  By computing spin-polarized electronic transport across a finite zigzag
graphene ribbon bridging two metallic graphene electrodes, we demonstrate, as a
proof of principle, that devices featuring 100% magnetoresistance can be built
entirely out of carbon. In the ground state a short zig-zag ribbon is an
antiferromagnetic insulator which, when connecting two metallic electrodes,
acts as a tunnel barrier that suppresses the conductance. Application of a
magnetic field turns the ribbon ferromagnetic and conducting, increasing
dramatically the current between electrodes. We predict large magnetoresistance
in this system at liquid nitrogen temperature and 10 Tesla or at liquid helium
temperature and 300 Gauss.
