Quantized resonant tunneling effect in Josephson junctions with ferromagnetic bilayers

We study the Josephson effect in one-dimensional SF$_1$F$_2$S junctions, which consist of conventional s-wave superconductors (S) connected by two ferromagnetic layers (F$_1$ and F$_2$). At low temperatures, the potential barrier at the F$_1$/F$_2$ interface can induce a quantized resonant tunneling effect. This effect not only modifies the amplitude of the critical current but also affects the phase of the Josephson current. As the exchange fields ($h_1$, $h_2$) and thicknesses ($d_1$, $d_2$) of the F$_1$ and F$_2$ layers vary, the critical current displays periodic resonance peaks. These peaks occur under the quantization conditions $Q_{1(2)} d_{1(2)} = \left(n_{1(2)} + 1/2\right) \pi$, where $Q_{1(2)} = 2h_{1(2)}/(\hbar v_F)$ is the center-of-mass momentum carried by Cooper pairs, with $v_F$ being the Fermi velocity, and $n_{1(2)} = 0, 1, 2, \cdots$. It can be inferred that the potential barrier suppresses the transport of spin-singlet pairs while allowing spin-triplet pairs with zero spin projection along the magnetization axis to pass through. As these spin-triplet pairs traverse the F$_1$ and F$_2$ layers, the total phase they acquire determines the ground state of the Josephson junction. At the resonance peaks, the Josephson current primarily arises from the first harmonic in both the parallel and antiparallel magnetization configurations. However, in perpendicular configurations, the second harmonic becomes more significant. In scenarios where both ferromagnetic layers have identical exchange fields and thicknesses, the potential barrier selectively suppresses the current in the 0-state while allowing it to persist in the $\pi$-state for parallel configurations. Conversely, in antiparallel configurations, the current in the 0-state is consistently preserved.