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

We consider trilayer ${\mathrm{F}}_{1}{\mathrm{F}}_{2}{\mathrm{F}}_{3}$ Josephson junctions that are finite in two dimensions and have arbitrary magnetizations each ferromagnet ${\mathrm{F}}_{i}\phantom{\rule{0.28em}{0ex}}(i=1,2,3)$. The trilayers sandwiched between $s$-wave superconductors with a macroscopic phase difference $\mathrm{\ensuremath{\Delta}}\ensuremath{\varphi}$. Our results reveal when the three orthogonal components, supercurrent can flow at $\mathrm{\ensuremath{\Delta}}\ensuremath{\varphi}=0$. With our generalized theoretical numerical techniques, we study planar spatial profiles $\mathrm{\ensuremath{\Delta}}\ensuremath{\varphi}$ dependencies of charge supercurrents, spin torques, density states. Remarkably, upon increasing magnetization strength central layer up to half-metallic limit, self-biased current induced second harmonic component become dramatically enhanced while critical reaches its maximum value. Additionally, for broad range exchange-field strengths orientations, ground state system be tuned an ${\ensuremath{\varphi}}_{0}$. For intermediate middle ${\mathrm{F}}_{2}$, ${\ensuremath{\varphi}}_{0}$ arise creates superconducting diode effect, whereby create one-way dissipationless flow. currents effective magnetic moments long-ranged torque phase. Moreover, states unveils emergence zero-energy peaks mutually configurations. suggest this simple junction excellent candidate producing experimentally accessible signatures supercurrents effects.