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.

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The magnetic moment, flux and current penetration, and creep in type-II superconductors of nonzero thickness in a perpendicular applied magnetic field are calculated. The presented method extends previous onedimensional theories of thin strips and disks to the more realistic case of arbitrary thickness, including as limits the perpendicular geometry ͑thin long strips and circular disks in a perpendicular field͒ and the parallel geometry ͑long slabs and cylinders in a parallel field͒. The method applies to arbitrary cross section and arbitrary current-voltage characteristics E(J) of conductors and superconductors, but a linear equilibrium magnetization curve Bϭ 0 H and isotropy are assumed. Detailed results are given for rectangular cross sections 2aϫ2b and power-law electric field E(J)ϭE c (J/J c) n versus current density J, which includes the Ohmic (nϭ1) and Bean (n→ϱ) limits. In the Bean limit above some applied field value the lens-shaped fluxand current-free core disconnects from the surface, in contrast to previous estimates based on the thin strip solution. The ideal diamagnetic moment, the saturation moment, the field of full penetration, and the complete magnetization curves are given for all side ratios 0Ͻb/aϽϱ. ͓S0163-1829͑96͒01429-4͔

The magnetization curves M (H) for ideal type-II superconductors and the maximum, minimum, and saddle point magnetic fields of the vortex lattice are calculated from Ginzburg-Landau theory for the entire ranges of applied magnetic fields Hc1 ≤ H ≤ Hc2 or induction 0 ≤ B ≤ µ0Hc2 and Ginzburg-Landau parameters 2 −1/2 ≤ κ ≤ 1000. Results for the triangular and square flux-line lattices are compared with the results of the circular cell approximation. The exact magnetic field B(x, y) and magnetization M (H, κ) are compared with often used approximate expressions, some of which deviate considerably or have limited validity. Useful limiting expressions and analytical interpolation formulas are presented.

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The current density and local magnetic field are calculated analytically for a strip of a type-I1 superconductor in perpendicular magnetic field Ha for constant critical current density. The penetrating flux front has vertical slope and the initial penetration depth, magnetization change and hysteretic losses are -H:, -H,8 and -H:, respectively. The analytical results differ from the widely used Bean model and explain numerous experiments in a natural way without the assumption of a surface barrier.

We show that an ac external magnetic field can generate a dc voltage in type-II superconductors carrying a constant transport current. This rectifying effect occurs even at low temperatures where flux creep may be disregarded and even for arbitrarily small applied current. The dc signal appears when the magnitude of the applied ac field exceeds a threshold value which depends on the shape of the superconductor and on the applied current. Experiments on this subject are discussed.