Helical edge states and edge-state transport in strained armchair graphene nanoribbons

Zheng-Fang Liu,Ai-Xi Chen,Guo-Xing Miao,Nian-Hua Liu,Qing-Ping Wu,Xian-Bo Xiao

doi:10.1038/s41598-017-08954-3

Abstract

A helical type edge state, which is generally supported only on graphene with zigzag boundaries, is found to also appear in armchair graphene nanoribbons in the presence of intrinsic spin-orbit coupling and a suitable strain. At a critical strain, there appears a quantum phase transition from a quantum spin Hall state to a trivial insulator state. Further investigation shows that the armchair graphene nanoribbons with intrinsic spin-orbit coupling, Rashba spin-orbit coupling, effective exchange fields and strains also support helical-like edge states with a unique spin texture. In such armchair graphene nanoribbons, the spin directions of the counterpropogating edge states on the same boundary are always opposite to each other, while is not conserved and the spins are canted away from the -direction due to the Rashba spin-orbit coupling, which is different from the case of the zigzag graphene nanoribbons. Moreover, the edge-state energy gap is smaller than that in zigzag graphene nanoribbons, even absent in certain cases.

Highlights

We further investigate the fate of the quantum spin Hall (QSH)-like effect in the armchair GNR in the presence of intrinsic spin-orbit coupling (SOC), Rashba SOC, strains and an exchange field
We studied the edge-state properties of armchair GNRs with intrinsic SOC, Rashba SOC, exchange fields (EEF) and strains
Within the continuum Dirac model, we predicted that the armchair GNR, when strained along the armchair direction, can support edge states even for very small intrinsic SOC

Summary

System Hamiltonian

We first consider a graphene honeycomb lattice in the x − y plane in the presence of uniaxial strains with homogeneous Rashba SOC, intrinsic SOC, and an effective exchange fields (EEF) interaction. We assume that electrons in the bulk of the graphene sheet are described by the real space π-orbital tight-binding Hamiltonian with nearest-neighbor hopping:. The last term represents the Rashba SOC with coupling strength tR. For uniaxial tension along the armchair direction, the two Dirac points move to the new positions. The system can of tension, the Dirac points become gapped at a critical tensile strain ε > 0.2342, which means a phase transition is triggered here. Unless many-body effects are of crucial importance, in the vicinity of Dirac points K or K′, the low-energy electronic properties of a monolayer graphene are well described by the Dirac-type equation.

SOC are given by νF

ΨA ΨB ΨB ΨA

Chern number can be calculated from

Summary

Additional Information