Correlation between hardness and bond orientation of vanadium borides

Abstract

The relationship between hardness and bond characteristic of vanadium borides was investigated by first-principles approach. The calculated lattice parameters of V–B system are in good agreement with previous experimental data. The convex hull indicates that the VB are most stable at ground state. The vanadium borides have higher bulk modulus, shear modulus and Young's modulus, and lower B/G ratio. These vanadium borides are brittle. We predict that the V₅B₆ and VB₂ are potential superhard materials. The nature of hardness is related not only to covalent bonding but also to bond orientation. The B–B and V–B covalent bonds parallel to the load plane are the origin of high levels of hardness.

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1. Introduction

Transition metal borides (TMBs) as potential superhard materials have attracted much attention in recent years.^1,2 Several TMBs, such as OsB₂, RuB₂, RuB_1.1, IrB_1.1 and CrB₄, have been synthesized under ambient pressure.^3–5 Theoretical calculations show that these borides all have high elastic modulus, high levels of hardness, ultra-incompressibility and exhibit a degree of metallic behavior.⁶ However, the majority of TMBs are not superhard materials.^7–9 On the other hand, Kaner et al.¹⁰ pointed out that the new design principle of TMBs superhard material is mainly determined by two design parameters: high valence electron density and bond covalency. However, the nature of superhard materials remains a conundrum. Therefore, detailed understanding of the nature of hardness is necessary in the search for superhard materials.

Owing to the outstanding physical properties of vanadium borides, they have been used in various applications such as high-temperature, surface protection and wear-resistant materials.^11,12 Moreover, the structures of vanadium borides of various stoichiometries can be obtained at ambient pressure: VB₂ (space group: P6/mmm with AlB₂ structure),¹³ V₂B₃ (space group: cmcm),¹⁴ V₃B₄ (space group: Immm with Cr₃B₄),¹² V₅B₆ (space group: Ammm with Ti₅B₆),¹⁵ VB (space group: cmcm with CrB structure)¹⁶ and V₃B₂ (space group: P4/mbm with Ta₃B₂ structure).¹⁷ However, only the structural, elastic modulus and electronic structure of VB₂ have been studied by the first-principles approach. It was found that the bulk modulus of VB₂ is 308 GPa, which is close to the value of CrB₄ (306 GPa),^18,19 indicating that these vanadium borides are potential candidates for superhard material. Unfortunately, the mechanical properties of other vanadium borides have seldom been reported.

In order to search for novel superhard materials and to reveal the correlation between hardness and bond characteristic, structural information, elastic modulus, hardness and electronic structure of vanadium borides including V₃B₂, VB, V₅B₆, V₃B₄, V₂B₃ and VB₂ were studied systematically by the first-principles approach in this work. We found that vanadium borides are potential superhard materials and bond orientation plays an important role in the hardness of TMBs. The main purpose of this work is to propose some helpful directions in the study novel transition metal borides for superhard materials.

2. Computational details

Six different vanadium borides, including V₃B₂, VB, V₅B₆, V₃B₄, V₂B₃ and VB₂, were considered in this paper. The crystal structures are shown in Fig. 1, where the blue and orange spheres represent the V and B atoms, respectively. In order to obtain the mechanical and physical properties at ground state, all calculations were performed using density functional theory with the CASTEP code.²⁰ To estimate our calculated results, the exchange-correlation function was taken into account through the local density approximation (LDA) with Ceperley–Alder (CA–PZ)²¹ and the general gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE).²² The 3p⁶3d³4s² and 2s²2p¹ were considered as the valence electrons for V and B atoms, respectively. A plane-wave basis set for electron wave function with a cutoff energy of 320 eV was used. Integrations in the Brillouin zone were performed using special k-points generated with 9 × 9 × 16, 11 × 4 × 11, 5 × 11 × 11, 11 × 5 × 11, 10 × 4 × 10 and 12 × 12 × 12 mesh grids for V₃B₂, VB, V₅B₆, V₃B₄, V₂B₃ and VB₂, respectively. During structural optimization, no symmetry and no restriction were constrained for the unit-cell shape, volume and atomic positions. Elastic stiffness constants were calculated by strain–stress method. From the calculated elastic constants C_ij, the polycrystalline bulk modulus (B) and shear modulus (G) were estimated by the Voigt–Reuss–Hill (VRH) approximation method.²³

Fig. 1 Crystal structure of V–B system. (a) V₃B₂, (b) VB, (c) V₅B₆, (d) V₄B₅, (e) V₂B₃, (f) VB₂, respectively. The blue and orange spheres represent the V and B atoms.

3. Results and discussion

The calculated lattice parameters, densities and volumes of V–B system with LDA and GGA are listed in Table 1, together with their corresponding experimental results. It can be seen that the calculated lattice parameters of six different vanadium borides and pure vanadium are in good agreement with the experimental data. Moreover, the calculated lattice parameters of V, V₃B₂, VB, V₅B₆, V₂B₃ and VB₂ are smaller than that of experimental data, the results of which are independent of LDA or GGA. However, we found that the lattice parameters of V₃B₄ by LDA are also smaller than experimental data in contrast to the lattice parameters by GGA which are larger than experimental data. Compared with the calculated results, we conclude that the calculated lattice parameters by GGA are better than LDA at ground state.

Table 1 Calculated lattice parameters, a (Å), b (Å), c (Å), densities ρ (g cm⁻³) and unit-cell volumes, V (ų) of V–B system

Phase	Method	Space group	Structure	a	b	c	ρ	V
V	LDA	Im-3m	Cubic	2.930			6.724	25.16
	PBE			3.000			6.261	27.02
	Exp^24			3.040			5.680
V3B2	LDA	P4/mbm	Tetra	5.637		2.962	6.155	94.13
	PBE			5.730		3.021	5.842	99.16
	Exp^17			5.755		3.038	5.830	100.62
VB	LDA	cmcm	Ortho	2.997	7.920	2.922	5.913	69.37
	GGA			3.053	8.044	2.965	5.632	72.83
	Exp^16			3.100	8.170	2.980	5.434	75.47
V5B6	LDA	ammm	Ortho	2.931	20.918	2.993	5.783	183.53
	GGA			2.974	21.225	3.050	5.512	192.54
	Exp^12			3.058	21.250	2.974	5.492	193.26
V3B4	LDA	Immm	Ortho	2.987	13.019	2.935	5.704	114.15
	GGA			3.044	13.207	2.977	5.440	119.70
V2B3	LDA	cmcm	Ortho	2.983	18.148	2.937	5.611	158.99
	GGA			3.042	18.406	2.978	5.349	166.79
	Exp^14			3.060	18.429	2.984	5.302	168.26
VB2	LDA	P6/mmm	Hexa	2.954		2.966	5.376	22.41
	GGA			2.993		3.029	5.127	23.50
	Exp^25			2.997		3.056	5.060
	Theo^26			3.008		3.068

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In order to estimate structural stability, the formation enthalpy of V–B system as a function of B concentration was calculated and is shown in Fig. 2. It can be seen that the VB displays the lowest negative formation enthalpy with minimum values of about −0.890 eV per atom with LDA and −0.849 eV per atom with GGA, respectively. The convex hull indicates that the VB is more stable than other vanadium borides. There is no doubt that the discrepancy is due to the localized hybridization and crystal structure.

Fig. 2 Formation enthalpy of V–B system.

The elastic modulus and plastic deformation of a solid are estimated by elastic constants. Here, the elastic constants of these borides will be discussed. Table 2 summarizes the calculated elastic constants of V–B system. The calculated elastic constants of VB₂ are in good agreement with previous theoretical results. We observed that these borides are mechanically stable because their elastic constants obey the Born stability criteria.²⁷ Moreover, the calculated elastic constant C₁₁ of V₃B₂, V₅B₆ and VB₂ is larger than C₃₃, indicating that these borides have strong ultra-incompressibility along the a-axis. Our previous work pointed out that the Vickers hardness is in the a–c plane and the direction of applied load is the b-direction. That is to say, the higher the C₁₁ and C₃₃, the stronger the hardness. Therefore, the higher elastic constants, C₁₁ and C₃₃, are the origin of the high elastic modulus and hardness. However, the calculated elastic constants C₂₂ and C₃₃ of VB, V₃B₄ and V₂B₃ are larger than C₁₁, implying that these borides exhibit excellent resistance to deformation along the b- and c-axis. On the other hand, the elastic constant C₄₄ is related to the resistance to shear deformation. As shown in Table 2, the calculated elastic constant C₄₄ of V–B system is larger than 200 GPa and the elastic constant C₄₄ of V₅B₆ (295 GPa by LDA and 265 GPa by GGA) is larger than other vanadium borides. There result shows that the V₅B₆ has strong resistance to shear deformation.

Table 2 Calculated elastic constants C_ij (GPa) of V–B system

Phase	Method	C11	C12	C13	C22	C23	C33	C44	C55	C66
V3B2	LDA	596	105	147			501	228		199
	PBE	552	95	132			451	211		185
VB	LDA	545	141	168	701	89	672	237	304	240
	PBE	492	123	146	628	77	608	218	277	226
V5B6	LDA	700	94	145	674	153	530	295	253	249
	PBE	630	81	130	615	137	484	265	230	227
V3B4	LDA	529	149	157	698	101	677	254	291	255
	PBE	479	133	137	629	91	620	236	261	232
V2B3	LDA	544	139	148	706	112	694	272	283	262
	GGA	488	124	129	639	101	632	250	251	236
VB2	LDA	740	115	135			538	254
	PBE	680	107	120			478	222
	Theo^18	681	110	125			460	230

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A low G means a low resistance to shear deformation, hence ductility; a low 1/B indicates a weak resistance to fracture, hence brittleness. Therefore, ductile and brittle behavior of a solid can be estimated by B/G ratio. According to the Pugh rule,²⁸ the critical value of B/G ratio which separates ductile and brittle materials has been evaluated to be equal to 1.75. If B/G is >1.75, a material behaves in a ductile manner; if B/G is <1.75, a material behaves in a brittle manner. In fact, the value of B/G ratio indirectly determines the hardness of a solid. The general trend is: the lower the B/G ratio, the higher the hardness. For example, the B/G ratio of diamond is only about 0.826.¹⁹

As we known, the nature of hardness of a solid is very complex, and is related to intrinsic factors such as electronic structure, bond orientation and crystal structure, and external factors such as defects and deformation. In order to obtain theoretical hardness, in this paper, the hard model of V–B system is adopted by intrinsic hard model²⁹ and semi-empirical hard model,³⁰ respectively.

The calculated bulk modulus, shear modulus, Young's modulus, Poisson's ratio, B/G ratio and theoretical hardness of V–B system within both LDA and GGA are listed in Table 3. We observed that the calculated bulk modulus of these vanadium borides is 2.0 times higher than that of pure V, and the shear modulus is 4–5 times larger than the pure V, implying that these vanadium borides have strong resistance to shape and shear deformation. There is no doubt that the B–B and V–B covalent bonds in V–B system can enhance the resistance to deformation upon the introduction of light element, B, into the lattice of the transition metal, and is expected to have profound influence on their chemical and mechanical properties.

Table 3 Calculated bulk modulus, B (GPa), shear modulus, G (GPa), Young's modulus, E (GPa), Poisson's ratio, σ, B/G ratio and hardness, and H (GPa) of V–B system

Phase	Method	B	G	E	σ	B/G	HGao	Hchen
V3B2	LDA	276	216	514	0.190	1.278	43.0	31.8
	PBE	252	200	474	0.186	1.260	40.7	30.9
VB	LDA	301	255	597	0.170	1.180	35.2	41.7
	PBE	269	234	544	0.163	1.149	36.9	38.3
V5B6	LDA	298	258	601	0.164	1.155	44.0	40.5
	PBE	269	234	544	0.163	1.150	47.6	38.3
V3B4	LDA	301	257	600	0.168	1.171	30.1	39.7
	PBE	271	235	547	0.164	1.153	32.0	38.2
V2B3	LDA	303	265	616	0.161	1.143	30.8	41.7
	PBE	273	240	557	0.160	1.137	33.5	39.5
VB2	LDA	307	270	626	0.160	1.137	53.4	42.8
	PBE	278	242	563	0.163	1.149	50.0	39.5
	Theo^31	175
V	Theo^32	155	54	144	0.340	2.870

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Moreover, the calculated bulk modulus of these borides are close to the diborides, which is in the contrary to the shear modulus which are much higher than that of diborides.^33,34 In addition, the calculated bulk modulus of V–B system is not affected by the B concentration. However, the shear and Young's modulus are slightly increased with the increase in B concentration. It is worth noticing that the calculated bulk modulus, shear modulus and Young's modulus of VB₂ at 307 GPa, 270 GPa and 626 GPa by LDA, respectively, are higher than other vanadium borides. These results can be correlated to the B/G ratio and Poisson's ratio because the calculated B/G ratio and Poisson's ratio of these borides relate to the B concentration. The calculated B/G ratio of V–B systems is smaller than 1.75, independent of LDA or GGA, indicating that these vanadium borides exhibit brittle behavior and have high levels of hardness. Note that the B/G ratio and Poisson's ratio of VB₂ are lower than other vanadium borides. We suggest that this discrepancy is related to bond covalency and atomic arrangement.

As shown in Table 3, the vanadium borides have high levels of hardness, regardless of the use of either intrinsic model or experimental model. We note that the hardness of V₃B₂, V₅B₆ and VB₂ by intrinsic model is higher than 40 GPa and the hardness of VB₂ by semi-empirical model is higher than 40 GPa. The hardness discrepancy is determined by the choice of the hardness model. Based on the analysis of hardness, we predict that the V₅B₆ and VB₂ are potential superhard materials. Furthermore, we suggest that the hardness of V–B system is determined by the B–B and V–B covalent bonds, which are demonstrated by their electronic structures (see Fig. 3 and 4).

Fig. 3 The total and partial density of states of vanadium borides. (a) V₃B₂, (b) VB, (c) V₅B₆, (d) V₄B₅, (e) V₂B₃, (f) VB₂, respectively.

Fig. 4 The difference charge density contour plots of V–B system in (100) plane. (a) V₃B₂, (b) VB, (c) V₅B₆, (d) V₄B₅, (e) V₂B₃, (f) VB₂, respectively.

To gain insight into the mechanical properties of the V–B system, the total and partial density of states (DOS) of V₃B₂, VB, V₅B₆, V₃B₄, V₂B₃ and VB₂ were calculated and presented in Fig. 3, while the black vertical dashed of DOS represents the Fermi level (E_F). Clearly, the DOS profiles show that all vanadium borides exhibit metallic behavior due to their finite values at E_F. Moreover, the DOS profiles below E_F are mainly contributed by V-3d state, B-2p state and B-2s state, reflecting significant hybridization between V and B atoms forming the V–B bond. In addition, the B-2s state stretches into the B-2p state below E_F so as to form B–B covalent bond. Obviously, the high elastic modulus and high hardness are derived from the strong B–B and V–B covalent bonds.

Moreover, the deep valley near E_F is denoted the pseudogap, which separates the bonding and antibonding states. As shown in Fig. 3, the deep valley of V₃B₂ near E_F is different from the valleys of VB, V₅B₆, V₃B₄, V₂B₃ and VB₂, indicating that the V(d)–B(p) bonding states of VB, V₅B₆, V₃B₄, V₂B₃ and VB₂, to be saturated. The nearly full occupation of the bonding states and a vacant antibonding state leads to high elastic modulus, smaller Poisson's modulus and high hardness.

To further comprehend the bond mechanism and mechanical properties, the calculated charge density distribution on the (100) plane of V–B system within LDA are shown in Fig. 4. It can be seen that there are some electrons between V and B atoms, indicating a strong directional V–B covalent bonds exist in these vanadium borides. Moreover, two neighboring B atoms form the B–B covalent bonds. It is emphasized that high elastic modulus and hardness are determined by bond strength and bonding direction.

The hardness of a solid is related to the bond state in the a–c plane. In order to reveal the nature of hardness, the bond state in a–c plane of V–B system will be discussed and analyzed next.

From Fig. 4(a), the B atom of V₃B₂ is located at the octahedral interstitial site (OIS) and each B atom is surrounded by eight V atoms, which form a 3D-network structure (see Fig. 1(a)). On the other hand, it has one type of B–B covalent bond (1.761 Å) and two types of V–B bonds (2.237 Å and 2.282 Å). In particular, the weak V–B covalent bond is compensated by B–B covalent bond in the a–c plane, which resists the load applied.

The structural feature and bond states of VB, V₅B₆, V₃B₄ and V₂B₃ with orthorhombic structure are similar, and the slight discrepancy is found with the number of B–B and V–B covalent bonds in the a–c plane. As seen in Fig. 4(b)–(e), the alternative stacked V and B layers can be viewed along the b-direction. In the a–c plane, we observed that the VB has one type of B–B (1.762 Å) and V–B (2.200 Å) covalent bonds, respectively. V₅B₆, V₃B₄ and V₂B₃ have one type of B–B (1.730 Å for V₅B₆, 1.733 Å for V₃B₄ and 1.736 Å for V₂B₃) covalent bond and two types of V–B (2.200 Å and 2.262 Å for V₅B₆, 2.198 Å and 2.261 Å for V₃B₄, 2.195 Å and 2.261 Å for V₂B₃), respectively. For VB₂ with hexagonal structure, it has one type of B–B (1.705 Å) and V–B (2.260 Å) covalent bonds. Obviously, the discrepancy of elastic modulus and hardness comes from the bond strength and bond orientation along a–c plane. Therefore, we conclude that bond orientation also plays an important role in the hardness of TMBs.

4. Conclusion

In summary, we have presented first-principle density functional theory for the study of the structural information, elastic modulus, hardness and electronic structure of V–B system. In order to estimate the calculated results, the exchange-correlation function was considered by LDA and GGA. The conclusions are as follows:

(1) The calculated lattice parameters and volumes of these borides are in good agreement with the experimental data, and the calculated results by GGA are better than those of LDA. The calculated formation enthalpy of VB is about −0.890 eV per atom with LDA and −0.849 eV per atom with GGA, which is smaller than other vanadium borides.

(2) These borides all have high bulk modulus, high shear modulus, low Poisson's ratio, and small B/G ratio. The calculated bulk, shear modulus and Young's modulus of VB₂ are higher than other vanadium borides. The calculated B/G ratios of these borides are smaller than 1.75. Therefore, they exhibit brittle behavior. The B/G ratio of VB₂ is smaller than other vanadium borides, indicating that VB₂ probably has high hardness.

(3) The calculated intrinsic hardness of V₃B₂, V₅B₆ and VB₂ is higher than 40 GPa and the semi-empirical hardness of VB₂ is also higher than 40 GPa, indicating that the V₅B₆ and VB₂ are potential superhard materials.

(4) The high elastic modulus and hardness originate from the strong hybridization between V and B atoms. We found that bond orientation plays an important role in hardness for these borides.

Acknowledgements

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

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