Abstract

The existence of vortices in type II superconductors was predicted first by Alexei A. Abrikosov when he discovered a two-dimensional (2D) periodic solution of the Ginzburg–Landau (GL) equations. Abrikosov correctly interpreted this solution as a periodic arrangement of flux lines, the so-called flux-line lattice (FLL). Each flux line (or, alternatively, fluxon or vortex line) carries one quantum of magnetic flux Ф₀ = h/2e = 2.07 × 10⁻¹⁵ Tm², generated by a vortex of circulating supercurrents (see Chapter A2.6). Consequently, the magnetic field peaks at the vortex positions. The vortex core is a tube in which the superconductivity is weakened with its centre defined by the line at which the superconducting order parameter vanishes. For well-separated or isolated vortices, the radius of the tube of magnetic flux is the magnetic penetration depth, λ, and the core radius approximates to the superconducting coherence length, [1, 2] (see Chapter A2.3). The spacing a of the vortices decreases with increasing applied magnetic field and the average flux density, B ¯ = 2 Φ 0 / ( 3 a 2 ) , increases for the triangular FLL (see Figure A3.1.1 and Chapter A2.6). The flux tubes begin to overlap with further increase in B ¯ such that the periodic induction B(x,y) is nearly constant, with only a small relative modulation around its average value. Eventually, the vortex cores begin to overlap such that the amplitude of the order parameter decreases until it vanishes when B ¯ reaches the upper critical field, B _c2 = Ф₀/(2πξ²), where the superconductivity disappears. Figure A3.1.2 shows profiles of the induction B(x,y) and of the superconducting order parameter |ψ(x,y)|² for two values of B ¯ corresponding to flux-line spacings a = 4λ and a = 2λ.