fe 2 NiSi and Ni 2 FeSi under both uniform and tetragonal strains. Overall, this study provides a detailed theoretical analysis and can give valuable reference for further experimental research.
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