Analysis of band structures of phosphorene and bismuthene based on the double group theory

Abstract

Abstract We study band structures of group-V two-dimensional materials, i.e. phosphorene and bismuthene, by carrying out first-principles calculations including spin–orbit coupling (SOC). We propose a method to identify irreducible representations (IR) of both symmorphic and nonsymmorphic systems. We find for the α structures that all the non-SOC bands are doubly degenerated on the first Brillouin zone edge due to sticking or pairing of bands and that the SOC slightly splits the bands in most of the cases. We evaluate Z 2 invariants based on identified IR. We find that the Z 2 invariant of 1 in the case of β bismuthene is due to the strong SOC that reverses the highest occupied and the lowest unoccupied bands at the Γ point.