Manipulating Abrikosov vortices with soft magnetic stripes

Tuning the polarization of a periodic array of magnetic stripes on top of a superconducting film allows control of Abrikosov vortex motion. Using direct magneto-optical imaging of the vortex patterns, we demonstrate that the proximity of the magnetic stripe ends to the edges of the superconducting film can strongly alter the vortex dynamics. We observe qualitatively different vortex behavior when the stripes overlap with the film edges. From the resulting unique magnetic flux patterns, we calculate the magnetic pinning strength of our stripe array and study effects of the modified edge barrier on vortex guidance and gating that result from different polarizations of the stripes.

Figure 1

Schematic of the sample. $2\times 2\mathrm{mm}$ 100-nm-thick Nb film is covered with 15-nm-thick $\mathrm{Si}{\mathrm{O}}_2$ and an array of 30-$\mu \mathrm{m}$-wide, 40-nm-thick Py stripes with 2-$\mu \mathrm{m}$ gaps is deposited on top as shown. The bottom end of the Py pattern coincides with the Nb film edge. Nb film bands, 200 $\mu \mathrm{m}$ wide, remain bare at the three other sides.

Figure 2

Schematic of magnetic charges $\left({\rho }_m\right)$ and stray fields $\left({\mathbf{H}}_s\right)$ at the edges of the transversely polarized FM stripes. The vortex distribution corresponds to the initial application of a small normal field $H_z$. M is the magnetization vector of the stripe. Edges of the thin stripes perpendicular to M act as magnetically charged filaments, such that the ${\mathbf{H}}_s$ vectors are directed perpendicular to these filaments in the vicinity of the edges. Cooling below $T_c$ produces stray field generated $H_s$ vortices. In addition, the normal field ($H_z$) applied below $T_c$ will generate $H_z$ vortices. $H_{\parallel }\ll M$.

Figure 3

(a) Schematic of magnetic charges at the edges of the transversely polarized FM strips and appropriate forces on the positive Abrikosov vortex. (b) Calculated vortex/edge interaction forces $F_{mv}$ [Eq. (1)] and $F_{ex}$ exact solution accounting for Meissner screening [Eq. (2)]. The numerical data for $F_{ex}$ are multiplied by 2.5 to match curves at $R-D/2=0$. (c) Interaction potentials $U_{mv}$ (for $F_{mv}$) and $U_{ex}$ (for $F_{ex}$).

Figure 4

Successive images of the normal flux entry and exit upon increasing (a–e) and decreasing (f–k) perpendicular field $H_z$ at $T=5\mathrm{K}$ in the sample with transversely polarized stripes. Black arrow in (e) points to the sharp current turn (SCT) line referred to in the text.

Figure 5

Arrangement of the sharp current turn (SCT) lines in the total critical state: (a) current flow scheme in the sample and (b) the MO image near the right sample side. Angles of the SCT lines are defined by the ratios of the critical currents in the bare SC film $\left(J_{c1}\right)$, currents flowing in the Py stripe patterned area across the stripes $\left(J_{c2}\right)$, and currents along the stripes $\left(J_{c3}\right)$. Distances between the current lines in different regions are inversely proportional to the current densities. From the geometry of the continuous current flow, one finds relations between $J_{ci}$ and angles of the SCT lines as shown in the picture. Angles $\alpha$ and $\delta$ are not independent, which allows estimates of the current ratios using both angles obtained from the experimental MO pictures.

Figure 6

Position of the SCT lines near the right sample side at different temperatures with increasing (top row) and decreasing (bottom row) normal fields, $H_z$. The total critical state (critical flux gradients form over the entire sample area) is achieved at different applied fields $H_z$ decreasing with $T$. Images at fields slightly larger than the total penetration field (top row) are chosen for illustration of the SCT lines. Position of the horizontal SCT line is marked by the small green circle, and crossing of the tilted SCT lines in the patterned area is marked by the red circle. Yellow arrows point to the crossing of the tilted SCT line in the bare Nb film with the edge of the patterned region.

Figure 7

Ratios of the critical currents in the bare Nb film $\left(J_{c\mathrm{Nb}}\right)$, and in the patterned area flowing across $\left(J_{\perp }\right)$ and along the Py stripes $\left(J_{\parallel }\right)$ at different temperatures. Symbols “Up” and “Down” correspond to ratios from MO patterns obtained with increasing and decreasing $H_z$.

Figure 8

Vortex entry and exit in the sample with longitudinally polarized stripes at $T=5\mathrm{K}$. $H_z$ increases in (a–d) and decreases in (e,f).

Figure 9

Normal flux entry at different polarizations of the Py stripes. Top row (a–d) shows images for four diagonal in-plane fields, and bottom row (e–h) illustrates two horizontal and two vertical polarizations indicated by red arrows. For each polarization, there are two successive images taken at $H_z=8$ (left) and 11.5 Oe (right), respectively. $T=5\mathrm{K}$.

Figure 10

Trajectories $X\left(t\right)$ of the vortex motion across the attractive (left) and repulsive (right) 1D potentials, $U\left(x\right)=\pm k/(1+x^2)$, under the action of a constant drive (a) $c=0.1$, and (b) above the critical drive $c>c_{\mathrm{cr}}=0.32476$ when vortex motion becomes continuous. At $c<c_{\mathrm{cr}}$ the vortex is delayed near the center of the potential and then can jump to the second continuous motion branch. The delay [horizontal segment of the lower branch of $X\left(t\right)$] and the jump (shown by arrow) are essentially longer for the repulsive potential. The intermediate segment with $\mathrm{d}X/\mathrm{d}t<0$ is not physical. $X$ is in arbitrary units and $t$ is in units of $c/\alpha$.