Vortices in a wedge made of a type-I superconductor

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.


Introduction
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. In type-II superconductors, with 1 2 κ > , 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 1 2 κ < , the vortex-vortex interaction, still long-range repulsive, becomes attractive at short intervortex distances. As a result, in the intermediate state of a type-I superconductor, vortices merge into macroscopic multiquanta flux domains of different shape, for instance bubble-like. The size and magnetic flux of those bubbles are shown to increase with increasing the superconductor slab thickness [2]. 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]. More recently, 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].
The vortex-vortex interactions in type-I superconductor films are known to strongly depend on the film thickness d [5][6][7]: at d λ < these interactions are determined by an effective penetration depth d 2 Λ λ ∼ rather than by the bulk value λ. Therefore, in sufficiently thin films of type-I superconductors (and also thin wires-see e.g. [8]), where the effective Ginzburg-Landau parameter Λ ξ exceeds 1 2 , vortex behaviour is type-II-like, so that only singly-quantized vortices are stable there. In the present paper, we consider a type-I superconductor with a uniform thickness gradient (superconducting wedge). Our goal is to analyze the coexistence of types-I and II vortex behaviours within a single sample as well as transformations of vortices moving from the effectively Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. type-II region to the type-I region of the wedge. The vortex states and vortex dynamics are numerically calculated using the time-dependent Ginzburg-Landau (TDGL) formalism for superconducting films with variable thickness [12].
The remaining part of the paper is organized as follows. In section 2 we describe the system under consideration and the TDGL formalism, used to calculate static and dynamic distributions of the order parameter as well as the corresponding magnetic and electric fields. The simulation results are given in section 3 where we first discuss the equilibrium vortex patterns in a superconductor wedge and then focus on the dynamics and transformations of vortices, which move in the direction of the thickness gradient. The main conclusions are drawn in section 4.

Theoretical model
We consider a type-I superconductor with the shape shown in figure 1. The lateral sizes of the thin superconducting 'wedge', L x and L y , are much larger than the thickness d(x), which varies linearly from d min at An external homogeneous magnetic field B 0 is applied along the z axis. When studying the vortex dynamics in the wedge, an external current is applied in the y direction. In order to ensure the coexistence of both types-I and II vortex behaviours within the wedge, its thickness has to vary from a small fraction of the bulk penetration depth λ to a value significantly larger than λ. For the structures with lateral sizes 400λ ≲ , which are analysed below, this requirement leads to a relatively large thickness gradient d 0.01  | | > . As a result, the so-called type-II/1 phase (see e.g. [13][14][15] and references therein), which could be expected when the effective Ginzburg-Landau Λ ξ parameter lies in the close vicinity of 1 2 , appears to be irrelevant: for the wedges under consideration, the width of the corresponding region is comparable to or even smaller than the vortex size.
Vortex states and vortex dynamics are described using the TDGL formalism. Although the applicability of the Ginzburg-Landau theory can be rigorously substantiated only when the temperature of a superconductor is close to the critical temperature T c [9], in practice this theory is known to provide correct qualitative or even semi-quantitative predictions for a much wider temperature range. This concerns, in particular, an adequate description of stable vortex states (see e.g. [10] and references therein) and vortex dynamics (see e.g. [11]) in relatively small samples with sizes comparable to those in our present calculations.
Assuming that the thickness d max is smaller than or comparable to the coherence length ξ of the superconductor, an effectively two-dimensional TDGL equation for the order parameter ψ, normalized to 1 and averaged over d(x), in a superconductor with varying thickness can be written as [12] Here, φ and A are the scalar and vector potentials, respectively, averaged over the superconductor thickness d(x), and x y e e x y 2  = ∂ ∂ + ∂ ∂ . All the relevant quantities are made dimensionless by expressing lengths in Here, e 0 Φ π =  is the magnetic flux quantum, 0 μ is the vacuum permeability, λ is the penetration depth at a given temperature T, GL τ is the Ginzburg-Landau time, H c2 is the second critical field, and j c is the critical (depairing) current density of a thin wire or film [16].
The distribution of the scalar potential φ is determined from the condition j · 0  = , which reflects the continuity of currents in the superconductor. The total current density j is given by the sum of the normal and superconducting components: where σ is the normal-state conductivity, which is taken as 1 12 σ = in our units [17]. For a thin superconductor with varying thickness the aforementioned condition of current continuity can be expressed [12] in a 2D form: The averaged vector potential A that enters in equation (1) can be represented as Here the contribution x y e 0 = − + corresponds to the externally applied magnetic field B , 0 while A s describes the averaged magnetic field, which is induced in the superconductor by the currents given by equations (2)-(4). For a thin superconducting layer with an x-dependent thickness, the latter contribution takes the form where κ λ ξ = is the Ginzburg-Landau parameter and the time-independent kernel Equation (1) with scalar and vector potentials defined by equations (5)-(7) are solved numerically using the approach described in [18][19][20]. The superconductor-insulator boundary conditions which assure zero values for both the superconducting and normal components of the current across the boundary, are assumed in the x-direction. Analogous boundary conditions are imposed in the y direction when simulating equilibrium vortex states. When studying vortex dynamics, at y = 0 and y L y = we take the normal metal-superconducor boundary conditions for the order parameter and the scalar potential where j e is the average transport current density in the superconductor.

Equilibrium vortex states
When calculating the equilibrium vortex patterns, we start with a random distribution of the order parameter with 1 ψ | | ≪ and alow the solution of equation (1)  Since the vortex-core size increases with L, for giant vortices it significantly exceeds the thickness of the superconductor under consideration (see figure 2(a)). For this reason, a giant vortex here can be seen as a flat 'current ring' rather than a long 'current tube' in relatively thick superconductor samples. The z component of the magnetic field, induced by such a current ring, has a circularly symmetric 'volcano-like' distribution (see figure 2(b)) instead of a bell-shaped magnetic field distribution, which corresponds to vortex tubes with radii much smaller than their length. With decreasing superconductor thickness, the giant vortices become less stable. As a result, in the left half of the superconductor sample the giant vortices are transformed into densely packed vortex clusters ( figure 2(a)), which produce asymmetric 'volcanoes' in the magnetic field distribution shown in figure 2(b).
As follows from figure 2, due to the long-range vortex-vortex repulsion, some of singly-quantized vortices (L = 1) can remain stable even when they are nucleated in the thickest part of the sample, where the vortex behaviour corresponds to a type-I superconductor. This result is in line with the recent experimental observations [4] on a Pb film. In figure 3(a) we show a scanning Hall probe microscopy (SHPM) image taken on the same sample that was studied in [4]. Due to a relatively large thickness of the film (5 μm), the magnetic field distribution measured in the vicinity of the film surface can be adequately approximated by the monopole model [21,22], which allows, in particular, to unambiguously quantify the winding number L for observed vortices(see e.g. [4,23]). Thus, by fitting the field profiles of the vortices, shown in figure 3(a), with the monopole model (solid lines in figure 3(b)), we can see that giant vortices II and III coexist there with a singly quantized vortex labelled by I.

Vortex dynamics
When simulating the vortex dynamics in a superconductor wedge in the presence of a dc transport current, we assume that the current is applied in the positive y direction. The Lorentz force, caused by this current and the applied field, tends to move vortices in the positive x direction, i.e. from the thinner part of the superconductor to its thicker part. If the external current densities j e are high enough, vortices can leave the superconductor through its rhs edge, while new vortices enter the wedge through the lhs edge. We focus on the stationary regime at those current densities. Figure 4 provides an example of the time evolution of the magnetic field pattern induced by the superconductor wedge in this regime. Vortices, which enter the wedge through its lhs edge, where the superconductor thickness is minimal, are singly quantized (L = 1). When propagating to the thicker part of the superconductor, vortices merge with each other, forming giant vortices with gradually increasing winding number L. As a result of this vortex coalescence process, individual vortices with L = 1 cannot reach the thicker (rhs) half of the of wedge and only giant vortices are present there-in contrast to the equilibrium vortex patterns in the absence of the transport current, which are described in the previous subsection. Singlyquantized vortices reappear only near the rhs edge of the wedge due to the fact that a giant vortex does not leave the superconductor at once: instead, singly-quantized vortices are split from a giant vortex and removed one by one, so that the vorticity of the remaining object step by step decreases down to 1.
At relatively low applied current densities j e the vortex merger processes can result in a regime where the pattern of moving vortices has the form of bands of definite vorticity, which increases with increasing the local thickness of the superconductor. As an example, in figure 5 we show few snapshots of the order parameter distribution for a wedge with the thickness increasing from d 20 min = nm to d 200 max = nm in the x direction at j 0.05 e = (see supporting information for the corresponding animations). As follows from figure 5, while in the vicinity of the lhs superconductor edge only vortices with L = 1 are present, the middle part of the superconductor contains only vortices with L = 2, which merge into vortices with L = 4 in the rhs part of the wedge (as discussed above, the latter step by step loose their vorticity when approaching the rhs edge closely). Remarkably, no vortices with L = 3 appear as a result of vortex coalescence. The reason is that only vortices with L = 2 reach the region where triply quantized vortices will be energetically stable. Therefore, formation of vortices with L = 3 from vortices with L = 2 would be possible either with the appearance of a singly-quantized vortex, that is not energetically advantageous in this region, or through a relatively unlikely 'many-particle process', where e.g. three doubly quantized vortices simultaneously transform into two vortices with L = 3. The obtained results together with the above arguments suggest an obvious generalization: in the regime under consideration the winding number, resulting from vortex coalescence, should preferentially (or even exclusively) take the values 2 n with n 1, 2, 3, = …Our simulations imply that, although at high current densities the vortex dynamics becomes less regular and the aforedescribed regime can be violated, the range of j e where this regime persists is relatively wide (see supporting information).

Conclusions
The results of our TDGL simulations imply that a type-I superconductor wedge can provide a convenient playground to study a gradual transition between the effectively type-II and type-I superconducting behaviours. The simulated equilibrium vortex patterns demonstrate, in particular, the existence of stable singly quantized vortices even in the thickest part of the wedge-in a qualitative agreement with the experimental SHPM data for thick films of type-I superconductors. We have also analysed the vortex dynamics and the processes of vortex merging in the presence of an externally applied current, which causes vortex flow from the thinner part of the wedge to its thicker part. It is shown that these processes can result in a regime where only vortices with winding numbers equal to powers of two are present in the sample (except for its thickest edge, where the winding number of a giant vortex decreases with time step by step). While a direct experimental monitoring of the described vortex dynamics seems to be a rather challenging task, it should be possible to experimentally verify the predicted regime by studying (e.g., with SHPM) the corresponding static vortex patterns after switching off the external current.