  We study in this paper the edge polarizations and their consequences for a
biased Bernal stacked bilayer graphene nanoribbon with zigzag termination. The
magnetic states are classified according to the interlayer and intralayer
couplings between the edge polarizations, and the magnetic phase diagram of
doping versus bias voltage is given. Coplanar magnetic phase is found and the
variation of its magnetic structure with the bias voltage is investigated. For
all the magnetic states, we also discuss the possibility of the
half-metallicity, and for a ribbon with perfect zigzag edges we predict seven
kinds of the half-metallic states, which are characterized by their distinct
magnetic structures and quantized electrical conductances along the ribbon.
