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

The structure of vortices within an infinite stack of thin superconducting layers is considered and examined in detail in the limit of zero interlayer Josephson coupling The basic building block for the description of three-dimensional (3D) vortex lines is shown to be the 2D pancake vortex, which is a vortex located in only one of the layers; the other layers contain no vortices, but have an important effect in screening the magnetic field generated by currents in the first layer It is shown that 3D vortex lines can be built up by superposing the contributions of stacks of 2D pancake vortices Thermal excitation is shown to break up a single 3D vortex line at a temperature corresponding to the Kosterlitz-Thouless temperature of a single superconducting layer The effect of thermally induced decoupling of the 2D vortex solids in different layers, corresponding to melting only in the direction perpendicular to the layers, is also considered It is shown that Josephson coupling can be neglected in the high-temperature superconductors only under very stringent conditions Although these conditions evidently are not met in ${\mathrm{Bi}}_{2}$${\mathrm{Sr}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{8}$ and ${\mathrm{Tl}}_{2}$${\mathrm{Ba}}_{2}$${\mathrm{Ca}}_{1}$${\mathrm{Cu}}_{2}$${\mathrm{O}}_{8}$, they should be satisfied in superconducting-insulating multilayer systems, such as ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$/${\mathrm{PrBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$

Two-dimensional vortices in a stack of thin superconducting films: A model for high-temperature superconducting multilayers

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

For H‖c-axis, the magnetic field induced broadening of the resistive transitions of high-Tc superconductors (HTS) is shown to depend strongly on the Cu-O layer spacing. For the highly anisotropic HTS, we show experimental evidence that flux motion results from a thermally activated crossover from three dimensional (3D) vortex lines to 2D independent pancake-like vortices in the Cu-O layers, which is intrinsic to the material and occurs when kBT exceeds the Josephson coupling energy of these layers. At low temperatures, however, thermally activated conventional depinning (which can be sample dependent) or melting in the uncoupled 2D Cu-O layers is also required for flux motion. For YBa2Cu3O7, this dimensional crossover does not occur belowHc2, presumably because the conducting Cu-O chains short-circuit the Josephson interlayer coupling, leading to better superconducting properties in a magnetic field. These results show that strong interlayer coupling is a key to finding good alternatives.

Effect of cu-o layer spacing on the magnetic field induced resistive broadening of high-temperature superconductors

An unusual loss mechanism has been commonly observed in the high-temperature superconductors in a magnetic field near the zero-field transition temperature {ital T}{sub {ital c}}. The resistive transitions associated with this loss in highly {ital c}-axis-oriented Tl{sub 2}Ba{sub 2}CaCu{sub 2}O{sub {ital x}} thin films are presented as a function of field perpendicular to the {ital c} axis, for the current {ital both} {ital perpendicular} {ital and} {ital parallel} {ital to} {ital the} {ital field}. The data for these orientations are essentially the same over a span of 5 orders of magnitude below the normal-state resistance. Since the {ital macroscopic} Lorentz force is zero for the current parallel to the magnetic field, it can play, at most, a significantly reduced role in this loss mechanism, and our results seriously question any explanation of these losses based on magnetic flux motion.

Lorentz-force independence of resistance tails for high-temperature superconductors in magnetic fields near Tc.

Electrical transport properties of {ital c}-axis-oriented Tl{sub 2}Ba{sub 2}CaCu{sub 2}O{sub 8} thin films have been examined within the context of the Kosterlitz-Thouless (KT) model. The nonlinear current versus voltage and resistivity versus magnetic field characteristics below the KT transition temperature ({ital T}{sub {ital c}}=99.0 K), together with the exponential inverse-square-root temperature dependence of the resistivity just above {ital T}{sub {ital c}}, were consistent with each other and with the theory. The parametrization of the resistivity data well above the transition by Aslamazov-Larkin theory was also consistent with that of the KT theory.

Kosterlitz-Thouless transition in Tl2Ba2CaCu2O8 thin films.

Superconductivity in thin films of the Bi−Ca−Sr−Cu−O system

The upper critical field Bc2 provides the most direct measure of the intrinsic anisotropy of a superconductor. For highly oriented, sputtered thin films of the high‐temperature superconductor Tl2Ba2CaCu2Ox, we find a Bc2 anisotropy of at least 70, which exceeds similar measurements on thin films and single crystals of the high‐temperature superconductors Bi‐Sr‐Ca‐Cu‐O and YBa2Cu3Oy. We discuss why the midpoints of the resistive transitions, used in these measurements, may be appropriate for defining the intrinsic Bc2.

R. T. Kampwirth

Papers

Lorentz-force independence of resistance tails for high-temperature superconductors in magnetic fields near Tc.

Kosterlitz-Thouless transition in Tl2Ba2CaCu2O8 thin films.

Superconductivity in thin films of the Bi−Ca−Sr−Cu−O system

Large anisotropy in the upper critical field of sputtered thin films of superconducting Tl‐Ba‐Ca‐Cu‐O

High resolution magneto-optical studies of the intermediate state in thin film superconductors