Ginzburg-Landau Theory: The Order Parameter That Predicts Superconducting Coherence

What Ginzburg-Landau Theory Is: The Order Parameter

Ginzburg-Landau theory replaces the microscopic mess of ~10²³ electrons with a single smooth field: the complex order parameter ψ(r) = |ψ|e^{iφ}. Its interpretation is remarkably physical:

• |ψ(r)|² = local density of superconducting carriers, n_s. In the microscopic picture (Gor'kov, 1959) these are Cooper pairs, so ψ carries charge e* = 2e and mass m* = 2m_e.
• φ(r) = the macroscopic quantum phase. Gradients of φ drive the supercurrent, giving the Josephson effect and flux quantization.

Above the critical temperature Tc the equilibrium value is ψ = 0 (normal metal). Below Tc, ψ grows continuously from zero — the hallmark of a second-order phase transition. This is the deep idea GL borrowed from Landau's 1937 theory of phase transitions: expand the free energy in powers of a small order parameter that spontaneously breaks a symmetry (here, U(1) gauge symmetry). The genius was recognizing that superconductivity's order parameter is complex, so that its phase — not just its magnitude — becomes a physical, current-carrying degree of freedom.

The Free-Energy Functional and the Two GL Equations

GL wrote the free-energy density as an expansion in ψ and its gradients, plus the magnetic field energy:

f = f_n + α|ψ|² + (β/2)|ψ|⁴ + (1/2m*)|(−iħ∇ − e*A)ψ|² + B²/2μ₀

Here α(T) = α₀(T − Tc) changes sign at Tc (negative below, so ψ ≠ 0 is favored), β > 0 keeps the energy bounded, A is the vector potential and B = ∇×A. Minimizing the total free energy ∫f d³r with respect to ψ* and A gives the two Ginzburg-Landau equations:

• αψ + β|ψ|²ψ + (1/2m*)(−iħ∇ − e*A)²ψ = 0 — a nonlinear Schrödinger-like equation for ψ.
• J_s = (e*/m*)|ψ|²(ħ∇φ − e*A) — the supercurrent, the source term in Ampère's law.

In the uniform, field-free case the first equation gives the equilibrium value |ψ_∞|² = −α/β = α₀(Tc − T)/β, so the condensate grows linearly in (Tc − T). These two equations, plus boundary conditions, contain the entire phenomenology — from the Meissner effect to Abrikosov's vortex lattice.

Two Lengths, One Ratio: ξ, λ, and κ

Linearizing the GL equations yields two natural length scales that dominate everything:

• Coherence length ξ = ħ/√(2m*|α|) — the distance over which |ψ| can vary (e.g. how fast it recovers near a defect or surface). It diverges as (Tc − T)^(−1/2) near Tc.
• Penetration depth λ = √(m*/(μ₀ e*² |ψ_∞|²)) — the depth an external magnetic field leaks into the superconductor (the Meissner screening length). It also diverges as (Tc − T)^(−1/2).

Their ratio is temperature-independent near Tc and is the single most important number in the theory: the GL parameter κ = λ/ξ. Because both lengths scale the same way with temperature, κ is essentially a material constant. Worked example, elemental lead: ξ ≈ 83 nm, λ ≈ 37 nm, so κ ≈ 0.45 < 1/√2 → type-I. Nb₃Sn: ξ ≈ 3 nm, λ ≈ 65 nm, κ ≈ 22 → strongly type-II. The upper critical field follows directly: Hc2 = Φ₀/(2πξ²), where Φ₀ = h/2e ≈ 2.07×10⁻¹⁵ Wb is the flux quantum; a 3 nm coherence length gives Hc2 ≈ 20–30 T.

How It's Observed and Applied

GL predictions are directly measurable, which is why the theory has survived 75 years:

• Flux quantization: the phase φ must be single-valued, forcing magnetic flux through a superconducting ring to come in units of Φ₀ = h/2e ≈ 2.07×10⁻¹⁵ Wb. Measured by Deaver-Fairbank and Doll-Näbauer in 1961 — the factor of 2e confirmed pairing.
• Vortex lattices: Abrikosov (1957) solved the GL equations for κ > 1/√2 and predicted a triangular lattice of flux tubes, each carrying exactly one Φ₀, with cores of radius ~ξ and circulating currents out to ~λ. Directly imaged by Bitter decoration, STM, and small-angle neutron scattering.
• Critical fields and thin films: Hc2(T), Hc3 = 1.695·Hc2 (surface superconductivity, Saint-James & de Gennes 1963), and the temperature dependence Hc2 ∝ (Tc − T) are routine lab measurements.

Technologically, GL underpins every high-field magnet. Type-II superconductors — NbTi (κ ≈ 40), Nb₃Sn, and cuprate/iron-based tapes — carry current in the vortex state up to tens of teslas, powering MRI scanners (1.5–3 T), the LHC dipoles (8.3 T), and tokamak fusion coils (13+ T). Flux pinning of Abrikosov vortices is what lets these wires carry large lossless currents.

Where GL Sits Among Its Cousins

GL is one member of a family of order-parameter theories, and knowing the boundaries matters:

• vs. London theory (1935): London gives only λ and the Meissner effect, treating n_s as fixed. GL adds the second length ξ and lets |ψ| vary in space — essential for interfaces, vortices, and the type-I/II distinction London cannot capture.
• vs. BCS theory (1957): BCS is the microscopic mechanism (phonon-mediated Cooper pairing) and predicts Tc, the energy gap Δ, and the isotope effect. GL is macroscopic and phenomenological. Gor'kov (1959) proved GL is the rigorous limit of BCS near Tc, fixing e* = 2e and relating α, β to microscopic quantities.
• vs. Landau theory of phase transitions: GL is Landau's general mean-field framework specialized to a complex, gauge-coupled order parameter. The same functional reappears across physics — superfluid helium, liquid crystals, and the Higgs mechanism (the Abelian Higgs model is mathematically the GL free energy in relativistic form).

Its regime of validity is set by mean-field theory: GL is quantitatively reliable close to Tc, where ψ is small and fluctuations are weak. In the critical region very near Tc — narrow in conventional superconductors, wider in high-Tc cuprates — fluctuation corrections beyond mean field become important.

Significance, Legacy, and Open Questions

Ginzburg-Landau theory is one of the great triumphs of phenomenological physics: a correct, quantitative theory written down before the underlying mechanism was known. Its influence is enormous. Alexei Abrikosov used it in 1957 to predict type-II superconductivity and the vortex lattice — initially dismissed by Landau, later vindicated and central to all high-field magnet technology. In 2003 the Nobel Prize in Physics went to Abrikosov, Ginzburg, and Anthony Leggett for pioneering contributions to superconductors and superfluids; Landau (d. 1968) had already won in 1962 for related work.

Beyond superconductivity, the GL functional is the template for order-parameter physics everywhere: superfluidity, cosmic strings, the Standard Model Higgs field, and pattern formation. Open frontiers include the correct order-parameter symmetry in unconventional superconductors (d-wave cuprates, multi-band and possible spin-triplet states), where a single complex scalar ψ is insufficient and multi-component GL functionals are needed. The time-dependent GL equation (Gor'kov-Eliashberg) models vortex dynamics and dissipation but has subtle limits of validity. And whether a clean GL-style description exists for the pseudogap and strange-metal regimes of high-Tc materials remains an open question 75 years on.

Type-I vs type-II superconductors under the Ginzburg-Landau classification, with representative materials and length scales

Property	Type-I	Type-II
GL parameter κ = λ/ξ	κ < 1/√2 ≈ 0.707	κ > 1/√2 ≈ 0.707
Response to field	Full Meissner state, then abrupt to normal at Hc	Meissner below Hc1, vortex (mixed) state Hc1→Hc2
Surface energy of N-S boundary	Positive (interfaces cost energy)	Negative (interfaces favored → vortices)
Example material	Pb (ξ≈83 nm, λ≈37 nm, Tc=7.2 K)	Nb3Sn (ξ≈3 nm, λ≈65 nm, Tc=18 K)
Typical critical field	Hc ~ 0.01–0.08 T	Hc2 up to 20–60+ T (Nb3Sn, YBCO)
Technological role	Rare in magnets (quenches at low field)	MRI, LHC, fusion magnets, all high-field use