We study the transport properties of a diffusive Josephson junction between two spin-split superconductors (SCs) made of SC-ferromagnetic insulator bilayers on top of a three-dimensional topological insulator. We derive the corresponding Usadel equation describing the quasiclassical Green's functions in these systems and first solve the equation analytically in the weak-proximity case. We demonstrate the appearance of an anomalous phase in the absence of an external magnetic field. We also explore nonreciprocal electronic transport. Specifically, we calculate the diode efficiency of the junction $\eta$ by solving the Usadel equation. We obtain a sizable diode effect even at zero applied magnetic field. We discuss how the diode efficiency $\eta$ depends on the different parameters and find a nonmonotonic behavior of $\eta$ with temperature.

• Received 4 October 2022
• Accepted 28 November 2022
• Corrected 23 December 2022

DOI:https://doi.org/10.1103/PhysRevB.106.214504