Topological-like states in nano ribbons induced by an electric field

The energy spectra of multilayer nano structures made of materials with strongly anisotropic honeycomb lattice in the presence of a perpendicular electric field and an in-plane magnetic field are investigated by means of the tight-binding method. An inherent property of thin layer nano structures with anisotropic honeycomb lattice is a nonplanar single layer, a bandgap that is often wider than in their bulk counterparts, and common regular and skewed armchair and zig-zag edges depending on the termination. Skewed and regular zig-zag edges support states with the wave functions exponentially decaying from the edges that have almost zero energy and are virtually dispersion less, and as such, they split the bandgap. The regular armchair edges usually do not support this type of edge state, so there is a wide bandgap in the energy spectrum. We analyze nano ribbons composed of multilayer phosphorene with regular armchair edges, that have a wide direct energy bandgap in the absence of external fields. When a perpendicular electric field is applied, the band gap decreases and, for a certain critical field value, the Dirac -like cone emerges at the zone center. Unlike conventional topological insulators, the inversion between the conduction and the valence band states at t he zone center is induced by the electric field. From the density of states, one may find that the conduction and valence band states near the anticrossing are peculiarly distributed along the upper and the lower ribbon surface. A thorough analysis of the interlayer coupling integrals shows that for sufficiently thin phosphorene slabs there is a strong interaction between the electron and the hole states, despite the tendency of the external electric field to push them to the opposite sides of the slab. When an arbitrary small in-plane magnetic field is applied, there is an opposite shift in the energy of the conduction and the valence band that occurs in the k-space turning the gap into an indirect one that closes for the critical value of the applied field. A similar effect is observed in quantum spin Hall insulators where an in -plane magnetic field induces a semiconductor-to-semimetal transition. (author)