Structure, Elastic Stiffness, and Hardness of Os1âxRuxB2 Solid

Article pubs.acs.org/JPCC

Structure, Elastic Stiffness, and Hardness of Os1−xRuxB2 Solid Solution Transition-Metal Diborides Mohammed Benali Kanoun,*,† Patrick Hermet,‡ and Souraya Goumri-Said*,† †

PSE Division, KAUST, Thuwal 23955-6900, Saudi Arabia Laboratoire Charles Coulomb (UMR CNRS 5221), Université Montpellier 2, 34095 Montpellier Cédex 5, France

‡

ABSTRACT: On the basis of recent experiments, the solid solution transitionmetal diborides were proposed to be new ultra-incompressible hard materials. We investigate using density functional theory based methods the structural and mechanical properties, electronic structure, and hardness of Os1−xRuxB2 solid solutions. A difference in chemical bonding occurs between OsB2 and RuB2 diborides, leading to significantly different elastic properties: a large bulk, shear moduli, and hardness for Os-rich diborides and relatively small bulk, shear moduli, and hardness for Ru-rich diborides. The electronic structure and bonding characterization are also analyzed as a function of Ru−dopant concentration in the OsB2 lattice.

I. INTRODUCTION Research on superhard materials is motivated by the technological importance to find new robust and chemically stable materials for, e.g., cutting tools and wear-resistant coatings. Although diamond and cubic boron nitride are well known to be intrinsically superhard materials, their technological applicability is limited.1 Introduction of light and covalent-bond-forming elements into the transition-metal (TM) lattices is expected to have a significant influence on their chemical, mechanical, and electronic properties. In this context, recent design of new intrinsically superhard materials has been concentrated on light-element (e.g., B, C, N, and O) TM compounds with high elastic moduli,1−8 such as OsB27 and RuB2.8 Although these ultra-incompressible compounds are expected to be intrinsically superhard,7,8 their measured loadinvariant hardnesses are below 30 GPa.1 These findings can be attributed to different physical origins of elastic moduli and hardness. In particular, elastic moduli describe the reversible elastic deformation at small strains close to equilibrium, whereas plastic deformation (a process of hardness tests) occurs in an irreversible manner at large shear strains where the electronic structure may undergo instability and transform to some phases with lower shear resistance. For example, OsB2 has high zero-pressure elastic moduli but a low hardness which is attributed to the presence of soft metallic Os−Os layers and hence its low ideal shear strengths.9−11 Kaner and co-workers5,12 suggested designing new superhard materials to overcome the dream of exceeding the hardness of diamond. In particular, the hardness of materials could be increased by intermixing two dissimilar phases that hinders the migration of dislocations. The possibility of enhancing the mechanical properties of transition-metal diborides, specifically by introducing a third element into their structures, has shown © 2012 American Chemical Society

some promise due to solid solution hardening. Thus, Weinberger et al.13 recently synthesized the Os1−xRuxB2 solid solutions using in situ high-pressure X-ray diffraction and Vickers hardness testing techniques. Although the mechanical and electronic properties of OsB2 and RuB2 have been intensively investigated from first-principles calculations,14−25 those associated to the Os1−xRuxB2 solid solutions have not been investigated. Thus, to support the experimental findings reported by Weinberger et al.,13 we used first-principles-based methods to study the structural and mechanical properties of these solid solutions. Our ambition is to contribute to a better understanding of the relationships between the mechanical properties and the electronic structure as a function of Ru− dopant concentration in the OsB2 lattice. The elastic properties are of particular interest as they determine the mechanical stability of the material and some important macroscopic properties, such as hardness, lubrification, friction, and machinability. The electronic structure and chemical bonding are also investigated because they provide an overall view of the electrostructural information to tailor and improve the electronic features of these solid solutions.

II. THEORETICAL APPROACHES AND COMPUTATIONAL DETAILS Our calculations were performed within the density functional theory (DFT) framework as implemented in the VASP26 and ABINIT27 packages. Although these two DFT codes use a planewave basis set, their main difference is in the treatment of the electronic core interactions of an atom and its nucleus. Received: February 16, 2012 Revised: April 26, 2012 Published: April 26, 2012 11746

dx.doi.org/10.1021/jp3015657 | J. Phys. Chem. C 2012, 116, 11746−11751

The Journal of Physical Chemistry C

Article

Table 1. Structural Properties of Os1−xRuxB2 Solid Solutions and Their Parent Compoundsa OsB2

Os0.75Ru0.25B2

Os0.5Ru0.5B2

Os0.25Ru0.75B2

RuB2

4.686 2.883 4.077 55.08 10.08 −0.748

4.679 2.882 4.069 54.87 8.77 −0.793

4.670 (4.645) 2.877 (2.885) 4.064 (4.046) 54.60 7.46 −0.825

4.674 2.893 4.072 55.06 10.09

4.659 2.889 4.060 54.65 8.81

4.645 2.882 4.048 54.19 7.52

PAW a (Å) b (Å) c (Å) volume (Å3) density (mg/m3) ΔEf (eV)

4.705 (4.684) 2.890 (2.873) 4.092 (4.077) 55.64 12.64 −0.618

4.695 2.887 4.084 55.36 11.37 −0.683

a (Å) b (Å) c (Å) volume (Å) density (Mg/m3)

4.701 2.904 4.098 55.94 12.58

4.687 2.900 4.085 55.52 11.34

PP

a

Experimental values are only reported for the parent compounds and they are between brackets.32

Table 2. Calculated Elastic Constants, Cij (GPa), of Os1−xRuxB2 with 0 ≤ x ≤ 1 PAW OsB2 Os0.75Ru0.25B2 Os0.5Ru0.5B2 Os0.25Ru0.75B2 RuB2 PP OsB2 Os0.75Ru0.25B2 Os0.5Ru0.5B2 Os0.25Ru0.75B2 RuB2

C11

C22

C23

C33

C44

C55

C12

C13

C66

566.1 555.6 551.6 538.2 523.4

534.4 510.2 497.0 480.1 463.5

122.6 122.0 124.7 128.7 120.9

770.3 752.4 739.7 715.2 701.4

90.1 93.4 96.3 100.2 94.3

200.2 203.6 211.1 213.1 220.3

175.7 177.1 179.6 184.6 178.0

188.1 180.8 173.4 166.0 162.1

195.7 201.8 195.3 184.7 180.5

551.7 544.8 538.7 529.5 521.4

515.6 501.0 486.2 468.5 450.7

135.9 134.2 132.1 131.4 130.3

754.3 746.1 734.8 725.4 713.4

116.7 121.7 123.8 126.9 127.7

207.2 213.6 217.8 221.9 224.4

182.1 184.9 186.7 190.5 193.2

176.7 169.1 161.8 154.7 148.2

188.6 185.2 181.9 176.4 171.2

Indeed, ABINIT code uses pseudopotentials (PP),28 whereas VASP code uses the projector-augmented wave (PAW) method. 29 The PAW method employs an augmented planewave basis for the electronic valence wave functions and frozen atomic wave functions for the core states. Thus, this method is able to produce the correct wave functions and densities close to the nucleus, including the correct nodal structure of the wave functions. The PAW method also has two advantages with respect to the PP one: (i) transferability problems are largely avoided and (ii) calculations require a smaller basis set. The reliability of the PAW and PP methods has been studied to predict the mechanical properties and electronic structure of the Os1−xRuxB2 solid solutions. The electronic wave functions are expanded in planewaves up to a kinetic energy cutoff of 90 Ha, and integrals over the Brillouin zone are approximated by sums over a 20 × 20 × 20 k-points mesh according to the Monkhorst−Pack scheme30 in ABINIT, whereas we use 20 Ha and a 15 × 23 × 17 grid in VASP, respectively. The exchange-correlation energy functional is evaluated in the two DFT codes within the generalized gradient approximation (GGA) as proposed by Perdew, Burke, and Ernzerhof.31 Geometry optimization is performed by conjugategradient minimization of Hellmann−Feynman forces on the atoms and the stresses in the unit cell. Atomic coordinates and axial ratios have been relaxed for different volumes of the unit cell. Convergence is assumed for an energy difference between two successive iterations less than 10−6 eV per unit cell and forces on the atoms less than 0.01 eV/Å. To build Os1−xRuxB2 solid solutions, we consider a 2 × 2 × 2 supercell derived from

the orthorhombic RuB2-type structure. Within this supercell, only the given Ru compositions x = 0, 1/4, 1/2, 3/4, and 1 are allowed.

III. GROUND STATE PROPERTIES The two ultra-incompressible OsB2 and RuB2 compounds crystallize in an orthorhombic rhenium diboride-type structure at ambient conditions (Pmmn space group). The B-atom sheets consist of boat-like six rings, again conjugated. Thus, each Os atom is surrounded by 2 + 4 + 2 B atoms, forming an irregular polyhedron which resembles a distorted trigonal trapezohedron.32 Both the Os−B and the Ru−B distances are calculated within the 2.18−2.21 Å range. These values are in excellent agreement with the experimental distances reported in the literature between 2.16 and 2.32 Å.32 Table 1 collects the lattice parameters, unit-cell volume, and density of the Os1−xRuxB2 (0 ≤ x ≤ 1) equilibrium structures calculated at the PAW and PP levels. Within these two levels of theory, the structural properties of the Os1−xRuxB2 solid solutions are quite close, highlighting a negligible improvement of the PAW method with respect to the PP one. The calculated OsB2 and RuB2 lattice constants are slightly overestimated by around 0.5% with respect to the experimental ones. This slight overestimate is consistent with the usual behavior of the GGA exchangecorrelation functional.33 The a, b, and c lattice parameters predicted at the PAW (PP) level decrease by around 0.74% (respectively 1.19%), 0.44% (respectively 0.75%), and 0.68% (respectively 1.22%), respectively, from OsB2 to RuB2. Thus, a gradual decrease of the lattice parameters and therefore of the 11747



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Table 3. Calculated Bulk (B, GPa), Shear (G, GPa), and Young (Y, GPa) Moduli along with Poisson Ratio (ν), Elastic Anisotropy (%), and Vickers Hardness (Hv) of the Os1−xRuxB2 Solid Solutionsa PAW OsB2 Os0.75Ru0.25B2 Os0.5Ru0.5B2 Os0.25Ru0.75B2 RuB2 PP OsB2 Os0.75Ru0.25B2 Os0.5Ru0.5B2 Os0.25Ru0.75B2 RuB2 a

B

G

Y

ν

AB

AG

Hv

312.7 (297b, 365c) 305.3 301.6 (311d) 296.3 287.0 (281b, 265b)

177.2 (161b) 177.7 177.5 174.2 170.8 (144b)

447.2 (410b) 446.6 445.2 437.1 427.6 (366b)

0.261 0.256 0.254 0.254 0.251

1.033 1.083 1.051 0.949 1.052

6.910 6.321 5.858 5.136 5.878

28.4 (34.8d) 28.6 28.5 (22.5d) 27.9 27.3 (24.2d)

309.3 304.6 299.4 294.5 289.1

184.1 184.6 183.7 181.9 179.2

460.7 460.7 457.5 452.6 445.5

0.252 0.248 0.245 0.243 0.243

0.983 0.979 0.971 0.997 1.027

3.749 3.481 3.421 3.419 3.568

29.7 29.7 29.6 29.3 28.8

Experimental values are between brackets when available. bReference 8. cReference 7. dReference 13.

function of the Ru concentration in the OsB2 lattice is displayed in Figure 1. According to the experiment, the calculated bulk

equilibrium volume and density is predicted as the Ru concentration increases in the OsB2 lattice. However, this decrease is slightly faster at the PP level. The formation energy of the Os1−xRuxB2 solid solutions is also reported in Table 1. It has been calculated from the difference between their total energies and the sum of the isolated atomic energies of the pure constituents. The formation energy of the solid solutions is always negative, indicating that these compounds should be stable. Furthermore, it is very close for all Ru concentrations.

IV. ELASTIC CONSTANTS AND RELATED MECHANICAL PROPERTIES In orthorhombic crystals, the elastic tensor is determined by nine elastic constants (Cij).34,35 They have been calculated from the stress−strain finite difference method at the PAW and PP levels. These nine elastic constants are reported in Table 2, and they satisfy the Born−Huang mechanical stability expected in orthorhombic structures.36 The variation of the elastic constants from OsB2 to RuB2 is calculated between 4% and 16%. We unusually observe high incompressibilities along the c axis, as demonstrated by the extremely large C33 value for all concentrations. The values of the calculated elastic constants also show a gradual decrease at the PAW level when we move from Os- to Ru-rich diborides except for C44, C55, and C12. The significant increase of these three latter elastic constants suggests the high resistance to deformation with respect to a shearing stress. The same behavior of the elastic constants is observed at the PP level. The bulk, shear, and Young moduli obtained of Os1−xRuxB2 solid solutions obtained the Voigt−Reuss−Hill approximation37 are summarized in Table 3. These structural parameters predicted at the PAW and PP levels show small differences. Thus, only the PAW values will be discussed in the following. Our calculated value of the OsB2 bulk modulus is underestimated by around 15% with respect to the experimental obtained from X-ray. A better agreement with our calculation is found for the OsB2 bulk modulus derived from the isotropic model where less than 1% discrepancy is observed between the two models. By contrast to OsB2, our calculated RuB2 bulk modulus (∼288 GPa) is in better agreement with the experimental value obtained from X-ray (281 GPa8) than that derived from the isotropic model (265 GPa8). Only the bulk modulus of the Os0.5Ru0.5B2 solid solution has been reported in the literature, and this value is consistent with ours. The variation of the calculated and experimental bulk modulus as a

Figure 1. Bulk modulus as a function of Ru−dopant concentration in the OsB2 lattice calculated at the PP and PAW levels. Experimental values are also reported for comparison.

moduli of the solid solutions scale linearly with composition (i.e., a linear interpolation of the binaries) for the two models. However, the slope between the experiment and our calculations is significantly different. This difference could be related to the used value of the bulk modulus derivative (B′) to fit the experimental data from the Birch−Murnaghan equation of state.38 The calculated values of the OsB2 and RuB2 shear moduli are also in agreement with the previously reported theoretical10−20 and experimental data.7 Our OsB2 and RuB2 calculated Young moduli are overestimated with respect to the experimental values obtained from nanoindentation with a Berkovich diamond indenter.8 Nevertheless, the agreement between our calculations and the experiments is still satisfactory. The elastic anisotropy of crystals is also important for their applications. For low-symmetry crystals, it can be described by the anisotropy percentage in compressibility (AB) and shear (AG).37 For AB and AG, the values of 0 and 1 (100%) represent the elastic isotropy and the largest anisotropy, 11748



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respectively. The comparison between the average elastic anisotropy in compressibility and the shear for the Os1−xRuxB2 solid solutions shows that all compounds are less anisotropic in compression and shear. Bulk and shear moduli are still the most important parameters for identifying the material hardness.39,40 High hardness requires that bulk modulus and shear modulus must be as large as possible. It has been suggested that a relationship exists between the shear modulus and hardness for many of the known high-strength materials.41 According to a linear relation and using the calculated shear moduli reported in Table 3, we estimated the Vickers hardness of the Os1−xRuxB2 solid solutions and their parent compounds. These values are also reported in Table 3. The OsB2 and RuB2 Vickers hardness are in good agreement with the experimental ones. However, this agreement may be coincidental since the strains associated with the hardness measurements are beyond the elastic limit. In fact, shear moduli are routinely used for a comparative measure of the stress needed for plastic flow, in particular, among materials with the same lattice structure.13 Furthermore, in accordance to the experimental results,13 our theoretical results show also that the solid solution formation is not an appropriate method to improve the hardness of the OsB2.

V. ELECTRONIC PROPERTIES AND BONDING CHARACTERIZATION In order to gain a deeper insight into the origin of the mechanical properties of Os1−xRuxB2 solid solutions, their total and projected electronic density of states (DOS) are displayed in Figure 2a and 2b. The Fermi levels have been fixed at zero. A wide pseudogap separates the bonding and antibonding states in the total DOS of all Os1−xRuxB2 solid solutions and their parent compounds (see Figure 2a). This pseudogap should increase the stability of the materials. From the projected DOS displayed in Figure 2b, we observe noticeable states at Fermi levels. These main orbital occupancies stem from Os 5d and/or Ru 4d electrons and indicate the metallic behavior of the diborides. For all solid solutions, the Os 5d and/or Ru 4d states have significant hybridizations with B 2p states. We therefore expect a strong Os−B and/or Ru−B covalent bonding nature, leading to a strengthening of the Os−B bond with respect to Ru−B. This hybridization difference between OsB2 and RuB2 diborides is responsible for the significant difference of their elastic properties. In particular, most of the elastic constants of OsB2 are greater than RuB2. The shape and height of the Os 5d peaks are slightly lower than the Ru 4d ones, indicating that the delocalization of 5d states is stronger that of the 4d states.42 To further improve the understanding of the bonding characterization of the alloys and their parent materials we turn to our attention to Hirshfeld population analyses using CASTEP calculations.43 It is well known that the Mulliken population analyses44 strongly depend on the used basis set in the calculation and the method of partitioning the overlap populations of each pair of atoms to the individual atoms in the pair. Thus, Hirshfeld suggested another exploiting approach based on spatial numerical integration of the electronic density,45 where the system is divided into well-defined atomic fragments. Then a natural choice will be to share the charge density at each point between the atoms, in proportion to their free-atom densities at the corresponding distances from the nuclei.43 The Hirshfeld charges are reported in Figure 3 for OsB2, RuB2, and their alloys. Because Os1−xRuxB2 solid solutions contain inequivalent atoms, the charge redistribution

Figure 2. (a) Total and (b) partial density of states for Os1−xRuxB2 solid solutions. Vertical dashed line denotes the Fermi level.

Figure 3. Hirshfeld charges of Os1−xRuxB2 solid solutions.

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(7) Cumberland, R. W.; Weinberger, M. B.; Gilman, J. J.; Clark, S. M.; Tolbert, S. H.; Kaner, R. B. J. Am. Chem. Soc. 2005, 127, 7264− 7265. (8) Chung, H. Y.; Weinberger, M. B.; Levine, J. B.; Kavner, A.; Yang, J. M.; Tolbert, S. H.; Kaner, R. B. Appl. Phys. Lett. 2008, 92, 261904/ 1−261904/3. (9) Yang, J.; Sun, H.; Chen, C. F. J. Am. Chem. Soc. 2008, 130, 7200− 7201. (10) Luo, X.; Li, J.; Hu, P.; Dong, S. Sci. China Ser. E: Tech. Sci. 2010, 53, 1877−1885. (11) Ivanovskii, A. L.; Medvedeva, N. I.; Russian, J. Inorg. Chem. 1999, 44, 1633−1641. (12) Gilman, J. J.; Cumberland, R. W.; Kaner, R. B. Int. J. Refract. Met. Hard Mater. 2006, 24, 1−5. (13) Weinberger, M. B.; Levine, J. B.; Chung, H. Y.; Cumberland, R. W.; Rasool, H. I.; Yang, J. M.; Kaner, R. B.; Tolbert, S. H. Chem. Mater. 2009, 21, 1915−1921. (14) Gou, H.; Hou, L.; Zhang, J.; Li, H.; Sun, G.; Gao, F. Appl. Phys. Lett. 2006, 88, 221904/1−221904/3. (15) Chiodo, S.; Gotsis, H. J.; Russo, N.; Sicilia, E. Chem. Phys. Lett. 2006, 425, 311−314. (16) Chen, Z. Y.; Xiang, H. J.; Yang, J.; Hou, J. G.; Zhu, Q. Phys. Rev. B 2006, 74, 012102/1−012102/4. (17) Hebbache, M.; Stuparevic, L.; Zivkovic, D. Solid State Commun. 2006, 139, 227−231. (18) Gou, H.; Hou, L.; Zhang, J.; Li, H.; Sun, G.; Gao, F. Appl. Phys. Lett. 2006, 88, 221904/1−221904/3. (19) Suh, B. J.; Zong, X.; Singh, Y.; Niazi, A.; Johnston, D. C. Phys. Rev. B 2007, 76, 144511/1−144511/4. (20) Hao, X.; Xu, Y.; Wu, Z.; Zhou, D.; Liu, X.; Meng, J. J. Alloys Compd. 2008, 453, 413−417. (21) Wang, J.; Wang, Y-.J. J. Appl. Phys. 2009, 105, 083539/1− 083539/6. (22) Chen, X.-Q.; Fu, C. L.; Krcmar, M.; Painter, G. S. Phys. Rev. Lett. 2008, 100, 196403/1−196403/4. (23) Lin, F.; Wu, K.; He, J.; Sa, R.; Li, Q.; Wei, Y. Chem. Phys. Lett. 2010, 494, 31−36. (24) Kanoun, M. B.; Shein, I. R.; Goumri-Said, S. Solid State Commun. 2010, 150, 1095−1098. (25) Luo, F.; Cheng, Y.; Ji, G.-F.; Chen, X.-R. Chin. Phys. Lett. 2009, 26, 097101/1−097101/4. (26) Kresse, G.; Furthmüller, J. Comput. Mater. Sci. 1996, 6, 15−50. (27) Gonze, X.; et al. Comput. Mater. Sci. 2009, 180, 2582−2615. (28) Troullier, N.; Martins, J.-L. Phys. Rev. B 1991, 43, 1993−2006. (29) Blöchl, P. E. Phys. Rev. B 1994, 50, 17953−17979. (30) Monkhorst, H. J.; Pack, J. D. Phys. Rev. B 1976, 13, 5188−5192. (31) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865−3868. (32) Frotscher, M.; Hölzel, M.; Albert, B. Z. Anorg. Allg. Chem. 2001, 636, 1783−1786. (33) Kanoun, M. B.; Jaouen, M. J. Phys.: Condens. Matter 2008, 20, 085211/1−085211/4. (34) Le Page, Y.; Saxe, P. Phys. Rev. B 2001, 63, 174103/1−174103/ 8. (35) Hermet, P.; Goumri-Said, S.; Kanoun, M. B.; Henrard, L. J. Phys. Chem. C 2009, 113, 4997−5003. (36) Born, M.; Hang, K. Dynamical Theory and Experiments I; Springer Verlag Publishers: Berlin, 1982. (37) Hill, R. Proc. Phys. Soc. London A 1952, 65, 349−354. (38) Birch, F. Phys. Rev. 1947, 71, 809−824. (39) Haines, J.; Leger, J. M.; Bocquillon, G. Annu. Rev. Mater. Res. 2001, 31, 1−23. (40) Gao, F. M.; Gao, L. H. J. Superhard Mater. 2010, 32, 148−166. (41) Teter, D. M. MRS Bull. 1998, 23, 22−26. (42) Vajeeston, P.; Ravindran, P.; Ravi, C.; Asokamani, R. Phys. Rev. B 2001, 63, 045115/1−045115/12. (43) Segall, M. D.; Lindan, P. J. D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C. J. Phys.: Condens. Matter 2002, 14, 2717−2744.

only occurs for these compounds. As the difference between these charges is of the same order, we preferred to display in Figure 3 the effective charge average for each atom B, Os, and Ru. We observe that the boron has a negative effective charge for all compounds, whereas Os and Ru have a positive charge. The Hirshfeld charge is found to be greater for Ru than for Os, especially in the Os0.25Ru0.75B2. Moreover, we expect that the largest electronic redistribution should be between the Ru and the Os neighboring atoms, especially for the Os0.5Ru0.5B2. Ideally, Hirshfeld charges would be primarily determined by the electron-donating or electron-withdrawing strengths of an atom, relative to its neighboring atoms. However, this is only partially true because there are other effects that influence the Hirshfeld charges such as the environment effect and the density deformation due to electron excess in the atomic environment.45, In our systems, the more we introduce Ru in OsB2 the more the Hirshfeld charge differences between Os and B expands, especially for Os0.5Ru0.5B2.

VI. CONCLUSIONS We investigated the structural and mechanical properties, electronic structure, and hardness of Os1−xRuxB2 solid solutions using density functional theory. First, our calculations show that the Os 1−xRu xB2 solution transition-metal diborides are mechanically stable and have a very high bulk modulus. Then their Vickers hardness has been calculated, and their values are in good agreement with experimental data. These diborides are less anisotropic in compression and shear, and their hardness is between the values reported for their parent compounds. We found that formation of solid solution is not a good method to boost the incompressibility of the OsB2. Finally, the bonding characterization was analyzed from the electronic structure and the Hirshfeld population. The electronic structure suggests (i) a metallic behavior for Os1−xRuxB2 and their parent compounds and (ii) stronger covalent Os−B bonds in OsB2 compared to RuB2.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Prof. R. B. Kaner (University of California) for his critical reading of this manuscript and for stimulating discussions. Calculations with ABINIT and VASP codes have been performed on ISCF computers of FUNDP University (Belgium). CASTEP calculations were carried out partially in the Beowulf class heterogeneous computer cluster at KAUST.

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REFERENCES

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(44) Hirshfeld, F. L. Theor. Chim. Acta 1977, 44, 129−138. (45) Saha, S.; Roy, R. K.; Ayers, P. W. Int. J. Quantum Chem. 2009, 109, 1790−1806.

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Structure, Elastic Stiffness, and Hardness of Os1âxRuxB2 Solid

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