Highpressure crystal structures and superconductivity of Stannane (SnH_{4})
 ^{a}National Laboratory of Superhard Materials, Jilin University, Changchun 130012, P. R. China;
 ^{b}Department of Geosciences, Department of Physics and Astronomy, and New York Center for Computational Sciences, Stony Brook University, Stony Brook, NY 117942100;
 ^{c}Geology Department, Moscow State University, Moscow 119992, Russia;
 ^{d}Materia Kondentsatuaren Fisika Saila, Zientzia eta Teknologia Fakultatea, Euskal Herriko Unibertsitatea, 644 Postakutxatila, 48080 Bilbo, Basque Country, Spain;
 ^{e}Donostia International Physics Center (DIPC), Paseo de Manuel Lardizabal, 20018, Donostia, Basque Country, Spain;
 ^{f}Centro de Fisica de Materiales Consejo Superior de Investigaciones Cientificas Universidad del Pais Vasco/ Euskal Herriko Unibertsitatea (CSICUPV/EHU), 1072 Posta kutxatila, E20080 Donostia, Basque Country, Spain; and
 ^{g}Computational Astrophysics Laboratory, RIKEN, 21 Hirosawa, Wako, Saitama 3510198, Japan
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Edited by Russell J. Hemley, Carnegie Institution of Washington, Washington, DC, and approved December 16, 2009 (received for review July 24, 2009)
Abstract
There is great interest in the exploration of hydrogenrich compounds upon strong compression where they can become superconductors. Stannane (SnH_{4}) has been proposed to be a potential hightemperature superconductor under pressure, but its highpressure crystal structures, fundamental for the understanding of superconductivity, remain unsolved. Using an ab initio evolutionary algorithm for crystal structure prediction, we propose the existence of two unique highpressure metallic phases having space groups Ama2 and P6_{3}/mmc, which both contain hexagonal layers of Sn atoms and semimolecular (perhydride) H_{2} units. Enthalpy calculations reveal that the Ama2 and P6_{3}/mmc structures are stable at 96–180 GPa and above 180 GPa, respectively, while below 96 GPa SnH_{4} is unstable with respect to elemental decomposition. The application of the AllenDynes modified McMillan equation reveals high superconducting temperatures of 15–22 K for the Ama2 phase at 120 GPa and 52–62 K for the P6_{3}/mmc phase at 200 GPa.
Relatively hightemperature superconductivity is now documented in lightelement metals such as Li under pressure (1 ⇓–3) and MgB_{2} (4), where transition temperatures T _{c} up to 20 K and 39 K, respectively, are observed. There is great interest in exploration of unique superconducting phases in other lightelement materials because their high phonon frequencies can enhance electronphonon coupling (see ref. 5). As the lightest element, hydrogen at very high densities is also predicted to be a superconductor with high transition temperatures (6 ⇓–8). Experiments indicate that the predicted metallic and superconducting states of hydrogen remain above ∼300 GPa (9 ⇓–11). It has been proposed that hydrogenrich compounds (e.g., group IVa hydrides (12)) are expected to metallize at pressures considerably lower than pure hydrogen due to the chemical “precompression” caused by heavier elements; these metallization pressures may fall within the range of current capabilities of static compression techniques. The exploration of potential superconductivity in these hydrogenrich compounds (e.g., SiH_{4}, GeH_{4}, and SnH_{4}) is thus desirable and numerous studies have been performed (13 ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ ⇓–25). Strikingly, recent experiments (15, 18) show that SiH_{4} transforms to a metallic phase near 50–60 GPa with a superconducting T _{c} of 17 K at 96 and 120 GPa, though debate remains (26). We have recently predicted (17) that GeH_{4} becomes a hightemperature superconductor with a T _{c} of 64 K at 220 GPa. A theoretical study of SnH_{4} (21) predicts that its T _{c} can be even higher, reaching the value of 80 K. Using simulated annealing and geometry optimization, that study found that the highpressure phase of SnH_{4} has P6/mmm symmetry with a layered structure intercalated by molecular H_{2} units, wherein the nearest HH distance, 0.84 Å, is short enough to be considered as covalent bonding, but significantly longer than the 0.74 Å in the free H_{2} molecule. This prediction of extremely high T _{c} is fully dependent on the correctness of the structural model. Therefore, if some other structures are more stable, the picture of superconductivity in SnH_{4} may be completely altered.
Here we undertake a different route to explore the highpressure structures of SnH_{4} and use our newly developed ab initio evolutionary simulation method (27 ⇓–29) to address the above question. This method has proven its reliability and efficiency in predicting stable structures with knowledge only of the chemical composition (17, 30 ⇓ ⇓ ⇓ ⇓ ⇓–36). We then performed the calculations on the total energy, band structure, phonons, and electronphonon coupling of the predicted highpressure structures. Our results reveal the appearance of unique chemistry of SnH_{4} at highpressure and suggest the high superconductivity of SnH_{4} under pressures that are accessible by current experimental technique.
Results and Discussion
We performed evolutionary variablecell structure prediction simulations with one to four SnH_{4} formula units per cell at 30, 70, 120, 200, and 250 GPa. Analysis of the predicted structures gave us a shortlist of candidate structures with space groups Cmcm * (4 molecules/cell), P2_{1}/m ^{†} (2 molecules/cell), Ama2 (4 molecules/cell) and P6_{3}/mmc (2 molecules/cell), respectively. In the Cmcm and P2_{1}/m structures predicted at 30 and 70 GPa, the Sn atoms are packed in zigzag chains between which nonbonded H_{2} molecules are located. This strongly indicates a tendency to decomposition into Sn + 2H_{2}. Indeed, enthalpy calculations (Fig. 2) show that decomposition occurs below 96 GPa.
In the Ama2 phase, Sn atoms form a simple hexagonal packing, where the trigonal prismatic holes are filled with semimolecular H_{2} units (HH distance 0.79 Å, which is longer than the 0.74 Å in the isolated H_{2} molecule). The H_{2} units are aligned either along the pseudohexagonal axis, or perpendicular to it, and these two orientations alternate (Fig. 1 A). Note that the simple hexagonal structure is a highpressure form of silicon (P6/mmm, ref. (37)), but it is not particularly dense. The P6_{3}/mmc phase here is based on the much denser hexagonal close packing of the Sn atoms (c/a = 1.84 at 200 GPa, relatively close to the ideal value of 1.63). In this structure, the ordered H atoms are clearly split into two categories. One sort forms semimolecular H_{2} units (the magenta atoms in Fig. 1 B) occupying hexagonal channels of the hexagonal close packing structure, whereas the other sort of H atoms occupies positions just below and above Sn atoms, forming chains SnH…HSnH…HSnH running along the c axis (Fig. 1 B).
In both the Ama2 and P6_{3}/mmc phases SnH distances are about 1.8–1.9 Å, and SnSn distances about 2.9 Å. The main differences are in the topology of the Sn framework and the relative proportion of the H_{2} semimolecular units and monatomic hydrogen species. The transition Ama2 → P6_{3}/mmc is accompanied by a major electronic reorganization due to the appearance of monatomic hydrogen and the likely change of Sn valence. In the bond valence model the bond order is v _{ij} = exp[(R_{ij}  R_{0,ij})/b], where R_{0} is a bondspecific parameter, R the bond length, and b a constant. Taking b = 0.25 Å (typical of strong covalent bonds (38)), we find that the HH bond order in the H_{2} semimolecular units is 0.82. Thus, hydrogen atoms in these H_{2} units are underbonded and tend to acquire the missing electrons from the metal atoms by charge transfer. We propose to call these chemically active H_{2} units perhydride groups; they have so far not been observed at ambient conditions, but predicted to exist at high pressures in GeH_{4} (ref. 17) and unique lithium hydrides LiH_{2}, LiH_{6} and LiH_{8} (39). Perhydride groups were also predicted in the structures of SnH_{4} calculated in refs. 21 (P6/mmm) and 40 (Cccm), which are otherwise different from our structures. The need for charge transfer implies that perhydrides can only be formed with rather electropositive metals, such as Ge, Sn, and Li, but not Si or C. We believe that perhydrides of alkali and alkali earth metals are likely to exist and are worth exploring. Coming back to SnH_{4} and using the estimated HH bond order (0.82), we find that each Sn atom transfers to all four H atoms 0.72 e in the Ama2 phase and 2.36 e in the P6_{3}/mmc phase. The remaining valence is used for SnSn bonds (∼0.15 e per bond, and there are eight and 12 such bonds in the Ama2 and P6_{3}/mmc phases, respectively), leading us to propose the Sn(II) valence state in Ama2 and Sn(IV) in P6_{3}/mmc.
Enthalpy curves of our predicted structures as a function of pressure are presented in Fig. 2. It is noteworthy that the currently predicted structures are energetically much more favorable than the P6/mmm (21) and Cccm (40) structures. Considering the tendency to decomposition found in structure searches performed at low pressure, the enthalpy of the decomposition (Sn + 2H_{2}, with structures of pure elements taken from refs. (41, 42) as a function of pressure) is also shown in Fig. 2. A wide region of decomposition into Sn + 2H_{2} is indeed confirmed below 108 GPa, above which Ama2 becomes stable in the pressure of 108–158 GPa and then P6_{3}/mmc structure takes over above 158 GPa. As divalent Sn(II) species are quite stable, decomposition of SnH_{4} might also yield SnH_{2}, the structure of which is unknown. We thus searched for the most stable highpressure phases of SnH_{2} at 50 and 100 GPa using the same evolutionary methodology. However, we found that SnH_{2} + H_{2} is 0.15 eV (0.17 eV) per SnH_{4} unit higher in enthalpy than Sn + 2H_{2} at 50 GPa (100 GPa). This eliminates the possibility that SnH_{4} decomposes into SnH_{2} + H_{2}. One should keep in mind, however, the importance of dynamical effects stemming from the high vibrational frequencies of light hydrogen atoms, as can be seen from the phonon density of states (DOS) (Fig. 3 B, D), which implies a large zeropoint (ZP) energy. This has already been found to affect the relative stabilities of hydrogen and hydrides (14, 17, 41). Therefore, we estimated the ZP vibrational energies for C2/c H_{2} (41), I4/mmm Sn (42), Ama2 and P6_{3}/mmc SnH_{4} at 120 GPa using the quasiharmonic approximation (43). It turns out that the inclusion of ZP effects does not change the topology of the phase diagram but extends the stability fields of the Ama2 and P6_{3}/mmc structures to be 96–180 GPa and above 180 GPa, respectively (insets of Fig. 2).
The calculated electronic band structures and DOS for the Ama2 structure at 120 GPa (Fig. 3 A) and P6_{3}/mmc structure at 200 GPa (Fig. 3 C) reveal that both phases are metallic. We found that the calculated valence bandwidths are very broad and show strong hybridization between the Sn and H orbitals. In particular, both H and Sn atoms participate in common overlapping bands, which is in agreement with previous theoretical prediction (12). Note that the P6_{3}/mmc structure has less dispersive band structure and more bands crossing E _{f}, which lead to a larger electronic DOS (2.18 × 10^{2} states/eV/Å ^{3} at the Fermi level N(E _{f})) than that (1.48 × 10^{2} states/eV/Å ^{3}) of the Ama2 structure. This larger N(E _{f}) contributes to a higher T _{c}. The phonon DOS for Ama2 SnH_{4} at 120 GPa and P6_{3}/mmc SnH_{4} at 200 GPa are depicted in Fig. 3 B, D, respectively. The absence of imaginary frequencies in the phonon DOS and phonon dispersion curves suggests that the Ama2 and P6_{3}/mmc structures are both dynamically stable. Three separate regions of phonon bands are clearly recognized. The heavy Sn atoms dominate the low frequencies, the HSnH bending vibrations contribute significantly to the intermediatefrequency region and the high frequencies are mainly due to intramolecular vibrations of perhydride H_{2} units.
We now discuss superconductivity in both the Ama2 and P6_{3}/mmc structures. The calculated phonon DOS projected on Sn and H atoms for Ama2 at 120 GPa is compared to Eliashberg phonon spectral function α ^{2} F(ω) (44) and the integrated electronphonon coupling (EPC) parameter λ in Fig. 3 B (Fig. 3 D for P6_{3}/mmc phase at 200 GPa). The contribution from the lowfrequency Sn translational vibrations constitutes 23% (also 23% for P6_{3}/mmc) of the total λ. The intermediatefrequency vibrational modes make up a significant section of 74% (73.3% for P6_{3}/mmc phase) and the remaining 3% (3.7% for P6_{3}/mmc phase) is derived from the intramolecular vibrational modes. The calculated EPC parameter λ is 0.61, which indicates that the EPC is strong, and the phonon frequency logarithmic average ω _{log} calculated directly from the phonon spectrum is 905 K. To estimate T _{c}, the effective Coulomb repulsion parameter μ ^{∗} must be known. It is defined as , where μ is the direct Coulomb repulsion between paired electrons,ω _{el} corresponds to a plasma frequency, while ω _{ph} the highfrequency cutoff in electronphonon coupling spectral function α ^{2} F(ω) (45). Due to the difficulty in calculating μ and ω _{ph}, it remains challenging to directly derive an accurate μ ^{∗} by theory. An upper bound on μ ^{∗} is estimated to be 0.25 (45). Ashcroft (12) has suggested that for dense metal hydrides μ ^{∗} can be best chosen as 0.1 or in the range of 0.1–0.13 and later on these choices have been widely used for SiH_{4}, GeH_{4} and SnH_{4} (13, 16, 17, 21). Using μ ^{∗} = 0.13–0.1, the critical temperature T _{c} for the Ama2 structure, estimated from the AllenDynes modified McMillan equation (45), is in the range of 15–22 K. In the same manner, the calculated T _{c} for the P6_{3}/mmc phase at 200 GPa reaches very high values of 52–62 K, much higher compared to Ama2. The larger T _{c} in the P6_{3}/mmc structure is mainly attributed to the stronger λ of 0.87 and the larger ω _{log} of 1,135 K. The larger N(E _{f}) and higher intermediate frequencies in the phonon spectrum of the P6_{3}/mmc structure are probably responsible for the stronger λ.
In summary, crystal structures of SnH_{4} at pressures up to 250 GPa have been extensively investigated using ab initio evolutionary simulation methods. Two unique metallic phases with structures having space groups Ama2 and P6_{3}/mmc are found to be energetically much more favorable than the earlier proposed structures. These phases are predicted to be stable in the pressure range of 96–180 GPa and above 180 GPa, respectively, while below 96 GPa SnH_{4} is predicted to decompose into Sn + 2H_{2}. The newly proposed structures for SnH_{4} both contain unique perhydride H_{2} units. Further electronphonon coupling calculations predict that both phases are superconductors with high T _{c} of 15–22 K for the Ama2 structure at 120 GPa and 52–62 K for P6_{3}/mmc at 200 GPa. Our calculation suggests that once synthesized at high pressure, these SnH_{4} phases can be decompressed to low pressures because kinetic barriers are likely to be sufficiently large to prevent the SnH_{4} → Sn + 2H_{2} decomposition reaction. Experiments (e.g., xray diffraction, electrical resistivity, and magnetic susceptibility) are needed to further explore the highpressure metallization and superconductivity of SnH_{4}. Our findings support the conjecture that hydrogenrich hydrides are a way to achieve a metallic phase with parallel to metallic hydrogen at readily accessible experimental pressures, and the twocomponent plasma state predicted for pure hydrogen, might also exist in these compounds.
Materials and Methods
Ab initio evolutionary simulation for crystal structure prediction done with the Universal Structure Predictor: Evolutionary Xtallography code (27 ⇓–29) searches for the structure possessing the lowest free energy at given PressureTemperature conditions and is capable of predicting the stable structure of a given compound for a given composition. The ﬁrst generation of structures is produced randomly. Each subsequent generation is produced from 65% of the lowestenthalpy structures of the preceding generation; in addition, the lowestenthalpy structure always survived into the next generation. The variation operators used for producing offspring included heredity (65% structures), lattice mutation (20%), and atomic permutation (15%). The underlying structure relaxations were performed using densityfunctional theory (46, 47) within the PerdewBurkeErnzerhof (PBE) generalized gradient approximation (GGA) (48), as implemented in the Vienna Abinitio Simulation Package code (49). The frozen core allelectron projectoraugmented wave (50) method was adopted. The use of a planewave kinetic energy cutoff of 600 eV and dense kpoint sampling were shown to give excellent convergence of the energy differences and stress tensors. EPC have been explored using the pseudopotential planewave method within the PBEGGA, through the QuantumopEn Source Package for Research in Electronic Structure, Simulation, and Optimization (ESPRESSO) package (51). Forces and stresses for the converged structures are optimized and checked to be within the error between the VASP and QuantumESPRESSO code. Pseudopotentials for H and Sn were generated by a TroullierMartins normconserving scheme (52) and tested by comparing the electronic band structures and phonon spectra with the results calculated from VASP code. In these calculations we used a kinetic energy cutoff of 50 Rydberg and a 12 × 12 × 6 MonkhorstPack (53) kpoint grids for Ama2 and P6_{3}/mmc, for which tests showed excellent convergence of the computed properties. The phonon spectra were calculated within densityfunctional perturbation theory using the QuantumESPRESSO package. A 4 × 4 × 2 grid of q points in the ﬁrst Brillouin zone was used in the interpolation of the force constants for the phonon dispersion curve calculation. The technique for the calculation of EPC has been described in detail in our previous publication (17).
Acknowledgments
A.B. thanks M. MartinezCanales for discussions. G.Y.G. thanks L. J. Zhang for his help in the calculation. G.Y.G. and Y.M.M. thank China 973 Program under Grant 2005CB724400, Natural Science Allied Foundation of China under Grant 10676011, the Program for 2005 New Century Excellent Talents in University, and the 2007 Cheung Kong Scholars Program of China. G.Y.G. is also grateful to the Project 20092004 supported by Graduate Innovation Fund of Jilin University. Part of the calculation was performed with Riken SuperCombined Cluster. A.R.O. acknowledges funding from Stony Brook Research Foundation and from Intel Corp. and access to supercomputers at New York Center for Computational Sciences (U.S.A.) and Joint Supercomputer Center (Russian Academy of Sciences). A.B. acknowledges funding from the Spanish Ministry of Education (Grants BFM200304428 and Grant BES20058057)
Footnotes
 ↵ ^{1}To whom correspondence should be addressed. Email: mym{at}jlu.edu.cn.

Author contributions: G.G., P.L., Z.L., H.W., and Y.M. performed research; G.G., A.R.O., T.C., Y.M., A.B., A.O.L., T.I., and G.Z. analyzed data; G.G., A.R.O., and Y.M. wrote the paper; and Y.M. designed research.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵ ^{*}For Cmcm structure at 20 GPa, the lattice parameters are a = 3.63 Å, b = 9.88 Å, and c = 4.12 Å with atomic positions of Sn at 4c (0, 0.29142, 0.25) and H at 4c (0, 0.50191, 0.75), (0, 0.42112, 0.75) and 8f (0 0.96117, 0.49332)

↵ ^{†}For P2_{1}/m structure at 70 GPa, the lattice parameters are a = 4.89 Å, b = 3.17 Å, c = 3.48 Å and β = 89.19°, with atomic positions of Sn at 2e (0.64826, 0.25, 0.74596) and H at 2e (0.92488, 0.25, 0.32702), (0.08651, 0.25, 0.38317), (0.02506, 0.25, 0.89266), and (0.18168, 0.25, 0.82398).
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