Electronic structure, magnetic properties, and elastic properties of full-Heusler alloys Cr_2-xFe_xMnSi (x=0, 1, and 2)

Full-Heusler Cr₂MnSi and Fe₂MnSi alloys will have two possible structure types, which are Cu₂MnAl- and Hg₂CuTi-type crystal structures. Quaternary Heusler compound CrFeMnSi is a LiMgPdSn-type compound which crystallizes into Y-type structure with three possible nonequivalent atomic configurations for the compounds: type 1, type 2, and type 3. The crystal structure of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys was shown in fig. 1.

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In order to find the most stable structure among the compound types, we carried out geometry optimizations for Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2). Considering the lattice parameters of 5.6586 Å and 5.654 Å for Fe₂MnSi [27] and Co₂MnSi [28], respectively, the experimental lattice constant $a_{0} = 5.65$ Å was used for the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloy. The crystal structure types of the Cr₂MnSi and Fe₂MnSi alloys and each possible structure type in the FeMnCrSi alloy were considered in our calculations. The calculated energy, lattice parameters, total and each atom magnetic moments for the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloy are shown in table 1.

Table 1:. Lattice parameter, energy, total and each atom magnetic moment for possible configurations of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys.

E (eV) $m_{\textit{total}}$$m_{\mathrm{Cr}_{1}/\mathrm{Fe}_{1}}$$m_{\mathrm{Cr}_{2}/\mathrm{Fe}_{2}}$$m_{\mathrm{Mn}}$$m_{\mathrm{Si}}$

			Magnetic moments $(\mu \mathrm{B})$
	a (Å)
Cr2MnSi
Cu2MnAl	5.707	−22787.5757	2.2	0.62	0.62	0.90	0.06
Hg2CuTi	5.718	−22790.3718	1	1.50	−1.44	0.88	0.06
Fe2MnSi
Cu2MnAl	5.579	−9972.7645	3	0.26	0.26	2.44	0.04
Hg2CuTi	5.581	−9971.6957	3	2.24	0.88	−0.16	0.04
Cr-Fe-Mn-Si
Type 1	5.709	−16381.1621	1.02	−1.76	0.26	2.54	−0.02
Type 2	5.715	−16379.5597	2.42	−1.30	2.22	1.48	0.02
Type 3	5.594	−16382.4777	−1	−1.32	−0.06	0.38	0

For full-Heusler Cr₂MnSi and Fe₂MnSi alloys, there are two possible structure types which Hg₂CuTi-type and Cu₂MnAl-type. Table 2 shows that the Hg₂CuTi-type Cr₂MnSi alloy and Cu₂MnAl-type Fe₂MnSi alloy have low energy. Therefore, we believe that the stable structures of the Cr₂MnSi and Fe₂MnSi alloys belong to the Hg₂CuTi-type and Cu₂MnAl-type, respectively. For CrFeMnSi alloy, we adopt a so-called Y-type structure with the prototype LiMgPdSn. For the CrFeMnSi alloy we have studied, the structure of type 3 has the lowest total energy so that the CrFeMnSi should crystallize in type 3 structure. Therefore, in the discussion below, we only adopt the stablest structure and its equilibrium lattice parameter to calculate the electronic structures and the magnetic properties.

Table 2:. The elastic constants, bulk modulus B, shear modulus G and Poisson's ratio ${\nu}$ of Cr_2-x Fe_x MnSi (x = 0, 1 and 2) alloy.

Compound	C11	C12	C44	B	G	B/G	${\nu}$	A
Cr2MnSi	280.1	155.5	113.0	197.0	82.4	2.39	0.316	1.81
FeMnCrSi	404.6	165.0	137.5	244.9	87.1	2.81	0.341	1.15
Fe2MnSi	307.3	224.2	173.3	251.9	101.8	2.47	0.322	4.17

Half-metals have a band structure in which one spin-polarized sub-band with a band gap at the Fermi level resembles the band structure of a semiconductor, whereas the other sub-band exhibits metallic properties. The band structures of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys are shown in fig. 2.

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As shown in fig. 2, the Cr₂MnSi and FeMnCrSi alloys have band gaps in the majority-spin direction, and the Fe₂MnSi alloy has a band gap in the minority-spin direction. The band gap energies for Cr₂MnSi, FeMnCrSi, and Fe₂MnSi are 0.136, 0.160, and 0.502 eV, respectively. Thus, the half-metallic band gap of the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloy was increased as the number of Fe atoms increases.

Table 1 shows that the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys have good magnetic properties. The total magnetic moments are an integral value for Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2), which is a typical property of half-metals [29,30]. To investigate the cause of magnetic properties, we calculated the density of states (DOS) and partial density of states (PDOS) for the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys, as shown in fig. 3. The majority-spin (spin-up) states are expressed by the positive DOS values and the minority-spin (spin-down) states are expressed by negative DOS values. The total density of states shows that Cr₂MnSi and CrFeMnSi alloys exhibit an energy gap in the majority-spin channel, but the Fe₂MnSi alloy exhibits an energy gap in the minority-spin channel. Therefore, the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys can be considered as half-metals.

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The DOS and PDOS of alloys can also be used to analyze the source of the alloy's magnetism. Figure 3(a), (b) shows that the 3d electrons of Cr and Mn contribute mostly to the DOS near the Fermi level for the Cr₂MnSi alloy, and at approximately 1.5 eV, Cr₁ and Cr₂ provide small d-peaks which are superposed with the DOS peak of Mn; the contributions from Cr₁ and Cr₂ to the total DOS are identical. This behavior shows that the magnetism of Cr₂MnSi mainly comes from the spin contribution of Cr- and Mn-d orbital electrons and the strong hybridization between them. A similar case is made for Fe₂MnSi and CrFeMnSi alloys in fig. 3(c)–(f). Therefore, the magnetism of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys has two reasons, one is that the spin contribution of the d electrons of the transition metal elements near the Fermi level are not equal in the two spin directions, and the other is strong hybridization between d electrons of transition metal elements.

The spin polarization can be calculated by the following equation:

$$\begin{equation} P=\frac{N_{\uparrow }\left(E_{F}\right)- N_{\downarrow }\left(E_{F}\right)}{N_{\uparrow }\left(E_{F}\right)+N_{\downarrow }\left(E_{F}\right)}, \end{equation} \tag{ 1 }$$

where $N_{\uparrow }\left(E_{F}\right)$ and $N_{\downarrow }\left(E_{F}\right)$ match the majority-spin and minority-spin DOS at the Fermi level, respectively [31]. From the total DOS of the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys, the majority-spin DOS at the Fermi level is zero for the Cr₂MnSi and FeMnCrSi alloys, while the minority-spin DOS at the Fermi level is zero for the Fe₂MnSi alloy; therefore, there is 100% spin polarization in Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2).

In material applications, the physical properties of alloys will be affected by changes in lattice constants [32]. Experimental results have shown that the change of stress results in the lattice parameter deviating from the equilibrium parameter. Therefore, a study considering the influence of lattice parameter changes on the half-metallic and magnetic properties of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys is necessary. A large number of studies have shown that half-metallic materials usually have an integer number of total magnetic moments. In order to check the stability of the half-metallic and magnetic properties of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2), we calculated the total moments with the lattice parameter being changed within the range from −3.5% to +3.5%. The results are displayed in fig. 4.

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Figure 4 shows that the total moment of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys is essentially constant across the entire scale, with only slight fluctuations. For the Cr₂MnSi alloy, the magnetic moments of Cr₁ and Cr₂ change significantly, but owing to the opposite trend of change in Cr₁ and Cr₂ atomic moments, the total magnetic moment remains constant at 1 μB when the lattice parameter is changed within the range from −1.5% to +3%. This behavior illustrates that the compressed volume will affect the half-metallic and magnetic properties of the Cr₂MnSi alloy. A similar observation is made for CrFeMnSi and Fe₂MnSi, the total magnetic moment remains an integer value when the lattice parameter is changed within the range from −2% to +3% and −3.5% to +3.5%, respectively.

Mechanical properties are important for the use of materials in various applications. Elastic constants indicate the mechanical properties of materials, such as brittleness and ductility, rigidity and resistance, and isotropy and anisotropy. For a cubic system with C₁₁, C₁₂, and C₄₄ as the three independent elastic constants, the mechanical stability is typically assessed based on the following conditions [33]:

$$\begin{equation} \left\{\begin{array}{l} C_{11}>0,\\ C_{44}>0,\\ C_{11}- C_{12}>0,\\ C_{11}+2C_{12}>0. \end{array}\right. \end{equation} \tag{ 2 }$$

For stable materials, C₁₁, C₁₂, and C₄₄ satisfy all the above conditions. Other parameters also indicate the elastic properties of the material, such as the bulk modulus (B), the Voigt-Reuss-Hill averaged shear modulus (G), Poisson's ratio $({\nu})$, and Young's modulus (E); they are expressed as follows [34]:

$$\begin{eqnarray} B&= \frac{C_{11}+2C_{12}}{3}, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} G_{V}&= \displaystyle\frac{C_{11}- C_{12}+3C_{44}}{5},\quad G_{R}=\frac{5C_{11}(C_{12}- C_{44})}{4C_{44}+3(C_{11}- C_{12})},\nonumber\\ \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} G&= \displaystyle\frac{G_{V}+G_{R}}{2}, \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \nu &= \frac{3B- Y}{6B}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} E&= \frac{9BG}{3B+G}, \end{eqnarray} \tag{ 7 }$$

where G_V is the Voigt shear modulus, and G_R is the Reus shear modulus. The results obtained are summarized in table 2.

B characterizes the ability of a material to resist pressure, and a large B indicates that the material has a strong ability to resist pressure. G characterizes the ability of a material to resist shear strain, and a large G implies that the material has strong rigidity. From table 2, the resistance and rigidity of the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys are observed to increase with the increase of Fe. According to Pugh's theory, the value of B/G reflects the brittleness and ductility of the material, i.e., ratios >1.75 and ratios <1.75 correspond to ductile and brittle materials, respectively. According to the Frantsevich rule, ${\nu}$ of a material can also be used to evaluate the brittleness and ductility of the material; for ductile materials, ${\nu}> 0.26$ and for brittle materials, ${\nu}< 0.26$. Therefore, the Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys are ductile materials.

The Zener anisotropy parameter A indicates the degree of elastic anisotropy. For cubic crystal materials, A can be calculated using the following equation:

$$\begin{equation} A=\frac{2C_{44}}{C_{11}- C_{12}}. \end{equation} \tag{ 8 }$$

In general, materials with large values of A deviate from the cubic structure. If $A = 1$, the material is isotropic, and if A > 1, the materials are anisotropic. Table 2 presents the elastic anisotropy calculations for Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys; the values of A are greater than 1, indicating that the compounds are elastic anisotropic.

Furthermore, the surface contours of Young's modulus may reflect the anisotropic properties of the material. The 3D surface of E is defined by the following equation [35,36]:

$$\begin{equation} \frac{1}{E}=S_{11}- 2\left(S_{11}- S_{12}- \frac{S_{44}}{2}\right)\left(l_{1}^{2}l_{2}^{2}+l_{2}^{2}l_{3}^{2}+l_{1}^{2}l_{3}^{2}\right)\!.\quad \end{equation} \tag{ 9 }$$

In eq. (9), S_ij represents companion matrix constants. The parameters l₁, l₂, and l₃ are directional cosines. An isotropic material is characterized by spherical surface contours of Young's modulus or circular plane projections. Figure 5(a)–(c) provides the surface contours of Young's modulus of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys, and fig. 5(d), (e) shows the (001) and (110) plane projections of Cr_2-x Fe_x MnSi ($x = 0$, 1, and 2) alloys. The anisotropy of Young's modulus is noticeable in both planes for Cr₂MnSi and Fe₂MnSi alloys. The directions where the maxima appear correspond to the high-fracture energy directions, which are along (111) in the (110) plane and along (110) in the (001) plane [37].

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