Ultra-incompressible Orthorhombic Phase of Osmium Tetraboride

Article pubs.acs.org/JPCC

Ultra-incompressible Orthorhombic Phase of Osmium Tetraboride (OsB4) Predicted from First Principles Meiguang Zhang,*,† Haiyan Yan,‡ Gangtai Zhang,† and Hui Wang*,§ †

Department of Physics and information Technology, Baoji University of Arts and Sciences, Baoji 721016, P. R. China Shaanxi Key Laboratory for Phytochemistry, Department of Chemistry and Chemical Engineering, Baoji University of Arts and Sciences, Baoji 721013, P. R. China § National Laboratory of Superhard Materials, Jilin University, Changchun 130012, P. R. China ‡

ABSTRACT: Using the ab initio particle swarm optimization algorithm for crystal structure prediction, we have predicted an orthorhombic Pmmn structure for OsB4, which is energetically much superior to the previously proposed WB4-type structure. The Pmmn structure consists of irregular OsB10 dodecahedrons connected by edges and is stable against decompression into a mixture of Os and B at ambient pressure. OsB4 within this orthorhombic phase is found to be an ultra-incompressible and hard material due to its high bulk modulus (294 GPa) and large hardness (28 GPa), originating from the strong and directional covalent B−B and B−Os bonds.

applied to many other systems.18−20 Indeed, an orthorhombic Pmmn (No. 59, Z = 2) structure is uncovered for OsB4 to be energetically much more preferable than the earlier proposed WB4-type structure under ambient conditions. First principles calculations were then performed to investigate the total energy, lattice parameters, bulk modulus, hardness, and density of states for this novel orthorhombic phase. In addition, other osmium borides with various stoichiometries were also studied for comparison.

1. INTRODUCTION Superhard materials are particularly useful in a variety of industrial applications because of their superior properties of higher compressional strength, thermal conductivity, refractive index, chemical inertness, and high hardness. Previously, it was generally accepted that the superhard materials are those strongly covalent bonded compounds formed by light elements, such as diamond,1 c-BN,2 BC2N,3 B6O,4 and BC5.5 Recently, it was reported that partially covalent heavy transition metal (TM) boride, carbide, nitride, and oxide are found to be good candidates for superhard materials, such as ReB2,6 WB4,7 RuO2,8 etc. Therefore, these pioneering studies9−13 open up a novel route for the search of new superhard materials. Since the pure osmium (Os) has been found to possess an extremely high bulk modulus (>395 GPa),14,15 its borides, carbides, oxides, and nitrides have, thus, been a topic of much interest to scientists, and are expected to be good candidates for superhard materials. Experimentally, the structures of osmium borides with various stoichiometries (OsB, Os2B3, and OsB2) were confirmed at ambient pressure and some related mechanical properties were investigated.7,10,14 The obtained results indicated that they are all ultra-incompressible and hard materials. However, the attempts to synthesize compounds with higher boron contents did not reveal any new phases so far. A promising material, osmium tetraboride (OsB4) within WB4-type structure, was proposed to be superhard with a claimed hardness of 46.2 GPa,16 although it has not been synthesized. However, this proposed WB4-type structure is based on the knowledge of known information. There is a possibility that hitherto unknown structures are stable instead. Here, we have extensively investigated the ground-state structure of OsB4 by the ab initio particle swarm optimization (PSO) algorithm on crystal structural prediction,17 unbiased by any known information. This method has been successfully © 2012 American Chemical Society

2. COMPUTATIONAL METHODS Our PSO methodology on crystal structural prediction has been implemented in CALYPSO code21 at 0 GPa with 1−4 formula units (f.u.) per simulation cell. The underlying ab initio calculations were performed using density functional theory within the generalized gradient approximation (GGA),22 as implemented in the VASP package.23 The all-electron projector augmented wave method24 was adopted with 2s22p1 and 4p65d66s2 treated as valence electrons for B and Os, respectively. The energy cutoff 520 eV and appropriate Monkhorst−Pack k meshes25 were chosen to ensure that enthalpy calculations are well converged to better than 1 meV/ f.u. The phonon calculations were carried out by using a supercell approach as implemented in the PHONOPY code.26 Single crystal elastic constants were determined from evaluation of stress tensor generated small strain, and the bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio were thus estimated by using the Voigt−Reuss−Hill approximation.27 The theoretical Vickers hardness was estimated by using the Šimůnek model.28 Received: November 5, 2011 Revised: January 8, 2012 Published: January 11, 2012 4293

dx.doi.org/10.1021/jp2106378 | J. Phys. Chem. C 2012, 116, 4293−4297

The Journal of Physical Chemistry C

Article

3. RESULTS AND DISCUSSION At 0 GPa, the variable cell simulation revealed the orthorhombic Pmmn structure as the most stable phase, as shown in Figure 1a. The Pmmn structure contains two OsB4 f.u.

Figure 2. Phonon dispersion curves for Pmmn-OsB4 at 0 GPa.

which indicates that it is indeed the ground state. However, it is noted that the much higher positive formation enthalpy of WB4−OsB4 (3.28 eV/f.u.), compared with the Pmmn-OsB4 and MoB4−OsB4, is unfavorable to its stability. Therefore, the present calculations give direct evidence that the Pmmn-OsB4 can be synthesized at ambient pressure. Further experimental work is thus strongly recommended. The mechanical properties (elastic constants and elastic moduli) of the Pmmn phase are important for the potential technological and industrial applications. We calculated the zero-pressure elastic constants Cij of the Pmmn-OsB4 by the strain−stress method. The calculated elastic constants Cij are listed in Table 1, along with the theoretical values and available experimental data of other osmium borides, MoB4 and WB4.7,16,29,30 The elastic stability is a necessary condition for a stable crystal. For a stable orthorhombic structure, Cij has to satisfy the elastic stability criteria:31,32 C11 > 0, C22 > 0, C33 > 0, C44 > 0, C55 > 0, C66 > 0, C11 + C22 + C33 + 2(C12 + C13 + C23) > 0, (C11 + C22 − 2C12) > 0, (C11 + C33 − 2C13) > 0, and (C22 + C33 − 2C23) > 0. As shown in Table 1, these conditions are clearly satisfied for Pmmn-OsB4, confirming that it is mechanically stable. Using the calculated elastic constants Cij, the polycrystalline bulk modulus and shear modulus are thus determined by the Voig−Reuss−Hill approximation method.27 In addition, the Young’s modulus E and Poisson’s ratio v are obtained by the equations E = (9BG)/(3B=G) and v = (3b − 2G)/(6B + 2G). The calculated bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio of Pmmn-OsB4 and the reference materials mentioned above are given in Table 1. The calculated bulk modulus of the Pmmn phase is 294 GPa, which is close to the experimental data of WB4 (304 GPa) and OsB2 (348 GPa),7 indicating the ultra-incompressible structural nature. Moreover, the calculated bulk modulus (B) agrees well with that (BEOS) directly obtained from the fitting results of the third-order Birch−Murnaghan equation of state,33 which further verifies the good accuracy of our elastic calculations. To further compare the incompressibility of Pmmn-OsB4, OsB2, and WB4 under pressure, the volume compressions as a function of pressure are plotted in Figure 3. One can see that the incompressibility of Pmmn-OsB4 is similar to those of WB4 and OsB2. Compared to the bulk modulus, the shear modulus of a material quantifies its resistance to the shear deformation and is a better indicator of potential hardness. Remarkably, Pmmn-OsB4 possesses the largest shear modulus of 218 GPa in

Figure 1. Crystal structures of the predicted Pmmn-OsB4. The large and small spheres represent Os and B atoms, respectively. (a) Polyhedral view of the Pmmn phase. (b) The B−Os−B sandwiches stacking order along the a-axis.

in a unit cell (a = 7.119 Å, b = 2.896 Å, and c = 4.015 Å), in which three inequivalent atoms Os, B1, and B2 occupy the Wyckoff 2b (0, 0.5, 0.5051), 4f (0.2017, 0, 0.3074), and 4f (0.3461, 0, 0.9841) sites, respectively. Each metal Os has 10 neighboring B atoms, forming irregular OsB10 dodecahedrons connected by edges. In the OsB10 dodecahedron, the Os−B bond distances are calculated to be 2.188 (×4), 2.249 (×2), 2.253 (×2), and 2.325 (×2) Å. Figure 1b reveals an intriguing B−Os−B sandwiches stacking order along the a-axis, and each B has three neighbors in B layers which consist of a series of quadrangle B rings. In the quadrangle B ring, the average B−B bond distance is 1.774 Å, which is smaller than those in OsB2 (1.880 Å) and Os2B3 (1.876 Å). At zero temperature, a stable crystalline structure requires all phonon frequencies to be positive. Therefore, we performed the full phonon dispersion calculation for Pmmn-OsB4 at 0 GPa. As shown in Figure 2, no imaginary phonon frequency was detected in the whole Brillouin zone, indicating the dynamical stability of the Pmmn structure. Our computational approach is based on constant-pressure static quantum mechanical calculations at T = 0 K, so the Gibbs free energy is reduced to enthalpy. In order to explore the thermodynamic stability for further experimental synthesis, the formation enthalpy of the Pmmn phase with respect to the separate phases is quantified by the reaction route ΔH = H(OsB4) − H(Os) − 4H(B). The Os (space group: P63/mmc) and α-B (space group: R-3m) were chosen as the reference phases. In addition, the WB4-type and MoB4-type29 structures were also considered. Results obtained for Pmmn-OsB4 (−0.31 eV/f.u.) and MoB4−OsB4 (−0.11 eV/f.u.) in this reaction route have demonstrated the stability against the decomposition into the mixture of Os and α-B. Compared to the MoB4-type phase, the predicted Pmmn phase possesses a lower formation enthalpy, 4294



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Table 1. The Calculated Elastic Constants Cij, Bulk Modulus B, EOS Fitted Bulk Modulus BEOS, Shear Modulus G, and Young’s Modulus E in Units of GPaa OsB4 OsB2

Os2B3 OsB

MoB4 WB4 a

this work this work theor.30 expt.7 this work expt.7 this work theor.30 expt.7 theor.29 theor.16 expt.7

C11

C12

C13

C22

C33

C44

C55

C66

B

BEOS

G

E

G/B

v

612 570 546

128 178 166

245 188 184

576 540 553

630 753 753

152 68 64

349 191 209

178 192 207

293 310

218 164 168

524 419 426

0.744 0.529

0.204 0.277 0.269

536

178

189

885

210

165

331

205

506

0.619

0.238

624 632

203 201

190 200

801 804

190 193

213 211

352

216 231

534 572

0.608

0.250 0.236

505 389

141 280

103 224

936 437

189 151

294 312 307 348 322 396 355 360 431 285 297 304

282

210 104

506

0.734 0.350

0.20

Also shown are Poisson’s ratio v and the G/B ratio.

Sij =

where Nij is the number or multiplicity of the binary system; Sij is the strength of the individual bond between atoms i and j; the reference energy ei is defined as ei = Zi/ri, where Zi is the number of valence electrons and ri is the radius of the sphere (centered at atom i) in which Zi electrons are contained; ni and nj are coordination numbers of atoms i and j, respectively; dij is the interatomic distance, and k corresponds to the number of different atoms in the system. In the present work, the values C = 1450, σ = 2.8, and the atomic radii ri of elements from Kittel’s textbook are used.35 In this binary orthorhombic phase, there are three inequivalent atoms Os, B1, and B2. Each Os atom has 10 B neighbors, n1(Os) = 10; each B1 atom has three B and three Os neighbors, n2(B) = 6; and each B2 atom has three B and two Os neighbors, n3(B) = 5. In the B layer, as shown in Figure 1b, each B atom has three neighbors with two different distances, n4(B) = n5(B) = 3. The bond strengths s12, s13, and s45 correspond to Os−B1, Os−B2, and B1−B2 bonds, respectively. In the Pmmn structure, the Os−B and B−B distances are not the same, respectively, d12(Os−B1) = 2.188−2.253 Å, d13(Os− B2) = 2.249−2.326 Å, and d45(B1−B2) = 1.656−1.893 Å; the average bond lengths d12 = 2.222 Å, d13 = 2.287 Å, and d45 = 1.774 Å are used here. Applying r1(Os) = 1.35 Å, r2(B) = 0.98 Å, Z1(Os) = 6, Z2(B) = 3, e1 = Z1/r1, e2 = Z2/r2, e3 = e4 = e5 = e2, and Ω = 82.78 Å3, we get H(OsB4) = 28.0 GPa, close to the hardness of β-Si3N4 (30.3 GPa). For Pmmn-OsB4, there are no available experimental data; however, taking account of the case of the molybdenum borides in our recent work,29,36 we believe our predicted values should be reliable. In order to understand the mechanical properties of the Pmmn phase, the total and partial densities of states (DOS) were calculated, as shown in Figure 4. Clearly, the Pmmn-OsB4 exhibits metallic behavior by evidence of the finite electronic DOS at the Fermi level (EF). From the partial DOS, it reveals that the peaks from −15 to −9 eV are mainly attributed to B-2s and B-2p states with a small contribution from Os-5d. The states from −9 to 0 eV mainly originate from Os-5d and B-2p orbitals with slight contributions of Os-5p and B-2s. Moreover, the partial DOS profiles for both Os-5d and B-2p are very similar in the range of −9 to 0 eV, reflecting the significant hybridization between these two orbitals. This fact also shows a strong covalent interaction between the Os and B atoms. The typical feature of the total DOS is the presence of a so-called pseudogap,37 which is considered as the borderline between the

Figure 3. The pressure dependence of unit cell volume for PmmnOsB4.

Table 1, and it is expected to withstand shear strain to a large extent. In addition, the relative directionality of the bonding in the material also has an important effect on its hardness and can be determined by the G/B ratio.34 Among these osmium borides, Pmmn-OsB4 has the largest G/B ratio of 0.741, which suggests the strongest directional bonding between Os and B atoms. All of these excellent mechanical properties strongly suggest that Pmmn-OsB4 is a potential candidate to be ultraincompressible and hard. In view of the high bulk and shear moduli of the Pmmn phase, the hardness calculations are of great interest. According to the Šimůnek model,28 the hardness of the ideal single crystal is proportional to the bond strength and to their number in the unit cell. More specifically, the hardness of crystals having different bond strengths is given by the following expressions: n

H = (C /Ω) ·n·[

∏

(NijSij)]1/ n e−σfe

i,j=1 k

k

i=1

i=1

eiej /(ninjdij)

fe = 1 − [k( ∏ ei)i / k / ∑ ei]2 4295



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Figure 5. Contours of the electronic localization function (ELF) of Pmmn-OsB4 on the (010) plane.

Figure 4. Total (a) and partial (b) densities of states for Pmmn-OsB4. The vertical line denotes the Fermi level.

dynamically stable and synthesizable under ambient conditions. Moreover, the electronic densities of states and electronic localization function analysis have demonstrated that the strong covalent B−B bonding in B layers and B−Os bonding in OsB10 dodecahedrons play a key role in the incompressibility and hardness of Pmmn-OsB4. These findings will inevitably stimulate extensive experimental works on synthesizing this technologically important material.

bonding and antibonding states. It should be pointed out that the EF is perfectly lying on the pseudogap, suggesting the p−d bonding states started to be saturated. This full occupation of the bonding states and without filling on the antibonding states enhances the stability of orthorhombic OsB4. To gain more detailed insight into the bonding character of Pmmn-OsB4, we have calculated the electronic localization function (ELF),38 which is based on a topological analysis related to the Pauli exclusion principle. The ELF is a contour plot in real space where different contours have values ranging from 0 to 1. A region with ELF = 1 is where there is no chance of finding two electrons with the same spin. This usually occurs in places where covalent bonds or lone pairs (filled core levels) reside. An area where ELF = 0 is typical for a vacuum (no electron density) or areas between atomic orbitals. This is where electrons of like spin approach each other the closest. ELF = 0.5 for a homogeneous electron gas, with values of this order indicating regions with bonding of a metallic character. It should be noted that ELF is not a measure of electron density but is a measure of the Pauli principle, and is useful in distinguishing metallic, covalent, and ionic bonding. The contours of ELF domains for the Pmmn phase on its (010) plane are shown in Figure 5. The high electron localization can be seen in the region between adjacent B and B atoms indicative of covalent bonding, with nearly identical B−B covalent “point attractors” at ELF = 0.83. Meanwhile, the ELF is negligible at the Os sites, whereas it attains local maximum values at the B sites, manifesting another covalent interaction between Os and B atoms at ELF = 0.577. Moreover, the region around B has an overall higher ELF value than the region around the Os atom, reflecting the ionicity in the bond with B withdrawing charge from Os. Therefore, the strong covalent interaction between B−B bonds and B−Os bonds is the main driving force for its higher bulk and shear modulus.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.Z.); [email protected] (H.W.).

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ACKNOWLEDGMENTS This work is supported by Natural Science Foundation of China (No. 91022029), Baoji University of Arts and Sciences Key Research (Grant Nos. ZK1032, ZK11060, ZK11061, and ZK11146), and the Phytochemistry Key Laboratory of Shaanxi Province (Grant No. 11JS008).

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REFERENCES

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4. CONCLUSIONS In summary, an orthorhombic Pmmn structure is unraveled to be the ground-state structure for OsB4 through the PSO algorithm, and it is energetically much superior to the earlier proposed WB4-type structure. The phonons and formation enthalpy calculations have confirmed that the Pmmn phase is 4296



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Ultra-incompressible Orthorhombic Phase of Osmium Tetraboride

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