Structural, electronic, magnetic and thermodynamic properties of full-Heusler compound Co2VSi: Ab initio study

Structural, electronic, magnetic and thermodynamic properties of full-Heusler compound Co2VSi: Ab initio study

Journal of Magnetism and Magnetic Materials 381 (2015) 65–69 Contents lists available at ScienceDirect Journal of Magnetism and Magnetic Materials j...

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Journal of Magnetism and Magnetic Materials 381 (2015) 65–69

Contents lists available at ScienceDirect

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Structural, electronic, magnetic and thermodynamic properties of fullHeusler compound Co2VSi: Ab initio study Ali Bentouaf a,n, Fouad El Haj Hassan b,c a

Département de Physique, Faculté des Sciences, Université de Hassiba Ben Bouali, Chlef 02000, Algeria Université Libanaise, Faculté des Sciences (I), Laboratoire de Physique et d'Electronique (LPE), Elhadath, Beirut, Lebanon c Condensed Matter Section, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste, Italy b

art ic l e i nf o

a b s t r a c t

Article history: Received 24 March 2014 Received in revised form 16 August 2014 Accepted 23 December 2014 Available online 24 December 2014

Density functional theory based on full-potential linearized augmented plane wave (FP LAPW) method is used to investigate the structural, electronic and magnetic properties of Co2VSi Heusler alloys, with L21 structure. It is shown that calculated lattice constants and spin magnetic moments using the general gradient approximation method are in good agreement with experimental values. We also presented the thermal effects using the quasi-harmonic Debye model, in which the lattice vibrations are taken into account. Temperature and pressure effects on the structural parameters, heat capacities, thermal expansion coefficient, and Debye temperatures are determined from the non-equilibrium Gibbs functions. & 2014 Elsevier B.V. All rights reserved.

Keywords: Heusler compounds Magnetic properties Ab-initio calculation Thermal properties

1. Introduction Heusler alloys [1] received tremendous experimental and theoretical interest. This attention is due essentially to their new possible applications in engineering science. One of the unmatched properties of Heusler alloys is the half-metallic (HM) demeanor. Interestingly, they have high Curie temperature above room temperature, the measured Curie temperatures for Co2MnSi, Co2MnGe and Co2FeSi are 985 K, 905 K [2] and 1100 K [3], respectively sowing a structural analogy with semiconductors. Another appealing aspect properties of some compounds in this family is the existence of martensitic transition in this material which shows thermoplastic and reversible characteristics leading to the shape memory effect. Such a shape memory has attracted considerable attention towards application as actuation devices and smart materials. Up to recently, the research has been practically performed on many Co2YZ L21(full-Heusler)-type half-metallic ferromagnets (HMFs). The cause of these investigations is that they show a potential application in spin-dependent devices, such as magnetic random access memories (MRAMs) and currentperpendicular-to-plane (CPP) spin-valve GMR heads [4]. Co2VSi full-Heusler alloys present such physical attitude [5]. Despite the strong demand motivated by the above applications, accurate calculation of mechanical properties on this kind of material n

Corresponding author. E-mail address: [email protected] (A. Bentouaf). 0304-8853/& 2014 Elsevier B.V. All rights reserved.

remains challenging. While there have been a number of works focused on the characterization of the magnetics properties of these alloys, there has been little effort [6], to the best of these our knowledge, to investigate thermal properties and do a topological study of the electronic densities. Other very significant objective is to investigate the derivation of the parameters that define the equation of state (EOS) and related fundamental solid state properties, such as thermal expansion, melting temperature and specific heat. A study of the thermodynamic properties for materials is well motivated by the understanding one thereby gains about the chemical bonds and the cohesion of material. Moreover, the elastic constants are also for a current interest; they are related to thermal properties according to the Debye theory. Thermal properties can be unambiguously determined from the quasi-harmonic Debye model [7]. To undertake such investigation, ab-initio calculations will be performed with a state of the art electronic structure method, namely the full potential LAPW methodology, to study the structural, bonding, elastic,magnetic and thermodynamics properties of Co2VSi in its L21 phase. The paper is divided in three parts. In Section 2, we briefly describe the computational techniques used in this study. We present the theoretical results and discussion of our work in Section 3. Finally, in Section 4 we summarize the main conclusions of our work.


A. Bentouaf, F.E.H. Hassan / Journal of Magnetism and Magnetic Materials 381 (2015) 65–69

2. Computational details Electronic structure calculations of our full-Heusler compound were performed using the scalar relativistic FP-LAPW method within the density functional theory [8,9] as implementedin Wien2K [10] code. As exchange and correlation potential, we use the generalized gradient approximation (GGA) [11,12]. In the FPLAPW method, the wave function, charge density and potential were expanded by spherical harmonic functions inside non-overlapping spheres surrounding the atomic sites (muffin-tin spheres) and by plane waves basis set in the remaining space of the unit cell (interstitial region). The maximum l quantum number for the wave function expansion inside atomic spheres was confined to lmax ¼10. Nonspherical contributions to the charge density and potential within the MT spheres were considered up to lmax ¼6. The cutoff parameter was RMT, Kmax ¼ 7. In the interstitial region, the charge density and the potential were expanded as a Fourier series with wave vectors up to Gmax ¼12 a.u.  1. The muffin-tin radius was assumed to be 2.21, 2.21 and 2.06 a.u. for Co, V and Si atoms, respectively. Using the energy eigenvalues and eigenvectors at these points, the density of states was determined by the tetrahedral integration method [13]. Both the plane wave cutoff and the number of k-points were varied to ensure total energy convergence. Our calculations for valence electrons were performed in a scalar-relativistic approximation, while the core electrons were treated fully relativistic. Study of thermodynamic properties of materials is of great importance in order to extend our knowledge about their specific behaviors when they are put under severe constraints such as highpressure and high temperature environment. Therefore, to investigate the thermodynamic properties under high temperature and high pressure of Co2VSi Heusler alloys, we have applied the quasi-harmonic Debye model which has been successfully applied to similar compounds [14]. We apply here the quasi-harmonic Debye model, implemented in the pseudo-code Gibbs [15]. The non-equilibrium Gibbs energy G* (V ; P , T ) can be written in the form

G* (V ; P , T ) = E (V ) + PV + A vib (T , θ (V ))


where E (V ) is the total energy per unit cell, θ (V ) is the Debye temperature and A vib is the vibrational Helmholtz free energy that can be written as [16,17]

⎡9 θ ⎛ θ ⎞⎤ A vib (θ ; T ) = nKT ⎢ + 3 ln(1 − e θ / T ) − D ⎜ ⎟ ⎥ ⎝T ⎠⎦ ⎣8 T


where D (θ /T ) is the Debye integral. The model of Debye, with the isotropic approximation, is used to determine the Debye temperature given by [16]

θ = h¯ (6π 2V 1/2n)1/3f (σ)

Bs (3)

K 2M

where V is the molecular volume, M is the molecular mass of the compound, n is the number of atoms per formula unit, Bs is the adiabatic bulk modulus, f (σ) is the scaling function [18,19], which depends on Poisson's ratio σ of the isotropic solid, and K is the Boltzmann constant. The scaling function f (σ) is given by [16,17] 1/3 ⎤−1⎫

⎧ ⎡ ⎪ ⎛ 2 1 + σ ⎞3/2 ⎛ 1 1 + σ ⎞3/2 ⎟ +⎜ ⎟ ⎥ f (σ) = ⎨3 ⎢2 ⎜ ⎝ 3 1 − σ ⎠ ⎥⎦ ⎪ ⎢⎣ ⎝ 3 1 − 2σ ⎠ ⎩

⎪ ⎬ ⎪ ⎭


The Poisson ratio σ is taken from Ref. [17]. The adiabatic bulk modulus can be approximated by the static compressibility [17]

⎛ d2E (V ) ⎞ ⎟⎟ Bs ≈ B (V ) = V ⎜⎜ ⎝ dV 2 ⎠


Thus, the non-equilibrium Gibbs function G* (V ; P , T ) can be minimized with respect to volume V

⎛ * ⎞ ⎜ ∂G (V ; P , T ) ⎟ = 0 ⎜ ⎟ ∂V ⎝ ⎠P, T


The thermal equation of state (EOS) V (P , T ) can be obtained by solving Eq. (5). The isothermal bulk modulus is given by

⎛ 2 * ⎞ ∂ G (V ; P , T ) ⎟ BT (P , T ) = V ⎜⎜ ⎟ 2 ∂V ⎝ ⎠P, T


The heat capacity Cv and the thermal expansion ∝ are expressed as [20]

⎡ ⎛θ ⎞ 3θ / T ⎤ Cv = 3nK ⎢4D ⎜ ⎟ − ⎥ ⎣ ⎝T ⎠ eθ/ T − 1 ⎦ ∝=




where γ is the Gruneisen parameter defined as


d ln θ (V ) d ln V


Through the quasi-harmonic Debye model, one could calculate the thermodynamic quantities of any temperatures and pressures of compounds from the calculated E–V data at T ¼0 K and P¼ 0 GPa.

3. Results and discussion 3.1. Crystal structure Heusler alloy [21] crystallizes in the L21 structure and stoichiometric composition of X2YZ(X ¼Co, Y ¼V and Z¼ Si) was used, where X and Y are transition metal elements, and Z is a group III, IV or V element. The full-Heusler structure consists of four penetrating fccsublattices with atoms in which the X atoms occupy the (0,0,0) and (1/2,1/2,1/2) sites, the Y atom in the (1/4,1/4,1/4) site, and the Z atom occupy the (3/4,3/4,3/4) site in Wyckoff coordinates. 3.2. Total energy and lattice parameter To determine the equilibrium lattice constant and find how the total energy varies with respect to the cell volume, we performed structural optimizations on Co2VSi Heusler compound. The total energy dependence on the cell volume is fitted to the Murnaghan [22] equation of state (EOS) given by

E T (V ) =

⎡ ⎤ B0 V ⎢ (V0/V )B0′ V0 B0 ⎥ + + E0 − 1 ⎢ ⎥ ′ ′ B0 ⎣ B0 − 1 B0′ − 1 ⎦


where B0 is the bulk modulus, B0′ is the bulk modulus derivative and V0 is the equilibrium volume. We summarized our results and the available experimental and other theoretical values in Table 1. A small difference could be observed between our calculated lattice constant and experimental one which can be attributed to the general trend that GGA usually overestimates this parameter. Fig. 1 displays the total

A. Bentouaf, F.E.H. Hassan / Journal of Magnetism and Magnetic Materials 381 (2015) 65–69


Table 1 Equilibrium parameters of Co2VSi: lattice constant a, and bulk modulus B. x

Lattice parameter, a (Å)

Co2VSi a b c


Bulk modulus, B (GPa)

Our work


Other cal.

Our work

Other cal.






Ref. [23]. Ref. [24]. Ref. [1].

3.4. Density of states In order to understand the bulk electronic structure, we have calculated the total density of states DOS for Co2VSi using GGA as presented in Fig. 2. The upper part of each panel displays the majority spin densities and the lower one the minority spin densities. We note that the gap mainly originates from the hybridization of the d–d orbitals of the transition metal atoms. The DOS around the Fermi level is heavily dominated by the 3d states of the V and Co atoms, and the majority spin states are nearly fully occupied. The DOS curves for the minority spins exhibit two peaks above the Fermi level which are due to both V and Co 3d contributions. 3.5. Thermal properties

Fig. 1. The total energy Etot (relative to that of the L21 phase) of the alloys studied as afunction of the volume with GGA calculation adjusting by Murnaghan equation.

energies in dependence on cell volume for our Heusler compounds. For our knowledge, there is no experimental data to compare the obtained results of bulk modulus. 3.3. Magnetic properties Starting with the compounds under investigation, all the information regarding the partial, total, and the previously calculated magnetic moments are summarized in Table 2. In most cases, the calculated magnetic moments are in good agreement with the predictions of other computational methods. We have found that the Co sites contribute much more compared with the V sites; this is because V-d states show exchange splitting. The Si atom carry a negligible magnetic moment, which does not contribute much to the overall momentwith respect to the Co atom occupying the X sites in the lattice constant. The calculated total magnetic moment of our Heusler compounds Co2VSi is 2.967 μB per unit cell. It should be noted that Mtot is the calculated total spin magnetic moment of the compound found by integration over the entire cell. Therefore, it is not just the combination of the moments at the Co(2 times), V and Si sites but respects also the moment of the interstitial between the sites. The interstitial is due to the calculation scheme using non-overlapping spheres to define the sites. That is the reason why the site resolved values MCo, MV and MSi alone are not summing up to result in Mtot. The missing or excess of the total moment is found in the interstitial between the sites where the wave functions are expanded as plane Table 2 Total and partial magnetic moments of Co2VSi. Compound





Co2VSi Our work Other cal.

2.967 3a, 3b

1.082 1.1a, 1.03a

0.743 0.82c, 0.8a, 0.82b

0.002 –

a b c

Ref. [24]. Ref. [1]. Ref. [25].

Thermodynamics is the key component of materials science and engineering. Therefore, we apply the quasi-harmonic Debye model [26] to obtain the thermodynamic properties of the Co2VSi through calculating E–V. As a first step, a set of total energy calculation versus primitive cell volume (E–V), in the static approximation, was carried out and fitted with a numerical EOS in order to determine its structural parameters at P ¼0 and T ¼0, and then derive the macroscopic properties as function of P and T from standard thermodynamic relations. The thermal properties are determined in the temperature range from 0 to 1300 K where the quasi-harmonic model remains fully valid. While the pressure effect is studied in the 0–20 GPa range, the variation of the volume with the temperature at different pressures is shown in Fig. 3. The volume increases with increasing temperature but the rate of increase is very moderate. The variation of the bulk modulus B with the temperature at a given pressure is shown in Fig. 4. The bulk modulus decreases with increasing temperature at a given pressure and increases with pressure at a given temperature. The Debye temperature is an important fundamental parameter and closely related to many physical properties of solids, such as specific heat and melting temperature. It is well known, that below Debye temperature, quantum mechanical effects are very important in understanding the thermodynamic properties, while above Debye temperature quantum effects can be neglected [27]. In Fig. 5 we remark that the effect of temperature on the Debye temperature is not as important as the pressure because it is seen that the decrease is nearly linear for each pressure. The current investigations demonstrated that we can classify our compounds among the hard materials grace to their high Debye temperature. The Debye temperature of our full Heusler alloy at 0 GPa is 719.9 K and at ambient temperature. Another thermodynamic parameter, heat capacity, is also very important. Through heat capacity Cv, one can obtain information about lattice vibrations, energy band structure, density of state, transition of phase of the solid, etc. The investigation of the heat capacities Cv versus temperature at pressures of 0, 4, 8, 12, 16 and 20 GPa is shown in Fig. 6. At high temperatures, the constant volume heat capacity Cv tends to the Petit and Dulong limit which is common to all solids [28]. At sufficiently low temperatures, Cv is proportional to T3. The specific heat capacity for Co2VSi is 85.70 J mol  1 K  1 in normal condition. Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. Fig. 7 represents the variation of the thermal expansion coefficient α of Co2VSi as function of temperature and pressure. It is shown that, for a given pressure, α decreases at low temperature especially at zero pressure and gradually tends to a linear behavior at higher temperatures. As pressure increases, the increase of α with temperature


A. Bentouaf, F.E.H. Hassan / Journal of Magnetism and Magnetic Materials 381 (2015) 65–69

6 4 2 0 -2 -4 -6 4



DOS (States/eV)

2 0 -2 -4 3


2 1 0 1 2 0.4


0.2 0.0 0.2 0.4 -18












Energy(eV) Fig. 2. Calculated spin-projected total DOS plots for Co2VSi.

Fig. 3. The temperature effects on the lattice parameter for Co2VSi.

Fig. 5. Variation of the Debye temperature with the temperature at different pressures from 0 to 20 GPa.

becomes smaller while for a given temperature α decreases strongly with increasing pressure. We conclude that the increasing temperature dependence of thermal expansion coefficient α is as important as the decreasing pressure dependence of the volume thermal expansion coefficient α.

4. Conclusion

Fig. 4. The temperature effects on bulk modulus for Co2VSi.

Using the density functional theory based on full-potential linearized augmented plane-wave (FP-LAPW) method, within generalized-gradient approximation we have studied structural, magnetic and thermal properties of Co2VSi. The agreement between our results and the available experimental and theoretical data is found to be generally satisfactory. Our calculated results show that the d–d hybridization between the transition atoms (Co–Co and Co–V) in our Heusler compounds is essential for the

A. Bentouaf, F.E.H. Hassan / Journal of Magnetism and Magnetic Materials 381 (2015) 65–69


experimental data available for these quantities we think that the ab-initio theoretical estimation is the only reasonable tool for obtaining such important information and our calculated results provide the reference for future experimental work.


Fig. 6. The Heat capacity depending on the temperature for different pressures from 0 to 20 GPa.

Fig. 7. The thermal expansion coefficient versus temperature at different pressures from 0 to 20 GPa.

formation of the gap at EF. Our calculated total magnetic moment is very close to 2.96 μB to be in good agreement with recent prediction results. On the basis of the quasi-harmonic Debye model, the thermodynamic properties of Co2VSi Heusler compounds such as the variation of volume, bulk modulus, the heat capacity, the thermal expansion and the Debye temperature are predicted in the whole pressure range from 0 to 20 GPa and the temperature range from 0 to 1300 K. Since there are no enough

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