Structural stability, electronic structure and magnetic properties of the new hypothetical half-metallic ferromagnetic full-Heusler alloy CoNiMnSi

We first optimized the unit cell volume by calculating the total energy as a function of volume of the quaternary Heusler alloy in its three structural phases, namely the Hg₂CuTi-type, Cu₂MnAl-type and LiMgPdSb-type structures for the ferromagnetic states. The calculated total energies within the GGA approximation as a function of the volume are displayed in Fig. 2. From the figure, one can see that the LiMgPbSb-type structure is the most stable due to its lowest total energy. The calculated total energies were fitted to Murnaghan’s equation of state [24] to obtain ground state properties, such as the equilibrium lattice constant, the bulk modulus B and its pressure derivative B’. The theoretically calculated ground state properties of this compound at zero pressure and ambient temperature for its different phases are given in Table 1. Because there are, to the best of our knowledge, no comparable studies of this compound in the literature, we compare our results to previously reported theoretical data on the Co₂MnSi compound [4, 25, 26]. Based on the good agreement between the present calculated values and the theoretical results for Co₂MnSi obtained from the literature, we believe that our calculations of the structural properties of the three structural phases are reliable, and we encourage future experimental work to corroborate our calculated results.

Fig. 2.

We now turn our attention to discuss the mechanical properties of the CoNiMnSi compound. In particular, we will confirm its mechanical stability in the LiMgPbSb-type structure via calculation of the elastic constants C_ij. These constants are fundamental and indispensable for describing the mechanical properties of materials because they are closely related to various fundamental solid-state phenomena, such as interatomic bonding, equations of state and phonon spectra. Elastic properties are also linked thermodynamically with specific heat, thermal expansion, Debye temperature, and the Grüneisen parameter. Most importantly, knowledge of the elastic constants is essential for many practical applications related to the mechanical properties of a solid. The elastic constants C_ij are the proportionality coefficients relating the applied strain ∊_i to the stress σ_i, where σ_i = C_ij∊_i. Therefore, the C_ij determine the response of the crystal to external forces. For a completely asymmetrical system, there are 21 independent elastic constants C_ij; however, the cubic symmetry of the studied crystal reduces this number to only three (C₁₁, C₁₂ and C₄₄) independent elastic constants. The elastic constants C_ij are obtained by calculating the total energy as a function of volume, while conserving strains that break the symmetry. Further details of the calculation can be found in the literature [30]. The presence of mechanically stable phases or macroscopic stability depends on the positive definiteness of the stiffness matrix [31]. The following conditions are known as the Born Huang criteria [32], and they are defined as C₁₁ > 0, C₄₄ > 0, C₁₁ > C₁₂ and (C₁₁ + 2C₁₂) > 0 for a cubic structure. The elastic constants C_ij are estimated from first-principles calculations for a single crystal of our compound. However, the prepared materials are in general polycrystalline, therefore, it is important to evaluate the corresponding moduli for the polycrystalline species. For this purpose, we have derived other mechanical parameters, namely the shear modulus G, Young’s modulus E and Poisson’s ratio ν, which are the important elastic moduli for applications, using the following standard relations [33, 34]: (1)G=C11−C12+3C145$G = {{{C_{11}} - {C_{12}} + 3{C_{14}}} \over 5}$(2)E=9BG3B+G$E = {{9BG} \over {3B + G}}$(3)v=3B−2G2(3B+G)$v = {{3B - 2G} \over {2\left( {3B + G} \right)}}$(4)A=2C44C11−C12$A = {{2{C_{44}}} \over {{C_{11}} - {C_{12}}}}$

Table 2 lists the elastic constants and elastic moduli for polycrystalline materials calculated within the GGA approximation. One can see that the LiMgPbSb-type structure for CoNiMnSi is mechanically stable at ambient conditions because the Born Huang criteria are satisfied. The C₁₁ value, which is related to the unidirectional compression along the principal crystallographic direction, is much greater than that of C₄₄, indicating a weak resistance to shear deformation compared with the resistance to unidirectional compression. We also note that this compound possesses a high bulk modulus and Young’s modulus, what classifies this compound as a strong incompressible material. The typical value of Poisson’s ratio (ν) for ionic materials is 0.25 [35]. The calculated value of Poisson’s ratio for the studied compound is less than 0.25, indicating a covalent contribution in the interatomic bonding. The calculated value of the shear anisotropic factor A for this compound obtained from GGA calculations is given in Table 2. The value of A must be one for an isotropic crystal, while any value smaller or greater than unity is a measure of the degree of elastic anisotropy possessed by the crystal [38]. The calculated value is slightly below unity, which means that this compound has a lower anisotropy and possesses a low probability to develop micro-cracks or structural defects during its growth process.

Once we have calculated the Young’s modulus E, bulk modulus B and the shear modulus G, we may estimate the Debye temperature from the average sound velocity ν_m [36]: (5)θD=hkB(3n4πVa)−13vm${\theta _D} = {h \over {{k_B}}}{\left( {{{3n} \over {4\pi {V_a}}}} \right)^{ - {1 \over 3}}}{v_m}$ where h is Plank’s constant, k_B is Boltzmann’s constant and V_a is the atomic volume. The average sound velocity in the polycrystalline material is given by [36]: (6)vm=[13(3vt3+1vl3)]−13${v_m} = {\left[ {{1 \over 3}\left( {{3 \over {v_t^3}} + {1 \over {v_l^3}}} \right)} \right]^{ - {1 \over 3}}}$ where ν₁ and ν_t are, respectively, the longitudinal and transverse sound velocities obtained using the shear modulus G and the bulk modulus B from Navier’s equation [37]: (7)vl=(3B+G3ρ)12${v_l} = {\left( {{{3B + G} \over {3\rho }}} \right)^{{1 \over 2}}}$(8)vt=(Gρ)12${v_t} = {\left( {{G \over \rho }} \right)^{{1 \over 2}}}$

The calculated sound velocities and Debye temperature for the CoNiMnSi compound in the LiMgPbSb-type structure are listed in Table 3. To the best of our knowledge, the sound velocities, Debye temperature, elastic constants and their related properties have not yet been established for this compound. We hope that our calculations can be used to make up for the lack of data on this compound.

In the following paragraph we shed more light on the electronic properties of the CoNiMnSi compound in its stable phase by calculating the total and atomic site-projected 1-decomposed densities of states (TDOS and PDOS) for the spin-up and spin-down electrons using the GGA and GGA + U approximations. The electronic valence states for the studied compound are 4s/4p/3d for Co, Ni and Mn and 3s/3p for Si. The plots of the TDOS and PDOS functions computed with both GGA and GGA + U are displayed in Fig. 3 and Fig. 4, respectively. One can see, no energy gap at the Fermi level for the minority spin states with the GGA approximation, indicating that this compound has ferromagnetism-false half-metallic character at the equilibrium state. However, the GGA + U approximation yielded a gap of 0.17 eV in the minority spin, as shown in Fig. 4. This gap is due to the strong hybridization between the Co/Ni/Mn-d states, thus, confirms that our compound exhibits half-metallic ferromagnetism. From a visual inspection, the TDOS spectrum is divided into two regions. The first one situated between −5 eV and 2.5 eV is due to the strong hybridization of the 3d-Co, 3d-Ni and 3d-Mn states. The second is the region outside of the energy range of −5 to 2.5 eV, which is dominated by Si-s,p states.

Fig. 3.

Fig. 4.

To understand the nature of the chemical bonding, we have also calculated the contours of the charge densities in the (1 1 0) plane with both the GGA and GGA + U approximations, and the results are displayed in Fig. 5 and Fig. 6, respectively. From the figures it can be observed that the charge contours are similar and that the near spherical charge distributions around the Si atom sites are negligible, indicating that the Si–Mn, Si–Ni and Si–Co bonding is expected to be ionic in character. However, between the transition metal elements, the Co and Ni atoms hybridize with an Mn atom for minority and majority spin, suggesting a covalent interaction between Co and Ni with Mn atoms. Therefore, the bonding character may be described as a mixture of covalent and ionic character.

Fig. 5.

Fig. 6.

We will now also report on the magnetic properties of the studied compound. For this purpose, we have evaluated the total and the local magnetic moments, which are presented in Table 4. We have found that the magnetic moment of the Mn atom is much higher than those of the other atomic constituents. We have also observed that the magnetic moments calculated with GGA + U are greater than those calculated with GGA as well as those previously obtained for the Co₂MnSi compound [4, 25, 26, 29]. The local moment of the Si atom is negligibly small and aligned anti-parallel to the Co, Ni and Mn moments. It originates from the hybridization of the transition metals and is caused by the overlap of the electron wave functions, as reported by Kandpal et al. [29]. In Table 5, we report the computed spin-polarization, magnetic exchange energy between ferromagnetic FM and anti-ferromagnetic AFM states (∆E_(FM−AFM)) and Curie temperature (T_c). The electron spin polarization (P) at the Fermi energy (E_F) of a material is defined by: (9)P=ρ↑(EF)−ρ↓(EF)ρ↑(EF)+ρ↓(EF)$P = {{\rho \uparrow \left( {{E_F}} \right) - \rho \downarrow \left( {{E_F}} \right)} \over {\rho \uparrow \left( {{E_F}} \right) + \rho \downarrow \left( {{E_F}} \right)}}$ where ρ ↑(E_F) and ρ ↓(E_F) are the spin dependent densities of states at the Fermi level (E_F). The ↑ and ↓ symbols denote the majority and the minority states, respectively. We note from the results of the calculated electron spin polarization at E_F that the 100 % spin polarization characteristic of HM materials is observed in the studied compound when Coulomb interactions are considered. To find the ground state magnetic structure, we have performed total-energy calculations of different magnetic configurations, i.e. the ferromagnetic (FM) and anti-ferromagnetic (AFM) phases, to verify the magnetic stability of the investigated compound. The anti-ferromagnetic AFM configuration is shown in Fig. 7. By comparing the total energies, we find that the FM configuration for our compound is more stable than the AFM configuration. The calculated total energy difference of the FM state is lower than that of the AFM state by about −0.011046 mRy/atom, as shown in Table 5. The Curie temperatures T_c for CoNiMnSi were estimated according to the model presented in the literature [4, 39] by applying the linear relation T_c = 23 + 181 μ^Total, where μ^Total is the total magnetic moment. The values obtained are 871.89 K and 1107.19 K for the GGA and GGA + U approximations, respectively.

Fig. 7.