Superconductive sodalite-like clathrate calciumhydride at high pressuresHui Wanga, John S. Tseb, Kaori Tanakab, Toshiaki Iitakac, and Yanming Maa,1

aState Key Lab of Superhard Materials, Jilin University, Changchun 130012, Peoples Republic of China; bDepartment of Physics and Engineering Physics,University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E2, Canada; and cComputational Astrophysics Laboratory, Rikagaku Kenkyūjo ,2-1 Hirosawa, Wako, Saitama 351-0198, Japan

Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved February 17, 2012 (received for review November 5, 2011)

Hydrogen-rich compounds hold promise as high-temperaturesuperconductors under high pressures. Recent theoretical hydridestructures on achieving high-pressure superconductivity are com-posed mainly of H2 fragments. Through a systematic investigationof Ca hydrides with different hydrogen contents using particle-swam optimization structural search, we show that in the stoichio-metry CaH6 a body-centered cubic structure with hydrogen thatforms unusual “sodalite” cages containing enclathrated Ca stabi-lizes above pressure 150 GPa. The stability of this structure isderived from the acceptance by two H2 of electrons donated byCa forming an “H4” unit as the building block in the constructionof the three-dimensional sodalite cage. This unique structure has apartial occupation of the degenerated orbitals at the zone center.The resultant dynamic Jahn–Teller effect helps to enhance electron–phonon coupling and leads to superconductivity of CaH6. A super-conducting critical temperature (Tc) of 220–235 K at 150 GPaobtained from the solution of the Eliashberg equations is the high-est among all hydrides studied thus far.

calcium hydride ∣ sodalite structure

Current studies of the hydrides of simple elements at high pres-sures are motivated by the proposition that a high Tc may be

achieved from “precompressed” H2 in dense hydrogen-dominanthydrides (1). Superconductivity has been reported in SiH4 (2),although controversy surrounds models of the origin of supercon-ductivity and the structure of the superconducting state (3–5).Many theoretical structures have the predicted Tc values rangingfrom 10 to 139 K (6–12). Except for AlH3 (13), in which super-conductivity was predicted but not observed, a common featureof these structures is the ubiquitous presence of molecular hydro-gen fragments. At high pressures, unfavorable electron–electronrepulsion in the atomic valence shells can be reduced by migra-tion and localization in the empty interstitial regions (14, 15).Introduction of an electronegative element, as was demonstratedin the case of K–Ag (16), increases the ability of the electrone-gative Ag atoms to accommodate electrons from K into the out-ermost 5s and 5p valence shells, thereby forming a bondingnetwork and stabilizing the alloy structures that otherwise wouldnot be observed under ambient pressures. Recently, Zurek et al.(17, 18) demonstrated that charge transfer from lithium or so-dium to hydrogen molecules under high pressures could lead tothe formation of new metallic lithium- or sodium-hydride alloys.Depending on the stoichiometry, the structures consist of “pre-dissociated” molecular H2 and/or monoatomic hydrogen.

Results and DiscussionThe known calcium hydrides have a stoichiometry of CaH2 atambient pressure. Here, we explored other calcium hydrides withlarger hydrogen contents by compressing a mixture containingCaþ hydrogen or CaH2 þ hydrogen. The search for stable CaH2n(n ¼ 2–6) structures at high pressures was performed using theparticle-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 can-didate structures relative to the products of dissociation intoCaH2 þ solid H2 at the appropriate pressures are summarizedin Fig. 1A. The essential information can be summarized asfollows: (i) except for CaH10, stable structures began to emergeat pressures <50 GPa; (ii) CaH4 was the most stable phase atpressures between 50 and 150 GPa, whereas at 200 GPa CaH6

had the lowest enthalpy of formation; (iii) the breakup of hydro-gen molecules depended on the Ca/H ratio, and a higher suscept-ibility to dissociation was observed at higher ratios. Notably,calcium hydrides with stoichiometry having an odd number ofhydrogen (e.g., CaH3, CaH5, and CaH7, etc.) were found to beenergetically very unfavorable and were excluded in the discus-sions (Supporting Information).

The structures of the stable phases for each stoichiometry at150 GPa are shown in Fig. 1 B)–(D. Three types of hydrogenspecies, “H4” units, monatomic HþH2, and molecular H2, wereobserved. CaH4 had a tetragonal (I4/mmm, Pearson symboltI10) structure and included a body-centered arrangement ofCa and two molecular and four monatomic hydrogen units(Ca2ðH2Þ2ðHÞ4). The structure of CaH6 adopted a remarkablecubic form (Im3̄m, Pearson symbol cI14), with body-centeredCa atoms and, on each face, squared H4 units tilted 45° withrespect to the plane of the Ca atoms. These H4 units were inter-linked to form a sodalite framework with a Ca atom enclathratedat the center of each cage. The next stable polymorph, CaH12,had a rhombohedral (R3̄, Pearson symbol hR13) structure con-sisting entirely of molecular H2.

The presence of different types of hydrogen can be rationa-lized based on the effective number of electrons contributedby the Ca atom and accepted by each H2 molecule. Assumingthat the two valence electrons of each Ca atom were completely“ionized” and accepted by H2 molecules, the “formal” effectivelyadded electron (EAE) per H2 for CaH4 was 1e∕H2, for CaH6 wasð2∕3Þe∕H2, and was ð1∕3Þe∕H2 for CaH12. Because H2 alreadyhad a filled σ bond, the added electrons resided in the antibond-ing σ� orbitals, which weakened the H-H bond (i.e., lengthenedthe H-H bond length) and eventually resulted in complete disso-ciation. The presence of H and H2 units in the CaH2n structuresdepended on the number of EAEs. Two formulae were present ineach unit cell of CaH4. Because half (two) of each H2 moleculewas retained, the remaining two H2 molecules had to accommo-date four “excess” electrons into their σ� orbital, which broke upthe molecules into monatomic hydrides (H−). The formation ofH4 units in CaH6 was not accidental. If molecular hydrogenatoms were present, each σ� orbital of H2 had to accommodate

Author contributions: Y.M. designed research; H.W., J.S.T., K.T., T.I., and Y.M. performedresearch; H.W., J.S.T., T.I., and Y.M. analyzed data; and H.W., J.S.T., and Y.M. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: mym@jlu.edu.cn.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1118168109/-/DCSupplemental.

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ð2∕3Þe. This led to a physically unfavorable structure with signif-icant weakening of the intramolecular H-H bonds. There is, how-ever, an alternative lower energy structure, the one predictedhere. Frommolecular orbital theory, a square H4 unit should pos-sess a half-filled degenerated “nonbonding” orbital (Fig. 2A).This orbital, in principle, can accommodate up to two electronswithout detrimentally weakening the H-H bond (Fig. 2B). Itshould be noted that at 150 GPa, the H…H distance of H4 inthe cI14 structure of CaH6 was 1.24 Å, which was substantiallyshorter than both the monoatomic H…H distance of 1.95 Åand 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, respec-tively. This suggested the presence of a weak covalent H…H in-teraction in CaH6. As will be described below, the H4 unit is thefundamental building block of the sodalite cage.

In CaH12, the EAE was ð1∕3Þe∕H2, which could be taken up byH2 without severing the bond. Extending this concept furtherthen predicted that CaH12 and hydride alloys with a high Hcontent (smaller EAE) would be composed predominantly ofmolecular H2. An important observation in support of the EAEdescription given above is that the H-H bond in CaH4 lengthenedfrom 0.81 Å at 100 GPa to 0.82 Å at 150 GPa, whereas the H-Hbond in CaH12 shortened from 0.81 Å to 0.80 Å. Here, more Cavalence electrons in CaH4 were available for transfer to the H2 σ�orbitals than were available in CaH12, thereby more severelyweakening the bond in CaH4. This effect was relatively smallfor CaH12, in which the H-H bond was shortened due to com-pression.

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

The three-dimentional sodalite cage in CaH6 is the result ofinterlink of other H4 units via each H atom at the corner ofone H4 unit. So, what is the electronic factor promoting the for-mation of these H4 units? To answer this question, the electronlocalization functions (ELF) of a hypothetical bare bcc Ca latticewith the H atoms removed and CaH6 hydride (Fig. 3A and 3B)were examined. In bare body-centered cubic (bcc) Ca, regionswith ELF values of 0.58 were found to localize at the H atomsites in the H4 units on the faces of the cube. The ELF of CaH6

hydride suggested that no bonds were present between the Caand H. A weak “pairing” covalent interaction with an ELF of0.61, however, was found between the H atoms that formed asquare H4 lattice. Their formation resulted from the accommo-dation by H2 of excess electrons from the Ca. An electron topo-logical analysis also showed the presence of a bond-criticalpoint (21) along the path connecting neighboring H atoms.The integrated charge within the H atomic basin was 1.17 e, whichcorresponded to a charge transfer of 1.02 e from each Ca. A par-tially ionized Ca was also clearly supported by the band structureand the density of states as reported in Fig. 3C and Fig. S15. At150 GPa, Ca underwent an s–d hybridization with an electrontransferred from the 4 s to the 3 d orbital. In CaH6, the Ca sitesymmetry was m3̄m (Oh) and the Ca 3 d manifold was clearlysplit into the eg and t2g bands, with the lower energy eg bandpartially 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 thehypothetical H6 (Fig. 3D) was 15.2 eV, comparable to 16.4 eVfor CaH6 (Fig. 3C). In comparison, the valence band width ofthe bare bcc Ca was only 4.3 eV (Fig. 3E). The band structureof CaH6 near the Fermi level was modified from H6 due tothe the hybridization between Ca 3 d and H 1 s orbital; however,the trend in the electronic band dispersions from −2 to −16.4 eVwas remarkably similar to that of H6 from 3 to −15.2 eV.

Fig. 1. Enthalpies of formation (ΔH, with respect to CaH2 and H2) of CaH2n

(n ¼ 2–6) and crystal structures. (A) The abscissa x is the fraction of H2 in thestructures. The open, solid, and half-filled symbols indicate that the structuresare composed of H4 units, molecular H2, and the coexistence of H2 and H,respectively. The metastable structures are indicated by circles. The stablepressure ranges for CaH4, CaH6, and CaH12 are 50–200 GPa, 150–200 GPa,and 100–200 GPa, respectively. The EAE (per H2) is shown in brackets. Theestimated stability fields were determined according to the static enthalpiesand may shift upon inclusion of dynamic effects (the zero-point motion ofthe nuclei). (B) Structure of tI10-CaH4. (C) Structure of cI14-CaH6. (D) Structureof hR13-CaH12. Monatomic H, molecular H2, and Ca atoms are shown as cyan,green, and royal blue, respectively. The green cylinders and gray dashed linesare drawn to represent molecular H2 and the sodalite cage, respectively.

Fig. 2. Hückel energy-level diagrams of H4 and H2−4 units. (A) Hückel energy-

level diagram of H4. (B) Hückel energy-level diagram of H2−4 . The partial oc-

cupation of electrons on the degenerate orbitals of H4 units can lead to a JTdistortion, but no JT distortion is expected for a closed shell electronic struc-ture as shown in B.

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Sodalite cage was constructed from linking of H4 units andthis topologically resulted in the formation of H6 faces. In fact,a primitive cell of CaH6 can be seen as composed of a H6 hexagonand a Ca. Band structure analysis suggested that four conductionbands of “Ca0H6” sodalite cage (Fig. 3D) were half-filled andthe addition of maximal two electrons to H6 gave rise to partialoccupancy of the degenerate bands at Γ as indicated in CaH6

(Fig. 3C). Partial occupancy of a degenerate orbital results inan orbitally degenerate state and is subject to Jahn–Teller (JT)distortion (22) (Fig. 2). The JT effect involves coupling betweenthe electron and nuclear degrees of freedom, leading to distor-tions in the structure, and it lifts the orbital degeneracy. If thedistortion is dynamic, JT vibrations can contribute to supercon-ductivity. The same mechanism has been invoked to explain thesuperconductivity in B-doped diamond (23). To investigate thispossibility, electron–phonon coupling (EPC) calculations on thesodalite structure of CaH6 at 150 GPa were performed. The pho-non dispersion curves, phonon linewidth γðωÞ, EPC parameter λ,and Eliashberg spectral function α2FðωÞ were calculated. A gapat 430 cm−1 (Fig. 4) separated the phonon spectrum into tworegions: The lower frequency branches were associated withthe motions of both Ca and H, whereas the higher frequencybranches were mainly associated with H atoms. The combinedcontribution (19% and 81%, respectively) gave an EPC para-meter λ of 2.69. The calculated phonon linewidths (Fig. 4)showed that the EPC was derived primarily from the T2g andEg modes at the zone center Γ. Incidentally, these two bands, re-spectively, were the in-plane breathing and rocking vibrations ofthe H atoms belonging to the H4 unit. The corresponding atomicvibrations led to distortions in the square planar structure. More-over, both phonon branches showed significant phonon softeningalong all symmetric directions. A large EPC also benefited fromthe high density of states at the Fermi level caused by a Van Hovesingularity at Γ (Fig. 3C). The very large EPC was unprecedentedfor the main group hydrides. Previous calculations on a varietyof systems predicted an EPC in the range of 0.5–1.6 (11). Themechanism suggested here is not inconsistent with the mechan-isms found in JT-induced superconductivity in alkali intercalatedC60, in which the intramolecular vibrations are responsible for

distortions that lower the symmetry of the molecules, which isfavorable for electron–phonon processes (24, 25).

Tc was calculated based on the spectral function α2FðωÞ bynumerically solving the Eliashberg equations (26), which consistof coupled nonlinear equations describing the frequency-depen-dent order parameter and renormalization factor. The Coulombrepulsion is taken into account in terms of the Coulomb pseudo-potential, μ�, scaled to a cutoff frequency (typically six times themaximum phonon frequency) (27). At 150 GPa, the predicted Tc

values were 235 K and 220 K using typical values for μ� of 0.1 and0.13, respectively. EPC calculations were also performed for200 GPa and 250 GPa, in which the calculated Tc was found todecrease with pressure (201 K at 200 GPa and 187 K at 250 GPa

Fig. 3. ELF and band structure. (A) ELF of CaH0 (cI14 structure with H removed). (B) ELF of CaH6. (C) Band structure of CaH6. (D) Band structure of Ca0H6 (cI14structure with Ca removed). (E) Band structure of CaH0. The horizontal dotted lines indicate the Fermi level.

Fig. 4. Phonon band structure and Eliashberg spectral function. Phonon dis-persion curves of cI14 at 150 GPa (Left). Circles indicate the phonon linewidthwith a radius proportional to the strength. Phonons with a larger linewidth atΓ belong to the t2g and eg modes, as indicated by the circles at 960 cm−1 and1;960 cm−1, respectively. Eliashberg electron–phonon coupling spectral func-tion α2FðωÞ at 150 GPa (Right). Dashed line is the integration of the electron–phonon coupling strength as a function of phonon frequency. The horizonlines are drawn as a guide.

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for μ� ¼ 0.13), with a pressure coefficient (dTc∕dP) of−0.33 K∕GPa. Tc of the order of 200 K is among the highestfor all reported hydrides.

The predicted high Tc for CaH6 is very encouraging, but itmust be viewed with caution. Formally, there is no upper limitto the value of Tc within the Midgal–Eliashberg theory of super-conductivity. Two practical factors must be considered. The cal-culation of the EPC is based on the harmonic approximation andwithout consideration of electron correlation effects. Becausestrong electron–phonon coupling in CaH6 arises from the proxi-mity of the electronic and structural instabilities, anharmonicityof the atomic motions can lead to renormalization of the vibra-tional modes, as demonstrated in a study of AlH3. In that study,lower renormalized frequencies were found to reduce the EPCand suppress superconductivity (28). On the other hand, the elec-tron–phonon matrix elements may be enhanced by anharmonicvibrations, as in the case of disordered materials (29). In a recenthybrid functional study of C60 anions, the inclusion of Hartree–Fock exchange contributions was shown to have little effect onthe structural properties and phonon frequencies but resultedin a strong increase in the electron–phonon coupling (24).

The formation of a hydrogen sodalite cage with enclathratedcalcium in CaH6, reported here for hydrogen-rich compounds,provides an unexpected example of a good superconductor cre-ated by the compression of a mixture of elemental calciumþhydrogen or CaH2 þ hydrogen. This superconductor can also beviewed as consisting of unique square H4 units and electron-donating calcium atoms subject to JT effects. Dense supercon-ductive states, such as those reported here, may be favored inother mixtures of elemental metalsþ hydrogen or any hydrideþhydrogen upon compression. This work highlights the major role

played by pressure in effectively overcoming the kinetic barrier toformation in the synthesis of hydrides.

MethodsOur structure prediction approach is based on a global minimization of freeenergy surfaces merging ab initio total-energy calculations via particle swarmoptimization technique as implemented in CALYPSO (crystal structure analy-sis by particle swarm optimization) code (19). Our CALYPSOmethod unbiasedby any known structural information has been benchmarked on variousknown systems 19 with various chemical bondings and had several successfulprediction of high-pressure structures of Li, Mg, and Bi2Te3 (30–32), amongwhich the insulating orthorhombic (Aba2, Pearson symbol oC40) structure ofLi and the two low-pressure monoclinic structures of Bi2Te3 have been con-firmed by independent experiments (32, 33). The underlying ab initio struc-tural relaxations were carried out using density functional theory within thePerdew–Burke–Ernzerhof exchange-correlation (34) as implemented in theVienna Ab Initio Simulation Package (VASP) code (35). The all-electron pro-jector-augmented wave method (36) was adopted with 1s and 3p64s2 trea-ted as valence electrons for H and Ca, respectively. Electronic properties,lattice dynamics and electron-phonon coupling were studied by density func-tional (linear-response) theory as implemented in the QUANTUM ESPRESSOpackage (37). More computational details can be found in the SupportingInformation.

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

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