Tunable 0-$\pi$ Josephson devices with ferromagnetic insulator barrier: The role of spin-orbit interaction and lattice impurities.

Josephson $\pi-junctions$ ($\pi$-JJs) are currently subject to intense research due to their applicability in superconducting quantum circuits and spintronics. The possible integration of $\pi-JJs$ in quantum circuits for superconducting qubits is quite promising, paving the way to scalable self-biased devices with increased robustness against magnetic noise and a more compact design. Superconductor-ferromagnet-superconductor JJs (SFS JJs) are promising platforms to implement $\pi-JJs$, exhibiting temperature induced $0-\pi$ transitions. Much less is known when the ferromagnet is insulating and thus more suitable for quantum circuits, due to its low dissipation. We investigate the transport properties of ferromagnetic-insulator barrier junctions (SFIS JJs) with the aim of finding an effective way to control the $0-\pi$ transitions, through a direct action on the temperature behavior of the critical current ({I_c $(T)$), that may be used as a fingerprint of the junction. We study the role of impurities as well as of spin orbit in this task. $0-\pi$ transitions can be properly tuned, thus achieving stable $\pi-JJs$ up to the critical temperature, that may be possibly employed in superconducting quantum circuits.