Superconducting Quantum Interference Device (SQUID)

A Superconducting Quantum Interference Device (SQUID) uses the properties of electron-pair wave coherence and Josephson Junctions to detect very small magnetic fields. The central element of a SQUID is a ring of superconducting material with one or more weak links. An example is shown in Figure 3.10, with weak-links at points W and X whose critical current, $i_{c}$, is much less than the critical current of the main ring. This produces a very low current density making the momentum of the electron-pairs small. The wavelength of the electron-pairs is thus very long leading to little difference in phase between any parts of the ring.

Figure 3.10: Superconducting quantum interference device (SQUID) as a simple magnetometer.

If a magnetic field, $B_{a}$, is applied perpendicular to the plane of the ring, a phase difference is produced in the electron-pair wave along the path XYW and WZX. A small current, $i$, is also induced to flow around the ring, producing a phase difference across the weak links. Normally the induced current would be of sufficient magnitude to cancel the flux in the hole of the ring but the critical current of the weak-links prevents this.

The quantum condition that the phase change around the closed path must equal $n2\pi$ can still be met by large phase differences across the weak-links produced by even a small current. An applied magnetic field produces a phase change around a ring, as shown in Equation 3.23, which in this case is equal to

$$\Delta\phi(B) = 2\pi\frac{\Phi_{a}}{\Phi_{0}}$$ (3.28)

where $\Phi_{a}$ is the flux produced in the ring by the applied magnetic field.[12] $\Phi_{a}$ may not necessarily equal an integral number of fluxons so to ensure the total phase change is a multiple of $2\pi$ a small current flows around the ring, producing a phase difference of $2\Delta\phi(i)$ across the two weak-links, giving a total phase change of

$$\Delta\phi(B) + 2\Delta\phi(i) = n2\pi$$ (3.29)

The phase difference due to the circulating current can either add to or subtract from that produced by the applied magnetic field but it is more energetically favourable to subtract: in this case a small anti-clockwise current, $i^{-}$.[12]

Substituting values from Equations 3.27 and 3.28, the magnitude of the circulating current, $i^{-}$, can be obtained

$$\vert i^{-} \vert = i_{c} \sin \pi \frac{\Phi_{a}}{\Phi_{0}}$$ (3.30)

As the flux in the ring is increased from 0 to $\nicefrac{1}{2}\Phi_{0}$ the magnitude of $i^{-}$ increases to a maximum. As the flux is increased greater than $\nicefrac{1}{2}\Phi_{0}$ it is now energetically favourable for a current, $i^{+}$, to flow in a clockwise direction, decreasing in magnitude to 0 as the flux reaches $\Phi_{0}$. The circulating current has a periodic dependence on the magnitude of the applied field, with a period of variation of $\Phi_{0}$, a very small amount of magnetic flux. Detecting this circulating current enables the use of a SQUID as a magnetometer.