Spin-polarized and valley helical edge modes in graphene nanoribbons

We investigate the electronic properties of zigzag-terminated graphene nanoribbons in the presence of a staggered sublattice potential. We show that due to the edge ferromagnetism, spin-polarized dispersive edge modes with well-defined valley indices can appear inside the bulk band gap opened by the inversion symmetry breaking. These edge modes are helical with respect to their valley indices, hence are robust against scattering from smooth disorder potentials. We further propose a concrete system with a zigzag graphene nanoribbon grown on top of a hexagonal boron-nitride substrate to realize such edge modes. These edge states could be utilized as perfect spin filters or analyzers in spintronics applications.