■ INTRODUCTION

The search for room-temperature superconductivity has been a long-standing scientific endeavor ever since Heike Kamerlingh Onnes discovered in 1911 that mercury could conduct electricity without resistance below a critical temperature around 4 K. Twenty-four years after the discovery of this phenomenon known as superconductivity, Wigner and Huntington 1 proposed that molecular hydrogen could undergo a phase transition into an atomic metallic hydrogen (AMH) phase at high pressures, which attracted great interest, especially after the conjecture of its ability to host very hightemperature superconductivity. [2][3][4][5] Creating AMH, however, has proven extremely challenging due to the stringent synthesis and characterization conditions at very high pressures, currently recognized to be above 500 GPa. [6][7][8] An early study found that hydrogen-rich metal hydride Th 4 H 15 was superconducting with a critical temperature T c of 8 K at ambient pressure; 9 later studies found that chemical substitution or doping could stabilize hydrogen-containing compounds that may host metallic and superconducting states approaching that of AMH. 10,11 This line of work has been actively pursued after the conceptual development of using chemical precompression to facilitate the formation of hydrogen-dominant metallic alloys 12 that host phonon-mediated high-temperature superconductivity.

Recent years have witnessed concerted efforts of exploring routes to achieving high-T c superconductivity in pressurestabilized metal hydrides (e.g., refs 13-16). Following the theoretical predictions of stabilizing H 2 S 17 and H 3 S 18 at high pressures, a high T c of 203 K at 150 GPa was documented experimentally. 19 A different class of binary high-T c metal (M) hydrides based on clathrate structure having stoichiometry MH 6 [13][14][15][16][20][21][22][23] was predicted, led by the theoretical prediction on CaH 6 . 23 Further efforts to reach higher T c led to the prediction and discovery of superhydrides with stoichiometries MH n with n > 6. 21,22,24 These efforts culminated in the experimental realization of superconducting LaH 10 with a T c of 250-260 K at pressures of 170-190 GPa, 25,26 followed by YH 9 with a T c of 243 K at 201 GPa 27 and YH 6 with a T c of 220 K at 166 or 237 GPa. 27,28 Very recently, the long-missing clathrate structured CaH 6 predicted in 2012 was successfully synthesized in the laboratory with a measured T c of 215 K at 170 GPa. 29,30 Other notable cases of clathrate structured high-T c hydrides include YH 10 , 21,22 from theory, and ThH 9 and ThH 10 34 from experiment. A recent work claimed possible room-temperature superconductivity in the C-S-H system, 35 but the structure and stoichiometry of the synthesized material remain unclear and require further investigation. An overriding idea underlying all of these efforts is to raise the hydrogen content while at the same time ensuring that the combination of compression and chemical interactions are favorable to promote the H 2 molecules in the dense structure to create material configurations closely resembling AMH in bonding environments. In this regard, a major task has thus been to find superhydrides with proper electron-rich heavy atoms that are capable of stabilizing hydrogen in a nonmolecular form to generate T c exceeding that of the established superhydrides such as LaH 10 . 13,31,36,37 Recent studies 15,16,21,22,34 showed that rare-earth (RE) and actinide (An) elements are capable of maintaining significant hydrogen by forming high-pressure clathrate compounds with a very high T c . These results raise the exciting prospects of finding RE/An superhydrides containing even higher hydrogen content and hence further increased T c that may approach or even exceed room temperature. Based on these considerations, we have explored electron-rich rare-earth and actinide elements that provide electrons to dissociate molecular hydrogen, thereby creating an AMH-like environment conducive to harboring high-T c superconductivity. Using our developed structural search algorithms, 38,39 we uncovered a new class of hydrogen-superrich compounds with stoichiometry MH 18 (M = RE or An atom). These extreme superhydrides exhibit unique H 36 clathrate atomic cages with bonding environments similar to that predicted for AMH. First-principles calculations for different MH 18 superhydrides predict diverse T c values among the same stoichiometry that offer insights into the effect and nature of chemical precompression. Notable among these extreme superhydrides are CeH 18 and ThH 18 that exhibit above-room-temperature T c near 330 K at 350 GPa and 321 K at 600 GPa, respectively, which represent the highest T c values among all hitherto known thermodynamically stable superhydrides. The results of this study provide a wealth of information on material behaviors and key physics and chemistry insights that allow an in-depth study of the factors with major influence on achieving ultrahigh-temperature phonon-mediated superconductivity in hydrides, thereby opening a path for creating binary and multinary high-T c superhydrides that can operate at or even above room temperature.

■ RESULTS AND DISCUSSION

We performed structure searches in binary hydrides MH m (m = 2-24) over a wide range of hydrogen contents to predict stable structures, with a sharp focus on hydrogen-superrich phases. Particularly noteworthy was the finding of new stoichiometric Ce and Th extreme superhydrides Ce/ThH 18 stable at experimentally accessible pressures 8,40 around 300 GPa after the zero-point energy (ZPE) was considered. Specifically, CeH 18 and ThH 18 become stable at reduced pressures of 315 and 281 GPa, respectively (see Figure 1a,b as well as Figure S1), making the experimental synthesis and characterization feasible. It is noted that these newly predicted extreme superhydrides possess higher T c with competing synthesis pressure compared to the previously proposed high-T c binary superhydride YH 10 , which was not successfully synthesized in a recent experiment. 27 A recent work reported that materials synthesis could be achieved near terapascal pressures, 40 making it feasible to realize the materials predicted in this work. Further calculations for a broad range of RE and An hydride compounds up to 700 GPa reveal ubiquitous stability of the MH 18 stoichiometry among diverse RE and An hydrides for RE/An = Y, La, Ce, Ac, and Th (Figure 1b and Tables S1 andS2). Details of the computational methods for the structure search and property calculations are provided in the Supporting Information.

Conspicuous in the predicted crystal structure of the MH 18 extreme superhydrides is the H 36 clathrate cage, which contrasts with the H 24 , H 29 , and H 32 structural units found in previously identified hydrides with lower hydrogen contents. The peculiar three-dimensional hydrogen clathrate structure crystallizes in a unit cell with space group Fddd (Figure 2a) in which H 36 cages are linked by a 6H 6 ribbon-ring structure with two wrinkled H 6 hexagons above and below with bridge bonds connecting the H 6 hexagons to the 6H 6 ribbon ring (Figure 2b,c). At higher pressures, depending on the choice of the M atom, the H 36 clathrate units rearrange and stabilize in another structure with space group Fmmm (Figure 2b). Below we focus our analysis mainly on CeH 18 to showcase its prominent properties while also discussing key data and trends involving other MH 18 compounds. We first examine the high-pressure electronic structure of CeH 18 . The results clearly indicate the CeH 18 phase is metallic with several bands crossing the Fermi level (Figure 3a, left), and hydrogen atoms make a substantial contribution to the electronic density of states (DOS) near the Fermi level, which is almost identical to the DOS contributed by the electrons from Ce (Figure 3a, right). The DOS is essentially flat around the Fermi energy, which is notably different from the DOS of LaH 10 , which exhibits a van Hove singularity near the Fermi energy. 21 The calculated band structures at different pressures reveal changes in band filling; notably, the hole-bands at the Xpoint, along the Z-X path, and at the Y-point move down with increasing pressure (see Figure S2), indicating a systematic pressure-driven electron transfer that suppresses T c on further compression.

Turning to the phonons and electron-phonon coupling in CeH 18 , the calculated phonon dispersion results are shown in Figure 3b, left. No imaginary phonon modes are present in the entire Brillouin zone, indicating the dynamic stability of this crystal structure. We then computed the Eliashberg spectral function α 2 F (ω), from which the electron-phonon coupling parameter can be obtained via a simple integration in the frequency domain. The resulting integrated electron-phonon coupling parameter λ = 2.3 is quite large and comparable to that found for H 3 S (λ = 2.2). 18 Such strong electron-phonon coupling makes various approximate weak-coupling T c formulas generally unreliable, and an accurate description necessitates direct numerical solutions to the Eliashberg equations. 41,42 Adopting the latter method, we calculated superconducting energy gap and transition temperature and varied the Coulomb pseudopotential from μ* = 0.10 (as is typically used) to μ* = 0.13 to estimate a reasonable range of T c values. The resulting T c of 309-329 K (for μ* = 0.10 and 0.13) at 350 GPa, where CeH 18 is stable, is well above the room temperature. These results are significantly higher than the highest T c predicted for previously reported Ce superhydrides with lower hydrogen contents (e.g., T c = 117 K for CeH 9 ) 43 and represent the highest predicted T c among all known thermodynamically stable superhydrides. 22 To elucidate the underlying mechanisms for the predicted superconductivity of these extreme superhydrides, we calculated electronic, phonon, and electron-phonon coupling parameters, and the resulting critical temperatures using the Eliashberg equations of other selected MH 18 compounds. We also performed identical calculations for pure AMH in the I4 1 / amd structure, which is predicted to be stable or metastable in the same pressure range. 44 First, we note that systematic examination of the energetics and dynamic stability of the MH 18 structures beyond their stability fields reveal metastability to lower pressures of phases that maintain high superconducting temperatures (Figure 1b). The pressure dependence of T c for the identified stable and metastable phases of MH 18 and I4 1 /amd hydrogen is given in Figure 4.

The T c values are distributed over a large range, from 50 K for Fddd-AcH 18 at 700 GPa up to near 330 K for Fddd-CeH 18 at 350 GPa (μ* = 0.10). Moreover, the resulting critical temperatures vary considerably among the MH 18 superhydrides at the same pressure. The associated microscopic properties of each phase listed in Table 1 detail the differences in superconducting behavior among the compounds. The disparities within the same stoichiometry and clathrate structures stem from their characteristic vibrational frequencies, electron-phonon coupling parameters λ, and electronic DOS at the Fermi level N(E f ). We also have calculated the charge transfer for CeH 18 and LaH 18 , and the results (Table S1) show charge transfer from metal Ce/La to hydrogen atoms. The T c values for different MH 18 structures may also be related to the different f electron fillings, e.g., there are different T c values for different MH 10 superhydrides as previously reported. 22 The present findings offer important clues for understanding the trends in superhydride T c values that approach that of AMH at high pressures. Among the extreme superhydrides, CeH 18 exhibits the highest T c over the entire pressure range, with nearest-neighbor H-H distances of 0.85-1.17 Å, close to that of I4 1 /amd structured AMH (1.0 Å) at 500 GPa. The calculated projected phonon DOS from the Ce and H atoms (Figure 3b) shows that the main contributions to the EPC come from the mid-and high-frequency hydrogen vibrations at 500-2500 cm -1 . Further, most of MH 18 compounds exhibit monotonically decreasing T c with increasing pressure, the exception being both Fddd-and Fmmm-ThH 18 for which T c slightly increases on compression. The decrease in T c with pressure of most superhydrides 21,22 is reminiscent of the behavior of AMH at higher pressures (700-1,000 GPa), 5 which is consistent with the idea that chemical precompression in hydrogen compounds shifts their material behavior toward lower pressures. These insights are helpful for continued exploration and rational design of optimal superconducting superhydrides.

We consider additional effects such as spin-orbit coupling (SOC), magnetism, and electron correlation that may affect the estimated T c of the predicted MH 18 compounds. We take the highest T c extreme superhydrides CeH 18 and ThH 18 as examples to assess these effects. Our calculations reveal that the T c values are insensitive to SOC (Table 1), which is consistent with the nearly identical band structures calculated with and without the SOC (Figure 3a). We examined the energetics of the magnetic structures of the predicted MH 18 compounds by considering six possible magnetic configurations (one ferromagnetic and five antiferromagnetic configurations). The results show that the nonmagnetic state is the most stable among all of the predicted MH 18 compounds. A recent work quantified the correlation between the f orbital states at the Fermi energy with T c for the lanthanide hydrides. 45 These results offer insights into the reason for CeH 18 and ThH 18 hosting high T c values due to the fact that both Ce and Th elements have less partially filled f electron states. Although we have not considered electron correlation effects on the EPC due to a lack of available computational tools, we note that previous studies 18,[21][22][23] have shown that the experimental results on several hydrides and superhydrides (e.g., those of La, Y, Th, and Ca [25][26][27][28][29][30]34,46 ) are well described by the theoretical results obtained within the current EPC computation scheme without considering the electron correlation effects. On this basis, it is expected that the results in the present work offer a reasonably accurate description of the superconductivity in the newly predicted MH 18 compounds. For other RE/An hydrides that are not discussed in this work, especially those with more filled f electron states, it is expected that the magnetic, SOC, and strong correlation effects play more prominent roles, and other hydrogen-rich super-conducting hydrides cannot be ruled out. However, phononmediated superconductivity is unlikely to be the underlying mechanism in these systems, and related studies are beyond the scope of this work.

■ CONCLUSIONS

Extreme superhydrides represent a class of hydrogen-rich materials containing the highest atomic hydrogen content among binary metal hydrides reported to date. The structures of these MH 18 compounds consist of unique H 36 clathrate units stabilized under pressure starting around 300-400 GPa. Superconducting critical temperatures exceeding room temperature arise from favorable electronic density of states near the Fermi level and the large phonon energy scale of the vibration modes, as well as the strong electron-phonon coupling, as demonstrated for CeH 18 and ThH 18 . Meanwhile, other MH 18 compounds exhibit a wide range of T c with variable combinations of electronic, phonon, and electron-phonon coupling. Recent focus in this research field has been mostly on reaching higher critical temperatures in ternary and higher multinary hydrides; the present work raises remarkable prospects that the bonding and electronic properties of the newly identified extreme binary superhydrides parallel those of atomic metallic hydrogen, giving rise to the highest T c values among all known thermodynamically stable superhydrides. Furthermore, we have examined the influence of various heavy elements in hydrogen-rich superconductors over a range of pressures, and the resulting insights extend and refine the original conjecture of reducing the pressure for achieving AMH-like superconductivity by chemical doping and chemical precompression. Additional considerations of electron-31 or hole- 47 doping effects in these extreme superhydrides may provide a promising platform for further exploration and optimization of superconductors that may host extraordinary above-room-temperature superconductivity.

■ COMPUTATIONAL METHODS

Our structure search is based on the PSO algorithm 48,49 as implemented in the CALYPSO methodology. 38,39 In an attempt to investigate the stable and metastable structures of MH x (x = 2-24; M stands for rare-earth/actinide metals), we have performed structure searches of MH x at 400 GPa. We have first performed structure searches for the crystal structures of the ThH x (x = 2-24) series at 400 GPa with simulation cells containing one to four formula units up to 50 atoms per cell. These structure searches found that ThH 18 holds the highest atomic hydrogen content among all known/predicted hydrides reported to date, and its superconducting T c is very high (the estimated T c of ThH 18 -Fmmm is in the range of 296-321 K in the pressure range of 400-600 GPa), which motivated us to focus on the MH 18 chemical composition. The other structures of MH x stoichiometry were established by substituting the elements into the known structures of ThH x (x = 2, 4, 6, 9, 10, 12, 16, 17, and 18). Our searches found that MH 18 (M = Sc, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Pa, U, Np, Pu, Am, Cm, and Cf) are unstable in the pressure range of 0-700 GPa. We then performed further variable-stoichiometry structure searches that are able to explore the most stable structure among a broad range of stoichiometries selected for the stable MH x (M = Y, La, Ce, and Ac; x = 2-24) system at 400 GPa, and the results of these additional searches confirm the aboveidentified stability of MH 18 (M = Y, La, Ce, Ac, and Th). Structural optimization and computations of enthalpy, phonon, electronic structures, and spin-orbit coupling (SOC) were all performed in the framework of density functional theory (DFT) as implemented in the VASP code. 50 The Perdew-Burke-Ernzerhof 51 generalized gradient approximation (GGA) 52 was employed, and a kinetic cutoff energy of 500 eV was adopted. The zero-point energy (ZPE) of predicted compounds was obtained from lattice dynamic calculations as implemented in the PHONOPY code. 53 The electron-phonon coupling calculations were carried out using the QUANTUM ESPRESSO code. 54 Ultrasoft pseudopotentials for RE/An and H elements were used with a kinetic energy cutoff of 80 Ry. To reliably calculate the electron-phonon coupling in these metallic systems, we employed k-meshes of 2π × 0.045 Å -1 for the electronic Brillouin zone integration and q-meshes of 2π × 0.09 Å -1 for all of the phonon calculations.

■ ASSOCIATED CONTENT

* sı Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.2c05834.

Computational details of the structure prediction and electron-phonon coupling simulations for predicted MH 18 structures; stability; electronic structures; and detailed structure information of predicted MH 18 structures (PDF)

Structure.zip (ZIP) Calculated pressure (P (GPa)) variation of λ, ω log (K), N (E f ) (states/Ry/atom/spin) and T c (K) for μ* = 0.10(0.13) for selected MH 18 compounds in Fmmm or Fddd phase and hydrogen in the I4 1 / amd structure. 5,44,55