The Flux-Line Lattice in Superconductors

Magnetic flux can penetrate a type-II superconductor in form of Abrikosov vortices. These tend to arrange in a triangular flux-line lattice (FLL) which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-$T_c$ supercon- ductors (HTSC's) also by thermal fluctuations. Many properties of the FLL are well described by the phenomenological Ginzburg-Landau theory or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb alloys and HTSC's the FLL is very soft mainly because of the large magnetic penetration depth: The shear modulus of the FLL is thus small and the tilt modulus is dispersive and becomes very small for short distortion wavelength. This softness of the FLL is enhanced further by the pronounced anisotropy and layered structure of HTSC's, which strongly increases the penetration depth for currents along the c-axis of these uniaxial crystals and may even cause a decoupling of two-dimensional vortex lattices in the Cu-O layers. Thermal fluctuations and softening may melt the FLL and cause thermally activated depinning of the flux lines or of the 2D pancake vortices in the layers. Various phase transitions are predicted for the FLL in layered HTSC's. The linear and nonlinear magnetic response of HTSC's gives rise to interesting effects which strongly depend on the geometry of the experiment.Comment: Review paper for Rep.Prog.Phys., 124 narrow pages. The 30 figures do not exist as postscript file

The Beltrami equations and prime ends

We first study the boundary behavior of ring Q-homeomorphisms in terms of Carath´eodory’s prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation ∂f = μ∂f in arbitrary bounded finitely connected domains D of the complex plane C