Non-local effect of a varying in space Zeeman field on the supercurrent and helix state in a spin-orbit-coupled s-wave superconductor

TL;DR

This paper investigates how spatial variations in a Zeeman field influence supercurrents and helix states in spin-orbit-coupled s-wave superconductors, revealing long-range effects and potential for coupling magnetic and superconducting circuits.

Contribution

It introduces a theoretical framework showing that inhomogeneous Zeeman fields cause significant long-range effects on supercurrents and helix states in dirty superconductors.

Findings

Weak inhomogeneity induces strong long-range effects.

Inhomogeneity can switch between helix and uniform states.

Supercurrents can be induced far from magnetic islands.

Abstract

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 $\exp(i\mathbf{Qr})$. 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.