Time-dependent Ginzburg-Landau treatment of rf magnetic vortices in superconductors: Vortex semiloops in a spatially nonuniform magnetic field

We apply time-dependent Ginzburg-Landau (TDGL) numerical simulations to study the finite frequency electrodynamics of superconductors subjected to an intense rf magnetic field. Much recent TDGL work has focused on spatially uniform external magnetic fields and largely ignores the Meissner state screening response of the superconductor. In this paper, we solve the TGDL equations for a spatially nonuniform magnetic field created by a point magnetic dipole in the vicinity of a semi-infinite superconductor. A two-domain simulation is performed to accurately capture the effect of the inhomogeneous applied fields and the resulting screening currents. The creation and dynamics of vortex semiloops penetrating deep into the superconductor domain are observed and studied, and the resulting third-harmonic nonlinear response of the sample is calculated. The effect of pointlike defects on vortex semi-loop behavior is also studied. This simulation method will assist our understanding of the limits of superconducting response to intense rf magnetic fields.

Figure 1

Schematic view of the superconductor and vacuum domains and boundary conditions in our TDGL simulations.

Figure 2

Plot of the TDGL two-domain solution for the $z$ component of the magnetic field in a plane through the center of the sphere in and around a superconducting sphere in the Meissner state subjected to a uniform static external magnetic field in the $z$ direction. The dashed lines show the boundaries of the spheres, with the smaller sphere being the superconducting sphere with diameter $10{\lambda }_0$ and the larger sphere being the vacuum domain with diameter $40{\lambda }_0$. The solution is obtained for temperature $T=0$, GL parameter $\kappa =1$, and external magnetic field ${\vec{B}}_{\text{applied}}=10\times {10}^{-3}{B}_{c2}\widehat{z}$. Black lines show the streamline plot of the magnetic field, while the color represents the value of magnetic field component $B_z$. The white line indicates the equator, and the magnetic field along the white line is shown in Fig. 3.

Figure 3

Top: Magnetic field $\widehat{z}$-component ($B_z$) profile through the center of the sphere (white line in Fig. 2). The results of a single-domain TDGL model are shown in red, those of a two-domain TDGL model are shown in green, and the analytic solution is shown as a blue solid line. Bottom: The difference between a two-domain TDGL model and the analytic solution is shown in green and the difference between the single-domain TDGL model and the analytic solution is shown in red. The biggest difference is observed at the surface.

Figure 4

The magnitude of the superconducting screening current density at the surface $J_{\text{screening}}$ as a result of a perpendicular magnetic dipole placed $h_{\text{DP}}=1{\lambda }_0$ above the superconductor vs the horizontal distance from the dipole location obtained from TDGL simulation (blue $\times$) and numerical solution for the same scenario obtained from Ref. [74] (red solid line). The left inset shows a schematic of the dipole over the superconductor, while the right inset shows the top view of the surface current distribution calculated by TDGL, which is azimuthally symmetric. The parameters of the simulation are listed in Table 1.

Figure 5

TDGL simulation setup for an oscillating horizontal magnetic dipole ${\vec{M}}_{\text{DP}}$ at height $h_{\text{DP}}$ above the superconductor surface. The magnetic probe is approximated as an oscillating point magnetic dipole parallel to the surface. Red arrows: Surface currents on the horizontal (xy) superconductor/vacuum interface as calculated from the self-consistent TDGL equations. Black arrows: Externally applied magnetic field on a vertical plane (xz) perpendicular to the superconductor surface and including the dipole.

Figure 6

Snapshot of three vortex semiloops at time $t=73{\tau }_0$ during the rf cycle of period $200{\tau }_0$. In this view, one is looking from inside the superconducting domain into the vacuum domain. Plots of ${\left|\mathrm{\Psi }\right|}^2$ are evaluated at the superconductor surface for an oscillating parallel magnetic dipole above the superconductor. The three-dimensional silver surfaces (corresponding to ${\left|\mathrm{\Psi }\right|}^2=0.005$) show the emergence of vortex semiloops. The simulation parameters are given in Table 1.

Figure 7

Summary of the TDGL solution for an oscillating parallel magnetic dipole above a superconducting surface. (a)–(f) Plots of ${\left|\mathrm{\Psi }\right|}^2$ evaluated at the superconductor surface at different times for an oscillating parallel magnetic dipole above the superconductor. In the top part of each panel, one is looking from inside the superconducting domain into the vacuum domain, whereas in the bottom part of each panel one is looking at the $x\text{−}z$ cross-section plane towards the $+y$ axis. $\overrightarrow{M_{\text{DP}}}\left(t\right)$ is chosen such that $\overrightarrow{B_0}\left(t\right)=0.55\text{sin}\left(\omega t\right)\widehat{x}$. The three-dimensional silver surfaces (corresponding to ${\left|\mathrm{\Psi }\right|}^2=0.005$) show the emergence of vortex semiloops. (g) $\left|\overrightarrow{B_0}\right|$ at the surface vs time during the first half of the rf cycle. Red crosses correspond to field values for snapshots (a)–(f).

Figure 8

Plots of ${\left|\mathrm{\Psi }\right|}^2$ (color) and ${\vec{J}}_{\text{surf}}$ (arrows) evaluated at the two-dimensional $x=0$ plane inside the superconductor at $t=50{\tau }_0$ for an oscillating parallel magnetic dipole above the superconductor. White arrows indicate the currents induced inside the superconducting domain. The three-dimensional silver surfaces (corresponding to ${\left|\mathrm{\Psi }\right|}^2=0.005$) show the emergence of vortex semiloops and the suppressed superconducting domain. All model parameters are listed in Table 1.

Figure 9

(a)–(h) Plots of ${\left|\mathrm{\Psi }\right|}^2$ evaluated at the superconductor surface at $t=50{\tau }_0$ for an oscillating parallel magnetic dipole above the superconductor as a function of dipole strength. In this view, one is looking from inside the superconducting domain into the vacuum domain. The maximum amplitude of the applied rf field is shown as $\left|{\vec{B}}_0\right|$. The silver three-dimensional surfaces correspond to ${\left|\mathrm{\Psi }\right|}^2=0.005$ and show the suppressed order-parameter domain and the vortex semiloops. The parameters of the simulation are listed in Table 1.

Figure 10

Summary of TDGL solutions for an oscillating parallel magnetic dipole above a superconducting surface in the presence of a localized defect at ${\vec{r}}_d=0\widehat{x}+y_d\widehat{y}-12\widehat{z}$, where $y_d$ is varied from zero to $16{\lambda }_0$. (a)–(e) Plots of vortex semiloops in the $y\text{−}z$ cross-section plane below the dipole illustrated with a three-dimensional silver surface (corresponding to ${\left|\mathrm{\Psi }\right|}^2=0.003$) at time $t=150{\tau }_0$, when the applied magnetic field reaches its peak amplitude. (f)–(j) Plots of vortex semiloops at time $t=180{\tau }_0$. The defect is denoted by the red dot to the right of the center. $\overrightarrow{M_{\text{DP}}}\left(t\right)$ is chosen such that $\overrightarrow{B_0}\left(t\right)=0.30\text{sin}\left(\omega t\right)\widehat{x}$. The full list of simulation parameters is given in Table 1.

Figure 11

(a)–(h) Plots of vortex semiloops illustrated with a silver surface (corresponding to ${\left|\mathrm{\Psi }\right|}^2=0.005$) at different times for a parallel rf magnetic field in the $\widehat{x}$ direction above the superconductor. A localized defect is placed at the origin (${\vec{r}}_d=0\widehat{x}+0\widehat{y}+0\widehat{z}$) with ${\sigma }_x=6$ and ${\sigma }_y={\sigma }_z=1$ and $T_{\mathrm{cd}}=0.1$. The color shows the order-parameter magnitude ${\left|\mathrm{\Psi }\right|}^2$ on the superconducting surface. (i) $\left|\overrightarrow{B_0}\right|$ at the surface vs time during the first half of the rf cycle. Red crosses correspond to field values for snapshots (a)–(h). The full list of simulation parameters is given in Table 1. Note that this is a transient solution rather than a steady-state solution.