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Condensed Matter Group

UR PAS

Vortex Fluctuations in Superconducting Systems

We have been carrying out numerical simulations to study the behavior of vortices in models of bulk (three dimensional) high temperature superconductors and planar (two dimensional) superconducting networks. Simulations are carried out using the 2D and 3D uniformly frustrated XY model or its dual Coulomb gas (in 2D) and interacting loop (in 3D) models.

When a magnetic field is applied to a conducting material, circulating eddy currents are induced. In a superconductor, these eddy currents are quantized in discrete integer units, corresponding to allowed twists of 2 in the phase of the superconducting condensate wavefunction, as one circulates around the closed path of the eddy current. Shrinking the path down to a point, about which there remains a net circulation of current, defines a topological singularity in the wavefunction known as a vortex. Since the circulating eddy currents of a vortex induce magnetic fields of their own, the magnetic flux density in the superconductor is not uniform; it is greater in the vicinity of a vortex than in regions between vortices. When a vortex moves, it carries with it this increased magnetic flux; the moving magnetic flux induces an electric field, and hence voltage drop, transverse to the motion of the vortex. Such “flux flow resistance”, with its associated electrical dissipation, can destroy the superconductivity of the material. The static and dynamic behavior of vortices thus determines all the macroscopic electromagnetic properties of a superconductor.

Vortices in superconductors have a well defined mutual interaction. The range of the interaction is the magnetic penetration length, which is set by the material parameters. Since the density of vortices is determined by the applied magnetic field, one can continuously tune the system from weakly to strongly interacting. Depending on the degree of thermal fluctuations and random impurities, vortices may exist in either solid, liquid, or glassy states. Applying a d.c. electrical current to the system produces a net force on the vortices that drives them into nonequilibrium steady states. Thus, in addition to its intrinsic interest for understanding the behavior of superconductors in potential applications, the system of interacting vortices also serves as a convenient tunable system in which one can investigate many more general questions of statistical physics concerning the effects of quenched randomness and nonequilibrium driving forces on phase transitions and critical phenomenon in strongly fluctuating many particle systems.

High Tc Superconductors

The discovery of the new class of “high Tc” copper-oxide superconductors in 1987, Crystal structure of LSCO, one of the first high Tc superconductors. in which superconductivity may exist up to ~ 100°K, crossed a key technological barrier in that liquid nitrogen at 77°K, a relatively inexpensive technology, may now be used as a coolant to achieve the superconducting state. However it was soon realized that, due to the significantly higher temperatures as well as the anisotropic layered nature of the materials, thermal fluctuations are dramatically more important than in conventional superconductors. Such fluctuations generally degrade the superconducting performance of the material. For a bulk sample in an applied magnetic field H, vortex singularities form continuous lines that thread the system parallel to the applied field. In a pure sample at low temperatures and magnetic fields, these vortex lines crystallize into a solid state known as the Abrikosov vortex line lattice. Upon increasing temperature, however, thermal fluctuations lead to a melting of this vortex line lattice into a liquid of freely diffusing lines. The flux flow resistance associated with such a line liquid leads to undesirably small critical currents over much of the H-T phase diagram. This has led researchers to try and understand the transitions between different vortex phases in pure single crystal samples, as well as in samples with artificially introduced random or patterned defect structures designed to pin the vortex lines and reduce flux flow resistance. We are currently carrying out numerical simulations of models of interacting vortex lines in three dimensional systems in order to clarify the nature of the vortex phases and the transitions between them.

Simulated melting of a vortex line lattice to a tangled vortex line liquid.

Josephson Junction Arrays and Superconducting Networks

Josephson junction arrays are two dimensional superconducting structures created by modern lithographic techniques. Superconducting islands placed at the nodes of a network are connected by "weak links" along the bonds of the network. Electrons may coherently tunnel across the weak links from island to island, creating a "Josephson junction" on each bond. A magnetic field applied transverse to the plane of the array induces circulating supercurrents across the junctions, described by a distribution of point vortices located at the centers of a fraction of the cells of the network. The density of vortices is proportional to the strength of the applied magnetic field. Since vortices are constrained to sit on the discrete sites defined by the cells of the network, and since they feel strong mutual repulsions, the specific configuration of vortices that minimizes the total energy results from a delicate balance of these two effects. For a periodic network, properties in principle change discontinuously, due to complex commensurability effects, as the magnetic field (and hence vortex density) is continuously varied. Because superconducting networks can be fashioned in any desired geometry, with considerable control over the sample parameters, they form a useful model system in which one can study the effects of pinning and commensurability on vortex behavior in superconductors. We have been carrying out numerical simulations to study the intricate pattern of vortex structures that form as vortex density varies, and the transitions between these structures and less ordered vortex phases. Most recently we have begun investigations on the non-equilibrium steady state behavior of such arrays, when vortices are driven by uniform external currents.

Simulated diffraction patterns for a dilute system of vortices in a honeycomb Josephson array. From left to right are: pinned vortex lattice, floating vortex lattice, vortex liquid.

Left: experimental measurement of kinetic inductance showing strong variations as magnetic flux density is varied (from Théron et al., PRL 1994); right: experimental sample of a superconducting wire network with complex geometry (from Santhanam and Chi, Physica B 1998)