A New Superhard Phase of C₃N₂ Polymorphs

1 Introduction

Searching new superhard material has been concentrated by researchers in recent years. Usually, borides, nitrides, and covalent of light elements (Be, B, C, N, etc.) are regarded as candidates of superhard materials [1–5]. Among these materials, carbon nitrides are a typical group. Theoretically, some superhard phases of carbon nitrides are predicted by first principles calculations before experimental preparation. Liu and Cohen [6, 7] presented a hexagonal superhard β-C₃N₄ structure, which has been obtained experimentally later [8, 9]. Then, C₃N₄ and C₁₁N4 are theoretically proved to be superhard materials [10]. The theoretical hardness of C₃N₄ is 85.7 GPa [11], which is close to that of diamond. In addition, a group of tetragonal crystalline carbon mononitrides (CN) have been predicted [12]. By replacing part of the nitrogen atoms in the cubic gauche of nitrogen, a new superhard CN phase (cg-CN) has been proposed [13]. It is interesting that the cg-CN is found to be a metallic compound, although most superhard carbon nitrides are insulators or semiconductors. For example, Weihrich et al. proposed a Pa3̅ phase CN₂ with a large bulk modulus of 405 GPa [14]. Using crystal structure prediction technique, a body-centered tetragonal CN₂ has been uncovered [15]. The ideal strength and hardness calculation show that bct-CN₂ is superhard with the hardness of 77.4 GPa. Tian et al. proposed Pm3̅m phase C₃N₂ (α-C₃N₂) at pressure below 1.0 GPa and P4̅3m phase (β-C₃N₂) at high pressure [16]. The Vickers hardness of both phases are approximately 86 GPa.

In the present work, we present a new metastable C₃N₂ phase by replacing part of the nitrogen atoms in the cagelike diamondoid nitrogen N₁₀ [17] with carbon atoms. In N₁₀ crystal, there are two inequivalent sites 12e and 8c. We replaced the N atoms in the Wyckoff position 12e (C₃N₂) and 8c (C₂N₃) with carbon atoms, respectively. Both C₃N₂ and C₂N₃ have the same space group I4̅3m. Phonon spectra calculations show that C₂N₃ is not stable, whereas C₃N₂ is found to be mechanically and dynamically stable up to at least 50 GPa, with a high hardness of 72.9 GPa.

2 Computational Details

Our calculations are performed using the VASP code [18] with the generalised gradient approximation (GGA) [19] and local density approximation (LDA) [20] for exchange correlation functional. The integration in the Brillouin zone is employed using the Monkhorst–Pack scheme (11×11×11), and the energy cut-off of 900 eV was chosen to ensure that energy calculations are converged to better than 1 meV/atom. The electron–core interactions are included by using the frozen-core all-electron projector augmented wave (PAW) method [21], with C:2s²2p² and N: 2s²2p³ treated as the valence electrons. The elastic constants are determined from evaluation of stress tensor-generated small strain. Bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio are estimated by using Voigt–Reuss–Hill approximation [22].

3 Results and Discussions

After relaxing the crystal structure at 0 GPa, we obtain that this structure has a lattice parameter of 4.9202 Å within GGA. The optimised parameters for C₃N₂ at 0 GPa within GGA and LDA are listed in Table 1, and the unit cell structure is shown in Figure 1. In this structure, there are two inequivalent C6N4 cages. One is at body centered site, and the other is at the vertex site. Each cage connectes with six equivalent cages by C–C bonds, whereas there is no covalent bonding between center and vertex cages. The C6N4 cage is formed by 12 C–N bonds, and the bond angles of C–N–C and N–C–N are 110.0 and 106.4 degree, respectively. The P–V relations of α-C₃N₂, β-C₃N₂, and I4̅3m-C₃N₂ are plotted in Figure 2, as well as those of c-BN and diamond. We can see that the incompressibility of I4̅3m-C₃N₂ is larger than β-C₃N₂ and almost the same as α-C₃N₂ and c-BN. We can deduce that the bulk modulus of these three phases of C₃N₂ should follow the same order. To verify this, we obtain the values of equilibrium bulk modulus and its pressure derivative by fitting the E–V data at different pressure into the third-order Birch–Murnaghan equation of state (EOS) [23], and the results are listed in Table 1. The fitted bulk modulus for I4̅3m-C₃N₂ is 358 GPa within GGA, which is larger than that of β-C₃N₂ (328 GPa) and slightly less than that of α-C₃N₂ (373 GPa).

Figure 1:

Crystal structure of I4̅3m-C₃N₂. The N atoms are represented in blue and C atoms black.

Figure 2:

Variation of ratio V/V₀ with pressures for I4̅3m-C₃N₂, α-C₃N₂, β-C₃N₂, c-BN, and diamond.

In order to check the stability of I4̅3m-C₃N₂, we calculate the elastic constants (see Tab. 2) and phonon frequencies (Fig. 3). For cubic crystal, the mechanical stability criteria at 0 GPa are as follows [24]: C₁₁+2C₁₂>0, C₄₄>0, and C₁₁–C₁₂>0. When a hydrostatic pressure is applied on crystals, the mechanical stability criteria for cubic crystals are as follows [25]: C₁₁+2C₁₂+P>0, C₄₄–P>0, and C₁₁–C₁₂–2P>0. One can see from Table 2 that the calculated elastic constants are satisfied above mechanical stability criteria, which shows the mechanical stability of I4̅3m-C₃N₂ up to 50 GPa. The phonon dispersion curves for I4̅3m-C₃N₂ at 0 and 50 GPa are shown in Figure 3. No imaginary phonon frequency is observed in the whole Brillouin zone, indicating its dynamical stability in the range from 0 to at least 50 GPa. In the pressure range, we studied that the enthalpy of I4̅3m-C₃N₂ is always higher than that of α-C₃N₂ or β-C₃N₂, so it is a metastable phase of C₃N₂. The calculated bulk modulus (see Tab. 2) by means of the Voigh–Reuss–Hill approximation method is consistent with the EOS fitting results. The shear modulus quantifies its resistance to shear deformation and is a better indicator of potential hardness for ionic and covalent materials [26]. Young’s modulus E is defined as the ratio between stress and strain and is used to provide a measure of the stiffness of the solid. Compared to α-C₃N₂ and β-C₃N₂, the less E of I4̅3m-C₃N₂ means that α-C₃N₂ and β-C₃N₂ are stiffer than I4̅3m-C₃N₂. The G/B ratio can be used to determine the relative directionality of the bonding in the material. The calculated ratio G/B for I4̅3m-C₃N₂ is 0.64, which is smaller than that of α-C₃N₂ and β-C₃N₂ at the GGA level. This shows that the directionality of C–N bond in I4̅3m-C₃N₂ is weaker than that of α-C₃N₂ and β-C₃N₂.

Figure 3:

Phonon dispersion curve of I4̅3m-C₃N₂ at 0 GPa (a) and 50 GPa (b).

Elastic anisotropy can give a prediction of the arrangement of the atoms in each direction, the bonding properties, and some chemical characters in different directions of materials. To illustrate the elastic anisotropy in detail, it is worthy to study the variation of Young’s modulus and shear modulus with direction. The variation of Young’s modulus along an arbitrary [hkl] direction for orthorhombic symmetry can be written as

where α, β, and γ are the direction cosines of the tensile stress direction, and s₁₁, s₁₂, and s₄₄ are elastic compliance constants. The relations between s_ij and C_ij are s11 = (C11 + C12)/(C112 + C11C12 − 2C122),s12 = −C12/(C112 + C11C12 − 2C122), and s₄₄=1/C₄₄. The directional dependence of Young’s modulus for I4̅3m-C₃N₂ is shown in Figure 4. β-C₃N₂ has almost the same elastic constants as α-C₃N₂ (see Tab. 2); thus, they should have the same anisotropy properties. So, we only illustrate the directional dependence of Young’s modulus for α-C₃N₂ in Figure 4 for comparison. It can be seen that the nonspherical nature of I4̅3m-C₃N₂ and α-C₃N₂ in Figure 4a and b shows the clear anisotropy of Young’s modulus. The maximum and minimum of Young’s modulus of I4̅3m-C₃N₂ in all directions are 589 and 547 GPa, respectively. The ratio E_max/E_min=1.08 is slightly less than that of α-C₃N₂ (E_max/E_min=820/747=1.10). The mean value of I4̅3m-C₃N₂ in all directions is 568 GPa, which matches well with the value calculated by the Voigt–Reuss–Hill approximation (572 GPa, see Tab. 2). In order to get a better understanding of the origin of the changes in Young’s modulus along different directions, the orientation dependences of Young’s modulus for I4̅3m-C₃N₂ and α-C₃N₂ are calculated when the tensile axis is within specific planes, as shown in Figure 5. From (1), we can obtain E⁻¹=s₁₁+sin²2θ(2s₁₂+s₄₄–2s₁₁)/4 for Young’s modulus in (100) plane, where θ is the angle between tensile stress and [001], and E⁻¹=sin⁴θ(2s₁₁+2s₁₂+s₄₄)/4+s₁₁cos⁴θ+sin²2θ(2s₁₂+s₄₄)/4 for Young’s modulus in (11̅0) plane, where θ is the angle between tensile stress and [001]. The maximum of Young’s modulus of I4̅3m-C₃N₂ is along [111] direction, whereas the minimum is along [100]. The order of Young’s modulus as a function of the principal crystal tensile for I4̅3m-C₃N₂ is E_[111]>E_[110]>E_[100]. For α-C₃N₂, the ordering is E_[100]>E_[110]>E_[111]. Using the Lyakhov–Oganov model [27], the hardness of I4̅3m-C₃N₂, as well as the hardness of α-C₃N₂ and β-C₃N₂, is calculated and listed in Table 2. The hardness of I4̅3m-C₃N₂ is 72.9 GPa within GGA. This shows that I4̅3m-C₃N₂ is a superhard material. Since C–C bond is a strong covalent bond, the shorter C–C bonds usually correspond to harder materials. For I4̅3m-C₃N₂, the C–C bond length is shorter than that of α-C₃N₂ and β-C₃N₂, so the hardness of I4̅3m-C₃N₂ is larger than that of α-C₃N₂ and β-C₃N₂.

Figure 4:

Illustrations of directional dependence of Young’s modulus for I4̅3m-C₃N₂ (a) and α-C₃N₂ (b), and projections in xy plane of directional dependent Young’s modulus for I4̅3m-C₃N₂ (c) and α-C₃N₂ (d) at 0 and 50 GPa.

Figure 5:

Plots of Young’s modulus for different crystallographic directions for I4̅3m-C₃N₂ (a) and α-C₃N₂ (b).

To obtain a deeper insight into the hardness of the I4̅3m-C₃N₂, we calculate the band structure and density of state (DOS) and the atom resolved partial density of state (PDOS) of I4̅3m-C₃N₂ at 0 GPa, as shown in Figure 6. It is found that I4̅3m-C₃N₂ is metallic due to the finite electronic DOS at the Fermi level. The gap above fermi level shows that I4̅3m-C₃N₂ is a hole conductor, similar to BC₅ [28] and BC₇ [29]. From inspection of PDOS curves, it can be seen that C-p orbitals have a significant hybridisation with N-p orbitals near the Fermi level, signifying the strong C–N covalent bonding nature.

Figure 6:

Band structure and DOS for I4̅3m-C₃N₂.

4 Conclusions

A new I4̅3m-C₃N₂ phase has been uncovered by replacing nitrogen atoms of N₁₀ with carbon atoms. Elastic and phonon calculations show that this phase is mechanically and dynamically stable up to at least 50 GPa. We have investigated the structural, mechanical, and electronic properties of I4̅3m-C₃N₂ using first principles calculations. The tensile directional dependence of Young’s modulus obeys the following trend: E_[111]>E_[110]>E_[100]. The calculated results demonstrate that I4̅3m-C₃N₂ has large bulk and shear moduli. The hardness of 72.9 GPa illustrates its superhard character.