Designing superhard metals: The case of low borides Open Access

Hardness is the resistance of materials against localized plastic deformation, corresponding to the formation and movement of dislocations, and thus hard and superhard materials are very useful for a wide range of applications from reducing wear to creating machining tools.¹ Traditional superhard materials like diamond and cubic boron nitride (cBN) are made up of light elements. Although these pure covalent solids possess exceptional hardness, they are thermodynamically metastable phases that must be synthesized under high pressures and temperatures, making them expensive. Moreover, diamond is nonresistant to oxidation and reactive to ferrous metals at moderate temperatures. These limitations stimulate the search for new superhard phases that exhibit high stability and are easy to synthesize. A great success has been achieved with the discovery of many hard or superhard transition-metal light-element compounds.^2–6 Among them, transition-metal borides (TM_1-xB_x) including diborides of Os and Re,^7,8 triborides (originally reported as tetraborides) of W and Mo,^9–20 tetraborides of Cr, Mn and Fe,^21–28 and dodecaborides of Zr, Y and Sc,^29,30 have currently been the focus of promising superhard materials. Implicit in such efforts is that the dense TM atoms provide high valence electron density to resist volume compression while the rich boron atoms form strong covalent network to counteract shape deformation, both of which contribute to the high hardness. As such, the hardness of TM_1-xB_x is enhanced by increasing the boron content in their lattices. It is not surprising that the aforementioned superhard materials are the highest borides in the respective TM-B systems, and they thus are strong covalent crystals.

Most transition metals, however, form borides with low boron content, which results in a greatly reduced level of covalent bonding and a significantly high degree of metallic bonding. Such metallic bonds are favorable for electron transport (good metallicity) but not for high hardness due to their low resistance against the formation and movement of dislocations. In this sense, it is a huge challenge to synthesize superhard metals from low borides. A typical example is that the highest boride of tungsten WB₃ (nonstoichiometric WB₄) is superhard (43.3-46.3 GPa under the load of 0.49 N),^9,10 and this is attributed to its strong covalent network formed by hexagonal boron sheets interconnected with zigzag W-B bonds.³¹ In contrast, the low boride of tungsten WB is not naturally superhard due to the presence of metal bilayers.^32,33 Surprisingly, Yeung et al.³⁴ have recently synthesized W_0.5Ta_0.5B by substituting Ta onto the W sites of WB, and its measured hardness touches 42.8 GPa under the load of 0.49 N, marking it as the first known superhard boride with relatively low boron content. Through systematic first-principles calculations, we have found that such an anomalous hardening from WB to W_0.5Ta_0.5B is attributed to the relief of the fierce dd antibonding interaction.³⁵ This exciting material has raised great expectations for a new generation of superhard metals with a good balance between hardness and ductility, but its asymptotic hardness is still less than the superhard threshold (H ≥ 40 GPa). It is therefore highly desired to develop a new strategy to further enhance hardness.

The systems of boron with the IVth to VIth group TMs have been synthesized experimentally.^36–40 Many first-principles studies^41–44 have been performed on their elastic properties and these calculations provide useful information for the understanding of mechanical characters. In this work, using these well-known and accessible boron-low TM_1-xB_x as illustrating examples, we propose a microscopic mechanism to enhance hardness by seeking highly stable thermodynamic phases and locking their weak slip planes. Of particular interest is the correlation between hardness and boron content. Our results show that the hardness of TM_1-xB_x (TM=V, Nb and Ta) first increases then decreases with the rise of boron content, contrasting with the common sense that the high boron content gives rise to the high hardness. Such an anomalous phenomenon originates from their peculiar electronic mechanisms. In addition, we suggest the possibility of enhancing the extrinsic hardness and achieving true superhard metals by creating a multiphase material with a large number of interfaces among different phases.

All calculations within the framework of density functional theory (DFT) are performed using the plane-wave basis set as implemented in the Vienna Ab Initio Simulation Package (VASP) code⁴⁵ and the electron-ion interactions are described by means of projector augmented wave (PAW). The exchange-correlation functionals within the generalized gradient approximation (GGA)⁴⁶ are employed. A large cutoff energy of 500 eV and the dense k-point meshes with a grid of 0.02 Å^-1 are adopted for the considered phases to ensure the energy convergence of about 1 meV per atom.

After having fully relaxed the studied phases with respect to their lattice constants and internal atomic positions, we calculate the formation energy defined as ΔE = E(TM_1−xB_x)−(1−x)E(TM)−xE(B). According to this definition, the negative formation energy means that the phase is energetically stable. The lower the formation energy is, the more stable the phase is. The elastic constants, bulk moduli, shear moduli, Young’s moduli, and Poisson’s ratios (v) are obtained by the efficient strain-energy method.^47–49 Since the hardness is a relatively complex quantity to describe the resistance of materials to plastic deformation and can be dramatically affected by extrinsic conditions, we first use Chen’s model⁵⁰ to estimate the hardness, then do Tian’s model⁵¹ to cross-check the overall trends.

We initially search for highly stable thermodynamic phases by calculating the formation energy. Based on the superhard W_0.5Ta_0.5B structure [see Fig. 1(a)], fifteen candidate phases (TiB, Ti_0.5Cr_0.5B, VB, V_0.5Cr_0.5B, and CrB in the 3d series; ZrB, Zr_0.5Nb_0.5B, NbB, Nb_0.5Mo_0.5B, and MoB in the 4d series; HfB, Hf_0.5Ta_0.5B, TaB, Ta_0.5W_0.5B, and WB in the 5d series) are chosen to examine their stability trends. Figure 2(a) shows our obtained formation energies of these TM_1-xB_x phases as a function of valence electron density. When the average valence electron number equals to 4 electrons per atom (e/atom), this type of monoborides (i.e., VB, NbB, and TaB) has the lowest formation energy among the 3d, 4d and 5d series, respectively.

After having obtained the highly stable VB, NbB, and TaB phases, we fine-tune their boron contents and broaden them to other compositions. Six composition phases (e.g., TM₃B₂, TMB, TM₅B₆, TM₃B₄, TM₂B₃ and TMB₂) are considered since they are well documented experimentally.³⁶ Figure 2(b) gives our calculated convex curves of the formation energy, which are a bit different from previous trends⁴¹ but in agreement with the trend of the V_1-xB_x system.⁴⁴ Interestingly, the six structures are thermodynamic ground states at different boron compositions. Besides concrete values, the three systems share nearly same stability trends, but the V_1-xB_x phases are more thermodynamically stable than the corresponding Nb_1-xB_x, and Ta_1-xB_x ones. The formation energy quantitatively characterizes the comprehensive information of a crystal, such as strong bonding, optimal band filling, and favorable bond topology, thus the highly thermodynamic stable solid tends to own the high resistance against dislocations.⁵² For x from 0.5 to 0.6, the V_1-xB_x phases have very low formation energies, and this implies that they could be promising superhard materials.

We next evaluate the mechanical properties of the TM_1-xB_x phases. Although several previous works predicted the mechanical properties of some TM_1-xB_x phases,^39,41,42 their results are quite scattered and in parts contradict each other due to the inconsistency in different techniques, and thus we reexamine these properties. Figure 3 shows their bulk moduli, shear moduli, and hardness as a function of the boron content. Although their mechanical properties vary widely with different boron contents, the overall trends of the three systems are very similar. This is not unexpected because V, Nb, and Ta lie in the same group on the periodic table. Most remarkable is that the changing trend of incompressibility is obviously different from that of hardness. As shown in Fig. 3(a), the bulk modulus increases with the rise of boron contents. The Ta_1-xB_x phases are more incompressible than the corresponding Nb_1-xB_x and V_1-xB_x ones, and TaB₂ becomes the most incompressible material among them. In contrast, in Fig. 3(b)–3(c), the shear modulus and hardness ascend gradually with the boron content from x=0.4 (TM₃B₂) to x=0.6 (TM₂B₃), but then descend from x=0.6 (TM₂B₃) to x=2/3 (TMB₂). Hence, the hardness maximum occurs at the boron contents x=0.6 (TM₂B₃). A positive correlation between hardness and boron content is widely expected, driving many attempts to hunt for superhard materials from the high-boron-content borides. It is rather surprising that the TM_1-xB_x (TM=V, Nb and Ta) systems do not follow such a monotonic correlation. A similar anomalous stress response has recently been reported in the WB_x compounds.⁵³ 

On the other hand, a large bulk modulus is often found in traditional superhard materials. For example, the hardest material, diamond, also has the largest bulk modulus. A theory of bulk modulus of covalent solids is useful for suggesting schemes to increase hardness.⁵⁴ The Ta_1-xB_x phases possess the highest bulk modulus among the same-composition phases of the three systems, but their hardness values are not the highest. The V_1-xB_x phases with lower bulk moduli are much harder than the corresponding phases of the other two systems. Completely opposite to the traditional superhard materials in which the bulk modulus and hardness share the same trend, this class of TM_1-xB_x (TM=V, Nb and Ta) exhibits obviously different trends between the hardness and bulk modulus.

Through seeking highly stable phases and fine-tuning their boron contents, the four low borides (VB, V₅B₆, V₃B₄ and V₂B₃) are identified to possess the highest hardness among these materials. Table I lists their lattice parameters and mechanical properties. For comparison, the related results of W_0.5Ta_0.5B are also listed. Calculated elastic constants manifest that these four phases are mechanically stable. Their shear moduli (≥ 232GPa), Young’s moduli (≥ 543GPa), Poisson’s ratios (≤ 0.171), and hardness values (≥ 36.6GPa) are by far superior to the corresponding values (201 GPa, 497 GPa, 0.239, 32.6 GPa, respectively)³³ of the recently synthesized W_0.5Ta_0.5B phase, and even match that of the high boride WB₃ (252 GPa, 588 GPa, 0.168, 38.9 GPa, respectively).^15,31 To the best of our knowledge, the four borides are the hardest borides with the same TM/B composition ratios.

As widely accepted, the high hardness usually exists in strong covalent solids like high borides. It therefore is a bit puzzling that the four low borides are so hard, and several fundamental questions remain to be answered: (i) Why does VB exhibit higher stability and hardness than the newly reported W_0.5Ta_0.5B? (ii) Why do the hardness values of the four borides (VB, V₅B₆, V₃B₄, and V₂B₃) gradually enhance with the rise of boron content? (iii) What limits the hardness of VB₂, the increase of boron content leading to an unexpected reduction of hardness? The answers to these questions must lie in the structural and electronic properties of this class of systems.

As illustrated in Fig. 1(a), VB is an orthorhombic structure (space group Cmcm) that is adopted by many transition-metal monoborides. Its metal sublattice consists of trigonal prisms. Each boron atom is situated in the interstitial centre of a trigonal prism and in contact with six metal atoms at the corners of the prism. The central holes of these trigonal prisms are connected to channels, thus the boron atoms are linked into zigzag chains. Since the boron concentration in VB is relatively low, its structure contains vanadium bilayers. Although these V-V bonds contribute to the good metallicity, they are mechanically weaker than the V-B and B-B bonds. As such, the metal bilayers form the weak slip planes. These easy-glide planes govern the formation and movement of dislocations and thus dominate the intrinsic hardness. Indeed, the high-pressure radial diffraction experiment of W_0.5Ta_0.5B³⁴ and the calculated study of CrB³⁸ have corroborated the presence of the weak slip planes. Similar behaviors also occur in other structural systems. For example, the Os-Os metallic bonding layers reduce the resistance of OsB₂ against large shear stresses in the easy-slip directions and thus limit the hardness.⁵⁵ As a result, if the slide dislocations within the metal bilayers can be prevented from moving in the crystals, the overall hardness of this type of monoborides should be able to be enhanced.

We now study the electronic structures of VB to reveal its underlying hardening mechanism. Its band structures, total density of states (DOS), and the V- and B-projected DOS are plotted in Figs. 4(a)–4(d), respectively. Several bands cross its Fermi level, indicating the electrical conductivity in VB. This good metallicity is uncommon since traditional superhard materials like diamond and cBN are insulators. We thus expect this class of superhard metals to find exciting applications (e.g., large volume press anvils, hard electromechanical coatings). The bonding mechanism of VB can be understood as follows: First, the crystal field effect splits the V-3d state into the t_2g bonding states (derived from d_xy, d_yz, and d_zx orbitals) and e_g antibonding states (derived from d_z² and d_x²_-y² orbitals). Then, a strong hybridization between the B-2p and t_2g states forms the pd bonding states and the pd antibonding states. This bonding picture is confirmed by our calculated electronic structures. As shown in Figs. 4(a)–4(d), the lowest two bands in the range (-12, -6) eV have predominantly B-2s character that is responsible for the B-B covalent bonds of boron chains. The six bands in the range (-6, 0) eV are basically derived from the pd bonding states, while the six bands in the unoccupied high region above 2 eV are the corresponding pd antibonding part. The four bands in the range (0, 2) eV belong to the dd antibonding states. Interestingly, the dd antibonding states and the pd bonding states near the Fermi level respond oppositely to the sliding within the metal bilayers. On one hand, the dd antibonding states mainly originate from the metallic bonding within the metal bilayers and are unfavorable. Moreover, the filling of the dd antibonding states leads to the electrostatic repulsion within the metal bilayers. These are beneficial for the slide dislocations within the metal bilayers and thus provide a negative contribution to the hardness. On the other hand, the pd bonding states, which have strong directional character, are extremely resistive to shear stresses. Since each boron atom is surrounded by six neighbor metal atoms, the filling of the pd bonding states restricts the dislocations between the metal and boron layers. Moreover, each boron atom also bonds to a metal atom in the second-neighbor layer, and this pd bonding offers additional locking effect that impedes the slipping within the metal bilayers. Hence, these give rise to a positive contribution to the hardness. Importantly, VB has the optimal valence electron number to fully occupy the pd bonding states, while leaving the dd antibonding states unoccupied. This optimal band filling plays a key role in high stability and hardness of VB. For W_0.5Ta_0.5B, the dd antibonding states start to be populated, leading to a drop in its formation energy and hardness.³⁵ These results not only explain why VB is much harder than W_0.5Ta_0.5B, but also present the main arguments that why we pursue the highly stable thermodynamic phases in the search for superhard metals.

Through optimizing the band filling, VB has the highest hardness among this class of monoborides. To further enhance the hardness, the weak metal bilayers are strengthened by the increase of boron contents. In VB, V₅B₆, V₃B₄ and V₂B₃ structures, their metal skeletons of trigonal prisms are nearly the same. However, the boron atoms accommodate in the central holes of different trigonal prisms, forming single boron chains in VB [Fig. 1(a)], single and double alternate boron chains in V₅B₆ [Fig. 1(b)], double boron chains in V₃B₄ [Fig. 1(c)], and triple boron chains in V₂B₃ [Fig. 1(d)]. It is clear that the number of metal bilayers gradually decreases with the increase of boron concentrations from VB to V₂B₃. Importantly, the increasing boron contents tend to form an increasing number of covalent B-B and V-B bonds. These factors continuously strengthen the mechanical properties from VB to V₂B₃, especially for an enhanced trend of their hardness.

When further increasing boron contents to form VB₂, it becomes the AlB₂ structure (space group P6/mmm), even though all trigonal prisms of metal atoms are filled by the boron atoms. As shown in Fig. 1(e), the boron atoms form rigid hexagonal sheets. The metal atoms sit directly in the interstices above and below the centers of boron hexagons and form planar metal layers. Our calculated band structures, total DOS, V-projected DOS, and B-projected DOS of VB₂ are plotted in Figs. 4(e)–4(h), respectively. Its bonding mechanism is completely different from that of VB. The local environment breaks the degeneracy of V-3d orbitals, forming three types of orbitals, namely, degenerate 3d_xy and 3d_x²_-y², degenerate 3d_yz and 3d_xz, and 3d_z². Since the V-3d states lie very close in energy to the B-2sp states, there is a strong hybridization among these states. The lowest five bands in the range (-15, -1.5) eV can be viewed as the bonding states of the V-B hybridization, among which the lower three (upper two) bands are mainly derived from the degenerate 3d_xy and 3d_x²_-y² (3d_yz and 3d_xz) states and the B-2sp² bonding (B-2p_z nonbonding) states, while the region above -1.5 eV is the corresponding antibonding part. The sixth band ranging from -1.5 to 1.5 eV is mainly derived from the antibonding V-3d_z² states. This analysis naturally leads to the conclusion that this structure shows the highest hardness when the five bonding bands are completely filled. That is to say, the hardness maximum occurs at 10 electrons per formula unit. For VB₂, each formula unit has 11 electrons and the antibonding states start to be populated. It is this strong antibonding interaction that leads to the anomalous hardness reduction of VB₂. This bonding mechanism responsible for the unusual hardness reduction is transferable to other systems or properties. As a matter of fact, the hardness maximum of cubic TMC_xN_1-x at 8.4 electrons per formula unit² and the highest stability of hexagonal TMB₂ at 10 electrons per formula unit⁵⁶ have already confirmed this.

By applying the above atom-scale schemes (i.e., optimizing band filling and tuning boron content), the four borides (VB, V₅B₆, V₃B₄ and V₂B₃) become the competitive superhard metals, but their asymptotic hardness is still less than 40GPa. We further propose a nanoscale way of enhancing extrinsic hardness by creating a multiphase material. As discussed above, their compositions are adjacent and structures are similar. One common structural feature is the presence of the metal bilayers, which limits the hardness enhancement. We thus envisage that one creates a multiphase solid-solution where the easy-slip directions of several borides are essentially random. This particular material not only includes a large number of interfaces among the different phases but also introduces some lattice strains. These interfaces form nanoscale interlocks that strongly restrict the glide dislocations within the metal bilayers, thus drastically enhancing the extrinsic hardness and achieving the true superhard metal. In fact, the recently synthesized nanotwin diamond and cBN have well illustrated that the creation of nanoscale interfaces indeed increase the extrinsic hardness,^57,58 and the nature of strengthening is the effective blockage of dislocation motion by numerous grain boundaries.

According to the above calculations, VB has the lowest formation energy (-0.849 eV/atom), and the formation energies of V₅B₆ (-0.830 eV/atom), V₃B₄ (-0.819 eV/atom), and V₂B₃ (-0.798 eV/atom) also are extremely low. Hence, the four phases are thermodynamically viable. It is interesting that their energy differences are very small, and the maximum span is as low as 0.051 eV/atom. Considering the temperature effect on the formation energy, such a small energy separation indicates that the four borides are energetically degenerate. Also, they share close compositions and similar structures. These factors thus provide a useful indication that this multiphase material should be able to synthesizable under appropriate conditions. One good example is the recently successful synthesis of the superhard and metallic W_0.5Ta_0.5B nanowires through flux growth.⁵⁹ We therefore expect future efforts to synthesize the particular multiphase material of V_1-xB_x. If realized, this should be record making true superhard metals from low borides.

In summary, we have proposed an atom- and nanoscale method of hardness enhancement in low borides by pursuing high stable thermodynamic phases and restricting weak slip planes. Using the typical TM_1-xB_x systems as illustrative cases, we find that the four low borides (VB, V₅B₆, V₃B₄ and V₂B₃) are harder than superhard W_0.5Ta_0.5B and exhibit the electrical conductivity. The four borides share close compositions, similar structures, and small energy differences. We thus suggest that one produces a multiphase material with a large number of interfaces among different phases. These interfaces form nanoscale interlocks that strongly hinder the glide dislocations within the metal bilayers of each boride, accordingly enhancing the extrinsic hardness and achieving the true superhard metal.

Furthermore, our results show that the low borides have high hardness whereas the high borides exhibit an unusual reduction of hardness, which brings an exception to the conventional knowledge that the high boron content improves the hardness. Such an anomalous hardness phenomenon stems from the peculiar bonding mechanisms in these borides. Therefore, the present work not only reveals the unique mechanism responsible for the anomalous hardness trend of this class of TM_1-xB_x, but also provides a multiscale avenue to creating novel superhard metals with rich functionalities.

We thank Prof. Changfeng Chen for his valuable suggestions. We acknowledge the financial support from the National Natural Science Foundation of China (No. 51671126). Chun Tang acknowledges the support from Jiangsu University start-up funding.