Strong pinning theory of thermal vortex creep in type-II superconductors

Abstract

We study thermal effects on pinning and creep in type-II superconductors where vortices interact with a low density $n_p$ of strong point-like defects with pinning energy $e_p$ and extension $\xi$, the vortex core size. Defects are classified as strong if the interaction between a single pin and an individual vortex leads to the appearance of bistable solutions describing pinned and free vortex configurations. Extending the strong pinning theory to account for thermal fluctuations, we provide a quantitative analysis of vortex depinning and creep. We determine the thermally activated transitions between bistable states using Kramer's rate theory and find the non-equilibrium steady-state occupation of vortex states. The latter depends on the temperature $T$ and vortex velocity $v$ and determines the current--voltage (or force--velocity) characteristic of the superconductor at finite temperatures. We find that the $T=0$ linear excess-current characteristic $v \propto (j-j_c) \, \Theta(j-j_c)$ with its sharp transition at the critical current density $j_c$, keeps its overall shape but is modified in three ways due to thermal creep: a downward renormalization of $j_c$ to the thermal depinning current density $j_\mathrm{dp}(T) < j_c$, a smooth rounding of the characteristic around $j_\mathrm{dp}(T)$, and the appearance of thermally assisted flux flow (TAFF) ${v \propto j \exp(-U_0/k_{\rm \scriptscriptstyle B} T)}$ at small drive $j \ll j_c$, with the activation barrier $U_0$ defined through the energy landscape at the intersection of free and pinned branches. This characteristic emphasizes the persistence of pinning of creep at current densities beyond critical.