Vortex depinning frequency in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ superconducting thin films: Anisotropy and temperature dependence

We have used the parallel-plate resonator technique to study the microwave surface resistance ${\mathit{R}}_{\mathit{s}}$ and the penetration depth λ of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ thin epitaxial films at 5.5 GHz in the presence of magnetic field H≤0.8 T and at 30 K<T<80 K. The magnetic-field dependence of ${\mathit{R}}_{\mathit{s}}$ and λ allows determination of the vortex viscosity η, pinning constant α, and depinning frequency ${\mathrm{\omega }}_0$ for different field orientations. To the best of our knowledge, this is among the first measurements of the angular dependence of η, α, and ${\mathrm{\omega }}_0$. We find that the angular dependences of η, α, and ${\mathrm{\omega }}_0$ are fairly well described by the scaling model of Blatter, Geshkenbein, and Larkin. We demonstrate that (i) the temperature dependence of η is well described by the Bardeen-Stephen model with reduced normal-state resistivity, (ii) the pinning constant is close to its upper limit at lowest temperatures and decreases exponentially with increasing temperature, and (iii) the depinning frequency is of the order of 10 GHz, almost orientation independent and weakly depends on temperature. We analyze the values of the depinning frequency for different type-II superconductors and demonstrate that it is directly related to the quasiparticle scattering rate.