Design, Fabrication and Characterization of a Microwave Resonator ...

2 Theoretical Description of a CPW Resonator2.3.5 Superconducting ResonatorFinally, we will discuss the temperature dependencies of the resonance frequency and thequality factor of superconducting resonators.From (2.24) and (2.21) one can directly observe that superconductivity influences thephase velocity v ph through the kinetic inductance defined by (2.22). The kinetic inductancedepends on the temperature dependent London penetration depth λ L (see (2.23)). In Figure2.12 the temperature dependence of the resonance frequency is shown. One can see thatwith constant length the resonant frequency decreases with increasing temperature.Figure 2.12: Dependence of the resonance frequency calculated with the kinetic inductancemodel for the Yale design on Silicon and Sapphire for a fixed lengthof 8.991 mm.The quality factor Q is also temperature dependent, according to (2.32) it consists of thetemperature dependent quantities β and α. In Figure 2.13 the quality factor dependence ontemperature is shown for Silicon and Sapphire.For Silicon one obtains quality factors of around 260. In our measurements the qualityfactor is one order of magnitude or more larger. The reason for this mismatching qualitiescould be that at lower temperatures the loss tangent is much lower than 0.004 (see Table 2.2).A value of around 0.00007 gives better matching results.22