Improving the performance of superconducting qubits and resonators generallyresults from a combination of materials and fabrication process improvementsand design modifications that reduce device sensitivity to residual losses. Oneinstance of this approach is to use trenching into the device substrate incombination with superconductors and dielectrics with low intrinsic losses toimprove quality factors and coherence times. Here we demonstrate titaniumnitride coplanar waveguide resonators with mean quality factors exceeding twomillion and controlled trenching reaching 2.2 $\mu$m into the siliconsubstrate. Additionally, we measure sets of resonators with a range of sizesand trench depths and compare these results with finite-element simulations todemonstrate quantitative agreement with a model of interface dielectric loss.We then apply this analysis to determine the extent to which trenching canimprove resonator performance.
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