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The two-dimensional nature of graphene Josephson junctions offers thepossibility of creating effective superconductor-ferromagnet-superconductorjunctions with tunable Zeeman splitting caused by an in-plane magnetic field.Such junctions would be able to alternate between a conventionalsuperconducting ground state and a ground state with an intrinsic phasedifference, making them controllable $0-\pi$ Josephson junctions. However, inaddition to the Zeeman splitting, an in-plane magnetic field will in generalalso produce an orbital effect because of height variations in graphene,colloquially known as ripples. Both the Zeeman and orbital effect will thusaffect the critical current, so to be able to identify $0-\pi$ transitions itis necessary to understand their combined effect. From both analytical andnumerical solutions of the Usadel equation we find that ripples can in factproduce a current response similar to that which is characteristic of a $0-\pi$transition. Hence, additional analysis is required in order to reveal thepresence of a $0-\pi$ transition caused by spin-splitting in graphene withripples. We provide a closed form analytical expression for the criticalcurrent in the presence of exchange field and ripple effects as well as anexpression for the scaling of critical current zeroes with junction parameters.

Combined Zeeman and orbital effect on the Josephson effect in rippled graphene