Pressure-stabilized superconductive yttrium hydrides

The search for high-temperature superconductors has been focused on compounds containing a large fraction of hydrogen, such as SiH4(H2)2, CaH6 and KH6. Through a systematic investigation of yttrium hydrides at different hydrogen contents using an structure prediction method based on the particle swarm optimization algorithm, we have predicted two new yttrium hydrides (YH4 andYH6), which are stable above 110 GPa. Three types of hydrogen species with increased H contents were found, monatomic H in YH3, monatomic H+molecular “H2” in YH4 and hexagonal “H6” unit in YH6. Interestingly, H atoms in YH6 form sodalite-like cage sublattice with centered Y atom. Electron-phonon calculations revealed the superconductive potential of YH4 and YH6 with estimated transition temperatures (Tc) of 84–95 K and 251–264 K at 120 GPa, respectively. These values are higher than the predicted maximal Tc of 40 K in YH3.

, ScH 3 8 , YH 3 9 , GaH 3 10 , H 2 S 11 et al.) at high pressures with predicted T c ranging from 17 to 86 K. Remarkably, our previously prediction of high T c (80 K at 160 GPa) in H 2 S 11 has been proven recently from experiments 12 . Recently, a new type of hydrogen solvated molecular complex SiH 4 (H 2 ) 2 has been synthesized 13,14 and was predicted to have a T c of ~100 K at 250 GPa 15 , which is much higher than the 17 K observed in SiH 4 2 .The experimental and theoretical studies have led to further investigations on hydrides with large hydrogen fraction that may provide a pathway to better superconductors. Using first-principle structure predictions, Zurek et al. 16 first predicted three new lithium hydrides (LiH 2 , LiH 6 and LiH 8 ) above 130 GPa, which are stabilized by charge transfer from Li to H. Subsequent studies have revealed a number of hydrides with large hydrogen fractions at high pressures, such as Na-H 17 , K-H 18,19 , Rb-H 20 , Cs-H 21 , Ca-H 22 , GeH 4 -H 2 23 and H 2 S-H 2 24 . Remarkably, some hydrides were predicted to possess high T c , e.g., ~82 K in LiH 6 (300 GPa) 25 , ~70 K in KH 6 (166 GPa) 18 and strikingly ~235 K in CaH 6 (150 GPa) 22 .
Among the different hydrides, yttrium hydrides (YH n ) are of special interest because each Y atom has three valence electrons and, in principle, could be shared with three H atoms. Experimentally, that during the continuous absorption of H atoms, a reversible transition of YH n between the reflecting YH 2 and optically transparent YH 3 was observed. This interesting phenomenon offers a great potential for practical application as a "switchable mirror". Raman 26 and infrared 27 studies found that the semiconducting YH 3 transforms to a metallic fcc structure above 10 GPa. Singnificantly, fcc-YH 3 was predicted to be a superconductor with T c of 40 K at 17.7 GPa, the lowest reported pressure for hydrides to date 9 . The prediction, however, has not been confirmed by experiment.
At high pressure, it is expected that the valence electronic state of Y atom will change and therefore provides a possibility of bonding with more H atoms. Here, we focus on the formation of Y hydrides with larger H concentration at high pressures of YH n (n = 2, 3,4,5,6,8). Two new thermodynamically stable

Results
The enthalpies of the candidate structures of YH n found in structure predictions relative to the products of dissociation into Y + solid H 2 and YH 3 + solid H 2 at selected pressures are summarized in Fig. 1 (a,b), respectively. Fig. 1 (a) shows all the stoichiometries considered here possess negative formation enthalpies with respect to Y + solid H 2 . Among those, YH 3 has the lowest energy. Two thermodynamically stable polymorphs, YH 4 and YH 6 , are to be thermodynamically more stable than the decomposition into YH 3 + sold H 2 were found at 140 and 160 GPa (Fig. 1b). Although YH 2 , YH 5 and YH 8 have negative formation enthalpies with respect to Y + solid H 2 , they are expected to decompose at all pressure. For example, YH 2 decomposes into YH 3 + Y since the enthalpy is above the tie-line connecting YH 3 and Y. Similarly, YH 5 and YH 8 decompose into YH 4 + YH 6 and YH 6 + H 2 , respectively (Fig. 1b). Therefore,YH 2 , YH 5 and YH 8 are excluded in the discussions hereafter.
So far, YH 3 is the only experimentally known yttrium hydrides at high pressure. The structure search readily reproduced the observed fcc structure 26,27 at 100 and 150 GPa. As can be seen from Fig. 2 (a), there exist only one type of Y atom occupying the fcc site and two nonequivalent, H1 and H2, atoms at the octahedral and tetrahedral sites. The H-H separation is of 1.9 Å at 120 GPa, clearly indicates no interaction between the H atoms (Fig. 2d). Therefore, the H atoms in fcc-YH 3 are monoatomic. fcc-YH 3 was previously predicted to be stable in a large pressure region of 20 GPa 28 to 197 GPa 29 and undergo a superconductor -metal -superconductor transition under pressure 9 . Figure 1 (c,d) show the formation enthalpies of YH 4 and YH 6 with respect to YH 3 + solid H 2 as functions of pressure. The formation enthalpy of YH 4 becomes negative relative to YH 3 + solid H 2 near 128 GPa (Fig. 1c). It is important to include the quantum nuclear zero-point energies (ZPE) when considering the energetics of systems containing light atoms. We therefore calculated the ZPEs of YH 4 , YH 3 and H 2 phases within the quasi-harmonic approximation. When ZPE corrections are included, the predicted pressure for the onset of stability of YH 4 is lowered to 112 GPa (inset in Fig. 1c). The stable YH 4 has a tetragonal structure (space group I4/mmm, denoted tI10 hereafter, Fig. 2b) with two formula units per unit cell. We found that the tI10-YH 4 has the same structure type with tI10-CaH 4 22 . The tI10 structure at 120 GPa consists of body-centered arranged Y atoms and two nonequivalent H1 and H2 atoms with H1-H2 and H2-H2 distances of 1.58 and 1.33 Å, respectively. Valence electrons localization was found between the two neighbouring H2 atoms while absent between H1 and H2 atoms (Fig. 2e). This indicates the presence of both molecular "H 2 " and monoatomic H in tI10-YH 4 .
YH 6 becomes thermodynamically more stable than YH 3 + solid H 2 above 122 GPa (Fig. 1d). The onset pressure of the stability of YH 6 is reduced to 110 GPa when considering the ZPE effect (inset in Fig. 1d). In the thermodynamically stable pressure region, YH 6 adopts a cubic structure with space group Im-3m (2 f.u./unit cell, denoted cI14 hereafter, Fig. 2c). The "H 6 " hexagons are forming a corner-shared sodalite-like cage with a Y atom at the center, the same sodalite structure found in CaH 6 above 150 GPa 22 . In this case, the H-H distance of 1.31 Å at 120 GP is longer than in CaH 6 . Despite the longer distance, covalent interaction between H atoms is clearly visible from the localized valence electrons between the H atoms ( Fig. 2f).

Discussion
We found three types of H species in YH n compounds, monatomic H in YH 3 , monatomic H+molecular "H 2 " in YH 4 and hexagonal "H 6 " in YH 6 . Since molecular H 2 has a filled covalent σ bond, the additional electrons donated from Y will occupy the antibonding σ* bond, resulting in a stretched or even dissociated H-H bond. For example, the formation of YH 3 can be described by the reaction 2Y + 3H 2 → 2YH 3 . Assuming all 6 valence electrons (3 from each Y atom) were transferred to the H 2 , then, each H 2 would accommodate two additional electrons into the σ* orbital and, thus, breaking the H 2 molecule into monatomic H. Integration of the electron density shows that each H1 (H2) atom in fcc-YH 3 have accommodated an additional 0.54 (0.47) electrons. A similar description can be used for the formation of YH 4 . In this the reaction is Y + 2H 2 → YH 4 . There are three electrons available to two H 2 molecules. Therefore, one H 2 bond is completely broken into two monoatomic H and the remaining electron occupied the σ* one the second H 2 , thereby weakening the bond and resulted in a longer H-H distance of 1.33 Å. The additional charge of the monatomic H1 in tI10-YH 4 is calculated to be 0.42 electrons. Only 0.29 electrons were added to H2 in tI10-YH 4 . Finally, the formation of YH 6 can be described as Y + 3H 2 → YH 6 . In this case only, one electron is added to each H 2 and the H-H bond is elongated to 1.31 Å, in close agreement with the "molecular" H 2 in YH 4 . In cI14-YH 6 , each H atom has accepted 0.25 electrons, which is not enough to dissociate the H 2 molecule.
A previous study 22 has shown that 4s-3d charge transfer turns Ca from s-dominant into s-d dominant at high pressure, similar to the electronic configuration of Y atom. Therefore, the presence of same structure types in YH 4 (YH 6 ) and CaH 4 (CaH 6 ) is not accidental. However, the H-H distance of "H 2 " molecule in YH 4 (1.33 Å) is much longer than that (0.81 Å) in CaH 4 at 120 GPa as Y transfer one more valence electron to H 2 than Ca resulting in a longer H-H distance in YH 4 (YH 6 ). The empirical consideration is support from quantitative calculations of the difference of the electron density of tI10-YH 4 to that of a hypothetical structure consisting only H sublattice. It is clearly shown in Fig. 3(a,b), that there is no electron density (covalent bond) between the two H2 atoms in the pure H structure. However, when the Y atoms were present, localized electrons are found between two H2 atoms. Therefore, the charge transfer from Y to H2 is responsible to the formation of molecular "H 2 " in YH 4 . Similarly, the foramtion of "H 6 " hexagons in YH 6 results from the accommodation by H of excess electrons from Y atom (Fig. 3c,d). (iii) strong Y-H hybridization derived from the significant overlap of Y-and H-DOS. Note that in a preivous study 9 it was demonstrated that the Y-H hybridization is responsible for the superconductivity in YH 3 . Figure 5 shows the calculated phonon dispersions, phonon density of states (PHDOS), Eliashberg spectral function (α 2 F(ω)/ω)and EPC integrated (λ(ω))for tI10-YH 4 and cI14-YH 6 at 120 GPa. The absence of any imaginary phonon modes proves the dynamical stabilities of both compounds. As expected, both phonon spectra are separeted into two frequency regions, with the low frequencies (<10 THz) dominated by the vibrations of Y atom while the high end of the spectra by H atoms. In tI10-YH 4 , the resulting EPC parameter λ is 1.01 at 120 GPa, which is comparable to the maximum value (~1.4 9 ) predicted for fcc-YH 3 . Note that the low-frequency vibrations contribute to 18% of the total λ while the ramaining 82% comes from H vibrations. Circles with radius proportional to the EPC were also plotted in Fig. 6 to illustrate the contributions associated with different phonon modes. One can observe that nearly all phonon modes contribute to the overall λ, reflecting a three-dimensional nature of the structure.
Surprisingly, according to the calculation, the EPC parameter λ of cI14-YH 6 reaches 2.93 at 120 GPa, even larger than that (2.69 at 150 GPa 22 ) in cI14-CaH 6 . However, the Eliashberg phonon spectral functions of cI14-CaH 6 and cI14-YH 6 are quite different. The EPC in cI14-CaH 6 was derived primarily from the two phonon modes (T 2g and E g ) at the zone center Γ point. However, we observed an overall contribution of different modes to λ along N-P-Γ-N directions. Moreover, 90% of the total λ is contributed by H vibrations. The superconductivity in YH 6 is associated with the Kohn anomalies observed in the phonon dispersion of the phonon branch Γ-H and H-N. The calculation of the nesting function (Fig. 5c) confirms this expectation and clearly show strong nesting along Γ-H and H-N directions. Compare to the other 5 bands crossing the Fermi level, the Fermi surface of strongly nested band (Fig. 6c) shows a complex "vase"-like topology with strong nesting along Γ-H.
T c was estimated from the spectral function (α 2 F(μ)) by numerically solving the Eliashberg equations 31 with typical choice of Coulomb pseudopotential μ* = 0.1−0.13. The Coulomb repulsion is taken into account in terms of the μ* scaled to a cutoff frequency 32 . At 120 GPa, the calculated T c is 84-95 K for tI10-YH 4 , much higher than the maximal 40 K predicted for fcc-YH 3 9 . Note that tI10-YH 4 has a much larger logarithmic average frequency of 1119 K than fcc-YH 3 (350 K 9 ) due to the presence of molecular "H 2 ", which helps to enhance the superconductivity. For cI14-YH 6 , T c value of 251-264 K was estimated. This value is comparable to the predicted T c (220-235 K at 150 GPa) in CaH 6 22 . Although, in principle, there is no upper limit to the T c value within the Midgal-Eliashberg theory, remarks on the very high T c value of cI14-YH 6 must be view with caution. The EPC calculations were based on the harmonic approximation and without the consideration of electron correction effects. A previous study 33 had shown that anharmonicity of atomic motion may reduce or even suppress the superconductivity of AlH 3 due to the renormalization of the lower vibration modes by anharmonicity 34 . However, this suggestion is contrary to the observation that anharmonic vibraions will significantly enhance T c in case of disordered compounds 35 . Another important, but often neglected, situation is that the Fermi level topology may be altered in improved electronic band structure including corrections to self interaction and electron correlation effects. In AH 3 , the parallel bands favouring nesting disappeared in the GW calculated band structure 36 . Here, GW band struture calculations were performed for tI10-YH 4 and cI14-YH 6 . No significant change in the band structures, particularly for the bands near or crossing the Fermi level, was found in both case (Fig. 4). Therefore, the discussion presented above will still be valid and we expect YH 4 and YH 6 are good superconductors.

Methods
Structure predictions for YH n were performed using the particle swarm optimization technique implemented in the CALYPSO code 37,38 . In recent studies, it was shown that the approach was successful on the prediction of high pressure structures on both elemental and binary compounds, such as N 33 , Ca-H 22 , H 2 S 11 and BeH 2 39 . In this work, systematic structure search were performed on six stoichiometries (YH 2 , YH 3 , YH 4 , YH 5 , YH 6 and YH 8 ) at 100 and 150 GPa. Model cells up to 4 formula units (f.u.) for each stoichiometry were used. The structure search was considered converged when~1000 successive structures were generated after a lowest energy structure was found.
ab initio structure relaxations were performed using density functional theory within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) as implemented in the Vienna ab initio simulation package (VASP) 40 . The band structures were calculated with both PBE-GGA and GW methods 41 . The GW interpolated band structures were computed using WANNIER90 42 . The all-electron projector augmented wave (PAW) 43 method was adopted with 1s and 4s 2 4p 6 4d 1 5s 2 treated as valence electrons for H and Y, respectively. An energy cutoff of 700 eV and a Monkhorst-Pack Brillouin zone sampling grid with a resolution of 0.5 Å −1 were used in the structure searches. Selected low energy structures were then re-optimized with a denser grid better than 0.2 Å −1 and a higher energy cutoff of 1000 eV. Phonon dispersion and electron-phonon coupling (EPC) calculations were performed with density functional perturbation theory using the Quantum-ESPRESSO package 44 . Norm-conserving pseudopotentials for Y and H were considered with a kinetic energy cutoff of 140 Ry. 8 × 8 × 8 (59 q-points) and 10 × 10 × 10 (47 q-points) q-meshes in the first Brillouin zones were used in the EPC calculations for YH 4 and YH 6 , respectively. Monkhorst-Pack grids of 32 × 32 × 32 and 40 × 40 × 40 were used to ensure k-points sampling convergence with Gaussians of width 0.03 Ry for YH 4 and YH 6 , respectively, in order to approximate the zero-width limit in the calculations of the EPC parameter, λ.

Conclusion
In conclusion, structure predictions have demonstrated that yttrium atom can react with more than three hydrogens under pressure. Two high-hydride phases, YH 4 and YH 6 , were predicted to be thermodynamically stable relative to YH 3 and H 2 above 110 GPa. At the stable pressure ranges, YH 4 has a bct strucure containing both moniatomic H and molecular "H 2 " while YH 6 adopted a bcc structure with a H sodalite-like cage. Electron-phonon coupling calculations show that both YH 4 and YH 6 are supercodncutive with T c higher than YH 3 . The results presented here support the suggestion that compressing the mixture of elements (compounds) and hydrogen is a way to search high-temperature superconductors. In addition, in principle, YH 4 and YH 6 can be synthesized by compressing the mixture of YH 3 and H 2 above 110 GPa.