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Anomalous Josephson current through a ferromagnetic trilayer junction

We studied the anomalous Josephson current appearing at zero phase difference in junctions coupled with a ferromagnetic trilayer which has noncoplanar magnetizations. A $π/2$ junction with an equilibrium phase difference $π/2$ is obtained under suitable conditions. The equilibrium phase difference and the amplitude of the supercurrent are all tunable by the structure parameters. In addition to calculating the anomalous current using the Bogoliubov-de Gennes equation, we also developed a clear physical picture explaining the anomalous Josephson effect in the structure. We show that the triplet proximity correlation and the phase shift in the anomalous current-phase relation all stem from the spin precession in the first and third ferromagnet layers.