Nucleation and Arrangement of Abrikosov Vortices in Hybrid Superconductor-Ferromagnetic Nanostructure

This study investigates the nucleation, dynamics, and stationary configurations of Abrikosov vortices in hybrid superconductor-ferromagnetic nanostructures exposed to inhomogeneous magnetic fields generated by a ferromagnetic nanodot. Using time-dependent Ginzburg-Landau simulations and Maxwell’s equations, we observe and provide an explanation for the evolution of curved vortex structures that undergo creep-like deformation while reaching a steady state. Spatial variations in the Lorentz force, along with the interaction between geometric constraints and vortex interactions, give rise to unusual stationary vortex configurations that gradually change with increasing field strength, a behavior not seen in homogeneous magnetic fields. These findings reveal complex pinning mechanisms, providing valuable insights for the optimization and further advancement of nanoscale superconducting systems.

💡 Research Summary

This paper presents a comprehensive numerical investigation of Abrikosov vortex nucleation, dynamics, and stationary configurations in a hybrid superconductor‑ferromagnet (SC‑FM) nanostructure subjected to the highly inhomogeneous stray field of a ferromagnetic nanodot. Using the time‑dependent Ginzburg‑Landau (TDGL) formalism coupled self‑consistently with Maxwell’s equations, the authors solve for the complex order parameter ψ and the vector potential A with a finite‑element method (FEM) implemented in COMSOL Multiphysics®. The superconducting element is modeled as a square‑cross‑section prism (side a, height h) with GL parameter κ = 3, penetration depth λ = 60 nm, and coherence length ξ = 20 nm (λ = κξ). The ferromagnetic nanodot has the same lateral dimensions, a height h_M = 700 nm, and a saturation magnetization M_s = 1.35 MA/m, producing a static out‑of‑plane stray field B_FM that decays with the separation d between the two components. By varying d from 10 nm to 50 nm the average stray field ⟨|B_FM|⟩ over the superconductor is tuned from a few hundred millitesla up to ≈ 0.5 T.

The simulations start from a uniform Meissner state (ψ = 1) and abruptly switch on the external field (either the inhomogeneous B_FM or a homogeneous reference field B_H). The system is then allowed to evolve until the time derivatives of ψ and A become negligible. Two quantitative observables are introduced: (i) the filling fraction f_fN, defined as the volume fraction where |ψ|² < 0.3, which measures the total volume occupied by vortex cores and normal‑phase indentations; and (ii) the average magnetization ⟨|M|⟩ = (1/μ₀V_SC)∫|∇×A − B_a| dV, which quantifies the overall diamagnetic response and demagnetizing effects.

The time evolution under the inhomogeneous field reveals four distinct stages, in contrast to three stages observed for the homogeneous reference case. Initially the whole prism is in the Meissner state. Magnetic flux first penetrates at the mid‑points of the bottom edges, creating low‑|ψ| “indentations” that expand laterally. This stage corresponds to a rapid rise of f_fN, reaching a maximum of ≈ 0.25 at t ≈ 5τ (τ = ξ²/D, the characteristic diffusion time). Subsequently, the vortex cores elongate along the local direction of B_FM, forming curved, filament‑like structures that creep‑like deform as the Lorentz force varies spatially. The curvature is a direct consequence of the gradient of the stray field; regions of stronger field exert larger forces, pulling vortex lines toward the nanodot, while weaker regions allow relaxation. By t ≈ 35τ the system settles into a mixed state where curved vortices coexist with pockets of normal phase. The final steady‑state configuration, visualized at t = 900τ, shows a complex three‑dimensional network of curved vortices that follow the magnetic field lines, rather than the straight, triangular lattice typical for uniform fields.

When the separation d is reduced, the average stray field increases, leading to higher f_fN values and a more pronounced reduction of ⟨|M|⟩. The vortex density grows, but more importantly the curvature and branching of vortex lines become more extreme, indicating that the inhomogeneous field creates a rich pinning landscape. The authors compare these results with three reference systems: (a) an infinite SC wire under a uniform field, (b) a finite SC prism under a uniform field, and (c) a SC sphere under a uniform field. Only the hybrid SC‑FM system exhibits the additional intermediate stage and the curved vortex morphology, underscoring the unique role of the stray field’s spatial variation.

The paper concludes that the interplay of geometry (finite size comparable to ξ and λ), the non‑uniform magnetic environment, and vortex‑vortex interactions yields novel pinning mechanisms that cannot be captured by conventional homogeneous‑field models. These mechanisms could be exploited to engineer desired vortex arrangements in nanoscale superconducting devices, potentially improving performance in superconducting spintronic elements, quantum bits, and magnetic‑field‑tunable superconducting circuits. The authors suggest future work involving experimental verification, exploration of dynamic (RF) stray fields, and coupling to more complex ferromagnetic textures such as skyrmions, which may further enrich the vortex landscape.