Modelling Realistic Multi-layer devices for superconducting quantum electronic circuits

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.

💡 Research Summary

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The paper introduces a comprehensive three‑dimensional numerical framework for modelling superconducting quantum devices that contain multilayer structures, with particular emphasis on nanobridge Josephson junctions and coplanar waveguides (CPWs). Unlike earlier approaches that either simplify the geometry to two dimensions or restrict the number of material layers, the presented model solves the full Usadel equations in the dirty limit without approximations to the layout. By employing a Φ‑parametrisation of the Green’s function, separating real and imaginary components, and using a Newton‑based nonlinear solver within the SfePy finite‑element environment, the authors are able to treat arbitrary cross‑sectional shapes (including realistic rounded edges) and arbitrary stacks of superconducting, normal‑metal, or insulating layers. Boundary conditions incorporate both the suppression parameter γ (ratio of resistivities and coherence lengths) and the interface resistance γ_B, allowing precise control over proximity‑effect coupling at each interface.

The authors validate the model against two experimental datasets. First, they compare simulated critical currents of aluminium nanobridge junctions (both planar and variable‑thickness designs) with measurements reported in Ref. 24. By adjusting the coherence length from the reported 40 nm to a more realistic 110 nm (derived from room‑temperature resistivity), the simulated I_C values match the experimental data within a few percent. Second, they benchmark the current‑phase relationship (CPR) against the analytical KO‑1 one‑dimensional solution. The planar junctions exhibit a reduced critical current and a left‑shifted CPR peak when the edges are rounded, reflecting a local suppression of the superconducting gap in the bridge region. Variable‑thickness (VTB) bridges, by contrast, show a less skewed CPR because the thick electrodes act as robust phase reservoirs, concentrating the phase drop across the bridge rather than the leads.

A systematic study of geometry shows that rounded edges lower the critical current by roughly 12 % compared with ideal rectangular cross‑sections, even after normalising for the reduced cross‑sectional area. Moreover, VTB designs improve qubit anharmonicity by 15–25 % relative to planar bridges, a consequence of the more sinusoidal CPR. The authors then embed the simulated CPRs into the Hamiltonians of transmon and fluxonium qubits (Eqs. 9‑10) using custom functions in scqubits and Qiskit‑Metal. By solving the resulting eigenvalue problem with SciPy’s dense eigensolver, they compute the anharmonicities for a range of E_J/E_C ratios. The results confirm that while the anharmonicity of a nanobridge‑based transmon is modestly reduced compared with a conventional tunnel junction, the multilayer VTB configuration recovers most of the lost nonlinearity, making it a viable alternative for high‑coherence qubits.

The paper also explores the proximity effect in multilayer CPWs capped with normal metals (Au, Al, TiN, Ta). By varying the suppression parameter γ (ρ_S ξ_S / ρ_N ξ_N) from 0.1 to 10, the authors demonstrate that larger γ values strongly suppress the superconducting order parameter Φ inside the Nb core, leading to a reduced energy gap Δ and an increased kinetic inductance L_k. Table I quantifies the gap reduction and the ratio L_k^cap/L_k^0 for each capping material. Materials with low γ (e.g., Al, TiN) have a minimal impact on Δ, whereas high‑γ caps (e.g., Au) can lower the gap by up to 5 % and increase kinetic inductance by a comparable amount, potentially raising quasiparticle densities and degrading resonator Q‑factors and qubit T₁ times.

In summary, the work delivers three major contributions: (1) a flexible, fully three‑dimensional numerical solver for dirty‑limit superconducting multilayers that respects realistic geometries and arbitrary material stacks; (2) a quantitative analysis showing how bridge shape, thickness variation, and multilayer CPW design affect critical currents, CPR skewness, and qubit anharmonicity; and (3) practical guidelines for selecting capping layers in superconducting microwave circuits based on the suppression parameter γ and its impact on the energy gap and kinetic inductance. These results provide a powerful design tool for next‑generation superconducting quantum processors, enabling engineers to optimise junction geometry and multilayer packaging to achieve higher coherence, better control of anharmonicity, and reduced loss from proximity‑induced quasiparticles.