Figure 1 plots the total energy per atom against the lattice constant a (the lattice constant c is fixed) for g-B₃N₃C. It can be seen that the total energy of g-B₃N₃C as a function of lattice constant a has a single minimum, meaning that the geometrical structure would be stable. Particularly, the charge population analysis reveals that the electron density around the C-N bond in α-g-B₃N₃C is much higher than that around the C-B bond in β-g-B₃N₃C, showing that the C-N bond is relatively strong, which is also consistent with the experimental results 1 . This strong interaction between C and N atoms in α-g-B₃N₃C directly results in a short C-N bond length (see the following paragraph) and can balance the strain of the monolayer graphite-like structure. Thereby, α-g-B₃N₃C could be a more thermodynamically stable topological phase against β-g-B₃N₃C, which has a lower total energy of around 0.43 eV/atom compared with β-g-B₃N₃C.

To explore further the mechanical stability of α-g-B₃N₃C, the optimized lattice constant a = 4.67Å is first obtained, as depicted in the left panel of Figure 1. Importantly, the C-N bond length converged to 1.31 Å, which is considerably reduced from the typical bond lengths of 1.37 to 1.48 Å in the related materials 28 . This strongly suggests the nature of the higher binding energy of the C-N bond 1 , which also influences the B-N bonding and extends its length. The obtained B-N bond length is 1.48 Å, which is slightly bigger than the value of 1.45 Å in h-BN. The calculated partial density of states (PDOS) is shown in Figure 2. The valence band is dominated by B p and C p states, while the conduction band is only dominated by B p states. There, one can find the majority of C p states to be semicore, lying 6 to 9 eV below the Fermi level. These states interact with those comprising the valence band with the same symmetry. As a result, there is a small admixture of C p and N p states close to the Fermi level. However, in α-g-B₃N₃C, one finds significant admixture of C p and N p states in the semicore energy window with 6 to 9 eV below the Fermi level. This suggests that the C p-N p interaction in the semicore region contribution to the C-N bonding is significantly more important in α-g-B₃N₃C than the C p-N p interaction close to the Fermi level.

Now, let us look at the band structures of α-g-B₃N₃C, as given in Figure 3. It explicitly demonstrates that all three XC potentials give similar band structures. The highest occupied crystalline orbitals are located at the K point of the reciprocal space, while the lowest unoccupied crystalline orbitals appear at the M point. This leads to an indirect bandgap semiconductor. To obtain the bandgap more accurately (see Figure 3c), the MBJ XC functional is used 24 25 . The bandgap of α-g-B₃N₃C within MBJ is obtained to be 1.22 eV, which nicely locates the middle region between 0.59 and 1.80 eV for the BNC samples with 12.5% and 50% C contents, respectively 4 . Note that the C content in α-g-B₃N₃C is 25%. In the experiments 1 , the absorption edges are redshifted as the C concentrations increase, which shows a tunable mechanism of an optical bandgap in actual applications. By comparing with 1 where BNC with around 65% C concentration shows an optical bandgap of 1.62 eV, we infer that such a higher energy absorption edge (take into account the bandgap of 1.22 eV in our case with 25% C concentration) arises from the formation of individual BN and graphene domains. In this way, the even distribution of C in BNC systems might serve as a good guide to find alternative solutions to existing bandgap-engineered applications. Future research can test this prediction directly.

In addition, the bandgaps based on the LDA and PBE methods are equal to 0.83 eV (see Figure 3a,b). The relative bandgap correction from MBJ with respect to LDA, R = ( Δ MBJ − Δ LDA ) / Δ MBJ with Δ _MBJ/LDA being the bandgap, is about 32%. We can see that our calculated R for α-g-B₃N₃C ( shows an excellent agreement with the 16 sp semiconductors) lies within the range of 16.0% to 100.0% (see Table 1 for details). The correction value is particularly very close to that in BN (25%), GaN (42%), AlP (37%), and AlN (25%). This finding is not surprising because the listed four solids have at least one element close to that of α-g-B₃N₃C in the periodic table of elements. It means that the similar chemical circumstances in these materials can be well described by the same XC functionals. This underlyingly confirms our prediction validity and that the α-g-B₃N₃C might be carried out experimentally.

The theoretical and experimental bandgaps (in eV) of the 16 sp semiconductors are directly taken from 25 . The relative bandgap correction R (in %) is calculated from the equation R = ( Δ MBJ − Δ LDA ) / Δ MBJ , where Δ _MBJ and Δ _LDA are the calculated bandgaps using XC potentials MBJ and LDA.

From Figure 2, which shows the DOS of α-g-B₃N₃C, we can see that the effect of the MBJ potential is to shift up (with respect to LDA/PBE) the unoccupied B 2s and 2p states. Here, three major differences between the LDA/PBE and MBJ methods can be extracted: (a) Another obvious effect of MBJ potentials is to shift down the middle of the valence band at around −4.0 eV. (b) The hybridization of s and p states of dominant B 2s and other atoms’ 2p states at the bottom of the valence band is very strong in MBJ calculations (denoted as down arrows in Figure 2). (c) The correction character of α-g-B₃N₃C is much more pronounced with the MBJ than with the LDA/PBE, which narrows the valence band just below the Fermi level. We would like to stress that the MBJ potentials open a bandgap of 0.39 eV in the α-g-B₃N₃C model compared to the result of LDA, which is consistent with the orbital-dependent potential principle 24 .

The optical absorption spectrum for α-g-B₃N₃C within MBJ potentials (see Figure 4c) shows a big absorption packet with two adjacent peaks in the range of 2.5 to 5.0 eV, which originates from the band structure as shown in Figure 3c. As an indirect bandgap material, the transition from the K point to Γpoint is very weak as the momentum conservation rule is not satisfied here, and thus, the corresponding characteristic photoluminescence in the optical absorption spectrum can be negligible. This is also true in our case for α-g-B₃N₃C. The first peak corresponds to the direct bandgap transition at the M point. The second peak comes from the larger direct gap from higher energy states located at the Γ point. More importantly, these two peaks are not separated distinctly. This feature can be attributed to the even distribution of C atoms in α-g-B₃N₃C as mentioned above, which shows a different formation mechanism compared with the hybrid BNC in the experiment 1 .

Correction effects were taken into account by adding a LDA correction potential in MBJ 25 . This important physical effect opens an additional bandgap by mimicking very well the behavior of orbital-dependent potentials and causes a rigid blueshift of the absorption spectrum compared with the LDA/PBE curves as shown in Figure 4. This explains the excellent qualitative agreement of the hybrid exchange-potential optical absorption spectrum seen in Figure 4, due to a compensation of significant errors within the standard DFT methods.

The predicted structure of β-g-B₃N₃C is shown in the right panel of Figure 1. The lattice constant of β-g-B₃N₃C is obtained to be 5.26 Å as shown in Figure 1. The results show the equilibrium value d _BC = 1.52 Å, which is close to the value of graphite-like BC₃, which is 1.55 Å 29 . It is to be noticed that all the equilibrium values, d _BN, in α-g-B₃N₃C are equal to 1.42 Å, which is slightly less than the value of 1.45 Å in the pristine BN sheet. This implies the stronger B-N bonds formed in β-g-B₃N₃C. Our calculations show that the β g-B₃N₃C leads to a ground state with a magnetic moment of 0.68 μ _B. The nonmagnetic state is 0.07 eV higher than this ground state.

From the band structures, we see that although both the pristine BN sheet and graphene are nonmagnetic, the β-g-B₃N₃C model can be spin-polarized. It is necessary to discuss magnetism in more detail from its electronic structures. Figure 5a,b,c presents the band structure and spin-resolved total density of state (TDOS). Remarkably, two bands cross the Fermi level (black and red circles in Figure 5a) and make the Fermi energy level occupied completely along the entire high-symmetry lines. In contrast, the spin-down one does not possess such a strongly localized feature, just one band (black circles) crosses the Fermi level monolithically as shown in Figure 5c. The calculated magnetic moment in β-g-B₃N₃C should originate from this asymmetric spin-dependent localization. The corresponding strong spin splitting can be further confirmed from TDOS as shown in Figure 5b. In addition, close examination of the top valence bands (see Figure 6a) indicates that the strong localization mainly comes from the 2p atomic orbitals of C and N atoms. Here, we also show the Fermi surfaces of β-g-B₃N₃C in the first Brillouin zone. The unique feature of the Fermi surface which is almost parallel to the high-symmetry lines (see Figure 5d,e,f) is the direct manifestation of the bands near the Fermi level in Figure 5a,c.

It should be noticed that such magnetism is induced without transition metals and without external perturbations so that β-g-B₃N₃C behaves as the first, theoretically predicted, metal-free magnetic material in the hybrid BNC system. Apparently, our finding points out a new direction for further related experimental investigations in spintronics.

We now address the possibility of the origin of magnetism in β-g-B₃N₃C. Based on the orbital-resolved density of state as shown in Figure 6a, the magnetic moment is mainly ascribed to the 2p orbitals of C and three N atoms. The spin polarization of C atom offers a magnetic moment of 0.18 μ _B. A total magnetic moment of 0.18 μ _B is shared equally by the 2p orbitals of three N atoms. The remaining magnetic moments distribute evenly in the interstitial region among N atoms. This is conceivable because β g-B₃N₃C, if compared with other BNC systems, has large interspaces between N atoms (see the 2×2 supercell in Figure 1 or Figure 6b). The remarkable feature is that the equilibrium surface density of β-g-B₃N₃C is around 1.27 times larger than that of α-g-B₃N₃C. Owing to the large interspaces between atoms, the hydrogen storage in β-g-B₃N₃C may be expected 30 . The detailed description of hydrogen storage related to β-g-B₃N₃C is beyond the scope of this work.

Concerning the special carbon atom in β-g-B₃N₃C, the counting of four valence electrons is as follows: Three electrons participate in the s p ² hybrid orbital, which forms a planar structure. The remaining one electron is then redistributed in the whole unit cell due to the enhanced B-N covalent bond (with shorter bond length compared with the value in pristine h-BN), which makes the magnetic properties more complicated. From the fourth electron, only 18% of the electron still fills the Π-orbital of the C atom and contributes a magnetic moment of 0.18 μ _B. This is in fairly good agreement with theoretical description of the Π-orbital state to the local magnetic moment of approximately 0.3 μ _Bin the graphene-like systems 31 32 . Around 32% of the fourth electron mainly resides at the interstitial region, which plays a crucial role in β g-B₃N₃C as follows: (a) enhances the B-N covalent bond, (b) provides the main interstitial magnetic moment of 0.32 μ _B, and (c) promotes the 2p _z of the N atom spin-polarized slightly with a magnetic moment of 0.06 μ _Bper N atom. The remaining percentage of the fourth electron acts as the conduction electrons and makes the system metallic, which dominates the mechanism of ferromagnetic ordering in β g-B₃N₃C. In the case of β g-B₃N₃C, we can see that the electron spin at the localized Π-orbital state of C and N atoms as well as the interstitial region compels two energy bands localized strictly along the entire high-symmetry lines, i.e., the Γ-M-K-Γline (see Figure 5a). Thus, the RKKY interaction 10 33 among the magnetic sites through the residual conduction electrons forms a spin ordering in these orbitals, which is the physical origin of ferromagnetism in β g-B₃N₃C. Figure 6b,c respectively plot the top and side views of the 3D iso-surfaces for net magnetic charge density in the xy plane. This finding is insightful, and three major points deserve comment: (a) The C site is more spin-polarized as compared with each N site. (b) The dumbbell-like magnetic moment distribution along the z direction implies that the 2p _z orbital becomes partially filled with one spin-up electron. (c) The induced moments are ferromagnetic coupled between the N and C sites based on the RKKY exchange interaction model as mentioned above.