Supercurrent diode effect, spin torques, and robust zero-energy peak in planar half-metallic trilayers

We consider trilayer F₁F₂F₃ Josephson junctions that are finite in two dimensions and have arbitrary magnetizations in each ferromagnet F₈0. 28em{0ex} (i=1, 2, 3). The trilayers are sandwiched between two s-wave superconductors with a macroscopic phase difference. Our results reveal that when the magnetizations have three orthogonal components, a supercurrent can flow at =0. With our generalized theoretical and numerical techniques, we study the planar spatial profiles and dependencies of the charge supercurrents, spin supercurrents, spin torques, and density of states. Remarkably, upon increasing the magnetization strength in the central ferromagnet layer up to the half-metallic limit, the self-biased current and induced second harmonic component become dramatically enhanced while the critical supercurrent reaches its maximum value. Additionally, for a broad range of exchange-field strengths and orientations, the ground state of the system can be tuned to an arbitrary phase difference ₀. For intermediate exchange-field strengths in the middle layer F₂, a ₀ state can arise that creates a superconducting diode effect, whereby can be tuned to create a one-way dissipationless current flow. The spin currents and effective magnetic moments reveal a long-ranged spin torque in the half-metallic phase. Moreover, the density of states unveils the emergence of zero-energy peaks for the mutually orthogonal magnetization configurations. Our results suggest that this simple trilayer Josephson junction can be an excellent candidate for producing experimentally accessible signatures for long-ranged self-biased supercurrents and supercurrent diode effects.