A semi-empirical model for two-level system noise in superconducting microresonators

We present measurements of the low--temperature excess frequency noise of four niobium superconducting coplanar waveguide microresonators, with center strip widths $s_r$ ranging from 3 $\mu$m to 20 $\mu$m. For a fixed internal power, we find that the frequency noise decreases rapidly with increasing center strip width, scaling as $1/s_r^{1.6}$. We show that this geometrical scaling is readily explained by a simple semi-empirical model which assumes a surface distribution of independent two-level system fluctuators. These results allow the resonator geometry to be optimized for minimum noise.