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

Since the first days of high-Tc superconductivity, the materials science and the physics of grain boundaries in superconducting compounds have developed into fascinating fields of research. Unique electronic properties, different from those of the grain boundaries in conventional metallic superconductors, have made grain boundaries formed by high-Tc cuprates important tools for basic science. They are moreover a key issue for electronic and large-scale applications of high-Tc superconductivity. The aim of this review is to give a summary of this broad and dynamic field. Starting with an introduction to grain boundaries and a discussion of the techniques established to prepare them individually and in a well-defined manner, the authors present their structure and transport properties. These provide the basis for a survey of the theoretical models developed to describe grain-boundary behavior. Following these discussions, the enormous impact of grain boundaries on fundamental studies is reviewed, as well as high-power and electronic device applications.

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Grain boundaries in high- T c superconductors

We review experimental studies of the time decay of the nonequilibrium magnetization in high-temperature superconductors, a phenomenon known as magnetic relaxation. This effect has its origin in motion of flux lines out of their pinning sites due to thermal activation or quantum tunneling. The combination of relatively weak flux pinning and high temperatures leads to rich properties that are unconventional in the context of low temperature superconductivity and that have been the subject to intense studies. The results are assessed from a purely experimental perspective and discussed in the context of present phenomenological theories. [S0034-6861(96)00403-5]

Magnetic relaxation in high-temperature superconductors

In the past few years magneto-optical flux imaging (MOI) has come to take an increasing role in the investigation and understanding of critical current densities in high-Tc superconductors (HTS). This has been related to the significant progress in quantitative high-resolution magneto-optical imaging of flux distributions together with the model-independent determination of the corresponding current distributions. We review in this article the magneto-optical imaging technique and experiments on thin films, single crystals, polycrystalline bulk ceramics, tapes and melt-textured HTS materials and analyse systematically the properties determining the spatial distribution and the magnitude of the supercurrents. First of all, the current distribution is determined by the sample geometry. Due to the boundary conditions at the sample borders, the current distribution in samples of arbitrary shape splits up into domains of nearly uniform parallel current flow which are separated by current domain boundaries, where the current streamlines are sharply bent. Qualitatively, the current pattern is described by the Bean model; however, changes due to a spatially dependent electric field distribution which is induced by flux creep or flux flow have to be taken into account. For small magnetic fields, the Meissner phase coexists with pinned vortex phases and the geometry-dependent Meissner screening currents contribute to the observed current patterns. The influence of additional factors on the current domain patterns are systematically analysed: local magnetic field dependence of jc(B), current anisotropy, inhomogeneities and local transport properties of grain boundaries. We then continue to an overview of the current distribution and current-limiting factors of materials, relevant to technical applications like melt-textured samples, coated conductors and tapes. Finally, a selection of magneto-optical experiments which give direct insight into vortex pinning and depinning mechanisms are reviewed.

https://iopscience.iop.org/article/10.1088/0034-4885/65/5/202/pdf

Magneto-optical studies of current distributions in high-Tc superconductors

In this paper, magneto-optical observations and analyses of skeleton structures of flux distributions in type-II superconductors are presented. It will be shown that these structures are formed by lines where the critical current is bending. Such lines are a characteristic feature of a vortical vector field if the absolute value of the vector is constant as for the current distribution in a type-II superconductor in the full critical state. Thin samples offer enhanced sensitivity to observe these current discontinuity lines magneto-optically due to two facts: The Meissner phase contracts to these lines and remains there up to fields, which are higher than the full penetration field ${\mathit{H}}^{\mathrm{*}}$. At much higher magnetic fields, when the Meissner phase has vanished, the logarithmic infinity of the magnetic flux density at these lines leads to sufficiently high contrast. The positions of the discontinuity lines are not affected directly by particular defects, but are determined by the shape of the sample. Because the vortices are not able to cross these lines, the current discontinuity lines determine the flux distribution in the superconductor like a magnetic skeleton. An analysis of the observed structures yields characteristic parameters of the sample as anisotropy, inhomogeneity, or irradiation-induced enhancement of the critical current density. Additionally, the presence of defects allows the observation of the direction of flux motion and, thus, to visualize directly the current distribution. A strong concentration of flux motion near defects without any pecularities of the current and magnetic flux distribution can be predicted using the consideration presented in this paper.

Observation of current-discontinuity lines in type-II superconductors.

The penetration and exit of magnetic flux in type-II superconductors is investigated for the realistic situation where a transverse magnetic field is applied to a square or rectangular plate or film. In rectangular specimens the pattern of the sheet current and of the density of the perpendicular flux has some common features with the one-dimensional distributions in circular disks or long strips. Other features, however, are characteristic for the rectangular shape, e.g., the starlike pattern of the penetrating flux and, in the fully penetrated critical state, the discontinuity lines at which the current stream lines perform sharp bends and at which the perpendicular magnetic field ${\mathit{H}}_{\mathit{z}}$(x,y) exhibits sharp ridges. These typical features have to be calculated from a genuine two-dimensional theory. Such a theory based on a highly nonlinear current-voltage law is outlined. The field patterns obtained by this general theory are compared with patterns observed magneto-optically at the surface of square and rectangular single crystals or films of high-${\mathit{T}}_{\mathit{c}}$ superconductors with homogeneous and inhomogeneous critical-current distribution. It is shown that the analysis of the current-discontinuity lines is essential to understand the flux dynamics in superconductors. In samples with inhomogeneous critical current density ${\mathit{j}}_{\mathit{c}}$(r), a strong concentration of flux motion and electric field can occur along the lines where ${\mathit{j}}_{\mathit{c}}$ changes abruptly. This may trigger flux jumps.

Current and field pattern in rectangular and inhomogeneous superconductors

Flux penetration and overcritical currents in flat superconductors with irradiation-enhanced edge pinning: Theory and experiment.

The penetration of magnetic flux into flat type-II superconductors of various shapes in a perpendicular magnetic field is investigated in detail. The magnetic field distribution at the sample surface is observed by the magneto-optical Faraday effect and calculated from first principles. The investigations are performed on ${\mathrm{DyBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ and ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ samples which were shaped into a cross or an indented rectangle by a laser-cutting technique. Magnetic and electric field and current distributions are calculated from Maxwell's equations treating the superconductor as a conductor with a highly nonlinear current-voltage law and zero reversible magnetization. A large concentration of magnetic flux and electric field and a high flux-line velocity occur at concave sample corners. This results from the fact that the flux lines can penetrate into regions of the sample which are bounded by the extensions of the sample edges only at these points. This large electric field and related energy dissipation are particularly relevant for superconducting tapes, in which ``sausaging'' effects (variations of the filament cross section) reduce their performance as an ideal conductor. Huge jumps of the electric field occur where the current flow changes from a straight to a circular path. This jump diverges as one over the distance to the corner at sharp indents or concave corners. \textcopyright{} 1996 The American Physical Society.

Th. Schuster

Papers

Observation of current-discontinuity lines in type-II superconductors.

Current and field pattern in rectangular and inhomogeneous superconductors

Flux penetration and overcritical currents in flat superconductors with irradiation-enhanced edge pinning: Theory and experiment.

Flux penetration into flat superconductors of arbitrary shape: Patterns of magnetic and electric fields and current

EuSe as magneto-optical active coating for use with the high resolution Faraday effect