Exploration of stable compounds, crystal structures, and superconductivity in the Be-H system

Using first-principles variable-composition evolutionary methodology, we explored the high-pressure structures of beryllium hydrides between 0 and 400 GPa. We found that BeH$_2$ remains the only stable compound in this pressure range. The pressure-induced transformations are predicted as $Ibam$ $\rightarrow $ $P\bar{3}m1$ $\rightarrow $ $R\bar{3}m$ $ \rightarrow $ $Cmcm$ $ \rightarrow $ $P4/nmm$, which occur at 24, 139, 204 and 349 GPa, respectively. $P\bar{3}m1$ and $R\bar{3}m$ structures are layered polytypes based on close packings of H atoms with Be atoms filling octahedral voids in alternating layers. $Cmcm$ and $P4/nmm$ structures have 3D-networks of strong bonds, but also feature rectanular and squre, respectively, layers of H atoms with short H-H distances. $P\bar{3}m1$ and $R\bar{3}m$ are semiconductors while $Cmcm$ and $P4/nmm$ are metallic. We have explored superconductivity of both metallic phases, and found large electron-phonon coupling parameters of $ \lambda $=0.63 for $Cmcm$ (resulting in a $T_c$ of 32.1-44.1 K) at 250 GPa and $ \lambda $=0.65 for $P4/nmm$ ($T_c$ = 46.1-62.4 K) at 400 GPa. The dependence of $T_c$ on pressure indicates that $T_c$ initially increases to a maximum of 45.1 K for $Cmcm$ at 275 GPa and 97.0 K for $P4/nmm$ at 365 GPa, and then decreases with increasing pressure for both phases.


I. INTRODUCTION
The search for new high-temperature superconductors has attracted great enthusiasm in both fundamental and applied research. Owing to its low mass and high electron density, "metallic hydrogen" has been predicted to possess a high superconducting transition temperature (T c > 200 K) [1][2][3] . However, hydrogen remains insulating at extremely high pressure (> 320 GPa 4 ), which are too high for any applications. Another feasible method of obtaining the properties of metallic hydrogen is to form hydrogen-rich alloys with other elements 5 . Due to "chemical precompression", the pressure of metallization may be reduced significantly.
Inspired by the elusive state of matter, theoretical and experimental research has made considerable progress towards exploring superconductivity in hydrogen-rich compounds, e.g.
for group IVa hydrides, calculations predicted that SiH 4 6-8 , GeH 4 9,10 , SnH 4 11,12 and PbH 4 13 may become superconductors at high (yet lower than pure H) pressure. The origin of highpressure superconductivity can be derived from the particular "H 2 " units, which are a feature common to hydrides of alkali metals 14 , alkaline earth metals 15,16 and group IVa elements 9,12 .
Experiments suggested metallization of SiH 4 at ∼60 GPa 17 and its superconducting transition temperature (T c ) is 17 First-principles variable-composition evolutionary simulations were performed at 0, 50,   100, 150, 200, 250, 300 and 400 GPa using the USPEX code [29][30][31][32] , which has the capability of discovering possible stoichiometries and the corresponding stable and metastable structures at given pressure-temperature conditions, and has successfully predicted a large number of stable structures [33][34][35] . The initial generation of structures and compositions was produced randomly with the use of space groups picked randomly from the total list of 230 groups.
50% of the lowest-enthalpy structures were used as parents for the next generation. In addition, 20% of structures in each new generation were produced by lattice mutation, 15% by atomic transmutation and 15% were produced randomly. Each generation contained 60 structures and runs proceeded for up to 50 generations.
The underlying structure relaxations were carried out using the Vienna Ab-initio Simulation Package (VASP) code 36 , in the framework of density functional theory (DFT) 37,38 within the Perdew Burke Ernzerhof generalized gradient approximation (PBE-GGA) 39 . The frozen all-electron projected augmented wave approach (PAW) 40 was adopted to describe the core electrons and their effects on valence orbitals. A plane-wave kinetic energy cutoff of 600 eV and dense Monkhorst-Pack k-point grids 41 with a resolution higher than 2π×0.06Å −1 were used for all structures. The most stable structures were studied further at increased accuracy using a reciprocal-space grid better than 2π×0.03Å −1 .
Phonon calculations were carried out using the supercell approach as implemented in the PHONOPY code 42 . Electron-phonon coupling (EPC) calculations were explored using the pseudopotential plane-wave method within PBE-GGA, as implemented in the Quantum-Espresso package 43 . In these calculations, we used the kinetic energy cutoff of 60 Ry and Monkhorst-Pack k-point grids of 20×20×12 for the Cmcm phase and 16×16×8 for the P4/nmm phase with a Methfessel-Paxton 44 smearing factor of 0.05 Ry. Additionally, qmeshes of 5×5×3 for Cmcm and 4×4×2 for P4/nmm were used to calculate the electronphonon coupling matrix elements, respectively. We used the Allen-Dynes-modified McMillan equation 45 to estimate T c , as follows: where ω log is the logarithmic average frequency, λ is the electron-phonon coupling constant and µ * is the Coulomb pseudopotential, which is assumed to be between 0.10-0.13 5 .  We found a similar high-pressure structure in the B-H system 19  P3m1 to R3m (Fig. 3b). The structures in ref. 24 are metastable with respect to the P3m1 structure.
At 204 GPa, the layered R3m structure transforms into an orthorhombic Cmcm structure   predicted in literature. Note that the Allen-Dynes formula is expected to be reliable when is less than 1-1.5 49 , which is the case here.

IV. CONCLUSIONS
In summary, using variable-composition evolutionary simulations for crystal structure prediction, we investigated the high-pressure phases of solid beryllium hydrides in the pressure range of 0-400 GPa. BeH 2 is found to be the only stable beryllium hydride. The