Supercurrent rectification effect in graphene-based Josephson junctions

We report a theoretic study of the supercurrent rectification effect in the graphene-based superconductor (GS) junction without invoking any spin-related interaction. By introducing the valley coupling effect through the Kekul\'e lattice distortion as well as the valley polarization interaction, we demonstrate that the GS/G/GS Josephson junction (JJ) can exhibit not only a $\ensuremath{\pi}$-state JJ but also a ${\ensuremath{\phi}}_{0}$-state one, in which a nonzero supercurrent can flow without the superconducting phase difference between the two GS electrodes. A supercurrent rectification effect, that the critical supercurrent is asymmetric between the two opposite flowing directions, is identified. The conditions for the optimal supercurrent rectification effect are also discussed.