Cubic C₃N: A New Superhard Phase of Carbon-Rich Nitride

Abstract

Using the particle swarm optimization technique, we proposed a cubic superhard phase of C₃N (c-C₃N) with an estimated Vicker’s hardness of 65 GPa, which is more energetically favorable than the recently proposed o-C₃N. The c-C₃N is the most stable phase in a pressure range of 6.5–15.4 GPa. Above 15.4 GPa, the most energetic favorable high pressure phase R3m-C₃N is uncovered. Phonon dispersion and elastic constant calculations confirm the dynamical and mechanical stability of c-C₃N and R3m-C₃N at ambient pressure. The electronic structure calculations indicate that both c-C₃N and R3m-C₃N are indirect semiconductor.

1. Introduction

Covalent light elements and their compounds are regarded as candidates of superhard materials. In light of the extensive applications of c-BN as superhard material, boron nitride compounds, such as Z-BN, Pbca-BN, w-BN, O-BN, and C222₁-B₃N₅, have been found to be superhard [1,2,3,4,5,6]. Except boron nitrides, carbon nitrides are also expected to be potential superhard materials [7,8,9,10,11,12,13,14,15]. Since the prediction of hexagonal β-C₃N₄ with extraordinary hardness, extensive experimental, and theoretical investigations have been carried out to search for new C–N materials with novel properties. A following work [9] proposed that the hardness of c-C₃N₄ is close to that of diamond using the microscopic model of hardness. In recent years, several different crystal structure prediction technique-based algorithms have been the most prevalent strategies to explore and predict new material structures. Using the swarm optimization technique, Li et al. [13] found a novel low compressible and superhard bct-CN₂ phase. Zhang et al. predicted that the stable CN is P4₂/m phase [11]. A phase diagram of carbon nitrides was predicted by Dong et al. [12] in the pressure range of 0–300 GPa. Replacing atoms is another common way to find new phases. By substituting all B sites and three N sites out of five in C222₁-B₃N₅ by C atoms, Hao et al. proposed a C222₁ superhard phase o-C₃N with a Vicker’s hardness of 76 GPa [16], and this o-C₃N is more energetically favorable than the previously proposed C₃N phases [17,18]. Although different kinds of structures have been proposed, the quest for a new energetically stable or metastable C₃N phase is a hot topic in superhard materials. In the present work, we performed an extensive structure search for the most energetically stable high pressure C₃N phases using the prevalent developed crystal structure analysis by particle swarm optimization (CALYPSO) package [19]. This method has been successfully applied to predict structures that have been confirmed by experiments [8,20,21,22]. Two high-pressure phases with group symmetry Fd-3m and R3m phases were uncovered, and the pressure-induced phase transition was then found. The crystal structures, electronic structures, and mechanical behaviors of these two phases were systematically studied.

2. Results and Discussion

At ambient pressure, the ground state of C₃N revealed by our structural search is confirmed to be the P6/mmm phase (h-C₃N) proposed in [16]. For high pressures at 10 GPa and 20–100 GPa, our structure searches found the most stable structures to be a cubic Fd-3m phase (hereafter denoted as c-C₃N, see Figure 1a) and a layered hexagonal R3m phase (see Figure 1c), respectively. Meanwhile, the recent proposed orthorhombic C222₁ phase (o-C₃N) was also explored as a metastable phase. As shown in Figure 1a, the c-C₃N contains 128 atoms in its unit cell, in which the C and N atoms occupying the 96g and 32e positions. In more detail, for c-C₃N, each N atom is sp²-hybridized and connects with three C atoms with a C–N bond length of 1.429 Å. Meanwhile, each C atom is connected with three C atoms and one N atom with two 1.551 Å C–C bonds, one 1.623 Å C–C bond, and one 1.429 Å C–N bond. Interestingly, as seen in Figure 1b, the c-C₃N consists of a fundamental building block: a C24N4 cage is composed of four C6 regular hexagon and twelve C4N pentagons. These cages are connected by sharing a C6 hexagon. The crystal structure of R3m-C₃N is shown in Figure 1c. There are 24 atoms in one unit cell, including three layers. Each layer contains two N atoms and six C atoms. N atoms are bonded with three C atoms, and the C–N bond length is 1.503 Å. Detailed structural information of the predicted c-C₃N and R3m-C₃N at ambient pressure is presented in

Table 1. The calculated equilibrium structural parameters of c-C₃N and R3m-C₃N at ambient pressure.

Phase	a, b, c (Å)	Atomic Coordinates	dC–C (Å)	dC–N (Å)
c-C3N	a = 9.5342	C 96g (0.93983, 0.93983, 0.75924) N 32e (0.35488, 0.35488, 0.35488)	1.551 1.623	1.429
R3m-C3N	a = 2.4519 c = 31.7508	C 3a (0.33333, 0.66667, 0.07030) C 3a (0.33333, 0.66667, 0.02080) C 3a (0.33333, 0.66667, 0.28862) C 3a (0.66667, 0.33333, 0.08675) C 3a (0.66667, 0.33333, 0.00436) C 3a (0.66667, 0.33333, 0.13582) N 3a (0.66667, 0.33333, 0.27269) N 3a (0.00000, 0.00000, 0.15174)	1.558 1.572 1.553 1.509	1.503

.

To determine the phase transition pressure of C₃N, the enthalpies differences of the predicted c-C₃N, R3m-C₃N, and o-C₃N relative to h-C₃N as a function of pressure are illustrated in Figure 2 up to 20 GPa. It should be noted that, in the 20–100 GPa pressure range, the enthalpies of these phases follow the same order as that at 20 GPa. To clearly show the phase transition pressure, in Figure 2, the pressure range is taken as 0–20 GPa. It can be seen from Figure 2 that the predicted c-C₃N becomes more stable than h-C₃N above 6.5 GPa, whereas the transition pressures from h-C₃N to R3m-C₃N and o-C₃N are 7.2 and 9.6 GPa, respectively. In the 6.5–15.4 GPa pressure range, c-C₃N is the most stable phase. Above 15.4 GPa, it transforms to the R3m phase. As seen, the previous proposed o-C₃N is a metastable phase in the whole considered pressure range. To verify the stability of the newly proposed phases, we have performed the phonon calculations of c-C₃N and R3m-C₃N at 0 and 100 GPa, and the results are illustrated in Figure 3. As shown in Figure 3, the absence of imaginary phonon frequency in the whole Brillouin zone indicates their dynamical stability up to at least 100 GPa.

To study the mechanical properties of c-C₃N, the elastic constants were calculated using the strain–stress method. The average bulk modulus, shear modulus, and Young’s modulus of c-C₃N were further estimated using the Voigt–Reuss–Hill approximation. The calculated results are listed in

Table 2. The calculated elastic constants C_ij (GPa), Young’s modulus E (GPa), bulk modulus B (GPa), shear modulus G (GPa), and the G/B ratios of o-C₃N, c-C₃N, and diamond.

Crystal	Reference	C11	C22	C33	C44	C55	C66	C12	C13	C23	E	B	G	G/B
o-C3N	This work	1105	806	804	484	424	424	122	125	222	-	404	409	1.01
-	[16]	1092	797	796	482	398	392	131	132	222	918	406	401	0.99
c-C3N	This work	766	-	-	403	-	-	133	-	-	810	344	365	1.06
Diamond	[4]	1052	-	-	563	-	-	120	-	-	-	431	522	1.2

, together with the elastic parameters of o-C₃N and diamond for comparison. Although the bulk modulus of c-C₃N (344 GPa) is less than that of o-C₃N (404 GPa) and diamond (433 GPa), it is comparable to that of P4₂/m-CN (341 GPa), Pnnm-CN (369 GPa), and BeC₂N (315 GPa for CH and 311 GPa for WU structure) [11,23], indicating the ultra-incompressibility of c-C₃N. The elastic constants of c-C₃N satisfy the criteria for a cubic crystal [C₁₁ − C₁₂ > 0, C₁₁ > 0, C₄₄ > 0], showing that it is mechanically stable at ambient pressure. By using Chen’s model [24], the Vicker’s hardness is 64.6 GPa estimated by $H_V=2{(G^3/B^2)}^{0.585}-3$, which means c-C₃N is a potential superhard material.

The ideal strength in a specified direction is microscopically determined by bond strength and breaking nature under strain. Thus, the comprehensive analysis of the bonding nature and of the ideal strength can provide a deeper insight into mechanical behavior and the hardness of materials. The tensile strain–stress relations can be obtained through the projection of the unitcell onto the corresponding crystal axes with one axis parallel to the stain direction for tension deformation. Figure 4 shows the ideal tensile and shear strength of c-C₃N. The ideal tensile strength for c-C₃N are calculated in the main high symmetry directions, including [100], [110], and [111]. The lowest ideal tensile strength appears in the [110] direction with 76.8 GPa. This means, under tensile loadings, c-C₃N would first cleave in the (110) plane. The ideal shear strength was then calculated by applying the [1–10] and [001] shear deformations in the (110) easy cleavage plane perpendicular to the weakest tensile direction. The ideal shear deformations in the (100)[010], (111)[1–10], and (111)[–1–12] are also shown in Figure 4 for comparison. The lowest peak shear stress occurs in the (110)[1–10] direction with 66.4 GPa, which is lower than the lowest tensile strength. This means the failure mode in c-C₃N is dominated by the shear type with 66.4 GPa. This value is also close to the hardness calculated by Chen’s model.

Figure 5 shows the band structure and density of states (DOS) of c-C₃N and R3m-C₃N. As seen in Figure 5a, c-C₃N is an indirect band gap semiconductor with 3.76 eV gap. The conduction band minimum (CBM) is located at Г point, whereas the valence band maximum (VBM) is located at (0.2 0.2 0.0) along the M-Г direction. As indicated by the partial DOS in Figure 5b, the peaks at low-energy range (below −12 eV) are mainly attributed to C-2s and N-2s states. In the energy range of −12 to 0 eV, the contribution from C-2p and N-2p states are much larger than that of C-2s and N-2s states. Near the Fermi level, the contributions from C-2p and N-2p are almost the same, which indicates the significant hybridization between these two orbitals. For R3m-C₃N, the CBM is located at M point and the VBM is at A point, which leads to an indirect band gap semiconductor with 3.85 eV gap, as seen in Figure 5c. Similar to c-C₃N, there is a significant hybridization between C-2p and N-2p orbitals due to the nearly equal values near the Fermi level (see Figure 5d).

3. Computational Methods

The crystal structure prediction is based on the global minimization of energy surfaces merging ab initio total energy calculations as implemented in CALYPSO code [19]. We performed variable cell structure predictions at 0, 10, 20, 50, and 100 GPa with one to eight formula units (f.u.) per simulation cell. At each pressure point, we found 30 generations and 40 structures generated via particle swarm optimization (PSO) per generation. For structural relaxations and electronic calculations, various first-principles calculation methods have been used in carbon-based systems [25,26,27]. In the present work, we performed structural relaxations and electronic calculations within the density functional theory, carried out within the Vienna ab initio simulation package (VASP) [28], with the projector augmented wave method [29]. The electronic wave functions were expanded in a plane-wave basis set with a well converged cutoff energy of 900 eV. Monkhorst–Pack k point meshes with a grid of 0.02 A⁻¹ for Brillouin zone sampling were chosen to ensure that total energies converged were better than 1 meV/atom. The phonon frequencies were calculated by a supercell approach as implemented in PHONOPY code [30], and the forces were calculated from VASP. The elastic constants were calculated by the strain–stress method, and Young’s modulus, the shear modulus, the bulk modulus, and Poisson’s ratio were derived from the Voigt–Reuss–Hill approximation [31].

4. Conclusions

In summary, by using unbiased structure searching techniques in combination with first-principles calculations, we have predicted two new high pressure phases of C₃N: c-C₃N and R3m-C₃N. Both c-C₃N and R3m-C₃N are dynamically stable in the pressure range from 0 to 100 GPa. The pressure induced a phase transition in the following order: h-C₃N → c-C₃N → R3m-C₃N. The transition pressure was 6.5 and 15.4 GPa, respectively. The high bulk modulus of c-C₃N shows that c-C₃N is ultra-incompressible. The Vicker’s hardness of c-C₃N was estimated to be about 65 GPa. The ideal strength calculations indicate that c-C₃N is intrinsically superhard. The failure mode in c-C₃N was dominated by the shear type with 66.4 GPa. Both c-C₃N and R3m-C₃N are indirect band gap semiconductors at ambient pressure with 3.76 and 3.85 eV gaps, respectively. Such theoretical predictions can guide future synthesis experiments of carbon-rich nitrides.