Geometry and temperature dependence of low-frequency flux noise in dc SQUIDs

Measurements on dc SQUIDs reveal a flux noise spectral density $S_\Phi(f) = A^2/(f/1~Hz)^\alpha$. An analytic model assuming non-interacting spins localized at the surface of the SQUID loop predicts that the mean square noise scales as R/W---the radius and width of the loop, respectively. However, there are no established theories for the scaling of $\alpha$ with geometry or the dependences of A and $\alpha$ on temperature T. To test the predicted geometric scaling of this model experimentally, we measured flux noise in ten SQUIDs with systematically varying geometries. We find that, at fixed T, $A^2$ scales approximately as R. From the measured values of A and $\alpha$, we estimate the mean square flux noise, which does not scale with R. As T is lowered, $\alpha$ increases significantly and in such a way that the spectra ``pivot'' about an approximately fixed frequency. This phenomenon implies that the mean square noise is temperature-dependent, an effect not predicted by the analytic model. We discuss our attempts to reconcile these discrepancies by considering the locking together of spins to form clusters.