Comparison of the Physical Properties and Electronic Structure of Nb₂B₃ and Ta₂B₃

Abstract

Based on the first-principles method, the effects of pressure and temperature on the physical properties of Nb₂B₃ and Ta₂B₃ were discussed. The approximate linear increase in B, G, and E with pressure was observed for Nb₂B₃ and Ta₂B₃ with a minor difference for bulk modulus and similar values for shear and Young’s modulus. Nb₂B₃ shows higher Vickers hardness and similar fracture toughness as compared with Ta₂B₃. An abnormal phenomenon of the simultaneous increase in hardness and B/G (σ) with the increased pressure was observed. The strong anisotropies of bulk, Young’s, and shear modulus were observed, and the differences of anisotropy between Nb₂B₃ and Ta₂B₃ increased with pressure. At low temperatures, the α of Nb₂B₃ is smaller than that of Ta₂B₃, but is larger than that of Ta₂B₃ at high temperatures. The Θ of Nb₂B₃ are larger than those of Ta₂B₃ under the same conditions. The combination of relatively high Vickers hardness and fracture toughness is determined by the metallic bond and covalent bond. With the increased pressure, Nb₂B₃ possesses the greater strength of B–B bonds than Ta₂B₃, which leads to its high hardness and Debye temperature.

1. Introduction

As a potential superhard metal, transition metal borides (TMBs) have received extensive interest due to their excellent mechanical, thermal, and chemical properties [1,2,3,4,5,6,7,8,9,10,11,12]. Especially, compared with the traditional superhard material, their metallic character provides outstanding electrical (thermal) conductivity, thus it is easily shaped post-synthesis by electric discharge machining. For TMBs, the high hardness is attributed to the high valence-electron concentration of TM atoms and strong covalent bonds of B atoms formed [8]. Thus, some researchers have focused on the effect of valence electron concentration (VEC) and B atom content for the hardness [9,10,11,12]. Contrary to the common opinion that there is a positive correlation between hardness and VEC, Liang et al. found the maximum hardness approaching 40 GPa when the VEC was ~8 electrons per formula unit for transition metal monoborides [9]. Li et al. found that boron content did not positively correlate to hardness in WB_n (n = 2,3,4) compounds; this intriguing behavior originated from the different bonding configurations of B atoms formed [10]. By replacing the W of WB with Ta sites, the high hardness ~42.8 GPa under 0.49 N was achieved in W₀.₅Ta₀.₅B [11]. It was also found that the low B content increased the hardness, whereas the high B content reduced the hardness of TM_1−xB_x (TM = V, Nb and Ta) [12]. Based on these studies, it is meaningful to study the TMBs with low B content for superhard metals.

As an important family of TMBs, it is meaningful to study the physical properties of 5d-series transition metal borides with low B content. Thus, in this study, the mechanical and thermal properties (such as the polycrystalline elastic moduli, hardness, fracture toughness, Poisson’s ratio σ, the volume thermal expansion coefficient, and Debye temperature) and elastic anisotropy of Nb₂B₃ and Ta₂B₃ under high pressure and temperature are investigated and compared. For a further understanding the physical properties, the electronic structure is also discussed to investigate the underlying micro-mechanism.

2. Results and Discussion

2.1. Equilibrium Structure

The space group of Nb₂B₃ and Ta₂B₃ with an orthorhombic structure is Cmcm (No. 63); see Figure 1. The optimized lattice constants are in good agreement with the available results (see

Table 1. Calculated lattice constants a, b, c (Å) of Nb₂B₃ and Ta₂B₃ with the available data for comparison.

		a	b	c
Nb2B3	present	3.310	19.507	3.159
	Cal. [13]	3.310	19.504	3.128
Ta2B3	present	3.337	19.687	3.159
	Cal. [13]	3.337	19.685	3.159

), indicating that the parameters used in our calculations are reliable.

2.2. Physical Properties

The stress–strain relationship is used to calculate the elastic constants with six strain steps for the optimized structures [14]. Nb₂B₃ and Ta₂B₃ are mechanically stable under the considered pressure based on the mechanical stability criteria [15]. The polycrystalline bulk modulus B and shear modulus G are determined using the elastic constants using the Voigt–Reusse–Hill (VRH) approximation [16], and the Young’s modulus E and Poisson’s ratio σ are obtained according to the bulk and shear modulus [17]. An empirical model proposed by Chen et al. is used to predict the Vickers hardness of the material [18]. The fracture toughness is estimated according to Niu’s model $K_{\mathrm{IC}}=\alpha V^{1/6}G{\left(B/G\right)}^{1/2}$, α = 1 for transition metal borides, the unit of volume V is m³, and that of shear and bulk modulus G and B are MPa [19], which measures the resistance-to-crack propagation of a material. The variation trends of polycrystalline bulk modulus B, shear modulus G, Young’s modulus E, Vickers hardness H_V, fracture toughness K_IC, Poisson’s ratio σ, and ratio B/G of Nb₂B₃ and Ta₂B₃ are shown in Figure 2. The bulk modulus expresses the resistance to volume change under hydrostatic pressure and reflects the incompressibility of materials. The shear modulus correlates the shape change under shear force and reflects the ability of resistance to the shape change of materials. Young’s modulus provides the resistance to tensile (compressive) deformations and reflects the stiffness of materials. It can be seen that the approximate linear dependence of pressure of B, G, and E for Nb₂B₃ and Ta₂B₃ is observed. Under pressure, the increase in G is slower than that of B and E. The small shear modulus is determined by the small elastic constants C₄₄ and C₆₆, indicating the low resistance to shear deformation. The difference in bulk modulus between Nb₂B₃ and Ta₂B₃ is very small, while that between shear and Young’s modulus is nearly negligible.

It is important to characterize hardness and fracture toughness for the wear-resistance of materials. The hardness increases slowly with the increased pressure, but the increased speed of Nb₂B₃ is slower than that of Ta₂B₃. In addition, the values of hardness of Nb₂B₃ are larger than those of Ta₂B₃ under the same pressure. The fracture toughness increases rapidly with pressure. At low pressures, the values of Nb₂B₃ and Ta₂B₃ are equal, but minor differences appear with the increased pressure. When P~100 GPa, the fracture toughness exceeds the experimental values of diamond 5.3–6.7 MPa·m^0.5 [19]. The high hardness and fracture toughness make it suitable for drill bits or ballistic vests.

The values of B/G and Poisson’s ratio σ are used to estimate the ductile (brittle) behavior of a material [20,21]. For brittle materials, B/G < 1.75 or σ < 0.26. As can be seen from Figure 2c, Nb₂B₃ and Ta₂B₃ show brittle behavior, though σ and B/G increase with pressure. The ductile feature of Ta₂B₃ is superior to that of Nb₂B₃. The counterintuitive phenomenon of the simultaneous increase in hardness and ductility with the increased pressure makes it applicable in the mechanical processing area, which is different from the previously reported results [22,23,24].

Figure 3 shows the effects of pressure on elastic anisotropy of Nb₂B₃ and Ta₂B₃ based on the ratio of maximum to minimum of directional bulk and Young’s and shear modulus B_max/B_min, G_max/G_min, E_max/E_min. As a representative 3D orientation dependence on bulk, shear and Young’s modulus of Nb₂B₃ and Ta₂B₃ at 100 GPa are given. Due to the same atom arrangement, 3D plots of Nb₂B₃ and Ta₂B₃ present similar shapes. The blue expresses the shrink and the red expresses the bulge. The obvious shrink meant larger axial compressibility along the [100] direction (see the inset of Figure 3a) is attributed to small C₁₁, and appearance of small difference along the [010] and [001] directions is due to the relatively small C₂₂ compared with C₃₃. The bulge along the [100] direction (see the inset of Figure 3b) makes it difficult to slip along x direction. These results are correlated to the layer structure along the x axis with B–B covalent bond in the yz plane. It can be seen that the anisotropy of the bulk modulus decreases, while the anisotropy of shear and Young’s modulus increases with pressure. Furthermore, the differences of anisotropy between Nb₂B₃ and Ta₂B₃ increase with pressure.

The thermodynamic properties are calculated using the quasi-harmonic Debye model implemented in the Gibbs program [25]. Figure 4 shows the temperature dependences of volume thermal expansion coefficient α at different pressures for Nb₂B₃ and Ta₂B₃. The pressure influences the increased speed of α with temperature. At fixed pressure, α increases rapidly above ~100 K, and gradually slows down at high temperatures. Furthermore, the increased speed of α with the temperature of Nb₂B₃ and Ta₂B₃ is different. Thus, the α of Nb₂B₃ is smaller than that of Ta₂B₃ at low temperatures and gradually larger than that of Nb₂B₃ at high temperatures. When the pressure increases, α decreases for a given temperature.

Figure 5 shows the pressure dependences of Debye temperature for Nb₂B₃ and Ta₂B₃ at different temperatures. The Debye temperature Θ of Nb₂B₃ and Ta₂B₃ at 0 GPa and 0 K is 823.2 K and 605.9 K, in line with the results 833.4 K and 610.4 K based on the equation using the first principles [26]:

$${\Theta }_D=\frac{h}{k_B}{\left(\frac{3n}{4\pi }\left(\frac{\rho N_A}{M}\right)\right)}^{1/3}{v}_m$$

where h is Planck’s constant, n is the number of atoms per molecule, ρ is the density, N_A is the Avogadro number, k_B is Boltzmann’s constant, M is the molecular molar mass, and $v_m={\left[\left(2/v_t^3+1/v_l^3\right)/3\right]}^{-1/3}$ is the average sound velocity with $v_t=\sqrt{G/\rho }$ (transverse acoustic wave velocity) and $v_l=\sqrt{\left(3B+4G\right)/3\rho }$ (longitudinal acoustic wave velocity).

At a given temperature, the increased speed with pressure of Θ for Nb₂B₃ is faster than that of Ta₂B₃. At a given pressure, the values decrease slightly with temperature. The effect of temperature is small and gradually diminishes with the increased pressure. For the same conditions, the values of Nb₂B₃ are larger than Ta₂B₃. Based on the Callaway–Debye theory, the minimum thermal conductivity is correlated to the Debye temperature [26]. So, Nb₂B₃ will possess a larger minimum thermal conductivity than that of Ta₂B₃.

2.3. Electronic Structure

To uncover the origin of physical properties, the bonding nature of materials is investigated. The energies of Fermi level E_F are 2.91 (2.19) electrons/eV and 2.67 (2.00) electrons/eV at 0 (100) GPa for Nb₂B₃ and Ta₂B₃, respectively. The nonzero values point to a metallic feature. Previous works indicated that the hardness of TM borides was mainly determined by the B–B and B–TM bonds [27]. Figure 6 shows the electron density differences in the (100) plane of Nb₂B₃ and Ta₂B₃ under different pressures. The electrons (red region) distribute in the interstitial region around the Nb(Ta) atom, indicating a typical metallic-like bond. The electron accumulations between B–B atoms and B–Nb (Ta) atoms indicate the covalent interaction. According to the structural feature, the ring structure formed with six B atoms shows three different bonds. With the increased pressure, the electron accumulations between B–B atoms and B–Nb (Ta) atoms increase, meaning the increased strength of B–B bonds and B–Nb (Ta) bonds with pressure. Furthermore, the strength of B–Nb (Ta) covalent bonds is weaker than that of covalent B–B bonds. The larger electron accumulation between B–Nb atoms than that of B–Ta atoms means that the strength of the B–Nb bond is greater than that of the B–Ta bond. The positive bond populations of B–B atoms gradually increase with pressure, and the bond lengths decrease; see Figure 7. These results also show that pressure enhances the strength of B–B covalent bonds. Under the same pressure, Nb₂B₃ shows relatively larger bond populations and shorter bond lengths than that of Ta₂B₃, thus possesses strong covalent interactions between B–B atoms.

Table 2. Atomic bond population (P), bond length of B–Nb(Ta) bonds, and atomic Mulliken charge for Nb₂B₃ and Ta₂B₃.

		0 GPa				100 GPa
	Bond	P	Bond Length (Å)	Atom	Charge (e)	P	Bond Length (Å)	Atom	Charge (e)
Nb2B3	B5--Nb4	−0.09	2.36768			−0.45	2.20505
	B2--Nb2	−0.09	2.36768			−0.45	2.20505
	B6--Nb2	0.77	2.39143	B1	−0.53	0.21	2.21945	B1	−0.65
	B3--Nb4	0.77	2.39143	B2	−0.53	0.21	2.21945	B2	−0.65
	B4--Nb3	0.03	2.43817	B3	−0.54	−0.11	2.26022	B3	−0.67
	B1--Nb1	0.03	2.43817	B4	−0.53	−0.11	2.26022	B4	−0.65
	B4--Nb1	0.18	2.44751	B5	−0.53	−0.53	2.26572	B5	−0.65
	B1--Nb3	0.18	2.44751	B6	−0.54	0.53	2.26572	B6	−0.67
	B2--Nb3	0.32	2.45273	Nb1	1.03	−0.21	2.27156	Nb1	1.25
	B5--Nb1	0.32	2.45273	Nb2	0.58	−0.21	2.27156	Nb2	0.72
	B6--Nb3	0.16	2.50015	Nb3	1.03	0.19	2.31677	Nb3	1.25
	B3--Nb1	0.16	2.50015	Nb4	0.58	0.19	2.31677	Nb4	0.72
	B6--Nb4	0.12	2.58615			0.02	2.37148
	B3--Nb2	0.12	2.58615			0.02	2.37148
Ta2B3	B5--Ta4	−0.05	2.39181			−0.35	2.23146
	B2--Ta2	−0.05	2.39181			−0.35	2.23146
	B6--Ta2	1.14	2.41141	B1	−0.61	0.98	2.24347	B1	−0.63
	B3--Ta4	1.14	2.41141	B2	−0.59	0.98	2.24347	B2	−0.60
	B4--Ta3	−0.01	2.46156	B3	−0.64	−0.35	2.28940	B3	−0.69
	B1--Ta1	−0.01	2.46156	B4	−0.61	−0.35	2.28940	B4	−0.63
	B4--Ta1	0.54	2.46975	B5	−0.59	0.42	2.29339	B5	−0.60
	B1--Ta3	0.54	2.46975	B6	−0.64	0.42	2.29339	B6	−0.69
	B2--Ta3	0.62	2.47436	Ta1	1.23	0.45	2.29746	Ta1	1.30
	B5--Ta1	0.62	2.47436	Ta2	0.61	0.45	2.29746	Ta2	0.62
	B6--Ta3	−0.00	2.52862	Ta3	1.23	−0.30	2.34801	Ta3	1.30
	B3--Ta1	−0.00	2.52862	Ta4	0.61	−0.30	2.34801	Ta4	0.62
	B6--Ta4	0.25	2.59587			0.28	2.40293
	B3--Ta2	0.25	2.59587			0.28	2.40293

gives the atomic bond populations, bond lengths, and atomic Mulliken charge of B–Nb (Ta) bonds for Nb₂B₃ and Ta₂B₃. It can be found that there is positive a bond population between B and Nb (Ta), but the values are small and the bond lengths are relatively longer compared with B–B bonds. These results explain the weaker strength of B–Nb (Ta) covalent bonds than that of covalent B–B bonds. With the increased pressure, the number of covalent bonds decreases, and the bond lengths and bond populations decrease. So, the increased strength of B–Nb(Ta) with pressure is attributed the decreased bond length. Compared to B–Nb bonds, the number of covalent bonds in B–Ta bonds is small, and the bond lengths and bond populations increase. So, the weak strength of B–Ta covalent bonds is attributed to the longer bond lengths and lower number of covalent bonds.

In addition, there is electron accumulation (red region) around B atoms and electron depletion (blue region) around Nb (Ta) atoms, creating electron transportation from Nb (Ta) atoms to B atoms; see Figure 6. Thus, there is an ionic feature of B–Nb(Ta) bonds, which can also be obtained from Table 2. B atoms gain electrons, while Nb(Ta) atoms lose electrons. With the increased pressure, the increased transferred electrons and short bond length indicate that the ionic feature becomes stronger. Furthermore, the number of transferred electrons from Nb to B is smaller than that from Ta to B at 0 GPa, indicating that the ionic bonding strength of B–Ta bonds is stronger than that of B–Nb bonds. But, with the increased pressure, the difference of transferred electrons disappears.

Based on the above discussions, the interatomic interaction of Nb₂B₃ and Ta₂B₃ is mainly determined by metallic bonds, covalent bonds, and some ionic bonds, which lead to high Vickers hardness and fracture toughness. The covalent interaction of B–B bonds is stronger than that of B–Nb(Ta) bonds. With the increased pressure, the strength of interactions increases. Nb₂B₃ possesses stronger interatomic interaction than Ta₂B₃, which leads to the high hardness and Debye temperature.

3. Methods and Computation Details

The space group of Nb₂B₃ and Ta₂B₃ is Cmcm (No. 63). In this work, the first-principles method implemented in MedeA VASP code was used [28]. The Perdew-Burke-Ernzerh (PBE) form within the generalized gradient approximation (GGA) was adopted to describe the exchange correlation energy in the electron–electron interaction [29]. The valence electrons considered in the atom pseudo potential calculation were Nb:4p⁶4d⁴5s¹, Ta:5p³5d³6s², and B:2s²2p¹, respectively. In all calculations, the cutoff energy for the plane wave expansion was selected as 550 eV with Monkhorst-Pack k point meshes of less than 0.015Å⁻¹ for Brillouin zone sampling [30]. The structure reached the convergence state with the total energy 5.0 × 10⁻⁶ eV/atom, maximum force per atom 0.01 eV/Å, maximum stress 0.02 GPa, and maximum displacement 5.0 × 10⁻⁴Å, respectively.

4. Conclusions

The effects of pressure and temperature on the physical properties of Nb₂B₃ and Ta₂B₃ are investigated based on the first-principles method. The approximate linear increase in B, G, and E with pressure is observed for Nb₂B₃ and Ta₂B₃ with a minor difference for bulk modulus and similar values for shear and Young’s modulus. Nb₂B₃ is more suitable for drill bits or ballistic vests due to its high Vickers hardness and similar fracture toughness as compared to Ta₂B₃. Although Nb₂B₃ and Ta₂B₃ show the brittle behavior, the values of B/G and Poisson’s ratio σ increase as pressure increases. The simultaneous increase in hardness and ductility with the increased pressure makes it applicable in some mechanical processing area. The strong anisotropy of the bulk, Young’s, and shear modulus is observed, and the differences in anisotropy between Nb₂B₃ and Ta₂B₃ increase with pressure. The effect of temperature is opposite to that of pressure for the volume thermal expansion coefficient α and Debye temperature Θ. At low temperatures, the α of Nb₂B₃ is smaller than that of Ta₂B₃, but is larger than that of Ta₂B₃ at high temperatures. The values of Θ for Nb₂B₃ are larger than that of Ta₂B₃ under the same conditions. The interatomic interaction of Nb₂B₃ and Ta₂B₃ is determined by metallic bonds, covalent bonds, and some ionic bonds. The high hardness and Debye temperature of Nb₂B₃ are mainly derived from the strong covalent interaction of B–B atoms. We expect that these results are meaningful for the further study of superhard metals.