Current phase relation in a planar graphene Josephson junction with spin–orbit coupling

We study a graphene Josephson junction where the inner graphene layer is subjected to spin–orbit coupling by proximity effect. This could be achieved, for example, by growing the graphene layer on top of a transition-metal dichalcogenide, such as WS2. Here, we focus on the ballistic, wide, and short junction limits and study the effects of the spin–orbit interaction on the supercurrent. In particular, we analyze the current-phase relation using an analytical approach based on the continuum model. We find combinations of types of spin–orbit coupling that significantly suppress the supercurrent by opening a gap in the graphene band structure. At the same time, other combinations enhance it, acting as an effective spin–valley resolved chemical potential. Moreover, we find that a strong Rashba spin–orbit coupling leads to a junction with a highly voltage tunable harmonic content.