Spin Controlled Coexistence of 0 and {\pi} States in SFSFS Josephson Junctions

Using the Keldysh-Usadel formalism, we theoretically study the $0$-$\pi$ transition profiles and current-phase relations of magnetic $SFSFS$ and $SFSFFS$ Josephson nanojunctions in the diffusive regime. By allowing the magnetizations of the ferromagnetic layers to take arbitrary orientations, the strength and direction of the charge supercurrent flowing through the ferromagnetic regions can be controlled via the magnetization rotation in one of the ferromagnetic layers. Depending on the junction parameters, we find opposite current flow in the ferromagnetic layers, revealing that remarkably such configurations possess well-controlled $0$- and $\pi$-states simultaneously, creating a three-terminal $0$-$\pi$ spin switch. We demonstrate that the spin-controlled $0$-$\pi$ profiles trace back to the proximity induced odd-frequency superconducting correlations generated by the ferromagnetic layers. It is also shown that the spin-switching effect can be more pronounced in $SFSFFS$ structures. The current-phase relations reveal the important role of the middle $S$ electrode, where the spin controlled supercurrent depends crucially on its thickness and phase differences with the outer $S$ terminals.