#  Potential high-$T_{\mathrm{c}}$ superconductivity in YCeHx and LaCeHx under
pressure

Peng Song  School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa
923-1292, Japan  Zhufeng Hou  State Key Laboratory of Structural Chemistry,
Fujian Institute of Research on the Structure of Matter, Chinese Academy of
Sciences, Fuzhou 350002, China  Kousuke Nakano  School of Information Science,
JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan  International School for
Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy  Kenta Hongo
Research Center for Advanced Computing Infrastructure, JAIST, Asahidai 1-1,
Nomi, Ishikawa 923-1292, Japan  Ryo Maezono  School of Information Science,
JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan

###### Abstract

Lanthanum, yttrium, and cerium hydrides are the three most well-known
superconducting binary hydrides (La-H, Y-H, and Ce-H systems), which have
gained great attention in both theoretical and experimental studies. Recent
studies have shown that ternary hydrides composed of lanthanum and yttrium can
achieve high superconductivity around 253 K. In this study we employ the
evolutionary-algorithm-based crystal structure prediction (CSP) method and
first-principles calculations to investigate the stability and
superconductivity of ternary hydrides composed of (Y, Ce) and (La, Ce) under
high pressure. Our calculations show that there are multiple stable phases in
Y-Ce-H and La-Ce-H systems, among which $P4/mmm$-YCeH8, $P\bar{6}m2$-YCeH18,
$R\bar{3}m$-YCeH20, $P4/mmm$-LaCeH8, and $R\bar{3}m$-LaCeH20 possessing H18,
H29 and H32 clathrate structures can maintain both the thermodynamic and
lattice-dynamic stabilities. In addition, we also find that these phases also
maintain a strong resistance to decomposition at high temperature. Electron-
phonon coupling calculations show that only three of these five phases can
exhibit high-temperature superconductivity. The superconducting transition
temperatures ($T_{\mathrm{c}}$) of $R\bar{3}m$-YCeH20, $R\bar{3}m$-LaCeH20,
and $P\bar{6}m2$-YCeH18 are predicted using the Allen-Dynes-modified McMillan
formula to be 122 K at 300 GPa, 116 K at 250 GPa, and 173 K at 150 GPa,
respectively. Moreover, the pressure to stabilize $P\bar{6}m2$-YCeH18 can be
lowered to 150 GPa, suggesting an accessible condition for its high-pressure
synthesis.

##  I Introduction

Superconductivity has gotten a remarkable progress and attention owing to its
unique phenomenon and wide range of industrial applications. Tinkham (2004);
Gennes (2018); Linder and Robinson (2015); Josephson (1962); Bergen  _et al._
(2019); Islam  _et al._ (2014); Thomas  _et al._ (2016) On the other hand, the
extremely low temperature environment required for the manifestation of these
properties severely limits the breadth of their application. Therefore, the
pursuit for high(room)-temperature superconductivity has historically been one
of the most competitive and sought goals in superconductivity. Tinkham (1988);
Kalsi (2011); Flores-Livas  _et al._ (2020) According to the Bardeen-Cooper-
Schrieffer (BCS) theory, the superconducting transition temperature
($T\rm_{c}$) significantly correlates with the phonon vibration frequency.
Bardeen (1955); Cooper (1956); Bardeen  _et al._ (1957a, b) In this sense,
metallic hydrogen is considered to be the most ideal candidate for
high-$T\rm_{c}$ superconductivity due to its ultra-high phonon vibration
frequency. Ashcroft (1968) However, hydrogen metallization, also known as the
Wigner-Huntington transition, typically requires extremely high pressures,
making it difficult to study its superconductivity-related electrical
properties at these pressures. Wigner and Huntington (1935); Loubeyre  _et
al._ (1996); McMinis  _et al._ (2015); Eremets and Troyan (2011); Dalladay-
Simpson  _et al._ (2016); Dias and Silvera (2017); Castelvecchi (2017);
Celliers  _et al._ (2018) Ashcroft proposed the introduction of other elements
into hydrogen can provide the necessary pre-compression for the entire system,
allowing it to maintain metallicity and superconductivity at lower pressures.
Ashcroft (2004) Thus, metal hydrides are one of the most ideal candidates for
high-$T\rm_{c}$ superconductivity.

According to Ashcroft, almost all binary hydrides have been theoretically
screened out by density functional theory (DFT) calculations, and some
predictions have been verified by experiments. Semenok  _et al._ (2020); Peng
_et al._ (2017); Zhang  _et al._ (2020); Li and Peng (2017); Yu  _et al._
(2015); Einaga  _et al._ (2016); Drozdov  _et al._ (2015, 2019); Salke  _et
al._ (2019); Chen  _et al._ (2021a); Shao  _et al._ (2021a, b); Troyan  _et
al._ (2021); Chen  _et al._ (2021b); Kong  _et al._ (2021); Li  _et al._
(2014); Somayazulu  _et al._ (2019) Compared to binary superconducting
hydrides, research on multi-component superconducting hydrides appears to be
slightly less. Since multi-component hydrides contain more combinations of
elements, however, they are more promising in the search for
high(room)-temperature superconductivity. Song  _et al._ (2021); Ma  _et al._
(2017a, b); Shao  _et al._ (2019); Zheng  _et al._ (2021); Rahm  _et al._
(2017); Liang  _et al._ (2021); Sun  _et al._ (2019); Liang  _et al._ (2019a);
Wei  _et al._ (2020); Shi  _et al._ (2021); Song  _et al._ (2022a); Semenok
_et al._ (2021); Liang  _et al._ (2019b); Jiang  _et al._ (2021); Cui  _et
al._ (2020); Cataldo  _et al._ (2020); Song  _et al._ (2022b); Cataldo  _et
al._ (2022); Vocaturo  _et al._ (2022); Somayazulu  _et al._ (2019) For
example, the DFT calculations by Sun et al. Sun  _et al._ (2019) predicted
that magnesium hydride can dissociate the H2 molecule into atomic H by lithium
doping. This doping increases its electron concentration at the Fermi level
($E_{\mathrm{F}}$), enabling it to reach a $T\rm_{c}$ value of 473 K at 250
GPa. Sun  _et al._ (2019) Furthermore, a recent high-pressure experiment has
revealed that a small amount of carbon doping in H3S can result in room-
temperature superconductivity ($T\rm_{c}$ = 287.7 $\pm$ 1.2 K at 267 $\pm$ 10
GPa), Snider  _et al._ (2020) although the anomaly of AC susceptibility in low
temperature for the carbonaceous sulfur hydride has been questioned by Hirsch
et al. Hirsch and Marsiglio (2021); Hirsch (2022) and more strong experimental
data are needed to support the claim of the observation of room-temperature
superconductivity. Therefore, increasing the density of electronic states at
$E_{\mathrm{F}}$ by electrons doping in the binary hydride can affect the
$T\rm_{c}$ value; however this modification also can result in a decrease in
$T\rm_{c}$ in some cases. Nakanishi  _et al._ (2018); Amsler (2019); Guan  _et
al._ (2021)

Furthermore, most of the high-$T\rm_{c}$ superconducting hydrides reported
thus far are concentrated in the Mendeleev table’s ”lability belt”, and many
of these hydrides have unusual structural properties, such as a cage-like
structure surrounded by H. Semenok  _et al._ (2020) Some ternary
superconducting hydrides (e.g., ScYH6, ScCaH8, CaYH12, and so on) can be
easily obtained by substituting one of the metal atoms in the binary parent
compound. Liang  _et al._ (2019a); Wei  _et al._ (2020); Shi  _et al._ (2021);
Song  _et al._ (2022a); Semenok  _et al._ (2021) The crystal structure,
pressure range of stability, unit cell volume, and other properties of the
parent binary superconducting hydrides that correspond to these ternary
superconducting hydrides resemble each other. Semenok  _et al._ (2020);
Flores-Livas  _et al._ (2020) For instance, by combining two types of binary
hydrides with comparable structural features, it is possible to produced a
more stable ternary hydride with similar structural properties. To test this
hypothesis, we noticed that CeH9, YH9, and LaH10, CeH10 have comparable
qualities in a variety of properties, and their structural properties have
been proven in high-pressure experiments. Salke  _et al._ (2019); Chen  _et
al._ (2021b); Kong  _et al._ (2021); Drozdov  _et al._ (2019) The related
ternary superconducting hydrides, one of which is (La,Y)H10, has been
successfully synthesized experimentally, and the $T\rm_{c}$ observed under
high pressure also matched the theoretically predicted $T\rm_{c}$ quite well.
Semenok  _et al._ (2021) During this work, we are aware that a very recent
high-pressure experiment study shows that La-Ce-H may have stable
$I4/mmm$-(La,Ce)H4 and $Im\bar{3}m$-(La,Ce)H6 around 100 GPa. Chen  _et al._
(2022a)

Therefore, this work will concentrate on the YCeHx and LaCeHx systems, which
have not yet been well investigated, to explore whether there exist stable
compounds under high pressure. By employing the evolutionary algorithm for
crystal structure prediction, we have theoretically discovered
thermodynamically stable compounds of YCeH5,YCeH7,YCeH8, YCeH18, YCeH20,
LaCeH8, LaCeH18, and LaCeH20 in pressure range of 100-400 GPa. In addition, we
have investigated the dynamic stability and superconductivity of these
hydrogen-rich ternary hydrides within the harmonic approximation. All of them
are dynamically stable with increasing pressure, More importantly, YCeH18
exhibits extremely strong electron-phonon coupling with superconducting
transition temperatures of 173 K at 150 GPa.

##  II Method

The crystal-structure search for YCeHx and LaCeHx ($x$ = 2$\sim$10, 12, 14,
16, 18, and 20) at 100, 200, and 300 GPa was performed using the USPEX
(Universal Structure Predictor: Evolutionary Xtallography) Glass  _et al._
(2006); Lyakhov  _et al._ (2013) code. In the USPEX run, 200 structures with
1-2 formula units were randomly built for the initial generation, and then 100
structures for each of the subsequent generations were produced by 40%
heredity, 40% random, 10% mutation, and 10% soft mutation. Each structure was
optimized through 4 times of relaxation from low to high accuracy levels. The
structure relaxation was carried out by the density functional theory
calculations within the generalized gradient approximation (GGA) with Perdew-
Burke-Ernzerhof (PBE) exchange-correlation functional Perdew  _et al._ (1996)
as implemented in the VASP (Vienna ab initio simulation package) Kresse and
Furthmüller (1996); Kresse and Joubert (1999) code. The electron-ion
interaction was treated using projector-augmented-wave (PAW) potentials.
Kresse and Furthmüller (1996); Kresse and Joubert (1999) The Ce 4f electrons
were explicitly considered into the valence electrons. The cutoff energy for
plane waves was set to 600 eV and the smallest allowed spacing between the
$k$-points in the irreducible Brillouin zone was set to be 0.2 $\AA^{-1}$. To
study the chemical bonding and electronic structures of some selected stable
YCeHx and LaCeHx, we have carried out the static calculations using the VASP
code and then performed the Crystal Orbital Hamilton Population (COHP) and
Crystal Orbital Bond Index (COBI) analyses using the Lobster 4.1.0 package
Müller  _et al._ (2021) together with the pbeVaspFit2015 basis set. The
electronic band structures were also checked by taking into account the spin-
orbit coupling (SOC) interaction and the correction of the on-site Coulomb
interaction of Ce 4f orbitals. The DFT+U method of Dudarev et al. Dudarev  _et
al._ (1998) was employed in the latter case. The Hubbard U parameter for Ce 4f
orbitals was determined by the density-functional perturbation theory
approach, as implemented in the HP package Timrov  _et al._ (2022). The
obtained U for Ce 4f orbitals is 4.2 eV, which is very close to the value
(i.e., 4.5 eV) adopted by Wang et al. Wang  _et al._ (2021) for their study on
CeH9. The exchange interaction parameter $J$ for Ce 4f orbitals was taken from
Ref. 77 and set to 0.5 eV.

The formation enthalpies of La-Ce-H and Y-Ce-H ternary systems (herein
abbreviated to $A$-Ce-Y with $A$ = La and Y), relative to the elemental solids
Y Chen  _et al._ (2012), Ce Vohra  _et al._ (1999), La Chen  _et al._ (2022b),
and H Pickard and Needs (2007), were calculated at each pressure defined as
below:

$\Delta\mathcal{H}=\frac{1}{x+y+z}\left[\mathcal{H}({A}_{x}\mathrm{Ce}_{y}\mathrm{H}_{z})-x\mathcal{H}({A})-y\mathcal{H}(\mathrm{Ce})-z\mathcal{H}(\mathrm{H})\right],$
(1)

where $\mathcal{H}(A_{x}\mathrm{Ce}_{y}\mathrm{H}_{z})$ is the enthalpy per
formula unit (f.u.) of the A-Ce-H compound; $\mathcal{H}(A)$,
$\mathcal{H}(\mathrm{Ce})$, and $\mathcal{H}(\mathrm{H})$ are the reference
enthalpies of $A$, Ce, and H single phases, respectively. The thermodynamic
stability of the predicted phase was determined by comparing the formation
enthalpies with respect to the convex hull energy:

$E_{\mathrm{above\\_hull}}=\Delta\mathcal{H}-\Delta\mathcal{H}_{f},$ (2)

where $\Delta\mathcal{H}_{f}$ is a convex hull energy obtained by constraining
the minimum value of the total enthalpies of a linear combination of stable
phases Akbarzadeh  _et al._ (2007), which can be computed using the ConvexHull
module in scipy. Virtanen  _et al._ (2020) $E_{\mathrm{above\\_hull}}$ is
energy above the convex hull. $E_{\mathrm{above\\_hull}}=0$ means that the
corresponding ternary phase is stable, namely, such a phase would not
decompose into any combination of elementary, binary, or other ternary phases.
The Gibbs free energies at finite temperatures were calculated within the
quasiharmonic approximation using the PHONOPY code. Togo and Tanaka (2015)
After incorporating the Gibbs free energies, we can drive the stability for
the whole system at high temperatures by computing the convex hull.

The phonon dispersion and electron-phonon coupling (EPC) of the stable ternary
phases were carried out using the Quantum ESPRESSO (QE) code Giannozzi  _et
al._ (2009) with the PAW method and the GGA-PBE exchange-correlation
functional. In particular, the Ce PAW potential was generated with a valence
configuration of 4d105s25p64f0.55d1.56s2, as provided by the PSLibray of Dal
Corso Corso (2014). The cutoff energy for plane-wave basis sets in the QE
calculations was set to 180 Ry. The $q$-point mesh (and integral $k$-points
mesh) in the first Brillouin-zone for the EPC calculations was set as follows:
$5\times 5\times 2$ ($20\times 20\times 8$) for $A$CeH8, $5\times 5\times 3$
($20\times 20\times 12$) for $A$CeH18 and $4\times 4\times 4$ ($20\times
20\times 20$) for $A$CeH20 ($A$ = La and Y). The the Allen-Dynes-modified
McMillan formula (AD) and the Eliashberg function derived from the EPC
calculation were used to predict the superconducting critical temperature as
follows McMillan (1968); Allen and Dynes (1975):

$T_{\mathrm{c}}=\frac{\omega_{\rm{log}}f_{1}f_{2}}{1.2}\rm{exp}\left(\frac{-1.04(1+\lambda)}{\lambda(1-0.62\mu^{\ast})-\mu^{\ast}}\right),$
(3)

with

$\displaystyle f_{1}f_{2}$ $\displaystyle=$
$\displaystyle\sqrt[3]{1+\left[\frac{\lambda}{2.46(1+3.8\mu^{\ast}))}\right]^{\frac{3}{2}}}$
(4)
$\displaystyle\times\left[1-\frac{\lambda^{2}(1-\omega_{2}/\omega_{\rm{log}})}{\lambda^{2}+3.312(1+6.3\mu^{\ast})^{2}}\right].$

When $f_{1}f_{2}$ = 1, Eq. (3) is restored to the original McMillan formula
(McM). The $\mu^{\ast}$ is the Coulomb pseudopotential, which is defined as
below Allen and Dynes (1975)

$\mu^{\ast}=\frac{\mu}{1+\mu\rm{log}(\omega_{el}/\omega_{ph})},$ (5)

where $\mu$ is instantaneous repulsion, the
$\rm{log}(\omega_{el}/\omega_{ph})$ is a ratio of propagation time. For most
metal compounds, the value of $\mu^{\ast}$ is roughly in the range of 0.1 to
0.16 by incorporating previous experimental and calculated data. The widely
accepted value of 0.1 for $\mu^{\ast}$ was used herein. The electron-phonon
coupling constant ($\lambda$), logarithmic average phonon frequency
($\omega_{\rm{log}}$), and mean square frequency ($\omega_{2}$) are defined as
below

$\lambda=2\int{\frac{\alpha^{2}F(\omega)}{\omega}}d{\omega},$ (6)
$\omega_{\rm{log}}={\rm{exp}}\left[\frac{2}{\lambda}\int{\frac{d\omega}{\omega}\alpha^{2}F(\omega){\rm{log}}\omega}\right],$
(7)

and

$\omega_{2}=\sqrt{\frac{1}{\lambda}\int{\left[\frac{2\alpha^{2}F(\omega)}{\omega}\right]\omega^{2}d\omega}},$
(8)

respectively.

##  III Results and discussion

###  III.1 Phase stability and crystal structural of YCeHx and LaCeHx

Because predicting the ternary phase diagram in its entirety is
computationally expensive, and our main target is to determine the stability
and superconductivity of the hydrogen-rich phase, this research focuses on
stable compounds and crystal structures at 100 GPa, 200 GPa, and 300 GPa along
a line chosen by YCeHx and LaCeHx ($x$ = 2-8, 10, 12, 14, 16, 18, and 20) for
the fixed composition search. This approach is currently seen to be effective
for alkaline earths and rare earth metal ternary hydrides. Liang  _et al._
(2019a); Shao  _et al._ (2021a); Cataldo  _et al._ (2020) For the stability
analysis of YCeHx and LaCeHx, the corresponding Y-H, La-H, and Ce-H binary
hydrides have been extensively studied. Drozdov  _et al._ (2019); Chen  _et
al._ (2021a); Kong  _et al._ (2021) Theoretical calculations show that there
is a hydrogen-rich structure $M$H9 ($M$ = Y, La, and Ce) that can be
stabilized below 100 GPa in the three binary systems. In addition, YH10 is
capable of exhibiting room temperature superconductivity. Peng  _et al._
(2017) Therefore, this study mainly used these stable binary hydrides as well
as single elements as a reference for thermodynamic stability assessment.

The constructed convex hulls of YCeHx and LaCeHx at 200 GPa are presented in
Fig. 1. Their stable and metastable phases are highlighted by the symbols of
circles and red diamond in Fig. 1, respectively. The more detailed results for
the stabilites of YCeHx and LaCeHx at difference pressures are presented in
Figs. S1-S6 in the Supplementary Material (SM). YCeH5, YCeH7, YCeH8, LaCeH8,
and LaCeH18 are found to be stabilized at 100 GPa. When the pressure increases
up to 200 GPa, YCeH5, YCeH7, and LaCeH18 vanish from the ternary convex, while
LaCeH20 and YCeH18 appear in the stable phases of convex hull. In the pressure
range of 100-300 GPa, YCeH20 fails to achieve stability. But further
increasing the pressure above 300 GPa, $R\bar{3}m$-YCeH20 becomes stable, and
its stable pressure interval is very close to that of YH10. Peng  _et al._
(2017) In addition, in the pressure interval of 300-400 GPa, LaCeH8, LaCeH20,
and YCeH20 undergo a phase transition to the more stable $Pmmn$, $P6_{3}/mmc$,
and $P4/mmm$ phases, respectively. The ranges of the stabilization pressures
of these new predicted phases have been summarized in Fig. 2. To further
explore the resistance of these phases to decomposition at high temperatures,
we calculated the ternary convex hull at a finite temperature at 200 GPa. We
note that the stable phase (YCeH8, YCeH18, LaCeH8, and LaCeH20) at this point
is also stable at high temperatures, and its specific energy values have been
placed in the SM. Therefore, the above-mentioned can be synthesized in the
same way as (La,Y)Hx Semenok  _et al._ (2021) by heating LaCe and YCe alloys
with NH3BH3 at high pressure.

It is interesting to note that most of the above predicted stable phases,
except YCeH5 and YCeH7, are composed of clathrate structures enclosed by H.
For instance, the clathrate structures of stable YCeH8, LaCeH8, YCeH18,
LaCeH18, and YCeH20 are shown in Fig. 3. Since the hydrogen-rich compounds are
more likely to achieve high-$T\rm_{c}$ superconductivity Flores-Livas  _et
al._ (2020), the hydrogen-less ones such as YCeH5 and YCeH7 will be discussed
elsewhere. The stable hydrides of YCeHx and LaCeHx with clathrate structures
can be roughly divided into three groups.

(i) The first one consists of H18 cages, as found in $P4/mmm$-YCeH8,
$P4/mmm$-LaCeH8, and $Pmmn$-LaCeH8, which are derived from their corresponding
parent compounds ($I4/mmm$-YH4, $I4/mmm$-CeH4, and $I4/mmm$-LaH4) Peng  _et
al._ (2017) with the same cage structure. If half of Ce atoms at the site $2a$
of the $I4/mmm$-CeH4 phase are orderly replaced by Y or La atoms, the $P4/mmm$
ternary phase would be formed. Peng  _et al._ (2017) The relative enthalpies
of these two structures of LaCeH4 and YCeH4 are shown in Figs. S1 and S4. We
find that the most stable candidate structures prefer the H18 cages. In
addition, LaCeH8 would be transformed into the $Pmmn$ phase when the pressure
is above 300 GPa. (ii) The second group consists of H29 cages, as found in the
$P\bar{6}m2$-YCeH18 and $Amm2$-LaCeH18 phases. The $P\bar{6}m2$-YCeH18 phase
can be derived by replacing half of Ce atoms at the site 2$d$ of
$P6_{3}/mmc$-CeH9 with Y atoms. As for the parent compounds of
$P\bar{6}m2$-YCeH18 and $Amm2$-LaCeH18, the previous study by Peng et al. Peng
_et al._ (2017) has shown that only stable $P6_{3}/mmc$-YH9 and
$P6_{3}/mmc$-CeH9 phases exist in the pressure range of 100-400 GPa, however
LaH9 does not. For the $Amm2$-LaCeH18 phase, only its internal energy plays a
superior role to hinder its possible decomposition pathway of 1/6 LaH4 \+ 1/6
LaH10 \+ CeH9, and thus it cannot be stabilized at higher pressures. Liu  _et
al._ (2017); Peng  _et al._ (2017) (iii) The third group consists of the H32
cages, as found in YCeH20 and LaCeH20. The $R\bar{3}m$-YCeH20 and
$R\bar{3}m$-LaCeH20 phases exhibit the same spatial structure with LaYH20,
Semenok  _et al._ (2021) and all of them are the supercell structures of their
parent phases $Fm\bar{3}m$-$A$H10. More specifically speaking, the equivalent
primitive unit cell of $R\bar{3}m$-YCeH20 can be obtained by replacing half of
Ce atoms with Y atoms in the $1\times 1\times 2$ extension of the primitive
unit cell of $Fm\bar{3}m$-CeH10. LaCeH20 undergoes a phase transition from
$R\bar{3}m$ to $P6_{3}/mmc$. For YCeH20, it also undergoes a phase transition
at 370 GPa and the new high-pressure phase can be derived by replacing half of
Ce atoms in the site $4b$ of the parent phase $Fm\bar{3}m$-CeH10 with Y atoms.
Considering the chemical similarity of elements, one may choose the stable
phases of binary hydrides as the starting structures to accelerate the search
for multi-component hydrides.

To further check the lattice dynamic stability of the above predicted YCeHx
and LaCeHx phases, we have employed the Phonopy Togo and Tanaka (2015) code to
initially calculate their phonon band structures, which are presented in Fig.
S7 in the SM. Among them, the $Amm2$-LaCeH18 phases cannot maintain dynamic
stability in the pressure range of their thermodynamic stability. To make the
$R\bar{3}m$-LaCeH20 phase dynamically stable, the pressure needs to be above
200 GPa. In addition, the high pressure applied in the current mainstream
experimental studies of metal hydrides falls into a range of 100-300 GPa. As
for the phases stabilized above 300 GPa, both $R\bar{3}m$-YCeH20 and
$R\bar{3}m$-LaCeH20 have a certain degree of similarity. Therefore, we pay
much attention to the $P4/mmm$-YCeH8, $P\bar{6}m2$-YCeH18, $R\bar{3}m$-YCeH20,
$P4/mmm$-LaCeH8, and $R\bar{3}m$-LaCeH20 phases for their electronic
properties and superconductivity.

###  III.2 Electronic Properties and Superconductivity

The electronic band structure, the electronic density of states (eDOS), of
$P4/mmm$-YCeH8, $P\bar{6}m2$-YCeH18, $R\bar{3}m$-YCeH20, $P4/mmm$-LaCeH8, and
$R\bar{3}m$-LaCeH20 at high pressures are presented in Figs. S8-S10 in the SM,
respectively. All these compounds in the pressure range of their
thermodynamical stabilities possess the metallic features in their energy band
structures. For $P4/mmm$-YCeH8 and $P4/mmm$-LaCeH8, there are very steep
conduction bands along the $Z\rightarrow\Gamma$ direction crossing the Fermi
level ($E_{\mathrm{F}}$), resulting in electron pockets near the $\Gamma$
point. For $R\bar{3}m$-YCeH20 and $R\bar{3}m$-LaCeH20, a band inversion can be
found around the $\Gamma$ point and near the $E_{\mathrm{F}}$, which is mainly
attributed to the H 1s and Ce 4f orbitals, as seen from the atom- and orbital-
weighted band structures in Figs. S8 and S9 in the SM, respectively. The eDOS
at $E_{\mathrm{F}}$ ($N_{E_{\mathrm{F}}}$) for $P4/mmm$-YCeH8,
$P\bar{6}m2$-YCeH18, $R\bar{3}m$-YCeH20, $P4/mmm$-LaCeH8, and
$R\bar{3}m$-LaCeH20 are 1.72, 1.39, 1.10, 1.34, and 1.26 states/eV/f.u.,
respectively. As further analyzed by the atom-projected eDOS, the difference
in the aforementioned values of $N_{E_{\mathrm{F}}}$ mainly comes from the
different contribution of H and Ce atoms to the $N_{E_{\mathrm{F}}}$. The
contribution of Ce atoms at $E_{\mathrm{F}}$ is continuously suppressed as the
hydrogen content increases, while the opposite trend is observed for H.
However, the decrease of Ce is higher than the increase of H, leading to a
constant decrease of the total DOS with increasing H content. According to the
arguments of Belli et al., Belli  _et al._ (2021) the superconductivity of
hydrides maintains a strong positive correlation with the contribution of H in
the total DOS at $E_{\mathrm{F}}$ ($H_{\mathrm{DOS}}$). The $H_{\mathrm{DOS}}$
for these five phases are 9.5%, 34.4%, 38.2%, 11.2%, and 34.7%, respectively.
This indicates that the H-rich phase has a higher H-derived DOS at
$E_{\mathrm{F}}$ although the corresponding total DOS is low, which may
suggest that the possibility of high-$T\rm_{c}$ in H-rich phases to some
extent. Belli  _et al._ (2021)

From the orbital-weighted electronic band structures of $P\bar{6}m2$-YCeH18,
$R\bar{3}m$-YCeH20 and, $R\bar{3}m$-LaCeH20, as shown in Fig. S9, it can be
seen that the bands around $E_{\mathrm{F}}$ are mainly contributed by the
hybridization between the Ce 4f and H 1s orbitals. We should point out that
they were predicted by the GGA-PBE functional. This raises a question how
strong the SOC interaction and the correlated effect would be associated with
the Ce 4f orbitals in these compounds. So, we further checked the electronic
band structures with the SOC interaction, the correction of the on-site
Coulomb interaction of Ce 4f orbitals, and their combination, which are
presented in Fig. S10 in the SM. We find that both the SOC interaction and the
correction of the on-site Coulomb interaction have much weak effect on the
bands around $E_{\mathrm{F}}$, which could be neglected. The superconducting
transition temperature is largely affected by the electronic properties at the
$E_{\mathrm{F}}$. Therefore, the computational methodology employed in the
present study would be acceptable to give a reasonable prediction on the
superconducting properties of Ce-containing ternary hydrides. The similar
treatment with ignoring the SOC interaction and the correlated effect has also
been employed in several of the previous theoretical studies Peng  _et al._
(2017); Wang  _et al._ (2021); Li  _et al._ (2019) on the electron-phonon
coupling of binary CeHx. Unfortunately, the combination of the DFT+U method
and the density-functional perturbation theory approach in the latest version
(v7.1) of QE code does not support the electron-phonon interaction
calculations. Because of the above reasons, our superconductivity calculations
did not consider the SOC interaction and the strongly-correlated effect.

The phonon dispersions with the mode-resolved electron-phonon coupling (EPC)
constant $\lambda_{q\nu}$, atom-projected phonon density of states (PHDOS),
and EPC spectra of $P4/mmm$-YCeH8, $P\bar{6}m2$-YCeH18, $R\bar{3}m$-YCeH20,
$P4/mmm$-LaCeH8, and $R\bar{3}m$-LaCeH20 at high pressures are presented Fig.
4 and Fig. S12 in the SM. For these five phases, the phonons with frequencies
above 10 THz are dominated by the H contribution, accounting for more than
70%. The phonons with frequencies below 10 THz are mainly contributed by La,
Ce, and Y atoms. Because of the difference in the atomic masses of Y, La, and
Ce, the low-frequency range in the phonon spectrum of YCeHx is slightly wider
than that of LaCeHx. The EPC constants of YCeH8 at 100 GPa, YCeH18 at 150 GPa,
YCeH20 at 300 GPa, LaCeH8 at 100 GPa, and LaCeH20 at 250 GPa are 0.41, 2.51,
1.02, 0.35, and 0.99, respectively. The EPC constant of YCeH18 is so
significant compared to the other four phases. From the mode-resolved EPC
constant $\lambda_{q\nu}$, we find that the largest contribution to the total
EPC constant of YCeH18 comes from the optical phonon branch in the frequency
range of 10 to 20 THz, which contributes about 0.64 (i.e., 25.7% to the total
EPC constant). The rare-earth-atom associated vibrational modes with the
frequencies of $<$10 THz for YCeH18 also have moderate contribution to the
total EPC constant, as compared with the other four phases. In contrast, the
contribution of rare-earth atoms to the EPC for YCeH20 with higher H content
is only 0.09. When the pressure is increased from 150 GPa to 300 GPa, the
phonons dominated by the H-atom-associated vibration in YCeH18 keep hardening,
which can been seen from Fig. 4(a) and Fig. S11(c). The increase of pressure
leads to a decrease of the EPC constant of YCeH18, so that the EPC constant of
YCeH18 at 300 GPa is very close to that of LaCeH20 at 300 GPa. The EPC
constants of $A$CeH8 ($A$ = Y and La) are so small, compared to $A$CeH18 and
$A$CeH20, since both the highest phonon frequencies and the H-derived
electronic density of states in the former cases are lower than those in the
latter cases. These results suggest that once the hydrogen atoms in metal
hydrides simultaneously provide sufficient electron density of states at
$E_{\mathrm{F}}$ and sufficient phonon density of states in the high-frequency
region, a strong EPC strength would be achieved.

Based on the Allen-Dynes-modified McMillan formula Allen and Dynes (1975) with
a widely accepted value of the Coulomb pseudopotential $\mu^{*}$ ($\mu^{*}$ =
0.1 used herein), the $T_{c}$ values of $P4/mmm$-YCeH8 at 100 GPa,
$P\bar{6}m2$-YCeH18 at 150 GPa, $R\bar{3}m$-YCeH20 at 300 GPa, $P4/mmm$-LaCeH8
at 100 GPa, and $P4/mmm$-LaCeH8 at 250 GPa are predicted to be 4.8, 173.8,
122.1, 1.8, and 116.1 K, respectively. We note that very recently three
independent high-pressure experiments Chen  _et al._ (2022a); Bi  _et al._
(2022); Huang  _et al._ (2022) on the La-Ce-H system have been reported, in
which the measurement pressures by Chen et al. Chen  _et al._ (2022a) and Bi
et al. Bi  _et al._ (2022) were both less than 130 GPa and the observed
$T_{\mathrm{c}}$ was around 180 K. One interesting thing is that even though
these two research groups Chen  _et al._ (2022a); Bi  _et al._ (2022) used
different ratios of the starting materials for synthesis, the final
measurement results were very close, suggesting that both of them synthesized
probably the same high-temperature superconducting phase. Our calculation
results show that at this pressure (130 GPa) the thermodynamically stable
phase is LaCeH18, which is not explored further here because the stabilization
of this phase may involve anharmonic approximation effect not the focus of
this work. The measurement pressure on La-Ce-H by Huang et al. Huang  _et al._
(2022) was in the range of 140-160 GPa. As mentioned above, LaCeH20 can be
stabilized in this pressure range. A high-pressure experiment study Chen  _et
al._ (2022c) on the Y-Ce-H system has also been reported very recently, in
which the observed $T_{c}$ values are between 97-140 K. The predicted
$T_{\mathrm{c}}$ value of YCeH18 by either the McM or AD formula shows a good
agreement with this experiment data. To a certain extent, these experiment
results show strong support to the validity of our theoretical prediction. In
short, the high-pressure phases of LaCeH18, LaCeH20, and YCeH18 discovered
computationally in this work have all been confirmed in the very recent high-
pressure experiments, among which LaCeH20 and YCeH18 are theoretically
predicted for the first time.

###  III.3 Discussion

As presented above, $A$CeH8, YCeH18 and $A$CeH20 ($A$ = La and Y) have the
space groups of $P4/mmm$, $P\bar{6}m2$ and $R\bar{3}m$, respectively, and the
rare earth atoms are encased in the H18, H29 and H32 cages, respectively.
Herein we performed the COHP and integrated COHP (ICOHP) analyses to further
investigate the chemical bonding in these cage structures. The critical bond
lengths of some representative H-H, Ce-H, and $A$-H ($A$ \- La, Y) atom pairs
and their corresponding ICOHP values that can reflect the bonding strength are
shown in Figure 5. According to the convention of COHP analysis, the positive
and negative values of $-$COHP indicate the bonding and anti-bonding
characteristics, respectively. As seen from Fig. 3, the H18 cage in YCeH8 can
be viewed as consisting of eight quadrangles and four hexagons. Accordingly,
the H-H bond in the H18 cages can be roughly classified into three groups: i)
H-H bonds shared by two neighboring quadrangles, ii) H-H bonds shared by two
neighboring hexagons, and iii) H-H bonds shared by a quadrangle and a hexagon.

The H-H bond lengths in the first and second groups are denoted as D1 and D2,
respectively, as indicated in Figs. S14(a) and S14(d) in the SM. These two
distinct bond lengths D1 and D2 in YCeH8 and LaCeH8 at 100 GPa can be sorted
in the following descending order: D2(LaCeH8) $>$ D1(YCeH8) $\gtrsim$
D1(LaCeH8) $>$ D2(YCeH8). Although the H-H bonds with D1 in YCeH8 and LaCeH8
have the very close bond lengths, the projected COHPs for them shown in Figs.
5(a) and 5(c) indicate that this H-H bond in LaCeH8 exhibits a stronger
bonding interaction than in YCeH8 because of the less anti-bonding states
appearing just below $E_{\mathrm{F}}$ and a slightly greater absolute value of
the corresponding ICOHP in LaCeH8. However, compared to the $A$-H and Ce-H
bonds from their corresponding ICOHP values, the interaction of this H-H bond
would be significantly weaker. For YCeH18 and $A$CeH20 with higher H content,
the H-H bonds are significantly shrunken because of a higher pressure required
to stabilize YCeH18 and $A$CeH20, the ICOHP values of the H-H bonds in YCeH18
and ACeH20 are almost three times than those in $A$CeH8, and much less (and
even no) anti-bonding states appear below $E_{\mathrm{F}}$ for the H-H bonds
in YCeH20 (and LaCeH20). From the H18 cages in $A$CeH8 to the H29 cages in
YCeH18 and H32 cages in $A$CeH20, the cage size increases and the number of
quadrangles (hexagons) decreases (increases) due to incorporating more H
atoms. These results suggest that the H-H bonds within anti-bonding states in
$A$CeH8 could be much stabilized in YCeH18 and $A$CeH20 because they seem
ready to be hybridized with the incorporated additional H atoms. In addition,
YCeH20 has some remarkable anti-bonding states appearing around -11 eV for the
Ce-H atom pair, as seen from Fig. 5(c), which mainly come from the
hybridization between the H-1s and Ce-5px orbitals, suggesting that the semi-
core states (such as 5p orbitals) of Ce in YCeH20 at the high pressure may
contribute to the Ce-H bonding.

The crystal orbital bond index (COBI) can be used to quantify the strength of
covalent bonds in covalently bonded solid materials. Müller  _et al._ (2021)
The integrated-COBI (abbreviated to ICOBI) values for the H-H bonds in these
caged structures of $A$CeH8, YCeH18 and $A$CeH20 are less than 0.15,
indicating very weak bonding interactions between H-H. The overall trend is
that the ICOBI of H-H bonds increases with the increase of pressure, but the
change is less than 0.02, suggesting that the effect of pressure on the H-H
bonding is small. The similar trend is also observed from the analysis of
ICOHP. One interesting thing is that the bonding strength of the H-H bonds in
$R\bar{3}m$-YCeH20 in the pressure range of 300-400 GPa is slightly stronger
than the one in $R\bar{3}m$-LaCeH20 according to both the electron
localization function (ELF) and ICOHP analyses, although the corresponding
ICOBI values in these two structures are quite close to each other and also
exhibit a very weak dependence on the pressure.

The five superconducting compounds reported above, which are both
thermodynamically and dynamically stable, are mainly of the $A$CeH8, $A$CeH18
and $A$CeH20 types. The previously reported ScCaH8 Shi  _et al._ (2021) and
YMgH8 Song  _et al._ (2022a) also have the same structure as $A$CeH8. For the
$A$CeH8 type, there are three distinct types of H-H bonds, as mentioned above,
in which the D1 decreases as the pressure increases, while the D2 exhibits a
weaker dependence on the pressure. In additional, the bonding strength of the
H-H bonds can also be described as the ELF.

Recently, Belli et al. Belli  _et al._ (2021) have proposed a strong
correlation between the $T_{\mathrm{c}}$ value of superconducting hydride and
the weak covalent H-H bonds. They also proposed a possible networking value
$\phi$, which is defined as the highest value of the ELF that creates an
isosurface spanning through the whole crystal in all three Cartesian
directions, to describe the nature of the H-H bond in hydrides. As noted by
Belli et al., Belli  _et al._ (2021) the precise acquisition of the $\phi$
value is not uniquely defined. From the radial distribution function of H
atoms in our studied YCeHx and LaCeHx phases as a function of pressure, as
presented in Fig. S14 in the SM, we note that within the radial range of $3$ Å
there is a main peak representing the most important distribution of H-H bond
lengths. In this way, we obtain the $\phi$ value by taking the average of all
ELF values at the valley points of the lines plots of ELF along different H-H
bonds that correspond to such a main peak. And according to the model proposed
by Belli et al., Belli  _et al._ (2021) the $T_{\mathrm{c}}$ of metal hydride
can be predicted by the following equation with an accuracy in the range of 60
K:

$T_{\mathrm{c}}=750\phi H_{f}\sqrt[1/3]{H_{\mathrm{DOS}}}-85,$ (9)

where $H_{f}$ is the hydrogen fraction and $H_{\mathrm{DOS}}$ is the hydrogen
fraction of the total DOS at $E_{\mathrm{F}}$. As presented in Table 1, the
network values $\phi$ of our studied phases are all less than 0.5 and they are
very close. Consequently, the $T_{\mathrm{c}}$ values predicted by Eq. (9) are
more significantly affected by the contribution of $H_{\mathrm{DOS}}$. We also
noted that the $T_{\mathrm{c}}$ values predicted by Eq. (9) are still somewhat
deviation from those predicted by the AD formula, i.e., Eq. (3). Through the
examination on YCeH18 at 150 and 300 GPa, it can be seen that when the
pressure increases, the H-H distance decreases, the overall network $\phi$
value exhibits an increasing trend, and the contribution of H to the density
of states at $E_{\mathrm{F}}$ also increases. And thus the Eq. (9) based on
ELF predicts that the $T_{\mathrm{c}}$ for YCeH18 increases with the increase
of pressure, which is opposite to the trend in the prediction of the AD
formula. This indicates that one possible limitation in the use of Eq. (9) to
predict $T_{\mathrm{c}}$ may be the pressure-dependence of $T_{\mathrm{c}}$.
From the predicted results of Peng et al. Peng  _et al._ (2017) for all binary
rare-earth metal hydrides, a trend can be seen that the elements containing
more f electrons exhibit a suppressing effect on the $T_{\mathrm{c}}$ value.
However, the dataset to build Eq. (9) in the work of Belli et al. Belli  _et
al._ (2021) contains too few the f-electron-containing cases (i.e., only CeH9
and PrH9 included therein). Therefore, the $T_{\mathrm{c}}$ does exhibit a
certain relationship with the $\phi$ and $H_{\mathrm{DOS}}$ values, however,
the generalization of Eq. (9) to the pressure-dependence and the f-electron-
containing systems needs a further improvement.

##  IV Conclusion

In summary, we employed the evolutionary algorithm and first-principles
calculations to study the ternary phase diagrams of YCeHx and LaCeHx systems
under the pressure range of 100-400 GPa. The clathrate structure surrounded by
H18, H29, and H32 can maintain thermodynamic stability in the searched
pressure range. These cage structures correspond to YCeH8, YCeH18, YCeH20,
LaCeH8, LaCeH18, LaCeH20, respectively. However, under the harmonic
approximation, LaCeH18 cannot achieve dynamic stability. We further studied
the superconductivity of YCeH8, YCeH18,YCeH20, LaCeH8, and LaCeH20. The
results show that all four can achieve high temperature superconductivity, and
$P\bar{6}m2$-YCeH18 can have the highest 173 K at 150GPa. In addition, in
terms of the structure of these phases, $P4/mmm$-YCeH8, $P4/mmm$-LaCeH8,
$P\bar{6}m2$-YCeH18, $P4/mmm$-YCeH20 can all be obtained by element
replacement in binary superconducting hydrides. In other words, the search for
new superconducting hydrides can start from the combination of binary
superconducting hydrides. When two kinds of binary superconducting hydrides
with similar properties are combined together, it is possible to form a novel
stable three ternary hydride.

## Acknowledgments

The computations in this work have been performed using the facilities of
Research Center for Advanced Computing Infrastructure (RCACI) at JAIST. P.S.
is grateful for financial support from Grant-in-Aid for JSPS Research Fellow
(JSPS KAKENHI Grant No. 22J10527). K.H. is grateful for financial support from
the HPCI System Research Project (Project ID: hp190169) and MEXT-KAKENHI
(JP16H06439, JP17K17762, JP19K05029, and JP19H05169). R.M. is grateful for
financial supports from MEXT-KAKENHI (19H04692 and 16KK0097), FLAGSHIP2020
(project nos. hp1 90169 and hp190167 at K-computer), Toyota Motor Corporation,
I-O DATA Foundation, the Air Force Office of Scientific Research (AFOSR-
AOARD/FA2386-17-1-4049;FA2386-19-1-4015), and JSPS Bilateral Joint Projects
(with India DST).

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Table 1:  Superconducting transition temperature ($T_{\mathrm{c}}$) predicted by the original McMillan formula (abbreviated to McM), McMillan (1968) the Allen-Dynes-modified McMillan (McM) formula (abbreviated to AD), Allen and Dynes (1975) and the model (i.e., Eq. (9)) proposed by Belli et al., Belli  _et al._ (2021) (abbreviated to ELF) for $A$-Ce-H ($A$ = Y and La) at some selected pressures. Accordingly, the values of some key parameters contained in these formulas and model are also listed.  Phase | Space group | $P$ | Natom | $d_{H-H}$ | $\phi$ | $H_{\mathrm{DOS}}$ | $\lambda$ | $\omega_{\mathrm{log}}$ |  $T_{\mathrm{c}}$ (ELF) |  $T_{\mathrm{c}}$ (McM) |  $T_{\mathrm{c}}$ (AD)
---|---|---|---|---|---|---|---|---|---|---|---
|  | (GPa) |  | Å |  |  |  | (K) | (K) | (K) | (K)
YCeH8 | $P4/mmm$ | 100 | 20 | 1.52 | 0.40 | 9.5% | 0.41 | 987.5 | 24.5 | 4.7 | 4.8
YCeH18 | $P\bar{6}m2$ | 150 | 22 | 1.07 | 0.43 | 34.4% | 2.51 | 812.2 | 118.3 | 133.8 | 173.8
YCeH18 | $P\bar{6}m2$ | 300 | 22 | 1.05 | 0.46 | 39.2% | 0.97 | 1547.7 | 142.2 | 103 | 110
YCeH20 | $R\bar{3}m$ | 300 | 66 | 1.05 | 0.45 | 38.2% | 1.03 | 1595.2 | 137.6 | 114.2 | 122.1
LaCeH8 | $P4/mmm$ | 100 | 20 | 1.56 | 0.38 | 11.2% | 0.35 | 975.8 | 24.9 | 1.73 | 1.76
LaCeH20 | $R\bar{3}m$ | 250 | 66 | 1.06 | 0.48 | 34.7% | 1.00 | 1566.9 | 145.0 | 109.1 | 116.1
Figure 1: Ternary convex hulls of YCeHx and LaCeHx at 200 GPa. The stable and
metastable phases are highlighted by the symbols of circles and red diamond,
respectively. The right panels are the convex hull at 0 K (with the correction
of zero-point energy) and 1000 K.  Figure 2: Phase diagram of YCeHx and LaCeHx
in the pressure range from 100 to 400 GPa. Figure 3: The clathrate structures
of stable YCeHx and LaCeHx.  Figure 4: The phonon spectra with a projection of
mode-resolved EPC constant $\lambda_{q\nu}$ (red circle), atom-projected
phonon density of states (PHDOS), and Eliashberg function of (a)
$P\bar{6}m2$-YCeH18 at 150 GPa, (b) $R\bar{3}m$-YCeH20 at 300 GPa, and (c)
$R\bar{3}m$-LaCeH20 at 250 GPa. The mode-revolved EPC constant
$\lambda_{q\nu}$ is defined as $\lambda_{q\nu}=\frac{\gamma_{q\nu}}{\pi\hbar
N(E_{F})\omega_{q\nu}^{2}}$, where $\gamma_{q\nu}$ is phonon linewidth,
$N(E_{F})$ is the DOS at the Fermi level, and $\omega_{q\nu}$ is the mode-
resolve phonon frequency. Figure 5: The Crystal Orbital Hamilton Population
(COHP) of some representative atom pairs (H-H, Y-H, La-H, and Ce-H) in
$P4/mmm$-YCeH8, $P\bar{6}m2$-YCeH18, $R\bar{3}m$-YCeH20, $P4/mmm$-LaCeH8, and
$R\bar{3}$m-LaCeH20 at the pressure of 100, 150, 300, 100, and 250 GPa,
respectively.