Superconductor Ginzburg-Landau model

[vidi]

Homework Statement

Why is it that for a superconductor \alpha<0,\beta>0 in the Ginzburg-Landau model wit free energy formulation
E=\int d^3x\,\, \frac{\vec B^2}{2\mu}+\frac{1}{2m}|(-i\hbar\nabla-e\vec A)\psi|^2+\alpha|\psi|^2+\beta|\psi|^4

Homework Equations

E=\int d^3x\,\, \frac{\vec B^2}{2\mu}+\frac{1}{2m}|(-i\hbar\nabla-e\vec A)\psi|^2+\alpha|\psi|^2+\beta|\psi|^4

The Attempt at a Solution

I have no idea how to start! perhaps it has to do with some critical temperature?

 

[Hypersphere]

\beta >0, otherwise the free energy could be minimized by |\psi |\rightarrow \infty. This may not be obviously unphysical, but the Ginzburg-Landau model does after all aim to model the phase transition around |\psi |=0.

\alpha <0, then. The parameter \alpha is indeed related to the critical temperature, but you don't need to look at the temperature dependence here. Just consider the simpler case with \vec{A}=0 and \nabla \psi \approx 0 and minimize the free energy. This will give you the sign. Of course, you'll probably have to extend the arguments a bit, to account for the inhomogeneous case and non-zero vector potentials, but that's the basic idea anyway.

 

[vidi]

Thanks, Hypersphere.