Phase stability, hardness and bond characteristics of ruthenium borides from first-principles

Abstract

The structural stability, elastic modulus, hardness and electronic structure of RuB_2−x (0 ≤ x ≤ 2) borides are systematically investigated using a first-principles approach. The calculated results indicate that the boron-poor region is more stable than the boron-rich region. Ru₂B₃ has a higher bulk modulus, shear modulus and Young's modulus compared with RuB₂ and RuB. Moreover, the calculated intrinsic hardness of Ru₂B₃, with a hexagonal structure (space group: P6₃/mmc), is 49.2 GPa, and is therefore a potential superhard material. The high hardness of Ru₂B₃ originates from triangular pyramid bonds, composed of a B–B covalent bond as the base and Ru–B covalent bonds as the two sides. The B–B and Ru–B covalent bonds in the a–c plane resist the applied load, this is the origin of the high elastic modulus and hardness.

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1. Introduction

In recent years, transition metal borides (TMBs) have received considerable attention due to their high bulk modulus, high hardness, ultra-incompressible nature, good thermal stability, degree of metallic behavior, etc.^1–6 For example, the average hardness of ReB₂, WB₄ and Os_0.5W_0.5B₂ is about 48 GPa, 46.2 GPa and 40.4 GPa, respectively.^7,8 However, numerous TMBs are not superhard materials. Therefore, exploring novel TMB superhard materials is necessary.

For Ru-based borides, although the calculated bulk modulus of RuB₂ is about 334.8 GPa,⁹ the average hardness of RuB₂ rapidly decreases from 24.4 GPa to 14.4 GPa when increasing the applied load.^10,11 The calculated intrinsic hardness of RuB₂ is 36.1 GPa, which is lower than 40 GPa.¹² Moreover, our previous research shows that the average measured hardness of RuB_1.1 is only about 10.6 GPa and the calculated bulk modulus is 346 GPa.¹³ Therefore, these results suggest that Ru-based borides are not superhard materials. In 2009, Rau et al. reported that the hardness of a biphasic ruthenium boride film is 49 GPa, which may be a potential superhard material.¹⁴ They pointed out that the high hardness originates from its microstructure which is composed of two Ru-based boride phases: Ru₂B₃ (main phase) and RuB₂ (second phase). However, the structural stability, elastic modulus, hardness and electronic structure of only RuB₂ are studied in detail. Unfortunately, reports of other Ru-based borides (Ru₂B₃, RuB, Ru₈B₁₁, etc.) are scarce.

On the other hand, numerous theoretical calculations show that the high hardness of TMBs is derived from bond covalency. In fact, the hardness is related not only to the bond covalency but also to other factors such as bond orientation, the arrangement of the bond, etc. To reveal the hard nature and to search for novel superhard materials, in this paper, the structural stability, elastic modulus, intrinsic hardness and electronic structure of RuB_2−x (0 ≤ x ≤ 2) borides are systematically investigated by a first-principles approach. Finally, we predict that the calculated intrinsic hardness of Ru₂B₃, with a hexagonal structure, is 49.2 GPa, which is a potential superhard material.

2. Computational detail

As we know, RuB₂ has an orthorhombic structure (space group: Pmmn, no: 59) with lattice parameters: a = 4.645 Å, b = 2.865 Å and c = 4.045 Å.¹⁵ The Ru and B atoms occupy the 2a (0.0114, 0.2500, 0.8773) and 4f (0.1489, 0.0776, 0.3940) sites (see Fig. 1), respectively. To reveal the correlation between hardness and boron concentration, in this paper, we began with a supercell of Ru₈B₁₆ representing the host RuB₂. The cases when x= 0, 0.125, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75 and 2.00, have been investigated. The main purpose of this work is to understand the relationship between the structural stability and hardness of Ru-based borides and stimulate future experimental study.

Fig. 1 The model of RuB₂. The green and pink spheres represent Ru and B atoms, respectively.

All calculations were performed using the CASTEP code.¹⁶ The exchange correlation functional was treated by the generalized gradient approximation (GGA),¹⁷ with the Perdew–Burke–Ernzerhof-functionals (PBE)¹⁸ we proved that these ruthenium borides are not spin polarized. The electron–ion interaction was described through ultrasoft pseudopotentials. A plane-wave basis set for the electron wave function with a cut-off energy of 360 eV was used. Integrations in the Brillouin zone were performed using a special k-point generated using 6 × 17 × 12 for these structures. During the structural optimization, no symmetry and no restriction were constrained for the unit-cell shape, volume and atomic position. The structural relaxation is stopped when the total energy, the max force and the max displacement were less than 1 × 10⁻⁵ eV per atom, 0.001 eV Å⁻¹, and 0.001 Å, respectively. In addition, the actual spacing of the DOS calculation was less than 0.015 Å⁻¹.

3. Results and discussion

To estimate the structural stability of each B concentration, the short-range ordered structure should be considered, as large as possible, and the total energy of all of the configurations should be calculated and discussed. According to the symmetrical operation, all 55 distinct RuB_2−x configurations are designed, corresponding to x = 0, 0.125, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75 and 2.00.

The formation energies with respect to RuB_2−x are calculated by:

ΔE(x) = E(RuB_2−x) − [E(Ru) + (2 − x)E(B)] (1)

where E(RuB_2−x), E(Ru) and E(B) are the first-principles calculated total energies of RuB_2−x borides, Ru with a hexagonal structure and pure B with a B₁₂ structure, respectively.

Fig. 2 shows the calculated formation energy of RuB_2−x as a function of boron concentration. For each boron concentration, the most stable structure is obtained by first-principles calculations. As seen in Fig. 2, the calculated formation energies of RuB_2−x are negative, indicating that these borides are stable in the ground state. Moreover, the calculated formation energies of Ru, RuB and Ru₂B₃ are lower than RuB₂ by 1.62 eV per atom, 0.37 eV per atom and 0.15 eV per atom, respectively. That is to say, the boron-poor region is more stable than the boron-rich region. In addition, we note that there is a convex hull at x= 0.25. This convex hull suggests the existence of an ordered metastable structure in this Ru-based boride.

Fig. 2 Calculated formation energies of RuB_2−x borides as a function of boron concentration, x = 0, 0.125, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75 and 2.00.

The elastic constants, bulk modulus, shear modulus, Young's modulus and Poisson's ratio are essential for understanding the mechanical properties of a solid. The calculated elastic constants of RuB₂, RuB (x= 1) and Ru₂B₃ (x= 0.5) are listed in Table 1. It is obvious that the elastic constants of these Ru-based borides satisfy the Born stability criteria, indicating that they are mechanically stable in the ground state. Also, the calculated elastic constants of RuB₂ are in good agreement with previous theoretical results. Unfortunately, there are neither experimental data nor theoretical studies available on the elastic modulus for RuB and Ru₂B₃. Therefore, we hope that the obtained results of RuB and Ru₂B₃ in this work may provide useful information for further experimental and theoretical studies.

Table 1 The calculated elastic constants, C_ij (in GPa), of RuB₂, RuB and Ru₂B₃

Type	Method	C11	C12	C13	C22	C23	C33	C44	C55	C66
RuB2	GGA	518	188	146	458	125	706	118	230	176
	Theo^19	540	174	154	484	120	719	116	225	183
RuB	GGA	541	187	171	541	171	774	168	168	178
Ru2B3	GGA	516	228	222	516	222	831	257	257	114

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The elastic constants, C₁₁, C₂₂ and C₃₃, measure the a-, b- and c-direction resistance to linear compression, respectively. The larger the value of C₁₁, C₂₂ and C₃₃, the higher the resistance to deformation along the corresponding direction. From Table 1, the calculated C₃₃ of the Ru-based borides is larger than the C₁₁ and C₂₂, implying that the resistance to deformation of the Ru-based borides along the c-direction is stronger than the a-direction and b-direction. This shows that the origin of the c-direction incompressibility is related not only to the strong B–B and Ru–B covalent bonds but also to the bond orientation (the discussion will be given in the following sections).

Moreover, the calculated C₁₁ of Ru₂B₃ is close to the value for RuB₂. However, the C₂₂ and C₃₃ of RuB and Ru₂B₃ are larger than the values for RuB₂. These results indicate that RuB and Ru₂B₃ have a high resistance to shear deformation along the b- and c-direction. This discrepancy is due to the fact that the structural type of RuB and Ru₂B₃ is different to RuB₂. For a hexagonal structure such as Ru₂B₃, the atomic arrangement along the b-direction results in strong hybridization between B and B atoms. This forms strong B–B covalent bonds which compensate for the weak Ru–B covalent bonds. For an orthorhombic structure such as RuB₂, the Ru–B and B–B covalent bonds in the a–c plane are in the load direction. Therefore, the Ru–B bonds play an important role in the measured hardness. Moreover, the calculated C₃₃ of Ru₂B₃ is larger than that of RuB₂ and RuB by 125 GPa and 57 GPa, respectively. The calculated C₄₄ for the former is larger than the latter by 139 GPa and 89 GPa, respectively, meaning that Ru₂B₃ has a larger elastic modulus and high hardness.

In this paper to estimate the elastic modulus, the Voigt–Reuss–Hill approximation is used.²⁰ Table 2 shows the calculated bulk modulus, shear modulus, Young's modulus and Poisson's ratio of RuB₂, RuB and Ru₂B₃. We found that the calculated bulk and shear modulus of RuB₂ are in good agreement with previous theoretical results. Moreover, the bulk and shear modulus of Ru₂B₃ are large than those of RuB and RuB₂ and the bulk and shear modulus of RuB are larger than those of RuB₂. These results suggest that the boron-poor region may have a high resistance to shape and shear deformation compared with the boron-rich region. This result is different from previous theoretical predictions, where the hardness of boron-rich TMBs is higher than boron-poor TMBs because the boron-rich TMBs have more covalent bonds. Therefore, we suggest that the hardness of TMBs is related not only to the bond covalency but also to other factors such as the bond arrangement. This is demonstrated by the overlap population and bond characteristics shown in Table 3 and Fig. 3. In addition, the Young's modulus is calculated to be in the following sequence, Ru₂B₃ > RuB > RuB₂. The high Young's modulus of Ru₂B₃ shows a rather smaller stiffness.

Table 2 The calculated Voigt bulk modulus, B_V (in GPa), Voigt shear modulus, G_V (in GPa), Reuss bulk modulus, B_R (in GPa), Reuss shear modulus, G_R (in GPa), bulk modulus, B (in GPa), shear modulus, G (in GPa), Young's modulus, E (in GPa), and Poisson's ratio δ of RuB₂, RuB and Ru₂B₃, respectively

Type	Method	BV	GV	BR	GR	B	G	E	δ
RuB2	GGA	289	186	284	172	286	179	444	0.241
	Theo^19	293	191	288	177	290	184
RuB	GGA	324	191	317	185	321	188	472	0.255
Ru2B3	GGA	355	210	342	193	349	202	508	0.257

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Table 3 The calculated bond length d^μ (in Å), Mulliken overlap population P^μ, bond volume of μ type v^μ (in ų), intrinsic hardness H_vcal (in GPa) and measured Vickers hardness H_vexp of RuB₂, RuB and Ru₂B₃, respectively

Type	Bond	d^μ	P^μ	v^μ	Hvcal	Hvexp
RuB2	B–B	1.817	0.64	1.841		10.9–28.9 (ref. 22)
	B–B	1.880	1.29	2.040
	Ru–B	2.190	0.35	3.224	36.8
RuB	Ru–B	2.173	0.25	3.345	24.7
Ru2B3	B–B	1.804	2.08	1.929	49.2
	Ru–B	2.175	0.88	3.381
	Ru–B	2.187	0.52	3.437
	B–B^22	1.840
	Ru–B^23	2.190

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Fig. 3 The difference charge density contour plots of the chemical bonds in RuB_2−x borides. (a) RuB₂ (110) plane, (b) RuB (110) plane and (c) Ru₂B₃ (010) plane.

Due to the high bulk and shear modulus, Ru₂B₃ is expected to be the harder material compared with other Ru-based borides. Here, the intrinsic hardness of the Ru-based borides is calculated using the hard model by Gao et al.²¹ The calculated intrinsic hardness, bond length, bond volume and Mulliken overlap population of RuB₂, RuB and Ru₂B₃ are presented in Table 3. It can be seen that the calculated intrinsic hardness of RuB₂ is 36.8 GPa, which is in good agreement with the previous theoretical data (36.1 GPa).¹² It is worth noting that the intrinsic hardness of Ru₂B₃ is about 49.2 GPa. This is very close to the average measured hardness of ruthenium boride film (49 GPa). Therefore, we predict that Ru₂B₃ is a potential superhard material.

To reveal the origin of high hardness of Ru-based borides, the bond characteristics and electronic structure of these Ru-based borides are studied in detail. As shown in Table 3, the calculated bond lengths of the B–B and Ru–B covalent bonds of these Ru-based borides are in good agreement with previous theoretical results. However, the bond lengths of B–B and Ru–B covalent bonds of Ru₂B₃ are shorter than those corresponding to RuB₂ and RuB, respectively. On the other hand, we know that the positive and negative values of overlap population indicate the bonding and anti-bonding states, respectively. Therefore, the calculated overlap population of the B–B and Ru–B covalent bonds of Ru₂B₃ are larger than those of RuB₂ and RuB. These results imply that the local hybridization between Ru and B atoms in Ru₂B₃ is stronger than in RuB₂ and RuB, and therefore forms strong B–B and Ru–B covalent bonds. This is demonstrated by the calculated bond strength (see Table 3).

The bond arrangement also plays an important role in the intrinsic hardness. To understand the bond arrangement of RuB_2−x borides, the charge densities of the RuB₂, RuB and Ru₂B₃ chemical bonds are discussed here. Fig. 3 shows the valence electron density along the RuB₂ (110), RuB (110) and Ru₂B₃ (010) planes, where the critical features are labeled. Similar to other TMBs, covalent bonding is observed, and the strong and directional Ru–B covalent bonds are present in these Ru-based borides. Note that the charge transition between Ru and B atoms in Ru₂B₃ (0.63) is larger than in RuB₂ (0.51) and RuB (0.28), indicating that the local hybridization for the former is stronger than the latter.

For RuB₂, the network bonds are composed of Ru–B covalent bonds with zigzag covalent chains, and directional B–B covalent bonds along the b-direction. The Ru–B covalent bonds have two dimensions and are formed in the a–c plane. Therefore, the shear fracture of RuB₂ occurs at the weak Ru–B covalent bonds. For RuB, we observe that there is no charge accumulation between B and B atoms. The RuB only has the network of Ru–B bonds and the network bond states, with a synergistic effect, can enhance the resistance to deformation.

For Ru₂B₃, each Ru atom is surrounded by seven B atoms, and each B atom is surrounded by four Ru atoms. This atomic arrangement can be viewed as alternatively stacked Ru and B layers along the c-direction. Moreover, the B layer is composed of two sub-boundary B layers. Therefore, the staggered B and Ru layers form two types of Ru–B bonds including the Ru–B (1) bonds (2.175 Å) and Ru–B (2) bonds (2.187 Å) and one type of B–B covalent bond (1.804 Å), which is in good agreement with experimental values.²⁴ It is interesting to find that the B–B and Ru–B covalent bonds form triangular pyramid bonds in Ru₂B₃, with the B–B covalent bond as the base and the Ru–B covalent bonds as the two sides. Therefore, the B–B and Ru–B covalent bonds in the a–c plane and the B–B covalent bonds compensate for the bonding energy of the weak Ru–B covalent bonds, which is the origin of the larger elastic modulus and high hardness.

The calculated electronic density of states (DOS) of RuB₂, RuB and Ru₂B₃ are shown in Fig. 4, in which the black vertical dashed line represents the Fermi level (E_F). It can be seen that there are some bands across the E_F, indicating that these Ru-based borides exhibit metallic behavior. From Fig. 4(a)–(c), the DOS profiles of RuB₂, RuB and Ru₂B₃ are contributed by Ru-4d states and B-2p states, implying that the local hybridization between Ru and B atoms form strong Ru–B bonds along the d–p direction. The covalent interaction between B and Ru atoms is demonstrated by the difference charge density (see Fig. 3).

Fig. 4 The total and partial density of states of ruthenium borides. (a) RuB₂, (b) RuB and (c) Ru₂B₃.

As we know, Ru₂B₃ may be a potential superhard material, in the following, the DOS profile of Ru₂B₃ is discussed. From Fig. 4(c), the DOS profile could be mainly divided into three parts. The first part, extending from the bottom up to −0.57 eV, consists mainly of Ru-4d, B-2s and B-2p states. The second section, from −0.57 eV to 3.58 eV, is mainly the contribution of the Ru-4d and B-2p states. The last part, from 3.58 eV to 7.60 eV, mainly contains mixtures of Ru-4d, B-2p and B-2s states. The DOS at the E_F is controlled by the overlap between the Ru-4d and B-2p states. Comparing RuB₂, RuB and Ru₂B₃, the main differences between the DOS profiles are that the Ru₂B₃ profile has a smooth valley near the E_F. This may be because the Ru–B covalent bonds of Ru₂B₃ are stronger than those of RuB₂ and RuB. Our calculated results show that the average nearest Ru–B bond length of Ru₂B₃ is shorter than the Ru–B bond lengths within the B–Ru–B bonds of the RuB₂ and RuB structures. Also the Mulliken overlap population of the Ru–B and B–B covalent bonds of Ru₂B₃ is larger than the corresponding bonds for RuB₂ and RuB (see Table 3 and Fig. 3). This is the reason why Ru₂B₃ has strong hybridization between B and Ru atoms and has high hardness.

4. Conclusions

In summary, using first-principles density-functional theory, we have investigated the structural stability, elastic properties, hardness and electronic structure of RuB_2−x (0 ≤ x ≤ 2) borides. All of the possible symmetrical configurations with different boron concentrations are discussed in detail. The calculated results show that the formation energies of RuB_2−x borides decreased rapidly with the decrease in boron concentration when x > 0.25, indicating that the boron-poor region is more stable than the boron-rich region.

The calculated bulk and shear modulus values of Ru₂B₃ are 349 GPa and 202 GPa, respectively, which are larger than those of RuB₂ and RuB. The Young's modulus is calculated in the following sequence, Ru₂B₃ > RuB > RuB₂. Therefore, Ru₂B₃ has a smaller stiffness. The calculated intrinsic hardness of RuB₂ is 36.8 GPa, which is in good agreement with previous theoretical results. We note that the intrinsic hardness of Ru₂B₃ is about 49.2 GPa.

The analysis of the structural features and electronic structure show that the high hardness of Ru₂B₃ is derived from the layered structure and bond characteristics. The sub-boundary B and Ru layers form two types of Ru–B and B–B covalent bonds along the c-direction, with Ru–B bonds as the two sides and the B–B covalent bond as the base. These triangular pyramid bonds improve the resistance to the shape and shear deformation, and enhance the elastic modulus and hardness. Therefore, we predict that the intrinsic hardness of Ru₂B₃, with a hexagonal structure (space group: P6₃/mmc), is about 49.2 GPa which is a potential superhard material.

Acknowledgements

Financial support from the National Natural Science Foundation of China (Grant no. 50525204, 50832001 and 50902057) and the important project of Nature Science Foundation of Yunnan (no. 2009CD134) are gratefully acknowledged.

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