Magnetic flux can penetrate a type-II superconductor in the form of Abrikosov vortices (also called flux lines, flux tubes, or fluxons) each carrying a quantum of magnetic flux phi 0=h/2e. These tiny vortices of supercurrent tend to arrange themselves in a triangular flux-line lattice (FLL), which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-Tc superconductors (HTSCs) 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 HTSCs the FLL is very soft mainly because of the large magnetic penetration depth lambda . The shear modulus of the FLL is c66~1/ lambda 2, and the tilt modulus c44(k)~(1+k2 lambda 2)-1 is dispersive and becomes very small for short distortion wavelengths 2 pi /k<< lambda . This softness is enhanced further by the pronounced anisotropy and layered structure of HTSCs, which strongly increases the penetration depth for currents along the c axis of these (nearly 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 ofthe two-dimensional `pancake vortices` in the layers. Various phase transitions are predicted for the FLL in layered HTSCs. Although large pinning forces and high critical currents have been achieved, the small depinning energy so far prevents the application of HTSCs as conductors at high temperatures except in cases when the applied current and the surrounding magnetic field are small.

https://iopscience.iop.org/article/10.1088/0034-4885/58/11/003/pdf

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.

The Flux-Line Lattice in Superconductors

Magnetization oscillations of a superconducting disk

In a simple bilayer geometry, a thin, microstructured ferromagnetic film spans a superconducting strip. Locally strong magnetic fringe fields at the edge of the ferromagnet quench the superconductivity, creating a weak link. The strength of the weak link can be varied between on and off by controlling the orientation of the in-plane magnetization of the ferromagnet. The observation of Shapiro steps demonstrates the ac Josephson effect in the “quenched” state. In this switchable Josephson junction, energy is required only to change states, which are thereafter maintained in thermodynamic equilibrium.

Nonvolatile switchable josephson junctions

Direct, high-resolution observations of magnetic structures in evaporated films of the type I superconductors lead, tin, and indium are presented for a range of applied fields. The films are polycrystalline and range in thickness from 200 to 20,000 A. the transition from the intermediate state to the mixed state is observed. The observation of the single-fluxoid vortex mixed state in the thin films with Ginzburg-Landau parameter κ less than$1/\sqrt 2$ confirms theoretical prediction by Tinkham, Maki, and Lasher. No ordered array of vortices is observed and this is shown to be caused by the effects of the grain boundaries on the magnetic structures.

Direct observations of the magnetic structure in thin films of Pb, Sn, and In

Recent direct observations of the magnetic structure in thin films of Pb, Sn, and In are discussed relative to resistance measurements on the same films and to earlier experimental and theoretical work on thin films. Comparison to the theoretical results of Maki and Lasher is made and it is found that these results predict reasonably accurately the critical thickness separating intermediate-and mixed-state behavior in the thin films studied. This implies that the critical thickness has been overestimated in earlier work on similar films. It is suggested that this accounts, in part, for the high values obtained for the pure, bulk Ginzburg-Landau parameter in those experiments.

G. J. Dolan

Papers

Direct observations of the magnetic structure in thin films of Pb, Sn, and In

Critical thicknesses for the transition from intermediate- to mixed-state behavior in superconducting thin films of Pb, Sn, and In