A New BCN Compound with Monoclinic Symmetry: First-Principle Calculations

In this study, we predicted and investigated a new light-element compound B-C-N in Pm phase, denoted as Pm-BCN, using density functional theory. Pm-BCN is mechanically, dynamically, and thermodynamically stable. The elastic moduli of Pm-BCN are larger than those of other B-C-N and light-element compounds, such as P213 BN, B2C3, P4/m BN, Pnc2 BN, and dz4 BN. By studying the mechanical anisotropy of elastic moduli, we proved that Pm-BCN is a mechanically anisotropic material. In addition, the shear anisotropy factors A2 and ABa of Pm-BCN are smaller than those of the seven B-C-N compounds mentioned in this paper. Pm-BCN is a semiconductor material with an indirect and wide band gap, suggesting that Pm-BCN can be applied in microelectronic devices.

Three new carbon allotropes with an orthogonal structure, oP-C 16 , oP-C 20 , and oP-C 24 , were proposed based on first-principles calculations [1], all of which showed metallicity. The hardness of oP-C 16 , oP-C 20 , and oP-C 24 are 47.5, 49.6, and 55.3 GPa, respectively. The ideal shear strengths of oP-C 16 , oP-C 20 , and oP-C 24 are higher than those of Cu and Fe, and Al. Yu et al. [23] predicted and studied a new sp 2 hybrid BN polymorph, Pnc2 BN, which showed mechanical and dynamic stability, and found that the elastic properties of Pnc2 BN are better than those of dz4 BN. The indirect band gap of Pnc2 BN calculated using the Heyd-Scuseria-Ernzerhof (HSE06) functional is 3.543 eV, indicating that Pnc2 BN has semiconductor properties. On the basis of density functional theory (DFT) calculations [24,25], m-B 3 CN 3 and m-B 2 C 3 N 2 , two new superhard BCN compounds, were designed by Xing et al. [5]. The shear modulus B, bulk modulus G, and Young's modulus E of m-B 3 CN 3 and m-B 2 C 3 N 2 are 345, 778, and 346, respectively; the B, G, and E for m-B 3 CN 3 and m-B 2 C 3 N 2 are little bit larger than those of o-BC 6 N [2], t-BC 6 N-1 [2], and t-BC 6 N-2 [2]. Both m-B 3 CN 3 and m-B 2 C 3 N 2 are superhard materials because both compounds have a hardness in excess of 40 GPa. The structural properties, anisotropy characteristics, elastic characteristics, and electronic properties, as well as the stability, of P4/m BN were investigated by Yu et al. [26]. By adopting DFT, Xing et al. established and studied CN and BCN 2 compounds with superhard characteristics and a space group of C2/m [4]. The hardness of CN is 58.63 GPa, and it is a semiconductor material, whereas BCN 2 is metallic. A superhard material, t-C 8 B 2 N 2 , was designed by Zhu et al. [10] and Wang et al. [11]. The bulk modulus of t-C 8 B 2 N 2 was found to be 383.4 [10] and 383.0 GPa [11], and the hardness was 64.7 [10] and 63.2 GPa [11].
In this study, we predicted a BCN polymorph, Pm BCN, which is mechanically and dynamically stable. We analyzed the structural, mechanical, and electronic characteristics of Pm BCN through first-principles calculations.

Theoretical Methods
On the basis of DFT calculations [24,25], we proposed and investigated a new lightelement compound using the Cambridge Serial Total Energy Package (CASTEP) [27]. We adopted the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) [28] and local density approximation (LDA) [29] functionals to describe the exchange and correlation potentials. To ensure that the crystal structure of Pm-BCN was optimal, we used Broyden-Fletcher-Goldfarb-Shanno (BFGS) [30] for geometry optimization. The convergence accuracy during optimization was less than 0.001 eV. We described the valence electrons by ultrasoft pseudopotentials [31]. We adopted the Monkhorst-Pack k-points for the k-points separation of 6 × 16 × 7 and found that the plane wave cut-off energy E cut-off is 500 eV for Pm-BCN. For the phonon spectra of Pm-BCN, we used the density functional perturbation theory (DFPT) approach [32], and for the electronic band structures of Pm-BCN, we adopted the HSE06 hybrid functional [33]. In addition, we used the Voigt-Reuss-Hill (VRH) approximations [34][35][36] to calculate the bulk modulus and shear modulus.

Results and Discussion
The crystal structure of Pm-BCN and its structure along the b-axis are shown in Figure 1a,b, respectively. Blue, gray, and purple spheres represent the B, N, and C atoms, respectively. In addition to the common rings such as the 4-and 6-membered rings in the crystal structures of Pm-BCN, two larger rings, a 10-and a 16-membered ring, are present in the crystal structure, the structures of which are depicted in Figure 1c,d. The 4membered ring consists of one B, one N, and two C atoms; the 6-membered ring consists of one B, one N, and four C atoms; the 10-membered ring consists of three B, three N, and four C atoms. The 16-membered ring consists of five B, five N, and six C atoms. The conventional Pm-BCN cell contains 12 atoms. Because Pm-BCN has a monoclinic system, the crystal structure of Pm-BCN is not symmetrical, and the position of each atom is different. Boron atoms occupy four positions: B1 1a (0.11803, 0.00000, and 0.20845), B3 1a (0.22107, 0.00000, and 0.73817), B10 1b (0.80293, 0.50000, and 0.62435), and B12 1b (0.59805, 0.50000, and 0.04251); nitrogen atoms occupy four positions: N2 1a (0.78254, 0.00000, and 0.74717), N7 1b (0.18971, 0.50000, and 0.61515), N8 1b (0.87536, 0.50000, and 0.36710), and N11 1b (0.40657, 0.50000, and 0.04285); and carbon atoms occupy four positions: C4 1a (0.88003, 0.00000, and 0.20041), C5 1a (0.71892, 0.00000, and 0.01576), C6 1a (0.29294, 0.00000, and 0.01183), and C9 1b (0.11385, 0.50000, and 0.37443). Table 1 shows the crystal lattice parameters of the B-C-N compounds. The crystal lattice parameters of Imm2 BCN and I-4m2 BCN are close to those previously reported [14]; therefore, the crystal lattice parameters of Pm-BCN reported in this manuscript are both convincing and reliable.
The stability of Pm-BCN through phonon spectra (Figure 2a), relative enthalpy (Figure 2b), and elastic parameters. In Figure 2a, no curve appears below zero, so Pm-BCN is dynamically stable. We calculated the formation energies of B-C-N compounds as: ∆H = H BxCyNz / m-xH c-BN /n-yH diamond /p. As several B-C-N compounds have an equal number of nitrogen and boron atoms, here, x is equal to z; m, n, and p are the B x C y N z unit and atom numbers of the conventional cell for B-C-N compounds, c-BN, and diamond. The formation energy of Pm-BCN is 0.7182 eV/atom, which is slightly lower than those of o-BC 6 N-1, t-BC 6 N-2 [2], B 2 C 2 N 2 -2, B 2 C 2 N 2 -3, B 2 C 2 N 2 -4, and B 2 C 2 N 2 -5 [38]. We found that the Pm-BCN is a metastable phase. Notably, B-C-N compounds with positive formation energies are not unusual [2,4,5,7,8,38].   [11], b [10], c [14], d [37].
The stability of Pm-BCN through phonon spectra (Figure 2a), relative enthalpy (Figure 2b), and elastic parameters. In Figure 2a, no curve appears below zero, so Pm-BCN is dynamically stable. We calculated the formation energies of B-C-N compounds as: ΔH = HBxCyNz/m-xHc-BN/n-yHdiamond/p. As several B-C-N compounds have an equal number of nitrogen and boron atoms, here, x is equal to z; m, n, and p are the BxCyNz unit and atom   [11], b [10], c [14], d [37].
For the monoclinic structure, the Born mechanical stability conditions are [39]: 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, Table 2 lists the C ij values of Pm-BCN and other B-C-N compounds. Table 2 shows that the elastic constants of Pm-BCN determined by LDA are slightly higher than those determined by GGA. All the elastic constants of Pm-BCN satisfy the above equation for a monoclinic system, proving that Pm-BCN is mechanically stable. We calculated the G and B of B-C-N compounds by using the Voigt-Reuss-Hill approximation [34][35][36]. The B V , B R , G V , and G R are given by [38]: Young's modulus E is calculated by E = 9BG/(3B + G), and Table 2 lists the calculated elastic moduli of B-C-N compounds. The elastic moduli of Pm-BCN are less than those of other B-C-N compounds, and larger than those of other light element compounds, such as Pnc2 BN [23], P4/m BN [26], P213 BN [40], B2C3 [41], dz4 BN [42], etc.
According the ElAM codes [43], we investigated the anisotropic elastic properties of Pm-BCN. The G, v, and E are illustrated in Figure 3a-e, respectively. The three-dimensional (3D) graphics of G, v, and E of Pm-BCN are not regular spheres, as shown in Figure 3. If a material possesses isotropic properties, its 3D diagram should be a regular sphere, and any shape deviating from a sphere indicates anisotropy [44][45][46][47][48][49][50]. So, we found that the G, v, and E of Pm-BCN exhibit anisotropic elastic properties. The Gmax/Gmin and Emax/Emin ratios are used to characterize the anisotropic elastic properties of G and E, which are 207.12/75.11 = 2.76 and 755.31/221.89 = 3.40 for Pm-BCN, respectively. As shown by the Gmax/Gmin and Emax/Emin ratios, the anisotropic elastic properties of Pm-BCN show that it has a greater shear modulus than B2C3N2 and B2CN2 [5], but a smaller one than BCN2 [4]. BCN2 has the largest Young's modulus among Pm-BCN, B2C3N2, and B2CN2; B2C3N2 shows the weakest anisotropy in E. Young's modulus E is calculated by E = 9BG/(3B + G), and Table 2 lists the calculated elastic moduli of B-C-N compounds. The elastic moduli of Pm-BCN are less than those of other B-C-N compounds, and larger than those of other light element compounds, such as Pnc2 BN [23], P4/m BN [26], P2 1 3 BN [40], B 2 C 3 [41], dz4 BN [42], etc.
According the ElAM codes [43], we investigated the anisotropic elastic properties of Pm-BCN. The G, v, and E are illustrated in Figure 3a-e, respectively. The three-dimensional (3D) graphics of G, v, and E of Pm-BCN are not regular spheres, as shown in Figure 3. If a material possesses isotropic properties, its 3D diagram should be a regular sphere, and any shape deviating from a sphere indicates anisotropy [44][45][46][47][48][49][50]. So, we found that the G, v, and E of Pm-BCN exhibit anisotropic elastic properties. The G max /G min and E max /E min ratios are used to characterize the anisotropic elastic properties of G and E, which are 207.12/75.11 = 2.76 and 755.31/221.89 = 3.40 for Pm-BCN, respectively. As shown by the G max /G min and E max /E min ratios, the anisotropic elastic properties of Pm-BCN show that it has a greater shear modulus than B 2 C 3 N 2 and B 2 CN 2 [5], but a smaller one than BCN 2 [4]. BCN 2 has the largest Young's modulus among Pm-BCN, B 2 C 3 N 2 , and B 2 CN 2 ; B 2 C 3 N 2 shows the weakest anisotropy in E.  [14]. Young's modulus E is calculated by E = 9BG/(3B + G), and Table 2 lists the calculated elastic moduli of B-C-N compounds. The elastic moduli of Pm-BCN are less than those of other B-C-N compounds, and larger than those of other light element compounds, such as Pnc2 BN [23], P4/m BN [26], P213 BN [40], B2C3 [41], dz4 BN [42], etc.
According the ElAM codes [43], we investigated the anisotropic elastic properties of Pm-BCN. The G, v, and E are illustrated in Figure 3a-e, respectively. The three-dimensional (3D) graphics of G, v, and E of Pm-BCN are not regular spheres, as shown in Figure 3. If a material possesses isotropic properties, its 3D diagram should be a regular sphere, and any shape deviating from a sphere indicates anisotropy [44][45][46][47][48][49][50]. So, we found that the G, v, and E of Pm-BCN exhibit anisotropic elastic properties. The Gmax/Gmin and Emax/Emin ratios are used to characterize the anisotropic elastic properties of G and E, which are 207.12/75.11 = 2.76 and 755.31/221.89 = 3.40 for Pm-BCN, respectively. As shown by the Gmax/Gmin and Emax/Emin ratios, the anisotropic elastic properties of Pm-BCN show that it has a greater shear modulus than B2C3N2 and B2CN2 [5], but a smaller one than BCN2 [4]. BCN2 has the largest Young's modulus among Pm-BCN, B2C3N2, and B2CN2; B2C3N2 shows the weakest anisotropy in E.    [51,52]. The A 1 , A 2 , and A 3 of carbon allotropes of seven B-C-N compounds are illustrated in Figure 4a. We found that the A 1 , A 2 , and A 3 of isotropic materials should be one; however, as shown in Figure 4a, the A 1 of Pm-BCN is much greater than one, whereas the A 2 of Pm-BCN is much lower than one, so Pm-BCN exhibits a larger anisotropy at the (100) and (010) shear plane. Among these seven B-C-N compounds, the (100), (010), and (001) shear planes of t-C 8 B 2 N 2 show minimal differ-ences, implying that the anisotropies at the three planes of the shear modulus of t-C 8 B 2 N 2 are similar. The B along the a, b, and c axes, B a , B b , and B c , were calculated as [51,53]: B a = Λ/(1 + α + β), B b = B a /α, and B c = B a /β, and Λ = C 11 + 2C 12 + C 22 α 2 + 2C 13 β + C 33 β 2 + 2C 23 The anisotropy of the bulk moduli along the a and c directions with respect to the b directions are described by: Figure 4b,c shows the B a , B c , A Ba , and A Bc of the seven B-C-N compounds. The B a and B c of the seven B-C-N compounds differ. The B a of BCN 2 is the largest, and Pm-BCN exhibits the smallest linear bulk modulus B a . Although BCN 2 has the largest linear bulk modulus B a , its linear bulk modulus B c is the smallest. Figure 4c shows that the anisotropy of B along the a direction, A Ba , of t-C 8 B 2 N 2 and I-4m2 BCN, and the A Bc of B 2 N 2 C 3 and B 3 N 3 C are very close to one, indicating that the B of these materials is less anisotropic along the a and c axes. Additionally, Pm-BCN exhibits the largest anisotropy in A Ba , and BCN 2 has the largest anisotropy in A Bc . the shear anisotropic factor for the (100) shear plane between [011] and [010] directions, the (010) shear plane between [101] and [001] directions, and the (001) shear plane between [110] and [010] directions, respectively. A1 = 4C44/(C11 + C33 − 2C13), A1 = 4C55/(C22 + C33 − 2C23), A3 = 4C66/(C11 + C22 − 2C11) [51,52]. The A1, A2, and A3 of carbon allotropes of seven B-C-N compounds are illustrated in Figure 4a. We found that the A1, A2, and A3 of isotropic materials should be one; however, as shown in Figure 4a, the A1 of Pm-BCN is much greater than one, whereas the A2 of Pm-BCN is much lower than one, so Pm-BCN exhibits a larger anisotropy at the (100) and (010) shear plane. Among these seven B-C-N compounds, the (100), (010), and (001) shear planes of t-C8B2N2 show minimal differences, implying that the anisotropies at the three planes of the shear modulus of t-C8B2N2 are similar. The B along the a, b, and c axes, Ba, Bb, and Bc, were calculated as [51,53]: Ba = Λ/(1 + α + β), Bb = Ba/α, and Bc = Ba/β, and Λ = C11 The anisotropy of the bulk moduli along the a and c directions with respect to the b directions are described by: ABa = Ba/Bb, ABc = Bc/Bb. Figure 4b,c shows the Ba, Bc, ABa, and ABc of the seven B-C-N compounds. The Ba and Bc of the seven B-C-N compounds differ. The Ba of BCN2 is the largest, and Pm-BCN exhibits the smallest linear bulk modulus Ba. Although BCN2 has the largest linear bulk modulus Ba, its linear bulk modulus Bc is the smallest. Figure 4c shows that the anisotropy of B along the a direction, ABa, of t-C8B2N2 and I-4m2 BCN, and the ABc of B2N2C3 and B3N3C are very close to one, indicating that the B of these materials is less anisotropic along the a and c axes. Additionally, Pm-BCN exhibits the largest anisotropy in ABa, and BCN2 has the largest anisotropy in ABc.  The electronic band structure and the PDOS of Pm-BCN obtained by the HSE06 function are shown in Figure 5, where the dashed line of zero energy (0 eV) indicates the Fermi level (E F ). The valence band maximum (VBM) of Pm-BCN is located at Z (0.0, 0.0, 0.5), and its conduction band minimum (CBM) appears at A (0.5, 0.5, 0.0). Pm-BCN has an indirect and wide band gap of 2.458 eV, therefore it is clearly a semiconductor. The PDOS can be divided into three parts: the first region ranges from −23 to −18 eV, the second region ranges from approximately −16 eV to the Fermi level, and the third region ranges approximately from 2.5 to 10 eV. The first region is dominated from the p orbital, which is primarily from the N-s, C-s, and C-p orbitals. The N-p state and C-s orbitals provide a major contribution to the −16 to −12 eV of the second region. From −12 eV to the Fermi level, the distributions of the B-p, C-p, and N-p orbitals are much greater than that of s orbitals. From 2.5 to 10 eV, the distribution of N-p orbitals is slightly smaller than that of B-p and C-p orbitals. To further understand the chemical bonds, Figure 6 plots the electron localization function (ELF) of Pm-BCN. ELF is an excellent measure of the strength of covalent bonds. Here, B-C and B-N bonding are strongly covalent, whereas the C-N bonding is weakly covalent. The band decomposed charge densities of VBM and CBM of Pm-BCN are depicted in Figure 6b,c, respectively. The B atom is the main contributor to the CBM; the C atom contributes a small amount to the CBM but is the main contributor to the VBM; and the N atom makes a small contribution to the VBM. divided into three parts: the first region ranges from −23 to −18 eV, the second region ranges from approximately −16 eV to the Fermi level, and the third region ranges approximately from 2.5 to 10 eV. The first region is dominated from the p orbital, which is primarily from the N-s, C-s, and C-p orbitals. The N-p state and C-s orbitals provide a major contribution to the −16 to −12 eV of the second region. From −12 eV to the Fermi level, the distributions of the B-p, C-p, and N-p orbitals are much greater than that of s orbitals. From 2.5 to 10 eV, the distribution of N-p orbitals is slightly smaller than that of B-p and C-p orbitals. To further understand the chemical bonds, Figure 6 plots the electron localization function (ELF) of Pm-BCN. ELF is an excellent measure of the strength of covalent bonds. Here, B-C and B-N bonding are strongly covalent, whereas the C-N bonding is weakly covalent. The band decomposed charge densities of VBM and CBM of Pm-BCN are depicted in Figure 6b,c, respectively. The B atom is the main contributor to the CBM; the C atom contributes a small amount to the CBM but is the main contributor to the VBM; and the N atom makes a small contribution to the VBM.   and wide band gap of 2.458 eV, therefore it is clearly a semiconductor. The PDOS can be divided into three parts: the first region ranges from −23 to −18 eV, the second region ranges from approximately −16 eV to the Fermi level, and the third region ranges approximately from 2.5 to 10 eV. The first region is dominated from the p orbital, which is primarily from the N-s, C-s, and C-p orbitals. The N-p state and C-s orbitals provide a major contribution to the −16 to −12 eV of the second region. From −12 eV to the Fermi level, the distributions of the B-p, C-p, and N-p orbitals are much greater than that of s orbitals. From 2.5 to 10 eV, the distribution of N-p orbitals is slightly smaller than that of B-p and C-p orbitals. To further understand the chemical bonds, Figure 6 plots the electron localization function (ELF) of Pm-BCN. ELF is an excellent measure of the strength of covalent bonds. Here, B-C and B-N bonding are strongly covalent, whereas the C-N bonding is weakly covalent. The band decomposed charge densities of VBM and CBM of Pm-BCN are depicted in Figure 6b,c, respectively. The B atom is the main contributor to the CBM; the C atom contributes a small amount to the CBM but is the main contributor to the VBM; and the N atom makes a small contribution to the VBM.

Conclusions
Based on DFT calculations, in this study, we designed and predicted a new lightelement compound, Pm-BCN. First, by analyzing the phonon spectrum, we found that the elastic constants and relative enthalpy of Pm-BCN are theoretically stable. Second, we found that Pm-BCN has an indirect and wide band gap and is a semiconductor material. Third, we showed that the B10 position is the main contributor to the CBM, the C9 position provides a small contribution to the CBM but the main contribution to the VBM, and the N8 position is a minor contributor to the VBM. Finally, we found that the elastic anisotropy in E and the G of Pm-BCN are slightly smaller than those of BCN 2 according to E max /E min and G max /G min , whereas the shear anisotropy factor A 2 and the anisotropy of B along the a direction with respect to the b direction A Ba of Pm-BCN are smaller than those of t-C 8 B 2 N 2 , I-4m2 BCN, Imm2 BCN, B 2 N 2 C 3 , BNC 2 , and B 3 N 3 C.