Geometry dependence of two-level-system noise and loss in $a$-$\mathrm{Si}C$:$H$ parallel-plate capacitors for superconducting microwave resonators

Parallel-plate capacitors (PPC) significantly reduce the size of superconducting microwave resonators, reducing the pixel pitch for arrays of single-photon energy-resolving kinetic inductance detectors (KIDs). The frequency noise of KIDs is typically limited by tunneling two-level systems (TLS), which originate from lattice defects in the dielectric materials required for PPCs. How the frequency noise level depends on the PPC's dimensions has not been experimentally addressed. We measure the frequency noise of 56 resonators with $a$-$\mathrm{Si}C$:$H$ PPCs, which cover a factor of 44 in PPC area and a factor of 4 in dielectric thickness. To support the noise analysis, we measure the resonators' TLS-induced power-dependent intrinsic loss and temperature-dependent resonance frequency shift. From the TLS models, we expect a geometry-independent microwave loss and resonance frequency shift, which is set by the TLS properties of the dielectric. However, we observe a thickness-dependent microwave loss and resonance frequency shift; this is explained by surface layers that limit the performance of PPC-based resonators. For a uniform dielectric, the frequency noise level should scale directly inversely with the PPC area and thickness. We observe that an increase in PPC size reduces the frequency noise, but the exact scaling is, in some cases, weaker than expected. Finally, we derive engineering guidelines for the design of KIDs based on PPC-based resonators.