Modelling of triplet proximity effects in conically magnetized NbN/Ho/NbN Josephson junctions

This study presents a theoretical analysis of the proximity effect in superconductor/ferromagnet (S/F) hybrid structures, specifically focusing on NbN/Ho/NbN and NbN/F1/Ho/F2/NbN multilayers. We use self-consistent solutions of the Usadel equations to investigate the energy-resolved density of states (DOS) and superconducting critical temperature ( $T_c$ ) as functions of ferromagnetic layer thickness, misorientation angles, interfacial roughness, and phase differences between the superconducting NbN layers. The thickness of the ferromagnetic Ho layer is shown to significantly influence superconducting properties, including a reduction in the superconducting gap due to the inverse proximity effect. For thin Ho layers, both DOS and $T_c$ are slightly suppressed, while intermediate thicknesses exhibit non-monotonic behavior, corresponding to complex superconducting-ferromagnetic interactions, and thicker Ho layers lead to diminished superconductivity. The interplay between the magnetization inhomogeneity in Ho and the phase coherence of the superconducting order parameter is emphasized, revealing the appearance of zero-energy peaks (ZEPs) in the DOS. These unique features, such as controllable singlet and triplet states, are sensitive to the phase difference between the NbN electrodes. Additionally, triplet generation is also sensitive to the misalignment angle between the F1, F2, and Ho layers. The phase difference modulates DOS and $T_c$ , with certain alignments favoring triplet pairing, enabling external control over triplet generation similar to a spin valve device. Furthermore, the effect of interfacial roughness is systematically analyzed, with its implications for experimental realizations providing practical insights into optimizing S/F systems for future applications in superconducting spintronics. Our findings offer a comprehensive understanding of the proximity effect in S/F hybrid structures and its potential for tailored superconducting spintronic applications.