Superconductivity of novel tin hydrides (SnnHm) under pressure

With the motivation of discovering high-temperature superconductors, evolutionary algorithm USPEX is employed to search for all stable compounds in the Sn-H system. In addition to the traditional SnH4, new hydrides SnH8, SnH12 and SnH14 are found to be thermodynamically stable at high pressure. Dynamical stability and superconductivity of tin hydrides are systematically investigated. Im2-SnH8, C2/m-SnH12 and C2/m-SnH14 exhibit higher superconducting transition temperatures of 81, 93 and 97 K compared to the traditional compound SnH4 with Tc of 52 K at 200 GPa. An interesting bent H3–group in Im2-SnH8 and novel linear H in C2/m-SnH12 are observed. All the new tin hydrides remain metallic over their predicted range of stability. The intermediate-frequency wagging and bending vibrations have more contribution to electron-phonon coupling parameter than high-frequency stretching vibrations of H2 and H3.

While SnH 4 was shown to be a relatively high-T c superconductor, the possibility of existence of other tin hydrides were not explored so far. At the same time, by now it is proven 22 that totally unexpected compounds become stable under pressure, and this gives hope of finding even better superconductors. Hence, in this study, we systematically search for the stable compounds using the highly efficient variable-composition evolutionary searches (VCES). Apart from the previously reported phases of SnH 4 , there is one metastable tetragonal phase of stannane with higher superconducting critical temperature. Other stable superconducting compounds, SnH 8 , SnH 12 and SnH 14 , are found to become stable at high pressure. Moreover, we found a semi-molecular group H 3in the I4m2 structure of SnH 8 . Novel H 4 -is also present in C2/m-SnH 12 . We calculate a high T c of 81 K at 220 GPa in the newly predicted compound SnH 8 , 93 K for SnH 12 at 250 GPa, 97 K for SnH 14 at 300 GPa and 91 K for the metastable phase of SnH 4 at 220 GPa.

Results
Evolutionary variable-composition searches for stable compounds and their structures with up to 20 atoms in the primitive unit cell were performed at 150, 200, 250 and 300 GPa. To further investigate the newly found compounds, fixed-composition structure predictions for the most promising compounds were performed, with one to three formula units per cell. Candidate low-enthalpy structures are metastable I4/mmm-SnH 4 , stable I4 m2-SnH 8 , C2/m-SnH 12 and C2/m-SnH 14 . In the I4m2-SnH 8 structure predicted to be stable at pressures above 220 GPa, Sn atoms are packed between H 2 and H 3 molecular groups, in which the bent H 3 units are perpendicular to one another and sepated by 1.35 Å. In C2/m-SnH 12 , Sn atoms form well-separated close-packed layers intercalated with blocks of H 2 and H 4 semi-molecules. Figure 1(a). shows the enthalpy of formation (ΔH) of Sn-H compounds at selected pressures. Significantly, in addition to reproducing various structures of solid SnH 4 13,21 , Sn 23 and H 2 24 , novel compounds SnH 8 , SnH 12 and SnH 14 are found to be stable in a wide pressure range in our systematic evolutionary structure search. It can be seen from Fig. 1(a). that at around 200 GPa the tetragonal SnH 8 with the space group of I4m2 lies above the convex hull, therefore, is metastable with respect to decomposition to P6 3 /mmc-SnH 4 and C2/c-H 2 . Between 150 to 300 GPa, we predict stable phases of H 2 , SnH 4 , SnH 8 , SnH 12 , SnH 14 and Sn 23 . Some metastable forms of SnH 6 , SnH 9 and SnH 16 are also shown in Fig. 1(a) by open symbols. SnH 4 is thermodynamically stable at pressures above 108 GPa as was predicted in previous report 13 . It goes through a phase transition at 160 GPa. Upon increasing pressure, at 220 GPa we predict stabilization of SnH 8 . SnH 12 and SnH 14 reach stability at the pressures of 250 GPa and 280 GPa, respectively, and remain stable at least up to 300 GPa. The structures of SnH n compounds are found to be dynamically stable within pressure ranges of their stability. In the I4m2-SnH 8 structure, Sn atoms occupy the 2a Wyckoff site and the H atoms are on the 4e, 8i and 4f sites (detailed structural information is provided in Table 1).
We checked the effects of zero-point energy using phonon calculations 25 at 250 GPa. The inclusion of zero-point noticeably lowered the formation enthalpy of SnH 8 with respect to SnH 4 and H 2 ( Fig. 1(a)), implying that this compound can be formed at lower pressure. At the same time, SnH 12 moves above the convex hull at 250 GPa, suggesting that higher pressure is needed to stabilize C2/m-SnH 12 .
In accord with what we expect, zero-point energy does not change the topology of the phase diagram, but shifts transition pressures.
In I4m2-SnH 8 structure, the H atoms are split into two categories. Some H atoms form H 3 groups, which were previously observed in solid phases of BaH 6 27 , in an unstable structure of H 5 Br ([H 3 ]Br[H 2 ]) 28 , and in an intriguing linear form with the bond length of 0.92 Å in H 5 Te 2 29 . In contrast to H 5 Br, which has approximately an equilateral triangle form of H 3 , here we report the formation of H 3 in a bent geometry with the angle of 146.2° and bond length of 0.86 Å at 220 GPa in the I4m2 structure. The other type of H atoms form H 2 groups which are aligned parallel to each other.
I4m2 structure can be presented as Sn[H 2 ][H 3 ] 2 as shown in Fig. 2(a,b). The bond length in H 3 unit is 0.86 Å, whereas H 2 has a longer bond length of 0.87 Å. Contrary to isolated H 2 molecule, which only has a filled σ bonding orbital, in the H 2 and H 3 semi-molecules, population of anti-bonding orbitals causes lengthening and  Analysis of electron localization function (ELF) shows a high ELF value of 0.88 between H atoms within the unit, indicating strong covalent bonding features (Fig. 2(e)). At the same time, the ELF value on the Sn-H bond is very low, reaching just 0.37.
In C2/m-SnH 12 , intriguing formation of novel H 4 semi-molecules are observed; at 250 GPa, they can be represented as two H 2 -groups separated by just 0.99 Å. Higher pressure of 300 GPa decreases the distance to 0.88 Å, leading to a strong covalent bond in the − H 4 units. Fig. 2(f). demonstrates covalent bonds in the linear H 4 units with the ELF magnitude of 0.85.
The calculated phonon dispersion curves and phonon density of states for I4m2 structure of SnH 8 at 220 GPa are shown in Fig. 3(a). Dynamical stability is clearly evidenced by the absence of any imaginary frequencies in the whole Brillouin zone. The low-frequency bands below 250 cm −1 are mainly from the vibrations of Sn atoms. Modes between 300 and 1700 cm −1 are mainly from the H-H wagging vibrations, and higher frequency vibrations above 2300 cm −1 are due to H-H stretching vibrations in H 2 and H 3 units.
Low-frequency translational modes, mostly from Sn atom, contribute 23.7% (9.2%) to the total λ. Intermediate H-H wagging vibrations make a significant contribution of 65.7% (67.9%), and the rest is from stretching H vibrations, which contribute 10.6% (22.9%) for SnH 8 (SnH 12 ). This is different from superconductivity in Cmcm-H 2 Br 28 , where Br translational modes make the largest contribution to the total λ and similar to the R3 m-H 4 Te 29 and P4/mmm-BaH 6 27 , where medium-frequency H-wagging and bending modes contribute the most to the EPC. In accord with our expectation, λ increases almost linearly with hydrogen content, where we found 60.2%, 72.2% and 77.1% contribution of H vibrations to the total λ of SnH 4 , SnH 8 and SnH 12 , respectively. This highlights the dominant role of H in the superconductivity of H-rich compounds.
Electronic band structure of I4m2-SnH 8 is depicted in Fig. 4. Occurrence of flat and steep bands near the Fermi level has been suggested as a condition for enhancing electron-phonon coupling (EPC) and the formation of Cooper pairs.
We can calculate T c based on the spectral function α 2 F(ω) and taking advantage of Allen-Dynes modified McMillan equation (Eq. 1.) by using Coulomb pseudo-potential μ * of 0.10 and 0.13 as widely accepted values (see Table 2). At 220 GPa, the predicted T c values for I4m2-SnH 8 are 81 K and 72 K using μ * values of 0.10 and 0.13, respectively. The calculated T c slightly decreases with pressure (82 K at 200 GPa and 79 K at 300 GPa using μ * = 0.10) with a pressure coefficient of − 0.023 K/GPa ( ) dT dP c . Reported λ is comparable to H 3 Se (λ = 1.09) at 200 GPa 11 , but in I4m2-SnH 8 structure, we have a lower ω log of 919 K (1477 K for H 3 Se), resulting in a lower T c value.
In conclusion, we explored the energetically stable/metastable high-pressure phases of the Sn-H system in detail by means of ab initio evolutionary structure prediction. The results demonstrate that SnH 8 , SnH 12 and SnH 14 , reported for the first time in this work, are thermodynamically stable compounds that coexist stably with solid Sn, H 2 and SnH 4 in a wide pressure range starting from 220 to at least 300 GPa.

Methods
To find stable and low-enthalpy metastable structures, we took advantage of evolutionary algorithm implemented in USPEX code [30][31][32] , which has been extensively used to predict stable crystal structures with just a knowledge of the chemical composition and without any experimental information 15,33-35 . In this method, the initial generation of structures and compositions is produced with the random symmetric algorithm 34 , and subsequent generations are produced by carefully designed variation operators. In order to find all stable stoichiometric compounds and the corresponding stable and metastable structures in the Sn-H binary system, we used VCES method implemented in USPEX 30,31 .
Structure relaxations were carried out using VASP package 36 in the framework of density functional theory (DFT) and using PBE-GGA (Perdew-Burke-Ernzerhof generalized gradient approximation) 37 . The projector-augmented wave approach (PAW) 38 was used to describe the core electrons and their effects on valence orbitals. The plane-wave kinetic energy cutoff was chosen as 1000 eV for hard PAW potentials, and we used Γ -centered uniform k-points meshes to sample the Brillouin zone.
Phonons and thermodynamic properties of Sn-H compounds are calculated using the PHONOPY package 25,39 . The supercell approach is used with supercell dimensions greater than 10 Å (typically 3 × 3 × 3 replication of the primitive cell). We used valence electron configurations of 4d 10 5s 2 5p 2 and 1s 1 for tin and hydrogen, respectively. Phonon frequencies and electron-phonon coupling (EPC) coefficients are calculated using DFPT as implemented in the Quantum ESPRESSO (QE) code 40 . In the QE calculations, we employed ultrasoft pseudopotentials and PBE-GGA exchange-correlation functional 37 . A plane-wave basis set with a cutoff of 70 Ry gave a convergence in energy with a precision of 1 meV/atom. The EPC parameter was calculated using 4 × 4 × 3, 5 × 5 × 4 and 5 × 5 × 4 q-point meshes for I4m2-SnH 8 , C2/m-SnH 12 and C2/m-SnH 14 , respectively. Denser k-point meshes, 8 × 8 × 6, 10 × 10 × 8 and 10 × 10 × 8 were used for convergence checks for the EPC parameter λ. The superconducting T c , was estimated using the Allen-Dynes modified McMillan equation 41 : where ω log is the logarithmic average frequency and μ * is the Coulomb pseudopotential, for which we used 0.10 and 0.13 values, which often give realistic results. The EPC constant and ω log were calculated as: