Subgap modes in two-dimensional magnetic Josephson junctions

We consider two-dimensional superconductor/ferromagnet/superconductor junctions and investigate the subgap modes along the junction interface. The subgap modes exhibit characteristics similar to the Yu-Shiba-Rusinov states that originate form the interplay between superconductivity and ferromagnetism in the magnetic junction. The dispersion relation of the subgap modes shows qualitatively different profiles depending on the transport state (metallic, half-metallic, or insulating) of the ferromagnet. As the spin splitting in the ferromagnet is increased, the subgap modes bring about a $0$-$\pi$ transition in the Josephson current across the junction, with the Josephson current density depending strongly on the momentum along the junction interface (i.e., the direction of the incident current). For clean superconductor-ferromagnet interfaces (i.e., strong coupling between superconductors and ferromagnet), the subgap modes develop flat quasi-particle bands that allow to engineer the wave functions of the subgap modes along an inhomogeneous magnetic junction.