High‐Pressure Stability and Superconductivity of Clathrate Thorium Hydrides

Recently, thorium hydride ThH9 possessing a H‐rich clathrate structure has been experimentally synthesized to exhibit a superconducting transition temperature Tc of 146 K at 170–175 GPa, while the more H‐rich clathrate thorium hydride ThH18 is theoretically predicted to reach a Tc of 296 K at 400 GPa. Using first‐principles calculations, it is found that ThH9 has a more ionic character between Th atoms and H cages than ThH18 and that the latter has a more substantial hybridization of the Th 6p semicore and H 1s states than the former. These different bonding characteristics of ThH9 and ThH18 reflect the very large difference in their stabilization pressures. Furthermore, it is revealed that the H‐derived density of states at the Fermi level EF is about two times larger in ThH18 than in ThH9, which in turn leads to the significant large differences in the electron–phonon coupling (EPC) constant and Tc between the two thorium hydrides. The findings not only present the different bonding and EPC characteristics of ThH9 and ThH18 but also have important implications for the design of H‐rich, high‐Tc clathrate metal hydrides.


Introduction
3] An early theoretical proposal [4] of high-temperature SC began with the atomic metallic hydrogen that can be transformed from molecular insulating hydrogen under high pressures. [5,6]owever, the experimental realization of atomic metallic hydrogen has been very difficult, [1][2][3] because it requires careful characterization of the material at very high pressures (%500 GPa and above), which is currently challenging for diamond anvil cells. [7,8]To achieve such a hydrogen-driven metallic state at relatively lower pressures, an alternative approach has been devised using hydride materials [9] in which hydrogen atoms can be "chemically precompressed" through interactions with other constituent atoms. [10,11]14][15][16][17] Motivated by the previous theoretical predictions of high superconducting transition temperatures T c in compressed hydrides, [18][19][20][21][22][23][24][25][26][27][28][29] intensive experimental efforts have been made for the synthesis of various hydrides such as sulfur hydride H 3 S, [12] rare-earth/actinide hydrides MH n (M = La, [13,14] Y, [15][16][17] Ce, [30][31][32] and Th [33] ), and alkaline earth hydride CaH 6 . [34,35]Specifically, the latter two families of metal hydrides have H-rich clathrate structures, where each metal atom is surrounded by the H cage composed of a large number of H atoms, that is, 24, 29, and 32 H atoms for n = 6, 9, and 10, respectively.These H-rich clathrate metal hydrides exhibit a wide range of T c depending on metal elements.For example, LaH 10 was experimentally observed to exhibit a T c of 250-260 K at pressures of 170-190 GPa [13,14] and subsequently YH 6 was measured to exhibit T c = 220 K at 166 GPa, [15,16] YH 9 , T c = 243 K at 201 GPa, [16,17] ThH 10 , T c = 159 K at 174 GPa, [33] and CaH 6 , T c = 215 K at 170 GPa. [34,35]ecently, Zhong et al. [36] used a crystal structure search method to find a new class of extremely H-rich clathrate rare-earth/actinide hydrides MH 18 (M = Y, La, Ce, Ac, and Th).Among them, the heaviest element Th hydride ThH 18 was predicted to exhibit a T c of 296 K at 400 GPa.Interestingly, these values of T c and stabilization pressure in ThH 18 are much higher than the experimental data (T c = 146 K at 170 GPa) in ThH 9 having half of hydrogen content. [33]Zhong et al. [36] explained the existence of the higher T c in ThH 18 in terms of the increased electronic density of states (DOS) at the Fermi level E F , the large phonon energy scale of the vibration modes, and the resulting enhanced electron-phonon coupling (EPC).Although these factors are essential for increasing T c in conventional phonon-mediated Bardeen-Cooper-Schrieffer [37] superconductors, more quantitative analysis is desired to explain the significant T c difference between ThH 9 and ThH 18 , as discussed below.Furthermore, understanding the significantly different bonding characteristics of ThH 9 and ThH 18 , which may reflect the very large difference in their stabilization pressures, is valuable in searching for H-rich, high-T c clathrate metal hydrides.
In this article, based on first-principles calculations, we investigate the origin of the very large differences in the stabilization pressure and T c between ThH 9 and ThH 18 .For this, we compare the structural, electronic, phononic, and superconducting properties of ThH 9 and ThH 18 compressed at 130 and 400 GPa, respectively.Compared to ThH 18 , ThH 9 has a greater ionic character with more charge transfer from Th to H atoms, while ThH 18 has substantially delocalized Th 6p semicore states at a high pressure of 400 GPa, leading to their strong hybridization with the H 1s state.These different bonding characteristics of ThH 9 and ThH 18 reflect a significant difference in their stabilization pressures.Furthermore, we reveal that ThH 18 has about two times larger H-derived DOS at E F and %29% higher averagesquared H-derived phonon frequency than ThH 9 , but the two hydrides have similar Fermi-surface average-squared electronphonon matrix element.As a result, the EPC constant λ and T c of ThH 18 are estimated as 2.08 and 269 K, respectively, which are greater than those (1.55 and 151 K) of ThH 9 .

Results and Discussion
We begin by optimizing the structures of ThH 9 [33] and ThH 18 [36]   using first-principles DFT calculations.Figure 1a,b shows the optimized structures of ThH 9 and ThH 18 at 130 GPa and 400 GPa, respectively.Hereafter, we focus on the comparison of the structural, electronic, phononic, and superconducting , which are identified by Wyckoff positions. [33,36]The (110) and (100) planes are drawn in the hcp and centered rectangular lattices, respectively.
properties of these two compressed structures.For ThH 9 , the Th sublattice forms the hexagonal closed packed (hcp) lattice with the lattice constants a = b = 3.726 Å and c = 5.565 Å (see Figure 1a), where the H 29 clathrate cage surrounding a Th atom consists of six tetragon, six pentagon, and six hexagon rings.Meanwhile, the Th sublattice in ThH 18 forms the centered rectangular lattice with a = 3.357 Å, b = 5.826 Å, and c = 7.180 Å (see Figure 1b), where the H 36 clathrate cage surrounding a Th atom consists of the strip of six hexagon rings and two wrinkled hexagon rings above and below the strip.It is worth noting that ThH 9 has four different H─H bond lengths of 1.096, 1.218, 1.253, and 1.446 Å (see Figure 1a).In contrast, the H─H bond lengths in ThH 18 are relatively shortened with a range of 0.995-1.188Å (see Figure 1b), which in turn increases the maximum H-derived phonon frequency compared to that of ThH 9 , as will be shown later.
Figure 2a,b shows the calculated band structures of ThH 9 and ThH 18 , respectively.It is seen that the band projections onto Th and H atoms in ThH 9 (ThH 18 ) represent their strong hybridization over the entire energy range between À21.4 (À28.2) eV and E F .To explore the Th─H hybridization in detail, we present the partial DOS (PDOS) of ThH 9 and ThH 18 in Figure 2c,d, respectively.In contrast to ThH 9 having the rather localized Th 6p semicore states around À20 eV below E F , ThH 18 has the substantially delocalized Th 6p semicore states that strongly hybridize with the H 1s state in a wide energy range from %À28 to %À17 eV.Such a large delocalization of Th 6p semicore electrons in ThH 18 is likely due to the increased interactions between Th atom and its surrounding H atoms at a high pressure of 400 GPa. [48,49]Note that the distances between Th and H atoms in ThH 18 are shortened with a range of 1.88-2.04Å, compared to those (2.08-2.23 Å) in ThH 9 .Therefore, the strong hybridization of the Th semicore and H 1s states in ThH 18 could be an essential ingredient for the stabilization of the large H 36 clathrate cage. [50]The right panels in Figure 2c,d display a closeup of the Th-and H-derived DOS around E F .For ThH 18 , we find a van Hove singularity at E F , giving rise to an H-derived DOS of 0.715 states/ eV at E F .This magnitude of H-derived DOS is %1.87 times larger than 0.382 states/eV for ThH 9 . [51]Such a large difference of the H-derived DOS at E F between ThH 9 and ThH 18 mainly contributes to a significant difference in their T c values, as discussed below.
It is well established that H-rich clathrate metal hydrides have an ionic character between metal and H atoms due to their charge transfer. [52,53]To estimate the charge transfer from Th to H atoms, we analyze their Bader charges [54] in ThH 9 and ThH 18 .Figure 3a,b shows the calculated total charge densities of ThH 9 and ThH 18 with the Th/H Bader basins, [54] respectively.Here, the Bader basins are obtained from a computation of the gradient of the total charge density. [55,56]The estimated cationic/ anionic charges within the Th/H Bader basins are summarized in Table 1.We find that the Th cationic charge in ThH 9 is 1.68e, larger than that (1.44e) in ThH 18 .Accordingly, the sum of the anionic charges of different H atoms in ThH 9 is larger in magnitude than that in ThH 18 .Therefore, ThH 9 has a more ionic character between Th and H atoms, compared to ThH 18 .Meanwhile, ThH 18 having a less ionic character between Th and H atoms would stabilize the extremely H-rich H 36 cage via the strong hybridization between the Th 6p semicore and H 1s states at a high pressure of 400 GPa.
Next, we examine the phonon spectra of ThH 9 and ThH 18 using density functional perturbation theory calculations. [40]igure 4a,b shows the calculated phonon dispersions of ThH 9 and ThH 18 , respectively. [57]The projected DOS onto Th and H atoms shows that, for both ThH 9 and ThH 18 , the acoustic phonon modes of Th atoms are well separated from the optical phonon modes of H atoms.We find that the H-derived phonon modes in ThH 9 are distributed between 496 and 1655 cm À1 (see Figure 4a), while those in ThH 18 reside in a wide frequency range from 397 to 2587 cm À1 (see Figure 4b).Therefore, the H-derived phonon modes in ThH 18 exhibit not only hardening of the high-frequency regime but also softening of the low-frequency regime, compared to those in ThH 9 .The former (latter) hardening (softening) is likely caused by the shorter H─H (Th─H) bond lengths in ThH 18 (see Figure 1b).As a result, we find that the logarithmic average ω log of H-derived phonon frequencies in ThH 18 is 1034 cm À1 , higher than that (885 cm À1 ) in ThH 9 (see Table 2).
Using the isotropic Migdal-Eliashberg formalism, [58][59][60] we calculate the Eliashberg spectral function α 2 FðωÞ and integrated EPC constant λðωÞ as a function of phonon frequency.Figure 4a, b shows the comparison of α 2 FðωÞ and λðωÞ between ThH 9 and ThH 18 .We find that for the two hydrides, all the phonon modes including the Th-derived acoustic and H-derived optical modes, contribute to increasing λðωÞ.For ThH 9 (ThH 18 ), the Th-derived acoustic modes are estimated to contribute to %15 (10)% of the total EPC constant λ = λ(∞), while the H-derived optical modes contribute to %85 (90)% of λ.Specifically, λðωÞ arising from H-derived phonon modes monotonically increases with    increasing frequency.Therefore, ThH 18 having higher H content reaches a much larger value of λ = 2.08, [61,62] compared to that (1.55) of ThH 9 (see Figure 4a,b).By numerically solving the isotropic Eliashberg equations [59] with the typical Coulomb pseudopotential parameter of μ Ã = 0.1, [36] we obtain the temperature dependence of superconducting gap Δ (see Figure 5).For ThH 9 , we find that Δ closes at a T c of %151 K at 130 GPa, in good agreement with the measured value of 146 K at 170-175 GPa. [33]eanwhile, for ThH 18 , the estimated T c is as high as %269 K, being comparable with that (296 K) of a previous theoretical calculation. [36]Here, we note that the anisotropy in Δ enhances T c due to the existence of multiple Fermi surfaces [63] in ThH 9 and ThH 18 . [64]Furthermore, we calculate the Δ versus T relation with respect to μ Ã .Figure S9, Supporting Information, shows that the estimated values of T c in ThH 9 (ThH 18 ) are 151 (269), 139 (250), and 128 (235) K with μ Ã = 0.1, 0.13, and 0.16, respectively, indicating that T c decreases monotonically with increasing μ Ã .Therefore, T c of ThH 9 estimated with μ Ã = 0.1 agrees well with the experimental data. [33]o understand why there is a large difference in λ between ThH 9 and ThH 18 , we compare the contributions of the underlying components that determine λ.Since the frequency ranges of the Th-derived acoustic and H-derived optical modes are well separated (Figure 4a,b), λ can be expressed as a sum of its atom-specific components λ j ( j = Th, H) according to the McMillan-Hopfield theory [65][66][67] that is where N j ðE F Þ is the Th-or H-derived DOS at E F , I 2 j is the atom-specific Fermi-surface average-squared electron-phonon matrix element, M is the atomic mass, and ω 2 2,j is the atom specific average-squared phonon frequency.Therefore, λ j is composed of the electronic part N j ðE F ÞI 2 j in the numerator of Equation ( 1) and the phonon part M j ω 2 2,j in the denominator.For ThH 9 and ThH 18 , each component contributing to λ j is listed in Table 2. Using the McMillan-Allen-Dynes formula, [68] we can estimate T c from λ j and ω log associated with each atom. [65]lthough the McMillan-Allen-Dynes formula usually underestimates T c , [69][70][71][72][73] its estimation may provide a qualitative aspect of how largely certain phonon modes contribute to T c .As shown in Table 2, the acoustic phonon modes of Th atoms contribute to an increase in λ, but they hardly increase T c , similar to the cases of other high-T c hydrides. [65,71]Since λ Th is insensitive to an increase in T c , we examine how Finally, we examine the pressure effect on the bonding characteristics of ThH 9 and ThH 18 by comparing their PDOS values at the same pressure of 400 GPa.As shown in Figure S10a, Supporting Information, the Th 6p semicore states in ThH 9 are substantially delocalized to hybridize with the H 1s state in a wide energy range between À26 eV and E F , like those in ThH 18 (see Figure 2d).It is thus likely that the large delocalization of Th 6p semicore electrons in ThH 18 is associated with the increased interactions between Th and H atoms at a high pressure of 400 GPa.Meanwhile, the band dispersion of ThH 9 at 400 GPa (see Figure S10b, Supporting Information) shows a little change around E F compared to that at 130 GPa (see Figure 2a).We find that the magnitude of H-derived DOS N H ðE F Þ in ThH 9 becomes 0.348 states/eV at 400 GPa, which is comparable with that (0.382 states/eV) at 130 GPa.The former value of N H ðE F Þ in ThH 9 is %2.05 times smaller than that (0.715 states/eV) in ThH 18 .Using the Allen-Dynes equation, we predict that ThH 9 has a T c of %126 K at 400 GPa, slightly lower than that (139 K) at 130 GPa.It is thus likely that pressure is insensitive to the T c difference between ThH 9 and ThH 18 , while stoichiometry plays an important role in their large difference in T c .The estimated values for ω log and T c arising from Th and H atoms are also given.Here, T c values are estimated using the McMillan-Allen-Dynes formula. [68]For comparison, the T c value estimated using the isotropic Eliashberg equations is given in the last column.

Conclusion
Using first-principles calculations, we have investigated the structural, electronic, phononic, and superconducting properties of ThH 9 and ThH 18 which showed the very large differences in the stabilization pressure and T c . [33,36]We found that ThH 9 has a greater ionic bonding character due to more charge transfer from Th to H atoms.Meanwhile, ThH 18 has a more substantial hybridization of the Th 6p-semicore and H 1s states, which leads to the stablization of the extremely H-rich H 36 clathrate cage at higher pressures.Furthermore, we revealed that ThH 18 has about two times larger H-derived DOS at E F than ThH 9 , which dominantly contributes to enhancing λ and T c .However, the electron-phonon matrix element was found to make minor contributions to the change of λ between ThH 9 and ThH 18 .We thus demonstrated that the differences in the bonding character between Th atom and its surrounding H cage reflects the different stabilization pressures of ThH 9 and ThH 18 and that the H-derived DOS at E F plays a crucial role in the very large difference in T c betweeen the two Th hydrides.

Figure 1 .
Figure 1.Optimized structures of a) ThH 9 at 130 GPa and b) ThH 18 at 400 GPa.The Th sublattice in ThH 9 (ThH 18 ) forms the hcp (centered rectangular) lattice with the H 29 (H 36 ) cage surrounding each Th atom.The H 29 cage is composed of six tetragon, six pentagon, and six hexagon rings, while the H 36 cage is composed of the strip of six hexagon rings and two wrinkled hexagon rings.There are three (five) different types of H atoms in ThH 9 (ThH 18 ), which are identified by Wyckoff positions.[33,36]The (110) and (100) planes are drawn in the hcp and centered rectangular lattices, respectively.

Figure 2 .
Figure 2. Calculated band structures of a) ThH 9 at 130 GPa and b) ThH 18 at 400 GPa.The projected bands onto Th and H atoms are represented by circles whose radii are proportional to the weights of the corresponding atoms.The energy zero represents E F .The Brillouin zones corresponding to the primitive cells of ThH 9 and ThH 18 are included in (a,b), respectively.The PDOS results (in the unit of states/eV per unit cell) for the Th and H orbitals of ThH 9 and ThH 18 are displayed in (c,d), respectively.The right panels in (c,d) enlarge the PDOS sums for Th and H atoms around E F .

Figure 3 .
Figure 3. Calculated total charge densities of a) ThH 9 and b) ThH 18 , together with the Bader basins of Th and H atoms.The green and red circles represent Th and H atoms, respectively.The charge densities in (a,b) are plotted on the (110) and (100) planes (see Figure1a,b) with the contour spacing of 0.1 e Å À3 , respectively.

Table 1 .
Estimated cationic/anionic charges (in unit of e) within the Bader basins of th/h atoms in ThH 9 and ThH 18 (see Figure3a,b).The numbers in parentheses represent the number of different H atoms in each formula unit.
2nd M H ω 2 2,H for H atom contribute to λ H that mostly determines T c .In Table2, we find that 1) N H ðE F Þ is %1.87 times larger in ThH 18 than in ThH 9 , 2) M H ω22,H is %29% larger in ThH 18 than in ThH 9 , and 3) I 2 H in ThH 18 is similar between the two hydrides.Consequently, we can say that for ThH 18 , λ H is dominantly increased by N H ðE F Þ despite its lowering via M H ω 2