Josephson anomalous vortices

TL;DR

The paper introduces Josephson anomalous vortices (JAVs) in Josephson junctions where a ferromagnetic weak link with strong spin-orbit coupling supports circulating currents without an applied phase bias. Through symmetry analysis, it identifies a rotary invariant in the free energy that yields magnetoelectric coupling even when the Lifshitz invariant vanishes, and a microscopic Usadel-model calculation provides expressions for $d_k$ and $e_{jk}$ and demonstrates control of the vortex by gate-tuned Rashba SOC islands. The circulating currents are described by ${\bm J}_{\rm rot} = 2 e \zeta ( f_s \nabla \times f_z^* + f_s^* \nabla \times f_z )$ and are accompanied by textures in the triplet components ${\bm f}_t$. The work suggests detection by scanning magnetometry and gate-based manipulation of Rashba islands to realize and control the JAVs, advancing superconducting spintronics and topological vortex matter.

Abstract

We show that vortices with circulating current, related with odd-frequency triplet pairing, appear in Josephson junctions where the barrier is a weak ferromagnet with strong spin-orbit coupling. By symmetry analysis we show that there is an additional term - a rotary invariant - in the superconducting free energy which allows for magnetoelectric effects even when the previously considered Lifshitz invariant vanishes. Using a microscopic model based on a modified Usadel equation incorporating those effects, we show that the size, shape, and position of these vortices can be controlled by manipulating Rashba spin-orbit coupling in the weak link, via gates, and we suggest that these vortices could be detected via scanning magnetometry techniques. We also show that the transverse triplet components of the pairing amplitudes can form a texture.

Figures (4)

• Figure 1: (a) Sketch of a superconductor-ferromagnetic metal-superconductor device studied here. There is strong spin-orbit coupling in the ferromagnetic layer. (b) Circulating current density in the weak link of (a) due to the rotary invariant, computed numerically using the microscopic model. Here no current is applied across the junction. Parameters: $(L_x, L_y, J, \alpha) = (\xi_0, \xi_0, \Delta_0, 2.35/\xi_0)$.
• Figure 2: (a) Texture formed from the anomalous Green's functions $f_x, f_y$, reminiscent of a magnetic texture. (b) Phase winding of the anomalous Green's functions $f_x + i f_y$, reminiscent of an Abrikosov vortex. (c) Phase texture of the superconducting correlation $f_z$. (d) Representative antivortex pattern in the current density. (e) Representative mixture between a current antivortex and a vortex patterns. (f) Representative vortex. Parameters for all panels: $(L_x, L_y, J) = (\xi_0, \xi_0, \Delta_0)$; for (a-c): $\alpha = 2/\xi_0$; for (d): $\alpha = 1/\xi_0$; for (e): $\alpha = 1.55/\xi_0$; for (f): $\alpha = 2/\xi_0$;
• Figure 3: (a-d) Total Poincaré index $\mathcal{P}$ for various junction sizes, over a range of exchange energies $J$ and spin-orbit coupling strengths $\alpha$. Antivortex patterns in the current correspond to $\mathcal{P} = -1$ and vortices to $\mathcal{P} = +1$, as indicated by the inset sketches. Junction size is indicated in the top right hand corner, $L_x \times L_y \xi_0^2$. (e-h) Magnetic field strength along the $\hat{z}$ direction at the centre of the junction, for the same parameter spaces. The Poincaré index and magnetic field strength are computed from the numerically calculated current densities.
• Figure 4: The size (a), position (b), and shape (c) of a JAV could be manipulated by defining an island with strong Rashba SOC $\alpha_{\rm island}$ via gates, within a weak link with weak SOC $\alpha_{\rm back}$. Inset: sketches of the gate defining $\alpha_{\rm island}$ (yellow patch) within the weak link (blue square). Parameters: $(L_x, L_y, J, \alpha_{\rm island}, \alpha_{\rm back})$ = $(5\xi_0, 5\xi_0, \Delta_0, 2/\xi_0, 1/(2\xi_0))$.