  We present an elaborate and systematic study of the conductance properties of
a zigzag bilayer graphene nanoribbon modeled by a Kane-Mele (KM) Hamiltonian.
The interplay of the Rashba and the intrinsic spin-orbit couplings with the
edge states, electronic band structures, charge and spin transport are explored
in details. We have analytically derived the conditions for the edge states for
a bilayer KM nanoribbon and show how these modes decay for lattice sites inside
the bulk. It is particularly interesting to note that for a finite-size ribbon
an even number of zigzag ribbon hosts a finite energy gap at the Dirac points,
while the odd ones do not. This asymmetry is present both in presence and
absence of a bias voltage that may exist between the layers. The interlayer
Rashba spin-orbit coupling, along with the intralayer intrinsic spin-orbit and
intralayer Rashba spin-orbit couplings seem to destroy the quantum spin Hall
(QSH) phase where the QSH phase is identified by the presence of a conductance
plateau (of magnitude 4e/h) in the vicinity of zero Fermi energy. The plateau
is sensitive to the values of the spin-orbit coupling parameters. Further, the
spin polarized conductance data reveal that a bilayer KM ribbon is found to be
more efficient for spintronic applications compared to a monolayer graphene.
Finally, a quick check with experiments is done via computing the effective
mass of electrons.
