Ab initio modeling of nonequilibrium dynamics in superconducting detectors and qubits

TL;DR

The paper addresses nonequilibrium quasiparticle and phonon dynamics in superconducting devices by introducing an ab initio framework that couples kinetic equations for $f(E)$ and $n(\Omega)$ with density functional perturbation theory to obtain the full-bandwidth electron-phonon coupling $\alpha^2F(\Omega)$ and $F(\Omega)$. The approach integrates these microscopic inputs with a dirty-limit Usadel-based description of the superconducting density of states and self-consistent $|\Delta|$, enabling parameter-free predictions for device behavior. Demonstrations on NbN-based SNSPDs and Ta-based transmons show that the method improves upon Debye-model predictions and aligns with experimental data without fitting, highlighting material-dependent resilience to quasiparticle poisoning and detection performance. The framework generalizes to other superconducting materials and devices, offering a path toward computational material discovery and optimization for quantum technologies.

Abstract

Nonequilibrium quasiparticle (QP) and phonon dynamics are central to the operation of superconducting devices. Superconducting detectors, such as the superconducting nanowire single-photon detector, perform best when a large QP population is generated in response to small perturbations. Conversely, for superconducting qubits and topologically protected Majorana fermions, even relatively small QP densities can lead to significant performance degradation, and thus, ideal materials are less susceptible to QP poisoning. However, existing models of these devices lack a rigorous description of the QP and phonon dynamics, relying on approximations and phenomenology. In this article, we combine kinetic equations with density functional theory to model the nonequilibrium dynamics of a superconducting film ab initio. To demonstrate the universality of our model, we illustrate two examples: (1) we develop a model for the detection of single photons in superconducting nanowires, and (2) we calculate the energy-relaxation rate of a transmon qubit due to the presence of excess QPs. Our examples demonstrate from first principles that NbN is well-suited for single-photon detection and that Ta transmon qubits possess reduced sensitivity to QP poisoning relative to other materials, which is likely in part responsible for their longer coherence times. In contrast to previous models, our ab initio approach makes these predictions without experimental input and thus can be used to accelerate progress in device development. Moreover, by considering the full-bandwidth electron-phonon coupling, our approach can incorporate strong-coupling effects. Our methods effectively integrate ab initio materials modeling with nonequilibrium theory of superconductivity to perform practical modeling of superconducting devices, providing a comprehensive approach that connects fundamental theory with device applications.

Figures (8)

• Figure 1: High-level overview of the modeling of nonequilibrium devices. (a) Typical SNSPD geometry consisting of a thin ($d \sim \xi_{\mathrm{c}}$) narrow ($w \ll \Lambda$) superconducting wire that is patterned in a meander to increase the active area of the detector. The device is single-photon sensitive when a sufficiently large bias current $I_{\mathrm{B}}$ is applied. (b) Microscopic picture of SNSPD detection. A photon is absorbed, generating an excited quasiparticle consisting of an electron-hole pair. This excitation triggers an energy-relaxation cascade and the generation of a phonon bubble. The resulting quasiparticles and phonons scatter and break pairs, locally suppressing the superconductivity and weakening the barrier for quantum and thermal fluctuations that may fully destroy the superconductivity across the strip. Due to the bias current, this normal strip produces a nonzero voltage across the terminals of the device, which is read out as a detection event. (c) A cartoon depiction of the Manhattan-style Josephson junction commonly used in transmon qubit fabrication. A cosmic ray interacts with the substrate, generating a large population of phonons that perturb the superconductivity in the Josephson junction. (d) Cross-sectional view of the process in (c). The quasiparticle recombination, scattering, and generation processes are displayed inside the superconducting layer of the Josephson junction. These processes are identical to those in the phonon bubble and thermalization steps shown in (b). (e) Ab initio approach to modeling nonequilibrium dynamics in superconductors. Beginning with the crystal structure, we obtain the electron-phonon coupling with density functional theory, which allows us to obtain the interaction probabilities for the quasiparticle and phonon systems. The evolution of the quasiparticle and phonon distributions is then modeled using a set of kinetic equations.
• Figure 2: (a) Normalized quasiparticle density of states $\rho(E)$ for $\delta$-$\mathrm{NbN}$ with $T_{\mathrm{c}} = 10\,\mathrm{K}$ and different ratios of the bias current normalized to the depairing current $I_{\mathrm{B}}/I_{\mathrm{dep}}(T)$. (b) Acoustic phonon modes for $\delta$-$\mathrm{NbN}$ calculated using density functional perturbation theory compared to experimental data for the longitudinal and transverse modes christensen1979phonon. The corresponding Eliashberg spectral function $\alpha^2F(\Omega)$ (solid grey) and phonon density of states $F(\Omega)$ (black) are displayed on the right compared to the Debye approximation (dashed and solid blue). Density functional perturbation theory calculated results are also displayed for (c) aluminum, (d) niobium, (e) titanium nitride, (f) lead, (g) tantalum. Spin-orbit coupling effects were included for lead and tantalum.
• Figure 3: Solutions to the kinetic equations for $\delta f(E)$, $\delta n(\Omega)$, and $\epsilon(t)$ for NbN. (a) Nonequilibrium excess quasiparticle $\delta\!f(E)$ and (b) excess phonon $\delta \! n(\Omega)$ distribution generated by the absorption of a photon with wavelength $\lambda_{\mathrm{ph}} = 1064\,\mathrm{nm}$ at $t=\tau_{\Delta}$ for $\mathrm{NbN}$ with electronic diffusion coefficients of $D=0.5\,\mathrm{cm^2/s}$ (solid black lines) and $D = 1.5\,\mathrm{cm^2/s}$ (dashed blue lines). In (b), $\alpha^2(\Omega)$ for NbN is displayed on the right axis. (c) Quasiparticle-induced suppression parameter $\varepsilon(t)$. A value of $\varepsilon = 0$ implies no suppression of $\Delta$.
• Figure 4: A comparison of the ab initio model (solid lines) against the phenomenological diffusive hotspot model Semenov2001Semenov_2021 (dotted lines), Debye model (dashed lines), and experimental data (large open markers) marsili2012efficientGourgues:1910.1063/5.0018818. (a) Determination of the detection current $I_{\mathrm{det}}$ for a given photon wavelength $\lambda_{\mathrm{ph}}$ and wire widths of $w = 30\,\rm{nm},\,50\,\rm{nm},$ and $85\,\rm{nm}$ with $D=0.5\,\mathrm{cm^2/s}$ compared to the experimental data of Ref. marsili2012efficient. A value of $\eta = 0.2$ for the diffusive hotspot model gives the best fit (See Ref. Semenov_2021 for a definition of $\eta$). (b) $I_{\mathrm{det}}$ as a function of $\lambda_{\mathrm{ph}}$ for $w=20\,\mathrm{nm}$ and parameters consistent with the material used in the experimental data of Ref. 10.1063/5.0018818. Results using both the full-bandwidth electron-phonon coupling (ab initio) and Debye model are displayed. (c) Ab initio predictions of the normalized detection current $I_{\mathrm{det}}/I_{\rm{c}}$ as a function of the reduced temperature $T/T_\mathrm{c}$ for $\lambda_{\mathrm{ph}} = 515\,\rm{nm}$ compared to experimental data for the temperature-dependence of $\mathrm{NbTiN}$ from Ref. Gourgues:19. Since the data in this figure is not for $\mathrm{NbN}$, only qualitative agreement of the temperature dependence is expected. Calculations from the ab initio model are in arbitrary units.
• Figure 5: The energy-relaxation rate $\Gamma_1 = 1/T_1$ of a transmon qubit. (a) $\Gamma_1$ as a function of the radiation power density of a $\rm{Cu}^{64}$ source as described in Ref. Will-Oliver. The ab initio prediction is depicted with the experimental data. (b) Predictions from the ab initio model of the dependence of $\Gamma_{\rm{qp}}$ on absorbed phonon energy density $\Omega_{\rm{tot}}/V$. Aluminum is found to be the most susceptible to quasiparticle poisoning, whereas niobium is the least.