Vortex contribution to the critical temperature oscillations in hybrid superconductor-ferromagnet coaxial cylinders

Abstract

The s-wave triplet Usadel equations which include the odd frequency triplet condensate are employed to study the oscillatory behavior of the superconducting critical temperature Tc in multiply connected hybrid systems e.g., a ferromagnetic core surrounded by a superconducting shell. This configuration allows us to treat the angular momentum number L of the Cooper pair as the vorticity parameter. The spiral exchange field in the ferromagnet characterized by a spiral wave vector Q, which rotates in the z-plane and its direction varies within an in-plane of the ferromagnetic core. The induced superconductivity in the ferromagnet core is controlled by the electrical contact at the boundary for which we are interested in the switching of the superconducting states among the various vorticity numbers. The peculiar superconducting states with L≠0 bear resemblance to the pi-phase state in the bilayer systems. We demonstrate the switching phenomena by solving the Usadel equations together with the self-consistent order parameter subject to the Kupriyanov-Lukichev boundary conditions at the contact surface to obtain the secular equation in the multimode method to determine the oscillatory behavior of Tc in the ferromagnet core. The Abrikosov–Gorkov like-formula is also obtained within the single mode approximation.