  We consider intrinsic contributions to the spin Hall and spin Nernst effects
in a bilayer graphene. The relevant electronic spectrum is obtained from the
tight binding Hamiltonian, which also includes the intrinsic spin-orbit
interaction. The corresponding spin Hall and spin Nernst conductivities are
compared with those obtained from effective Hamiltonians appropriate for states
in the vicinity of the Fermi level of a neutral bilayer graphene. Both
conductivities are determined within the linear response theory and Green
function formalism. The influence of an external voltage between the two atomic
sheets is also included. We found transition from the topological spin Hall
insulator phase at low voltages to conventional insulator phase at larger
voltages.
