Tru Physics

Ginzburg-Landau Theory

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Introduction

The Ginzburg-Landau (GL) theory, named after physicists Vitaly Ginzburg and Lev Landau, is a phenomenological theory that describes superconductivity and superfluidity. It was originally developed to explain the behavior of superconductors near their critical temperature.

Ginzburg-Landau Free Energy

The GL theory is based on the concept of a complex order parameter \Psi, which characterizes the superconducting state. The Ginzburg-Landau free energy functional is given by:

F = \int d^3x \left[\alpha |\Psi|^2 + \dfrac{\beta}{2} |\Psi|^4 + \dfrac{1}{2m^*} \left| \left(-i\hbar\nabla - \dfrac{2e}{c}\vec{A}\right)\Psi \right|^2 + \dfrac{(\nabla \times \vec{A})^2}{2\mu_0}\right]

where \alpha and \beta are phenomenological parameters, m^* is the effective mass of the superconducting electrons, \vec{A} is the magnetic vector potential, \hbar is the reduced Planck’s constant, e is the elementary charge, and \mu_0 is the vacuum permeability.

Ginzburg-Landau Equations

Variation of the free energy with respect to \Psi^* and \vec{A} leads to the Ginzburg-Landau equations:

\alpha \Psi + \beta |\Psi|^2 \Psi - \dfrac{1}{2m^*} \left(-i\hbar\nabla - \dfrac{2e}{c}\vec{A}\right)^2\Psi = 0

\nabla \times (\nabla \times \vec{A}) = \dfrac{2e}{\mu_0 c} \vec{j}_s

where \vec{j}_s = \dfrac{i\hbar e}{2m^} (\Psi^* \nabla \Psi - \Psi \nabla \Psi^*) - \dfrac{2e^2}{m^*c} |\Psi|^2 \vec{A} is the supercurrent.

Coherence Length and Penetration Depth

From the GL theory, two fundamental lengths can be defined: the coherence length \xi, which describes the size of the superconducting wave function, and the penetration depth \lambda, which characterizes the distance over which an external magnetic field can penetrate the superconductor. These are given by:

\xi = \sqrt{\dfrac{\hbar^2}{2m^*|\alpha|}}

\lambda = \sqrt{\dfrac{m^*c^2}{4\mu_0 e^2 |\Psi_0|^2}}

where \Psi_0 is the equilibrium value of the order parameter.

Type I and Type II Superconductors

Depending on the ratio \kappa = \lambda/\xi, superconductors are classified into Type I (\kappa < 1/\sqrt{2}) and Type II (\kappa > 1/\sqrt{2}). Type I superconductors expel all magnetic fields (perfect diamagnetism) below their critical temperature, while Type II superconductors allow magnetic fields to penetrate through quantized vortices.

Conclusion

The Ginzburg-Landau theory is a powerful tool for understanding superconductivity, providing important insights into the behavior of superconductors near magnetic fields.

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