Threading dislocation lines in two-sided flux-array decorations

Two-sided flux decoration experiments indicate that threading dislocation lines ~TDL’s !, which cross the entire film, are sometimes trapped in metastable states. We calculate the elastic energy associated with the meanderings of a TDL. The TDL behaves as an anisotropic and dispersive string with thermal fluctuations largely along its Burger’s vector. These fluctuations also modify the structure factor of the vortex solid. Both effects can, in principle, be used to estimate the elastic moduli of the material. @S0163-1829~97!05842-6# Recent two-sided flux decoration experiments have proven an effective technique to visualize and correlate the positions of individual flux lines on the two sides of Bi2Sr2CaCu2O8 ~BSCCO! thin superconductor films. 1,2 This material belongs to the class of high-Tc superconductors ~HTSC’s !, whose properties have drawn considerable attention in the last few years. 3 Due to disorder and thermal fluctuations, the lattice of rigid lines, representing the ideal behavior of the vortices in clean conventional type-II superconductors, is distorted. Flux decoration allows one to quantify the wandering of the lines as they pass through the sample. The resulting decoration patterns also include different topological defects, such as grain boundaries and dislocations, which in most cases thread the entire film. Decoration experiments are typically carried out by cooling the sample in a small magnetic field. In this process, vortices rearrange themselves from a liquidlike state at high temperatures, to an increasingly ordered structure, until they freeze at a characteristic temperature. 4,5 Thus, the observed patterns do not represent equilibrium configurations of lines at the low temperature where decoration is performed, but metastable configurations formed at this higher freezing temperature. The ordering process upon reducing temperature requires the removal of various topological defects from the liquid state: Dislocation loops in the bulk of the sample can shrink, while threading dislocation lines ~TDL’s ! that cross the film may annihilate in pairs, or glide to the edges. However, the decoration images still show TDL’s in the lattice of flux lines. The concentration of defects is actually quite low at the highest applied magnetic fields H ~around 25G!, but increases as H is lowered ~i.e., at smaller vortex densities !. Given the high-energy cost of such defects, it is most likely that they are metastable remnants of the liquid state. ~Metastable TDL’s are also formed during the growth of some solid films. 6 ! Generally, a good correspondence in the position of individual vortices and topological defects is observed as they cross the sample. Nevertheless, differences at the scale of a few lattice constants occur, which indicate the wandering of the lines. Motivated by these observations, we calculate the extra energy cost associated with the deviations of a TDL from a straight line conformation. The meandering TDL behaves like an elastic string with a dispersive line tension which depends logarithmically on the wave vector of the distortion. By comparing the experimental data with our results for mean-square fluctuations of a TDL, it is in principle possible to estimate the elastic moduli of the vortex lattice. Hence, this analysis is complementary to that of the hydrodynamic model of a liquid of flux lines, used so far to quantify these coefficients. 10 On the other hand, the presence of even a single fluctuating TDL considerably modifies the density correlation functions measured in the decoration experiments. The contribution of the fluctuating TDL to the longwavelength structure factor is also anisotropic and involves the shear modulus, making it a good candidate for the determination of this coefficient. In the usual experimental setup, the magnetic field H is oriented along to the z axis, perpendicular to the CuO planes of the superconductor. The displacements of the flux lines from a perfect triangular lattice at a point ( r,z), are described in the continuum elastic limit, by a two-dimensional ~2D! vector field u(r,z). The corresponding elastic free-energy cost is