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

Chorng-Haur Sow¹, Ken Harada², Akira Tonomura², George Crabtree³, and David G. Grier¹

Phys. Rev. Lett. 80, 2693-2696 (1998)

Abstract:

We describe the first direct measurement of the pair interaction potential for magnetic flux lines in a Type-II superconductor. Our approach relies on quantitative analysis of Lorentz microscope images of vortices creeping through a superconducting thin film. Time-resolved 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 Type-II superconductivity, tremendous effort has been directed toward understanding the behavior of magnetic flux quanta in Type-II 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 real-time 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 Type-II 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 free-standing Nb film 100 nm thick was prepared by chemically etching a rolled film and annealing at 2200$^\circ$C for 10 minutes to increase the grain size to roughly 300 $\mu$m. The sample was zero-field cooled to T = 4.5 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$^\circ$ 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 ${\bf r}_i(t)$ of each vortex in a digitized image at time t with sub-pixel accuracy using techniques developed for conventional digital video microscopy [5]. We then link the resulting locations into trajectories,

$$\rho({\bf r},t) = \sum_{j=1}^{N(t)} \delta( {\bf r} - {\bf r}_j(t))$$ (1)

using a maximum likelihood algorithm [5]. The trajectory superimposed on Fig. 1 indicates the direction and magnitude of typical flux flow in the 4 x 6 $\mu$m² field of view over 11 seconds.

The areal number density of vortices in Fig. 1 is estimated to be $\rho_0 = 2.6 \pm 0.1~\mu{\rm m}^{-2}$ from the Voronoi diagram of the vortex distribution [6]. This corresponds to a magnetic field of $B = 53 \pm 2$ 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 $\hat v$ of average flux motion, is $\vec \nabla \rho \cdot \hat v = (2 \pm 1) \times 10^{-3} \rho_0~\mu{\rm m}^{-1}$.

The distribution of flux lines shows no long-range ordering. The six-fold bond-orientational order parameter

$$\psi_6 ( {\bf r}_i, t) = {1 \over n_i} \sum_{j=0}^{n_i} \exp \left( 6i \theta_{ij} \right)$$ (2)

measures the degree of instantaneous local crystalline ordering between a vortex at ${\bf r}_i$ and its n_i nearest neighbors arrayed at angles $\theta_{ij}$ with respect to a reference axis [7]. Its magnitude achieves the maximum value of $\vert\psi_6\vert = 1$ for a perfect Abrikosov lattice. The value of $\vert\psi_6\vert = 0.41 \pm 0.03$ 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.

Figure 1: Lorentz microscope image of flux line distribution in a Nb thin film at 4.5 K. The scale bar indicates 1 $\mu$m. Circles mark measured vortex locations in the field of view. One vortex's trajectory over 330 time steps is superimposed on the image to indicate the direction and magnitude of flux line flow.

Pinned flux lines are most likely to be found in regions where the local pinning energy, $V({\bf r})$, is most negative. Thus, we can use the time-averaged vortex distribution, $\rho({\bf r}) = {1 \over T} \int_0^T \rho({\bf r},t) \, dt$, to map out the spatial distribution of pinning centers. To do this, we coarse grain $\rho({\bf r})$ 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 $P({\bf r}) \, d{\bf r}$ of finding a vortex in a region of extent $d{\bf r}$ centered at r is a measure of the local depth of the pinning potential:

$$P({\bf r}) \, d{\bf r} = \exp \left( - {V({\bf r}) \over U_0} \right) \, d{\bf r},$$ (3)

where U₀ is a characteristic energy scale. This energy scale is determined by the distribution of pinning energies rather than the thermal energy scale, k_B T, at the low temperatures of this study. If k_B T 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.

Figure 2: (a) Spatially resolved pinning potential $V({\bf r})$ for the lower left region of Fig. 1 calculated from 990 video frames using Eq. (3). The spatial bins are 18 nm on a side and the pinning energy is scaled according to the color bar on the right. (b) Number N(V) of spatial bins in (a) characterized by pinning potential V. The dashed line is a fit to Eq. (4).

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

$$N(V) = N_0 \, \exp( -V / U_0),$$ (4)

where N(V) is the number of spatial bins after coarse graining which contain a pinning center of depth V. 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, U₀, 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 U₀ become possible once the scale for vortex pair interactions is determined.

While the ensemble-averaged vortex locations map out the pinscape, instantaneous pair correlations

$$g(r,t) = \frac{\langle \int_{-\infty}^\infty \rho({\bf x} -... ...[ \int_{-\infty}^\infty \rho( {\bf x} ) \: d{\bf x} \right]^2}$$ (5)

also reflect inter-vortex interactions. Here, $\langle \cdots \rangle_\theta$ 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:

$$g(r) = \frac{1}{T} \int_0^T g(r,t) \, dt.$$ (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 multiple-vortex interactions can be ignored, the pair correlation, g(r) is related to the vortex pair potential U(r) through the Boltzmann relation

$${U(r) \over U_0} = - \ln \left[ g(r) \right].$$ (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 many-body effects.

Figure 3: Pair distribution function averaged over 990 video frames.

The structure of a homogeneous isotropic distribution of interacting particles is described by the Ornstein-Zernike equation [8],

$$\tilde c(k) = {\tilde h(k) \over 1 + \rho_0 \tilde h(k)},$$ (8)

where $\tilde h(k)$ is the Fourier-Bessel transform of the total correlation function, h(r) = g(r) - 1, and $\rho_0$ is the mean number density of flux lines. The direct correlation function, c(r), is the inverse Fourier transform of $\tilde c(k)$ 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

$${U(r) \over U_0} = - \ln \left[ g(r) \right] + h(r) - c(r)$$ (9)

is found to be reasonably accurate and straightforward to implement [9].

Having measured g(r) experimentally, we use Eq. (8) to calculate the direct correlation function c(r) and use the HNC closure to extract U(r). The result, which appears in Fig. 4, is the first direct measurement of the pair interaction potential for flux lines in a Type-II superconductor.

Correlations between consecutive video frames tend to introduce correlated artifacts into U(r). The weak minimum at $r = 0.7~\mu{\rm m}$ 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 U(r) at small r. These systematic errors would be minimized in larger data sets sampled at greater time intervals than was possible in the present study.

Figure 4: The pair interaction potential extracted from g(r) using Eqs. (8) and (9). The curve is a two-parameter fit to Eq. (10) for the London penetration depth $\lambda$ and the characteristic energy scale U₀. The arrow indicates the mean inter-vortex separation. Inset: The pair potential replotted to emphasize the functional form in Eq. (10).

Since Eqs. (5), (8) and (9) do not rely on an assumed form for the interaction potential, the measured potential U(r) may be used to test theories for vortex interactions. For example, the solid line in Fig. 4 is a two-parameter fit to the London potential

$$\frac{U(r)}{U_0} = \frac{{\varphi_0}^2d}{8 \pi^2 \lambda^2 U_0} \, K_0 (r/\lambda),$$ (10)

for the London penetration depth $\lambda$, and U₀. The magnetic flux quantum is given by $\varphi_0 = hc / 2e \approx 2 \times 10^{-7}$ G cm², d = 100 nm is the sample thickness, and K₀(x) is the modified Bessel function of zeroth order [11]. The extracted penetration depth $\lambda = 39.1 \pm 0.7^{+ 8}_{- 1}$ nm is consistent with the accepted value $\lambda = 45 \pm 1$ nm [12] for Nb at T = 4.5 K. The error estimate for $\lambda$ 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, $U_0 = 11 \pm 2$ 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 $\exp(-x)/x^{1 \over 2}$ behavior of K₀(x) for large x.

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$_{\rm c}$ superconductors for which outstanding questions remain. For example, direct measurements of vortex interactions in NbSe₂ or Bi₂Sr₂CaCu₂O₈ 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 vortex-chain-vortex-lattice 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. Carbajal-Tinoco, 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 DMR-9400379 and in part through the Science and Technology Center for Superconductivity under Award Number DMR-9120000. One of us (GCW) acknowledges support by the US Department of Energy Office of Basic Energy Sciences - Material Sciences, under contract #W-31-109-ENG-38. C.H.S was supported by an Overseas Graduate Scholarship from the National University of Singapore.