Chiral Bloch states in single layer graphene with Rashba spin-orbit coupling: Spectrum and spin current density

We study the Bloch spectrum and spin physics of 2D massless Dirac electrons in single layer graphene subject to a one dimensional periodic Kronig-Penney potential and Rashba spin-orbit coupling. The Klein paradox exposes novel features in the band dispersion and in graphene spintronics. In particular it is shown that: (1) The Bloch energy dispersion $\veps(p)$ has unusual structure: There are {\it two Dirac points} at Bloch momenta $\pm p \ne 0$ and a narrow band emerges between the wide valence and conduction bands. (2) The charge current and the spin density vector vanish. (3) Yet, all the non-diagonal elements of the spin current density tensor are finite and their magnitude increases linearly with the spin-orbit strength. In particular, there is a spin density current whose polarization is perpendicular to the graphene plane. (4) The spin density currents are space-dependent, hence their continuity equation includes a finite spin torque density.