###### Abstract

It was recently demonstrated the possibility to tune, through the application
of a control gate voltage, the superconducting properties of mesoscopic
devices based on Bardeen-Cooper-Schrieffer metals. In spite of the several
experimental evidence obtained on different materials and geometries, a
description of the microscopic mechanism at the basis of such unconventional
effect has not been provided yet. This work discusses the technological
potential of gate control of superconductivity in metallic superconductors and
revises the experimental results which provide information regarding a
possible thermal origin of the effect: in the first place, we review
experiments performed on high critical temperature elemental superconductors
(niobium and vanadium) and show how devices based on these materials can be
exploited to realize basic electronic tools such as, e. g., a half-wave
rectifier. In a second part, we discuss the origin of the gating effect by
showing the gate-driven suppression of the supercurrent in a suspended
titanium wire and by providing a comparison between thermal and electric
switching current probability distributions. Furthermore, we discuss the cold
field-emission of electrons from the gate by means of finite element
simulations and compare the results with experimental data. Finally, the
presented data provide a strong indication regarding the unlikelihood of
thermal origin of the gating effect.

###### keywords:

superconductivity, Josephson effect, gate-controlled

1 1 0 Gate control of superconductivity in mesoscopic all-metallic devices
Electrostatic control of superconductivity in mesoscopic all-metallic devices
Claudio Puglia ${}^{1,2,^{*}}$, Giorgio De Simoni 2 and Francesco Giazotto 2
Claudio Puglia, Giorgio De Simoni and Francesco Giazotto Puglia, C.; De
Simoni, G.; Giazotto, F. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

In the last two years, the impact of gate voltage on the superconducting
properties of Bardeen-Cooper-Schrieffer (BCS) Bardeen (1962) metallic
superconductors has been investigated De Simoni et al. (2018); Paolucci et al.
(2018, 2019a, 2019b, 2019c); Bours et al. (2020). In these studies the authors
analyze the effect of the electrostatic gating, generating an electrostatic
field of the order of $10^{8}$ V/m and, at the same time, modifying negligibly
the concentration of the surface charge carriers. Particularly, ampbipolar
suppression of supercurrent has been demonstrated in all-metallic
superconductor wires De Simoni et al. (2018), nano-constriction Josephson
junctions (JJs) Paolucci et al. (2018, 2019a), fully-metallic Superconducting
Quantum Interference Devices (SQUID) Paolucci et al. (2019c), and proximity
nanojunctions De Simoni et al. (2019). Such unconventional gating effect in
BCS superconductor systems is the first step for the realization of easy
fabrication and high-scalable technologies in both the environments of
classical and quantum electronics. The aim of this work is to review the most
recent advances on the modification of superconducting properties in
mesoscopic structures via the application of a control gate voltage. For such
an effect, a fulfilling microscopic theory has not been provided yet. Indeed,
it is not possible to take into account for experimental observations through
the conventional BCS framework, in which the superconducting properties are
negligibly affected by electric fields Virtanen et al. (2019). Although, some
theories have been proposed, including the electric field driven Rashba
orbital polarization Mercaldo et al. (2020), and the gate-driven Schwinger
excitation of quasiparticles from the BCS vacuum Solinas et al. (2020), they
have not been experimentally verified yet. The injection of high-energy field-
emitted cold-electrons into the weak-link was also hypothesized to be at the
origin of the gating effect Alegria et al. (2021); Ritter et al. (2020).
Nevertheless, even in presence of the latter mechanism, several experimental
results are not compatible with a mere power injection resulting in an
overeating of the superconductor De Simoni et al. (2018); Paolucci et al.
(2018); Puglia et al. (2020); Rocci et al. (2020).

The article is organized as follows: Section 2 displays evidence of a gate-
driven supercurrent suppression in vanadium and niobium Dayem bridges (DBs).
Moreover, different technological implementations based on these materials are
presented. In the same section, two further topics are faced. The former is
the modification of the switching current probability distribution as a
function of the electric field. The latter is the influence of the substrate
on the gating effect in titanium weak-links. Section 3 analyses the evidence
against a thermal origin of the supercurrent suppression. Finally, Section 4
provides a summary of the results presented in this review resuming the main
achievements and proposing new experiments to increase the understanding of
the gating effect.

##  2 Gate-driven supercurrent suppression in Nb and V nanojunctions

In this section we present a series of experiments, performed on niobium and
vanadium superconducting weak-link devices, aimed at extending the range of
materials suitable for gated-superconductor applications at elemental
superconductors with critical temperature higher than the liquid helium
temperature $\sim 4.2$ K. The presented results demonstrate the possibility to
implement gate-controlled all-metallic superconducting electronics Likharev
(2012) compatible with industrial standards.

###  2.1 Niobium gate-controlled transistor

All-metallic supercurrent transistors consist of a superconducting mesoscopic
channel realized with BCS metals, equipped with gate electrodes
lithographically fabricated at a distance of a few nanometer from the channel.
The gate electrode is polarized through the application of an either positive
or negative control gate voltage. Niobium gate-controlled transistors
typically consist of a 8-$\mu$m-long, 2.5-$\mu$m-wide wire interrupted by a
50-nm-wide, 120-nm-long constriction. Aligned with the DB weak-link it a was
realized a co-planar, 60-nm-far, 80-nm-wide metallic gate. The thin film was
deposited on a sapphire Al2O3 substrate via DC magneto-sputter deposition of a
10/40-nm-thick Ti/Nb bylayer. The former metal was necessary to increase the
adhesion and the mechanical strength of the metallic film. A pseudo-color
scanning electron micrograph (SEM) is shown in Figure 1(a).

Figure 1:  (a) Pseudo-color SEM of a representative niobium gate-tunable
transistor with the four-probes bias scheme. The junction and the wire are in
false-colored orange, the gate in blue. (b) $I$ vs $V$ curves for different
gate voltages $V_{G}$ at a bath temperature of 20 mK. Characteristics are
horizontally shifted for clarity. A clear symmetric suppression of $I_{S}$ is
visible as the modulus of the gate voltage $|V_{G}|$ is increased. (c) $I_{S}$
vs $V_{G}$ for several bath temperatures $T$ in a range from $20$ mK to $3$ K.
The switching current $I_{S}$ values were acquired by acquiring $50$
repetitions of the $I$ vs $V$ characteristics. Error bars are smaller than the
data symbol.

The device shows a Dayem bridge normal state resistance $R_{DB}\simeq 30\
\Omega$ and a critical temperature $T_{DB}\simeq 3$ K De Simoni et al. (2020).
On the other hand, the niobium banks inherit the critical temperature of the
pristine thin film $T_{Nb}\simeq 7.9$ K De Simoni et al. (2020). The smaller
critical temperature of the Dayem bridge is due to its lateral size which is
comparable with the niobium coherence length Stromberg and Swenson (1962);
Finnemore et al. (1966). $T_{Nb}$, instead, is about $80$% of the conventional
critical temperature for Nb because of the proximity effect of the adhesion
titanium layer. The conventional dissipationless transport is displayed by
plotting the current $I$ vs voltage $V$ curves acquired at a bath temperature
$T=20$ mK and a gate voltage $V_{G}=0$ V, as shown in Figure 1(b). The Dayem
bridge switching current is $I_{S}\simeq 30\ \mu$A. The forward and backward
$I$ vs $V$ characteristics shows the characteristic hysteretic behavior due to
the retrapping current $I_{R}$ Giazotto et al. (2006); Tinkham (2004).

The suppression of the switching current via the application of a gate voltage
was demonstrated by measuring the current $I$ vs voltage $V$ characteristics
with a gate voltage in the range from $-40$ to $40$ V at a bath temperature of
$T=20$ mK. Figure 1(b) displays $V(I)$ curves at several gate voltages. A
shadow grey area is drawn to underline the suppression region. The quenching
of the supercurrent is symmetric for $V_{G}\longrightarrow-V_{G}$ for bath
temperatures between 20 mK and 3 K as shown in Figure 1(c). As reported in
conventional experiments De Simoni et al. (2018, 2019); Paolucci et al. (2018,
2019a), the widening of the plateau in which the gate voltage is not effective
is clearly visible as the temperature rises. The suppression of the
supercurrent can be observed up to a temperature of $3$ K with complete
suppression at $|V_{G}|=40$ V for $T>2$ K.

####  2.1.1 Rectification properties

Basing on the peculiar shape of the $R$ vs $V_{G}$ characteristic De Simoni et
al. (2020), it is possible to implement a superconducting diode scheme. In
particular, an alternate gate voltage $V_{AC}$ can be rectified exploiting the
sharpness of the super-to-normal transition when the junction is current
biased. In this configuration, a sinusoidal gate voltage is transformed into a
square wave voltage-drop across the junction. Such a peculiar system response
is shown in Figure 2(a). The gate voltage signal is the sum of $V_{AC}$ and of
a DC pre-bias voltage used to define a switching current working range
$I_{S}(V_{G}^{max})<I_{B}<I_{S}(V_{G}^{min})$. The oscillation of $I_{S}$
above and below $I_{B}$ results in time dependent normal-to-super and vice
versa transitions that generate a time dependent voltage drop $V(t)$ across
the junction as shown in Figure 2(b). The output signal maintains the same
periodicity of $V_{AC}$ and the duty cycle is defined by the time fraction for
which the condition $I_{S}<I_{B}$ is satisfied.

Figure 2: (a) Sketch of the operation principle exploited by the niobium-based
rectifier. The system is DC current biased with $I_{B}$ (horizontal red dashed
line in (a)) and the side-gate is carries a signal consisting of an alternate
part $V_{AC}$ (blue/green line) and a DC part $V_{DC}$. The resulting
characteristic $I_{S}(t)$ (purple/yellow curve) depends on the amplitude of
$V_{AC}$ and on the value of $V_{DC}$ and generates recurring super-to-normal
and normal-to-super state transitions. In this way, the voltage signal across
the junction $V$ oscillates between a low and a high state with the same
period of $V_{AC}$. (b) Voltage $V$ acquired across constriction in a lock in
four-wires scheme. The alternate voltage $V_{AC}$ is the reference signal of
the lock-in amplifier. The system is current biased with $I_{B}=2.5\ \mu$A.
$V$ is almost zero until $I_{S}(V_{G})<I_{B}$. The peak represent the
rectification effect of the AC gate voltage signal.

Notably, the output voltage depends directly on the amplitude of the AC input
signal thanks to the behavior of $R$ vs $V_{G}$ characteristic. In the
configuration shown in Figure 2 our system acts as a half-wave rectifier which
could be used in a vast range of devices such as diodes and detectors. In the
next paragraph further evidences of the rectification properties of such
systems are provided, with emphases on the versatility of the technology.

###  2.2 Vanadium gate-controlled transistor

The vanadium gate-controlled transistors typically consist of a planar 60-nm-
thick, 160-nm-long, 90-nm-wide Dayem bridge Josephson junction flanked by
70-nm-far, 120-nm-wide side-gate. Our exploited bridge geometry is similar to
those already discussed for Nb-based devices. The system was realized by means
of a one-step electron beam lithography performed on a silicon dioxide
substrate (SiO2) followed by the electron beam evaporation of high purity
vanadium at a rate of 0.36 nm/s in an ultra-high vacuum chamber. Figure 3
shows the pseudo-color SEM of a typical vanadium gate-controlled transistor.

Figure 3: (a) Pseudo-color SEM of a representative vanadium gate-tuned
transistor. The junction and the wire are colored in orange, the gate in blue.
(b) Current $I$ vs voltage $V$ curves of a typical device acquired at several
bath temperatures from $2.0$ to $3.8$ K. The characteristics are horizontally
shifted for clarity. (c) Switching current $I_{S}$ vs gate voltage $V_{G}$
curves for several bath temperatures in the range between $2.0$ and $3.3$ K.
$I_{S}$ values were obtained averaging $25$ acquisitions.

The device shows a normal state resistance $R_{N}\simeq 106\ \Omega$, a
switching current at 2 K of $I_{S}=1.42$ mA and a critical temperature
$T_{C}\simeq 3.6$ K Puglia et al. (2020) as displayed in Figure 3(b).

The suppression of the supercurrent as a function of the gate voltage was
demonstrated by measuring the $I_{S}$ vs $V_{G}$ characteristics. Figure 3(c)
shows the bilateral suppression of the supercurrent down to total quenching
for $|V_{G}|\simeq 10$ V in a range of bath temperature from $2$ to $3.2$ K.
Notably, the sharper suppression of $I_{S}$ observed for positive values of
the gate voltage is in contrast with a possible cold field-emission origin of
the quenching effect. Indeed, the device geometry could facilitate the
electron extraction from the gate that occurs at negative gate bias values
Alegria et al. (2021); Ritter et al. (2020). This consideration is deeply
discussed in Section 3.

####  2.2.1 Half-wave rectifier

A time-resolved characterization of the device was carried out using both
sinusoidal and square -wave gate voltages. Figure 4(a) shows the bias scheme
of the measurement setup consisting in a DC bias current, a DC voltage
generator and an ADC/DAC digital board providing the AC gate voltage signal.
The latter voltage generators provides a $V_{G}(t)=V_{DC}+V_{AC}(t)$ gate
signal, setting the right operation point in the parameters space (see Figure
4(a)).

Figure 4: (a) Bias scheme for AC measurements. The gate voltage is generated
adding up a DC $V_{DC}$ and a AC $V_{AC}$ arbitrary waveform voltages. The
ADC/CAD board that provides the AC signal performs a real-time measurements of
the voltage drop $V$ across the junction. (b) Voltage $V$ vs current $I$
curves for chosen values of $V_{G}$. The two dot couples displays the
operation points of the system for two bias currents $I_{B}=18$, $71\ \mu$A.
The time dependent response of the system $V(t)$ is represented with the same
color of the dot pairs. The gate voltage signal $V_{G}(t)$ consists of a DC
voltage $V_{DC}=10$ V and an AC square-wave voltage with amplitude $V_{AC}=5$
V. The bath temperature for these measurements was $T=3$ K.

The time dependent voltage drop $V(t)$ across the junction was acquired as a
function of the time-dependent gate voltage $V_{G}(t)$ obtaining a zero-signal
(low-state) when $I_{S}\left[V_{G}(t)\right]>I_{B}$ (superconducting state).
By contrast, when $I_{S}\left[V_{G}(t)\right]<I_{B}$ the junction switches to
the dissipative state and a voltage drop different from zero is obtained
across the constriction (high-state). We measured the system response to a
transistor–transistor logic-like (TTL) square-wave gate signal composed by a
$V_{DC}=10$ V voltage DC bias plus a $V_{AC}=5$ V square-wave with a frequency
up to $\sim 50$ Hz, as shown in Figure 4(b).

The low and high states are highlighted on top of the $I$ vs $V$
characteristics (obtained for $V_{G}$ ranging from $10$ to $15$ V) with
couples of dots of the same colors in Figure 4(b). The $V(t)$ signal resulting
from the $V_{G}(t)$ excitation is shown in Figure 4(d) for two current bias
($I_{S}=18,\ 71\ \mu$A). We note that the output voltage is proportional to
the bias current. It is worth to emphasize again that $V(t)$ maintains the
shape of the input voltage signal with frequencies, in principle, limited only
by $f_{\Delta}$ Barone and Paternò (1982); Tinkham (2004).

Finally, we show the response of the system to a sinusoidal gate voltage
signal. The measurement setup is the same of the square-wave characterization
shown in Figure 4(a). The excitation consists in a $V_{AC}$ sine-wave with
amplitude in the range between $1.0$ and $3.5$ V summed with a $V_{DC}=11$ V
DC voltage bias shown in Figure 5(b). The bias current for this experiment was
chosen to be $I_{B}=72\ \mu$A in order to have a sharp super-to-normal
transition and a linear dependence between $R$ and $V_{G}$. The continuous
variation of the gate voltage provides a continuous variation of the junction
resistance accordingly with the $R(V_{G})$ curves Puglia et al. (2020). Due to
the former behavior, the system lies in the zero-voltage state for
$I_{S}\left[V_{G}(t)\right]>I_{B}$, and a voltage drop across the junction
appears when the condition $I_{S}\left[V_{G}(t)\right]\simeq I_{B}$ is
satisfied. When $I_{S}\left[V_{G}(t)\right]<I_{B}$, the voltage drop across
the nanojunction increases because of the gate-driven evolution of the
constriction resistance and, eventually, reaches the saturation level at the
asymptotic value of the resistance of the normal state. Figure 5(a) displays
the voltage drop across the junction as a function of both the bias current
and the gate voltage. The transition edge is highlighted by a dashed red line.
The working point set by $I_{B}$ and $V_{DC}$ is shown by white dashed lines.
The time resolved $V(t)$s for $V_{AC}$ equal to $1.0$ and $3.5$ V are reported
in Figure 5(c), (d). Notably, the rectification threshold and the portion of
the input signal rectified can be selected setting both $I_{B}$ and $V_{AC}$.

Figure 5: (a). Color-plot of the voltage drop $V(t)$ across the Dayem bridge
constriction versus the gate voltage $V_{G}$ (X-axis) and the bias current
$I_{B}$ (Y-axis). The three dots represent the zero signal state (yellow), the
transition to the normal-state (green) and the maximum $V(t)$ signal (blue),
respectively. The vertical dashed red line shows the operating curve defined
by the bias current. The yellow-to-blue sinusoidal represents the AC + DC gate
voltage excitation $V_{G}(t)$. (b) and (c) Time dependent voltage drop across
the constriction for two amplitude of the alternate component of the gate
voltage, $V_{AC}=\ 3.5,\ 1.0$ V. The color-map is all the panels. The
measurements were performed at a bath temperature of $T=3$ K.

The former characteristics, typical of a half-wave rectifier, here are
realized for the first time exploiting an all-metallic gated superconducting
Dayem bridge. We speculate that the described rectifying behavior can be
suitably exploited to rectify an incoming radiation coupled to the gate
through an antenna, realizing a gate-controlled version of a transition edge
sensor Ullom and Bennett (2015); Paolucci et al. (2021); Gol’tsman et al.
(2001). This device could operate in an extremely wide frequency range,
spanning from below 1 GHz to about 1 THz. This interval is particularly
relevant for both technological applications and for fundamental research
(e.g. in astrophysics for cosmic microwave background detection).

####  2.2.2 Amplification properties

The vanadium Dayem bridge, thanks to the peculiar $R(V_{G})$ characteristics,
is suitable for the realization of an amplifier. The gain parameter of a gated
DB device is defined as the ratio between peak-to-peak amplitudes of gate
voltage input and output voltage drop across the junction
$g=\frac{V_{out}}{V_{in}}$. For our system $V_{out}=R(V_{G})I_{B}$ is the
voltage drop across the junction and is directly proportional to the
resistance and the current bias. The $V_{in}$ is obtained by the ratio of the
width of SCPD Puglia et al. (2020) and the transcoductance
$\tau=\frac{dI_{S}}{dV_{G}}$. For the devices taken into account in this
section $g\sim 7$ with a typical power consumption of $\sim 40$ nW. It is
worth to highlight, on the one hand, that $g$ is of the same order of
magnitude of the conventional semiconductor cold amplifier Ivanov et al.
(2011); Oukhanski et al. (2003). Such a result, on the other hand, was
obtained with a power consumption of about three order of magnitude smaller
than the typical semiconducting counterpart. Furthermore, a chain of $N$
rectifiers can be obtained by deliver the gate electrode of the n-th device
with the output voltage of the ($n-1$)-th DB, obtaining a total gain equal to
$g^{N}$.

The possibility to tune the critical supercurrent via conventional gating
paves the way to a wide range of applications. Indeed, gate-controlled devices
could be exploited to realize tunable magnetometers Clarke and Bragisnki
(2004); Giazotto et al. (2010) and heat control systems Giazotto and Martínez-
Pérez (2012); Fornieri and Giazotto (2017). Furthermore, by exploiting the
gating effect a voltage-controlled version of the nanocryotron McCaughan
(2018); McCaughan and Berggren (2014) can be implemented. The latter is a
three terminal superconducting device in which a localized switching-current
suppression (triggered by injecting a control-current which generates a
localized high temperature region, an hotspot, by Joule heating) destroys the
superconducting properties of the nearby system. The gated version of the
nancryotron, the so-called (EF-Tron) Paolucci et al. (2019c), is implemented
through the parallel of a resistor and a gated superconductor. Differently
from current-driven devices McCaughan and Berggren (2014); Morpurgo et al.
(1998), the EF-tron is expected not to be limited by the characteristic time
scale of thermal effects, that does not allows to use signals with frequencies
larger than about hundreds of MHz at cryogenic temperatures Giazotto et al.
(2006). In this view, it is worth to discuss the role of an eventual direct
power injection into the gated device (driven, e.g., by a gate-superconductor
leakage current) that could produce an increase of the electronic temperature,
detrimental for device performance.

##  3 Non-thermal origin of supercurrent suppression in gated all-metallic
superconducting devices

###  3.1 SCPDs in a titanium gate-controlled transistor

In a current biased JJ the transition between the superconducting and the
normal state for fixed values of external parameters, e.g. temperature,
electric and magnetic field, is triggered by a phase slip events, i.e., local
random $2\pi$ jumps of the macroscopic quantum phase $\phi$ Bezryadin (2012).
The accidental nature of such events leads to a non-univocal definition of the
switching current, whose value is distributed according to the switching
current probability distribution (SCPD). The investigation of the SCPD of a JJ
is, therefore exploited to probe dynamics of the phase slips. Here, we discuss
an experiment, where a well-established technique is adopted to probe the
impact of gate voltage on the frequency of phase slips events in gate-
controlled metallic titanium (Ti) Josephson weak-links, with the conventional
theory Kurkijärvi (1972); Bezryadin and Goldbart (2010); Fulton et al. (1972).

The device chosen to study the evolution if the SCPD under gating action
consists of a titanium Dayem-bridge. Such JJs consist of $30$-nm-thick,
$10$-$\mu$m-long, $2.5$-$\mu$m-wide wire interrupted by a constriction. This
$30$-nm-thick, $150$-nm-long, $120$-nm-wide narrow structure was aligned with
a planar, $80$-nm-far, $140$-nm-wide metallic gate. The sample was fabricated
by means of single step electron beam lithography on a single-crystal sapphire
(Al2O3) with a typical resistivity larger than $10^{10}\ \Omega$/cm. The metal
was deposited by means of an electron beam evaporation at a rate of $1.2$
nm/s. Figure 6(a) shows a pseudo-color scanning electron micrograph.

Figure 6: (a) Pseudo-color scanning electron micrograph and bias scheme of a
representative Ti gate controlled transistor. The superconducting wire and the
Dayem bridge constriction are colored in orange and the gate electrode in
blue. (b) Evolution of the switching current $I_{S}$ as a function the gate
voltage. The switching current were acquired 50 times in order to obtain the
average value shown in the picture.

The device shows a normal state resistance $R_{N}\simeq 550\ \Omega$, a
switching current at 20 mK of $I_{S}=6.0\ \mu$A and a critical temperature
$T_{C}\simeq 310$ mK Puglia et al. (2020).

The dependence of $I_{S}$ Paolucci et al. (2018, 2019a, 2019c, 2019b); De
Simoni et al. (2018, 2019, 2020); Puglia et al. (2020, 2020) on the gate
voltage is showed acquiring the $I_{S}$ vs $V_{G}$ characteristics as a
function of bath temperature. Figure 6(b) shows that supercurrent vanishes for
$\left|V_{G}\right|\simeq 34$ V and such value appears to be independent from
the bath temperature. By increasing the values of the temperature
$I_{S}^{0}=I_{S}(V_{G}=0)$ lowers and a greater ineffectiveness range of the
gate voltage on $I_{S}$ is observed. This latter behavior is the same obtained
on Ti and Al superconducting gate-controlled devices Paolucci et al. (2018);
De Simoni et al. (2018).

To characterize the effect of the temperature on a superconducting Dayem-
bridges JJ, SCPDs were measured at different values of the thermal bath
temperature. The distributions were reconstructed drawing an histogram based
on $10^{4}$ switching current acquisitions for each value of the bath
temperature $T$.

Figure 7(a), (d), (c) shows the evolution of these so-called thermal-SCPDs in
a bath temperature range between $20$ and $300$ mK. The dependence of the
shape of thermal-SCPDs is analyzed through the conventional Kurkijärvi-Fulton-
Dunkleberger (KFD) theory Kurkijärvi (1972); Fulton and Dynes (1971) with a
fit procedure Puglia et al. (2020). First of all the different phase slip
regimes were identified thanks to the evolution of distribution standard
deviation $\sigma$ as a function of $T$. The evolution of the $\sigma$ vs $T$
curve Bezryadin (2012); Puglia et al. (2020) is flat at low temperature in the
Quantum Phase Slip (QPS) regime, linear in $T$ in Thermal Activated Phase Slip
(TAPS) regime, and decreasing for the Multiple Phase Slip (MPS) regime
Bezryadin et al. (2000); Bezryadin and Goldbart (2010); Bezryadin (2012);
Giordano (1988); Giordano and Schuler (1989); Kurkijärvi (1972); Fulton and
Dunkleberger (1974); Fulton and Dynes (1971). The temperature $T_{Q}$,
separating the QPS and the TAPS regime occurs at about $T\simeq 110$ mK while
the crossover between TAPS and MPS regimes appears for $T_{M}\simeq 160$ mK.

Figure 7: (a) SCPDs vs current $I$ in the QPS regime for temperatures between
$20$ and $90$ mK. The inset shows the evolution of the $\sigma$ vs $T$
characteristic in the QPS regime. (b) SCPDs vs current $I$ in the TAPS regime
for temperatures between $120$ and $150$ mK. The inset shows the evolution of
the $\sigma$ vs $T$ characteristic in the TAPS regime. (c) SCPDs vs current
$I$ in the MPS regime for temperatures between $160$ and $300$ mK. The inset
shows the evolution of the $\sigma$ vs $T$ characteristic in the MPS regime.
For each SCPD the total sampling number of $I_{S}$ is $10^{4}$. Dotted lines
represent the best-fit curves obtained with KFD model.

Although these devices show the conventional evolution of the phase slips
dynamic as a function of the temperature, the gate voltage drives the junction
in a different regime. Figure 8(a) shows vertically shifted SCPDs collected
for several values of the gate voltage at $T=20$ mK. In particular, the SCPDs
are overlapping for $V_{G}<8$ V, and a low current tail appears for
$8<V_{G}<14$ V. In addition, the distributions greatly widens for
$14<V_{G}<24$ V and for high gate voltage values, i.e. $V_{G}>24$ V, the SCPDs
turns out to narrow. In this electrostatically driven scenario, the $\sigma$
vs $V_{G}$ curve takes the place of the conventional $\sigma$ vs $T$
characteristic. As shown in Figure 8(b), $\sigma(V_{G})$ curve exhibits a
region where $\sigma$ is constant, thereby showing a marginal effect of the
gate voltage to the phase slips for small $V_{G}$ values. This behavior is
similar to the QPS regime. Therefore, we identify a crossover gate voltage
$V_{Q}\simeq 8$ V between the former and the Electric Activated Phase Slip
(EAPS) regime, where the $\sigma$ grows up to $\sim 200\ nA$ as the gate
voltage increases. Notably, $\sigma$ starts to increase at the same voltage at
which the switching current begins to be affected by the electric field. Such
evidence seems to suggest that, regardless of the origin of $I_{S}$
suppression, the latter is associated with a corresponding raise of the number
of phase slip events. Finally, for greater values of the gate voltage (i.e.,
$V_{G}>V_{E}\simeq 14$ V), the standard deviation$\sigma$ decreases and
saturates to $\sim 75\ nA$.

Figure 8: (a) SCPDs vs current $I$ in the EAPS regime with gate voltages
ranging from $0$ V to $21$ V. The inset shows the evolution of the $\sigma$ vs
$V_{G}$ characteristic in the EAPS regime. (b) SCPDs vs current $I$ in the MPS
regime with gate voltages ranging from $24$ V to $30$ V. The inset shows the
evolution of the $\sigma$ vs $V_{G}$ characteristic in the MPS regime. For
each SCPD the total sampling number of $I_{S}$ is $10^{4}$. The curves are
horizontally and vertically offset for clarity.

The starkly different behavior between thermal- and electric- SCPDs is clearly
displayed in Figure 9(a) where three $I_{S}$-matched couples of thermal and
electric distributions are plotted in the same graph for comparison. The
$I_{S}$-paired SCPDs highlight remarkably different widths and shapes. Such
behavior seems to stem from a gate-driven strong non-equilibrium state induced
in the junction. Concerning the standard deviation of the distributions, on
the one hand, the comparison between the $\sigma$ vs $I_{S}$ curves extracted
from the two thermal- and electric- SCPDs series, shown in Figure 9(b),
displays a qualitatively-similar behavior. On the other hand, the electric
driven SCPDs present a $\sigma$ on average around one order of magnitude
larger.

Figure 9: (a) $I_{S}$-matched SCPDs. Red/orange SCPDs were acquired with a
negligible electric field $V_{G}=0$ V at chosen temperatures and blue/green
SCPDs were obtained at a bath temperature of $T=20$ mK for different gate
voltages. The values of $I_{S}$ are respectively from left to right $2.2,\
2.8,\ 4.0\ \mu$A. (b) $\sigma$ vs $I_{S}$ characteristics obtained for thermal
(lower curve) and electric (upper curve) distributions.

Indeed, if we assume that the voltage-driven broadening of the SCPD is due to
an increase of the electronic temperature, e.g. a trivial Joule heating due to
a gate-DB current, we run into the absurdity of obtaining an electronic
temperature higher than the critical temperature of the superconductorGiazotto
et al. (2006); Bezryadin (2012). This observation reflects into the
impossibility to fit the gate-activated distributions by means of a
conventional KFD transform, since the obtained parameters would be
nonphysical. Therefore, these data demonstrate on the one hand a strong link
between phase slip events and electric field. On the other hand, they suggest
a non-thermal origin of the switching current suppression: the action of the
gate voltage drives the DB in a state whose description is incompatible with
that of a superconductor heated through a voltage-driven power injection at a
thermal steady state with electronic temperature higher than that of the
thermal bath.

###  3.2 Suspended titanium gate-controlled transistor

We have shown that it is not possible to describe the modification of the
SCPDs as a consequence of a trivial overheating. In other words, the effect of
the gate is unlikely that of driving the superconductor into a higher
electronic-temperature steady state by a mere power injection (driven by a
current injection). To further investigate the role of a possible injection of
current between the gate and superconducting channel, fully-suspended gated
superconducting nanobridges were tested. In conventional gated devices two
gate-channel charge transport mechanisms might be present: the diffusive
current injection through the substrate and the ballistic cold-electron field-
emission (CFE) through the vacuum. The suspended geometry permits to exclude
the first leaving the CFE as the only possible charge transport mechanism.

This experiment was performed on titanium gated suspended wires. The devices
consist of $70$-nm-thick and 1.7-$\mu$m-long nanobridge deposited on top of an
undoped $130$-nm-suspended crystalline InAs Li et al. (2014); Iorio et al.
(2019) nanowire set on two Ti/Au-coated ($5/15$ nm) pillars of cross-linked
insulating PMMA. The wires were gated through two $\sim$350-nm-wide, $40$-nm-
far symmetric Ti side electrodes. Titanium was evaporated at a rate of about
1.2 nm/s. Figure 10(a),(b) shows a 3D-sketch and a pseudo-color scanning
electron micrograph of a typical device.

Such a suspended nanojunction show four different superconducting transition
Rocci et al. (2020), that can be interpreted as the switch of the
superconducting banks for $I_{B}\simeq 1.8\ \mu$A and of the series of three
junction with switching current respectively $I_{S_{1}}\simeq 25\ nA$,
$I_{S_{2}}\simeq 150\ nA$ and $I_{S_{3}}\simeq 180\ nA$. The geometry of the
junction and the multi-step fabrication process induced to the existence of
such series of three junction in the nanobridge due to inhomogeneities of the
titanium layer thickness, covering a InAs nanowire. Moreover, the switching
current difference resides in a the variation of the cross-section of Ti film
coating the wire and on the inhomogeneous anti-proximization effect of the
superconducting film due to the bottom gold layer.

$I$ vs $V$ shifted characteristics at selected gate voltage from $-20$ to $20$
V of the bridge are shown in Figure 10(c) at a temperature of 20 mK. Figure
10(d),(e),(f) shows the evolution of the switching currents $I_{S_{i}}$ of the
three junctions as a function of the voltage gate, extracted from the $I$ vs
$V$ curves measured at several bath temperatures.

Figure 10: (a) SEM image of a representative device. (b) The wire (blue) is
connected in a typical four-wires configuration. The switching current $I_{S}$
was decreased via the application of the same gate voltage
$V_{G}=V_{S_{1}}=V_{S_{2}}$ to the symmetric gate electrodes (green). (c) Back
and forth current $I$ vs voltage $V$ curves for selected gate voltage $V_{G}$
values acquired with a bath temperature of $T=20$ mK. The curves are
horizontally shifted for clarity. Grey regions highlight the decrease of the
switching currents $I_{S_{1}}$, $I_{S_{2}}$ and $I_{S_{3}}$. (d) $I_{S_{1}}$
vs gate voltage $V_{G}$ acquired at different bath temperatures $T$. (e) and
(f) show the gate voltage $V_{G}$ dependence of $I_{S_{2}}$ and $I_{S_{3}}$,
respectively. Error bars represents the standard deviation of $I_{S_{i}}$ with
$i=1,\ 2,\ 3$ calculated over $25$ repetitions.

Notably, as the temperature increases, the ineffectiveness plateau shrinks.
The former observations is in starkly contrast with previous works De Simoni
et al. (2018); Paolucci et al. (2018, 2019a), in which the plateaux broadened
and the quenching gate voltages were constant as the temperatue increases. We
attribute such discrepancy to a smaller substrate-to-bridge thermal coupling
compared with devices located on a substrate. Certainly, independently of the
microscopic origin of the gate effect, the quenching of the switching current
could be connected to a sizable increase of the quasi-particles number in the
superconducting nanowire Puglia et al. (2020); Alegria et al. (2021). Such
enlargement is expected to be more efficient in a suspended geometry in which
the relaxation of quasiparticle excitations via electron-phonon interaction is
reduced compared to conventional devices. These results demonstrate that the
presence of an interface between the substrate and the superconducting
junction is not necessary for the gate effect to occur. This is a unequivocal
evidence against the hypothesis of a Joule heating origin of the supercurrent
suppression due to a diffusive current injected into the substrate.

###  3.3 Leakage current finite element method simulations

The suspended geometry experiment allows to eliminate any Joule overheating
conveyed to the nanowire through phonon coupling due to a leakage current
injected via the substrate. In this framework, a current flowing from the
bridge to the gate (and vice versa) could be possible only via the field
emission of cold electrons (CFE) in the vacuum. Such emission is expected to
arise because of the application of a strong electrostatic field between the
nanowire and the gate electrodes Simmons (1963); Fowler and Nordheim (1928).

To investigate the role of a possible field-emitted current in the $I_{S}$
suppression, the cold emitted current ($I_{FE}$) can be quantified with a
3-dimensional finite-element method simulations implemented on the same
geometry of the suspended titanium gate-controlled transistor, and then
compared with the measured leakage current $I_{L}$ Rocci et al. (2020).
$I_{FE}$ is calculated via the integration over the surface of the cathode,
i.e., the gate (wire) for negative (positive) $V_{G}$, of the Fowler Nordheim
tunnel current density that at the cathode is written as Fowler and Nordheim
(1928); Simmons (1963)

$\vec{J}_{FE}(\vec{F},h_{0})=\frac{2.2e^{3}}{8\pi
hh_{0}}\vec{F}^{2}\exp\left[-\frac{8\pi}{2.96he|\vec{F}|}(2m_{e})^{1/2}h_{0}^{3/2}\right],$

where $\vec{F}(x,y,z)$ is the modulus of the electrostatic field at the
surface of the cathode, $m_{e}$ is the electron mass, $e$ is the electron
charge, $h$ is the Plank’s constant, and $h_{0}=4.3$ eV is the work function
of titanium Wilson (1966). The electrostatic field vector
$\vec{F}(x,y,z;V_{G})$ was computed in the 3D vacuum space zone, around the
bridge and the side gates, via the Maxwell equation $\vec{F}=-\nabla
V(x,y,z;V_{G})$. The electric potential $V(x,y,z;V_{G})$ was calculated thanks
to the numerical integration of the Poisson equation
$\nabla^{2}V(x,y,z;V_{G})=0$ (see Figure 11(a)). As boundary condition, the
bridge and the gate electrode surfaces were simulated with perfect
equipotential conductor at $V=0$ and $V=V_{G}$, respectively.

The propagation of the electric field module $\vec{|E|}$ in the space obtained
for the entire 3D domain of the simulation is color-plotted in Figure 11(b)
from a top view ( X-Y ) for $V_{G}=-15$ V. The electrostatic field is confined
in the region between the titanium bridge and the lateral gate surfaces and it
quickly fades elsewhere, so not influencing the superconducting banks.

Figure 11: (a) Electric potential $V$ in a XY cut-plane which intersects the
suspended nanowire at half height. Applied gate voltage $V_{G}=15$ V. (b)
Color-plot showing the derived electric field amplitude $|F|$ distribution
from the solution of the electric potential. Values obtained for $V_{G}=-15$
V. The simulation shows that the electric field is extremely localized.

The electric field reaches the maximum intensity of $0.2\ GV/m$ in
correspondence of the center of the gate and it is localized near the side
gate surfaces. Moreover, $|\vec{F}(x,y,z)|$ drops more than one order of
magnitude between $500$ nm from the lateral edge of the gate electrode. Such a
field geometry let us to conclude that the banks are unlikely to be affected
by the gate voltage.

By solving the ballistic trajectories of the electron emitted by the
electrode, it is possible to compute the current density $|\vec{J}_{FE}|$ in
the region between the gate and the wire. Figure 12(a), (b) shows the induces
charge density $\rho$ and the current density $|\vec{J}_{FE}|$ evaluated on a
X-Y plane. It is worth to notice the extreme localization of the electrons in
a region of about $500$ nm centered on the electrodes that affect only a small
portion of the nanobridge. Such evidence prove that for CFE the the entire
number of the electrons emitted/absorbed from the wire is absorbed/emitted
from the gate.

Figure 12: a Color-plot of the induced charge density $\rho$. (b) Modulus of
the propagating current density $|J|$obtained solving the ballistic
trajectories of the emitted electrons. Values obtained for $\phi_{0}=4.3$ eV
and $V_{G}=-15$ V. The distribution of the current shows that the injection
mechanism influences only a $400$ nm section of the nano-bridge.

Finally, the integration of the current density $\vec{J}_{FE}$ over the
surface of the electrodes returns the current $I_{FE}$, a quantity that can be
directly compared with the gate-wire current measured in the experiment (shown
in Figure 13(a)) Rocci et al. (2020). Figure 13(b) shows the results of the
simulation for $I_{FE}$.

Figure 13: Comparison between the measured leakage current $I_{L}$ (orange
points) and the field emission current $\left(I_{FE}\right)$ (violet points)
as a function of the gate voltage. The latter quantity wad obtained
integrating numerically the Fowler-Nordheim current density $\vec{J}_{FE}$
with the titanium work function $\phi_{0}=4.3$ eV.

Notably, $I_{FE}$ is more than $20$ orders of magnitude smaller than the
greatest value of the gate-bridge $I_{L}$ measured in the experiment.
Furthermore, a current of about $10^{-40}$ A corresponds to the emission, on
average, of a single electron in $10^{28}$ years. Such emission rate was
computed consistently with an electrostatic field at the surface of the that
is not intense enough to start a true CFE current. As a matter of fact, cold-
emitted electrons typically needs an electrostatic field of the order of
$1-10$ GV/m Bhushan (2012), but in our situation the greatest electrostatic
field is one order of magnitude smaller, at most. Moreover, the simulation
shows an intrinsic asymmetry, of several orders of magnitude, of $I_{FE}$ when
$V_{G}\longrightarrow-V_{G}$ due to the geometry difference between gate and
wire. This seems to suggest that, if the field emitted current was the leading
mechanism in determining the $I_{S}$ suppression, a strongly asymmetric
behavior should be observed for positive and negative gate voltages. Such a
feature was never reported in experiments on gated metallic superconductors De
Simoni et al. (2018); Paolucci et al. (2018, 2019a).

###  3.4 Heating through single cold-electron field emission or absorption

If we admit that a certain number of electrons are emitted or absorbed by the
gate and absorbed/emitted by the wire the expected experimental phenomenology
should be totally different compared to those observed: an electron with an
energy of the order of $10$ eV, and ballistically hitting the junction going
through the vacuum, is expected to release its entire energy increasing
abruptly the system electronic temperature. A straightforward calculation for
the heat capacitance due to electronic contribution $C_{e}$ of a weak-link in
the normal-state is:

$C_{e}=\Omega\gamma T_{e}$

where $\Omega$ is the volume occupied by the junction, $\gamma$ is the
Sommerfeld constant for titanium, $T_{e}$ is the electronic temperature of the
weak-link.

The energy $E(V)$ released in the junction is directly proportional to the
acceleration gate voltage $V$ between the weak-link and the gate electrode:

$E\left(V\right)=qV,\ \ \ \ \ \ P\left(t\right)=E\ \delta\left(t\right)$

where $q$ is the electron charge, $\delta$ is Dirac’s delta distribution and
$P(t)$ is the power as a function of $t$. According to conventional theory of
the heat transport, the electronic temperature in the system is conveniently
described by the differential equation Giazotto et al. (2006), where $T_{B}$
is the lattice temperature:

$C_{e}\frac{\partial T_{e}}{\partial t}=P\left(t\right)\longrightarrow
T_{e}=\sqrt{\frac{2E}{\Omega\gamma}+T_{B}^{2}}$

We wish to stress that by this approach we obtain final electronic temperature
$T_{e}$ which is an underestimate of its real value in the weak-link since we
considered $C_{e}$ to be the normal-state heat capacitance, that is generally
exponentially greater than in the superconducting state Giazotto et al.
(2006). The, already mentioned, calculation demonstrates that one electron
accelerated with a $30$ V gate voltage and injected into a typical DB
constriction at $10$ mK of bath temperature would increase starkly its
electronic temperature $T_{e}$ that would reach a value that is more than $20$
times greater than its critical temperature ($T_{C}\simeq 500$ mK). Such a
conclusion permits us to make the statement that the heat generating from the
absorption of a field-emitted electron in the junction in incompatible with an
equilibrium state with a definite gate-tunable $T_{e}$ and $I_{S}$.

###  3.5 Continuous power injection

In principle, we can assume that, the repeated absorption of high-energy
electrons, leads the system to bouncing continuously between the normal and
superconducting states as shown in Figure 14.

Figure 14: Electronic temperature $T$ vs time $t$ of a mesoscopic
superconducting weak-link that periodically absorb electrons with energy of
the order 10 eV. The red horizontal line represents the critical temperature
of the superconductor. Each electron starkly increase the electronic
temperature of the system driving it in the normal state. $\tau$ is the
measurement time.

If this were the case, the bridge is suddenly driven into the dissipative
state for every electron absorbed. Then, the system relaxes back to the
dissipationless state. The recurrence of such events is due to the emission
rate of the electrons, and the super-to-normal relaxation time is typically
given by the electron-phonon coupling related time, that is of the order of
$\tau\simeq 1$ ns Giazotto et al. (2006). Such time scale is much smaller than
the typical integration time of our voltage measure setup ($\simeq 20$ ms). In
this scheme, in the time of a voltage $V$ vs current $I$ measurement, every
time an electron is absorbed by the junction, if the bias current $I$ is below
the $I_{R}$ the transition that lead to a finite resistance value is likely
too fast to be detected by our setup. On the contrary, if $I>I_{R}$, the
absorption of an electron by the constriction drives the system to the normal
state, and it should persist in such state until the flowing current $I$ is
lowered to $0$. Such consideration suggest that when electron are field-
emitted, $I_{R}$ and $I_{S}$ should always correspond. Since this does not
occur, we conclude that any hot-electron injection-related mechanism related
to field emission as the main cause of our experimental results, should be
excluded.

###  3.6 Unconventional sum rule

Another strong evidence against a trivial heating or a direct power-injection
origin of the supercurrent suppression comes from the evolution of the
switching current of a superconducting wire under the action of a pair of
lateral side gates. By means of a titanium Dayem bridge consisting of a double
gate flanked nanoconstrictions interrupting a Ti strip Paolucci et al.
(2019c), it was possible to asses the mutual influence and the spatial
extension of two opposite gate electrodes effect in the suppression of the
switching current. Figures 15(a),(b) show two color-plot of the normalized
switching current $I_{S}/I_{S}^{0}$ vs $V_{G_{1}}$ (X axis) and $V_{G_{2}}$ of
two representative devices $A$ and $B$. The difference of the two systems in
the complete quenching gate voltage $V_{G}^{C}$ was ascribed to the different
in the gate-DB distance ($\simeq 80$ nm for the sample $A$ and $\simeq 120$ nm
for the sample $B$). The observed square-like shape indicates the existence of
a voltage threshold $V_{th}$: when one of the two gates is biased above
$V_{th}$ the critical supercurrent is suppressed by a fraction which is not
dependent of the voltage applied to the other gate. In other word, the effect
of the two gates on $I_{S}$ are independent and no obvious sum rule exists
between the action of the two voltages. Such evidence suggests that the gate-
driven suppression of the supercurrent is likely related to a surface effect,
which affects non-locally superconductivity, i.e., once that the electric
field established at one of the surface of the superconductor overcomes a
critical value, its effect is propagated inside the superconducting body over
a distance at least comparable with the device width. Accordingly, with
previous works De Simoni et al. (2018) and theoretical calculations Ummarino
et al. (2017), the surface perturbation could affect the superconductor for a
thin depth of the order of the superconducting coherence length $\xi$.
Furthermore, the aforementioned behavior is hardly comparable with the picture
of a direct heat/power injection due to charge transport from/to the gate.
Indeed, in the latter case a sum rule for the total power $P_{sum}$ is
expressed as $V^{2}_{G_{1}}/R_{1}+V^{2}_{G_{2}}/R^{2}$, where $R_{1,2}$ are
the gate-superconductor leakage resistance respectively for the gate 1 and 2,
respectively.

Figure 15: (a) and (b)Effect of the electric field generated in a double gate
flanked Dayem bridge symmetric side gate electrodes. Contour-plots show the
normalized supercurrent as a function of the two gate voltages $V_{G_{1}}$ (x
axis) and $V_{G_{2}}$ (y axis) for two representative devices (A and B).

##  4 Summary and further research

Along this review we showed the electrostatic control of the superconducting
properties in several all-metallic Josephson weak-links: niobium and vanadium
Dayem bridges and titanium suspended wires (Section 2). On the one hand, we
focused on the technological application of the effect, demonstrating
supercurrent suppression on the material that represents the industrial
standard for superconducting electronics, the niobium. Moreover, the vanadium
Dayem bridge experiments showed the potential for electric signal
rectification of such geometry. On the other hand, we investigated the dynamic
of the phase slip in a titanium Dayem bridge JJ under the effect of an
electrostatic field. The results demonstrated that it is impossible to ascribe
the modifications of the shape of the SCPDs to a conventional heating effect.
In particular, in the framework of the established KFD theory, it was not
possible to interpret the width of the distribution with the usual parameters.
Furthermore, the experiment carried out with a suspended titanium wire
demonstrated that the presence of the substrate is not critical to the
occurrence of the effect. Such evidence confutes any possible contribution to
the superconductivity quenching due to the existence of an injection current
that flows in the substrate.

In the second part of this review, we faced the hypothesis of the thermal
origin of the electrostatic effect on BCS superconductors. Thanks to finite
element simulations performed on a system with the same geometry of the
suspended titanium transistor, we demonstrated that the cold field emission
cannot be a satisfactory explanation for the suppression of the supercurrent.
In fact, even assuming that single electrons are emitted from the gate and
absorbed by the junction, the local increase of the electronic temperature of
the system is incompatible with the superconducting state. Additionally, the
non-trivial summation rule of two side gate in the quenching of the
supercurrent is a further evidence that a trivial thermal effect is no able to
explain our unconventional gating effect.

To shed light on these experiments and to progress towards the understanding
of the origin of the effect a set of complementary experiments are required.
For example, SQUID microscopy could provide useful information on the
distribution of the supercurrent in the field-affected region of the
superconductor. In addition, scanning tunnelling spectroscopy and scanning
gate experiments are critical to investigate the spatial variation of the
superconducting gap. Moreover, radio-frequency-based experiments are crucial
to acquire information on the characteristic time scale of the effect and on
the role of the quasiparticle excitation in the quenching of the supercurrent.

Besides a theoretical model able to explain the observed phenomenology has not
been provided yet. From a technological point of view, the unconventional
field-effect promises to be suitable for a wide range of applications. We
already demonstrated the rectification properties of a Dayem bridge system and
more complex devices such as gate- controlled radiation detector Zmuidzinas
and Richards (2004); Gousev et al. (1994); Lösch et al. (2019), signal routers
and computational systems Likharev (2012); Mukhanov (2011) are at reach.

2

The authors acknowledge the European Union’s Horizon 2020 research and
innovation program under Grant Agreement No. 800923 (SUPERTED) for partial
financial support.

The authors declare no conflict of interest.

References

## References

  * Bardeen (1962) Bardeen, J.  Critical Fields and Currents in Superconductors.  Rev. Mod. Phys. 1962, 34, 667–681.  doi:black10.1103/RevModPhys.34.667.
  * De Simoni et al. (2018) De Simoni, G.; Paolucci, F.; Solinas, P.; Strambini, E.; Giazotto, F.  Metallic supercurrent field-effect transistor.  Nat. Nanotechnol. 2018, 13, 802–805.  doi:black10.1038/s41565-018-0190-3.
  * Paolucci et al. (2018) Paolucci, F.; De Simoni, G.; Strambini, E.; Solinas, P.; Giazotto, F.  Ultra-Efficient Superconducting Dayem Bridge Field-Effect Transistor.  Nano Lett. 2018, 18, 4195–4199.  doi:black10.1021/acs.nanolett.8b01010.
  * Paolucci et al. (2019a) Paolucci, F.; De Simoni, G.; Solinas, P.; Strambini, E.; Ligato, N.; Virtanen, P.; Braggio, A.; Giazotto, F.  Magnetotransport Experiments on Fully Metallic Superconducting Dayem-Bridge Field-Effect Transistors.  Phys. Rev. Appl. 2019, 11, 024061.  doi:black10.1103/PhysRevApplied.11.024061.
  * Paolucci et al. (2019b) Paolucci, F.; Vischi, F.; De Simoni, G.; Guarcello, C.; Solinas, P.; Giazotto, F.  Field-Effect Controllable Metallic Josephson Interferometer.  Nano Lett. 2019, 19, 6263–6269, [1904.08349].  doi:black10.1021/acs.nanolett.9b02369.
  * Paolucci et al. (2019c) Paolucci, F.; De Simoni, G.; Solinas, P.; Strambini, E.; Puglia, C.; Ligato, N.; Giazotto, F.  Field-effect control of metallic superconducting systems.  AVS Quantum Sci. 2019, 1, 016501.  doi:black10.1116/1.5129364.
  * Bours et al. (2020) Bours, L.; Mercaldo, M.T.; Cuoco, M.; Strambini, E.; Giazotto, F.  Unveiling mechanisms of electric field effects on superconductors by a magnetic field response.  Phys. Rev. Res. 2020, 2, 033353, [2003.01655].  doi:black10.1103/PhysRevResearch.2.033353.
  * De Simoni et al. (2019) De Simoni, G.; Paolucci, F.; Puglia, C.; Giazotto, F.  Josephson Field-Effect Transistors Based on All-Metallic Al/Cu/Al Proximity Nanojunctions.  ACS Nano 2019, 13, 7871–7876.  doi:black10.1021/acsnano.9b02209.
  * Virtanen et al. (2019) Virtanen, P.; Braggio, A.; Giazotto, F.  Superconducting size effect in thin films under electric field: Mean-field self-consistent model.  Phys. Rev. B 2019, 100, 224506.  doi:black10.1103/PhysRevB.100.224506.
  * Mercaldo et al. (2020) Mercaldo, M.T.; Solinas, P.; Giazotto, F.; Cuoco, M.  Electrically Tunable Superconductivity Through Surface Orbital Polarization.  Phys. Rev. Appl. 2020, 14, 034041.  doi:black10.1103/PhysRevApplied.14.034041.
  * Solinas et al. (2020) Solinas, P.; Amoretti, A.; Giazotto, F.  Schwinger effect in a Bardeen-Cooper-Schrieffer superconductor.  arXiv 2020, [2007.08323].
  * Alegria et al. (2021) Alegria, L.D.; Bøttcher, C.G.L.; Saydjari, A.K.; Pierce, A.T.; Lee, S.H.; Harvey, S.P.; Vool, U.; Yacoby, A.  High-energy quasiparticle injection into mesoscopic superconductors.  Nat. Nanotechnol. 2021, [2005.00584].  doi:black10.1038/s41565-020-00834-8.
  * Ritter et al. (2020) Ritter, M.F.; Fuhrer, A.; Haxell, D.Z.; Hart, S.; Gumann, P.; Riel, H.; Nichele, F.  A superconducting switch actuated by injection of high energy electrons 2020.  [2005.00462].
  * Puglia et al. (2020) Puglia, C.; De Simoni, G.; Giazotto, F.  Electrostatic Control of Phase Slips in Ti Josephson Nanotransistors.  Phys. Rev. Appl. 2020, 13, 054026.  doi:black10.1103/PhysRevApplied.13.054026.
  * Rocci et al. (2020) Rocci, M.; De Simoni, G.; Puglia, C.; Degli Esposti, D.; Strambini, E.; Zannier, V.; Sorba, L.; Giazotto, F.  Gate-Controlled Suspended Titanium Nanobridge Supercurrent Transistor.  ACS Nano 2020, 14, 12621–12628.  doi:black10.1021/acsnano.0c05355.
  * Likharev (2012) Likharev, K.K.  Superconductor digital electronics.  Phys. C Supercond. its Appl. 2012, 482, 6–18.  doi:black10.1016/j.physc.2012.05.016.
  * De Simoni et al. (2020) De Simoni, G.; Puglia, C.; Giazotto, F.  Niobium Dayem nano-bridge Josephson gate-controlled transistors.  Appl. Phys. Lett. 2020, 116, 242601.  doi:black10.1063/5.0011304.
  * Stromberg and Swenson (1962) Stromberg, T.F.; Swenson, C.A.  Negative Surface Free-Energy Effects in Superconducting Niobium.  Phys. Rev. Lett. 1962, 9, 370–374.  doi:black10.1103/PhysRevLett.9.370.
  * Finnemore et al. (1966) Finnemore, D.K.; Stromberg, T.F.; Swenson, C.A.  Superconducting Properties of High-Purity Niobium.  Phys. Rev. 1966, 149, 231–243.  doi:black10.1103/PhysRev.149.231.
  * Giazotto et al. (2006) Giazotto, F.; Heikkilä, T.T.; Luukanen, A.; Savin, A.M.; Pekola, J.P.  Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications.  Rev. Mod. Phys. 2006, 78, 217–274.  doi:black10.1103/RevModPhys.78.217.
  * Tinkham (2004) Tinkham, M.  Introduction to Superconductivity; Dover Publications: Mineola, 2004; p. 454.
  * Puglia et al. (2020) Puglia, C.; De Simoni, G.; Ligato, N.; Giazotto, F.  Vanadium gate-controlled Josephson half-wave nanorectifier.  Appl. Phys. Lett. 2020, 116, 252601.  doi:black10.1063/5.0013512.
  * Barone and Paternò (1982) Barone, A.; Paternò, G.  Physics and Applications of the Josephson Effect; Wiley, 1982\.  doi:black10.1002/352760278X.
  * Ullom and Bennett (2015) Ullom, J.N.; Bennett, D.A.  Review of superconducting transition-edge sensors for x-ray and gamma-ray spectroscopy.  Supercond. Sci. Technol. 2015, 28, 084003.  doi:black10.1088/0953-2048/28/8/084003.
  * Paolucci et al. (2021) Paolucci, F.; Ligato, N.; Germanese, G.; Buccheri, V.; Giazotto, F.  Fully Superconducting Josephson Bolometers for Gigahertz Astronomy.  Appl. Sci. 2021, 11, 746.  doi:black10.3390/app11020746.
  * Gol’tsman et al. (2001) Gol’tsman, G.N.; Okunev, O.; Chulkova, G.; Lipatov, A.; Semenov, A.; Smirnov, K.; Voronov, B.; Dzardanov, A.; Williams, C.; Sobolewski, R.  Picosecond superconducting single-photon optical detector.  Appl. Phys. Lett. 2001, 79, 705–707.  doi:black10.1063/1.1388868.
  * Ivanov et al. (2011) Ivanov, B.I.; Trgala, M.; Grajcar, M.; Il’ichev, E.; Meyer, H.G.  Cryogenic ultra-low-noise SiGe transistor amplifier.  Rev. Sci. Instrum. 2011, 82, 104705.  doi:black10.1063/1.3655448.
  * Oukhanski et al. (2003) Oukhanski, N.; Grajcar, M.; Il’ichev, E.; Meyer, H.G.  Low noise, low power consumption high electron mobility transistors amplifier, for temperatures below 1 K.  Rev. Sci. Instrum. 2003, 74, 1145–1146.  doi:black10.1063/1.1532539.
  * Clarke and Bragisnki (2004) Clarke, J.; Bragisnki, A.I.  The SQUID Handbook; Wiley: Weinheim, 2004.
  * Giazotto et al. (2010) Giazotto, F.; Peltonen, J.T.; Meschke, M.; Pekola, J.P.  Superconducting quantum interference proximitytransistor.  Nat. Phys. 2010, 6, 254–259.  doi:black10.1038/nphys1537.
  * Giazotto and Martínez-Pérez (2012) Giazotto, F.; Martínez-Pérez, M.J.  The Josephson heat interferometer.  Nature 2012, 492, 401–405, [1205.3353].  doi:black10.1038/nature11702.
  * Fornieri and Giazotto (2017) Fornieri, A.; Giazotto, F.  Towards phase-coherent caloritronics in superconducting circuits.  Nat. Nanotechnol. 2017, 12, 944–952.  doi:black10.1038/nnano.2017.204.
  * McCaughan (2018) McCaughan, A.N.  Readout architectures for superconducting nanowire single photon detectors.  Supercond. Sci. Technol. 2018, 31, 040501.  doi:black10.1088/1361-6668/aaa1b3.
  * McCaughan and Berggren (2014) McCaughan, A.N.; Berggren, K.K.  A Superconducting-Nanowire Three-Terminal Electrothermal Device.  Nano Lett. 2014, 14, 5748–5753.  doi:black10.1021/nl502629x.
  * Morpurgo et al. (1998) Morpurgo, A.F.; Klapwijk, T.M.; van Wees, B.J.  Hot electron tunable supercurrent.  Appl. Phys. Lett. 1998, 72, 966–968.  doi:black10.1063/1.120612.
  * Bezryadin (2012) Bezryadin, A.  Superconductivity in Nanowires: Fabrication and Quantum Transport; Wiley-VCH, 2012; p. 282.
  * Kurkijärvi (1972) Kurkijärvi, J.  Intrinsic Fluctuations in a Superconducting Ring Closed with a Josephson Junction.  Phys. Rev. B 1972, 6, 832–835.  doi:black10.1103/PhysRevB.6.832.
  * Bezryadin and Goldbart (2010) Bezryadin, A.; Goldbart, P.M.  Superconducting Nanowires Fabricated Using Molecular Templates.  Adv. Mater. 2010, 22, 1111–1121.  doi:black10.1002/adma.200904353.
  * Fulton et al. (1972) Fulton, T.A.; Dunkleberger, L.N.; Dynes, R.C.  Quantum Interference Properties of Double Josephson Junctions.  Phys. Rev. B 1972, 6, 855–875.  doi:black10.1103/PhysRevB.6.855.
  * Fulton and Dynes (1971) Fulton, T.; Dynes, R.  Switching to zero voltage in Josephson tunnel junctions.  Solid State Commun. 1971, 9, 1069–1073.  doi:black10.1016/0038-1098(71)90465-0.
  * Bezryadin et al. (2000) Bezryadin, A.; Lau, C.N.; Tinkham, M.  Quantum suppression of superconductivity in ultrathin nanowires.  Nature 2000, 404, 971–974.  doi:black10.1038/35010060.
  * Giordano (1988) Giordano, N.  Evidence for Macroscopic Quantum Tunneling in One-Dimensional Superconductors.  Phys. Rev. Lett. 1988, 61, 2137–2140.  doi:black10.1103/PhysRevLett.61.2137.
  * Giordano and Schuler (1989) Giordano, N.; Schuler, E.  Macroscopic quantum tunneling and related effects in a one-dimensional superconductor.  Phys. Rev. Lett. 1989, 63, 2417–2420.  doi:black10.1103/PhysRevLett.63.2417.
  * Fulton and Dunkleberger (1974) Fulton, T.A.; Dunkleberger, L.N.  Lifetime of the zero-voltage state in Josephson tunnel junctions.  Phys. Rev. B 1974, 9, 4760–4768.  doi:black10.1103/PhysRevB.9.4760.
  * Li et al. (2014) Li, Q.; Huang, S.; Pan, D.; Wang, J.; Zhao, J.; Xu, H.Q.  Suspended InAs nanowire gate-all-around field-effect transistors.  Appl. Phys. Lett. 2014, 105, 113106.  doi:black10.1063/1.4896105.
  * Iorio et al. (2019) Iorio, A.; Rocci, M.; Bours, L.; Carrega, M.; Zannier, V.; Sorba, L.; Roddaro, S.; Giazotto, F.; Strambini, E.  Vectorial Control of the Spin–Orbit Interaction in Suspended InAs Nanowires.  Nano Lett. 2019, 19, 652–657, [1807.04344].  doi:black10.1021/acs.nanolett.8b02828.
  * Simmons (1963) Simmons, J.G.  Generalized Formula for the Electric Tunnel Effect between Similar Electrodes Separated by a Thin Insulating Film.  J. Appl. Phys. 1963, 34, 1793–1803.  doi:black10.1063/1.1702682.
  * Fowler and Nordheim (1928) Fowler, R.H.; Nordheim, L.  Electron emission in intense electric fields.  Proc. R. Soc. London. Ser. A, Contain. Pap. a Math. Phys. Character 1928, 119, 173–181.  doi:black10.1098/rspa.1928.0091.
  * Wilson (1966) Wilson, R.G.  Vacuum Thermionic Work Functions of Polycrystalline Be, Ti, Cr, Fe, Ni, Cu, Pt, and Type 304 Stainless Steel.  J. Appl. Phys. 1966, 37, 2261–2267.  doi:black10.1063/1.1708797.
  * Bhushan (2012) Bhushan, B.  Encyclopedia of Nanotechnology; Springer Netherlands: Dordrecht, 2012; pp. 824–837.
  * Ummarino et al. (2017) Ummarino, G.A.; Piatti, E.; Daghero, D.; Gonnelli, R.S.; Sklyadneva, I.Y.; Chulkov, E.V.; Heid, R.  Proximity Eliashberg theory of electrostatic field-effect doping in superconducting films.  Phys. Rev. B 2017, 96, 064509.  doi:black10.1103/PhysRevB.96.064509.
  * Zmuidzinas and Richards (2004) Zmuidzinas, J.; Richards, P.  Superconducting detectors and mixers for millimeter and submillimeter astrophysics.  Proc. IEEE 2004, 92, 1597–1616.  doi:black10.1109/JPROC.2004.833670.
  * Gousev et al. (1994) Gousev, Y.P.; Gol’tsman, G.N.; Semenov, A.D.; Gershenzon, E.M.; Nebosis, R.S.; Heusinger, M.A.; Renk, K.F.  Broadband ultrafast superconducting NbN detector for electromagnetic radiation.  J. Appl. Phys. 1994, 75, 3695–3697.  doi:black10.1063/1.356060.
  * Lösch et al. (2019) Lösch, S.; Alfonsov, A.; Dobrovolskiy, O.V.; Keil, R.; Engemaier, V.; Baunack, S.; Li, G.; Schmidt, O.G.; Bürger, D.  Microwave Radiation Detection with an Ultrathin Free-Standing Superconducting Niobium Nanohelix.  ACS Nano 2019, 13, 2948–2955.  doi:black10.1021/acsnano.8b07280.
  * Mukhanov (2011) Mukhanov, O.A.  Energy-Efficient Single Flux Quantum Technology.  IEEE Trans. Appl. Supercond. 2011, 21, 760–769.  doi:black10.1109/TASC.2010.2096792.