Graphene with Structure-Induced Spin-Orbit Coupling: Spin-Polarized States, Spin Zero Modes, and Quantum Hall Effect

Spin splitting of the energy spectrum of single-layer graphene on Au/Ni(111) substrate has been recently reported. I show that eigenstates of spin-orbit coupled graphene are polarized in-plane and perpendicular to electron momentum $\mathbf{k}$; the magnitude of spin polarization $\mathbf{S}$ vanishes when $k\to 0$. In a perpendicular magnetic field $\mathbf{B}$, $\mathbf{S}$ is parallel to $\mathbf{B}$, and two zero modes emerge in the Landau level spectrum. Singular $\mathbf{B}$-dependence of their magnetization suggests existence of a novel magnetic instability. They also manifest themselves in a new unconventional quantum Hall effect.