First principle investigation of the structural, electronic and elastic properties of the Laves phase compounds SrX2 (X = Pd and Pt)

Accepted Manuscript

First principle Investigation of the Structural, electronic and elastic properties of the Laves Phase Compounds SrX2 (X=Pd and Pt) Ahmad A. Mousa , Raed Jaradat , Said M. Azar , Mohammed Abu-Jafar , Emad K. Jaradat , J.M. Khalifeh , K.F. Ilaiwi PII: DOI: Reference:

S0577-9073(19)30030-9 https://doi.org/10.1016/j.cjph.2019.03.004 CJPH 803

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

6 January 2019 15 February 2019 11 March 2019

Please cite this article as: Ahmad A. Mousa , Raed Jaradat , Said M. Azar , Mohammed Abu-Jafar , Emad K. Jaradat , J.M. Khalifeh , K.F. Ilaiwi , First principle Investigation of the Structural, electronic and elastic properties of the Laves Phase Compounds SrX2 (X=Pd and Pt), Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.03.004

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Highlights

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Our results for elastic stiffness verify the Born’s mechanical stability criteria for cubic structures. The SrPt2 compound is found to be stiffer than SrPd2 compound. The Poisson’s ratio value makes sure that SrPd2 and SrPt2 in a ductile manner and has ionic bond. Both compounds possess metallic characteristics

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First principle Investigation of the Structural, electronic and elastic properties of the Laves Phase Compounds SrX2 (X = Pd and Pt) Ahmad A. Mousa1,*, Raed Jaradat2, Said M. Azar1, Mohammed Abu-Jafar2,*, Emad K. Jaradat3 J. M. Khalifeh4 and K. F. Ilaiwi2,5 Department of Basic Sciences, Middle East University, Amman, 11831, Jordan

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Physics Department, An-Najah National University, P. O. Box 7, Nablus, Palestine

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Department of Physics, The University of Jordan, Amman, 11942, Jordan

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Physics Department, Mutah University, Jordan

Arab Open University, Ramallah, Palestine

Corresponding authors: [email protected] (Ahmad Mousa), [email protected] (Mohammed Abu-Jafar)

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Keywords: Laves phase; DFT; GGA; Elastic properties; Electronic properties; FP-LAPW.

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Abstract

Structural, electronic and elastic properties of the cubic Laves phases SrX2 (X=Pd

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and Pt) with Fd-3m (No. 227) space group and crystallize in the MgCu2 structure are studied using the all-electrons full potential linearized augmented plane wave

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(FP-LAPW) method as implemented in the Wien2k code. The exchangecorrelation potential (Exc) was treated within the scheme of Perdew, Burke, and Ernzerhof generalized gradient approximation (PBE-GGA). In present calculations, studied compounds show a metallic characteristic. When the palladium element was replaced with the platinum, the Fermi level was observed to be increased, indicating an increase in the additional valence electrons that filled the hybridized 2

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bonding states. The elastic constants were calculated for both compounds and show that the two compounds suitably verify the Born’s mechanical stability criteria and have the ductile manner behaviors.

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1. Introduction

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The Laves phases are intermetallic compounds that have a general chemical formula of AB2. There are three structural types of Laves phases, namely: cubic C15 (MgCu2), hexagonal C14 (MgZn2) and hexagonal C36 (MgNi2) [1,4]. In these compounds, the A atoms (are an electropositive elements such as an alkali metals, an alkaline earth metals) take up positions as in diamond, hexagonal diamond or related structure while the B atoms (are a transition elements less electropositive than A atoms) take up tetrahedral positions around A atoms [5].

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Laves phases are potential materials for diverse applications. For instances, Zr(Cr, Fe)2 and TiCr2compounds had been used as hydrogen storage materials because of their high hydrogen absorbing capabilities and favorable hydridingdehydriding kinetics[6,7]. Also, (Tb, Dy)Fe2 and (Al, Fe)Zr2 have been considered for magneto elastic transducer applications due to their massive magneto restriction [8,9]. The (Hf, Zr)V2 and NbBe2 [10,11]alloys have shown superconducting characteristics with a mixture of high critical temperature, perfect magnetic strength, and high current density. Moreover, the high melting points and good retention of mechanical characteristics at elevated temperatures are the reasons for using HfCr2, NbCr2, and TiCr2 based on two-phase alloys for a structure of high-temperature [6-17].

The structural and electronic properties of BaRh2, BaPd2, and BaPt2 compounds are discussed by analyzing the total and partial DOS which give knowledge of chemical bonding nature [2]. Some Laves phase’s compounds have a very hard and brittle at ambient temperatures, like NbCr2 compound [14]. By introducing ductile, soft particles the ductility and toughness will improve substantially. 3

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Furthermore, due to the complicated crystal structure, the high hardness indicates difficulty in nucleation and glide of dislocation.

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In this contribution, we have investigated of the structural, electronic and mechanical properties of the SrX2 (X=Pd and Pt) intermetallic compounds in cubic C15 (MgCu2) phase using first principle calculations. The band structure and density of states (DOS) were calculated to discuss the electronic structure and the metallicity behavior for SrX2 compounds. Furthermore, we estimated the elastic constants where the stability of SrX2 compounds is discussed. Present work is organized as follows: Sec. 2 is devoted to the computational details; Sec. 3 deals with the results and discussions and the conclusion is presented in Sec. 4. 2. Computational details

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The computation has been performed by using full potential linearized augmented plane wave (FP-LAPW) method implemented in Wien2k code [18]. The unit cell in this method is partitioned into two types of regions: (I) atomic muffin-tin centered spheres with radii RMT, and (II) the interstitial region [19]. The RMT values used in this work are 2.5 a.u for Sr atom and 2.3 au for Pd and Pt atoms. The core cutoff energy is −81.66 eV and the plane wave’s cutoff Kmax = 8/RMT. Inside the muffin-tin spheres, Fourier expanded up to Gmax=12 with a cutoff lmax = 10 and 145 k-points in the irreducible Brillion zone with grid 16Χ16Χ16 meshes are used to obtain self-consistency for SrX2 compounds. Exchange correlation potential (Exc) is computed within the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) [20]. The self-consistence calculations are considered to be converged only when the convergence of energy and charge are less than 0.1 mRy and 0.001 electron charges, respectively. The FP-LAPW method has proven to be one of the accurate methods for calculating the electronic structure of solids within DFT [21-25].

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3. Results and discussions 3.1 Structural properties and formation energy

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In this subsection the crystalline structure and stability of the SrX2 (X=Pd and Pt) intermetallic compounds will be displayed. The SrX2 compound shown in Figure 1, has the Laves phases, which has a general chemical formula AB2 with space group Fd-3m (No. 227). It is shown that the conventional and the primitive unit cell of SrPt2 crystal in cubic laves phase C15, and one can see that the Sr positions occupy [(1/8,1/8,1/8), (7/8,7/8,7/8)], and X positions occupy [(1/2,1/2,1/2), (1/2,3/4,3/4), (3/4,1/2,3/4), (3/4,3/4,1/2)] [26]. In this study we used Murnaghan’s equation of state [27] and a third-order Birch equation of state [28] to calculate the equilibrium lattice parameter (a) and the bulk modulus (B), respectively. In figure 2, the relationship between the lattice constant and formation energy is shown. The optimized lattice constant (a), bulk modulus (B) and the pressure derivative of the bulk modulus (B′) for SrPd2 and SrPt2 are listed in Table I next to the previous calculated [29,30] and measured [31,32] data for ease of comparison.

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The comparison between our calculations of the lattice parameters with the previous study are shown in Table I. This table clearly shows the compatibility with the previous calculations [29, 30] and a slight difference with the previous measurements [31, 32] by 1.53% and 1.78% for SrPd2 and SrPt2, respectively. The bulk modulus (B) for SrPt2 compound (118.62GPa) is much larger than that of the SrPd2 compound (81.82GPa), this indicates that the SrPt2 compound is more hardness than the SrPd2 compound. We also calculated the formation energies for SrX2 intermetallic compounds, to study phase stability. The formation energy (ΔE) is defined as [33]:

where and are the total energies of the elements Sr and X(Pt,Pd) atoms in their stable bulk structures, and is the equilibrium total energy of the SrPt2 and SrPd2 compounds. 5

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3.2 Band structure and density of state

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The calculated formation energies and the previous theoretical calculations [29, 30] are listed in table I. According to our knowledge, there is no experimental data has been reported for the formation energy for SrPd2 and SrPt2 compounds, yet. From this table one can see that our calculations agree well with the previous theoretical calculation, moreover, we can see that the SrPt2 has more stability than SrPd2. On the other hand, the two compounds have a negative formation energy, which indicates that they could exist in nature.

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The electronic properties such as electronic band structure, partial density of state (PDOS), and the total density of state (TDOS) of cubic laves phase SrPd2 and SrPt2 have been calculated and discussed in this section. The electronic band structure analysis of the materials gives us information about the materials electric types: conductor, semiconductor or insulator. The atomic-resolved, the lcomponent and the TDOS give a detailed data about the number of states at occupied orbital levels in condensed matter and the bonding energy [34].

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The calculated band structure and TDOS, at optimized lattice constants, for SrPd2 and SrPt2 along the high symmetry lines in the first Brillouin zone is shown in figure 3. The band structure shape of SrPd2 and SrPt2 are almost identical with some differences. As we see from band structure with TDOS diagram (Figure. 3), the valance and conduction bands overlapped together at Fermi level EF which alludes that both materials possess metallic characteristics. The metallicity of these two compounds SrPd2 and SrPt2 confirmed by their band structures. The immense flat bands, mainly d character around EF, are dominated by palladium 4d and platinum 5d orbitals. The flat bands of the two compounds SrPd2 and SrPt2 are well-separated from the other bands along - L direction for both compounds. In two compounds, SrPd2 and SrPt2, the flat band appears below the EF near the  point, as displayed in figure 3a. SrPt2 has more valance electrons in unit cell than SrPd2. This leads more occupation bonding states near EF. Furthermore, EF 6

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moves from lower energy level to a higher one when Pd is substituted by Pt. So, the hybridized bonding states are filled by extra valance electrons. This filled band increases the bulk modulus (B) of SrPt2 compared to SrPd2. The electronic configuration for both elements Pd and Pt are [Kr] 4d10 and [Xe] 4f14 5d9 6s1, respectively. In Palladium case, the completely filled d state (4d10) has lower energy than the partially filled states 4d9s1 and 4d8s2.

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The nature of the bond of cubic lave compounds SrX2 (X=Pd and Pt) could be studied by analysis the DOS. The comparison between TDOS of SrPd2 and SrPt2 laves phase is displayed in figure 4. The TDOS is quite similar for the two compounds. The peaks and their relative values in TDOS details of the two compounds are rather similar, where the supreme peaks occur in SrPd2. The main differences are occurred around EF. In SrPt2, EF falls on a pronounced peak, whereas in SrPd2 it occurs on the low-energy side.

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The Electronic density of states (total and partial) of C15-SrX2 are shown in figure 5. They give details about the impact of electronic configurations of constitutional elements on their chemical bonding. The occupied bonding and unoccupied antibonding are separated by EF. The d orbitals in Pd and Pt dominate the bonding and near EF, whereas the d orbit in Sr and some hybridization orbitals between Srd and Pd or Pt d orbitals dominate the antibonding states. This supports the covalent bonding nature of SrPd2 and SrPt2. In more details, strong covalent bonds are between Sr atoms and metallic bonds are between Sr and Pd or Pt atoms.

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The electrons near Fermi level EF, play a main role in the chemical bonds formation. The calculated TDOS at EF for SrPd2 is 1.0525 states/eV and for SrPt2 is 2.569 states/eV, which means that the SrPd2 is more stable than SrPt2. 3.3 Elastic properties The elastic behavior for SrPd2 and SrPt2 compounds are discussed in this subsection using calculating elastic stiffness. Where the elastic stiffness provides

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relevant information about material stability, material stiffness and bonding characteristic. The calculated stiffness is are listed in Table II. Our results for elastic stiffness as given in Table II suitably verify the Born’s mechanical stability criteria for cubic structures [35,36]: C11> 0; C44> 0; C11 +2‫‏‬C ‫ ‏‬12> 0; C11-C12 > 0. In addition, the C11 values are more than 60% higher than C44 for both SrPd2 and SrPt2 compounds. This indicates that the compounds are likely to experience pure shear deformation as to resistance to uniaxial compression. Moreover, C44 values give us an indication about the stiffness, where the big value of C44 means that the compound is stiffer. From our data, we can conclude that the SrPt2 compound is expected to be stiffer than SrPd2 compound.

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The bulk modulus was calculated by using the elastic constants, where B = (C11 + 2C12)/3 and the results are summarized in Table II. One can see that the good agreement for B0 values in this calculations with the results from Birch equationof-state calculations (see Table 1). This confirms the validity of our calculations of the elastic constants for the studied compounds.

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Another important factor used for discussed the structural stability called an elastic anisotropy factor (A). This factor has a Great importance in engineering science [37], where it has highly mutual relationship with the possibility of inducing micro-cracks in the materials [38,39]. The elastic constants are used to calculate the anisotropy factor according to the following equation:

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When A equals to 1 then the material is reckoned completely isotropic, otherwise, the material is reckoned anisotropy, i.e., the deviation of A values from 1 is the measure of what the crystal possesses from the amount of elastic anisotropy [38]. Anisotropic factors for SrPd2 and SrPt2 compounds are tabulated in table II, it is clearly shown that they are completely anisotropic compounds. Young’s modulus (E) (the ratio between linear stress and strain) is usually used as a material stiffness indicator. The larger values of Young’s modulus mean the

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material is stiffer. The Young’s modulus (E) can be calculated according to the equation:

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where GH is the Hill’s shear value [40] has been calculated by the average value of Reuss (GR) [41] and Voigt (GV) [42] approximations,

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and we can use the elastic stiffness to calculate GR and GV as:

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and

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From our results of E as listed in table II, we can see that the Young’s modulus of the SrPt2 compound is around 50% greater than that of the SrPd2 compound. So, the SrPt2 compound is stiffer than SrPd2.

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The value of Poisson's ratio (ν) gives us a good information about the nature of bonding forces, where if this value is greater than 0.25, the material possesses ionic bond; otherwise the material possesses covalent bond [43]. From Table II, Poisson’s ratio for the SrPd2 and SrPt2 compounds are 0.31 and 0.304, respectively, i.e. there is a higher ionic bond between Sr atom and X atom for both SrPd2 and SrPt2 compounds. One can estimate the ductile/ brittle nature of materials using Cauchy’s pressure ( C12-C44), Pugh’s index of ductility (B/G) and Poisson’s ratio (ν). The Cauchy’s pressure value sign is an indication of ductile (brittle) for the material, where if the pressure positive the material is expected to be ductile, otherwise the material is brittle [44]. The calculations for Cauchy’s pressure are listed in table II, we can conclude that the SrPd2 and SrPt2 9

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compounds have the ductile nature. The critical value for Pugh’s index of ductility is 1.75 [35]; if the value of the Pugh’s index is greater than the critical value (1.75), the material will be ductile, but if it is less, it will be brittle. It is observed that B/G> 1.75 (see table II), thereby classifying these compounds as ductile materials. In addition, the Poisson’s ratio values (ν) make sure that SrPd2 and SrPt2 compounds are in a ductile manner. For brittle materials, ν is less than 0.26, otherwise the material is considered a ductile material [45].

3.4 Variation of elastic constants and their related constants with lattice constant

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Figure 6, a and b shows the lattice constant dependence of the elastic constants and bulk modulus for Srpd2 and Srpt2 compounds, respectively. It is noticeable that elastic constants C11, C12, C44 and bulk modulus decrease as the lattice parameter increases, which means as the pressure decreases the elastic constants decrease. Decreasing of elastic constants and B mean the compounds become low mechanically stable. Also we can notice that elastic constants and B decrease in the same manner, and even compounds become low mechanically stable, they remain stable. Debye temperature (θD) of a certain compound defines as, the temperature of a crystal's highest normal mode of vibration. Debye temperature is a fundamental

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parameter and can be estimated from the average sound velocity equation [46] ⁄

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by the

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where h is Plank’s constant, kB is Boltzmann’s constant, NA is Avogadro’s number, n is the number of atoms per unit formula, ρ is the mass density per unit volume and M is the molecular weight. Here

is average sound velocity and can be

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respectively and given from [46,47].

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where

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estimated from

√ ⁄

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Figs. 7 and 8 show the predicted lattice constant dependent of Debye's temperature and average wave velocity, respectively. As the lattice constant increases (pressure decreases) the average wave velocity decreases, as a result the Debye temperature decreases because of the linear dependence of on . This result is in accordance with the fact that Debye temperature is proportional to the bulk modulus, as the lattice constant increase the B and decrease. 4. Conclusion

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Disclosure statement

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Ab-initio study of cubic Laves phases SrX2 (X=Pd and Pt) with space group Fd-3m (No. 227) and crystallize in the MgCu2, were carried out using the all-electron fullpotential linear augmented plane wave method within the generalized gradient approximation (GGA). We have studied the structural, electronic and elastic properties of SrPd2 and SrPt2 compounds. Our calculated ground state properties agree well with the available previous calculated and measured values. Our results for elastic stiffness suitably verify the Born’s mechanical stability criteria for cubic structures. From our data we can conclude that the SrPt2 compound is expected to be stiffer than SrPd2 compound. On the other hand, it is observed that the B/G > 1.75, thereby classifying these compounds as ductile materials. In addition, the Poisson’s ratio value (ν) makes sure that SrPd2 and SrPt2 in a ductile manner and has ionic bond. From band structure diagram the valance and conduction bands are overlapped with each other at Fermi level which indicates that both materials possess metallic characteristics. SrPt2 has more valance electrons in unit cell than SrPd2. This leads more occupation bonding states near EF. Furthermore, EF moves from lower energy level to a higher one when Pd is substituted by Pt. So, the hybridized bonding states are filled by extra valance electrons. The calculated total density of states (TDOS) at Fermi level DOS (EF) for SrPd2 is 1.0525 states/eV and for SrPt2 is 2.569 states/eV.

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No potential conflict of interest was reported by the authors.

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[44] Pettifor, D. G., “Theoretical Predictions of Structure and Related Properties of Intermetallics”,‫‏‬Mater. Sci. Technol. 8, (1992) 345.

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[45] I. N. Frantsevich, F. F. Voronov, S. A. Bokuta, I. N. Frantsevich, “Elastic constants and elastic moduli of metals and insulators handbook”, (Naukuva Dumka, Kiev) pp. 60–180, Chapter 2, (1983) Mechanical Properties.

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CE

[46] G. Sin'Ko, N. Smirnov, “Ab Initio Calculations of Elastic Constants and Thermodynamic Properties of Bcc, Fcc, and Hcp Al Crystals Under Pressure”, J. Phys. Condens. Matter 14 (2002) 6989. [47] O. Anderson, E. Schreiber, N. Soga, “Elastic Constants, and Their Measurements”, McGraw-Hill, New York, 1973.

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Figure captions:

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Figure 1: Crystal structure of SrPt2 in cubic laves phase C15. (a) Conventional unit cell and (b) primitive cell.

-560

a)

b)

PT

-780

-540

ED

-760

-580

SrPt2

SrPd2

-820

-600

CE

E(meV)

-800

AC

-840

7.7

-620

7.8

7.9

8.0

8.1 7.7

7.8

7.9

8.0

8.1

0

lattice constant:a (A )

Figure 2: Formation energy per atom as a function of lattice constant of (a) SrPd2 and (b) SrPt2. 17

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Figure 3: Electronic band structures and total DOS of (a) SrPd2 and (b) SrPt2.

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Figure 4: The comparison of total DOS for SrPd2 and SrPt2 cubic structure.

Figure 5: The total and partial DOS of (a) SrPd2 cubic structure (b) SrPt2 cubic structure.

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Figure 6: Variation of elastic constants and bulk modulus with lattice constant for a) SrPd2 and for b) SrPt2 compounds.

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Figure 7: Variation of Debye temperature θD (K) with lattice constants for SrPd2 and SrPt2 compounds.

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Figure 8: Variation of average wave velocity vm (m/s) with lattice constant for SrPd2 and SrPt2 compounds.

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Table captions:

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SrPd2

SrPt2

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ΔE(meV)

a(Ǻ)

B(GPa)

Present work

7.92

81.82

4.95

-635.15

Other theoretical work

7.949a

…

…

-651c

Experimental work

7.826b 7.800c

…

…

…

Present work

7.88

118.62

5.00

-849.68

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Compound

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Other theoretical work Experimental work

7.893d

118 d

7.777 a 7.742 b

…

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a) Ref. 29 b) Ref. 31 c) Ref. 32 d) Ref. 30

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Table I: Lattice constant a(A°), bulk modulus B(GPa), pressure derivative of bulk modulus B’ and the formation energy ΔE(meV) for the SrPd2 and SrPt2 compounds.

C12(GPa)

C44(GPa)

B(GPa)

SrPd2

116.9221

65.5961

44.9645

82.704

SrPt2

177.3246

91.5683

63.0157

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C11(GPa)

ED

120.153

G(GPa) E(GPa)

v

A

B/G

35.904

94.095

0.31

1.75211394

2.303476

54.004

140.902

0.304

1.46964596

2.224891

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CE

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Table II: The calculated elastic constants, bulk modulus B(GPa), shear modulus G(GPa), Young’s modulus E(GPa) , Poisson’s ratio v, anisotropic ratio A and Pugh’s index B/G of the C15-SrX2.

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