Formation mechanism of chemically precompressed hydrogen clathrates in metal superhydrides

In recent years, hydrides have attracted much attention theoretically and experimentally because of their promising possibility for the realization of room-temperature superconductivity (SC) [1, 2]. Motivated by the pioneering idea of Neil Ashcroft on high-temperature SC in metallic hydrogen [3] and the incessant theoretical predictions of high superconducting transition temperature $T_{c}$ in a number of hydrides [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], experiments have confirmed that sulfur hydride H₃S and lanthanum hydride LaH₁₀ exhibit $T_{\rm c}$ around 203 K at ${\sim}$155 GPa [21] and 250$-$260 K at ${\sim}$170 GPa [22, 23], respectively. Subsequently, such a conventional Bardeen-Cooper-Schrieffer type SC has also been experimentally observed in various compressed hydrides at high pressures [25, 26, 27, 24, 28, 30, 29]. For examples, ThH₁₀ (ThH₉) exhibits $T_{c}$ = 159$-$161 (146) K between 170 and 175 GPa [24], while CeH₉ exhibits a $T_{c}$ of ${\sim}$100 K at 130 GPa [27]. More recently, carbonaceous sulfur hydride was observed to exhibit a room-temperature SC with a $T_{\rm c}$ of 288 K at ${\sim}$267 GPa [30]. Therefore, the experimental observations of high-temperature SC in either surfur-containing hydrides [30, 21] or superhydrides containing an abnormally large amount of hydrogen [22, 23, 25, 26, 27, 28, 24] has launched a new era of high-$T_{\rm c}$ superconductors.

Compared to the existence of metallic hydrogen at high pressures over ${\sim}$400 GPa [31, 32], the syntheses of superhydrides with H-rich clathrate structures have been achieved at relatively much lower pressures, because H atoms can be “chemically precompressed” by chemical forces between metal atoms and H cages [33]. Using density-functional theory (DFT) calculations, the high-$T_{c}$ superconducting phases of various superhydrides have been predicted to be metastable at higher pressures than a critical pressure $P_{c}$ [34, 35, 36, 37]. It is noticeable that the magnitude of $P_{c}$ reflects the strength of chemical precompression in superhydrides. Experimentally, the $P_{c}$ value of ThH₁₀ having a fcc-Th framework [see Fig. 1(a)] was measured to be ${\sim}$85 GPa [24], much lower than $P_{c}$ ${\approx}$ 170 of an isostructural superhydride LaH₁₀ [22, 23]. Furthermore, ThH₉ (CeH₉) having a hcp metal framework [see Fig. 1(b)] was observed to exhibit a $P_{c}$ of ${\sim}$86 (80) GPa [24, 25, 26, 27]. Based on these existing experimental data [22, 23, 24], it is most likely that $P_{c}$ changes with respect to metal species: i.e., Group-4 metal hydrides ThH₁₀, ThH₉, and CeH₉ with occupied $f$-subshell electrons have lower $P_{c}$ values or larger chemical precompressions compared to a Group-3 metal hydride LaH₁₀. Our recent DFT calculations for LaH₁₀ [38] revealed that the isolated La framework without H atoms behaves as an electride at high pressures, where some electrons detached from La atoms are well localized in interstitial regions. These interstitial excess electrons are easily captured to H atoms, forming a H clathrate structure in LaH₁₀. In the present study, such an electride feature in the La framework of LaH₁₀ is compared with other metal frameworks of the above-mentioned superhydrides ThH₁₀, ThH₉, and CeH₉. By the estimation of Coulomb attractions between metal atoms and H cages, we provide an explanation for the different chemical precompressions observed in such superhydrides [22, 23, 24, 25, 26, 27], as will be discussed below.

In this paper, using first-principles DFT calculations, we perform a comparative study of chemical precompressions in ThH₁₀, ThH₉, CeH₉, and LaH₁₀. We find that the isolated metal frameworks of ThH₁₀, ThH₉, and CeH₉ possess more interstitial excess electrons than that of LaH₁₀ at an equal pressure of 300 GPa. Such loosely bound electrons can be easily captured to form H clathrate structures with attractive Coulomb interactions between cationic metal atoms and anionic H cages. Using the calculated Bader charges [39] and positions of metal and H atoms in each superhydride, we estimate a chemical pressure acting on H cage around a metal atom. As a result, ThH₁₀, ThH₉, and CeH₉ are found to have larger chemical precompressions than LaH₁₀, consistent with the experimentally observed $P_{c}$ values in these superhydrides [22, 23, 24, 25, 26, 27]. It is thus demonstrated that Group-4 metal hydrides occupying $f$ electrons can be more chemically precompressed compared to Group-3 metal hydride, thereby contributing to lower $P_{c}$. The present findings illuminate that interstitial excess electrons in the metal frameworks of superhydrides are of importance to generate the chemical precompression of H cages around metal atoms.

Our first-principles DFT calculations were performed using the Vienna ab initio simulation package with the projector-augmented wave method [40, 41, 42]. Here, we treated Th 6$s^{2}$6$p^{6}$5$f^{1}$6$d^{1}$7$s^{2}$, Ce 5$s^{2}$5$p^{6}$4$f^{1}$5$d^{1}$6$s^{2}$, La 5$s^{2}$5$p^{6}$5$d^{1}$6$s^{2}$ and H 1$s^{1}$ as valence electrons, including 6$s^{2}$6$p^{6}$, 5$s^{2}$5$p^{6}$, and 5$s^{2}$5$p^{6}$ semicore electrons for Th, Ce, and La, respectively. For the exchange-correlation energy, we employed the generalized-gradient approximation functional of Perdew-Burke-Ernzerhof [43, 44]. A plane-wave basis was used with a kinetic energy cutoff of 500 eV for ThH₁₀ and ThH₉. The ${\bf k}$-space integration was done with 24${\times}$24${\times}$24 and 18${\times}$18${\times}$12 $k$ points for ThH₁₀ and ThH₉, respectively. All atoms were allowed to relax along the calculated forces until all the residual force components were less than 0.005 eV/Å. We calculated phonon frequencies with the 6${\times}$6${\times}$6 (5${\times}$5${\times}$3) $q$ points for ThH₁₀ (ThH₉) using the QUANTUM ESPRESSO package [45]. For CeH₉ and LaH₁₀, we chose the calculation parameters used in our previous works [38, 46].

We first optimize the structures of experimentally synthesized superhydrides ThH₁₀, ThH₉, CeH₉, and LaH₁₀ as a function of pressure using DFT calculations. These superhydrides have hydrogen sodalitelike clathrate structures with high-symmetry space groups: i.e., $Fm$$\overline{3}m$ (No. 225) for ThH₁₀ and LaH₁₀, while P6₃/$mmc$ (No. 194) for ThH₉ and CeH₉. As shown in Fig. 1(a), ThH₁₀ (LaH₁₀) is constituted by the fcc metal framework, where each Th (La) atom is surrounded by the H₃₂ cage consisting of 32 H atoms. Meanwhile, ThH₉ (CeH₉) have the hcp metal framework with the H₂₉ cage surrounding a Th (Ce) atom [see Fig. 1(b)]. Note that there are two (three) species of H atoms composing the H₃₂ (H₂₉) cages in ThH₁₀ and LaH₁₀ (ThH₉ and CeH₉). The optimized structures of these superhydrides show that the lattice constants decrease monotonously with increasing pressure (see Fig. S1 in the Supplemental Material). Accordingly, the averaged bond lengths $d_{\rm M-H}$ between metal and H atoms decrease with increasing pressure [see Fig. 2(a)]. We find that $d_{\rm M-H}$ for ThH₁₀ having the H₃₂ cage is longer than ThH₉ having the H₂₉ cage at a given pressure. Meanwhile, $d_{\rm M-H}$ for CeH₉ is shorter than that for ThH₉, possibly because of the smaller size of Ce atom with the atomic number of $Z$ = 58 compared to Th atom with $Z$ = 90. Interestingly, despite the larger atomic number of Th than La ($Z$ = 57), the $d_{\rm M-H}$ values for ThH₁₀ and ThH₉ are close to that for LaH₁₀ at a given pressure, implying that the former superhydrides have larger chemical precompressions than the latter one. We note that the charges of cationic metal and anionic H atoms are also essential ingredients for determining chemical precompression, as discussed below. Figure 2(b) displays the averaged H$-$H bond lengths $d_{\rm H-H}$ for ThH₁₀, ThH₉, CeH₉, and LaH₁₀, which also decrease monotonously with increasing pressure. It is seen that the $d_{\rm H-H}$ values for ThH₁₀ and LaH₁₀ are shorter than those for ThH₉ and CeH₉, indicating that the H₃₂ cages composed of larger number of H atoms give rise to shorter $d_{\rm H-H}$ compared to the H₂₉ cages.

In order to examine the dynamical stability of ThH₁₀, ThH₉, CeH₉, and LaH₁₀, we calculate their phonon spectra as a function of pressure. As shown in Fig. 3(a), the phonon spectrum of ThH₁₀, calculated at 130 GPa, exhibits the softening of low-energy phonon modes (marked by arrows) along the $\Gamma-L$ and $\Gamma-K$ lines. Such H-derived phonon modes finally have imaginary frequencies at 120 GPa [see Fig. 3(b)]. This indicates that the fcc-ThH₁₀ phase becomes unstable as pressure decreases. Therefore, the phonon spectra as a function of pressure allow us to predict the $P_{c}$ values of about 130, 110, 100, and 220 GPa for ThH₁₀, ThH₉, CeH₉ [46], and LaH₁₀ [36], respectively (see Fig. S2 in the Supplemental Material). We find that ThH₁₀ and ThH₉ have much lower $P_{c}$ than LaH₁₀, while their $P_{c}$ values are close to that of CeH₉. The overall trend of these predicted $P_{c}$ values in four superhydrides are well consistent with the experimentally measured ones of about 85, 86, 80, and 170 GPa for ThH₁₀ [24], ThH₉ [24], CeH₉ [25, 26, 27], and LaH₁₀ [22, 23], respectively. It was previously pointed out that for LaH₁₀, the anharmonic effects on phonons and the quantum ionic zero-point energy are of importance for a proper prediction of $P_{c}$ [47]. Therefore, the above overestimation of predicted $P_{c}$ values is likely due to the ignorance of anharmonic and quantum effects [48, 49, 50] in the present theory. Nevertheless, we can say that ThH₁₀, ThH₉, and CeH₉ have significantly larger chemical precompressions compared to LaH₁₀.

We next explore the electride-like characteristics of the isolated metal frameworks of ThH₁₀, ThH₉, CeH₉, and LaH₁₀. Here, the structure of each metal framework is taken from the optimized structure of the corresponding superhydride. The valence charge densities ${\rho_{\rm M}}$ (without including of semicore electrons) of the metal frameworks of ThH₁₀, ThH₉, CeH₉, and LaH₁₀, calculated at an equal pressure of 300 GPa, are displayed in Figs. 4(a), 4(c), 4(e), and 4(g), respectively. It is seen that some electrons detached from metal atoms are localized in the interstitial regions around the $A_{1}$ and $A_{2}$ sites. These interstitial excess electrons of the so-called $A_{1}$ and $A_{2}$ anions are well confirmed by the corresponding electron localization function (ELF) [51]. Figures. 4(b), 4(d), 4(f), and 4(h) represent the calculated ELFs of the metal frameworks of ThH₁₀, ThH₉, CeH₉, and LaH₁₀, respectively. For the Th framework of ThH₁₀, the number of electrons $Q_{A_{1}}$ ($Q_{A_{2}}$) within the muffin-tin sphere of the $A_{1}$ ($A_{2}$) anion is $-$0.281 ($-$0.234)$e$, larger in magnitude than $-$0.204 ($-$0.194)$e$ for the La framework of LaH₁₀ [see Figs. 4(a) and 4(g)]. Similarly, as shown in Figs. 4(c) and 4(e), $Q_{A_{1}}$ ($Q_{A_{2}}$) in ThH₉ is $-$0.124 ($-$0.080)$e$, larger in magnitude than the corresponding value of $-$0.118 ($-$0.072)$e$ in CeH₉. Therefore, the metal framework of ThH₁₀ (ThH₉) exhibits a more electride feature than that of LaH₁₀ (CeH₉). It is noted that the magnitudes of $Q_{A_{1}}$ and $Q_{A_{2}}$ change as a function of pressure (see Figs. S3 and S4 in the Supplemental Material), showing that the electride-like characteristics of metal frameworks are enhanced with increasing pressure. Indeed, the localization of interstitial excess electrons also emerges in compressed alkali metals at high pressures [52, 53, 54], in order to reduce Coulomb repulsions arising from the overlap of atomic valence electrons. Such loosely bound anionic electrons residing in the metal frameworks of compressed superhydrides can be easily captured to H atoms, forming H cages with attractive Coulomb interactions between cationic metal and anionic H atoms. It is remarkable that the anionic electrons in H cages are mostly supplied because of the electride nature of metal frameworks at high pressures, rather than due to the different electronegativities of metal and H atoms [11].

Figures 5(a) and 5(b) show the calculated total charge densities of ThH₁₀ and ThH₉ at 300 GPa, respectively. It is seen that the H atoms in each H cage are bonded to each other with covalent bonds, where each H$-$H bond has a saddle point of charge density at its midpoint, similar to the C$-$C covalent bond in diamond [55]. For ThH₁₀ (TH₉), the charge densities ${\rho}_{\rm H-H}$ at the midpoints of the H$-$H bonds are 0.911 and 0.729 (0.992, 0.769, and 0.601) $e$/ų: see the arrows in Figs. 5(a) and 5(b). In order to confirm that the interstitial excess electrons of the Th framework of ThH₁₀ (ThH₉) are captured to form the H₃₂ (H₂₉) cages, we calculate the charge densities of the isolated H₃₂ (H₂₉) cages without Th atoms. Here, the structure of each isolated H cage is taken from the optimized structure of the corresponding superhydride. As shown in Fig. 5(c) [5(d)], we find that ${\rho}_{\rm H-H}$ decreases as 0.742 and 0.621 (0.843, 0.580, and 0.479) $e$/ų, smaller than those in ThH₁₀ (TH₉). This indicates that the H$-$H covalent bonds in ThH₁₀ and TH₉ are strengthened by capturing the interstitial excess electrons of isolated Th frameworks.

To provide an explanation for the variation of $P_{c}$ in ThH₁₀, ThH₉, CeH₉, and LaH₁₀, we estimate chemical precompression by calculating the attractive Coulomb forces between a metal atom and its surrounding H atoms [see the lower panel in Figs. 1(a) and 1(b)]. This simple estimation is based on the complete screening of the electric field arising from metal atoms within H cages. Using the Bader [39] analysis, we calculate the cationic charge $Q_{\rm M}$ inside the Bader basin [see Figs. 5(a) and 5(b)] of metal atom in each superhydride. For ThH₁₀, ThH₉, CeH₉, and LaH₁₀, we obtain $Q_{\rm M}$ values of 1.486, 1.464, 1.199, and 1.036$e$, respectively (see Fig. 6). Assuming that $Q_{\rm M}$ is the point charge at the position of corresponding metal atom and the anionic charge ($-$$Q_{\rm M}$) of H atoms is uniformly distributed on the spherical shell with a radius of $d_{\rm M-H}$, we evaluate the magnitudes of Coulomb forces acting on the H atoms composing the H₃₂ or H₂₉ cage, and divide it by the surface area of the spherical shell. Figure 6 shows such estimated chemical pressures of ThH₁₀, ThH₉, CeH₉, and LaH₁₀ at 300 GPa, with ratios relative to the value of LaH₁₀. We find that the chemical pressures of ThH₁₀ and ThH₉ (CeH₉) are about two (one and half) times higher than that of LaH₁₀, indicating that the former Group-4 metal hydrides have larger chemical precompressions to attain lower $P_{c}$ values than the latter Group-3 metal hydride. Considering that the $d_{\rm M-H}$ values of ThH₁₀ and ThH₉ are close to that of LaH₁₀ [see Fig. 2(a)], the higher chemical pressures in Th superhydrides are likely attributed to more cationic and anionic charges compared to LaH₁₀. As shown in Fig. 6, the chemical pressures of four superhydrides are well consistent with their variations of $Q_{\rm M}$. It is noted that the estimated chemical pressure of CeH₉ is lower than that of isostructural ThH₉ at 300 GPa (see Fig. 6), while the predicted value of $P_{c}$ = 100 GPa for the former is lower than that (110 GPa) for the latter. This inconsistency of chemical precompression and $P_{c}$ between CeH₉ and ThH₉ may be due to the delocalized nature of Ce 4$f$ electrons [46], which could lower $P_{c}$ via a more hybridization with the H 1$s$ state. Nevertheless, despite their crude simulations, the estimated relative chemical pressures of ThH₁₀, ThH₉, CeH₉, and LaH₁₀ are in reasonable agreement with the variation of experimentally measured $P_{c}$ values [22, 23, 24, 25, 26, 27].

Using first-principles DFT calculations, we have conducted a comparative study of chemical precompressions in experimentally synthesized superhydrides including ThH₁₀, ThH₉, CeH₉, and LaH₁₀. We found that these superhydrides form H clathrates by capturing excess electrons in interstitial regions of their isolated fcc- and hcp-metal frameworks. By taking into account the attractive Coulomb interactions between cationic metal atom and its surrounding H atoms, we estimated chemical precompressions in ThH₁₀, ThH₉, CeH₉, and LaH₁₀. It was found that ThH₁₀, ThH₉, and CeH₉ have larger chemical precompressions than LaH₁₀, consistent with the variation of experimentally measured $P_{c}$ values [22, 23, 24, 25, 26, 27]. Our findings not only demonstrated that interstitial excess electrons in the metal frameworks of superhydrides play an importnt role in generating the chemical precompression of H cages around metal atoms, but also have important implications for the exploration of new superhydrides which can be synthesized at moderate pressures below ${\sim}$100 GPa.

See Supplemental Material for the lattice constants of ThH₁₀, ThH₉, CeH₉, and LaH₁₀, the phonon spectrum of ThH₉, and the valence charge densities of the Th frameworks of ThH₁₀ and ThH₉.

S. Y., C. W., and S. L. contributed equally to this work.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (Grants No. 2019R1A2C1002975, No. 2016K1A4A3914691, and No. 2015M3D1A1070609). The calculations were performed by the KISTI Supercomputing Center through the Strategic Support Program (Program No. KSC-2020-CRE-0163) for the supercomputing application research.

The data that support the findings of this study are available from the corresponding author upon reasonable request.