Moving Pearl vortices in thin-film superconductors

The magnetic field $h_z$ of a moving Pearl vortex in a superconducting thin-film in $(x,y)$ plane is studied with the help of time-dependent London equation. It is found that for a vortex at the origin moving in $+x$ direction, $h_z(x,y)$ is suppressed in front of the vortex, $x>0$, and enhanced behind ($x<0$). The distribution asymmetry is proportional to the velocity and to the conductivity of normal quasiparticles. The vortex self-energy and the interaction of two moving vortices are evaluated.


I. INTRODUCTION
The time-dependent Ginzburg-Landau equations (GL) are the major tool in modeling vortex motion. Although this approach, strictly speaking, is applicable only for gapless systems near the critical temperature [1], it reproduces qualitatively major features of the vortex motion.
A much simpler London approach had been successfully employed through the years to describe static or nearly static vortex systems. The London equations express the basic Meissner effect and can be used at any temperature for problems where vortex cores are irrelevant. The magnetic structure of moving vortices is commonly considered the same as that of a static vortex displaced as a whole.
It has been shown recently, however, that this is not so for moving vortex-like topological defects in, e.g., neutral superfluids or liquid crystals [2]. Also, this is not the case in superconductors within the Time Dependent London theory (TDL) which takes into account normal currents, a necessary consequence of moving magnetic structure of a vortex [3,4]. In this paper we consider the magnetic field distribution of moving Pearl vortices in thin films. We show that the self-energy of a moving vortex decreases with increasing velocity. Moreover, the interaction energy of two vortices moving with the same velocity becomes anisotropic, it is enhanced when the vector R connecting vortices is parallel to the velocity v and suppressed if R ⊥ v. The magnetic flux carried by moving vortex is equal to flux quantum, but this flux is redistributed so that the part of it in front of the vortex is depleted whereas the part behind it is enhanced.
In time dependent situations, the current consists, in general, of normal and superconducting parts: where E is the electric field and Ψ is the order parameter. The conductivity σ approaches the normal state value σ n when the temperature T approaches T c in fully gapped s-wave superconductors; it vanishes fast with decreasing temperature along with the density of normal excitations. This is, however, not the case for strong pair-breaking when superconductivity becomes gapless while the density of states approaches the normal state value at all temperatures. Alas, there is little experimental information about the T dependence of σ. Theoretically, this question is still debated, e.g. Ref. [5] discusses possible enhancement of σ due to inelastic scattering.
Within the London approach |Ψ| is a constant Ψ 0 and Eq. (1) becomes: where λ 2 = mc 2 /8πe 2 |Ψ 0 | 2 is the London penetration depth. Acting on this by curl one obtains: where r ν (t) is the position of the ν-th vortex, z is the direction of vortices. Equation (3) can be considered as a general form of the time dependent London equation.
As with the static London approach, the time dependent version (3) has the shortcoming of being valid only outside vortex cores. As such it may produce useful results for materials with large GL parameter κ in fields away of the upper critical field H c2 . On the other hand, Eq. (3) is a useful, albeit approximate, tool for low temperatures where GL theory does not work and the microscopic theory is forbiddingly complex.

II. THIN FILMS
Let the film of thickness d be in the xy plane. Integration of Eq. (3) over the film thickness gives for the z component of the field for a Pearl vortex moving with arXiv:2102.00073v1 [cond-mat.supr-con] 29 Jan 2021 velocity v: Here, φ 0 is the flux quantum, g is the sheet current density related to the tangential field components at the upper film face by 2πg/c =ẑ × h; Λ = 2λ 2 /d is the Pearl length, and τ = 4πσλ 2 /c 2 . With the help of divh = 0 this equation is transformed to: As was stressed by Pearl [6], a large contribution to the energy of a vortex in a thin film comes from stray fields. In fact, the problem of a vortex in a thin film is reduced to that of the field distribution in free space subject to the boundary condition (5) at the film surface. Since outside the film curlh = divh = 0, one can introduce a scalar potential for the outside field: The general form of the potential satisfying Laplace equation that vanishes at z → ∞ of the empty upper halfspace is Here, k = (k x , k y ), r = (x, y), and ϕ(k) is the twodimensional Fourier transform of ϕ(r, z = 0). In the lower half-space one has to replace z → −z in Eq. (7). As is done in [3], one applies the 2D Fourier transform to Eq. (5) to obtain a linear differential equation for h zk (t). Since h zk = −kϕ k , we obtain: In fact, this gives distributions of all field components outside the film, its surface included. In particular, h z at z = +0 (the upper film face) is given by We are interested in the vortex motion with constant velocity v = vx, so that we can evaluate this field in real space for the vortex at the origin at t = 0: It is convenient in the following to use Pearl Λ as the unit length and measure the field in units φ 0 /4π 2 Λ 2 : (we left the same notations for h z and k in new units; when needed, we indicate formulas written in common units).

A. Evaluation of hz(r)
With the help of identity one rewrites the field as To evaluate the last integral over k, we note that the 3D Coulomb Green's function can be written as To make here the last step, we used R = (r, z), q = (k, q z ) and It follows from Eq. (14) After integrating by parts, one obtains: For the Pearl vortex at rest s = 0, ρ = r, and the known result follows, see e.g. Ref. [7]: Y 0 and H 0 are second kind Bessel and Struve functions. Hence, we succeeded in reducing the double integral (11) to a single integral over u. Besides, the singularity at r = 0 is now explicitly represented by 1/r, whereas the integral over u is convergent and can be evaluated numerically.
The results are shown in Fig. 1. The field distribution is not symmetric relative to the singularity position: the field in front of the moving vortex is suppressed relative to the symmetric distribution of the vortex at rest, whereas behind the vortex it is enhanced. This is an interesting consequence of our calculations: the magnetic flux of the moving vortex is redistributed so that it is depleted in front of the vortex and enhanced behind it. We can characterize this redistribution by calculating the magnetic flux Φ + in front of the vortex: The integral over y gives 2πδ(k y ), whereas integrating over k x we use where P indicates that the integral over k x in Eq. (20) should be understood as the principal value. Hence, we have The integration now is straightforward and we obtain Note that the total flux carried by vortex is given by Fourier component h z (k = 0) = φ 0 , see Eq. (9), i.e. φ 0 /2 is the flux through the half-plane x > 0 of the vortex at rest. The flux behind the moving vortex is therefore B. Potential and London energy of moving vortex The potential ϕ introduced above is useful not only as an intermediate step in evaluation of magnetic field, it is directly related to the London energy (the sum of the magnetic energy outside the film and the kinetic energy of the currents inside) [8].

Self-energy of moving vortex
This energy is given by whereas the integral (26) in this limit is logarithmically divergent. As is commonly done, we can approach the singularity at r = 0 from any side, e.g. setting x = 0 and y = ξ, the core size: for the small dimensionless ξ c = ξ/Λ. Compare this with the energy of a vortex at rest, see e.g. [8]: Hence, the vortex self-energy decreases with increasing velocity, the result qualitatively similar to that of moving vortices in the bulk [4].

Interaction of moving vortices
It has been shown in Ref. [8] that in infinite films the interaction is given by int = (φ 0 /8π)[ϕ 1 (2) + ϕ 2 (1)], ϕ 1 (2) is the potential of the vortex at the origin at the position r of the second. Using Eq. (26) we obtain Clearly, int (x, y) = int (−x, y). This energy can be evaluated numerically and the result is shown in Fig. 2 for s = 2. It is worth noting that in thin films the interac- tion is not proportional to the field of one vortex at the position of the second. In our case the field of one vortex, see Fig. 1, is not symmetric relative to x → −x, whereas the interaction energy is.

C. Electric field and dissipation
Having the magnetic field (9) of a moving vortex, one gets for two vortices, one at the origin and the second at R: (in common units). The moving nonuniform distribution of the vortex magnetic field causes an electric field E out of the vortex core, which in turn causes the normal currents σE and the dissipation σE 2 . Usually this dissipation is small relative to Bardeen-Stephen core dissipation [9], but for fast vortex motion and high conductivity of normal excitations [5] it can become substantial [3].
The field E is expressed in terms of known h with the help of the Maxwell equations i(k × E k ) z = −∂ t h zk /c and k · E k = 0: For the stationary motion, one can consider the dissipation at t = 0. The dissipation power is: .
The integral here is divergent at large k, but the London theory anyway breaks down in the vortex core of a size ξ, so one can introduce a factor e −k 2 ξ 2 to truncate this divergence. We then calculate the reduced quantity w(x, y) = W (πc 2 Λ 2 /φ 2 0 σdv 2 ) shown in Fig. 3. An interesting feature of this result is that the dissipation w(x, y) develops a shallow ditch along the x axis. Hence, for a fixed separation of vortices in the pair, the dissipation is minimal if they are aligned along the velocity.

III. DISCUSSION
We have shown that in thin films the magnetic structure of the moving Pearl vortex is distorted relative to the vortex at rest. The flux quantum of a moving vortex is redistributed, the back side part of the flux is enhanced, whereas the in-front part is depleted. Physically, the distortion is caused by normal currents arising due to changing in time magnetic field at each point of space, the electric field is induced and causes normal currents. Naturally, it leads to suppression of the flux where it is increasing (in front of the moving vortex) and to enhancement where it is decreasing (behind the vortex). We characterize this asymmetry by the difference of fluxes behind (x < 0) and in front (x > 0) the moving vortex ∆Φ = Φ − − Φ + = (2φ 0 /π) arctan s. For a realistic situation s = vτ /Λ 1, although the relaxation time τ ∝ σλ 2 where σ is the poorly-known conductivity of above-the-gap normal excitations. Measuring ∆Φ one can extract σ, an important physical characteristics of superconductors. There is an experimental technique which, in principle, could probe the field distribution in moving vortices [10]. This is highly sensitive SQUIDon-tip with the loop small on the scale of possible Pearl lengths.
Recent experiments have traced vortices moving in thin superconducting films with extremely high velocities well exceeding the speed of sound [10,11]. Vortices crossing thin-film bridges being pushed by transport currents have a tendency to form chains directed along the velocity. The spacing of vortices in a chain is usually exceeds by much the core size, so that commonly accepted reason for the chain formation, namely, the depletion of the order parameter behind moving vortices is questionable. But at distances r ξ the time dependent London theory is applicable.
In this paper, we consider only properties of a single vortex and of interaction between two vortices moving with the same velocity, The problem of interaction in systems of many vortices is still to be considered.

IV. ACKNOWLEDGEMENTS
The work of V.K. was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Ames Laboratory is operated for the U.S. DOE by Iowa State University under contract # DE-AC02-07CH11358.