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Structural, magnetic, electronic and elastic properties of half-metallic ferromagnetism full-Heusler alloys: Normal-Co2TiSn and inverse- Zr2RhGa using FP-LAPW method
Materials Chemistry and Physics 240 (2020) 122122
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Structural, magnetic, electronic and elastic properties of half-metallic ferromagnetism full-Heusler alloys: Normal-Co2TiSn and inverse- Zr2RhGa using FP-LAPW method Doha N. Abu Baker a, Mohammed S. Abu-Jafar a, *, Ahmad A. Mousa b, Raed T. Jaradat a, Khaled F. Ilaiwi a, c, R. Khenata d a
Physics Department, An-Najah N. University, P.O. Box 7, Nablus, Palestine Department of Basic Sciences, Middle East University, Amman, 11831, Jordan c Arab Open University, Ramallah, Palestine d Laboratoire de Physique Quantique et de Mod�elisation Math�ematique de la Mati�ere (LPQ3M), Universit�e de Mascara, 29000, Algeria b
H I G H L I G H T S
� Normal Co2TiSn and inverse Zr2RhGa compounds possess half-metallic behavior. � Normal Co2TiSn and inverse Zr2RhGa compounds have an indirect energy gaps. � Normal Full Heusler Co2TiSn and inverse Full Heusler Zr2RhGa are mechanically stable. � B/S results show that normal Co2TiSn is brittle, while inverse Zr2RhGa is ductile. � Poisson’s ratio (ν) values show that both compounds have ionic bonds. A R T I C L E I N F O
A B S T R A C T
Keywords: Full-heusler Elastic properties Magnetic properties FP-LAPW mBJ
The equilibrium structural parameters, electronic, magnetic and elastic properties of the inverse Zr2RhGa and normal Co2TiSn Full-Heusler compounds have been studied using density-functional theory (DFT) and fullpotential linearized augmented plane-wave (FP-LAPW) method as implemented in the WIEN2k package. The Generalized Gradient Approximation (GGA) has been used for the exchange-correlation potential (Vxc) to compute the equilibrium structural parameters; lattice constant (a), bulk modulus (B) and bulk modulus pressure derivative (B0 ). In addition to the GGA approach, the modified Becke Johnson (mBJ) scheme has been used to calculate the band gap energies. The normal Heusler Co2TiSn compound and the inverse Heusler Zr2RhGa compound within GGA and mBJ approaches are found to have a half-metallic behavior, with an indirect energy gap in the spin down configuration. The calculated total magnetic moments for both compounds are to some extent compatible with the available experimental and theoretical results. Elastic constants and their related properties were obtained by using the IRelast package. These compounds are mechanically stable. The elastic constants satisfy the Born mechanical stability criteria. B/S ratio shows that the normal Co2TiSn has a brittle nature, while the inverse Zr2RhGa has a ductile nature; Poisson’s ratio (ν) values show that both compounds have an ionic bond nature.
1. Introduction In recent years, Heusler alloys received growing attention due to their interesting physical properties [1–4], especially the half metallic character, half metallic materials (HM) exhibiting a 100% spin polari zation around the Fermi surface [1–3] (a half metal is a ferromagnetic
with a gap in one of the spin directions at the Fermi energy (FE)). Half metals can be used as spin injectors for magnetic random access mem ories and other spin dependent devices [2]. A lot of alloys were pre dicted to be half metallic materials. In fact, investigating and searching for new (HM) materials are mainly focusing on the Heusler alloys [3–17]. From structural point of view, Heusler family can be described
* Corresponding author. E-mail address: [email protected] (M.S. Abu-Jafar). https://doi.org/10.1016/j.matchemphys.2019.122122 Received 1 May 2019; Received in revised form 3 August 2019; Accepted 3 September 2019 Available online 4 September 2019 0254-0584/© 2019 Elsevier B.V. All rights reserved.
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by two variants: Full Heusler X2YZ phases, which typically crystallize in Cu2MnAl (L21)-type structure and the half Heusler XYZ which typically crystallize in NiMnSb (C1b) type structure, X and Y are transition ele ments (3d, 4d or 5d elements) and Z is s-p element. The atoms in normal Heusler compounds are lined up in X2:(1/4,1/4,1/4), (3/4,3/4,3/4), Y (1/2,1/2,1/2), and Z (0,0,0), while the atoms in inverse Heusler com pounds are lined up in X2:(1/4,1/4,1/4), (1/2,1/2,1/2), Y (3/4,3/4, 3/4), and Z (0,0,0) [18]. There are a lot of previous studies that have been done on Heusler compounds and other half-metallic compounds by using different methods [19–24]. In 2007, Kandpal et al. [19] measured the lattice parameter for inverse Heusler compound Co2TiSn and found it to be 7.072 Å, they also calculated the magnetic moment using LMTO-ASA code [20], SPRKKR code [21], FP-LAPW method [22] and FPLMTO method [23], and found to be 1.40μB, 1.55μB, 2μB and 2μB, respectively. It is clear that the LMTO-ASA and SPRKKR codes fail to get the correct value of the total magnetic moment, while Wien2k and FPLMTO codes do correctly obtain minority gap in this compound and also calculated correctly the magnetic moment 2 μB per formula unit. In 2014, Birsan and Kuncser [24] studied the electronic, structural and magnetic prop erties of Zr2CoSn full-Heusler compound by using FP-LAPW method. They calculated the energy band gap (Eg ¼ 0.543 eV), total magnetic moment (Mtot ¼ 3 μB) and the lattice parameter (a ¼ 6.76 Å). In 2014, Birsan [25] investigated the structural and magnetic properties of the full-Heusler compound, Zr2CoAl using FP-LAPW method. He calculated the lattice parameter (a ¼ 6.54 Å), energy band gap (Eg ¼ 0.48 eV) and total magnetic moment (Mtot ¼ 2 μB). In 2015, Wang et al. [26] studied the half-metallic state and magnetic properties versus the lattice con stant in Zr2RhZ (Z ¼ Al, Ga, In) Heusler alloys by using the density functional theory (DFT) pseudo-potential method incorporate in CASTEP code [27,28]. The computed lattice parameters for Zr2RhZ (Z ¼ Al, Ga, In) are found to be 6.66, 6.64 and 6.81 Å, respectively and magnetic moment for all alloys was found to be 2 μB [26]. In 2017, Jain et al. [29] studied the electronic structure, magnetic and optical prop erties of Co2TiZ (Z ¼ B, Al, Ga, In) Heusler alloys by using FP-LAPW approach. They found the lattice parameters to be 5.494, 5.842, 5.845
Table 1 Calculated lattice parameter a (in Ǻ), bulk modulus B (in GPa), pressure deriv ative for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds. Compound Normal Co2TiSn Inverse Zr2RhGa
Present Experimental Present Theoretical
Lattice parameter
B
Bʹ
6.094 6.072 [19] 6.619 6.64 [26]
166.932 – 129.319 –
4.627 – 5.073 –
Fig. 1. The crystal structure of Full-Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds.
Fig. 2. Total energy as function of the volume for (a) normal Heusler Co2TiSn and (b) inverse Heusler Zr2RhGa compounds. 2
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Table 2 Total magnetic moment for normal and inverse Co2TiSn compound. Compound Normal Co2TiSn Inverse Co2TiSn
Magnetic Moment (μB) Present Experimental Present
Co
Co
Ti
Sn
1.32398 – 0.46092
1.32398 – 1.54481
0.19866 – 0.25961
0.00709 – 0.01579
Interstitial 0.87511 –
0.64087
Total magnetic moment (μB) 1.9786 2 [19] 1.64026
Table 3 Total magnetic moment for normal and inverse Zr2RhGa compound. Compound Normal Zr2RhGa Inverse Zr2RhGa
Magnetic Moment (μB) Present Experimental Present Theoretical
Zr
Zr
Rh
Ga
Interstitial
Total magnetic moment (μB)
0 – 0.88019 –
0 – 0.44447 –
0 – 0.15415 –
0 – 0.00905 –
0 –
0 – 1.99 2 [26]
0.50214
–
Fig. 3. PBE-GGA-Band structure spin-up for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
and 6.087 Å, respectively. The bulk modulus is found to be 233,182,184 and 161 GPa, respectively. The energy band gap is found to be 0.1, 0.44, 0.16 and 0.06 eV, respectively. The total magnetic moment is found to be 1, 0.99,1 and 1.03 μB, respectively. In 2017, Wei et al. [30] estimated the electronic, Fermi surface, Curie temperature and optical properties of Zr2CoAl compound by using FPLO code [31,32]. They computed the lattice parameter (a), bulk modulus (B) and total magnetic moment (Mtot) are found to be 6.629 Å, 106.8 GPa and 2 μB for the normal full-Heusler structure (Cu2MnAl). In present work we investigated the electronic, structural, magnetic and elastic properties of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds by using the full potential-linearized augmented plane wave method [FP-LAPW] method within the Perdew-BurkeErnzerhof generalized gradient approximation (PBE-GGA) [33]
integrated in Wien2k code [34]. 2. Computational method The calculations reported in present work were carried out by using the full potential linearized augmented plane wave, implemented in WIEN2k package [34]. The exchange correlation potential (VXC) was treated by two approximations, the generalized gradient approximation and the modified Becke–Johnson (mBJ-GGA) approach [35]. For the compound Co2TiSn, the muffin-tin radii (RMT) of Co, Ti and Sn atoms are 2.22 a. u., 2.17 and 2.22 a. u., respectively. For Zr2RhGa the RMT of Zr, Rh and Ga atoms are 2.37 a. u., 2.49 a. u. and 2.37 a. u., respectively. The number of plane waves was restricted by KMAX � RMT ¼ 8 and the expansions of the wave functions was set to l ¼ 10 inside the muffin-tin 3
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Fig. 4. PBE-GGA Band structure spin-down by for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
spheres. 35k points in the irreducible Brillouin zone (BZ) with a grid of 10 � 10 �10 Monkhorst-Pack (MP) [36] meshes (equivalent to 1000k points in the full Brillion zone (BZ)) are used to obtain self-consistency for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds. The self-consistent calculations are considered to converge when the calculated total energy of the crystal is stable within 10 5 Ryd. The elastic constants Cij determine the stiffness of a crystal against an externally applied strain. For small deformations we expect a quadratic dependence of the crystal energy on the strain (Hooke’s law) following this expression; !! 6 6 X 1X E V; εk ¼ E0 þ V0 σi εi þ Cij εi εj (1) 2 i¼1 ij¼1
must be applied to the starting crystal. Following the method developed by Morteza Jamal and integrated as the IRelast package in Wien2k code [37], the elastic constants for cubic system are computed by using some strain tensors. Further details of the calculation can be found elsewhere [38,39]. The first elastic components C11 and C12 are computed using a distortional orthorhombic deformation, Dortho and a volumetric cubic deformation, Dcubic given by the following tensors; 2 3 1þε 0 0 4 5 0 1 ε 0 DOrtho ¼ � � 0 0 1 1 ε2 2
DCubic
whereðεk Þdenotesε1 ; ε2 ; ::::::ε6 , V0(V) is the volume and E0(E) is the en ergy of the unstrained (strained) cubic system and Cij are the elastic constants. The elastic constants Cij are calculated by taking the second-order derivatives of the total energy of the strained crystal, Etot ðV; εÞ :Cij ¼ 2 1 ∂ Etot ðV;εÞ V0 ∂εi εj ,
3 1þε 0 0 4 ¼ 0 1 þε 05 0 0 1þε
And the elastic moduli, C44 is determined by the application of a distortional monoclinic deformation, Dmonoc 2 3 0 1 ε 5 Dmonoc ¼ 4 ε 1 �0 � 0 0 1 1 ε2
where εi are the strain tensor elements, V0 is being the vol
ume of the unstrained lattice. For the cubic systems, there are three independent elastic constants, namely, C11, C12 and C44. A set of three equations is needed to determine all the constants. This means that three types of strain εðIÞ (deformations)
Application of these deformations has an effect on the total energy from its unstrained value as follows: � �� (2) EðV; εÞ ¼ E0 þ V0 ðC11 C12 Þε2 þ O ε4
Table 4 Energy band gap for normal Co2TiSn and inverse Zr2RhGa Compounds.
EðV; εÞ ¼ E0 þ V0 ε ð σ 1 þ σ 2 þ σ3 Þ þ V0
Compound
Normal Co2TiSn Inverse Zr2RhGa
Band gap type
High symmetry points
Energy gapPBE-GGA (eV)
Energy gapmBJ-GGA (eV)
Indirect
Γ-X
0.482
1.430
Indirect
L-Δ
0.573
0.641
� � � 3 ðC11 þ 2C12 Þε2 þ O ε3 2 (3)
� �� EðV; εÞ ¼ E0 þ V0 ð2C44 Þε2 þ O ε4
(4)
We carried out calculations for five values of epsilon (strain) equal to 0.02, 0.01, 0.0, 0.01 and 0.02 with a dense mesh of uniformly distributed k-points equals to 4000 for the integration of the Full 4
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Fig. 5. mBJ-GGA- Band structure spin-up for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
Fig. 6. mBJ-GGA- Band structure spin-down for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
5
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Fig. 7. (a)Total and partial density of states of spin-up for normal Co2TiSn compound and for (b) Co atom (c) Ti atom (d) Sn atom by using PBE-GGA method.
Brillouin Zone. An increase in the number of k-points to 4000 is rec ommended for the Cubic/Hex-elast package [38] as elastic constants require a denser k-mesh to converge with a small percentage error.
inverse Zr2RhGa compound has space group F-43 m (216). We have calculated the structural properties for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds. The estimated lattice parameter (a), bulk modulus (B) and derivative pressure (Bʹ) at zero pressure are tabulated in Table 1, along with experimental and other theoretical re sults [19,26]. The crystal structure of full-Heusler Co2TiSn and Zr2RhGa compounds are shown in Fig. 1. The total energy as function of the volume for normal Heusler Co2TiSn and inverse Heusler. Zr2RhGa compounds are shown in Fig. 2a and b. Table 1 shows that our computed values of the lattice parameter of the normal Heusler Co2TiSn compound is slightly overestimated within 0.36% to the corresponding experimental values [19]. This is attributed to our use of the generalized gradient approximation (GGA), which is known to overestimate the lattice constant value compared to the measured one. The calculated parameter of inverse Heusler Zr2RhGa compound is found to be in good agreement to previous theoretical result [26].
3. Results and discussion 3.1. Structural properties The optimized lattice constants, bulk modulus, and its pressure de rivative were calculated by fitting the total energy to Murnaghan’s equation of state (EOS) [40,41]. Murnaghan’s equation of state (EOS) is given by: " # ’ BV ðV0 =VÞB BV0 EðVÞ ¼ E0 þ ’ þ 1 (5) B B’ 1 B’ 1 where E0 is the minimum energy, B is the bulk modulus at the equilib rium volume and Bʹ is the pressure derivative of the bulk modulus. dE dP Pressure (P) and bulk modulus (B) are given by P ¼ V dV ¼ dV , B ¼ 2
dE V dV 2 . Normal Co2TiSn compound has space group Fm-3m (225), and
6
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Materials Chemistry and Physics 240 (2020) 122122
Fig. 8. (a) Total and partial density of states of spin-down for normal Co2TiSn compound and for (b) Co atom (c) Ti atom (d) Sn atom by using PBE-GGA method.
3.2. Magnetic properties
that the calculated value of the total magnetic moment of normal Co2TiSn compound is slightly smaller than the experimental value [19]. Our calculated total magnetic moment of inverse Zr2RhGa compound is found to be closer to the other theoretical result [26]. Present total magnetic moment results are to some extent compatible with experi mental and theoretical results. The calculated total spin magnetic mo ments are clearly integral values except for inverse Co2TiSn and are in agreement with the Slater-Pauling rule [42]. The values of total and partial magnetic moments calculated in present study are reported in Table 2, we can easily notice that Co and Zr atoms are the mainly magnetic moment contributors.
In this part, the total and partial magnetic moment for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds were calcu lated and listed in Tables 2 and 3. Results from earlier experimental and theoretical works are quoted for comparison. The integer value of the total magnetic moment (Mtot) is character istic of half-metallic materials. We noticed that the total magnetic mo ments are found to be in the range from 1.64026 to 2 μB. We found that the normal and inverse Co2TiSn compounds are ferromagnetic, while only the inverse Zr2RhGa compound is ferromagnetic. The total mag netic moment for normal Zr2RhGa compound is found to be zero (0.0μB), which means it does not have magnetic behavior. Present results show 7
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Materials Chemistry and Physics 240 (2020) 122122
Fig. 9. (a)Total and partial density of states of spin-up for inverse Zr2RhGa compound and for (b) Zr atom (c) Rh atom (d) Ga atom by using PBE-GGA method.
3.3. Electronic properties
Zr2RhGa compounds using PBE-GGA method. The energy band gaps of normal Co2TiSn and inverse Zr2RhGa compounds are found to be 0.482 eV and 0.573 eV, respectively as shown in Table 4. The band structure of normal Co2TiSn and inverse Zr2RhGa are also calculated by using mBJ-GGA method. Fig. 5a and b also shows that the spin-up of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds both have metallic behavior using mBJGGA method. Fig. 6a and b shows the energy band gap within mBJGGA method for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds is still indirect band gap with an increase of the value of the band gap to 1.430 eV for normal Co2TiSn and to 0.641 eV for inverse Zr2RhGa for spin-down. Total and partial density of states for spin-up and spin-down for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds are
In this part, the band structure, the total and partial density of states for normal Co2TiSn compound and inverse Zr2RhGa compound were calculated. It is clear from the band structure and density of states that the normal Co2TiSn and inverse Zr2RhGa compounds both have a half -metallic behavior, which means at spin up the material behaves as a metal, while at spin down the material behaves as a semiconductor. Fig. 3a and b shows that the spin-up band structure of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds have metallic nature by using PBE-GGA method. Fig. 4a and b shows that the spindown band structure of normal Co2TiSn and inverse Zr2RhGa com pounds have an indirect energy band gap. The values of the energy band gaps of spin-down are calculated for normal Co2TiSn and inverse 8
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Materials Chemistry and Physics 240 (2020) 122122
Fig. 10. (a)Total and partial density of states of spin-down for inverse Zr2RhGa compound and for (b) Zr atom (c) Rh atom (d) Ga atom by using PBE-GGA method.
shown in Figs. 7 – 10. Densities of states Figs. 7–10, also confirm the half metallic property for normal Co2TiSn and inverse Zr2RhGa compounds with existing small energy band gap in the spin-down direction, and this means that these compounds have half metallic property. In the spin-up of normal Co2TiSn (Fig. 7), the valence band is due to the D-state of Co, s-state and p-state of Sn and small contribution from Ti D-state, while the conduction band is due to the D-state of Ti with few contribution from Sn p-state. In spin-down of normal Co2TiSn (Fig. 8), the valence band is due to the Co D-state, Ti D-state and Sn s-state and p-
state, while the conduction band is due to the Co D-state and Ti D-state and small contribution from s-state and p-state. In spin-up of inverse Zr2RhGa (Fig. 9), the valence band is due to the Rh D-state, Ga D-state, and small contribution from Zr D-state near to Fermi energy level, and the conduction band is due to the Zr D-state near to Fermi energy level. In spin-down of inverse Zr2RhGa (Fig. 10), the valence band is due to Rh D-state, Ga D-state and small contribution from Zr D-state, and the conduction band is due to Zr D-state near to Fermi energy level, and small contribution from Rh and Ga d-states.
Table 5 Elastic constants for normal Co2TiSn and inverse Zr2RhGa Full Heusler Compounds. Compounds
C11 (GPa)
C12 (GPa)
C44 (GPa)
B (GPa)
SV (GPa)
B/SV
Y (GPa)
V
A
Normal Co2TiSn Inverse Zr2RhGa
246.976 145.279
136.962 116.519
109.226 70.458
173.633 126.105
87.53 48.026
0.841 2.626
224.828 127.848
0.284 0.331
1.986 4.8997
9
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3.4. Elastic properties
4. Conclusions
In this part, the elastic constants (Cij), bulk modulus (B), shear modulus (S), B/S ratio, Young’s modulus (Y), Poisson’s ratio (ν) and anisotropic factor (A) of the normal Co2TiSn and inverse Zr2RhGa compounds were computed. For a cubic crystal, the standard mechani cal stability is
In this work, the structural, electronic, magnetic and elastic prop erties for normal Co2TiSn and inverse Zr2RhGa full Heusler compounds have been studied. We found that normal Co2TiSn compound and the inverse Zr2RhGa compound have half-metallic behavior, for their half metallic nature, these compounds are candidate for spintronics appli cations. The normal Co2TiSn and inverse Zr2RhGa compounds have an indirect energy gap of 0.482 eV and 0.573 eV. The band structure of normal Co2TiSn and inverse Zr2RhGa were calculated by using mBJGGA. It was shown that the energy band gap within mBJ-GGA for Co2TiSn and for Zr2RhGa are still indirect band gap and the energy gap increases for normal Co2TiSn to be 1.430 eV and for inverse Zr2RhGa to be 0.641 eV. The calculated total magnetic moment for these two compounds are in the range from 1.64 to 2 μB, which means that present results are to some extent compatible with the experimental and theoretical results. The elastic properties indicate that the normal full Heusler Co2TiSn compound and the inverse full Heusler Zr2RhGa compounds are me chanically stable. B/S results show that the normal Co2TiSn has a brittle nature, while the inverse Zr2RhGa has a ductile nature, on the other hand Poisson’s ratio (ν) values show that both compounds have ionic bonds. The elastic anisotropy (A) values show that the normal Co2TiSn and the inverse Zr2RhGa compounds are elastically anisotropies.
C11 > 0, C11
C12 > 0,C11 þ2 C12 > 0 and C44 > 0 [43].
In our calculations we focused on normal Co2TiSn and inverse Zr2RhGa compounds. The present calculations on normal Co2TiSn and inverse Zr2RhGa are presented in Table 5, they satisfied all the above conditions. This indicates that the normal Co2TiSn and inverse Zr2RhGa compounds are found to be mechanically stable. For the face center cubic crystal, the bulk modulus and shear modulus were calculated using Voigt approximation [44]. Bulk modulus for cubic structure can be calculated from the following equations: 1 B ¼ ðC11 þ 2C12 Þ 3
(6)
Voigt Shear modulus Sv is given by the following: 1 Sv ¼ ðC11 5
C12 þ 3C44 Þ
(7)
Young’s modulus (Y) is defined as the ratio of the stress to strain, and given by: Y¼
9Sv B ðSv þ 3BÞ
Acknowledgments
(8)
This work has been carried out in the Advanced Computational Physics Laboratory, Physics Department, An-Najah N. University.
Poisson’s ratio and anisotropic factor can be computed by using bulk and shear moduli, Poisson’s ratio and anisotropic factor can be given by: v¼
3B 2Sv 2ð3B þ Sv Þ
(9)
A¼
2c44 C11 C12
(10)
Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.matchemphys.2019.122122. References
Elastic constants, Voigt bulk modulus (B), Voigt shear modulus (Sv ), B/Sv ratio, Voigt Young’s modulus (Y), Voigt Poisson’s ratio (V) and anisotropic factor (A) are presented in Table 5. The Bulk (B) or shear modulus (S) measures the hardness of materials [45]. The ratio B/S measures the ductility and brittleness of the mate rials. When B/S > 1.75, the material behaves in a ductile nature, otherwise it behaves in a brittle nature [46]. In the present calculations, the B/Sv ratio of normal Co2TiSn and inverse Zr2RhGa compounds are 0.841 and 2.6257, respectively. Depending on the B/Sv ratio results, the normal Co2TiSn has brittle nature, while inverse Zr2RhGa has ductile nature. Young modulus (Y) is measures the stiffness of materials. The highest the value of Young modulus (Y), the stiffer the material is. The Poisson ͐s ratio (V) measures the stability of the material and provides useful in formation about the nature of the bonding. When Poisson’s ratio (ν) is greater than 1/3, the material behaves in a ductile nature; otherwise it behaves in a brittle nature [46]. Poisson’s ratio for covalent bonds compounds is too lower than 0.25; 0.1 for pure covalent bonds; while for ionic bonds compounds ν ¼ 0:25 to 0.5. In the present calculations, the ν of normal Co2TiSn and inverse Zr2RhGa compounds are found to be 0.284 and 0.331, respectively, ν indicates that the normal Co2TiSn and inverse Zr2RhGa compounds have ionic bonds. Likewise, the elastic anisotropy is an important parameter to measure the degree of anisot ropy of materials [47]. For an isotropic material, the value of A is unity. Otherwise, the material has an elastic anisotropy [48]. In the present calculations, the value of A for normal Co2TiSn and inverse Zr2RhGa compounds are found to be 1.985 and 4.899, respectively, which meansthe normal Co2TiSn and inverse Zr2RhGa compounds are elastic anisotropy.
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Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Structural, magnetic, electronic and elastic properties of half-metallic ferromagnetism full-Heusler alloys: Normal-Co2TiSn and inverse- Zr2RhGa using FP-LAPW method Doha N. Abu Baker a, Mohammed S. Abu-Jafar a, *, Ahmad A. Mousa b, Raed T. Jaradat a, Khaled F. Ilaiwi a, c, R. Khenata d a
Physics Department, An-Najah N. University, P.O. Box 7, Nablus, Palestine Department of Basic Sciences, Middle East University, Amman, 11831, Jordan c Arab Open University, Ramallah, Palestine d Laboratoire de Physique Quantique et de Mod�elisation Math�ematique de la Mati�ere (LPQ3M), Universit�e de Mascara, 29000, Algeria b
H I G H L I G H T S
� Normal Co2TiSn and inverse Zr2RhGa compounds possess half-metallic behavior. � Normal Co2TiSn and inverse Zr2RhGa compounds have an indirect energy gaps. � Normal Full Heusler Co2TiSn and inverse Full Heusler Zr2RhGa are mechanically stable. � B/S results show that normal Co2TiSn is brittle, while inverse Zr2RhGa is ductile. � Poisson’s ratio (ν) values show that both compounds have ionic bonds. A R T I C L E I N F O
A B S T R A C T
Keywords: Full-heusler Elastic properties Magnetic properties FP-LAPW mBJ
The equilibrium structural parameters, electronic, magnetic and elastic properties of the inverse Zr2RhGa and normal Co2TiSn Full-Heusler compounds have been studied using density-functional theory (DFT) and fullpotential linearized augmented plane-wave (FP-LAPW) method as implemented in the WIEN2k package. The Generalized Gradient Approximation (GGA) has been used for the exchange-correlation potential (Vxc) to compute the equilibrium structural parameters; lattice constant (a), bulk modulus (B) and bulk modulus pressure derivative (B0 ). In addition to the GGA approach, the modified Becke Johnson (mBJ) scheme has been used to calculate the band gap energies. The normal Heusler Co2TiSn compound and the inverse Heusler Zr2RhGa compound within GGA and mBJ approaches are found to have a half-metallic behavior, with an indirect energy gap in the spin down configuration. The calculated total magnetic moments for both compounds are to some extent compatible with the available experimental and theoretical results. Elastic constants and their related properties were obtained by using the IRelast package. These compounds are mechanically stable. The elastic constants satisfy the Born mechanical stability criteria. B/S ratio shows that the normal Co2TiSn has a brittle nature, while the inverse Zr2RhGa has a ductile nature; Poisson’s ratio (ν) values show that both compounds have an ionic bond nature.
1. Introduction In recent years, Heusler alloys received growing attention due to their interesting physical properties [1–4], especially the half metallic character, half metallic materials (HM) exhibiting a 100% spin polari zation around the Fermi surface [1–3] (a half metal is a ferromagnetic
with a gap in one of the spin directions at the Fermi energy (FE)). Half metals can be used as spin injectors for magnetic random access mem ories and other spin dependent devices [2]. A lot of alloys were pre dicted to be half metallic materials. In fact, investigating and searching for new (HM) materials are mainly focusing on the Heusler alloys [3–17]. From structural point of view, Heusler family can be described
* Corresponding author. E-mail address: [email protected] (M.S. Abu-Jafar). https://doi.org/10.1016/j.matchemphys.2019.122122 Received 1 May 2019; Received in revised form 3 August 2019; Accepted 3 September 2019 Available online 4 September 2019 0254-0584/© 2019 Elsevier B.V. All rights reserved.
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by two variants: Full Heusler X2YZ phases, which typically crystallize in Cu2MnAl (L21)-type structure and the half Heusler XYZ which typically crystallize in NiMnSb (C1b) type structure, X and Y are transition ele ments (3d, 4d or 5d elements) and Z is s-p element. The atoms in normal Heusler compounds are lined up in X2:(1/4,1/4,1/4), (3/4,3/4,3/4), Y (1/2,1/2,1/2), and Z (0,0,0), while the atoms in inverse Heusler com pounds are lined up in X2:(1/4,1/4,1/4), (1/2,1/2,1/2), Y (3/4,3/4, 3/4), and Z (0,0,0) [18]. There are a lot of previous studies that have been done on Heusler compounds and other half-metallic compounds by using different methods [19–24]. In 2007, Kandpal et al. [19] measured the lattice parameter for inverse Heusler compound Co2TiSn and found it to be 7.072 Å, they also calculated the magnetic moment using LMTO-ASA code [20], SPRKKR code [21], FP-LAPW method [22] and FPLMTO method [23], and found to be 1.40μB, 1.55μB, 2μB and 2μB, respectively. It is clear that the LMTO-ASA and SPRKKR codes fail to get the correct value of the total magnetic moment, while Wien2k and FPLMTO codes do correctly obtain minority gap in this compound and also calculated correctly the magnetic moment 2 μB per formula unit. In 2014, Birsan and Kuncser [24] studied the electronic, structural and magnetic prop erties of Zr2CoSn full-Heusler compound by using FP-LAPW method. They calculated the energy band gap (Eg ¼ 0.543 eV), total magnetic moment (Mtot ¼ 3 μB) and the lattice parameter (a ¼ 6.76 Å). In 2014, Birsan [25] investigated the structural and magnetic properties of the full-Heusler compound, Zr2CoAl using FP-LAPW method. He calculated the lattice parameter (a ¼ 6.54 Å), energy band gap (Eg ¼ 0.48 eV) and total magnetic moment (Mtot ¼ 2 μB). In 2015, Wang et al. [26] studied the half-metallic state and magnetic properties versus the lattice con stant in Zr2RhZ (Z ¼ Al, Ga, In) Heusler alloys by using the density functional theory (DFT) pseudo-potential method incorporate in CASTEP code [27,28]. The computed lattice parameters for Zr2RhZ (Z ¼ Al, Ga, In) are found to be 6.66, 6.64 and 6.81 Å, respectively and magnetic moment for all alloys was found to be 2 μB [26]. In 2017, Jain et al. [29] studied the electronic structure, magnetic and optical prop erties of Co2TiZ (Z ¼ B, Al, Ga, In) Heusler alloys by using FP-LAPW approach. They found the lattice parameters to be 5.494, 5.842, 5.845
Table 1 Calculated lattice parameter a (in Ǻ), bulk modulus B (in GPa), pressure deriv ative for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds. Compound Normal Co2TiSn Inverse Zr2RhGa
Present Experimental Present Theoretical
Lattice parameter
B
Bʹ
6.094 6.072 [19] 6.619 6.64 [26]
166.932 – 129.319 –
4.627 – 5.073 –
Fig. 1. The crystal structure of Full-Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds.
Fig. 2. Total energy as function of the volume for (a) normal Heusler Co2TiSn and (b) inverse Heusler Zr2RhGa compounds. 2
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Table 2 Total magnetic moment for normal and inverse Co2TiSn compound. Compound Normal Co2TiSn Inverse Co2TiSn
Magnetic Moment (μB) Present Experimental Present
Co
Co
Ti
Sn
1.32398 – 0.46092
1.32398 – 1.54481
0.19866 – 0.25961
0.00709 – 0.01579
Interstitial 0.87511 –
0.64087
Total magnetic moment (μB) 1.9786 2 [19] 1.64026
Table 3 Total magnetic moment for normal and inverse Zr2RhGa compound. Compound Normal Zr2RhGa Inverse Zr2RhGa
Magnetic Moment (μB) Present Experimental Present Theoretical
Zr
Zr
Rh
Ga
Interstitial
Total magnetic moment (μB)
0 – 0.88019 –
0 – 0.44447 –
0 – 0.15415 –
0 – 0.00905 –
0 –
0 – 1.99 2 [26]
0.50214
–
Fig. 3. PBE-GGA-Band structure spin-up for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
and 6.087 Å, respectively. The bulk modulus is found to be 233,182,184 and 161 GPa, respectively. The energy band gap is found to be 0.1, 0.44, 0.16 and 0.06 eV, respectively. The total magnetic moment is found to be 1, 0.99,1 and 1.03 μB, respectively. In 2017, Wei et al. [30] estimated the electronic, Fermi surface, Curie temperature and optical properties of Zr2CoAl compound by using FPLO code [31,32]. They computed the lattice parameter (a), bulk modulus (B) and total magnetic moment (Mtot) are found to be 6.629 Å, 106.8 GPa and 2 μB for the normal full-Heusler structure (Cu2MnAl). In present work we investigated the electronic, structural, magnetic and elastic properties of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds by using the full potential-linearized augmented plane wave method [FP-LAPW] method within the Perdew-BurkeErnzerhof generalized gradient approximation (PBE-GGA) [33]
integrated in Wien2k code [34]. 2. Computational method The calculations reported in present work were carried out by using the full potential linearized augmented plane wave, implemented in WIEN2k package [34]. The exchange correlation potential (VXC) was treated by two approximations, the generalized gradient approximation and the modified Becke–Johnson (mBJ-GGA) approach [35]. For the compound Co2TiSn, the muffin-tin radii (RMT) of Co, Ti and Sn atoms are 2.22 a. u., 2.17 and 2.22 a. u., respectively. For Zr2RhGa the RMT of Zr, Rh and Ga atoms are 2.37 a. u., 2.49 a. u. and 2.37 a. u., respectively. The number of plane waves was restricted by KMAX � RMT ¼ 8 and the expansions of the wave functions was set to l ¼ 10 inside the muffin-tin 3
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Fig. 4. PBE-GGA Band structure spin-down by for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
spheres. 35k points in the irreducible Brillouin zone (BZ) with a grid of 10 � 10 �10 Monkhorst-Pack (MP) [36] meshes (equivalent to 1000k points in the full Brillion zone (BZ)) are used to obtain self-consistency for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds. The self-consistent calculations are considered to converge when the calculated total energy of the crystal is stable within 10 5 Ryd. The elastic constants Cij determine the stiffness of a crystal against an externally applied strain. For small deformations we expect a quadratic dependence of the crystal energy on the strain (Hooke’s law) following this expression; !! 6 6 X 1X E V; εk ¼ E0 þ V0 σi εi þ Cij εi εj (1) 2 i¼1 ij¼1
must be applied to the starting crystal. Following the method developed by Morteza Jamal and integrated as the IRelast package in Wien2k code [37], the elastic constants for cubic system are computed by using some strain tensors. Further details of the calculation can be found elsewhere [38,39]. The first elastic components C11 and C12 are computed using a distortional orthorhombic deformation, Dortho and a volumetric cubic deformation, Dcubic given by the following tensors; 2 3 1þε 0 0 4 5 0 1 ε 0 DOrtho ¼ � � 0 0 1 1 ε2 2
DCubic
whereðεk Þdenotesε1 ; ε2 ; ::::::ε6 , V0(V) is the volume and E0(E) is the en ergy of the unstrained (strained) cubic system and Cij are the elastic constants. The elastic constants Cij are calculated by taking the second-order derivatives of the total energy of the strained crystal, Etot ðV; εÞ :Cij ¼ 2 1 ∂ Etot ðV;εÞ V0 ∂εi εj ,
3 1þε 0 0 4 ¼ 0 1 þε 05 0 0 1þε
And the elastic moduli, C44 is determined by the application of a distortional monoclinic deformation, Dmonoc 2 3 0 1 ε 5 Dmonoc ¼ 4 ε 1 �0 � 0 0 1 1 ε2
where εi are the strain tensor elements, V0 is being the vol
ume of the unstrained lattice. For the cubic systems, there are three independent elastic constants, namely, C11, C12 and C44. A set of three equations is needed to determine all the constants. This means that three types of strain εðIÞ (deformations)
Application of these deformations has an effect on the total energy from its unstrained value as follows: � �� (2) EðV; εÞ ¼ E0 þ V0 ðC11 C12 Þε2 þ O ε4
Table 4 Energy band gap for normal Co2TiSn and inverse Zr2RhGa Compounds.
EðV; εÞ ¼ E0 þ V0 ε ð σ 1 þ σ 2 þ σ3 Þ þ V0
Compound
Normal Co2TiSn Inverse Zr2RhGa
Band gap type
High symmetry points
Energy gapPBE-GGA (eV)
Energy gapmBJ-GGA (eV)
Indirect
Γ-X
0.482
1.430
Indirect
L-Δ
0.573
0.641
� � � 3 ðC11 þ 2C12 Þε2 þ O ε3 2 (3)
� �� EðV; εÞ ¼ E0 þ V0 ð2C44 Þε2 þ O ε4
(4)
We carried out calculations for five values of epsilon (strain) equal to 0.02, 0.01, 0.0, 0.01 and 0.02 with a dense mesh of uniformly distributed k-points equals to 4000 for the integration of the Full 4
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Fig. 5. mBJ-GGA- Band structure spin-up for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
Fig. 6. mBJ-GGA- Band structure spin-down for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds.
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Fig. 7. (a)Total and partial density of states of spin-up for normal Co2TiSn compound and for (b) Co atom (c) Ti atom (d) Sn atom by using PBE-GGA method.
Brillouin Zone. An increase in the number of k-points to 4000 is rec ommended for the Cubic/Hex-elast package [38] as elastic constants require a denser k-mesh to converge with a small percentage error.
inverse Zr2RhGa compound has space group F-43 m (216). We have calculated the structural properties for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds. The estimated lattice parameter (a), bulk modulus (B) and derivative pressure (Bʹ) at zero pressure are tabulated in Table 1, along with experimental and other theoretical re sults [19,26]. The crystal structure of full-Heusler Co2TiSn and Zr2RhGa compounds are shown in Fig. 1. The total energy as function of the volume for normal Heusler Co2TiSn and inverse Heusler. Zr2RhGa compounds are shown in Fig. 2a and b. Table 1 shows that our computed values of the lattice parameter of the normal Heusler Co2TiSn compound is slightly overestimated within 0.36% to the corresponding experimental values [19]. This is attributed to our use of the generalized gradient approximation (GGA), which is known to overestimate the lattice constant value compared to the measured one. The calculated parameter of inverse Heusler Zr2RhGa compound is found to be in good agreement to previous theoretical result [26].
3. Results and discussion 3.1. Structural properties The optimized lattice constants, bulk modulus, and its pressure de rivative were calculated by fitting the total energy to Murnaghan’s equation of state (EOS) [40,41]. Murnaghan’s equation of state (EOS) is given by: " # ’ BV ðV0 =VÞB BV0 EðVÞ ¼ E0 þ ’ þ 1 (5) B B’ 1 B’ 1 where E0 is the minimum energy, B is the bulk modulus at the equilib rium volume and Bʹ is the pressure derivative of the bulk modulus. dE dP Pressure (P) and bulk modulus (B) are given by P ¼ V dV ¼ dV , B ¼ 2
dE V dV 2 . Normal Co2TiSn compound has space group Fm-3m (225), and
6
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Materials Chemistry and Physics 240 (2020) 122122
Fig. 8. (a) Total and partial density of states of spin-down for normal Co2TiSn compound and for (b) Co atom (c) Ti atom (d) Sn atom by using PBE-GGA method.
3.2. Magnetic properties
that the calculated value of the total magnetic moment of normal Co2TiSn compound is slightly smaller than the experimental value [19]. Our calculated total magnetic moment of inverse Zr2RhGa compound is found to be closer to the other theoretical result [26]. Present total magnetic moment results are to some extent compatible with experi mental and theoretical results. The calculated total spin magnetic mo ments are clearly integral values except for inverse Co2TiSn and are in agreement with the Slater-Pauling rule [42]. The values of total and partial magnetic moments calculated in present study are reported in Table 2, we can easily notice that Co and Zr atoms are the mainly magnetic moment contributors.
In this part, the total and partial magnetic moment for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds were calcu lated and listed in Tables 2 and 3. Results from earlier experimental and theoretical works are quoted for comparison. The integer value of the total magnetic moment (Mtot) is character istic of half-metallic materials. We noticed that the total magnetic mo ments are found to be in the range from 1.64026 to 2 μB. We found that the normal and inverse Co2TiSn compounds are ferromagnetic, while only the inverse Zr2RhGa compound is ferromagnetic. The total mag netic moment for normal Zr2RhGa compound is found to be zero (0.0μB), which means it does not have magnetic behavior. Present results show 7
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Materials Chemistry and Physics 240 (2020) 122122
Fig. 9. (a)Total and partial density of states of spin-up for inverse Zr2RhGa compound and for (b) Zr atom (c) Rh atom (d) Ga atom by using PBE-GGA method.
3.3. Electronic properties
Zr2RhGa compounds using PBE-GGA method. The energy band gaps of normal Co2TiSn and inverse Zr2RhGa compounds are found to be 0.482 eV and 0.573 eV, respectively as shown in Table 4. The band structure of normal Co2TiSn and inverse Zr2RhGa are also calculated by using mBJ-GGA method. Fig. 5a and b also shows that the spin-up of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds both have metallic behavior using mBJGGA method. Fig. 6a and b shows the energy band gap within mBJGGA method for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds is still indirect band gap with an increase of the value of the band gap to 1.430 eV for normal Co2TiSn and to 0.641 eV for inverse Zr2RhGa for spin-down. Total and partial density of states for spin-up and spin-down for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds are
In this part, the band structure, the total and partial density of states for normal Co2TiSn compound and inverse Zr2RhGa compound were calculated. It is clear from the band structure and density of states that the normal Co2TiSn and inverse Zr2RhGa compounds both have a half -metallic behavior, which means at spin up the material behaves as a metal, while at spin down the material behaves as a semiconductor. Fig. 3a and b shows that the spin-up band structure of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds have metallic nature by using PBE-GGA method. Fig. 4a and b shows that the spindown band structure of normal Co2TiSn and inverse Zr2RhGa com pounds have an indirect energy band gap. The values of the energy band gaps of spin-down are calculated for normal Co2TiSn and inverse 8
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Materials Chemistry and Physics 240 (2020) 122122
Fig. 10. (a)Total and partial density of states of spin-down for inverse Zr2RhGa compound and for (b) Zr atom (c) Rh atom (d) Ga atom by using PBE-GGA method.
shown in Figs. 7 – 10. Densities of states Figs. 7–10, also confirm the half metallic property for normal Co2TiSn and inverse Zr2RhGa compounds with existing small energy band gap in the spin-down direction, and this means that these compounds have half metallic property. In the spin-up of normal Co2TiSn (Fig. 7), the valence band is due to the D-state of Co, s-state and p-state of Sn and small contribution from Ti D-state, while the conduction band is due to the D-state of Ti with few contribution from Sn p-state. In spin-down of normal Co2TiSn (Fig. 8), the valence band is due to the Co D-state, Ti D-state and Sn s-state and p-
state, while the conduction band is due to the Co D-state and Ti D-state and small contribution from s-state and p-state. In spin-up of inverse Zr2RhGa (Fig. 9), the valence band is due to the Rh D-state, Ga D-state, and small contribution from Zr D-state near to Fermi energy level, and the conduction band is due to the Zr D-state near to Fermi energy level. In spin-down of inverse Zr2RhGa (Fig. 10), the valence band is due to Rh D-state, Ga D-state and small contribution from Zr D-state, and the conduction band is due to Zr D-state near to Fermi energy level, and small contribution from Rh and Ga d-states.
Table 5 Elastic constants for normal Co2TiSn and inverse Zr2RhGa Full Heusler Compounds. Compounds
C11 (GPa)
C12 (GPa)
C44 (GPa)
B (GPa)
SV (GPa)
B/SV
Y (GPa)
V
A
Normal Co2TiSn Inverse Zr2RhGa
246.976 145.279
136.962 116.519
109.226 70.458
173.633 126.105
87.53 48.026
0.841 2.626
224.828 127.848
0.284 0.331
1.986 4.8997
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3.4. Elastic properties
4. Conclusions
In this part, the elastic constants (Cij), bulk modulus (B), shear modulus (S), B/S ratio, Young’s modulus (Y), Poisson’s ratio (ν) and anisotropic factor (A) of the normal Co2TiSn and inverse Zr2RhGa compounds were computed. For a cubic crystal, the standard mechani cal stability is
In this work, the structural, electronic, magnetic and elastic prop erties for normal Co2TiSn and inverse Zr2RhGa full Heusler compounds have been studied. We found that normal Co2TiSn compound and the inverse Zr2RhGa compound have half-metallic behavior, for their half metallic nature, these compounds are candidate for spintronics appli cations. The normal Co2TiSn and inverse Zr2RhGa compounds have an indirect energy gap of 0.482 eV and 0.573 eV. The band structure of normal Co2TiSn and inverse Zr2RhGa were calculated by using mBJGGA. It was shown that the energy band gap within mBJ-GGA for Co2TiSn and for Zr2RhGa are still indirect band gap and the energy gap increases for normal Co2TiSn to be 1.430 eV and for inverse Zr2RhGa to be 0.641 eV. The calculated total magnetic moment for these two compounds are in the range from 1.64 to 2 μB, which means that present results are to some extent compatible with the experimental and theoretical results. The elastic properties indicate that the normal full Heusler Co2TiSn compound and the inverse full Heusler Zr2RhGa compounds are me chanically stable. B/S results show that the normal Co2TiSn has a brittle nature, while the inverse Zr2RhGa has a ductile nature, on the other hand Poisson’s ratio (ν) values show that both compounds have ionic bonds. The elastic anisotropy (A) values show that the normal Co2TiSn and the inverse Zr2RhGa compounds are elastically anisotropies.
C11 > 0, C11
C12 > 0,C11 þ2 C12 > 0 and C44 > 0 [43].
In our calculations we focused on normal Co2TiSn and inverse Zr2RhGa compounds. The present calculations on normal Co2TiSn and inverse Zr2RhGa are presented in Table 5, they satisfied all the above conditions. This indicates that the normal Co2TiSn and inverse Zr2RhGa compounds are found to be mechanically stable. For the face center cubic crystal, the bulk modulus and shear modulus were calculated using Voigt approximation [44]. Bulk modulus for cubic structure can be calculated from the following equations: 1 B ¼ ðC11 þ 2C12 Þ 3
(6)
Voigt Shear modulus Sv is given by the following: 1 Sv ¼ ðC11 5
C12 þ 3C44 Þ
(7)
Young’s modulus (Y) is defined as the ratio of the stress to strain, and given by: Y¼
9Sv B ðSv þ 3BÞ
Acknowledgments
(8)
This work has been carried out in the Advanced Computational Physics Laboratory, Physics Department, An-Najah N. University.
Poisson’s ratio and anisotropic factor can be computed by using bulk and shear moduli, Poisson’s ratio and anisotropic factor can be given by: v¼
3B 2Sv 2ð3B þ Sv Þ
(9)
A¼
2c44 C11 C12
(10)
Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.matchemphys.2019.122122. References
Elastic constants, Voigt bulk modulus (B), Voigt shear modulus (Sv ), B/Sv ratio, Voigt Young’s modulus (Y), Voigt Poisson’s ratio (V) and anisotropic factor (A) are presented in Table 5. The Bulk (B) or shear modulus (S) measures the hardness of materials [45]. The ratio B/S measures the ductility and brittleness of the mate rials. When B/S > 1.75, the material behaves in a ductile nature, otherwise it behaves in a brittle nature [46]. In the present calculations, the B/Sv ratio of normal Co2TiSn and inverse Zr2RhGa compounds are 0.841 and 2.6257, respectively. Depending on the B/Sv ratio results, the normal Co2TiSn has brittle nature, while inverse Zr2RhGa has ductile nature. Young modulus (Y) is measures the stiffness of materials. The highest the value of Young modulus (Y), the stiffer the material is. The Poisson ͐s ratio (V) measures the stability of the material and provides useful in formation about the nature of the bonding. When Poisson’s ratio (ν) is greater than 1/3, the material behaves in a ductile nature; otherwise it behaves in a brittle nature [46]. Poisson’s ratio for covalent bonds compounds is too lower than 0.25; 0.1 for pure covalent bonds; while for ionic bonds compounds ν ¼ 0:25 to 0.5. In the present calculations, the ν of normal Co2TiSn and inverse Zr2RhGa compounds are found to be 0.284 and 0.331, respectively, ν indicates that the normal Co2TiSn and inverse Zr2RhGa compounds have ionic bonds. Likewise, the elastic anisotropy is an important parameter to measure the degree of anisot ropy of materials [47]. For an isotropic material, the value of A is unity. Otherwise, the material has an elastic anisotropy [48]. In the present calculations, the value of A for normal Co2TiSn and inverse Zr2RhGa compounds are found to be 1.985 and 4.899, respectively, which meansthe normal Co2TiSn and inverse Zr2RhGa compounds are elastic anisotropy.
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