Electronic structure, elasticity, magnetism of Mn₂X In (X = Fe, Co) full Heusler compounds under biaxial strain: First-principles calculations

Heusler compounds, with the general formula X_nYZ (n = 1, 2), where X and Y represent transition metal elements, and Z denotes a main group element, have garnered significant attention from both theoretical and experimental perspectives for their potential applications in spintronics, energy, and magnetocaloric technologies.^[1–3] Owing to their inherent compositional flexibility, Heusler compounds exhibit a diverse array of physical properties, making them viable candidates for a wide range of applications.^[4,5] These include their utilization as high-temperature ferromagnetic and ferrimagnetic materials, multiferroic shape memory alloys, and topological insulators.^[6–9] The current focus on the spintronic applications of Heusler compounds is primarily attributed to the observation of half-metallicity in select compounds, a phenomenon that enables the achievement of remarkable tunneling magnetoresistance at room temperature. Since the pioneering prediction by de Groot et al. that NiMnSb is a half-metallic ferromagnet, Heusler compounds have garnered immense interest in spintronics.^[10–16]

One advantage of Heusler compounds is that most of them do not contain rare earth elements. Numerous tetragonal Heusler compounds are Mn-, Fe- or Co-based, and their magnetism is commonly ferromagnetic or ferrimagnetic. Another advantage is that a significant number of magnetic Heusler compounds exhibits gap or pseudo-gap in one spin channel, classifying them as the half-metallic materials with high spin polarization. These properties make them highly suitable for use in spintronic devices, particularly in spin valves and magnetic tunnel junctions, due to their significant potential in these research domains.^[17–20] As an example, Hamaya and colleagues epitaxially grew an Fe_3−xMn_xSi/Ge layer, which exhibited a high degree of spin polarization. This layer was then utilized as a spin injector and detector in spintronic devices based on Ge material.^[21] The inherent characteristics of Heusler compounds have been shown to be modifiable, particularly through structural manipulation via the application of external strain.^[22] Moreover, it has been shown that the inherent characteristics of Heusler compounds are modifiable, especially through structural manipulation via the application of external strain.^[23,24] Notably, in the context of epitaxial thin films, the disparity in lattice parameters between the substrate and the film holds significant implications for the properties of the film.^[25–28] Understanding the properties of Heusler compounds under different strain is of great significance for designing epitaxial thin films with specific functions. Additionally, achieving control of physical properties through strain is also beneficial for the application of Heusler compounds in spintronics. Therefore, the change of electronic structure and magnetic properties under biaxial strain for Heusler compounds is worth exploring.

The Heusler compounds Mn₂X In (X = Fe, Co) were first predicted by Faleev recently.^[29] In the report by Faleev, the lattice constants and magnetic moments of Mn₂X In (X = Fe, Co) were examined. However, the elasticity, spin polarization and magnetic anisotropy were not explored in detail. Particularly, the tuning of these properties under strain remains poorly understood or has not been thoroughly studied. In this paper, we use first-principles calculations to determine the stable crystal structure of Mn₂X In (X = Fe, Co). The magnetic configuration and optimal lattice constant are ascertained through calculations. The total magnetic moment of the compound with different applied biaxial strain is also predicted. The origin of magnetic moment is investigated by orbital-resolved magnetic moments. Band structure analysis is conducted to investigate the changes of spin polarization with biaxial strain. Finally, differences in magnetic anisotropy energy under various strains are calculated and analyzed using second-order perturbation theory.

We used the first-principles calculations package based on density functional theory in Vienna ab-initio simulation package (VASP) to obtain the electronic structure, elasticity and magnetism of Mn₂X In (X = Fe, Co) Heusler compounds.^[30] The density functional theory (DFT) was employed utilizing the generalized gradient approximation (GGA) framework for calculations. The plane-wave augmented wave (PAW) pseudopotential method was utilized in conjunction with the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional. The grid division of k-point describing Brillouin zone is 9×9×9, by taking 729 k-points. The plane wave cutoff energy was set at 500 eV. The convergence criterion for the calculations was 5×10⁻² eV/Å, ensuring that the results meet the required accuracy standards. Then, the charge density was utilized in the subsequent non-collinear calculations with spin–orbit coupling (SOC) taken into account for the magnetic anisotropy.

According to the report by Faleev, structure of Mn₂X In (X = Fe, Co) belongs to the inverse Heusler compounds consisted of four interpenetrating face-centered cubic (FCC) sublattices.^[29] Figure 1 illustrates the two potential crystal structures of Mn₂X In (X = Fe, Co) in cubic symmetry. Correspondingly, Table 1 presents the Wyckoff positions for both two possible configurations.^[31,32] In theoretical calculations, there are numerous methods to assess the stability of materials, including binding energy, formation energy, phonon spectrum, molecular dynamics, and Born stability criteria, among others.^[33–36] To determine whether it would be possible to synthesize the full-Heusler compound Mn₂X In, we calculated the formation energy using the formula^[37]

In this formula, E_T (Mn₂X In) represents the energy of the chemical formula Mn₂X In, while E(Mn), E(X), and E(In) represent the energies of the free atoms Mn, X, and In, respectively. The lower the E_F, the more stable the compound is. Based on the information shown in Table 2, the formation energy of the XA–I structure is lower than that of the other type of structures for both Mn₂X In (X = Fe, Co) compounds. This means that the XA–I structure has better thermal stability and it can be possible to prepare experimentally. All calculations will be carried out for the XA–I stable crystal structure under the F43¯m space group (No. 216).

To determine the stable magnetic state of Mn₂X In (X = Fe, Co), we systematically explored numerous potential magnetic configurations. This exploration involved considering diverse combinations of magnetic moment orientations for the transition metal Mn atoms. We calculated the self-consistent total energy of the compound across these varied magnetic states.^[38–40] If all Mn atoms exhibit identical magnetic moment orientations, the compound is categorized as ferromagnetic (FM), as depicted in Fig. 2(a). We also evaluated anti-ferromagnetic (AFM) phases, considering two potential spin arrangements within the unit cell: the A-type (A-AFM) and G-type (G-AFM) antiferromagnetic structures, illustrated in Figs. 2(b) and 2(c), respectively.^[41–43]

To ascertain the equilibrium lattice constant and elucidate the relationship between the total energy and lattice constant, we undertake structural optimizations on Mn₂X In (X = Fe, Co) compounds across three distinct magnetic structures: the FM, A-AFM, and G-AFM. Through these optimizations, we aim to pinpoint the lattice constant at which the system attains equilibrium, while concurrently discerning how the total energy varies as a function of the lattice constant. This comprehensive approach affords insights into the structural stability and energetics of Mn₂X In compounds across different magnetic configurations. Figure 3 shows the total energy as a function of lattice parameters for the FM, A-AFM, and G-AFM magnetic structures for Mn₂X In (X = Fe, Co). The equilibrium lattice constants, the calculated magnetic moments, and spin polarization ratios are listed in Table 3. It is clear that in all the magnetic structures, the A-AFM state shows the lower energy. Therefore, the optimal structure of Mn₂X In (X = Fe, Co) is A-AFM structure. The optimal lattice constant of Mn₂X In (X = Fe, Co) are 6.105 Å and 6.1 Å, which is similar to the report of Faleev.^[29]

Elastic constants provide crucial insights into the stability of any compound in response to distortion or deformation in terms of stress or strain. These constants are typically calculated using well-defined mathematical expressions^[44]

where, V₀ is volume in the equilibrium state, E denotes the internal energy, and ε_i and ε_j are the applied strains. The value of C_ij helps determine the moduli (bulk, shear, Young’s) and related parameters, which relate hardness to tensile strength and indicate ductility, brittleness, plasticity, or elasticity, among other properties.^[45] Born stability condition plays an important role in the determination of mechanical stability.

To calculate these elastic constants, we use the following method: apply a series of small strains from −0.015 to 0.015 in 0.005 steps to the optimized unit cell, and then calculate the corresponding stresses. This process is designed to simulate the response of a material to a small deformation in order to accurately determine its elastic properties.

In the present study, we have employed a method pioneered by Thomas Charpin and integrated into VASKPIT to calculate elastic constants C₁₁, C₁₂, and C₄₄.^[46] Bulk modulus (B), shear modulus (G), Young modulus (E), Poisson’s coefficient (P₀), anisotropy (A), longitudinal, transverse, and mean velocities (v_t, v_l, and v_m), and Debye temperature (θ_D) have been computed using the following equations:^[44,47]

The symbols ρ, h, k_B, n, and V_a, respectively, represent the density, Planck’s constant, Boltzman’s constant, number of atoms per formula unit, and the atomic volume of the alloy.^[48–50]

In the current investigation, the cubic symmetry of the alloys constrains the number of independent elastic parameters to three (C₁₁, C₁₂, and C₄₄). These parameters, along with other elastic properties derived from them, are computed using Eqs. (2)–(10) and summarized in Table 4. It is noted that the Mn₂X In (X = Fe, Co) compounds satisfy the mechanical stability criteria (named as Born–Huang criteria) for a cubic crystal in the F43¯m configuration, which are C₁₁ > 0, C₁₂ > 0, C₁₁ – C₁₂ > 0, and C₁₁ + 2 C₁₂ > 0.^[51]

The Poisson’s ratio (ν), defined as the ratio of transverse strain to longitudinal strain along the direction of elastic loading, provides profound insights into the underlying bonding properties of solids.^[52,53] Generally, covalent materials exhibit a Poisson’s ratio of roughly 0.1, whereas ionic materials possess a ratio of 0.25. Metals, on the other hand, occupy a range from 0.28 to 0.42.^[54] Remarkably, the Mn₂X In compound demonstrates a Poisson’s ratio spanning from 0.29 to 0.33, indicating the presence of distinctive metallic bonding characteristics within its structure.

To further analyze the ductile or brittle nature of materials, there are three criteria considered: Cauchy’s pressure P, Pugh’s ratio B₀/G, and Poisson’s coefficient P₀. Cauchy’s pressure (P = C₁₂–C₄₄) for Mn₂X In (X = Fe, Co) is positive, suggesting a ductile tendency in these compounds.^[55] The values of Pugh’s ratio (B₀/G), calculated as 3.38 for Mn₂FeIn and 2.73 for Mn₂CoIn, exceed the critical value of 1.75, which serves as a threshold for distinguishing ductile and brittle materials.^[56–58] Additionally, the Poisson’s coefficients of 0.37 for Mn₂FeIn and 0.34 for Mn₂CoIn are higher than the critical value of 0.26, further confirming the ductile nature of Mn₂X In.

The calculated values of the anisotropy factor A for Mn₂X In increase when X changes from Fe to Co, attributed to the increasing atomic radius.^[59] This increase in A values also indicates a high degree of elastic anisotropy in these materials.

Based on the analyses, we can confidently assert that the Mn₂X In (X = Fe, Co) compound possesses superior ductility. This characteristic suggests that the lattice mismatch encountered during the fabrication of heterojunction and spintronic devices using Mn₂X In is unlikely to cause significant damage to the devices. To gain a deeper understanding of how lattice mismatch affects the material’s properties, we will proceed to investigate the changes in Mn₂X In’s electronic structure, spin characteristics, and magnetic anisotropy under biaxial strain conditions.

As the Slater–Pauling curve (SPC), the magnetic moment per unit cell, expressed in multiples of Bohr magneton (in units of μ_B/f.u), is delineated by a relationship^[60,61]

The variable N represents the count of valence electrons contained within the unit cell. For Mn₂FeIn, there are a total of 25 (2 × 7 + 8 + 3) valence electrons. For Mn₂CoIn, there are a total of 26 (2 × 7 + 9 + 3) valence electrons. For this reason, the magnetic moment expected from SPC is 1 μ_B/f.u for Mn₂FeIn and 2 μ_B/f.u. for Mn₂CoIn. The calculated values of total and individual magnetic moments at the equilibrium lattice constants for Mn₂X In (X = Fe, Co) are shown in Fig. 4. The calculated magnetic moment of Mn₂FeIn is 1.40 μ_B/f.u. and Mn₂CoIn is 1.69 μ_B/f.u. Faleev et al. have also reported the magnetic moment of Mn₂X In (X = Fe, Co) to be 1.26 μ_B/f.u. and 1.87 μ_B/f.u., respectively.^[29] To further elucidate this phenomenon, we have analyzed and presented the trends in total and partial magnetic moments under varying strain conditions, as depicted in Fig. 4. The strain is calculated as (a₀ – a)/a₀ × 100%, where a represents the current lattice constant on the biaxial axis, and a₀ denotes the equilibrium lattice constant.

As the strain decreases, magnetic moment of Fe and Mn2 in Mn₂FeIn increases, while the magnetic moment of Mn1 decreases. Based on the simplified rigid band model, the emergence of magnetic moments stems from the difference in density of states (DOS) between the spin-majority and spin-minority bands. The PDOS (partial density of states) for the Fe, Mn2, and Mn1 atoms were calculated and plotted in Figs. 5 and 6.

Combined with data of the DOS, there is degeneracy in the d_yz and d_xz orbitals of the two atoms (Fe and Mn2). As shown in Fig. 5, With the decrease of biaxial strain, the proportion of the spin-up d_xy, d_yz (d_xz), and d_z² orbitals of Fe in the occupied state increases with the trend of the movement to the state of lower energy and there is no significant change in the proportion of spin- down d_xy and d_yz (d_xz) orbitals which results in an increase in the magnetic moment of these orbitals. Meantime, the proportion of the spin-down d_z² orbital in the occupied state decreases with the movement to the state of higher energy which results in the increase of the total magnetic moment of the d_z² orbitals. As a result, the magnetic moment of the Fe atom increases with the decrease of strain. In terms of magnetic moment, the Mn2 atom has a similar tendency with the Fe atom. As shown in Fig. 5 (right), the spin-up d_x²–y² and d_yz (d_xz) orbitals of Mn2 atoms move towards the state of lower energy, resulting in the increasing proportion of occupied states and the increasing magnetic moment of the spin-up. The proportion of spin-down occupied states diminishes due to the movement of the spin-down d_x²–y² orbitals to the higher energy state. This results in the increase in the proportion of the spin-down unoccupied orbitals near the Fermi surface, while the proportion of occupied states decreases. As a result, the orbital magnetic moment contributed by the spin-up contribution of the Mn2 atom increases and the magnetic moments contributed by the spin-down decreases, and finally the total magnetic moment of the Mn2 atom increases.

Similarly, in Fig. 6, the proportion of spin-up unoccupied state increases due to the movement toward the state of the higher energy of d_xy, d_z², d_xz, and d_x²–y² orbitals of Mn1 for Mn₂CoIn with the decrease of biaxial strain, while the spin-down occupied proportion of d_xy, d_z², d_xz, and d_x²–y² orbitals decreases which result in an increase of the magnetic moments contributed by the spin-down contribution and the decrease of the magnetic moments contributed by the spin-up, and finally lead to the decrease of total magnetic moment of the Mn1 atom.

According to the formula for calculating the orbital magnetic moment:

where E_f is the energy of Fermi level, E_MAX and E_MIN refer to lowest and highest energies of density of states, and N(E) refers to the distribution of density of states with energy. We calculated the magnetic moments of the orbitals in atoms based on data of DOS, and organized to Table 5 as shown below.

Based on the comprehensive analysis and calculation results, the variation of atomic magnetic moments under strain originates from changes in the proportion of d-orbital states.

Based on the total DOS data, we calculated the spin polarizability exhibited by the material under different strains. The electron spin polarization (SP) at Fermi surface of a material is defined as follows:^[22,62–66]

where ρ^↑ (E_F) and ρ^↓ (E_F) are the majority and minority densities of states at Fermi surface. Compounds are considered to be genuine half-metals when the SP value reaches 100%, which occurs when either the density of states (DOS) from the majority or minority spins is precisely zero at the energy level of the Fermi surface, while the other DOS remains non-zero. Calculated SP for Mn₂FeIn and Mn₂CoIn with the strain from −2% to 5 % is shown in Fig. 7.

It becomes evident that the SP of Mn₂FeIn can be enhanced by tensile strain. As strain varies from −2% and 5%, Mn₂FeIn exhibits a remarkable SP variation from −2% to 74% and Mn₂CoIn exhibits a relatively gradual SP variation from −8% to 13%.

We depicted the total and partial DOS under different biaxial strain, as shown in the Fig. 8. Fe and Co atoms play a dominant role in the total spin down state on the Fermi level. Under the strain of 5%, the spin-down DOS exhibits a very low proportion on the Fermi surface in Mn₂FeIn. Furthermore, according to the detailed data of DOS, it can be known that the proportion of the density of states here is about 0.9, Thus, the pseudo-band gap here is the main reason for the large change of spin polarization in Mn₂FeIn with strain.

We further explore, from the band structure, the variation of the spin polarization of Mn₂FeIn with strain of 5% and −2%. As shown in Fig. 9, under the strain of −2%, the band structures are observed to show metallic behavior for majority spins and minority spins, which are in good agreement with the DOS (Fig. 8(a)). With the increase of strain, the energy bands at the Fermi surface for minority spin move towards the occupied state, and the band shapes at the K points of 1/2 of L–Γ and 1/2 of Γ–X become steeper and show the tendency to be more close to K point of Γ. Meanwhile the band gap at other K points increases, resulting in a decrease in the proportion of electron energy states at the Fermi surface and the appearance of pseudo-band gap (still metallic) in DOS.

Then, to further understand the d-orbital distribution around Fermi surface and the effect of strain, we plotted the d-orbital splitting and partial DOS of Mn and Fe atoms in the Mn₂FeIn compound, as shown in Figs. 10 and 11, respectively.

In Fig. 10, we have elucidated the plausible hybridization of d electrons occurring between Fe and Mn atoms within Mn₂FeIn. The d orbitals associated with Fe and Mn are partitioned into threefold-degenerate t_2g states (d_xy, d_yz, d_zx) and twofold-degenerate e_g states (d_x²–y², d_z²).^[51] To explicate the hybridization process, we initially examine the d states of Mn1 and Mn2 atoms. Following the hybridization of these states, a set of five bonding and antibonding states emerge, comprising (2 × e_g and 3 × t_2g) and (2 × e_u and 3 × t_1u), respectively. Subsequently, the newly formed states resulting from the Mn1–Mn2 hybridization interact with the t_2g and e_g states of Fe, thereby yielding ten states (comprising five bonding and five antibonding states). However, it is noteworthy that the antibonding states (2 × e_u and 3 × t_1u) originating from the Mn1–Mn2 hybridization cannot effectively couple with the d orbitals of Fe. This limitation arises from the fact that Fe solely possesses e_g and t_2g states, thus underscoring its role in establishing the d–d band gap.^[67]

Figure 11 shows the DOS for each atom with and without strain. From Fig. 11, it is evident that in majority spin PDOS, e_g electrons and t_2g electrons of Fe and Mn1 highly contribute at E_F for Mn₂FeIn. In minority spin PDOS, the proportion at the Fermi surface is mainly due to the contribution of t_2g electrons of the Fe atom. With the effect of the strain, the spin-down d-e_g electrons of Fe move toward the lower energy of the state and lead to the decrease for the proportion the total spin-down DOS, thus leading to a very large variation in SP.

The magnetic anisotropic energy (MAE) is defined as the energy difference for the magnetization along the arbitrary direction [abc] and perpendicular [0 0 1] direction.^[68,69] The SOC included in the noncollinear calculations using the second variation method is considered for calculating the MAE:^[70–73]

According to the second-order perturbation theory, the energy state distribution of atomic orbital at the Fermi surface will cause changes in the magnetic anisotropy of the material. The MAE under different strains for Mn₂X In is shown in Table 6.

From the results in the above table, it can be concluded that the easy axis corresponding to the lowest energy point of Mn₂FeIn under different strains is transformed between [001] and [100], indicating that the magnetic orientation of Mn₂FeIn is sensitive to strain, while Mn₂CoIn exhibits a stable in-plane MA, indicating that it is not strain-sensitive. Subsequently, we began to explore the principle of the easy axial transition of Mn₂FeIn under different strains. Firstly, we explored the change in the MAE_{[1 0 0]} of each atom under different strains through atom-resolution as shown in Fig. 12.

The change in MA is mainly due to the Fe and Mn1 atoms in layer II. Under the strain of 5% to −2%, both Fe and Mn1 atoms undergo a transition from in-plane MA to out-of-plane MA.

We have calculated the orbital-resolved MAE of the Fe and Mn1 atom located in the layer II of Mn₂FeIn. As shown in Fig. 13, when the strain is 5%, the interaction between the d_x²–y² and d_xy orbitals (d_x²–y²/d_xy) of the Fe atom results in a significant positive contribution to its total MAE, while the MAE from the d_z2 and d_yz orbitals (d_z2/d_yz) contributes negatively. Due to the fact that the magnitude of the positive contribution surpasses the negative one, the Fe atom exhibits a positive MAE under the strain of 5%. When the strain is −2%, the positive MAE from the d_x²–y²/d_xy orbitals decreases, and the negative MAE from d_z2/d_yz orbitals reduces, resulting in the rapid decrease of the total MAE of the Fe atom. Under the strain of 5%, the MAE of Mn1 atom is mainly caused by the hybridization of d_x²–y²/d_xy orbitals and d_z2/d_yz orbitals, and when the strain is −2%, the hybridization is weakened, so that the value of the total MAE changes from the positive to the negative.

Based on numerous theoretical investigations, the alterations in MAE are closely tied to variations in the electron occupancy of d orbitals near the Fermi level. Therefore, to deeply understand the influence of the strain on the MAE of the Fe and Mn1 atom in the layer-II, we explored the atomic d-orbital resolved PDOS, as shown in Fig. 14.

Magnetic anisotropy stems from the SOC effect, and the MAE is the consequence of the competition between in-plane and perpendicular contributions of the SOC Hamiltonian. Within the framework of the second-order perturbation theory, the MAE can be approximately expressed in terms of angular momentum operators L_x (or L_y) and L_z as^[74]

Here ξ is the SOC constant; E_uα and E_oβ are the energy levels of an unoccupied state with spin α (|uα〉) and an occupied state with spin β (|oβ〉), respectively. As shown in Fig. 14, The spin conservation term of spin-up occupied d_x²–y² orbital and the spin-up unoccupied d_xy orbital produce hybridization at the Fermi surface, resulting in an out-of-plane MAE contribution (positive contribution) to Fe atoms. With the decrease of strain, the d_x²–y² orbital moves to the lowest energy state of the occupied state, resulting in a decrease in the proportion at the Fermi surface. And the d_xy orbital moves to the occupied state of lower-energy, resulting in a small proportion increase in the proportion at the Fermi surface. According to the second-order perturbation theory, the MAE produced by the hybridization of d_x²–y²/d_xy orbitals decreases. When the strain is 5%, the hybridization of d_z²/d_yz occurs mainly at the spin-up Fermi surface and results in the negative contribution. As the decrease of the strain, the proportion of d_yz in the spin-up occupied state and d_z² in the spin-down unoccupied state increases, resulting in a positive MAE contribution.

Similarly, for Mn1 atoms, the spin flip term of SOC from d_x²–y² and the d_xy orbital produces negative MAE, which is mainly due to the hybridization of the spin-up occupied d_x²–y² orbital in the near the Fermi surface and the spin-down unoccupied d_xy orbital. As the strain decreases, the proportion of the spin-down unoccupied d_xy orbital decreases, resulting in a decrease in the negative contribution to the MAE. The spin flip term d_z² and d_yz orbitals have a positive contribution to the overall MAE, which is mainly derived from the hybridization of the spin-up occupied d_yz orbital in the near the Fermi surface and the spin-down unoccupied d_z² orbital. As the proportion of d_z² orbitals at the Fermi surface decreases under strain, the resulting positive contribution to MAE decreases.

It is evident from the band structure shown in Figs. 15 and 16 that the spin-up d_x²–y² orbital state tends to move away from the Fermi level at the L point, while the spin-down d_z² orbital state shows the same trend at the Γ point as the strain decreases for Fe in Mn₂FeIn. And according to the information of bubble diagram in the band structure, the proportion of spin-down unoccupied d_yz orbital of Mn1 atom along the k-point path of Γ–X decreases with the decrease of strain. These d orbitals state shifts in the energy bands are responsible for the decrease of MAE.

The electronic structure and magnetism of the Mn₂X In (X = Fe, Co) Heusler compounds were studied over the range of strain between −2% and 5% by density functional calculations. Firstly, we predicted the crystal structure and magnetic properties, and found that both Mn₂FeIn and Mn₂CoIn belong to XA–I structures of the inverse Heusler compounds and have A-AFM ferromagnetic magnetism. Mn atoms have antiferromagnetic coupling between them. In addition, we calculated the structural stability of Mn₂X In, determined the toughness of the two compounds, and found that their total magnetic moments remained at about 1.40 μ_B/f.u. and 1.69 μ_B/f.u., respectively. Under the biaxial strain, we observed that the DOS of Mn₂FeIn is sensitive to the biaxial strain, which is mainly manifested in the increase of spin polarization from −2% to 74%. The change of MA from [001] to [100] direction under strain can be ascribed to the hybridization of the d_x²–y²/d_xy and d_z²/d_yz orbitals.