Collective flux creep in high-Tcsuperconductors

We develop a scaling approach to flux-line pinning in high-${\mathit{T}}_{\mathit{c}}$ superconductors. Our main result is a nonlinear relation V(${\mathit{j}}_{\mathrm{ex}}$)∼exp(-${\mathit{Cj}}_{\mathrm{ex}}^{\mathrm{-}\mathrm{\mu }}$) between the voltage V and the external current density ${\mathit{j}}_{\mathrm{ex}}$, where the exponent μ is related to the roughness exponent and has been estimated in an earlier paper [T. Nattermann, Phys. Rev. Lett. 64, 2454 (1990)]. For the low-frequency ac resistivity we find two contributions ${\mathrm{\rho }}_1$(ω)∼${\mathrm{\omega }}^2$[1-(T/${\mathit{T}}^{\mathrm{*}}$)lnω${\mathrm{\tau }}_0$ ${]}^{4/\mathrm{\psi }}$ and ${\mathrm{\rho }}_2$(ω)∼ω[1-(T/${\mathit{T}}^{\mathrm{*}}$)lnω${\mathrm{\tau }}_0$ ${]}^{2/\mathrm{\psi }}$, where ψ≊1 in three dimensions, and where ${\mathit{T}}^{\mathrm{*}}$ is a characteristic temperature. Similar power laws are obtained for the dynamic susceptibility χ’ ’(ω), whereas the magnetization due to the change of the applied field varies in time as (lnt${)}^{\mathrm{-}1/\mathrm{\mu }}$.