Film-thickness dependence of 10 GHz Nb coplanar-waveguide resonators

We have studied Nb lambda/2 coplanar-waveguide (CPW) resonators whose resonant frequencies are 10-11 GHz. The resonators have different film thicknesses, t=0.05, 0.1, 0.2, and 0.3 um. We measured at low temperatures, T=0.02-5 K, one of the scattering-matrix element, S_21, which is the transmission coefficient from one port to the other. At the base temperatures, T=0.02-0.03 K, the resonators are overcoupled to the input/output microwave lines, and the loaded quality factors are on the order of 10^3. The resonant frequency has a considerably larger film-thickness dependence compared to the predictions by circuit simulators which calculate the inductance of CPW taking into account L_g only, where L_g is the usual magnetic inductance determined by the CPW geometry. By fitting a theoretical S_21 vs. frequency curve to the experimental data, we determined for each film thickness, the phase velocity of the CPW with an accuracy better than 0.1%. The large film-thickness dependence must be due to the kinetic inductance L_k of the CPW center conductor. We also measured S_21 as a function of temperature up to T=4-5 K, and confirmed that both thickness and temperature dependence are consistent with the theoretical prediction for L_k.