Ginzburg–Landau Vortex & Abrikosov Lattice

Type-II superconductor: |ψ|² order parameter & vortex lattice forming above Hc1

|ψ(x,y)|² — superconducting order parameter
B(x,y) — magnetic field (flux quanta at vortex cores)
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Vortex count
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Hc1/Hc2
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Phase
GL free energy: F = ∫[α|ψ|² + β/2|ψ|⁴ + |(−iℏ∇ − 2eA)ψ|²/2m + B²/2μ₀] d³r
Abrikosov lattice: Each vortex carries flux Φ₀ = h/2e ≈ 2.07 × 10⁻¹⁵ Wb. At H > Hc1, vortices penetrate in a triangular lattice (minimizes repulsion).
κ > 1/√2: Type-II (vortex phase exists); κ < 1/√2: Type-I (Meissner only). Hc1/Hc2 ≈ ln(κ)/(2κ²)