Evidence for Long-Range Correlations within Arrays of Spontaneously Created Magnetic Vortices in a Nb Thin-Film Superconductor

We have imaged spontaneously created arrays of vortices (magnetic flux quanta), generated in a superconducting film quenched through its transition temperature at rates around ${10}^9\mathrm{K}/\mathrm{s}$. The spontaneous appearance of vortices is predicted by the Kibble-Zurek and by the Hindmarsh-Rajantie models of phase transitions under nonequilibrium conditions. Differentiating between these models requires a measurement of the internal correlations within the emerging vortex array. In addition to short-range correlations predicted by Kibble and Zurek, we found unexpected long-range correlations which are not described by any of the existing models.

Figure 1

Typical images of spontaneously created vortices in a superconductor cooled at (a) $2\times {10}^9\mathrm{K}/\mathrm{s}$ and (b) $4\times {10}^8\mathrm{K}/\mathrm{s}$. The intensity is proportional to the local magnetic field. Bright and dark spots represent vortices with opposite polarity.

Figure 2

Images of magnetic flux in the superconducting film cooled in the presence of an external magnetic field. The external field in (a) is 0.2 mT and in (b) is 4 times larger, 0.8 mT. The cooling rate is ${10}^9\mathrm{K}/\mathrm{s}$ in both panels. The majority of the vortices (dark spots) are associated with the external magnetic field. Bright spots are vortices with polarity opposite to the field. At higher external fields the number of vortices with opposite sign is reduced.

Figure 3

Density-density correlation function $D(r)$. The cooling rate is $2\times {10}^9\mathrm{K}/\mathrm{s}$. Distances are in units of $r_{\mathrm{av}}=8.2\mu \mathrm{m}$. The solid line is a fit to $G(r)\propto r^2\exp (-r^2/{\widehat{\xi }}^2)$ [21]. The statistical error bars are smaller than the point size.

Figure 4

First moment of the vortex-vortex correlation function $G(r)$. The negative peak at short distances reflects vortex-antivortex correlations predicted by the KZ model. The solid line is a fit to the theory of 21. The statistical error bars are smaller than the point size.