Motion of the vortex lattice in a dirty type II superconductor

We reexamine here microscopically the transport properties of the vortex state in a dirty type II superconductor in the high-field region. We show that the previous theory by Caroli and Maki predicts the entropy carried by a single vortex line, which is consistent with recent experiments, provided the expression of the heat current is properly corrected. Furthermore, we show that a large heat delivered by vortex lines together with the normal thermoelectric effect, which persists even in the vortex state, results in a large Hall current in the vortex state. This may account for a large Hall effect observed by Niessenet al. in Nb-Ta alloys.     

Rotation and inversion symmetry properties of flux-line lattice in type II superconductors

Magnetic flux lines are arranged regularly in the mixed state of a type II superconductor and form a two-dimensional triangular lattice. This is unaltered on rotation through an angle /3, one of the flux lines being an axis of symmetry of the sixth order. It is also symmetric with respect to the inversion. The center of symmetry is any point on a flux line. The symmetry properties of the magnetic field, the order parameter, and Eilenberger's integrated Green's function are derived. The magnetic field is developed into Fourier series with the help of a two-dimensional reciprocal lattice. The normalized Fourier coefficient is the form factor in neutron diffraction and is shown to have a well-known property of the rotation symmetry. The Fourier transform of the normal Green's function satisfies a similar symmetry relation with an additional phase factor. When we expand the anomalous Green's function in terms of plane wave functions along the direction of one of the nearest-neighbor flux lines, the symmetry properties give useful conditions which the expansion coefficient satisfies as a function of the orthogonal coordinate and wave number vector. In addition to the plane wave expansion, the order parameter can be developed in terms of the harmonic oscillator eigenfunctions of the orthogonal coordinate with 6n quanta. The first term withn = 0 reproduces Abrikosov's solution. The symmetry properties help us effectively simplify the Eilenberger equation. An example of the simplification is given.     

The flux-line lattice in high-T c superconductors

The flux-line lattice in type-II superconductors has unusual nonlocal elastic properties which make it soft for short wavelengths of distortion. This softening is particularly pronounced in the highly anisotropic high-T _c superconductors (HTSC) where it leads to large thermal fluctuations and to thermally activated depinning of the Abrikosov vortices. Numerous transitions are predicted for these layered HTSC when the temperatureT, magnetic inductionB, or current densityJ are changed. In particular, the flux lines are now chains of two-dimensional (2D) “pancake vortices” which may “evaporate” by thermal fluctuations or may depin individually. At sufficiently highT, ohmic resistivityρ(T, B) is observed down toJ → 0. This indicates that the flux lines are in a “liquid state” with no shear stiffness and with small depinning energy or that the 2D vortices can move independently. At lowerT, ρ(T, B, J) is nonlinear since the pinning energy of an elastic vortex lattice or “vortex glass” increases with decreasingJ as predicted by theories of collective pinning and by “vortex glass” scaling.     

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.     

The statistical summation of weak flux-line pins in type II superconductors

A random array of flux-line-pinning defects is shown to produce instabilities in the flux-line lattice and a significant bulk pinning-force density even if the pins individually fail to satisfy the threshold criterion. A simplified model defect-flux-line interaction is used to obtain approximate expressions for the critical pinning-force densityF _c.The theory predicts that in materials with a dilute array of weak pinsF _cdepends only weakly on magnetic field except within narrow peaks near the upper and lower critical fields. The width of the peaks increases as the pins are made denser or stronger. A broad, dome-shaped peak inF _cand scaling-law behavior, both of which are usually associated with large-F _cmaterials, is the strong-pinning limit of this theory. This limit may be reached either by making the pins very strong or very dense. The results forF _cagree at least qualitatively with a number of experimental observations.     

Flux-flow instability in a type II superconductor

The stability of the flux-flow state in a superconducting slab is studied. The flux-flow kinematics is based on the Boltzmann equation. To investigate the interaction between the flux lines, the relaxation function is derived. The instability of the flux-flow state is associated with the growth of the autocorrelation function at the characteristic frequency of the flux-line collective mode. Thus the upper critical current value is obtained.     

Properties of the distorted flux-line lattice near a planar surface

The magnetic field, current density, and energy of an arbitrary array of curved or straight flux lines in a type II superconductor with a planar surface are calculated from the London theory. The general expressions and their expansion with respect to displacements from the equilibrium flux-line positions are given. The elastic energy of the distorted flux-line lattice near a planar surface is presented and discussed. The equilibrium configuration becomes unstable to the growth of helical perturbations if a current exceeding a critical value is applied parallel to the external magnetic field.     

Green's function for the elastic field of an edge dislocation in a finite orthotropic medium

A fundamental solution of plane elasticity in a finite domain is developed in this paper. A closed-form Green's function for the elastic field of an edge dislocation of arbitrary Burger's vector at an arbitrary point in an orthotropic finite elastic domain, that is free of traction, is presented. The method is based on the classical theory of potential fields, with an additional distribution of surface dislocations to satisfy the free traction boundary condition. A solution is first developed for a dislocation in a semi-infinite half-plane. The resulting field is composed of two parts: a singular contribution from the original dislocation, and a regular component associated with the surface distribution. The Schwarz-Christoffel transformation is then utilized to map the field quantities to a finite, polygonal domain. A closed form solution containing Jacobi elliptic functions is developed for rectangular domains, and applications of the method to problems of fracture and plasticity are emphasized.This material is based upon work supported by the Department of Energy under Award number DE-FG03-91ER54115 at UCLA.     

Dielastic interaction of the flux line lattice with internal stresses of crystal imperfections. I. The change of elastic constants in an inhomogeneous,type II superconductor

Analytical expressions for the change of the elastic constants due to the superconducting phase transition are derived within the framework of the micromagnetic theory of superconductivity. In particular, the variation of the shear modulus in the environment of a single vortex is discussed, which is important for the second-order interaction (E effect) between flux lines and crystal dislocations. Furthermore, the change of elastic moduli in the mixed state, as caused by the periodic flux line lattice, is investigated, and its field dependence is studied. Numerical values for the material parameters that describe the strain dependence of the critical fieldH _c and the critical temperatureT _c are presented for Nb and Nb ₃ Sn.     

Evaluation of the three-dimensional elastic Green's function in anisotropic cubic media

By using the Fourier transforms method, the three-dimensional Green's function solution for a unit force applied in an infinite cubic material is evaluated in this paper. Although the elastic behavior of a cubic material can be characterized by only three elastic constants, the explicit solutions of Green's function for a cubic material are not available in the literatures. The central problem for explicitly solving the elastic Green's function of anisotropic materials depends upon the roots of a sextic algebraic equation, which results from the inverse Fourier transforms and is composed of the material constants and position vector parameters. The close form expression of Green's function is presented here in terms of roots of the sextic equation. The sextic equation for an anisotropic cubic material is discussed thoroughly and specific results are given for possible explicit solutions.     

Green's function for a pre-stressed thin plate on an elastic foundation under axisymmetric loading

Green's functions are obtained for an infinite pre-stressed (compressed or stretched) thin plate on an elastic foundation under axisymmetric loading. The analytical procedure for the solution of the derived fundamental differential equation of a pre-stressed thin plate on an elastic foundation is based on Hankel's integral transforms and generalized functions theory. Numerical examples are included.     

Application of the Green's function method to thin elastic polygonal plates

Summary Recently, the Green's function method has been applied successfully to problems of plane elasticity, using influence functions of some finite basic domain of simple geometrical shape, which contains the given one as a subdomain. The result of this formulation is a pair of integral equations, which have to be defined only along that part of the boundary not coinciding with the border of the basic domain.A rather general formulation for the solution of bending of plates of arbitrary convex planform and loading is presented, where, for the sake of brevity, plates of polygonal shape are considered. The polygonal plate is embedded in a rectangular domain, thereby applying coincidence of boundaries as far as possible. Those boundary conditions in the actual problem, which are not already satisfied, lead to a pair of coupled integral equations for a density function vector with components to be interpreted as line loads and moments distributed in the basic domain along the actual boundary. Thus, the kernel is the corresponding Green's matrix. Hence, having solved the integral equations, deflections and stresses in the actual problem are explicitly known.Solution of the integral equations is generally achieved by a numerical procedure.The method is tested in example problems by considering a trapezoidal plate under various boundary conditions.

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Anwendung der Methode der Greenschen Funktion auf dünne elastische Polygonplatten

Zusammenfassung Die Methode der Greenschen Funktion wurde in jüngster Zeit erfolgreich auf Probleme der ebenen Elastizitätstheorie angewendet. Dabei fanden Einflußfunktionen eines endlichen Grundbereiches einfacher geometrischer Form, der den gegebenen Bereich einschließt, Verwendung. Das Resultat dieser Formulierung ist ein Integralgleichungspaar, welches entlang dem Teil des Randes zu erstrecken ist, der nicht mit dem Rand des Grundbereiches bereits zusammenfällt.Eine allgemein gehaltene Formulierung der Biegelösung von Platten mit konvexem Grundriß unter beliebiger Belastung wird angegeben, wobei allerdings der Kürze halber eine Beschränkung auf polygonplatten erfolgt. Die Polygonplatte wird in einen Rechteckbereich eingebettet, wobei soviele Ränder wie möglich zusammenfallen sollen. Jene Randbedingungen des wirklichen Problems, welche dann noch nicht erfüllt sind, führen auf ein gekoppeltes Integralgleichungspaar für den Dichtefunktionsvektor, dessen Komponenten als im Grundbereich entlang der wirklichen Berandung verteilte Linienlasten und Linienmomente gedeutet werden. Damit wird der Kern zur korrespondierenden Greenschen Matrix. Weiters sind, nach Lösung der Integralgleichungen, Biegefläche und Spannungen des wirklichen Problems explizit bekannt. Die Lösung der Integralgleichungen erfolgt im allgemeinen numerisch.Die Methode wird an Beispielen getestet, wobei eine Trapezplatte unter verschiedenen Randbedingungen untersucht wird.

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With 5 Figures

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Presented at the XVth Int. Congr. Theor. and Appl. Mechanics (IUTAM), Toronto 1980, by F. Z.     

The flux-line lattice in high-Tc superconductors

The flux-line lattice in type-II superconductors has unusual nonlocal elastic properties which make it soft for short wavelengths of distortion. This softening is particularly pronounced in the highly anisotropic high-T _c superconductors (HTSC) where it leads to large thermal fluctuations and to thermally activated depinning of the Abrikosov vortices. Numerous transitions are predicted for these layered HTSC when the temperatureT, magnetic inductionB, or current densityJ are changed. In particular, the flux lines are now chains of two-dimensional (2D) pancake vortices which may evaporate by thermal fluctuations or may depin individually. At sufficiently highT, ohmic resistivity(T, B) is observed down toJ 0. This indicates that the flux lines are in a liquid state with no shear stiffness and with small depinning energy or that the 2D vortices can move independently. At lowerT, (T, B, J) is nonlinear since the pinning energy of an elastic vortex lattice or vortex glass increases with decreasingJ as predicted by theories of collective pinning and by vortex glass scaling.     

Curved flux line near the edge of a type II superconductor

Applying the London theory, we show that the energy due to the interaction between the ends of the flux lines near their outlet points on the surface of a type II superconductor may be neglected. The interaction of a straight or curved flux line with the edge of a superconductor is considered. At the edge of the sample the surface barrier opposing the entry of the vortices has a value lower than that at the planar surface. The energy barrier is maximum for a planar surface, whereas it is zero at the edge of a superconductor with two perpendicular surfaces.     

Fractal vortex structure in a superconductor lattice model

The vortex structure is considered within the framework of a superconductor lattice model with a kinetic term of the Harper type. The problem is reduced to the analysis of discrete maps typical of the theory of fractal structures.