The calculated equilibrium lattice parameters a₀, c₀, and elastic constants C_ij of 5d transition metal diborides with hexagonal ReB₂ structure are shown in Table 1. The values of bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio ν are calculated in terms of the five independent elastic constants of C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄, which are also shown in Table 1. The mechanical stability criteria are C₄₄ > 0, C₁₁ > |C₁₂|, .

Table 1.

Table 1.

Table 1. Calculated values of equilibrium lattice parameters a0 (Å), c0 (Å), elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), and Poisson’s ratio ν for 5d transition metal diborides with ReB2 structure. . a0 c0 C11 C12 C13 C33 C44 B G E ν HfB2 3.084 8.535 194 246 87 469 69 187 – – – TaB2 3.021 8.153 143 408 118 600 125 239 – – – Theor. 2.993a 8.081a – – – – – 245a – – – WB2 2.927 7.743 601 156 95 938 276 311 269 626 0.164 ReB2 2.908 7.490 633 173 118 1034 263 341 271 643 0.186 Expt. 2.900b 7.478b – – – – – 360b – – – Theor. 2.872c 7.405c 716c 151c 133c 1108c 290c 359c 310c 725c 0.17c 2.880d 7.420d 675d 147d 115d 1081d 278d 347.7d 295.4d 690.6d 0.17d 2.8809e 7.4096e 667.9e 136.7e 147.4e 1062.7e 273.2e 354.5e 289.4e 682.5e 0.1791e 643f 159f 129f 1035f 263f 344f 304f 642f 0.21f OsB2 2.930 7.330 465 171 227 869 211 323 187 469 0.258 Theor. 2.941g 7.338g 461g 177g 236g 873g 209g 326g 207g 513g 0.24g IrB2 3.092 6.999 364 202 236 748 126 296 113 302 0.330 Theor. 3.072a 7.074a 315a 197a 254a 695a 126a 280a 109a 289a 0.33a PtB2 3.103 7.258 354 150 191 560 24 251 62 172 0.386 a Ref. [9] b Ref. [1] c Ref. [17] d Ref. [18] e Ref. [19] f Ref. [20] g Ref. [4]

Table 1. Calculated values of equilibrium lattice parameters a0 (Å), c0 (Å), elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), and Poisson’s ratio ν for 5d transition metal diborides with ReB2 structure. .

For hexagonal phase,^[15] the Voigt bulk modulus B_V and Reuss bulk modulus B_R are calculated from the following formulas in terms of the elastic constants:

Hill^[16] proved that the Voigt and Reuss equations represent upper and lower limits of the true polycrystalline constants, respectively. Hence, the bulk modulus and shear modulus of hexagonal crystal can be approximated by

It can be seen from Table 1 that the calculated equilibrium lattice parameters and elastic properties for TaB₂, ReB₂, OsB₂, and IrB₂ compare favorably with the previous theoretical values.^{[4,9,17–20]} For the only experimentally synthesized 5d transition metal diborides, ReB₂, the calculated theoretical lattice parameters, a₀ = 2.908 Å, c₀ = 7.490 Å, and bulk modulus, B = 341 GPa, are in good agreement with the available experimental values,^[1] a₀ = 2.900 Å, c₀ = 7.478 Å, B = 360 GPa. The calculated elastic constants as shown in Table 1 indicate that all 5d transition metal diborides are mechanically stable except HfB₂ and TaB₂. Among the mechanically stable diborides, ReB₂ has the largest bulk modulus (341 GPa), shear modulus (271 GPa), Young modulus (643 GPa), followed by WB₂, which has the smallest Poisson’s ratio (0.164).

The calculated total and partial densities of states (TDOSs and PDOSs) of 5d transition metal diborides at zero pressure are shown in Fig. 2, where the vertical line is the Fermi level (E_F). A strong hybridization between TM-5d and B-2p orbitals results in covalent bonding. The metallic behaviors for 5d transition metal diborides are indicated by the finite TDOSs at the Fermi level, namely N(E_F). It is found that the electrons from both TM-5d and B-2p orbitals contribute to the DOSs at the Fermi level. The DOSs at the Fermi level for HfB₂ and TaB₂ are 4.631 electrons/eV and 4.988 electrons/eV, respectively, which are much larger than those for other 5d transition metal diborides.

Fig. 2.

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	Fig. 2. Total and partial densities of states for 5d transition metal diborides with ReB2 structure. Vertical dotted line at zero indicates Fermi energy level. The position of Ep is indicated by an arrow.

The position of pseudogap (E_p) is determined by the PDOS of B-2p.^[11] The structural stability is related to the position of E_F and the pseudogap.^[21–25] For HfB₂ and TaB₂, the Fermi level falls below the pseudogap, indicating that not all the bonding states are filled. As the atomic number increases, all the bonding states are just filled for WB₂. For ReB₂, OsB₂, IrB₂, and PtB₂, the Fermi level falls above the pseudogap, which suggests that all the bonding states and partial antibonding states are filled. On the basis of the band filling theory,^[22,23] the cohesion (or stability) will be enhanced or reduced by filling bonding or antibonding/nonbonding states. Thus, HfB₂ and TaB₂ require some extra electrons to reach maximum stability, while ReB₂, OsB₂, IrB₂, and PtB₂ can improve the stability by extracting electrons. The nearly saturated bonding states and just unoccupied antibonding/nonbonding states^[26] for WB₂ result in larger bulk modulus, shear modulus and the smallest Poisson’s ratio.

The charge density distributions in plane for WB₂ is shown in Fig. 3 in order to have an insight into the bonding behavior for 5d transition metal diborides. The electron density around W atoms is much larger than that near B atoms, indicating that W has a higher valence electron density. The charge density between W and B atoms indicates a strong covalent effect, resulting in favorable elastic properties. In addition, the covalent bonding between B and B atoms is much stronger than that of W–B, which will enhance the hardness for WB₂.

Fig. 3.

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	Fig. 3. (color online) Charge density distribution in plane for WB2 with ReB2 structure.

In general, the hardness values of μ-type bond for complex multibond compounds are calculated according to the following formulas proposed by Gao:^[27]

Since the electrons occupying the levels above E_p become delocalized and not directly related to hardness, a correction to formula (12) should be considered for crystals with partial metallic bonding as shown below

The formulas proposed by Gou et al.^[10] only give a method of calculating the metallic population for single bond compounds with metallic bonding by using the total number of effective free electrons defined as

But for multibond compounds with metallicity, it is difficult to determine the partial number of effective free electrons for each bond. Segall et al.^[28] found the correlations of overlap population with bond strength, so for multibond compounds with metallicity, the partial number of effective free electrons for each bond can be calculated by the total and partial Mulliken overlap population. The calculation formulas are as follows:

For multibond compounds, the weakest bond will play a determinative role in the hardness of multibond material.^[27] In other words, if there are differences in strength among different bonds, the bonds will start to break from a softer bond. The hardness of 5d transition metal diborides are shown in Table 2. The hardness calculated from B–B bond is much larger than that from TM–B bond, so the smaller hardness, that is the hardness of TM–B bond, will be taken as the hardness of 5d transition metal diborides.

Table 2.

Table 2.

Table 2. Calculated values of bond distance dμ (Å), Mulliken overlap population Pμ, bond volume (Å3), partial number of effective free electrons , metallic population P′μ, metallicity , and hardness (GPa) of TM–B and B–B bonds for 5d transition metal diborides with ReB2 structure. . dμ Pμ P′μ Hv HfB2 Hf–B 2.594 0.173 4.962 1.047 0.015 0.086 8.1 8.1 B–B 1.848 0.790 1.792 2.385 0.034 0.043 211.5 TaB2 Ta–B 2.478 0.243 4.468 0.804 0.012 0.051 14.1 14.1 B–B 1.831 0.710 1.803 1.173 0.018 0.026 191.7 WB2 W–B 2.342 0.220 3.898 0.029 0.001 0.002 16.8 16.8 B–B 1.802 0.690 1.776 0.046 0.001 0.001 195.9 ReB2 Re–B 2.262 0.243 3.621 0.853 0.016 0.064 19.7 19.7, 30.1 ± 1.3 ~ 48.0 ± 5.6a, 20.7 ~ 31.1b, 37.0 ± 1.2c, 16.9 ± 0.6 ~ 18.8 ± 2d B–B 1.824 0.647 1.897 1.133 0.021 0.032 159.3 OsB2 Os–B 2.229 0.243 3.525 0.654 0.012 0.049 21.0 21.0 B–B 1.855 0.617 2.031 0.829 0.015 0.025 136.6 IrB2 Ir–B 2.219 0.267 3.560 1.289 0.022 0.083 21.8 21.8 26.65e B–B 1.983 0.577 2.539 1.394 0.024 0.042 86.6 PtB2 Pt–B 2.322 0.183 3.940 1.634 0.027 0.147 11.8 11.8 B–B 1.914 0.703 2.206 3.134 0.052 0.074 128.9 a Ref. [1] b Ref. [29] c Ref. [30] d Ref. [33] e Ref. [9]

Table 2. Calculated values of bond distance dμ (Å), Mulliken overlap population Pμ, bond volume (Å3), partial number of effective free electrons , metallic population P′μ, metallicity , and hardness (GPa) of TM–B and B–B bonds for 5d transition metal diborides with ReB2 structure. .

As shown in Table 2, Ir–B bond possesses the largest Mulliken overlap population, indicating the strongest covalency. So IrB₂ has the largest hardness (21.8 GPa), which is slightly smaller than the corresponding theoretical value (26.65 GPa) calculated by Zhao and Wang,^[9] which maybe because the metallicity is considered in our calculations. Among all 5d transition metal diborides, only experimental hardness values from 16.9 GPa to 48.0 GPa are obtained for ReB₂. The microhardness of ReB₂ measured by Chung et al.^[1] are 30.1 ± 1.3 GPa and 48.0 ± 5.6 GPa under applied loads of 4.9 N and 0.49 N, respectively, while lower hardness values of 20.7 GPa and 31.1 GPa are obtained by Locci et al.^[29] under the same loads. This may be due to the different synthesized methods while Chung et al.^[1] took advantage of a casting technique and Locci et al.^[29] obtained bulk ReB₂ through sintering. It is currently in dispute over the hardness of ReB₂. The measured hardness usually increases with load decreasing. The noticed inverse relationship of load to hardness results from the well-known indentation size effect (ISE).^[29] The ISE makes the hardness results larger under smaller loads, and this scenario is just the opposite under larger loads. That is to say, the smaller loads applied to ReB₂ by Locci et al.^[29] and Chung et al.^[1,30] will result in larger hardness results. Dubrovinskaia et al.^[31] also suggested that the load which is used to measure the hardness of ReB₂ is not in the asymptotic-hardness region. Gao F M and Gao L H^[32] suggested that a comparative load should be chosen to test the accurate Vickers hardness. The load is taken approximately as , where is the calculated hardness of compoud and is the Vickers hardness of diamond under 9.8-N load, about 96 GPa. So we should choose larger load while measuring compound with larger hardness to avoid the ISE. Qin et al.^[33] adopted larger loads, 2.94 N, 4.9 N, 9.8 N, 29.4 N, and 49 N, and the hardness of ReB₂ are 18.8 ± 2 GPa, 18.4 ± 1 GPa, 18 ± 1.2 GPa, 17.2 ± 0.8 GPa, 16.9 ± 0.6 GPa, respectively, in agreement with our calculations (19.7 GPa), which implies that our calculated results are reliable.

Thus, the hardness values of 5d transition metal diborides with ReB₂ structure are all smaller than 40 GPa, indicating that they are not superhard materials. However, besides instable HfB₂ and TaB₂, the mechanical properties and hardness of 5d transition metal diborides with ReB₂ structure are both remarkable, meaning that they are good candidates for hard materials.