  The intrinsic spin-orbit coupling (ISOC) in bilayer graphene is investigated.
We find that the largest ISOC between $\pi$ electrons origins from the
following hopping processes: a $\pi$ electron hops to $\sigma$ orbits of the
other layer and further to $\pi$ orbits with the opposite spin through the
intra-atomic ISOC. The magnitude of this ISOC is about $0.46meV$, 100 times
larger than that of the monolayer graphene. The Hamiltonians including this
ISOC in both momentum and real spaces are derived. Due to this ISOC, the
low-energy states around the Dirac point exhibit a special spin polarization,
in which the electron spins are oppositely polarized in the upper and lower
layers. This spin polarization state is robust, protected by the time-reversal
symmetry. In addition, we provide a hybrid system to select a certain spin
polarization state by an electric manipulation.
