Ginzburg-Landau Theory and Vortices

Ginzburg-Landau theory describes superconductivity through a complex order parameter, and the ratio of its two characteristic lengths divides superconductors into type-I and the technologically vital type-II that admit quantized flux vortices.

Definition

Ginzburg-Landau theory describes the superconducting state by a complex order parameter whose magnitude measures the local density of the condensate; the ratio of the magnetic penetration depth to the coherence length, the Ginzburg-Landau parameter, distinguishes type-I superconductors from type-II superconductors that allow magnetic flux to enter as quantized vortices.

Scope

This topic covers the Ginzburg-Landau phenomenological theory: the complex order parameter and free-energy expansion, the coherence length and penetration depth, and the Ginzburg-Landau parameter that classifies superconductors as type-I or type-II. It treats the mixed state of type-II superconductors, the quantized flux line (Abrikosov vortex) and its lattice, the lower and upper critical fields, and flux pinning. It bridges the London electromagnetic theory and the BCS microscopic theory.

Core questions

• What does the Ginzburg-Landau order parameter represent, and how is the free energy built from it?
• How do the coherence length and penetration depth define the Ginzburg-Landau parameter?
• What distinguishes type-I from type-II superconductors?
• What is an Abrikosov vortex, and why does flux enter type-II superconductors in quantized lines?

Key concepts

• Complex order parameter and free-energy expansion
• Coherence length and penetration depth
• Ginzburg-Landau parameter
• Type-I versus type-II superconductors
• Abrikosov vortex lattice and flux pinning

Key theories

Ginzburg-Landau order-parameter theory

Ginzburg and Landau expanded the free energy in a complex order parameter and its gradients, capturing spatial variations of the condensate, surface energies, and the critical fields, with the order parameter later shown by Gor'kov to follow from BCS theory.

Abrikosov vortex state

Abrikosov predicted that type-II superconductors admit magnetic field as a lattice of quantized flux vortices, each carrying one flux quantum with a normal core, allowing superconductivity to survive to very high fields, the basis of practical superconducting magnets.

Clinical relevance

Type-II superconductors and the physics of vortex pinning make high-field superconducting magnets possible, enabling MRI, NMR spectrometers, particle accelerators, and fusion devices; controlling vortex motion is essential to carrying large supercurrents without dissipation.

History

Ginzburg and Landau proposed their order-parameter theory in 1950; Abrikosov used it in 1957 to predict the vortex lattice of type-II superconductors, and Gor'kov soon derived the theory from BCS, work recognized with the 2003 Nobel Prize to Ginzburg and Abrikosov.

Key figures

• Vitaly Ginzburg
• Lev Landau
• Alexei Abrikosov

Seminal works

• abrikosov1957
• tinkham2004

Frequently asked questions

What is the difference between type-I and type-II superconductors?

Type-I superconductors expel magnetic field completely until they abruptly lose superconductivity at a single critical field; type-II superconductors instead let field penetrate as quantized vortices over a range of fields, remaining superconducting to a much higher upper critical field.

Why must magnetic flux enter as quantized vortices?

The superconducting order parameter is a single-valued complex function, so its phase must wind by a multiple of two pi around any flux line; this constraint forces the enclosed flux to come in discrete quanta, each forming one Abrikosov vortex.