Topological edge and corner states and fractional corner charges in blue phosphorene

TL;DR

This paper investigates edge and corner states in monolayer blue phosphorene, revealing their origins from Wannier orbitals, structural effects, and the emergence of fractional corner charges, with implications for topological properties.

Contribution

It introduces a theoretical framework linking Wannier orbitals to edge and corner states in blue phosphorene, highlighting fractional charges and structural influences.

Findings

Edge states exist in zigzag and armchair nanoribbons due to Wannier orbitals.

Corner states appear at the Fermi energy in armchair corners, causing fractional charges.

Blue and black phosphorene share similar topological edge and corner properties.

Abstract

We theoretically study emergent edge and corner states in monolayer blue phosphorus (blue phosphorene) using the first-principles calculation and tight-binding model. We show that the existence of the Wannier orbitals at every bond center yields edge states both in zigzag and armchair nanoribbons. The properties of the edge states can be well described by a simple effective Hamiltonian for uncoupled edge orbitals, where the structural relaxation near the boundary significantly affects the edge band structure. For corner states, we examine two types of corner structures consisting of zigzag and armchair edges, where we find that multiple corner states emerge in the bulk gap as a consequence of hybridization of edge and corner uncoupled orbitals. In the armchair corner, in particular, we demonstrate that corner states appear right at the Fermi energy, which leads to the emergence of fractional corner charge due to filling anomaly. Finally, we discuss the relationship between blue phosphorene and black phosphorene, and show that two systems share the equivalent Wannier orbital positions and similar edge/corner state properties even though their atomic structures are totally different.