First principles study on the structural, electronic, and elastic properties of Na–As systems

Solid State Communications 151 (2011) 1349–1354

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First principles study on the structural, electronic, and elastic properties of Na–As systems H.B. Ozisik a,b,∗ , K. Colakoglu a , E. Deligoz b , H. Ozisik b a

Gazi University, Faculty of Science, Physics Department, 06500, Teknikokullar, Ankara, Turkey

b

Aksaray University, Faculty of Arts and Science, Physics Department, 68100, Campus, Aksaray, Turkey

article

info

Article history: Received 5 February 2011 Received in revised form 5 May 2011 Accepted 14 June 2011 by D.D. Sarma Available online 23 June 2011 Keywords: D. Mechanical properties D. Electronic band structure D. Phase transition

abstract We have performed the first principles calculation by using the plane-wave pseudopotential approach with the generalized gradient approximation for investigating the structural, electronic, and elastic properties Na–As systems (NaAs in NaP, LiAs and AuCu-type structures, NaAs2 in MgCu2 -type structure, Na3 As in Na3 As, Cu3 P and Li3 Bi-type structures, and Na5 As4 in A5 B4 -type structure). The lattice parameters, cohesive energy, formation energy, bulk modulus, and the first derivative of bulk modulus (to fit to Murnaghan’s equation of state) of the related structures are calculated. The second-order elastic constants and the other related quantities such as Young’s modulus, shear modulus, Poisson’s ratio, sound velocities, and Debye temperature are also estimated. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Alkali metal pnictides are member of the family of Zintl phase which has drawn attention of scientists in inorganic research for many years [1,2]. These materials show different physical and chemical properties depending on the structure and chemical composition [2–4]. They possess a band gap range from 0.1 eV up to 2.5 eV for different stoichiometry and structures. This important property is used in thin film and photocathode application [5,6]. Some alkali pnictides are, also used in crystal growth techniques [7–9]. There are many studies on alkali pnictides, but most of them are on the compounds of the alkali metals with Sb and Bi elements. It has been reported only few recent theoretical or experimental works on Na–As compounds [2,10–18]. The structural parameters, melting points, standard enthalpies, and entropies of formation of the synthesized Na–As compounds are investigated in orthorhombic NaP, hexagonal Na3 As, and Cu3 P structures [10]. Brauer and Zintl [11] analyzed the crystal structure of Na3 As using the X-ray method. Mansmann [12] studied crystal structure of Na3 As in Cu3 P-type structure. In early works, the melting point and enthalpy of formation for Na3 As were determined calorimetrically [13,14]. In a very recent work,

∗ Corresponding address: Gazi University, Department of Physics, 06500, Ankara, Turkey. Tel.: +90 312 202 1458; fax: +90 312 212 2279. E-mail addresses: [email protected], [email protected] (H.B. Ozisik). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.06.019

Burtzlaff et al. [2] obtained the structural parameters of monoclinic NaAs compound. The phase transition from the Na3 As structure to the Li3 Bi structure was estimated by Beister et al. [15]. Hafner and Range [16] synthesized the Na3 As compound in the P63 cm space group and investigated structural properties and revisions proposed by Brauer and Zintl [11], experimentally. Theoretically, less attention has been paid to these compounds and there are only few works on the half-metallic ferromagnetism of NaAs in zinc blende [17] and rock salt structures [18]. Here, we aim to investigate the structural, electronic, and elastic properties of Na–As compounds for different stoichiometries and structures (NaAs in NaP, LiAs and AuCu-type structures, NaAs2 in MgCu2 -type structure, Na3 As in Na3 As, Cu3 P and Li3 Bi-type structures, and Na5 As4 in A5 B4 -type structures) in detail and interpret the salient results. Detailed crystallographic parameters of the considered structures are given in Table 1. The layout of this paper is as follows: The method of calculation is given in Section 2. The results and overall conclusion are presented and discussed in Sections 3 and 4, respectively. 2. Method of calculation All calculations have been carried out using the Vienna ab initio simulation package (VASP) [19–22] based on the density functional theory (DFT). For the exchange and correlation terms in the electron–electron interaction, Perdew–Burke–Ernzerhof (PBE) [23] was used within the generalized gradient approximation (GGA). The electron–ion interaction was considered in the form

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Table 1 Crystallographic data for considered structures of Na–As compounds and the calculated atomic positions after local optimizations on the ab initio level (GGA–PBE). Compound

Space group (number)

Prototype

Atom

Site

GGA–PBE (present)

k-grid mesh

NaAs

P21 21 21 (19) orthorhombic

NaP

Na1 Na2 As1 As2

4a (x1 , y1 , z1 ) 4a (x2 , y2 , z2 ) 4a (x3 , y3 , z3 ) 4a (x4 , y4 , z4 )

4a (0.4086, 0.9025, 0.0450) 4a (0.1487, 0.6370, 0.3277) 4a (0.2875, 0.1349, 0.2904) 4a (0.4182, 0.4025, 0.1235)

8×8×5

NaAs

P21 /c (14) monoclinic

LiAs

Na1 Na2 As1 As2

4e (x1 , y1 , z1 ) 4e (x2 , y2 , z2 ) 4e (x3 , y3 , z3 ) 4e (x4 , y4 , z4 )

4e (0.2189, 0.3904, 0.3300) 4e (0.2346, 0.6634, 0.0323) 4e (0.3197, 0.8938, 0.2895) 4e (0.3120, 0.1610, 0.1188)

8×8×5

NaAs

P4/mmm (123) tetragonal

AuCu

Na As

1a (0, 0, 0) 1d (1/2, 1/2, 1/2)

1a (0, 0, 0) 1d (0.5, 0.5, 0.5)

8 × 8 × 11

NaAs2

Fd − 3m (227) cubic

MgCu2

Na As

8a (1/8, 1/8, 1/8) 16d (1/2, 1/2, 1/2)

8a(0.125, 0.125, 0.125) 16d (0.5, 0.5, 0.5)

6×6×6

Na3 As

P63 /mmc (194) hexagonal

Na3 As

Na1 Na2 As

2b (0, 0, 1/4) 4f (1/3, 2/3, z1 ) 2c (1/3, 2/3, 14)

2b (0, 0, 0.25) 4f (0.3333, 0.6667, 0.5801) 2c (0.3333, 0.6667, 0.25)

10 × 10 × 6

Na3 As

Fm − 3m (225) cubic

Li3 Bi

Na1 Na2 As

4b (1/2, 1/2, 1/2) 8c (1/4, 1/4, 1/4) 4a (0, 0, 0)

4b (0.5, 0.5, 0.5) 8c (0.25, 0.25, 0.25) 4a (0, 0, 0)

7×7×7

Na3 As

P63 cm (185) hexagonal

Cu3 P

Na1 Na2 Na3 Na4 As

2a (0, 0, z1 ) 4b (1/3, 2/3, z2 ) 6c (x1 , 0, z3 ) 6c (x2 , 0, z4 ) 6c (x3 , 0, z5 )

2a (0, 0, 0.2871) 4b (0.3333, 0.6667, 0.2231) 6c (0.3015, 0, 0.5762) 6c (0.3607, 0, 0.9154) 6c (0.3308, 0, 0.2460)

6×6×6

Na5 As4

C 2/m (12) monoclinic

A5 B4

Na1 Na2 Na3 As1 As2

4i (x1 , 0, z1 ) 4i (x2 , 0, z2 ) 2a (0, 0, 0) 4i (x3 , 0, z3 ) 4i (x4 , 0, z4 )

4i (0.1005, 0, 0.3866) 4i (0.7549, 0, 0.1630) 2a (0, 0, 0) 4i (0.4562, 0, 0.1704) 4i (0.3930, 0, 0.3926)

6×6×6

of the projector–augmented–wave (PAW) method [20,24] with plane wave up to energy of 450 eV. The k-mesh values determined by choosing k-point separation 0.02 Å−1 are given in Table 1 were found sufficient for the total energy calculations. Structural optimization was performed for each structure for all lattice constants, angles, and internal atomic coordinates until the difference in total energy and the maximum force were within 1.0×10−6 eV and 1.0×10−4 eV/Å, respectively. For a given volume of the unit cell, the lattice parameters are determined in such a way that the total energy becomes a minimum. The data of total energy versus volume were fitted using Murnaghan’s equation of state (eos) [25]. Besides the equilibrium volume, this fitting procedure gives the minimum value of the total energy E, the isothermal bulk modulus B and its pressure derivative B′ at the equilibrium state. 3. Results and discussion 3.1. Structural properties First, we have been optimized the volume of the cell and the ionic positions of atoms. The obtained parameters were used to calculate other physical properties. The calculated atomic positions and lattice parameters are listed in Tables 1 and 2 along with the available works for comparison. The crystal total energy is calculated for different values of lattice constant (at constant b/a and c /a), and fitted in terms of Murnaghan’s eos [25]. (The energy–volume curves are not shown here to save space.) The bulk modulus and its pressure derivative have also been calculated based on the same Murnaghan’s eos and the results are listed in Table 2. It can be seen that the present lattice parameters are in good agreement (around 1%) with experimental ones except for Na3 As and Li3 Bi structures. For these latter structures the lattice constants are overestimated about 0%–4.4% from Refs. [10,15] and 4.8% from Ref. [15], respectively. As a fundamental physical property, the bulk modulus of solids provides valuable information including in the average bond

strengths of atoms for the given crystals [26]. In the present case, the largest value of bulk modulus (50.9 GPa) is obtained for the MgCu2 -type structure, and it implies that this structure is the least compressible one among the others. The cohesive energy is known as a measure of the strength of the forces, which bind atoms together in the solid state. In this context, the cohesive energies of Na–As compounds are calculated in the considered structures. The cohesive energies (Ecoh ) of given phases are defined as Nax Asy

Ecoh

Na Asy

= Etotalx

 Na  As − xEatom + yEatom

(1)

Nax Asy

where Etotal is the total energy (in formul unit) of the compound Na As at equilibrium lattice constant and Eatom and Eatom are the atomic energies of the pure constituents calculated with a large cutoff rmax = 15 Å (free atoms). Also from the total energy of the compound and the constituent elemental solids (BCC W-type (Im3m, #229, A2) for Sodium and rhombohedral α -As-type (R-3m, #166, A7) for the Arsenic element), one can find the formation energy using the relation Nax Asy

Eform

Na Asy

= Etotalx

 Na  As − xEsolid + yEsolid .

(2)

The calculated cohesive and formation energies are listed in Table 2 and they are in good agreement with the values of Refs. [13,14]. These results imply that present compounds possess the negative formation energy and can easily been synthesized in various stable phases. The ranking order of cohesive energy with formation energy is the same. Formation energy and cohesive energy of NaP (LiAs), MgCu2 , Na3 As (Cu3 P), and A5 B4 phases are, relatively, higher than the other type of stoichiometry. It is seen from Table 2 that the NaP and LiAs have the same formation energy (0.795 eV/f.u.) and similarly, the formation energy for Na3 As and Cu3 P phases (1.716 eV/f.u.) are equal. The value of cohesive energy also exhibits the similar trend. The phase transition pressures of Na–As compounds are calculated from the Gibbs free energy at 0 K. The value of phase

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Table 2 The calculated equilibrium lattice parameters (a, b, and c in Å), bulk modules (B in GPa), and its pressure derivatives (B′ ), band gap (Eg in eV), cohesive energy (Ecoh in eV/f.u.), and formation enthalpy (1Hf in eV/f.u.) Compound

Prototype

NaAs

NaP

NaAs

LiAs

NaAs NaAs2 Na3 As

AuCu MgCu2 Na3 As

Na3 As

Li3 Bi

Na3 As

Cu3 P

Na5 As4

A5 B4

a b c d e f

a 6.2882 6.240a 6.2820 6.242b 4.2533 8.0645 5.0867 5.098a 4.874c 7.1639 6.835c 8.8221 8.813a 8.7838d 10.5460

β (°)

b

c

5.9954 5.910a 5.9460 5.849b

10.5879 10.510a 11.6358 11.55b 4.5587

116.8 117.1b

9.0690 9.000a 8.515c

4.6572

9.0678 8.982a 8.9990d 11.5627

111.13

1Hf

B

B′

Eg

34.3

4.033

0.545

−6.755

35.3

4.052

1.013

−6.755

40.0 50.9 21.9

4.207 4.174 4.045

0.108

−6.437 −11.201 −10.290

23.7

4.151

0.134

−10.108

−0.477 −0.586 −1.716 −2.305e −2.259e −2.255f −1.535

22.2

4.140

0.072

−10.290

−1.716

32.0

4.289

−28.783

−3.634

Ecoh

−0.795 −1.032a −0.795

Exp. Ref. [10]. Exp. Ref. [2]. Exp. Ref. [15]. Exp. Ref. [16]. Exp. Ref. [13]. Exp. Ref. [14].

Fig. 1. Estimation of phase transition pressure from Na3 As to Li3 Bi structure for Na3 As compound.

transition pressure is obtained from the same eos data where the phase transition occurs once the enthalpy difference of the two phases becomes zero. It is well known that the interatomic and intermolecular distances of crystals change with the increased hydrostatic pressure. The shape and orientation of structure under a very small displacement of atoms in the crystal becomes distorted. So, the phase transition occurs in reconstruction of the initial structure [27]. In this context the phase transition pressure from NaP and LiAs to AuCu structure is calculated and found to be 6.9 and 8.3 GPa with a 15.8% and 13.7% volume collapse, respectively. Also, the phase transition pressure from Na3 As and Cu3 P to Li3 Bi structure is evaluated to be 3.2 and 3.3 GPa with a 9.0% and 9.4% volume collapse, respectively (see Figs. 1 and 2 similar curves showing the transitions between other structures are obtained, but not shown here due to the page limitation) and it is in agreement with the experimental result of 3.6 GPa in Ref. [15] for transition from Na3 As to Li3 Bi structure. We have also calculated the band structures of Na–As compounds along the high symmetry directions by using the calculated equilibrium lattice parameters. To the best of our knowledge, no experimental and theoretical data seem to exist to compare with our calculated results. It can be seen from Fig. 3 that a 0.108, 0.134, and 0.072 eV direct band gap exists

Fig. 2. Volume versus pressure curves of Na3 As compound in Na3 As and Li3 Bi structures.

at Γ point for Na3 As, Li3 Bi and Cu3 P phases, respectively, and a 0.545 and 1.013 eV indirect band gap exists from a point (about mid-point) between Γ and X to Γ and a point (close to Y ) between Γ and Y to Y direction for NaP and LiAs phases. NaP and LiAs exhibit semiconductor character and AuCu, MgCu2 , and A5 B4 phases exhibit metallic character. The Na3 As, Li3 Bi and Cu3 P phases have a semiconductor character with very narrow bandgap (108, 134, 72 meV, respectively). It is known that the narrow band gap materials are important for mid-infrared optoelectronic applications [28–30]. In this context, these compounds may be viable alternatives for infrared applications. 3.2. Elastic properties The elastic constants of solids provide a link between the mechanical and dynamical behavior of crystals, and give important information concerning the nature of the forces operating in solids. In particular, they provide information on the stability and stiffness of materials, and their ab initio calculation requires precise methods. Since the forces and the elastic constants are functions of the first-order and second-order derivatives of the potentials, their calculation will provide a further check on the

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Fig. 3. The calculated band structures of Na–As compounds along high symmetry directions. Table 3 The calculated elastic constants (Cij in GPa) of Na–As compounds in considered phases. Compound

Prototype

C11

C12

C13

NaAs NaAs NaAs NaAs2 Na3 As Na3 As Na3 As Na5 As4

NaP LiAs AuCu MgCu2 Na3 As Li3 Bi Cu3 P A5 B4

55.2 60.0 82.8 87.2 53.0 30.4 51.7 54.2

26.6 27.3 34.7 34.7 14.3 21.8 13.9 15.2

20.1 21.5 12.3

C15

C22

C23

−8.4

56.2 57.8

27.6 26.0

C25

C33

−1.6

59.0 58.6 101.1

5.9 6.6 17.3

C35

C44 27.2 24.3 20.3 41.6 11.2 28.7 11.9 14.7

2.0

55.6

−6.4

52.9

accuracy of the calculation of forces in solids. The elastic constants are computed at zero temperature and zero pressure by using the ‘‘stress–strain’’ method [31], and the obtained results are listed in Table 3. For the stability of lattice, Born’s stability criteria should be satisfied. The used relations for polycrystalline elastic moduli and the mechanical stability criteria for the considered structures (cubic, hexagonal, tetragonal, orthorhombic, and monoclinic) are taken from Refs. [32,33]. It is known that the Voigt bound is obtained by the average polycrystalline moduli based on an assumption of uniform strain throughout a polycrystalline and is the upper limit of the actual effective moduli while the Reuss bound is obtained by assuming

11.2

1.4

53.9 103.1

12.2

C46

C55

C66

−0.7

20.8 21.8

24.7 27.1 −75.9 19.4

−0.1

18.2

18.9 17.1

a uniform stress and is the lower limit of the actual effective moduli [33]. The bulk modulus (B), shear modulus (G), Young’s modulus (E ), and Poisson’s ratio (ν), which are the most interesting elastic properties for applications, are often measured for polycrystalline materials when investigating their hardness. We use the Hill average [34] to calculate Young’s modulus and Poisson’s ratio as follows (V and R subscript denotes Voigt and Reuss Bound [35,36], respectively): B=

BV + BR 2

and

G=

GV + GR 2

(3)

H.B. Ozisik et al. / Solid State Communications 151 (2011) 1349–1354 Table 4 The calculated linear compressibility values (βa , βb , and βc in TPa−1 ) for Na–As systems along the principles axes (a-, b-, c-axis). Compound

Structure

βa

NaAs

NaP LiAs AuCu MgCu2 Cu3 P Li3 Bi Na3 As A5 B4

10.8 9.9 7.7 6.5 13.7 13.5 13.5 12.5

NaAs2 Na3 As

Na5 As4

E=

9BG 3B + G

and ν =

3B − 2G 2 (3B + G)

βb

The present value of the Poisson’s ratio is close to 0.25, therefore the ionic contributions in inter atomic bonding are dominant for these compounds. Poisson’s ratio also, provides more information dealing with the characteristic of the bonding forces than any of the other elastic constants. 0.25 and 0.5 are the lower and upper limits for central force solids, respectively [40]. Inter-atomic forces are predominantly central forces for some phases (NaP, LiAs, Li3 Bi and, A5 B4 ) than the other ones (MgCu2 , Na3 As and Cu3 P phases). According to the criterion [41,42], a material is brittle (ductility) if the B/G ratio is less (high) than 1.75. Therefore, LiAs, Li3 Bi phases are classified as ductile and the other phases are classified as brittle in nature. Moreover, for covalent and ionic materials the typical relations between bulk and shear moduli are G ≈ 1.1B and G ≈ 0.8B, respectively [39]. Here, the G/B ratio for all compounds strongly supports the ionic contribution to inter atomic bonding. The Debye temperature (ΘD ) is closely related to many physical properties of solids such as the specific heat and melting temperature. At low temperatures the vibrational excitations arise solely from acoustic vibrations. Hence, at low temperatures the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements. One of the standard methods for calculating the Debye temperature is to use elastic constant data since ΘD may be estimated from the average sound velocity (vm ) using the following equation [43]:

βc

8.0 8.5

9.5 9.6 8.0 15.2 15.2 6.2

14.0

.

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(4)

The calculated elastic constants are given in Table 3. Our results show that Na–As compounds are mechanically stable at zero temperature and zero pressure except for the AuCu phase. The linear compressibility β of the material could be calculated with known elastic constants [37]. The calculated linear compressibility’s along the three directions are given in Table 4 for the considered structures. From Table 4, one can see the MgCu2 structure is less compressible than other structures. Na3 As has almost the same linear compressibility values in Cu3 P and Na3 As structures for all three directions. The linear compressibility of NaAs in NaP and LiAs structures are almost same (max. 9% deviation in a-axis) as in the case of energetic behavior. The linear compressibility for the Na5 As4 compound along the a-and b-axes is much higher than that for c-axis. It is known that, the elastic constant C44 is the most important parameter indirectly governing the indentation hardness of a material. The calculated C44 for NaAs2 compound has the highest value. The large C44 means a strong ability of resisting the monoclinic shear distortion in (100) plane. The calculated bulk modulus, isotropic shear modulus, Young’s modulus, and Poisson’s ratio are given in Table 5. It is known that isotropic shear modulus and bulk modulus are a measure of the hardness of a solid. The bulk modulus is a measure of resistance to volume change by applied pressure, whereas the shear modulus is a measure of resistance to reversible deformations upon shear stress [38]. Therefore, isotropic shear modulus is better predictor of hardness than the bulk modulus. The calculated bulk modulus from the eos and elastic constants are almost the same (see Tables 2 and 5). The shear modulus for NaAs2 phase is higher than the other ones. Young’s modulus is defined as the ratio of the tensile stress to the corresponding tensile strain and it is a measure of the stiffness of the solid. The material is stiffer if the value of Young’s modulus is high. Here, the highest value of Young’s modulus (84.8 GPa) is found for MgCu2 -type structure, and it is stiffer than the other phases. According to the Table 5, these compounds are almost in soft character. The value of Poisson’s ratio is indicative of the degree of directionality of covalent bonds. The typical value of Poisson’s ratio is about 0.1 for covalent materials and 0.25 for ionic materials [39].

ΘD =

h¯

[

k

3n



NA ρ

4π

M

] 31

vm

(5)

where h¯ is Plank’s constant, k is Boltzmann’s constant, NA is Avogadro’s number, n is the number of atoms per formula unit, M is the molecular mass per formula unit, ρ is the density and, vm is given as

vm =

[  1

2

3

vt3

+

1

]− 31 (6)

vl3

vl and vt are the longitudinal and transverse elastic wave velocities, respectively, which are obtained from Navier’s equations as follows [44]:

 vl =

3B + 4G 3ρ

 and vt =

G

(7)

ρ

where B is the bulk modulus and G is the shear modulus. The calculated Debye temperatures are listed in Table 5 for studied phases. The lowest value of Debye temperature is that of Li3 Bi phases. The values of Debye temperature are nearly same for the other phases. Generally, the Debye temperature for soft (hard) materials is lower (higher) [45]. These compounds posses low Debye temperature, so again these are soft materials.

Table 5 The calculated bulk modulus (B in GPa), Young’s modulus (E in GPa), Shear modulus (G in GPa), Poisson’s ratio (ν ), Debye temperature (ΘD in K) and sound velocities (vt , vl , vm in m/s) of considered phases of Na–As compounds. Compound

Prototype

B

G

E

v

B/G

ΘD

vl

vt

vm

NaAs NaAs NaAs2 Na3 As Na3 As Na3 As Na5 As4

NaP LiAs MgCu2 Na3 As Li3 Bi Cu3 P A5 B 4

35.3 36.0 52.2 23.7 24.6 23.4 31.8

20.3 20.5 34.5 16.5 13.8 16.5 19.4

51.1 51.6 84.8 40.1 34.8 40.0 48.3

0.2587 0.2610 0.2291 0.2174 0.2636 0.2145 0.2467

1.738 1.759 1.513 1.436 1.782 1.418 1.641

283 283 331 297 268 297 284

4375 4346 4736 4409 4067 4400 4317

2496 2472 2807 2649 2304 2653 2503

2774 2748 3109 2930 2562 2933 2778

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4. Summary and conclusions We have calculated the structural, electronic, and mechanical properties for Na–As systems using the first-principles calculations based on the generalized gradient approximation implemented in VASP. The calculated lattice parameters are in accord with the experimental values. The phase transition from NaP and LiAs structure to AuCu structure and from Na3 As and Cu3 P structure to AuCu structure are found to be 6.9, 8.3, 3.2, 3.3 GPa with 15.8%, 13.7%, 9.0% and 9.4% volume collapse, respectively. NaP, LiAs, Na3 As, Li3 Bi and Cu3 P phases exhibit the semiconductor character with the gap-energy 0.545, 1.013, 0.108, 0.134, and 0.072 eV, respectively. AuCu, MgCu2 , and A5 B4 phases possess a metallic nature. These compounds except for NaAs in the AuCu phase are mechanically stable. The MgCu2 structure is less compressible than the others. Moreover, these compounds have ionic in nature and LiAs, Li3 Bi phases can be classified as ductile and the other phases classified as brittle in nature. Consequently, we conclude that our theoretical predictions on the considered properties of these compounds would serve as a reliable reference for further experimental and theoretical investigations in the future. References [1] W.G. Xu, B. Jin, J. Mol. Struct. Theochem. 759 (2006) 101–107. [2] S. Burtzlaff, M. Holynska, S. Dehnen, Z. Anorg. Allg. Chem. 636 (2010) 1691–1693. [3] M. Tegze, J. Hafner, J. Phys.: Condens. Matter 4 (1992) 2449–2474. [4] G. Derrien, M. Tillard, A. Manteghetti, C. Belin, Z. Anorg. Allg. Chem. 629 (2003) 1601–1609. [5] A.R.H.F. Ettema, R.A. de Groot, J. Phys.: Condens. Matter 11 (1999) 759–766. [6] H. Hirt, H. Deiseroth, Z. Anorg. Allg. Chem. 630 (2004) 1357–1359. [7] J.-Q. Yan, S. Nandi, J.L. Zarestky, W. Tian, A. Kreyssig, B. Jensen, A. Kracher, K.W. Dennis, R.J. McQueeney, A.I. Goldman, R.W. McCallum, T.A. Lograsso, Appl. Phys. Lett. 95 (2009) 222504. [8] Z.G. Chen, R.H. Yuan, T. Dong, N.L. Wang, Phys. Rev. B 81 (2010) 100502. [9] T. Dong, Z.G. Chen, R.H. Yuan, B.F. Hu, B. Cheng, N.L. Wang, Phys. Rev. B 82 (2010) 054522. [10] J. Sangster, A.D. Pelton, J. Phase Equilib. Diff. 14 (1993) 240–242.

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