A new type of vanadium carbide V₅C₃ and its hardening by tuning Fermi energy

Abstract

Transition metal compounds usually have various stoichiometries and crystal structures due to the coexistence of metallic, covalent and ionic bonds in them. This flexibility provides a lot of candidates for materials design. Taking the V-C binary system as an example, here we report the first-principles prediction of a new type of vanadium carbide, V₅C₃, which has an unprecedented stoichiometry in the V-C system and is energetically and mechanically stable. The material is abnormally much harder than neighboring compounds in the V-C phase diagram and can be further hardened by tuning the Fermi energy.

Introduction

Transition metal carbides have attracted continuing interest due to their excellent physical properties and wide engineering applications^1,2,3,4,5,6. Because of the coexistence of the covalent, ionic and metallic bonding types between the transition metals and carbon, the transition metal carbides usually have various stoichiometries. The flexibility in stoichiometry leads to rich chemical and physical behaviors and provides a lot of candidates for materials design.

The V-C system is a typical binary system which has many different stoichiometries. V₂C, V₄C₃, V₆C₅, V₈C₇ and VC have been synthesized and investigated for many years^7,8,9,10,11. T₅M₃ is a common stoichiometry composed of transition metals T and main-group elements M. There are several structure types for this specific stoichiometry, including D8₈ (Mn₅Si₃, hexagonal, P6₃/mcm, No.193), D8_l (Cr₅B₃, tetragonal, I4/mcm, No.140) and D8_m (W₅Si₃, tetragonal, I4/mcm, No.140), with their prototypes and space groups given in the parentheses. For silicides of group VB transition metals, both Ta₅Si₃ and Nb₅Si₃ have the Cr₅B₃-type structure and V₅Si₃ has the Mn₅Si₃-type structure^{12,13,14,15,16,17}. But carbides with this stoichiometry have never been synthesized nor theoretically studied.

In this work, we take the V-C system as a model system to explore the possibility to design new materials by changing the stoichiometry. The calculations were performed to investigate the crystal structure, phase stability, electronic structure and mechanical properties of V₅C₃. The results show that the Cr₅B₃-type V₅C₃ is stable mechanically, dynamically and thermodynamically and can be synthesized at high pressures. The hardness of the hard material can be enhanced further through tuning the Fermi energy.

Results and Discussion

As mentioned above, three typical structure types for T₅M₃, i.e., Mn₅Si₃, Cr₅B₃ and W₅Si₃ types are considered in this work, as shown in Fig. 1. For comparison, the known vanadium carbides in the V-C phase diagram, i.e. VC (cubic, Fm-3m), V₂C (orthorhombic, Pbcn), V₄C₃ (hexagonal, R-3m), V₆C₅ (hexagonal, P3₁) and V₈C₇ (cubic, P4₃32) are also included in the calculations.

Figure 1

Structure models of V₅C₃: (a) Cr₅B₃-type, (b) W₅Si₃-type, (c) Mn₅Si₃-type. The large and small spheres represent V and C, respectively.

The formation enthalpy was calculated using the following equation,

where E_total(V_xC_y) was the obtained total energies for the considered vanadium carbide, E_total(V) and E_total(C) were the total energy of the pure metal V and the graphite, respectively. The calculated lattice parameters and formation enthalpy ∆H at zero pressure are listed in Table 1. For the known vanadium carbides, the calculated values are in good agreement with previous calculation values.

Table 1 Calculated lattice parameters a, b and c (Å) and formation enthalpy ∆H (eV/atom).

The total energies of V₅C₃ as a function of volume for the three structure types are plotted in Fig. 2(a). The Cr₅B₃-type V₅C₃ has the lowest energy at all the volumes. Hereafter, only the Cr₅B₃-type V₅C₃ is considered unless stated otherwise. It is worth noting that the formation enthalpies of these vanadium carbides are all negative at zero pressure. The negative formation enthalpies indicate that the carbides are more stable than the mixture of elemental V and C.

Figure 2

(a) Energy-volume relationships for the Cr₅B₃-type, W₅Si₃-type and Mn₅Si₃-type V₅C₃. (b) The relative enthalpy-pressure diagram of the Cr₅B₃-type V₅C₃ and its respective competing phases.

For a compound to be synthesized experimentally, it is more reliable to compare its enthalpy with the known compounds of neighboring stoichiometries. In the V-C phase diagram, V₅C₃ would locate in the two-phase region bounded by V₂C and V₄C₃. Therefore, we need to compare the formation enthalpy of V₅C₃ with the mixture of V₂C and V₄C₃. The formation enthalpies as a function of pressure have been calculated for both V₅C₃ and the mixture of V₂C and V₄C₃, as shown in Fig. 2(b). The mixture is more stable than V₅C₃ under pressures below 9.2 GPa, above which V₅C₃ becomes more stable. It indicates that V₅C₃ is thermodynamically more stable than that of the mixture at high pressures.

The elastic properties of a material are very important as they determine the mechanical stability, strength, hardness and ductile or brittleness behavior. The calculated elastic constants C_ij, the minimum elastic eigenvalue λ₁ ¹⁸, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν and hardness H_ν of these vanadium carbides are listed in Table 2. The calculated values of V₂C, V₄C₃, V₆C₅, V₈C₇ and VC in this work are in good agreement with the previous calculation values.

Table 2 Calculated elastic constants C_ij (GPa), the minimum elastic eigenvalue λ₁ (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa),Poisson’s ratio ν and hardness H_ν (GPa).

The Cr₅B₃-type V₅C₃ is tetragonal. For a tetragonal system, the mechanical stability criteria are given by C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, C₁₁ − C₁₂ > 0, C₁₁ + C₃₃ − 2 C₁₃ > 0 and 2(C₁₁ + C₁₂) + C₃₃ + 4 C₁₃ > 0¹⁹. The elastic constants of the Cr₅B₃-type V₅C₃ satisfy these stability conditions, indicating that it is mechanically stable.

The phonon dispersions were calculated to verify the dynamical stability of the Cr₅B₃-type V₅C₃. A dynamically stable crystal structure requires that all phonon frequencies should be positive²⁰. As shown in Fig. 3 for the Cr₅B₃-type V₅C₃ at zero pressure, it is clear that no imaginary phonon frequency can be found in the whole Brillouin zone, indicating that the Cr₅B₃-type V₅C₃ is dynamically stable under ambient conditions.

Figure 3

Phonon dispersions of the Cr₅B₃-type V₅C₃ at zero pressure along high symmetry directions of the Brillouin zone.

Because the hardness measurement involves complex deformation processes, including elastic deformations, plastic deformations and fracture, it is difficult to obtain directly the hardness value of a material from first-principles calculations. Therefore, correlations between elastic moduli and hardness have been suggested as indirect indicators of materials hardness. A hard material should have a high bulk modulus to resist the volume contraction in response to an applied load and a high shear modulus to resist shear deformation. Recently, the softest elastic mode has been shown to correlate better to the hardness number than the other elastic moduli¹⁸, indicating that elastic anisotropy is essential in determining the hardness. The elastic properties (B, G, E and λ₁) of V₅C₃ and the other previously known vanadium carbides as a function of the V/C ratio are plotted in Fig. 4. For the know vanadium carbides, the general trend is that the elastic moduli decrease with the V/C ratio. An abnormal increase occurs at V₅C₃, the elastic moduli of which are higher than both the neighboring V₂C and V₄C₃.

Figure 4

(a) The minimum elastic eigenvalue λ₁, (b) shear modulus G, (c) bulk modulus B and (d) Young’s modulus E of vanadium carbides as a function of the V/C ratio. The lines are guide to the eye.

In order to explain the origin of the stability and the abnormal mechanical properties of the Cr₅B₃-type V₅C₃, the electronic structure of V₅C₃, V₂C and V₄C₃ has been analyzed. Their densities of states (DOS) are plotted in Fig. 5(a). They are metallic with non-zero DOS values at the Fermi level. There are valleys (sometimes called pseudogap) close to the Fermi level for all the three compounds. In general, the electronic states with lower energies than the valley are bonding orbitals and those with higher energies are antibonding orbitals²¹. To clarify the nature of the chemical bonding near the Fermi level, we performed the Crystal Orbital Hamilton Population (−COHP) analysis²², which gives an idea about the participating orbital pair. The positive value represents the bonding states and negative value represents the antibonding states. As shown in Fig. 5(c) for V₅C₃, it is clear that the pseudogap separates the bonding and antibonding states appears. A deeper valley means that the bonding orbitals are more stabilized and the antibonding orbitals are more destabilized, forming strong chemical bonds. Among the three compounds, V₅C₃ has the deepest valley close to the Fermi level. Therefore, the stability and the abnormal mechanical properties of V₅C₃ can be attributed to the pseudogap effect^23,24.

Figure 5

(a) Densities of states of V₅C₃, V₂C and V₄C₃; (b) Densities of states; and (c) Crystal Orbital Hamilton Population (−COHP) analysis of V₅C₃. The red vertical dashed lines denote the Fermi level at zero and the black vertical dashed lines correspond to the energy valley.

The electronic structure of V₅C₃ suggests an interesting method to improve its hardness. The Fermi level of V₅C₃ has a higher energy than the valley, indicating that some antibonding orbitals are occupied. Since the antibonding orbitals would weaken the chemical bonds, once they are made empty, the material could be further strengthened. We consider alloying V₅C₃ with Ti, which has one less valence electron than V. Since Ti is neighboring to V in the periodic table, it should be relatively easy to enter the lattice of V₅C₃. According to the rigid band model, the alloying element normally generates small changes in the nature of chemical bond in the host materials. The Cr₅B₃-type V₅C₃ with the alloying contents of 5 at.%, 10 at.%, 20 at.%, 25 at.% and 30 at.% Ti were investigated. The supercells for the calculations are shown in Fig. 6. In order to minimize the interactions between the alloying atoms, they were placed as far as allowed in the supercells.

Figure 6

The supercells of (a) V₅C₃, (b) (V_0.95Ti_0.05)₅C₃, (c) (V_0.9Ti_0.1)₅C₃, (d) (V_0.8Ti_0.2)₅C₃, (e) (V_0.75Ti_0.25)₅C₃ and (f) (V_0.7Ti₀₃)₅C₃.

The DOS curves of V₅C₃ and its alloys (V_1−xTi_x)₅C₃ were illustrated in Fig. 5(b). As expected, the Fermi level shifts to lower energies with increasing content of Ti from x_Ti = 0 to x_Ti = 0.3. The Fermi level is located at the valley for x_Ti = 0.2.

The calculated elastic constants are listed in Table 3. All the alloys are mechanically stable because the elastic constants of these alloys satisfy the mechanical stability criteria and there is no negative elastic eigenvalue. For the Cr₅B₃-type V₅C₃ and its alloys, the smallest elastic eigenvalue λ₁ is C₆₆, which represents the shear deformation in xy planes. The smallest elastic eigenvalue λ₁, the hardness H_ν, shear modulus G and Young’s modulus E are plotted in Fig. 7. A general trend is that λ₁, H_ν, G and E increase with the content of Ti from x_Ti = 0.05 to x_Ti = 0.2, where they reach their maxima and then decrease as x_Ti increases further. The trend is exactly what we expect from the electronic structure analysis. At x_Ti = 0.2, the Fermi level is located at the valley in DOS. In this case, all of the bonding orbitals are occupied and the antibonding orbitals empty, leading to the strongest chemical bonds.

Table 3 Calculated elastic constants C_ij (GPa), the minimum elastic eigenvalue λ ₁ (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Poisson’s ratio ν and hardness H_ν (GPa) of (V_1−xTi_x)₅C₃.

Figure 7

(a) The smallest elastic eigenvalue λ₁, (b) hardness H_ν, (c) shear modulus G and (d) Young’s modulus E of V₅C₃ alloys.

Conclusions

In summary, the crystal structure, phase stability, electronic structure and mechanical properties of V₅C₃ have been studied. It is demonstrated that the Cr₅B₃-type V₅C₃ is thermodynamically, mechanically and dynamically stable and can be synthesized under pressures above 9.2 GPa.

The elastic properties and electronic structures of (V_1−xTi_x)₅C₃ alloys have also been investigated. When 20 at.% V is substituted by Ti, the Fermi level is tuned to the valley in DOS, giving the maximum hardness of V₅C₃ alloys. While V₅C₃ itself is not a superhard material, the electronic structure and the hardness optimization based on it suggest an interesting way for searching hard materials. The Fermi energy of a material can be tuned to maximize the occupation of bonding orbitals and minimize the occupation of antibonding orbitals, thus strengthening the material.

Computational Methods

In this work, the density functional theory (DFT) calculations were performed using the projector-augmented wave (PAW) method^25,26,27, as implemented in the Vienna Ab-initio Simulation Package (VASP) code²⁸. The generalized gradient approximation (GGA)²⁹ with the Perdew-Burke-Ernzerhof (PBE) scheme was used to describe the exchange-correlation function. Geometry optimization was carried out using the conjugate gradient algorithm. The plane-wave cutoff energy was 500 eV. The k-points were generated using the Monkhorst-Pack mesh³⁰. Lattice parameters and atomic positions were optimized simultaneously. In order to obtain equilibrium volume of the materials, the total-energies were calculated at several fixed volume with the ionic positions and the cell shape allowed to vary. These total energies were then fitted with the Birch-Murnaghan equation of state^31,32,33. The elastic constants were calculated using the universal-linear-independent coupling-strains (ULICS) method³⁴, which is computationally efficient and has been widely used in calculations of single-crystal elastic constants^{35,36,37,38,39}. Based on the single-crystal elastic constants, the bulk modulus B and the shear modulus G were calculated according to the Voigt-Reuss-Hill approximation⁴⁰. Young’s modulus E and Poisson’s ratio ν were obtained by the following equation:

The hardness (H_ν) of V₅C₃ is relative to G and B through the empirical formulabased on the Pugh modulus ratio k = G/B^41,42:

Phonon dispersions were calculated using the direct supercell method, as implemented in the PHONOPY code^43,44. The Crystal Orbital Hamilton Population (−COHP) analysis have been performed to determine the bonding properties of the electronic states close to the Fermi level. Density functional method with LCAO basis sets, as implemented in the SIESTA code⁴⁵, has been used to calculate the COHP. The PBE parameterization of GGA was used. The DZP basis sets were employed. The norm-conserving Troullier-Martins pseudopotentials⁴⁶ were used for the core-valence interactions. The mesh cut-off value was set at 200 Rydberg and the Brillouin zone was sampled using Monkhorst-Packset of k points.