Creep effects on the Campbell response in type-II superconductors

Applying the strong pinning formalism to the mixed state of a type-II superconductor, we study the effect of thermal fluctuations (or creep) on the penetration of an ac magnetic field as quantified by the so-called Campbell length lambda(C). Within strong pinning theory, vortices get pinned by individual defects, with the jumps in the pinning energy (Delta e(pin)) and force (Delta f(pin)) between bistable pinned and free states quantifying the pinning process. We find that the evolution of the Campbell length lambda(C)(t) as a function of time t is the result of two competing effects, the change in the force jumps Delta f(pin)(t) and a change in the trapping area S-trap(t) of vortices; the latter describes the area around the defect where a nearby vortex gets and remains trapped. Contrary to naive expectation, we find that during the decay of the critical state in a zero-field cooled (ZFC) experiment, the Campbell length lambda(C)(t ) is usually nonmonotonic, first decreasing with time t and then increasing for long waiting times. Field cooled (FC) experiments exhibit hysteretic effects in lambda(C); relaxation then turns out to be predominantly monotonic, but its magnitude and direction depend on the specific phase of the cooling-heating cycle. Furthermore, when approaching equilibrium, the Campbell length relaxes to a finite value, different from the persistent current, which vanishes at long waiting times t, e.g., above the irreversibility line. Finally, measuring the Campbell length lambda(C)(t) for different states, zero-field cooled, field cooled, and relaxed, as a function of different waiting times t and temperatures T, allows to spectroscopyse the pinning potential of the defects.