  Superconducting resonators are now found in a broad range of applications
that require high-fidelity measurement of low-energy signals. A common feature
across almost all of these applications is the need for increased numbers of
resonators to further improve sensitivity, and the ability to read out large
numbers of resonators without the need for additional cryogenic complexity is a
primary motivation. One of the major limitations of current resonator arrays is
the observed scatter in the resonator frequencies when compared to the initial
design. Here we present recent progress toward identifying one of the dominant
underlying causes of resonator scatter, inductor line width fluctuation. We
designed and fabricated an array of lumped-element resonators with inductor
line width changing from 1.8um to 2.2um in step of 0.1um defined with
electron-beam lithography to probe and quantify the systematic variation of
resonance frequency across a 6-inch wafer. The resonators showed a linear
frequency shift of 20MHz (140FWHM) and 30MHz (214FWHM), respectively, as they
are connected to two different capacitors. This linear relationship matches our
theoretical prediction. The widely used MLA photon lithography facility for
MKID fabrication has a resolution on the order of 600nm, which could cause
frequency fluctuation on the order of 100MHz or 710FWHM.
