Dirac-Weyl electrons in a periodic spin-orbit potential

Graphene super-structures have been widely studied but the original structure of the SU(2) Hamiltonian was never modified. We study SU(2)xSU(2) super-structures arising from spatial modulation of spin-orbit interactions and derive an analytic band condition for a lattice vector along the direction of modulation of the spin-orbit interactions. The simple form of this band condition enables us to estimate the size of gaps due to avoided band crossings and gives insight into the dependence of the band structure on the width of the potential. We also investigate band structures for the case where the lattice momentum forms a finite angle with respect to the modulation direction of the spin-orbit interactions.