The fundamental differences between Quantum Spin Hall edge-states at zig-zag and armchair terminations of honeycomb and ruby nets

TL;DR

This study compares the properties of Quantum Spin Hall edge states at zigzag and armchair edges in honeycomb and ruby nets, revealing distinct behaviors in Fermi velocity and decay length related to spin-orbit coupling.

Contribution

It provides a combined analytical and numerical analysis of edge-state dispersion, highlighting fundamental differences based on edge type and material structure.

Findings

Fermi velocity on ZZ edges increases linearly with SOC strength.

Fermi velocity on AC edges is independent of SOC.

Decay length of edge states varies inversely with SOC, differing between AC and ZZ edges.

Abstract

Combining an analytical and numerical approach we investigate the dispersion of the topologically protected spin-filtered edge-states of the Quantum Spin Hall state on honeycomb and ruby nets with zigzag (ZZ) and armchair (AC) edges. We show that the Fermi velocity of the helical edge-states on ZZ edges increases linearly with the strength of the spin-orbit coupling (SOC) whereas for AC edges the Fermi velocity is independent of the SOC. Also the decay length of edge states into the bulk is dramatically different for AC and ZZ edges, displaying an inverse functional dependence on the SOC.