Numerical Analysis of the Kinetic Inductance Fraction of Superconducting Thin-film Resonators

Superconducting transmission line resonators play a crucial role in the measurement of solid-state quantum bits and the detection of weak astrophysical signals. Robust design of resonator arrays requires a thorough understanding of the modification of propagation velocity and mode impedance by the kinetic inductance of the superconducting thin film. Here, we describe a numerical approach to the calculation of kinetic inductance fraction in superconducting transmission lines that can be applied to structures with arbitrary 2D cross section and arbitrary penetration depth. We use fluxoid conservation and matrix inversion to access the current distribution in the structure; from this, we obtain the kinetic inductance contribution of the signal and ground traces along with the modification of the geometric inductance due to finite penetration depth. We compare our numerical results against experimental data from aluminum and niobium coplanar waveguide resonators.