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Revista mexicana de física

versão impressa ISSN 0035-001X

Rev. mex. fis. vol.64 no.2 México Mar./Abr. 2018

https://doi.org/10.31349/revmexfis.64.135 

Research

Structural, elastic, electronic and magnetic properties of quaternary Heusler alloy Cu2MnSi1−xAlx (x = 0 − 1): First-principles study

B. Benichoua  b  * 

Z. Nabic 

B. Bouabdallahd 

H. Bouchenafaa  e 

aDepartment of Physics, Faculty of Sciences Exact, Djillali Liabès University, Sidi Bel Abbès 22000, Algeria.

bDepartment of Electronics, Faculty of Technology, Hassiba Benbouali University, Chlef 02000, Algeria.

cLaboratory of Catalysis and Reactive Systems, Physics Department, Djillali Liabès University, Sidi Bel Abbès 22000, Algeria.

dCondensed Matter and Sustainable Development Laboratory, Djillali Liabès University, Sidi Bel Abbès 22000, Algeria.

eDepartment of Physics, Faculty of Sciences Exact and Informatics, Hassiba Benbouali University, Chlef 02000, Algeria.


Abstract

We investigate the structural, elastic, electronic and magnetic properties of the Heusler compounds Cu2 MnSi, Cu2 MnAl and Cu2 MnSi1−x Alx quaternary alloys, using the full-potential linear-augmented plane-wave method (FP-LAPW) in the framework of the density functional theory (DFT) using the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE). Our results provide predictions for the quaternary alloy Cu2MnSi1−xAlx (x = 0.125, 0.25, 0.375, 0.5) in which no experimental or theoretical data are currently available. We calculate the ground state’s properties of Cu2MnSi1−xAlx alloys for both nonmagnetic and ferromagnetic configurations, which lead to ferromagnetic and metallic compounds. Also, the calculations of the elastic constants and the elastic moduli parameters show that these quaternary Heusler alloys are ductile and anisotropic.

Keywords: Electronic structure; elastic properties; ab-initio calculations; quaternary Heusler alloys

PACS: 75.10.Hk; 71.45.Nt

1. Introduction

So far, the Heusler alloys are still interesting due to their potential application in spintronics, such as giant magnetoresistance (GMR), tunneling magnetoresistance (TMR) 1, superconductors 2, ferromagnetic shape memory alloys 3 and magnetic actuator 4. Besides of Heusler alloys, the intermetallic compounds are also promising materials for automobile, aviation, aerospace and advanced thermoelectric applications.

Many works on Cu2 MnZ, especially Cu2MnAl alloys, have been investigated in both cases, experimentally 5-7 and theoretically 8-10. Rai et al., 11 showed that Cu2MnAl is an interesting ferromagnetic and metallic compound in spite of its non-ferromagnetic elements. Hamri et al., 12 illustrated that all the studied ferromagnetic systems X2MnSn (X = Cu, Ni, Pd) exhibit a metallic character and possess an interesting elastic constants. Also, Ghosh et al., 13 found that Cu2MnGa has metallic and ferromagnetic properties and is thermodynamically as well as mechanically stable alloy. In addition to ternary Heusler studies, there exist several searches on quaternary alloys. Galaknakis 14 investigated quaternary alloys as X2Y1−x Y’x Z, (X1−x X’x)2YZ and X2YZ1−x Z'x, he found that there is a possibility of obtaining half-metallic systems. The spin polarization of Co2 Cr 1−x Fe x Al quaternary alloys have been reported by Karthik et al., 15. Nanto et al., 16 have studied the magnetic properties of nanocrystalline Fe2Mn0.5Cu0.5Al using mechanical alloying technique.

This paper is arranged as follows: In the next section, we give a brief description of the calculation method. Section 3 deals with the crystal structural aspects. In Sec. 4, the results and their comments are presented and the paper is ended after by a conclusion summarizing the study.

2. Computational Details

The first principles calculations performed within the FPLAPW method (17) which is implemented in the WIEN2k code 18, based on the DFT theory 19,20, where the GGA approximation 21 has employed to describe the exchange and correlation potential. For the numerics, we estimate the plane wave parameter RMT × Kmax as 7.0, and to ensure the correctness of the calculations, we have taken lmax = 12. The Gmax parameter was 12.0. The separation energy between the core and the valence states has chosen as 6.0 Ry. The self consistent potentials calculated on a 21 × 21 × 21 k-mesh in the Brillouin Zone for ternary alloys and 2 × 2 × 2 k-mesh for quaternary alloys, which correspond respectively, to 286 and 4 k-points in the irreducible BZ. The muffin-tin sphere radii were 2.2, 2.0, 2.0 and 1.9 for Cu, Mn, Si and Al, respectively. The energy convergence criterion was taken as 10−5 Ry.

3. Crystal structure

Full-Heusler alloys have the chemical formula X2YZ, where X and Y denote transition metals and Z is an s-p element. The atomic positions for X (Cu) atoms are (1/4, 1/4, 1/4), (3/4, 3/4, 3/4), while (1/2, 1/2, 1/2) for Y (Mn) and for Z (Si, Al) it is (0, 0, 0). In Fig. 1, we show the crystal structure of Cu2MnSi 1−x Al x (x = 0, 0.125, 0.25, 0.375, 0.5, 1) alloys, where the present structures consist of four interpenetrating face-centered-cubic sublattices, with L21 phase and Fm-3m, space group no. 225. To simulate Cu2MnSi1−x Alx quaternary alloys, we consider a (2 × 2 × 2) supercell eight times bigger than L21 unit cell. The supercell is then constituted of 32 atoms; 16 Cu, 8 Mn and 8 Si/Al, as shown in Fig. 1.

Figure 1 Crystal structure of Cu2MnSi1−x Alx (x = 0, 0.125, 0.25, 0.375, 0.5, 1). 

4. Results and Discussion

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 MnSi1−x Alx 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 Cu2MnSi1−x Alx (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.

Cu2MnSi1-xAlx :a=5.86×1-x+5.93×x (1)

Table I Calculated lattice parameter (a), bulk modulus (B) and its pressure derivative (B') for Cu2MnSi1−x Alx (x = 0, 0.125, 0.25, 0.375, 0.5, 1). 

Compound a(Å) B(GPa) B'
Cu2MnSi 5.8645 136.137 5.147
Cu2MnSi1-0-125Al0.125 5.87 118.758 6.136
Cu2MnSi1-0.25Al0.25 5.88095 124.910 3.450
Cu2MnSi1-0-375Al0.375 5.8879 126.467 4.229
Cu2MnSi1-0.5Al0.5 5.8918 118.944 6.261
Cu2MnAl 5.9274 126.689 4.066
Theo. 8,11,27 5.9628
5.95711 115.6411
5.91527
Exp.24 5.948(24)

Figure 2 Calculated total energy as a function of volume curves for Cu2MnSi1−x Alx (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 Cu2MnSi1−x Alx 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 Cu2MnSi1−x Alx 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.

Table II Calculated elastic constants (in GPa), and G, B/G, E, υ, A, ξ of Cu2MnSi1−x Al x (x = 0, 0.125, 0.25, 0.375, 0.5, 1). 

Compound C11 C12 C44 B G B/G E υ A ξ
Cu2MnSi 133.423 138.671 108.124 137.029 63.824 2.146 165.740 0.319 -41.201 1.025
Cu2MnSi1-0.125Al0.125 155.607 129.243 103.390 137.994 47.485 2.90 173.682 0.312 7.843 0.882
Cu2MnSi1-0.25Al0.25 154.061 129.182 99.346 137.417 45.382 3.027 167.508 0.318 7.986 0.888
Cu2MnSi1-0.375Al0.375 200.123 115.834 111.110 143.903 70.485 2.017 209.953 0.279 2.636 0.690
Cu2MnSi1-0.5Al0.5 202.463 124.960 111.640 150.976 73.096 2.065 209.332 0.291 2.880 0.721
Cu2MnAl 137.867 122.284 108.790 127.645 68.391 1.866 174.081 0.295 13.962 0.922
Theo. 11,25 137.6811 104.6111 460.4111 11511 0.8211
13725 11525 11225 12225 0.8925
Exp. 23 135.523 97.323 9423

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:

G=GR+GV2 (2)

GV=C11-C12+3C445 (3)

GR=5C11-C12C444C44+3(C11-C12 (4)

E=9BG3B+G (5)

v=3B-2G23B+2G (6)

A=2C44C11-C12 (7)

ξ=C11+8C127C11+2C12 (8)

where GV and GR 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 Cu2MnSi1−x Alx 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 Alx 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 MnSi1−x - Alx compounds are anisotropic.

4.3. Electronic Properties

To determine the electronic structure’s nature of Cu2 MnSi1−x Alx 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 Cu2MnSi1−x Alx 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 Cu2MnSi1−x Alx (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 Alx 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 Cu2MnSi1−x Alx (x = 0, 0.125, 0.25, 0.375, 0.5, 1) alloys versus the composition x, are non-linear.

Table III Calculated total and partial spin magnetic moments (in μB) of Cu2MnSi1−x Alx (x = 0, 0.125, 0.25, 0.375, 0.5, 1). 

Compound MCu MMn MSi MAl Minterstitial Mtot
Cu2MnSi 0.067 3.305 -0.0062 -------- 0.302 3.735
Cu2MnSi1-0.125Al0.125 0.073 3.332 -0.00002 -0.028 0.314 3.790
Cu2MnSi1-0.25Al0.25 0.070 3.330 -0.00429 -0.030 0.302 3.762
Cu2MnSi1-0.375Al0.375 0.063 3.297 -0.0071 -0.032 0.279 3.686
Cu2MnSi1-0.5Al0.5 0.060 3.297 -0.0085 -0.033 0.262 3.636
Cu2MnAl 0.047 3.273 -------- -0.034 0.200 3.502
Theo. 1,11,23,33 0.058 3.242 -0.078 3.478
3.408 3.5611
3.5311 3.5123
0.07333 3.4933 -0.04633 3.7333
Exp.34 3.634

Figure 4 Magnetic moment for Cu, Mn, Si, Al and total magnetic moment within the concentration x for Cu2MnSi1−x Alx (x = 0, 0.125, 0.25, 0.375, 0.5, 1). 

5. Conclusion

In conclusion, using the FP-LAPW based on GGA approximations calculations to predict the structural parameters, elastic, electronic and magnetic properties of the Cu2MnSi1−x Alx quaternary alloy, we have found that the lattice constants are in excellent agreement with the estimated values by Vegard’s law. The analysis of the electronic band structures and density of states of Cu2MnSi1−x Alx alloys reveal that they are ferromagnetic and metallic compounds by nature. The large magnetic moment is located on Mn sites. We have also found that the Cu2MnSi does not fulfill the mechanical stability conditions, where Cu2MnAl and Cu2MnSi1−x Alx Heusler alloys are stable and have a ductile behavior. The Young’s modulus, Shear modulus, Poisson’s ratio anisotropy factor and Kleinman parameters, often measured for polycrystalline samples, were also derived. We hope that this simulation may be a guideline for the experimentalists.

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Received: November 27, 2017; Accepted: December 19, 2017

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