Meissner shielding currents and magnetic flux penetration in thin-film superconductors

We have observed the magnetic flux penetration at 4.2 K into superconducting lead films of about 4 µm thickness using a high-resolution magnetooptical technique. The specimens had the shape of small squares or long strips. In increasing magnetic field it was found that magnetic flux starts to penetrate into the bulk of the specimens in the form of flux tubes containing about 60 flux quanta. For the samples with the strip geometry the magnetic fieldH*, at which flux penetration into the bulk sets in, was investigated in the presence of an external electrical currentI along the strip superimposed on the Meissner shielding currents. The fieldH* was found to decrease linearly with increasing transport currentI. The magnetic field near a long superconducting cylinder in the presence of both a transverse applied magnetic field and an applied electrical current parallel to the cylinder axis has been calculated. The calculation is valid for a cylinder of elliptical cross section, which approximates a long, flat strip. The experimental observations for the slope H*/I are in good agreement with the calculations.Work performed in part in the Argonne National Laboratory and in part in the Ames Laboratory of the U.S. Atomic Energy Commission.     

Viscous flux motion in anisotropic magnetic superconductors

Magnetic field penetrates in the form of flux lines or vortex thread into type-II magnetic superconductors (MSC) and induces magnetization of magnetic subsystem over a distance of an order of the London penetration depth surrounding the normal cores. When a flux line moves by, surrounding magnetization moves as a whole through the sample and a free motion of vortices is subjected to magnetic viscous drag, giving rise to dissipation. The flux flow resistance in the mixed state of anisotropic MSC has been studied on the basis of the London theory. Expressions for the dissipation and viscosity coefficient associated with the change of the magnetic subsystem as a vortex moves about are derived.     

Estimation of intragrain magnetic flux motion viscosity in high-temperature superconductors

A simple method for evaluation of the lower limit of intragrain magnetic flux motion viscosity is presented. We have also shown that in our experiment the energy loss in an alternating magnetic field in a granular YBa₂Cu₃O_7–x superconductor prepared by the standard sintering method is determined by thermally assisted magnetic flux flow into superconducting grains and does not depend on the intergrain space.     

Improvement of the superconducting magnetic levitation system forthe determination of the magnetic flux quantum

An improvement of the preliminary superconducting magnetic levitation system for the absolute determination of the magnetic flux quantum is described. This improvement includes the development of the flux-up method to determine the flux in terms of the Josephson voltage. The improvement is essential for the determination of the magnetic flux quantum as well as of the coil current, in terms of the Josephson voltage and quantized Hall resistance     

Flux-line cutting in hard superconductors

The effect of flux-line cutting upon the magnetic behavior of hard superconductors subjected to a dc bias magnetic field H_z and a transverse field H_y undergoing half-wave oscillations of large amplitude is investigated theoretically. We have applied both the generalized double critical-state model and the elliptic flux-line-cutting one to interpret available experimental results for VTi ribbons. The predictions of these models are compared and discussed.     

Experimentally realizable devices for controlling the motion of magnetic flux quanta in anisotropic superconductors

A new generation of microscopic ratchet systems is currently being developed for controlling the motion of electrons and fluxons, as well as for particle separation and electrophoresis. Virtually all of these use static spatially asymmetric potential energies to control transport properties. Here we propose completely new types of ratchet-like systems that do not require fixed spatially asymmetric potentials in the samples. As specific examples of this novel general class of ratchets, we propose devices that control the motion of flux quanta in superconductors and could address a central problem in many superconducting devices; namely, the removal of trapped magnetic flux that produces noise. In layered superconductors there are two interpenetrating perpendicular vortex lattices consisting of Josephson vortices (JVs) and pancake vortices (PVs). We show that, owing to the JV-PV mutual interaction and asymmetric driving, the a.c. motion of JVs and/or PVs can provide a net d.c. vortex current. This controllable vortex motion can be used for making pumps, diodes and lenses of quantized magnetic flux. These proposed devices sculpt the microscopic magnetic flux profile by simply modifying the time dependence of the a.c. drive, without the need for samples with static pinning--for example, without lithography or irradiation.     

Stable thermomagnetic waves in hard superconductors

The problem of the stability of a nonlinear thermomagnetic wave with respect to small thermal and electromagnetic perturbations in hard superconductors was studied. It is shown that spatially bounded solutions may correspond only to the perturbations decaying with time, which implies stability of the nonlinear thermomagnetic wave.     

Kinetic equations for clean superconductors: Application to the flux flow hall effect

The kinetic equations for clean superconductors (l) are derived. Expanding the equations for the time dependent Green functions in the quasiclassical parameter, the new contributions are found which contain the derivatives of the distribution functions with respect to the quasiparticle momentum. The transition from the ultra-clean case (no relaxation) to a relaxation-dominated behavior, for which the kinetic equations coincide with the usual quasiclassical approximation, occurs for the relaxation time of the order of E_F/². The kinetic equations can be used for various dynamic processes in superconductors including the flux-flow Hall effect. The derived equations, after necessary modifications for the p-wave pairing, are especially suitable for nonstationary problems in the theory of superfluidity of³He.     

Asymmetry between flux penetration and flux expulsion in Tl-2212 superconductors

dc magnetic hysteresis as well as flux penetration and flux expulsion were investigated in Tl_2–y Ba₂CaCu₂O_8–x polycrystals and monocrystals. All measurements were performed at 35 K and in the 0–5 T field range. Hysteresis measurements revealed an irreversibility field of about 2 T. Existing models predict identical field-cooled (fc) and zero-field-cooled (zfc) magnetizations and vanishing time dependence above this field. Although the identical fc and zfc magnetizations are in fact observed, the time dependence vanishes only for flux penetration after zero-field cooling; a remanence is preserved after field cooling and decays with a finite relaxation rate. Activation energies calculated on the basis of the thermal activation model display a pronounced field dependence, and arelower for flux penetration than for flux expulsion in high fields (H3 T) for all orientations. This behavior of extreme layered superconductors contradicts classical theoretical models and questions the original definition of the irreversibility line as well. All of our results are consistent with the recent theory of lock-in transition, and can be well interpreted by using those principles.     

Order parameter and magnetic field of the distorted vortex lattice and their application to flux pinning in type II superconductors. II. Curved flux lines

From the Ginzburg-Landau equations, the order parameter, local magnetic field, supervelocity, and elastic energy are derived for an arbitrarily distorted lattice of curved or straight flux lines. Exact solutions are given for small strains and for fields close to the upper critical fieldH _c2 . These solutions are generalized to large strains and arbitrary inductions. The general expressions approximately reproduce all previous results. They apply even to isolated and to crossed flux lines and to flux line loops. As a first application of the distorted-lattice solutions the pinning force density exerted by a small inclusion on the flux line elements is calculated. It turns out that the force may be concentrated on a flux line section of length smaller than even the coherence length. The response of the flux line lattice to such localized forces requires treatment by a nonlocal theory of elasticity and is much larger than expected so far.     

Optical levitation and trapping of a micro-optic inclined end-surface cylindrical spinner

Cylindrical micro-objects with inclined end surfaces have been observed to be optically trapped in a focused laser beam and to rotate about the beam's propagation direction. A simplified vector model is developed that accounts for the observed effect; the enhanced ray-optics model is used to simulate the behavior of time evolution for the cylindrical object in the beam. Experimental laser trap configurations are presented, along with a video image sequence of the cylinder's precession.     

Order parameter and magnetic field of the distorted vortex lattice and their application to flux pinning in type II superconductors. I. Parallel flux lines

A program for the rigorous calculation of the volume pinning force exerted by-imperfections on the flux line lattice in type II superconductors is outlined. As one step of this program, the order parameter, local magnetic field, supervelocity, and elastic energy of the distorted lattice of still parallel flux lines are calculated from the Ginzburg-Landau theory. These general solutions provide a transformation of the Ginzburg-Landau free energy functional of an imperfect material into a mere function of the flux line positions, and thus allow a definition of elastic and pinning force densities which for the first time apply also to large inductions. First, exact solutions for small strains and for fields close toH _c2 are derived. Then, approximate solutions are presented applying to larger strains and to arbitrary inductions. The forces on each flux line exerted by a small needle-shaped inclusion parallel to the flux lines are calculated.     

The wall-thickness dependence of magnetic shielding or trapping in a low-field superconductor,Pb40Sn60

The measured internal magnetic field H₂ of superconducting cylindrical tubes made of soft solder depends on the external magnetic field H₁ and the wall thickness W. An empirical equation relating H₁, H₂, and W is obtained. This empirical equation suggests that the distribution of the magnetic field in the wall of the tubes is expressed as 1/(h′)²=1+1/X², X₁≤X≤X₂; where h′ is the magnetic field normalized with respect to the critical field, and X₁ and X₂ are the dimensionless locations of the external and inner surfaces of the wall, respectively.     

A study of the flux density distribution in type II superconductors rotating in a magnetic field

Torque measurements on type II superconductors rotating in a magnetic field H ₀ directed perpendicular to the rotational axis can be used to determine pinning and viscous friction of vortices. The exact analysis of these measurements requires a knowledge of the vortex configuration in the rotating specimen, which is studied for the case of a circular cylinder of infinite length. A method is developed to calculate numerically the vortex configuration if the dependence of critical current density J _c and flux flow conductivity _f on flux density B and if the equilibrium magnetization curve are known. An inverse procedure also allows us to determine these material parameters from the measured torque and magnetization of the rotating sample. Results obtained by applying this procedure to different materials are reported.     

Activated flux creep in layered high-temperature superconductors

We study thermally activated flux creep in layered superconductors in the case when the magnetic field is parallel to the layers and vortices move across the layers. We assume that the sample is without imperfections, so that the pinning is due to an interaction between vortices and the layered microstructure of the superconductors. The height of the energy barrier that vortices overcome during each activation process is calculated in the case when the vortex lattice period is commensurate with the interlayer spacing. This energy barrier depends on the transport current: it grows at small currents and goes to zero linearly whenjj _c wherej _c is the depinning current.     

Magnetic flux noise in copper oxide superconductors

We report on the magnetic flux noise in thin films of YBa₂Cu₃O_7-x (YBCO), Tl₂Ca₂Ba₂Cu₃O_x, and TlCa₂Ba₂Cu₃O_x and in crystals of YBCO and Bi₂Sr₂CaCu₂O_8+x, measured with a Superconducting QUantum Interference Device (SQUID). We ascribe the noise to the motion of flux vortices. In the low magnetic fields in which the experiments are performed the average vortex spacing always exceeds the superconducting penetration depth. The spectral density of the noise usually scales as 1/f (f is frequency) from 1 Hz to 1 kHz and increases with temperature to a peak which is of the same magnitude in all samples, at the transition temperature. Furthermore, the noise power increases with the magnitude of the magnetic field in which the sample is cooled, with a power-law dependence over several decades, whereas a supercurrent well below the critical current density applied to YBCO films suppresses the noise power by an order of magnitude. Most of the measurements were made on YBCO films, and for this set of samples the noise decreases dramatically as the crystalline quality is improved. A model of thermally activated vortex motion is developed which explains the dependence of the noise on frequency, temperature, magnetic field, and current. The pinning potential is idealized as an ensemble of symmetrical double wells, each with a different activation energy separating the two states. From the noise measurements, this model yields the distribution of pinning energies, the vortex hopping distance, the number density of mobile vortices, and the restoring force on a vortex at a typical pinning site. The distribution of pinning energies in YBa₂Cu₃O_7-x shows a broad peak below 0.1 eV. Over narrow temperature intervals, most samples exhibit random telegraph signals in which the flux switches between two discrete levels, with activation energies and hopping distances much greater than those deduced from the 1/f noise measurements.     

Radiation-induced flux pinning in type II superconductors

In monocyrstalline foils of oxygen-doped niobium and niobium—zirconium alloys, statistically distributed or regularly arranged voids were created during irradiation with high-energy ⁵⁸Ni⁺ ions (3.5MeV, up to 8.1 × 10¹⁶ ions/cm²) at temperatures between 750 and 900°C. The voids exhibit a strong interaction with flux lines, which was determined from measurements of the (anisotropic) critical currents as a function of transverse magnetic field, temperature, and defect geometry. The experimentally determined volume pinning forces obey scaling laws and lead to elementary interaction forces between voids and fluxoids that are larger than theoretical values calculated for various possible mechanisms of interaction. The validity of the statistical summation of elementary forces is discussed.Research supported by the U.S. Energy Research and Development Administration, the Deutsche Forschungsgemeinschaft, and the Akademie der Wissenschaften in Göttingen. For numerical computations the facilities of the Gesellschaft für Wissenschaftliche Datenverarbeitung mbH Göttingen were used. This investigation is part of a Habilitation Thesis presented at the University of Göttingen.     

Non-isothermal flux penetration in type II superconductors

The development of thermal instabilities in a single niobium crystal has been studied experimentally under various external conditions. The observed flux jump behaviour cannot be explained in terms of existing theoretical models. A new model, taking account of surface currents only, is introduced. The results of model calculations agree well with the experimental results.     

Adaptive control of magnetic levitation system subject to external disturbances

In this paper, an adaptive controller is designed for a magnetic levitation system to cope with internal time-varying uncertainties and external disturbances. Since, in experimental studies, in experiments involving the traditional magnetic levitation design it is not easy to realize external disturbances to the system, a horizontal configuration is constructed in this paper. To facilitate the analysis and controller design, the equation of motion is derived in detail. Due to the asymmetric nature of the magnetic loop, there is a big challenge in the controller design process. In addition, since some of the uncertainties enter the system in a mismatched manner, few control strategies are feasible. A multiple-surface sliding control law is proposed with the function approximation technique to stabilize the closed loop system under various uncertainties and disturbances. A rigorous mathematical proof is given to verify the feasibility of the design. Experimental studies are conducted including comparisons with conventional PID design to clarify the performance of the proposed controller.