Questions: Ginzburg-Landau Theory

Question 1

The Ginzburg-Landau order parameter ψ(r) is a complex function. What is the physical significance of its phase?

• A. The phase has no physical meaning and can be set to zero
• B. The gradient of the phase determines the supercurrent: J_s = (e*ħ/m*)|ψ|²∇φ - (e*²/m*c)|ψ|²A. A uniform phase gives zero current; a spatially varying phase drives a supercurrent. Phase coherence across the sample is the defining property of the superconducting state
• C. The phase determines the energy gap
• D. The phase encodes the crystal structure of the superconductor

Answer: (B)

The supercurrent is proportional to the gauge-invariant phase gradient (∇φ - e*A/ħc). This has profound consequences: a superconducting ring must have the phase change around a loop equal to 2πn (single-valuedness of ψ), leading to flux quantization Φ = nΦ₀ = nhc/2e. The phase rigidity — the energy cost of varying the phase — is what protects supercurrents from decay. The Josephson effect arises from the phase difference between two superconductors connected by a weak link.

Question 2

The GL parameter κ = λ/ξ determines whether a superconductor is Type I or Type II. What is the physical reason for the threshold at κ = 1/√2?

• A. It is an arbitrary convention
• B. At κ = 1/√2, the surface energy of a normal-superconducting interface changes sign. For κ < 1/√2, the interface has positive energy (unfavorable — Type I prefers complete flux expulsion or complete penetration). For κ > 1/√2, the interface has negative energy (favorable — Type II spontaneously creates flux-carrying vortices in the mixed state)
• C. It marks where the critical field exceeds the Earth's magnetic field
• D. It depends on whether the material is an element or a compound

Answer: (B)

The surface energy of a normal-superconducting boundary depends on the competition between the condensation energy (gained over a distance ξ as ψ recovers from zero to its bulk value) and the magnetic energy (saved over a distance λ as B decays). When ξ > λ (κ < 1/√2, Type I), the condensation energy gain dominates and the surface energy is positive — the system minimizes interfaces. When λ > ξ (κ > 1/√2, Type II), the magnetic energy saving dominates, the surface energy is negative, and the system maximizes interface area by creating quantized vortices.

Question 3

Ginzburg-Landau theory was developed phenomenologically in 1950, before BCS theory. Gor'kov later showed that GL theory is derivable from BCS theory near T_c.

• T. True
• F. False

Answer: True

GL theory was originally a phenomenological extension of Landau's general theory of second-order phase transitions, applied to superconductivity with a complex order parameter and minimal coupling to the electromagnetic field. In 1959, Gor'kov showed that the GL equations can be derived rigorously from BCS theory in the limit T → T_c (where Δ(r) varies slowly in space). The GL order parameter ψ is proportional to the BCS gap function Δ, the GL coefficients (α, β, m*, e*) are expressed in terms of microscopic BCS parameters, and e* = 2e confirms that the fundamental charge carriers are Cooper pairs. This connection validated GL theory and extended its credibility.

Question 4

Explain why Ginzburg-Landau theory is particularly powerful for describing spatially inhomogeneous superconductivity (vortices, boundaries, thin films) where BCS theory is difficult to apply.

Model answer: BCS theory is formulated in momentum space for a uniform system — the gap equation involves integrals over k-space assuming translational invariance. Spatial inhomogeneities (vortex cores, surfaces, interfaces, applied field gradients) require solving the Bogoliubov-de Gennes equations, which are computationally expensive. GL theory is formulated in real space as a differential equation for ψ(r) coupled to Maxwell's equations, making it naturally suited to inhomogeneous problems. The two GL equations (analogous to a nonlinear Schrodinger equation) can be solved analytically for simple geometries and numerically for complex ones. Near T_c where GL is exact, it provides a complete description of vortex structure, surface superconductivity, critical fields, and the mixed state.

This is why GL theory remains the standard tool for understanding vortex physics, even though BCS theory is more fundamental. The Abrikosov vortex lattice, the upper and lower critical fields, and surface superconductivity (the Saint-James-de Gennes effect) were all predicted using GL theory.