The areal number density of vortices in Fig. This corresponds to a magnetic field of G and is constant to within 3 G over the 33 second observation period. Although flux flow is driven by gradients in the vortex density, such gradients also are negligibly small over the field of view. The time-averaged maximum linear density gradient, directed along the direction of average flux motion, is.

The distribution of flux lines shows no long-range ordering. The six-fold bond-orientational order parameter. Its magnitude achieves the maximum value of for a perfect Abrikosov lattice. The value of averaged over our sample's field of view and over time is appropriate for points randomly distributed in the plane. These observations suggest that the flux line distribution in the field of view is homogeneous, isotropic and uniform over the course of our observations. In these respects, it resembles the distribution of atoms in a simple fluid 8. The major difference is that disorder in fluids is driven by random thermal fluctuations while disorder in the flux line distribution reflects the quenched disorder of the pinning potential.

Nevertheless, an ensemble average of statistically independent flux line distributions on the fixed pinning potential closely resembles the more familiar annealed average and may be interpreted to reveal similar aspects of the system's interactions.

The Lorentz force redistributing vortices among the pinning centers is sufficiently strong that none of the vortices remains completely pinned throughout the experiment. Rather, the flux lines sample the potential surface over the entire field of view. The driving is weak enough, however, that the vortices are limited to creeping flow.

For this reason, their configurations in a sequence of snapshots may be viewed as an ensemble of static distributions near equilibrium. Pinned flux lines are most likely to be found in regions where the local pinning energy, , is most negative.

### Acknowledgments

Thus, we can use the time-averaged vortex distribution, , to map out the spatial distribution of pinning centers. To do this, we coarse grain into regions comparable to the area subtended by a single flux line. Were the pinning potential featureless, such regions would be occupied with equal probability. The probability of finding a vortex in a region of extent centered at r is a measure of the local depth of the pinning potential:. This energy scale is determined by the distribution of pinning energies rather than the thermal energy scale, , at the low temperatures of this study.

If were larger than typical pinning energies, the vortices would have diffused randomly rather than undergoing punctuated bursts of motion. Similarly, if the mean pair potential were larger than typical pinning energies, the vortices would have formed a regular Abrikosov lattice. Instead, the pinning energy distribution distorted the vortex ensemble away from the ideal Abrikosov lattice while thermal energy activated hopping among pinning centers.

Sample motion would introduce artifacts into the pinning potential map. We determined the vibration amplitude and tracking errors to be no greater than 10 nm, or roughly one pixel, by following the apparent motions of the dark feature in the lower right quadrant of Fig.

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This defect was created by ion bombardment before the sample was mounted in the Lorentz microscope and serves as a fiducial mark. The dark bands, on the other hand, result from interference between electrons scattered by different atomic layers and so indicate slight sample curvature.

Even our relatively small field of view contains enough pinning centers that we can estimate the distribution of pinning energies. This distribution appears in Fig. While details of the distribution are not important for the discussion which follows, the measured distribution is reasonably well described by the form. The resemblance may be even better than indicated in Fig. We identify the scale energy, , from this distribution with the characteristic energy scale in Eq.

The resemblance to the Boltzmann distribution renders more compelling the analogy we draw between the vortex distribution in the quenched pinning potential and the distribution of particles under the influence of thermal forces. Numerical estimates for become possible once the scale for vortex pair interactions is determined. While the ensemble-averaged vortex locations map out the pinscape, instantaneous pair correlations.

Here, indicates an average over angles.

Because a single video frame captures only about 50 vortices, many independent distributions must be averaged together to provide adequate statistics:. The data in Fig.

### References

We have confirmed that our results are insensitive to the choice of binning resolution at least over the range 0. At low magnetic inductions for which multiple-vortex interactions can be ignored, the pair correlation, is related to the vortex pair potential through the Boltzmann relation. Without invoking a priori knowledge of the interaction's range, we cannot assume that this is the case for our data set. Fortunately, results from the theory of liquid structure enable us to correct Eq.

The structure of a homogeneous isotropic distribution of interacting particles is described by the Ornstein-Zernike equation 8 ,. The direct correlation function, , is the inverse Fourier transform of and is effectively defined by the Ornstein-Zernike equation.

It can be related at least approximately to the pair interaction potential through a so-called closure relation 8. Extensive work in the theory of liquid structure has identified closure relations which work best in systems such as the present vortex distribution which are characterized by long-ranged repulsive interactions. Of these, the hypernetted chain HNC relation. Having measured experimentally, we use Eq. The result, which appears in Fig. Correlations between consecutive video frames tend to introduce correlated artifacts into. The weak minimum at is likely to be such an artifact.

Monte Carlo simulations of the analytical techniques introduced in this study 10 revel that such correlations tend to systematically increase the measured slope of at small. These systematic errors would be minimized in larger data sets sampled at greater time intervals than was possible in the present study. Since Eqs. For example, the solid line in Fig. The magnetic flux quantum is given by G cm , nm is the sample thickness, and is the modified Bessel function of zeroth order The extracted penetration depth nm is consistent with the accepted value nm 12 for Nb at K.

The scale of pinning energies, meV is consistent with values obtained from flux lattice depinning studies in Nb thin films The data's functional agreement with Eq. The present study focused on a well-understood conventional superconductor whose behavior affords verification for the techniques described in this Letter.

The same techniques can be applied also to layered, anisotropic, and high-T superconductors for which outstanding questions remain. For example, direct measurements of vortex interactions in NbSe or Bi Sr CaCu O could reveal the van der Waals attraction recently predicted 14 for vortices in layered superconductors.

## Jon Chapman's Homepage

Straightforward extensions to the analysis presented above would shed light on the origin of the mixed vortex-chainâ€”vortex-lattice state observed for layered superconductors in oblique fields Ongoing advances in Lorentz microscopy also will make possible studies over a range of applied fields and temperatures in the near future. We are grateful to M. Carbajal-Tinoco, S. Coppersmith and M. Mungan for enlightening conversations. We then link the resulting locations into trajectories, 1 using a maximum likelihood algorithm 5.

The six-fold bond-orientational order parameter 2 measures the degree of instantaneous local crystalline ordering between a vortex at and its nearest neighbors arrayed at angles with respect to a reference axis 7. Figure 1. Lorentz microscope image of flux line distribution in a Nb thin film at 4.

The scale bar indicates 1 m. Circles mark measured vortex locations in the field of view. One vortex's trajectory over time steps is superimposed on the image to indicate the direction and magnitude of flux line flow. The probability of finding a vortex in a region of extent centered at r is a measure of the local depth of the pinning potential: 3 where is a characteristic energy scale. Concepts such as depinning transition and creep motion in vortex systems have been proposed and studied extensively.

However, many aspects of glassy behaviors, in particular the interplay between the different dynamic regimes and their connection to particular underlying glassy phases are still far from being fully understood. The matter is complicated by the fact that detecting and identifying fine details for example, inferring the reliable exact values of the universal critical exponents from experimental data poses considerable experimental challenge.

This makes numerical simulations an irreplaceable tool for studying vortex creep dynamics critical to our understanding the nature of glasses. By computer simulation based on the Langevin dynamics, we have clarified the scaling properties around the zero-temperature depinning force among the velocity or resistance , driving force and temperature with two universal exponents for vortex motions. From the exponents and the scaling curve, the Arrhenius law for the creep motion with a linearly suppressed energy barrier is derived for vortex glass. A non-Arrhenius type creep motion is observed for Bragg glass.

Very recently, we evaluated in high accuracy the creep exponents both for Bragg glass and vortex glass. We also found a new phase boundary inside the Bragg glass phase associated with a glass transition, which occurs at the temperature about half of the melting point of Bragg glass. Figure: Traces of vortex lines subject to random point pins driven by Lorentz force. The form of the lattice of interlayer Josephson vortices is determined by the competition between the inter-vortex repulsion and the intrinsic pinning from the layered structure.

In certain regimes of magnetic field, the intrinsic pinning enhances the order by suppressing fluctuations along the c axis, and in other regimes it tends to weaken the order established by the repulsion when the lattice determined solely by the magnetic field is not naturally commensurate with the underlying layer structure. New vortex phases and new melting processes are therefore possible in contrast to the case of magnetic field parallel to the c axis where there is a single first-order melting transition.

By using the density functional theory, we exposed explicitly the interference between the tendency of crystallization driven by the inter-vortex repulsions and the underlying layer structure.

## Measurement of the Vortex Pair Interaction Potential in a Type-II Superconductor

As a result, we derived an intermediate vortex smectic of period two layers, with more vortices in every other layers, which freezes into lattice via a first-order transition and melts into liquid via a second-order phase transition in 2D Ising universality class. This smectic phase of period two layers has been verified by a very recent cutting-edge experiment. In the extremely high magnetic field, the intra-layer vortex repulsion becomes much stronger than the inter-layer one since the minimal c-axis vortex separation is fixed by the period of layer structure.

This unbalance results in another intermediate phase called sliding phase characterized by short-range positional order in c-direction and quasi-long-range order in ab-plane. This BKT type sliding phase freezes into lattice via a second-order phase transition in 3D XY universality class, and transforms into vortex liquid via a BKT phase transition. In this intermediate phase, the dissipation caused by an in-plane current is shown to be independent of the angle between the applied current and the magnetic field, which explains a famous experiment in this field performed long-time ago and posing a challenge to our theoretical understanding.

In low magnetic fields, the effect of layer pinning is suppressed by thermal fluctuations, where the melting process becomes as simple as the case of c-axis magnetic field and can be captured by the anisotropy scaling. Figure: B-T phase diagram for magnetic field parallel to the CuO layers of high-Tc superconductors, including vortex lattice and vortex liquid phases, as well as novel intermediate phases such as smectic and sliding phases.