Motivated by recently searching for topological states in organic materials as well as successful experimental synthesis of a graphitelike metal-organic framework ${\mathrm{Ni}}_3\left({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_6\right){}_2$ [Sheberla et al., J. Am. Chem. Soc. 136, 8859 (2014)], we systematically investigated the electronic and topological properties of the ${\mathrm{Ni}}_3\left({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_6\right){}_2$ monolayer using an ab initio method combined with a tight-binding model. Our calculations demonstrate that the material can be in a quantum spin Hall or $Z_2$ metallic state in different electron-doped concentrations, which are experimentally accessible with currently electrostatic gating technologies. The tight-binding model also shows that the real next-nearest-neighbor interaction is essential to drive the $Z_2$ metallic phase in ${\mathrm{Ni}}_3\left({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_6\right){}_2$-type lattices.

• Received 26 September 2014

DOI:https://doi.org/10.1103/PhysRevB.90.201403