Phase Stability, Physical Properties, and Hardness of Transition-Metal

Article pubs.acs.org/JPCC

Phase Stability, Physical Properties, and Hardness of TransitionMetal Diborides MB2 (M = Tc, W, Re, and Os): First-Principles Investigations Ming-Min Zhong,† Xiao-Yu Kuang,†,* Zhen-Hua Wang,† Peng Shao,† Li-Ping Ding,† and Xiao-Fen Huang‡ †

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China Physics Department, Sichuan Normal University, Chengdu 610068, China

‡

S Supporting Information *

ABSTRACT: Using first-principles calculations, the structural stability, elastic strength, and formation enthalpies of four diborides MB2 (M = Tc, W, Re, and Os) are investigated by means of the pseudopotential plane-waves method, as well as the roles of covalency and bond topology in the phase incompressibility. Three candidate structures of known transition-metal diborides are chosen to probe. The calculated lattice parameters, elastic properties, Poisson’s ratio, and B/G ratio are derived. It is observed that the ReB2-type structure containing well-defined zigzag covalent chains exhibits an unusual incompressibility along the c axis comparable to that of diamond. Formation enthalpy calculations demonstrate that the ground-state phase is synthesizable at low pressure, whereas the other phase can be achieved through the phase transformation. Moreover, according to Mulliken overlap population analysis, a semiempirical method to evaluate the hardness of multicomponent crystals with a partial metallic bond is presented. The predicted hardness of WB2−WB2, ReB2−ReB2, and OsB2−OsB2 is in reasonable agreement with experiment data. Both strong covalency and a zigzag topology of interconnected bonds underlie the ultraincompressibilities. In addition, the superior performance and largest hardness of ReB2−ReB2 indicate that it is a superhard material. This work provides a useful guide for designing novel borides materials having excellent mechanical performances.

1. INTRODUCTION Ultraincompressible hard materials have been a subject of inherent interest due to their excellent performance in both fundamental science and technological applications, such as abrasives, cutting tools, and wear-resistant coatings. To search for novel superhard materials, great efforts have been devoted to two kinds of materials. The first kind contains the lightelement compounds in the B−C−N−O system with strong covalent and short bonds. Diamond-like BC2N has been synthesized with Vickers hardness of 76 GPa ranking second only to diamond.1 The Vickers hardness of synthesized cubic BC5 has been reported to be up to 71 GPa.2 However, such materials are expensive because they should be synthesized under extreme high pressure and high temperature. The second kind is developed by introducing the light-element forming strong covalent bonds into transition metal that has high electron density, such as IrN2, OsN2, PtN2, ReB2, OsB2, Ru2C, and PtC and so on.3−8 Recently, the new transition-metal borides such as ReB2,9 OsB2,10 TaB2,11 and WB412 have been successfully synthesized. These transition-metal borides have high bulk moduli that are not very far from diamond. What is more, they can be synthesized under ambient pressure, which © XXXX American Chemical Society

leads to the low cost synthesis condition and is beneficial to their application. Great progress has been made in synthesizing the superhard material, such as transition-metal nitrides, carbides, and borides. To design novel materials having excellent mechanical performance, it is necessary to have a detailed understanding of the hardness. To date, ReB2 has been synthesized by many scientists,4,9,13−15 but there is still substantial controversy over whether ReB2 is a superhard material. With the arc-melting techniques at ambient pressure, Chung et al.9 have reported that ReB2 has an average hardness of 48 GPa and scratch marks left on a diamond surface confirmed its superhard nature. However, as pointed out by Dubrovinskaia et al.,13 the loadinvariant hardness of this material is clearly below 30 GPa, that is, ReB2 is not superhard. In response to the comment of Dubrovinskaia et al.,13 Chung et al. has presented an atomic force microscopy profile of scratch marks on the diamond surface.14 More recently, Gu et al. have reported hardness of Received: January 7, 2013 Revised: March 18, 2013

A

dx.doi.org/10.1021/jp400204c | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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chosen in the form of rhenium diboride-type hexagonal lattice (No. 194, P63/mmc space group). Second, the osmium diboride-type (OsB2) orthorhombic lattice (No. 59, Pmmn space group) for all four structures (TcB2, WB2, ReB2, and OsB2) was chosen as initial structure. Third, the tungsten diboride-type (WB2) hexagonal lattice (No. 194, P63/mmc space group) was only performed as initial structure of WB2. To further verify the mechanical stability of these polymorphs, the elastic constants were calculated with the strain−stress method. The elastic constants of a given crystal should satisfy the generalized elastic stability criteria.22 The bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio ν were estimated via the Voigt−Reuss−Hill (VRH) approximations.23 Hexagonal phase (C11, C33, C44, C12, and C13)

ReB2 to be about 39.3 GPa at 0.49N, with a load-invariant hardness of only 26.2 GPa.4 Using the spark plasma sintering techniques, the measured Vickers hardness of ReB2 is within the limit (40 GPa).16 Furthermore, these researchers are unable to reproduce the scratching of diamond by ReB2 as reported by Chung et al.9 All of these experimental results show that although ReB2 has a high zero-pressure bulk modulus, that is, low compressibility, it is not intrinsically superhard. Similar controversy occurs at WB2 and OsB2. The ultraincompressible OsB2 has been synthesized, possessing a high modulus of 365− 395 GPa,10 which is not far from diamond (443 GPa). However, the hardness values as measured in experiment are in the range of 24.8−17.8 GPa,16 which is less than the superhard limit (40 GPa). Theoretically, on the basis of the atomic properties and bond strengths, Šimůnek has reported the hardness of ReB2 as 35.8 GPa.17 Using the calculation method involving hardness anisotropies in different dimensions, the hardnesses of TcB2 and ReB2 are 37.0 and 35.9 GPa, respectively.18 Unfortunately, their asymptotic hardness values are all lower than 40 GPa. In general, bonds in novel boride materials are strongly directional having a large covalent component with partial ionicity and metallicity. The partial ionicity and metallicity decrease the hardness and should be considered in discussing the origin of ultraincompressibility. This is achieved in this article. According to Mulliken overlap population analysis, an analytical method to evaluate the hardness of multicomponent crystals with a partial metallic bond is presented. The calculated hardness of ReB2, WB2, and OsB2 is in agreement with the experimental results. In this article, we systematically explore the structural and elastic properties, formation enthalpy, and hardness for transition-metal diborides MB2 focusing on elements (M = Tc, W, Re, and Os) that have the lowest compressibilities among all metals. The purpose of the present work is 3-fold. First, it is to give a comprehensive and complementary investigation on the structural and mechanical properties of these four diborides. Second, it is to provide a further understanding on phase and thermodynamic stability of these four diborides MB2 with different structures under high pressure. Third, it is to provide powerful guidelines for future experimental investigations and hope that such an investigation might contribute some further understanding to the hardness and the origin of ultraincompressibility, especially for the metallic bond and interconnected bonds.

B V = (1/9)[2(C11 + C12) + 4C13 + C33] G V = (1/30)(M + 12C44 + 12C66] BR = C 2 / M G R = (5/2)[C 2C44C66]/[3B V C44C66 + C 2(C44 + C66)]

M = C11 + C12 + 2C33 − 4C13 2 C 2 = (C11 + C12)C33 − 2C13

The mechanical stability criteria were given by 2 C44 > 0, C11 > |C12| , (C11 + 2C12)C33 > 2C13

Orthorhombic phase (C11, C22, C33, C44, C55, C66, C12, C13, and C23) B V = (1/9)[C11 + C22 + C33 + 2(C12 + C13 + C23)] G V = (1/15)[C11 + C22 + C33 + 3(C44 + C55 + C66) − (C12 + C13 + C23)] BR = Δ[C11(C22 + C33 − 2C23) + C22(C33 − 2C13) − 2C33C12 + C12(2C23 − C12) + C13(2C12 − C13) + C23(2C13 − C23)]−1

G R = 15{4[C11(C22 + C33 + C23) + C22(C33 + C13)

2. COMPUTATIONAL METHODS In the first stage of the calculations, the equilibrium geometries of the systems (TcB2, WB2, ReB2, and OsB2) were obtained, and then the mechanical and electronic properties of them were calculated by using density functional theory method within the CASTEP code.19 The exchange-correlation functional was taken into account through generalized gradient approximation (GGA) and Perdew Burk Ernzerhof functional (PBE).20 The interactions between the ions and the electrons were described by using the pseudopotentials plane-wave method. Pseudoatomic calculations were done by employing the reference configuration B 2s22p1, Tc 4d56s2, W 5d46s2, Re 5d56s2, and Os 5d66s2, respectively. The plane-wave cutoff was 450 eV. The Brillouin zone sampling was performed using Monkhorst-pack grid.21 The self-consistency tolerance was set to 1 × 10−6 eV for the total energy per atom and 5 × 10−6 eV for eigenvalues. There were three different strategies in the optimization stage. First, the initial structure of TcB2, WB2, ReB2, and OsB2 was

+ C33C12 − C12(C23 + C12) − C13(C12 + C13) − C23(C13 + C23)]/Δ + 3[(1/C44) + (1/C55) + (1/C66)]}−1 Δ = C13(C12C23 − C13C22) + C23(C12C13 − C23C11) 2 + C33(C11C22 − C12 )

The criteria for mechanical stability were given by 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, (C22 + C33 − 2C23) > 0.

In terms of the Voigt−Reuss−Hill approximations, B



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hope our results can provide powerful guidelines for further experimental and theoretical investigations. 3.1.2. Mechanical Stability and Elastic Properties. To be mechanically stable, the elastic stiffness constants of a given crystal should satisfy the generalized elastic stability criteria. Accurate elastic constants can help us to understand the mechanical properties and also provide very useful information to estimate the hardness of the material. The calculated elastic constants using the strain−stress method are listed in Table 2. Meanwhile, in Figure 2 the calculated elastic constants are presented for MB2 (M = Tc, W, Re, and Os) compounds in the ReB2-type and OsB2-type structures. As shown in Table 2, all studied compounds satisfy the mechanical stability criteria22 indicating that they are mechanically stable. As is well-known, elastic constants represent the ability to resist elastic deformation. From Figure 2, it is found that the elastic constants C11, C22, and C33 in different diborides are nearly same, and C33 values are extremely larger than others, which induce unusually high incompressibilities along the c axis. Among these diborides, the ReB2−ReB2 phase has the highest C33 value of 1008 GPa comparable with that of diamond (C33 = 1079 GPa). Measurements reported for ReB2 (c axis) also approach that of diamond.9 Also, it is interesting to note that the C33 values of the OsB2-type structure are only slightly lower (by about 100 GPa) than those of the ReB2-type structure suggesting that a common feature in the electronic structure contributes to their high C33 values. The C44 is an important parameter indirectly governing the indentation hardness. The C44 values of the studied diborides are also large. Especially, the diborides in ReB2-type structure have comparative large C44 (in Figure 2) indicating their relatively strong shear strength. It is worthy pointing out our calculated results are in accordance with the previous theoretical studies (Table 2). Generally speaking, superhard materials should possess high bulk modulus to support the volume decrease caused by an applied load and high shear modulus to restrict deformation in a direction different from the applied load. According to Voigt− Reuss−Hill (VRH) approximation, the calculated elastic properties, including bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio ν of four diborides with different structures are presented in Table 3. Apparently, it can be seen that the listed compounds all have large bulk moduli demonstrating that these materials are difficult to compress. For the ReB2−ReB2, B from elastic constants is 336 GPa, and BE from the third order Birch−Murnaghan equation of states is 360 GPa, in good agreement with the experimental result (360 GPa).9 As to OsB2−OsB2, the bulk moduli as measured in the experiments are in range from 343 to 395 GPa.16 Our B and BE are 314 and 340 GPa, respectively, which are in reasonable agreement with the experiment. For other crystals, the bulk moduli are more than 300 GPa and close to ReB2−ReB2. All studied compounds have high bulk moduli indicating their low compressibility. As seen in Table 3, the bulk moduli B are in consistent with BE demonstrating the reliability of the present theoretical method. It is worthy of noting that calculated values of B0′ are consistent with value of 4.0 characteristic of many materials. As is well-known, shear modulus provides a much better correlation with hardness than bulk modulus. Clearly, the ReB2−ReB2 has the largest shear modulus (291 GPa) suggesting that it can withstand the shear strain to the largest extent. In general, the ReB2-type structure possesses more excellent performance than the OsB2-type one because of the

B = (1/2)(BR + B V ), G = (1/2)(G R + G V )

Young’s modulus E and Poisson’s ration ν were obtained by the following formulas: E = 9BG /(3B + G), ν = (3B − 2G)/[2(3B + G)]

The bulk modulus BE was obtained by fitting the third order Birch−Murnaghan equation of states. BE and B from elastic constants were nearly the same demonstrating the reliability of the theoretical method. Formation enthalpies were calculated from ΔH = E(MB2) − E(solid M) − 2E(solid B). M represents transition-metal Tc, W, Re, and Os. The solid phase of boron was from its α phase.

3. RESULTS AND DISCUSSION 3.1. Determination of Crystal Structure. 3.1.1. Total Energies and Lattice Parameters. After full geometry optimization, all structures keep the same symmetry as the initial symmetries as shown in Figure 1. The calculated

Figure 1. Crystal structures of (a) OsB2-type, (b) ReB2-type, and (c) WB2-type.

equilibrium lattice parameters, volumes, densities and total energies for the four diborides MB2 (M = Tc, W, Re, and Os) are listed in Table 1. According to the calculated total energies, the relative stability order of these four diborides is ReB2−TcB2 > OsB2−TcB2, ReB2−WB2 > OsB2−WB2 > WB2−WB2, ReB2− ReB2 > OsB2−ReB2, and OsB2−OsB2 > ReB2−OsB2. The ReB2-type structure is favorite structure for the TcB2, WB2 and ReB2 crystals, and OsB2 prefers the OsB2-type structure. For the synthesized crystals WB2−WB2, ReB2−ReB2, and OsB2− OsB2, the calculated lattice parameters are very close to the experimental values with deviations less than 1%. For other crystals like ReB2−TcB2, ReB2−WB2, and OsB2−ReB2, our calculations also agree well with the previous predictions. Unfortunately, there are no available data to compare with our calculations for OsB2−TcB2, OsB2−WB2, and ReB2−OsB2. We C



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Table 1. Calculated Structural Parameters and Total Energies per Formula Unit of MB2 (M = Tc, W, Re, and Os) with Different Structures, Compared with Experimental and Other Theoretical Data diboride

a (Å)

ReB2−TcB2

2.875 2.877a 4.571 2.907 2.898b 2.994 2.982,b 2.986c 4.627 2.881 2.879,d 2.900e 4.582 4.578f 2.906 4.653 4.684g

OsB2−TcB2 ReB2−WB2 WB2−WB2 OsB2−WB2 ReB2−ReB2 OsB2−ReB2 ReB2−OsB2 OsB2−OsB2

b (Å)

2.869

2.897

2.869 2.869f 2.859 2.872g

ρ (g/cm3)

E0 (Hartree)

53.09

7.53

−87.140

53.52 56.27

7.48 12.12

−87.135 −77.041

108.10

12.62 12.77c 12.13 12.96 12.95e 12.88

−77.039

13.17 13.01 12.67g

−98.882 −98.883

V (Å3)

c (Å) 7.418 7.421a 4.081 7.690 7.681b 13.926 13.874,b 13.896c 4.199 7.410 7.436,d 7.478e 4.077 4.078f 7.301 4.066 4.077g

56.28 53.25 55.18,d 53.29e 53.59 53.40 54.08

−77.040 −96.471 −96.463

a

Reference 18, CASTEP. bReference 24, VASP. cReference 4, experiment. dReference 25, VASP. eReference 9, experiment. fReference 26, CASTEP. Reference 27, experiment.

g

Table 2. Calculated Elastic Constants (GPa) of MB2 (M = Tc, W, Re, and Os) with Different Structures diboride

C11

ReB2−TcB2

609 595a 532 536b 573 619c 607 570d 515 672 643e 595 590 487 560 570

OsB2−TcB2 ReB2−WB2 WB2−WB2 OsB2−WB2 ReB2−ReB2 OsB2−ReB2 ReB2−OsB2 OsB2−OsB2 a

C22

549 545b

559

606 589 573 540

C33

C44

943 937a 835 821b 944 993c 695 672d 883 1008 1035e 931 899 880 786 753

256 251a 199 274b 277 298c 234 202d 173 273 263e 221 293 215 121 68

C55

206 183b

327

331 198 227 191

C66

C12

C13

238 227a 231 208b 189

133 142a 183 173b 195 193c 144 145d 232 141 159e 208 185 180 171 178

98 96a 145 103b 106 115c 199 200d 151 105 129e 173 112 229 176 188

232 213d 264 265 244e 282 242 154 212 192

C23

87 144b

66

100 165 115

Reference 18, CASTEP. bReference 25, VASP. cReference 24, VASP. dReference 28 CASTEP. eReference 25, VASP.

Poisson’s ratio is an important parameter to describe the degree of directionality of the covalent bonding. The smallest Poisson’s ratio of 0.16 for ReB2−TcB2 and ReB2−ReB2 implies their strong degree of covalent bonding. The ratio value of B/G is commonly used to describe the ductility or brittleness of materials with 1.75 as the critical value.22 A B/G value higher (or lower) than the criteria is considered to be ductile (or brittle). Markedly, all diborides have smaller B/G values than critical value implying their brittle nature. In a word, the large elastic modulus, low Poisson’s ratio, and small B/G ratio show that the four studied diborides would be potential superhard materials. 3.2. Formation Enthalpy. To obtain a deep insight into the phase stability of these diborides, we have carried out calculations of the formation enthalpy. At zero pressure, the negative values of the calculated formation enthalpy indicate that all studied diborides are thermodynamic stable and can be synthesized easily. The obtained results are summarized in Table 4. Among these diborides, the ReB2−ReB2 phase has the lowest formation enthalpy of −1.29 eV, in consistent with

Figure 2. Calculated elastic constants of diborides MB2 (M = Tc, W, Re, and Os).

larger shear moduli. Totally, for the four crystals with different structures, the shear moduli are relatively large. In addition, D



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Table 3. Calculated Bulk Modulus B (GPa), Shear Modulus G (GPa), Young’s Modulus E (GPa), B/G, and Poisson’ Ratio ν of MB2 (M = Tc, W, Re, and Os) with Different Structures diboride

B

BE

B0 ′

G

ReB2−TcB2

309 314a 302 301a 319 320b 331 315 309d 336 360,e 347−377b 315 330a 334 314 348,g 312h

318

4.07

314

4.07

322

3.97

332 323

4.01 3.99

360

4.07 4.00e 4.05 4.18a 4.03 4.10 4.40g

268 277a 240 247a 252 208,b 273c 231 239 230d 291 302(18),f 285−299b 269 270a 192 200 164h

OsB2−TcB2 ReB2−WB2 WB2−WB2 OsB2−WB2 ReB2−ReB2 OsB2−ReB2 ReB2−OsB2 OsB2−OsB2

354 354a 346 340 310h

B/G

ν

623

1.15

0.16

569

1.26

0.18

598

1.26

0.28

562 572

1.43 1.32

0.22 0.20

678 712(43),f 683−699b 639 582a 485 496 419h

1.15

0.16 0.18b 0.19 0.25k 0.26 0.24 0.28h

E

1.26 1.74 1.57

a

Reference 25, VASP. bReference 28 CASTEP. cReference 30, VASP. dReference 26, CASTEP. eReference 9, experiment. fReference 31, experiment. Reference 4, experiment. hReference 29, CALYPSO.

g

Table 4. Calculated Formation Enthalpy ΔH and Mulliken Population Analysis of MB2 (M = Tc, W, Re, and Os) with Different Structures

a

diboride

ΔH

B1

B2

M

ReB2−TcB2 OsB2−TcB2 ReB2−WB2 WB2−WB2 OsB2−WB2 ReB2−ReB2 OsB2−ReB2 ReB2−OsB2 OsB2−OsB2

−1.28 −1.13 −1.06, −1.15a −1.01, −1.12,a −0.88b −1.02 −1.29, −1.34c −1.07 −0.55 −0.58

−0.34 −0.34 −0.36 −0.35 −0.34 −0.32 −0.34 −0.32 −0.30

−0.34 −0.34 −0.36 −0.49 −0.34 −0.32 −0.34 −0.32 −0.30

0.69 0.68 0.72 0.84 0.69 0.65 0.68 0.63 0.61

Reference 24, VASP. CASTEP.

b

Reference 28, CASTEP.

c

the 5d transition-metal diborides MB2 (M = W, Re, and Os), the transferred charge decrease with the increase of total valence electrons. When the transition metal has the same valence electrons, the net charges are nearly the same. For a different crystal structure, the ionicity is different. Therefore, the charge transfer effect is more influenced by the total valence electrons and the crystal structure. The trends in structural stability and the bonding of MB2 compounds can be understood from the electronic structure. The total and partial density of states (DOSs) of TcB2, WB2, ReB2, and OsB2 with different structures are calculated at zero pressure and presented in Figure 4. As seen in this figure, they are all metallic due to their finite electron DOS at Fermi level. It is found that the B-s electron in these compounds is localized and naturally its effect on bonding is very small. In the vicinity of the Fermi level, the DOSs are mainly composed of the M-d (Tc-4d and W, Re, Os-5d) and B-2p states. The typical feature of the total DOSs of all of these compounds is the presence of what is termed as a pseudogap (a sharp valley around the Fermi level) as shown in Figure 4, a borderline between the bonding and antibonding states. The presence of pseudogap will surely increase the stability of these compounds. There are two mechanisms to propose for formation of a pseudogap in the binary alloy: one is of ionic origin, and the other is hybridization effects.36 The electronegativity difference between M (M = Tc, W, Re, and Os) and B atoms is small, and, hence, the ionicity does not play a major role on the bonding behavior of these compounds. Consequently, the present pseudogap is mainly due to strong hybridization between the metal and boron atoms. Indeed, it is clearly seen that the DOSs of M-d and B-2p have a similar shape in all diborides, which indicates that there are strong hybridizations between the M-d and B-2d in them. Therefore, strong covalent bonding exists in these compounds. The strong covalent bonding would be beneficial to their high bulk and shear moduli. In a word, the chemical bond between metal (Tc, W, Re, and Os) and B atoms is mainly strong covalent bonding with partial ionic and metallic contributions. The role of metallic components should be considered in the evaluation of the hardness.

Reference 32,

previous results (−1.34 eV).31 The second lowest formation enthalpy of −1.28 eV is calculated to be the ReB2−TcB2 phase. Furthermore, the thermodynamic stability of various structural modifications as a function of pressure is investigated. As shown in Figure 3, it is confirmed that the ReB2-type structure is most stable for TcB2, WB2, and ReB2 under low pressure. Results further reveal that the WB2−WB2 becomes favorable above 67 GPa. Moreover, it is interesting to note that the OsB2−OsB2 is stable at zero pressure and transforms to ReB2−OsB2 above 81 GPa. As discussed above, for different transition metals the same structure has different stability. The difference could be intimately related to the largest atomic radius in the Os elements. It is previously suggested that the atomic radius is the key to describe the variation of the crystal structure in transition diborides, which is the same as transition nitrides. 3.3. Electronic Structures. The electronic properties of four diborides MB2 (M = Tc, W, Re, and Os) are investigated by analyzing their net charges and density of states. The atomic population giving the net charges on transition-metal and boron atoms is summarized in Table 4. It is clearly seen from Table 4 the net charges on transition-metal atoms are positive and on the boron atoms are negative. Therefore, the chemical bonding between M and B has some character of ionicity. For E



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Figure 3. Relative formation enthalpy−pressure diagram of diborides (a) TcB2, (b) WB2, (c) ReB2, and (d) OsB2.

3.4. Hardness. From the above investigation, the studied diborides have large bulk and shear moduli, low Poisson’s ratio, small B/G ratio, and strong covalent bonding demonstrates the large possibility of high hardness. However, according to the electronic structure analysis, the metallic components exist in these diborides. Next, we will conclusively estimate their theoretical hardness according to our semiempirical hardness model, which includes the role of metallic bond. Because metallic bonding is delocalized and not directly related to hardness,34 for hardness calculation of crystals with partial metallic bonding a correction of metallic bonding in the formula should be considered. The screening effect of the metallic component may be described by introducing a correction factor of exp(−βf nm), where β and n are two constants. Thus, the expression of hardness for the type of crystals similar to transition-metal carbides, nitrides, and borides can be written as Hν(GPa) =

APv b−5/3

exp( −βf mn )

where vb is the volume of bond, P is Mulliken population, and f m is the metallicity. The metallicity f m can be estimated as nm/ ne, where nm and ne are the numbers of electrons that can be excited at the ambient temperature and the total number of the valence electrons in a unit cell, respectively. According to the electronic Fermi liquid theory, f m can be written as fm =

= ln A −

βf mn

μ

μ

Hν = [∏ (Hνμ)n ]1/ ∑ n

(1)

μ

(5)

where nμ is the number of μ-type bond in the complex compound. The volume of bond is expressed as35 v bμ = (d μ)3 /∑ [(d ν)3 Nbν]

(2)

ν

(6)

Using eq 3 and the data in Table S1 of the Supporting Information, we calculate the hardness of the 13 rocksaltstructured transition-metal carbides or nitrides. Considering the metallic components, our calculated Vickers hardness values agree well with the experimental data. Meanwhile, the hardness of semiconductor crystals with a zinc blende structure are also

ln(Hνv5/3 b /P) is plotted against f m using a test of materials with known hardness. Finally, the obtained exponential regression equation of hardness can be written as: Hν(GPa) = 699Pv b−5/3 exp( −3005fm1.553 )

(4)

where DF is the electron density of states at the Fermi level. But for the compounds like the recently synthesized p-PtN2 and mOsN2, in fact, they are complex transition-metal compounds. Namely, they contain at least two types of chemical bonds in their unit cell. In this case, the hardness can be expressed as the average of hardness of all hypothetical binary systems in the complex compound34

To determine the values of A, β, and n, the equation is rewritten in the following form: ln(Hνv b5/3/P)

nm 0.026DF = ne ne

(3) F



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their calculated hardness values are in good agreement with the experimental results. Also, the hardness values in Tables S1 and S2 of the Supporting Information are in consistent with results from Chen’s model,36 and Gao’s model,34 indicating the reliability of our model. Using the obtained explicit expression of hardness for covalency-dominant solids, we predict the theoretical Vickers hardness of the diborides with different structures. The calculated bond parameter and hardness of four diborides MB2 (M = Tc, W, Re, and Os) are listed in Table 5. Both ReB2type and OsB2-type structures contain 18 bonded bonds in single crystal, 6 B−B bonds, and 12 M−B bonds. In the ReB2type structure, the 6 B−B bonds have same bond length, but the OsB2-type does not. A similar situation can also be found with M−B bonds. Meanwhile, the shortest B−B and M−B bonds as well as the Vickers hardness are plotted in Figure 5. It is noted that the shortest B−B bond distance of MB2 falls in the single bond (1.72 Å for B2F4),38 and the shortest M−B bond is about 2.2 Å. According to our calculations, the obtained hardness values of TcB2, WB2, ReB2, and OsB2 in ReB2-type structure are 34.0, 35.7, 39.1, and 34.5 GPa, respectively. In the OsB2-type structure, there are 14.7, 31.8, 29.3, and 30.1 GPa. These crystals are hard materials, and ReB2−ReB2 is nearly the superhard material. The origin of the ultraincompressibility correlates not only with the strong covalency of B−B and M−B bonds but also with the local buckled structure of interconnected covalent bonds. From Figure 5, it is found that the hardness of ReB2-type structure is larger than the OsB2-type. In the ReB2-type structure, there exist well-defined zigzag covalent chains along the c direction, interconnected by shared B and Re atoms, and there are also Re−B covalent bonds directly along the c axis, although not as strong as the Re−B bonds in the chains. The B−B bonding is strongly complemented by the Re−B covalent bonds. For the OsB2-type structure, the diverse bond length in B−B and B−M would decrease the hardness. Interestingly, the Vickers hardness of WB2 is estimated to be similar to that of superhard ReB2. Three considered structures are investigated for WB2, that is, ReB2-, OsB2-, and WB2-type structures. OsB2−WB2 is the hardest one in OsB2-type structure. We attribute the origin of the high hardness coming from the W−W bond whose population is 0.05. The W−W bonds are characterized by the metallic bonds, but there is also some covalence character, which has high mechanical strength. The hardness of WB2−WB2 is 25.6 GPa, which also possesses excellent mechanical properties. So, the crystal may have larger hardness under high pressure when considering the phase transition. These features make WB2 a good candidate along with other hard materials for possible application under extreme conditions. We hope that the microscopic models of hardness would play an important role in the search for new hard materials.

4. CONCLUSIONS Using the pseudopotential plane-waves based on the density functional theory, the structural, elastic properties and hardness of four diborides MB2 (M = Tc, W, Re, and Os) are investigated. From the above studies, we have the following conclusions: (i) The structural and elastic properties for the four diborides with different structures have been derived. The calculated results are found to be in good agreement with experimental and theoretical values. For all studied

Figure 4. Total and partial density of states for diborides MB2 (M = Tc, W, Re, and Os). The vertical dashed line at the zero is at the Fermi energy level.

calculated and listed in Table S2 of the Supporting Information. It is found that for these semiconductor crystals f m = 0, and G



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Table 5. Calculated Bond Parameters and Vickers Hardness of MB2 (M = Tc, W, Re, and Os) with Different Structures diboride

bond

d (Å)

vb (Å3)

P

f m (10−3)

Hv. calcd

ReB2−TcB2

B−B Tc−B B−B(1) B−B(2) Tc−B(1) Tc−B(2) B−B W−B B−B(1) B−B(2) W−B(1) W−B(2) W−W B−B(1) B−B(2) W−B(1) B−B Re−B B−B(1) B−B(2) Re−B(1) Re−B(2) B−B Os−B B−B(1) B−B(2) Os−B(1) Os−B(2)

1.801 2.241 1.788 1.814 2.191 2.238 1.787 2.330 1.774 1.818 2.254 2.318 2.911 1.727 1.838 2.335 1.807 2.240 1.790 1.815 2.191 2.241 1.833 2.221 1.796 1.883 2.154 2.193

1.825 3.515 1.831 1.913 3.368 3.593 1.728 3.827 1.124 1.211 2.309 2.510 4.968 2.521 3.038 6.229 1.846 3.515 1.836 1.912 3.365 3.602 1.952 3.474 1.926 2.218 3.321 3.508

0.693 0.190 0.610 0.710 0.010 0.145 0.693 0.217 0.540 0.695 0.180 0.150 0.050 0.757 0.650 0.260 0.643 0.247 0.530 0.650 0.160 0.180 0.630 0.233 0.580 0.560 0.120 0.240

0 1.233 0 0 1.711 1.711 0 0.919 0 0 0.998 0.998 0.998 0 0 1.787 0 1.311 0 0 1.508 1.508 0 2.002 0 0 1.213 1.213

34.0

OsB2−TcB2

ReB2−WB2 OsB2−WB2

WB2−WB2

ReB2−ReB2 OsB2−ReB2

ReB2−OsB2 OsB2−OsB2

a

Hv. exptl

14.7

35.7 31.8

25.6

27.7a 20.6 ± 2b

39.1

39.3,a 30.1−48c 31.1−20.7d

29.3

34.5 30.1

23.5a 21.6 ± 3e 24.8−17.8f

Reference 4. bReference 37. cReference 9. dReference 15. eReference 31. fReference 16.

Figure 5. Vickers hardness Hv (GPa) and the shortest B−B and M−B bonds lengths dB−B and dM−B (Å) of the diborides MB2 (M = Tc, W, Re, and Os).

H



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(5) Wang, M.; Li, Y. W.; Cui, T.; Ma, Y. M.; Zou, G. T. Origin of Hardness in WB4 and its Implications for ReB4, TaB4, MoB4, TcB4, and OsB4. Appl. Phys. Lett. 2008, 93, No. 101905. (6) Ono, S.; Kikegawa, T.; Ohishi, Y. A High-Tressure and HighTemperature Synthesis of Platinum Carbide. Solid State Commun. 2005, 133, 55−59. (7) Zhao, Z. S.; Wang, M.; Cui, L.; He, J. L.; Yu, D. L.; Tian, Y. J. Semiconducting Superhard Ruthenium Monocarbide. J. Phys. Chem. C 2010, 114, 9961−9964. (8) Li, Y. W; Wang, H.; Li, Q.; Ma, Y. M; Cui, T.; Zou, G. T. Twofold Coordinated Ground-State and Eightfold High-Pressure Phases of Heavy Transition Metal Nitrides MN2 (M = Os, Ir, Ru, and Rh). Inorg. Chem. 2009, 48, 9904−9909. (9) Chung, H. Y.; Weinberger, M. B.; Levine, J. B.; Kavner, A.; Yang, J. M.; Tolbert, S. H.; Kaner, R. B. Synthesis of Ultra-Incompressible Superhard Rhenium Diboride at Ambient Pressure. Science 2007, 316, 436−439. (10) Cumberland, R. W.; Weinberger, M. B.; Gilman, J. J.; Clark, S. M.; Tolberks, S. H.; Kamer, R. B. Osmium Diboride, an UltraIncompressible, Hard Material. J. Am. Chem. Soc. 2005, 127, 7264− 7265. (11) Zhang, X. H.; Hilmas, G. E.; Fahrenholtz, W. G. Synthesis, Densification, and Mechanical Properties of TaB2. Mater. Lett. 2008, 62, 4251−4253. (12) Mohammadi, R.; Lech, A. T.; Xie, M.; Weaver, B. E.; Yeung, M. T.; Tolbert, S. H.; Kaner, R. B. Tungsten Tetraboride, an Inexpensive Superhard Material. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 10958− 10962. (13) Dubrovinskaia, N.; Dubrovinsky, L.; Solozhenko, V. L. Comment on “Synthesis of Ultra-Incompressible Superhard Rhenium Diboride at A mbient Pressure. Science 2007, 318, 1550c. (14) Chung, H. Y.; Weinberger, M. B.; Levine, J. B.; Cumberland, R. W.; Kavner, A.; Yang, J. M.; Tolbert, S. H.; Kaner, R. B. Response to Comment on “Synthesis of Ultra-Incompressible Superhard Rhenium Diboride at A mbient Pressure. Science 2007, 318, 1550d. (15) Locci, A. M.; Licheri, R.; Orru, R.; Cao, G. Reactive Spark Plasma Sintering of Rhenium Diboride. Ceram. Int. 2009, 35, 397− 400. (16) Ivanovskii, A. L. Mechanical and Electronic Properties of Diborides of Transition 3d−5d Metals from first Pinciples: Toward Search of Novel Ultra- Incompressible and Superhard Materials. Prog. Mater. Sci. 2012, 57, 184−228. (17) Šimůnek, A. How to Estimate Hardness of Crystals on a Pocket Calculator. Phys. Rev. B 2007, 75, No. 172108. (18) Aydin, S.; Simsek, M. First-Principles Calculation of MnB2, TcB2, and ReB2 with in the ReB2-Type Structure. Phys. Rev. B 2009, 80, No. 134107. (19) MATERIALS STUDIO, version 4.1; Accelrys Inc.: San Diego, CA., 2006. (20) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (21) Monkhorst, H. J.; Pack, J. D. Special Points for Brillouin-zone Integrations. Phys. Rev. B 1976, 13, 5188−5192. (22) Wu, Z. J.; Zhao, E. J.; Xiang, H. P.; Hao, X. F.; Liu, X. J.; Meng, J. Crystal Structures and Elastic properties of Superhard IrN2 and IrN3 from First Principles. Phys. Rev. B 2007, 76, No. 054115. (23) Hill, R. The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc., London, Sect. A 1952, 65, 349−354. (24) Zhao, E.; Meng, J.; Ma, Y.; Wu, Z. Phase Stability and Mechanical Properties of Tungsten Borides from First Principles Calculations. Phys. Chem. Chem. Phys. 2010, 12, 13158−13165. (25) Wang, Y. X. Elastic and Electronic Properties of TcB2 and Superhard ReB2: First-Principles Calculations. Appl. Phys. Lett. 2007, 91, No. 101904. (26) Hao, X. F.; Wu, Z. J.; Xu, Y. H.; Zhou, D. F.; Liu, X. J.; Meng, J. Trends in Elasticity and Electronic Structure of 5d Transition Metal Diborides: First-Principles Calculations. J. Phys.: Condens. Matter 2007, 19, No. 196212.

diborides, the formation enthalpies are negative indicating that all are thermodynamically stable and can be synthesized in experiment. The high pressure structural phase transition is found for the WB2 and OsB2 crystals, whereas the TcB2 and ReB2 have the structural stabilities in the range of 0−100 GPa. (ii) Seen from the calculated DOSs, all studied diborides are metallic. The bond nature of four diborides is described as covalent-like due to the hybridization of M-d and B-2p states, but there is also some ionic character with electron transfer from the transition-metal to boron atom as well as the obvious metallic character. Both strong covalency and a zigzag topology of interconnected bonds underlie the ultraincompressibility. (iii) On the basis of the first-principle calculations, a semiempirical method to evaluate the hardness of multicomponents crystals with partial metallic bonding is presented. The present theoretical results of WB2− WB2, ReB2−ReB2, and OsB2−OsB2 are in reasonable agreement with experimental data. Moreover, the ReB2− ReB2 has the largest hardness (39.1 GPa) among the studied diborides. In addition, the larger shear modulus, larger Young’s modulus, lower Poisson’s ratio, and smaller B/G ratio of ReB2−ReB2 indicate that it is a potential superhard material. We hope that these calculations will stimulate extensive experimental work on these technologically important diborides.

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ASSOCIATED CONTENT

S Supporting Information *

Calculated bond parameters and Vickers Hardness of transition-metal monocarbides and mononitrides as well as crystals with the zinc blende structure. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Nos. 11274235 and 11104190) and the Doctoral Education Fund of Education Ministry of China (Nos. 20100181110086 and 20110181120112).

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REFERENCES

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