Spin Hall and spin Nernst effects in graphene with intrinsic and Rashba spin-orbit interactions

Spin Hall and spin Nernst effects in graphene are studied based on Green's function formalism. We calculate intrinsic contributions to spin Hall and spin Nernst conductivities in Kane-Mele model with various structures. When both intrinsic and Rashba spin-orbit interactions are present, their interplay leads to some characteristics of the dependence of spin Hall and spin Nernst conductivities on the Fermi level. When Rashba spin-orbit interaction is smaller than intrinsic spin-orbit coupling, a weak kink in the conductance appears. The kink disappears and a divergence appears when the Rashba spin-orbit interaction enhances. When the Rashba spin-orbit interaction approaches and is stronger than intrinsic spin-orbit coupling, the divergence becomes more obvious.