Superconductivity Within a Slab

exb2
Model of a superconducting slab in a parallel magnetic field.


Variables and parameters:

$\lambda$: square of external field (bifurcation parameter)

d: thickness of the slab (d=5)

x: space variable, $0\le x\le d$

$\kappa$: Ginzburg-Landau parameter ($\kappa=1)$

$\Theta$: an order parameter characterizing different superconducting states

$\Psi$: potential of the magnetic field


The Ginzburg-Landau equations are equivalent to the boundary-value problem of two second-order ODEs

\begin{displaymath}\Theta'' & = \Theta(\Theta^2 -1+\lambda \Psi^2 )\kappa^2
\cr\end{displaymath}


\begin{displaymath}\Psi''& = \Theta^2 \Psi \cr\end{displaymath}




\begin{displaymath}\Theta'(0) = \Theta'(d) = 0, \ \ \Psi '(0) = \Psi '(d) = 1.\end{displaymath}



Comments


Figure 1
Bifurcation diagram $\Psi(0)$ versus $\lambda$. Two branches intersecting in a pitchfork bifurcation.

Figure 2
Two antisymmetric solutions $\Theta(x)$ for $\lambda=0.6647$.


Figure 3
Two antisymmetric solutions $\Psi(x)$ for $\lambda=0.6647$.


Figure 4
Symmetric solution $\Theta(x)$ for $\lambda=0.91196$.


Figure 5
Symmetric solution $\Psi(x)$ for $\lambda=0.91196$.


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