  We investigate the effects of Rashba spin-orbit interactions on the
electronic band-structure and corresponding wave-functions of graphene. By
exactly solving a tight-binding model Hamiltonian we obtain the expected
splitting of the bands -due to the SU(2) spin symmetry breaking- that is
accompanied by the appearance of additional Dirac points. These points are
originated by valence-conduction band crossings. By introducing a convenient
gauge transformation we study a model for zigzag nanoribbons with RSO
interactions. We show that the RSO interactions lifts the quasi-degeneracy of
the edge band while introducing a state-dependent spin separation in real
space. Calculation of the average magnetization perpendicular to the ribbon
plane suggest that RSO could be used to produce spin-polarized currents.
Comparisons with the intrinsic spin-orbit (I-SO) interaction proposed to exist
in graphene are also presented.
