Disorder-induced power-law response of a superconducting vortex on a plane

We report drive-response experiments on individual superconducting vortices on a plane, a realization for a (1+1)-dimensional directed polymer in random media. For this we use magnetic force microscopy to image and manipulate individual vortices trapped on a twin boundary in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ near optimal doping. We find that when we drag a vortex with the magnetic tip, it moves in a series of jumps. As theory suggests, the jump-size distribution does not depend on the applied force and is consistent with power-law behavior. The measured power is much larger than widely accepted theoretical calculations.

Figure 1

Vortices on and off a twin boundary (TB). (a) Illustration of vortices in a superconductor. The left vortex (V) can meander in $d_{\perp }=2$ dimensions perpendicular to an external magnetic field $\vec{H}$ while the vortex trapped on the TB (TBV), a common planar defect, can meander only on the plane, i.e., in $d_{\perp }=1$ dimensions. $\mathrm{\Delta }$ measures how much a vortex meanders from the field axis. (b) Polarized-light microscopy photograph of our $80-\mu \text{m}$-thick sample, revealing two TBs (white arrows). (c) MFM scan of vortices (black disks) that form a high density chain along a TB and an Abrikosov lattice around it ($z\approx 1.15\mu \text{m}, \mathrm{\Delta }f$ spans 0.93 Hz). (d) MFM scan at $0\le H\le 10\mu \text{T}$. Vortices (blue disks) accumulate on a TB and exhibit ($1+1$)-dimensional physics ($z\approx 0.28\mu \text{m}, \mathrm{\Delta }f$ spans 1.6 Hz). (e) Many vortices in the chain in (d) are isolated because their separation is much larger than ${\lambda }_{ab}\approx 120\text{nm}$ (here, $z\approx 0.4\mu \text{m}, \mathrm{\Delta }f$ spans 0.6 Hz).

Figure 2

Manipulation scans of TB vortices at $T=15\text{K}$. (a) Forward (Fwd) and backward (Bwd) line scans (taken along the dashed lines from the scans in the insets) containing a tip-induced vortex jump of size ${\mathrm{\Delta }}_{\text{jump}}=|x^{*}-x_{\text{jump}}|$ that we associate with an abrupt change in the position of the upper part of the vortex. (b) Fwd and Bwd line scans taken along the dashed lines from the scans in the insets. Numerous vortex jumps with a variety of ${\mathrm{\Delta }}_{\text{jump}}$'s are apparent. The difference between the overall shapes of the Fwd and Bwd line scans suggests that nonequilibrium effects may be involved. Insets: The scans from which the line scans in (a) and (b) were taken. The scan height and the span of $\mathrm{\Delta }f$ is indicated for each panel. The horizontal double arrows indicate the back or forth scan direction along the TB (the $x$ axis) and the large vertical arrows indicate the direction we step the tip after each back and forth cycle (the $y$ axis).

Figure 3

Histograms binning all measured jump lengths (${\mathrm{\Delta }}_{\text{jump}}$) for different ranges of the force exerted along the TB by the tip ($F_x$). Inset: Normalized distributions of ${\mathrm{\Delta }}_{\text{jump}}$ for the different $F_x$ ranges. All the distributions collapse onto each other, revealing the independence of ${\mathrm{\Delta }}_{\text{jump}}$ from $F_x$.

Figure 4

Measured vortex jump lengths (${\mathrm{\Delta }}_{\text{jump}}$) and fits to the data. Although the data are consistent with a power-law distribution, the exponent we obtain does not match ${\alpha }_{\text{theory}}=3/2$ predicted for a system in equilibrium. Inset: Values of a power-law exponent ${\alpha }_{\text{fit}}$ obtained by fitting the data in the main panel for different values of the lower cutoff $a_x$. ${\alpha }_{\text{fit}}$ saturates (arrow) for $a_x>a_0=49\pm 3$, a clear indication that ${\alpha }_{\text{meas}}=2.75\pm 0.06$ is the best-fit exponent for the distribution.