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AbrikosovLattice

< The magnetic fluxoid | Index | The field distribution in the flux lattice >

Abrikosov first noticed that a solution of the linearized G-L Eq. 3 is:

{$ (1)\qquad\qquad \psi(x,y)=\sum_n \left(e^{i2n\frac {2\pi} b y}\, e^{\frac{(x-2n\Phi_0/bB)^2} {2\xi^2}} \,+\,e^{i(2n+1)\frac {2\pi} b y+\pi}\, e^{\frac{(x-(2n+1)\Phi_0/bB)^2} {2\xi^2}} \right) $}

where {$\mathbf{B}=B \hat z$}. One can check that this is a periodic function in both {$y$}, with period {$b$}, and {$x$}, with period {$a=\frac {\Phi_0} {Bb} $}, defining a lattice such that each cell contains a flux quantum

{$\qquad\qquad abB=\Phi_0$}

One can further check that the phase changes by {$2\pi$} circulating around lattice sites, i.e. the circulation of {$\mathbf{\nabla}\phi$} signals (Eq. 2, prev. page) the presence of the flux quantum, or fluxon at each site. In Eq. 1 we have chosen one particular solution, the triangular lattice, with

{$ (2) \qquad\qquad a=\sqrt{\frac {\phi_0} B \sqrt{\frac 4 3}}, $}

but a square lattice, as well as more complex ones are possible. The actual choice relies on the effects of the neglected non-linear terms.

We shall next describe the simplest condition for the magnetic field inside a type II superconductor in the Abrikosov flux lattice state. We restrict ourselves to the high {$\kappa$} approximation (extreme type II), where {$l\lambda\gg \xi$}. As we mentionad before, the main difficulty in the interaction of the superconductor with electromagnetic fields lies in the non-local effects, that are well described by Ginzburg-Landau equations. In the extreme type II case we may neglected these effects in first approximation. This brings us back to the London treatment, which, surprisingly, was originally thought for the regime of {$\xi\gg\lambda$}, but instead works pretty well in the opposite case.

When the applied field is the minimum, {$H_{c1}$}, to let the field penetrate the superconductor, there will be just one fluxon and we can conveniently reproduce the circulation of the phase gradient by writing {$\psi=\psi_\infty f(r)e^{i\theta}$}, where the complex wavefunction phase coincides with the cylindrical angular coordinate, {$\phi=\theta$}. It can be shown that in this case a good approximation is {$f(r)=\tanh \frac {\nu r} \xi$} with {$\nu\sim 1$} and that the magnetic field around the fluxon is {$ (3) \qquad\qquad B(r)= \frac {\Phi_0} {2\pi\lambda^2} K_0\left(\frac r \lambda\right) $} where {$K_0(x)$} is the zero order Hankel function of imaginary argument (computed by besselk(0,abs(x)) in matlab).

Internal field {$B$} and spatial variation of the wavefunction {$\frac \psi {\psi_\infty}=f$}, along a cut through the fluxon core. Both functions vary dramatically inside a region of radius {$\xi$}, the core. {$B$} varies approximately as {$e^{-r/\lambda}$} outside the core.

< The magnetic fluxoid | Index | The field distribution in the flux lattice >