Vortex-antivortex pair generation by an in-plane magnetic dipole on a superconducting film

We apply the time-dependent Ginzburg-Landau approach to describe the nucleation of vortex-antivortex pairs in a hybrid system, formed by a thin superconducting film with a single in-plane magnetic dipole on top of it, and to simulate the vortex dynamics in the presence of a direct transport current. At current densities, sufficient to depin vortices and antivortices from a magnetic dipole, it periodically generates vortex-antivortex pairs, producing vortex and antivortex streams which eventually coalesce into phase-slip lines with further increasing current. We demonstrate the possibility to efficiently control the generation rate and trajectories of emitted vortices by the applied current density and an additional weak homogeneous magnetic field.

Figure 1

(Color online) Sketch of the structure under consideration. Shaded areas correspond to the superconductor with sizes $L_x$, $L_y$, and $d$, and to the magnetic bar with sizes $a_x$, $a_y$, and $a_z$, and uniform magnetization $\mathbf{M}$. The external current with average density $j_e$ is applied along the $y$ axis.

Figure 2

(Color online) Distributions of (a) the $z$ component of the magnetic field induced by the magnetic bar in the superconducting film, (b) the $x$ component of the current density in the film at $j_e=0,B_0=0$, and the square modulus of the order parameter at $B_0=0$ in the cases of (c) $j_e=0$ and (d) $j_e=0.1$. The parameters of the magnetic bar with shape and position, schematically shown by the white dashed line, are $a_x=10$, $a_y=3$, $a_z=3$, $d_1=3$, and $M_x=-10$. The dashed (dotted) vertical black lines in panels (c) and (d) correspond to the positions of the vortex (antivortex) core at $j_e=0$ and $j_e=0.1$.

Figure 3

(Color online) Panel (a): time-averaged voltage drop $⟨V⟩$ as a function of the density of the externally applied current $j_e$ in the case of $B_0=0$. The vertical up (down) arrow labels the part of the curve for increasing (decreasing) applied currents. The insets show few snapshots of the order-parameter distribution, which illustrate the vortex and antivortex dynamics at $j_e=0.36$ and the coalescence of vortices and antivortices into a phase-slip line at $j_e=0.4$. The position of the magnetic bar is schematically shown by the white dashed line. Panel (b): generation rate $\Gamma$ for V-AV pairs in the vortex-flow regime as a function of $j_e$ at $B_0=0$ (solid line) and $B_0=0.03$ (dashed line). Both panels correspond to $L_x=L_y=50$.

Figure 4

(Color online) Equilibrium distributions of the square modulus of the order parameter at different densities $j_e$ of the externally applied current in the case of $B_0=0.03$. The calculations are performed for $L_x=L_y=50$ and the magnetic-bar position, schematically shown by the white dashed line.

Figure 5

(Color online) Four snapshots of the $z$ component of the magnetic field induced in the superconductor by the currents, which flow in it at $j_e=0.2$ and $B_0=0$. The time moment $\tau =0$ is chosen at random within the periodic V-AV generation process. The calculations are performed for $L_x=L_y=50$ and the magnetic-bar position, schematically shown by the white dashed line.

Figure 6

(Color online) Distribution of the parameter $S$, used to visualize time-averaged trajectories of vortices and antivortices, in the cases of no homogeneous external magnetic field $B_0=0$ [panel (a)] and $B_0=0.03$ [panel (b)]. The calculations are performed for $L_x=L_y=50$, $j_e=0.2$, and the magnetic-bar position, schematically shown by the white dashed line.

Figure 7

(Color online) (a) Distribution of the parameter $S$ used to visualize trajectories of vortices and antivortices, (b) a snapshot of the order-parameter distribution, and (c) the corresponding distribution of the $x$ component of the supercurrent density. The density of the externally applied current is $j_e=0.2$ and the homogeneous external magnetic field is $B_0=0.02$. The calculations are performed for $L_x=L_y=100$ and the magnetic-bar position, schematically shown by the white dashed line.