## Main

It is generally thought that metallic nanostructures are not affected by electric fields, as long as their size is larger than the corresponding screening length, which is typically below 1 nm. Recent experiments1,2,3,4,5,6 have, however, shown that gate voltages can have a dramatic impact on the superconducting properties of metallic devices, including the ambipolar quenching of the critical current. The microscopic mechanism responsible for this behaviour has sparked debate. First, it was suggested that an electric field can penetrate a superconducting film up to the London penetration depth1. Second, it was proposed that an electric field might perturb the polarization of atomic orbitals at the metal surface, and this would affect the superconducting properties in the bulk7,8. Third, studies of the switching probability distribution (SPD) in metallic nanowires suggested an interplay between an electric field and superconducting phase slips5.

We have previously reproduced the most distinctive features of these experiments using titanium nitride (TiN), niobium and titanium nanowires9. In our samples, the critical current suppression was always accompanied by a current flowing between the gate and nanowire. In these experiments, the gate current is carried by electrons with energies of several electronvolts, which is orders of magnitude larger than the superconducting energy gap in the nanowires. We concluded that the emission of relatively few electrons leads to an avalanche of quasiparticles, which effectively quench the critical current10. This hypothesis was supported by tunnelling spectroscopy experiments11, which highlighted a non-thermal increase in quasiparticle population as a gate voltage was applied. Further work also demonstrated a correlation between the onset of gate currents and suppression of superconducting properties12,13. However, open questions remain. For example, in a scenario where the injection of high-energy electrons controls the critical current suppression, a marked asymmetry would naively be expected between injecting high-energy electrons into the nanowire (negative gate voltage) and extracting electrons from the nanowire at the Fermi energy (positive gate voltage), as well as having them relax either in the substrate or in the gate electrode. Unravelling the microscopic mechanisms behind these observations could prove valuable in the development of technological applications of the phenomenon, such as the realization of voltage-controlled superconducting switches and resonators.

In this Article, we show that the quenching of superconductivity in metallic nanowires can be linked to the relaxation of high-energy electrons, and not to the presence of electric fields at the superconductor surface. In particular, we examine the effect of high-energy electrons flowing into the nanowire, out of the nanowire and between two remote gate electrodes in the vicinity of the nanowire. Detailed measurements reveal that superconductivity is most efficiently suppressed when a current is injected into the nanowire. However, a qualitatively similar critical current suppression is observed when high-energy electrons flow near the nanowire, without any current or electric field directly reaching the nanowire itself. The non-local nature of the observed effect is consistent with the energy relaxation of electrons by phonon emission in the substrate. Due to their relatively high energy, phonons generate quasiparticles in the superconductors and efficiently quench the critical current in our devices. At cryogenic temperatures, phonons can propagate over considerable distances in the crystalline silicon substrate before thermalizing. The effect is, thus, distinct from the situation where a local temperature increase is produced by a resistive heater. Our observations question the existing interpretations and theories based on electric fields, and provide an insight into the complex interactions between out-of-equilibrium phenomena in solids and the performance of superconducting hardware.

## Critical current suppression and electric fields

Seven TiN nanowires on Si substrates were investigated during this work. All the nanowires had a length of 2 μm, width of 80 nm and height of 20 nm. At low temperatures, the devices showed critical currents IC between 42 and 45 μA, retrapping current IR = 1.0 μA and normal-state resistance RN ≈ 1,750 Ω, consistent with a previous work9. The uniformity of these values demonstrates that the nanowires were homogeneous and not characterized by accidental weak links. The large difference between IC and IR indicates substantial self-heating in the normal state, together with limited heat extraction via the leads or the substrate, typical of metallic nanowires14,15. Further details on sample fabrication and basic characterization are reported in another study9 and in Methods. Here we present the results from four devices, referred to as devices A1, A2, B and C. Extended data and three additional devices, used as references, are shown in more detail in Supplementary Figs. 13.

Figure 1a shows a false-coloured scanning electron micrograph of device A1, together with the schematic of the measurement configuration. Device A1 consists of a nanowire (blue) and three gates (red). Gate 1, controlled by voltage VG1, was separated from the nanowire by a gap of 80 nm. Gates 2 and 3, controlled by voltages VG2 and VG3, respectively, were separated from each other by 80 nm and from the nanowire by a distance d = 1 μm. A similar device, named device A2, had d = 80 nm (Supplementary Fig. 1).

We first discuss the response of device A1 to a side-gate voltage VG1, similar to previous work1,9,11. The electric-field distribution in this configuration was calculated using three-dimensional (3D) finite element simulations (Methods). Figure 1b shows the field magnitude E on a plane 10 nm above the substrate for VG1 = 6 V. Figure 1c represents E on a plane perpendicular to both substrate and wire axis, and intersecting the gate (Fig. 1b, red arrow). To better highlight the field distribution, the colour scale was saturated to E = 70 MV m−1. The highest E in our simulations was below E = 500 MV m−1, which is several orders of magnitude smaller than typical electric fields required to perturb superconductivity in a metallic device16,17,18. Figure 1d shows the experimentally measured IC as a function of VG1, for temperatures ranging from 20 mK (blue) to 3 K (red). Figure 1e shows the gate current IG1 simultaneously measured to the data in Fig. 1d. Consistent with previous observations9,12, the decrease in IC was correlated to the onset of IG1, and the initial decrease in IC took place for IG1 < 1 pA. Furthermore, IG1 was found to exponentially increase with VG1 and to be approximately symmetric around VG1 = 0.

We now discuss the dependence of IC on a differentially applied voltage VG2 − VG3, with VG2 = −VG3. Figure 1f shows the numerically computed electric field for VG2 − VG3 = 7 V. As expected, E is strongly confined between gates 2 and 3. If superconductivity in the nanowire were controlled by the electric fields, this configuration should result in negligible effects on IC. Strikingly, quenching of the supercurrent occurred even in this situation (Fig. 1g). Figure 1h shows the current IG2 flowing from gate 2 (we found that IG2 = −IG3 within the experimental error). Remarkably, the suppression of IC was strongly correlated to the onset of IG2, despite no measurable gate current reaching the nanowire and negligible electric fields between the gate and nanowire.

To test whether residual electric fields were relevant, we also measured IC with gates 2 and 3 biased at the same voltage (VG2 = VG3). In Fig. 1g, we plot IC as a function of the quantity VS = 2VG2 = 2VG3 (Fig. 1g, solid grey line) as, at any one point in this plot, the absolute voltages VG2 and VG3 on the gate electrodes are identical and the absolute value of the electric field E reaching the nanowire is similar. More specifically, we estimate E(VG2 = VG3)E(VG2 = −VG3) at the nanowire surface. Nevertheless, no current was detected between the gates and nanowire for symmetrically applied gate voltages (Fig. 1h, grey curve) and IC was not perturbed. These results further corroborate our findings that high-energy electrons, and not electric fields, are responsible for the suppression of IC. Similar results obtained with device A2 are presented in Supplementary Fig. 1.

Overall, experiments and numerical simulation presented in Fig. 1 demonstrate that the suppression of superconductivity takes place irrespective of the electric fields at the nanowire surface. Instead it requires the flow of high-energy electrons in the surroundings of the device. This is the first conclusion of our work.

## Role of substrate

The remote action of VG2 − VG3 on IC points to the existence of an efficient energy transfer mechanism triggered by the flow of IG2. We now analyse the origin of this remote action more carefully using devices B and C (Fig. 2a,b, respectively). Device B is identical to device A1, except for the presence of a 510-nm-deep, 200-nm-wide, 80-μm-long trench etched into the substrate between the remote gates and nanowire. Device C consists of two parallel TiN nanowires separated by a distance of 80 nm. Each nanowire was controlled by a nearby gate (Fig. 2b, red). We measured the critical current of one of the two nanowires (Fig. 2b, blue), whereas the second one (Fig. 2b, purple) was set in the resistive state and was traversed by a d.c. current IH, resulting in Joule heating, similar to another work19.

Figure 2c,d summarizes the behaviour of our devices in terms of IC as a function of IG1 and IG2, respectively. The full dataset is presented in Supplementary Figs. 1 and 2. The dependence on IG1 (Fig. 2c) is similar in all the devices, with a faster suppression of IC for IG1 < 0. Due to the exponential dependence of IG1 on VG1, this asymmetry is hard to spot in Figs. 1d,e. We further notice that device B (grey diamonds) exhibited a particularly slow decay of IC for IG1 > 0. We will discuss the possible causes for this asymmetry below. Figure 2d reveals that IG2 is considerably less effective in suppressing IC than IG1; furthermore, device A2 (blue squares; d = 80 nm) was six times more efficient than device A1 (red circles; d = 1 μm), which was six times more efficient than device B (grey diamonds; d = 1 μm plus an etched trench). In the case of device B, the maximum IG2 allowed in our setup (100 nA) was not sufficient to reach IC = 0. Altogether, these results demonstrate that most of the remote action of IG2 on IC is mediated by the substrate, that is, the high-energy electrons relax by emitting phonons, which travel through the substrate and affect superconductivity in the nanowire. This is the second main conclusion of our work.

## Comparison to Joule heating

We now discuss the properties of the generated phonons in more detail. In particular, we compare their effect on IC with that of the heat generated by a resistive conductor placed 80 nm from the superconducting nanowire. These experiments were performed with device C (Fig. 2b). The dependence of IC on heater current IH is shown in Fig. 2e for various temperatures. As expected, Joule heating eventually resulted in the suppression of IC. However, the current required to reach IC = 0 was several orders of magnitude higher than in the configurations where a gate voltage was applied.

Figure 3 provides a comparison between the devices presented above in terms of the suppression of normalized critical currents IC as a function of dissipated power. For each measurement configuration, we distinguished the case of a positive and negative voltage bias with full and empty markers, respectively. Curves at higher temperatures are shown in Supplementary Fig. 4. The critical current is more efficiently suppressed when voltage bias VG1 is applied to a gate directly facing the nanowire (red dots). In this case, the dissipated power is calculated as IG1VG1. When a remote current IG2 flows, the power is calculated as IG2(VG2 − VG3). Suppressing IC by means of Joule heating with a resistive conductor (Fig. 3, purple line) required a considerably higher power $${I}_{{{{\rm{H}}}}}^{2}{R}_{{{{\rm{N}}}}}$$ than the other configurations. As noted above, the dependence on IG1 (Fig. 3, red circles) shows a difference between the positive and negative gate polarity, with the negative polarity being 2.5 times more power efficient in suppressing IC compared with the positive one.

We have shown that Joule heating is orders of magnitude less efficient in suppressing IC of our nanowires than a current of high-energy electrons. In addition to these quantitative differences, we gain a further insight from the SPDs of our devices. The SPD is the probability of switching from the superconducting to resistive state to occur per unit of source–drain current. The SPD has proven to be a powerful tool to study Josephson junction and metallic nanowire properties that are hard to access with standard transport measurements5,20. Figure 4a,b shows the SPDs of devices A1 and C, respectively, under various experimental conditions. For these experiments, the source–drain current was swept 20,000 times from 0 to 49 μA. For each sweep, the source–drain current value at which a switch to the resistive state occurred was recorded. At low temperature and zero gate voltage, device A1 exhibited a sharp SPD (Fig. 4a, blue markers), with a standard deviation σI = 47 nA. At a temperature of 2.2 K (Fig. 4a, green markers), the SPDs had their maximum at half of the low-temperature IC value, with σI = 100 nA. A more detailed analysis (Supplementary Fig. 5) revealed that the switching mechanisms at 20 mK and 2.2 K are consistent with quantum phase slips and thermal fluctuations, respectively. Much broader SPDs were obtained by applying a gate leakage current IG1 = 10 pA (Fig. 4a, red markers), with σI = 2.0 μA. The finding that the application of a gate voltage results in much broader SPDs than increasing the bath temperature (for equal suppression of IC) is consistent with the observations elsewhere5. However, we show that a similarly broad SPD is also obtained by applying a remote current IG2 = 2.5 nA (Fig. 4a, orange circles; σI = 1.2 μA), which is without any electric field or current reaching the nanowire. Using device C (Fig. 4b), we compare the SPD obtained when IC is suppressed by 50% either by Joule heating (solid purple line) or by increasing the bath temperature to 2.1 K (green triangles). The two results are indistinguishable, indicating that a resistive heater indeed affects the superconductivity in the same way as an increase in the bath temperature, but in a totally different manner than a current of high-energy electrons (grey triangles). Because of the difference between the SPDs obtained at a high temperature (green markers) and finite gate voltage (red markers), another study5 excluded the presence of electrical currents. This conclusion was, however, reached under the assumption that a gate current causes heating similar to an increase in the bath temperature. Our results demonstrate, instead, that a current of high-energy electrons perturbs the superconducting properties of nanowires in a way that is qualitatively and quantitatively distinct from a bare temperature increase, even if the current does not flow into the nanowire but only in its surroundings. This is the third main conclusion of our work.

## Nature of generated phonons

Our observations are consistent with the phenomenology of phonon generation by hot electrons in the substrate. First, we note that phonons with energies above the superconducting gap (500 μeV for TiN (ref. 21)) are well known to affect superconducting devices22,23,24,25. Second, electrons accelerated by high electric fields in Si undergo a series of relaxation events over timescales below 1 ns and on mean free paths below 10 nm. Such relaxation most probably happens by the emission of optical and acoustic phonons26,27,28,29. Phonons in Si have a maximum energy of the order of 50 meV, which means that a single electron with an energy of a few electronvolts can generate a large amount of phonons30,31 as it travels between two metallic electrodes. At temperatures below 3 K, phonons in Si have long mean free paths (up to 1 μm (refs. 32,33)) and even longer thermalization lengths. It is, therefore, expected that the emitted phonons reach the nanowire in an out-of-equilibrium state34,35.

The electronic mean free path in Si decreases as E increases26,27, resulting in intense phonon emission close to the metal electrodes, independent of the gate voltage polarity (Fig. 1c)30. This may be the reason for the more efficient suppression of IC when a current is either injected or extracted from the nanowire (Fig. 2c) compared with the case where a current flows between two gates near the nanowire (Fig. 2d, device A2). Furthermore, the suppression of IC by a fixed factor requires 2.5 times less power for VG1 < 0 (Fig. 3) compared with VG1 > 0. This could indicate that electrons reaching the nanowire are not completely thermalized, and can still generate a sizeable number of quasiparticles via electron–electron interactions in the nanowire10,36,37. Assuming the phononic contribution is similar for both gate polarities (that is, phonon emission is isotropic), we estimate that more than half the suppression of IC for VG1 < 0 is due to electron–electron interaction. Future work might use more complex geometries to map out angular anisotropies in the phonon emission and absorption processes.

The broadening of SPDs with gate voltage is consistent with the nanowire being subject to extremely energetic events, capable of suppressing superconductivity even at small source–drain currents. The characteristic energy spread of such events can be quantified by the Kurkijavi power law20, which allows one to relate the width of the SPD to the effective energy Eeff (Supplementary Section 5). As shown by the red dots in Fig. 4a, we obtain Eeff ≈ 8.6 meV, consistent with the idea that the energy of a leakage electron (7 eV) dissipates in successive scattering events in the substrate before reaching the nanowire. In the case of a remote current (Fig. 4a, orange dots), we get Eeff ≈ 6.3 meV, indicating that, on average, the phonons thermalize more over the longer distance. A possible framework for analysing such SPDs in Josephson junctions subject to high-energy electrons was recently proposed elsewhere38.

In device B, we noticed an anomalously large asymmetry in the parametric plot of IC versus IG1 (Fig. 2c). With three reference devices (Supplementary Fig. 3a), we confirmed that such an asymmetry is a robust feature that arises following the fabrication steps required to etch trenches into the substrate (Methods). Similarly, the efficiency of the remote action of IG2 slightly decreased after additional fabrication, even when trenches were not etched (Supplementary Fig. 3b). Interestingly, no other sample parameters were affected by the additional fabrication steps. These results suggest that some of the out-of-equilibrium processes taking place in our device are sensitive to the surface treatment of the samples. Measuring device B, we have shown that out-of-equilibrium phonons are primarily responsible for the remote action of IG2 on IC. However, our work does not exclude the presence of additional energy relaxation mechanisms that contribute, together with phonons, to the suppression of IC (such as photon emission). Previous works detected photons in a variety of devices as a result of tunnelling events39,40,41,42,43 as well as bremsstrahlung and carrier recombination of high-energy electrons44,45. It is also well known that superconducting nanowires46 and Josephson junctions47 are highly sensitive to the impact of high-energy photons. Both phonon and photon transport may be affected by the additional fabrication steps for trenching, for example, by a change in surface roughness or dielectric properties. The relative contribution of phonons and photons is estimated by comparing the response of devices A1 and B to VG2 − VG3 (Fig. 3). Device B required a six times higher power to reach the same IC/I0, indicating that the trench blocks five-sixth of the power that would have been otherwise absorbed by the nanowire. If we assume that any photonic contributions are unaffected by the trench, we can calculate an upper bound on such a contribution in that it must be smaller than one-fifth of the phononic contribution. Note that the reduced power reaching the nanowire in device B could also be carried by phonons, which—if travelling deep in the substrate—are also not affected by the trench.

## Conclusions

We have reported a comprehensive study of the mechanism responsible for the suppression of critical currents in metallic nanowires in the presence of large gate voltages. We have shown that previously reported features, which were attributed to the electric field on the superconductor, can be obtained in the absence of electric fields. Our data indicate that critical currents are suppressed as a consequence of the relaxation of high-energy electrons, either in the substrate or in the electrodes. Our results also elucidate the mechanism behind the ambipolar suppression of IC as a function of gate voltage (Fig. 1d), which was not fully explained in previous works9,11,12. The ambipolar suppression of IC requires an approximately symmetric gate current (which is experimentally observed (Fig. 1e)) as well as an efficient energy equilibration mechanism between the gate and nanowire. Energy equilibration is dominated in our devices by energetic phonons spreading through the substrate over distances in excess of 1 μm. Although this remote action may pose a limit to the device integration density, it could also open new paths for device design. For example, it could be used to develop efficient superconducting switches48,49,50,51 that do not require the injection of electrons into the switching element, but are instead mediated by high-energy phonons that are guided towards a switching element. It also opens new possibilities to investigate the interplay between out-of-equilibrium phenomena, resulting quasiparticle generation and superconducting quantum hardware.

## Methods

### Sample fabrication

A 20-nm-thick TiN film was sputtered on a Si substrate. The Si substrate used for this work was intrinsic and became insulating at temperatures below 100 K. Before TiN deposition, the Si chip was immersed in a buffered hydrofluoric acid (HF) solution for the removal of native oxides. The TiN film showed a critical temperature of 3.7 K and a resistivity of 68 Ω sq–1. Devices were defined by electron-beam lithography on a negative hydrogen silsesquioxane resist and dry etching in HBr plasma. The resist was then removed by immersion in HF. The devices were contacted by Ti/Au bond pads defined by optical lithography and metal evaporation. Some devices were further processed after the deposition of bond pads. In this case, 2 nm Si3N4 and 210 nm SiO2 hard mask were deposited by atomic layer deposition and plasma-enhanced chemical vapour deposition, respectively. A trench was defined in the hard mask with electron-beam lithography and CSAR AR-P 6200.09 resist, standard development and reactive ion etching of the SiO2 layer. The Si substrate was further etched in inductively coupled HBr plasma. Finally, the hard mask was etched in buffered HF.

### Electrical measurements

Measurements were performed in a dilution refrigerator with a base temperature of 20 mK. Critical currents IC were measured by applying a sawtooth wave ISD signal with an amplitude of 49 μA and repetition rates between 33 and 133 Hz, whereas voltages V across the nanowires were recorded by a digital oscilloscope. The measurement setup was synchronized so that a switch from zero to a finite voltage in the oscilloscope could be related to the source–drain current at which the switch occurred. This technique allowed us to reliably extract critical currents down to 700 nA. The critical currents presented in Fig. 1 were obtained by averaging 108 such switching events. Sporadic fluctuations of IC visible at T ≥ 1.5 K are associated with the instabilities of the temperature controller. The SPDs presented in Fig. 4 were obtained by recording 20,000 switches over a time interval of 10 min. To keep the nanowire potential constant as ISD varies, ISD was generated by sourcing two synchronized sawtooth waves with opposite polarities into 163 kΩ resistors placed at both ends of the nanowire (which add to the existing line resistance of 2.2 kΩ). Gate voltages were applied via high-precision source-measure units, which recorded the current flowing into the gate contacts. The gate current data shown in Fig. 1e,h were obtained after subtracting the linear components ranging between 1 and 5 pA V−1, as discussed elsewhere9. Such resistive contributions are attributed to spurious leakage paths in our setup.

### Electrostatic simulations

The electric-field distributions presented in Fig. 1 were produced with finite-element 3D electrostatic simulations performed with Ansys Maxwell version 2019R2. A substrate permittivity of 12 was assumed to resemble the electromagnetic properties of silicon, and its thickness was set to 1 μm. The metallic layer comprising the nanowire and gate electrodes was modelled as a 20-nm-thick perfect conductor. The upper edges of the structures were filleted with a radius of 3 nm. The geometry of the nanowire and gates was generated from the same layout file used for the electron-beam lithography of the devices. The fields shown in Fig. 1b,f are slices of the 3D simulation taken at half the height of the nanowire. Figure 1c shows the image taken perpendicular to the substrate and intersecting the gate electrode. The colour scale was saturated to a maximum value of 70 MV m−1 to evidence the field distribution, whereas the full scale reached up to 500 MV m−1.