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Developed in 1950 by Vitaly Ginzburg and Lev Landau, this is a phenomenological theory that describes superconductivity near the phase transition. It introduces a complex order parameter, [latex]\Psi[/latex], to represent the density of superconducting electrons. The theory successfully describes effects like the Meissner effect and predicts the distinction between Type I and Type II superconductors based on a single parameter, [latex]\kappa[/latex].
The Ginzburg-Landau theory is a macroscopic theory, meaning it does not explain the microscopic origin of superconductivity (which was later done by BCS theory), but it brilliantly describes the behavior of superconductors. It is based on Landau’s general theory of second-order phase transitions. The central idea is to write the free energy of the system as an expansion in powers of the order parameter [latex]\Psi[/latex] and its gradient. The order parameter is zero in the normal state and non-zero in the superconducting state. The free energy density is given by [latex]f = f_n + \alpha|\Psi|^2 + \frac{\beta}{2}|\Psi|^4 + \frac{1}{2m^*}|(-i\hbar\nabla – e^*\mathbf{A})\Psi|^2 + \frac{|\mathbf{B}|^2}{2\mu_0}[/latex], where [latex]\alpha[/latex] and [latex]\beta[/latex] are phenomenological parameters, [latex]\mathbf{A}[/latex] is the magnetic vector potential, and [latex]e^*[/latex] and [latex]m^*[/latex] are the effective charge and mass of the superconducting charge carriers. Minimizing this free energy yields the Ginzburg-Landau equations, which describe the spatial variation of [latex]\Psi[/latex] and the supercurrents. The theory defines two characteristic length scales: the coherence length [latex]\xi[/latex], over which [latex]\Psi[/latex] can vary significantly, and the London penetration depth [latex]\lambda[/latex]. Their ratio, the Ginzburg-Landau parameter [latex]\kappa = \lambda/\xi[/latex], determines the type of superconductor. If [latex]\kappa 1/\sqrt{2}[/latex], it’s Type II. This prediction of a second type of superconductor, which allows partial magnetic field penetration in a vortex lattice, was a major success of the theory, confirmed experimentally by Abrikosov in 1957.
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