Quantum spin Hall phase in honeycomb nanoribbons with two different atoms: edge shape effect to bulk-edge correspondence

Abstract

In this study, we consider a quantum spin Hall (QSH) phase in both the zigzag and the armchair type of honeycomb nanoribbons with two different atoms from the viewpoint of bulk-edge correspondence. Generally, the QSH phase in honeycomb nanoribbons is determined by the topology of the bulk Hamiltonian. However, the armchair type of nanoribbons seems to become the QSH phase in a very different region compared with bulk materials. On the other hand, the zigzag type of nanoribbons seems to become the QSH phase in almost the same region as bulk materials. We study the reason why the QSH phase in nanoribbons seems to be different from that of bulk materials using the extended Kane-Mele Hamiltonian. As a result, there is a clear difference in the edge states in the QSH phase between the zigzag and the armchair type of nanoribbons. We find that the QSH phase region in nanoribbons is actually different from that of bulk materials. This is because the coherence lengths of edge wave functions of nanoribbons are extremely influenced by their edge-shapes. We can conclude that the bulk-edge correspondence does not hold for relatively narrow nanoribbons compared with their coherence lengths and that the edge shapes of nanoribbons make their coherence lengths of edge wavefunctions different, which largely influences the QSH phase.