Interference of Topologically Protected Edge States in Silicene Nanoribbons

Silicene is a two-dimensional quantum spin-Hall insulator. We study the edge channels of silicene nanoribbons from the viewpoint of the topological protection and the interference between the two edges. It is found that the behaviors of the helical edge channels (HECs) are completely different between the armchair and zigzag edges. The penetration depths $\xi$ of the HEC is antiproportional to the spin-orbit gap for the armchair edge $\xi_{\rm arm} \sim \hbar v_\text{F}/ \Delta$ ($v_\text{F}$: the Fermi velocity, $\Delta$: the gap due to the spin-orbit interaction), while it remains as short as the lattice constant for the zigzag edge. Zero-energy states disappear in armchair nanoribbons due to an interference of two edge states, while they remains in zigzag nanoribbons even if the width $W$ is quite narrow. The gap $\delta$ of HECs behaves as $\delta \sim \hbar v_\text{F}/W$ for $\Delta=0$, and as $\delta \sim \Delta \exp[- W/\xi_{\rm arm}]$ for $\Delta\not=0$ for armchair nanoribbons, while it remains essentially zero irrespectively of $\Delta$ for zigzag nanoribbons.