Spin Polarized and Valley Helical Edge Modes in Graphene Nanoribbons

Inspired by recent progress in fabricating precisely zigzag-edged graphene nanoribbons and the observation of edge magnetism, we find that spin polarized edge modes with well-defined valley index can exist in a bulk energy gap opened by a staggered sublattice potential such as that provided by a hexagonal Boron-Nitride substrate. Our result is obtained by both tight-binding model and first principles calculations. These edge modes are helical with respect to the valley degree of freedom, and are robust against scattering, as long as the disorder potential is smooth over atomic scale, resulting from the protection of the large momentum separation of the valleys.