Quantitative Modeling of Superconducting Planar Resonators for Electron Spin Resonance

We present three designs for planar superconducting microwave resonators for electron spin resonance (ESR) experiments. We implement finite-element simulations to calculate the resonance frequency and quality factors as well as the three-dimensional microwave magnetic field distribution of the resonators. One particular resonator design offers an increased homogeneity of the microwave magnetic field while the other two show a better confinement of the mode volume. We extend our model simulations to calculate the collective-coupling rate between a spin ensemble and a microwave resonator in the presence of an inhomogeneous magnetic resonator field. Continuous-wave ESR experiments of phosphorus donors in Si-nat demonstrate the feasibility of our resonators for magnetic resonance experiments. We extract the collective-coupling rate and find good agreement with our simulation results, corroborating our model approach. Finally, we discuss specific application cases for the different resonator designs.