Boron and nitrogen are the two nearest neighbors of carbon in the periodic table of elements, and thus boron nitride (BN) has some similar structures with carbon.^[1–3] For example, three-dimensional (3D) hexagonal BN (h-BN) is like graphite. They can be cut into single-layer h-BN and graphene, respectively, and the two-dimensional (2D) monolayers can be further rolled to one-dimensional nanotubes.^[4–7] Beside the similar structures, both single-layer h-BN and graphene have high thermal conductivities, high chemical stabilities, and excellent mechanical properties.^[8–12] However, the electronic properties of single-layer h-BN and graphene are completely different. It is known that graphene is a Dirac material with Dirac cones locating at the K point in the momentum space.^[13–15] The super-fast Dirac electrons lead to graphene being a star material in electronics. On the contrary, h-BN is an insulator with a band gap larger than 4 eV, and thus in most cases h-BN is only used as an insulating substrate.^[16–21] The insulating of h-BN originates from the electronegativity difference between boron and nitrogen atoms which breaks the symmetry of the honeycomb lattice.

In recent years, many 2D carbon monolayers other than graphene have been proposed, such as T-graphene,^[22,23] phagraphene and tetra-penta-hepta (TPH) graphene,^[24] and Kagome graphene,^[25] and some of them have been synthesized successfully. Although these 2D carbon allotropes are still flat monolayers, they are made of triangular, tetragonal, and pentagonal carbon rings rather than only hexagonal rings. Moreover, these carbon monolayers show different electronic properties with graphene: T-graphene is a metal; Kagome graphene has a flat band around the Fermi level, which induces rich physical phenomena such as ferromagnetism, Wigner crystallization, superconducting, and anomalous quantum Hall effect.^[26–34] These new physical properties extremely extend the applications of carbon materials.

Motivated by the fast development of studies on carbon materials, one could expect that, 2D BN materials should also have many allotropes, consequently there may be unusual physical properties hidden behind the structures. To the best of our knowledge, only a few works have studied new 2D BN materials. For example, Shahrokhi et al. studied the electronic and optical properties of five new BN allotropes,^[35] and Li et al. studied the electronic properties of BN structures like graphyne including sp hybridization.^[36] However, all the proposed new BN allotropes are still insulators or semiconductors. A question arises: Is it possible to find BN structures having special electronic properties?

In this work, we propose two 2D BN monolayers and study their electronic properties by first-principles calculations. One of the monolayers is made of five- and seven-membered rings, and thus is named as 5–7 BN structure. The other monolayer is named as Kagome BN because it is somewhat similar to the Kagome lattice. The two new BN structures not only show good stability but also exhibit interesting electronic properties. The 5–7 BN is a Dirac semimetal, whose conduction and valence bands cross on the Fermi level and thus form Dirac points. The Kagome BN has two flat bands just below the Fermi level. The flat bands generate heavy fermions and there is strongly correlated effect between the fermions. After one-hole doping, the Kagome BN becomes a half metal because the flat band splits to spin-up and spin-down bands. In addition, the possibility of synthesizing 5–7 BN on PbO₂ (111) substrate and Kagome BN on CdO (111) substrate is also discussed, respectively. With the development of the synthesis technologies, the two BN structures could be obtained soon. Their unusual electronic properties will expand the applications of BN materials to new fields.

Figures 1(a) and 1(b) show the two new BN monolayers, respectively. The ratios of boron and nitrogen atoms in both structures are 1 : 1, and both structures have B–N, B–B, and N–N bonds. The monolayer in Fig. 1(a) is made of five- and seven-membered rings, and thus is named as 5–7 BN. The monolayer in Fig. 1(b) consists of boron triangular rings and nitrogen dimers, and the rings are linked together by the dimers. When a nitrogen dimer and two boron atoms at the two ends of the dimer are contracted to one lattice, the structure becomes a standard Kagome lattice.^[37,38] Therefore, the structure in Fig. 1(b) is named as Kagome BN.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (a) Top view and (c) side view of 5–7 BN, blue is for N and pink is for B atoms. Its primitive cell is shown by the dashed lines in (a). (b) Top view and (d) side view of Kagome BN. Its primitive cell is shown by the dashed lines in (b).

Our calculations were performed within the density-functional theory (DFT) as implemented with the Perdew–Burke–Ernzerhof (PBE) approximation to the exchange–correlation functional.^[39,40] The core-valence interactions were described by the projector augmented wave (PAW) potentials as carried under the Vienna ab initio simulation package (VASP) code.^[41] Plane waves with a kinetic energy cutoff of 600 eV were used as the basis set. For the Brillouin zone integration, 5–7 BN and Kagome BN were done with 11 × 11 × 1 and 5 × 5 × 1 Γ-centered Monkhorst–Pack k-point meshes, respectively.^[42] The atomic positions were fully optimized by the conjugate gradient method,^[43] and the energy and force convergence criteria were set to be 10^–6 eV and 10^–3 eV/Å, respectively. Periodic boundary conditions (PBC) were used and empty space of 20 Å was introduced in the direction perpendicular to the monolayers, which ensures that the interaction between the periodic images of the sheet is negligible. To investigate the thermal stability, we carried out ab initio molecular dynamics (AIMD) simulations based on a canonical ensemble,^[44] a 3 × 3 supercell containing 72 atoms was used for 5–7 BN and a 2 × 2 supercell containing 48 atoms was used for Kagome BN, and the AIMD simulations were performed with a Nose–Hoover thermostat from 300 K to 1800 K.

After being fully optimized, the two BN monolayers in Fig. 1 are still planar structures. The space groups of 5–7 BN in Fig. 1(a) and Kagome BN in Fig. 1(b) are Cmmm (D2H-19) and P6/mmm (D6H-1), respectively. The primitive cells of the two structures are shown in the dashed-line boxes. The lattice constants of 5–7 BN are a₁ = b₁ = 4.90 Å, while those of Kagome BN are a₂ = b₂ = 9.97 Å. In the 5–7 BN, there are two types of B–N bonds whose bond lengths are 1.42 Å and 1.50 Å, respectively. The two bonds can be comparable with the B–N bond in h-BN sheet whose bond length is 1.45 Å. In the Kagome BN, there is only one type of B–N bond whose bond length is 1.33 Å. The shorter B–N is similar to the acetylene bond in graphyne.^[45] The calculated cohesive energies of 5–7 BN and Kagome BN are 7.55 eV and 6.90 eV, respectively. These cohesive energies are lower than that of the h-BN monolayer (7.90 eV), which means that the two new monolayer structures are metastable.

To confirm the structural stabilities of the two BN structures, their dynamic stabilities are examined by phonon spectrum calculations. One can find that, in Figs. 2(a) and 2(b), the phonon dispersions of both structures have no soft mode, which indicates that they are dynamic stable. The thermal stabilities of the two structures are also examined by utilizing AIMD simulation in a canonical ensemble. For the 5–7 BN structure, we do not observe any structural decomposition after heating up to the targeted temperature of 1500 K for 20 ps (see Fig. S1(a)). The structure reconstruction only occurs at 1700 K during the 20 ps simulation, as shown in Fig. S1(b). For the Kagome BN structure, we do not observe any structural decomposition when heated up to 1200 K, but reconstruction occurs at 1300 K (see Figs. S1(c) and S1(d)). Therefore, the 5–7 BN and Kagome BN have rather high thermodynamic stability and outstanding dynamic and thermal stabilities.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Phonon dispersions of (a) 5–7 BN and (b) Kagome BN, where no soft mode is found.

The band structure of the 5–7 BN is calculated, and the result is shown in Fig. 3(a). Along the high-symmetry line Γ–X and high-symmetry point Y, the conduction and valence bands cross at D1 and D2, and result in two Dirac points whose 3D band structures are shown in Fig. 3(d). Because of the time reversal and inversion symmetries,^[46,47] there are total four Dirac points in the first Brillouin zone (BZ) (see Fig. 3(c)). The calculated velocities of the Dirac electrons are approximate 4 × 10⁵ m/s. To the best of our knowledge, all the former proposed 2D BN structures are not Dirac materials. This is the first BN material in which Dirac electrons can be found.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (a) Projected band structure of 5–7 BN calculated by DFT, inset: the wave functions for the states from B1 and B2. (b) PDOS for B atoms and N atoms. The colors of the energy bands in (a) correspond to the colors of the projected atomic orbitals in (b). (c) Total Dirac points in the first Brillouin zone. (d) The 3D band structures for D1 and D2.

To reveal the origin of the Dirac cones in the 5–7 BN, we first calculate its partial density of states (PDOS), as shown in Fig. 3(b). One can find that the energy bands around the Fermi level are mainly contributed by the p_z orbitals of the B and N atoms. The projected energy bands in Fig. 3(a) further illustrate that the two crossing bands of the Dirac points result from different atoms: the band with red dots is mainly induced by the p_z orbital of the B atoms, while the band with blue dots is mainly induced by the p_z orbital of the N atoms. In fact, the conduction and valence bands of h-BN are contributed by the p_z orbitals of the B and N atoms, respectively (see Fig. S2). Although the B–N bond in h-BN is a covalent bond, it is more like an ionic bond in which electrons localize on the N atoms. The energy of the occupied state in N atoms is much lower than that of the unoccupied state in B atoms, and thus a wide gap appears, as shown in Fig. S2(c). In the 5–7 BN structure, there exist B–B and N–N bonds. The insets in Fig. 3(a) illustrate that a bonding state is formed in the B–B bonds while an anti-bonding state is formed in the N–N bonds. This results in the energy of the B atoms decreasing while that of the N atoms increasing (see Fig. 3(a)). In this case, band crossing occurs and Dirac points form.^[48,49]

The band structure of the Kagome BN is shown in Fig. 4(a). One can find that there are two flat bands just below the Fermi level. Some other bands cross at K point, but the crossing points are far from the Fermi level. Figures 4(b) and 4(c) present PDOS of Kagome BN. It illustrates that, the top flat band is attributed by the p_x/y orbitals of the B and N atoms, while the bottom flat band is attributed by the p_z orbitals of the B and N atoms.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (a) Projected band structure of Kagome BN calculated by DFT. PDOS for B atoms (b) and N atoms (c), respectively. The colors of the energy bands in (a) correspond to the colors of the projected atomic orbitals in (b) and (c). Inset: the wave functions for the states from B3 and B4 in (a).

In a flat band, the electron velocities are approximately equal to zero, i.e., the electrons are heavy fermions that are inverse to Dirac fermions.^[50] The flat band will give rise to a series of many-body phenomena, such as ferromagnetism,^[51] superconducting,^[52] Wigner crystallization,^[53] and anomalous quantum Hall effect.^[54] In general, spin polarization of common bands is typically not favored because the kinetic energy cost is often larger than the exchange energy gain.^[55] However, the flat-band physics gives another picture: when electrons fill in the flat band, the kinetic energy penalty of spin polarization does not exist anymore, hence the exchange interaction stabilizes the polarized state. In Fig. 5, we calculate the band structure of the Kagome BN when a hole is doped in the structure. One can find that the spin-up and spin-down energy bands split. The splitting of the top flat band is the largest (∼ 0.6 eV). The Kagome BN becomes ferromagnetic. The magnetic moments of the atoms are calculated, and the results indicate that magnetic moments of the B and N atoms are 0.68μ_B and –0.04 μ_B, respectively. The B atoms possess strong magnetism.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Band structure of Kagome BN after one-hole doping, where the solid and dashed lines represent spin-up and spin-down energy bands, respectively. The flat band has the biggest spin splitting at about 0.57 eV.

The above discussions on the stabilities of the two new BN structures have demonstrated the feasibility to synthesize the structures. Considering that many 2D monolayers have been synthesized on substrates,^[56,57] we also investigate the possibilities to grow the two new monolayers on some suitable substrates. It is found that the lattice parameters of a supercell of a PbO₂ (111) substrate ( Å, Å) are very close to those of 5–7 BN structure’s conventional cell (a₁ = 7.56 Å, b₁ = 6.23 Å). The mismatches between them are −2.77% and 2.08% in x and y orientations, respectively (the negative percentage means the lattice constant of the substrate is shorter than the corresponding lattice parameter of 5–7 BN structure). A 5–7 BN placing on the PbO₂ (111) substrate is shown in Figs. 6(a) and 6(c). The adhesion energy between the 5–7 BN and substrate is −0.98 eV/atom. Similarly, the lattice parameters of a 3 × 3 supercell of a CdO (111) substrate ( Å ) are very close to those of the primitive cell of the Kagome BN. The mismatch between them is smaller than 0.5%. A Kagome BN placing on the CdO (111) substrate is shown in Figs. 6(b) and 6(d). The adhesion energy between the Kagome BN and substrate is −0.42 eV/atom. These calculated energies can be comparable with those of other monolayers on oxide substrates (−0.8 eV/atom and −0.26 eV/atom) in Refs. [58,59], and thus we believe that the two predicted BN structures can be synthesized successfully.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. (a) Top view and (c) side view of 5–7 BN on the PbO2 (111) substrate. Its primitive cell is shown by the dashed line in (a). (b) Top view and (d) side view of Kagome BN on the CdO (111) substrate. Its primitive cell is shown by the dashed line in (b).

In summary, we propose two 2D BN structures by using first-principles calculations. One is 5–7 BN made of five- and seven-membered rings, and the other is Kagome BN whose lattice is like the Kagome lattice. Both structures show good dynamical and thermal stabilities. More interestingly, the two structures reveal unusual electronic properties different from those in the former 2D BN allotropes. Most former BN allotropes are insulators or semiconductors. The 5–7 BN structure is a Dirac material, whose electron velocities are approximate 10⁵ m/s. The band crossing is originated from the bonding states in the B–B bonds and antibonding states in the N–N bonds. The Kagome BN structure is a flat-band system in which two flat bands exist just below the Fermi level. There are heavy fermions with very low velocities in the structure. After one-hole doping, the top flat band splits to spin-up and spin-down flat bands, and thus ferromagnetism occurs. Dirac fermions and flat-band-induced ferromagnetism in BN allotropes have never been reported in the previous literatures. We also propose a promising approach to prepare the two new monolayers by epitaxial growth on a PbO₂ (111) substrate and a CdO (111) substrate, respectively. Our work extends people’s understanding on the BN structures, and will also extend the applications of BN allotropes in different fields.