Frequency and temperature dependence of the microwave 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 the results of a study of the complex microwave surface impedance ${\mathit{Z}}_{\mathit{S}}$ resulting from vortex motion in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ thin films in a dc magnetic field applied parallel to the film c axis. Using the technique of stripline resonators we have measured ${\mathit{Z}}_{\mathit{S}}$ at frequencies from 1.2 to 22 GHz and at temperatures from 5 to 65 K in magnetic fields from 0 to 4 T. We find that both the surface resistance ${\mathit{R}}_{\mathit{S}}$ and the surface reactance ${\mathit{X}}_{\mathit{S}}$ increase almost linearly with the magnetic field. In zero applied magnetic field we find the frequency dependence of the surface resistance to be ${\mathit{f}}^2$. In the mixed state, however, there is a significant increase in ${\mathit{R}}_{\mathit{S}}$, particularly at lower frequencies, causing ${\mathit{R}}_{\mathit{S}}$ to be approximately proportional to ${\mathit{f}}^{1.2}$ at all measured temperatures. We show that fits of these data to models which include only a single pinning energy and a single characteristic pinning frequency are not able to explain our results. We propose that these data indicate the existence of a large number of metastable bound vortex states separated by energy barriers ${\mathit{U}}_{\mathit{b}}$ whose magnitudes extend from ${\mathit{U}}_{\mathit{b}}$∼0 K to several hundred K, and that the dominant part of ${\mathit{R}}_{\mathit{S}}$ arises from vortex transitions between these states. © 1996 The American Physical Society.