We consider the spin-orbit-induced spin Hall effect and spin swapping in diffusive superconductors. By employing the nonequilibrium Keldysh Green's function technique in the quasiclassical approximation, we derive coupled transport equations for the spectral spin and particle distributions and for the energy density in the elastic scattering regime. We compute four contributions to the spin Hall conductivity, namely, skew scattering, side jump, anomalous velocity, and the Yafet contribution. The reduced density of states in the superconductor causes a renormalization of the spin Hall angle. We demonstrate that all four of these contributions to the spin Hall conductivity are renormalized in the same way in the superconducting state. In its simplest manifestation, spin swapping transforms a primary spin current into a secondary spin current with swapped current and polarization directions. We find that the spin-swapping coefficient is not explicitly but only implicitly affected by the superconducting gap through the renormalized diffusion coefficients. We discuss experimental consequences for measurements of the (inverse) spin Hall effect and spin swapping in four-terminal geometries. In our geometry, below the superconducting transition temperature, the spin-swapping signal is increased an order of magnitude while changes in the (inverse) spin Hall signal are moderate.

Conventional s-wave superconductors repel external magnetic flux. However, a recent experiment [A. Di Bernardo et al., Phys. Rev. X 5, 041021 (2015)] has tailored the electromagnetic response of superconducting correlations via adjacent magnetic materials. We consider another route to alter the Meissner effect where spinorbit interactions induce an anisotropic Meissner response that changes sign depending on the field orientation. The tunable electromagnetic response opens new paths in the utilization of hybrid systems comprised of magnets and superconductors.Introduction. The Meissner effect in superconductors is the expulsion of magnetic fields and it is one of its two defining properties, the other being the absence of electrical resistance. Experiments have shown [1] that a non-superconducting material can also exhibit a Meissner response when it is in proximity to a superconductor. Via the proximity effect, superconducting correlations leak into the neighbouring metal. Intuitively, one might expect stronger superconducting correlations to give rise to a stronger Meissner response of the normal metal. The modelling of such systems via quasiclassical theory largely confirms this picture [2], except the observation that the magnetic susceptibility has a puzzling re-entrant behavior as a function of temperature [3][4][5].When superconductors are placed in contact with ferromagnets, triplet Cooper pairs emerge that carry a net spin [6][7][8][9]. Such pairs are additionally characterized by an odd-frequency symmetry [10] which influences several physical properties, such as the electronic density of states and electromagnetic response. Very recently, an experiment [11] observed a paramagnetic Meissner effect in an Nb/Ho/Au structure. In this system, superconductivity enhanced the magnetic signal rather than expelling it. Such a finding is of a fundamental interest, since it questions the hallmark property of perfect diamagnetism in superconductors. From a practical point of view, a paramagnetic Meissner effect could lead to an integration of magnetic and superconducting materials in a way that has not been possible previously. Moreover, the recent demonstration of remotely induced magnetism via a superconductor reported in Ref. [12] suggests that the study of how superconductivity influences magnetic signals is particularly timely.Motivated by these experimental advances, we show in this Letter that by combining superconductors with spin-orbit coupled materials, the Meissner effect can be modulated by the orientation of an external magnetic field. Not only does the Meissner response of the system become anisotropic as a function of field orientation, but it can even change sign. This offers a way to control the electromagnetic response of a superconducting system in situ. In addition, we demonstrate that magnetic exchange fields h that are much smaller than the superconducting gap ∆ 0 , e.g. induced via the Zeeman-effect of an external field, can lead to a similar re-entrant behavior of the susceptibility as in ...

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