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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.