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Superconductor Ginzburg-Landau model

  1. May 9, 2013 #1
    1. The problem statement, all variables and given/known data

    Why is it that for a superconductor [itex]\alpha<0,\beta>0[/itex] in the Ginzburg-Landau model wit free energy formulation
    [tex]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[/tex]

    2. Relevant equations

    [tex]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[/tex]

    3. The attempt at a solution

    I have no idea how to start! perhaps it has to do with some critical temperature?
     
  2. jcsd
  3. May 9, 2013 #2
    [itex]\beta >0[/itex], otherwise the free energy could be minimized by [itex]|\psi |\rightarrow \infty [/itex]. This may not be obviously unphysical, but the Ginzburg-Landau model does after all aim to model the phase transition around [itex]|\psi |=0 [/itex].

    [itex]\alpha <0[/itex], then. The parameter [itex]\alpha [/itex] 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 [itex]\vec{A}=0[/itex] and [itex]\nabla \psi \approx 0[/itex] 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.
     
  4. May 13, 2013 #3
    Thanks, Hypersphere.
     
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