From a ℤ2 topological insulator to a square ribbon: Tuning quantum spin Hall states and conductivity

Abstract

By introducing a new type of strained lattice, one dimensional arm-distorted ribbons, we demonstrate the possibility of opening a bandgap from the originally gapless graphene nanoribbons in the ℤ2 topological class. Typically, a gap opens up due to time-reversal/twofold rotational symmetry breaking; however, our approach leads to a bandgap at the edge of the Brillouin zone while preserving the above symmetries. The calculated gap opening is due to a properly scaled extra hopping interaction, compared to the Kane-Mele Hamiltonian where this hopping is omitted since it is a third neighbor interaction in graphene. For square ribbons with a variable number of legs, we discuss Rashba-related spin-dependent transport properties in the presence and absence of a magnetic flux. In such ribbons, opposite spins travel in opposite directions along the edges, while the spin current in the center leg turns out to be smaller by at least an order of magnitude. In addition, the spin difference between the left and right (say) edges shows plateaus as a function of the magnetic flux. We also discuss transport properties resulting from a non-spin-orbit coupled Hamiltonian which includes third neighbor hopping during a systematic transformation from honeycomb to square ribbons.