Edge and sublayer degrees of freedom for phosphorene nanoribbons with twofold-degenerate edge bands via electric field

Abstract

For the pristine phosphorene nanoribbons (PNRs) with edge states, there exist two categories of edge bands near the Fermi energy (E F ), i.e., the shuttle-shaped twofold-degenerate and the near-flat simple degenerate edge bands. However, the usual experimental measurement may not distinguish the difference between the two categories of edge bands. Here we study the varying rule for the edge bands of PNRs under an external electrostatic field. By using the KWANT code based on the tight-binding approach, we find that the twofolddegenerate edge bands can be divided into two separated shuttles until the degeneracy is completely removed and a gap near E F is opened under a sufficiently strong in-plane electric field. Importantly, each shuttle from the ribbon upper or lower edge outmost atoms is identified according to the local density of states. However, under a small off-plane field the shuttle-shaped bands are easily induced into two near-flat bands contributed from the edge atoms of the top and bottom sublayers, respectively. The evidence provides the edge and sublayer degrees of freedom (DOF) for the PNRs with shuttle-shaped edge bands, of which is obviously different from another category PNRs intrinsically with near-flat edge bands. This is because that the former category of ribbons solely have four zigzag-like atomic configurations at the edges in each unit cell, which also results in that the property is robust against the point defect in the ribbon center area. As an application, furthermore, based on this issue we propose a homogenous junction of a shuttle-edge-band PNR attached by two electric gates. Interestingly, the transport property of the junction with field manipulation well reflects the characteristics of the two DOFs. These findings may provide a further understanding on PNRs and initiate new developments in PNR-based electronics.