  We theoretically study self-consistent proximity effects in finite-sized
systems consisting of ferromagnet ($\rm F$) layers coupled to an $s$-wave
superconductor ($\rm S$). We consider both $\rm SF_1F_2$ and $\rm SH$
nanostructures, where the $\rm F_1 F_2$ bilayers are uniformly magnetized, and
the ferromagnetic $\rm H$ layer possesses a helical magnetization profile. We
find that when the $\rm F_1 F_2$ layers are weakly ferromagnetic, a hard gap
can emerge when the relative magnetization directions are rotated from parallel
to antiparallel. Moreover, the gap is most prominent when the thicknesses of
$\rm F_1$ and $\rm F_2$ satisfy $\rm d_{F1}\leq d_{F2}$, respectively. For the
$\rm SH$ configuration, increasing the spatial rotation period of the exchange
field can enhance the induced hard gap. Our investigations reveal that the
origin of these findings can be correlated with the propagation of
quasiparticles with wavevectors directed along the interface. To further
clarify the source of the induced energy gap, we also examine the spatial and
energy resolved density of states, as well as the spin-singlet, and
spin-triplet superconducting correlations, using experimentally accessible
parameter values. Our findings can be beneficial for designing magnetic hybrid
structures where a tunable superconducting hard gap is needed.

Superconductivity