Spin-orbit coupling assisted magnetoanisotropic Josephson effect in ferromagnetic graphene Josephson junctions

The spin-flipping effect can be induced by the Rashba spin-orbit coupling (RSOC), leading to triplet equal-spin pairs in a superconducting hybrid structure. Herein, by combining the Dirac--Bogoliubov--de Gennes equation and the Furusaki-Tsukada formalism at a finite temperature, we theoretically investigate the Josephson effect in graphene-based superconductor-ferromagnet-R-superconductor junctions, where R refers to a region with the RSOC. It is demonstrated that as a result of the RSOC, one $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition can be attained by tuning the orientation of the exchange field $\stackrel{P\vec}{h}$, which is determined by its magnitude $h$ that could be periodically taken, and the periodical $0\text{\ensuremath{-}}\ensuremath{\pi}$ transitions are also caused by manipulating $h$ during a considerable scope of the orientation. More interestingly, although varying the RSOC strength $\ensuremath{\lambda}$ cannot give rise to the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition in itself, not only is it a necessary condition for the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition induced by modulating the orientation of $\stackrel{P\vec}{h}$ but also can produce the shift of the crossover point. Furthermore, two different kinds of anomalous Josephson current effect are exhibited by controlling the orientation of $\stackrel{P\vec}{h}$. Particularly, the out-of- and in-plane magnetoanisotropic Josephson currents always exist, varying monotonically with $\ensuremath{\lambda}$ while nonmonotonically with $h$. The characteristics may provide more insights into the proximity-induced RSOC and pave the way to a new class of tunable superconducting spintronic devices based on large-scale graphene.