Abstract
Using the time-dependent Ginzburg–Landau approach, we analyze vortex states and vortex dynamics in type-I superconductor films with a thickness gradient in one direction. In the thinnest part of the structures under consideration, the equilibrium states manifest the typical type-II vortex patterns with only singly-quantized vortices. At the same time, in the regions with larger thickness the singly-quantized and giant vortices coexist, in a qualitative agreement with our scanning Hall probe microscopy measurements on relatively thick Pb films. In the presence of an external current applied perpendicularly to the thickness gradient direction, the singly-quantized vortices, which enter the wedge through its thinnest edge, merge into giant vortices when propagating to the thicker parts of the structure. Remarkably, the results of our simulations imply that at moderate external current densities a regime is possible where the winding number of giant vortices, formed as a result of vortex coalescence, takes preferentially (or even exclusively) the values given by positive integer powers of two.
Highlights
Single-band superconductors are commonly subdivided into two types with qualitatively different vortex behaviour depending on the value of the Ginzburg–Landau parameter κ = λ ξ, where ξ is the coherence length and λ is the penetration depth of the superconducting material
Equilibrium vortex states When calculating the equilibrium vortex patterns, we start with a random distribution of the order parameter with ∣ψ ∣ ≪ 1 and alow the solution of equation (1) together with equations (5)–(7) to evolve towards a stable state in the presence of a constant applied magnetic field B0
A typical example of the resulting equilibrium distributions of the order parameter and the z component of the magnetic field in a type-I superconductor wedge is illustrated by figure 2
Summary
In type-II superconductors, with κ > 1 2 , purely repulsive vortex–vortex interaction leads to the formation of the famous Abrikosov lattice of singlyquantized vortices at magnetic fields above the first critical field [1]. In type-I superconductor slabs, characterized by κ < 1 2 , the vortex–vortex interaction, still long-range repulsive, becomes attractive at short intervortex distances. In the intermediate state of a type-I superconductor, vortices merge into macroscopic multiquanta flux domains of different shape, for instance bubble-like. As concerns singly-quantized vortices in a type-I superconductor, the possibility of their coexistence with multiquanta fluxoids (giant vortices) was theoretically predicted for mesoscopic samples [3]. It was experimentally demonstrated that singly-quantized vortices can be stabilized in the intermediate state of a relatively thick macroscopic type-I superconductor film [4]
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