Topological and superconducting properties of two-dimensional $C_{6-2x}{\left(\mathrm{BN}\right)}_x$ biphenylene network: First-principles investigation

First-principles calculations have been used to investigate the electronic and topological properties of the two-dimensional $C_{6-2x}{\left(\mathrm{BN}\right)}_x$ biphenylene network, a graphenelike structure composed of not only hexagonal ring but also octagonal and square rings. Nontrivial topological properties have been found in two of them, with a stoichiometry of $C_4\mathrm{BN}$ and $C_2{\left(\mathrm{BN}\right)}_2$. The former $C_4\mathrm{BN}$ is predicted to be a type-II Dirac semimetal with a superconducting critical temperature $T_c=0.38$ K, which is similar to the pure carbon biphenylene network (C-BPN). The latter shows an isolated edge state exists between the conduction and valence bands. By regulation of strains and virtual-crystal approximation calculations, we found the annihilation of two pairs of Dirac points (DPs) in the nonhigh-symmetric region causes the two corresponding edge states to stick together to generate this isolated edge state. In addition, we found that one pair of DPs arises from the shift of DPs in the C-BPN, while another pair of DPs emerges around the time-reversal invariant momenta point $X$ due to the doping of boron and nitrogen. We constructed a tight-binding model to reveal the mechanism of forming the isolated edge state from the C-BPN to $C_2{\left(\mathrm{BN}\right)}_2$. This study not only demonstrates the existence and mechanism of forming the isolated edge state in semimetals, but also provides an example in which the DPs can move away from the high-symmetry region.