Helical edge states in silicene and germanene nanorings in perpendicular magnetic field

TL;DR

This paper investigates the topological edge states in silicene and germanene nanorings under a perpendicular magnetic field, revealing the nature of quantum spin Hall phases and the behavior of helical edge states.

Contribution

It provides a detailed analysis of the energy spectra, electron density, and current density in silicene and germanene nanorings, highlighting differences between zigzag and armchair edges.

Findings

Helical edge states are present at the topological phase boundaries.

Zigzag nanorings exhibit edge states well described by classical magnetic moments.

Armchair nanorings show eigenfunctions spread throughout the ring, unlike zigzag edges.

Abstract

Due to nonzero intrinsic spin-orbit interaction in buckled honeycomb crystal structures, silicene and germanene exhibit interesting topological properties, and are therefore candidates for the realization of the quantum spin Hall effect. We employ the Kane-Mele model to investigate the electron states in hexagonal silicene and germanene nanorings having either zigzag or armchair edges in the presence of a perpendicular magnetic field. We present results for the energy spectra as function of magnetic field, the electron density of the spin-up and spin-down states in the ring plane, and the calculation of the probability current density. The quantum spin Hall phase is found at the edges between the nontrivial topological phase in silicene and germanene and vacuum. We demonstrate that the helical edge states in zigzag silicene and germanene nanorings can be qualitatively well understood by means of classical magnetic moments. However, this is not the case for comparable-sized armchair nanorings, where the eigenfunctions spread throughout the ring. Finally, we note that the energy spectra of silicene and germanene nanorings are similar and that the differences between the two are mainly related to the difference in magnitude of the spin-orbit coupling.