Ginzburg–Landau Theory: Macroscopic Quantum Description of Superconductivity

Dive into the Ginzburg–Landau theory, a bridge between quantum mechanics and macroscopic superconducting behavior through the concept of an order parameter and energy minimization.

Written by: Ajay Kumar

Posted: 6/19/2025

Superconductivity
Visualization of the Ginzburg–Landau order parameter and superconducting coherence length

🔁 Previous Post Recap

In Part 6, we studied the London equations, which describe how electromagnetic fields behave in superconductors, particularly how magnetic fields decay over a length scale called the London penetration depth.

While insightful, these equations are phenomenological — they don’t explain why superconductivity occurs or how it emerges from microscopic principles.

This brings us to the Ginzburg–Landau theory — a more powerful, quantum-inspired model that captures both thermodynamic and electromagnetic behavior.

🌌 What is Ginzburg–Landau Theory?

The Ginzburg–Landau (GL) theory, proposed in 1950 by Vitaly Ginzburg and Lev Landau, is a macroscopic quantum theory of superconductivity. It introduces a complex order parameter that varies smoothly through space, offering a quantum-mechanical yet spatially averaged view of superconductivity.

The order parameter is denoted as:

ψ(r)=∣ψ(r)∣eiθ(r)\psi(\mathbf{r}) = |\psi(\mathbf{r})| e^{i\theta(\mathbf{r})}

Here:

  • ∣ψ∣2|\psi|^2 represents the density of superconducting electrons
  • θ\theta is the quantum phase

This wavefunction-like parameter becomes nonzero in the superconducting state and vanishes above the critical temperature TcT_c.

🧮 Free Energy Functional

GL theory builds on the idea of minimizing the free energy of a superconductor. The total free energy is expressed as:

F=Fn+α∣ψ∣2+β2∣ψ∣4+12m∗∣(−iℏ∇−2eA)ψ∣2+∣B∣22μ0F = F_n + \alpha |\psi|^2 + \frac{\beta}{2} |\psi|^4 + \frac{1}{2m^*} \left| \left( -i\hbar \nabla - 2e\mathbf{A} \right) \psi \right|^2 + \frac{|\mathbf{B}|^2}{2\mu_0}

Where:

  • FnF_n is the free energy of the normal state
  • α,β\alpha, \beta are phenomenological parameters
  • A\mathbf{A} is the magnetic vector potential
  • B=∇×A\mathbf{B} = \nabla \times \mathbf{A} is the magnetic field

This functional includes:

  • Thermodynamic terms (α∣ψ∣2+β∣ψ∣4\alpha |\psi|^2 + \beta |\psi|^4)
  • Kinetic energy of supercurrents
  • Electromagnetic energy

🧾 Ginzburg–Landau Equations

By minimizing the free energy with respect to ψ∗\psi^* and A\mathbf{A}, we obtain two coupled differential equations — the GL equations:

1️⃣ First Ginzburg–Landau Equation:

αψ+β∣ψ∣2ψ+12m∗(−iℏ∇−2eA)2ψ=0\alpha \psi + \beta |\psi|^2 \psi + \frac{1}{2m^*} \left( -i\hbar \nabla - 2e\mathbf{A} \right)^2 \psi = 0

This governs the spatial variation of the order parameter under electromagnetic influence.

2️⃣ Second Ginzburg–Landau Equation:

Js=2eℏm∗Im[ψ∗(∇−2ieℏA)ψ]\mathbf{J}_s = \frac{2e\hbar}{m^*} \text{Im} \left[ \psi^* \left( \nabla - \frac{2ie}{\hbar} \mathbf{A} \right) \psi \right]

This expresses the supercurrent density in terms of the wavefunction and magnetic vector potential.

📏 Coherence Length and Penetration Depth

GL theory naturally introduces two important characteristic lengths:

1. Coherence Length (ξ\xi):

Describes the length scale over which ψ\psi can vary:

ξ=ℏ22m∗∣α∣\xi = \sqrt{\frac{\hbar^2}{2m^* |\alpha|}}

It characterizes how quickly the superconducting state can recover spatially from a disturbance (e.g., near a boundary or defect).

2. London Penetration Depth (λ\lambda):

Still defined as the scale over which magnetic fields decay, but now explicitly tied into the GL framework.

⚖️ The GL Parameter κ and Superconductor Types

The Ginzburg–Landau parameter κ\kappa is defined as:

κ=λξ\kappa = \frac{\lambda}{\xi}

This single ratio determines whether a material is:

  • Type I: κ<12\kappa < \frac{1}{\sqrt{2}}
  • Type II: κ>12\kappa > \frac{1}{\sqrt{2}}

So, GL theory unifies classification and field behavior — a major leap forward over the London model.

🧪 Physical Predictions and Successes

Ginzburg–Landau theory can:

  • Describe the interface between normal and superconducting regions
  • Model vortex structures in Type II superconductors
  • Predict critical field behavior
  • Bridge thermodynamics and quantum mechanics
  • Enable computational simulations of superconducting domains

In 2003, Ginzburg shared the Nobel Prize in Physics (with Abrikosov and Leggett) for his foundational contributions to superconductivity.

📘 Conclusion

The Ginzburg–Landau theory offers a conceptual bridge between macroscopic thermodynamics and microscopic quantum mechanics. Its introduction of a spatially varying order parameter allows us to model and predict complex superconducting phenomena — from magnetic field penetration to vortex lattice formation.

🔮 Coming Up Next…

In Part 8, we’ll dive into the BCS theory, which finally explained the microscopic origin of superconductivity. We’ll understand Cooper pairs, energy gaps, and the true nature of the superconducting quantum ground state.

Stay tuned as we zoom into the quantum world even further!