Microwave hysteretic losses in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ and NbN thin films

Measurements of the nonlinear surface impedance of high-quality epitaxial YBCO and granular NbN films as a function of temperature (4.2–91 K) and frequency (1–13 GHz) are presented. The microwave-field-dependent surface impedance ${\mathit{Z}}_{\mathit{s}}$(${\mathit{H}}_{\mathrm{rf}}$) for both YBCO and NbN films increases quadratically with increasing ${\mathit{H}}_{\mathrm{rf}}$ in the low- and intermediate-rf-field region (${\mathit{H}}_{\mathrm{rf}}$<50 Oe for YBCO at 77 K and for NbN at 13.4 K). In the high-rf-field region, ${\mathit{Z}}_{\mathit{s}}$(${\mathit{H}}_{\mathrm{rf}}$) changes to a different functional dependence of ${\mathit{H}}_{\mathrm{rf}}$. Through the use of a modified Bean critical-state model, the results for the high-rf-field region are explained quantitatively by hysteresis losses due to penetration of microwave vortices. The value of the microwave-vortex penetration field ${\mathit{H}}_{\mathit{p}}$(T) for YBCO thin films, which is nearly frequency independent, is low compared with the dc lower critical fields. The critical- current density values obtained from fits to the hysteresis model are consistent with those measured via dc transport.