Singlet-Triplet Conversion and the Long-Range Proximity Effect in Superconductor-Ferromagnet Structures with Generic Spin Dependent Fields

The long-range proximity effect in superconductor-ferromagnet (S/F) hybrid nanostructures is observed if singlet Cooper pairs from the superconductor are converted into triplet pairs which can diffuse into the ferromagnet over large distances. It is commonly believed that this happens only in the presence of magnetic inhomogeneities. We show that there are other sources of the long-range triplet component (LRTC) of the condensate and establish general conditions for their occurrence. As a prototypical example, we consider first a system where the exchange field and spin-orbit coupling can be treated as time and space components of an effective SU(2) potential. We derive a SU(2) covariant diffusive equation for the condensate and demonstrate that an effective SU(2) electric field is responsible for the long-range proximity effect. Finally, we extend our analysis to a generic ferromagnet and establish a universal condition for the LRTC. Our results open a new avenue in the search for such correlations in $S/F$ structures and make a hitherto unknown connection between the LRTC and Yang-Mills electrostatics.

Figure 1

Different geometers considered in the text. (a) The transversal $S/F/S$ junction. (b) The lateral $S/F$ structure and space dependence of $F=\sum _{\omega }|f{|}^2$ for the singlet and LRTC normalized with the asymptotic value at $x=-\infty$. Here, ${\xi }_0=\sqrt{D/\Delta }$, $h=10\Delta$, $T=0.05\Delta$, and $\Delta$ is the order parameter in the superconductor. (c) The lateral Josephson junction.