Abstract
We investigate theoretically a Josephson junction made of two conventional superconductors and a sandwitched zigzag graphene nanoribbon in between. An anomalous atomic-scale 0-π state transition is found in the magnetized or nonmagnetic junction when the zigzag-chain number of the ribbon is odd and only the edge states are involved in transport. The mechanism for the π state is not the usual ferromagnetic exchange splitting that results in a nonzero momentum of the traveling Cooper pair, but the different pseudoparities of quasiparticles at different valleys, which gives rise to an inherent phase shift between the wave numbers of the hole-like and electron-like quasiparticles traveling in the ribbon. When the magnetization increases, the critical Josephson current is enhanced significantly, but the 0-π oscillating period remains nearly unchanged; moreover, the oscillation itself becomes much more regular and outstanding.