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We theoretically study self-consistent proximity effects in finite-sizedsystems consisting of ferromagnet ($\rm F$) layers coupled to an $s$-wavesuperconductor ($\rm S$). We consider both $\rm SF_1F_2$ and $\rm SH$nanostructures, where the $\rm F_1 F_2$ bilayers are uniformly magnetized, andthe ferromagnetic $\rm H$ layer possesses a helical magnetization profile. Wefind that when the $\rm F_1 F_2$ layers are weakly ferromagnetic, a hard gapcan emerge when the relative magnetization directions are rotated from parallelto 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 exchangefield can enhance the induced hard gap. Our investigations reveal that theorigin of these findings can be correlated with the propagation ofquasiparticles with wavevectors directed along the interface. To furtherclarify the source of the induced energy gap, we also examine the spatial andenergy resolved density of states, as well as the spin-singlet, andspin-triplet superconducting correlations, using experimentally accessibleparameter values. Our findings can be beneficial for designing magnetic hybridstructures where a tunable superconducting hard gap is needed.

Induced Energy Gap in Finite-Sized Superconductor/Ferromagnet Hybrids