Frequency dependence of the surface impedance of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ thin films in a dc magnetic field: Investigation of vortex dynamics

We report measurements of the surface impedance ${\mathit{Z}}_{\mathit{s}}$=${\mathit{R}}_{\mathit{s}}$+iωλ’ of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ thin films in an externally applied dc magnetic field B (parallel to the c axis) using a stripline resonator. At T=4.3 K we obtain the surface resistance ${\mathit{R}}_{\mathit{s}}$ and the microwave penetration depth λ’ as a function of applied dc field up to 5 T and as a function of microwave frequency f from 1.2 to 20 GHz. While λ’ increases linearly with B for B>1 T at all frequencies, ${\mathit{R}}_{\mathit{s}}$ is found to be roughly ∝${\mathit{B}}^{\mathrm{\alpha }\left(\mathit{f}\right)}$, where α(f)<1 for f≤10 GHz and α(f)≊1 for f≥10 GHz. In zero dc field, ${\mathit{R}}_{\mathit{s}}$∝${\mathit{f}}^2$. For B>1 T, ${\mathit{R}}_{\mathit{s}}$ shows a much weaker dependence on f. The results for ${\mathit{Z}}_{\mathit{s}}$(f,T,B) have been quantitatively explained using a model developed by Coffey and Clem, based on a self-consistent treatment of vortex dynamics that includes the influence of vortex pinning, viscous drag, and flux creep. The pinning force constant ${\mathrm{\alpha }}_{\mathit{p}}$, the pinning frequency ${\mathrm{\omega }}_{\mathit{p}}$, and the pinning activation energy ${\mathit{U}}_0$(T,B) are obtained through the fitting procedure. We find that the effects of thermally activated flux creep at 4.3 K upon the surface resistance are significant. The low ${\mathit{U}}_0$≊35 K that we determine is interpreted as arising from the interaction of the vortex lattice with a dense random pinning potential as described in the collective-pinning theory.