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Spin Hall effect in AA-stacked bilayer graphene

Intrinsic spin Hall effect in the AA-stacked bilayer graphene is studied theoretically. The low-energy electronic spectrum for states in the vicinity of the Dirac points is obtained from the corresponding $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian. The spin Hall conductivity in the linear response regime is determined within the Green function formalism. Conditions for the existence of spin Hall insulator phase are also analyzed, and it is shown that the spin Hall insulator phase can exist for a sufficiently large spin-orbit coupling, which opens a gap in the spectrum. The electric field perpendicular to the graphene plane leads then to reduction of the gap width and suppression of the spin Hall insulator phase. The low temperature spin Nernst effect is also calculated from the zero temperature spin Hall conductivity.