Mechanical, electronic and thermodynamic properties of full Heusler compounds Fe 2 VX(X = Al,Ga)

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1 International Journal of Modern Physics B Vol. 29 (2015) (15 pages) c World Scientific Publishing Company DOI: /S X Mechanical, electronic and thermodynamic properties of full Heusler compounds Fe 2 VX(X = Al,Ga) M. Khalfa, H. Khachai,, F. Chiker, N. Baki, K. Bougherara, A. Yakoubi, G. Murtaza, M. Harmel, M. S. Abu-Jafar,,, S. Bin Omran and R. Khenata Laboratoire d Étude des Matériaux & Instrumentations Optiques, Département Matériaux et Développement Durable, Faculté des Sciences Exactes, Université Djillali Liabès de Sidi Bel Abbès 22000, Algérie Department of Physics, Islamia College Peshawar, KPK, Pakistan Dipartimento di Fisica Universita di Roma La Sapienza, Roma, Italy Physics Department, An-Najah N. University, P. O. Box 7, Nablus, Palestine Department of Physics and Astronomy, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia Laboratoire de Physique Quantique et de Modélisation Mathématique, Université de Mascara, Algeria h khachai@yahoo.fr mabujafar@najah.edu Received 8 April 2015 Revised 17 September 2015 Accepted 22 September 2015 Published 3 November 2015 The electronic structure, mechanical and thermodynamic properties of Fe 2 VX, (with X = Al and Ga), have been studied self consistently by employing state-of-theart full-potential linearized approach of augmented plane wave plus local orbitals (FP-LAPW + lo) method. The exchange-correlation potential is treated with the local density and generalized gradient approximations (LDA and GGA). Our predicted ground state properties such as lattice constants, bulk modulus and elastic constants appear more accurate when we employed the GGA rather than the LDA, and these results are in very good agreement with the available experimental and theoretical data. Further, thermodynamic properties of Fe 2 VAl and Fe 2 VGa are predicted with pressure and temperature in the ranges of 0 40 GPa and K using the quasi-harmonic Debye model. We have obtained successfully the variations of the heat capacities, primitive cell volume and volume expansion coefficient. Keywords: DFT; electronic structure; mechanical properties; thermodynamic properties; heusler compounds. PACS numbers: De, Mb, Ci, Corresponding authors

2 M. Khalfa et al. 1. Introduction The ternary Fe 2 VZ (Z = Al, Ga) intermetallics represent part of Heusler family X 2 YZ type compounds 1 where X and Y belong to transition metals and Z is a main group element which crystallizes in cubic L 2 1 structure. The Fe 2 VZ (Z = Al, Ga) intermetallics are known for their unusual magnetic, transport, electrical and thermal properties, 2 5 nuclearmagneticresonancemeasurementsindicate these two compounds to be nonmagnetic semimetals and reveal a small density of states (DOS) at Fermi level (E f ) within the pseudogap, 6,7 in agreement with theoretical works. 8 In order to learn more about the electronic structures of Fe 2 VAl and Fe 2 VGa compounds, we have performed a systematic study of the different physical properties including structural, elastic, electronic, as well as thermal properties on these compounds, in the framework of the full-potential linearized augmented plane wave method plus local orbitals (FP-LAPW+lo) 9 11 based on density functional theory (DFT), 12 within the generalizedgradient approximation(gga) 13 and local density approximation (LDA) Computation Details All calculations in this work were performed by means of FP-LAPW+lo as implemented in WIEN2K package. 15 GGA and LDA were used for the exchange correlation potential. In this method, the space is divided into nonoverlapping muffin tin (MT) spheres and an interstitial region (IR). In the IR region, the basis functions are expanded into plane waves, while the region of the nonoverlapping spheres is treated by linear combination of the spherical harmonics Y lm (r). The MT sphere radii R MT are 2.17, 2.1, 2.04 and 2.00 a.u. for Fe, V, Al and Ga, respectively. The valence wavefunctions inside the spheres are expanded up to l max = 10. The cutoff parameter was set to R MT K max = 9. In the IR, the potential and the charge density were expanded as a Fourier series with wave vectors up to G max = 12 (a.u.) 1. The k-points mesh in the whole Brillouin zone turns out to be satisfactory with a k sampling The self-consistency was achieved within energy limit 10 5 Ry. In order to investigate the thermodynamic properties, we used the quasiharmonic Debye model, 16 in which the nonequilibrium Gibbs function G (V;P;T) is written as follows: G (V;P;T) = E(V)+PV +A Vib [θ D (V);T], (1) where E(V) is the total energy per unit cell, PV corresponds to the constant hydrostatic pressure condition, θ D (V) is the Debye temperature and A vib is the vibrational Helmholtz free energy that can be written as 17,18 : [ ( )] 9θD θd A vib (θ D ;T) = nkt 8T +3ln(1 e θdt ) D, (2) T

3 Mechanical, electronic and thermodynamic properties of full Heusler compounds wherenisthenumberofatomsperformulaunit andd(θ D /T)representsthe Debye integral. For an isotropic solid Debye temperature, θ D is expressed as: 16 θ D = ħ k [6π2 V 1/2 n] 1/3 f(σ) BS M, (3) where V is the molecular volume, M is the molecular mass, B s is the adiabatic bulk modulus, f(σ) is the scaling function, which depends on Poisson s ratio σ, and k is the Boltzmann constant. The scaling function f(σ) is given by 19,20 : [ ( ) 3/2 ( ) ] 3/2 1 1/3 2 f(σ) = 3 1+σ 1 1+σ 2 +, (4) 31 2σ 31 σ where σ is the Poisson ratio and the explicit expression for the f(σ) is taken from Refs. 16 and 21. The adiabatic bulk modulus Bs, isothermal bulk modulus BT, heat capacity CV and the thermal expansion αv are expressed as: 22 B S d 2 E(V) = B(V) = V dv 2, (5) ( 2 G ) (V;P,T) B T (P,T) = V V 2 (6) [ C V = 3nk 4D ( θd where γ is the Gruneisen parameter defined as: T P,T ) 3θ ] D/T, (7) e θd/t 1 α V = γc V B T V, (8) γ = dlnθ D(V) dlnv. (9) 3. Results and Discussion 3.1. Structural and elastic properties The Heusler alloys with general chemical formula X 2 YZ (X = Fe, Y = V, Z = Al, Ga) crystallize in the cubic L2 1 structure, belonging to the 225(Fm3m) space group, the X atom is placed at the Wyckoff position 8c (1/4,1/4,1/4); Y is situated at 4a (0,0,0) and Z at 4b (1/2,1/2,1/2). The structural properties were calculated using GGA and LDA. The optimized lattice constants (a 0 ), bulk moduli (B 0 ), its pressure derivative (B ) at P = 0 GPa are calculated by fitting the total energy to the Murnaghan s equation of state. 23 The calculated lattice constants and other properties related to the equilibrium structures are given in Table 1 together with

4 M. Khalfa et al. Table 1. Lattice constant a, bulk modulus B 0 and its pressure derivative B. Experimental and other theoretical values are also quoted for comparison. Present Fe 2 Val Present Fe 2 VGa LDA GGA Other Expt. LDA GGA Other Expt. a o (Å) a 5.76 a a 5.77 a b b B 0 (GPa) a a b b B b b a Ref. 24. b Ref. 25. the available experimental and theoretical data. Our computed value of the lattice constants using LDA and GGA are smaller than the experimental value within 2.9% (3%) and 0.8% (0.8%) for Fe 2 Val (Fe 2 VGa), respectively. We note that the better theoretical results are obtained with the GGA, when comparing the results obtained with the LDA. The results of the structural optimization are shown in Fig. 1. In the present work, we have also calculated the elastic constants C 11, C 12 and C 44 of Fe 2 VAl and Fe 2 VGa using the Mehl method. 26,27 The calculated values of the elastic constants for the Fe 2 VAl and Fe 2 VGa are listed in Table 2, together with the available results. Our calculated values of the elastic constants are in good agreement with other theoretical results. It is noticeable that the elastic stability criteria for a cubic crystal are C C 12 > 0, C 44 > 0 and C 11 C 12 > 0, 28 the present result of elastic constants satisfies these stability conditions. We have calculated Young s modulus E, shear modulus G poisson s ratio σ and anisotropic ratio A from the theoretical elastic constants by the following equations. 29 E = 9BG/(3B +G), (10) G = (C 11 C 12 +3C 44 )/5, (11) σ = (3B E)/(6B), (12) A = 2C 44 /C 11 C 12. (13) The computed values of E, G, σ and A are summarized in Table 2. The obtained results agree well with theoretical data reported in Ref. 25. According to the empirical relationship proposed by Pugh, 30 the B/G is a critical value of ductility/brittleness of material, the shear modulus G represents the resistance to the plastic deformation, while B represents the resistance to the volumetric deformation. If B/G > 1.75 the material behaves in a ductile, otherwise the material behaves in a brittle manner, a Cauchy pressure (C 12 C 44 ) can be used as another criterion of the brittle ductile characteristic of the materials, if the Cauchy pressure

5 Mechanical, electronic and thermodynamic properties of full Heusler compounds Fig. 1. The total energies versus the volume for the Fe 2 VAl and Fe 2 VGa at P = 0 GPa. is positive (negative) the material is expected to be ductile (brittle). Our results show that B/G ratios are 1.50 and 1.89 for Fe 2 VAl and Fe 2 VGa, respectively and Cauchy pressure is negative for Fe 2 VAl and positive for Fe 2 VGa, we can classify the Fe 2 VAl in brittle material and Fe 2 VGa in ductile materials. We have also examined the variation of elastic constants C ij with pressure range from 0 to 40 GPa presented in Fig. 2. In Table 3, we list the results for the pressure derivatives dc 11 /dp, dc 12 /dp and dc 44 /dp. It can be seen that the elastic constants increase when the pressure increases. All the three elastic constants show a linear variation with pressure. We observe that C 12 > C 44 when P > 13.5 GPa for Fe 2 VAl then a Cauchy pressure (C 12 C 44 > 0) is positive, therefore Fe 2 VAl is ductile up to 13.5 GPa

6 M. Khalfa et al. Table 2. Calculated elastic constants C ij (GPa), Young s modulus E (GPa), shear modulus G (GPa), Poisson s ratio σ and anisotropy factor A for Fe 2 VAl and Fe 2 VGa. Fe 2 VAl C 11 C 12 C 44 E G σ A B/G Present Other a Fe 2 VGa Present Other a a Ref. 25. Elastic constants(gpa) Elastic constants(gpa) Fe 2 VAl Fe 2 VGa Pressure(GPa) Pressure(GPa) C11 C12 C44 C11 C12 C44 Fig. 2. Pressure dependence of the elastic constants (C 11, C 12 and C 44 ) for Fe 2 VAl and Fe 2 VGa

7 Mechanical, electronic and thermodynamic properties of full Heusler compounds Table 3. The pressure derivatives of the elastic constants. dc 11 /dp dc 12 /dp dc 44 /dp Fe 2 VAl Fe 2 VGa Electronic properties In Fig. 3, we have shown the spin polarized electronic band structure of Fe 2 VAl and Fe 2 VGa along the higher symmetry directions in the Brillouin zone calculated at equilibrium lattice parameters within the GGA approximation. The materials in which valence band and conduction band cross each other but does not overlap are called the semimetals. The band gaps in the semimetals are represented by negative values. For better clarification of the semimetallic nature in these compounds the bands are plotted in a small energy scale ranging from 2 to 2 ev. The analysis Fig. 3. Spin polarized electronic band structure for Fe 2 VAl (leftpanel) and Fe 2 VGa (rightpanel)

8 M. Khalfa et al. Fig. 4. TDOS and PDOS of Fe 2 VAl and Fe 2 VGa

9 Mechanical, electronic and thermodynamic properties of full Heusler compounds of the band structure shows that the Fe 2 VAl and Fe 2 VGa are semimetals. Our calculations are in good agreement with other theoretical calculation. 31,32 The total and partial density of states (TDOS and PDOS) for these compounds are plotted in Fig. 4. The band situated in the energy range 9.4 to 6 ev ( to 7.4 ev), not shown in the band structure, is due to Al-s (Gas) for Fe 2 VAl (Fe 2 VGa). The structure located between 5.6 ev ( 6 ev) below E f is mainly formed from Fe-d, V-d and Al-p (Ga-p) with small contributions for Fe 2 VAl (Fe 2 VGa), at higher energies peaks correspond mainly to the Ga-d states for Fe 2 VGa. The conduction band is composed of Fe-d and V-d for Fe 2 VAl and Fe 2 VGa. Fig. 5. Temperature dependence of the volume V 0 at various pressures for Fe 2 VAl and Fe 2 VGa

10 M. Khalfa et al Thermodynamic properties The study of thermodynamic properties at high pressure and temperature was done within the quasi-harmonic Debye model. In this work, the thermal properties are calculated in the temperature range from 0 to 1500 K, where the quasi-harmonic model is probably valid, the pressure is studied in 0 40 GPa range. In Fig. 5, we present the evaluation of volume V 0 of Fe 2 VAl and Fe 2 VGa as function of temperature at different pressures. It can be seen that when the temperature increases the volume V 0 (a lattice parameter) increases, on the other side Fig. 6. Temperature dependence of the Bulk modulus B 0 at various pressures for Fe 2 VAl and Fe 2 VGa

11 Mechanical, electronic and thermodynamic properties of full Heusler compounds Fig. 7. Variation of the expansion coefficient α v versus temperature at various pressures for Fe 2 VAl and Fe 2 VGa. as the pressure P increases the volume decreases at a given temperature. The calculated lattice constant values for Fe 2 VAl and Fe 2 VGa at room temperature and zero pressure are Å, Å, respectively. Figure 6 displays the dependence of Bulk modulus versus temperature at given pressure, it is noticed that the Bulk modulus is almost constant from 0 to 120 K and decreases linearly with increasing temperature for T > 120 K at given pressure and increase with P at given temperature. At 300 K and zero pressure, the bulk modulus for Fe 2 VAl and Fe 2 VGa are and GPa, respectively. The variation of the thermal expansion coefficient with temperature and pressure for Fe 2 VAl and Fe 2 VGa are shown in Fig. 7. The results indicated that, for a given pressure α of the two compounds enhanced as the temperature increased up to 600 K, when T > 600 K α gradually tends to increase linearly. At given tem

12 M. Khalfa et al. Fig. 8. Variation of the heat capacities C v with temperature at various pressures for Fe 2 VAl and Fe 2 VGa. perature, the thermal expansion coefficient decreased strongly with the increase of pressure. This decrease in the expansion coefficient may be due to the decrease in the volume of the crystal unit cell with the increase in the pressure. Heat capacity parameters are important in predicting thermodynamic properties. Thus, in this work we also present the pressure and temperature dependences of the heat capacity in Fig. 8. We note that when T < 1000 K (Cv is proportional to T 3 ), 33 the heat capacity Cv increased very quickly, and is dependent on both temperature and pressure. At higher temperature (T > 1000 K), Cv increases slowly and tends to the Petit and Dulong limit 34 (Cv J mol 1 K 1 ) in both materials under discussion. We display in Fig. 9 the relationship between the Debye temperature θ D and temperature in the pressure range between 0 40 GPa for both Fe 2 VAl and Fe 2 VGa compounds. One can notice that θ D is nearly constant from 0 to 80 K and decreases linearly with increasing temperature for T > 80 K. Our calculated θ D at zero

13 Mechanical, electronic and thermodynamic properties of full Heusler compounds Fig. 9. Variation of the Debye temperature θ D as function of temperature at various pressures for Fe 2 VAl and Fe 2 VGa. pressure and zero temperature are K and K for Fe 2 VAl and Fe 2 VGa, respectively. 4. Conclusion In summary, we have performed first-principles density functional calculations to study the structural, elastic, electronic as well as some thermodynamic properties under pressure of two Heusler compounds Fe 2 VAl and Fe 2 VGa, by using the FP- LAPW + lo method within GGA. The calculated structural parameters of these compounds are in good agreement with the available experimental and theoretical

14 M. Khalfa et al. data. Further, our results for the elastic constants and elastic moduli E, G, σ, A agree well with theoretical data and obey to a stability criteria. Then, from the calculatedelectronicpropertiesweconcludedthatfe 2 VAlandFe 2 VGaaresemimetals. Finally, the quasi-harmonic Debye model is used to investigate the thermal properties, in which the variations of the volume, bulk modulus, thermal expansion heat capacity and Debye temperature with pressure and temperature have been well detailed for both compounds under discussion Fe 2 VAl and Fe 2 VGa. Acknowledgment The authors (R. Khenata and S. Bin-Omran) acknowledge the financil support by the Deanship of Scientific Research at the King Saud University for funding the work through the research group project, No. RPG-VPP-088. References 1. F. Heusler and V. Dtsch, Phys. Ges. 5, 219 (1903). 2. K. Endo et al., J. Magn. Magn. Mater , 1437 (1998). 3. Y. Nishino, Mater. Trans. 42, 902 (2001). 4. M. Kato et al., J. Phys. Condens. Matter. 12, 1769 (2000). 5. Y. Nishino et al., Phys. Rev. B 63, (2001). 6. C. S. Lue and J. H. Ross Jr., Phys. Rev. B 58, 9763 (1998). 7. C. S. Lue and J. H. Ross, Jr.. Phys. Rev. B 63, (2001). 8. A. Bansil et al., Phys. Rev. B 60, (1999). 9. G. K. H. Madsen et al., Phys. Rev. B 64, (2001). 10. K. Schwarz, P. Blaha and G. K. H. Madsen, Comput. Phys. Commun. 147, 71 (2002). 11. P. Blaha et al., WIEN2k, An augmented plane wave + local orbitals program for calculating crystal properties (Karlheinz Schwarz, Techn. Universitat Wien, Austria, 2001). 12. P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964). 13. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 14. J. P. Perdew and Y. Wang, Phys. Rev. B 45, (1992). 15. M. A. Blanco, E. Francisco and V. Luana, Comput. Phys. Commun. 158, 57 (2004). 16. M. A. Blanco and A. Martín Pendás, J. Mol. Struct. Theochem 368, 245 (1996). 17. M. Flórez et al., Phys. Rev. B 66, (2002). 18. E. Francisco et al., J. Phys. Chem. 102, 1595 (1998). 19. E. Francisco, M. A. Blanco and G. Sanjurjo, Phys. Rev. B 63, (2001). 20. A. Bouhemadou, R. Khenata and M. Chegaar, Eur. Phys. J. B 56, 209 (2007). 21. R. Hill, Proc. Phys. Soc. Lond. A 65, 349 (1952). 22. F. D. Murnaghan, Proc. Nat. Acad. Sci. USA 30, 244 (1944). 23. L.-S. Hsu et al., Phys. Rev. B 66, (2002). 24. V. Kanchana et al., Phys. Rev. B 80, (2009). 25. M. J. Mehl et al., Phys. Rev. B 41, (1990). 26. M. J. Mehl, B. M. Klein and D. A. Papaconstantopoulos, Intermetallic Compd. 1, 195 (1994). 27. J. Wang et al., Phys. Rev. Lett. 71, 4182 (1993). 28. E. Schreiber, O. L. Anderson and N. Soga, Elastic Constants and their Measurement (McGraw-Hill, New York, 1973). 29. S. F. Pugh, Philos. Mag. 45, 823 (1954)

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