4.1. Structural Properties

To obtain the lattice constant, the bulk modulus and its first pressure derivative which are listed in Table I, we have fitted the computed energies to the empirical Murnaghan’s equation of state ²². In this respect, the optimization of the geometrical structure parameters of Cu₂ MnSi_1−x Al_x alloys has performed by using nonmagnetic (NM) and ferromagnetic (FM) configurations. The total energy variation, which is taken as function of volume for both non-magnetic and ferromagnetic states with different concentrations, is illustrated in Fig. 2. It can be seen that these alloys are ferromagnetic. The results for Cu₂MnAl compound, which are given in Table I, agree with the experimental results ^23,24 and other theoretical works ^8,10,11,25. To the best of our knowledge, no comparable studies in literature on Cu₂MnSi_1−x Al_x (x = 0.125, 0.25, 0.375, 0.5) alloys. The estimated lattice parameters (Å), from the Vegard’ slaw ²⁶ in Eq. (1), for the selected concentrations 0.125, 0.25, 0.375, 0.5, are 5.86875, 5.8775, 5.88625 and 5.895 respectively.

Figure 2 Calculated total energy as a function of volume curves for Cu₂MnSi_1−x Al_x (x = 0.125, 0.25, 0.375, 0.5) alloy. 

4.2. Elastic Properties

In order to discuss the mechanical stability of the parent compounds Cu₂MnSi, Cu₂MnAl and Cu₂MnSi_1−x Al_x quaternary alloy, we have calculated the three independent elastic constants for cubic crystals C₁₁, C₁₂ and C₄₄, by using a numerical first-principles method. The traditional mechanical stability conditions in cubic crystal are expressed as follows: C₁₁ -C₁₂ >0, C₁₁ >0, C₄₄ >0, C₁₁ + 2C₁₂ > 0 and C₁₂ < B < C₁₁²⁸. The calculated elastic constants C _ij , given in Table II, show that with the exception of Cu₂MnSi which does not fulfill the stability criteria, Cu₂MnAl and Cu₂MnSi_1−x Al_x alloys are elastically stable. The obtained results for Cu₂MnAl alloy agree with experimental results in Ref. 23 and those of Rai et al., ¹¹ and Jalilian ²⁵.

In addition, other parameters have been calculated as the Shear modulus (G), Young’s modulus (E), Poisson’s ratio (ν), anisotropy factor (A), and Kleinman parameter (ξ), listed in Table II too. For these calculations, we have used the following equations:

where G_V and G_R are Voigt’s shear modulus and Reuss’s shear modulus corresponding to the upper and the lower bound of G values respectively.

Pugh ²⁹ proposed an approximate criterion by the ratio B/G to predict the ductility of materials. If B/G ratio is higher than the critical value that separates brittle and ductile materials which is about 1.75, this corresponds to ductile behavior; else, the material is brittle. The calculated values in Table II indicate that the B/G ratios are between 1.866 and 3.027, suggesting the ductile nature of the studied alloys. The Cauchy pressure C₁₂ C₄₄ identifies the type of bonding ³⁰. Negative Cauchy pressure corresponds to more directional and non-metallic character, while positive value indicates predominant metallic bonding. According to Cauchy pressure, the predominant bonding for Cu₂MnSi_1−x Al_x Heusler alloys is metallic. The Young’s modulus (E) characterizes the stiffness of a material. A higher value of E, stiffer is the material. It can be seen, from Table II, that Cu₂MnAl is stiffer than Cu₂MnSi. Poisson’s ratio (ν) indicates the degree of directionality of the covalent bonds. Its value for covalent materials is small (ν < 0.1), whereas the typical value for ionic materials is 0.25 ³¹. Our calculated Poisson’s ratios are from 0.279 to 0.319, so the contribution in the intra atomic bonding for Cu₂ MnSi _1−x Al_x alloys is ionic. For an isotropic material, the anisotropy factor (A) is equal to one, while any different value shows anisotropy. The calculated anisotropy factor indicates that Cu₂ MnSi_1−x - Al_x compounds are anisotropic.

4.3. Electronic Properties

To determine the electronic structure’s nature of Cu₂ MnSi_1−x Al_x compounds, we have calculated the total and partial densities of states for spin-up and spin-down, as displayed in Fig. 3. From this figure, one can see that there is no energy gap at Fermi level in both minority and majority spin states, proving the metallic character of the system. The TDOS spectrum of the parent compounds is divided into two main regions. The lowest valence bands below −9 eV for Cu₂MnSi (below −6 eV for Cu₂MnAl) are entirely due to Si and Al s-states, while the bands from −7 to 3 eV for Cu₂MnSi (−5.5 to 3 eV for Cu₂MnAl) are chiefly governed by the Cu and Mn 3d states. A comparison with other studies obtained by Kulkova et al., ⁸ and Rai et al., ¹¹, our results for Cu₂MnAl show quite good agreement. For Cu₂MnSi_1−x Al_x quaternary alloy, the principally parts of the total densities of states situated between −5 to 3 eV, are contributed by the 3d states of Cu and Mn atoms.

Figure 3 Total and partial density of states for Cu₂MnSi_1−x Al_x (x = 0, 0.125, 0.25, 0.375, 0.5, 1) Heusler alloy. 

4.4. Magnetic Properties

The calculated total and partial spin magnetic moments for Cu₂ MnSi, Cu₂MnAl and Cu₂ MnSi _1−x Al_x quaternary alloy are quoted in Table III. Obviously, for Cu₂MnSi and Cu₂MnAl alloys, the total magnetic moment, which includes the contribution from the interstitial region, originates mainly from the Mn atom, with a small contribution of Si, Al and Cu sites. Our results agree well with Kulkova et al., ⁸, Ghosh et al., ¹³ and Kandpal et al., ³². For all the considered concentrations x, the positive spin magnetic moment of Cu and Mn means a ferromagnetic coupling between Cu and Mn atoms. The negative magnetic moment for Si and Al leads to an antiferromagnetic alignment of Si and Al ones. The obtained total and partial spin magnetic moment for the Cu₂MnAl alloy are in good agreement with previous theoretical results ^8,11,32,33 and experimental ones ³⁴, which are also reported in Table III. To our knowledge, there are no values of the magnetic moments in the literature for the quaternary alloy. In Fig. 4, the variation of the total and partial magnetic moments for Cu₂MnSi_1−x Al_x (x = 0, 0.125, 0.25, 0.375, 0.5, 1) alloys versus the composition x, are non-linear.

Figure 4 Magnetic moment for Cu, Mn, Si, Al and total magnetic moment within the concentration x for Cu₂MnSi_1−x Al_x (x = 0, 0.125, 0.25, 0.375, 0.5, 1).