# Why Mercury is a Superconductor

Cesare Tresca  Department of Physical and Chemical Sciences, University of
L’Aquila, Via Vetoio 10, I-67100 L’Aquila, Italy  Dipartimento di Fisica,
Sapienza Università di Roma, 00185 Roma, Italy  Gianni Profeta  Department of
Physical and Chemical Sciences, University of L’Aquila, Via Vetoio 10, I-67100
L’Aquila, Italy  SPIN-CNR, University of L’Aquila, Via Vetoio 10, I-67100
L’Aquila, Italy  Giovanni Marini  Department of Physical and Chemical
Sciences, University of L’Aquila, Via Vetoio 10, I-67100 L’Aquila, Italy
Giovanni B. Bachelet  Dipartimento di Fisica, Sapienza Università di Roma,
00185 Roma, Italy  Antonio Sanna  Max-Planck-Institut für Mikrostrukturphysik,
Weinberg 2, D-06120 Halle, Germany  Matteo Calandra  Sorbonne Université,
CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252, Paris, France
Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo,
Trento, Italy  Lilia Boeri  Dipartimento di Fisica, Sapienza Università di
Roma, 00185 Roma, Italy

###### Abstract

Despite being the oldest known superconductor, solid mercury is mysteriously
absent from all current computational databases of superconductors. In this
work, we present a critical study of its superconducting properties based on
state-of-the-art superconducting density-functional theory. Our calculations
reveal numerous anomalies in electronic and lattice properties, which can
mostly be handled, with due care, by modern ab-initio techniques. In
particular, we highlight an anomalous role of (i) electron-electron
correlations on structural properties (ii) spin-orbit coupling on the
dynamical stability, and (iii) semicore $d$ levels on the effective Coulomb
interaction and, ultimately, the critical temperature.

###### pacs:

75.70.Tj, 74.20.Pq, 74.25.Dw

##  I Introduction

In 1911 Kamerlingh Onnes,Kamerlingh Onnes (1911) investigating the transport
properties of mercury at low temperatures, observed for the first time a
superconducting (SC) transition: below a critical temperature TC= 4.15 K, the
electrical resistivity dropped to zero. The discovery marked a milestone in
physics history. The first microscopic theory of this phenomenon was
formulated only fifty years later by Bardeen, Cooper, and Schrieffer
(BCS).Bardeen  _et al._ (1957) Their theory, refined through the
Migdal–Éliashberg (ME)Migdal (1958); Éliashberg (1960) Green’s function
formalism and the Morel and Anderson Coulomb pseudopotential $\mu^{*}$,Morel
and Anderson (1962) permitted to draw an accurate picture of the normal and
the SC phase of conventional (phonon-mediated) superconductors.

In the 60’s and 70’s, when an ab-initio solution of the Éliashberg equations
was beyond available computational capabilities, mercury, among others, served
as a benchmark to derive approximate analytical expressions for various
superconducting properties, whose main ingredients were extracted from
experiments. Normal-state electronic structure was inferred from de Haas van
Alphen,Brandt and Rayne (1966) magnetoresistance, and cyclotron-resonance
measurements;Dishman and Rayne (1968); Dixon and Datars (1968) phonon
dispersion curves from neutron inelastic scattering;Kamitakahara  _et al._
(1977) while the Éliashberg function $\alpha^{2}F(\omega)$ and the SC gap from
tunneling experiments.Bermon and Ginsberg (1964); Hubin and Ginsberg (1969)
Notable examples are the McMillan-Allen-DynesMcMillan (1968); Allen and Dynes
(1975) approximate formulas for TC.

Towards the end of the century, progress in Density Functional (Perturbation)
Theory Baroni  _et al._ (2001) allowed first-principles calculations of the
electron-phonon spectral functionSavrasov and Savrasov (1996), superconducting
TC’s and gaps.Giustino (2017); Sanna  _et al._ (2020) These methods, combined
with modern crystal-structure prediction algorithms, Pickard and Needs (2011);
Curtarolo  _et al._ (2013); Pickard (2019); Oganov  _et al._ (2019); Flores-
Livas  _et al._ (2020) and with Ashcroft’s intuition of high-TC SC in
hydrogen-rich metallic alloys,Ashcroft (2004) were the driving force behind
the hydride rush of the last five years. Flores-Livas  _et al._ (2020); Roa

Following these achievements, Density Functional Theory (DFT) based methods
are rapidly becoming the tool of choice to guide new superconductor
discoveries. The field is evolving in the direction of high-throughput
material design; this requires extensive benchmarks on known materials to
vouch the accuracy of current theoretical/computational methods. The first
results seem encouraging: for most conventional superconductors, the agreement
between different theoretical approaches and experiment is remarkable.Savrasov
and Savrasov (1996); Kawamura  _et al._ (2020); Sanna  _et al._ (2020)
However, a few notable exceptions exist. In particular, a close inspection of
the available literature reveals that mercury is unexplicably absent from all
currently accessible benchmark calculations.

This paper reports our attempt to fill this gap. In particular, we address the
following questions: if Onnes had not discovered superconductivity in mercury,
could we predict it today? And, even more importantly, can state-of-the-art
theoretical and computational approaches completely describe superconductivity
in mercury? We will show that the answers are not straightforward, since, in
mercury, all physical properties relevant for conventional superconductivity,
i.e. the electronic structure, phonon dispersions, electron-phonon coupling
and Coulomb matrix elements, are anomalous in some respect.

In the following, we will discuss each of these aspects separately, and show
how they concur to determine a consistent picture of superconductivity in this
fascinating element.

##  II Results and discussion

Crystal Structure:  At ambient conditions mercury is liquid, but below 235 K
it crystallizes in a monoatomic rhombohedral lattice, the so-called $\alpha$
phase, Pauling (1947); Barrett (1957); CRC (1997) shown in Fig. 1, which is
commonly accepted as the actual superconducting phase of mercury. Schulte and
Holzapfel (1993); Swenson (1958); Pauling (1947); Barrett (1957); Brandt and
Rayne (1965); Keeton and Loucks (1966); Kamitakahara  _et al._ (1977); CRC
(1997); Schulte and Holzapfel (1993, 1996, 1993, 1996); Moriarty (1988);
Biering and Schwerdtfeger (2011)

Figure 1: Different views of the experimental $\alpha$-Hg structure of
superconducting solid mercury in the monoatomic rhombohedral lattice (space
group $R3m$ or 160 Pauling (1947); Barrett (1957); CRC (1997)). The three
primitive lattice vectors are indicated by blue lines; the angle between each
pair of them, $\theta$ , equals to 60o in the fcc structure, but deviates from
this value in the rhombohedral phase.

The structure may be seen as the compression of the fcc structure along a
three-fold axis, causing the six equatorial distances ($d^{\prime}$, red in
Fig.1) to become greater than the six others ($d$, blue in Fig.1).Pauling
(1947) The interatomic distances $d$ and $d^{\prime}$ are uniquely determined
by the lattice parameter, $a$ and the rhombohedral angle, $\theta$, which
would equal 60o in the undistorted fcc structure. Total energy structural
optimization using Generalized Gradient Approximation (GGA)Perdew  _et al._
(1996) functional yields $a=d=3.12$ Å and $\theta=78.51^{o}$
($d^{\prime}=4.83$ Å)111The Local Density Approximation (LDA) gives in
$a_{LDA}=d_{LDA}=2.94$ Å and $\theta_{LDA}=74.54^{o}$ ($d^{\prime}_{LDA}=4.68$
Å), to be compared with the experimental values $a$=$d$=3.005 Å and
$\theta=70.53^{o}$CRC (1997) ($d^{\prime}=4.907$ Å).

The $\sim 5\%$ discrepancy between theory and experiment, already reported in
literature, signals that semi-local energy functionals cannot properly
reproduce the experimental lattice parameters.Gaston  _et al._ (2006); Calvo
_et al._ (2013) A $\sim 5\%$ accuracy on the structural parameters would be
considered acceptable for most materials. However, in mercury even minor
structural differences cause dramatic effects on the electronic and dynamical
properties, which are instead perfectly reproduced assuming the experimental
lattice structure. Hence, in the following, all calculations of the
electronic, dynamical and superconducting properties will employ the
experimental lattice crystal structure, and the PBE-GGA exchange and
correlation functional. Further computational details can be found in the
Appendix.

Electronic structure. We start from the electronic band structure, shown in
Fig.2. In agreement with previous literature,Keeton and Loucks (1966); Jansen
_et al._ (1983); Deng  _et al._ (1998) we find a well-dispersed parabolic
band, derived from $s$-states, partially hybridized with unoccupied $p$
states. In the region between 5.5 eV and 9 eV below the Fermi level ($E_{F}$),
the $s$-parabola is tangled with the $d$-states. Including relativistic spin-
orbit coupling (SOC) causes sizable effects in the $d$ band region and, to a
lesser extent, in the vicinity of $E_{F}$. In particular, SOC removes several
band degeneracies, for example around the $L$ point and along the
$K\rightarrow X$ path, – compare full and dashed lines in Fig. 2.

The resulting density of states (DOS) has a rather interesting shape: a broad
feature, corresponding to $s$-states, extends from $\simeq-10$ eV to $E_{F}$,
and two high, narrow peaks, due to the two groups of spin-orbit splited $d$
bands, are centered around 6 and 9 eV below $E_{F}$.

Figure 2: (Top panel) Electronic band structure and density of states (DOS) of
$\alpha$-Hg, with (continuous line) and without (dashed line) spin-orbit
coupling. (Bottom panel) from left to right, 3D plot of the Fermi surface from
fully relativistic calculations; 2D cuts along the BZ boundary are compared to
the corresponding experimental cuts from Ref. Keeton and Loucks, 1966.

To the best of our knowledge, the band structure of mercury has never been
measured by Angle-Resolved-Photo-Emission Spectroscopy, but indirect evidence
of the Fermi surface shape can be inferred from de-Haas-van-Alphen, magneto-
resistance, and cyclotron-resonance measurementsBrandt and Rayne (1965);
Keeton and Loucks (1966); Deng  _et al._ (1998). In the bottom-left panel of
Fig.2, we show a three-dimensional view of the calculated Fermi Surface, which
comprises two disconnected parts: a tubular network extending throughout the
Brillouin zone (BZ), and a disk enclosing the $L$-point. Our calculations
reporoduce the experimental measurements with striking accuracy: not only the
main features, but also finer details, such as the small circular hole pockets
around the $X$ and $T$ points, and an elongated hole pocket around the
$K$-point, are perfectly reproduced.

Figure 3: Vibrational frequencies of $\alpha$-Hg. We highlight the role of
spin-orbit coupling reporting both results with (continuous line) and without
(dashed line) the SOC contribution. For comparison the experimental
measurements from Ref.9 are reported as empty black circles.

Vibrational Properties. Besides the low-energy electronic structure, DFT-GGA
calculations also reproduce with excellent accuracy the phonon dispersions,
provided that spin-orbit coupling is included and the experimental structure
is considered.

In Fig. 3, calculated phonon dispersions are compared to the neutron
scattering data from Ref. Kamitakahara  _et al._ , 1977.

The phonon spectrum extends up to 120 cm-1 with a pseudo-gap around 75 cm-1
separating transverse and longitudinal modes. The lower transverse branch is
very soft and almost flat throughout the whole BZ. Around the $L$-point, a
further softening occurs: here $\omega_{L}$ is only $6.5$ cm-1. We find that
including relativistic (SOC) effects is crucial for correctly capturing the
experimental dispersion in the low-frequency region and obtaining a
dynamically stable structure. In fact, without SOC the frequencies of the
transverse branch around the $L$ point are imaginary (Fig. 3). This result is
consistent with the recent report that relativistic effects are required to
explain also the low melting temperature of mercury Calvo  _et al._ (2013).

Electron-phonon coupling. In the left panel of Fig. 4 we report the branch-
and momentum-dependence of the electron-phonon linewidth
$\gamma_{\mathbf{q}\nu}$ (half-width at half-maximum):

$\gamma_{\mathbf{q}\nu}=\frac{2\pi\omega_{\mathbf{q}\nu}}{N_{k}}\sum_{\mathbf{k}nm}|g^{\mathbf{q}\nu}_{\mathbf{k+q}m,\mathbf{k}n}|^{2}\delta(\varepsilon_{\mathbf{k+q}m}-\varepsilon_{F})\delta(\varepsilon_{\mathbf{k}n}-\varepsilon_{F})$
(1)

In Eq.1 the summation of electron-phonon matrix elements,
$g^{\nu}_{\mathbf{k+q}n,\mathbf{k}m}$Calandra and Mauri (2008), is performed
on electronic states ($\varepsilon_{\mathbf{k}n}$) at the Fermi level
($\varepsilon_{F}$) using $N_{k}$ $k$-points in the BZ. The phonon linewidths,
and hence the electron-phonon coupling, are rather constant over all phonon
modes. However, the presence of a soft and weakly dispersive phonon branch
causes a pronounced peak at about 15 cm-1 in the phonon DOS and in the
Éliashberg electron-phonon coupling spectral function (Fig. 4b):

$\alpha^{2}F(\omega)=\frac{1}{2\pi
N(\varepsilon_{F})N_{q}}\sum_{\mathbf{q}\nu}\frac{\gamma_{\mathbf{q}\nu}}{\hbar\omega_{\mathbf{q}\nu}}\delta(\omega-\omega_{\mathbf{q}\nu}),$
(2)

obtained summing over $N_{q}$ phonons with wavevectors $\mathbf{q}$ and mode
index ($\nu$) with frequency $\omega_{\mathbf{q}\nu}$. The shape of the
$\alpha^{2}F(\omega)$ results in a large electron-phonon coupling parameter
$\lambda(\omega\to\infty)=2\int_{0}^{\omega}\alpha^{2}F(\omega^{\prime})/\omega^{\prime}d\omega^{\prime}$
= 1.57 and a rather small logarithmic-averaged phonon frequency:
$\omega_{\textrm{log}}=27.3$ K.

The agreement with tunneling measurementsKamitakahara  _et al._ (1977) is
excellent: both the soft-phonon peak at low frequencies (below 25 cm-1), and
the reduced coupling of the longitudinal mode at higher frequencies are well
reproduced by our calculations. The calculated $\omega_{\textrm{log}}$ and
$\lambda$ are also in agreement with the corresponding experimental values
from tunneling ($\omega_{\textrm{log}}=29$ K and $\lambda$=1.6 Hubin and
Ginsberg (1969); Allen and Dynes (1975)) and with the specific-heat
renormalization data for $\lambda$ 222We derived the $\lambda$ parameter from
the experimental Sommerfield coefficients $\gamma$ according to the following
relation: $\lambda^{exp}=\frac{\gamma^{exp}}{\gamma^{th}}(1-\lambda^{th})-1$
($\lambda^{sh}\sim$1.56van der Hoeven and Keesom (1964),1.66Phillips  _et al._
(1964),1.58Finnemore and Mapother (1965) and $\sim 2.0$ Finnemore  _et al._
(1960)).

Figure 4: Panel (a) Phonon dispersion. The thickness of the light-blue shading
is proportional to the phonon linewidths.
Panel (b), Calculated Éliashberg function (blue), and $\lambda(\omega)$
(light-blue, top side scale). Experimental $\alpha^{2}F(\omega)$ from
tunneling (Ref. Hubin and Ginsberg, 1969), red.

Superconducting Properties. Superconducting Density Functional Theory (SCDFT),
Davydov  _et al._ (2020) is an extension of DFT to the SC phase, which was
developed with the explicit purposeOliveira  _et al._ (1988); Lüders  _et al._
(2005); Marques  _et al._ (2005); Floris  _et al._ (2005) of treating both the
electron-phonon and the Coulomb interaction on an equal footing, eliminating
any adjustable parameters, such as the empirical Coulomb pseudopotential
$\mu^{*}$.

The solution of the SCDFT gap equationSanna  _et al._ (2020); Flores-Livas
_et al._ (2020) for $\alpha$-Hg in the static and isotropic approximation,
including both electron-phonon and electron-electron interactions, reproduces
experimental data with remarkable accuracy, as shown in Fig. 5, where the
temperature dependence of the SC gap (at $E_{F}$) obtained in SCDFT (light
blue open circles) is compared with tunneling data from Ref. Bermon and
Ginsberg, 1964 (red squares).

The two curves follow each other rather closely. The critical temperature
obtained extrapolating the calculated low-T data is TCSCDFT=3.84 K, to be
compared with the experimental value TCexp=4.15 K. For the BCS ratio
$2\Delta(0)/k_{B}$TC, SCDFT predicts a value of 4.70, to be compared with
experimental values of 4.6$\pm 0.2$Richards and Tinkham (1960) and 4.60$\pm
0.11$Bermon and Ginsberg (1964). This value places Hg in the strong-coupling
regime; the low TC results essentially from the extremely low phonon
frequencies.

Figure 5: Comparison between theoretical (linearly connected light blue
circles) and experimental gapBermon and Ginsberg (1964) (red squares) plotted
as a function of temperature. The blue (linearly connected) points are
obtained with a 1.25 energy scaling as described in the main text.

Coulomb Interaction. In addition to the electron-phonon interaction, SCDFT
gives a microscopic insight also into the residual Coulomb scattering, an
aspect disregarded in most studies of SC, which typically employ the popular
Morel-Anderson approximation. Also this aspect is strongly anomalous in
mercury, as we will show in the following.

In SCDFT, in the absence of SOC, the Coulomb interaction between electrons is
described by the iso-energy surface average
$V(\varepsilon,\varepsilon^{\prime})$ of the screened Coulomb matrix elements
$V_{i\mathbf{k},j\mathbf{k}^{\prime}}$:Sanna  _et al._ (2018); Massidda  _et
al._ (2009)

$V(\varepsilon,\varepsilon^{\prime})=\sum_{i\mathbf{k},j\mathbf{k^{\prime}}}V_{i\mathbf{k},j\mathbf{k}^{\prime}}\frac{\delta(\varepsilon-\varepsilon_{i\mathbf{k}})}{N(\varepsilon)}\frac{\delta(\varepsilon^{\prime}-\varepsilon_{j\mathbf{k}\prime})}{N(\varepsilon^{\prime})}$
(3)

The effect of $V(\varepsilon,\varepsilon^{\prime})$ depends crucially on the
energies $\varepsilon,\varepsilon^{\prime}$ of the two electrons involved in
the SC pairing: Coulomb interaction will in fact suppress superconductivity if
both states lie in energy regions where the SC gap is positiveMorel and
Anderson (1962); Scalapino  _et al._ (1966), i.e. close to $E_{F}$, but can
also favor it, if one of the two electrons occupies a state at high energies,
where the SC gap is negative. In this case, high-energy states will cause a
net renormalization (reduction) of the effective Coulomb interaction Flores-
Livas  _et al._ (2020).

A two-dimensional plot of the calculated $V(\varepsilon,\varepsilon^{\prime})$
function for mercury is shown in Fig.6 – here and in the following, energies
are measured from $E_{F}$.

Figure 6: Two-dimensional plot of the Coulomb potential as a function of the
energy with respect to the Fermi level. The color scale expresses the
intensity of $V(\varepsilon,\varepsilon^{\prime})$ in Ry.

Due to the different nature and dispersion of the $s$ and $d$ bands the
diagonal elements of $V(\varepsilon,\varepsilon^{\prime})$ show an hot-spot
(blue) around $-10$ eV, corresponding to the bottom of the $s$ parabola in
Fig. 1, a square-like feature, with moderate coupling (green) from -8 to -5
eV, related to $d$ states, and an extended region from zero to high energies
(green), related to $s$ and $p$ states. The off-diagonal $\varepsilon=0$ row,
$V(0,\varepsilon^{\prime})$, which accounts for interband contributions
involving the $s$ states at the Fermi level and all the other states, is non-
zero in the low-energy $s$ region, is very low (nearly zero) for $d$ states
($\varepsilon^{\prime}<-5$ eV), and different from zero again only for
$\varepsilon^{\prime}\sim-10$ eV, i.e. near the bottom of the $s$ parabola in
Fig. 1.

Based on this energy structure, we expect that the net effect of Coulomb
interactions on TC will be rather weak, due both to low diagonal matrix
elements in the low-energy (repulsive) region, and large inter- and intra-band
contributions in the high-energy (attractive) regions.

The diagonal part of the $V(\varepsilon,\varepsilon^{\prime})$ kernel,
evaluated at the Fermi energy ($\varepsilon=0$), yields the so-called $\mu$
parameter. For mercury the calculated $\mu=0.159$ is in line with its
neighbours in the periodic table, like Au and Cd ($\mu=$0.136 and 0.142,
respectivelyKawamura  _et al._ (2020)), but much smaller than the average
value $\mu=0.25$ found in most other elemental superconductors like Pb or Al
Kawamura  _et al._ (2020). Together with the large bandwidth of the $s$ band,
and the extremely small characteristic frequency of mercury, this translates
into a Morel-Anderson pseudopotential $\mu^{*}$=0.07, significantly smaller
than the standard value $\mu^{*}=0.10$ – details in the Supplementary
Material.

However, the most interesting anomaly in the Coulomb screening, which cannot
be captured by the standard Morel-Anderson approach, is connected to the off-
diagonal part of the $V(\varepsilon,\varepsilon^{\prime})$ kernel.

Influence of $d$ states on TC. Due to the presence of non-negligible off-
diagonal $s-d$ Coulomb matrix elements, the calculated TC in mercury turns out
to depend in a critical way on the position of the high-energy $d$-states.

This was verified through a simple gedanken experiment, in which we solved
again the SCDFT equations, leaving all terms unchanged, apart from a scaling
of the electronic spectrum, necessary to bring the energy position of the
calculated DFT-GGA-SOC $d$ bands with experimental X-Ray photoemission
data.Svensson  _et al._ (1976) The physical origin of the energy shift between
DFT-GGA calculations and experiments is the lack of non-local exchange and
correlation terms Rangel  _et al._ (2012); Marini  _et al._ (2002); Svane  _et
al._ (2011); in fact, the shift can be easily removed employing non-local
functionals, such as the Heyd-Scuseria-Ernzerhof (HSE06) functional Heyd  _et
al._ (2003, 2006) – upper panel of Fig. 7.

As shown in the lower panel of the same figure, a simple linear scaling of the
whole DFT-GGA-SOC spectrum f$=1.25$ is sufficient to mimick this effect and
produce an almost perfect agreement between spectra and with experiments.

Figure 7: Top panel: Electronic spectra calculated in GGA (blue) and hybrid
HSE06Heyd  _et al._ (2003, 2006) (light blue), and X-Ray Experiments from Ref.
Svensson  _et al._ , 1976.(red) Bottom panel: Linearly-scaled GGA DOS
(f$=1.25$).

Solving the SCDFT equations as a function of temperature, we obtain the data
shown as blue filled circles in Fig.5: the TC, obtained from extrapolation of
the low-T data, is 4.05 K, with a clear improvement with respect to our
previous GGA-SOC result (see Supplementary Materials). Although the almost
perfect agreement with experiment may be fortuitous, this numerical experiment
demonstrates that shifting the position of apparently inert high-energy states
can lead to a 10$\%$ effect on TC.

##  III Conclusions

In this work we carried out a critical study of the superconducting properties
of $\alpha$-mercury, aimed at understanding whether this material, which
played an essential role in superconductivity history, can be described by
state-of-the-art computational methods. Our first-principles calculations,
validated with an extensive comparison with available experimental literature,
demonstrate that state-of-the-art SCDFT can describe the superconducting state
of Hg, provided that special care is taken to handle several anomalous
electronic and lattice properties.

In particular, ($i$) due to strong non-local exchange and correlation effects
structural properties are so poorly described by standard density functional,
that in order to obtain meaningful electronic and phonon spectra, all
calculations have to be performed in the experimenetal crystal structure;
($ii$) SOC effects are also extremely strong, and crucially affect dynamical
stability; ($iii$) due to anomalously large off-diagonal $s-d$ matrix
elements, the effective Coulomb potential is strongly affected by the energy
position of the low-energy $d$ states. Taken as a whole, our results
demonstrate that, even for an apparently simple compound like mercury, common
approximations cannot be applied blindly, as this may cause severe qualitative
and quantitative errors. This aspect is crucial for future high-throughput
calculations. We would also like to stress that some of the effects discussed
here, may appear spectacularly enhanced in high-TC conventional
superconductors, such as the recently discovered superhydrides, where
renormalization of the Coulomb interaction has been invoked to justify
differences as large as 100 K in the calculated TC’s Troyan  _et al._ (2021).

## Appendix: Computational details

All calculations were performed using the plane-wave pseudopotential DFT
Quantum-Espresso package Giannozzi  _et al._ (2009, 2017) including
relativistic effects. We used Optimized Norm-Conserving Vanderbilt
PseudopotentialsHamann (2013, 2017); van Setten  _et al._ (2018) including
5$s$, 5$p$, 5$d$ and 6$s$ states in valence, and the Generalized Gradient
Approximation (GGA) for the Exchange and correlation term, with an energy cut-
off of 70 Ry.

Integrations over the BZ were carried out using uniform 18$\times$18$\times$18
Monkhorst and Pack gridsMonkhorst and Pack (1976) and a 0.02 Ry Gaussian
smearing.

Phonon frequencies and electron-phonon matrix elements were calculated using
linear response theory Giannozzi  _et al._ (2009, 2017), on a
8$\times$8$\times$8 grid to which correspond 65 $q$-points in the irreducible
BZ and a dense 24$\times$24$\times$24 mesh for electronic wavevectors.

Total electron-phonon coupling parameter is calculated Wannier interpolating
the electron-phonon matrix elementsMarzari and Vanderbilt (1997); Souza  _et
al._ (2001); Calandra  _et al._ (2010) on a denser phononic and electronic
meshes of $12\times 12\times 12$ and $36\times 36\times 36$, respectively.

The SC critical temperature mediated by electron-phonon interaction has then
been calculated fully ab-initio in the SCDFT frameworkOliveira  _et al._
(1988); Lüders  _et al._ (2005); Marques  _et al._ (2005); Floris  _et al._
(2005), using the most accurate available functional Sanna  _et al._ (2020).
In this picture, the Coulomb interaction is treated self-consistently at the
same level as the electron-phonon interaction, in the static isotropic
approximation. Screened Coulomb matrix elements were calculated in the random
phase approximation as in Ref.Sanna  _et al._ , 2018; Marini  _et al._ , 2019
without relativistic effects (being the spin-orbit effects negligible around
the Fermi energy).

The HSE06 DOS in Fig.5 was obtained using the VASPKresse and Furthmüller
(1996a, b); Kresse and Joubert (1999) code.

## Data availability

All the data that support the findings of this study are available from the
corresponding authors (C.T. and G.P.) upon reasonable request.

## References

  * Kamerlingh Onnes (1911) H. Kamerlingh Onnes, Commun. Phys. Lab. Univ. Leiden 12, 120 (1911).
  * Bardeen  _et al._ (1957) J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
  * Migdal (1958) A. B. Migdal, Sov. Phys. JETP 7, 996 (1958).
  * Éliashberg (1960) G. M. Éliashberg, Sov. Phys. JETP 11, 696 (1960).
  * Morel and Anderson (1962) P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962).
  * Brandt and Rayne (1966) G. B. Brandt and J. A. Rayne, Phys. Rev. 148, 644 (1966).
  * Dishman and Rayne (1968) J. M. Dishman and J. A. Rayne, Phys. Rev. 166, 728 (1968).
  * Dixon and Datars (1968) A. E. Dixon and W. R. Datars, Phys. Rev. 175, 928 (1968).
  * Kamitakahara  _et al._ (1977) W. A. Kamitakahara, H. G. Smith, and N. Wakabayashi, Ferroelectrics 16, 111 (1977).
  * Bermon and Ginsberg (1964) S. Bermon and D. M. Ginsberg, Phys. Rev. 135, A306 (1964).
  * Hubin and Ginsberg (1969) W. N. Hubin and D. M. Ginsberg, Phys. Rev. 188, 716 (1969).
  * McMillan (1968) W. L. McMillan, Phys. Rev. 167, 331 (1968).
  * Allen and Dynes (1975) P. B. Allen and R. C. Dynes, Phys. Rev. B 12, 905 (1975).
  * Baroni  _et al._ (2001) S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
  * Savrasov and Savrasov (1996) S. Y. Savrasov and D. Y. Savrasov, Phys. Rev. B 54, 16487 (1996).
  * Giustino (2017) F. Giustino, Rev. Mod. Phys. 89, 015003 (2017).
  * Sanna  _et al._ (2020) A. Sanna, C. Pellegrini, and E. K. U. Gross, Phys. Rev. Lett. 125, 057001 (2020).
  * Pickard and Needs (2011) C. J. Pickard and R. J. Needs, Journal of Physics: Condensed Matter 23, 053201 (2011).
  * Curtarolo  _et al._ (2013) S. Curtarolo, G. L. W. Hart, M. B. Nardelli, N. Mingo, S. Sanvito, and O. Levy, Nature Materials 12, 191 (2013).
  * Pickard (2019) C. J. Pickard, Phys. Rev. B 99, 054102 (2019).
  * Oganov  _et al._ (2019) A. R. Oganov, C. J. Pickard, Q. Zhu, and R. J. Needs, Nature Reviews Materials 4, 331 (2019).
  * Flores-Livas  _et al._ (2020) J. A. Flores-Livas, L. Boeri, A. Sanna, G. Profeta, R. Arita, and M. Eremets, Physics Reports 856, 1 (2020).
  * Ashcroft (2004) N. W. Ashcroft, Phys. Rev. Lett. 92, 187002 (2004).
  * (24) .
  * Kawamura  _et al._ (2020) M. Kawamura, Y. Hizume, and T. Ozaki, Phys. Rev. B 101, 134511 (2020).
  * Pauling (1947) L. Pauling, Journal of the American Chemical Society 69, 542 (1947).
  * Barrett (1957) C. S. Barrett, Acta Crystallographica 10, 58 (1957).
  * CRC (1997) _CRC Handbook of Chemistry and Physics_ (CRC Press, New York, 1997).
  * Schulte and Holzapfel (1993) O. Schulte and W. B. Holzapfel, Phys. Rev. B 48, 14009 (1993).
  * Swenson (1958) C. A. Swenson, Phys. Rev. 111, 82 (1958).
  * Brandt and Rayne (1965) G. Brandt and J. Rayne, Physics Letters 15, 18 (1965).
  * Keeton and Loucks (1966) S. C. Keeton and T. L. Loucks, Phys. Rev. 152, 548 (1966).
  * Schulte and Holzapfel (1996) O. Schulte and W. B. Holzapfel, Phys. Rev. B 53, 569 (1996).
  * Moriarty (1988) J. A. Moriarty, Physics Letters A 131, 41 (1988).
  * Biering and Schwerdtfeger (2011) S. Biering and P. Schwerdtfeger, Theoretical Chemistry Accounts 130, 455 (2011).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Note (1) The Local Density Approximation (LDA) gives in $a_{LDA}=d_{LDA}=2.94$ Å and $\theta_{LDA}=74.54^{o}$ ($d^{\prime}_{LDA}=4.68$ Å).
  * Gaston  _et al._ (2006) N. Gaston, B. Paulus, K. Rosciszewski, P. Schwerdtfeger, and H. Stoll, Phys. Rev. B 74, 094102 (2006).
  * Calvo  _et al._ (2013) F. Calvo, E. Pahl, M. Wormit, and P. Schwerdtfeger, Angewandte Chemie International Edition 52, 7583 (2013), https://onlinelibrary.wiley.com/doi/pdf/10.1002/anie.201302742 .
  * Jansen  _et al._ (1983) H. Jansen, A. Freeman, M. Weinert, and E. Wimmer, Physical Review B-Condensed Matter 28, 593 (1983).
  * Deng  _et al._ (1998) S. Deng, A. Simon, and J. Köhler, Angewandte Chemie International Edition 37, 640 (1998).
  * Calandra and Mauri (2008) M. Calandra and F. Mauri, Phys. Rev. Lett. 101, 016401 (2008).
  * Note (2) We derived the $\lambda$ parameter from the experimental Sommerfield coefficients $\gamma$ according to the following relation: $\lambda^{exp}=\frac{\gamma^{exp}}{\gamma^{th}}(1-\lambda^{th})-1$.
  * van der Hoeven and Keesom (1964) B. J. C. van der Hoeven and P. H. Keesom, Phys. Rev. 135, A631 (1964).
  * Phillips  _et al._ (1964) N. E. Phillips, M. H. Lambert, and W. R. Gardner, Rev. Mod. Phys. 36, 131 (1964).
  * Finnemore and Mapother (1965) D. K. Finnemore and D. E. Mapother, Phys. Rev. 140, A507 (1965).
  * Finnemore  _et al._ (1960) D. K. Finnemore, D. E. Mapother, and R. W. Shaw, Phys. Rev. 118, 127 (1960).
  * Davydov  _et al._ (2020) A. Davydov, A. Sanna, C. Pellegrini, J. K. Dewhurst, S. Sharma, and E. K. U. Gross, Phys. Rev. B 102, 214508 (2020).
  * Oliveira  _et al._ (1988) L. N. Oliveira, E. K. U. Gross, and W. Kohn, Phys. Rev. Lett. 60, 2430 (1988).
  * Lüders  _et al._ (2005) M. Lüders, M. A. L. Marques, N. N. Lathiotakis, A. Floris, G. Profeta, L. Fast, A. Continenza, S. Massidda, and E. K. U. Gross, Phys. Rev. B 72, 024545 (2005).
  * Marques  _et al._ (2005) M. A. L. Marques, M. Lüders, N. N. Lathiotakis, G. Profeta, A. Floris, L. Fast, A. Continenza, E. K. U. Gross, and S. Massidda, Phys. Rev. B 72, 024546 (2005).
  * Floris  _et al._ (2005) A. Floris, G. Profeta, N. N. Lathiotakis, M. Lüders, M. A. L. Marques, C. Franchini, E. K. U. Gross, A. Continenza, and S. Massidda, Phys. Rev. Lett. 94, 037004 (2005).
  * Richards and Tinkham (1960) P. L. Richards and M. Tinkham, Phys. Rev. 119, 575 (1960).
  * Sanna  _et al._ (2018) A. Sanna, J. A. Flores-Livas, A. Davydov, G. Profeta, K. Dewhurst, S. Sharma, and E. K. U. Gross, Journal of the Physical Society of Japan 87, 041012 (2018).
  * Massidda  _et al._ (2009) S. Massidda, F. Bernardini, C. Bersier, A. Continenza, P. Cudazzo, A. Floris, H. Glawe, M. Monni, S. Pittalis, G. Profeta, A. Sanna, S. Sharma, and E. K. U. Gross, Superconductor Science and Technology 22, 034006 (2009).
  * Scalapino  _et al._ (1966) D. J. Scalapino, J. R. Schrieffer, and J. W. Wilkins, Phys. Rev. 148, 263 (1966).
  * Svensson  _et al._ (1976) S. Svensson, N. Martensson, E. Basilier, P. Malmqvist, U. Gelius, and K. Siegbahn, Journal of Electron Spectroscopy and Related Phenomena 9, 51 (1976).
  * Rangel  _et al._ (2012) T. Rangel, D. Kecik, P. E. Trevisanutto, G.-M. Rignanese, H. Van Swygenhoven, and V. Olevano, Phys. Rev. B 86, 125125 (2012).
  * Marini  _et al._ (2002) A. Marini, R. Del Sole, and G. Onida, Phys. Rev. B 66, 115101 (2002).
  * Svane  _et al._ (2011) A. Svane, N. E. Christensen, M. Cardona, A. N. Chantis, M. van Schilfgaarde, and T. Kotani, Phys. Rev. B 84, 205205 (2011).
  * Heyd  _et al._ (2003) J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal of Chemical Physics 118, 8207 (2003).
  * Heyd  _et al._ (2006) J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal of Chemical Physics 124, 219906 (2006).
  * Troyan  _et al._ (2021) I. A. Troyan, D. V. Semenok, A. G. Kvashnin, A. V. Sadakov, O. A. Sobolevskiy, V. M. Pudalov, A. G. Ivanova, V. B. Prakapenka, E. Greenberg, A. G. Gavriliuk, I. S. Lyubutin, V. V. Struzhkin, A. Bergara, I. Errea, R. Bianco, M. Calandra, F. Mauri, L. Monacelli, R. Akashi, and A. R. Oganov, Advanced Materials 33, 2006832 (2021), https://onlinelibrary.wiley.com/doi/pdf/10.1002/adma.202006832 .
  * Giannozzi  _et al._ (2009) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, Journal of Physics: Condensed Matter 21, 395502 (2009).
  * Giannozzi  _et al._ (2017) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. D. Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, Journal of Physics: Condensed Matter 29, 465901 (2017).
  * Hamann (2013) D. R. Hamann, Phys. Rev. B 88, 085117 (2013).
  * Hamann (2017) D. R. Hamann, Phys. Rev. B 95, 239906 (2017).
  * van Setten  _et al._ (2018) M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete, D. Hamann, X. Gonze, and G.-M. Rignanese, Computer Physics Communications 226, 39 (2018).
  * Monkhorst and Pack (1976) H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
  * Marzari and Vanderbilt (1997) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
  * Souza  _et al._ (2001) I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001).
  * Calandra  _et al._ (2010) M. Calandra, G. Profeta, and F. Mauri, Phys. Rev. B 82, 165111 (2010).
  * Marini  _et al._ (2019) G. Marini, P. Barone, A. Sanna, C. Tresca, L. Benfatto, and G. Profeta, Phys. Rev. Materials 3, 114803 (2019).
  * Kresse and Furthmüller (1996a) G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996a).
  * Kresse and Furthmüller (1996b) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996b).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).

## Acknowledgements

L. B., C. T. and G. B. Bachelet acknowledge support from Bando Ateneo
Sapienza, 2017-2020. G. P. acknowledges financial support from the Italian
Ministry for Research and Education through PRIN-2017 project “Tuning and
understanding Quantum phases in 2D materials - Quantum 2D” (IT-MIUR Grant No.
2017Z8TS5B).