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First-principles study of structural stabilities, elastic and electronic properties of transition metal monocarbides (TMCs) and mononitrides (TMNs) H. Rached a, D. Rached a, S. Benalia a, A.H. Reshak b, c, *, M. Rabah a, R. Khenata d, S. Bin Omran e a

Laboratoire des Matériaux Magnétiques, Faculté des Sciences, Université Djillali Liabès de Sidi Bel-Abbès, Sidi Bel-Abbès 22000, Algeria Institute of Complex Systems, FFPW, CENAKVA, University of South Bohemia in CB, Nove Hrady 37333, Czech Republic Center of Excellence Geopolymer and Green Technology, School of Material Engineering, University Malaysia Perlis, 01007 Kangar, Perlis, Malaysia d Laboratoire de Physique Quantique et de Modélisation Mathématique de la Matière (LPQ3M), université de Mascara, Mascara 29000, Algeria e Department of Physics and Astronomy, Faculty of Science, King Saud University, Riyadh 11451, Saudi Arabia b c

h i g h l i g h t s Structural stabilities, elastic, electronic properties of 5d TMNs XN are investigated. 5d TMCs XC with (X ¼ Ir, Os, Re and Ta) were investigated. The ground state properties for the six considered crystal structure are calculated. The elastic constants of TMNs and TMCs in its different stable phases are determined. The elastic modulus for polycrystalline materials, G, E, and n are calculated.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 February 2013 Received in revised form 24 May 2013 Accepted 16 August 2013

The structural stabilities, elastic and electronic properties of 5d transition metal mononitrides (TMNs) XN with (X ¼ Ir, Os, Re, W and Ta) and 5d transition metal monocarbides (TMCs) XC with (X ¼ Ir, Os, Re and Ta) were investigated using the full-potential linear mufﬁn-tin orbital (FP-LMTO) method, in the framework of the density functional theory (DFT) within the local density approximation (LDA) for the exchange correlation functional. The ground state quantities such as the lattice parameter, bulks modulus and its pressure derivatives for the six considered crystal structures, Rock-salt (B1), CsCl (B2), zinc-blend (B3), Wurtzite (B4), NiAs (B81) and the tungsten carbides (Bh) are calculated. The elastic constants of TMNs and TMCs compounds in its different stable phases are determined by using the total energy variation with strain technique. The elastic modulus for polycrystalline materials, shear modulus (G), Young’s modulus (E), and Poisson’s ratio (n) are calculated. The Debye temperature (qD) and sound velocities (vm) were also derived from the obtained elastic modulus. The analysis of the hardness of the herein studied compounds classiﬁes OsN e (B4 et B81), ReN e (B81), WN e (B81) and OsC e (B81) as superhard materials. Our results for the band structure and densities of states (DOS), show that TMNs and TMCs compounds in theirs energetically and mechanically stable phase has metallic characteristic with strong covalent nature MetaleNonmetal elements. Ó 2013 Elsevier B.V. All rights reserved.

Keywords: A. Carbides A. Nitrides C. Ab initio calculations C. Hardness D. Band-structure D. Elastic properties

1. Introduction

* Corresponding author. Institute of Complex Systems, FFPW, CENAKVA, University of South Bohemia in CB, Nove Hrady 37333, Czech Republic. Tel.: þ420 777 729 583; fax: þ420 386 361 219. E-mail address: [email protected] (A.H. Reshak). 0254-0584/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matchemphys.2013.08.020

The super hard materials are used in a variety of industrial applications; these include abrasives, cutting tools, coatings where wear prevention, and scratch resistance [1,2]. The diamond and cubic boron-nitride (c-BN) are super hard materials; the diamond is not effective for cutting ferrous metals, including steel because of the chemical reaction that produces iron carbide. Concerning the c-

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H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

BN, it does not occur naturally and must be synthesized under conditions of extreme pressure and temperature, making it quite expansive. Hence in the recent years, several studies were made to design the new super hard materials like OsB2 [3], C3N4 polymorphs [4], Si3N4 [5], BC2N [6] and B6O [7]. In this context the transition metal carbides (TMCs) and nitrides (TMNs) have been the most studied and investigated compounds since the beginning of the use of hard coatings to improve the performance of mechanical components; they have recently attracted much attention in the search for new super hard materials [8e13]. In 2005, ﬁrst principles calculations have been performed by Jin-Cheng Zheng [14] to study the compressibility of Os, OsC, and OsN both in their cubic and hexagonal structural forms, and OsO2 in its rutile phase. The study reported by Zang et al. [8] reveals that the hexagonal structure of Os, OsC, and OsN compounds is less compressible than the cubic phase. One year afterwards, J. C. Crowhurst et al. [15] have synthesized platinum nitride (PtN) that was shown to have a large bulk modulus, and they propose a structure that is isostructural with pyrite and has a PtN2 stoichiometry. Also, Patil and co-workers [16] studied the mechanical stability of platinum nitride (PtN), in four different crystal structures, the rock-salt (rs-PtN), zinc-blende (zbPtN), cooperite, and a face-centered orthorhombic phase by using ﬁrst-principles calculations (VASP code). Out of these phases only the rs-PtN phase is found to be stable and has the highest bulk modulus B ¼ 284 GPa. Two years afterwards, D. Aberg and collaborators [17], studied the thermodynamic stabilities of various phases of the nitrides of the platinumemetal elements by using the density functional theory. In the same year, Y. Liang and collaborators [18] performed ﬁrst principles calculations on the structural stability, mechanical and electronic properties of twelve systems for OsB, OsC, OsN in the WC, NaCl, CsCl and ZnS structures. They found that OsB(WC), OsB(CsCl), OsC(WC), and OsC(ZnS) are mechanically stable, and none of them is superhard. In this brief report, we are interested in continuing the way to search for new superhard materials, we have chosen to study the class of transition metal 5d of mononitrides (TMNs) and monocarbides (TMCs) focusing on the structural, mechanical and electronic properties of XN (X ¼ Ir, Os, Re, W and Ta) and XC (X ¼ Ir, Os, Re and Ta) compounds. Our calculations were performed using the full-potential linear mufﬁn-tin orbital (FP-LMTO) method, in the framework of the density functional theory (DFT) within the LDA approximation. The paper is organized as follows; the computational method was given in Section 2. The results for the structural, elastic, hardness and electronic properties were presented and discussed in Section 3, and a brief conclusion was drawn in Section 4.

within the local density approximation (LDA) [26]. The present method is an improved one compared to previous LMTO methods. The FP-LMTO method treats mufﬁn-tin spheres and interstitial regions on the same footing, leading to improvements in the precision of the eigen values. At the same time, the FP-LMTO method, in which the space is divided into an interstitial regions (IR) and nonoverlapping mufﬁn-tin spheres (MTS) surrounding the atomic sites, uses a more complete basis than its predecessors. In the IR regions, the basis functions are represented by Fourier series. Inside the MTS spheres, the basis functions are represented in terms of numerical solutions of the radial Schrödinger equation for the spherical part of the potential multiplied by spherical harmonics. The charge density and the potential are represented inside the MTS by spherical harmonics up to lmax ¼ 6. The k-points integration over the Brillouin zone is performed using the tetrahedron method [27]. The values of the sphere radii (MTS), the number of plane waves (NPLW) and cut-off energy (Ecut in Rydbergs) used in our calculation are listed in Tables 1-1 and 1-2.

3. Results and discussions 3.1. Structural properties Fig. 1 presents a variety of structure types which were taken into account for the study of TMNs and TMCs compounds. We have chosen three cubic structures namely; B1 structure space group Fm3m (N 225), B2 structure space group Pm3m (N 221) and B3 structure space group F43m (N 216); and three hexagonal structures, B81 structure space group P63/mmc (N 194), B4 structure space group P63/mc (N 186) and Bh structure space group P6m2 (N 187). The electronics states for these compounds are as follows:

Table 1-1 Parameters used in the calculations for TMNs compound: number of plane wave (NPLW), cut-off energy (Ecut in Rydbergs) and the mufﬁn-tin radius (MTS in atomic units). Material

Structure

NPLW

Ecut (Ryd)

MTS (u.a) A

B

IrN

B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh

2974 3070 5064 13,464 7630 3440 2974 3070 5064 13,464 7630 3440 2974 3070 5064 13,780 7632 3440 2974 3070 5064 19,002 7330 3440 2974 3070 5064 12,346 7330 3528

118.9319 119.4423 149.7432 282.3341 221.5593 125.0131 118.9319 119.4423 153.1112 282.3341 221.5593 125.0131 119.7211 124.7417 152.4816 170.2696 143.7949 133.1706 111.8310 116.5104 142.2773 218.5002 137.1930 129.4506 117.3081 122.6977 147.7358 169.9378 134.3139 132.0054

2.322 2.509 2.15 1.703 1.844 2.371 2.322 2.554 2.126 1.733 1.844 2.413 2.355 2.531 2.168 2.141 2.378 2.386 2.436 2.619 2.245 1.946 2.388 2.399 2.338 2.552 2.203 2.132 2.414 2.411

1.751 1.971 1.622 1.284 1.412 1.863 1.751 1.926 1.604 1.254 1.412 1.82 1.704 1.853 1.57 1.55 1.722 1.728 1.764 1.917 1.625 1.409 1.729 1.737 1.763 1.868 1.595 1.543 1.748 1.746

OsN

2. Calculation method It is well known that most of the transition-metal monocarbides and mononitrides usually crystallize in the cubic chloride sodium (NaCl-B1) structure (space group Fm3m) [19], whereas tungsten carbide (WC-Bh) is stable in the hexagonal WC-type structure (space group P-6m2) at room temperature [20]. In addition, the cubic Zinc blende (ZB-B3) structure (space group F-43m) and the hexagonal nickel arsenide (NiAs-B81) structure (space group P63mc) have previously been reported as possible stable phases of several transition-metal monocarbides and mononitrides [21]. Therefore, six possible structures e namely; WC-Bh, NiAs-B81, NaCl-B1, CsCl-B2, ZnS-B3 and Wûrtzite-B4 types are chosen for XN with (X¼ Ir, Os, Re, W and Ta) and XC with (X ¼ Ir, Os, Re and Ta). In this work, the full potential linear mufﬁn-tin orbital (FPLMTO) method [22,23] within the density-functional theory (DFT) [24,25] has been employed. Perdew and Wang parameterization scheme has been used for the exchange-correlation (XC) potential

ReN

WN

TaN

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108 Table 1-2 Parameters used in the calculations for TMCs compound: number of plane wave (NPLW), cut-off energy (Ecut in Rydbergs) and the mufﬁn-tin radius (MTS in atomic units). Material

Structure

NPLW

Ecut (Ryd)

MTS (u.a) A

B

IrC

B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh

2974 3070 5064 15,960 8136 3818 2974 3070 5064 13,464 6992 3766 2974 3070 5064 13,804 8522 3440 2974 3070 5064 14,626 7100 3468

116.4294 123.3589 150.0323 187.3407 142.4013 137.2959 118.5969 125.2607 152.1396 180.4779 134.6017 140.1792 117.9772 124.9786 149.0820 171.3771 152.1998 133.1414 114.3868 118.3056 143.9989 186.7565 126.3477 125.4931

2.305 2.425 2.11 2.083 2.319 2.338 2.325 2.494 2.133 2.126 2.332 2.372 2.331 2.496 2.155 2.098 2.323 2.341 2.367 2.566 2.193 2.031 2.418 2.406

1.22 1.984 1.658 1.636 1.837 1.837 1.754 1.881 1.609 1.604 1.759 1.792 1.758 1.883 1.626 1.583 1.753 1.766 1.786 1.936 1.654 1.532 1.824 1.815

OsC

ReC

TaC

95

Ir, Os and Re: [core] 6s,6p,5d; W and Ta: [core] 6s,6p,5d,4f; N and C: [core] 2s,2p [28e30]. First, we present the results of our calculated total energy versus volume for different structures proposed, using the FP-LMTO method within the local density approximation. Fig. 2-1 and 2-2 shows the total energy versus volume of TMNs and TMCs compounds chosen in this study, which are ﬁtted by Murnaghan’s equation of state [31] to obtain their equilibrium structural properties such as equilibrium volume, bulk modulus, and its pressure derivative. One can see from Fig. 2-1 and 2-2 that B3, B81, and Bh phases are energetically more stables for these binary compounds. The structural parameters (equilibrium lattice constants, bulk modulus, and its ﬁrst pressure derivative) at zero pressure and zero temperature are listed in Tables 2-1 and 2-2. Our calculated bulk moduli and lattice constants agree well with the available theoretical data. The normalized lattice constant (a/a0) and (c/c0) (where a0; c0 are the zero-pressure equilibrium lattice constant) as a function of pressure for our compounds are shown in Fig. 3-1 and 3-2. It is seen that when the pressure is enhanced the compression along the (a,c)-axis decreases. To compare the compressibility of our compounds, under pressure for different phases, the volume compressions versus pressure are plotted in Fig. 4-1 and 4-2. From Fig. 3-1, 3-2, 4-1 and 4-2, one can observe that the OsN-B81, ReN-B81, TaC-B1 and IrC-B81 are less compressible. 3.2. Elastic properties In this section we turn our attention to study the mechanical properties of mononitrides XN with (X ¼ Ir, Os, Re, W and Ta) and monocarbides XC with (X ¼ Ir, Os, Re and Ta) and conﬁrm theirs

Fig. 1. The different crystalline structures considered in this work.

Fig. 2. 1. The calculated total energy vs. volume for TMNs in the different structures considered in this work. 2. The calculated total energy vs volume for TMCs in the different structures considered in this work.

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

97

Fig. 2. (continued).

mechanical stabilities in its various structures via calculating the elastic constants Cij. These constants are fundamental and indispensable for describing the mechanical properties of materials because they are closely related to various fundamental solid-state

phenomena such as interatomic bonding, equations of state, and phonon spectra. Elastic properties are also linked thermodynamically with speciﬁc heat, thermal expansion, Debye temperature, and Grüneisen parameter. Most importantly, knowledge of elastic

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H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

Table 2-1 The calculated equilibrium lattice parameters a ( A), axial ratio c/a, bulk modulus B0 (GPa), and its pressure derivative (B00 ) for TMNs in the different structures considered in this work, compared to the available theoretical data. Material

Structure

a0( A)

c/a

B0 (GPa)

B00

IrN

B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh

4.3116(4.328a) 2.7381 4.6111(4.573a) 2.5749 2.4605 2.9040 4.3113(4.287a, 4.335b, 4.32c) 2.6954 4.5595 (4.527a) 3.1078 2.9240 2.7330 (2.743b, 2.753d) 4.3123 (4.274c, 4.297f) 2.6815 (2.664c, 2.679f) 4.5961 (4.546e, 4.569f) 3.0573 (2.75f) 2.7724 (2.747e, 2.759f) 2.8271 (2.747e, 2.748f) 4.3279 (4.446f) 2.6933 (2.772f) 4.6252 (4.73f) 3.5493 (3.535f) 2.8383 (2.913f) 2.8418 (2.922f) 4.3409 (4.326a, 4.395f) 2.7011 (2.731f) 4.6417 (4. 659a, 4.734f) 3.3186 (3.435f) 2.8686 (2.895f) 2.8935 (2.922f)

e e e 1.6865 1.5869 1.024 e e e 2.0274 1.9078 1.130 (1.13b, 1.47b) e e e 2.2058 (2.414f) 2.114 (2.117e,f) 1.019 (1.08e, 1.09f) e e e 1.3307 (1.335f) 2.0231 (2.004f) 1.0196 (0.997f) e e e 1.5588 (1.505f) 2.0247 (2.015f) 0.9897 (0.980f)

381.8(346a) 351.0 314.9(289.3a) 297.7(260.2g) 366.8 334.3 395.8(381.4a, 342b, 372c) 333.9 352.8(327.2a) 280.5 406.2 383.3(367b, 351d) 424.9(406e, 396f) 435.9(412e, 382f) 350.3(340e, 310f) 280.9(349f) 474.7(453e, 418f) 446.2(433e, 398f) 423.8 (319f) 421.2 (306f) 333 (271f) 346.2 (236f) 450.9 (349f) 444.7 (349f) 402.7 (379.6a, 375f) 388.1 (375f) 297.1 (274.2a, 280f) 304.9 (280f) 397.7 (373f) 405 (384f)

4.18 4.8 4.36 4.59 4.49 3.96 4.84(4.15b) 6.61 4.68 4.12 4.67 5.7(5.08b, 4.79d) 4.78(4.729e) 4.59(4.542e) 4.45(4.241e) 4.40 4.43(4.203e) 4.51(4.319e) 4.05 4.69 4.61 3.84 4.64 4.71 4.87 4.49 4.24 4.88 4.12 4.18

OsN

ReN

WN

TaN

a b c d e f g

Ref. Ref. Ref. Ref. Ref. Ref. Ref.

[32]. [14]. [33]. [18]. [34]. [35]. [65].

constants is essential for many practical applications related to the mechanical properties of a solid: load deﬂection, thermoelastic stress, internal strain, sound velocities, and fracture toughness. The elastic constants Cij are the proportionality coefﬁcients relating the applied strain (3 i) to the stress (si), si ¼ Cij3 i. So, the Cij determine the response of the crystal to external forces. There are 21 independent elastic constants Cij, but the symmetry of cubic and hexagonal crystal reduce this number to only three (C11, C12, and C44) and ﬁve (C11, C12, C13, C33, and C44) independent elastic constants, respectively. The elastic constants Cij, are obtained by calculating the total energy as a function of volume conserving strains that break the symmetry. Further details of the calculation can be found elsewhere [39,40]. The mechanically stable phases or macroscopic stability depends on the positive deﬁniteness of stiffness matrix [41]. The following conditions are known as the Born Huang criteria [42]. These criteria are deﬁned as C11 > 0, C44 > 0, C11 > C12, (C11 þ 2C12) > 0 for the cubic structure, and as C44 > 0, C11 > C12, (C11 þ 2C12)C33 > 2C213 for the hexagonal structure. The elastic constants Cij are estimated from ﬁrst-principles calculations for our compounds single crystal. However, the prepared materials are in general polycrystalline, and therefore it is important to evaluate the corresponding moduli for the polycrystalline species. For this purpose, we have applied the VoigteReusseHill [43] approximations. In this approach, the actual effective modulus for polycrystals could be approximated by the arithmetic mean of the two well-known bounds for monocrystals according to Voigt [44] and Reuss [45]. In this way the above moduli for the cubic structure are deﬁned as

G ¼

GV þ GR 2

(1)

C11 þ 2C12 3

(2)

GV ¼

C11 C12 þ 3C44 5

(3)

GR ¼

5ðC11 C12 ÞC44 ½4C44 þ 3ðC11 C12 Þ

(4)

BV ¼ BR ¼

while, for the hexagonal structure they are deﬁned as

BV ¼

1 ½2ðC11 þ C12 Þ þ 4C13 þ C33 9

(5)

GV ¼

1 ðM þ 12 C44 þ 12 C66 Þ 30

(6)

BR ¼

C2 M

(7)

GR ¼

5 C 2 C44 C66 2 2 3BV C44 C66 þ C ðC44 þ C66 Þ

(8)

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

99

Table 2-2 0 The calculated equilibrium lattice parameters a ( A), axial ratio c/a, bulk modulus B0 (GPa), and its pressure derivative (B0) for TMCs in the different structures considered in this work, compared to the available theoretical data. Material

Structure

A) a0 (

c/a

B0 (GPa))

B0

IrC

B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh

4.3572 2.6939 4.6061 3.1044 3.0687 2.8372 (3.012a, 3.024a) 4.3172 (4.333c, 4.33d) 2.6734 4.5741 (4.63e) 3.2202 2.8655 2.7199 (2.955e, 2.928c) 4.3286 2.6764 4.6207 3.0474 2.9683 (2.853f) 2.8191 (2.85f) 4.396 (4.525g, 4.406g) 2.7508 4.7016 3.4058 2.9576 2.9677

e e e 2.1125 1.6812 1.045(1.14a, 1.139a) e e e 1.6891 1.9504 1.1356 (0.926e, 0.928c) e e e 2.213 1.7657 (1.96f) 1.022 (0.977f) e e e 1.4723 1.9704 0.9658

376.1 397.1 325.3 259.5 384.6 386.8 (373b) 418.1 (366c, 392d) 413.5 339 (292e) 322.7 442.7 439.9 (438b) 439.7 363.4 331.5 290.5 468.4 (422f) 482.7 (444f, 464b) 380.3 (365g, 318g) 335.9 267.5 278.1 353.7 347.4

4.82 5.19 4.82 4.72 4.59 5.10 4.62 (3.91c) 4.49 4.56 (4.36e) 4.41 4.83 4.82 4.50 4.59 4.29 4.33 4.43 4.49 4.19 4.23 4.12 4.44 4.06 4.07

OsC

ReC

TaC

a b c d e f g

Ref. Ref. Ref. Ref. Ref. Ref. Ref.

[36]. [37]. [14]. [33]. [18]. [21]. [38].

Fig. 3. (a): Lattice parameters vs. pressure of TMNs in the more stable phase. (b): Lattice parameters vs. pressure of TMCs in the more stable phase.

0

100

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

Fig. 4. 1: Volume compressions as a function of pressure for TMCs in their various structures. 2: Volume compressions as a function of pressure for TMNs in their various structures.

Fig. 4. (continued).

102

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

Table 3-1 The calculated single crystal elastic constants Cij (in GPa) and polycrystalline elastic modulus (shear modulus G (in GPa), Young’s modulus E (in GPa), and Poisson’s ratio n) for TMNs in various structures. Material Structure C11

C12

C13

C33

C44

G

E

v

IrN

e 163(168.4b) 295.4(271a, 248.6b) 275.8(256a, 245b)

e e e

e e e

e 68,3 22.9(31a)

e 192.4 67.3(91a)

e 0.408 0.46(0.44a)

215.5(158a, 172.6b) e e e e e 160.7(144a) 317.1 e e e e 62 216.9(245a) 270.1 e e e e 243.4(218a) 285.3(214a) e e e e 173.5(172a) 208.6(179a)

791.2(669a, 641.4b) e e e e e 903.3(788a) 633.6 e e e e 212.8 814.9(852a) 844.8 e e e e 865.8(720a) 786.5(712a) e e e e 908.5(890a) 945.7(911a)

e 11.8(-7.5b) 19.6(43a, 32.9b) 95.9(88a, 71.6b) e e e e e 151.3(139a) 30.7 e e 141.8(128a) e 86.1 217.8(216a) 32.6 e e e e 317.6(238a) 132.1(85a) 61.7(74a) e e e 332.2(331a) 312.8(287a)

72.6(60a, 59.3b) e e e e e 231.6(204a) 105.1 e e 249.1(204a) e 58.6 244.1(238a) 160.8 e e e e 287.67(211a) 199.84(148a) 188.9(153a) e e e 168.2(167a) 294.6(261a)

201.6(167a, 165.5b) e e e e e 544.9(507a) 290.2 e e 627.87(519a) e 164.4 617.3(599a) 430.8 e e e e 637(526a) 512.4(390a) 490.2(404a) e e e 442.2(437a) 650.6(638a)

0.38(0.39a, 0.394b) e e e e e 0.23(0.24a) 0.24 e e 0.26(0.27a) e 0.31 0.28(0.26a) 0.28 e e e e 0.25(0.25a) 0.3(0.31a) 0.297(0.32a) e e e 0.189 0.2(0.22a)

B1 B2 B3

Unstable 714.7(595.8b) 353.4(311a, 292.8b) 331.9(292a, 293.2b) Unstable Unstable Unstable Unstable Unstable 697.4(651a) 587 Unstable Unstable 911.9(900a) Unstable 87.2 705.1(727a) 720.5 Unstable Unstable Unstable Unstable 743.7(576a) 719.9(610a) 886.8(881a) Unstable Unstable Unstable 513.4(508a) 759.4(672a)

B4

OsN

ReN

WN

TaN

a b

B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh

e e e e e 182.2(162a) 264.8 e e 91.5(122a) e 12.65 197.2(247a) 243.7 e e e e 229(217a) 212.3(189a) 127.2(122a) e e e 400(394a) 257.4(256a)

Ref. [35]. Ref. [65].

Table 3-2 The calculated single crystal elastic constants Cij (in GPa) and polycrystalline elastic modulus (shear modulus G (in GPa), Young’s modulus E (in GPa), and Poisson’s ratio n) for TMCs in various structures. Matériaux

Structure

C11

C12

C13

C33

C44

G

E

v

IrC

B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh B1 B2 B3 B4 B81 Bh

Unstable Unstable 284.87 Unstable 470.88 512.74(487a) Unstable Unstable Unstable Unstable 541.29 Unstable Unstable Unstable Unstable Unstable 778.85(726b) 825.68(796b) 888.5(621c) Unstable Unstable Unstable 518.72 554.48

e e 279.98 e 350.15 345.71(323a) e e e e 290.19 e e e e e 231.97(229b) 253.61(242b) 96.4(155.3c) e e e 123.48 182.36

e e e e 206.76 240.41(234a) e e e e 295.16 e e e e e 231.42(225b) 196.99(192b) e e e e 269.40 177.22

e e e e 992.79 862.13(852a) e e e e 808.83 e e e e e 1004.41(993b) 1126.20(1123b) e e e e 672.67 815.86

e e 70.79 e 148.49 84.21(79a) e e e e 180.30 e e e e e 290.91(284b) 196.56(195b) 200.8(166.8c) e e e 175.06 102.37

e e 43.45 e 149.53 121.12(115a) e e e e 164.62 e e e e e 295.53(274b) 277.83(223b) 278.9 e e e 179.40 170.69

e e 124.79 e 397.12 329.01 e e e e 439.39 e e e e e 732.53 699.31 672.4(550c) e e e 460.37 440

e e 0.436 e 0.328 0.20(0.36a) e e e e 0.335 e e e e e 0.24 0.26 0.205(0.21c) e e e 0.283 0.29

OsC

ReC

TaC

a b c

Ref. [37]. Ref. [21]. Ref. [38].

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108 Table 4-1 The calculated longitudinal, transverse, and average sound velocity (vl, vt, and vm, in m s1) and Debye temperature (qD, in K) for TMNs compound in their more stable phase. (T ¼ 0) (P ¼ 0)

Structure

VS (m s1)

V1 (m s1)

Vm (m s1)

qD (K)

IrN OsN ReN WN TaN

B3 B81 B81 B81 Bh

2579.081 1849.64 5337.86 5902.54 4327.35

9924.044 4634.75 9677.84 9025.89 7136.60

2943.725 2095.34 5948.43 6468.28 4782.11

189.3 159.41 256.26 271.66 234.15

Table 5-1 The calculated hardness HV in (GPa) for TMNs in their mechanically stable phase. Structure (AB) IrN OsN

B3 B4 B81 B2 B81 Bh B81 Bh B1 B81 Bh

ReN

WN TaN

M ¼ C11 þ C12 þ 2C33 4C13

(9)

2 C 2 ¼ ðC11 þ C12 ÞC33 2C13

(10)

We have calculated Young’s modulus, E, and Poisson’s ratio, which are frequently measured for polycrystalline materials when investigating their hardness. These quantities are related to the bulk modulus B and the shear modulus G by the following equation:

E ¼

9BG 3B þ G

(11)

Table 4-2 The calculated longitudinal, transverse, and average sound velocity (vl, vt, and vm, in m s1) and Debye temperature (qD, in K) for TMCs compound in their more stable phase. (T ¼ 0) (P ¼ 0)

Structure

VS (m s1)

Vl (m s1)

Vm (m s1)

qD (K)

IrC OsC ReC TaC

B3 B81 Bh B1

1769.02 4359.60 4121.84 4298.3

5254.02 8815.52 7307.12 7058.84

2012.30 4893.77 4585.07 4747.99

205.24 230.13 210.18 507.48

HV (in GPa) 40.96 44.39 46.41 21.63 45.78 16.74 40.03 14.76 20.22 29.53 11.05

1 h 3n 3 vm kB 4pVa

(12)

We listed in Tables 3-1 and 3-2 the elastic constants and elastic moduli for polycrystalline materials calculated within the LDA approximation. We note that we have chooses only the values which assure the criteria of mechanical stability. We remark from these tables and according to the criteria of stability that the B81 and Bh phases are mechanically stable in the ambient conditions for our compounds except for IrN that is stable in B2, B3 and B4 phases. From the results we observe that C11 and C33 elastic constants of ours compounds in theirs hexagonal structure are signiﬁcantly larger than other elastic constants, resulting in pronounced elastic anisotropy in these structure. C33 elastic constants always remain much larger than C11, indicating that the a-axis is more compressible than the c-axis. The C11 values, which is related to the unidirectional compression along the principal crystallographic direction, is much greater than that of C44, indicating the weaken resistance to shear deformation compared to the resistance to the unidirectional compression. We note that these compounds in theirs mechanically stable phases possess the higher bulk modulus and young’s modulus, suggesting a strong incompressibility for these compounds. The typical value of Poisson’s ratio (n) for ionic materials is 0.25 [46]. The calculated values of the Poisson’s ratio for our compounds are larger than 0.25 for the majority structure, indicating a considerable ionic contribution in inter-atomic bounding except for IrC(Bh structure), ReC(B1 structure) and TaN(B81 and Bh structures) have covalent contribution. A high B/G ratio may then be associated with ductility whereas a low value would correspond to a more brittle nature. The critical value, which separates ductile and brittle materials is around 1.75, if B/G > 1.75,

d (A-B) (in A) 1.997 1.79 2.189 2.322 2.169 2.176 2.178 2.188 2.17 2.202 2.200

the materials behaves in a ductile manner, otherwise the materials behaves a brittle manner. In our compounds the B/G ratio are higher than 1.75 for the majority of structure, indicating the ductility of these compounds. Once we have calculated the Young’s modulus E, bulk modulus B and the shear modulus G, we may estimate the Debye temperature from the average sound velocity vm [47]:

qD ¼

3B 2G v ¼ 2ð3B þ GÞ

103

(13)

where h is Plank’s constant, kB Boltzmann’s constant and Va the atomic volume. The average sound velocity in the polycrystalline material is given by [47]:

" vm ¼

1 2 1 þ 3 v3t v3 l

!#1 3

(14)

where nl and nt are the longitudinal and transverse sound velocity obtained using the shear modulus G and the bulk modulus B from Navier’s equation [48]:

vl ¼

3B þ 4G 3r

1 2

and

vt ¼

1 G 2

(15)

r

The calculated sound velocity and Debye temperature for TMNs and TMCs compounds in its more stable phase are given in Tables 4-1 and 4-2. Unfortunately, as far as we know, there are no data available related to these properties in the literature for these compounds. Future experimental work will testify our calculated results. 3.3. Hardness Hardness is a measure of a material’s resistance to being scratched or dented, a measure to resist penetration, deformation, abrasion, and wear. These properties, important in a variety of industrial applications, drive the contemporary effort focused on the synthesis and characterization of hard or superhard materials Table 5-2 The calculated hardness HV in (GPa) for TMCs in their mechanically stable phase. Structure (AB) IrC

OsC ReC TaC

B3 B81 Bh B81 B81 Bh B1 B81 Bh

d (A-B) (in A)

HV (in GPa)

1.994 2.191 2.209 2.165 2.157 2.173 2.198 2.244 2.233

35.38 37.40 14.16 40.02 38.89 14.59 19.20 27.13 10.32

Fig. 5. 1: Total density of states (TDOS) and partial (PDOS) of TMNs in their more stable phase. 2: Total density of states (TDOS) and partial (PDOS) of TMCs in their more stable phase.

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

105

Fig. 5. (continued).

[49e58]. We have calculated the hardness of the investigated TMNs and TMCs binary compounds in the mechanically stable phases using the empirical formula [59e62]. Following these formulas, an alternate scheme for hardness prediction has been proposed by Simunek and Vackar [59]. Instead of relating the resistance to the bond energy gap, the resistance is assumed to be proportional to the bond strength Sij between atoms i and j as:

Sij ¼

pﬃﬃﬃﬃﬃﬃﬃﬃ ei ej dij nij

(16)

where ei ¼ Zi =Ri is a reference energy, Zi is the valence electron number of atom i, and nij is the number of bonds between atom i and its neighboring atoms j at the nearest neighbor distance dij. The radius Ri for each atom in a crystal is determined such that a sphere around an atom with radius Ri contains exactly Zi valence electrons. For a binary compound with two different atoms i and j, hardness is expressed as:

H ¼

pﬃﬃﬃﬃﬃﬃﬃﬃ ei ej sf C e e U dij nij

(17)

where the exponential factor phenomenologically describes the difference between ei and ej. For a multicomponent system, hardness can be calculated as:

0 11 n n C @ Y n H ¼ Nij Sij A esfe

U

(18)

i;j ¼ 1

2 k

k Y

6 6 fe ¼ 1 6 i ¼ 1 4 Pk

!1k ei

i ¼ 1 ei

32

7 7 7 5

(19)

Fig. 5. (continued).

H. Rached et al. / Materials Chemistry and Physics 143 (2013) 93e108

where Nij is the multiplicity of the binary system ij, and k is the number of different atoms in the system. To address some issues rooted in the above model (e.g., different coordination number for constituent atoms) [60], and “to estimate hardness of crystals on a pocket calculator”, a generalization has been proposed [61,52]. Bond strength is redeﬁned as:

Sij ¼

pﬃﬃﬃﬃﬃﬃﬃﬃ ei ej ni nj dij

(20)

where ni and nj are the coordination number of atom i and j, respectively. To use sij in calculations, a number bij for counting individual bonds of the ij-type in the unit cell is introduced. Subsequently, all that remain to be performed in Eq. (18) is the replacement of Sij and Nij with sij and bij, respectively. The radius Ri, which is determined from the ﬁrst-principles calculation in the original formula, is also replaced by the atomic radius. Constants C and s are chosen to be 1450 and 2.8 respectively for the hardness calculation. Consequently, no constant determined by ab initio methods are required for the estimation of hardness [61]. Given that constants C and s are determined from experimental data, this model is also a semi-empirical one. This bond strength model works well for the hardness estimation of covalent, polar covalent and ionic crystals. The calculated Hardness (HV) and the inter-atomic distance (d12) for TMNs and TMCs binary compounds in its mechanically stable phase are given in Tables 5-1 and 5-2. The OsN e (B4 et B81), ReN e (B81) WN e (B81) and OsC e (B81) are classiﬁed as superhard compounds since theirs hardness exceeds the value of 40 GPa which deﬁne superhard materials [63]. 3.4. Electronic properties In the following paragraph we shed more light on the electronic properties of TMNs and TMCs binary compounds in theirs energetically and mechanically stable phase via calculating the energy band structure and density of states. Given that there’s no energy gap near the Fermi level, the calculated band structure in equilibrium volume for TMNs and TMCs compounds within LDA show metallic character in their stable phase. We have presented the calculated the total and atomic site-projected l-decomposed densities of states (TDOS and PDOS) for these compounds in order to further elucidate the nature of the electronic band structure. These are displayed in Fig. 5-1 and 5-2. It is clear that the bonding nature of these compounds is metallic since the DOS has a large ﬁnite value at the Fermi level N(EF). From the partial PDOS we are able to identify the angular momentum character of the different structures. The DOS shows three main regions, the energetically lower region expanded between 20 and 10 eV is due mainly to the s-N for TMNs and s-C for TMCs states. This region is followed by a second region expanded up to the Fermi level arises mainly from a mixture of p-N for TMNs and p-C for TMCs states and d states of transition metals. The third region which represents the high of conduction band contains a strong contribution from p states of transition metals and p-N for TMNs and p-C for TMCs states with low contribution from d states of transition metals. The strong hybridization between d states of transition metals and p-N for TMNs and p-C for TMCs states, indicates the strong covalent bonding between transition metals X and Nitrogen for TMNs binary and carbon for TMCs binary. This covalent nature of XeN and XeC bond is mainly responsible for the high value of the bulk modulus and the hardness of TMNs and TMCs compounds [32,64,65].

107

4. Conclusion In this study, we have investigated the structural stabilities, elastic and electronic properties of 5d transition metal mononitrides (TMNs) XN with (X ¼ Ir, Os, Re, W and Ta) and 5d transition metal monocarbides (TMCs) XC with (X ¼ Ir, Os, Re and Ta) by using the FP-LMTO method within LDA approximation. The ground state properties, including lattice parameter, bulk modulus, and its pressure derivatives for the six considered crystal structures (B1, B2, B3, B81, Bh, and B4) are determined. The elastic constants were obtained using the volume conserving strain technique. Shear modulus, Young’s modulus, Poisson’s ratio, sound velocities, and Debye temperature were also derived from the obtained elastic moduli. Analysis of the hardness classiﬁed OsN e (B4 et B81), ReN e (B81), WN e (B81) and OsC e (B81) as superhard materials. Our results for the energy band structure and DOS, show that TMNs and TMCs compounds has metallic characteristic. Acknowledgements For author D. Rached this work has been supported by the Algerian National Project PNR (Céramiques Piézoélectriques Propriétés et Applications). For the author A.H. Reshak this work was supported from the institutional research concept of the project CENAKVA (No. CZ.1.05/2.1.00/01.0024), School of Material Engineering, Malaysia University of Perlis, Malaysia. Authors (R. Kh, & S. B. O) extend their appreciation to the Deanship of Scientiﬁc Research at King Saud University for funding the work through the research group project No. RGP-VPP-088. References [1] V.L. Solozhenko, D. Andrault, G. Fiquet, M. Mezouar, D.C. Rubie, Appl. Phys. Lett. 78 (2001) 1385. [2] A.G. Thornton, J. Wilks, Nature 274 (1978) 792. [3] M. Hebbache, L. Stuparevic, D. Zivkovic, Solid State Commun. 139 (2006) 227. [4] T.S. Wang, D.L. Yu, Y.J. Tian, F.R. Xiao, J.L. He, D.C. Li, W.K. Wang, L. Li, Chem. Phys. Lett. 334 (2001) 7. [5] J. He, L. Guo, D. Yu, R. Liu, Y. Tiana, H.T. Wang, Appl. Phys. Lett. 85 (2004) 5571. [6] H. Sun, S.-H. Jhi, D. Roundy, M.L. Cohen, S.G. Louie, Phys. Rev. B 64 (2001) 094108. Y. Zhang, H. Sun, C. Chen, Phys. Rev. Lett. 93 (2004) 195504; Y. Zhang, H. Sun, C. Chen, ibid. 94, (2005) 145505. [7] D.W. He, Y.S. Zhao, L. Daemen, J. Qian, T.D. Shen, T.W. Zerda, Appl. Phys. Lett. 82 (2002) 643. [8] C. Zang, H. Sun, J.S. Tse, C. Chen, Phys. Rev. B 86 (2012) 014108. [9] Z. Li, F. Gao, Z. Xu, Phys. Rev. B 85 (2012) 144115. [10] Xing-Qiu Chen, H. Niu, C. Franchini, D. Li, Y. Li, Phys. Rev. B 84 (2011) 121405. [11] R.F. Zhang, Z.J. Lin, Ho-Kwang Mao, Y. Zhao, Phys. Rev. B 83 (2011) 060101. [12] F. Tian, J. Wang, Z. He, Y. Ma, L. Wang, T. Cui, C. Chen, B. Liu, G. Zou, Phys. Rev. B 78 (2008) 235431. [13] D. Errandonea, J. Ruiz-Fuertes, J.A. Sans, D. Santamaría-Perez, O. Gomis, A. Gómez, F. Sapiña, Phys. Rev. B 85 (2012) 144103. [14] Jin-Cheng Zheng, Phys. Rev. B 72 (2005) 052105. [15] J.C. Crowhurst, A.F. Goncharov, B. Sadigh, C.L. Evans, P.G. Morrall, J.L. Ferreira, A.J. Nelson, Science 311 (2006) 1275. [16] S.K.R. Patil, S.V. Khare, B.R. Tuttle, J.K. Bording, S. Kodambaka, Phys. Rev. B 73 (2006) 104118. [17] D. Aberg, B. Sadigh, J. Crowhurst, A.F. Goncharov, Phys. Rev. Lett. 100 (2008) 095501. [18] Y. Liang, J. Zhao, B. Zhang, Solid State Commun. 146 (2008) 450. [19] L.E. Toth, Transition Metal Carbides and Nitrides, Academic, New York, 1971. [20] A.Y. Liu, R.M. Wentzcovitch, M.L. Cohen, Phys. Rev. B 38 (1988) 9483. [21] Z. Chen, M. Gu, C.Q. Sun, X. Zhang, R. Liu, Appl. Phys. Lett. 91 (2007) 061905. [22] S. Savrasov, D. Savrasov, Phys. Rev. B 46 (1992) 12181. [23] S.Y. Savrasov, Phys. Rev. B 54 (1996) 16470. [24] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [25] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [26] J.P. Perdew, Y. Wang, Phys. Rev. B 46 (1992) 12947. [27] P. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. [28] A. Chaudhari, S.L. Lee, Int. J. Quantum Chem. 107 (2007) 212. [29] B. Kharat, S.B. Deshukh, A. Chaudhari, Int. J. Quantum Chem. 109 (2009) 1103. [30] V. Kalamse, S. Gaikwad, A. Chaudhari, Bull. Mater. Sci. 33 (2010) 233. [31] F.D. Murnaghan, Proc. Natl. Acad. Sci. U. S. A. 30 (1947) 244; J.R. Macdonald, D.R. Powell, J. Res. Natl. Bur. Stand. A 75 (1971) 441.

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