Vortex States of a Superconducting Film from a Magnetic Dot Array

Using Ginzburg-Landau theory, we find novel configurations of vortices in superconducting thin films subject to the magnetic field of a magnetic dot array, with dipole moments oriented perpendicular to the film. Sufficiently strong magnets cause the formation of vortex-antivortex pairs. In most cases, the vortices are confined to dot regions, while the antivortices can form a rich variety of lattice states. We propose an experiment in which the perpendicular component of the dot dipole moments can be tuned using an in-plane magnetic field. We show that in such an experiment the vortex-antivortex pair density shows broad plateaus as a function of the dipole strength. Many of the plateaus correspond to vortex configurations that break dot lattice symmetries. In some of these states, the vortex cores are strongly distorted. Possible experimental consequences are mentioned.

Figure 1

Phase diagram with metastable states. Stable states are indicated by numbers in parentheses; metastable states by numbers in brackets. These numbers indicate how many vortex-antivortex pairs per unit cell. Dotted (broken) lines indicate energies of metastable states for leftward (rightward) sweeps. “Bond” subscripts indicate states in which antivortices lie along nearest neighbor bonds; “striped” and “diagonal” subscripts indicate antivortices that lie on diagonals (next-nearest neighbor bonds).

Figure 2

${\rho }_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}}$ and $E_{\mathrm{G}\mathrm{L}}$ plots for sweeps to the left and to the right. The energy is given in units of $d{\xi }^3\Delta$, where $\Delta$ is the condensation energy per unit volume and $d$ is the film thickness.

Figure 3

Cooper pair density plots of stable phases. The images in (a), (b), (c), and (d) correspond to dot magnet strengths of $3.61$, $4.07$, $4.33$, and $4.95$ fundamental flux units, respectively. Cooper pair density is highest in red areas and lowest in blue regions. Antivortices appear as small blue spots, while the large blue spots indicate dot regions.

Figure 4

Cooper pair density plots of stable phases. Images in (a), (b), (c), and (d) correspond to magnetic dot strengths equal to $5.16$, $5.26$, $5.42$, and $6.04$ fundamental flux units, respectively.