Magnetic-flux dependence of the resistive transition in a square Josephson-junction array

We report measurements of the resistive behavior as a function of the perpendicular magnetic flux in a square proximity-coupled Josephson-junction array. The array was made by a novel ion-beam sputtering technique using the University of Chicago scanning ion microscope. The magnetic flux produces a uniform frustration of phase ordering between the superconducting islands in the array. We have measured both the Ohmic and non-Ohmic resistances as a function of temperature at several values of the frustration index f=φ/${\phi }_0$, the ratio of the applied flux φ to the superconducting flux quantum ${\phi }_0$=hc/2e. We compare the resistive transitions in the array with no frustration at f=0 and f=1, with weak frustration at f=0.05, with commensurate frustration at f=(1/2, and with incommensurate frustration at f=1-1/τ and at f=1/τ [τ=(1+ √5 )/2, the golden ratio]. We find that the transitions at both f=0 and f=1 resemble the Kosterlitz-Thouless transition of a two-dimensional superfluid. The transition at f=0.05 is much broader and is not described by the Kosterlitz-Thouless theory. Instead, the resistance has a pinned flux-flow behavior at low temperatures. The transitions at f=(1/2 and the irrational frustrations are similar to each other but different from the unfrustrated transitions, a result which differs from the expectations of arguments based on the ground-state properties of the array for those fluxes. Our data support the idea that some type of freezing occurs for incommensurate frustration; however, we did not observe the hysteretic effects one might expect to see when cooling and heating through a glass transition.