Flux-line cutting in type II superconductors

Experimental evidence for flux-line cutting in superconductors (intersection and cross-joining of singly quantized vortices) is briefly reviewed. The interaction energy between two straight vortices tilted at an angle ( 0)is then shown to be finite in the London model, i.e., in the limit of vanishing core radius. Next, the activation energy and maximum interaction force are calculated for the vortices in an analytic approximation to the Ginzburg-Landau theory. Here two competing interactions determine the behavior. Electromagnetic repulsion (0 < < /2) varies as cos and decays over distances scaled by the penetration depth , while core attraction is independent of and varies over distances scaled by the coherence length . The force is always repulsive at large flux-line separation (0 < < /2) and its maximum decreases rapidly as decreases, so that flux-line cutting isexpected to be more probable in low- materials. The calculations provide a basis for explaining longitudinal flux-flow resistance as well as some intriguing magnetization behavior in the same configuration.This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences Division, and by the Deutsche Forschungsgemeinschaft.On leave from Max-Planck-Institut für Metallforschung, Institut für Physik, Stuttgart, West Germany.On leave from New University of Ulster, Coleraine, Northern Ireland.     

Steady-state flux-line cutting in type II superconductors

Flux-line cutting (intersection and cross-joining of adjacent nonparallel vortices) has been suggested as a mechanism for steady-state dissipation in current-carrying, type II superconductors in longitudinal magnetic fields. In this paper a specific theoretical flux-line-cutting model is proposed which generates a constant steady-state electric field in a current-carrying, ideal, type II superconducting slab in a longitudinal field. The assumed model consists of parallel vortex planes at different angles which periodically shuttle back and forth between the regions in which flux-line cutting occurs. The resulting macroscopic electric current and magnetic flux density distributions are calculated. The model yields a nonlinear voltage-current characteristic and a longitudinal paramagnetic moment.This work was supported by the U.S. Department of Energy, contract No. W-7405-Eng-82, Division of Materials Sciences budget code AK-01-02-02-3.     

Longitudinal critical current in type II superconductors. II. Helical vortex instability near the surface

The calculation of the critical current for helical instability of the flux-line lattice, performed in a previous paper for a bulk superconductor, is extended to include the effect of a planar surface. The extension is of interest since an applied longitudinal current in general flows near the surface before the helical instability of the flux-line lattice triggers the transition to the flux-flow state. The magnetic stray field of the surface increases the critical current by a factor 1.41 (1.34) for weak (moderate) pinning, and it modifies the axis ratio of the helices. The pitch length and the extension of the helical mode into the specimen are decreased from their bulk values by surface pinning, but they are slightly increased if bulk pinning dominates.     

Irreversible thermodynamics for vortex motion in type II superconductors

Irreversible thermodynamics consideration is applied to give a systematic discussion of dissipative processes involving vortex motion in superconductors.     

Longitudinal critical current in type II superconductors. I. Helical vortex instability in the bulk

The vortex lattice in type II superconductors is unstable against the growth of helical perturbations if the current along the vortices exceeds a critical value. The longitudinal critical current, the pitch, and the spatially varying amplitude of the elliptically polarized helices are calculated from the London theory at the onset of instability in planar current distributions far from the surface. For weak pinning (_L² c ₆₆) the wavelength and width of the mode extend over the entire specimen, and the critical current is 2H(c ₆₆/c ₁₁)^1/4. For moderate pinning (c _{66 L}² c ₁₁) the wavelength and width are close to Campbell's pinning length (c ₁₁/_L)^1/2, and the critical current times its mean density is 2H ²(_L/c ₁₁)^1/2. For strong pinning (_L² c ₁₁) helical instability occurs at pin-free vortex sections, the helix wavelength is 2.2d, and the critical current density is 0.47Hd/² (H, d, c ₁₁ and c ₆₆), and _L are the magnetic field, spacing, elastic moduli, and pinning parameter of the vortex lattice, and is the magnetic penetration depth).     

Microscopic calculations on the vortex state of type II superconductors

Eilenberger's quasiclassical equations are solved numerically, for the first time without approximations, for a hexagonal vortex lattice. The numerical methods used in this computation are described in detail. Particular emphasis is laid on the thermodynamics of low-, type II superconductors, where a first-order transition at the lower critical fieldH _c1 occurs (type II/1 behavior). The phase boundary separating type II/1 from type II behavior is computed and compared with the predictions of the asymptotic theory. Besides the thermodynamic properties, the microscopic structure of the order parameter and magnetic field is discussed in detail. On the boundary of the Wigner-Seitz cell the magnetic field shows a new stationary point (saddle point). It should be possible to test this prediction by SR experiments.     

Computer simulation of vortex pinning in type II superconductors. II. Random point pins

Pinning of vortices in a type II superconductor by randomly positioned identical point pins is simulated using the two-dimensional method described in a previous paper (Part I). The system is characterized by the vortex and pin numbers (N _v ,N _p ), the vortex and pin interaction ranges (R _v ,R _p ), and the amplitude of the pin potentialA _p . The computation is performed for many cases: dilute or dense, sharp or soft, attractive or repulsive, weak or strong pins, and ideal or amorphous vortex lattice. The total pinning forceF as a function of the mean vortex displacementX increases first linearly (over a distance usually much smaller than the vortex spacing and thanR _p ) and then saturates, fluctuating about its average \(\bar F\) . We interpret \(\bar F\) as the maximum pinning forcej _c B of a large specimen. For weak pins the prediction of Larkin and Ovchinnikov for two-dimensional collective pinning is confirmed: \(\bar F\) =const· \(\bar W\) /R _p c ₆₆, where \(\bar W\) is the mean square pinning force andc ₆₆ is the shear modulus of the vortex lattice. If the initial vortex lattice is chosen highly defective (“amorphous”) the constant is 1.3–3 times larger than for the ideal triangular lattice. This finding may explain the often observed “history effect”. The function \(\bar F\) (A _p ) exhibits a jump, which for dilute, sharp, attractive pins occurs close to the “threshold value” predicted for isolated pins by Labusch. This jump reflects the onset of plastic deformation of the vortex lattice, and in some cases of vortex trapping, but isnot a genuine threshold. For strong pins \(\bar F\) ～(N _p \(\bar W\) )^1/2 approaches the direct summation limit. For both weak and strong pinningj _c B is related to the mean squareactual (not maximum) force of each pin. This mean square in general is not proportional toA _p ² but, due to relaxation of the vortex lattice, may be smaller or larger than its rigid-lattice limit. Therefore, simple power lawsj _c ～n _p A _p ² orj _c ～n _p A _p in general donot hold except for very weak or unphysically strong pinning.     

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

The elastic energy of a distorted lattice of flux lines interacting by an arbitrary two-body potential is calculated. At low inductions, where the London model applies, the elastic response to localized forces appears to be larger than expected from local elasticity theory. This effect is most pronounced in hard superconductors, where the effective moduli for compression and bending waves may decrease by a factor 2² B/H _c2 . Such a behavior agrees with that obtained previously from the Ginzburg-Landau theory at large inductions. Those results can be interpreted by an effective interaction with range /(1-B/H _c2 )^1/2 and amplitude (H _{c2 - B)} .     

Computer simulation of vortex pinning in type II superconductors. I. Two-dimensional simulation

The summation of pinning forces to a volume force exerted on the vortex lattice in type II superconductors allowing it to carry a loss-free current is not fully understood. In order to clarify this question we have started computer simulations of flux pinning. It is shown that for many experimental situations bending of vortices may be neglected since the vortices are too short or pinning is too weak, and thus pinning is two-dimensional. As a first step, two-dimensional pinning simulations will thus be instructive with regard to, say, ribbons of amorphous metals. A general expression for the energy of a vortex-pin system in two and three dimensions is given. The simulation method is presented and illustrated for the isolated pin (with a detailed discussion of the “threshold effect” and of elastic instabilities) and for pin “walls” (grain boundaries) and “nozzles.” Random point pins acting on a perfect or defective vortex lattice are treated in an accompanying paper.     

Temperature dependence of the vortex nucleation field of thin-film,type II superconductors

The critical supercurrent versus applied field parallel to the surface of Pb-20 wt % In thin films was measured at various temperatures. The temperature dependence of the first vortex nucleation field (the lower critical field of a thin-film superconductor) suggests that the vortex structure is distorted due to surface effects.     

Pinning in type II superconductors

Large and randomly arranged pinning centers cause a strong deformation of a flux line lattice, so that each pinning center acts on the lattice with a maximum force. The average force for such single-particle pinning can be inferred from a simple summing procedure and has a domelike dependence on magnetic field. Pinning centers of average force, such as clusters of dislocations, strongly deform the flux line lattice only in weak fields and in fields close to the critical field, where there is a peak in the dependence of the critical current on magnetic field. In the range of intermediate fields there is a weak collective pinning. A large concentration of weak centers leads to collective pinning in all fields. In this case, near the critical field a critical current peak should be observed. To explain this peak and to define the boundaries between the regions of collective and single-particle pinning the possible break-off of the flux line lattice from the lines of magnetic force should be taken into consideration, which leads to extra softening of the lattice.     

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 inductionsBH _c2 . They appear to be strongly nonlocal, i.e., the elastic modulic ₁₁ andc ₄₄ for homogeneous deformations do not apply if the strain field varies over distances /(1–B/H _c2 )^1/2 d ( is the penetration depth,d is the FL distance). For smaller strain wavelength,c ₁₁ andc ₄₄ are smaller by factors (1–B/H _c2 )²/2 ² and (1–B/H _c2 )/ 2 ² , 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.     

Vortex motion in type II superconductors with controllable type defects

A model that connects the nonlinearity of the voltage-current characteristicsV(I) in the mixed state with the statistical distribution of the pinning forces acting on moving vortices is considered. The role of the elastic properties of the vortex lattice in theV(I) behavior is discussed. A method for obtaining the statistical distribution parameters of the pinning force from experimentalV(I) data is considered, based on experimental data onV(I) dependences in Al-Ag. In alloys with controllable defects, the distribution parameters of the pinning force can be found and connected with typical parameters of the defect structure of the specimen.     

Flux-line-cutting threshold in type II superconductors

The threshold for flux-line cutting (intersection and cross-joining of adjacent nonparallel vortices) is considered theoretically for a model array of parallel, equally spaced vortex planes. The vortices within each plane are parallel to each other but are inclined at an angle with respect to vortices in adjacent planes. Forces between vortices are calculated numerically using the interaction force proposed by Brandt, Clem, and Walmsley. Although for small interplanar angles the array is stable, for sufficiently large angles infinitesimal perturbations of the array grow until flux-line cutting occurs.This work was supported by the U.S. Department of Energy, contract No. W-7405-Eng-82, Division of Materials Sciences, budget code AK-01-02-02-3.     

Vortex-induced rectification in type II superconductors

We report a large rectification effect in superconducting films in a parallel magnetic field. This rectification effect is manifested in two features in current-voltage characteristics: The critical current, I_c,is found to differ by as much as 40% for negative and positive currents, and beyond I_c,the magnitude of the voltage is different for positive and negative currents, ¦V(+I)¦ ¦V(–I)¦.Furthermore, the critical current difference ¦I_c+¦ – ¦I_c–¦shows complicated behavior, changing sign as temperature and magnetic field are varied. We discuss a model based on the Bean-Livingston surface barrier and inhomogeneous bulk pinning that accounts for all observed behavior.     

Shape — Effects in type II superconductors

The magnetisation behaviour M(H °) of ideal and non-ideal Type II superconductors in disc-geometry and in perpendicular applied field H ° is obtained by numerical calculation and is compared with experiment on niobium and BSCCO single crystal specimens.     

Magnetic pinning in inhomogeneous type II superconductors

The magnetic structure of a vortex in the presence of an anisotropic inhomogeneity distribution is calculated within the London model. The anisotropic supercurrent pattern consists of closed current loops, superimposed on a backflow pattern. The nonvanishing current density at the vortex core gives rise to an intrinsic Lorentz force identical with the pinning force resulting from the spatial variation of the vortex self-energy.     

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.