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Non-local effect of a varying in space Zeeman field on supercurrent and helix state in a spin-orbit-coupled s-wave superconductor
A weak parallel Zeeman field combined with the spin-orbit coupling can induce the supercurrent in
an s-wave two-dimensional superconductor. At the same time, the thermodynamically equilibrium
state of such a system is characterized by the helix phase where the order parameter varies in space
as exp(iQr). In this state the electric current that is induced by the Zeeman interaction is exactly
counterbalanced by the current produced by the gradient of the order-parameter. We studied the
interplay of the helix state and magnetoelectric current in the case of a varying in space Zeeman field,
as it might be realized in hybrid heterostructures with magnetic and superconducting layers. The
theoretical analysis was based on Usadel equations for Green functions in a dirty superconductor. It
is shown that even a weak inhomogeneity produces a strong long-range effect on the magnetoelectric
current and the order-parameter phase. Consequently, depending on the macroscopic shape of such
an inhomogeneity, either the helix state with the zero supercurrent, or a locally uniform state with
the finite supercurrent are realized. A mixture of these two extreme situations is also possible.
It is also shown that the current can be induced at a large distance from a ferromagnetic island
embedded into a superconductor. Quantum effects associated with the magnetoelectric effect are
briefly discussed for multiply connected systems. The theory proposes a new point of view on
interplay of the magnetoelectric effect and helix phase in spin-orbit coupled superconductors. It
also suggests an interesting method allowing to couple superconducting and magnetic circuits.