Quantized Flux Lines
When a type-II superconductor is placed in a magnetic field between Hc1 and Hc2, flux enters not uniformly but in discrete tubes — vortices — each carrying exactly one quantum of magnetic flux Φ₀ = h/2e. The vortex has a normal core of radius ξ where the superconducting order parameter vanishes, surrounded by a swirl of supercurrents extending to distance λ. Abrikosov predicted this in 1957; direct imaging came decades later with STM and magnetic decoration.
The Triangular Lattice
Vortices interact repulsively — the circulating currents of neighboring vortices push them apart. In equilibrium, they arrange into a regular triangular lattice that minimizes the total free energy. The lattice parameter a₀ = 1.075√(Φ₀/B) decreases as the field increases, packing more vortices into the sample. At Hc2, vortex cores overlap completely and superconductivity is destroyed.
Pinning and Critical Currents
A transport current exerts a Lorentz force on each vortex: F = J × Φ₀. If vortices move, the time-varying flux generates voltage and dissipation — destroying the zero-resistance state. Pinning sites — metallurgical defects, nano-precipitates, columnar tracks from heavy-ion irradiation — trap vortices and prevent motion. The critical current density Jc is set by the depinning threshold: Jc = Fp/B, where Fp is the pinning force density.
Vortex Matter Phase Diagram
Modern understanding recognizes that the vortex system has its own rich phase diagram. The ordered Abrikosov lattice can melt into a vortex liquid at high temperatures, pass through a Bragg glass phase with quasi-long-range order, or form a disordered vortex glass when pinning is strong. In high-Tc cuprates, the vortex phase diagram is particularly complex due to large thermal fluctuations and layered crystal structure.