Nonlocal 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 spin-orbit coupling can induce the supercurrent in an s-wave two-dimensional superconductor. At the same time, the thermodynamic equilibrium state of such a system is characterized by the helix phase in which the order parameter varies in space as exp(iQ . r). 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 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, is 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 an alternative point of view on the interplay of the magnetoelectric effect and helix phase in spin-orbit coupled superconductors. It also suggests an interesting method that enables us to couple superconducting and magnetic circuits.