A novel stable hydrogen-rich SnH₈ under high pressure

Abstract

A first-principles calculation is applied to perform a comprehensive study of the Sn–H system. Besides the common tetravalent hydride, a novel SnH₈ crystal with the space group I4̄m2 is reported with the most dominant enthalpy from structure searching techniques. All the H atoms of SnH₈ are in the form of H₂ or H₃ units with electrons localized around them, showing covalent bond character. The rich and multiple Fermi surface distribution displays a metallic feature. Further electron–phonon coupling calculations reveal the high T_c of 63–72 K at 250 GPa.

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Introduction

The metallization and superconductivity of H₂ is one of the most controversial and challenging subjects in the research of high pressure physics. The exploration has lasted for 80 years, since Winger and Hunter¹ gave the prediction that hydrogen could be metallic under sufficient pressure. Since then, many efforts have been devoted to exploring its structures and superconductivity.^2–5 However, there are still many uncertainties which have not been solved until now. The difficulty in producing metallic hydrogen by physical compression is that the required pressure exceeds the limit of the current static experimental capacity. The proposed bypass is “chemical precompression”⁶ method by doping with particular elements, which can reduce the pressure of metallization. Recently, several first-principles calculation^7–11 and an experiment¹² show a high superconducting transition temperature about 200 K in the sulfur hydride system. These results indicate that hydrogen-rich compounds are well worthy of exploration on their structures and peculiar properties by pressure effect.

The group IV hydrides are materials with lower metallic pressure than pure hydrogen and high T_c, which have been experienced many discussions. For one of Si–H compounds, SiH₄ has proven to be superconductor with T_c of 17 K at 96 GPa in experiment,¹³ though debate remains.¹⁴ Moreover, different proportions of the Si–H compounds predicted with high-T_c property, such as Si₂H₆ with T_c of 139 K at 275 GPa,¹⁵ SiH₈ with T_c of 107 K at 250 GPa,¹⁶ are also deeply explored. Recently, the various stoichiometries of GeH_n compounds have been calculated.¹⁷ Except for GeH₄ with T_c of 64 K at 220 GPa,¹⁸ three stable metallic crystal structures Ge₃H, Ge₂H, and GeH₃ are concomitant under high pressure.¹⁷ In Si–H and Ge–H system, there have several different components under high pressure. In addition to illustrating the diversity of the components, it is also displayed an effective method of “chemical precompression” by adding Si and Ge atoms to hydrogen. As one of the elements closely correlated to Si and Ge, only SnH₄ has been investigated up to now. Tse et al.¹⁹ predicted a layered SnH₄ structure with P6/mmm symmetry, which has a T_c close to 80 K at 120 GPa. Afterwards, Gao et al.²⁰ proposed two SnH₄ structures with Ama2 and P6₃/mmc symmetries stable between 96–158 GPa and above 158 GPa. Therefore, it is necessary to carry out the more comprehensive and accurate variable components prediction, owning to the diversity of the compounds in the system under high pressure. This prompts us to investigate an extensive research to update the Sn–H phase diagram with different Sn/H ratios and pressures.

In this work, various stoichiometries structures of SnH_n under pressure of 50–350 GPa are widely investigated. Besides the most studied tetravalent hydride, a novel proportion SnH₈ at pressure above 238 GPa is predicted. All H atoms in this structure form H₂ or H₃ units with electrons located around them. The complex electronic structure near the Fermi surface indicate metallic feature. The calculations show a superconducting transition temperature of 63–72 K at 250 GPa.

Computational methods

We searched for the thermodynamically stable structures of various stoichiometries of SnH_n (n = 1/3, 1/2, 1, 2, 3, 4, 5, 6, 7, 8) using ELocR code,²¹ and checked by the USPEX.²² The geometrical optimizations and total energy calculations are performed using the VASP code²³ with the Perdew–Burke–Ernzerhof (PBE) parameterization of generalized gradient approximation (GGA)²⁴ used to treat the exchange–correlation energy. The projector augmented wave (PAW) method is adopted with the PAW potentials where 1s¹ and 4d¹⁰5s²5p² are treated as valence electrons for H and Sn atoms, respectively. A plane-wave cutoff energy of 800 eV and a dense k-point grid with the spacing of 2π × 0.03 Å⁻¹ in the first Brillouin zone (BZ) are used for all structure relaxations.

The dynamic properties, electronic properties and electron–phonon coupling calculations are studied in the QUANTUM-ESPRESSO package.²⁵ The Troullier–Martins-type norm-conserving pseudopotentials are used, which have been carefully tested by comparing the calculated volume with VASP code. A cutoff energy selected as 80 Ry with the BZ grid of spacing 2π × 0.025 Å⁻¹ for the structural optimization. In addition, a more intensive k mesh 26 × 26 × 32 is adopted for the Fermi surface calculation. A 4 × 4 × 4 q-point mesh for I4̄m2 phase at 250 and 300 GPa, a 3 × 3 × 4 q-point mesh at 350 GPa in the first BZ is used in the electron–phonon coupling calculation.

Results and discussion

All possible structures with lower enthalpies have been relaxed, and the computed enthalpies with respect to the crystals of Sn and H₂ at the selected pressures are provided in the convex hull, as shown in Fig. 1(a). The well-known I4/mmm of Sn,²⁶ C2/c and Cmca1̄2 of hydrogen structures² are regarded. Due to the decomposition of SnH_n in the pressure range of 50–100 GPa (the enthalpies are all totally located above the convex hull), we only present the convex hull at 150–350 GPa in Fig. 1(a). So we speculate that there should be no SnH_n crystals in the pressure below 100 GPa. At 150 and 200 GPa, only previously reported stable tetravalent hydrides (Ama2 for 150 GPa, P6₃/mmc for 200 GPa)²⁰ locate on the convex hull, showing their thermodynamical stability. With the increase of pressure, a new stoichiometry of SnH₈ appears on the convex hull at 250–350 GPa, although SnH₄ remains the most dominant compound by the view of enthalpy. Similarly, the ratio of 1 : 8 has been predicted in the Si and Ge hydrides.^16,27 Fig. 1(b) provides the different enthalpy curves between SnH₈ (I4̄m2) and SnH₄ + 2H₂, and depicts the thermodynamic stability of I4̄m2 structure above 238 GPa. The phase diagram for SnH_n crystals are presented in Fig. 1(b) inset. Comparing with Ge–H system¹⁷ (the stable P3̄ Ge₃H start to form at 32 GPa), we can not find stable crystals in the low pressure range below 131 GPa among SnH_n compounds. As it was suggested by Tse et al.¹⁹ and Gao et al.¹⁸ that the atomic radius of Sn is larger than Ge element, which could displayed weak interaction between Sn and H atoms, leading to the decomposition at the lower pressure range. Above 131 GPa, the predicted SnH₄ with Ama2 and P6₃/mmc symmetries²⁰ appeared at 131–150 GPa and above 150 GPa, respectively. Afterwards, our predicted stoichiometry SnH₈ is uncovered thermodynamic stable above 238 GPa.

Fig. 1 (a) Enthalpy difference curves of SnH_n with respect to Sn and H₂ at selected pressures. The SnH_n structures on the convex hull (solid lines) are thermodynamically stable relative to decomposition into other SnH_n compounds and elements, whereas these located above the convex hull (dashed lines) are unstable. (b) Enthalpy difference curves of SnH₈ (I4̄m2) with respect to SnH₄ and H₂ at selected pressures. Inset: calculated thermodynamically stable ranges for SnH_n compounds.

Fig. 2 depicts the crystal structure of I4̄m2 SnH₈ at 250 GPa. This tetragonal structure consist of two SnH₈ units in a conventional cell with the lattice parameters a = 2.999 Å, c = 5.389 Å. Sn atoms occupy at crystallographic 2b site (0.0, 0.0, 0.5), arranging in the center of the square and the middle of the c axis, respectively. Three inequivalent H atoms occupy on the crystallographic 4e (0.0, 0.0, 0.118), 8i (0.727, 0.0, 0.169) and 4f (0.0, 0.5, 0.169) sites. We note that the crystal structure of I4̄m2 SnH₈ is substantially different from the I4̄m2 phases in SiH₈ and GeH₈, which have been proposed in low pressure range. The primary difference is the sites of the heavier atoms: Si and Ge atoms take over 2a sites in the I4̄m2 crystals of SiH₈ and GeH₈, while Sn atoms occupy on 2b site in I4̄m2 SnH₈. In addition, all of the H atoms in SnH₈ connect to either H₂ units or H₃ ones, which strongly distinguish the only formation of H₂ units by parts of H atoms in SiH₈ and GeH₈ systems. The rich distributions of H₃ and H₂ units are also discovered in the other high hydrogen-rich stoichiometry compound, e.g. R3̄m SnH₇. In the pressure range of 100–200 GPa, an interesting structure R3̄m also contains H₂ and linear H₃ units with the lowest enthalpy among SnH₇ structures. By comparison of the thermodynamic property, the R3̄m structure eventually not present on the convex cell indicating its metastability.

Fig. 2 Structures of I4̄m2 SnH₈ at 250 GPa. The gray atoms depict Sn and the orange atoms represent H.

The bonding character in SnH₈ structure is explored by the electron localization functions (ELF).²⁸ Fig. 3(a) shows the isosurface value of 0.75 in yellow color, surrounding H₂ and H₃ units, which represent the strong electronic localization with the covalent bond character. 2D-ELF map is drawn with the selected coplanar H₂, H₃ units, and Sn atom in Fig. 3(b). Except the strong electronic localization of H–H bonds in the H₂ and H₃ units, the free electron channels with isosurface value of 0.5 surround the Sn atoms and H₂ units, showing the metallic character. Pressure induced charge transfer also has been observed in the I4̄m2 SnH₈, similar to the reported crystals with Ama2 and P6₃/mmc space group in SnH₄ compounds.²⁰ Bader analysis²⁹ demonstrates the charge transfer from Sn to H atoms with about 1.39 electrons at 250 GPa.

Fig. 3 The calculated ELF of I4̄m2 at 250 GPa. (a) The isosurface value of 0.75 colored by yellow. (b) 2D-ELF for (1, 0, 0) plane.

The electronic band structure and the projected density of states (PDOS) of SnH₈ are calculated, as shown in Fig. 4(a). Two bands marked with 1 and 2 crossing over the Fermi level contribute large total electronic density distribution and reveal the strong metallic character. Although band 3 is very close to the Fermi level at high symmetry point P in BZ, our calculations can not find any contributions to the free electrons on Fermi level until 350 GPa. Moreover, a flat band in the vicinity of Fermi level close to the G point is observed in the band structure, and the flat band near the Fermi level has been suggested as a favorable condition to promote electron pairing and superconducting behavior.^19,30 This complex electronic band structure near the Fermi energy brings the rich and multiple Fermi surface feature, as displayed in Fig. 4(b). The insets labeled the number of 1 and 2 in Fig. 4(b) represent the electrons distribution on the three-dimensional Fermi surface of the band 1 and 2 in BZ, respectively. The rich Fermi surface features will benefit to the strong electron–phonon interaction in a superconductor.

Fig. 4 (a) The electronic band structure and PDOS of the I4̄m2 structure at 250 GPa. (b) The BZ and the calculated three-dimensional Fermi surface.

Satisfying the dynamic stability is one of the basic conditions to determine the stability of crystal structure. The phonon band structure and projected phonon density of states (PHDOS) of I4̄m2 at 250 GPa are described in Fig. 5. From the phonon band structure, no negative frequency in the entire BZ shows the dynamic stability of the structure. From the PHDOS projected on elements in Fig. 5, we can see clearly that the lighter element H mainly contribute to the frequency upon 9 THz while the low frequency below 9 THz mainly come from the vibrations of heavier element Sn.

Fig. 5 Phonon dispersion curves, PHDOS projected on Sn and H atoms, Eliashberg spectral function α²F(ω) and the EPC λ for I4̄m2 at 250 GPa.

To explore the potential superconductivity of SnH₈, the electron–phonon coupling (EPC) strength λ, the logarithmic average phonon frequency (ω_log) and the Eliashberg phonon spectral function α²F(ω)³¹ of I4̄m2 at 250 GPa have been investigated. Eliashberg phonon spectral function α²F(ω) and the EPC parameter λ with the value of 1.14 are presented in Fig. 5. As compared with the PHDOS of Sn and H atoms, we can separate the contribution into two parts. The low frequency vibrations of Sn atoms (<9 THz) contribute approximately 28% to total λ, while the contribution of the high frequency vibration modes of H atoms (>9 THz) is 72%. The phonon frequency logarithmic average ω_log calculated directly from the phonon spectrum is 859 K. We now can analyze the superconductivity using the Allen–Dynes modified McMillan equation,³² , where μ* is the Coulomb pseudopotential representing coulombic repulsion. With the commonly accepted values μ* as 0.1–0.13 for hydrogen dominant metallic alloys,⁶ the estimated T_c is in the range of 63–72 K at 250 GPa. The value of SnH₈ is less than 98–107 K of SiH₈ (ref. 16) and 76–90 K of GeH₈ (ref. 27) at 250 GPa, but higher than 52–62 K of SnH₄ (ref. 20). So, it implied that the T_c of material decreases with the raising mass of the doped atoms in the XH₈ compounds (X = Si, Ge, Sn), and the increase of hydrogen content plays positive effect to T_c in Sn–H system. Then, we also calculated the variety of T_c as a function of pressure. The DOS at the Fermi level N(ε_f), ω_log, the EPC λ of I4̄m2 structure at 250, 300 and 350 GPa are summarized in Table 1. With the increase of pressure, the T_c gradually become higher, and it is consistent with the trend of the EPC λ. Therefore, we think that the increase of T_c with pressure is closely related to EPC λ in the structure.

Table 1 The calculated N(ε_f), ω_log, λ and T_c of I4̄m2 structure at selected pressures

Pressure (GPa)	N(εf) (states per spin per Ry per unit cell)	ωlog (K)	λ	Tc (K)
				μ* = 0.1	μ* = 0.13
250	2.837	859.001	1.139	72	63
300	2.978	724.997	1.354	75	67
350	2.857	762.179	1.392	81	73

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Conclusions

In conclusion, a high-pressure phase diagram of the hydrogen-rich Sn–H system was built from structure prediction simulations and first-principles calculation. According to our exploration, no Sn–H crystals appeared below the pressure point of 131 GPa. Afterwards, the previously predicted SnH₄ with Ama2 and P6₃/mmc symmetries appeared at 131–150 GPa and above 150 GPa, respectively. Moreover, a hitherto unknown stoichiometry SnH₈ with I4̄m2 symmetry is also uncovered stable at pressure above 238 GPa. All H atoms of SnH₈ are in the form of H₂ or H₃ units with electrons localized around them, showing covalent bond character. The rich and multiple Fermi surface distribution displays a metallic feature. Lattice dynamics and electron–phonon coupling calculations further indicate that the I4̄m2 phase is a superconductor with high T_c of 63–72 K at 250 GPa, which derives from strong EPC λ.

Acknowledgements

This work was supported by the National Basic Research Program of China (No. 2011CB808200), National Natural Science Foundation of China (No. 51572108, 11174102, 11404134), Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1132), National Found for Fostering Talents of basic Science (No. J1103202). Part of calculations were performed in the High Performance Computing Center (HPCC) of Jilin University.

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