Introduction

Magnetic flux lines in type II superconductors belong to a class of systems in which interacting particles are subjected to random pinning forces [1,2]. At very low temperatures, the resulting configurations are determined by a competition between elastic energy cost and pinning energy gain as the flux line ensemble attempts to accommodate the random medium. This competition gives rise to a rich and complex energy landscape of metastable states. One expects that structural properties of the particle configurations should contain information about the interparticle interactions as well as the statistical properties of the pinning potential landscape, the pinscape.

In an earlier publication [3] by some of us, henceforth referred to as SHTCG, the near-equilibrium configurations of flux lines in a Nb thin film obtained from Lorentz microscope images [4] were analyzed. SHTCG argued that a sequence of flux line configurations on the fixed pinscape resembles the instantaneous configurations of a classical simple liquid, with the underlying pinscape playing the role of an effective heat bath. This conjecture implies that in the low-density limit, the pair-correlation function g(r) should be of the form [5]

$$g(r) = e^{-U(r)/U_o} + \ldots,$$ (1)

where U(r) is the flux line pair interaction, U_o is an effective pinning energy scale, and the terms omitted correspond to higher order corrections due to many-body effects. For stiff magnetic flux lines, the interparticle potential U(r) (per unit length) is given in the low density limit by the asymptotic form [6,7]

$$U(r) = \frac{{\phi_{o}}^2}{8 \pi^2 \lambda^2} \sqrt{\frac{\pi \lambda }{2r}} \, e^{-r / \lambda},$$ (2)

where $\phi_{o}$ is the flux quantum and $\lambda$ is the London penetration depth. By treating U_o as an adjustable parameter, SHTCG showed that the small r behavior (lift-off) of the pair correlation function agrees well with the form given in Eq. (1), yielding a value for the penetration depth in good agreement with the accepted value for Nb (see Fig. 1).

Figure: Pair correlation function obtained from averaging over Lorentz microscope images of flux lines in Nb [3,4] at T = 5 K. The region of interest is where g(r) rises from zero ( lift-off). The inset shows a comparison of the lift-off region of g(r) with liquid theory behavior, Eq. (1) (refer to section II for further details).

Bitter decoration images of flux line configurations in the high temperature superconductor BSCCO analyzed in the same manner also yield results in good quantitative agreement with the accepted value of the penetration depth for BSCCO [8]. The fit value for U_o presumably describes the characteristic energy scale for the pinscape disorder.

The present paper aims at a more thorough understanding of how structural correlations arise from microscopic properties such as the particle interactions and potential landscape. Particular questions we address include: Under what conditions can the static configurations of an interacting system in a quenched random potential give rise to liquid-like correlations of the form Eq. (1)? How does Eq. (1) compare to a theory where the quenched nature of randomness is accounted for explicitly?

We report the results of numerical simulations and present a theory that captures the essential features observed both in experiments and in our simulations, providing thus some answers to the questions raised above. In particular our simulations show that a relation of the form Eq. (1) emerges in the limit of point-like pins, i.e. pins of small spatial extent, and for broad (e.g. exponential) pinning strength distributions. In contrast, pinscapes composed of identically strong or spatially extended pins are not well described by Eq. (1). Our theory is based on the observation that the behavior of the pair correlation function at small r is dominated by strongly pinned flux lines. In the low-density limit, the functional form of the pair correlation function therefore arises essentially from a convolution of the tails of the pinning energy distribution with the probability distribution for locating a flux line at a given position.

The paper is organized as follows. We present our model and its motivation in Section II. Section III contains details of the numerical simulations. Numerical results and their discussion are given in Section IV, while Section V is devoted to a description of the underlying theoretical picture. We conclude with a discussion and summary of our results in Section VI.