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 $\bf k$; the magnitude of spin polarization $\bf S$ vanishes when $k \to 0$. In a perpendicular magnetic field $\bf B$, $\bf S$ is parallel to $\bf B$, and two zero modes emerge in the Landau level spectrum. Singular $\bf B$-dependence of their magnetization suggests existence of a novel magnetic instability. They also manifest themselves in a new unconventional quantum Hall effect.