Proximity effect in asymmetric-gap superconducting bilayers and regularization of transition rates

TL;DR

Proximity coupling between two superconductors with different gaps intrinsically regularizes the BCS density-of-states singularities, removing unphysical divergences in observables tied to quasiparticles. The authors develop a quasiclassical Usadel framework for a weakly coupled bilayer, deriving analytical approximations for the DoS in each film and showing a Dynes-like broadening in the high-gap layer and a shifted square-root threshold in the low-gap layer. They apply these results to gap-asymmetric superconducting qubits, demonstrating how proximity suppresses resonant divergences in quasiparticle relaxation rates and smooths the discontinuity in the qubit frequency shift. The work provides practical insights and analytic tools for interpreting spectral properties of proximitized superconducting bilayers in devices such as transmons and kinetic-inductance detectors.

Abstract

The standard mean-field treatment of low-temperature superconductors leads to a square-root divergent density of states at the gap value. This feature can lead to unphysical logarithmic divergences in various quantities, such as currents and qubit transition rates. We revisit their possible regularization based on the proximity effect between two superconducting films with different gaps. We derive analytical approximations for the density of states in each superconducting film. We find that the smearing of the density of states grows with the gap asymmetry. As a concrete example, we discuss the regularization of transition rates in qubits with frequency close to resonance with the gap asymmetry between the two films, and the consequent smoothening of the jump discontinuity in the qubit frequency shift.

Figures (8)

• Figure 1: Band schematic for superconducting films. (a) Zero transparency (no proximity effect): the BCS quasiparticle DoSs are characterized by square-root singularities at the gap edge. (b) Small transparency (weak proximity): the gap $\epsilon_g\gtrsim\Delta_1$ globally characterizes the full structure, with finite DoS in the left electrode for $\epsilon_g<\epsilon<\Delta_2$. At the gap edge, the square root singularity is replaced by a square-root threshold (see discussion in the text), and each peak is broadened.
• Figure 2: Broadening of the BCS singularity and subgap states in the high-gap superconductor. (a) DoS of $\mathrm{S}_1$ and $\mathrm{S}_2$ computed solving numerically Eq. \ref{['eq:UsadelAngular']} (solid curves). The dashed line is the analytical approximations obtained using Eq. \ref{['eq:explicitGamma']} in Eq. \ref{['eq:DynesAnsatz']} for $\mathrm{S}_2$ and the dot-dashed one a BCS DoS with renormalized gap $\tilde{\epsilon}_g$ [Eq. \ref{['eq:tildeEpsWeak']}] for $\mathrm{S}_1$. These approximations are accurate for weakly-coupled superconductors except close to the gap (shaded aquamarine region). (b) Enlarged view for energies close to $\Delta_2$ (corresponding to the dashed rectangle in panel a) showing the broadening of the $\mathrm{S}_2$ DoS. The BCS DoS is also plotted for comparison. Parameters: $\tau_1=\tau_2=50/\Delta_2$ and $\Delta_1=0.9\Delta_2$.
• Figure 3: DoS in (a) $\mathrm{S}_1$ and (b) $\mathrm{S}_2$ close to the gap $\epsilon_g$. The solid curves, computed solving numerically Eq. \ref{['eq:UsadelAngular']}, are as in Fig. \ref{['fig:Figure2']}, since the parameters are the same; different vertical scales are used in the two panels for better readability. In panel (a), the light green dot-dashed curve gives the approximate BCS-like DoS for $\mathrm{S}_1$ with renormalized gap $\tilde{\epsilon}_g$ [Eq. \ref{['eq:tildeEpsWeak']}], already plotted in Fig. \ref{['fig:Figure2']}. The orange dotted curve is the analytical approximations obtained from the solution of Eq. \ref{['eq:depressedCubicWeak']}, while the yellow dashed one is calculated at the next order in perturbation theory [see Eq. \ref{['eq:DoShigherOrder']} in Appendix \ref{['app:expansionWeak']}]. In panel (b), the orange dotted curve is computed by substituting $\mathcal{N}_1$ from the the solution to Eq. \ref{['eq:depressedCubicWeak']} into Eq. \ref{['eq:relationDoSapp']}. The black dashed curve is the analytical approximation computed using Eq. \ref{['eq:explicitGamma']} in Eq. \ref{['eq:DynesAnsatz']}, and is also shown in Fig. \ref{['fig:Figure2']}.
• Figure 4: Schematic lateral view of a Josephson junction deposited with a double-angle evaporation technique. The junction is fabricated depositing a first superconducting film with thickness $d_2$ (dark grey) on top of an insulating substrate (light blue, typically silicon or sapphire). After an oxidation step to form the insulating barrier, a second film (light grey) is deposited on top of the first film with a different evaporation angle to form the junction. The thickness of the second film ($d_1$) is larger to ensure film continuity around the junction; this thickness mismatch determines a gap asymmetry for aluminum devices (see text). Due to the deposition technique, the two films form a bilayer everywhere, except for the two narrow regions close to the junction.
• Figure 5: Regularization of the divergence in the qubit relaxation rate. Quasiparticle spectral density function $S_1$ (divided by the thermal quasiparticle density) vs frequency for a gap-asymmetric transmon. The result of numerical computation of the rate obtained by solving the Usadel equation (solid yellow) is compared with the analytical approximation ignoring the proximity effect Eq. \ref{['eq:S_1PlusBCS']} (dashed black), and the leading order approximation including proximity Eq. \ref{['eq:regularizedDiv']} (dot-dashed green). Parameters: $\Delta_2/h=50~$GHz, $\Delta_1=0.9\Delta_2$, $\tau_1=10^5/\Delta_2$, $\tau_2=2\times10^4/\Delta_2$, $T=0.01~\Delta_2$.