Nonlinear electrodynamics of superconducting NbN and Nb thin films at microwave frequencies

Abstract

The surface resistance ${\mathit{R}}_{\mathit{s}}$ of NbN and Nb thin films with particular reference to nonlinear effects is reported at high values of the microwave magnetic field ${\mathit{H}}_{\mathrm{rf}}$\ensuremath{\le}500 Oe using a stripline resonator in the frequency range 0.6\ensuremath{\le}f\ensuremath{\le}12 GHz. In the stripline geometry, the microwave current is concentrated on the narrow center conductor. Thus a high microwave current density and, therefore, a high ${\mathit{H}}_{\mathrm{rf}}$ can be achieved with moderate power. For Nb thin films, ${\mathit{R}}_{\mathit{s}}$ does not increase with ${\mathit{H}}_{\mathrm{rf}}$ for low ${\mathit{H}}_{\mathrm{rf}}$ values, as expected from weak-link theory. However, for the NbN thin films, ${\mathit{R}}_{\mathit{s}}$ at temperature T follows a ${\mathit{R}}_{\mathit{s}}$(T,f)=${\mathit{R}}_{\mathit{s}0}$(T,f)+S(T,f)${\mathit{H}}_{\mathrm{rf}}$ dependence, for f6 GHz where ${\mathit{R}}_{\mathit{s}0}$(T,f) is the surface resistance at zero ${\mathit{H}}_{\mathrm{rf}}$, and the slope S(T,f) is proportional to ${\mathit{f}}^{2,3}$. This nonlinear effect is consistent with Halbritter's weak-link theory. For f>6 GHz, ${\mathit{R}}_{\mathit{s}}$ shows a plateau in the dependence on ${\mathit{H}}_{\mathrm{rf}}$ the magnitude of which varies as ${\mathit{f}}^{3,5}$, which is not predicted theoretically. When ${\mathit{H}}_{\mathrm{rf}}$ increases above a critical value ${\mathit{H}}_{\mathrm{rf}}^{\mathit{c}}$, the resonance curves for the strip-line resonator become asymmetrical and the intermodulation products saturate, indicating strong nonlinearities. The temperature dependence of ${\mathit{H}}_{\mathrm{rf}}^{\mathit{c}}$(T) for Nb, in contrast to NbN, thin films follows that of the dc ${\mathit{H}}_{\mathit{c}1}$. Information on the granularity of NbN, derived from the present study, provides insights into the surface impedance of the granular high-${\mathit{T}}_{\mathit{c}}$ copper oxide thin films.