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

Benichou, B.; Nabi, Z.; Bouabdallah, B.; Bouchenafa, H.; Benichou, B.; Nabi, Z.; Bouabdallah, B.; Bouchenafa, H.

doi:10.31349/revmexfis.64.135

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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.64 n.2 México Mar./Apr. 2018

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

Research

Structural, elastic, electronic and magnetic properties of quaternary Heusler alloy Cu₂MnSi_1−xAl_x (x = 0 − 1): First-principles study

B. Benichou^a^b^*

Z. Nabi^c

B. Bouabdallah^d

H. Bouchenafa^a^e

^{^a}Department of Physics, Faculty of Sciences Exact, Djillali Liabès University, Sidi Bel Abbès 22000, Algeria.

^{^b}Department of Electronics, Faculty of Technology, Hassiba Benbouali University, Chlef 02000, Algeria.

^{^c}Laboratory of Catalysis and Reactive Systems, Physics Department, Djillali Liabès University, Sidi Bel Abbès 22000, Algeria.

^{^d}Condensed Matter and Sustainable Development Laboratory, Djillali Liabès University, Sidi Bel Abbès 22000, Algeria.

^{^e}Department 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 MnSi_1−x Al_x 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 Cu₂MnSi_1−xAl_x (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 Cu₂MnSi_1−xAl_x 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 X₂Y_1−x Y’_x Z, (X_1−x X’_x)₂YZ and X₂YZ_1−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 Fe₂Mn_0.5Cu_0.5Al 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 R_MT × K_max as 7.0, and to ensure the correctness of the calculations, we have taken l_max = 12. The G_max 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 Cu₂MnSi_1−x Al_x 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 Cu₂MnSi_1−x Al_x (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 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.

Cu2MnSi1-xAlx :aÅ=5.86×1-x+5.93×x (1)

Table I Calculated lattice parameter (a), bulk modulus (B) and its pressure derivative (B') for Cu₂MnSi_1−x Al_x (x = 0, 0.125, 0.25, 0.375, 0.5, 1).

Compound	a(Å)	B(GPa)	B'
Cu₂MnSi	5.8645	136.137	5.147
Cu₂MnSi_1-0-125Al_0.125	5.87	118.758	6.136
Cu₂MnSi_1-0.25Al_0.25	5.88095	124.910	3.450
Cu₂MnSi_1-0-375Al_0.375	5.8879	126.467	4.229
Cu₂MnSi_1-0.5Al_0.5	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 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.

Table II Calculated elastic constants (in GPa), and G, B/G, E, υ, A, ξ of Cu₂MnSi_1−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
Cu₂MnSi_1-0.125Al_0.125	155.607	129.243	103.390	137.994	47.485	2.90	173.682	0.312	7.843	0.882
Cu₂MnSi_1-0.25Al_0.25	154.061	129.182	99.346	137.417	45.382	3.027	167.508	0.318	7.986	0.888
Cu₂MnSi_1-0.375Al_0.375	200.123	115.834	111.110	143.903	70.485	2.017	209.953	0.279	2.636	0.690
Cu₂MnSi_1-0.5Al_0.5	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 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.

Table III Calculated total and partial spin magnetic moments (in μB) of Cu₂MnSi_1−x Al_x (x = 0, 0.125, 0.25, 0.375, 0.5, 1).

Compound	M^Cu	M^Mn	M^Si	M^Al	M^interstitial	M^tot
Cu₂MnSi	0.067	3.305	-0.0062	--------	0.302	3.735
Cu₂MnSi_1-0.125Al_0.125	0.073	3.332	-0.00002	-0.028	0.314	3.790
Cu₂MnSi_1-0.25Al_0.25	0.070	3.330	-0.00429	-0.030	0.302	3.762
Cu₂MnSi_1-0.375Al_0.375	0.063	3.297	-0.0071	-0.032	0.279	3.686
Cu₂MnSi_1-0.5Al^0.5	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 Cu₂MnSi_1−x Al_x (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 Cu₂MnSi_1−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 Cu₂MnSi_1−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 Cu₂MnSi_1−x Al_x 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.

References

1. C. Felser, G.H. Fecher, and B. Balke, Angew. Chem. Int. Ed. 46 (2007) 668. [ Links ]

2. J. Winterlik, G.H. Fecher, and C. Felser, Solid State Commun. 145 (2008) 475. [ Links ]

3. K. Ullakko, J. K. Huang, C. Kantner, R. C. O’Handley, and V. V. Kokorin, Appl. Phys. Lett. 69 (1996) 1966. [ Links ]

4. A. Ayuela, J. Enkovaara, K. Ullakko, and R.H. Nieeminen, J. Phys.: Condens. Matter. 11 (1999) 2017. [ Links ]

5. R.A. Dunlap, G. Stroink, and K. Dini, J. Phys. F: Met. Phys. 16 (1986) 1083. [ Links ]

6. P.J. Webster, and K.R.A. Ziebeck, J. Phys. Chem. Solids. 34 (1973) 1647. [ Links ]

7. S. Ishida, J. Ishida, S. Asano, and J. Yamashita, J. Phys. Soc. Jpn. 45 (1978) 1239. [ Links ]

8. S.E. Kulkova, S.V. Eremeev, T. Kakeshita, S.S. Kulkov, and G.E. Rudenski, Mater. Trans. 47 (2006) 599. [ Links ]

9. K.H.J. Buschow, and P.G. van Engen, J. Magn. Magn. Mater. 25 (1981) 90. [ Links ]

10. Y. V. Kudryavtsev et al., J. Appl. Phys. 97 (2005) 113903-1. [ Links ]

11. D.P. Rai, and R.K. Thapa, J. Alloys. Compd. 612 (2014) 355. [ Links ]

12. B. Hamri et al., Comput. Condens. Matter. 3 (2015) 14. [ Links ]

13. S. Ghosh, and D.C. Gupta, J. Magn. Magn. Mater. 411 (2016) 120. [ Links ]

14. I. Galaknakis, J. Phys.: Condens. Matter. 16 (2004) 3089. [ Links ]

15. S.V. Karthik, and A. Rajanikanth, Appl. Phys. Lett. 89 (2006) 052505-1. [ Links ]

16. D. Nanto, D.S. Yang, and S.C. Yu, Physica B. 435 (2014) 54. [ Links ]

17. J.C. Slater, Adv. Quant. Chem. 1 (1964) 5564. [ Links ]

18. P. Blaha, K. Schwartz, G.K.H. Madsen, D. Kvasnicka and J. Liutz, WIEN2k An Augmented Plane Wave Plus Local Orbitals Program for calculating Cristal Properties (Vienna University of Technology, Vienna, Austria, 2001). [ Links ]

19. P. Hohenberg, and W. Kohn, Phys. Rev. B. 136 (1964) 864. [ Links ]

20. W. Kohn, and L. J. Sham, Phys. Rev. A. 140 (1965) 1133. [ Links ]

21. J.P. Perdew, K. Burke, and M. Ernzerhof, Phy. Rev. Lett. 77 (1996) 3865. [ Links ]

22. F. D. Murnaghan, Proc. Natl. Acad. Sci. USA. 30 (1944) 244. [ Links ]

23. E.D.T. de Lacheisserie, D. Gignoux, and M. Schlenker, Magnetism: Fundamentals (Springer Science, Boston, USA, 2005). [ Links ]

24. R. Topkaya et al., J. Supercond. Novel. Magn. 25(2012)2605. [ Links ]

25. J. Jalilian, J. Alloys. Compd. 626 (2015) 277. [ Links ]

26. L. Vegard, Z. Phys. 5 (1921) 17. [ Links ]

27. P. Entel, W.A. Adeagbo, A.T. Zayak, and M.E. Gruner, NIC Symposium; G. Münster, D. Wolf and M. Kremer, Eds. (NIC Series, Jon von Neumann Institute for Computing, Jülich, Germany, 2006), pp. 159-166. [ Links ]

28. G.V. Sin’ko, and N.A. Smirnov, J. Phys. Condens. Matter. 16 (2004) 8101-8104. [ Links ]

29. S.F. Pugh, Philos. Mag. 45 (1954) 823. [ Links ]

30. D.G. Pettifor, Mater. Sci. Technol. 8 (1992) 345. [ Links ]

31. J. Haines, J.M. Léger, and G. Bocquillon, Annu. Rev. Mater. Res. 31 (2001) 1. [ Links ]

32. H.C. Kandpal, G.H. Fecher, and C. Felser, J. Phys. D, Appl. Phys. 40 (2006) 1507. [ Links ]

33. S. Ishida, Y. Kubo, J. Ishida, and S. Asano, J. Phys. Soc. Jpn. 48 (1980) 814. [ Links ]

34. R. Caroli, and A. Blandin, J. Phys. Chem. Solids. 27 (1966) 503. [ Links ]

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

e-mail: boucif_benichou@yahoo.fr

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