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Moving Pearl Vortices in Thin-Film Superconductors Article Moving Pearl Vortices in Thin-Film Superconductors Vladimir Kogan 1,* and Norio Nakagawa 2 1 Ames Laboratory—DOE, Ames, IA 50011, USA 2 Center for Nondestructive Evaluation, Iowa State University, Ames, IA 50011, USA; [email protected] * Correspondence: [email protected] Abstract: The magnetic field hz of a moving Pearl vortex in a superconducting thin-film in (x, y) plane is studied with the help of the time-dependent London equation. It is found that for a vortex at the origin moving in +x direction, hz(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. Keywords: thin films; Pearl vortex 1. Introduction The time-dependent Ginzburg–Landau equations (GL) are the major tool in modeling vortex motion. Although this approach is applicable only for gapless systems near the critical temperature [1], it is gauge invariant and reproduces correctly major features of the vortex motion. A simpler linear London approach has been 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. Moving vortices are commonly considered the same as static which are displaced as a Citation: Kogan, V.; Nakagawa, N. whole. Moving Pearl Vortices in Thin-Film However, recently it has been shown that this is not the case for moving vortex- Superconductors. Condens. Matter like topological defects in, e.g., neutral superfluids or liquid crystals [2]. This is not so 2021, 6, 4. https://doi.org/10.3390/ in superconductors within the time-dependent London theory (TDL) which takes into condmat6010004 account normal currents, a necessary consequence of moving magnetic structure of a vortex [3,4]. In this paper, the magnetic field distribution of moving Pearl vortices in thin Received: 22 December 2020 films is considered. It is shown that the self-energy of a moving vortex decreases with Accepted: 20 January 2021 increasing velocity. The interaction energy of two vortices moving with the same velocity Published: 24 January 2021 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 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional clai- to flux quantum, but this flux is redistributed so that the part of it in front of the vortex is ms in published maps and institutio- depleted, whereas the part behind it is enhanced. nal affiliations. In time-dependent situations, the current consists, in general, of normal and supercon- ducting parts: 2e2jYj2 f J = sE − A + 0 rc , (1) mc 2p Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. where E is the electric field and Y is the order parameter. This article is an open access article The conductivity s approaches the normal state value sn when the temperature T distributed under the terms and con- approaches Tc; in s-wave superconductors it vanishes fast with decreasing temperature ditions of the Creative Commons At- along with the density of normal excitations. This is not the case for strong pair breaking tribution (CC BY) license (https:// when superconductivity becomes gapless, and the density of states approaches the normal creativecommons.org/licenses/by/ state value at all temperatures. Unfortunately, there is not much experimental information 4.0/). Condens. Matter 2021, 6, 4. https://doi.org/10.3390/condmat6010004 https://www.mdpi.com/journal/condensedmatter Condens. Matter 2021, 6, 4 2 of 9 about the T dependence of s. Theoretically, this question is still debated; e.g., [5] discusses the possible enhancement of s due to inelastic scattering. Within the London approach jYj is a constant Y0, and Equation (1) becomes: 4p 4ps 1 f J = E − A + 0 rc , (2) c c l2 2p 2 2 2 2 where l = mc /8pe jY0j is the London penetration depth. By acting on this via curling, one obtains: 1 4ps ¶h f − r2 + + = 0 ( − ) h 2 h 2 2 z ∑ d r rn , (3) l c ¶t l n where rn(t) is the position of the n-th vortex; z is the direction of vortices. Equation (3) can be considered as a general form of the time-dependent London equation. The time-dependent version of London Equation (3) is valid only outside vortex cores, similarly to the static London approach. As such, it may give useful results for materials with large GL parameter k values in fields away from the upper critical field Hc2. On the other hand, Equation (3) is a useful, albeit approximate tool for low temperatures where GL theory does not work and the microscopic theory is forbiddingly complex. 2. Thin Films Let the film of thickness d be in the xy plane. Integration of Equation (3) over the film thickness gives, for the z component of the field, a Pearl vortex moving with velocity v: 2pL ¶h curl g + h + t z = f d(r − vt). (4) c z z ¶t 0 Here, f0 is the flux quantum; g is the sheet current density related to the tangential field components at the upper film face by 2pg/c = zˆ × h; L = 2l2/d is the Pearl length; and t = 4psl2/c2. With the help of divh = 0, this equation is transformed to: ¶h ¶h h − L z + t z = f d(r − vt). (5) z ¶z ¶t 0 As was shown 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 supplied by solutions of Equation (4) at the film’s surface. Outside the film curlh = divh = 0, one can introduce a scalar potential for the outside field: h = rj, r2 j = 0 . (6) The general form of the potential satisfying Laplace equation that vanishes at z ! ¥ of the empty upper half-space is Z d2k j(r, z) = j(k)eik·r−kz . (7) 4p2 Here, k = (kx, ky), r = (x, y), and j(k) is the two-dimensional Fourier transform of j(r, z = 0). In the lower half-space, one has to replace z ! −z in Equation (7). As is done in [3], one applies the 2D Fourier transform to Equation (5) to obtain a linear differential equation for hzk(t). Since hzk = −kjk, we obtain: f e−ik·vt j = − 0 . (8) k k(1 + Lk − ik · vt) Condens. Matter 2021, 6, 4 3 of 9 In fact, this gives distributions for all field components outside the film, its surface included. In particular, hz at z = +0 (the upper film face) is given by f e−ik·vt h = −kj = 0 . (9) zk k 1 + Lk − ik · vt We are interested in the vortex’s motion with constant velocity v = vxˆ, so that we can evaluate this field in real space for the vortex at the origin at t = 0: f Z d2k eik·r ( ) = 0 hz r 2 . (10) 4p 1 + Lk − ikxvt It is convenient in the following to use Pearl L as the unit length and measure the field in 2 2 units f0/4p L : Z d2k eik·r vt hz(r) = , s = . (11) 1 + k − ikxs L (we left the same notations for hz and k in new units; when needed, we indicate formulas written in common units). 2.1. Evaluation of hz(r) With the help of identity Z ¥ −1 −u(1+k−ikxs) (1 + k − ikxs) = e du , (12) 0 one rewrites the field as Z ¥ Z −u 2 ik·r−uk hz(r) = du e d k e , 0 r = (x + us, y). (13) To evaluate the last integral over k, we note that the three-dimensional (3D) Coulomb Green’s function is 1 1 Z d3q 1 Z d2k = eiq·R = eik·r−kz. (14) 4pR (2p)3 q2 8p2 k To do here the last step, we used R = (r, z), q = (k, qz) and Z ¥ eiqzz p e−kjzj = dqz 2 2 . (15) −¥ k + qz k It follows from Equation (14) Z ¶ 1 2pz d2k eik·r−kz = −2p p = . (16) ¶z r2 + z2 (r2 + z2)3/2 p Replace now r ! r, z ! u, R ! r2 + u2 to obtain instead of Equation (13): Z ¥ u e−u hz(r) = 2p du . (17) 0 (r2 + u2)3/2 After integrating by parts, one obtains: " # 1 Z ¥ du e−u s(x + su) hz = 2p − p 1 + . (18) r 0 r2 + u2 r2 + u2 Condens. Matter 2021, 6, 4 4 of 9 For the Pearl vortex at rest s = 0, r = r, and the known result follows; see, e.g., [7]: 1 p h (r) = 2p + [Y (r) − H (r)] , (19) z r 2 0 0 Y0 and H0 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 Figure1. y 4 0.1 2 0.4 hz(x,0) 0.5 3 y 0 x 0.2 0.6 0.3 2 -2 1 -4 x -4 -2 0 2 4 -4 -2 2 4 x 2 2 Figure 1.
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