Unified low-energy effective Hamiltonian and the band topology of $p$-block square-net layer derivatives

Abstract

In recent years, low-dimensional materials with tetragonal $P4/nmm$ (orthorhombic $Pnma$) space group having square-net (chain-like) substructure of $p$-block elements have been studied extensively. By using a first-principles calculation and a two-sites $\otimes$ two-orbitals tight-binding model, we construct the unified low-energy effective Hamiltonian and the $\mathbb{Z}_{2}$ topological phase diagram for such materials with different filling factors. Near the chemical potential, we show that the staggered arrangement of ions at 2c (4c) site yields the virtual hopping that have the same form with the second nearest-neighbor hopping between the square-net (chain-like) ions. We show that this hybridization and low-symmetry of the chain-like structure protects the quantum spin Hall insulator phase. Finally, the second order spin-orbit coupling on top of the atomic spin-orbit coupling is considered to clarify the origin of the non-zero Berry phase signals reported in recent quantum oscillation experiments.