Structure, Relative Stabilities, Physical Properties, and Hardness of

Article pubs.acs.org/JPCC

Structure, Relative Stabilities, Physical Properties, and Hardness of Osmium Carbides with Various Stoichiometries: First-Principle Investigations Li-Ping Ding,*,† Peng Shao,† Fang-Hui Zhang,† Xiao-Fen Huang,‡ and Tao-Li Yuan† †

College of Science, Shaanxi University of Science & Technology, Xi’an 710021, China Physics Department, Sichuan Normal University, Chengdu 610068, China

Downloaded by TEXAS A&M INTL UNIV on September 8, 2015 | http://pubs.acs.org Publication Date (Web): September 8, 2015 | doi: 10.1021/acs.jpcc.5b06666

‡

ABSTRACT: Using the first-principles calculations, the structural features, mechanical properties, formation enthalpies, electronic properties and hardness of osmium carbides with various stoichometries have been investigated systematically. The structural stability, thermodynamic stability together with mechanical stability show that Re2N−Os2C, OsB2−OsC2, trigonal P3̅m1 OsC2, trigonal P3̅m1 OsC3, orthorhombic Cmmm OsC4 and Ru2Ge3−Os2C3 are the most stable structure for each kind of compounds. But, OsB2−OsC2 and Ru2Ge3−Os2C3 are dynamically unstable based on the calculation of phonon dispersion. The formation enthalpies under high pressure indicate that the Re2N−Os2C, P3̅m1 OsC3, Cmmm OsC4 and Os2Si3−Os2C3 (P4c2) have structural stabilities in the entire range of pressure. While for OsC2, there is a high pressure phase transition exists above 40 GPa. In addition, all the studied osmium carbides exhibit metallic behavior and strong covalent bonding. According to the calculated Vicker hardness based on a semiempirical method, we found that the OsC4 with Cmmm space group has the largest hardness value (28.4 GPa). Combined with its largest shear modulus and Yong’s modulus, smallest Poission’s ratio and low B/G ratio, we predict it is a potential superhard material. By the comparison between the crystal structure of OsC2 and OsC3, it is found that the increased C−C bonds in a cell increase their hardness, whereas the ionicity Os−Os bonds are unfavorable.

1. INTRODUCTION

For osmium element, OsC with WC structure was synthesized at ambient pressure and high temperature in 1960.6 However, from then to now, no other experimental synthesis of osmium carbide has been reported. Until recent years, a large number of theoretical works devoted to the investigation of OsC.7−12 For instance, Zheng et al.7 studied the compressibility of WC- and NaCl-structured OsC by performing the first-principle calculations. The results show that OsC in the WC structure is energetically more favorable than NaCl structure, but the mechanical stability of both structures was absent. In 2007, the study of Chen et al.8 indicated that the WC structure of OsC is mechanically unstable, suggesting that OsC with the WC structure cannot be synthesized. Next year, the same conclusion was obtained by Zemzemi et al.9 Later, Wang and co-workers10 studied the OsC with WC, FeSi, and OsSi structures. They concluded that OsC in the FeSi structure is energetically more stable than that in the WC-structure. Meanwhile, Guo et al.11 performed first-principle calculations on OsC with nine assumed structures. They found that only the FeSi-, NiAs-, and CoSn-structured OsC satisfy the mechanical stability criteria, and the NiAs structure is the most stable. All the above studies are limited to osmium carbides with 1:1

Superhard materials have attracted considerable attention due to their wide range of applications from cutting and polishing tools to wear-resistant coatings. Therefore, attempts to synthesize or theoretically predict new superhard materials with hardness exceeding or comparable to diamond are the subject of intensive current research activities. One common way to synthesize new superhard materials, besides the traditional B−C−N−O systems, is the combination of transition metals (TMs) with small covalent main elements (namely, boron, carbon, nitrogen, or oxygen) to form short strong covalent bond. As is well-known, TMs possess high bulk modulus because of their high valence electron densities, while they are soft in their pure elemental state. For example, pure osmium has one of the highest valence electron densities (0.572/Å3)1 and high bulk modulus in the range of 395−462 GPa2−4 which is very close to 440 GPa5 of diamond. However, its hardness is only of 4 GPa1 which is quite smaller than that of diamond (90 GPa).5 Thus, it is expected that superhard materials might be formed by inserting B, C, N atom into TMs and introducing the nondirectional metallic bonding in pure TMs to highly directional covalent bonding in their corresponding borides, carbides, or nitrides. Such as, tungsten can be combined with carbon to form WC. The hardness (9 GPa) of pure W may increase to 30 GPa of WC.1 © XXXX American Chemical Society

Received: July 11, 2015 Revised: August 31, 2015

A

DOI: 10.1021/acs.jpcc.5b06666 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Table 1. Optimized Equilibrium Lattice Parameters a, b and c (Å); Densities ρ, Cell Volume (V in Å3) per Formula Unit and Formation Enthalpies Ef (GPa) at Zero Pressure of Osmium Carbides with Various Stoichiometries Os2C


OsC2

OsC3

structure

space group

a

Re2N Re2P Ru2Si Rh2Si OsGe2 OsSi2 OsSi2a RuSi2 FeSi2 marcasite OsB2 OsB2a OsN2 OsN2a OsN2b RhN2

P63/mmc Pnma Pnma Pnma C12/m1 Cmca

2.79 5.05 6.56 5.46 7.06 8.13 7.701 2.65 8.22 5.92 5.67 5.548 4.47 4.403 4.41 12.40 2.62 2.619 3.21 12.94 5.58 2.68 2.689 11.97 12.0 5.10 7.16 5.30 9.98 4.13 5.00 5.01 9.51 6.24

LaC2 TcP3 RuP3

OsC4

Os2C3

a

WB4 OsB4 PtSn4 ReP4 FeP4 Ru2Ge3 Ru2Ge3b Os2Si3 Os2Si3

P4/mmm Cmca Pnnm Pmmn Pnnm

Pnnm P3̅m1 P3̅m1c I4/mmm Pnma P1̅ P3̅m1 P3̅m1c Cmmm Cmmmc P63/mmc Pmmn Ccca Pbca C2221 P4̅c2 Pbcn P4̅c2

b 2.79 3.54 3.99 2.78 5.88 6.068 5.77 4.88 2.92 2.789 4.78 4.747 4.87 4.88

2.74 6.70

2.62 2.627 3.46 7.68 8.98 9.47

9.69

c

ρ

Ef

10.16 9.90 6.17 6.45 6.36 9.97 10.117 3.84 10.23 2.88 4.86 4.333 2.93 2.743 2.75 2.92 9.04 9.059 5.57 3.88 5.07 10.57 10.594 2.51 2.519 7.32 5.80 5.31 11.60 4.13 7.91 7.94 7.17 6.52

18.96 18.62 18.18 18.50 12.99 8.40

−2.68 −2.27 −1.58 −2.10 −4.48 −4.60

13.15 11.74 4.75 4.17

−4.79 −4.43 −6.94 −6.91

12.31

−3.21

2.23 13.27

−6.95 −6.47

12.37 10.93 10.10 11.41

−3.01 −8.29 −8.46 −10.31

10.02

−12.86

9.61 9.62 13.17 2.97 9.78 13.97

1.95 −7.65 −4.81 −5.44 −10.96 −6.10

8.38 7.46

−5.04 −8.35

Reference12 bReference13 cReference14

composition. However, investigations for the osmium carbides with high carbon content are scare. Only few theoretical reports appeared in recent years. Such as, Cai et al.12 calculated OsC2 in seven hypothetical structures and the results indicated that OsSi2-structure is the energetically most stable among the considered structures. Du et al.13 studied the OsC, OsC2 and Os2C3 with different structures. They assumed that the structures P4̅c2 and Pbcn Ru2Ge3 for Os2C3, the WN2 and OsB2 structures for OsC2 are elastically stable. Li and coworkers14 predicted two trigonal P3̅m1 and an orthorhombic Cmmm structures for OsC2, OsC3 and OsC4, respectively, which are all energetically stable. But the most stable structures and the mechanism of super hardness for these osmium carbides are still open problem. Thus, studying Os−C with various stoichimetries, especially with high carbon content, is of great significant due to their excellent properties, such as high hardness and melting point, chemical stability, good wear, and oxidation resistance. Having these things in mind, we investigate the structures, relative stabilities, electronic properties, and hardness of the osmium carbides with various stoichimetries based on the known transition metal crystal structures. The structures of OsC have been extensive studied by many research groups, so

here we do not consider it in detail any more. We hope that our theoretical investigation may provide a guidance to better synthesize potential hard or ultraincompressible materials.

2. COMPUTATIONAL DETAILS Structure. As is well-known, the chemically related compounds may have a similar structure; i.e., Wang et al.15 have successfully predicted IrN2 to be IrP2 structure. We here apply this idea to osmium carbides to investigate potential structures. Four possible structures were adopted for Os2C: both experimental hexagonal Re2N (P63/mmc)16 and orthorhombic Re2P (Pnna)17 together with two orthorhombic Ru2Si (Pnma)18 and Rh2Si (Pnma).19 For OsC2, besides the experimental synthesized OsN 2 (Pnnm) 20 and OsB 2 (Pmmn),1 the orthorhombic OsSi2 (Cmca),21 RhN2 (Pnnm)22 and FeSi2 (Cmca),23 tetragonal LaC2 (I4/mmm),24 RuSi2 (P4/ mmm)25 and RhSn2 (I4/mcm),26 monoclinic OsGe2 (C12/ m1),1 as well as two types of FeS2 (marcasite: Pnnm; pyrite: Pa3)̅ 27,28 structures were also considered. As for OsC3 and Os2C3, the potential crystal structures for both OsC3 (TcP3 structure,29 RuP3 structure30) and Os2C3 (Ru2Ge331 and two Os2Si332 moieties with different space groups) were determined. With regard to OsC4, the considered structures are the B


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The Journal of Physical Chemistry C trigonal IrGe4 structure,33 tetragonal PtPb4 structure,34 hexagonal WB4 structure,35 orthorhombic structures (IrSn4,36 OsB4,37 PtSn4,38 ReP4,39 and FeP440) and monoclinic CdP4.41 In addition, since no experimental values, the predicted structures (two trigonal P3̅m1 structures, respectively, for OsC2 and OsC3 and one orthorhombic Cmmm structure for OsC4)14 via the developed particle swarm optimization algorithm approach were also considered as a comparison. Method. The geometry optimization and calculation of mechanical and electronic properties for osmium carbides (Os2C, OsC2, OsC3, OsC4, and Os2C3) were performed by using the CASTEP code42 based on density functional theory (DFT). The Vanderbilit US-PP43 was used with the same cutoff energy 330 eV for Os2C and OsC2, 450 eV for OsC3, 600 eV for OsC4, as well as 650 eV for Os2C3. The k-points in Brillouin zone sampling were degenerated using the Monkhorst-pack scheme.44 The exchange and correlation functional were treated by GGA-PBE.45 Lattice parameters and atomic position were realized by minimizing the forces and stress tensors. For the self-consistent field iterations, the convergence tolerances for geometry optimization were set as the difference in total energy being within 5 × 10−6 eV/atom, the maximum ionic Hellmann−Feynman force within 0.01 eV/Å, the maximum ionic displacement within 5 × 10−4 Å, and the maximum stress tensor within 0.02 GPa. To further verify the mechanical stability of these osmium carbides, elastic constants were calculated by strain−stress method. The elastic constants for a given crystal should satisfy the generalized elastic stability criteria.46 The bulk modulus B, shear modulus G, Young’s modulus Y, and Poisson’s ratio ν have been estimated via the Voight-Reuss-Hill (VRH) approximation.47 Formation enthalpies were calculated by the following formula:

Figure 1. Optimized energetically most stable crystal structures at zero pressure: (a) Re2N−Os2C, (b) RhN2−OsC2, (c) P3̅m1 of OsC2, (d) P3m ̅ 1 of OsC3, (e) Cmmm of OsC4 and (f) Ru2Ge3−Os2C3. The blue and gray balls represent Os and C atoms, respectively. For parts d and e, the different color balls represent the different layers which are composed by carbon atoms.

membered Os−C rings. In these six-membered rings, the interatomic distance of Os−C is 2.144 Å, which is much smaller than the sum (2.83 Å) of the covalent radii of Os atom (1.92 Å) and C atom (0.91 Å), suggesting a strong covalent bonding feature. Along the c axis, a connection between the adjacent 3D skeleton layers is absent due to the large distance (3.67 Å) of Os−C. For OsC2, the RhN2−OsC2 phase is the most thermodynamically stable phase against decomposition into Os + 2C. It has the similar structure with marcasite- and OsB2−OsC2. These three types of structures have the same space group Pnnm, in which each unit cell contains one OsC4 f.u. The calculated formation enthalpy of RhN2−OsC2 is very close to those of marcasite- and OsB2−OsC2 structures, while their relative energy order is RhN2−OsC2 > marcasite−OsC2 > OsB2−OsC2. In the structure of RhN2−OsC2 (Figure 1b), the Os atoms occupy the corners of the primitive orthorhombic cell, and the carbon atoms locate inside the cuboid formed by Os atoms. From the ab plane view, the Os and C atoms form a hyperbola in which the bond length of C−C and Os−C are calculated to be 1.242 and 1.964 Å, respectively. Among the other structures where each unit cell contains the different number of OsC2 f.u., the previously predicted OsC2 with P3m ̅ 1 space group is the most thermodynamically stable structure (Figure 1c). In addition, our calculated lattice parameters and volumes for OsSi2−OsC2, OsB2−OsC2, OsN2−OsC2, and P3m ̅ 1-OsC2 phases are in good agreement with the previous results,12−14 validating our used methodology is credible. Unfortunately, there is no available data for the other crystals to compare with our calculation. We hope that our results can provide powerful guidelines for further experimental and theoretical investigations. According to the calculated total energies, the relative stability order of these three type OsC3 is P3̅m1-OsC3 > RuP3− OsC3 > TcP3−OsC3. This suggests that the previous predicted OsC3 with P3̅m1 symmetry is the most stable structure, and it is also the most thermodynamically stable due to its smallest formation enthalpy (−10.31 eV). In Table 1, it can be seen that our calculated lattice parameters are very close to the previous

Ef = Etotal − (mEOs /4 + nEgra /6)

where Etotal is the total energy per f.u. of various osmium carbides, EOs is the total energy of the four-atoms hcp unit cell of Os metal, Egra is the total energy of the six-atoms unit cell of graphite at zero pressure.

3. RESULTS AND DISCUSSION 3.1. Structural Features. During the structural optimization procedures, PhSn2−OsC2, pyrite−OsC2, IrGe4−OsC4, IrSn4−OsC4, PtPb4−OsC4, and CdP4−OsC4 structures all cannot converge, thus they are ruled out. For the other osmium carbides with different carbon concentration, the calculated lattice parameters, densities and formation enthalpies are listed in Table 1. The optimized lowest energy structures of the studied osmium carbides are shown in Figure 1. From Table 1, one can see that all the formation enthalpies of considered structures for Os2C have negative values. As is wellknown, the negative formation enthalpies indicate that they are thermodynamically stable. The lowest formation enthalpy of Re2N−Os2C suggests that it is the most thermodynamically stable structure at ambient conditions. Moreover, this conclusion has been further confirmed by their total energies. In this structure (Figure 1a), the Os and C atoms respectively occupy the Wyckoff 4f (1/3, 2/3, 0.106) and 2d (1/3, 2/3, 3/ 4) sites, and each unit cell possesses two Os2C f.u. Outwardly, the Re2N−Os2C exhibits a packed three-dimension (3D) skeleton along the ab plane, which mainly composed of sixC


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Figure 2. Calculated formation enthalpy as a function of pressure for (a) Os2C, (b) OsC2, (c) OsC3, (d) OsC4, and (e) Os2C3.

results14 with deviations less than 1%. In the P3̅m1 OsC2 structure, each unit cell contains two OsC2 f.u., and there are four nonequivalent atoms Os, C1, C2 and C3 (for detail information, see Figure 1d). Each Os atom is coordinated by three C3 atoms. In this structure, the calculated Os−C distance is 2.024 Å, and the shortest C−C distance is 1.482 Å, which is shorter than that (1.530 Å) in diamond. With more C atom participated, we found that the OsC4 in Cmmm symmetry is still the most favorable structure among all the considered structures. Each Os atom is bonded to four C1 atoms (Figure 1e). The smallest C−C bonds and Os−C bonds are calculated to be 1.512 and 2.136 Å, respectively, which is in excellent agreement with the previous values (1.514 and 2.124 Å).14 The calculated formation enthalpy of FeP4−OsC4 is second to that of Cmmm-OsC4, illustrating its good structural stability. While for WB4−OsC4, although the hexagonal WB4 structure has been adopted by osmium tetraborides,35 the positive formation enthalpy indicates that it is not available for OsC4 at ambient condition. With regard to the Os2C3 crystal, our predicted Os2Si3-type Os2C3, which belongs to the tetragonal P4̅c2 space group, is more thermodynamically stable than the previous predicted Ru2Ge3-type structure (as shown in Table 1). Despite the fact that these structures have the same shapes. Os2Si3−Os2C3 contains one Os5C16 f.u. in a unit cell (a = 6.24 Å and c = 6.51 Å). A supercell structure of Os2Si3−Os2C3 is shown in

Figure 1f. It is seen that this structure contains two undulated layers, in which the carbon atoms form boatlike structures connected by one Os atom. Leaving osmium atoms are sandwiched between the two layers. These two layers along the bc and ac plane are perpendicular to each other. Within the corrugated layers, the C−C bond lengths are 1.583 and 1.793 Å, and the shortest Os−C bond is 2.205 Å. These short bonds may contribute to its elastic properties and strength. 3.2. Formation Enthalpy. Gaining the thermodynamic stability of all relevant phases is necessary for the application of a compound, especially the phase stability.48,49 Therefore, we have calculated the formation enthalpies of osmium carbides with different carbon content under the pressure up to 100 GPa. The calculated formation enthalpies of Os2C, OsC2, OsC3, OsC4, and Os2C3 with different structures are plotted in Figure 2. From Figure 2a, it can be seen that the Re2N−Os2C is the most stable phase among all the considered structures up to 100 GPa. Above 20 GPa, the stabilities of Re2P−Os2C, Ru2Si− Os2C, and Rh2Si−Os2C phases increase gradually with the pressure increasing, while the stability of Re2N−Os2C is decreased. All these four structures are thermodynamically stable with respect to reactant of 2Os + C in the whole range of pressure. More importantly, we found that the smallest value of the formation enthalpy occur at pressure of 10 GPa, indicating these four structures are more thermodynamically stable at 10 D


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Figure 3. Phonon dispersion for (a) Re2N−Os2C, (b) OsB2−OsC2, (c) P3̅m1 OsC2, (d) P3̅m1 OsC3, (e) Cmmm OsC4, and (f) Ru2Ge3−Os2C3.

Figure 2d shows the formation enthalpies of OsC4 crystal with various different structures. It is found that the orthorhombic Cmmm OsC4 is the most thermodynamically stable structures up to 100 GPa. The phase transitions occur between FeP4−OsC4 and OsB4−OsC4, ReP4−OsC4, and OsB4−OsC4, as well as WB4−OsC4 and PtSn4−OsC4, WB4− OsC4, and ReP4−OsC4 structures. In addition, it is clearly seen that OsB4−OsC4 structure is hardly affected by pressure. The curve of WB4-type OsC4 is very different from those of other structures. It is thermodynamically stable only in the pressure range of 7−12 GPa and 45−63 GPa. As for Os2C3, the considered structures all retained their initial crystal structures after the structural optimizations. Our predicted Os2Si3-type Os2C3 with space group P4c2 has the relative lower formation enthalpy up to 100 GPa compared with the previous predicted Ru2Ge3−Os2C3. Therefore, it is more stable than Ru2Ge3−Os2C3. The Ru2Ge3-type Os2C3 becomes thermodynamically unstable above 40 GPa, indicating that a stable phase under atmospheric pressure may be unstable under high pressure. The stability of Os2Si3-type Os2C3 structure with Pbcn space group is second to the Os2Si3− Os2C3 which belongs to the space group P4c2. Additionally, it is well-known that a stable crystalline structure requires all phonon frequencies to be positive at zero temperature. Thus, we have carefully performed the phonon dispersion calculation within the finite displacement theory for Re2N−Os2C, OsB2−OsC2, P3m ̅ 1 OsC2 and OsC3, and Cmmm OsC4, as well as Ru2Ge3−Os2C3 phases. The calculated results show that the OsB2−OsC2 and Ru2Ge3− Os2C3 are dynamically unstable due to the appearance of imaginary frequencies in the whole Brillouin zone (Figure 3b and 3f, respectively). On the contrary, none of imaginary phonon frequency is detected for Re2N−Os2C and P3̅m1 OsC2 and OsC3, as well as Cmmm OsC4 (Figure 3, parts a, c, d, and e, respectively), indicating that they are dynamical stabilities at ambient condition.

GPa. This may be due to the structure of carbon reactants is affected significantly by the change of pressure between 10 and 20 GPa (e.g., a = 2.93 Å, c = 5.16 Å, V = 38 Å3 at pressure 10 GPa; a = 3.45 Å, c = 3.10 Å, V = 32 Å3 at pressure 20 GPa). Furthermore, the enthalpy−pressure curves of Ru2Si−Os2C and Rh2Si−Os2C phases merge together above 20 GPa, which suggests that they belong to the same structure. For OsC2 (Figure 2b), all the considered carbides are thermodynamically stable with respect to the pure metal osmium and carbon reactants in the whole range of pressure. RhN2−OsC2 is the most stable phase at ambient pressure, but it transforms to the P3̅m1-OsC2 phase when the pressure exceeds 40 GPa. The high pressure phase transitions are also identified between P3̅m1 OsC2 and marcasite−OsC2, P3̅m1 OsC2 and OsB2−OsC2, as well as P3̅m1 OsC2 and RhN2−OsC2 at 40 GPa, above which P3̅m1-OsC2 is energetically more favorable. These transitions indicate that the OsC2 crystal with space group P3m ̅ 1 is a high pressure phase. The RhN2−OsC2 is energetically close to marcasite− and OsB2−OsC2 in the entire range of pressure, which further verify that all of them have the similar structure. Furthermore, a few other high phase transitions are also available at the different pressure point as is indicated by the enthalpy-pressure curve in Figure 2b. With more carbon atoms adding, the formation energy of OsC3 (space group P3̅m1), which is predicted using the developed particle swarm optimization algorithm approach previously, is far lower than those of RuP3−OsC3 and TcP3− OsC3 in the entire range of pressure. This relatively low formation energy suggests that OsC3 with P3̅m1 space group can be obtained easily. In addition, the formation enthalpies of RuP3−OsC3 and TcP3−OsC3 are very close to each other up to 60 GPa. There are three transformation points between RuP3− OsC3 and TcP3−OsC3 structures within the pressure range from 60 to 100 GPa, suggesting that the high pressure phases of both structures are unstable. Namely, both of the structures are affected significantly by the high pressure. E


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Table 2. Calculated Elastic Constants Cij (GPa), Bulk Modulus B (GPa), Shear Modulus G (GPa), the Young Modulus E (GPa), the B/G, Poission’s Ratio ν, and Elastic Anisotropy Index AU for Osmium Carbides with Various Stoichiometries Os2C

OsC2


OsC3

OsC4

Os2C3 a

structure

C11

C12

C13

C22

C23

Re2N Re2P Ru2Si Rh2Si OsSi2 OsSi2a RuSi2 OsB2 OsB2a P3̅m1 P3̅m1b RuP3 P3̅m1 P3̅m1b Cmmm Cmmmb PtSn4 Ru2Ge3 Ru2Ge3c

639 661 297 469 543 564 844 910 918 845 817 340 812 809 866 884 875 662 665

142 238 224 227 258 287 120

329 245 196 314 164 170 124

608 453 441 418 404

190 278 191 152

560 558

115 120

154 161 204 173 178 242 248 119 236 235

163 196 167 148 156 96 90 108 210 213

190

575

119 105 117

665 682 898

C33

C44

579 631 687 529 345 362 872 456 415 753 737 500 839 843 979 1001 869 424 431

111 127 82 176 59 58 198 116 146 136 128 178 232 224 418 418 184 170 176

C55

C66

B

G

B/G

E

ν

139 100 107 87 84

109 166 180 64 55 195 324 340 378 328 93 311 316 241 229 185 91 99

381 360 293 319 263 273 366 310 319 378 386 271 378 382 378 379 370 332 335

152 152 111 134 86 82 253 187 194 232 184 103 285 262 294 280 250 152 184

2.51 2.37 2.15 2.38 3.06

1.63

402 400 296 353 233 233 617 519 484 476

0.32 0.31 0.33 0.32 0.35 0.36 0.20 0.13 0.25 0.29

2.63 1.33

274 683

0.33 0.20

4.00 0.14

1.29

700

0.19

0.53

1.48 2.18

612 396 467

0.22 0.30 0.27

0.64 0.53

120 131

102

210 200 191

1.45 1.03

AU 0.91 0.35 0.99 0.49 0.62 0.48 −0.71 1.05

Reference12 bReference14 cReference13

Young’s modulus E, and Poisson’s ratios ν are tabulated in Table 2. Generally speaking, superhard material possesses high bulk modulus and shear modulus. The high bulk modulus supports the decrease of volume caused by an applied load and the high shear modulus restricts deformation in a direction. It is clearly seen from Table 2 that the calculated bulk modulus of all the predicted osmium carbides are large (above 263 GPa), indicating that these materials are difficult to compress. Namely, these materials have strong ability to resist volume deformation. On the other hand, the large Young’s moduli of these compounds indicate their strong ability of resisting tension and pressure in the range of elastic deformation. Among these materials, Os2C of Re2N structure possesses the largest bulk modulus (381 GPa), revealing that this structure is more difficult to compress than other structures. Meanwhile, it is surprising to find that the structure of Re2N−Os2C is not only the most thermodynamically stable, but also the energetically stable. For OsC2, the RhN2−OsC2 (each unit contains one OsC4 f.u.) is the most thermodynamically and energetically stable structure, while it is mechanically unstable. OsSi2-type OsC2 structure, which is the most energetically stable phase in previous theoretical study,52 is no longer energy favorable in our calculation. Its bulk modulus (263 GPa) and shear modulus (86 GPa) agree well with the previous results (273 and 82 GPa, respectively). The shear modulus, which quantifies the resistance to the shear deformation, is a much better indicator of hardness than bulk modulus. Most of the predicted phases have large shear moduli except for OsSi2−OsC2, marcasite-OsC2 and Ru2Ge3− Os2C3, illustrating their strong ability to resist shear change. Remarkably, Cmmm-OsC4 possesses the largest shear modulus of 294 GPa, which is comparable to that of CrB4 (261 GPa)50 and much larger than that of WB4 (104 GPa).51 It suggests that Cmmm-OsC4 can withstand the shear strain to a largest extent. By comparing P3̅m1 phases OsC2 and OsC3, it is obviously found that the shear modulus of osmium carbides increases with the carbon content increasing. Poisson’s ratio is an important parameter to describe the degree of directionality of

3.3. Elastic Properties. The mechanical stability of a phase has been investigated because it is a necessary condition for the existence of a crystal. The accurate elastic constants are helpful to well understand the mechanical properties and provide useful information to estimate the hardness of a material. In order to check the mechanical stability of osmium carbides, we calculated their elastic constants using the strain−stress method and the results are listed in Table 2. Since the WB4−OsC4 is thermodynamically unstable at ambient condition, thus, we will not discuss it any more. From Table 2, it can be seen that all the elastic constants of the structures satisfy the mechanical stability criteria,46 indicating that they are elastically stable. Moreover, the calculated eigenvalues of the elastic constant matrix for these materials are positive, which further prove that they are elastically stable. The large values of C11, C22, and C33 of these compounds indicate that they are extremely difficult to be compressed along the a-axis, b-axis and c-axis, respectively. The calculated elastic constants for OsSi2−OsC2, OsB2−OsC2, OsC2, and OsC3 with P3̅m1 space group, OsC4 with Cmmm space group, and Ru2Ge3−Os2C3 phases are in excellent agreement with the previous theoretical values.12−14 It is noted that the extreme large values of C11 (910 GPa) for OsB2−OsC2 and C33 (979 GPa) for OsC4 with Cmmm space group are comparable to C11 (1042 GPa) of diamond50 and C33 (1015 GPa) of superhard ReB2,51 suggesting their high linear incompressibility along a- and c-axis, respectively. In addition, it is well-known that the elastic constant C44 is an important parameter indirectly governing the indentation hardness. As can be seen from Table 2, most of the osmium carbides have large C44 values. The OsC4 with the space group Cmmm has the largest values of C44 (418 GPa), which indicates its relatively strong shear strength. On the basis of the calculated elastic constants, the bulk modulus and shear modulus are determined by the VoigetReuss−Hill (VRH) approximation. The Young’s modulus E and Poisson’s ratio ν are calculated by the equations E = (9GB)/(3B + G) and ν = (3B − 3G)/[2(3B + G)], respectively. The obtained bulk modulus B, shear modulus G, F


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Figure 4. Total and partial DOS. The Fermi level is at zero.

carbides with various different structures and listed them in Table 2. Where B and G denote the bulk and shear moduli, the superscripts V and R represent the Voigt and Reuss approximations. It is noted that Ru2Si−Os2C, marcasite− OsC2, RuP3−OsC3, PtSn4−OsC4, and Ru2Ge3−Os2C3 phases are significantly anisotropic in their respective species. The other structures also exhibit some anisotropy to a certain degree. For OsC3 with P3̅m1 space group, it has a superior elastic isotropic character as is indicated by its smaller AU value.37 3.4. Electronic Structure Analysis. To further understand the electronic properties of osmium carbides with various stoichiometries, the total and partial density of states (DOS) of the chosen structures Re2N−Os2C, P3̅m1-OsC2, OsB2−OsC2, P3̅m1-OsC3, Cmmm-OsC4 and Ru2Ge3-type Os2C3 were calculated at zero pressure. The reults are shown in Figure 4. According to Figure 4, it is found that all the calculated osmium carbides here exhibit metallic behavior due to the finite electronic DOS at the Fermi level. From the partial DOS, we can see that the peaks of P3̅m1-OsC2, P3̅m1-OsC3, CmmmOsC4 and Ru2Ge3−Os2C3 are mainly attributed to Os-d and Cp orbitals in the range of −5 to +5 eV. In the case of Re2N− Os2C, the Os-d orbital plays a dominate role with a small contribution of C-p orbital. While for OsB2−OsC2, the C-p orbital plays an important role, which may be related to their relative lower hardness. Moreover, the partial DOS profiles of both Os-d and C-p for these osmium carbides are very similar,

the covalent bonding. The relative directionality of the bonding in a material also has an important effect on its hardness and can be determined by the value of G/B.53 From Table 2, it obviously can be seen that Os2C in Re2P-type structure, OsC2 with OsB2-structured, OsC3 in P3̅m1 space group, OsC4 with Cmmm symmetry, and Os2C3 in Ru2Ge3 structure possess the smallest Poisson’s ratio in respective species, implying their strong degree of covalent bonding. These small Poisson’s ratios contribute to their high hardness. Moreover, these osmium carbides also have a relatively large G/B ratio, indicating their more pronounced directional bonding between the ions. Thus, in principle, they should be the potential superhardness materials. A high (low) B/G value54 corresponds to the ductility (brittleness) of a material, and the critical value is about 1.75. The calculated B/G ratio of RuSi2−OsC2, OsB2− OsC2, PtSn4−OsC4, tetragonal P3̅m1 OsC2, P3̅m1 OsC3 ,and orthorhombic Cmmm OsC4 are smaller than 1.75, indicating their brittle nature, while the other structures (listed in Table 2) are all ductile. By the comparation between OsC2 and OsC3 which have the same space groups P3̅m1, we found that the B/ G ratio gradually decreases with the carbon content increasing, resulting in the weak brittle property. The elastic anisotropy of materials has an important implication in engineering science because it is highly related with the possibility to induce microcracks in materials.53 On the basis of the universal elastic anisotropy index of AU = 5GV/GR + BV/BR − 6,55 we calculated the elastic anisotropy of osmium G


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The Journal of Physical Chemistry C Table 3. Calculated Bond Parameters and Vicker Hardness of the Stable Osmium Carbides crystals Re2N−Os2C P3̅m1-OsC2

P3̅m1-OsC3

Cmmm-OsC4


Ru2Ge3−Os2C3

bond type

dμ

νbμ

P

f m (×10−3)

Hνμ

Hνcalc

Os−C C−C C−C Os−C C−C C−C C−C Os−C C−C C−C Os−C C−C Os−C Os−C Os−C Os−C Os−C Os−C Os−C Os−C

2.144 1.582 1.598 2.158 1.482 1.535 1.633 2.024 1.512 1.609 2.136 2.510 2.062 2.101 2.119 2.168 2.679 2.313 2.325 2.653

8.593 3.306 3.407 8.391 5.643 6.268 7.548 14.371 4.204 5.371 12.587 3.551 1.969 2.083 2.137 2.288 4.318 2.779 2.822 4.193

0.96 2.62 0.45 0.11 0.74 0.59 2.38 1.14 1.71 1.36 0.56 0.04 0.63 0.45 0.57 0.31 0.22 0.11 0.34 0.01

4.741 0 0 2.269 0 0 0 1.027 0 0 1.573 0 1.835 1.835 1.835 1.835 1.835 1.835 1.835 1.835

8.9 249.7 40.8 1.6 28.9 19.4 57.3 8.8 109.2 57.7 5.0 3.4 120.2 78.2 94.9 46.1 11.3 11.8 35.6 0.5

8.9 23.7

μ

indicating the significant hybridization between these two orbitals. This phenomenon reveals a strong covalent interaction between Os and C atoms. Furthermore, the typical feature of these DOS is the appearance of a deep valley, namely, the pseudogap at the Fermi level leads to a separation between the bonding and antibonding states. The presence of a pseudogap will surely increases the stability of compounds and also implies that the strong Os−C covalent bonding exist in these carbides. For example, OsC2 with OsB2 structure (Figure 4b) is more stable than OsC2 with space group P3̅m1 at zero pressure, which may be explained by the position and the value of N(Ef) of DOS at the Fermi level.56 Compared with OsC2 structure of P3̅m1 space group, the value of N(Ef) of the DOS at the Fermi level for OsB2-structured OsC2 is smaller, indicating the stronger stability of the later. 3.5. Hardness. The Mulliken population of the thermodynamic and mechanical stable structures (Re2N−Os2C, OsC2, and OsC3 with space group P3̅m1, OsC4 in space group Cmmm, as well as Ru2Ge3−Os2C3) was calculated with distance cutoff for bond population 3.0 Å. From the population analysis, we found that the bonding is suggested to contain both ionic and covalent contributions. In OsC2, OsC3, OsC4, and Os2C3 structures, C−C bonding plays an important role, while Os−C bonding takes dominant position in Re2N−Os2C. It was postulated that small metallic component has the stronger negative effect on hardness than ionic component.57 Thus, their Vicker hardness was calculated using our group’s hardness formula, which takes the metallicity of bonds into crystal structure. The expression is written as follows Hv(Gpa) = 699Pvb−5/3 exp( −3005fm1.553 )

μ

Hv(Gpa) = [∏ (Hvμ)n ]1/ ∑ n

24.1

28.4

19.8

μ

(2)

where nμ denotes the number of μ-type bonds in the complex compound. Using the eqs 1 and 2, we theoretical predicted the Vicker hardness of the chosen osmium carbides. The calculated bond parameters and hardness are listed in Table 3. It is found that the hardness of carbides increase with the carbon content increasing (8.9 GPa for Re2N−Os2C, 23.7 GPa for P3̅m1-OsC2, 24.1 GPa for P3̅m1-OsC3, 28.4 GPa for Cmmm-OsC4, and 19.8 GPa for Ru2Ge3−Os2C3). In the Re2N−Os2C structure, only four Os−C bonds exist. On the basis of the semiempirical model, our predicted Vicker hardness value of Re2N−Os2C is 8.9 GPa. Although both OsC2 and OsC3 have the same space group P3̅m1 and two Os−C bonds in a cell, the hardness of OsC3 is larger than that of OsC2. Therefore, the increased hardness value of OsC3 is attributed to the increased C−C bonds. In addition, we inferred that the existence of antibonding Os−Os in P3̅m1-OsC2 and P3̅m1-OsC3 (with Mulliken overlap population −0.63 and −0.38, respectively) may result in the hardness values of OsC2 and OsC3 are smaller than that of OsC4. The hardness value of Cmmm-OsC4 is comparable to WB4 (31.8 GPa).59

4. CONCLUSIONS The structural feature, relative stability, electronic structure and hardness of osmium carbides with various stoichiometries have been investigated systematically. All the obtained results are summarized as follows. (1) By investigating the structure stability, formation enthalpy, elastic property and phonon dispersion, we found that Re2N-type Os2C, OsB2-type, and trigonal P3m ̅ 1-OsC3, and orthorhombic ̅ 1-OsC2, trigonal P3m Cmmm-OsC4, as well as Ru2Ge3-type Os2C3, are the ground states for each kind of compounds at zero and high pressures. But, the OsB2−OsC2 and Ru2Ge3−Os2C3 are dynamically unstable based on the phonon dispersion spectra.

(1)

where νb is the volume of bond (νμb = (dμ)3/∑[(dν)3Nνb]), P is Mulliken population, and f m is a factor of metallicity ( f m = (0.026N(EF)/ne)). A detailed description has been presented in our previous work.58 The estimation of the hardness on multicomponent systems which contains at least two types of chemical bonds in their unit cell. The hardness can be expressed as H


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(2) According to the calculated formation enthalpies up to 100 GPa, we found that the Re2N−Os2C, P3̅m1-OsC3, Cmmm-OsC4 and Os2Si3-type Os2C3 (space group P4c2) are the most stable structures with respect to reactants osmium and carbon. However, for the OsC2 crystal, the formation enthalpy of trigonal P3m ̅ 1-OsC2 is far lower than that of OsB2-type above 40 GPa, indicating that P3̅m1-OsC2 is a high pressure stable phase. (3) The analysis of density of states indicates that Re2N− Os2C, OsB2−OsC2, trigonal P3̅m1 OsC2, P3̅m1-OsC3, Cmmm-OsC4, and Ru2Ge3−Os2C3 all exhibit metallic behavior. In addition, the strong covalent C−C bonding and Os−C bonding with partial metallicity bonding exist in these carbides. (4) On the basis of the semiempirical method with allowance for metallic components, the OsC4 is is tested to have the largest Vicker hardness (28.4 GPa) among the studied carbides. Combined with its larger shear modulus, Yong’s modulus, lower Possion’s ratio and smaller B/G ratio, we predicted that it is a potential superhard material. Moreover, it is found that the covalent C−C bonds in a cell will increase carbides hardness, while the ionicity Os−Os bonds are unfavorable.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (L.-P.D.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (No. 11274235) and the Shaanxi University of Science & Technology Key Research Grant (No.BJ15-07).

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REFERENCES

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Structure, Relative Stabilities, Physical Properties, and Hardness of

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