The flux-line lattice in superconductors

Abstract

Magnetic flux can penetrate a type-II superconductor in form of Abrikosov vortices (also called flux lines, flux tubes, or fluxons) each carrying a quantum of magnetic flux φ 0 = h/2e. These tiny vortices of supercurrent 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 superconductors (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 λ: The shear modulus of the FLL is c 66 ∼ 1/λ 2 , and the tilt modulus c 44 (k) ∼ (1 + k 2 λ 2 ) −1 is dispersive and becomes very small for short distortion wavelength 2π/k ≪ λ. This softness 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 (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 of the two-dimensional "pancake vortices" in the layers. Various phase transitions are predicted for the FLL in layered HTSC's. Although large pinning forces and high critical currents have been achieved, the small depinning energy so far prevents the application of HTSC's as conductors at high temperatures except in cases when the applied current and the surrounding magnetic field are small.

Fluxon pinning in niobium films

“…The value taken by the upper critical field is essential to the understanding of the fluxon structure and of their dynamics [3,4]. Accurate measurements are available for pure niobium [18][19][20] showing that for t ≤ 0.5 the temperature dependence of the upper critical field H c2 and of the thermodynamical critical field H c agrees to within a few percent with the Gorter-Casimir expressions…”

“…As most models of fluxon dynamics [3,4] predict R fl ≅ R n H ext / H c2 , R n being the value of the surface resistance in the normal conducting state and H c2 the upper critical field, this difference between films and bulk has been blamed [5,6] on a similar difference between the values taken by H c2 . In Reference [1] R fl was measured for a large variety of films spanning a wide range of electron mean free paths and the evolution of R 0 fl anf R 1 fl was tracked for the first time from the dirty limit (films containing more than 10 atomic percent of neon) to the clean limit (deep-drawn bulk niobium subsequently annealed at 1100°C).…”

Magnetic penetration depth in unconventional superconductors

“…Since the advent of high temperature superconductivity, enormous attention has focused on the new features of the H-T phase diagram. [27,36] It is widely believed that over a substantial portion of the diagram the vortex lattice is melted or at least very weekly pinned, so vortices can be easily displaced. Also, the much higher temperatures that occur in copper oxide superconductors imply that flux flow may be thermally assisted.…”

Grain boundaries in vortex matter

“…Most high-T c materials behave in a magnetic field as type-II superconductors, with further complications due to the broader phase space-in terms of temperature T and field H-in comparison to conventional type-II superconductors. [1][2][3] This leads to several possibilities for the mixed phase, where magnetic flux penetration is incomplete. As first discussed by Abrikosov for conventional superconductors, 4 flux is quantized and carried by vortex lines which are arranged in the form of a lattice.…”