# Ionic gating in metallic superconductors: A brief review

Erik Piatti<EMAIL_ADDRESS>Department of Applied Science and
Technology, Politecnico di Torino, I-10129 Torino, Italy

###### Abstract

Ionic gating is a very popular tool to investigate and control the electric
charge transport and electronic ground state in a wide variety of different
materials. This is due to its capability to induce large modulations of the
surface charge density by means of the electric-double-layer field-effect
transistor (EDL-FET) architecture, and has been proven to be capable of tuning
even the properties of metallic systems. In this short review, I summarize the
main results which have been achieved so far in controlling the
superconducting (SC) properties of thin films of conventional metallic
superconductors by means of the ionic gating technique. I discuss how the
gate-induced charge doping, despite being confined to a thin surface layer by
electrostatic screening, results in a long-range “bulk” modulation of the SC
properties by the coherent nature of the SC condensate, as evidenced by the
observation of suppressions in the critical temperature of films much thicker
than the electrostatic screening length, and by the pronounced thickness-
dependence of their magnitude. I review how this behavior can be modelled in
terms of proximity effect between the charge-doped surface layer and the
unperturbed bulk with different degrees of approximation, and how first-
principles calculations have been employed to determine the origin of an
anomalous increase in the electrostatic screening length at ultrahigh electric
fields, thus fully confirming the validity of the proximity effect model.
Finally, I discuss a general framework – based on the combination of ab-initio
Density Functional Theory and the Migdal-Eliashberg theory of
superconductivity – by which the properties of any gated thin film of a
conventional metallic superconductor can be determined purely from first
principles.

Cite this article as: E. Piatti Nano Ex. 2, 024003 (2021).

ionic gating, metallic superconductors, proximity effect, electrostatic
screening, thin films

##  I Introduction

The electric field effect is the modulation of the electronic properties of a
material by the application of a transverse electric field. Its most
influential application is the field-effect transistor (FET), which forms the
backbone of modern semiconductor electronics. The possibility to control the
superconducting (SC) state by means of the electric field effect has been an
open issue in condensed matter physics since the seminal experiments of Glover
and Sherrill [1] and Stadler [2] in the Sixties. In the former, one electrode
of a capacitor was replaced with either a tin or an indium SC film, and upon
charging the transition temperature $T_{\rm c}$ of the films was shifted at
most by $0.08$ and $0.07$ mK respectively, corresponding to a faint $\approx
0.002\%$ shift in the original $T_{\rm c}$ [1]. In the latter, the maximum
$T_{\rm c}$ shift was enhanced to $\approx 0.035\%$ by substituting the solid-
oxide dielectric with a ferroelectric in the asymmetric capacitor [2]. These
first results showed that controlling the SC phase of metallic superconductors
via the electric field effect was feasible but of negligible impact for device
applications, and the topic was largely abandoned for over thirty years. The
discovery of high-$T_{\rm c}$ superconductors led to a rekindled interest in
the topic, since the much smaller carrier density of the cuprates with respect
to metallic superconductors allowed a much more effective tunability. Examples
included an electric-field-induced 10 K $T_{\rm c}$ shift in YBa2Cu3O7-δ by
gating through ultra-high-$\kappa$ dielectric SrTiO3 [3], and a full
superconductor-insulator transition (SIT) induced in GaBa2Cu3O7-δ [4] by
ferroelectric gating. Similarly, the low carrier density of ultrathin Bi films
was also exploited to achieve a SIT by gating through SrTiO3 [5]. Still, the
range of applicability remained limited.

A turning point was achieved in the late 2000s with the introduction of the
ionic gating technique, which rapidly became a very popular tool to
investigate and control the electric transport and electronic ground state in
a wide variety of different materials [6, 7, 8, 9, 10, 11, 12, 13, 16, 14, 15,
17, 18, 19, 21, 20, 22, 23, 24, 25, 26, 27]. Ionic gating operates by
exploiting the ultrahigh electric fields that build up at a voltage-polarized
solid-electrolyte interface in the so-called electric double layer (EDL), made
possible by the sub-nanometric spacing between the two charged layers in the
EDL [29, 28]. When incorporated in an EDL-FET architecture, this allows
attaining modulations of the surface charge density in the device channel
which are often comparable to those occurring in metallic systems [13, 30,
31]. Since then, the ionic gating technique has found large success in tuning
the properties of low- and moderate-carrier density superconductors. The SC
properties of intrinsically-superconducting transition metal dichalcogenide
NbSe2 have been efficiently tuned by ionic gating both in the electrostatic
[32] and in the electrochemical [33, 34] regime. High-$T_{\rm c}$ copper
oxides are also strongly tunable by ionic gating, as showcased first in
La1-xSrxCuO4 [35] and then in YBa2Cu3O7-δ [36, 37], although in oxides in
general care has to be taken to distinguish between electrostatic field-effect
doping and field-induced changes in the oxygen stoichiometry due to the high
mobility of the oxygen vacancies [38, 39, 40, 41, 42, 43]. Among iron-based
superconductors, low-carrier-density compound FeSe has been shown to develop a
high-temperature ($T_{\rm c}\gtrsim 40$ K) SC phase upon surface electron
accumulation [44, 45, 46, 47], and bulk gate-induced lithiation [48] and
protonation [49]. While these results have been shown to be somewhat
transferable to FeSe1-xTex [50, 51] and (Li,Fe)OHFeSe [52], other iron-based
compounds such as BaFe2As2 have been shown to be much less compliant [53],
indicating that the ionic-gate tunability of iron-based superconductors is
strongly dependent on the specific properties of each compound.

In contrast, the applicability of the ionic gating technique to conventional
metallic superconductors has received significantly less attention, likely
owing to their high intrinsic carrier density preventing the observation of
large gate-induced changes to the intrinsic $T_{\rm c}$ or other strong
changes to the ground state of the gated material. In this short review, I
present the main results obtained so far concerning the tuning of the SC
properties of metallic superconductors from both the experimental and the
theoretical points of view, showing how the comparable simplicity of
conventional metallic superconductors with respect to cuprates or iron-based
compounds makes them an ideal playground to investigate the general physics of
gate-induced perturbations to the SC state.

##  II Ionic-gate-tuned superconductivity in niobium and niobium nitride films

Figure 1:  a, Sketch of an EDL-FET with a metallic active channel, biased with
a positive gate voltage $V_{\rm G}$. Gate, source (S) and drain (D) electrodes
are indicated, as well as the thickness of the surface charge-doped layer
$d_{\rm s}$. b, Resistance vs. temperature in the vicinity of $T_{\rm c}$ of a
8nm-thick Nb thin film for different values of $V_{\rm G}$. Reprinted from
Ref. 54, with the permission of AIP Publishing. c, Normalized resistance vs.
temperature in the vicinity of $T_{\rm c}$ of a 18.3nm-thick NbN film for
different values of $V_{\rm G}$. d, Normalized resistance vs. temperature in
the vicinity of $T_{\rm c}$ of a NbN film for the extremal values of $V_{\rm
G}$, as a function of the total film thickness. In both c and d the
temperature scale is referenced to the midpoint of the transition at $V_{\rm
G}=0$ (see Ref. [56] for details). c,d are reprinted with permission from Ref.
56. Copyright 2017 by the American Physical Society.

The working principle of the EDL-FET architecture for an ion-gated metallic
superconductor is sketched in Fig. 1a. The active channel is separated from
the gate counter-electrode by an electrolyte, which is usually either an ionic
liquid [54] or an ion gel [55, 56]. When a positive/negative voltage $V_{\rm
G}$ is applied to the gate electrode, cation/anions in the electrolyte
accumulate at the surface of the sample and lead to the accumulation/depletion
of electrons at its surface to maintain charge neutrality, building up the
EDL. If no bulk electrochemical interactions are triggered (such as
intercalation, protonation or formation of mobile vacancies), the application
of an ionic gate thus results in a strong charge doping which is confined to a
thin layer from the surface, with a thickness $d_{\rm s}$ that is set by the
electrostatic screening length of the system, while the remaining bulk remains
unperturbed [57]. The resulting modification to the electric transport
properties of both the normal and SC state in the active channel can then be
monitored by standard four-point resistivity measurements [54, 55, 56].

The first metallic superconductor to be investigated by means of the ionic
gating technique was niobium (Nb), a standard electron-phonon superconductor
with a bulk $T_{\rm c}=9.26$ K. Specifically, Choi et al. reported that the
$T_{\rm c}$ of niobium thin films could be reversibly modulated by gating
through an ionic liquid [54]. The corresponding resistive transitions for an
8nm-thick film are shown in Fig. 1b. From a qualitative standpoint, the
results mirrored the early findings in indium thin films [1], with the
accumulation of extra electrons in the system reducing the $T_{\rm c}$ and the
depletion of electrons enhancing it. The magnitude of the effect was of course
much larger thanks to the huge capacitance of the ionic gate, with the largest
$T_{\rm c}$ shifts being around $\sim 35$ mK, i.e. $\approx 0.8\%$ of the
intrinsic $T_{\rm c}\simeq 4.3$ K [54]. The shape of the resistive transition
was unaffected by the applied gate voltage, indicating that the ionic gate did
not introduce additional inhomogeneity in the sample. Notably, the gate-
induced modulations of $T_{\rm c}$ were volatile, i.e. fully reversible by
simply removing the applied gate voltage and warming the sample above the
melting point of the ionic liquid [54]. Additionally, it was also shown that
the critical current $I_{\rm c}$ was also tunable by the ionic gate with the
same behavior of $T_{\rm c}$, indicating that not only the onset of the SC
state, but also the superfluid density, could be enhanced and suppressed by
electron depletion and accumulation respectively [54].

A second metallic superconductor, niobium nitride (NbN), was investigated via
ionic gating shortly afterwards [55, 56]. With respect to pure Nb, NbN
features a higher bulk $T_{\rm c}\simeq 16$ K (making it more appealing for
applications such as bolometers and single-photon detectors) and a shorter
coherence length that allows thinning NbN films to few nanometers in thickness
without major losses in $T_{\rm c}$, all the while being mechanically robust
and chemically stable [58]. Furthermore, preliminary ab initio calculations
suggested that its $T_{\rm c}$ could be nearly doubled if a sufficiently large
number of electrons were to be added to its first unit cell from the surface
[55]. In contrast with these predictions, it was experimentally shown that the
gate-induced modulations of $T_{\rm c}$ in NbN thin films followed the same
qualitative behavior observed in Nb thin films: As shown in Fig. 1c, electron
depletion and accumulation enhanced and suppressed $T_{\rm c}$ respectively,
without affecting the shape of the transition, and the gate-induced shifts
were fully volatile [55, 56]. From a quantitative standpoint, a larger maximum
$T_{\rm c}$ shift $\sim 85$ mK was reported in 9.5nm-thick NbN films, which
however corresponded to a smaller $\approx 0.6\%$ relative modulation due to
the larger intrinsic $T_{\rm c}\simeq 13.7$ K with respect to that of Nb [56].

In both ion-gated Nb and NbN thin films, the rigid shift of the resistive
transition and the volatility of the $T_{\rm c}$ modulation were strongly
suggestive of a (mostly) electrostatic origin of the charge doping. On the
other hand, the observation of $T_{\rm c}$ suppressions in films of both
materials with a thickness much larger than the electrostatic screening length
is hardly compatible with a perturbation of the SC state occurring only at the
surface, since the underlying unperturbed bulk would always reach a zero-
resistance state at the same original $T_{\rm c}$. Indeed, for $T_{\rm c}$
suppressions to be detected in the resistive transition, a suppression of the
SC pairing must extend across the whole thickness of the film. In Ref. 54, the
observation of $T_{\rm c}$ suppressions in NbN films as thick as 120 nm could
not be justified in terms of pure surface charge doping, and the authors
suggested that electro-mechanical effects due to the frozen ionic liquid may
be responsible for the bulk modulations of SC. Conversely, corroborated by the
fact that analogous bulk-like suppressions of $T_{\rm c}$ had been reported in
gated indium films in the absence of any measurable strain [1], in Ref. 55 it
was suggested that the electrostatic screening responsible for the surface
confinement of the charge doping may deviate strongly in a superconductor from
that found in a normal metal. Additionally, in Ref. 56 it was shown that the
gate-tunability of $T_{\rm c}$ was strongly enhanced upon reducing of the
thickness of the NbN thin films by means of ion milling (Fig. 1c). This ruled
out a pure bulk origin of the modulation, which would be thickness-independent
in the case of $T_{\rm c}$ suppressions, as well as a purely surface-bound
phenomenon, which would be thickness-independent in the case of $T_{\rm c}$
enhancements, and hinted at a more complex interplay between surface charge
doping and bulk perturbation of the SC pairing [56].

##  III Surface-to-bulk coupling via superconducting proximity effect

Figure 2:  a, Scheme of an ion-gated superconducting thin film. The EDL is
composed by the Helmholtz layer of cations and the charge-doped surface layer
(dark blue region). The unperturbed bulk of the film is indicated in light
blue color. The input parameters of the proximity-effect model are indicated
for both layers (see text for details). b, Sketch of the SC order parameter
profile (i.e. the SC energy gap) across a superconductor(S)/normal metal(N)
junction, based on Ref. 63, for three values of the Gor’kov coherence length
$\xi$. Dashed black line is the bulk regime ($\xi\ll l_{\rm s},l_{\rm n}$)
where proximity effect is negligible. Dotted blue line is the mesoscopic
regime ($\xi\sim l_{\rm s},l_{\rm n}$) where a sizeable penetration of the
superfluid in the normal metal occurs. Solid red line is the Cooper limit
($\xi\gg l_{\rm s},l_{\rm n}$) described in the text.

The apparent contradiction between surface charge doping and bulk modulation
of the SC pairing can be solved by considering the coherent nature of the SC
condensate. Close to $T_{\rm c}$, the superfluid density is small [59] and the
electrostatic screening is dominated by unpaired electrons [60]. In the limit
of weak and moderate electric fields, the thickness $d_{\rm s}$ within which
the gate-induced charge doping is confined should then be comparable to the
Thomas-Fermi length, i.e. few Å at most [57]. Conversely, perturbations to the
SC order parameter cannot occur over length scales much smaller than the
coherence length, which in metallic superconductors is at least of the order
of few nanometers and can be as large as few microns [61]. As sketched in Fig.
2a, gated SC thin films can then be conceptualized from the electronic point
of view as a stack of a doped surface layer, where the electronic and
vibrational properties are changed by the applied electric field, and an
underlying bulk which remains unperturbed. The gate-induced perturbation to
the SC pairing can then be described as a coupling between these two layers by
means of SC proximity effect [56, 62]. This is a phenomenon that occurs at any
interface between a superconductor and a normal metal, caused by the
tunnelling of the Cooper pairs from the superconducting to the normal side
(Andreev reflection) [63]. This leakage of Cooper pairs into the metallic bank
can be thought of as an “evanescent tail” of the SC order parameter, and close
to the interface causes the appearance of a finite SC gap in the metal while
suppressing it in the superconductor [61, 63], as sketched in Fig. 2b. If the
thicknesses of the two layers are small enough (Cooper limit [64]), as is
often the case in gated thin films [54, 56, 62], the interface can be
described as a single superconductor with an effective electron-phonon
coupling constant $\langle\lambda\rangle$ that is a weighted average of the
coupling constants in the normal ($\lambda_{\rm n}$) and superconducting
($\lambda_{\rm s}$) layers [61, 63].

Following the strong-coupling BCS theory for proximity effect [65], in Ref. 56
it was proposed that the $T_{\rm c}$ of a gated thin film could be obtained
by:

$T_{\rm
c}=\frac{\Theta}{1.45}\exp\left(-\frac{1+\left\langle\lambda\right\rangle}{\left\langle\lambda\right\rangle-\mu^{*}}\right)$
(1)

with

$\langle\lambda\rangle=\frac{\lambda_{s}N_{\rm s}d_{\rm s}+\lambda_{\rm
b}N_{\rm b}d_{\rm b}}{N_{\rm s}d_{\rm s}+N_{\rm b}d_{\rm b}}.$ (2)

where the subscripts $s$ and $b$ referred to the surface and bulk layers of
the film, $N_{\rm s,b}$ are the densities of states (DOS) at the Fermi level,
and $d_{\rm s,b}$ are the thicknesses of the layers. Of course $d_{\rm
s}+d_{\rm b}=t$ with $t$ the total thickness of the film. It was then assumed
that $\Theta$ (the representative temperature of the phonon spectrum) and
$\mu^{*}$ (the Coulomb pseudopotential) were unchanged in the doped surface
layer, and that $T_{c}$ was modulated by charge doping mainly by shifting the
Fermi level $E_{\rm F}$: This in turn changes the DOS ratio $N_{s}/N_{b}$,
calculated via density functional theory (DFT), so that
$\lambda_{s}=\lambda_{b}\cdot N_{s}/N_{b}$ [56]. Despite these rough
approximations, the experimentally-measured shifts of the $T_{\rm c}$ of gated
thin NbN films as a function of both the induced charge density and of the
film thickness could be accurately reproduced with the proximity effect model
[56]. These findings showed that, even when the gate-induced electric field is
confined in a thin layer at the surface by electrostatic screening, the
coherent nature of the SC condensate forces the gate-induced perturbation to
the SC state to extend in a region much larger than a single unit cell [56].
Indeed, critical-current measurements in gated mesoscopic samples of metallic
superconductors later showed that, at temperatures much lower than $T_{\rm
c}$, the gate-induced suppressions of the SC state extended even further
inside the bulk, approaching the London penetration depth of the material [66,
67, 68]. In these latter cases, however, it must be noted that the source of
the suppression appears to be unrelated with charge doping (which is
negligible due to the low capacitance of gating through a vacuum) and is still
debated in the literature, with some authors suggesting that the penetration
of the electric field may itself be detrimental to mesoscopic superconductors
[66, 67], while others propose instead that the SC state is being destroyed by
surface injection of high-energy electrons tunnelling from the gate [68, 69].
Critical current enhancements have also been recently reported in gated NbN
nanobridges [70].

##  IV Anomalous electrostatic screening at ultrahigh electric fields

Figure 3:  a, Thickness of the charge-doped surface layer (one atomic layer
$\approx 2.2$ Å) vs. the induced charge density per unit surface in gated NbN
thin films. Filled circles are the data from Ref. 56 for different film
thicknesses. Solid black line is the result of the non-linear Thomas-Fermi
model [56, 57]. Hollow circles are the results from the results from the DFT
calculations in Ref. 75. Dashed blue line marks the thickness of one unit
cell. b, Screening charge contributions per unit surface due to the induced
($\Delta n_{\rm 2D}$) and displaced ($\Delta n_{\rm disp}$) charge densities,
together with their ratio $\Delta n_{\rm 2D}/\Delta n_{\rm disp}$, as a
function of $\Delta n_{\rm 2D}$. The solid lines act as guides to the eye. c,
Ball-and-stick model of the NbN lattice, and color maps of the screening
charge in a gated NbN slab for two values of $\Delta n_{\rm 2D}$ ($1\cdot
10^{14}$ and $1\cdot 10^{15}$ cm-2). The color map is a linear scale between
$\pm 30\%$ of the maximum value in the surface accumulation layer. b,c are
reprinted from Ref. 75, with permission from Elsevier.

The good agreement between the experimental results on NbN thin films and the
strong-coupling BCS model for the proximity effect in Ref. 17 also allowed to
map the thickness of the electronically-perturbed surface layer $d_{\rm s}$
(i.e. the screening length of the electric field) as a function of the induced
charge density and of the film thickness. As shown in Fig. 3a (filled
circles), $d_{\rm s}$ was found to be independent on the film thickness and
smaller than one unit cell at low induced charge density [56]. These two
findings are important checks for the consistency of the model, since on the
one hand electrostatic screening is a purely surface property that does not
depend on the thickness of a material as long as the latter is larger than the
screening length [57], and on the other hand a good agreement is found with
the standard Thomas-Fermi theory of screening ($d_{\rm TF}\sim 2$ Å in NbN
[56]) which is valid in the limit of low electric fields [57]. Surprisingly,
however, $d_{\rm s}$ was found to strongly increase at large induced charge
density (thus, at large gate electric fields), extending for over 5 unit cells
for the highest induced values of charge doping [56], in contrast with the
simple Thomas-Fermi picture in which the thickness of a surface accumulation
layer is expected to decrease strongly at the increase of the electric field,
due to the gate potential well becoming deeper and narrower [57, 71]. This was
the first time that such an increase in the electrostatic screening length was
reported in a conventional metal, although anomalously large thicknesses of
the surface accumulation layer had previously been reported in ion-gated
YBa2Cu3O7 [72] and SrTiO3 [73]. Notably, a very similar behavior of $d_{\rm
s}$ vs. the induced charge density was later reported in SrTiO3 gated with a
standard solid dielectric [74], ruling out that such a phenomenon could be
associated to ion-gated surfaces only and suggesting that this would be a
general behavior of any gated material once sufficiently large electric fields
are achieved.

This unexpected feature could be partially accounted for by including non-
linear corrections to the standard Thomas-Fermi screening model (solid black
line in Fig. 3a), but even this extended model breaks down at larger electric
fields [56]. A more complete theoretical description was given in Ref. 75 by
employing a novel DFT method [76, 77] that treats the gate-induced charge
doping more accurately than a uniform background charge [78, 18, 27] by fully
accounting for the gate electric field of the FET configuration in a self-
consistent way [79, 80, 81, 82, 83, 84, 85]. The DFT calculations in the FET
configuration accurately reproduced the doping-dependence of the thickness of
the surface accumulation layer (hollow red circles in Fig. 3a), although they
became too computationally costly to investigate the largest values of induced
charge density [75]. Furthermore, a crucial physical insight on the origin of
the breakdown of the Thomas-Fermi approximation could be obtained by
considering the spatial dependence of the screening charge for increasing
field-induced electron doping.

As shown in Fig. 3c, the total screening charge was composed of two main
contributions, i.e. the accumulation layer where the extra induced carriers
were confined, and an additional polarization of the pre-existent electron
density [75]. The first contribution – analogous to the Thomas-Fermi screening
prediction – is always localized within the first atomic layer and dominates
the total screening charge at low and moderate fields. The second contribution
consists instead in the formation of charge dipoles arising from the
polarization of the pristine atomic orbitals, and extends for an increasing
number of atomic layers from the surface as the gate electric field is
increased, eventually reaching well beyond the first unit cell from the
surface. This displaced charge density was found to be negligible at low and
moderate applied fields, but became comparable to that confined within the
accumulation layer upon the application of sufficiently large electric fields
[75], as shown in Fig. 3b.

The findings reported in Ref. 75 thus fully confirmed the validity of the
proximity-effect model developed in Ref. 56 for how the gate-induced surface
perturbation of the electronic properties of a thin film of a metallic
superconductor results in a bulk perturbation of its SC properties.
Additionally, they also firmly established that a proper treatment beyond the
Thomas-Fermi approximation is mandatory to describe the electrostatic
screening in the presence of the ultrahigh electric fields typical of the
ionic gate technique, even in the case of conventional metallic systems.

##  V Eliashberg theory of gated superconducting films

Figure 4:  a, Electron-phonon spectral function vs. frequency for Pb, for
different values of charge doping per unit cell $x$. All curves are shifted
for clarity by 1. b, Corresponding electron-phonon coupling $\lambda$ (green
diamonds) and representative phonon frequency $\omega_{\rm ln}$ (brown
pentagons) vs. $x$. c, Corresponding SC critical temperature $T_{\rm c}$ vs.
$x$ in the absence of any proximity effect. d, Charge doping per unit cell $x$
and charge-doped surface layer thickness $d_{\rm s}$ vs. induced charge
density per unit surface used as input for the proximity Eliashberg equations
in Ref. 62. e, $T_{\rm c}$ vs. induced charge density per unit surface, for
different total thicknesses of the gated Pb film. “Direct” and “indirect”
gate-tuning regimes are highlighted in d,e (see text for details). All lines
in b-e are guides to the eye. All panels are reprinted with permission from
Ref. 62. Copyright 2017 by the American Physical Society.

Despite its success in describing the observed behavior of gated NbN thin
films, the proximity-effect model developed in Ref. 56 described the
modulations to the SC state purely in terms of doping-induced changes to the
electronic properties of the system, namely the density of states at the Fermi
level. While this was proven to be a good approximation for NbN, in general
field-effect doping affects both the electronic [76, 77] and the vibrational
[80, 81, 85] properties of a gated material, and a complete model must account
for both contributions. This challenge was tackled in Ref. 62 by developing a
method to determine the $T_{\rm c}$ of any gated thin film of a single-band
metallic superconductor purely from first principles, by combining DFT
calculations with the full Migdal-Eliashberg theory of proximity effect. In
this method, the electron-phonon spectral function $\alpha^{2}F(\omega)$ of
the gated material, which describes its frequency-dependent electron-phonon
coupling, is determined as a function of charge doping by DFT calculations.
The $\alpha^{2}F(\omega)$ of the doped surface layer and that of the
unperturbed bulk are then used as input for the proximity Migdal-Eliashberg
equations, which allow to determine $T_{\rm c}$ – and, in principle, also
other SC properties – for the complete structure with arbitrary thickness. If
the DFT calculations are performed using a uniform doping [78, 18, 27, 62], a
selected thickness of the perturbed surface layer must also be included as an
input parameter. Conversely, if the DFT calculations are performed with in the
FET geometry [80, 81, 85], the model has no free parameters except for the
total thickness of the film and the unperturbed value of $\mu^{*}$. The
Eliashberg formalism also allows to directly probe whether the gate-induced
tuning of the SC state is mostly caused by a modulation of the electron-phonon
coupling $\lambda$ or the representative phonon frequency $\omega_{\rm ln}$,
since they can both be easily obtained from the $\alpha^{2}F(\omega)$ [62]:

$\lambda=2\int_{0}^{+\infty}\frac{\alpha^{2}F(\omega)}{\omega}d\omega$ (3)

and

$\omega_{\rm
ln}=\exp\left[\frac{2}{\lambda}\int_{0}^{+\infty}\ln(\omega)\frac{\alpha^{2}F(\omega)}{\omega}d\omega\right]$
(4)

and their doping-dependence compared to that of $T_{\rm c}$.

In Ref. 62 this method was benchmarked by calculating the $T_{\rm c}$ of gated
lead (Pb) thin films, as a function of the thicknesses of both the film and
the perturbed surface layer. It was first shown that, unlike in In, Nb and NbN
thin films, but analogously to Sn thin films, electron accumulation lead to an
increase in $T_{\rm c}$ (see Fig. 4c). This increase was driven by a strong
softening of the phonon modes, as evidenced by the doping-dependence of the
$\alpha^{2}F(\omega)$ (Fig. 4a), which strongly increased $\lambda$ while only
weakly reducing $\omega_{\rm ln}$ (Fig. 4b) [62]. Most importantly, it was
found that proximity effect is strongly detrimental to the $T_{\rm c}$
tunability achieved by field-effect doping: In the absence of an unperturbed
bulk layer, the calculations indicated that the $T_{\rm c}$ of Pb could be
increased by nearly 2 K for a doping of 0.4 electrons per unit cell. In a 5nm-
thick film this increase was already reduced by $\sim 90\%$, and by $\sim
99.4\%$ in a 40nm-thick film [62].

Using a realistic dependence of the screening length on the induced charge
density, mirroring the one found in NbN (see Fig. 4d), it was also shown that
the mechanism of $T_{\rm c}$ tuning of a superconducting film could be
separated in two regimes [62]: At low electric fields, where Thomas-Fermi
screening holds and the screening length is nearly constant, $T_{\rm c}$ is
tuned by field-effect doping directly affecting the $\alpha^{2}F(\omega)$.
Whereas at high electric fields, where the Thomas-Fermi model breaks down and
the screening length increases with the electric field, $\alpha^{2}F(\omega)$
remains nearly unchanged and $T_{\rm c}$ is tuned via proximity effect only,
via the change in the thickness of the perturbed surface layer. As shown in
Fig. 4e, the calculations indicated that this second, indirect tuning allows
attaining nearly the same $T_{\rm c}$ shift achieved in the absence of the
unperturbed bulk layer, but requires nearly 20 times the amount of induced
charge density per unit surface [62]. In both regimes, increasing the film
thickness dramatically reduces the tunability of $T_{\rm c}$. As such, the
findings reported in Ref. 62 clearly indicated how ultrathin films are
mandatory to obtain sizeable modulations of $T_{\rm c}$ in metallic
superconductors, while also reducing the required values of induced charge
density – thus requiring smaller gate voltages to drive the EDL-FETs, in turn
allowing to much more easily maintain the electrostatic operation of the ionic
gate [55, 56]. These results are consistent with the experimental findings
reported in gated NbSe2, where the $T_{\rm c}$ of encapsulated bilayers could
be electrostatically tuned by nearly $\approx 40\%$ of its original value
[32].

Beyond the benchmark on gated Pb thin films, which is theoretically well-
established but for which no experimental data are yet available, the
proximity-Eliashberg method allowed to quantitatively reproduce the original
$T_{\rm c}$ shift observed in gated indium films [86], giving a solid
theoretical explanation to a problem which had otherwise remained unsolved for
nearly fifty years. Furthermore, an extension of this method to two-band
metallic superconductor MgB2 was later developed in Ref. 87, thus overcoming
the strong assumption of the single-band approximation.

##  VI Summary and outlook

In this short review, I have discussed the non-trivial way in which ionic
gating can affect the properties of thin films of conventional metallic
superconductors. Despite the efficient electrostatic screening of metallic
systems confining the gate-induced charge doping to a thin layer at the
surface of the material, a bulk-like modulation of the superconducting
properties is systematically observed across multiple independent experiments.
This coupling of the quasi-2D perturbation of the electronic structure to the
long-range “bulk” modulation of the SC condensate stems from the intrinsically
coherent nature of the latter, which cannot be perturbed over length scales
smaller than its coherence length. I have examined how this coupling can be
described in terms of a proximity effect between the perturbed surface layer
and the unperturbed bulk, and discussed how gated thin films of metallic
superconductors can be modelled to different degrees of approximation. I have
discussed how both experimental findings and ab-initio calculations allowed to
unveil an anomalous increase of the electrostatic screening length at the
ultra-high electric fields attainable via the ionic gating technique, and
reviewed a general framework by which the superconducting properties of any
gated thin film of a conventional metallic superconductor can be determined
purely from first principles.

A few promising avenues for future investigations may also be suggested. Since
the proximity-effect induced “smearing” of the modulations to the SC pairing
is a general phenomenon occurring in any gated superconductor, it could hinder
the possibility to obtain large $T_{\rm c}$ shifts in any film significantly
thicker than the screening length. It is then pivotal to maximize the
tunability of the SC state by minimizing the influence of the bulk, such as by
employing ultrathin films of metallic superconductors, or by downscaling the
devices in more than one direction and investigating gated nanowires,
nanobeams or nanoconstrictions, as evidenced by recent experimental findings.
It is also necessary to consider device geometries suitable for the
measurement of gated SC properties other than the critical temperature, such
as the critical current (related to the superfluid density) and the
superconducting energy gap. It may also be beneficial to expand the list of
metallic superconductors to which ionic gating can be applied, which is often
hampered by susceptibility to irreversible electrochemical interactions with
the electrolyte: In this context, the development of a scalable encapsulation
technique that would preserve a high gate capacitance would be needed.

Finally, in the perspective of potential applications the capability to
operate the gate at temperatures close or lower than the critical temperature
of the superconducting film would be highly desirable, demanding further
investigations towards the development of electrolytes with lower melting
point, or highly reliable ferroelectric gates. This is because the necessity
to warm up the devices above their transition temperature to change the ionic-
gate polarization limits applications to reconfigurable SC circuits and
devices such as resonators [88]. However, attaining sizeable gate modulations
below their transition temperature would open up a plethora of tantalizing
applications, including low-dissipation logic and memory elements [70], novel
architectures for qubits, interferometers, amplifiers and photodetectors [67],
as well as magnetometers and heat-mastering systems [89].

## References

  * [1] R. E. Glover and M. D. Sherrill. Changes in Superconducting Critical Temperature Produced by Electrostatic Charging. Phys. Rev. Lett. 5, 248 (1960).
  * [2] H. L. Stadler. Changing Properties of Metals by Ferroelectric Polarization Charging. Phys. Rev. Lett. 14, 979 (1965).
  * [3] J. Mannhart, J. Ströbel, J. G. Bednorz, and Ch. Gerber. Large electric field effects in YBa2Cu3O7-δ films containing weak links. Appl. Phys. Lett. 62, 630 (1993).
  * [4] C. H. Ahn, S. Gariglio, P. Paruch, T. Tybell, L. Antognazza, J.-M. Triscone. Electrostatic Modulation of Superconductivity in Ultrathin GdBa2Cu3O7-x Films. Science 284, 1152-1155 (1999).
  * [5] K. A. Parendo, K. H. Sarwa B. Tan, A. Bhattacharya, M. Eblen-Zayas, N. E. Staley, and A. M. Goldman. Electrostatic Tuning of the Superconductor-Insulator Transition in Two Dimensions. Phys. Rev. Lett. 94, 197004 (2005).
  * [6] J. T. Ye, S. Inoue, K. Kobayashi, Y. Kasahara, H. T. Yuan, H. Shimotani, and Y. Iwasa. Liquid-gated interface superconductivity on an atomically flat film. Nat. Mater. 9, 125-128 (2010).
  * [7] J. T. Ye, Y. J. Zhang, R. Akashi, M. S. Bahramy, R. Arita, and Y. Iwasa. Superconducting Dome in a Gate-Tuned Band Insulator. Science 338, 1193-1196 (2012).
  * [8] S. Jo, D. Costanzo, H. Berger, and A. F. Morpurgo. Electrostatically Induced Superconductivity at the Surface of WS2. Nano Lett. 15, 1197-1202 (2015).
  * [9] W. Shi, J. T. Ye, Y. Zhang, R. Suzuki, M. Yoshida, J. Miyazaki, N. Inoue, Y. Saito, and Y. Iwasa. Superconductivity Series in Transition Metal Dichalcogenides by Ionic Gating. Sci. Rep. 5, 12534 (2015).
  * [10] Y. Yu, F. Yang, X. F. Lu, Y. J. Yan, Y.-H. Cho, L. Ma, X. Niu, S. Kim, Y.-W. Son, D. Feng, S. Li, S.-W. Cheong, X. H. Chen, and Y. Zhang. Gate-tunable phase transitions in thin flakes of 1T-TaS2. Nat. Nanotechnol. 10, 270-276 (2015).
  * [11] Y. Saito and Y. Iwasa. Ambipolar Insulator-to-Metal Transition in Black Phosphorus by Ionic-Liquid Gating. ACS Nano 9, 3192-3198 (2015).
  * [12] R. S. Gonnelli, F. Paolucci, E. Piatti, K. Sharda, A. Sola, M. Tortello, J. R. Nair, C. Gerbaldi, M. Bruna, and S. Borini. Temperature Dependence of Electric Transport in Few-layer Graphene under Large Charge Doping Induced by Electrochemical Gating. Sci. Rep. 5, 9554 (2015).
  * [13] L. J. Li, E. C. T. O’Farrel, K. P. Loh, G. Eda, B. Özyilmaz, and A. H. Castro Neto. Controlling many-body states by the electric-field effect in a two-dimensional material. Nature 529, 185-189 (2016).
  * [14] D. Ovchinnikov, F. Gargiulo, A. Allain, D. J. Pasquier, D. Dumcenco, C.-H. Ho, O. V. Yazyev, and A. Kis. Disorder engineering and conductivity dome in ReS2 with electrolyte gating. Nat. Commun. 7, 12391 (2016).
  * [15] E. Piatti, S. Galasso, M. Tortello, J. R. Nair, C. Gerbaldi, M. Bruna, S. Borini, D. Daghero, and R. S. Gonnelli. Carrier mobility and scattering lifetime in electric double-layer gatedfew-layer graphene. Appl. Surf. Sci. 395, 37-41 (2017).
  * [16] Y. Wang, J. Xiao, H. Zhu, Y. Li, Y. Alsaid, K. Y. Fong, Y. Zhou, S. Wang, W. Shi, Y. Wang, A. Zettl, E. J. Reed, and X. Zhang. Structural phase transition in monolayer MoTe2 driven by electrostatic doping. Nature 550, 487-491 (2017).
  * [17] E. Piatti, Q. Chen, and J. Ye. Strong dopant dependence of electric transport in ion-gated MoS2. Appl. Phys. Lett. 111, 013106 (2017).
  * [18] R. S. Gonnelli, E. Piatti, A. Sola, M. Tortello, F. Dolcini, S. Galasso, J. R. Nair, C. Gerbaldi, E. Cappelluti, M. Bruna, and A. C. Ferrari. Weak localization in electric-double-layer gated few-layer graphene. 2D Mater. 4, 035006 (2017).
  * [19] J. Zeng, E. Liu, Y. Fu, Z. Chen, C. Pan, C. Wang, M. Wang, Y. Wang, K. Xu, S. Cai, X. Yan, Y. Wang, X. Liu, P. Wang, S.-J. Liang, Y. Cui, H. Y. Hwang, H. Yuan, and F. Miao. Gate-induced interfacial superconductivity in 1T-SnSe2. Nano Lett. 18, 1410-1415 (2018).
  * [20] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang. Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2. Nature 563, 94-99 (2018).
  * [21] E. Piatti, D. De Fazio, D. Daghero, S. R. Tamalampudi, D. Yoon, A. C. Ferrari, and R. S. Gonnelli. Multi-Valley Superconductivity in Ion-Gated MoS2 Layers. Nano Lett. 18, 4821-4830 (2018).
  * [22] E. Piatti, Q. Chen, M. Tortello, J. Ye, and R. S. Gonnelli. Possible charge-density-wave signatures in the anomalous resistivity of Li-intercalated multilayer MoS2. Appl. Surf. Sci. 461, 269-275 (2018).
  * [23] Z. Wang, T. Zhang, M. Ding, B. Dong, Y. Li, M. Chen, X. Li, J. Huang, H. Wang, X. Zhao, Y. Li, D. Li, C. Jia, L. Sun, H. Guo, Y. Ye, D. Sun, Y. Chen, T. Yang, J. Zhang, S. Ono, Z. Han, and Z. Zhang. Electric-field control of magnetism in a few-layered van der Waals ferromagnetic semiconductor. Nat. Nanotechnol. 13, 554-559 (2018).
  * [24] E. Piatti, F. Galanti, G. Pippione, A. Pasquarelli, and R. S. Gonnelli. Towards the insulator-to-metal transition at the surface of ion-gated nanocrystalline diamond films. Eur. Phys. J. Spec. Top. 228, 689-696 (2019).
  * [25] X. Ren, Y. Wang, Z. Xie, F. Xue, C. Leighton, and C. D. Frisbie. Gate-Tuned Insulator-Metal Transition in Electrolyte-Gated Transistors Based on Tellurene. Nano Lett. 19, 4738-4744 (2019).
  * [26] E. Piatti, D. Romanin, D. Daghero and R. S. Gonnelli. Two-dimensional hole transport in ion-gated diamond surfaces: A brief review. Low Temp. Phys. 45, 1143-1155 (2019).
  * [27] E. Piatti, A. Pasquarelli, and R. S. Gonnelli. Orientation-dependent electric transport and band filling in hole co-doped epitaxial diamond films. Appl. Surf. Sci. 528, 146795 (2020).
  * [28] T. Fujimoto and K. Awaga. Electric-double-layer field-effect transistors with ionic liquids. Phys. Chem. Chem. Phys. 15, 8983-9006 (2013).
  * [29] K. Ueno, H. Shimotani, H. Yuan, J. Ye, M. Kawasaki, and I. Iwasa. Field-Induced Superconductivity in Electric Double Layer Transistors. J. Phys. Soc. Jpn. 83, 032001 (2014).
  * [30] D. Daghero, F. Paolucci, A. Sola, M. Tortello, G. A. Ummarino, M. Agosto, R. S. Gonnelli, J. R. Nair, and C. Gerbaldi. Large Conductance Modulation of Gold Thin Films by Huge Charge Injection via Electrochemical Gating. Phys. Rev. Lett. 108, 066807 (2012).
  * [31] M. Tortello, A. Sola, K. Sharda, F. Paolucci, J. R. Nair, C. Gerbaldi, D. Daghero, and R. S. Gonnelli. Huge field-effect surface charge injection and conductance modulation in metallic thin films by electrochemical gating. Appl. Surf. Sci. 269, 17-22 (2013).
  * [32] X. Xi, H. Berger, L. Forró, J. Shan, and K. F. Mak. Gate Tuning of Electronic Phase Transitions in Two-Dimensional NbSe2. Phys. Rev. Lett. 117, 106801 (2016).
  * [33] Z. J. Li, B. F. Gao, J. L. Zhao, X. M. Xie, and M. H. Jiang. Effect of electrolyte gating on the superconducting properties of thin 2H-NbSe2 platelets. Supercond. Sci. Technol. 27, 015004 (2014).
  * [34] M. Yoshida, J. T. Ye, T. Nishizaki, N. Kobayashi, and Y. Iwasa. Electrostatic and electrochemical tuning of superconductivity in two-dimensional NbSe2 crystals. Appl. Phys. Lett. 108, 202602 (2016).
  * [35] A. T. Bollinger, G. Dubuis, J. Yoon, D. Pavuna, J. Misewich, and I Božovic. Superconductor-insulator transition in La2-xSrxCuO4 at the pair quantum resistance. Nature 472, 458-460 (2011).
  * [36] X. Leng, J. Garcia-Barriocanal, S. Bose, Y. Lee, and A. M. Goldman. Electrostatic Control of the Evolution from a Superconducting Phase to an Insulating Phase in Ultrathin YBa2Cu3O7-x Films. Phys. Rev. Lett. 107, 027001 (2011).
  * [37] X. Leng, J. Garcia-Barriocanal, B. Yang, Y. Lee, J. Kinney, and A. M. Goldman. Indications of an Electronic Phase Transition in Two-Dimensional Superconducting YBa2Cu3O7-x Thin Films Induced by Electrostatic Doping. Phys. Rev. Lett. 108, 067004 (2012).
  * [38] J. Jeong, N. Aetukuri, T. Graf, T. D. Schladt, M. G. Samant, and S. S. P. Parkin. Suppression of Metal-Insulator Transition in VO2 by Electric Field–Induced Oxygen Vacancy Formation. Science 339, 1402-1405 (2013).
  * [39] T. D. Schladt, T. Graf, N. B. Aetukuri, M. Li, A. Fantini, X. Jiang, M. G. Samant, and S. S. P. Parkin. Crystal-Facet-Dependent Metallization in Electrolyte-Gated Rutile TiO2 Single Crystals. ACS Nano 7, 8074–8081 (2013).
  * [40] J. Walter, H. Wang, B. Luo, C. D. Frisbie, and C. Leighton. Electrostatic versus Electrochemical Doping and Control of Ferromagnetism in Ion-Gel-Gated Ultrathin La0.5Sr0.5CoO3-δ. ACS Nano 10, 7799-7810 (2016).
  * [41] L. Zhang, S. Zeng, X. Yin, T. C. Asmara, P. Yang, K. Han, Y. Cao, W. Zhou, D. Wan, C. S. Tang, A. Rusydi, Ariando, and T. Venkatesan. The Mechanism of Electrolyte Gating on High‑$T_{c}$ Cuprates. ACS Nano 11, 9950-9956 (2017).
  * [42] S. W. Zeng, X. M. Yin, T. S. Herng, K. Han, Z. Huang, L. C. Zhang, C. J. Li, W. X. Zhou, D. Y. Wan, P. Yang, J. Ding, A. T. S. Wee, J. M. D. Coey, T. Venkatesan, A. Rusydi, and Ariando. Oxygen Electromigration and Energy Band Reconstruction Induced by Electrolyte Field Effect at Oxide Interfaces. Phys. Rev. Lett. 121, 146802 (2018).
  * [43] M. S. Saleem, B. Cui, C. Song, Y. Sun, Y. Gu, R. Zhang, M. U. Fayaz, X. Zhou, P. Werner, S. S. P. Parkin, and F. Pan. Field Control of Phase Transition and Tunable Resistive Switching in SrFeO2.5. ACS Appl. Mater. Interfaces 11, 6581-5688 (2019).
  * [44] J. Shiogai, Y. Ito, T. Mitsuhashi, T. Nojima and A. Tsukazaki. Electric-field-induced superconductivity in electrochemically-etched ultrathin FeSe films on SrTiO3 and MgO. Nat. Phys. 12, 42-46 (2016).
  * [45] B. Lei, J. H. Cui, Z. J. Xiang, C. Shang, N. Z. Wang, G. J. Ye, X. G. Luo, T. Wu, Z. Sun, and X. H. Chen. Evolution of high-temperature superconductivity from low-Tc phase tuned by carrier concentration in FeSe thin flakes. Phys. Rev. Lett. 116, 077002 (2016).
  * [46] K. Hanzawa, H. Sato, H. Hiramatsu, T. Kamiya, and H. Hosono. Electric field-induced superconducting transition of insulating FeSe thin film at 35 K. Proc. Natl. Acad. Sci. USA 113, 3986 (2016).
  * [47] T. Miyakawa, J. Shiogai, S. Shimizu, M. Matsumoto, Y. Ito, T. Harada, K. Fujiwara, T. Nojima, Y. Itoh, T. Aida, Y. Iwasa, and A. Tsukazaki. Enhancement of superconducting transition temperature in FeSe electric-double-layer transistor with multivalent ionic liquids. Phys. Rev. Materials 2, 031801(R) (2018).
  * [48] B. Lei, N. Z. Wang, C. Shang, F. B. Meng, L. K. Ma, X. G. Luo, T. Wu, Z. Sun, Y. Wang, Z. Jiang, B. H. Mao, Z. Liu, Y. J. Yu, Y. B. Zhang, and X. H. Chen. Tuning phase transitions of FeSe thin flakes by field effect transistor with solid ion conductor as gate dielectric. Phys. Rev. B 95, 020503(R) (2017).
  * [49] Y. Cui, Z. Hu, J.-S. Zhang, W.-L. Ma, M.-W. Ma, Z. Ma, C. Wang, J.-Q. Yan, J.-P. Sun, J.-G. Cheng, S. Jia, Y. Li, J.-S. Wen, H.-C. Lei, P. Yu, W. Ji, and W.-Q. Yu. Ionic-Liquid-Gating Induced Protonation and Superconductivity in FeSe, FeSe0.93S0.07, ZrNCl, 1$T$-TaS2 and Bi2Se3. Chin. Phys. Lett. 36, 077401 (2019).
  * [50] S. Kouno, Y. Sato, Y. Katayama, A. Ichinose, D. Asami, F. Nabeshima, Y. Imai, A. Maeda, and K. Ueno. Superconductivity at 38 K in an electrochemical interface between ionic liquid and FeSe0.8Te0.2 on various substrates. Sci. Rep. 8, 14731 (2018).
  * [51] C. S. Zhu, J. H. Cui, B. Lei, N. Z. Wang, C. Shang, F. B. Meng, L. K. Ma, X. G. Luo, T. Wu, Z. Sun, and X. H. Chen. Tuning electronic properties of FeSe0.5Te0.5 thin flakes by a novel field effect transistor. Phys. Rev. B 95, 174513 (2017).
  * [52] B. Lei, Z. J. Xiang, X. F. Lu, N. Z. Wang, J. R. Chang, C. Shang, A. M. Zhang, Q. M. Zhang, X. G. Luo, T. Wu, Z. Sun, and X. H. Chen. Gate-tuned Superconductor-Insulator transition in (Li,Fe)OHFeSe. Phys. Rev. B 93, 060501(R) (2016).
  * [53] E. Piatti, T. Hatano, D. Daghero, F. Galanti, C. Gerbaldi, S. Guastella, C. Portesi, I. Nakamura, R. Fujimoto, K. Iida, H. Ikuta, and R. S. Gonnelli. Ambipolar suppression of superconductivity by ionic gating in optimally doped BaFe2(As,P)2 ultrathin films. Phys. Rev. Materials 3, 044801 (2019).
  * [54] J. Choi, R. Pradheesh, H. Kim, H. Im, Y. Chong, and D. H. Chae. Electrical modulation of superconducting critical temperature in liquid-gated thin niobium films. Appl. Phys. Lett. 105, 012601 (2014).
  * [55] E. Piatti, A. Sola, D. Daghero, G. A. Ummarino, F. Laviano, J. R. Nair, C. Gerbaldi, R. Cristiano, A. Casaburi, and R. S. Gonnelli. Superconducting Transition Temperature Modulation in NbN via EDL Gating. J. Supercond. Novel Magn. 29, 587-591 (2016).
  * [56] E. Piatti, D. Daghero, G. A. Ummarino, F. Laviano, J. R. Nair, R. Cristiano, A. Casaburi, C. Portesi, A. Sola, and R. S. Gonnelli. Control of bulk superconductivity in a BCS superconductor by surface charge doping via electrochemical gating. Phys. Rev. B 95, 140501(R) (2017).
  * [57] J.-N. Chazalviel, Coulomb Screening by Mobile Charges: Applications to Materials Science, Chemistry, and Biology (Springer, New York, 1999).
  * [58] S. P. Chockalingam, M. Chand, J. Jesudasan, V. Tripathi, and P. Raychaudhuri. Superconducting properties and Hall effect of epitaxial NbN thin films. Phys. Rev. B 77, 214503 (2008).
  * [59] J. E. Hirsch. Reply to “Comment on ‘Charge expulsion and electric field in superconductors’ ”. Phys. Rev. B 70, 226504 (2004).
  * [60] T. Koyama. Comment on “Charge expulsion and electric field in superconductors”. Phys. Rev. B 70, 226503 (2004).
  * [61] P. G. de Gennes. Boundary Effects in Superconductors. Rev. Mod. Phys. 36, 225 (1964).
  * [62] G. A. Ummarino, E. Piatti, D. Daghero, R. S. Gonnelli, I. Yu. Sklyadneva, E. V. Chulkov, and R. Heid. Proximity Eliashberg theory of electrostatic field-effect doping in superconducting films. Phys. Rev. B 96, 064509 (2017).
  * [63] T. M. Klapwijk. Proximity Effect From an Andreev Perspective. J. Supercond. 17, 593 (2004).
  * [64] L. N. Cooper. Superconductivity in the Neighborhood of Metallic Contacts. Phys. Rev. Lett. 6, 689 (1961).
  * [65] W. Silvert. Superconducting proximity effect in the strong-coupling limit . Phys. Rev. B 12, 4870 (1975).
  * [66] G. De Simoni, F. Paolucci, P. Solinas, E. Strambini, and F. Giazotto. Metallic supercurrent field-effect transistor. Nat. Nanotechnol. 13, 802-805 (2018).
  * [67] F. Paolucci, G. De Simoni, P. Solinas, E. Strambini, N. Ligato, P. Virtanen, A. Braggio, and F. Giazotto. Magnetotransport Experiments on Fully Metallic Superconducting Dayem Bridge Field-Effect Transistors. Phys. Rev. Appl. 11, 024061 (2019).
  * [68] M. F. Ritter, A. Fuhrer, D. Z. Haxwell, S. Hart, P. Gumann, H. Riel, and F. Nichele. A superconducting switch actuated by injection of high-energy electrons. Nat. Commun. 12, 1266 (2021).
  * [69] S. Kafanov, Y. Pashkin, V. Tsepelin, A. Guthrie, and I. Golokolenov. On the origin of the controversial electrostatic field effect in superconductors. Nat. Commun. 12, 2747 (2021).
  * [70] M. Rocci, D. Suri, A. Kamra, G. Vilela, Y. Takamura, N. M. Nemes, J. L. Martinez, M. Garcia Hernandez, and J. S. Moodera. Large Enhancement of Critical Current in Superconducting Devices by Gate Voltage. Nano Lett. 21, 216–221 (2021).
  * [71] T. Ando, A. B. Fowler, and F. Stern. Electronic properties of two-dimensional systems. Rev. Mod. Phys. 54, 437-672 (1982).
  * [72] A. Fête, L. Rossi, A. Augieri, and C. Senatore. Ionic liquid gating of ultra-thin YBa2Cu3O7 films. Appl. Phys. Lett. 109, 192601 (2016).
  * [73] K. Ueno, T. Nojima, S. Yonezawa, M. Kawasaki, Y. Iwasa, and Y. Maeno. Effective thickness of two-dimensional superconductivity in a tunable triangular quantum well of SrTiO3. Phys. Rev. B 89, 020508 (2014).
  * [74] D. Valentinis, S. Gariglio, A. Fête, J.-M. Triscone, C. Berthod, and D. van der Marel. Modulation of the superconducting critical temperature due to quantum confinement at the LaAlO3/SrTiO3 interface. Phys. Rev. B 96, 094518 (2017).
  * [75] E. Piatti, D. Romanin, R. S. Gonnelli, and D. Daghero. Anomalous screening of an electrostatic field at the surface of niobium nitride. Appl. Surf. Sci. 461, 17-22 (2018).
  * [76] T. Brumme, M. Calandra, and F. Mauri. Electrochemical doping of few-layer ZrNCl from first principles: Electronic and structural properties in field-effect configuration. Phys. Rev. B 89, 245406 (2014).
  * [77] T. Brumme, M. Calandra, and F. Mauri. First-principles theory of field-effect doping in transition-metal dichalcogenides: Structural properties, electronic structure, Hall coefficient, and electrical conductivity. Phys. Rev. B 91, 155436 (2015).
  * [78] Y. Ge and A. Y. Liu. Phonon-mediated superconductivity in electron-doped single-layer MoS2: A first-principles prediction. Phys. Rev. B 87, 241408 (2013).
  * [79] T. Brumme, M. Calandra, and F. Mauri. Determination of scattering time and of valley occupation in transition-metal dichalcogenides doped by field effect. Phys. Rev. B 93, 081407 (2016).
  * [80] T. Sohier, M. Calandra, and F. Mauri. Density functional perturbation theory for gated two-dimensional heterostructures: Theoretical developments and application to flexural phonons in graphene. Phys. Rev. B 96, 075448 (2017).
  * [81] D. Romanin, Th. Sohier, D. Daghero, F. Mauri, R. S. Gonnelli, and M. Calandra. Electric field exfoliation and high-Tc superconductivity in field-effect hole-doped hydrogenated diamond (111). Appl. Surf. Sci. 496, 143709 (2019).
  * [82] E. Piatti, D. Romanin, R. S. Gonnelli, and D. Daghero. Mapping multi-valley Lifshitz transitions induced by field-effect doping in strained MoS2 nanolayers. J. Phys. Condens. Matter 31, 114002 (2019).
  * [83] D. Romanin. Field-effect-induced superconductivity in surfaces of tetrahedrally coordinated semiconductors: The case of (111) hydrogenated silicon. Il Nuovo Cimento C 43, 115 (2020).
  * [84] D. Romanin, T. Brumme, D. Daghero, R. S. Gonnelli, and E. Piatti. Strong band-filling-dependence of the scattering lifetime in gated MoS2 nanolayers induced by the opening of intervalley scattering channels. J. Appl. Phys. 128, 063907 (2020).
  * [85] D. Romanin, G. A. Ummarino, and E. Piatti. Migdal–Eliashberg theory of multi-band high-temperature superconductivity in field-effect-doped hydrogenated (111) diamond. Appl. Surf. Sci. 536, 147723 (2021).
  * [86] G. A. Ummarino and D. Romanin. Theoretical Explanation of Electric Field‐Induced Superconductive Critical Temperature Shifts in Indium Thin Films. Phys. Status Solidi B 257, 1900651 (2020).
  * [87] G. A. Ummarino and D. Romanin. Proximity two bands Eliashberg theory of electrostatic field-effect doping in a superconducting film of MgB2. J. Phys. Condens. Matter 31, 024001 (2019).
  * [88] J. Zmuidzinas. Superconducting microresonators: Physics and applications. Annu. Rev. Condens. Matter Phys. 3, 169–214 (2012).
  * [89] F. Paolucci, G. De Simoni, P. Solinas, E. Strambini, C. Puglia, N. Ligato, and F. Giazotto. Field-effect control of metallic superconducting systems. AVS Quantum Sci. 1, 016501 (2019).