# Signatures of gate-driven out of equilibrium superconductivity
in Ta/InAs nanowires

Tosson Elalaily  Department of Physics, Institute of Physics, Budapest
University of Technology and Economics, Müegyetem rkp. 3., H-1111 Budapest,
Hungary
MTA-BME Superconducting Nanoelectronics Momentum Research Group, Müegyetem
rkp. 3., H-1111 Budapest, Hungary
Department of Physics , Faculty of Science, Tanta University, Al-Geish St.,
31527 Tanta, Gharbia, Egypt
Martin Berke  Department of Physics, Institute of Physics, Budapest University
of Technology and Economics, Müegyetem rkp. 3., H-1111 Budapest, Hungary
MTA-BME Superconducting Nanoelectronics Momentum Research Group, Müegyetem
rkp. 3., H-1111 Budapest, Hungary
Máté Kedves  Department of Physics, Institute of Physics, Budapest University
of Technology and Economics, Müegyetem rkp. 3., H-1111 Budapest, Hungary
MTA-BME Correlated van der Waals Structures Momentum Research Group, Müegyetem
rkp. 3., H-1111 Budapest, Hungary
Gergő Fülöp  Department of Physics, Institute of Physics, Budapest University
of Technology and Economics, Müegyetem rkp. 3., H-1111 Budapest, Hungary
MTA-BME Superconducting Nanoelectronics Momentum Research Group, Müegyetem
rkp. 3., H-1111 Budapest, Hungary
Zoltán Scherübl  Department of Physics, Institute of Physics, Budapest
University of Technology and Economics, Müegyetem rkp. 3., H-1111 Budapest,
Hungary
MTA-BME Superconducting Nanoelectronics Momentum Research Group, Müegyetem
rkp. 3., H-1111 Budapest, Hungary
Thomas Kanne  Center for Quantum Devices and Nano-Science Center, Niels Bohr
Institute, University of Copenhagen, Universitetsparken 5, DK-2100,
Copenhagen, Denmark
Jesper Nygård  Center for Quantum Devices and Nano-Science Center, Niels Bohr
Institute, University of Copenhagen, Universitetsparken 5, DK-2100,
Copenhagen, Denmark
Péter Makk<EMAIL_ADDRESS>Department of Physics, Institute of Physics,
Budapest University of Technology and Economics, Müegyetem rkp. 3., H-1111
Budapest, Hungary
MTA-BME Correlated van der Waals Structures Momentum Research Group, Müegyetem
rkp. 3., H-1111 Budapest, Hungary
Szabolcs Csonka<EMAIL_ADDRESS>Department of Physics, Institute
of Physics, Budapest University of Technology and Economics, Müegyetem rkp.
3., H-1111 Budapest, Hungary
MTA-BME Superconducting Nanoelectronics Momentum Research Group, Müegyetem
rkp. 3., H-1111 Budapest, Hungary

###### Abstract

Understanding the microscopic origin of the gate-controlled supercurrent (GCS)
in superconducting nanobridges is crucial for engineering superconducting
switches suitable for a variety of electronic applications. The origin of GCS
is controversial, and various mechanisms have been proposed to explain it. In
this work, we have investigated the GCS in a Ta layer deposited on the surface
of InAs nanowires. Comparison between switching current distributions at
opposite gate polarities and between the gate dependence of two opposite side
gates with different nanowire$-$gate spacings shows that the GCS is determined
by the power dissipated by the gate leakage. We also found a substantial
difference between the influence of the gate and elevated bath temperature on
the magnetic field dependence of the supercurrent. Detailed analysis of the
switching dynamics at high gate voltages shows that the device is driven into
the multiple phase slips regime by high-energy fluctuations arising from the
leakage current.

††preprint: APS/123-QED

## Introduction

Since superconducting circuits have the potential to realize electronics with
short switching time and ultra-low power consumption, various architectures
have been developed for integrating semiconductor technology with
superconducting devices to reduce the high power consumption required for
cooling the high-density semiconductor-based microchips McCaughan and Berggren
(2014); McCaughan  _et al._ (2019); Frasca and Charbon (2019). The cryotron
Buck (1956), josephson cryotron Matisoo (1966), rapid single flux quantum
(RSFQ) device Likharev and Semenov (1991), and nanocryotron (nTron) McCaughan
and Berggren (2014) were all developed as building blocks for superconducting
switches, however, their scalability or even the difficulty of interfacing
with CMOS electronics limited their applications.

In the recent years, suppression of supercurrent by applying a voltage to a
gate electrode in the vicinity of superconducting metallic nanowire has
attracted much attention as a promising building block for highly scalable
superconducting switches De Simoni  _et al._ (2018, 2019); Paolucci  _et al._
(2019a, b, c); De Simoni  _et al._ (2020); Rocci  _et al._ (2020); Puglia  _et
al._ (2020a, b); Bours  _et al._ (2020, 2020); Puglia  _et al._ (2021); De
Simoni  _et al._ (2021); PUGLIA  _et al._ (2021); Orús  _et al._ (2021);
Paolucci  _et al._ (2021); Basset  _et al._ (2021); Elalaily  _et al._ (2021);
Alegria  _et al._ (2021); Ritter  _et al._ (2021); Golokolenov  _et al._
(2021); Catto  _et al._ (2021); Ritter  _et al._ (2022). In some works, the
effect is attributed to the large electric field ($10^{8}$ V/m) at the
superconducting surface De Simoni  _et al._ (2018, 2019); Paolucci  _et al._
(2019a, b, c); De Simoni  _et al._ (2020); Rocci  _et al._ (2020); Puglia  _et
al._ (2020a, b); Bours  _et al._ (2020, 2020); Puglia  _et al._ (2021); De
Simoni  _et al._ (2021); PUGLIA  _et al._ (2021); Orús  _et al._ (2021);
Paolucci  _et al._ (2021), which distorts the superconducting state and leads
to the quenching of the superconductivity Mercaldo  _et al._ (2020, 2021);
Solinas  _et al._ (2021); Chirolli  _et al._ (2021); Amoretti  _et al._
(2022). Other studies Alegria  _et al._ (2021); Ritter  _et al._ (2021);
Golokolenov  _et al._ (2021); Catto  _et al._ (2021); Elalaily  _et al._
(2021); Basset  _et al._ (2021); Ritter  _et al._ (2022) reported a
correlation between the gate controlled supercurrent (GCS) and the leakage
current flowing between the gate and the superconducting device. Some of these
studies suggest that the GCS results from ballistic injection of high-energy
quasiparticles Alegria  _et al._ (2021); Ritter  _et al._ (2021); Golokolenov
_et al._ (2021). In another work, the quenching of the supercurrent was
attributed to the absorption of phonons emitted in the relaxation process of
high-energy electrons injected from the gate electrode Ritter  _et al._
(2022). In order to engineer efficient superconducting switches for future
electronic applications, it is important to understand the dominant mechanism
behind the CGS effect.

In this work, we have studied the GCS in a superconducting Ta shell deposited
on the surface of InAs nanowires Carrad  _et al._ (2020). We chose Ta because
of its strong spin-orbit interaction Niimi and Otani (2015); Vélez  _et al._
(2016), so it is expected that the electric field has a strong influence on
the superconducting state. We investigated the influence of the distance
between the gate and the nanowire on the suppression of the supercurrent for
the fabricated devices. Also, the magnetic field dependence of the
supercurrent under the influence of the gate voltage and elevated temperatures
was investigated. In addition, the switching current distribution at opposite
gate polarities and at different current ramp speeds was studied. Furthermore,
we give a detailed analysis for the switching dynamics at high gate voltages.
Our findings contradict the proposed theoretical explanations based on
electric fields or ballistic injection of high-energy electrons, and they are
consistent with the nonequilibrium phonons picture as the origin of the GCS
effect.

## Experiments

###  .1 Device outline and characterization

In our device configuration, we used InAs nanowires with a 20-nm-thick Ta
shell layer deposited on only three facets of the nanowire Carrad  _et al._
(2020). In order to investigate the impact of the gate on the supercurrent
flowing in the Ta layer, four-terminal nanowire-based devices were fabricated
with the configuration shown in Fig. 1a, b. The Ta/InAs nanowires
(green/brown) were deposited on a doped Si wafer with a 290-nm-thick oxide
layer. Four Ti/Al contacts (blue) with a thickness of 10/80 nm were fabricated
for quasi-four terminal measurements. A two metallic Ti/Au side gates SG1
(orange) and SG2 (light blue) with a thickness of 7/33 nm were placed with
unequal spacings and on opposite sides of the nanowire. This provides a
possibility to study the GCS effect for the device with gates at different
spacings. The results presented in this paper are based on measurements
performed on three different devices, A, B and C with the same device
geometry, but with different values of nanowire$-$gate spacing $d$ in the
range from 30 to 120 nm. The results in Fig. 1 were measured on device A, in
Fig. 2 on device B, while the results in Fig. 3 and their analysis in Fig. 4
were performed on device C.

The current$-$voltage ($I-V$) characteristics measured at 35 mK show a clear
switching from the superconducting state to the normal state at the switching
current $I_{\mathrm{SW}}$ $\simeq$ 1.17 $\mu$A (see blue curve in Fig. 1c).
When the measurements are carried out in the opposite sweep direction (grey
curve), the device shows a hysteretic behaviour and switches back to the
superconducting state at two successive retrapping current values at $\simeq$
0.61 $\mu A$ and 0.4 $\mu A$. This hysteretic behaviour can be attributed to
large Joule heating dissipated in the resistive state Courtois  _et al._
(2008). The GCS is investigated by measuring the dependence of
$I_{\mathrm{SW}}$ under the influence of gates SG1 (orange) and SG2 (light
blue) with $d$ of $\simeq$ 65 and $\simeq$ 115 nm, respectively. Fig. 1d shows
$I_{\mathrm{SW}}$ as a function of $V_{\mathrm{sg}}$ for both gates, where
each of the plotted curves has the same colour as the corresponding gate in
Fig. 1a. The plot reveals that both gates completely switch the device to the
normal state at almost the same critical gate voltage $V_{\mathrm{sg,C}}$
$\simeq$ $\pm$13 V. Despite the nanowire$-$gate spacing for SG1 is about half
that for SG2, SG2 still suppresses $I_{\mathrm{SW}}$ at lower threshold gate
voltage $V_{\mathrm{th}}$ than SG1. Importantly, at $V_{\mathrm{th}}$, a
correspondingly large increase in the gate leakage current $I_{\mathrm{leak}}$
is observed for each of the gates (see Fig. 1e), which has also been reported
elsewhere Ritter  _et al._ (2021); Golokolenov  _et al._ (2021); Elalaily  _et
al._ (2021); Basset  _et al._ (2021); Ritter  _et al._ (2022).

Figure 1: Device geometry and gate dependence characterization (Device A). a A
false-colored SEM image and b schematic of the side view of the nanowire
device. c $I-V$ characteristics of the device measured at 35 mK. As the bias
current ramps from negative to positive values (blue arrow), the device
switches to finite-resistance state at the switching current
$I_{\mathrm{SW}}$= 1.17 $\mu A$. If the current ramps in the opposite
direction (grey arrow), the device switches back to the superconducting state
at two successive retrapping current values at $\simeq$ 0.61 $\mu A$ and
$\simeq$ 0.4 $\mu A$. d $I_{\mathrm{SW}}$ as a function of $V_{\mathrm{sg}}$
(at magnetic field $B$ = 0.1 T) applied to SG1 (orange curve) and SG2 (light
blue curve) with nanowire$-$gate spacings of $\simeq$ 65 and $\simeq$ 115 nm,
respectively. e The leakage current as a function of $V_{\mathrm{sg}}$ for
both gates. Figure 2: Magnetic field dependence and comparison between the GCS
effect and effect of bath temperature (device B, $d$ = 35 nm). a $I-V$ curve
as a function of out-of-plane magnetic field $B$ up to $\pm$ 2 T. b
$I_{\mathrm{SW}}$ as a function of $V_{\mathrm{sg}}$ at various values of
B-field up to 2 T. c $I_{\mathrm{SW}}$ as a function of the B-field at various
elevated temperatures and, d at various values of $V_{\mathrm{sg}}$. e
Magnification of the curves surrounded by the red rectangle in c and d. f
Comparison between the SCDs measured at T = 600 mK (blue) and
$V_{\mathrm{sg}}$ = 3 V (green).

###  .2 Magnetic field dependence

The dependence of the supercurrent in our device on the out-of-plane magnetic
field $B$ is shown in Fig. 2a, where the $I-V$ curves are measured as a
function of the $B$-field up to $\pm$2 T. The white region represents the
zero-resistance state, with a transition to and from the normal state (red and
blue regions) at the switching and retrapping current values in the positive
and negative bias current values, respectively. The magnitude of
$I_{\mathrm{SW}}$ shows a rapid suppression with increasing $B$-field below
100 mT and then slowly decreases with further increasing the magnetic field up
to 2 T. The sharp decrease in the critical current below 100 mT coincides with
the $B_{\mathrm{C}}$ of the Al electrodes contacting the nanowire Elalaily
_et al._ (2021), therefore we believe that this decrease is a result of the Al
contacts switching to normal state. Although the maximum $B$-field in our
setup (2 T) does not allow full suppression of the superconducting state in
the Ta shell, based on the measured trend, $B_{\mathrm{C}}$ is expected to be
about 3.5 T, which is consistent with earlier findings on identical Ta/InAs
nanowires Bjergfelt (2019).

The gate dependence of $I_{\mathrm{SW}}$ under the influence of B-field is
shown in Fig. 2b. $I_{\mathrm{SW}}$ is plotted as a function of
$V_{\mathrm{sg}}$ at different values of magnetic field up to 2 T. No
significant change in $V_{\mathrm{sg,C}}$ with increasing $B$-field was
observed, which is in contrast to the dependence observed for Ti and Al
nanostructures De Simoni  _et al._ (2018); Elalaily  _et al._ (2021). Fig. 2c
and d show the dependence of $I_{\mathrm{SW}}$ on $B$-field under influence of
temperature $T$ and $V_{\mathrm{sg}}$, respectively. In the former case,
$I_{\mathrm{SW}}$ decreases with increasing $T$, as expected, accompanied by a
suppression of $B_{\mathrm{C}}$, giving $B_{\mathrm{C}}$ = 2 T at 800 mK and
$B_{\mathrm{C}}$ = 1.5 T at 900 mK. In the case of the gate control,
$I_{\mathrm{SW}}$ also decreases with increasing $V_{\mathrm{sg}}$, but
surprisingly, $no$ $change$ $in$ $B_{\mathrm{C}}$ $was$ $observed$. For a
better comparison, Figs. 2e shows a zoom in of the curves in both dependencies
marked by the red rectangle and having almost the same magnitude of
$I_{\mathrm{SW}}$ (at $B$ = 0 T). It can be clearly seen that the
$B_{\mathrm{C}}$ dependence behaves differently under the influence of
temperature and gate voltage. While from 900 mK to 950 mK $B_{\mathrm{C}}$
further decreases from 1.5 T to 1 T, $I_{\mathrm{SW}}$ does not seem to be
suppressed by the magnetic field in the case of the gate (see also the
Supporting Information) in strong contradiction to other works Paolucci  _et
al._ (2019b); Bours  _et al._ (2020); Elalaily  _et al._ (2021).

Another remarkable difference between temperature and gate dependence is that
$I_{\mathrm{SW}}$ exhibits large fluctuations at finite gate voltages (see
green curves in Fig. 2e). In order to investigate this effect, the switching
current distribution (SCD) at finite temperatures and gate voltages is
measured by ramping the current at constant speed from 0 to 3 $\mu$A for
10,000 times and recording the corresponding $I_{\mathrm{SW}}$ value every
time (see Methods). A comparison between the SCDs obtained at 600 mK and 3 V
is shown in Fig. 2f. Despite the fact that both histograms have almost the
same mean value $\langle$$I_{\mathrm{SW}}$$\rangle$, the width of the
histogram obtained under influence of the gate voltage is an order of
magnitude larger than that obtained at elevated bath temperature. The large
gate-induced broadening is consistent with Refs. 14; 22; 26; 19; 28 and shows
that the gate voltage induces an out-of-equilibrium state in the
superconducting nanowire, which cannot be described with an effective
temperature.

Figure 3: SCD measurements and schematics for different proposed mechanisms of
the GCS effect (device C, $d$ = 30 nm). a SCDs measured at positive (orange)
and negative (blue) gate polarity and paired at the same $|V_{\mathrm{sg}}|$.
The SCDs are normalized to their maximum counts and shifted on the y-axis for
clarity. The inset shows $I_{\mathrm{SW}}$ as a function of $V_{\mathrm{sg}}$
for the investigated device measured at 0.1 T. b SCDs measured at positive
(orange) and negative (blue) gate polarity and paired at approximately the
same $P_{\mathrm{G}}$. The inset shows the mean value of
$\langle$$I_{\mathrm{SW}}$$\rangle$ of SCDs measured at both gate polarities
as a function of $P_{\mathrm{G}}$. c Schematic diagram of the electric field
$E$ applied from the metallic gate N to the superconducting nanowire SC at
positive gate polarity. The colored/uncolored parts represent
occupied/unoccupied states. d Schematic diagram of the ballistic electron
injection from the gate to the nanowire at negative gate polarity. The high-
energy electron (red circle) tunnels through the potential barrier of the
substrate S and relaxes to the lowest unoccupied state (close to the
superconducting gap edge), releasing heat on the SC-side. e Schematic diagram
of relaxation of high-energy electrons in the substrate when injected from the
SC/N side to N/SC at positive/negative gate polarity in the left/right panels.
In the case of positive gate polarity, the electrons relax close to the SC
side (superconducting nanowire) so that it is heated more than in the case of
negative gate polarity at the same $P_{\mathrm{G}}$.

###  .3 Gate dependence of the probability distributions

In the following, we will compare the SCDs measured at positive and negative
gate polarity, as they are expected to behave differently for different
microscopic origins of the GCS. The dependence of $I_{\mathrm{SW}}$ on
$V_{\mathrm{sg}}$ of the device is shown in the inset of Fig. 3a, where the
positive and negative gate polarities represented by the orange and blue
curves, respectively. Fig. 3a shows the SCDs measured at the same
$|V_{\mathrm{sg}}|$ but with opposite polarities are paired and shifted along
the y-axis for clarity. For simplicity, we made the measurements with the Al
leads in the normal state, at $B$ = 100 mT Elalaily  _et al._ (2021). There is
a clear difference in the shape and $\langle$$I_{\mathrm{SW}}$$\rangle$ of
SCDs paired at equal $|V_{\mathrm{sg}}|$. In addition, we also paired SCDs for
opposite gate polarities and with approximately the same power dissipated at
the gate $P_{\mathrm{G}}$ = $I_{\mathrm{leak}}$$\cdot$$V_{\mathrm{sg}}$ as
shown in Fig. 3b. Comparing Fig. 3a and b, one can conclude that the pairing
at the same power gives a better match between SCDs with opposite polarities.
We also found that the SCDs measured at positive polarity have a slightly
smaller $\langle$$I_{\mathrm{SW}}$$\rangle$ than those measured at negative
polarity at the same $P_{\mathrm{G}}$ (see inset in Fig. 3b).

Assuming that the electric field $E$ applied by the gate (Fig. 3c) is
responsible for the suppression of $I_{\mathrm{SW}}$ Mercaldo  _et al._ (2020,
2021); Solinas  _et al._ (2021); Chirolli  _et al._ (2021); Amoretti  _et al._
(2022), its effect should not depend on the sign of $E$. Therefore, we expect
the SCD obtained at a given voltage $V_{\mathrm{sg}}$ to be identical to the
SCD obtained at the same gate voltage with the opposite sign,
-$V_{\mathrm{sg}}$. Since the measured SCDs do not match at opposite
polarities (see Fig. 3a), our results contradict the electric field-based
explanation. Another possible microscopic picture is that the CGS is caused by
ballistic injection of high-energy quasiparticles, as shown in Fig. 3d. After
injection of these electrons, their energy is released by relaxation, heating
the side on which they end up. Therefore, for negative gate polarity (Fig.
3d), they heat the superconducting bridge, while for positive polarity they
heat the gate electrode instead. Thus, a stronger suppression of
superconductivity is expected for negative polarity. Therefore, at the same
$P_{\mathrm{G}}$ value, the mean value of the distribution is expected to be
significantly smaller for negative polarity than for positive polarity.
Comparing this prediction with the measured results in Fig. 3b, one can
conclude that the experimental findings are just opposite, so that ballistic
injection of electrons can also be excluded.

The most likely explanation for our results is the generation of phonons by
series of relaxation events of the high-energy electrons in the substrate
Ritter  _et al._ (2022). The small shift between the
$\langle$$I_{\mathrm{SW}}$$\rangle$ measured for the two polarities (see Fig.
3b) can be attributed to the short energy relaxation length of electrons in
SiO2 ($\leq 3$ nm) at high electric fields compared to nanowire$-$gate spacing
($d=30$ nm) Fischetti  _et al._ (1985); Brorson  _et al._ (1985); Fischetti
_et al._ (1987); DiMaria and Fischetti (1988). Thus, at positive gate
polarity, it is expected that the high-energy electrons will relax close to
the nanowire (Fig. 3e, left panel) and the generated phonons can heat the
superconducting nanowire more than at negative gate polarity (Fig. 3e, right
panel).

###  .4 Analysis of the switching dynamics

Figure 4: Analysis of the switching dynamics under influence of
$V_{\mathrm{sg}}$ (device C, $d$ = 30 nm). a Standard deviation $\sigma$ and
mean value $\langle$$I_{\mathrm{SW}}$$\rangle$ as a function of
$|V_{\mathrm{sg}}|$ for all SCDs measured at negative gate polarity in the
blue and green curves, respectively. b The calculated skewness as a function
of $|V_{\mathrm{sg}}|$ for SCDs measured with a step of 0.1 V in the interval
[-3.5, -2.7 V] (surrounded by the vertical grey dotted lines in panel a) where
a corresponding increase in $I_{\mathrm{leak}}$ is observed. c Logarithm of
escape rate $\Gamma$ as a function of $I_{\mathrm{SW}}$ (colored curves) for
different values of $V_{\mathrm{sg}}$ from -3.5 V (orange curve) up to -2.7 V
(purple curve) with a step of 0.1 V. The colored solid lines represent the
fitting of different portions of these curves with an exponential of higher
orders n of the slope $\alpha$. The inset shows the variation of the slope
$\alpha$ with increasing $V_{\mathrm{sg}}$.

The standard deviation $\sigma$ of SCDs measured under the influence of the
gate is represented by the blue curve in Fig. 4a. For small values of
$|V_{\mathrm{sg}}|$, where $I_{\mathrm{leak}}$ is negligible, $\sigma$ is
independent of $|V_{\mathrm{sg}}|$ and no significant change in the
$\langle$$I_{\mathrm{SW}}$$\rangle$ of SCDs (green curve) was observed. Beyond
$V_{\mathrm{th}}$ at $|V_{\mathrm{sg}}|$ = 2.7 V, $\sigma$ increases with
$|V_{\mathrm{sg}}|$ because the fluctuations assisted by $I_{\mathrm{leak}}$
become stronger and more frequent. This increases the probability of nanowire
switching at small $I_{\mathrm{SW}}$ values with a corresponding suppression
in the $\langle$$I_{\mathrm{SW}}$$\rangle$ of the SCD. This increase in the
width of the SCDs is analogous to the typical temperature dependence (see the
Supporting Information) Ejrnaes  _et al._ (2019); Puglia  _et al._ (2020a);
PUGLIA  _et al._ (2021) associated with thermally-activated phase slips
McCumber and Halperin (1970); Bezryadin (2013). However, the large width of
the SCDs obtained under the influence of the gate indicates that the system is
driven to a nonequilibrium state where the fluctuations are an order of
magnitude larger than expected from the bath temperature. With further
increasing $|V_{\mathrm{sg}}|$, $\sigma$ decreases and the SCDs become more
symmetric, as shown by their calculated skewness in Fig. 4b. This is analogous
with the picture that the switching of the system is due to multiple phase
slips (MPS) found at finite temperatures Ejrnaes  _et al._ (2019).

Interestingly, the SCD in Fig. 3a at $V_{\mathrm{sg}}$ = 2.8 V (orange curve)
shows two peaks, a sharp one at 1.57 $\mu$A and a broad one around 1.5 $\mu$A.
This distribution looks like the sum of two overlapping probability
distributions, similar to distributions shown in Refs. 14; 22. Since the
probability distribution in this transition region depends strongly on the
ramp speed of the bias current $\nu_{\mathrm{I}}$ (see the Supporting
Information), we could completely switch between the two distributions when
the ramp speed was changed from 300 (at which the SCDs in Fig. 3 are measured)
to 9.375 $\mu$A/s. For a more accurate evaluation, it is better to transform
the measured probability distributions into the speed-independent escape rate
$\Gamma(I,T)$ (see the Supporting Information) by the direct Kurkijärvi-
Fulton-Dunkleberger (KFD) transformation Kurkijärvi (1972); Fulton and
Dunkleberger (1974); Bezryadin (2013):

$\Gamma(I_{\mathrm{N}},T)=\frac{P(I_{\mathrm{N}},T)\nu_{\mathrm{I}}}{1-w\sum_{k=0}^{N}P(I_{\mathrm{k}},T)}$
(1)

where $w$ is the bin size in the current axis of the measured probability
distribution $P(I,T)$, and $P(I_{\mathrm{k}},T)$ is the switching probability
in the bias current interval $[kw,(k+1)w]$ with $k$ $\in$ $[0,N]$. Fig. 4c
shows the logarithm of the calculated $\Gamma(I_{\mathrm{N}},T)$ as a function
of current for SCDs measured under the influence of $V_{\mathrm{sg}}$ in the
interval [-3.5, -2.7 V]. As long as $V_{\mathrm{sg}}$ is small, the SCDs have
a sharp peak around $I_{\mathrm{SW}}$=1.57 $\mu$A, resulting in a large escape
rate around this value, which represents the escape rate due to quantum
tunnelling or thermal escape $\Gamma_{\mathrm{T}}$. As the influence of the
gate voltage sets in (see e.g., purple curve), a finite escape rate appears at
lower $I_{\mathrm{SW}}$ values, corresponding to the gate-assisted escape rate
$\Gamma_{\mathrm{L}}$. The latter contribution becomes the dominant escape
rate at higher gate voltages (see e.g., green curve).

For $V_{\mathrm{sg}}$ $>$ -3 V, where $I_{\mathrm{leak}}$ is negligible (the
first three curves from the right), $\Gamma_{\mathrm{L}}$ can be well fitted
with an exponential curve (black solid line) given by

$\Gamma_{L}(I_{\mathrm{sw}})=Ae^{n\alpha I_{\mathrm{sw}}}$ (2)

with $n=1$ and using $\alpha$ and $A$ as fitting parameters. The switching
dynamics in this region have been studied in detail in Ref. 22. On the other
hand, for $V_{\mathrm{sg}}$ $\leq$ -3 V $\Gamma_{\mathrm{L}}$ deviates from
the single exponential dependence described by Eq. 2. For example, the light
green curve measured at -3.2 V can be fitted at large current values using Eq.
2 with n=1 (see black solid lines). Interestingly, the measured curve for
$I_{\mathrm{SW}}$ $<$ 0.9 $\mu$A can be well fitted by adding extra higher
order terms with n=2,3,$\ldots$, keeping the same values of the fitting
parameters. Further increasing $V_{\mathrm{sg}}$ ($I_{\mathrm{leak}}$)
requires more higher order terms to fit the escape rate dependence (e.g.
orange curve) Ejrnaes  _et al._ (2019). The value of $\alpha$ required to fit
the escape rate dependence decreases sharply as $V_{\mathrm{sg}}$ increases,
and saturates at large values of $V_{\mathrm{sg}}$ as shown in the inset of
Fig. 3c.

The deviation of the escape rate dependence with current from a pure
exponential at higher gate voltages is similar to elevated temperatures in
Ref. 43. This can be attributed to the reduced impact for a single fluctuation
event triggered by the leakage current, since the dissipation during the
induced phase slip event is smaller at lower values of $I_{\mathrm{SW}}$
Bezryadin (2013); Ejrnaes  _et al._ (2019). Thus, several coincident
fluctuation events with corresponding induced multiple phase slips (MPS) are
required to trigger the switching of the nanowire to the normal state Ejrnaes
_et al._ (2019); Shah  _et al._ (2008); Pekker  _et al._ (2009). In this
regime, at large bias current values, the dissipation of a single MPS event
($n=1$) is sufficient to switch the nanowire into the resistive state. On the
other hand, at lower current values, the dissipation of a single MPS event is
reduced and higher orders ($n=2,3,...$) of the MPS event are required to
trigger the resistive switching of the nanowire Ejrnaes  _et al._ (2019).

## Comparison with the proposed microscopic pictures

In the following, we will compare our experimental results with the possible
microscopic pictures. Starting from the two gates (Fig. 1d,e), despite SG2 has
almost twice the nanowire$-$gate spacing of SG1, it suppresses the
$I_{\mathrm{SW}}$ at lower $V_{\mathrm{th}}$ than SG1. This contradicts the
electric field picture as a possible explanation for the origin of the GCS. On
the other hand, $I_{\mathrm{SW}}$ starts to be suppressed with the onset of
leakage current between the nanowire and each of the gates (see the Supporting
Information). In another cool-down, the influence of the two gates for the
same device shows an opposite situation, as SG1 shows a stronger influence on
$I_{\mathrm{SW}}$ than SG2 (see the supporting information). This excludes any
concerns arising from the quite large dielectric constant of the InAs nanowire
between SG2 and the Ta shell (see Fig. 1a), which may lead to a larger
influence of SG2 on $I_{\mathrm{SW}}$ than SG1. Interestingly, we found that
the influence of the two gates on $I_{\mathrm{SW}}$ gives better matching with
$P_{\mathrm{G}}$ in the two cool-downs (see the Supporting Information).

Accepting that the leakage current plays a key role in the GCS, a simple
explanation arises: that the leaking electrons increase the temperature of the
superconducting nanowire. We have investigated for the first time the
$B$-field dependence of superconducting nanowire with normal contacts, which
allows efficient cooling of the superconducting nanowire. The $B$-field
dependence at finite $T$ and finite gate voltage was strictly different,
indicating that the effect of leakage current cannot be described by simple
hot electron regime induced by elevated bath temperature. The highly non-
equilibrium state of the superconductor at finite gate voltage is further
supported by the broad SCDs in our work and in previous results Puglia  _et
al._ (2020a); Basset  _et al._ (2021); Golokolenov  _et al._ (2021); PUGLIA
_et al._ (2021); Ritter  _et al._ (2022). Our detailed comparison of the SCDs
for different gate polarities (Fig. 3) provided another important finding
which is inconsistent with electric-field-induced suppression of
superconductivity. Pairing of SCDs measured at opposite gate polarities at the
same leakage current dissipation, $P_{\mathrm{G}}$, provided a better matching
than at the same $|V_{\mathrm{sg}}|$ (Fig. 3a,b). This reveals that the
suppression of $I_{\mathrm{SW}}$ depends not only on the energy of the
injected electrons (e$V_{\mathrm{sg}}$) or the rate of their injection
($I_{\mathrm{leak}}$/e), but on the power dissipated at the gate
$P_{\mathrm{G}}$. Based on the $\langle$$I_{\mathrm{SW}}$$\rangle$ of the SCDs
for the two polarities, the ballistic injection of electrons from the gate
into the superconducting nanowire can be discarded. We conclude that the
phonon-mediated excitation of the superconductor remains a microscopic picture
consistent with the measured results.

Furthermore, we also noticed that the power dissipation at the gate required
to fully suppress $I_{\mathrm{SW}}$, $P_{\mathrm{G,C}}$ is comparable to the
power dissipation that occurs when the device switches to the resistive state
$P_{\mathrm{n}}$ = $I_{\mathrm{SW}}^{\mathrm{2}}$$\cdot$$R_{\mathrm{n}}$. For
example, for device A, in the case of SG1 (the closer to the Ta shell),
$P_{\mathrm{G,C}}$ $\simeq$ 1.5 nW (see the Supporting Information), while
$P_{\mathrm{n}}$ $\simeq$ 1 nW (using $R_{\mathrm{n}}$= 780 $\Omega$ and
$I_{\mathrm{SW}}$ $\simeq$ 1.17 $\mu$A).

Finally, a very large leakage current was required to quench $I_{\mathrm{SW}}$
when we investigated the GCS in similar Ta/InAs devices fabricated on a
sapphire substrate (see the Supporting Information). These results indicate
that the GCS depends mainly on the properties of the substrate and the leakage
pathway between the gate and nanowire.

## Conclusions

We investigated the origin of GCS in the Ta half-shell layer deposited on InAs
nanowires by various measurements. Devices with small nanowire-gate spacing
(specifically devices B and C) fully switch to the normal state below
$V_{\mathrm{sg}}$ = $\pm$5 V, which makes them promising for integration into
classical electronic circuits. When the wire is connected by electrodes in the
normal state, the critical magnetic field $B_{\mathrm{C}}$ is not suppressed
under the influence of the gate as for elevated temperatures. Moreover, the
comparison of the switching current distributions at opposite gate polarities,
as well as the gate dependence of two opposite side gates at different
nanowire-gate spacings show that the power dissipated at the gate
$P_{\mathrm{G}}$ is the relevant parameter for this effect. Analysis of the
switching dynamics under strong gate influence shows a deviation in the escape
rate dependence with the bias current from a pure exponential. This indicates
that the device is driven into the MPS regime by the high energy fluctuations
originating from the leakage current. Our findings contradict the microscopic
pictures proposing electric fields or ballistic injection of high-energy
electrons as the origin of the GCS effect, but they are consistent with the
non-equilibrium superconducting state resulting from the absorption of phonons
generated by the leakage current.

## Methods

InAs nanowires were grown by the VLS mechanism using Molecular Beam Epitaxy
and the Ta shell was deposited in-situ under UHV using electron beam
evaporation at a substrate temperature of about $25^{\circ}$ C. Based on the
TEM characterization, the morphology of the Ta shells was continuous but
granular on the InAs nanowires and was found to be non-crystalline Bjergfelt
(2019).

The Ta/InAs nanowires were deposited on the top of a doped Si wafer with 290
nm thick $\mathrm{SiO_{2}}$ layer by means of a hydraulic micromanipulator
along with a high magnification optical microscope. The nanowire device was
fabricated in two separate electron beam lithography (EBL) steps. In the first
step, four Ti/Al contacts with a thickness of 10/80 nm were fabricated. Prior
to the metal evaporation, Ta/InAs nanowires were exposed to Ar-ion plasma
milling for 8 minutes at 50 W to remove any oxides on the top of the Ta shell.
In the second step, two metallic gates of Ti/Au layers with a thickness of
7/33 nm were fabricated with unequal spacing and on opposite sides of the
nanowire. The metallic gates were fabricated in a separate lithography step,
since thin resist is used for precise alignment of the gates from the
nanowire.

The $I-V$ characteristics of the device were measured by a pure DC
measurements using a quasi 4-probe method in which the current was injected
through the nanowire via a pair of Al contacts by using a standard voltage
source (Basel DAC SP 927) with series resistor of 1 M$\Omega$, while the
voltage was measured across the other pair with a differential voltage
amplifierand a digital multimeter (Keithley 2001). The leakage current was
recorded by measuring the voltage across a 10 M$\Omega$ preresistor connected
to the gate and corrected according to the method reported in Ref. 23.

The SCD was measured using a NI-DAQ card (USB-6341), where a periodic current
wave signal was engineered. This signal is composed of a positive linear ramp
with an amplitude of 3 $\mu$A and a slope in the range from 9.375 to 300
$\mu$A/s followed by a 2.5 ms of zero-current plateau for cooling down the
superconducting device. This signal is repeated 10,000 times, and
$I_{\mathrm{SW}}$ is extracted each time. All SCDs are measured at 0.1 T to
switch the Al leads to the normal state. All measurements were carried out in
a Leiden Cryogenics CF-400 top-loading cryo-free dilution refrigerator system
with a base temperature of 30 mK.

## Author contributions

T.E., M.B. fabricated the devices, T.E., M.B., M.K., G.F. and Z.S. performed
the measurements and did the data analysis. T. K. and J. N. developed the
nanowires. All authors discussed the results and worked on the manuscript.
P.M. and S.C. guided the project.

## Acknowledgments

This work has received funding from Topograph FlagERA, the SuperTop QuantERA
network, SuperGate Fet Open, the FET Open AndQC, and from the OTKA FK-123894
grants. This research was supported by the Ministry of Innovation and
Technology and the National Research, Development and Innovation Office within
the Quantum Information National Laboratory of Hungary and by the Quantum
Technology National Excellence Program (Project Nr.
2017-1.2.1-NKP-2017-00001), by the ÚNKP-22-5 New National Excellence Program,
the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by
ÚNKP-22-2-I-BME-22 New National Excellence Program of the Ministry for Culture
and Innovation from the source of the National Research, Development and
Innovation Fund, by the Carlsberg Foundation, Innovation Fund Denmark and the
Danish National Research Foundation. We thank M. Bjergfelt, D. Carrad and T.S.
Jespersen for their contributions to the development of the hybrid nanowires.

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Lath, R. Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, D.
Sheffield, S. Somalwar, R. Stone, S. Thomas, P. Thomassen University of
Tennessee, Knoxville, USA
H. Acharya, A.G. Delannoy, J. Heideman, G. Riley, S. Spanier Texas A&M
University, College Station, USA
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Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock,
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S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z.
Wang, A. Whitbeck Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij, Q. Xu
University of Virginia, Charlottesville, USA
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Neu, Y. Wang, E. Wolfe, F. Xia Wayne State University, Detroit, USA
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N. Woods †: Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at Université Libre de Bruxelles, Bruxelles, Belgium
6: Also at University of Chinese Academy of Sciences, Beijing, China
7: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia
8: Also at Joint Institute for Nuclear Research, Dubna, Russia
9: Also at Helwan University, Cairo, Egypt
10: Now at Zewail City of Science and Technology, Zewail, Egypt
11: Also at Suez University, Suez, Egypt
12: Now at British University in Egypt, Cairo, Egypt
13: Also at Fayoum University, El-Fayoum, Egypt
14: Also at Department of Physics, King Abdulaziz University, Jeddah, Saudi
Arabia
15: Also at Université de Haute Alsace, Mulhouse, France
16: Also at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, Moscow, Russia
17: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
18: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
19: Also at University of Hamburg, Hamburg, Germany
20: Also at Brandenburg University of Technology, Cottbus, Germany
21: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary
22: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
23: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary
24: Also at Indian Institute of Technology Bhubaneswar, Bhubaneswar, India
25: Also at Institute of Physics, Bhubaneswar, India
26: Also at Shoolini University, Solan, India
27: Also at University of Visva-Bharati, Santiniketan, India
28: Also at Isfahan University of Technology, Isfahan, Iran
29: Also at Plasma Physics Research Center, Science and Research Branch,
Islamic Azad University, Tehran, Iran
30: Also at ITALIAN NATIONAL AGENCY FOR NEW TECHNOLOGIES, ENERGY AND
SUSTAINABLE ECONOMIC DEVELOPMENT, Bologna, Italy
31: Also at CENTRO SICILIANO DI FISICA NUCLEARE E DI STRUTTURA DELLA MATERIA,
Catania, Italy
32: Also at Università degli Studi di Siena, Siena, Italy
33: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
34: Also at Kyung Hee University, Department of Physics, Seoul, Korea
35: Also at Riga Technical University, Riga, Latvia
36: Also at International Islamic University of Malaysia, Kuala Lumpur,
Malaysia
37: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
38: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
39: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
40: Also at Institute for Nuclear Research, Moscow, Russia
41: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
42: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
43: Also at University of Florida, Gainesville, USA
44: Also at P.N. Lebedev Physical Institute, Moscow, Russia
45: Also at California Institute of Technology, Pasadena, USA
46: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
47: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
48: Also at University of Belgrade, Belgrade, Serbia
49: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy
50: Also at National and Kapodistrian University of Athens, Athens, Greece
51: Also at Universität Zürich, Zurich, Switzerland
52: Also at Stefan Meyer Institute for Subatomic Physics (SMI), Vienna,
Austria
53: Also at Beykent University, Istanbul, Turkey
54: Also at Istanbul Aydin University, Istanbul, Turkey
55: Also at Mersin University, Mersin, Turkey
56: Also at Piri Reis University, Istanbul, Turkey
57: Also at Gaziosmanpasa University, Tokat, Turkey
58: Also at Adiyaman University, Adiyaman, Turkey
59: Also at Ozyegin University, Istanbul, Turkey
60: Also at Izmir Institute of Technology, Izmir, Turkey
61: Also at Marmara University, Istanbul, Turkey
62: Also at Kafkas University, Kars, Turkey
63: Also at Istanbul University, Istanbul, Turkey
64: Also at Istanbul Bilgi University, Istanbul, Turkey
65: Also at Hacettepe University, Ankara, Turkey
66: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom
67: Also at School of Physics and Astronomy, University of Southampton,
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68: Also at Monash University, Faculty of Science, Clayton, Australia
69: Also at Bethel University, St. Paul, USA
70: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
71: Also at Purdue University, West Lafayette, USA
72: Also at Bingol University, Bingol, Turkey
73: Also at Sinop University, Sinop, Turkey
74: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
75: Also at Texas A&M University at Qatar, Doha, Qatar
76: Also at Kyungpook National University, Daegu, Korea
77: Also at University of Hyderabad, Hyderabad, India