Ernst Helmut Brandt

The flux-line lattice in superconductors

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.

Levitation in Physics

Several physical effects allow free floatation of solid and even liquid matter. Materials may be levitated by a jet of gas, by intense sound waves, or by beams of laser light. In addition, conductors levitate in strong radio-frequency fields, charged particles in alternating electric fields, and magnets above superconductors or vice versa. Although levitation by means of ferromagnets is unstable, supper-conductors may be suspended both above and below a magnet as a result of flux pinning. Levitation is used for containerless processing and investigation of materials, for frictionless bearings and high-speed ground transportation, for spectroscopy of single atoms and microparticles, and for demonstrating superconductivity in the new oxide superconductors.

Elastic energy of the vortex state in type II superconductors. I. High inductions

The elastic properties of the flux line lattice (FLL) in type II superconductors are calculated from the linearized Ginzburg-Landau (GL) theory for large inductionsB≈Hc2. They appear to be strongly nonlocal, i.e., the elastic modulic11 andc44 for homogeneous deformations do not apply if the strain field varies over distances λ/(1−B/Hc2)1/2 ≫d (λ is the penetration depth,d is the FL distance). For smaller strain wavelength,c11 andc44 are smaller by factors (1−B/Hc2)2/2κ2 and (1−B/Hc2)/ 2κ2, respectively. The order parameter and local field of a deformed FLL exhibit the expected spatial “frequency modulation,” but also a pronounced “amplitude modulation” whose degree of modulation increases with the strain wavelength. The results of further calculations avoiding the linearization of the GL theory are given.

Acoustic physics: Suspended by sound

Ultrasound waves can levitate heavy balls of tungsten. This contact-free method of keeping items suspended in the air can be applied to the investigation and processing of new materials.

Friction in levitated superconductors

A type I superconductor levitated above a magnet of low symmetry has a unique equilibrium position about which it may oscillate freely. In contrast, a type II superconductor has a continuous range of stable equilibrium positions and orientations where it floats rigidly without swinging or orbiting as if it were stuck in sand. A strong internal friction conspicuously indicates the existence and unpinning of flux lines in oxide superconductors levitated above liquid nitrogen. It is shown how these effects follow from the hysteretic magnetization curves and how the energy is dissipated.

Flux line lattice in high-Tc superconductors: anisotropy, elasticity, fluctuation, thermal depinning, AC penetration and susceptibility

Abstract Various properties of the flux-line lattice (FLL) in high- T c superconductors are compiled, derived, and discussed. Useful general and updated expressions for the energy and magnetic field of arbitrary flux-line arrangements in anisotropic and layered superconductors are presented. From these follow the nonlocal elastic moduli of the FLL which are then applied to the intricate problems of thermal fluctuations and thermal depinning of the FLL and to the linear response of the FLL to applied AC fields. Explicit expressions are derived for the complex AC penetration depth, surface impedance, AC resistivity and for the AC susceptibility of films and cylinders, taking full account of elastic pinning and viscous drag of vortices, and of thermally activated linear flux creep or tunneling. These AC properties are then applied to vibrating superconductors and to thermally activated flux diffusion, which gives a quantitative explanation of the depinning lines observed in various experiments. Finally, the shear-limited critical current density is estimated and FLL decoration experiments are discussed.

THERMAL DEPINNING AND “MELTING” OF THE FLUX-LINE LATTICE IN HIGH-Tc SUPERCONDUCTORS

In high-Tc superconductors (HTSC) the thermal fluctuation of the vortex lattice (VL) may become large since the vortex lattice is soft due to the strong overlap of the vortex fields and since the temperature T can be high. It was thus argued that the three-dimensional (3D) vortex lattice is thermally entangled and may “melt”. This type of transition and the consequences of melting are not clear as yet since the always present pinning of the vortex cores by material inhomogeneities may cause similar disorder. In HTSC the pinning energy may become comparable with kBT because the coherence length ξ (vortex radius) is small and T may be high. Therefore, thermally activated depinning competes with possible effects of “flux melting”, and the “irreversibility line” in the B-T-plane (B=magnetic field) should better be called “depinning line”. Due to the diffusive character of flux motion the depinning line of a given experiment, a line of constant flux diffusivity D(T, B), depends on the frequency or sweep rate, ...

Peak effect, vortex-lattice melting line, and order-disorder transition in conventional and high-Tc superconductors

We investigate the order - disorder transition line from a Bragg glass to an amorphous vortex glass in the H-T phase diagram of three-dimensional type-II superconductors with account of both pinning-caused and thermal fluctuations of the vortex lattice. Our approach is based on the Lindemann criterion and on results of the collective pinning theory and generalizes previous work of other authors. It is shown that the shapes of the order - disorder transition line and the vortex lattice melting curve are determined only by the Ginzburg number, which characterizes thermal fluctuations, and by a parameter which describes the strength of the quenched disorder in the flux-line lattice. In the framework of this unified approach we obtain the H-T phase diagrams for both conventional and high-Tc superconductors. Several well-known experimental results concerning the fishtail effect and the phase diagram of high-Tc superconductors are naturally explained by assuming that a peak effect in the critical current density versus H signalizes the order - disorder transition line in superconductors with point defects.

Optical conductivity of BCS superconductors with arbitrary purity

Abstract An explicit expression for the complex optical and AC conductivity of a homogeneous BCS superconductor with arbitrary electron mean free path is given. This compact expression and a fast self-contained FORTAN program may be used to fit experimental data. For comparison, we give also the complex AC conductivity of high- T c superconductors containing an elastically pinned, viscously moving flux-line lattice with flux creep.

Magnetic field density of perfect and imperfect flux line lattices in type II superconductors. I: Application of periodic solutions

The densityn(B) of the spatially varying magnetic fieldB inside a type II superconductor can be measured by nuclear magnetic resonance or muon-spin rotation (μ+SR). For a perfect flux-line latticen(B) exhibits van Hove singularities at the maximum, minimum, and saddle point values of the ideally periodicB(r). In a real superconductor, these singularities are smeared due to distortions of the flux-line lattice caused by, e.g., the interaction of flux lines with inhomogeneities in the material (pinning), structural defects in the flux-line lattice, the nonellipsoidal shape of the specimen, or fluctuations of the applied field and temperature. Such perturbations of the periodicity ofB(r) typically broaden the idealn(B) by convolution with a Gaussian whose width in general depends onB and which thus smears each singularity differently. Knowledge of the broadening is required for the interpretation of μ+SR experiments in the new ceramic superconductors and also in pure niobium, where it competes with the broadening caused by the diffusion of the positive muons. In this paper (Part I), the broadening ofn(B) is discussed in detail and some of its features are derived from the periodic solutions of the Ginzburg-Landau and BCS-Gorkov theories. Forthcoming parts will deal with the application of nonperiodic solutions and with computer simulations.