Now you are in the subtree of Special Topics in Many-Body Theory, Spring 2016 project. 

Ginzburg-Landau theory

A simple effective theory developed by Ginzburg and Landau gives a surprising amount of useful information about the behavior of all known superconductors, including those where the microscopic mechanism is unclear. More precisely, most of the phenomenology of superconductors follows from the Ginzburg-Landau equation. This equation follows from minimizing the so-called Landau free energy, which we will now construct from physical assumptions.

The Landau free energy is written in terms of a single-particle wavefunction. One major contribution of BCS theory was explaining why a single-particle wavefunction is a natural description for a system of fermions. With Bose-Einstein condensed bosons, description in terms of a single-particle wavefunction is more natural, since then most of the bosons are in the ground state. The magnitude of this wavefunction gives the fraction of electrons at a point that are superconducting, in a two-fluid model where some electrons are superconducting and some are normal:
$$ |\psi({\bf r})|^2 = {n_s({\bf r}) \over n}. $$
Here $n$ is the total electron density, assumed constant in space.

First suppose that the wavefunction is constant in space, and let $f(\psi,T)$ be the difference in free energy density between the superconducting and normal states if $\psi$ is uniform. That is, for a system of volume $V$, the free energy difference is
$$ \Delta F = F_N - F_S = V f(\psi,T). $$
Now we make another assumption. If the system is just below the superconducting transition $T_c$, then only a few electrons are superconducting, which means that we can expand $f$ in a power series in $\psi$, since $|\psi|^2 \ll 1$:
$$ f(\psi,T) \approx a(T) |\psi|^2 + \frac{1}{2}b(T) |\psi|^4. $$
We can make one more simplification. First, note that the free energy is minimized when
$$ {\partial f \over \partial |\psi|^2} = 0 \Rightarrow b(T) |\psi|^2 + a(T) = 0 \Rightarrow |\psi|^2 = - {a(T) \over b(T)}. $$
At this magnitude, the free energy difference is
$$ f(\psi,T) = -\frac{1}{2} {a^2(T) \over b(T)}. $$

Recall that one of the defining features of a superconductor is the expulsion of magnetic field (the Meissner effect). The magnetic field reenters and drives the system normal once it is energetically favorable to do so (here we are assuming that the superconductor is type I, so the magnetic field penetrates uniformly above the critical magnetic field $H_c$). The free energy difference per volume is therefore related to $H_c$.
$$ f(\psi,T) = -\frac{1}{2} {a^2(T) \over b(T)} = - {{H_c}^2 \over 8 \pi}. $$

We can obtain another relation involving $a$ and $b$ if we use the fact that the superfluid density goes as the inverse square of the penetration depth: then
$$ {\lambda^2(0) \over \lambda^2(T)} = {|\psi(T)|^2 \over |\psi(0)|^2} = |\psi(T)|^2 = -{a(T) \over b(T)}. $$
Here we assumed that at zero temperature, all the electrons participate in the superconductivity. These two equations involving $a$ and $b$ can be used to reexpress everything in the Ginzburg-Landau equation in terms of the experimental quantities $H_c$ and $\lambda$ (left as an exercise).

Now we want to allow for spatial variations in $\psi$. For slow variations, we can keep just the gradient-squared term, which introducing some constants becomes
$$ \int {n^* \over 2 m^*} \left | {\hbar \over i} \nabla \psi({\bf r}) + {e^* \over c} {\bf A}({\bf r}) \psi({\bf r})\right |^2 d{\bf r}. $$
Note that we have defined quantities $n^*, m^*, e^*$ with the units of number density, mass, and charge. The BCS theory will predict $e^* = 2e$ and $m^* = 2m$. We have also assumed that an external vector potential ${\bf A}$ enters in the same way as for a single particle.

Combining the constant and gradient terms, and the magnetic field energy, gives finally the Landau free energy
$$ \begin{eqnarray} F(\psi,T) &=& \int {n^* \over 2 m^*} \left | {\hbar \over i} \nabla \psi({\bf r}) + {e^* \over c} {\bf A}({\bf r}) \psi({\bf r})\right |^2 d{\bf r} \cr &&+ \int \left[ a(T) |\psi({\bf r})|^2 + \frac{1}{2} b(T) |\psi({\bf r})|^4 \right] d{\bf r} + \int {H({\bf r})^2 \over 8 \pi} d{\bf r}. \end{eqnarray} $$
Then the minimization of this functional over $\psi$ gives
$$ {\delta F \over \delta \psi(r)} = 0 \Rightarrow {\hbar^2 n^* \over 2 m^*} \left[ \nabla + {i e^* \over \hbar c} {\bf A}({\bf r})\right]^2 \psi({\bf r}) + a(T) \psi({\bf r}) + b(T) |\psi({\bf r})|^2 \psi({\bf r}) = 0. $$

Before trying to find a microscopic justification of this theory, let us review what it does contain and what it doesn't. It accounts for the existence of a supercurrent and the Meissner effect, and even for the existence of vortices in type II superconductors.
It also shows the importance of a good phenomenology: only seven years passed between the above "derivation" of the Ginzburg-Landau equation and its justification by BCS theory. Of course, the numbers $m^*$ and $e^*$ remain unexplained, and it gives no prediction of when the approximation by a single wavefunction breaks down: what happened to the original (fermionic) electrons?