Advanced Solid State Physics

Taking the curl of both sides of this equation yields theLondonequation:

∇×j=

q^2 n

m

∇×A, (330)

since∇×∇g= 0independent of the scalar functiong. Due to theMeissner - Ochsenfeldeffect
we know thatB= 0 in the interior of the superconductor. Therefore, we havej= 0 and it follows
from (329) that

qA=~∇θ. (331)

We now select a closed pathCthrough the interior of the superconductor and integrate the right hand
side of (331):

~

∮

C

dl∇θ=~(θ 2 −θ 1 ), (332)

i.e.~times the change of the phase when going once around the contourC. Sinceψmust not change
through the integration we obtain that

θ 2 −θ 1 ≡ 2 πs, (333)

withssome integer. For the left hand side of (331) we obtain throughStokes’ theorem

q

∮

C

dlA=q

∫

F

dF·∇×A=q

∫

F

dF·B=qΦ, (334)

the magnetic flux. Here,dFis an element of the area bounded by the contourC. Accordingly

Φ =

~ 2 πs

q

= Φ 0 s, (335)

with the flux quantum (referred to as fluxoid)Φ 0. Note that in the particular case ofCooperpairs
we have to replaceq→ 2 q.

In type II superconductors one observes these quantized magnetic flux tubes in theShubnikovphase.
The magnetic field penetrates the superconductor and is organized in vortices. These vortices interact
with each other and are rearranged in a periodic structure (like a lattice), see Fig. 165. The complete
theory of type II superconductors was given byGinzburg,Landau,AbrikosovandGorkov.