Measurement of the Vortex Pair Interaction Potential in a TypeII Superconductor
Abstract.
We describe the first direct measurement of the pair interaction potential for magnetic flux lines in a TypeII superconductor. Our approach relies on quantitative analysis of Lorentz microscope images of vortices creeping through a superconducting thin film. Timeresolved vortex distributions imaged in this way reflect both the vortices' mutual interaction and also the influence of random pinning forces. These influences are isolated and quantified by adapting results from the theory of simple liquids.
Since Abrikosov's initial description of TypeII superconductivity, tremendous effort has been directed toward understanding the behavior of magnetic flux quanta in TypeII superconductors. Flux lines collectively determine such material properties as a superconductor's critical current and upper critical field. This collective behavior in turn arises both from the flux lines' interactions with each other and with the underlying pinning potential. The microscopic mechanisms of flux pinning and flux line interactions have been treated theoretically and can be gauged indirectly through measurements of collective flux behavior. The advent of realtime vortex imaging through Lorentz microscopy (1); (2); (3); (4) makes possible direct measurements of both the pinning potential and of the vortex pair interaction in a TypeII superconductor. In this Letter, we describe the first such measurements on flux lines in a thin film of Nb.
Lorentz microscopy exploits the deflection of a transmission electron microscope's electron beam by magnetic fields to create images of individual flux quanta localized within a superconductor. Individual vortices appear in images such as Fig. 1 as adjacent bright and dark blurs. Their motions are captured to video tape at 30 frames per second before being digitized and analyzed on a computer. For the present experiment, a freestanding Nb film 100 nm thick was prepared by chemically etching a rolled film and annealing at 2200C for 10 minutes to increase the grain size to roughly 300 m. The sample was zerofield cooled to K on the electron microscope stage. Flux flow was initiated by ramping an applied magnetic field up to 80 G. Because the sample was mounted at 45 to both the applied field and the imaging plane, the component of the field normal to the samples surface was 56.6 G. Images such as Fig. 1 and the data taken from them were corrected for the resulting perspective distortion.
Vortex motion slowed to creeping flow as the local flux density approached the applied field. The punctuated motion in this regime provides insights into both the local pinning potential and also the vortex pair interaction potential.
We locate the centroid of each vortex in a digitized image at time with subpixel accuracy using techniques developed for conventional digital video microscopy (5). We then link the resulting locations into trajectories,
(1) 
using a maximum likelihood algorithm (5). The trajectory superimposed on Fig. 1 indicates the direction and magnitude of typical flux flow in the m field of view over 11 seconds.
The areal number density of vortices in Fig. 1 is estimated to be from the Voronoi diagram of the vortex distribution (6). 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 timeaveraged maximum linear density gradient, directed along the direction of average flux motion, is .
The distribution of flux lines shows no longrange ordering. The sixfold bondorientational 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). 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. Thus, we can use the timeaveraged 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:
(3) 
where is a characteristic energy scale. 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. The pinning energy landscape, or “pinscape,” calculated with Eq. (3) for the field of view appears in Fig. 2(a).
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. 1. 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. 2(b). While details of the distribution are not important for the discussion which follows, the measured distribution is reasonably well described by the form
(4) 
where is the number of spatial bins after coarse graining which contain a pinning center of depth . The resemblance may be even better than indicated in Fig. 2 since our short data set undersamples weakly pinned regions while coarse graining overemphasizes strongly pinned regions. We identify the scale energy, , from this distribution with the characteristic energy scale in Eq. (3). 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 ensembleaveraged vortex locations map out the pinscape, instantaneous pair correlations
(5) 
also reflect intervortex interactions. 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:
(6) 
The data in Fig. 3 represent an average over 990 video frames distributed into bins 4.4 nm, or half a pixel, on a side. We have confirmed that our results are insensitive to the choice of binning resolution at least over the range 0.1 to 1 pixel.
At low magnetic inductions for which multiplevortex interactions can be ignored, the pair correlation, is related to the vortex pair potential through the Boltzmann relation
(7) 
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. (7) for manybody effects.
The structure of a homogeneous isotropic distribution of interacting particles is described by the OrnsteinZernike equation (8),
(8) 
where is the FourierBessel transform of the total correlation function, , and is the mean number density of flux lines. The direct correlation function, , is the inverse Fourier transform of and is effectively defined by the OrnsteinZernike equation. It can be related at least approximately to the pair interaction potential through a socalled 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 longranged repulsive interactions. Of these, the hypernetted chain (HNC) relation
(9) 
is found to be reasonably accurate and straightforward to implement (9).
Having measured experimentally, we use Eq. (8) to calculate the direct correlation function and use the HNC closure to extract . The result, which appears in Fig. 4, is the first direct measurement of the pair interaction potential for flux lines in a TypeII superconductor.
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. (5), (8) and (9) do not rely on an assumed form for the interaction potential, the measured potential may be used to test theories for vortex interactions. For example, the solid line in Fig. 4 is a twoparameter fit to the London potential
(10) 
for the London penetration depth , and . The magnetic flux quantum is given by G cm, nm is the sample thickness, and is the modified Bessel function of zeroth order (11). The extracted penetration depth nm is consistent with the accepted value nm (12) for Nb at K. The error estimate for includes both the 95% confidence interval from the fit to the data in Fig. 4 and also the range of errors estimated from Monte Carlo simulations (10). The scale of pinning energies, meV is consistent with values obtained from flux lattice depinning studies in Nb thin films (13). The data's functional agreement with Eq. (10) can be judged from the inset to Fig. 4 which emphasizes the asymptotic behavior of for large .
The present study focused on a wellunderstood conventional superconductor whose behavior affords verification for the techniques described in this Letter. The same techniques can be applied also to layered, anisotropic, and highT superconductors for which outstanding questions remain. For example, direct measurements of vortex interactions in NbSe or BiSrCaCuO could reveal the van der Waals attraction recently predicted (14) for vortices in layered superconductors. Straightforward extensions to the analysis presented above would shed light on the origin of the mixed vortexchain–vortexlattice state observed for layered superconductors in oblique fields (15). 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. D. CarbajalTinoco, S. N. Coppersmith and M. Mungan for enlightening conversations. The work at The University of Chicago was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR9400379 and in part through the Science and Technology Center for Superconductivity under Award Number DMR9120000. One of us (GCW) acknowledges support by the US Department of Energy Office of Basic Energy Sciences – Material Sciences, under contract #W31109ENG38. C.H.S was supported by an Overseas Graduate Scholarship from the National University of Singapore.
References

(1)
K. Harada et al., Nature 360, 51 (1992).

(2)
A. Tonomura, Electron Holography, Vol. 70 of Springer Series in Optical Sciences (SpringerVerlag, New York, 1993).

(3)
K. Harada et al., Science 274, 1167 (1996).

(4)
T. Matsuda et al., Science 271, 1393 (1996).

(5)
J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298 (1996).

(6)
F. P. Preparata and M. I. Shamos, Computational Geometry (New York, Springer Verlag, 1985).

(7)
D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457 (1979).

(8)
J.P. Hansen and I. R. McDonald, Theory of Simple Liquids, 2nd ed. (Academic Press, London, 1986).

(9)
K.C. Ng, J. Chem. Phys. 61, 2680 (1974).

(10)
M. Mungan, C.H. Sow, S. N. Coppersmith, and D. G. Grier, in preparation (1997).

(11)
The Clem model for vortex interactions (J. R. Clem, J. Low Temp. Phys. 18, 427 (1975)) is more appropriate for Nb, whose superconducting coherence length is comparable to its London penetration depth. The present data set does not offer sufficient resolution to distinguish its predictions from those of the London model, however, and we present the simpler and quantitatively adequate model for clarity.

(12)
R. Meservey and B. B. Schwartz, in Superconductivity, ed. R. D. Parks (New York: Marcel Dekker, 1969) pp. 117191.

(13)
G. S. Park, C. E. Cunningham, B. Cabrerra, and M. E. Huber, Phys. Rev. Lett. 68, 1920 (1992).

(14)
G. Blatter and V. Geshkenbein, Phys. Rev. Lett. 77, 4958 (1996); S. Mukherji and T. Nattermann, Phys. Rev. Lett. 79, 139 (1997).

(15)
C. A. Bolle et al., Phys. Rev. Lett. 66, 112 (1991); L. L. Daemen et al., Phys. Rev. Lett. 70, 2948 (1993); M. M. Doria and I. G. de Oliveira, Phys. Rev. B 49, 6205 (1994); I. V. Grigorieva et al., Phys. Rev. B 51, 3765 (1995).