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 $\lambda_\textrm{C}$. Within strong pinning theory, vortices get pinned by individual defects, with the jumps in the pinning energy ($\Delta e_\mathrm{pin}$) and force ($\Delta f_\mathrm{pin}$) between bistable pinned and free states quantifying the pinning process. We find that the evolution of the Campbell length $\lambda_{\rm C}(t)$ as a function of time $t$ is the result of two competing effects, the change in the force jumps $\Delta f_\mathrm{pin}(t)$ and a change in the trapping area $S_\mathrm{trap}(t)$ 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 $\lambda_{\rm C}(t)$ is usually nonmonotonic, first decreasing with time $t$ and then increasing for long waiting times. Field cooled (FC) experiments exhibit hysteretic effects in $\lambda_\textrm{C}$; relaxation then turns out to be predominantly monotonic, but its magnitude and direction depends 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 $\lambda_\textrm{C}(t)$ 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.