⇦Chinese Physics Letters, 2019, Vol. 36, No. 3, Article code 036201 An Orthorhombic Phase of Superhard $o$-BC$_{4}$N * Nian-Rui Qu (屈年瑞), Hong-chao Wang (王洪超), Qing Li (李青), Zhi-Ping Li (李志平)**, Fa-Ming Gao (高发明)** Affiliations Key Laboratory of Applied Chemistry, Yanshan University, Qinhuangdao 066004 Received 21 October 2018, online 23 February 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 21671168 and 21875205, the Hebei Natural Science Foundation under Grant No B2015203096, and the Qinhuangdao Science and Technology Support Program under Grant No 201703A014.
**Corresponding author. Email: zpli@ysu.edu.cn; fmgao@ysu.edu.cn Citation Text: Qu N R, Wang H C, Li Q, Li Z P and Gao F M et al 2019 Chin. Phys. Lett. 36 036201    Abstract A potential superhard $o$-BC$_{4}$N with $Imm2$ space group is identified by ab initio evolutionary methodology using CALYPSO code. The structural, electronic and mechanical properties of $o$-BC$_{4}$N are investigated. The elastic calculations indicate that $o$-BC$_{4}$N is mechanically stable. The phonon dispersions imply that this phase is dynamically stable under ambient conditions. The structure of $o$-BC$_{4}$N is more energetically favorable than $g$-BC$_{4}$N above the pressure of 25.1 GPa. Here $o$-BC$_{4}$N is a semiconductor with an indirect band gap of about 3.95 eV, and the structure is highly incompressible with a bulk modulus of 396.3 GPa and shear modulus of 456.0 GPa. The mechanical failure mode of $o$-BC$_{4}$N is dominated by the shear type. The calculated peak stress of 58.5 GPa in the (100)[001] shear direction sets an upper bound for its ideal strength. The Vickers hardness of $o$-BC$_{4}$N reaches 78.7 GPa, which is greater than that of $t$-BC$_{4}$N and $bc$-BC$_{4}$N proposed recently, confirming that $o$-BC$_{4}$N is a potential superhard material. DOI:10.1088/0256-307X/36/3/036201 PACS:62.20.-x © 2019 Chinese Physics Society Article Text Superhard materials are very useful in various industrial applications, such as cutting tools, abrasives and protective coatings.[1] The best well-known superhard material is diamond with 96 GPa Vickers hardness in nature, but its utility is limited by several shortcomings, including brittleness, a tendency to react with iron, and oxidization in air at high temperature. Another well-known superhard material is artificially synthesized cubic boron nitride, which can overcome diamond's shortcomings, but its hardness is obviously weaker than diamond.[2] It is an urgent requirement to search for novel phases with superior mechanical properties. Ternary B–C–N crystals are predicted to exceed diamond in thermal stability and chemical stability and are harder than $c$-BN due to the unique construction combining diamond and $c$-BN.[3] Since superhard $c$-BC$_{2}$N and $c$-BC$_{4}$N crystals with a diamond-like structure were successfully synthesized using a diamond-anvil cell,[4-6] many investigations based on first-principles calculations[7-10] have been carried out on the possible crystal structure, ideal strength, and Vickers hardness of them. Zhao et al.[5] first reported that well-sintered BC$_{2}$N and BC$_{4}$N were translucent with light-yellowish color and that the measured Vickers indentation hardness values were about 62 GPa for the BC$_{2}$N samples and about 68 GPa for the BC$_{4}$N samples. Recently, novel $bc$-BC$_{4}$N[9] and $t$-BC$_{4}$N[10] were predicted, respectively. $bc$-BC$_{4}$N belongs to orthorhombic systems with a $Pmmm$ space group, which is a superhard material with 68 GPa Vickers hardness, and $t$-BC$_{4}$N belongs to the tetragonal system with a $I\bar{4}m2$ space group, which is highly incompressible with a bulk modulus of 383.4 GPa and shear modulus of 383.0 GPa. Actually, first-principles calculations and evolutionary methodology have been widely applied to predict the possible crystal structure and to explain the mechanical properties of potential superhard materials.[11,12] In the present work, we propose a novel superhard phase of BC$_{4}$N with orthorhombic structure (referred to as $o$-BC$_{4}$N hereafter) using first-principles calculations. The structural, electronic and mechanical properties of $o$-BC$_{4}$N were studied employing first-principles calculations. This phase is dynamically stable under ambient conditions and more thermodynamically favorable than the previous predicted $t$-BC$_{4}$N. The calculations demonstrate that this material has a large bulk modulus, shear modulus and hardness, between those of diamond and $c$-BN. The band structure indicates that the $o$-BC$_{4}$N phase is a semiconductor. The ideal strength and theoretical hardness of the $o$-BC$_{4}$N phase were also calculated in the present work, since they are effective and reliable to investigate the mechanical properties for a superhard material. Structural predictions were performed using the crystal structure analysis by particle swarm optimization (CALYPSO) algorithm,[13,14] which has successfully predicted many other structures at high pressures.[15,16] Subsequent structure optimizations and most property calculations were performed using density functional theory[17,18] as implemented in VASP with the projector augmented wave scheme[19] to describe electron–ion interaction. The exchange-correlation functional was treated by the generalized gradient approximation (GGA) using the functional of Perdew–Burke–Ernzerhof.[20] According to our convergence test, a plane wave cut-off energy of 550 eV and $5\times15\times12$ $k$-point sampling were sufficient to give fully converged results. The optimization of internal atomic positions and lattice parameters was achieved by minimization of forces and stress tensors. Then elastic constants were calculated and the bulk modulus and shear modulus were obtained based on Voigt–Reuss–Hill approximations.[21] Calculations of the phonon dispersions were carried out using the density functional perturbation theory[22] within the local density approximation[17] in a plane wave basis, as implemented in the quantum ESPRESSO code[23] with Troullier–Martins pseudopotentials.[24] A plane wave cut-off energy of 110 Ry was used in the calculations, and the estimated energy error in self-consistency was less than 10$^{-14}$ a.u. The calculations of stress–strain relations were performed using the method in Refs. [25,26]. This approach with a relaxed loading path has been successfully applied to the calculation of the strength of several strong solids.[27,28] Hardness calculations were based on microhardness models for covalent crystal.[29-31]
cpl-36-3-036201-fig1.png
Fig. 1. Structures of $o$-BC$_{4}$N: (a) three-dimensional topology, (b) along the [010] direction, and (c) along the [001] direction.
The optimized structure of orthorhombic $o$-BC$_{4}$N with $Imm2$ symmetry is shown in Fig. 1. As can be seen from Fig. 1, the unit cell of $o$-BC$_{4}$N contains twelve atoms. The B atoms locate at 2$b$ (0, 0.5, 0.329) sites, N atoms are situated at 2$a$ (0, 0, 0.589), and C atoms occupy two inequivalent crystallographic sites (C1: 4$c$ (0.168, 0, 0.822) and C2: 4$c$ (0.171, 0.5, 0.066)), respectively. The most prominent structural feature of $o$-BC$_{4}$N is that the alternative C layer and the BN layer are stacked along the [100] direction with an atomic ratio of 2:1. The bond lengths of C–C are 1.539 Å and 1.547 Å and the bond lengths of B–C and C–N are 1.618 Å and 1.535 Å, respectively. The bond lengths of C–C in $o$-BC$_{4}$N are longer than that of $t$-BC$_{4}$N but shorter than that of $c$-BC$_{4}$N. All atoms in $o$-BC$_{4}$N are four-coordinated with $sp^{3} $ hybridization, leading to chair/boat-like B–C$_{4}$–N or C$_{6}$ rings. The thermodynamic stability of $o$-BC$_{4}$N was assessed by calculating the relative enthalpy with respect to the other available BC$_{4}$N phases ($g$-BC$_{4}$N, $t$-BC$_{4}$N, $c$-BC$_{4}$N and $bc$-BC$_{4}$N) in the pressure range from 0 GPa to 40 GPa. Enthalpy calculations (Fig. 2(a)) suggest that $g$-BC$_{4}$N is the most stable phase at zero pressure, which is very consistent with previous results. The enthalpies of $o$-BC$_{4}$N, $t$-BC$_{4}$N, $c$-BC$_{4}$N and $bc$-BC$_{4}$N are above $g$-BC$_{4}$N with 0.36 eV, 0.76 eV, 0.24 eV and 0.73 eV per atom, respectively. Under hydrostatic compression, the transition pressures from $g$-BC$_{4}$N to $o$-BC$_{4}$N and $c$-BC$_{4}$N are 25.1 GPa and 14.2 GPa, respectively. Although its enthalpy is still higher than that of $c$-BC$_{4}$N, $o$-BC$_{4}$N is energetically much more stable than the previously proposed $t$-BC$_{4}$N and $bc$-BC$_{4}$N, suggesting that $o$-BC$_{4}$N should be accessed more easily than $t$-BC$_{4}$N and $bc$-BC$_{4}$N. Furthermore, the phonon dispersion curves were calculated to confirm the dynamical stability of $o$-BC$_{4}$N. No imaginary frequencies can be observed throughout the whole Brillouin zone from Fig. 2(b), indicating that $o$-BC$_{4}$N is dynamically stable at zero pressure.
cpl-36-3-036201-fig2.png
Fig. 2. (a) Enthalpy as a function of pressure for BC$_{4}$N allotropes relative to $g$-BC$_{4}$N. (b) The phonon-dispersion curves of $o$-BC$_{4}$N at a pressure of 0 GPa. (c) The band structure (left panel) and density of states (DOS) (right panel) of $o$-BC$_{4}$N. Zero denotes the energy measured from the valence top. (d) The conventional unit cell of $o$-BC$_{4}$N with ELF isosurface with an ELF value of 0.75.
Moreover, the calculated electronic band structure of $o$-BC$_{4}$N at 0 GPa is shown in the left panel of Fig. 2(c). The special $k$ points were extracted from the Bilbao Crystallographic Server, promoted by Cracknell, Davies, Miller and Love.[32] It can be seen that $o$-BC$_{4}$N is a semiconductor with an indirect band gap of 3.95 eV under ambient conditions, with the top of the valence band at the ${\it \Gamma}$ point and the bottom of the conduction band at the $S$ point. The band gap of 3.95 eV is remarkably larger than that of $t$-BC$_{4}$N (1.8 eV)[10] and approached the $c$-BC$_{4}$N value of 4.06 eV[7] under ambient conditions. To further understand the bonding and electronic nature, the partial DOS of $o$-BC$_{4}$N was calculated, and the dominating valence electron states for different atoms ($p$ states of B, C, N and $s$ states of B, C, N) are shown in the right panel of Fig. 2(c). It can be seen that the density of $p$ states of the nitrogen atom is larger than that of the carbon atom in the bonding orbitals, where it is smaller in the non-bonding orbitals. There is a harmonic 2$p$–2$s$ overlap between the B(C,N)-2$s$ and B(C,N)-2$p$ states at the whole-energy level (from 0 to $-$22 eV) in $o$-BC$_{4}$N, confirming the hybridization of $s$ and $p$ orbitals and the strong interaction between the B, C and N atoms. To further understand the bonding properties, the valence electron localization functions (ELF) of $o$-BC$_{4}$N were calculated. The structure with ELF=0.75 (a typically used number for the characterization of covalent bonding)[33] isosurfaces for $o$-BC$_{4}$N is presented in Fig. 2(d), which clearly illustrates the nature of strong covalent bonding in $o$-BC$_{4}$N. Moreover, a greater amount of charges are localized in the C–C bonding regions connecting carbon squares, indicating a strong covalent interaction.
Table 1. Calculated equilibrium lattice parameters $a$ (Å), $b$ (Å), $c$ (Å), density $\rho$ (g/cm$^{3}$), zero-pressure elastic constants $C_{ij}$ (GPa), bulk modulus $B$ (GPa), shear modulus $G$ (GPa), Young's modulus $E$ (GPa) and Poisson's ratio $\upsilon$ of $o$-BC$_{4}$N, compared with available data of $t$-BC$_{4}$N and $c$-BC$_{4}$N.
Phase $o$-BC$_{4}$N $t$-BC$_{4}$N $c$-BC$_{4}$N
This work This work Ref. [9] This work Ref. [6]
Symmetry $Imm2$ $I\bar{4}m2$ $I\bar{4}m2$ $P3m1$ $P3m1$
$a$ 7.651 2.552 2.547 2.538 2.507
$b$ 2.539 2.552 2.547 2.538 2.507
$c$ 3.614 10.964 10.947 6.267 6.196
$\rho$ 3.447 3.389 3.407 3.461 3.588
$C_{11}$ 1027.2 930.4 965.6 1056.0 1131.5
$C_{22}$ 1038.9
$C_{33}$ 933.2 854.6 872.0 1083.2 1116.9
$C_{44}$ 498.0 303.3 381.8 425.2 508.5
$C_{55}$ 502.3
$C_{66}$ 391.8 376.0 335.4 477.5 527.9
$C_{12}$ 24.4 39.1 40.8 101.0 75.8
$C_{13}$ 125.6 130.6 141.3 63.2 59.5
$C_{23}$ 133.7
$C_{15}$ 47.9 $-$6.7
$B_{V}$ 396.30 368.44 383.36 405.57 418.83
$B_{R}$ 396.28 368.41 383.35 405.56 418.77
$G_{V}$ 459.4 357.5 385.1 463.5 521.3
$G_{R}$ 452.6 350.1 380.9 461.0 521.1
$B$ 396.3 368.4 383.4 405.6 418.8
$G$ 456.0 353.8 383.0 462.2 521.2
$E$ 988.8 804.1 861.9 1004.9 1105.1
$B/G$ 0.87 1.04 1.00 0.88 0.80
$\upsilon$ 0.08 0.14 0.13 0.09 0.06
To understand the elastic properties of $o$-BC$_{4}$N, the single-crystal zero-pressure elastic constants $C_{ij}$ were calculated according to Hooke's law.[34] For $o$-BC$_{4}$N, there are nine independent single-crystal elastic constants as listed in Table 1. The calculated elastic constants of $o$-BC$_{4}$N satisfy the mechanical stability criteria for an orthorhombic structure ($C_{11}>0$, $C_{22}>0$, $C_{33}>0$, $C_{44}>0$, $C_{55}> 0$, $C_{66}>0$, $[C_{11}+C_{22}+C_{33}+2(C_{12}+C_{13}+C_{23})]>0$, $(C_{11}+C_{22}-2C_{12})>0$, $(C_{11}+C_{33}-2C_{13})>0$, $(C_{22}+C_{33}-2C_{23})>0$),[35] indicating the elastically stability of $o$-BC$_{4}$N under the ambient condition. The ideal superhard materials should have a high bulk modulus, high shear modulus and low Poisson's ratio, which may lead to a minimum of plastic deformation since hardness depends strongly on plastic deformation.[1,29] From the calculated $C_{ij}$ (Table 1), the polycrystalline elastic moduli were also estimated using the Voigt–Reuss–Hill approximation.[36] The calculated bulk modulus and shear modulus with the GGA in the present work are 396.3 and 456.0 GPa, respectively, which are between those of $c$-BN and diamond. Compared with $t$-BC$_{4}$N ($B=368.4$ GPa and $G=353.8$ GPa) and $c$-BC$_{4}$N ($B=405.6$ GPa and $G=462.2$ GPa), the values of the bulk modulus and the shear modulus of $o$-BC$_{4}$N are relatively higher than the former and slightly lower than the latter, respectively. The calculated density, elastic modulus and Poisson's ratio are listed in Table 1. The corresponding values of diamond, $c$-BN, $c$-BC$_{4}$N and $t$-BC$_{4}$N are also given in Table 1 to compare with $o$-BC$_{4}$N. One can see that the density of $o$-BC$_{4}$N is slightly smaller than $c$-BC$_{4}$N and larger than $t$-BC$_{4}$N. Interestingly, the calculated Young's modulus and Poisson's ratio of $o$-BC$_{4}$N are 998.3 GPa and 0.084, respectively, between those of $c$-BN and diamond. Further, $o$-BC$_{4}$N exhibits brittle behavior due the low ratio of the bulk-to-shear modulus ($B/G=0.87)$ according to Pugh's rule, which states that a larger value of $B/G$ indicates the ductile property of a material, with the transition from brittle to ductile behavior occurring at a $B/G$ value of around 1.75.[37] In addition, the calculated ratio of $G/B$ (1.15) of $o$-BC$_{4}$N indicates that the directionality of the bonding in $o$-BC$_{4}$N is very strong. Thus the above results provide a clear indicator that this novel boron-carbon-nitrogen phase of $o$-BC$_{4}$N is a potential superhard material.
cpl-36-3-036201-fig3.png
Fig. 3. Calculated tensile stress versus tensile strain of $o$-BC$_{4}$N(a), t-BC$_{4}$N(b), $c$-BC$_{4}$N(c)and $bc$-BC$_{4}$N(d) in principal symmetry directions.
cpl-36-3-036201-fig4.png
Fig. 4. Calculated shear stress versus shear strain for $o$-BC$_{4}$N in principal symmetry directions.
To explore the atomic and electronic deformation mechanism upon tension at permanent large strain, the ideal tensile strengths of $o$-BC$_{4}$N were calculated within the GGA scheme[38,39] in comparison with $t$-BC$_{4}$N, $c$-BC$_{4}$N and $bc$-BC$_{4}$N (see Fig. 3). The highest peak stress (161.8 GPa) arises in the [001] direction, which is comparable with that of $bc$-BC$_{4}$N (163.1 GPa), and larger than that of $c$-BC$_{4}$N (104.7 GPa) and $t$-BC$_{4}$N (156.4 GPa), but lower than that of $c$-BN (180.4 GPa) and diamond (204.3 GPa). Nevertheless, the weakest tensile strength of $o$-BC$_{4}$N is found to be 59.2 GPa in the [101] direction, which is lower than that of diamond (84.0 GPa) and $c$-BC$_{4}$N (71.4 GPa), but visibly larger than $t$-BC$_{4}$N (19.6 GPa), $bc$-BC$_{4}$N (54.9 GPa) and $c$-BN (56.1 GPa). The anisotropy ratio of ideal tensile strengths for $o$-BC$_{4}$N, $t$-BC$_{4}$N, $c$-BC$_{4}$N and $bc$-BC$_{4}$N are 2.73, 8.00, 1.47 and 2.01, respectively. We also make a comprehensive study of the stress–strain relations of $o$-BC$_{4}$N in various principal symmetry directions under shear deformation. The anisotropy ratio of ideal shear strengths for $o$-BC$_{4}$N is $\sigma_{(101)[\bar{1}{11}]}: \sigma_{(100)[010]}:\sigma_{(101)[010]}:\sigma_{(001)[010]}$ : $\sigma_{(001)[110]}:\sigma_{(100)[011]}:\sigma_{(001)[100]}:\sigma_{(101)[10\bar{1}]}:\sigma_{(100)[001]}=124.0:104.6:72.7:71.2:65.5: 64.5:62.3:59.1:58.5$$\approx$$2.12:1.79:1.24:1.22:1.12:1.10:1.06:1.01:1$. The weakest shear strength of 58.5 GPa of $o$-BC$_{4}$N is found along the (100)[001] shear deformation path at a strain of 0.14 (see Fig. 4). This reduction in the shear strength anisotropy results from the directional arrangements of bonds in $o$-BC$_{4}$N. The magnitude of the ideal shear strength of the $o$-BC$_{4}$N phase is larger than that of 58.3 GPa found for the (111)[11$\bar{2}$] slip system of $c$-BN. Moreover, it is found that the ideal weakest shear strength of $o$-BC$_{4}$N is smaller than its ideal weakest tensile strength. Thus the mechanical failure mode of $o$-BC$_{4}$N is dominated by the shear type.
Table 2. Calculated average hardness $H_{\rm v}$ for $o$-BC$_{4}$N, $t$-BC$_{4}$N, $c$-BC$_{4}$N, $bc$-BC$_{4}$N, $\beta$-BC$_{2}$N, $c$-BN and diamond, where $n$ is the number of $\mu$-type bonds per unit cell, $d^{\mu}$ is the bond length, $v_{{\rm b}}^{\mu}$ is the bond volume, $N_{\rm e}^{\mu}$ is the electron density expressed as the number of valence electrons per cubic angstroms, $E_{\rm h}^{\mu}$ is the homopolar gap, $E_{\rm g}^{\mu}$ is the valence–conduction band gap, $f_{\rm i}^{\mu}$ is the ionicity of the bond, respectively, compared to the hardness of $H_{\rm v}$ (Chen) and the experimental hardness $H_{\rm exp}$.
Phase Bond $n$ $d^{\mu}$ (Å) $v_{{\rm b}}^{\mu}$ (Å$^{3}$) $N_{\rm e}^{\mu}$ (Å$^{-3}$) $E_{\rm h}^{\mu}$ (eV) $E_{\rm g}^{\mu}$ (eV) $f_{\rm i}^{\mu}$ $H_{\rm v}^{\mu}$ (GPa) $H_{\rm vav}$ (GPa) $H_{\rm v}$ (Chen) (GPa) $H_{\rm exp}$ (GPa)
$o$-BC$_{4}$N $C_{2}$-N 4 1.535 2.782 0.652 13.603 16.421 0.314 62.1 78.7 81.7
C$_{1}$-C$_{2}$ 4 1.539 2.803 0.882 13.514 13.528 0.002 109.3
BN 4 1.579 3.025 0.662 12.684 15.025 0.287 60.3
B–C$_{1}$ 4 1.618 3.252 0.455 11.942 12.031 0.015 61.2
C$_{1}$-C$_{2}$ 8 1.547 2.844 0.756 13.352 13.364 0.002 97.5
$t$-BC$_{4}$N C$_{1}$-N 8 1.588 3.070 0.596 12.513 15.699 0.365 50.7 73.8 56.1
B–C$_{2}$ 8 1.605 3.169 0.445 12.184 12.261 0.013 61.7
C$_{1}$-C$_{2}$ 8 1.518 2.686 1.092 13.987 14.092 0.015 128.5
$c$-BC$_{4}$N B–C$_{3}$ 1 1.617 3.235 0.342 11.946 12.024 0.013 50.7 80.3 81.4 85$\pm$4$^{\rm a}$
B-N 3 1.566 2.938 0.735 12.944 15.031 0.258 68.3
C$_{4}$-C$_{3}$ 3 1.550 2.850 0.843 13.278 13.278 0.000 104.5
C$_{4}$-C$_{1}$ 1 1.554 2.872 0.349 13.193 13.193 0.000 57.7
C$_{2}$-C$_{1}$ 3 1.549 2.850 0.849 13.306 13.307 0.000 105.2
C$_{2}$-N 1 1.516 2.842 0.494 14.052 16.080 0.236 58.4
$bc$-BC$_{4}$N $C_{4}$-N 4 1.520 3.348 0.584 13.960 14.193 0.033 82.7 67.7 48.7
C$_{1}$-C$_{2}$ 4 1.533 3.439 0.767 13.650 16.781 0.338 67.4
C$_{2}$-C$_{3}$ 4 1.559 3.612 0.549 13.104 13.113 0.001 77.3
C$_{3}$-C$_{4}$ 4 1.563 3.642 0.584 13.014 13.064 0.008 79.5
B-N 4 1.596 3.612 0.249 12.358 12.385 0.004 42.9
B–C$_{1}$ 4 1.600 3.875 0.734 12.280 14.209 0.253 65.2
$\beta$-BC$_{2}$N C$_{1}$-C$_{2}$ 4 1.528 2.744 0.911 13.767 13.767 0 114.1 75.5 76.9$^{\rm b}$ 76$^{\rm c}$
B–C$_{1}$ 4 1.591 3.094 0.485 12.453 12.453 0 67.8
C$_{2}$-N 4 1.578 3.021 0.662 12.706 14.49 0.231 64.6
B-N 4 1.576 3.011 0.664 12.741 14.528 0.231 65.0
Diamond C–C 16 1.544 2.834 0.706 13.413 13.413 0.000 93.8 93.8 95.7$^{\rm b}$ 96$\pm$5$^{\rm c}$
$c$-BN B-N 16 1.569 2.973 0.673 12.889 14.691 0.230 66.4 66.4 65.2$^{\rm b}$ 63$\pm$5$^{\rm c}$
Here $^{\rm a}$Ref. [6], $^{\rm b}$Ref. [41], and $^{\rm c}$Ref. [29].
At present, the microscopic models of hardness proposed by Gao et al.[29-31] have been widely applied in much research and have achieved success. The hardness of $\mu$-type bonds in a pseudo-binary compound can be expressed as $$\begin{align} H_{\nu}^{\mu}\,({\rm GPa})=8.82(N_{{\rm e}}^{\mu} )^{2/3}E_{{\rm h}}^{\mu} e^{-1.191f_{{\rm i}}^{\mu}}.~~ \tag {1} \end{align} $$ Nevertheless, the bond strength could be varied with different bond types, whereby the softest one is usually the first to be broken. Henceforth, in the calculation of the Vickers hardness of multicomponent crystals, a geometric average of all bonds should be considered, and the hardness should have the following form $$\begin{align} H_{\rm vav.}=\Big[\prod\limits^{\mu}{(H_{\rm v}^{\mu})^{n^{\mu}}}\Big]^{1/\sum n^{\mu}}.~~ \tag {2} \end{align} $$ As listed in Table 2, the atomic bond hardness for $o$-BC$_{4}$N has been calculated according to the microscopic hardness model to compare with that of recently proposed BC$_{4}$N allotropes and $\beta$-BC$_{2}$N, $c$-BN and diamond. The theoretical Vickers hardness of $o$-BC$_{4}$N is predicted to be 78.7 GPa, which is evidently larger than $t$-BC$_{4}$N (73.8 GPa), $bc$-BC$_{4}$N (67.7 GPa), $\beta$-BC$_{2}$N (75.5 GPa) and $c$-BN (66.4 GPa), and comparable with $c$-BC$_{4}$N (80.3 GPa). In addition, the Vickers hardness values of the BC$_{4}$N allotropes obtained through the models of Chen et al.[40,41] are also listed in Table 2. The theoretical Vickers hardness of $o$-BC$_{4}$N, $t$-BC$_{4}$N, $c$-BC$_{4}$N and $bc$-BC$_{4}$N calculated by the models of Chen are 81.7 GPa, 56.1 GPa, 81.4 GPa and 48.7 GPa, respectively. Both models indicate that $o$-BC$_{4}$N is a superhard material with a high hardness of about 80 GPa. However, it can be found that the hardness values of BC$_{4}$N allotropes given by the two models differ greatly with the increase of anisotropy. Therefore, it is necessary to further clarify the relationship between hardness and anisotropy in the next work. In summary, an orthorhombic $o$-BC$_{4}$N has been proposed, and the structural, electronic and mechanical properties of $o$-BC$_{4}$N are investigated by first-principles calculations. Results show that $o$-BC$_{4}$N is structurally stable, and should be accessed more easily than $t$-BC$_{4}$N and $bc$-BC$_{4}$N proposed recently. The ideal shear strength of 58.5 GPa sets an upper bound for its ideal strength, which is evidently higher than the general hard materials. The calculated hardness of $o$-BC$_{4}$N is 78.7 GPa, which is harder than $t$-BC$_{4}$N and $bc$-BC$_{4}$N. The results suggest that $o$-BC$_{4}$N is a novel intrinsic superhard material. 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