Magnetic properties of the transition to localized superconductivity around columnar defects

An extreme type II superconductor is considered to have a dense square array of columnar defects of small radius and insulating core. An applied field parallel to these defects gives rise to a state of flux lines, and we analyze the region of high density near the upper critical field, and above, where superconductivity only exists on a thin ring around each columnar defect. We find that the ratio between the surface critical field and the upper critical field, which characterizes this region, decreases for increasing defect radius, a result opposite to that found in the single defect problem