Due to its broad industrial applications, superhard material is always of great scientific interest.^[1,2] As we all know, the diamond and cubic boron nitride (c-BN) are well-studied superhard materials with excellent physical properties, such as wide band gaps, high hardness and thermal conductivity, large shear and bulk modules. However, the adhibition of diamond has been conditioned by its own imperfections, including brittleness, reacting with iron easily, and generating oxidizing reaction in the air at a high temperature. Cubic boron nitride shows better performance than diamond above, but its hardness and mechanical properties are obviously weaker than those of diamond.^[3] B–C–N ternary system is considered to be harder and stronger than c-BN, chemically and thermally more stable than diamond.^[4] Therefore, searching superhard materials from the B–C–N ternary system has been a focus of scientific research. Zhao et al.^[5] successfully synthesized stoichiometric c-BC₂N and c-BC₄N crystals with diamond-like structures by using a diamond-anvil cell. Then Fan et al.^[6–15] did systematic investigations of these structures based on the first-principles calculations, such as crystal structure, band gap, optical spectrum, ideal strength and hardness, in order to establish facts and reach new conclusions.

As we know that B–C–N compounds can consist of the typical sp³ covalent bonds containing B–C, B–N, C–N, and C–C bonds, the mechanical properties thus would be better with the increase of carbon content.^[16–18] Graphite-like BC₆N (g-BC₆N)^[19] has been found as an n-type semiconductor, which was synthesized by the chemical vapor deposition. The diamond-like t-BC₆N and r-BC₆N were proposed by Luo et al.,^[20] which were obtained by replacing two carbon atoms in the eight-atom cubic diamond unit cell with a nitrogen atom and a boron atom. According to their results, the Vickers hardness of the two BC₆N phases is about 80 GPa, higher than that of c-BN (66 GPa) and c-BC₂N (76 GPa). In fact, the first-principles and their evolution methods are always used in predicting possible crystal structures or interpreting the mechanical properties of superhard materials.^[21,22]

In the present work, a C-centered monoclinic BC₆N (Cm-BC₆N) and a primitive-centered monoclinic BC₆N (Pm-BC₆N) are proposed by structure searching method based on first-principles calculations. The structural, electronic, and mechanical properties of the proposed BC₆N allotropes are investigated systematically. The Cm-BC₆N and Pm-BC₆N have dynamical and mechanical stability in ambient circumstance. We not only find their semiconductor nature from the band structures, but also can draw a conclusion that Cm-BC₆N is more favorable than the previous predicted t-BC₆N and r-BC₆N in thermodynamics. By computing the ideal strength and theoretical hardness, we find out the superhard nature structures.

Crystal structures of Cm-BC₆N and Pm-BC₆N were explored by using the evolutionary methodology through particle swarm optimization as implemented in CALYPSO package,^[23,24] which has successfully predicted various structures.^[25–28] Subsequent structural relaxation and property calculations were performed based on the density functional theory^[29,30] with the Vienna ab initio simulation software package (VASP) code, with a projector augmented wave (PAW) scheme^[31] to show the electrons–ion interaction. The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE)^[32] functional was employed to describe the exchange and correlation potentials. From our text data, a plane-wave kinetic energy cutoff of 500 eV and quantitative value of k-points sampling of 0.25 Å⁻¹ in Brillouin zone were used to give good convergence of the total energies. Optimization of the internal atomic positions and the lattice parameters was achieved by the minimization of forces and stress tensors. After calculating the elastic constants, the bulk modulus and shear modulus were figured out following the Voigt–Reuss–Hill (VRH) approximations.^[33] From the local density approximation in plane wave basis, the phonon dispersions were calculated by the density functional perturbation theory, which has been implemented the Quantum ESPRESSO code^[34] with the Troullier–Martins (TM) pseudopotentials.^[35] Static enthalpy differences calculated relative to the graphite-like g-BC₆N structure for the various structures are plotted in Fig. 2 as a function of pressure. The calculations of stress–strain relations were performed following the method of Refs. [36–38], which have calculated the strength of several strong solids successfully.^[39,40] By the micro hardness models for covalent crystal, we acquired the hardness.^[41–43]

The optimized structures of Cm-BC₆N and Pm-BC₆N are shown in Figs. 1(a) and 1(b). Both of Cm-BC₆N and Pm-BC₆N have irregular six membered rings along the [100] and [010] directions. There are B–N bonds in Cm-BC₆N structure, which is the key distinction to Pm-BC₆N. All atoms are four coordinated with sp³ hybridization, resulting in six membered rings like chair/boat. More detailed structure information of Cm-BC₆N and Pm-BC₆N is listed in Table 1. For comparison, the structure parameters of t-BC₆N and r-BC₆N are also provided. The unit cell of Cm-BC₆N contains sixteen atoms. Two B, two N, and four C atoms locate at four inequivalent crystallographic 2a sites, respectively. The other eight C atoms averagely locate at two inequivalent crystallographic 4b sites. The unit cell of Pm-BC₆N contains eight atoms. One B and three C (from C1 to C3) atoms locate at four inequivalent crystallographic 1a sites, respectively. One N and the other three C (from C4 to C6) atoms locate at four inequivalent crystallographic 1b sites, respectively. As shown in Table 1, the mass density of Cm-BC₆N (3.446 g/cm³) is the largest among the BC₆N allotropes, suggesting that the spatial structure of Cm-BC₆N is the most compact one among them.

Fig. 1.

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	Fig. 1. Structures of (a) Cm-BC6N and (b) Pm-BC6N; the yellow, orange, and green spheres represent B, C, and N atoms, respectively.

Table 1.

Table 1.

Table 1. The equilibrium lattice parameters of a, b, c, density ρ, volume, and atomic positions. . structure Cm-BC6N Pm-BC6N t-BC6N r-BC6N Symmetry Cm (No. 8) Pm (No. 6) P 2m (No. 111) R3m (No. 160) a/Å 5.088 4.232 3.611 3.574a 5.101 b/Å 5.102 2.54 3.611 3.574a 5.101 c/Å 4.407 4.406 3.603 3.564a 6.221 α/(°) 90 90 90 90 β/(°) 125.3 90.15 90 90 γ/(°) 90 90 90 120 3.446 3.396 3.424 3.535a 3.443 Volume/Å3 93.365 47.37 46.988 45.512a 140.188 Atomic positions B(0.281, 0, 0.94) B(0.102, 0, 0.959) / / C1(0.282, 0.5, 0.94) C1(0.978, 0, 0.616) C2(0.412, 0, 0.686) C2(0.484, 0, 0.957) C3(0.781, 0.251, 0.435) C3(0.609, 0, 0.626) C4(0.908, 0.243, 0.192) C4(0.978, 0.5, 0.128) N(0.399, 0.5, 0.686) C5(0.615, 0.5, 0.116) C6(0.089, 0.5, 0.449) N(0.473, 0.5, 0.453) a Ref. [20].

Table 1. The equilibrium lattice parameters of a, b, c, density ρ, volume, and atomic positions. .

To assess the thermodynamic stability of the BC₆N (Cm-BC₆N, Pm-BC₆N, t-BC₆N, and r-BC₆N) phases, the relative enthalpies (without temperature effect) are calculated at the pressure range of 0–60 GPa. The pressure–enthalpy relationship calculated for all BC₆N phases is shown in Fig. 2. At zero pressure, g-BC₆N is the most stable phase, and Cm-BC₆N, Pm-BC₆N, t-BC₆N, and r-BC₆N are higher than g-BC₄N by 0.42 eV, 0.75 eV, 0.60 eV, and 0.44 eV per atom, respectively. The enthalpy of Cm-BC₆N is the lowest among the other phases within the entire pressure range, suggesting that Cm-BC₆N should be accessed more easily than Pm-BC₆N, t-BC₆N, and r-BC₆N. Under hydrostatic compression, Cm-BC₆N becomes more favorable energetically than g-BC₆N above 20.6 GPa; and it has a lower transition pressure than 22.2 GPa for r-BC₆N, 35.8 GPa for t-BC₆N, and 55.5 GPa for Pm-BC₆N. Moreover we calculate the phonon dispersion to prove the dynamical stability of Cm-BC₆N and Pm-BC₆N. As shown in Fig. 3, there are no imaginary frequencies observed in the whole Brillouin zone, which indicates that the two phases keep dynamically stable at zero pressure.

Fig. 2.

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	Fig. 2. Enthalpy as a function of pressure for BC6N allotropes relative to g-BC6N.

Fig. 3.

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	Fig. 3. The phonon-dispersion curves of (a) Cm-BC6N and (b)Pm-BC6N at 0 GPa.

By the Hookeʼs law, we calculate the single-crystal zero-pressure elastic constants C_ij to comprehend Cm-BC₆N and Pm-BC₆N in elastic stability and mechanical properties.^[44] For monoclinic symmetry, there are thirteen independent single-crystal elastic constants. As shown in Table 2, the calculated elastic stiffness constants C_ij of Cm-BC₆N and Pm-BC₆N satisfy the mechanical stability criteria for a monoclinic structure,^[45] indicating the elastically stability at ambient condition.

Table 2.

Table 2.

Table 2. Calculated zero-pressure elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Youngʼs modulus E (GPa), and Poissonʼs ratio υ of Cm-BC6N and Pm-BC6N. . Phase Cm-BC6N Pm-BC6N t-BC6N r-BC6N C11 1085.3 1034.8 935.8 1007.2 C22 1026.3 983.7 C33 1014.9 979.4 913.8 1095.4 C44 474.8 442.1 481.9 439.8 C55 441.8 384.1 C66 436.4 368.4 488.0 457.7 C12 59.0 40.6 123.3 91.8 C13 62.0 26.5 105.5 61.9 C14 39.2 C15 –3.0 –0.5 C23 94.9 115.9 C25 46.0 –20.5 C35 –40.4 34.8 C46 41.8 –30.8 BV 395.3 373.8 383.8 393.4 BR 395.3 373.6 383.6 393.2 GV 464.6 426.6 453.8 460.4 GR 459.9 419.4 450.6 456.2 B 395.3 373.7 383.7 393.3 G 462.3 423.0 452.2 458.3 E 997.8 921.3 974.0 990.3 B/G 0.86 0.88 0.85 0.86 υ 0.08 0.09 0.08 0.08

Table 2. Calculated zero-pressure elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Youngʼs modulus E (GPa), and Poissonʼs ratio υ of Cm-BC6N and Pm-BC6N. .

Superhard materials not only manifest high bulk and shear moduli but also exhibit low Poissonʼs ratio, because of hardness strongly depending on the plastic deformation.^[41] Additionally, the Voigt–Reuss–Hill approximation is used to reckon the polycrystalline elastic moduli as the C_ij have been calculated, and the results are listed in Table 2. The calculated bulk modulus of Cm-BC₆N (395.3 GPa) within GGA in the present work is the greatest one, followed by r-BC₆N (393.3 GPa), t-BC₆N (383.7 GPa), and Pm-BC₆N (373.7 GPa). Note that the bulk moduli of t-BC₆N and r-BC₆N are 410.4 GPa and 399.9 GPa in Ref. [20]. The order of shear modulus is the same as that of bulk modulus: Cm-BC₆N (462.3 GPa) r-BC₆N (458.3 GPa) t-BC₆N (452.2 GPa) Pm-BC₆N (423.0 GPa). When describing the directional degree of covalent bonds about a material, we always choose another momentous parameter called Poissonʼs ratio υ. All of the proposed phases have low Poissonʼs ratio (0.08–0.09), which indicates the possibilities of directional covalent bonding. Furthermore, all the BC₆N phases exhibit the brittle behavior due to the low ratio of B/G (0.85–0.88) according to Pughʼs rule.^[46]

The Youngʼs module (E) is one of the most important parameters which measures the stiffness of a solid material. We show the three-dimensional (3D) representation and the corresponding two-dimensional (2D) projection of the Youngʼs modulus for Cm-BC₆N and Pm-BC₆N and in Fig. 4. The shape of the 3D curved surface of Cm-BC₆N is close to a sphere and the 2D projections in the , and xy planes are close to a circle, which show that Cm-BC₆N has slight anisotropy. The same thing happens to Pm-BC₆N. As can be seen in Fig. 4, the , and xy plane areas of Cm-BC₆N are larger than those of Pm-BC₆N, indicating that the Youngʼs modulus property of Cm-BC₆N is superior to that of Pm-BC₆N.^[47]

Fig. 4.

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	Fig. 4. The 3D representations (left panel) and 2D projections (right panel) of Youngʼs moduli for (a) Cm-BC6N and (b) Pm-BC6N. Note that the negative sign only denotes the direction corresponding to the positive one.

The calculated electronic band structure and density of states (DOS) of Cm-BC₆N and Pm-BC₆N at 0 GPa are presented in Fig. 5. It can be seen from the left panel of Fig. 5 that Cm-BC₆N and Pm-BC₆N are semiconductor with an indirect band gap of 2.66 eV and 0.36 eV, respectively. The band gap of Cm-BC₆N is larger than that of the previous proposed r-BC₆N (2.09 eV) and t-BC₆N (2.06 eV). The band gap of Pm-BC₆N (0.36 eV) is the smallest among them. From the data, Cm-BC₆N has the top of the valence band at point and the bottom of the conduct band at Z point. Pm-BC₆N has the top of the valence band at point and the bottom of the conduct band at E point.

Fig. 5.

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	Fig. 5. Electronic band structures (left panel) and density of states (right panel) of (a) Cm-BC6N and (b) Pm-BC6N. The energy is measured from the valence top.

To further understand the chemical bonding and electronic properties, we calculate the partial density of states (PDOS) of Cm-BC₆N and demonstrate the primary valence-electron states of B, C, and N atoms. From the right panel of Fig. 5, it can be find out that there are harmonic 2p–2s overlaps between B(C, N)-2s and B(C, N)-2p states over the whole valence bands in both Cm-BC₆N and Pm-BC₆N structures, which ascertains the hybridization of the s and p orbitals and the strong interaction between B, C, and N atoms. It is in accordance with the bonding behaviors in the previous reports of B–C–N system. In Cm-BC₆N, the B(C, N)-2s mainly contributes to the valence bands below about 10.0 eV, and the B(C, N)-2p mainly contributes to the valence bands between about 10.0 eV and 0 eV. We note the nonmetallic nature of Cm-BC₆N by the large energy band gap which is clear and near the Fermi level.

We also calculate the ideal strength to connect the atomic level strength under plastic deformation and permanent large strain.^[48,49] The calculated tensile stress versus tensile strain of Cm-BC₆N, Pm-BC₆N, t-BC₆N, and r-BC₆N are shown in Fig. 6. As shown in Fig. 6(a), the stress responses of Cm-BC₆N in the seven principal directions with the peak tensile stresses are within the range of 55.9–118.6 GPa. The anisotropy ratio of the ideal tensile strengths for Cm-BC₆N is . Nevertheless, the weakest tensile strength of Cm-BC₆N is 55.9 GPa in the [011] direction at a strain of 0.08. As shown in Fig. 66(b), the strong stress responses in the seven principal directions with the peak tensile stresses are between 32.9 GPa and 91.8 GPa for Pm-BC₆N. The anisotropy ratio of the ideal tensile strengths for Pm-BC₆N is . Nevertheless, the weakest tensile strength of Pm-BC₆N is 32.9 GPa in the [001] direction at a strain of 0.05. As shown in Fig. 6(c), the strong stress responses in the five principal directions with the peak tensile stresses are between 52.3 GPa and 170.6 GPa for t-BC₆N. The anisotropy ratio of the ideal tensile strengths for t-BC₆N is . Nevertheless, the weakest tensile strength of t-BC₆N is 52.3 GPa in the [111] direction at a strain of 0.08. From Fig. 6(d), we find that with the peak tensile stresses, the strong stress responses in the three prime directions are within 82.8–50.9 GPa for r-BC₆N. The anisotropy ratio of the ideal tensile strengths for r-BC₆N is . Nevertheless, the weakest tensile strength of r-BC₆N is 50.9 GPa in the [210] direction at a strain of 0.08. From the data above, the weakest tensile strength of Cm-BC₆N is the largest among the BC₆N allotropes, whereas the weakest tensile strength of Pm-BC₆N is the smallest. Both of the ideal tensile strengths of Cm-BC₆N and Pm-BC₆N are significantly larger than that of β-BC₂N.^[50] 7

Fig. 6.

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	Fig. 6. Calculated tensile stress versus tensile strain for (a) Cm-BC6N, (b) Pm-BC6N, (c) t-BC6N, and (d) r-BC6N in principal symmetry directions.

Fig. 7.

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	Fig. 7. Calculated shear stress versus shear strain for (a) Cm-BC6N and (b) Pm-BC6N in principal symmetry directions.

We also investigate the stress–strain relations of Cm-BC₆N and Pm-BC₆N in various principal symmetry directions under shear deformation. The anisotropy ratio of the ideal shear strengths for Cm-BC₆N is . The weakest shear strength of 53.9 GPa of Cm-BC₆N is found along the (011)[01 shear deformation path. The anisotropy ratio of the ideal shear strengths for Pm-BC₆N is . The weakest shear strength of 25.0 GPa of Pm-BC₆N is found along the (010)[101] shear deformation path. The magnitude of the ideal shear strength of Cm-BC₆N approaches to 58.3 GPa found for the (111)[11 slip system of c-BN.^[51] Besides, it is found that the lowest peak shear stresses of Cm-BC₆N and Pm-BC₆N are lower than their lowest peak tensile stress, respectively, indicating that the mechanical failure modes of these two BC₆N allotropes are dominated by the shear type.

Due to the connection of the hardness and the bulk modulus or shear modulus, we can acquire the hardness. The hardness calculations are performed based on the microscopic models of hardness constructed by Gao et al.^[41–43] and Chen et al.^[52,53] respectively, which have been widely applied currently. In Gaoʼs microscopic hardness models, the hardness of a covalent material is determined by two factors: the properties of bonding at equilibrium and the number of bonds per unit area. The later includes the bond length, charge density, iconicity, etc. The Vickers hardness of the μ-type bond can be calculated as follows: 1 where is the number of valence electrons of the μ-type bonds per cubic angstroms, and it is determined by the number of effective valence electrons per μ bond and the bond volume , which can be expressed as . is Phillipsʼs homopolar band gap of the μ-type bonds,^[54] which characterizes the strength of the covalent bond.^[41] is ionicity of chemical bond in crystal scaled by Phillips,^[55] and , where and .

In practical applications, we define the geometric average hardness of all kinds of chemical bonds in crystals as follows: 2 where is the number of the μ-type bonds composing the actual complex crystal.

We calculate the atomic bond hardness of BC₆N and show the result in Table 3, and the calculation results of c-BC₄N, β-BC₂N, c-BN, and diamond are also given for comparison. The theoretical Vickers hardness values of r-BC₆N (80.2 GPa) and t-BC₆N (80.0 GPa) are according with the calculation results of Ref. [20]. The theoretical Vickers hardness values of c-BC₄N(80.3 GPa) and β-BC₂N(75.5 GPa) agree well with experimental results of Refs. [15] and [6]. This proves the reliability. The hardness values of Cm-BC₆N and Pm-BC₆N are predicted to be 80.6 GPa and 77.4 GPa by Gaoʼs model and to be 84.0 GPa and 76.5 GPa by Chenʼs model, respectively. Coincidently, the hardness by Chenʼs model is compared favorably with for the two novel BC₆N allotropes. Both methods indicate that Cm-BC₆N is a superhard material with the high Vickers hardness above 80 GPa, which is evidently larger than that of c-BN (66.4 GPa), β-BC₂N (75.5 GPa), and c-BC₄N (80.3 GPa); furthermore, Cm-BC₆N is harder than r-BC₆N (80.2 GPa) and t-BC₆N (80.0 GPa); nevertheless, it is slightly inferior to diamond (93.8 GPa). The results show the superior mechanical properties of Cm-BC₆N and Pm-BC₆N in potential technological applications.

Table 3.

Table 3.

Table 3. Hardness calculations of Cm-BC6N, Pm-BC6N, t-BC6N, r-BC6N, c-BC4N, c-BN, and diamond, where is the bond length, is the electron density expressed in number of valence electrons per cubic angstroms, is the homopolar gap, is the valence-conduction band gap, is the ionicity of bond, is geometric average of all bonds vickers hardness, compared to the hardness of (Chen) and the experimental hardness , respectively. . Phase Bond /Å Population /Å3 /Å−3 /eV /eV /GPa /GPa /GPa (Chen)/GPa /GPa Cm–BC6N B–C3 2 1.599 0.77 3.145 0.445 12.286 12.294 0.001 63.1 80.6 84.0 B–C1 4 1.592 0.92 3.104 0.533 12.423 12.448 0.004 71.7 B–N 2 1.589 0.62 3.085 0.569 12.486 15.190 0.324 51.4 C1–C2 4 1.548 0.81 2.851 0.739 13.333 13.373 0.006 95.5 C3–C4 2 1.533 0.82 2.773 0.777 13.648 13.664 0.002 101.5 C1–C2 4 1.542 0.83 2.817 0.766 13.466 13.508 0.006 98.7 C1–C4 4 1.538 0.89 2.795 0.827 13.557 13.599 0.006 104.6 C2–C3 4 1.559 0.89 2.916 0.803 13.086 13.106 0.003 99.4 C4–N 2 1.542 0.63 2.820 0.632 13.457 16.346 0.322 59.6 C2–N 4 1.565 0.69 2.947 0.663 12.972 16.107 0.351 57.3 Pm-BC6N B–C1 1 1.616 0.43 3.241 0.267 11.970 11.974 0.001 43.8 77.4 76.5 B–C4 1 1.599 0.65 3.136 0.413 12.301 12.317 0.003 60.0 B–C6 2 1.566 0.92 2.946 0.614 12.959 13.016 0.009 81.7 C5–C6 1 1.534 0.62 2.772 0.724 13.634 13.671 0.005 96.3 C3–C4 1 1.563 0.66 2.933 0.759 13.007 13.086 0.012 94.1 C2–C6 1 1.490 0.74 2.542 0.943 14.654 14.707 0.007 123.2 C1–C3 1 1.550 0.81 2.861 0.966 13.280 13.341 0.009 113.2 C2–C4 2 1.540 0.90 2.805 0.843 13.500 13.518 0.003 105.9 C1–C5 2 1.554 0.90 2.880 0.829 13.208 13.211 0.000 102.8 C2–N 1 1.626 0.44 3.303 0.415 11.781 13.807 0.272 41.8 C5–N 1 1.601 0.57 3.153 0.563 12.246 15.122 0.344 48.9 C3–N 2 1.590 0.68 3.084 0.699 12.473 15.708 0.369 55.8 t-BC6N C1–N 4 1.571 0.60 2.986 0.631 12.838 15.583 0.321 56.8 80.0 79.9a 83.7 B–C1 4 1.574 0.72 3.000 0.504 12.791 12.792 0.000 71.4 C1–C2 8 1.553 0.76 2.880 0.809 13.230 13.274 0.007 100.5 c-BC6N B–N 3 1.576 0.64 3.013 0.335 12.740 13.004 0.040 51.7 80.2 79.1a 83.2 C1–C2 9 1.539 0.85 2.804 0.899 13.526 13.567 0.006 110.3 C1–N 9 1.572 0.68 2.991 0.633 12.818 15.787 0.341 55.5 B–C2 9 1.578 0.89 3.025 0.520 12.699 12.713 0.002 72.3 C1–C2 18 1.552 0.87 2.876 0.718 13.246 13.277 0.005 93.1 c-BC4N B–C3 1 1.617 0.51 3.235 0.342 11.946 12.0-24 0.013 50.7 80.3 84.3b 81.4 c B–N 3 1.566 0.69 2.938 0.735 12.944 15.031 0.258 68.3 C4–C3 3 1.550 0.83 2.850 0.843 13.278 13.278 0.000 104.5 C4–C1 1 1.554 0.50 2.872 0.349 13.193 13.193 0.000 57.7 C2–C1 3 1.549 0.83 2.850 0.849 13.306 13.307 0.000 105.2 C2–N 1 1.516 0.47 2.842 0.494 14.052 16.080 0.236 58.4 β-BC2N C1–C2 4 1.528 0.83 2.744 0.911 13.767 13.767 0 114.1 75.5 75.4 76d B–C1 4 1.591 0.72 3.094 0.485 12.453 12.453 0 67.8 C2–N 4 1.578 0.59 3.021 0.662 12.706 14.49 0.231 64.6 B–N 4 1.576 0.62 3.011 0.664 12.741 14.528 0.231 65.0 Diamond C–C 16 1.544 0.75 2.834 0.706 13.413 13.413 0 93.8 93.8 93.6 e c-BN B–N 16 1.569 0.65 2.973 0.673 12.889 14.691 0.23 66.4 66.4 63.5 e a Ref. [20], b Ref. [7], c Ref. [15], d Ref. [6], e Ref. [41].

Table 3. Hardness calculations of Cm-BC6N, Pm-BC6N, t-BC6N, r-BC6N, c-BC4N, c-BN, and diamond, where is the bond length, is the electron density expressed in number of valence electrons per cubic angstroms, is the homopolar gap, is the valence-conduction band gap, is the ionicity of bond, is geometric average of all bonds vickers hardness, compared to the hardness of (Chen) and the experimental hardness , respectively. .

In summary, two surperhard BC₆N with monoclinic structures were proposed. The structural, electronic, and mechanical properties were investigated by first-principles calculations. Cm-BC₆N and Pm-BC₆N are both mechanically and dynamically stable, and Cm-BC₆N is suggested to be more easily synthesized than Pm-BC₆N, t-BC₆N, and r-BC₆N. We found that the lowest stress peak of Cm-BC₆N is 53.9 GPa, which is higher than that of other general hard materials. Comparing with other BCN allotropes which have been proposed and Pm-BC₆N, Cm-BC₆N has a higher hardness from Gaoʼs model (80.6 GPa). In Cm-BC₆N, the B and N atoms connect each other by the prominent and strong C–C bond which results in its superior mechanical properties. Both of Cm-BC₆N and Pm-BC₆N are significantly harder than β-BC₂N and c-BN, suggesting the intrinsic superhard materials.