Anomalous supercurrent modulated by interfacial magnetizations in Josephson junctions with ferromagnetic bilayers

Based on the Bogoliubov-de Gennes equations, we investigate the transport of the Josephson current in a S/$f_L$-F$_1$-$f_C$-F$_2$-$f_R$/S junction, where S and F$_{1,2}$ are superconductors and ferromagnets, and $f_{L, C, R}$ are the left, central, and right spin-active interfaces. These interfaces have noncollinear magnetizations, and the azimuthal angles of the magnetizations at the $f_{L, C, R}$ interfaces are $\chi_{L, C, R}$. We demonstrate that, if both the ferromagnets have antiparallel magnetizations, the critical current oscillates as a function of the exchange field and the thickness of the ferromagnets for particular $\chi_L$ or $\chi_R$. By contrast, when the magnetization at the $f_C$ interface is perpendicular to that at the $f_L$ and $f_R$ interfaces, the critical current reaches a larger value and is hardly affected by the exchange field and the thickness. Interestingly, if both the ferromagnets are converted to antiparallel half-metals, the critical current maintains a constant value and rarely changes with the ferromagnetic thicknesses and the azimuthal angles. At this time, an anomalous supercurrent can appear in the system, in which case the Josephson current still exists even if the superconducting phase difference $\phi$ is zero. This supercurrent satisfies the current-phase relation $I=I_c\sin(\phi+\phi_0)$ with $I_c$ being the critical current and $\phi_0=2\chi_C-\chi_L-\chi_R$. We deduce that the additional phase $\phi_0$ arises from phase superposition, where the phase is captured by the spin-triplet pairs when they pass through each spin-active interface. In addition, when both the ferromagnets are transformed into parallel half-metals, the $f_C$ interface never contributes any phase to the supercurrent and $\phi_0=\chi_R-\chi_L+\pi$. In such a case, the current-phase relation is similar to that in a S/$f_L$-F-$f_R$/S junction.