Flux bundle interactions
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I. INTRODUCTION Flux creep, which is hard to detect in older type II superconductors, 12 is apparently the dominant effect in limiting the current-carrying capability of the ceramic oxide superconductors.3 Their shorter coherence length leads to a weaker flux pinning energy, while operation at higher temperatures gives the flux bundles considerably more thermal energy. Thermally activated hopping of flux bundles is therefore much more rapid than in the older materials. This phenomenon has been analyzed from two very different points of view. A spin glass model was originally proposed.4 Subsequently, a large number of people have applied the Anderson-Kim model of flux hopping.5 In the former model, the interaction between flux bundles is so strong that several bundles must move in a coordinated fashion during a relaxation event. The latter model assumes flux bundles interact very weakly and hop independently of their neighbors. The purpose of this paper is to define the regime for which flux bundle interactions are important, to describe their effect on flux creep resistance, and to delineate the boundaries between the independent hopping, interactive hopping, and coordinated hopping regimes. We conclude that the interaction is strong enough to significantly restrict flux creep over most of the range of conditions for which these superconductors will be used. At fields above 7 T it is sufficiently strong that cooperative multibundle hopping dominates. II. QUALITATIVE DESCRIPTION We will define the boundary between cooperative and independent flux hopping using a minimal array of flux bundles—one bundle (containing a quantum of flux) surrounded by a hexagon of six neighboring bundles at the equilibrium distance 6 (in the absence of current). r = a H 2802
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J. Mater. Res., Vol. 5, No. 12, Dec 1990
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where H is the applied field and a « 1. For the purposes of this discussion, r is written in terms of the lower critical field,7 Hci = o In K/4TTA2, and magnetic field penetration length, A. The Ginsberg-Landau parameter K is small enough so that it can be ignored, and one can write: (lb) The flux bundles are pinned, but we assume that the density of pinning sites is so high that that does not modify the position of the flux bundles (separation between pinning sites, d, much less than r) and their hexagonal symmetry is unaffected. We then turn on a transverse current density/. The pinning sites (with pinning energy U), current, and neighbors interact with the central flux bundle to produce a free energy which varies with position, as shown in Fig. 1. Thermally generated fluctuations cause the flux bundle to periodically jump out of its pinning site. On such a jump the Lorentz force from the current tends to drive the flux bundles to the right, reducing the energy at the right-hand side by Uj(x) = J x B • x, but interaction with the surrounding lattice, Ai/ int , resists those hops. The dynamics of the flux lattice depends on the relative sizes of these three energ
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