Potential Room Temperature Superconductivity in Clathrate Lanthanide/Actinides Octadechydrides at Extreme Pressures

Atomic metallic hydrogen (AMH) hosting high-temperature superconductivity has long been considered a holy grail in condensed matter physics and attracted great interest, but attempts to produce AMH remain in intense exploration and debate. Meanwhile, hydrogen-rich compounds known as superhydrides offer a promising route toward creating AMH-like state and property, as showcased by the recent prediction and ensuing synthesis of LaH 10 that hosts extraordinary superconducting critical temperatures (T c ) of 250-260 K at 170-190 GPa. Here we show via advanced crystal structure search a series of hydrogen-superrich clathrate compounds MH 18 (M: rare-earth/actinide metals) comprising H 36 -cage networks, which are predicted to host T c up to 329 K at 350 GPa. An in-depth examination of these extreme superhydrides offers key insights for elucidating and further exploring ultimate phonon-mediated superconductivity in a broad class of AMH-like materials.


Introduction
First predicted 86 years ago by Wigner and Huntington, 1 metallic hydrogen has been attracting great interest, especially after the conjecture of its ability to host high-temperature superconductivity. [2][3][4][5] The quest for ultimate atomic metallic hydrogen (AMH), however, has proven extremely challenging due to stringent synthesis requirements of ultrahigh pressures and supersensitive characterizations that push experimental limits. 6-8 An early study of Th 4 H 15 found this metal hydride superconducting with a critical temperature T c of 8 K at ambient pressure; 9 a proposal was also made to use chemical pre-compression to stabilize hydrogen-containing compounds that may host metallic and superconducting hydrogen states. 10 This line of work has been most actively pursued after Ashcroft's suggestion 11 that hydrogen dominant metallic alloys may serve as surrogate materials for AMH in probing ultimate phonon-mediated superconductivity. Recent years have seen concerted efforts in exploring compressed metal hydrides. [12][13][14] Hydrogen sul des with T c of 203 K at 150 GPa was discovered 15 following theoretical predictions of high-T c superconductivity in H 2 S 16 and H 3 S. 17 On the other route, a distinct class of clathrate superhydrides has been predicted to possess higher T c values 18-20 with CaH 6 being the rst of such examples ever predicted, 21 culminating with the experimental realization of LaH 10 that exhibit high T c of 250-260 K at pressures of 170-190 GPa, 22,23 YH 9 with a T c of 243 K at 201 GPa, 24 YH 6 with a T c of 220 K at 166 GPa or 237 GPa 24,25 and CaH 6 with a T c of 215 K at 170 GPa. 26 14,19,20,31 show that rare-earth (RE) and actinide (An) elements are capable of holding a large amount of hydrogen by forming clathrate compounds at high pressures, and the resulting hydrides host high-T c values. These results raise the prospects of nding RE/An hydrides containing even higher hydrogen contents, and such superhydrides may possess further increased T c approaching or even exceeding room temperature.
Based on these considerations, we have chosen a series of electron-rich rare-earth and actinide elements as hosts to provide electrons to dissociate molecular hydrogen pairs, thereby creating an AMH-like environment conducive to harboring higher T c superconductivity.
Here, we report on the nding of a series of hydrogen-superrich MH 18 compounds, where M stands for RE or An elements, identi ed using our developed structural search algorithm. 34 GPa, while other compounds possess widely variable T c from 50 K to 321 K. These intriguing results provide a wealth of information on rich material behaviors and key physics insights that allow an indepth study of factors with major in uence on approaching ultimate phonon-mediate superconductivity.
This work opens a promising path for further exploration of binary and higher order superhydrides that can meet or even exceed AMH in hosting superconducting states with higher T c values.

Results
Phase diagram. We have performed structure searches in binary hydrides MH m (m=2−24) over a wide range of hydrogen contents to predict stable structures at high pressures. This search process has led to the discovery of a series of hydrogen-superrich compounds. Particularly noteworthy among these are new stoichiometric Ce and Th superhydrides Ce/ThH 18 that stabilize at experimentally accessible but technically very challenging pressure range around 400 GPa without considering the zero-point energy (ZPE), as shown in Figure 1(a). Further calculations of the formation enthalpy with the inclusion of ZPE show that CeH 18 and ThH 18 become stable at notably reduced pressures of 315 and 281 GPa ( Figure   1(b)), respectively, making the experimental synthesis and characterization more feasible. A systematic comparison of the stability pressure ranges calculated without and with the inclusion of ZPE is given in Figure S1. We examined a broad range of RE and An superhydrides up to 700 GPa, and the results reveal that this MH 18 stoichiometry is ubiquitous among diverse RE and An superhydrides for RE/An=Y, La, Ce, Ac, and Th, which are stable over variable extended ranges of pressures as shown in Figure 1 Figure 2(a) shows the crystal structure that comprises a peculiar three-dimensional hydrogen clathrate structure of space group Fddd, where each M atom is located at the center of a clathrate H 36 cage that, as shown in Figure 2(c), consists of 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 structure. At higher pressures, which vary for different M atoms, this H 36 clathrate units rearrange and stabilize in another structure of space group Fmmm shown in Figure 2(b). 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.

Discussion
Band/Dos character and Superconductivity of CeH 18 . To assess superconducting properties of CeH 18 , we rst evaluate its electronic band structure, taking 400 GPa as a representative case study. Calculated results clearly indicate the metallic nature of CeH 18 with several band crossing the Fermi level, as shown in the left panel of Figure 3(a). Results in the right panel of Figure 3(a) show that the 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. It is seen that the DOS is essentially at around the Fermi energy, which is notably different from the DOS of LaH 10 that hosts van Hove singularity around the Fermi energy 19 . To examine the changes of the band-lling states, we have calculated the band structures at different pressures, and the results (see Figure S2) show that the holebands at the X-point, along the Z-X path, and at the Y-point move downward with rising pressure, indicating a systematic pressure driven electron transfer, which provides an explanation for the decreasing trend of T c with increasing pressure. We next examine the phonon and electron-phonon coupling in CeH 18 . Calculated phonon dispersion results are shown in the left panel of Figure 3(b). 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 electronphonon coupling parameter can be obtained via a simple integration in the frequency domain. 37 Table I with the T c data plotted in Figure 4.
Calculated results show that the electron-phonon coupling parameter λ and the resulting T c vary considerably among the MH 18 superhydrides at the same pressure points (see Table I and Figure 4). This phenomenon re ects the broadly variable lattice dynamics and electronic states in different MH 18 compounds, despite that they all share the same stoichiometry and clathrate structures. These contrasting properties indicate substantial differences in their bonding strengths and charge distributions, which are manifested in the large disparities of their characteristic vibrational frequencies and electronic density of states at the Fermi level as listed in Table I.
It is noted that the MH 18 compounds host T c values that distribute over a large range, from 50 K for Fddd- These results show that these newly identi ed extreme MH 18 superhydrides represent a series of compounds that approach AMH in superconducting properties, con rming Ashcroft's original conjecture. 11 We have calculated the projected phonon density of states (PHDOS) from Ce and H atoms. The results given in the middle panel of Figure  3(b) show that the main contributions to the EPC come from the mid-and high-frequency hydrogen vibrations in the range of 500-2500 cm −1 . It is also seen from the results in Figure 4 that most MH 18 compounds exhibit monotonically decreasing T c with increasing pressure with the exception of Fddd-ThH 18 that hosts a slightly upward trend in its T c versus rising pressure. This pressure induced reduction of T c is reminiscent of the behavior of solid hydrogen at higher pressures (700-1,000 GPa), 5 which is consistent with the idea that chemical pre-compression in hydrogen compounds would shift material behaviors toward lower pressures. These insights are helpful for rational exploration and elucidation of optimal superconducting superhydrides.
Taking potential effects into consideration. Additional effects, such as spin-orbit coupling (SOC), magnetism and electron correlation, may affect the estimated T c of the predicted MH 18 compounds. We have taken CeH 18 and ThH 18 as case studies to assess the effect of SOC. Our calculations reveal that the T c values are insensitive to SOC (see Table I), which is consistent with the nearly identical band structures with and without the SOC (Figure 3a). We have examined the energetics of the magnetic structures of the predicted MH 18 compounds by considering six possible magnetic con gurations (one ferromagnetic and ve antiferromagnetic con gurations). The results show that the nonmagnetic state is the most stable for all the predicted MH 18 com pounds. We have not considered electron correlation effects on the EPC due to a lack of available computational tools for this purpose. We note, however, that previous studies 17,19−21 have shown that the experimental results on several superhydrides (e.g., La, Y, Th, and Ca hydrides 15,[22][23][24][25]29,30,42 ) 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.
In summary, we have identi ed a series of MH 18 compounds containing the highest atomic hydrogen content among metal hydrides reported to date. These extreme superhydrides comprise unique H 36 clathrate units stabilized at high pressures starting around 300-400 GPa. Favorable conditions of electronic density of states near the Fermi level and lattice vibrational modes that set large phonon energy scale and strong electron-phonon coupling in CeH 18 generate high T c of 329 K at 350 GPa.
Meanwhile, other MH 18 compounds exhibit a wide range of T c with variable electronic, phonon and electron-phonon coupling combinations. These results offer insights into intricate mechanisms for superconductivity in superhydrides, establishing a promising platform for further exploration and optimization of diverse superhydrides that approach and may even exceed high-temperature superconductivity predicted for atomic metallic hydrogen.

Methods
Structural predictions. Our structure search is based on PSO algorithm 43,44 using the CALYPSO methodology. 34,35,45 We performed structure searches at 400 GPa with 1− 4 formula units (f. u.) per cell of MH x (x=2-24). Most searches converge in 50 generations with about 2,500 structures generated.
Ab initio calculations. Structural optimization and computations of enthalpy, phonon, electronic structures, and spin-orbital coupling (SOC) were all performed in the framework of density-functional theory (DFT) as implemented in the VASP code. 46 The Perdew-Burke-Ernzerhof 47 generalized gradient approximation 48 was employed, and a kinetic cutoff energy of 500 eV was adopted to ensure that the calculated enthalpy converges to better than 1 meV/atom. The ZPE of predicted compounds were obtained from lattice dynamic calculations as implemented the PHONOPY code. 49 EPC calculations. The electron-phonon coupling (EPC) calculations were carried out using the QUANTUM ESPRESSO code. 50 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 metallic systems, we have 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 the phonon calculations of MH 18 compounds. We have employed the Migdal-Eliashberg theory 51 to calculate superconducting energy gap and transition temperature.

Data availability
The authors declare that the main data supporting the ndings of this study are contained within the paper and its associated Supplementary Information. All other relevant data are available from the corresponding author upon reasonable request.