# In-silico synthesis of lowest-pressure high-$T_{\text{c}}$ ternary
superhydrides

Roman Lucrezi  Institute of Theoretical and Computational Physics, Graz
University of Technology, NAWI Graz, 8010 Graz, Austria  Simone Di Cataldo
Institute of Theoretical and Computational Physics, Graz University of
Technology, NAWI Graz, 8010 Graz, Austria  Dipartimento di Fisica, Sapienza
Università di Roma, 00185 Rome, Italy  Wolfgang von der Linden  Institute of
Theoretical and Computational Physics, Graz University of Technology, NAWI
Graz, 8010 Graz, Austria  Lilia Boeri<EMAIL_ADDRESS>Dipartimento di
Fisica, Sapienza Università di Roma, 00185 Rome, Italy  Enrico Fermi Research
Center, Via Panisperna 89 A, 00184 Rome, Italy  Christoph Heil
<EMAIL_ADDRESS>Institute of Theoretical and Computational Physics,
Graz University of Technology, NAWI Graz, 8010 Graz, Austria

###### Abstract

We report the theoretical prediction of two high-performing hydride
superconductors BaSiH8 and SrSiH8. They are thermodynamically stable above
pressures of 130 and $174\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$,
respectively, and metastable below that. Employing anharmonic phonon
calculations, we determine the minimum pressures of dynamical stability to be
around $3\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for BaSiH8 and
$27\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for SrSiH8, and using the fully
anisotropic Migdal-Eliashberg theory, we predict $T_{\text{c}}$’s around 71
and $126\text{\,}\mathrm{K}$, respectively. We also introduce a method to
estimate the lowest pressure of synthesis, based on the calculation of the
enthalpy barriers protecting the BaSiH8 $Fm\bar{3}m$ structure from
decomposition at various pressures. This kinetic pressure threshold is
sensibly higher than the one based on dynamic stability, but gives a much more
rigorous limit for synthesizability.

##  I Introduction

The unexpected discovery of high-temperature superconductivity in high-
pressure hydrides has revamped the hopes of ending the century-long quest for
superconductivity at ambient conditions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13]. In less than five years, superhydrides have established higher and
higher records for critical temperatures, starting with SH3
($203\text{\,}\mathrm{K}$) [3], LaH10 ($265\text{\,}\mathrm{K}$) [5, 4], and
C-S-H ($288\text{\,}\mathrm{K}$) [9].

While it is very exciting to strive for new superconductors with even higher
$T_{\text{c}}$’s attaining hot superconductivity [14, 15, 16], lowering the
required stabilization pressures while retaining $T_{\text{c}}$ above the
boiling point of nitrogen ($77\text{\,}\mathrm{K}$) is of even greater
importance [17, 18, 19, 20, 21, 22, 16]. In fact, the discovery of a
conventional ($s$-wave) superconductor with these properties would open up a
wide array of technological applications in key strategical sectors such as
energy conservation, climate change, and medicine.

After exhausting the search space of binary hydrides, the focus of
superhydride research is rapidly shifting to ternary hydrides, where the
parameter space is much larger [6, 17]. We have recently predicted, for
example, that a ternary hydride with LaBH8 composition could form an
$Fm\bar{3}m$ ternary sodalite-like clathrate (SLC) structure and remain stable
down to a critical pressure
$p_{\text{c}}\sim$35\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$, with a
$T_{\text{c}}=$126\text{\,}\mathrm{K}$$ [19, 16]. Our findings were later
confirmed by independent studies on the La-B-H system [21, 23]. The predicted
$p_{\text{c}}$ of LaBH8 is a factor of four lower than in binary hydrides,
where $p_{\text{c}}$’s are in the Megabar range, but still too high to
envision any large-scale applications for this particular compound. However,
through the identification of the $Fm\bar{3}m$ $XY$H8 structural template, the
discovery of LaBH8 paved the road to the study of a whole new family of
potential high-$T_{\text{c}}$ ternary hydrides, where $p_{\text{c}}$ and
$T_{\text{c}}$ may be improved even further.

In this work, using first-principles methods for crystal structure prediction
and superconductivity, we identify two high-$T_{\text{c}}$ alkaline-
earth/silicon superhydrides, BaSiH8 and SrSiH8. We predict that both compounds
will spontaneously form in the $Fm\bar{3}m$ $XY$H8 structure at high pressures
(130 and $174\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, respectively), and
remain dynamically stable down to much lower pressures ($p_{\text{c}}=3$ and
$27\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$), with superconducting
$T_{\text{c}}$’s of 71 and $126\text{\,}\mathrm{K}$, respectively.

We also succeed in deriving a rigorous limit for the stability of BaSiH8,
calculating explicitly the energy barrier protecting the metastable
$Fm\bar{3}m$ structure from decomposition (kinetic stability) as a function of
pressure, using the Variable-Cell Nudged Elastic Band (VCNEB) method [24]. We
find that, indeed, $Fm\bar{3}m$ BaSiH8 should remain metastable at pressures
well below $130\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. However, the kinetic
critical pressure $p_{\text{kin}}$ determined by the energy barrier is
significantly higher ($30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$) than the
value estimated from anharmonic lattice dynamics. Similar discrepancies
between dynamical and kinetic pressure of stability may explain the systematic
underestimation of the predicted pressures of (meta)stability found in other
high-pressure hydrides compared to experiments [25, 11, 26, 27].

##  II Results

Figure 1: Elemental composition and phase space search. a Dynamical stability
of the $XY$H8 structure at $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for
various combinations of $X$ and $Y$ elements. Dynamically stable and unstable
compounds are indicated by red crosses and green ticks, respectively. b
Crystal structure of $Fm\bar{3}m$ $XY$H8 in the fcc conventional unit cell; H
atoms are shown in blue, H-H bonds in grey, and the lighter (heavier) atom in
orange (green). c Ternary convex hull for BaSiH8 at
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ \- side view; sampled structures
are shown as orange dots. d Ternary convex hull for BaSiH8 at
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ \- top view. In c and d the color
scale (from orange to blue) represents the depth of the hull with respect to
the elemental composition, from 0 to
$-0.7\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$;
stable compositions are shown as blue dots and are labeled.

Initial screening: BaSiH8 and SrSiH8 were first identified through a high-
throughput screening of possible substitutions of La and B by neighbouring
elements into the $Fm\bar{3}m$ $XY$H8 structure, as shown in Fig. 1a. In this
structure – Fig. 1b – hydrogen atoms form rhombicuboctahedral cages around
lanthanum; boron, being much smaller, fills the six cubic voids surrounding
the cages. This structure permits a very efficient realization of a mechanical
precompression mechanism observed in SLC binary hydrides [28, 25]. To
elaborate: in addition to the large central atom (La), also the smaller atom
in the interstitials (B) exerts an additional pressure on the hydrogen
sublattice, lowering the minimum stabilization pressure $p_{\text{c}}$ [19].

We assumed that, considering different combinations of large ($X$) and small
($Y$) atoms, the superconducting properties of LaBH8 could be improved even
further. Aiming at identifying compounds with low critical pressures, we
performed structural relaxations at
$50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for all combinations of
$X=[\text{K, Rb, Cs, Ca, Sr, Ba, Sc, Y, and La}]$ and $Y=[\text{B, Al, Ga, C,
Si, and Ge}]$ elements in the $Fm\bar{3}m$ $XY$H8 template, and evaluated the
dynamical stability of the resulting compounds by calculating the harmonic
phonon dispersions on a 4$\times$4$\times$4 grid in reciprocal space.

As shown in Supplementary Figure 6 of the Supplementary Material (SM), the
lattice constants of the relaxed structures exhibit an almost perfect linear
dependence on the sum of the $X$ and $Y$ empirical atomic radii
$R=R_{X}+R_{Y}$ [29], with slopes determined by the total number of valence
electrons $N_{e}=N_{X}+N_{Y}$. Out of the 54 compounds investigated, only
three compounds with $N_{e}=6$ – green ticks in Fig. 1a – are dynamically
stable at $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$: Apart from the already
reported LaBH8 [19], we also found two silicon hydrides, BaSiH8 and SrSiH8.
Both compounds exhibit a larger $R$ than LaBH8, suggesting that mechanical
precompression in both compounds will be more efficient, and hence their
$p_{\text{c}}$ may be lower. In particular, BaSiH8, where $R$ is considerably
larger than in the other two compounds, looks very promising in this respect.
Indeed, our harmonic phonon calculations yield $p_{\text{c}}=5$ and
$30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for BaSiH8 and SrSiH8,
respectively, lower than the
$p_{\text{c}}=$40\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$ in LaBH8.

Convex hulls and phase diagrams: Having established that BaSiH8 and SrSiH8 are
dynamically stable in the $Fm\bar{3}m$ structure close to ambient pressure, we
determined the pressures at which the two compounds may spontaneously
crystallize in the $Fm\bar{3}m$ structure starting from appropriate
precursors. To this end, we computed the full ternary convex hulls for the two
ternary $X$-Si-H systems at ambient pressure and at
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, employing ab-initio variable-
composition evolutionary crystal structure prediction methods as implemented
in Uspex [30, 31]. To construct each hull, we sampled around 20000 structures
and 5000 compositions, including the contribution of the zero-point energy
(ZPE) from the ionic vibrations for structures close to the hull. Our results
are in good agreement with previously reported structural searches on the Ba-H
and Sr-H binary hydrides [32, 33]. Further computational details are given in
Supplementary Method 2 and Supplementary Note 2 in the SM. An example is
illustrated in Fig. 1c-d, where we show the ternary convex hull of the Ba-Si-H
system at $100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. (The analogous convex
hull for the Sr-Si-H system at 0 and 100 GPa is shown in Supplementary Figure
9 and 10 of the SM.)

In the Ba-Si-H system, the BaSiH8 composition is thermodynamically stable,
i.e., it lies on the hull at $100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ in a
distorted $P1$ phase. The $Fm\bar{3}m$ phase is slightly higher in enthalpy
($32\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$),
but becomes enthalpically favourable above
$130\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. In the Sr-Si-H system at
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, according to our calculations,
SrSiH8 should decompose into SrSiH6 and SrSiH12, which are the stable ternary
compositions along with Sr2SiH10. The corresponding structures are shown in
Supplementary Figure 12 of the SM. However, the $Fm\bar{3}m$ phase lies only
$54\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$
above the convex hull. From a comparison of the formation enthalpies, we
predict that it should become stable at
$174\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$.

Anharmonic lattice dynamics:

Figure 2: Vibrational structure and electron-phonon properties. a Phonon
dispersion of BaSiH8 at $5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$; the
thickness of the dark blue lines is proportional to the mode- and wave-vector
resolved coupling constant $\lambda_{\mathbf{q},\nu}$. Solid (dotted) lines
correspond to anharmonic (harmonic) results. b Phonon DOS shown as solid black
line, $\alpha^{2}F(\omega)$ as filled curve, and $\lambda(\omega)$ as solid
blue line. Panels c and d as a and b but for SrSiH8 at
$30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$.  Figure 3: Electronic band
structure and density of states. a Electronic band structure of BaSiH8 at
$5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ and b corresponding total DOS
(solid black) and projected H DOS (filled blue). Panels c and d as a and b but
for SrSiH8 at $30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$.

Preliminary phonon calculations in the harmonic approximation indicated that
both BaSiH8 and SrSiH8 experience remarkable phonon softening at lower
pressures. More specifically, we find soft phonon modes, mainly at $\Gamma$
($T_{2g}$, $E_{g}$, and $A_{1g}$) and $X$ ($E_{g}$, and additionally $A_{2g}$
in SrSiH8), as shown in Fig. 2. Eventually, imaginary frequencies appear for
the $E_{g}$ mode at $X$ below $5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for
BaSiH8 and $40\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for SrSiH8. (A full set
of phonon dispersions at all pressures considered is provided in the SM.)

Soft-mode behaviour, associated with strong anharmonicity, has been reported
for many hydrogen-rich materials [26, 34, 35, 36, 25]. Anharmonic lattice
effects have been shown to crucially affect the range of dynamical stability,
phonon frequencies and eigenvectors of superhydrides, and their inclusion is
essential to obtain accurate estimates of these quantities.

In order to account for anharmonic effects on the phonon dispersions of BaSiH8
and SrSiH8, we evaluated the adiabatic potential energy surface (APES) for
every soft mode of a 2$\times$2$\times$2 wave-vector grid, which includes
explicitly the special points $\Gamma$, $L$, and $X$, and solved the resulting
Schrödinger equations. (More information can be found in Supplementary Method
4 of the SM, and Ref. [37].)

The harmonic and the anharmonic phonon dispersions for BaSiH8 and SrSiH8 close
to their critical pressures of stability are shown as dashed and full lines in
Fig. 2. Anharmonicity causes a considerable hardening of the $T_{2g}$ (at
$\Gamma$) and $E_{g}$ (at $\Gamma$ and $X$) modes, leading to a decrease of
the critical pressure of dynamical stability $p_{\text{c}}$: Taking this
hardening into account, SrSiH8 is stable down to a pressure of
$27\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ and, even more excitingly, BaSiH8
down to $3\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$.

Electronic structure and superconductivity: Having determined that both BaSiH8
and SrSiH8 remain dynamically stable close to ambient pressure, we are left
with the question whether at these relatively low pressures they can still be
considered superhydrides. A first positive indication comes from the analysis
of their electronic structure.

In Fig. 3 we show the electronic dispersions and DOS for BaSiH8 (left) and
SrSiH8 (right), at 5 and $30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$,
respectively. Despite a scaling of the total bandwidth due to the different
pressures, the band structures are qualitatively very similar. The Fermi level
is located just above a large DOS shoulder; in this region and up to
$-5.5\text{\,}\mathrm{e}\mathrm{V}$ for BaSiH8
($-3.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ for SrSiH8), the bands are of
purely hydrogen character. This is an important prerequisite for
high-$T_{\text{c}}$ conventional superconductivity in hydrides, since it can
imply a strong electron-phonon ($ep$) coupling between electronic states to
the light hydrogen sublattice [38].

Similar conclusions can be inferred from the electron localization function
(ELF) of the two compounds (see Supplementary Figure 15 in the SM), where, in
line with what we observed in LaBH8 [19], we find an increased charge
localization in the vicinity of the H and Si atoms and between H-H, but not
between Sr/Ba and H or Si and H. This supports the idea that neither Sr/Ba,
nor Si form bonds with H, and thus act on the H sublattice essentially as
mechanical spacer, as in binary sodalite-like clathrate hydrides [25, 34, 6,
39].

The Fermi surface topology is the same in the two compounds: a large, spheric-
like electron pocket is centered around the Brillouin zone center, while a
more complicated hole-like network extends around the faces of the Brillouin
zone, enclosing the $X$ and $W$ points (see the insets of Fig.4).

Figure 4: Superconducting gap as a function of temperature. $\Delta(T)$ for
BaSiH8 (blue) at $5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ and for SrSiH8
(green) at $30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ calculated using the
fully anisotropic ME theory; dashed lines indicate the mean value as a guide
to the eye. The insets in the top right and bottom left corner show the Fermi
surface distribution of $\Delta_{\mathbf{k}}$ at $10\text{\,}\mathrm{K}$ for
BaSiH8 and SrSiH8 respectively; blue/red correspond to
$\text{min}(\Delta)$/$\text{max}(\Delta)$.

These qualitative electronic structure arguments are confirmed by actual
calculations of the superconducting properties. Fig. 4 shows the energy
distribution of the superconducting gap $\Delta$ as a function of temperature
$T$ for BaSiH8 at $5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ pressure and for
SrSiH8 at $30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, obtained by solving the
anisotropic Migdal-Eliashberg (ME) equations on a 30$\times$30$\times$30
$\mathbf{k}-$ and $\mathbf{q}$-grid using the anharmonically corrected phonon
dispersions with the Epw code [40, 41]. (The superconducting properties at all
other considered pressures and further computational details are provided in
the SM.) We observe two distinct superconducting gaps: The inset of the figure
shows that large $\Delta$ values correspond to the $\Gamma$-centered,
spherical electron pocket, while lower values occur on the hole-like tubular
network around the $X$ and $W$ points. The superconducting critical
temperatures are predicted to be 71 and $126\text{\,}\mathrm{K}$ for BaSiH8
and SrSiH8, respectively.

Figure 5: Superconducting properties as a function of pressure. a
$T_{\text{c}}$ from anisotropic ME theory, b $\omega_{\text{log}}$, and c
$\lambda$ (bottom panel) as functions of pressure for BaSiH8 (blue) and SrSiH8
(green). Circles and solid lines (crosses and dashed lines) indicate
anharmonically corrected (harmonic) calculations. The contributions to
$\alpha^{2}F(\omega)$ from the $E_{g}$ mode at $X$ are set to zero in the
harmonic calculations for SrSiH8 at 30 and $40\text{\,}\mathrm{GPa}$, as the
corresponding phonon frequencies are imaginary.

Further details on the origin of the remarkable $T_{\text{c}}$’s of BaSiH8 and
SrSiH8 can be obtained from an analysis of the distribution of their $ep$
coupling over phonon branches. In panel b of Fig. 2, the mode and wave-vector
resolved $ep$ coupling $\lambda_{\mathbf{q},\nu}$ are overlayed onto the
phonon dispersions as fat bands; panel d of the same figure shows the
Eliashberg spectral function $\alpha^{2}F(\omega)$, and the total frequency-
dependent $ep$ coupling parameter $\lambda(\omega)$.

Figure 6: VCNEB calculations for BaSiH8 at different pressures. Minimum energy
path (MEP) for BaSiH8 at various pressures, calculated using the VCNEB method.
Each point represents an individual crystal structure (image) along the path.
Particularly relevant intermediate states are shown.

The figure clearly shows that the largest contributions to the total coupling
come from low-energy modes; a substantial fraction is associated to soft
phonon modes around $\Gamma$ and $X$. In the case of BaSiH8, for example, we
estimate that the phonons related to $\Gamma-T_{2g}$ contribute roughly
$40\text{\,}\%$ to the total $\lambda$ and around $X-E_{g}$ about
$35\text{\,}\%$. These modes are of purely hydrogen character and their
patterns can be more easily visualized in terms of the H cubes around Si: The
$T_{2g}$ mode at $\Gamma$ is a centrosymmetric stretching and squeezing along
one of the space diagonals of the H cube, while the $E_{g}$ mode at $X$ is a
non-deforming rotation around a face normal of the H cube, with alternating
phase in two neighbouring cells.

Fig. 5 shows the evolution of the superconducting properties of BaSiH8 and
SrSiH8 as a function of pressure from the lowest dynamical stability pressure
up to $100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$; open circles and solid
lines correspond to anharmonic results, while dashed lines refer to harmonic
values. Interestingly, $T_{\text{c}}$ is approximately constant with pressure
– panel a; this is true even close to $p_{\text{c}}$, where one would expect a
strong deviation due to anharmonicity. An inspection of panels b and c shows
that this happens because of the compensating effect of phonon hardening on
$\omega_{\text{log}}$ and $\lambda$. The effect is more marked in SrSiH8 and
in BaSiH8 close to the instability point, where harmonic and anharmonic
dispersions diverge the most. For SrSiH8 at $30\text{\,}\mathrm{GPa}$, we find
an anharmonic (harmonic) $\lambda$ of 3.21 (3.96) and an $\omega_{\text{log}}$
of $33.1\text{\,}\mathrm{meV}$ ($22.3\text{\,}\mathrm{meV}$), and for BaSiH8
at $5\text{\,}\mathrm{GPa}$, we find a $\lambda$ of 1.61 (2.55) and an
$\omega_{\text{log}}$ of $33.3\text{\,}\mathrm{meV}$
($23.7\text{\,}\mathrm{meV}$).

Metastability: In all theoretical studies of high-pressure hydrides performed
so far, the range of metastability of high-pressure phases has been assumed to
coincide with the range of dynamical stability, eventually including quantum
corrections to lattice dynamics. However, a comparison of these predictions
and available experimental data suggests that theoretical estimates of the
critical pressure are systematically lower than experimental values [25, 11,
26, 27].

Indeed, dynamical stability is only a prerequisite for thermodynamic
metastability. The latter is determined by the existence of an energy
(enthalpy) barrier protecting a metastable phase from decomposition into other
phases [42]. Attempts to quantitatively estimate the barrier height are
extremely rare. For $Fm\bar{3}m$ BaSiH8, where the dynamical instability
pressure $p_{\text{c}}$ is extremely close to ambient pressure, this issue is
obviously crucial, since the presence of a sufficiently high barrier could be
considered the definitive proof of synthesizability.

In this work, we estimated the barrier height using the VCNEB as implemented
in the Uspex code [24]. In this method, a number of intermediate structures
(images) are created between the metastable structure of interest and the
ground-state structure at a given pressure; elastic forces between each image
are added to the ‘physical’ forces, and the whole chain of images thus created
is finally relaxed to obtain the energy/enthalpy profile of the transition.

We ran VCNEB simulations for BaSiH8 at six different pressures: 10, 25, 35,
50, 100, and $200\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. As end-members for
the VCNEB path, we chose the $Fm\bar{3}m$ phase and a $P1$ phase, identified
as the ground-state structure at $10\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$
through a fixed-composition structural search. Due to its large unit cell and
low symmetry, it can be assumed that the $P1$ phase, relaxed at the different
pressures, approximates quite well the true ground-state of the system;
indeed, at all pressures the $P1$ BaSiH8 phase lies only a few meV above the
hull. In practice, analyzing the VCNEB images in Fig. 6, we observe that the
transition from the $Fm\bar{3}m$ to the $P1$ phase corresponds to the
decomposition of BaSiH8 into BaSiH6 \+ H2, with the expulsion of hydrogen in
molecular form.

The potential barrier, shown in Supplementary Figure 3, decreases with
pressure from
$153\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$
at $100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ to
$57\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$
at $35\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$; a sharp transition is visible
at $25\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, where the kinetic barrier
abruptly drops to
$9\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$.
Although a small barrier survives down to
$10\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, this sharp decrease is the
signature of an impending kinetic instability, i.e., the metastable state will
be short-lived and most likely will not be observed in experiments.

Combining the convex hull results in Fig. 1 with the VCNEB analysis, we can
argue with confidence that an $Fm\bar{3}m$ BaSiH8 phase could be synthesized
above $100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, and retained down to
$\sim$$30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, where a clearly visible
enthalpy barrier exists. At lower pressures, metastable $Fm\bar{3}m$ BaSiH8
will decompose, even though (anharmonic) lattice dynamics calculations predict
it to be stable. Hence, kinetic stability poses a stricter bound for
synthesizability than dynamical stability.

In addition to the promising theoretical synthesizability, both materials
appear to be convenient in the experimental setting. As the ambient-pressure
convex hulls in Figs. S7 and S9 reveal, both systems feature known stable
orthorhombic binary monosilicides, namely BaSi and SrSi [43, 44], which serve
as adequate starting materials having the target composition of Ba/Sr and Si.
The monosilicide can be loaded in a diamond anvil cell together with the
common solid hydrogen storage medium ammonia borane (H3NBH3) which releases
the hydrogen upon heating and forms the refractory compound boron nitride
(BN).

Discussion: In summary, our study shows that indeed the superconducting
properties of the $XY$H8 template can be optimized by a suitable choice of the
$X$ and $Y$ elements. The two silicides identified in this work represent an
improvement compared to LaBH8: for SrSiH8 we predict a dynamical stability
pressure $p_{\text{c}}$ of $27\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, with a
$T_{\text{c}}$ of $127\text{\,}\mathrm{K}$. Using the figure of merit $S$
introduced by the authors of Ref. [17], this means passing from 1.3 in H3S and
LaH10 to 2.2 in LaBH8 to 2.7 in SrSiH8. Even more remarkably, BaSiH8 is
predicted to be dynamically stable down to pressures of
$3\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, with a critical temperature of
$71\text{\,}\mathrm{K}$, which is substantially higher than all established
[45] and claimed [46] experimental $T_{\text{c}}$ records for conventional
superconductors at ambient pressure.

VCNEB calculations demonstrate that the shape of the potential energy
landscape of BaSiH8 is favorable for its synthesis in a wide range of
pressures. In contrast to the binary La-B system, the Ba-Si system features
also the ideal starting material for the synthesis, namely the stable
monosilicide BaSi, and a possible path involves synthesis at high pressure
($p>$130\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$) and/or laser heating, and
rapid quenching of the resulting phase to lower pressure, i.e., down to
$\sim$$30\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, where the metastable
$Fm\bar{3}m$ crystal structure is protected by a sizable potential barrier.
This defines a kinetic threshold pressure, $p_{\text{kin}}$, which is
substantially higher than $p_{\text{c}}$.

By defining a concrete method to estimate the synthesizability of a proposed
structure, our study sets a new standard for the ab-initio design of new
superconductors at high pressures, based on the more rigorous concept of
kinetic stability, rather than dynamical stability. The existence of a
distinct kinetic stability criterion may also be invoked to explain why many
long-standing predictions of high-$T_{\text{c}}$ superconductors have not been
realized experimentally [47, 48, 49].

We strongly believe that the proposed method represents a major step forward
towards achieving high-$T_{\text{c}}$ conventional superconductivity at room
pressure.

##  III Methods

###  III.1 Crystal structure prediction

Crystal structure prediction runs were carried out using evolutionary
algorithms as implemented in the Uspex package [30, 31, 50]; the underlying
total energy and structural relaxation calculations were performed using
plane-waves and pseudopotentials as implemented in the Vienna ab-initio
Simulation Package Vasp [51]. Further computational details are provided in
Supplementary Method 2 of the SM.

###  III.2 Electronic and vibrational properties

All density-functional theory (DFT) and density-functional perturbation theory
(DFPT) calculations of electronic and vibrational properties were carried out
using the plane-wave pseudoptential code Quantum Espresso [52], scalar-
relativistic optimized norm-conserving Vanderbilt pseudopotentials (ONCV)
[53], and the PBE-GGA exchange and correlation functional [54]. Computational
details are provided in Supplementary Method 1 of the SM.

###  III.3 Phase transition paths

The phase transition path between the $P1$ and $Fm\bar{3}m$-BaSiH8 was
evaluated using the VCNEB method as implemented in Uspex [30, 24], using
variable elastic constants and a variable number of images between the
endpoints. The energy and forces were calculated using Vasp. Further
computational details are provided in Supplementary Method 3 of the SM.

###  III.4 Migdal-Eliashberg theory

The interpolation of the $ep$ matrix elements onto dense wave-vector grids and
the subsequent self-consistent solution of the fully anisotropic Migdal-
Eliashberg equations were done with Epw [41]. Based on our previous work on
LaBH8, where we calculated the Morel-Anderson pseudopotential $\mu^{*}$ from
first principles using $GW$ and found consistent values for $\mu^{*}$ of about
0.1 [25, 38, 19], we chose the same value in the current work. Further
computational details are provided in Supplementary Method 5 of the SM.

###  III.5 Phonon anharmonicity

Anharmonic corrections to phonon frequencies were obtained by explicitly
solving the Schrödinger equation for the APES of every soft mode on a
2$\times$2$\times$2 wave-vector grid. By calculating and solving 2D APES, we
also checked that phonon-phonon interactions for various modes and wave
vectors can be neglected in a good first approximation. The interpolation of
the real-space force constants obtained on the 2$\times$2$\times$2
$\mathbf{q}$-grid was performed using the corresponding harmonic support DFPT
solution as implemented in the CellConstructor package [55]. Further
computational details are provided in Supplementary Method 4 of the SM.

##  IV Data availability

The authors confirm that the data supporting the findings of this study are
available within the article and its supplementary materials. Further
information is available upon request.

##  V Acknowledgments

We thank Dmitrii Semenok (Skolkovo Institute of Science and Technology) for
pointing out the experimental convenience of the two proposed materials.

This research was funded by the Austrian Science Fund (FWF) P 30269-N36 and P
32144-N36. For the purpose of open access, the authors have applied a CC BY
public copyright license to any Author Accepted Manuscript version arising
from this submission. This work was supported by the dCluster of the Graz
University of Technology and the VSC-4 of the Vienna University of Technology.
L.B. acknowledges support from Fondo Ateneo-Sapienza 2017-2019. S.D.C.
acknowledges computational resources from CINECA, proj. IsC90-HTS-TECH_C

##  VI Author contributions

R.L. and S.D.C. performed the calculations, and L.B. and C.H. conceived and
supervised the project. All authors contributed to the discussion of the
results and participated in preparing the manuscript.

##  VII Competing interests

The authors declare no competing financial or non-financial interests.

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