Asymmetric crossing of the attractive and repulsive magnetic potential by Abrikosov vortices

We present studies of Abrikosov vortex motion across the magnetically charged edges of long thin ferromagnetic stripes placed above or under a thin superconducting film. The magnetic charges at the stripe edges form attractive or repulsive potential wells for vortices. Using a relatively small in-plane magnetic field to polarize the stripe edges and normal-to-plane magnetic field to induce up or down polarized vortices, we tune the attractive or repulsive stripe-vortex interactions. Imaging of the vortex dynamics reveals that the repulsive magnetic potential ${U_m}^{+}$ acts as a robust vortex pinning barrier, while the attractive ${U_m}^{-}$ has practically no effect on the vortex motion irrespective of the position of the stripes located above or underneath the superconducting film. We analyze the observed asymmetry using equations of the overdamped vortex motion. The formal analytical solution yields an asymmetry, but the numerical modeling with and without noise terms does not confirm it. Instead, we find that the asymmetry is caused by the creation of spontaneous vortices at the maxima of $U_m$, which depends on the edge polarity and suppress ${U_m}^{-}$. Furthermore, our experiment and time-dependent Ginzburg-Landau simulations demonstrate that magnetic pinning dominates over vortex pinning due to corrugations of the superconducting layer deposited on top of the ferromagnetic stripes.

Figure 1

Sketch of the magnetic fields and charges due to vortices polarized along $z$ and magnetic stripes with $\mathbf{M}\left|\right|x$. Long Py stripes are parallel to $y$ (not shown) and are placed above (a) or underneath (b) the Nb film. $U\left(x\right)$ is the magnetic potential for vortices induced by the stripes. The right panel shows the MO image of the enhanced field at the sample edges in a small applied field $H_z$ at $T<T_c$. The sample size is 2 × 2 mm. The blue square in the center outlines the area occupied by stripes, and green rectangles show areas of the MO images in Fig. 2.

Figure 2

MO images of the normal field distribution in areas outlined by green rectangles in Fig. 1 for Py stripes placed on top of the Nb film. (a) Stray fields of the stripes polarized along the horizontal arrow at $T>T_c$. Positively and negatively charged stripe edges are revealed by bright and dark contrast respectively. (b)–(e) normal flux entry with increasing applied field [ $T=4\mathrm{K}$, $H_z=9.9$, 14.6, 17. 6, 21 Oe in (b) to (e)]. (f) trapped flux at $H_z=0$ after application of 800 Oe. Double arrows show the position of the left and right sides of the stripe pattern. The dark negative (attractive) left edge does not hinder vortex motion. In contrast, vortices are delayed and accumulate at the bright (repulsive) positively charged edge on the right. When reducing $H_z$ to zero, negative vortices, induced by the reversed critical currents, start to enter from the Nb square boundaries and are impeded at the negative stripe edge [dark contrast along left arrow in (f)] while they pass the positive stripe edge [right arrow in (f)]. Narrowing of the flux penetration front towards the sample center, well observed in Fig. 2, is due to the finite size of the sample where currents flow along the sides of the square.

Figure 3

Same as Fig. 2 but the Py stripes are under the Nb film. The position of stripes is presented in the top part of each panel. Positively and negatively charged stripe edges are bright and dark, respectively. $T=4\mathrm{K}$, ${\mathrm{H}}_{\mathrm{z}}=4.6$, 11.1, 15.4, 17.6 Oe in (a) to (d). (e) Positive flux (bright contrast) exit and negative flux (dark contrast) entry with reducing field from 800 to 8 Oe. Red double arrows show the left edge of the first Py stripe in (b) and (e), and the blue double-arrow in (c) shows the right edge of the first stripe. Unlike in Fig. 2, the negatively charged Py stripe edges impede vortices and the positive stripe edges does not affect their motion [in accordance with the sketch in Fig. 1]. A slight contrast is observed along the positive Py stripe edge marked by red arrow in (b), which could be due to the Nb film corrugation. However, it is much weaker than the contrast on the negative stripe edge. In (e), the entry of negative vortices is inhibited (dark line along the double arrow) by the positively charged stripe edge.

Figure 4

Sketch of the geometry for estimating the stripe edge/vortex potential.

Figure 5

Real part of the solution of Eq. (7), for the attractive $K=-0.84$ (blue) and repulsive $K=+0.84$ (red) potential with $\mathrm{b}=0.42$. Retracting branches are unphysical and the system should transfer from lower to higher branches via jumps, as shown by arrows.

Figure 6

Trajectories of vortices in the vicinity of the stripe edges obtained by numerical modeling of Eq. (6). Dimensionless $x\left(t\right)$ are shown for decreasing driving forces (increasing $\left|K\right|$) for attractive $\left(K<0\right)$ and repulsive $\left(K>0\right)$ potentials. At a critical driving force, $|K_c|=0.42$, vortices halt in the stop points $+/-x_1$. At a larger force, $\left|K\right|<|K_c|$, they pass (although asymmetrically) both $K>0$ and $K<0$ barriers.

Figure 7

Vortex trajectories with driving force below critical $\left(1/\left|K\right|<1/|K_c|\right)$ in the presence of bipolar (a) and monopolar (b) noise. The random noise amplitude is defined as ${\zeta }_{\max }={(3\tilde{T}/h_t)}^{1/2}$ with an effective temperature $\tilde{T}$ and time step of the calculations $h_t$.