Anomalous Josephson current through a topological noncoplanar ferromagnetic trilayer

We present an anomalous Josephson effect in a TI-based Josephson junction with a ferromagnetic insulator (FI) trilayer which has noncoplanar magnetizations. It is shown that there exist equal fractional spin-singlet and -triplet correlations due to the magnetism-tuning chiral Dirac energy band structure. The consequent anomalous Josephson supercurrent is exhibited, in which a 0- or similar 0- state transition through phase shift is induced only by exchange field strength of the first FI region, while the supercurrent and the maximum one gradually drop with the increase of exchange field strengths of the second and third FI regions without relative state transitions. With the increase of the FI region length, the supercurrent and the maximum one are also both decreased, are found, which is different from the situation for increasing the exchange filed strength. In addition, the corresponding free energies are presented and discussed.