The pursuit of high-T c and even room-temperature superconductivity has been a grand challenge in physics since the 1911 discovery of superconductivity [1]. The Bardeen-Cooper-Schrieffer (BCS) theory [2] can describe well the mechanism for conventional superconductors, opening the door for the discovery and design of high-T c materials. Ashcroft [3] proposed that compressed hydrides are good candidates for high-T c superconductors due to chemical "precompression" effects. Subsequent theoretical studies provided explicit predictions of these phenomena, and they have led to the discovery of very high temperature superconductivity, first in H 3 S (T c of 203 K) [4] and more recently in LaH 10 (T c up to 260 K) [5][6][7]. These experimental results confirmed theoretical predictions for both the structures and basic mechanism of superconductivity [8][9][10][11], thereby marking a new era for both superconductivity and "materials by design" [5,[12][13][14].
Compared to binary hydrides, exploration of high-T c superconductors in ternary hydrides greatly enlarges the configurational space and opens up new chemistry and physics that could in principle enhance both T c and stability, including at modest pressures. This additional elemental degree of freedom in compressed ternary hydride systems has led to new predicted high-T c superconductors such as CaYH 12 (T c = 258 K at 200 GPa) [15], LiPH 6 (T c = 150-167 K at 200 GPa) [16], and most remarkably Li 2 MgH 16 (T c = 473 K at 250 GPa) [17]. On the other hand, existing experimental studies have uncovered only relatively low-T c materials such as in BaReH 9 (T c = 7 K) [18] and Li 5 MoH 11 (T c = 6.5 K) [19]. Returning to sulfur-containing systems, carbon disulfide (CS 2 ) has been shown to transform to a metal at 50 GPa, and a superconductor with a T c of ∼6 K at 60-170 GPa [20]. Several ternary hydrides based on the H 3 S structure are predicted to exhibit high-T c behavior, including H 6 SSe (i.e., SH 3 -SeH 3 , with an estimated T c of 195 K at 200 GPa) [21], Y(La)SH 6 [SH 3 -Y(La)H 3 , T c = 95 and 35 K at 210 and 300 GPa, respectively] [22], and even the noble gas Xe with H 3 S (H 3 SXe, T c = 89 K at 240 GPa) [23]. Moreover, several superconductors have also been reported in ternary F-S-H, Y-S-H, and B-S-H compounds [24][25][26].
There is also general interest in carbon-containing high-T c superconductors. Dense carbides that are isostructural with hydrides have been predicted to exhibit high-T c behavior and to be stable on decompression due to their rigid carbon sp 3 frameworks [27]. Various hydrocarbons (e.g., CH 4 , C 2 H 4 , and C 2 H 6 ) are stable in different molecular phases to megabar pressures [28][29][30]. CH 4 and Mg are predicted to form CH 4 Mg, a potential superconductor with a T c of 84-121 K at 75-120 GPa [31]. The related compound CH 4 K has a predicted T c of about 12 K at 80 GPa [32]. These results encouraged us to theoretically search for high-T c superconductors in the C-S-H ternary system. As shown in detail below, Cm, R3m, and Pnma symmetry phases were found, for which electronphonon coupling calculations reveal superconductivity with T c 's as high as 194 K at 150 GPa, as estimated by solving the Eliashberg equations numerically using the typical choice of the Coulomb potential, μ * = 0.10-0.13.
We began by identifying stable structures in the C-S-H system, focusing on C x S y H z (x = 1 -2; y = 1 -2; z = 1 -8) compositions at megabar pressures (100-200 GPa), The brown, blue, and yellow-green spheres denote C, S, and H atoms, respectively (see the supplemental material [37] for more detailed structural information).
using heuristic algorithms based on particle swarm optimization [33][34][35] and evolutionary algorithms [36] in combination with first-principles theory [computational details are provided in the supporting information (SI) [37]]. The CSH 7 stoichiometry emerged as being particularly stable relative to the elements in the pressure range considered. The enthalpies of formation from the elemental phases for CSH Although the enthalpies of formation of the CSH 7 phases are negative with respect to the elemental phases, these species do not lie on the 3D convex hull. In fact, they are metastable with respect to CH 4 + H 3 S (and CH 4 + H 2 S + 1/2H 2 ), but stable relative to H 2 S + C + 5/2H 2 below 188 GPa. The zero-point energy (ZPE) uncorrected enthalpy of R3m is 33 meV/at above the convex hull at 130 GPa. As shown in Table S1 of the SI, inclusion of the ZPE or finite-temperature effects does not stabilize the ternary phase. It should, however, be noted that a number of compressed hydrides found to be metastable via first-principles calculations have been synthesized in the laboratory. Examples include a Ca 2 H 5 phase, which was calculated to be 20 meV/at above the convex hull [38], and hydrides of phosphorus, which are at least 30 meV/at above the convex hull [39][40][41]. Moreover, these phases could potentially be stabilized by anharmonic or nuclear quantum effects as found for LaH 10 [42]. Therefore, it is expected that the CSH 7 phases could be realized experimentally, especially if they are calculated to be dynamically stable.
The relative enthalpies of the CSH 7 phases between 100 and 300 GPa are shown in Fig. S2. The small differences in the static lattice enthalpies calculated for these hydride perovskites suggests there may be many other phases, differing in the number of atoms in the formula unit and rotations of the methane molecules, that could be realized. Phonon calculations reveal that Cm is dynamically stable from 100 to 110 GPa, whereas R3m is dynamically stable from 130 to 300 GPa, and Pnma is dynamically stable at the current level of approximation from at least 150 to 300 GPa (Fig. S3). Molecular-dynamics simulations (Fig. S4), suggest that at finite temperature the methane molecules may become rotationally disordered.
Predicted 0 K structures of CSH 7 and cubic Im 3m H 3 S are shown in Fig. 1 and Fig. S1. In pure H 3 S, the S atoms form a bcc lattice with each S atom octahedrally coordinated by H atoms [Fig. 1(a)]. The CSH 7 phases are based on the H 3 S structure with the central SH 6 unit replaced by a CH 4 molecule, and a distortion of the cubic framework. In the Cm phase [Fig. S1(a)], the cubic H 3 S lattice is replaced by two layers of SH 5 . The H-S distance between neighboring SH 5 layers is too long (1.906 Å) to form an H-S covalent bond. In the R3m structure, for example, the H 3 S framework is retained but with a slight rhombohedral distortion such that the a, b, and c unit-cell vectors are equal in length, but the α, β, and γ angles measure 89.4 • (instead of 90 • ). The rotation barrier of the CH 4 unit in R3m-CSH 7 was estimated to be <1 meV/at at 150 GPa (Fig. S5), suggesting the molecule may exhibit rotational disorder, and that this may occur in other structures, consistent with the molecular dynamics simulations (Fig. S4).
To explore the bonding in the CSH 7 phases, their electronic localization functions (ELFs) were plotted (Fig. S6). The ELF is useful for visualizing covalent bonds and lone pairs; it maps values in the range from 0 to 1, where 1 corresponds to perfect localization of the valence electrons indicative of a strong covalent bond. The ELF values for the S-H and C-H bonds are close to 0.9, indicating their strong covalent character. The Cm structure is characterized by five S-H bonds (bond lengths of 1.38-1.55 Å) and one lone electron pair [Figs. S6(a) and S6(b)]. Similar to free methane, the CH 4 units contain four strong covalent C-H bonds, as shown in the three-dimensional ELF plots. In the R3m and Pnma structure, the cubic H 3 S framework (i.e., Pm 3m) is clearly evident.
The calculated pressure-volume equation of states (EOS) of CSH 7 and assemblages consisting of CH 4 and H 3 S, as well as C + S + 7/2H 2 , are shown in Fig. 2. The sum of the volume of the predicted CH 4 (Cmca) [43] and H 3 S (R3m and Im 3m) [9] phases is also plotted for comparison. The evolution of the volume under pressure of CSH 7 falls between the experimental and theoretical data obtained for CH 4 + H 3 S. As shown in Fig. S7, the PV contribution to the enthalpy favors the assemblages over CSH 7 within this pressure range. The pressure dependence of select S-H distances is plotted in Fig. S8. Under pressure the S1-H and S2-H bonds in the Cm phase approach each other, and the S3-H distance decreases, as the phase undergoes pressure-induced bond equalization. Because the CH 4 molecule breaks the symmetry, the S-H1 and S-H2 bonds in the R3m phase differ slightly. Both S-H bonds in R3m-CSH 7 are slightly shorter than those calculated for Im 3m H 3 S, whose highest T c , 203 K, was measured at 155 GPa [4]. If the S-H distance, which can be modulated by the identity of the intercalant molecule as well as the pressure, is an important factor in determining the T c , it is expected that the maximum value for CSH 7 would be at lower pressures, consistent with the explicit calculations of T c discussed below.
The band structures and partial densities of states (DOSs) for CSH 7 perovskite structures were calculated to further The inset shows the experimental EOS for H 2 [44], C [45], S [46], CH 4 [47], and H 3 S [48]; the points are the measured data.
explore their electronic properties [Fig. 3 and 63 THz) are derived from combinations of H-wagging, bending, and stretching modes; and the higher frequencies (∼100 THz, 3300 cm -1 ) are derived primarily from the C-H stretching modes of the molecular CH 4 units, close to the asymmetric stretching modes of free CH 4 (∼97 THz, 3200 cm -1 ).
The superconducting properties of the CSH 7 phases were estimated by the Allen-Dynes modified McMillan equation [49], using typical values of the Coulomb pseudopotential, μ * = 0.13-0.1. As shown in Table I, below 250 GPa, λ was calculated to be appreciable for all three phases. At 100 GPa, λ was 1.2 and ω log was 1091 K for the Cm phase, yielding a T c of 86-98 K. For R3m at 150 GPa, λ was 2.47, which is twice the value found for Cm and comparable to that calculated for cubic H 3 S at 200 GPa (λ = 2.19) [9], resulting in a T c of 143-152 K. The logarithmic average phonon frequency of R3m CSH 7 at 150 GPa is somewhat lower than that of Im 3m H 3 S at 200 GPa [9], 925 K versus 1335 K, which is the reason for the slightly lower T c of the ternary. The increase in λ for R3m as compared to Cm is not unreasonable in view of the differences in their phonon band structures. In R3m, TABLE I. The calculated electron-phonon coupling parameter (λ), logarithmic average phonon frequency (ω log ), and the estimated T c for selected CSH 7 structures using the Allen-Dynes modified McMillan (ADM) equation [49], and numerically solving the Eliashberg equations [50] with μ * = 0.10 (0.13). there are two obvious Kohn anomalies and softened modes at 18-32 THz located along the F -and -Z high-symmetry lines, which contribute a significant amount (∼36%) to λ. In addition, λ of R3m and Pnma was found to decrease with increasing pressure, which was not offset by the increase in ω log so that T c decreased from 150 to 300 GPa. The maximum T c for CSH 7 , therefore, occurs at a slightly lower pressure than for H 3 S. Noteably, it is larger than the highest value computed for H 3 SXe, 89 K at 240 GPa [23], because of the smaller mass of CH 4 relative to Xe. For strongly coupled superconductors (λ > 1.5), Eliashberg theory [50] gives a better estimate of T c , and therefore we also numerically solved the Eliashberg equations for select pressures, as shown in Table I, Table SII and Fig. 4. As expected, the estimated T c values were higher, 181-194 and 157-170 K for R3m and Pnma, respectively.
In summary, we have explored the stability and electronic properties of C x S y H z ternary compounds at high pressure.
Stoichiometric CSH 7 was found to be dynamically stable above 100 GPa, adopting phases with space-group symmetries based on distortions of the high-symmetry Pm 3m structure. These perovskite-type structures are characterized by CH 4 intercalated to form distorted H 3 S perovskite frameworks, which could represent a new class of high-temperature superconductors. Electron-phonon coupling calculations indicate that CSH 7 phases are promising conventional superconductors with T c 's estimated to be as high as ∼200 K. The results are expected to stimulate the search for other high-T c H 3 S intercalation compounds and more chemically complex hydride superconductors, especially in carbon-bearing systems. Just like pressure, the size of the intercalant molecule can be used to modulate the H-S framework, and thereby to tune its properties, such as T c . Additional theoretical studies would be useful to examine the effects of quantum and anharmonic dynamics on the calculated critical temperatures and stability of this class of materials, as well as mechanisms for enhancing their phase stabilities at lower pressures.