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Ginzburg-Landau (GL) theory describes superconductivity through a complex order parameter psi(r) — a macroscopic wavefunction whose magnitude |psi|^2 gives the superfluid density and whose phase determines the supercurrent. The free energy functional F = integral [alpha|psi|^2 + (beta/2)|psi|^4 + (1/2m*)|(-ihbar nabla - (e*/c)A)psi|^2 + B2/8pi] d3r, minimized over psi and A, yields two GL equations that describe the spatial variation of the order parameter and the supercurrent. GL theory introduces two fundamental length scales: the coherence length xi (over which psi varies) and the penetration depth lambda (over which B decays). Their ratio kappa = lambda/xi determines whether the superconductor is Type I (kappa < 1/sqrt(2)) or Type II (kappa > 1/sqrt(2)).
Ginzburg-Landau theory takes a different approach from BCS: instead of building superconductivity from microscopic electron pairing, it describes the superconducting state through a macroscopic order parameter psi(r) — a complex field that is zero in the normal state and nonzero in the superconducting state. The theory is built on Landau's general framework for second-order phase transitions: near T_c, the free energy is expanded as a functional of psi, keeping terms consistent with the symmetry (gauge invariance requires the coupling to the electromagnetic vector potential A).
The GL free energy functional contains four terms: the "potential energy" alpha|psi|^2 + (beta/2)|psi|^4 (with alpha changing sign at T_c to drive the transition), the "kinetic energy" |(−ihbar nabla − e*A/c)psi|^2/(2m*) (the gauge-invariant gradient of psi, measuring the supercurrent), and the magnetic field energy B2/8pi. Minimizing with respect to psi gives the first GL equation — a nonlinear Schrodinger equation for the order parameter. Minimizing with respect to A gives the second GL equation — the supercurrent in terms of psi and A, which serves as the source for Maxwell's equations.
Two length scales emerge naturally. The coherence length xi = hbar/sqrt(2m*|alpha|) is the distance over which psi can vary — it sets the size of vortex cores and the thickness of normal-superconducting boundaries. The penetration depth lambda is the distance over which magnetic fields decay, as in London theory. Their ratio kappa = lambda/xi is the Ginzburg-Landau parameter and determines the fundamental character of the superconductor. For kappa < 1/sqrt(2) (Type I), the interface between normal and superconducting regions has positive energy, and the material expels flux completely until a first-order transition at H_c. For kappa > 1/sqrt(2) (Type II), the interface energy is negative, and it becomes favorable to admit flux in quantized vortices above a lower critical field H_{c1}.
The most celebrated prediction of GL theory is the Abrikosov vortex lattice in Type II superconductors. Between H_{c1} and H_{c2}, magnetic flux penetrates as quantized vortices (each carrying flux quantum Phi_0 = hc/2e), arranged in a triangular lattice. The order parameter vanishes at each vortex core (over a distance ~xi) and the field decays over ~lambda from each core. At H_{c2} = Phi_0/(2pi xi2), the vortex cores overlap and superconductivity is destroyed. GL theory provides a complete quantitative description of this mixed state, surface superconductivity above H_{c2}, and the critical current at which vortex motion produces dissipation — all essential for superconductor applications.