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

3 Designs Procedure for the ResonatorsIn this thesis two substrate materials, namely Sapphire and Silicon, were used, as mentionedin the previous chapters.The Sapphire substrates had already been used by Andreas Emmmert [5] to fabricate asuperconducting beam splitter or hybrid ring. If it is possible to realize both resonator andbeam splitter on the same chip new measurements e.g. vacuum fluctuations in the resonatorwill be possible. (This idea is presented in [11])The production of nanostructured qubits by means of electron beam lithography is muchmore straight forward on Silicon substrates because of the high conductivity of Silicon atroom temperature. Hence, we fabricated resonators on Silicon for the future experiment ofplacing a qubit in the resonator.The (effective) dielectric constants and the (effective) thinknesses of the substrates werementioned in subsection 2.1.2. On top of the substrates the resonators were deposited withNiobium, which is weakly sensitive to an applied magnetic field (the upper critical field B C2is around 2 T) and has only a very small amount of quasiparticles at the future operationtemperature of 50 mK due to a high critical temperature of 9.2 K.For the design of the resonators you have to decide on the following parameters:1. substrate material (Sapphire or Silicon)2. gap W between center strip and ground planes3. resonance frequency4. coupling capacitance3.1 ResonatorAfter choosing the substrate material and the width of the gap W, the center strip width Sis calculated for Z cs to be 50 Ω, to match the resonator structure to the coaxial cables andavoid reflections. The approximations (3.1) and (3.2) of equation (2.25) for CPWs withinfinite box height H 1 are used for calculating the center strip width S corresponding to thechosen gap width W. In order to calculate the center strip width we used approximations forSapphire{W+0.3308S =0.492if W < 51(3.1)W+0.89080.4886if W > 51and for SiliconS ={W+0.37420.5908if W < 51W+0.95790.6006if W > 51.(3.2)27