Spin Hall and spin Nernst effects due to intrinsic spin-orbit coupling in monolayer and bilayer graphene.

We consider intrinsic contributions to the spin Hall and spin Nernst effects in monolayer and bilayer graphene. The spin Hall (Nernst) effect consists in the generation of transverse spin current by longitudinal electric field (temperature gradient). The relevant electronic spectrum for monolayer and bilayer graphene has been obtained from the corresponding effective Hamiltonians. Both spin Hall and spin Nernst conductivities have been determined within the linear response theory and Green function formalism. The influence of an external vertical voltage between the two atomic sheets in the case of a bilayer graphene is also analyzed and discussed. This voltage can generally lead to a phase transition between the topological insulator phase and conventional insulator. In the case of bilayer graphene, the main focuss is on an asymmetrical case, with different spin-orbit parameters in the two atomic sheets. Such a difference may be generated by different atomic planes adjacent to bilayer graphene on its both sides.