Clathrate metal superhydrides under high-pressure conditions: enroute to room-temperature superconductivity

ABSTRACT Room-temperature superconductivity has been a long-held dream of mankind and a focus of considerable interest in the research field of superconductivity. Significant progress has recently been achieved in hydrogen-based superconductors found in superhydrides (hydrides with unexpectedly high hydrogen contents) that are stabilized under high-pressure conditions and are not capturable at ambient conditions. Of particular interest is the discovery of a class of best-ever-known superconductors in clathrate metal superhydrides that hold the record for high superconductivity (e.g. Tc = 250–260 K for LaH10) among known superconductors and have great promise to be those that realize the long-sought room-temperature superconductivity. In these peculiar clathrate superhydrides, hydrogen forms unusual ‘clathrate’ cages containing encaged metal atoms, of which such a kind was first reported in a calcium hexa-superhydride (CaH6) showing a measured high Tc of 215 K under a pressure of 170 GPa. In this review, we aim to offer an overview of the current status of research progress on the clathrate metal superhydride superconductors, discuss the superconducting mechanism and highlight the key features (e.g. structure motifs, bonding features, electronic structure, etc.) that govern the high-temperature superconductivity. Future research direction along this line to find room-temperature superconductors will be discussed.


INTRODUCTION
A superconductor exhibits two characteristic physical properties when cooled below its superconducting critical temperature ( T c ) where electrical resistance vanishes [1 ] and magnetic flux fields are expelled from the bulk [2 ].Since the first discovery of superconductivity below 4.2 K in solid mercury in 1911, tremendous efforts have been paid to the goal of achieving superconductors that work at ever higher temperatures for practical applications (see Fig. 1 below).Room-temperature superconductors are yet to be achieved and remain a century-long-held dream of mankind.
Intensive superconductivity research has been devoted to the investigation of two families of socalled 'unconventional' cuprate and iron-based superconductors, whose superconducting mechanism on the electron pairing is not believed to be mediated by the exchange of phonons [3 -6 ].In these activities, the highest T c of 133 K at ambient pressure [3 ] was attained in a Hg-Ba-Ca-Cu-O cuprate material whose T c was further promoted to 164 K at a highpressure condition of 31 GPa [4 ,5 ], setting the record for T c at the time.
On the way to room temperature superconductivity, metallic hydrogen was proposed in 1968 [7 ] as a potential high-temperature superconductor  T c ) for various superconductors.The square, circle and rhombus color blocks represent conventional, cuprate and iron-based superconductors, respectively.In particular, blue stars represent the conventional clathrate metal superhydride superconductors.The pressures required to synthesize these superconductors are represented by blue labels.Inset: crystal structures of covalently bonded hydrogen sulfide superconductor H 3 S [22 ] (left panel) and clathrate superhydride CaH 6 [24 ] (right panel).The yellow, pink, black and green spheres represent the S, H, Ca and H atoms, respectively.
according to an equation within the superconductive picture of Bardeen-Cooper-Schrieffer (BCS) theory [8 ] showing that where D is the Debye temperature, N (0) is the electron density of states at the Fermi energy and V is an effective pairing potential dominated by the attractive electron-phonon coupling interaction.Hydrogen, the most abundant element in the universe having the lightest atomic mass, naturally provides the highest possible D and V parameters for the solids that are necessary for a high-temperature phonon-mediated superconductivity.At ambient pressure, solid hydrogen is a wide-gap insulator.It has been suggested that metallization of solid hydrogen would require a strong compression above a pressure of 500 GPa [9 -12 ].This raises a highly experimental challenge, especially when one deals with hydrogen, the number one element showing the most mobile behavior due to the smallest atomic core among the periodic table of elements.As a result, hydrogen atoms often go into the inside of diamond for a breakdown of the diamond anvil cell, a device for generation of high pressure.As a result, metallic hydrogen has not yet been obtained through a direct compression of solid hydrogen in experiments despite great efforts in the high-pressure research field [13 ,14 ].
As an alternative route, hydrogen-containing materials or hydrides play an important role in the pursuit of metallic hydrogen, and its high-temperature superconductivity.The idea was first proposed as early as 1971 [15 ] in a hypothetic system of Li-F-H and later in 2004 [16 ] in IV group hydrides (e.g.methane, silane and germane).It is believed that the introduction of non-hydrogen elements into the lattice inevitably causes a chemical pressure to be placed on hydrogen.The resultant metallization pressure of hydrides is significantly reduced compared to that needed for pure hydrogen.Highpressure experimental investigation of metallic hydrides in a lab becomes feasible at the current level of experimental technique.
In recent years, remarkable progress has been achieved in the discovery of high-temperature superconductors in superhydrides stabilized under high-pressure conditions with the established superconductivity T c value approaching 260 K ( −13 • C) for LaH 10 [17 -20 ], a record high T c among known superconductors.Major findings are organized into two catalogues: (i) covalently bonded hydrogen sulfide superconductors (e.g.H 3 S with T c at ∼200 K) [21 -23 ] and (ii) clathrate metal superhydride superconductors (e.g.CaH 6 [24 -26 ], YH 6 [17 ,27 -3 0 ], YH 9 [17 ,28 ,3 0 ,31 ] and LaH 10 [17 -20 ] with T c = 215, 220, 240 and 260 K, respectively), as represented by the blue stars in Fig. 1 .Particular interest has been placed on the latter class of superconductors found in clathrate metal superhydrides that hold the record high superconductivity and have great potential to be those that superconduct at room temperature.In this peculiar class of clathrate superhydride superconductors, the first such example was theoretically proposed in a calcium superhydride CaH 6 back to 2012 [24 ], which was successfully synthesized in 2022 in a lab with a measured high T c of 215 K under a pressure of 170 GPa [25 ,26 ].
In this article, we review recent progress on the findings of hydrogen-based high-temperature superconductors among superhydrides stabilized under high-pressure conditions, with a particular focus on the family of best-ever-known superconductors found in clathrate metal superhydrides.In the next two sections, we provide a brief overview of the discovery of covalently bonded hydrogen sulfide superconductors, and discuss the CALYPSO crystal structure prediction method and the role it plays in aiding the experimental discovery.Then in the following two sections we respectively discuss metal hydride superconductors at ambient pressure and at high pressures.The latter mainly focuses on clathrate superhydrides, and includes a discussion of the superconducting mechanism and the key features (e.g.structure motifs, bonding features, electronic structure, etc.) that govern the high superconductivity.In the final section, we discuss the future challenge and opportunity for roomtemperature superconductors among clathrate metal superhydrides.

HYDROGEN SULFIDE SUPERCONDUCTORS AT A GLANCE
H 2 S exists as a gas molecule in nature and smells of rotten eggs; it is the only stable stoichiometry compound found at ambient pressure in the hydrogen sulfide system.The work of Li et al. [21 ] is the original literature proposing sulfur hydride as a superconductor under high-pressure conditions.
Since 2014 when Li et al. made the first attempt to predict high-temperature superconductivity in sulfur dihydride [21 ], there has been much less interest in this system since it was believed to dissociate into its constituent elements of sulfur and hydrogen under pressure [32 ,33 ].Li et al. 's extensive structuresearching simulation on hydrogen sulfide via the CALYPSO method [34 ,35 ] found that earlier proposed elemental dissociation would not occur, and a structure with a high superconductive potential consisting of 2(SH 3 ) units was predicted for H 2 S with a theoretical T c reaching 80 K at a pressure of 160 GPa [21 ].This theoretical work [21 ] initiated the practical work by Eremets' group [23 ] where H 2 S compressed in a diamond anvi l cel l was found to exhibit two astonishing superconductive states: (i) the one prepared at low temperature has a T c of 30-150 K, in high accordance with the predicted H 2 S superconductor [21 ]; (ii) the one annealed at room temperature exhibits an unexpected high T c at 180-203 K, surpassing the earlier T c record of 164 K [5 ] set by cuprate.The latter one has been ascribed to be H 3 S through a stoichiometric change via a decomposition of H 2 S into H 3 S+S [36 ], where H 3 S is a known stoichiometry of (H 2 S) 2 H 2 that has already been synthesized at a pressure of 7 GPa [37 ] and was later theoretically predicted to be a 200-K superconductor with a cubic structure (space group Im 3 m ) at megabar pressures [22 ].In this cubic structure, each pair of S atoms symmetrically accommodates an atomic H between them, forming robust six-fold polar covalent S-H bonds (as shown in the inset of Fig. 1 ).The observed superconductivity in H 3 S shows a strong isotopic effect pointing toward a phonon-mediated pairing mechanism [23 ].
The findings of hydrogen sulfide superconductors at a megabar pressure condition mark a milestone in superconductivity history and are a result of joint theoretical and experimental efforts, where theory plays a critical role in guiding the experimental exploration.

CALYPSO CRYSTAL STRUCTURE PREDICTION METHOD AND ITS ROLE IN AIDING EXPERIMENTAL DISCOVERY
Research on unknown superhydrides presents a challenge, as they can only be produced under megabar pressure conditions where limited information is available regarding their compositions and crystal structures.Moreover, owing to the weak X-ray scattering of hydrogen, the exact position of hydrogen is hardly determined by X-ray diffraction experiments.All these difficulties call for advanced theory that can predict the crystal structures of superhydrides in a reliable manner with only the chemical composition given.Recently developed theoretical crystal structure prediction methods [38 ,39 ] have taken center stage for this purpose, due to their trustworthy predictive power regarding compositions and structures.Over the years, techniques such as random search, genetic algorithm and swarm intelligence methods have evolved to become the preferred choices for computational discovery of structures [38 ,39 ].
One may recall that crystal structure prediction presents a challenging, NP-hard problem in the minimization of the high-dimensional potential-energy surface (PES).To deal with this challenging problem, the CALYPSO (crystal structural analysis by particle swarm optimization) structure prediction method has been developed by the Ma group [34 ].The method adopts a heuristic numerical solution scheme that is an optimal compromise between global exploration and local exploitation of the potential energy surface, making it particularly suited for crystal structure prediction.A number of algorithms has been devised in the method: (i) a symmetry classification searching strategy for highcoverage sampling of the potential energy surface; (ii) a bond characterization matrix for fingerprinting the structures and dividing the entire PES into a set of simpler fragments that are easier to explore and, most importantly, (iii) a swarm intelligence algorithm for rapid location of the energetically most stable structure by swarm-directed smart learning of optimal structures.The CALYPSO method has been coded into the same-name software [35 ] that is freely available for academic users, and it has proven a highly efficient and accurate tool for crystal structure prediction.
Given chemical compositions, CALYPSO can predict structures, in an intelligent and automatic way, for two-or three-dimensional crystals, nanoclusters and nanoparticles, protein molecules, reconstructed surfaces and interfaces, etc.The proof of the method's generality and reliability is in its utilization: up to now ( June 2023), CALYPSO has been widely used in the world by more than 40 0 0 researchers from 74 countries.Use of CALYPSO has generated many groundbreaking discoveries dedicated to high-pressure science, including the longpuzzled oC 40 phase structure of semiconducting lithium, chemical compounds of Fe/Ni 3 Xe, the substitutional alloy structure of Bi 2 Te 3 , the atomic structure of solid oxygen, counterintuitive compounds of H 3 O and Ca 3 O, the polymeric N 10 structure, the superhard cubic phase of BC 3 , etc. [39 ].
CALYPSO has played an important role in the design of hydrogen-based superconductors under high-pressure conditions.Besides the abovementioned prediction of metallic and superconducting structures of H 2 S [21 ], CALYPSO has been used to make breakthrough predictions on a class of clathrate metal superhydride superconductors (e.g.CaH 6 [24 ], YH 6 [27 ], YH 9 [17 ], LaH 10 [17 ,18 ], etc.) that have received subsequent experimental confirmation.With the development of the crystal-structure searching methods, future design of room-temperature superhydride superconductors becomes feasible.

METAL HYDRIDE SUPERCONDUCTORS AT AMBIENT PRESSURE
The first hydride superconductor ever reported in 1970 was Th 4 H 15 with a T c of 8 K at ambient pressure [40 ].Later on, other binary metal hydride superconductors of PdH [41 ] and NbH 0.69 [42 ] were also reported with T c values of ∼10 K. Several ternary metal hydride superconductors (e.g.Hf V 2 H [43 ] and Pd 0.55 Cu 0.45 H 0.7 [44 ] with T c values of 4.8 and 16.6 K, respectively) were also synthesized.
It is worth noting that all these ambient-pressure metal hydride superconductors possess low superconductivity ( < 16.6 K).The top panel of Fig. 2 lists the most hydrogen-rich hydrides achieved at ambient pressure.It is seen that H contents in these ambient-pressure metal hydrides are generally low with a metal/H ratio larger than 1/3.In the crystal structures (Fig. 3 ), H atoms typically occupy the interstitial octahedral ( O ) or tetrahedral ( T ) sites of the closely packed metal lattice.In order to understand the physical origin for the low superconductivity at ambient pressure, the electron density of states of PdH [41 ], ScH 2 [45 ] and ScH 3 [45 ] are calculated, as shown in Fig. 3 .It is found that hydrogen electrons are localized at a low-lying energy level, and do not contribute to the Fermi level.As a result, hydrogens in these low-H-content hydrides are not expected to play an important role in the superconductivity.It is worth noting that two properties of metal hydrides are crucial to achieving H-dominated high-T c superconductivity: a large H-derived density of state at the Fermi level and large modifications of the electronic structure in response to the motions of H atoms (electron-phonon coupling) [17 ,38 ].It is apparent that these ambient-pressure metal hydrides are not good candidates for the H-dominated superconductors.
From a chemical point of view, it is not unexpected to see such a negligible hydrogen electron contribution to the Fermi level in these hydrides.Metal hydrides can be regarded as M + cation-doped solid hydrogen (H 2 ).Once there is a formation of hydrides, valence electrons of metal atoms wi l l transfer to H 2 molecules in the solid due to the large difference in electronegativities of hydrogen and metal atoms.The resultant electron occupancy of the antibonding σ * 1 s orbitals of H 2 molecules (Fig. 4 ) wi l l lead to the dissociation of molecules into hydrogen atoms forming ionic Pd + …H − , Sc given an orange background.The related data were collected from Table 1 , the Inorganic Crystal Structure Database [46 ] and other excellent reviews [47 -49 ].
respectively.As a result, hydrogen electrons are confined in H − states at deep energy levels far away from the Fermi level (Fig. 3 ).
Recently, a ternary N-doped lutetium hydride [50 ] was claimed to exhibit a remarkably high T c of 294 K (21 • C) at a near-ambient pressure of 1 GPa.However, a number of subsequent experiments [51 -53 ] that were able to reproduce the as-synthetic ternary product of N-doped lutetium hydride found no superconductivity at all for such a sample.From the current results achieved, it is suggested that such a claimed high superconductivity in an N-doped lutetium hydride is unlikely to be true.

CLATHRATE METAL SUPERHYDRIDE SUPERCONDUCTORS AT HIGH PRESSURES
Pressure as a thermodynamical variable can profoundly modify electronic orbitals and bonding patterns of hydrogen and metal atoms.It is thus a  Constructive combination leads to a bonding H 2 molecular orbital.
Nodal plane between nuclei.

Energy σ 1s
Electron density concentrared between nuclei.powerful tool for the creation of exotic metal superhydrides (Fig. 2 ) that are not accessible at ambient conditions.As can be seen from Fig. 2 , many new binary superhydrides appear with extremely high H content under high-pressure conditions.This opens a research avenue for finding high-temperature superconductors in these newly formed superhydrides.
A large number of superhydrides have been experimentally synthesized, some of which have been confirmed to be high-T c superconductors, as listed in Table 1 .Among various superhydrides, clathrate metal superhydrides take center stage since they exhibit the highest-T c values (up to 250-260 K) known thus far.

CaH 6 : the first clathrate superhydride
At ambient pressure, the only thermodynamically stable compound in the Ca-H system has a stoichiometry of CaH 2 that adopts a cotunnite-type structure and is not a superconductor (we refer the reader to our discussion on ambient-pressure hydrides).In 2012, through a structure search study on a mixture of Ca + H 2 using the CALYPSO method [34 ,35 ], we theoretically explored other calcium hydrides with higher hydrogen contents that can be stabilized under high-pressure conditions [24 ].It was predicted that, besides CaH 2 , three new stoichiometric calcium hydrides appear (Fig. 5 (b)), CaH 4 , CaH 6 and CaH 12 , that are thermodynamically stable at the megabar pressure regime.
CaH 4 is not a superconductor and has a tetragonal I 4/ mmm structure (Fig. 5 (b)) where Ca adopts a body-centered arrangement and H takes a mixed chemical form of monoatomic H and molecular H 2 [24 ].Following our prediction, this proposed CaH 4 has been subsequently synthesized by two independent experimental works [55 ,56 ].CaH 12 has a rhombohedral structure consisting entirely of molecular H 2 units (Fig. 5 (b)) that is not a good candidate for high-T c superconductors [24 ].
Of particular interest is the prediction of a peculiar clathrate-structured CaH 6 that adopts a body-centered cubic structure (space group Im 3 m ; Figs 5 (b) and 6 (a)) where 24 H atoms form a perfect cage (eight hexagons plus six squares) with Ca at the center of the cage [24 ].Within the H cage, there is clear charge localization (Fig. 6 (b)) between nearest-neighboring H atoms and analysis of the electron localization function indicates that there is a weak covalent interaction between H atoms.It is interesting to note that all the nearest H-H distances are equal to 1.24 Å at 150 GPa.From the calculated partial electron density of states (Fig. 6 (c)), we see that H electrons dominate the density of states at the Fermi level in this structure.As we have discussed in the above context, the dominant contribution of H electrons to the Fermi level is the most significant signature for the H-based superconductor.Here, CaH 6 must be the one that we really want in the hunt of a H-based high-T c superconductor.Indeed, a realistic electron-phonon coupling calculation on CaH 6 gave a predicted exceptionally high T c of 220 K at 150 GPa [24 ].As expected, the H cage is found to be crucial to the superconductivity since it contributes most (84%) to the electron-phonon coupling parameter (Fig. 6 (d)).
Although CaH 6 is the first-ever clathrate-type superhydride superconductor proposed in the field, its experimental synthesis has been regarded as one of the most challenging tasks.Previous attempts [55 ,56 ] on the synthesis use a mixture of Ca + H 2 or CaH 2 + H 2 as precursors.Unfortunately, this synthetic route leads to the easy formation of low-H content CaH 4 at a low-pressure region and further formation of high-H CaH 6 at the higher-pressure regime is kinetically hindered.Thanks to the concept of using ammonia borane [57 -59 ] as the hydrogen source where it decomposes into boron mononitride and hydrogen at high-temperature conditions, one can control the hydride synthesis at a desirable pressure condition.Very recently, after 10 years of continuous effort, two independent experimental works [25 ,26 ] reported the successf ul sy ntheses of as-predicted clathrate-structured CaH 6 by using a mixture of Ca and ammonia borane as precursors.
Since we have various calcium hydrides at hand, it is possible to derive a general picture of the evolution of the chemical bonding of hydrogen to understand the formation mechanism of various calcium hydrides under pressure, as depicted in Fig. 5 (b).Ca was introduced into solid molecular hydrogen as a dopant to achieve metallic hydrogen.As a result, there wi l l be inevitable charge transfer from Ca into H 2 molecules.The accepted electrons by each H 2 molecule wi l l occupy the antibonding σ * 1 s orbital (Fig. 4 ) since each H 2 molecule already has a fil led σ 1 s bond.The occupancy of this σ * 1 s orbital wi l l lead to a weakening of the H-H bond, which in turn lengthens the H-H bond length and ultimately results in the complete dissociation of the H 2 molecule.The existence of monatomic H and H 2 units in the structures is contingent on the number of electrons each H 2 molecule accepts.If we assume that the two valence electrons of each Ca atom are fully 'ionized' and taken up by H 2 molecules, then the accepted electrons per H 2 for CaH 12 , CaH 6 , CaH 4 and CaH 2 are 1  3 e , 2 3 e , 1 e and 2 e , respectively.As for CaH 12 , each H 2 molecule accepts 1  3 e without severing the bond, but rather results in a bond elongation from 0.74 to 0.8-1.0Å.In CaH 6 , acceptance of 2  3 e per H 2 molecule leads to the formation of the clathrate structure [24 ], where the H-H distance is in the range 1.0-1.3Å and H-H is weakly covalently bonded.In CaH 4 , there are two H 2 formulas in each unit cell and each H 2 molecule accepts 1 e .Given that two H 2 molecules are preserved, the remaining two H 2 molecules accommodate four 'excess' electrons into their σ * 1 s orbital.This process disrupts the molecules into monatomic hydrogen, with the H-H distances in the range 1. 3   conditions show excellent agreement with those in metal-inclusion H 2 complexes (LnM-H 2 ) at ambient pressure [54 ] (Fig. 5 (a)) that have been well established by nuclear magnetic resonance experiments, as depicted in Fig. 5 (a).This analogue for H-H bonding behaviors between calcium superhydrides stabilized under high-pressure conditions and H 2 complexes at ambient pressure is not accidental, but it rather reflects the true physics of hydrogen.We further emphasize that the current analysis of hydrogen bonding is also applied to the understanding of chemical bonding in other superhydrides and it is thus not necessary to discuss again in the context below to avoid any repetition.
Several CaH 6 -type alloyed clathrate superhydrides in ternary systems have been proposed by using the idea of partial substitution of Ca in CaH 6 or Y in YH 6 with alternative metal elements, e.g.CaYH 12 [69 ] and YMgH 12 superhydrides [70 ].It is worth noting that a ternary clathrate alloyed (La,Y)H 6 superconductor has been experimentally synthesized with a measured T c of 237 K at a pressure of 176 GPa [71 ].

YH 9 -type clathrate superhydrides
In 2017, Peng et al. [17 ] theoretically proposed that clathrate structures are commonly formed in rare-earth superhydrides with stoichiometries besides CaH 6 , with even higher hydrogen contents (e.g.YH 9 and LaH 10 ).The structure of YH 9 [17 ] in space group P 6 3 / mmc can be regarded as a variant of CaH 6 , although the number of hydrogen atoms in the hydrogen cage has been enlarged from 24 atoms in CaH 6 to 29 atoms in YH 9 .In this H 29 cage (Fig. 7 (b)) where one Y atom sits at the center of the cage, there are six quadrilaterals, six pentagons and six regular hexagons [17 ].YH 9 was computed to exhibit superconductiv ity w ith T c reaching room temperature at 269 K at a pressure of 150 GPa.Soon after this theoretical prediction, two inde- pendent experiments successf ully sy nthesized the clathrate-structured YH 9 and reported measured T c values of 262 K at 182 GPa [31 ] and 243 K at 201 GPa [28 ].
Recently, two YH 9 -type ternary alloyed superhydrides of (La,Ce)H 9 [94 ] and (Y,Ce)H 9 [93 ] have been synthesized via random substitution of half Ce by La and Y, respectively.These two works demonstrated that substitutional alloying could act as an effective tool for substantially enhancing superconductivity since giant T c enhancements are evidential.Superconductivity of 100 K in parent CeH 9 has been enhanced to 178 and 131 K in resultant child (La,Ce)H 9 and (Y,Ce)H 9 , respectively, after substitutional alloying.
It is worth noting that the notably low synthesis pressures of 100-130 GPa for CeH 9 , (La,Ce)H 9 and (Y,Ce)H 9 are believed to result from the particularly strong chemical pressure exerted by the delocalized 4 f electrons of Ce.This suggests that other rare-earth hydrides with a similar delocalized f character may also be stable at relatively low pressures [102 ].

LaH 10 -type clathrate superhydrides
In the same work for the proposal of YH 9 -type superhydrides, Peng et al. [17 ] also theoretically proposed a class of clathrate rare-earth superhydrides in a stoichiometry of LaH 10 with predicted T c values as high as 288 K for LaH 10 at 200 GPa and 303 K for YH 10 at 400 GPa.The structure of LaH 10 in space group F m 3 m can also be regarded as a variant of the CaH 6type structure, where the number of hydrogen atoms in the hydrogen cage has been enlarged from 24 atoms in CaH 6 to 32 atoms in LaH 10 .The H 32 cage (Fig. 7 (c)) is composed of six squares and 12 regular hexagons [17 ].Meanwhile, Liu et al. also independently predicted the clathrate-structured LaH 10 and YH 10 by a systematic exploration of the structures and superconductivities of La-H and Y-H systems under high-pressure conditions [18 ].The aspredicted LaH 10 has been successf ully sy nthesized in two independent experiments [19 ,20 ] with measured T c values of 250-260 K at pressures of 170-185 GPa.
The successf ul sy nthesis of LaH 10 at pressures < 200 GPa corroborated recent theoretical calculations by taking quantum effects into account [103 ] where the classical ab initio calculations [17 ,18 ] predicted structural distortions in the LaH 10 below ∼230 GPa.The inclusion of zero-point energy has a clear effect on the stabilization of the F m 3 m phase at a pressure regime as low as 129 GPa [103 ].
It is worth mentioning that the synthesis of LaH 10 created a T c record of 250-260 K (Fig. 1

Ba 8 Si 46 -type clathrate superhydrides
It is known that clathrate structures often appear in a variety of silicon/germanium-based materials.Ba 8 Si 46 is one such good example, in the structure (Fig. 7 (d)), and it is the first clathrate superconductor found in a bulk phase [104 ] with a measured T c of 8 K at ambient pressure.
Several Ba 8 Si 46 -type clathrate superhydrides including Ba 8 H 46 [97 ], Eu 8 H 46 [98 ], La 4 H 23 [99 ] and Lu 4 H 23 [100 ] [105 ] is interesting; their predicted T c at 200 GPa are 214 and 203 K, respectively.In addition, by performing a combination of high throughput screening and structural search, An et al. [106 ] predicted a thermodynamically stable Ba 8 Si 46 -type superhydride of LiNa 3 H 23 that exhibits an extraordinarily high T c of 310 K at 350 GPa.

CeH 18 -type clathrate superhydrides
Very recently, a new class of extremely hydrogenrich CeH 18 -type clathrate superhydrides with a stoichiometry of CeH 18 were theoretically proposed in rare-earth/actinide superhydrides [107 ].These peculiar superhydrides are composed of H 36 cages (Fig. 7 (e)), the largest cage for a known clathrate superhydride structure.This CeH 18 -type clathrate superhydride forms a crystal structure with either the Fddd or Fmmm space group.In this structure, H 36 cages are interconnected by a 6H 6 ribbon-ring structure.Two undulating H 6 hexagons are positioned above and below this structure, with bridge bonds linking the H 6 hexagons to the 6H 6 ribbon ring (as shown in Fig. 7 (e)).First-principles calculations [107 ] for different CeH 18 -type clathrate superhydrides predict diverse T c values among the same stoichiometry.Among these extreme superhydrides, CeH 18 and ThH 18 are par ticularly notewor thy.They display superconductivity above room temperature, with T c values peaking at 330 K at 350 GPa and 321 K at 600 GPa, respectively.These represent the high-est predicted T c values among all known thermodynamically stable superhydrides.Future experiments are highly desirable for the synthesis of this class of CeH 18 -type clathrate superhydrides to search for room-temperature superconductors.

Ternary clathrate superhydrides
Binary superhydrides have been exhaustively investigated both theoretically and experimentally [47 ].The quest for high-T c superconductors among superhydrides has recently evolved, moving beyond the realm of binary compounds and shifting focus towards ternary ones, which offer a vast array of material types and configurations.Ternary superhydrides can enjoy advantageous properties through tuning of two non-hydrogen elements in structures, leading to superior superconducting properties beyond those of binary hydrides.Earlier theoretical works [108 ,109 ] have revealed promising ternary superhydrides that superconduct at or above room temperature.However, the challenge remains to attain stoichiometric ternary compounds with a well-resolved crystal structure that can host high-temperature superconductivity above 100 K.Although several examples of ternary substitutional ly al loyed clathrate superhydrides (e.g.(La,Ce)H 9 and (Y,Ce)H 9 ) have been discussed in the aforementioned text where the primary focus is placed on enhancing superconductivity through elemental substitution on known binary clathrate hydride superconductors, these activities have not offered any new prototype ternary structure for high-temperature superconductivity.
On the way to finding room-temperature superconductors among ternary superhydrides, a significant research question in this field is how to stabilize a superhydride that incorporates as many hydrogen atoms as possible, without forming any H 2 molecules within the lattice, in order to achieve a hydrogen-dominated electron density of states at the Fermi surface [17 ,38 ].A useful strategy is to introduce extra electrons via metal doping into H 2 -rich binary superhydrides [110 ], as has been demonstrated in the design of clathrate-structured ternary Li 2 MgH 16 [110 ] and Li 2 Y/LaH 17 [111 ].Li 2 MgH 16 has been predicted to exhibit 'hot' superconductivity with a T c value of up to 473 K at 250 GPa, which is the highest predicted T c among all known superhydride superconductors [110 ]  The challenge of achieving room-temperature superconductivity at significantly lower pressures is a recognized issue.A strategy to address this challenge is to use small-radius elements (e.g.B or Be) and hydrogen to achieve ternary superhydride superconductors with alloy backbones [112 ].By employing this approach, a range of LaBH 8 -type ternary superhydrides with a 'fluorite-type' backbone have been proposed [112 -114 ], among which LaBeH 8 [112 ] is anticipated to be a thermodynamically stable phase above 98 GPa and dynamically stable down to 20 GPa with a high T c of 185 K. Very recently, LaBeH 8 has been successfully synthesized with a measured T c of up to 110 K at 80 GPa [101 ].LaBeH 8 is the first experimental realization of an archetype ternary prototype with exact stoichiometry, well-resolved structure and T c beyond 100 K.

Clathrate frameworks at ambient pressure
Hydrogen clathrate structures in superhydrides stabilized under high-pressure conditions are a class of peculiar structures quipped for hydrogen-based superconductors.As we have discussed above, the clathrate structure is the key to create the observed high superconductivity in superhydrides that holds the record high-T c values among superconductors known thus far.However, these hydrogen clathrate structures can only be stabilized at megabar pressures and are not capturable at ambient pressure upon the release of pressure.For any practical application, it is essential to discover superconductors at ambient pressure.
From a structure point of view, we find two kinds of analogous materials at ambient pressure: (i) CaH 6 -, Ba 8 H 46 -and Li 2 YH 17 -type superhydrides are structurally equivalent to the t ype-VII, t ype-I and type-II silicon-based clathrate structures, respectively; (ii) hydrogen sublattices in CaH 6 -, LaH 10 -, Ba 8 Si 46 -and Li 2 YH 17 -type superhydrides share the structure similarity to the known SOD-, AST-, MEPand MTN-type zeolite frameworks, respectively.
To understand how clathrate structures in superhydrides are related to zeolite frameworks, we take LaH 10 [17 ,18 ] as an example for the i l lustration (Fig. 8 ) whose hydrogen sublattice shares the structure similarity w ith the A ST-type zeolite framework in an aluminum phosphate AlPO 4 -16 material [115 ].It is noted that, when all non-framework cations are removed, AlPO 4 -16 turns out to be an open three-dimensional framework structure composed of corner-sharing TO 4 (T = Al and P) tetrahedra (Fig. 8 ).The network of T atoms is identical to the hydrogen sublattice in LaH 10 .
The structure similarity between clathrate superhydrides and silicon-based clathrate or zeolite materials may give a hint on future design of high-T c superconductors at ambient pressure.Encouragingly, there exists a large number of clathrate materials at ambient pressure that might be useful as templates for such a design.For example, over 200 zeolite framework types of material have been documented [116 ].

CONCLUSION AND OUTLOOK
In this review, we provide an up-to-date perspective on the research field of superconducting clathrate superhydrides under high-pressure conditions.Clathrate superhydrides have emerged as the most promising candidates for hunting high-temperature superconductors.Finding room-temperature superconductors along this direction might become true in a near future.
Multi-element superhydrides are the immediate targets for the discovery of high-temperature superconductors.As the number of elements increase in hydrides, the number of conceivable structures and potential superconducting compounds grows rapidly [108 ,117 ], suggesting that there are more open rooms for clathrate superhydrides in ternary and quaternary systems than those in binary ones.However, the exponential growth of chemical space with increasing element species makes the search a great challenge both theoretically and experimentally.Therefore, it might be useful to summarize several strategies to accelerate the design of hightemperature superconducting superhydrides based on successful experiences: (i) doping/substituting metal elements into known clathrate superhydrides; (ii) introducing extra electrons via metal doping into known superhydrides that contain abundant quasi-H 2 molecular units [110 ]; (iii) using small-radius elements to stabilize ternary hydrogen-based superconductors with alloy backbones [112 ]; (iv) atomic substitution of known clathrate materials at ambient pressure.
Given the rapid development of this field, we anticipate that certain aspects of this review might shortly become outdated.However, we trust that the methodological framework and the amalgamation of knowledge from both experiments and theories outlined here wi l l provide a useful reference for future research and inspire exciting future discoveries.

Figure 1 .
Figure 1.Chronological evolution of the superconducting critical temperature ( T c ) for various superconductors.The square, circle and rhombus color blocks represent conventional, cuprate and iron-based superconductors, respectively.In particular, blue stars represent the conventional clathrate metal superhydride superconductors.The pressures required to synthesize these superconductors are represented by blue labels.Inset: crystal structures of covalently bonded hydrogen sulfide superconductor H 3 S[22 ] (left panel) and clathrate superhydride CaH 6[24 ] (right panel).The yellow, pink, black and green spheres represent the S, H, Ca and H atoms, respectively.

Figure 3 .
Figure 3. Crystal structures (inset) and partial electron density of states (PDOS) of face-centered-cubic -structured (a) PdH, (b) ScH 2 and (c) ScH 3 at ambient pressure.The blue, purple and pink spheres represent Pd, Sc and H atoms, respectively.Most H atoms occupy the interstitial octahedral ( O ) or tetrahedral ( T ) sites of the metal lattice.Solid black and red lines represent PDOS of metal and H, respectively.Blue dotted lines represent Fermi levels.

Figure 4 .
Figure 4.The formation of H 2 molecular orbitals from two H atomic orbitals based on molecular orbital theory, where the lower-(higher-)energy molecular orbital σ 1 s ( σ * 1 s ) corresponds to the bonding (antibonding) behavior with electron density concentrated between (behind) the H nuclei.

Figure 5 .
Figure 5. (a) The entire reaction coordinate for the activation of H 2 on a metal as a function of the degree of backdonation within the large regime of hundreds of LnM-H 2 complexes, Adapted with permission from [54 ].Copyright 2007 American Chemical Society.(b) Several typical H-motifs of compressed metal.The corresponding crystal structures are illustrated with a Ca-H system as an example.H-H bond distance ( d H-H ) of H 2 complexes (from crystallography and NMR) or H-motifs varying from 0.74 to > 1.6 Å are shown.

Figure 6 .
Figure 6.(a) Crystal structure, (b) the electron localization function, (c) the partial electron density of states and (d) projected phonon densities of states as well as the isotropic Eliashberg spectral function α 2 F ( ω) and EPC parameter λ( ω) of CaH 6 at 150 GPa [24 ].The small and large spheres in (a, b) represent H and Ca atoms, respectively.

Figure 7 .
Figure 7. Crystal structures of several typical clathrate superhydrides and their H-motifs.(a) Im 3 m-CaH 6 , which is composed of Ca-centered H 24 cages.(b) P 6 3 / mmc -YH 9 , which is composed of Y-centered H 29 cages and distorted H 8 cubes.(c) Fm 3 m-LaH 10 , which is composed of La-centered H 32 cages and H 8 cubes.(d) Ba 8 Si 46 -type P m 3 n-Ba 8 H 46 , which is composed of Ba-centered H 24 and Ba-centered H 20 cages.(e) Fddd -and Fmmm -CeH 18 , which are composed of Ce-centered H 36 cages.(f) F d 3 m-Li 2 YH 17 , which is composed of Y-centered H 28 and Li-centered H 20 cages.The small and large spheres represent H and metal atoms, respectively.

Figure 8 .
Figure 8.Comparison of the H sublattice in clathrate LaH 10 and the AST-type zeolite framework in an AlPO 4 -16 material.

Table 1 .
Measured superconducting transition temperature values ( T c in kelvins) of experimentally confirmed hydrides under ambient pressure and high-pressure conditions (P in gigapascals).
We point out that H-H bonding features in calcium hydrides discovered under high-pressure Crystal structure and H-H bond distance of compressed Ca-H compounds have been experimentally synthesized, among which Lu 4 H 23 has an observed T c value up to 71 K at 218 GPa.No superconducting properties were experimentally studied for La 4 H 23 and Ba 8 H 46 , although they are expected to be high-T c superconductors.Eu 8 H 46 is not a superconductor since it was predicted to exhibit a ferromagnetic property caused by the local unpaired f electrons of the Eu atom.The theoretical proposal for thermodynamically stable Ca 8 H 46 and Sr 8 H 46 . The Li 2 Y/LaH 17 clathrate structure comprises Li-centered H 20 cages and Y/La-centered H 28 cages, where each H 20 or H 28 cage consists of 12 pentagons or 12 pentagons and four hexagons (Fig. 7 (f)).It is worth noting that the estimated T c values of Li 2 YH 17 and Li 2 LaH 17 are relatively low with maximum values of 108 K at 200 GPa and 156 K at 160 GPa, respectively [111 ].