Abstract
We study the possible valley supercurrent flowing in a valley Josephson junction based on the O-shaped Kekul\'e graphene superlattice (GS), in which the valley coupling term can make the valley degree of freedom condensate and open an energy gap around the Dirac point. It is found that the GS/G/GS formally resembles a conventional Josephson junction and a nonzero valley supercurrent flows through the junction when the arguments of the complex valley coupling terms in each GS are different. Unlike the conventional Josephson current, a local static potential in the middle G region via gate voltage can manipulate the valley supercurrent direction to form the 0-$\ensuremath{\pi}$ state transition and, moreover, the valley supercurrent magnitude as well as the current-phase relation can also be modulated by shifting the universal chemical potential of the system. Our findings provide a way to producing the pure valley current in the graphene system.
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