  In this study, based on the self-energy method and the total energy
calculation, the indirect exchange coupling between two semi-infinite
ferromagnetic strips (FM electrodes) separated by metallic graphene nanoribbons
(GNRs) is investigated. In order to form a FM/GNR/FM junction, a graphitic
region of finite length is coupled to the FM electrodes along graphitic zigzag
or armchair interfaces of width $N$. The numerical results show that, the
exchange coupling strength which can be obtained from the difference between
the total energies of electrons in the ferromagnetic and antiferromagnetic
couplings, has an oscillatory behavior, and depends on the Fermi energy and the
length of the central region.
