We consider dc supercurrents in SNS junctions. Spin-orbit coupling in combination with Zeeman fields can induce an effective vector potential in the normal conductor. As a consequence, an out-of-plane spin density varying along the transverse direction causes a longitudinal phase difference between the superconducting terminals. The resulting equilibrium phase-coherent supercurrent is analog to the nonequilibrium inverse spin Hall effect in normal conductors. We explicitly compute the effect for the Rashba spin-orbit coupling in a disordered two-dimensional electron gas with an inhomogeneous perpendicular Zeeman field.

Here, we study the flow of energy between coupled simulators in a co-simulation environment using the concept of power bonds. We introduce energy residuals which are a direct expression of the coupling errors and hence the accuracy of co-simulation results. We propose a novel EnergyConservation-based Co-Simulation method (ECCO) for adaptive macro step size control to improve accuracy and efficiency. In contrast to most other co-simulation algorithms, this method is noniterative and only requires knowledge of the current coupling data. Consequently, it allows for significant speed ups and the protection of sensitive information contained within simulator models. A quarter car model with linear and nonlinear damping serves as a co-simulation benchmark and verifies the capabilities of the energy residual concept: Reductions in the errors of up to 93 % are achieved at no additional computational cost.

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.

When simulators are energetically coupled in a co-simulation, residual energies alter the total energy of the full coupled system. This distorts the system dynamics, lowers the quality of the results, and can lead to instability. By using power bonds to realize simulator coupling, the Energy-Conservation-based Co-Simulation method (ECCO) [Sadjina et al. 2016] exploits these concepts to define non-iterative global error estimation and adaptive step size control relying on coupling variable data alone. Following similar argumentation, the Nearly Energy Preserving Coupling Element (NEPCE) [Benedikt et al. 2013] uses corrections to the simulator inputs to approximately ensure energy conservation. Here, we discuss a modification to NEPCE for when direct feed-through is present in one of the coupled simulators. We further demonstrate how accuracy and efficiency in non-iterative co-simulations are substantially enhanced when combining NEPCE with ECCO's adaptive step size controller. A quarter car model with linear and nonlinear damping characteristics serves as a co-simulation benchmark, and we observe reductions of the coupling errors of up to 98 % utilizing the concepts discussed here.