  Recent experiments indicate that the Bernal stacked graphene multilayer (BGM)
have an interaction induced gapped (or pseudo gapped) ground state. Here, we
propose that, due to the electron correlation, the BGM can be induced into a
half metallic phase by applying a vertical electric field and doping. The half
metallic states in even-layer and odd-layer BGMs have totally different
behaviors, due to their different band structures. We systematically calculate
the graphene tetralayer (4L-BGM) and trilayer (3L-BGM) as the typical examples
of the even-layer and odd-layer BGMs, respectively. In 4L-BGM, we find an
interesting phenomenon of electric field induced inversion of the
spin-polarized bands. Namely, in the half metallic phase, the spin polarization
of the conducting channel and the net magnetic moment are inversed when the
applied electric field exceeds a critical value. In 3L-BGM, a remarkable
feature is that the inequivalence of the the two sublattices will intrinsically
break the degeneracy of the the spin-up and spin-down bands even in the zero
electric field case. Our results suggest that 4L-BGM should be an ideal
platform to detect the proposed half metallic phase in BGM systems.
