Superconductive sodalite-like clathrate calcium hydride at high pressures

The known calcium hydrides have a stoichiometry of CaH₂ at ambient pressure. Here, we explored other calcium hydrides with larger hydrogen contents by compressing a mixture containing Ca + hydrogen or CaH₂ + hydrogen. The search for stable CaH_2n (n = 2–6) structures at high pressures was performed using the particle-swam optimization structure prediction algorithm (19) in combination with ab initio calculations. For each stoichiometry, calculations were performed at pressures of 50–200 GPa with up to 4 formula units (f.u.). in the model. The enthalpies of candidate structures relative to the products of dissociation into CaH₂ + solid H₂ at the appropriate pressures are summarized in Fig. 1A. The essential information can be summarized as follows: (i) except for CaH₁₀, stable structures began to emerge at pressures < 50 GPa; (ii) CaH₄ was the most stable phase at pressures between 50 and 150 GPa, whereas at 200 GPa CaH₆ had the lowest enthalpy of formation; (iii) the breakup of hydrogen molecules depended on the Ca/H ratio, and a higher susceptibility to dissociation was observed at higher ratios. Notably, calcium hydrides with stoichiometry having an odd number of hydrogen (e.g., CaH₃, CaH₅, and CaH₇, etc.) were found to be energetically very unfavorable and were excluded in the discussions (Supporting Information).

The structures of the stable phases for each stoichiometry at 150 GPa are shown in Fig. 1 B)–(D. Three types of hydrogen species, “H₄” units, monatomic H + H₂, and molecular H₂, were observed. CaH₄ had a tetragonal (I4/mmm, Pearson symbol tI10) structure and included a body-centered arrangement of Ca and two molecular and four monatomic hydrogen units (Ca₂(H₂)₂(H)₄). The structure of CaH₆ adopted a remarkable cubic form (Im, Pearson symbol cI14), with body-centered Ca atoms and, on each face, squared H₄ units tilted 45° with respect to the plane of the Ca atoms. These H₄ units were interlinked to form a sodalite framework with a Ca atom enclathrated at the center of each cage. The next stable polymorph, CaH₁₂, had a rhombohedral (R, Pearson symbol hR13) structure consisting entirely of molecular H₂.

The presence of different types of hydrogen can be rationalized based on the effective number of electrons contributed by the Ca atom and accepted by each H₂ molecule. Assuming that the two valence electrons of each Ca atom were completely “ionized” and accepted by H₂ molecules, the “formal” effectively added electron (EAE) per H₂ for CaH₄ was 1e/H₂, for CaH₆ was (2/3)e/H₂, and was (1/3)e/H₂ for CaH₁₂. Because H₂ already had a filled σ bond, the added electrons resided in the antibonding σ^∗ orbitals, which weakened the H-H bond (i.e., lengthened the H-H bond length) and eventually resulted in complete dissociation. The presence of H and H₂ units in the CaH_2n structures depended on the number of EAEs. Two formulae were present in each unit cell of CaH₄. Because half (two) of each H₂ molecule was retained, the remaining two H₂ molecules had to accommodate four “excess” electrons into their σ^∗ orbital, which broke up the molecules into monatomic hydrides (H^-). The formation of H₄ units in CaH₆ was not accidental. If molecular hydrogen atoms were present, each σ^∗ orbital of H₂ had to accommodate (2/3)e. This led to a physically unfavorable structure with significant weakening of the intramolecular H-H bonds. There is, however, an alternative lower energy structure, the one predicted here. From molecular orbital theory, a square H₄ unit should possess a half-filled degenerated “nonbonding” orbital (Fig. 2A). This orbital, in principle, can accommodate up to two electrons without detrimentally weakening the H-H bond (Fig. 2B). It should be noted that at 150 GPa, the H…H distance of H₄ in the cI14 structure of CaH₆ was 1.24 Å, which was substantially shorter than both the monoatomic H…H distance of 1.95 Å and the H…H₂ distance of 1.61 Å in CaH₄ but in good agreement with those of 1.27 and 1.17 Å in isolated and H₄ squares, respectively. This suggested the presence of a weak covalent H…H interaction in CaH₆. As will be described below, the H₄ unit is the fundamental building block of the sodalite cage.

In CaH₁₂, the EAE was (1/3)e/H₂, which could be taken up by H₂ without severing the bond. Extending this concept further then predicted that CaH₁₂ and hydride alloys with a high H content (smaller EAE) would be composed predominantly of molecular H₂. An important observation in support of the EAE description given above is that the H-H bond in CaH₄ lengthened from 0.81 Å at 100 GPa to 0.82 Å at 150 GPa, whereas the H-H bond in CaH₁₂ shortened from 0.81 Å to 0.80 Å. Here, more Ca valence electrons in CaH₄ were available for transfer to the H₂ σ^∗ orbitals than were available in CaH₁₂, thereby more severely weakening the bond in CaH₄. This effect was relatively small for CaH₁₂, in which the H-H bond was shortened due to compression.

The zero-point motion was not included in the calculation of the formation enthalpy of the various hydrides (Fig. 1A), although it is expected to be very influential due to the presence of large amounts of hydrogen. We estimated the zero-point energies of CaH₆ and CaH₄ using the quasiharmonic model (20) at 150 GPa. It was found that the inclusion of zero-point motion significantly lowered the formation enthalpy of CaH₆ with respect to CaH₄ (Fig. S11). As a consequence, CaH₆ became more stable at and above 150 GPa. The physical mechanism underlying this effect stemmed from the H₄ moieties, which included much longer H-H distances and led to significantly softened phonons. This contrasted with other Ca hydrides studied (e.g., CaH₄ and CaH₁₂), in which the presence of H₂ molecular units gave rise to higher frequency phonons and, thus, a larger zero-point energy.

The three-dimentional sodalite cage in CaH₆ is the result of interlink of other H₄ units via each H atom at the corner of one H₄ unit. So, what is the electronic factor promoting the formation of these H₄ units? To answer this question, the electron localization functions (ELF) of a hypothetical bare bcc Ca lattice with the H atoms removed and CaH₆ hydride (Fig. 3A and 3B) were examined. In bare body-centered cubic (bcc) Ca, regions with ELF values of 0.58 were found to localize at the H atom sites in the H₄ units on the faces of the cube. The ELF of CaH₆ hydride suggested that no bonds were present between the Ca and H. A weak “pairing” covalent interaction with an ELF of 0.61, however, was found between the H atoms that formed a square H₄ lattice. Their formation resulted from the accommodation by H₂ of excess electrons from the Ca. An electron topological analysis also showed the presence of a bond-critical point (21) along the path connecting neighboring H atoms. The integrated charge within the H atomic basin was 1.17 e, which corresponded to a charge transfer of 1.02 e from each Ca. A partially ionized Ca was also clearly supported by the band structure and the density of states as reported in Fig. 3C and Fig. S15. At 150 GPa, Ca underwent an s–d hybridization with an electron transferred from the 4 s to the 3 d orbital. In CaH₆, the Ca site symmetry was m (O_h) and the Ca 3 d manifold was clearly split into the e_g and t_2g bands, with the lower energy e_g band partially occupied.

A comparison of the band structures of CaH₆, “H₆” (Ca₀H₆), and bare Ca (CaH₀) provided additional supporting evidence. Even without the presence of Ca, the valence band width of the hypothetical H₆ (Fig. 3D) was 15.2 eV, comparable to 16.4 eV for CaH₆ (Fig. 3C). In comparison, the valence band width of the bare bcc Ca was only 4.3 eV (Fig. 3E). The band structure of CaH₆ near the Fermi level was modified from H₆ due to the the hybridization between Ca 3 d and H 1 s orbital; however, the trend in the electronic band dispersions from -2 to -16.4 eV was remarkably similar to that of H₆ from 3 to -15.2 eV.

Sodalite cage was constructed from linking of H₄ units and this topologically resulted in the formation of H₆ faces. In fact, a primitive cell of CaH₆ can be seen as composed of a H₆ hexagon and a Ca. Band structure analysis suggested that four conduction bands of “Ca₀H₆” sodalite cage (Fig. 3D) were half-filled and the addition of maximal two electrons to H₆ gave rise to partial occupancy of the degenerate bands at Γ as indicated in CaH₆ (Fig. 3C). Partial occupancy of a degenerate orbital results in an orbitally degenerate state and is subject to Jahn–Teller (JT) distortion (22) (Fig. 2). The JT effect involves coupling between the electron and nuclear degrees of freedom, leading to distortions in the structure, and it lifts the orbital degeneracy. If the distortion is dynamic, JT vibrations can contribute to superconductivity. The same mechanism has been invoked to explain the superconductivity in B-doped diamond (23). To investigate this possibility, electron–phonon coupling (EPC) calculations on the sodalite structure of CaH₆ at 150 GPa were performed. The phonon dispersion curves, phonon linewidth γ(ω), EPC parameter λ, and Eliashberg spectral function α²F(ω) were calculated. A gap at 430 cm^-1 (Fig. 4) separated the phonon spectrum into two regions: The lower frequency branches were associated with the motions of both Ca and H, whereas the higher frequency branches were mainly associated with H atoms. The combined contribution (19% and 81%, respectively) gave an EPC parameter λ of 2.69. The calculated phonon linewidths (Fig. 4) showed that the EPC was derived primarily from the T_2g and E_g modes at the zone center Γ. Incidentally, these two bands, respectively, were the in-plane breathing and rocking vibrations of the H atoms belonging to the H₄ unit. The corresponding atomic vibrations led to distortions in the square planar structure. Moreover, both phonon branches showed significant phonon softening along all symmetric directions. A large EPC also benefited from the high density of states at the Fermi level caused by a Van Hove singularity at Γ (Fig. 3C). The very large EPC was unprecedented for the main group hydrides. Previous calculations on a variety of systems predicted an EPC in the range of 0.5–1.6 (11). The mechanism suggested here is not inconsistent with the mechanisms found in JT-induced superconductivity in alkali intercalated C₆₀, in which the intramolecular vibrations are responsible for distortions that lower the symmetry of the molecules, which is favorable for electron–phonon processes (24, 25).

T_c was calculated based on the spectral function α²F(ω) by numerically solving the Eliashberg equations (26), which consist of coupled nonlinear equations describing the frequency-dependent order parameter and renormalization factor. The Coulomb repulsion is taken into account in terms of the Coulomb pseudopotential, μ∗, scaled to a cutoff frequency (typically six times the maximum phonon frequency) (27). At 150 GPa, the predicted T_c values were 235 K and 220 K using typical values for μ∗ of 0.1 and 0.13, respectively. EPC calculations were also performed for 200 GPa and 250 GPa, in which the calculated T_c was found to decrease with pressure (201 K at 200 GPa and 187 K at 250 GPa for μ ∗= 0.13), with a pressure coefficient (dT_c/dP) of -0.33 K/GPa. T_c of the order of 200 K is among the highest for all reported hydrides.

The predicted high T_c for CaH₆ is very encouraging, but it must be viewed with caution. Formally, there is no upper limit to the value of T_c within the Midgal–Eliashberg theory of superconductivity. Two practical factors must be considered. The calculation of the EPC is based on the harmonic approximation and without consideration of electron correlation effects. Because strong electron–phonon coupling in CaH₆ arises from the proximity of the electronic and structural instabilities, anharmonicity of the atomic motions can lead to renormalization of the vibrational modes, as demonstrated in a study of AlH₃. In that study, lower renormalized frequencies were found to reduce the EPC and suppress superconductivity (28). On the other hand, the electron–phonon matrix elements may be enhanced by anharmonic vibrations, as in the case of disordered materials (29). In a recent hybrid functional study of C₆₀ anions, the inclusion of Hartree–Fock exchange contributions was shown to have little effect on the structural properties and phonon frequencies but resulted in a strong increase in the electron–phonon coupling (24).

The formation of a hydrogen sodalite cage with enclathrated calcium in CaH₆, reported here for hydrogen-rich compounds, provides an unexpected example of a good superconductor created by the compression of a mixture of elemental calcium + hydrogen or CaH₂ + hydrogen. This superconductor can also be viewed as consisting of unique square H₄ units and electron-donating calcium atoms subject to JT effects. Dense superconductive states, such as those reported here, may be favored in other mixtures of elemental metals + hydrogen or any hydride + hydrogen upon compression. This work highlights the major role played by pressure in effectively overcoming the kinetic barrier to formation in the synthesis of hydrides.

Our structure prediction approach is based on a global minimization of free energy surfaces merging ab initio total-energy calculations via particle swarm optimization technique as implemented in CALYPSO (crystal structure analysis by particle swarm optimization) code (19). Our CALYPSO method unbiased by any known structural information has been benchmarked on various known systems 19 with various chemical bondings and had several successful prediction of high-pressure structures of Li, Mg, and Bi₂Te₃ (30–32), among which the insulating orthorhombic (Aba2, Pearson symbol oC40) structure of Li and the two low-pressure monoclinic structures of Bi₂Te₃ have been confirmed by independent experiments (32, 33). The underlying ab initio structural relaxations were carried out using density functional theory within the Perdew–Burke–Ernzerhof exchange-correlation (34) as implemented in the Vienna Ab Initio Simulation Package (VASP) code (35). The all-electron projector-augmented wave method (36) was adopted with 1s and 3p⁶4s² treated as valence electrons for H and Ca, respectively. Electronic properties, lattice dynamics and electron-phonon coupling were studied by density functional (linear-response) theory as implemented in the QUANTUM ESPRESSO package (37). More computational details can be found in the Supporting Information.

H. W. and Y. M. acknowledge Prof. Aitor Bergara for the valuable discussions and are thankful to the financial support by Natural Science Foundation of China (NSFC) under 11104104, the China 973 Program (2011CB808200), NSFC under 11025418 and 91022029, Changjiang Scholar and Innovative Research Team in University (IRT1132), and the research fund of Key Laboratory of Surface Physics and Chemistry (SPC201103). T. I. was supported by Ministry of Education, Culture, Sports, Science and Technology of Japan (20103001–20103005). The calculations were performed in the computing facilities at Rikagaku Kenkyūjo Integrated Cluster of Clusters system (Japan) and the High Performance Computing Center of Jilin University.