Exotic quantum states in multilayer phosphorene nanoribbons in electric and magnetic fields

Abstract

Using the tight-binding method, we modeled the energy spectra of multilayer phosphorene nanoribbons in a perpendicular electric field and in-plane magnetic field. Phosphorene nanosheets have a highly anisotropic honeycomb-like lattice. Their band gap is wider than that of their bulk counterparts, and armchair and zigzag edges of either skewed or regular type terminate the nanowire edges. Zigzag and various skewed edges support states whose wave functions decay exponentially from an edge. These states are virtually dispersionless and split the band gap. In principle, regular armchair edges do not host edge states. Thus, the energy spectrum in this case has a wide band gap. Here, we consider nanoribbons composed of multilayer phosphorene with regular armchair edges. A wide direct energy band gap exists when external fields are absent, but its width decreases when a perpendicular electric field is applied. The Dirac-like cones cross-section emerges at the zone center for a particular field value, named the lowest critical field. Although spin–orbit coupling was not included in the model, there is a small gap at the anticrossing site. The local density of states shows that the conduction- and valence-band states near the anticrossing are localized on the top and bottom surfaces of the nanoribbon. A thorough analysis of the interlayer coupling integrals indicates that for sufficiently thin phosphorene slabs, the electron and hole states at the opposite sides of the slab couple mutually strongly, despite the tendency of an external electric field to separate them. A further increase in the electric field induces an inversion between the conduction and valence band states in the zone center, which is inherent to topological insulators. However, sharp anticrossings at the zone center emerged for certain higher field values, named higher critical fields. Furthermore, when an in-plane magnetic field is applied, the conduction and valence band states shift, causing the dispersion to twist around the center of the k-space. Therefore, the band gap is indirect and closes for a sufficiently large magnetic field. A similar effect is observed in quantum spin Hall insulators, in which an in-plane magnetic field induces a semiconductor-to-semimetal transition. We conclude that the band inversion and topological-like features induced by external fields can be attributed to the strong interlayer coupling inherent to multilayered materials with anisotropic honeycomb lattices.