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We consider a Josephson junction with ${\rm F_1 F_2 F_3}$ ferromagnetictrilayers in the ballistic regime, where the magnetization in each ferromagnet${\rm F}_i (i=1,2,3)$, can have arbitrary orientations and magnetizationstrengths. The trilayers are sandwiched between two $s$-wave superconductorswith a macroscopic phase difference $\Delta \varphi$. A broad range ofmagnetization strengths of the central $\rm F_2$ layer are considered, from anunpolarized normal metal (N) to a half-metallic phase, supporting only one spinspecies. Our results reveal that when the magnetization configuration in ${\rmF_1 F_2 F_3}$ has three orthogonal components, a supercurrent can flow at$\Delta \varphi=0$, and a strong second harmonic in the current-phase relationappears. Upon increasing the magnetization strength in the central ferromagnetlayer up to the half-metallic limit, the self-biased current and secondharmonic component become dramatically enhanced, and the critical supercurrentreaches its maximum value. The higher harmonics in the current-phase relationscan be controlled by the relative magnetization orientations, with negligiblecurrent damping compared to the corresponding ${\rm F_1 N F_3}$ counterparts.For a broad range of exchange field strengths in the central ferromagnet ${\rmF}_2$, the ground state of the system can be tuned to an arbitrary phasedifference $\varphi_0$ by rotating the magnetization in the outer ferromagnet$\rm F_3$. For intermediate exchange field strengths in ${\rm F}_2$, a$\varphi_0$ state can arise that creates a superconducting diode effect,whereby $\Delta\varphi$ can be tuned to create a one-way dissipationlesscurrent flow. The density of states demonstrates the emergence of zero energypeaks for the mutually orthogonal magnetization configurations, which isstrongest in the half-metallic phase.

Supercurrent Diode Effect, Spin Torques, and Robust Zero-Energy Peak in Planar Half-Metallic Trilayers