  We theoretically study a finite size $SF_1NF_2$ spin valve, where a normal metal ($N$) insert separates a thin standard ferromagnet ($F_1$) and a thick half-metallic ferromagnet ($F_2$). For sufficiently thin superconductor ($S$) widths close to the coherence length $\xi_0$, we find that changes to the relative magnetization orientations in the ferromagnets can result in substantial variations in the transition temperature $T_c$, consistent with experiment [Singh et al., Phys. Rev. X 5, 021019 (2015)]. Our results demonstrate that, in good agreement with the experiment, the variations are largest in the case where $F_2$ is in a half-metallic phase and thus supports only one spin direction. To pinpoint the origins of this strong spin-valve effect, both the equal-spin $f_1$ and opposite-spin $f_0$ triplet correlations are calculated using a self-consistent microscopic technique. We find that when the magnetization in $F_1$ is tilted slightly out-of-plane, the $f_1$ component can be the dominant triplet component in the superconductor. The coupling between the two ferromagnets is discussed in terms of the underlying spin currents present in the system. We go further and show that the zero energy peaks of the local density of states probed on the $S$ side of the valve can be another signature of the presence of superconducting triplet correlations. Our findings reveal that for sufficiently thin $S$ layers, the zero energy peak at the $S$ side can be larger than its counterpart in the $F_2$ side. 
Half-Metallic Superconducting Triplet Spin Valve