First-principles study of mechanical, exchange interactions and the robustness in Co2MnSi full Heusler compounds

First-principles study of mechanical, exchange interactions and the robustness in Co2MnSi full Heusler compounds

Journal of Magnetism and Magnetic Materials 422 (2017) 13–19 Contents lists available at ScienceDirect Journal of Magnetism and Magnetic Materials j...

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Journal of Magnetism and Magnetic Materials 422 (2017) 13–19

Contents lists available at ScienceDirect

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First-principles study of mechanical, exchange interactions and the robustness in Co2MnSi full Heusler compounds A. Akriche a,n, H. Bouafia c, S. Hiadsi a, B. Abidri b, B. Sahli c, M. Elchikh a, M.A. Timaoui a, B. Djebour a a Laboratoire de Microscope Electronique et Sciences des Matériaux, Université d’Oran des Sciences et de la Technologie-USTO, Mohamed Boudiaf, Faculté de physique, Département de Génie Physique, Oran, Algeria b Laboratoire des Matériaux Magnétiques, Université Djillali Liabés, Sidi Bel-Abbes, Algeria c Laboratoire de Génie Physique, Université Ibn-Khaldoun, Tiaret 14000, Algeria

art ic l e i nf o

a b s t r a c t

Article history: Received 13 February 2016 Received in revised form 18 August 2016 Accepted 21 August 2016 Available online 22 August 2016

In this work we report the results of ab-initio studies of structural, mechanical, electronic and magnetic properties of Co based Co2MnSi Heusler compound in stoichiometric composition. All of which are accurately calculated by the full-potential (FP-LMTO) program combined with the spin polarized generalized gradient approximation in the density functional formalism (DFT). The total energy calculations clearly favor the ferromagnetic ground state. The lattice parameter, elastic constants and their related parameters were also evaluated and compared to experimental and theoretical values whenever possible. In this paper, the electronic properties are treated with GGAþ U approach. The magnetic exchange constants temperature has been calculated using a mean field-approximation (MFA). The half-metal to metal transition was observed around 40 GPa. Increasing pressure has no impact on the total magnetic moment or the overall shape of the band structure that indicates the robustness of the electronic structure of this system. & 2016 Elsevier B.V. All rights reserved.

Keywords: Heusler FP-LMTO Elastic properties Electronic structure Magnetic properties

1. Introduction Heusler alloys are well-positioned in the electronic field; this is both acknowledged by technological sources, and indicated by the fact that many studies have been conducted on the subject. When a particular Heusler feature is introduced, it will have a competitive advantage in terms of cost, quality and innovation once the feature is ready for use. They are known by their ternary systems with general configuration X2YZ where X and Y are transition metals and Z an item spotted in the (III–V) columns. There is a rising interest in using Heusler alloys for many applications, especially in the spintronic field [1–8]. Heuslers present in their great majority, an important property which is the half-metallicity and magnetization, particularly the ferromagnetic; what earned them the designation of half-metallic ferromagnets abbreviated in (HMFs) [9–12]. It should be pointed out that further investigations have shown that Co2MnSi alloy of L21 structure supports a Curie temperature higher than 985 K [13–15] with a magnetic moment of about 5 μB per unit cell. The Heusler compounds based on cobalt have been studied theoretically by the First-principles n

Corresponding author. E-mail address: [email protected] (A. Akriche). 0304-8853/& 2016 Elsevier B.V. All rights reserved.

method. This study indicated that most of them have a half-metallic behavior [16–21]. Among the Heusler systems, Co2MnSi compound has been widely used in the manufacture of thin films of various groups [22–26]. This compound is considered to be an ideal local moment system [27,28], also the total magnetic moment of the system follows the Slater-Pauling principle [29–31].

2. Computational details This article presents calculations of structural, mechanical, electronic and magnetic properties by using the FP-LMTO within the GGA and GGA þU method and shows, in the same time, the undeniable effectiveness of this method which has proven its success in giving results in very good agreement with other works. The power of this method is justified by the precision of the calculations it presents [32,33]. The space targeted by this method is divided into two areas which are the interstitial region (IR) and socalled Mufin-Tin region (MT) characterizing non-overlapping spheres centered on atomic sites. The basic functions of the first region are formulated by Fourier series. For a particle at fixed energy, the Schrödinger equation represents, within the spheres (MT), empirical solutions and their energy derivatives basis sets. Inside the spheres, the valence wave functions are developed to a


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certain value of lmax ¼ 6. Using Perdew-Wang parameterization, the generalized gradient approximation (GGA) [34] allows describing the energy exchange correlation of the electrons through the use of available computer code Lmtart [35]. The convergence of the total energy to a minimum value of 10  6 Ry determines the convergence of self-consistency calculations. The k integration over the Brillouin zone is performed up to (12, 12, 12) k mesh, yielding 32,768 FFT grid in the irreducible Brillouin zone (BZ), using the tetrahedron method [36]. We summarized in Table 1, the MTS radius taken for each atomic position, number of plane waves (NPLW) and the cut-off energies used in our calculations. The variations of total energy as function of cell volume are fitted to Murnaghan's equation [37] in their ferromagnetic (FM) phases.

3. Results and discussion 3.1. Structural properties It is known that the Heusler alloys are ternary systems of stoichiometric composition represented by the general formula X2YZ in the L21 phase for (Fm 3̅ m space group, #225). In this configuration, the X atoms have magnetic transition metal positioning in (¼ ¼ ¼) and (¾ ¾ ¾) in the general previsions of Wyckoff. While Y and Z are transition metals and III–V group elements occupying the positions (½ ½ ½) and (0 0 0) respectively [38,39]. Fig. 1 gives a representation of this configuration. Our Calculations are developed on the Co2MnSi compound in the various magnetic phases (PM, FM and AFM) to determine the most stable phase, and see to what extent are changes in total energy with the lattice parameter. As we have done previously, we adjusted to the empirical Murnaghan equation [37] the total energies calculated versus volume. For the ferromagnetic phase of this compound, Table 2 shows the equilibrium lattice constant, the bulk modulus, the bulk modulus pressure derivative, the equilibrium energy and the cohesive energy. It should be noted that all calculated values are in good agreement with the experimentally achieved values and those found theoretically using the first principle methods. As can be noted, the values of the bulk modulus (B) and its derivative (B′) are very close to the theoretical Table 1 The muffin-tin radius (RMT), the number of plane wave (NLPW) and cut-off energy used in our calculation for Co2MnSi compound. Atoms



Ecut-off (Ry)

Co Mn Si

2.263 2.356 2.356



values. The lack of experimental data, the less that is known, about the last parameter of the compound Co2MnSi pushes us to consider our results as a reliable data base until future measures will confirm them. 3.2. Elastic stiffness constants Given the importance of the elastic properties in hydrodynamic investigations, the use of simulation methods to calculate the elastic constants, is crucial and of proven importance. A tensor of order 4 [Cijkl] with 81 elastic coefficients allows properly to model the behavior of an elastic material. If one takes into account the symmetry of the stress and strain tensors as well as the energy of the stability of the tensor, the number of independent coefficients will be reduced to 21. Generally, the relationship is given in Refs. [42–44]. A cubic crystal has only three separate stiffness constants (C11, C12, and C44). Mehl method [45] is used to develop these constants. This method that we already used in our earlier work gave his evidence and showed its effectiveness [44,46,47]. As we know that Voigt and Reuss moduli represent the upper and lower limits of effective moduli [48], Voigt-Reuss-Hill modulus, when to him, is the average of Voigt and Reuss module [49]. Table 3 summarizes calculated elastic constants C11, C12 and C44, quantities GV, GR, G as well as other available values. Mechanical stability conditions (for the cubic polycrystal) are met when the elastic constants C11, C12 and C44 satisfy the conditions: C11  C12 40, C11 40, C44 40, C11 þ2C12 40, C12 oB oC11 with P ¼0 GPa [51]. According to the analysis of C11, C12 and C44 following the study of Heusler compounds, it appears that the Born criteria are met, confirming the stability of the cubic phase L21. It is noted that to our knowledge, there are no previous experimental work on elastic constants and shear modulus of the Co2MnSi compound, which leads us to compare the found results with only the existing theoretical work. Found values of the elastic constants C11, C12 and C44 are in perfect harmony with the existing theoretical values, against the values of GV, GR and G shear moduli are large compared to those found in [41,50]. The disparity of found values with those of the elastic constants available [41,50] is mainly due to the sensitivity of the methods used in predicting elastic constants and parameters such as shear modulus G for example. The Zener anisotropy factor A, which is an indicator of the degree of elastic anisotropy owned by the crystal. It is well known Table 2 Calculated equilibrium lattice a0, bulk modulus B, its derivative B′, total Energy (Ry) and cohesive energy of Co2MnSi compound, in (FM) phase. Refs.

a0 (Å)

B (GPa)


Etot (Ry)

ΔH (Ry/cell)

Present Exp [14] Exp [15] Theory [40] Theory [41]

5.645 5.673 5.654 5.633 5.642

205.8 – – 212.8 240.89

5.00 – – 4.68 4.983



– – –

– – –

Table 3 The calculated elastic constant C11, C12, C44 (GPa), shear modulus GV, GR, G (GPa) of Co2MnSi.

Fig. 1. The L21 structure of Heusler alloy Co2MnSi.








This study Theoretical [40] Theoretical [41] Theoretical [50]

363 311.6 313.91 290

126 164.75 203.44 179

170 153.26 101.58 158

149 – 83.03 –

144 – 76.05 –

147 – 79.54 104

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that completely isotropic compound is characterized by Zener factor equal to 1 but values smaller or larger than unity indicate the anisotropy. This parameter is calculated in terms of the computed data [43]. From the obtained results presented in Table 4, it is clear that the Co2MnSi compound is elasticity anisotropic.


Table 5 GGA þU parameter for Co2MnSi (All values are given in eV). Element






Co-3d Mn-3d

3.5 5

1 0.9

3.5 5

8.60 7.74

5.37 4.80

3.3. Compliance constants and Young modulus Hooks's law states that the elastic compliance tensor [S] has the same shape as the tensor [C] except the way that differs (opposite). Predicting certain mechanical parameters such as Young's modulus and Poisson's ratio generally uses the values of the elastic compliance tensor [S]. The Sij component that corresponds to the Cij term can be explained by the relationship [42]. The Young's modulus of a material which is the ratio of stress to strain is a parameter that is perfect for describing the elastic properties of materials that are either stretched or compressed. To produce the elongation or compression of a material used in the context of the Young's modulus; the stress shall not exceed the yield strength of the material. For an arbitrary crystallographic direction m, the Young's modulus can now be provided by [43]. Anisotropic materials have a Poisson ratio υ which also varies with orientation. The ratio υ can be obtained in Ref. [43]. Pogh [52] has proposed a simple empirical equation which connects the elastic properties of materials with their elastic modulus. This equation is based on the report of bulk modulus B to shear modulus G. Pugh has introduced this report as prediction of the brittleness and ductile behavior of materials. The critical value, which separates ductile and brittle materials, is approximately 1.75. The material behaves in a ductile manner when B/G 4 1,75; else the material adopt brittle behavior if B/G o1,75. The studied compound ratio is less than 1.75, which means that it is brittle. Unfortunately, there are no theoretical and experimental results to compare with the calculated Compliance Constants. However, it should be noted that related parameters such as Young's modulus Y and Poisson's ratio υ perfectly match with [50]. Table 4 gives the values of elastic compliance constants, Poisson's ratio υ, young's modulus Y and Zener anisotropic factor A and B/G report.

steps of 1.0 eV) and J (0.3–1.2 eV in steps of 0.1 eV). The values of U, J and the Slater integrals for Co 3d and Mn 3d electrons obtained from the above relations that give the total magnetic moment, which is the best representative of the experimental, are given in Table 5. The calculated energy bands structures of this compound along the higher symmetry direction in the Brilloin zone are presented in Fig. 2. The spin dependent band structure of Heusler alloy Co2MnSi is given in this figure. It is obvious that the spin-up (majority) bands structure has metallic intersections at the Fermi level, indicating clearly strong metallic nature, whereas, for spin down (minority) bands structure exhibit a semiconductor behavior, however we note an indirect energy band gap along Γ–X point. Therefore the material shows evidence of half-metallic ferromagnets material. The comparison of the found values, by the GGA and GGA þ U methods, with other theoretical results shows that they are in good agreement with other results achieved by other work. From the Table 6 we can see that there is a slight difference in the energy gap of about 0.03 eV. We can justify the difference in values between our results and those using the same method FPLMTO [58] by the value of U and J introduced in the calculated and used functional (GGA þ U or LDA þU). We summarized in Table 6 the results calculated with available data. To predict the type of bonding between atoms of this compound, we used a quantitative appreciation of bond between two atoms which amounts to calculate the factor of iconicity fi and this by several methods that measure this quantity [59–63]. According to the principle of Pauling [59], the ionicity is related to the difference in electro-negativity from which the ionicity factor of the bond between two atoms A and B can be calculated by the following formula:

3.4. Electronic properties

⎧ ⎪ ( χ − χB )2 ⎫ ⎪ ⎬ fi =1−exp⎨ − A ⎪ ⎪ 4 ⎩ ⎭

When the orbital spin interactions are important and D-electrons are perfectly situated, for such systems, GGA method is insufficient for their description, especially the electronic properties. To overcome this problem, the GGA þ U approach [53–56] was introduced in order to correctly describe these systems. Thus the addition of the Hubbard term (U) to energy GGA, allows to correctly describing the interaction d–d or f–f. It is certain that this approach has made its evidence and its effectiveness for systems strongly interacting like Mott insulators. Moreover, we had used this method before with satisfactory results [44]. In the GGA þU method three input parameters are required for 3d electrons. They are Slater integrals F°, F2, and F4. These integrals are directly related with on-site Coulomb interaction (U) and exchange interaction (J) by relations U¼ F° [57], J ¼(F2 þF4)/14, and F4/F2 0.625 [54]. We have tried several values of U (1.5–8.5 eV in Table 4 The elastic compliance constants Sij (GPa  1), Young's modulus Y (GPa), Poisson's ratio υ and the B/G ratio of Co2MnSi. Refs.








This study Theoretical [50]

0.003 –

0.0008 –

0.0058 –

303 269

0.26 0.29

1.43 –

1.39 –


where: χA and χB are the electro-negativities of A and B ion respectively. It is noted that FP-LMTO method provided the charge densities of interstitial and muffin-tin regions, but these values depend clearly on the choice of muffin-tin rays [64]. From our results, all values of the iconicity factor are near to 0% (fiCo–Co ¼0, fiCo–Mn ¼0,0392, fiCo–Si ¼0,0025 and fiMn–Si ¼0,0222) for the bond between the atoms of the studied compound and according to the electronic properties; this material has a metallic behavior which means that the bonds have a predominant metalcovalent character. 3.5. Magnetic properties The calculated total spin magnetic moment of Co2MnSi satisfies a simple relationship called the rule of 24 which is the analog of the Slater-Pauling rule in Full Heusler alloys [19,31,41,65]. In this rule, the total magnetic moment Mt per formula unit is related to the total number of valence electrons Zt in the unit cell by: Mt ¼Zt– 24. In Table 7 we have shown the calculated total and partial magnetic moments of Co2MnSi. The magnetic moment is mainly located on the Mn and Co atoms. Indeed, the great part of the total


A. Akriche et al. / Journal of Magnetism and Magnetic Materials 422 (2017) 13–19

Fig. 2. The up- and down-spin band structures for Co2MnSi along the high-symmetry axes of the Brillouin Zone from GGA þ U approach.

Table 6 Co2MnSi with minority band gaps, ΔE is the difference between Emin and Emax. All energies are given in eV. GGA þ U

GGA Refs.

Emin (X)

Emax (Γ)


Emin (X)

Emax (Γ)


Present work Others [20] Others [40] Others [58]

0.477 0.506 – –

 0.323  0.292 – –

0.8 0.798 0.77 –

1.327 1.307 – 1.307

 0.031 0.007 – 0.003

1.358 1.300 – 1.3

Table 7 Calculated total, partial magnetic moment (in mB) and spin polarization (P) for Co2MnSi. Refs.








Present Exp [66] Theory [20] Theory [40]

1.075 – 1.00 –

2.966 – 3.00 –

 0.079 – – –

5.00 5.01 5.03 5.00

1.22 – – 1.17

0 – – 0

100 – – 100

magnetic moment results from the Mn and the two Co atoms which have a magnetic moment value of 2.966 mB and 1.075 mB respectively. Therefore, the silicon atom has a negligible magnetic moment, which does not contribute a lot to the total magnetic moment. Generally, computed magnetic moments are in good agreement with the experimental and other theoretical results. The spin polarization P(E) of Co2MnSi alloy at the Fermi level (EF) is linked to the spin dependent DOS via the expression.


n ↑ (EF ) − n ↓ (EF ) n ↑ (EF ) + n ↓ (EF )

Fig. 3. The amplified curves of the DOS for Co2MnSi near the Fermi level.

comparing the first-principles total energy differences between FM and AFM calculations, the magnetic exchange parameters can be obtained. Fig. 4 shows a choice of three spin supercells suggested by MStudio MindLab for AFM configurations (AFMI, AFMII, AFM ΙΙΙ). The classical method [67] was used to map the firstprinciples total energies onto the Heisenberg Hamiltonian:

H=− (2)

Where n ↑ (EF ) and n ↓ (EF ) are the values of majority and minority DOS at Fermi level (EF). Our present works, are presented in Table 7. We can notice that, the Co2MnSi alloy illustrates 100% spin polarization at EF as can be clearly seen in the density of states (Fig. 3) which explained that this compound has half-metallic nature. 3.6. Curie temperature and exchange interaction By choosing several spin supercells for AFM configurations, and

∑ Jij Si. Sj i≠j


where Si and Sj are vectors pointing in the direction of the local magnetic moment on sites i and j respectively. Jij is the energy exchange interaction between two spins at i and j positions. As we mentioned in Section 3.6, the Co2MnSi alloy has two magnetic atoms per unit cell, so in this case the exchange parameters are: J1 is the interaction between Mn–Mn, J2 between Mn– Co and J3 between Co–Co atoms. Hence, we may express the total differences of energies between FM and AFM configurations as following:

A. Akriche et al. / Journal of Magnetism and Magnetic Materials 422 (2017) 13–19


Fig. 4. The ferromagnetic and antiferromagnetic ordering configurations: (a) FM, (b) AFMI, (c) AFMII, and (d) AFMΙΙΙ. Arrows indicate magnetic moment orientations on Co and Mn atoms.

ΔEΙ = EAFMΙ − EFM = 12J1 + 12J2 + 18J3 ΔEΙΙ = EAFMΙΙ − EFM = 12J1 + 8J2 + 12J3 ΔEΙΙΙ = EAFMΙΙΙ − EFM = 12J1 + 4J2 + 18J3


The Curie temperature for FM configuration is evaluated using the mean-field approximation (MFA) according to the following expression:


2 3

∑ J0j j≠0


where KB is the Boltzmann's constant, we summarized in Table 8, the computed results of the Curie temperature and exchange constants. From the Table 8 we can see that the interaction between Mn and Co atoms gives a primary contribution to the total effective coupling. Ref. [68] reported a similar result. Concerning the Curie temperature, our calculations overestimate the experimental values [14,15]. This is hollowing out for mean-field approximation (MFA). We note that our calculated Curie temperature is in good agreement with experimental and calculated values using the first theoretical methods [40,68].

Table 8 Calculated exchange constants J1, J2 and J3 in (mRy) and Curie temperatures Tc in (K) for Co2MnSi. Refs.





This work Exp [14] Exp [15] Theoretical [40] Theoretical [68]

2.01 – – – 1.83

7.6 – – – 10.2

0.95 – – – 1.57

1162 996 985 928 1251

Table 9 Minority spin energy band gap values as a function of pressure for Co2MnSi All energies are given in eV. P (GPa)


Eg (Γ)

Eg (X)

Eg (Γ–X)

0 20 40 60 80 100

0,48546 0,22335 0.09  0,21891  0,35816  0,4659

0,89 0,99 1,07 1,09 1,09 1,1

0,98 0,91 0,83 0,78 0,75 0,73

0,8216 0,74 0,68 0,62 0,61 0,6056

3.7. Pressure effect on the electronic and magnetic properties 3.7.1. Electronic properties In several crystal structures, significant physical properties are brought under the action of pressure. Numerous compounds which are notorious to indicate structural phase transitions pressurized. Occasionally, these structural transitions produce radical amendment in the electronic band structures. Half-metallic GdN offers an interesting example. This alloy suffers a transition phase from rocksalt to zincblende structure for prolonged lattice parameter and from Half-metallic rocksalt (B1) to a semiconducting hexagonal wurtzite (B4) structure hydrostatically pressurizing [69]. However, Co2MnSi alloy exhibits interesting behavior, wherein L21 crystallographic phase remains stable, whereas the system undergoes a transition from half metal to metal with the growth in pressure. Table 9 summarizes calculated results of different energy band gap values at various pressures for spin-down state of Co2MnSi. The fourth column in the table called CBM (conduction band minimum) measures the energy difference between the minimum of the conduction band and Fermi level at point X. The values of the various energy band gap (direct at Γpoint, X-point and indirect at X–Γ) are almost stable with increasing pressure, it means that the pressure does not affect much on these last. By against, the change of CBM with pressure may be used to define the half metal to metal transition pressure. From Table 9 and Fig. 5 we can determine the transition pressure which is above 40 GPa or CBM reaches the Fermi level. It should be mentioned a significant point concerning the general shape of the band structure for the two states spin-Up and spin-Down, this

shape does not change with increasing pressure, showing the robustness of the electronic structure of our compound. This property also has a major impact on thin films, in which the lattice parameter could be modified slightly by the increase over a substrate with the incompatibility of the lattice. For this function, Co2MnSi alloy becomes successful materials especially in tunneling magnetoresistive (TMR) devices. 3.7.2. Magnetic properties We study the robustness of the half-metallicity at high-pressure. In Fig. 6, we have shown the computed total and partial magnetic moments of Co2MnSi compound. As we can see, the total magnetic of Co2MnSi decreases slowly with the increase of pressure. This slight decrease is due to the decrease in the magnetic moment of the Mn atom; however, the magnetic moment of Co atom remains stable. The negative spin magnetic moment induced on Si atom increases very slowly with pressure. This slight decrease in the total magnetic of Co2MnSi indicates the rigidity of this compound at high-pressure. We notice that the spin polarization Fig. 7(a) is high in the pressure range of 0 GPa to 50 GPa and that this system keeps its spin polarization at the Fermi level of 100% for the value of the lattice parameter ranging from 4.65 Å to the value of 5.37 Å Fig. 7 (b). Indeed, the perfect half metallicity of this system decreases above the value of 50 GPa, this can be explained by the absence of the band gap of minority spins.


A. Akriche et al. / Journal of Magnetism and Magnetic Materials 422 (2017) 13–19

Fig. 5. Minority spin band structures for Co2MnSi under (a): 0 Gpa, (b): 20 Gpa, (c): 40 Gpa, (d): 60 Gpa, (e): 80 Gpa and (f): 100 GPa pressures.

Fig. 6. Total and partial magnetic moments (in mB) for Co2MnSi.

A. Akriche et al. / Journal of Magnetism and Magnetic Materials 422 (2017) 13–19

Fig. 7. Spin polarization (a) and lattice parameter and (b) of Co2MnSi at various pressures.

4. Conclusion In summary, the first-principles study of the structural, mechanical, electronic and magnetic properties in Co based Heusler compound Co2MnSi is presented. The lattice parameter, elastic constants and their related parameters were also evaluated. GGA þU method is used to describe more accurately the electronic properties of these systems. The band structure calculations reveal the obvious half-metallicity. Our calculated total magnetic moment is very close to 5 mB and is in very good agreement with recent experiments. The exchange coupling interactions were determined to be sufficiently strong to achieve a high Curie temperature above room-temperature. The mean-field approximation (MFA) was used to estimate the Curie temperature. The perfect halfmetallism of this system at equilibrium (i.e. zero pressure) is destroyed at elevated pressures yielding P¼ 87% spin polarization ratio at 50 GPa.

Acknowledgments We kindly acknowledge Mohammad I.A. AbuHamdieh Palestine Polytechnic University (PPU), G. Vergoten Lille France for their help.

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