Probing the low-frequency vortex dynamics in a nanostructured superconducting strip

We investigate by scanning susceptibility microscopy the response of a thin Pb strip, with a square array of submicron antidots, to a low-frequency ac magnetic field applied perpendicularly to the film plane. By mapping the local permeability of the sample within the field range where vortices trapped by the antidots and interstitial vortices coexist, we observed two distinct dynamical regimes occurring at different temperatures. At a temperature just below the superconducting transition, $T/T_c=0.96$, the sample response is essentially dominated by the motion of highly mobile interstitial vortices. However, at a slightly lower temperature, $T/T_c=0.93$, the interstitial vortices freeze up leading to a strong reduction of the ac screening length. We propose a simple model for the vortex response in this system which fits well to the experimental data. Our analysis suggests that the observed switching to the high mobility regime stems from a resonant effect, where the period of the ac excitation is just large enough to allow interstitial vortices to thermally hop through the weak pinning landscape produced by random material defects. This argument is further supported by the observation of a pronounced enhancement of the out-of-phase response at the crossover between both dynamical regimes.

Figure 1

Profiles of the in-phase (top) and out-of-phase (bottom) components of the ac magnetic permeability, $b_{ac}\left(x\right)/h_{ac}$ (in units of ${\mu }_0$) calculated using a full numerical inversion of Eq. (7) (dashes) and the analytic approximation Eq. (8) (full lines). The middle panel presents similar profiles for the vortex contribution to $b^{\prime }/h_{ac}$ (see text).

Figure 2

Schematic representation of the potential energies of (a) an interstitial vortex caged by artificially pinned vortices and (b) a vortex trapped in an artificial pinning center near the second matching field. In both panels, the blue (red) curve represents the potential energy with (without) the contribution from natural defects. Cartoons: gray (red)-shaded disks represent vortices in their (out-of-)equilibrium position.

Figure 3

(a) Schematic layout of the investigated transport bridge with a patterned area. Most of the scanning Hall probe microscopy images were obtained in a $16\times 16\mu {\mathrm{m}}^2$ area at the border of the sample. The dc and ac magnetic fields are applied perpendicularly to the plane of the Pb film. (b) Atomic force microscopy image of the sample surface.

Figure 4

Contour plots: scanning Hall probe microscopy images obtained after field cooling down to $T=4.2$ K in presence of the indicated dc magnetic field $H=0H_1,1/4H_1,1/3H_1,1/2H_1,2/3H_1$, and $H=3/4H_1$. The open circles represent the antidot positions. The theoretical prediction of Ref. [41] is shown schematically above the respective image, with filled circles representing the antidots occupied by a singly quantized vortex.

Figure 5

Scanning Hall probe microscopy images obtained after field cooling down to $T=4.2$ K. From left to right, for $H=H_1,1.06H_1$ and $H=1.31H_1$, respectively. The open circles indicate the position of the antidots.

Figure 6

SSM images showing the ac response (mapped in a $16\times 16\mu {\mathrm{m}}^2$ region near the sample edge) to a 77.123 Hz excitation field of amplitude ${\mu }_0{h}_{ac}=0.016$ mT for different field values at $T=6.7$ K (left) and $T=6.9$ K (right). The first row shows the dc (time-average) flux distributions. The in-phase and out-of-phase components of the total ac response are mapped in the second and fourth rows, respectively. The in-phase vortex response, defined as the difference between the in-phase and the Meissner responses, is shown in the third row. In all images, the white dots and the white line show schematically the position of the square antidots and the sample edge, respectively. The dashed circles highlight the position of selected interstitial vortices. All red-blue color bars are in units of ${\mu }_0{h}_{ac}$.

Figure 7

Contour plots of the inductive magnetic permeability, $b^{\prime }/h_{ac}$ (in units of ${\mu }_0$), measured at (a) $T=6.7$ K and (b) $T=6.9$ K and averaged over the direction along the strip, as a function of position across the strip and the applied dc field (in units of $H_1$). Panels (c) and (d) correspond to similar contour plots calculated using the model described in Sec. 2 and the values of the empirical parameters $c_v$ and $c_r$ appearing in Eqs. (17, 18, 19, 20). These values were obtained by fitting the model (lines) to the experimental data (symbols) corresponding to averaging the in-phase (e) and the out-of-phase (f) SSM images over the scan area.

Figure 8

Symbols: experimental cross-section profiles of the permeability averaged over lines parallel to the strip at 6.9 K for selected dc field values. Error bars are standard deviations from the mean. Lines: theoretical permeability profiles calculated using the fitting parameters $c_v=0.09$ and $c_r=0.019$.

Figure 9

Temperature dependence of the real (thick blue line) and imaginary (thick green line) components of the effective ac penetration depth calculated for the sample under excitation frequency 77.123 Hz using Eq. (16). Light dashed lines correspond to the approximate Eq. (21). For comparison, we also plot ${\mathrm{\Lambda }}_{ac}\left(T\right)$ for the case where random pinning is absent (thin red line) and the case where thermal fluctuations are ignored (thin black line).