Topologically trivial states induced by strong spin-orbit coupling and Chern insulators in doped X(C21N3H15) (X=Ta , Hf) metal-organic frameworks

Abstract

We study topological properties of two-dimensional (2D) bi-triangular lattices with multiple $p$ and $d$ orbitals. Tight-binding (TB) model calculations show that the multiple $p$ (${p}_{x}/{p}_{y}$) and $d$ (${d}_{xy}/{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ or ${d}_{xz}/{d}_{yz}$) bands in the 2D bi-triangular lattices, with ${C}_{3}$ symmetry, are all degenerate at the $\mathrm{\ensuremath{\Gamma}}$ point with quadratic non-Dirac band dispersions and gapped at the $K/{K}^{\ensuremath{'}}$ points. Spin-orbit coupling (SOC) is found not only resulting in the topologically nontrivial states, but also the topologically trivial states. In the 2D bi-triangular lattices, a topological phase transition from a topologically nontrivial state to a trivial state can be triggered by increasing the atomic SOC strength. Our first-principles calculations predict a large band-gap quantum anomalous Hall state (about 30 meV) and topological phase transitions in a 2D metal-organic frameworks $X({\mathrm{C}}_{21}{\mathrm{N}}_{3}{\mathrm{H}}_{15}$) ($X=\mathrm{Ta}$, Hf), rationalized well by the TB model built above. Our results are helpful to deeply understand the topological states and provide theoretical guidance for the search of 2D topological materials based on the multiple $p$ and $d$ orbitals.