SQUID Magnetometer

The circulating current produced by a flux change in the SQUID can be detected by the use of a measuring current, $I$, as shown in Figure 3.10. This current divides equally between both weak-links if the ring is symmetrical. Whilst the current through the weak-links is small there will be no voltage detected across the ring. As $I$ is increased it reaches a critical measuring current, $I_{c}$, at which voltages begin to be detected.

The magnitude of the critical measuring current is dependent upon the critical current of the weak-links and the limit of the phase change around the ring being an integral multiple of $2\pi$. For the whole ring to be superconducting the following condition must be met

$$\alpha + \beta + 2\pi\frac{\Phi_{a}}{\Phi_{0}} = n \centerdot 2\pi$$ (3.31)

where $\alpha$ and $\beta$ are the phase changes produced by currents across the weak-links and $2\pi\nicefrac{\Phi_{a}}{\Phi_{0}}$ is the phase change due to the applied magnetic field.

When the measuring current is applied $\alpha$ and $\beta$ are no longer equal, although their sum must remain constant. The phase changes can be written as

$$\alpha = \pi \left[ n - \left( \frac{\Phi_{a}}{\Phi_{0}} \right) \right] - \delta$$ (3.32)

$$\beta = \pi \left[ n - \left( \frac{\Phi_{a}}{\Phi_{0}} \right) \right] + \delta$$ (3.33)

where $\delta$ is related to the measuring current $I$. Using the relation between current and phase in Equation 3.27 and rearranging to eliminate $i$ we obtain an expression for $I$,

$$I = 2i_{c} \left\vert \cos \pi \frac{\Phi_{a}}{\Phi_{0}} \cdot \sin \delta \right\vert$$ (3.34)

As $\sin \delta$ cannot be greater than unity we can obtain the critical measuring current, $I_{c}$ from Equation 3.34 as

$$I_{c} = 2i_{c} \left\vert \cos \pi \frac{\Phi_{a}}{\Phi_{0}} \right\vert$$ (3.35)

which gives a periodic dependence on the magnitude of the magnetic field, with a maximum when this field is an integer number of fluxons and a minimum at half integer values as shown in Figure 3.11.

Figure 3.11: Critical measuring current, $I_{c}$, as a function of applied magnetic field.