Nonlinear surface impedance for ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ thin films: Measurements and a coupled-grain model

We present measurements of the surface impedance as a function of frequency (1–17 GHz), temperature (4.2–91 K), and peak rf magnetic field (0<${\mathit{H}}_{\mathrm{rf}}$<500 Oe) for high-quality epitaxial ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ thin films using a stripline-resonator technique. The lowest surface resistance at 1.5 GHz was 15 μΩ at 77 K and 3 μΩ at 4.3 K. The results for the low- and intermediate-field regions (${\mathit{H}}_{\mathrm{rf}}$<50 Oe at 77 K) are explained by a coupled-grain model, which treats the film as a network of superconducting grains connected by grain boundaries acting as resistively shunted Josephson junctions. Quantitative agreement has been obtained between the model and the measurements. The effective junction critical current density, grain size, and shunt resistance are used in the model to characterize the films. The high-field region may be explained by flux penetration into the grains but is not modeled in detail here.