The Fluxoid

One effect of the long range phase coherence is the quantisation of magnetic flux in a superconducting ring. This can either be a ring, or a superconductor surrounding a nonsuperconducting region. Such an arrangement can be seen in Figure 3.9 where region N has a flux density $\boldsymbol{B}$ within it due to supercurrents flowing around it in the superconducting region S.

Figure 3.9: Superconductor enclosing a non-superconducting region.

In the closed path XYZ encircling the nonsuperconducting region there will be a phase difference of the electron-pair wave between any two points, such as X and Y, on the curve due to the field and the circulating current as given by Equation 3.21.

The total phase change around the path XYZX can be written as

$$\Delta\phi = \frac{4\pi m}{hn_{s}e} \oint \boldsymbol{J}_{s} \cen... ...+ \frac{4\pi m}{h} \iint_{S} \boldsymbol{B} \centerdot \mathrm{d}\boldsymbol{S}$$ (3.24)

where $\boldsymbol{S}$ is the area enclosed by XYZX.[12]

If the superelectrons are represented by a single wave then at any point on XYZX it can only have one value of phase and amplitude. Due to the long range coherence the phase is single valued meaning around the circumference of the ring $\Delta\phi$ must equal $2\pi n$ where $n$ is any integer.[13] Rewriting Equation 3.24 using this condition we have a definition for $\Phi'$

$$\Phi' = \frac{m}{n_{s}e^{2}} \oint \boldsymbol{J}_{s} \centerdot ... ...2e}\iint_{S} \boldsymbol{B} \centerdot \mathrm{d}\boldsymbol{S} = n\frac{h}{2e}$$ (3.25)

with the central part of this equation named the fluxoid by F. and H. London. Due to the wave only having a single value the fluxoid can only exist in quantised units. This quantum is termed the fluxon, $\Phi_{0}$, given by

$$\Phi_{0} = \frac{h}{2e} = 2.07 \times 10^{-15} \ensuremath{\unskip\,\mathrm{Wb}}$$ (3.26)