Josephson π state induced by valley polarization

We theoretically explore possible pi-state Josephson junctions made from graphene-like two-dimensional materials (G) with the honeycomb lattice structure. It is shown that the valley polarization in the G sheet could lead to a 0-pi state transition of the Josephson junction because of the valley-singlet Cooper pairs acquiring a nonzero momentum. When the valley-mixing scattering exists in the interfaces of the junction due to lattice mismatch, an odd-frequency valley-triplet supercurrent flows in the system even though the G sheet is fully valley polarized, and the supercurrent is characterized by a rapid atomic-scale oscillation with a periodicity of three lattice constants.