Statistical Physics (PHY831): Part 4: Ginzburg-Landau theory, modeling of dynamics and scaling in complex systems

London theory was developed by Fritz London in 1935 to describe the Meissner effect. This theory leads to the introduction of the penetration depth to describe the extent of magnetic field penetration, λ into type I superconductors. The penetration depth is also important in type II superconductors and describes the extent of flux penetration near vortices as well as at surfaces. Prior to his studies of superconductivity, Landau had developed a simple mean field theory to describe phase transitions. Ginzburg added a term to describe fluctuations which also enables description of inhomogenious systems. Ginzburg-Landau (GL) theory is a field theory and provides a systematic phenomenological approach to many body systems. The GL theory introduces a second length, the healing length or coherence length ξ. Below we first introduce the G-L theory and we show how the length ξ emerges. Before proceeding it is important to note that the analysis of London theory and LG theory below uses q for charge, m for mass and ns for the number density of superconducting electrons. In all superconductors found so far q = 2e is the charge of the fundamental Bosons (Cooper pairs), nc = ns/2 is the number density of cooper pairs and m is the effective mass of Cooper pairs. In some materials m can be significantly different than 2me due to band structure effects. Ginzburg-Landau theory, which was published in 1950, does a good job of describing the electromagnetic properties of superconductors, including vortex effects and the effect of pinning on these vortices. One of the key successes of the Ginzburg-Landau theory is its prediction of the distinction between type I and type II superconductors that have very different electromagnetic properties. Flux penetrates type II superconductors in the form of quantized vortices with flux φ0 = h/2e. The reason for flux quantization is purely quantum mechanical, as it arises through the requirement that the wavefunction of the superconductor be single valued at every point in space.