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High-energy quasiparticle injection into mesoscopic superconductors


At non-zero temperatures, superconductors contain excitations known as Bogoliubov quasiparticles (QPs). The mesoscopic dynamics of QPs inform the design of quantum information processors, among other devices. Knowledge of these dynamics stems from experiments in which QPs are injected in a controlled fashion, typically at energies comparable to the pairing energy1,2,3,4,5. Here we perform tunnel spectroscopy of a mesoscopic superconductor under high electric fields. We observe QP injection due to field-emitted electrons with 106 times the pairing energy, an unexplored regime of QP dynamics. Upon application of a gate voltage, the QP injection decreases the critical current and, at sufficiently high electric field, a field-emission current (<0.1 nA in our device) switches the mesoscopic superconductor into the normal state, consistent with earlier observations6. We expect that high-energy injection will be useful for developing QP-tolerant quantum information processors, will allow rapid control of resonator quality factors and will enable the design of electric-field-controlled superconducting devices with new functionality.

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Fig. 1: Tunnel spectroscopy and critical current of a titanium nanowire at high gate voltage (VG) and 20 mK temperature.
Fig. 2: Device characterization in applied magnetic field.
Fig. 3: Injection in fixed high magnetic field, a second device and a model of QP injection due to field emission.

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Data availability

The transport data and device images that support the findings of this study are publicly available in the Harvard Dataverse with the identifier doi:10.7910/DVN/LHCDHV. Source data are provided with this paper.


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We thank P. Gumann and A. Annunziata for discussions. This work is supported by IBM Quantum, under the Q Network for Academics programme. This work is also partly supported by NSF grant DMR-1708688. L.D.A. acknowledges support from an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at Harvard University, administered by Oak Ridge Institute for Science and Education through an interagency agreement between the US Department of Energy and the Office of the Director of National Intelligence.

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L.D.A., A.K.S., C.G.L.B., A.T.P., S.H.L. and S.P.H. performed the low-temperature measurements. L.D.A. and A.K.S. fabricated the devices and performed the data analysis. L.D.A. and A.Y. designed the experiments. All authors discussed the results and commented on the manuscript.

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Correspondence to Loren D. Alegria.

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The authors declare no competing interests.

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Peer review informationNature Nanotechnology thanks A. Levy Yeyati and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Self-consistency calculation.

Comparison of the spectroscopy fitting to the self-consistency relation for weak-coupling superconductors. This relation links the zero temperature gap to the critical temperature as Δ0 = 1.764kBTc, and the temperature dependence of the gap implicitly through the relation \(\,\text{ln}\,(2{e}^{\gamma }\hslash {\omega }_{D}/{k}_{\text{B}}{T}_{c}\pi )=\mathop{\int}\nolimits_{0}^{\hslash {\omega }_{D}}\,\text{tanh}\,(\sqrt{{\xi }^{2}+{{\Delta }}{(T)}^{2}}/2{k}_{\text{B}}T)/\sqrt{{\xi }^{2}+{{\Delta }}{(T)}^{2}}d\xi\) where γ is Euler’s constant, and ωD is the Debye energy. For weak coupling ωDkBT, so that Δ(T) is parametrized only by Tc23. Above we plot the titanium gap energy versus TQP extracted from the spectroscopy model (that is from the data of Fig. 1c-d). The data are within the range corresponding to Tc = 0.24 - 0.31 K, the grey region, obtained by solving the foregoing equation numerically for those two bounds. The conformity to the quasiparticle population to the self-consistency relation is therefore quite good, which further excludes an exotic dissipationless gate effect. The slight departure of the data from a typical BCS dependence may be due to variations in the non-equilibrium state due to the energy of the impinging electrons, which varies dramatically over this data set. The error bars are calculated as described in the methods section.

Extended Data Fig. 2 Titanium wire magnetotransport.

Resistance of the Ti wire of Fig. 1 as a function of magnetic field applied along x, y, and z directions for zero applied gate voltage (top row) and 35 V (bottom row).

Extended Data Fig. 3 Broadening mechanism comparison.

Tunnel junction conductance vs bias data (dots) for gate voltages from 20 V (most peaked) to 43 V (least peaked) as compared to best fits with TQP as free parameter (lines) and best fits with αAl as free parameter (dashed lines).

Extended Data Fig. 4 Electric field calculation and current measurements.

The device of the main text (a) has a slight asymmetry in the y-direction. As a result, the left and right gates produce different electric field distributions, the magnitude of which we calculate numerically in (b) for 40 V applied to either gate. As a result of this asymmetry, we use the right gate in the data of the main text, since this most effectively applies field to the Ti. (c) To look for current flow through the gate, we measure gate current in devices identical to the devices in the text, but at 4.2 K. We observe breakdown at 45 - 60 V in such devices, above the region of stable emission in the text. Below breakdown, the small amount of current detected is ohmic and almost certainly takes place in the contacts in this measurement. In the devices of Extended Data Figure 6, measured in a separate dilution refrigerator with high line-isolation, current was measured to be less than 1 pA for VG = 52 V, but these devices had a different gate geometry (90 nm SiO2 back gate). The Fowler-Nordheim model described in the main text predicts current of ~ 0.1 nA at the highest voltage.

Extended Data Fig. 5 1D Quasiparticle diffusion model.

The minimal model of quasiparticle dynamics presented in the main text does not account for diffusion of QP along the Ti wire, away from the injection region. Correcting this can account for the departure of the data from equation (5) as follows: We assume that the QP distribute along the length of the Ti wire according to ρ(y) due to one-dimensional diffusion over a distance \({(2{D}_{\text{Ti}}{\tau }_{\text{eff}})}^{1/2}\) where DTi is the diffusion constant calculated from the magnetic field data. That is \(\rho (y)={\rho }_{0}\mathop{\int}\nolimits_{-{l}_{I}/2}^{{l}_{I}/2}\exp ({(s-y)}^{2}/2{D}_{\text{Ti}}{\tau }_{\text{eff}})ds\) where ρ0 is such that ∫ρ(y)dy = 1, and lI is the injector length. Moreover, we relax the assumption that \({\tau }_{\text{eff}} \sim {x}_{\text{Ti}\,}^{-1}\). The new model amounts to replacing A on the right hand side of equation (5) with Aρ(yTJ, τeff)xTiτeffΩ/(lItTJwTJ) where yTJ is the distance from the tunnel junction to the centre of the injection electrode, lI is the length of the injection electrode, tTJ is the thickness of the tunnel junction portion of the Ti wire, wTJ is the width of that portion, and now τeff = τ0xν. The values lI = 600 nm, tTJ = 5 nm, wTJ = 7 nm, and ν = 2.5 correspond to the curved line in Fig. 3f of the main text, which closely follows the data. However, the junction dimensions here are lower than expected, and a finite element model would be still more realistic. (a) ρ(y) plotted at the relevant gate voltages, with the sharpest distribution corresponding to the highest voltage. (b) ρ(y) evaluated at the location of the detector as a function of the gate voltages of Fig. 3f. (c) The effective lifetime of quasiparticles implied by this model (filled circles) as compared to the minimal model described in the main text (open circles).

Extended Data Fig. 6 Additional Al and Ti devices.

Wide, back-gated critical current devices are measured at 50 mK in the geometry shown in (a). In a device composed of a 10 nm Al film on a 90 nm SiO2 gate dielectric, the gate effect is observed only at elevated magnetic field. Critical current measurements are performed both at fixed gate voltage while sweeping magnetic field (b), or at fixed magnetic field while sweeping gate voltage (c). Oscillations observed at high Bx are likely to be related to Weber blockade and can be adjusted by the gate voltage. Hysteresis is observed in the gate effect, as can be seen in (d) in which the gate is swept from high to low voltage (over 20 minutes). A reversed behavior occurs when the gate is swept from low to high. (e) A similar Ti device shows a gate effect at zero magnetic field and the hysteresis takes the form of an overshoot of the gate effect (peak at positive VG). The thickness of the Ti here is 30 nm.

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Alegria, L.D., Bøttcher, C.G.L., Saydjari, A.K. et al. High-energy quasiparticle injection into mesoscopic superconductors. Nat. Nanotechnol. 16, 404–408 (2021).

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