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

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.