Vortex cores in narrow thin-film strips

We study vortex current distributions in narrow thin-film superconducting strips. If one defines the vortex core ``boundary"as a curve where the current reaches the depairing value, intriguing features emerge. Our conclusions based on the London approach have only qualitative relevance since the approach breaks down near the core. Still, the main observation which might be useful is that the core size near the strip edges is smaller than in the rest of the strip. If so, the Bardeen-Stephen flux-flow resistivity should be reduced near the edges. Moreover, at elevated temperatures, when the depairing current is small, the vortex core may extend to the whole strip width, thus turning into an edge-to-edge phase-slip line.


I. INTRODUCTION
Long thin-film strips are essential elements of various superconducting circuits, the current carrying properties of which are determined by vortices residing there or crossing strips and causing non-zero voltages and energy dissipation, see e.g. [1]. Many vortex effects can be described by studying the supercurrent distributions, which away of vortex cores are well represented by the London theory. However, within this theory, vortex cores are treated just as point singularities despite the fact that properties of cores and, in particular, their size and shape are relevant for evaluation the vortex self-energy and its dynamic properties, the flux-flow resistivity is just an example.
To describe properly the core structure is challenging even in uniform bulk materials far from sample boundaries. The question then whether one can extract some information about core shapes from the London current distribution. For a single vortex in isotropic bulk superconductors the current density distribution out of core is given by j = (cφ 0 /32π 2 λ 3 )K 1 (r/λ), where φ 0 is the flux quantum, λ is the penetration depth, and K 1 is the Modified Bessel function. At short distances r λ, the London current j = cφ 0 /32π 2 λ 2 r reaches the depairing value (defined as j d = cφ 0 /16π 2 λ 2 ξ near the critical temperature T c ) at r ∼ ξ with ξ being the coherence length. In fact, this is one of popular ways to define the vortex core size. "Sweeping under the rug" complexities of the vortex core physics, the simple core model as the circular normal state region of radius ξ provides correct estimates of the core energy as the condensation energy within the core, πξ 2 (H 2 c /8π), and of the flux-flow resistivity as the normal resistance within the core, ρ f ≈ ρ n 2πξ 2 B/φ 0 (ρ n is the normal resistivity, B is the magnetic induction) [2,3].
In thin films, the vortex current distribution differs substantially from the bulk since the stray fields out of the film affect the currents in the film [4]. Evaluation of these distributions are difficult in particular in finite samples, where the film edges may cause drastic modifications of currents for vortices situated at distances of the order of Pearl length Λ = 2λ 2 /d from the edges. Exceptions are the small samples, e.g., narrow thin-film bridges of a width W Λ where effects of self-fields can be disregarded [5]. Following the above qualitative prescription for determination of the core shape and size, one may expect different from circular core shapes for vortices in such bridges, the subject of our discussion below.

II. NARROW THIN-FILM STRIPS
Consider a thin-film strip in the plane (x, y), x axis is directed across the strip, 0 < x < W , whereas y is along it. For a vortex at x = a, y = 0, the London equations for the film interior read h is the magnetic field and j is the supercurrent density. Averaging this over the thickness d, one obtains where g(r) is the sheet current density, r = (x, y), a = (a, 0), and Λ = 2λ 2 /d. Other components of Eq. (1) turn identities after averaging. In strips of a width W Λ, the self-field of the current g, given by the Biot-Savart integral, is of the order g/c, whereas the second term on the left-hand side of Eq. (2) is of the order gΛ/cW . Hence, the self-field h z can be disregarded, unlike the applied field if it exists.
It is convenient to introduce a scalar "steam function" G(r) such that g = curl Gẑ [5], Then, we obtain for G: arXiv:2002.07090v1 [cond-mat.supr-con] 17 Feb 2020 The boundary conditions g x = 0 at the strip edges x = 0, W give G = 0 at the edges. Thus, the problem is equivalent to that of two-dimensional electrostatic potential of a linear "charge" q = cφ 0 /8π 2 Λ situated at r = a between two grounded metal plates parallel to zy at x = 0 and x = W . The solution of Eq. (4) is obtained by conformal mapping [6]: The alternative way to present this result as due to an infinite sum over ± vortex images out of the strip, see e.g. Ref. 7, is equivalent to Eq. (5), but having closed form, Eq. (5) is more convenient for numerical and analytic work.
Here J is the dimensionless sheet current density in units of cφ 0 /2πΛW and x, y, a are measured in units of W .
Stream lines of the current coincide with contours of where ds is the line element. Example of current stream lines is shown in Fig. 1 for a vortex close to the left edge of the strip. To have a better idea on the distribution of current values, one can plot contours of constant J (x, y) = J 2 x + J 2 y . This distribution near the vortex core affects the core shape. Using the standard estimate for the deparing current density one gets the sheet deparing density g d ≈ cφ 0 d/16π 2 λ 2 ξ = cφ 0 /8π 2 Λξ and the dimensionless depairing current It is worth noting that the depairing current value adopted here is not a universal quantity which may vary (slightly) with the sample geometry. This uncertainty may introduce an extra factor ∼ 1 in Eq. (7). If one takes data from Ref. [8] for WSi 4 nm-thick narrow bridges with T c = 3.4 K, the low temperature value ξ 0 ≈ 7.8 nm, and W = 2 µm, one estimates the low temperature J d (0) ≈ 20.4. Hereafter we use these data to compare with our model predictions.
On warming, the depairing current decreases according to empirical relation J d = J d (0)(1 − t 2 ) 3/2 , t = T /T c [9,10]. For our example, J d ≈ 2 at T ≈ 3 K. The contour J (x, y) = 2 is shown in Fig. 2 for a vortex at a = 0.1 W . Hence, the core shape, defined by J (x, y) = J d for a vortex penetrating the edge x = 0, has the shape of a liquid droplet stuck to the film edge. This unusual shape can be attributed to enhanced current density between the edge and the position of the current singularity for a vortex close to the edge seen in Fig. 1. It is worth noting that since J d decreases on warming, according to Fig. 2 the normal core expands on warming as it should.
A. Core shape dependence on vortex position For the above example of thin and narrow bridge of WSi, J d = 5 corresponds to T ≈ 0.78 T c . This value of J d is chosen here to demonstrate the dynamics of the core shape of a vortex moving away from the edge at x = 0. Figure 3 shows contours J (x, y) = J d = 5 for a set of vortex positions near the edge.
One sees, that when a 0.03, the core shape is close to semi-circles or ovals with a base at the edge, i.e. at the y axis, and with the size growing with increasing a. When the distance a from the edge increases further, the core acquires a shape reminiscent of a liquid droplet still attached to the edge, see curves for a = 0.05, 0.062. Eventually, the core droplet disengages from the edge and acquires a nearly circular shape, a = 0.07, 0.10. It is readily shown that in general the disengagement happens at One also observes that when the vortex proceeds up to a ≈ 0.03, the core expands in both directions. For larger a = 0.05, 0.062, the increase stops in the y direction along the strip, but the width of the core in x direction expands.
The behavior of so defined core is even more intriguing at higher temperatures and lower J d . An example of J d = 0.4, shown in Fig. 4, corresponds to T ≈ 3.3 K= 0.96 T c . One can see that up to a = 0.4 the core, being still attached to the left edge, ends up at some x * < 1. One readily verifies that x * satisfies J (x * , 0, a) = sin πa 2| cos πa − cos πx * | = J d .
However, for a = 0.45 and 0.5 the core turns into a normal state edge-to-edge channel. One may expect these channels to behave as line-type phase slips [12]. Recall that one-dimensional phase slips are responsible for dissipation in wires thinner than ξ [2]. When a localized vortex crosses a superconducting strip of length 2L with the width W ξ, the phase difference between edges y = −L and L also "slips" by 2π as in thin wires, see e.g. [11]. The line phase slips were proposed as another mechanism of dissipation in wide strips [12]. Hence, one may expect a similar behavior in thin-film strips of interest here. One can also look at the disengagement from the edge calculating the width 2y 1 of the droplet base at the left edge at x = 0 where y 1 is found from |J y (0, y 1 , a)| = J d . This gives cos πy 1 = sin πa 2J d + cos πa .
Similarly, for the edge at x = W , one obtains from |J y (W, y 2 , a)| = J d : at low temperatures with large J d the vortex cores are separated from the edges for most of vortex positions in the sample, except narrow belts near the edges. These belts expand on warming and at J d = 0.5, all vortices at a < 0.5 are stuck to the left edge, while those at a > 0.5 are attached to the right edge. With further warming J d < 0.5, the two domains overlap, as shown at the lower panel, in other words, in a finite interval of positions centered at the sample middle the cores are attached to both edges. That is the situation when the cores are expected to behave as line phase slips.

III. DISCUSSION
A word of caution: the definition of the "core boundary" as a curve where the current values in the London approximation reach the depairing level is rather artificial, notwithstanding reasonable results it leads to in isotropic bulk situation. In fact, the London approach breaks down near these boundaries. Hence, a better theory should be employed in the core vicinity. To reaffirm the contours of J(x, y) = J d in Figs. 2-4 as representing, at least qualitatively, vortex core shapes, one has to see whether or not the order parameter modulus |∆(x, y)| indeed decreases when one crosses these contours. This, of course, cannot be done within the London model where |∆| is assumed constant. One should turn to a microscopic theory or, for temperatures close to T c , to the Ginzburg-Landau theory (GL).
The problem of the order parameter distributions for vortices perpendicular to a thin film, is challenging because one has to take the stray fields into account. Given these difficulties, one could turn to other geometry, where the comparison between the current and order parameter distributions is easier to make. Such a case is a vortex parallel to faces of a thin superconducting slab [7].
It is straightforward to show that the London current distribution for a vortex parallel to a slab thinner than the London λ are, in fact, the same as in narrow thinfilm bridges discussed above, except now we have the length scale λ instead of Pearl's Λ and the slab thickness instead of the bridge width W . We are not aware of calculations describing the order parameter distribution within the core of such a vortex. Physically, however, such a distribution is similar to that of a vortex close to the surface of a bulk sample, the problem considered within the frame of time-dependent GL theory [13], and recently in the discussion of surface barriers near T c [14]. Contours of constant order parameter |ψ| shown in Fig. 1 of Ref. 14 are indeed qualitatively similar to contours of our Fig. 2 of constant current value.
To conclude, we have shown that along with the distorted vortex current distribution near the film edges, vortex cores are strongly affected as well. Close to the edge, the core is shaped as a liquid droplet attached to the edge, which grows when vortex singularity moves away from the edge. At some distance, depending on the depairing current value, the core disconnects from the edge, Eq. (8), and acquires a "normal" round shape. If temperatures are high, the depairing current is small, and the thin-film bridge is narrow, the core can be attached to both edges simultaneously thus forming a structure similar to line-type phase slips.
Our discussion could be relevant for interpretation of data on resistivity transition from the normal to superconducting state in narrow thin-film strips reported in [8]. In particular, these data were shown to be consistent with enhanced flux-flow conductivity near the strip side edges in applied fields on the Tesla order. The conductivity variation could be associated with changing core size of vortices crossing the strips.