Film-thickness dependence of 10 GHz Nb coplanar-waveguide resonators

Abstract: We have studied Nb lambda/2 coplanar-waveguide (CPW) resonators whose resonant frequencies are 10-11 GHz. The resonators have different film thicknesses, t=0.05, 0.1, 0.2, and 0.3 um. We measured at low temperatures, T=0.02-5 K, one of the scattering-matrix element, S_21, which is the transmission coefficient from one port to the other. At the base temperatures, T=0.02-0.03 K, the resonators are overcoupled to the input/output microwave lines, and the loaded quality factors are on the order of 10^3. The resonant frequency has a considerably larger film-thickness dependence compared to the predictions by circuit simulators which calculate the inductance of CPW taking into account L_g only, where L_g is the usual magnetic inductance determined by the CPW geometry. By fitting a theoretical S_21 vs. frequency curve to the experimental data, we determined for each film thickness, the phase velocity of the CPW with an accuracy better than 0.1%. The large film-thickness dependence must be due to the kinetic inductance L_k of the CPW center conductor. We also measured S_21 as a function of temperature up to T=4-5 K, and confirmed that both thickness and temperature dependence are consistent with the theoretical prediction for L_k.

• The paper demonstrates that film thickness critically influences the resonant frequency of Nb CPW resonators through modifications in kinetic inductance.
• Precise microwave transmission measurements at 0.02–0.03 K reveal a strong correlation between film thickness and phase velocity deviations in resonator performance.
• The research validates theoretical models and underscores the importance of accounting for kinetic inductance in the design of superconducting circuits.

Film-Thickness Dependence of 10 GHz Nb CPW Resonators

Introduction

The paper investigates the film-thickness dependence of Nb $\lambda/2$ coplanar-waveguide (CPW) resonators with resonant frequencies ranging from 10 to 11 GHz. The study is conducted across varying film thicknesses (0.05, 0.1, 0.2, and 0.3 μm) at low temperatures between 0.02 and 5 K. The principal focus is to elucidate how the kinetic inductance $L_k$ of the CPW center conductor influences resonant frequency, a crucial factor in high-frequency superconducting applications, such as qubits and multiplexing systems.

Experiment

The Nb CPW resonators were fabricated using standard photolithographic techniques and measured in a dilution refrigerator. The configuration consisted of microwave lines coupled through designed capacitors. Transmission measurements ($S_{21}$) were carried out to determine the frequency response and quality factors at the base temperature 0.02–0.03 K. The resonators exhibited a significant film-thickness dependence in resonant frequency, underscoring the role of kinetic inductance $L_k$ which deviates from predictions based solely on geometrical magnetic inductance $L_g$.

Results

The transmission coefficients ($S_{21}$) revealed loaded quality factors on the order of $10^3$, indicating overcoupling to input/output lines. Phase velocity $v_p$ was precisely determined for each film thickness, with variations in $v_p$ strongly correlated to $L_k$. Temperature-dependent measurements showed consistent agreement with theoretical models for $L_k$, reinforcing its dominance in CPW resonators with thinner films.

Discussion

The analysis demonstrates that resonant frequency dependence is predominantly due to kinetic inductance $L_k$, rather than geometric inductance $L_g$. Meservey and Tedrow's calculations for kinetic inductance in superconducting films provided a basis to quantitatively assess the relationship between film thickness $t$, temperature $T$, and inductance contributions in the CPW resonator design.

The study establishes that thin-film superconductors require careful consideration of kinetic inductance for accurate prediction of resonator behavior, particularly in superconducting circuits. The findings facilitate improved design strategies for high-frequency applications requiring precise frequency control.

Conclusion

The research delineates the importance of kinetic inductance $L_k$ in Nb CPW resonators across varying film thicknesses. A thorough understanding of the thickness-dependent and temperature-dependent behavior of $L_k$ enriches the design process for applications in superconducting qubits and microwave multiplexing circuits. This comprehensive characterization contributes to a broader understanding of superconducting microwave circuits, providing insights for future innovations in superconducting device applications.