Modelling Realistic Multi-layer devices for superconducting quantum electronic circuits

TL;DR

This work addresses the accurate simulation of realistic 3D multilayer superconducting devices, focusing on nanobridge Josephson junctions and coplanar waveguides. It introduces a Usadel-equation–based numerical model that handles complex multilayer geometries with self-consistent $\Delta$ and flexible boundary conditions, avoiding simplifications of materials or layouts. Validation against published data shows good agreement and reveals that multilayer films offer enhanced control over $I_C$, CPR, and the energy gap $\Delta$, with nanobridge-based qubits achieving improved anharmonicity; proximity effects and kinetic inductance in encapsulated CPWs are also characterized. The framework supports optimized design of superconducting qubits and CPW resonators, enabling more reliable and scalable quantum devices; future work includes experimental validation of the proposed designs.

Abstract

In this work, we present a numerical model specifically designed for 3D multilayer devices, with a focus on nanobridge junctions and coplanar waveguides. Unlike existing numerical models, ours does not approximate the physical layout or limit the number of constituent materials, providing a more accurate and flexible design tool. We calculate critical currents, current phase relationships, and the energy gap where relevant. We validate our model by comparing it with published data. Through our analysis, we found that using multilayer films significantly enhances control over these quantities. For nanobridge junctions in particular, multilayer structures improve qubit anharmonicity compared to monolayer junctions, offering a substantial advantage for qubit performance. For coated multilayer microwave circuits it allows for better studies of the proximity effect, including their effective kinetic inductance.

Figures (10)

• Figure 1: 3D design of the nanobridge junctions that will be analyzed later in the manuscript. (a) Planar Nanobridge junction (b) Variable-thickness Nanobridge (VTB) junction, (c) cross-section of simplified VTB junction with rectangular edges and (d) cross-section of VTB junction with realistically rounded edges. Colours are to indicate the presence of different materials
• Figure 2: (a) Simulated CPR of different Nanobridge Junctions for $L/\xi_{S'}=2$, $L'=2L$, $T/T_{CS'}=0.1$, $T_{CS}/T_{CS'}=5$, $\xi_S=5\xi_{S'}$, $\gamma_b=0.01$, $\gamma = 0.01$, $T_{CS}=5 K$ for the case of VTB. For the planar junction, there is only one material so the parameters are $L/\xi_{S}=2$, $T/T_{CS}=0.1$, $T_{CS}=5 K$. $A$ is the cross-sectional area of the nanobridge. In the inset, we show the normalized pair potential of the "planar rounded" Junction (simulated with the parameters aforementioned and $\delta=0.3\pi$), as expected it reduces in the weak link. (b) Comparison between simulated and experimentalVijaySquid critical currents at different nanobridges' lengths, specifically we show the difference in percentage between the values reported in the paper and the one obtained either with the simulations or the KO-1 model. The simulated data agree to a good extent with the experimental ones, the alignment is further improved if a higher coherence length than the one reported in the paper is used propertiesAl.
• Figure 3: Simulated Transmon (a) and Fluxonium (b) qubit anharmonicities with the CPR in Fig. \ref{['fig:simulationCPR1']} supposing the same Josephson energy for all of them. $E_C= 300$ MHz was used for both qubits and $E_L=0.53 E_C$Fluxonium was used for the Fluxonium. In the latter case smaller values of $E_J$ were explored due to the fact that Fluxonium qubit works in the regime $E_J<10 E_C$FluxoniumwithEjvalue.
• Figure 4: (a) 3D design (not to scale) of a multilayer CPW. Magnitude of function $\Phi$ in the CPW for (b) $\gamma=10$ (c) $\gamma=1$ (d) $\gamma=0.1$. The dotted line was added to the images to indicate the separation between the niobium and encapsulating layer. Other parameters used were $\xi_S=5\xi_{N}$, $\xi_{N}=30 nm$, $\gamma_b=0.01$, $T/T_{CS}=0.1$, and the thickness of the superconducting and normal metal is 200 and 20 nm respectively. It is evident that the proximity effect is highly sensitive to the value of $\gamma$. As this parameter increases, the magnitude of function $\Phi$ reduces even when the thickness of the encapsulating material is smaller than the coherence length.
• Figure 5: Simulated magnitude of the function $\Phi$ in a CPW (Not to scale for better visualization) capped with the materials in Tab. \ref{['tab:prox']}: a) Au b) Al c) TiN d) Ta. Again, the dotted line was added to the images to indicate the separation between the niobium and encapsulating layer. Materials with low value of $\gamma$ have low impact on $\Phi$ as expected.