High-temperature Superconductivity in compressed Solid Silane

Crystal structures of silane have been extensively investigated using ab initio evolutionary simulation methods at high pressures. Two metallic structures with P21/c and C2/m symmetries are found stable above 383 GPa. The superconductivities of metallic phases are fully explored under BCS theory, including the reported C2/c one. Perturbative linear-response calculations for C2/m silane at 610 GPa reveal a high superconducting critical temperature that beyond the order of 102 K.

383 GPa, see Figure 1 inset. The C2/c phase predicted in this work is identical with the one proposed by C. J. Pickard and R. J. Needs 8 , which forms three-dimensional networks. Above 383 GPa, P2 1 /c phase take over C2/c phase and becomes most competitive on enthalpy. As shown in Figure 2(a), Si atoms show a fold layered arrangement with H atoms site around them. There are four SiH 4 units with five unequivalent atoms in the conventional cell. One Si atom and four H atoms occupy the crystallographic 4e position with 1 symmetry in this monoclinic crystal. The shortest distance between H atoms is 1.045 Å at 400 GPa, which is longer than the 0.762 Å of the H-H bond length in ''H 2 '' unit in the Cmca-12 structure 13 of solid hydrogen at the same pressure. P2 1 /c phase keep stable on enthalpy at least to 606 GPa until another competitive phase with C2/m symmetry appears, as shown in Figure 1. C2/m phase obtains monoclinic base-centered lattice, and contains six formula units in the conventional cell. There are three unequivalent Si atoms occupy the 4i position with m symmetry, whereas nine H atoms sit on the 8j (site symmetry is 1), 4i (site symmetry is m) and 4g (site symmetry is 2) positions respectively. The shortest distance between two H atoms is 1.009 Å , and slightly shorter than 1.045 Å in the P2 1 /c phase. The coordination of Si atoms in C2/c, P2 1 /c, and C2/m are all eleven, i.e. each Si atom bond with eleven H atoms, which is quite different with the molecule crystal under low pressure 14 and implies some different physical characters. Detail parameters of the structures are listed in Table 1. The decomposition enthalpies reference to Cmca-12 13 structure of H 2 (below 500 GPa), I4 1 /amd 13,15 structure of H 2 (500-1000 GPa), Fm-3m 16 structure of Si and C2/c 17 structure of SiH 3 can also be seen in Figure 1 (The comparison with SiH 4 (H 2 ) 2 are shown in supplementary information). It is noteworthy that the enthalpies of the three structures are lower. Moreover, P2 1 /c and C2/m structures are found to be energetically much superior to previous structures [7][8][9]11,12 . Therefore, three monoclinic (C2/c, P2 1 /c, and C2/m) phases can be taken as energetically stable structures of silane under high pressure range, see Figure 1.
The mechanical stability of structure can provide insight into the stability of materials. To evaluate the mechanical stability of the C2/c, P2 1 /c and C2/m phases, elastic constants have been calculated and listed in Table 2. According to the mechanical stability criteria, the crystal deformation energy is positive, this means the determinants of elastic constants matrix C ij should be positive 18 . Considering the crystal symmetry, the mechanical stability of expression will be further simplified 19 . It can be found that the elastic constants of these three structures satisfy the mechanical stability criteria, indicating that these three structures are mechanically stable. The phonon band structure and projected phonon density of states (PHDOS) of the three phases at selected pressures are presented in Figure 3. Absence of any imaginary frequency in the Brillouin zone establishes the dynamical stability. The PHDOS of these three structures shows that the heavier Si atoms dominate the low-frequency vibrations, and the lighter H atoms contribute significantly to the high-frequency modes.
To analysis the electronic properties of C2/c, P2 1 /c and C2/m phases, we first calculated the electronic density of states (DOS). As shown in Figure 4, they are all metals with large total DOS at Fermi level. Specially, for C2/m structure, the total DOS of Fermi level is significantly higher than the other structures. These high DOS values might favor the superconducting behavior. From the Figure 4(a-c), H atoms contribute more to DOS than the Si atoms below Fermi level, and contribute less to DOS than the Si atoms above Fermi level. With increasing pressure, the contributions from atoms H and Si do not significantly change. The electron localization functions (ELF) 20 of the C2/c, P2 1 /c, and C2/m phases are calculated at 300 GPa, 400 GPa, and 610 GPa, respectively. The isosurface plots at ELF 5 0.5 are shown in Figure 4(d-f). The electron-gas-like  The electron-phonon coupling strength (EPC) l and the logarithmic average phonon frequency v log of the three structures were calculated to explore the possible superconductivity of SiH 4 . The Eliashberg phonon spectral function a 2 F(v) and the l as a function of frequency are shown in Figure 5. The l of C2/c structure at 300 GPa, P2 1 /c structure at 400 GPa and C2/m structure at 610 GPa are 0.69, 0.66 and 1.18, respectively. The superconducting critical temperature can be estimated from the Allen-Dynes modified McMillan equation 21 T c~v which has been found to be highly accurate for many materials with l , 1.5. The Coulomb pseudopotential m* is taken as 0.13 for hydrogen dominant metallic alloys by Ashcroft 22 , resulting that the estimated T c of C2/c, P2 1 /c and C2/m at 300 GPa, 400 GPa, and 610 GPa are 29.65 K, 31.57 K and 106.31 K, respectively. Subsequently, the contributions to EPC l of each atom are analyzed. As for C2/c, P2 1 /c and C2/m, Si vibrations provide a contribution of 36%, 34%, and 37% respectively, while the H translational vibrations contribute for nearly 64%, 66% and 63% respectively. The result shows that the element H plays a significant role in the EPC l.
The trend of T c with pressures for these three structures was explored for further investigation. T c , v log and l along with the changes of pressures are displayed in Figure 6. With increasing pres- where M is the atomic mass. The DOS at the Fermi level N( F ), the square of the electron-ion matrix element AEI 2 ae and the average pho-  We also realized that T c magnitude of C2/m is beyond 10 2 K which is higher than other two phases that promoted us to explore the underlying superconducting mechanism. From Figure 6, the  trend of l along with pressures is consistent with T c . So, l played an important role in the superconducting critical temperature. The calculated EPC l of C2/m phase at 610 GPa is 1.18, which is much higher than the values of 0.69 and 0.66 of two phases. The bigger EPC l can directly contribute to higher T c . Furthermore, formula (2) defined by MaMillan's strong coupling theory can be used to analyze the mechanisms. Comparing the contribution to l in the phase of C2/m and C2/c, Figure 6 reveals that N( F ) of C2/m contributes about four times than the one of C2/c, meanwhile the two factors of v 2 1 2 and AEI 2 ae together contribute about half of C2/c. Although there are two disadvantages of parameters in C2/m according formula (2), the higher N( F ) sustains the higher l as a whole. As for C2/m and P2 1 /c, there are no obvious differences on the values of v 2 1 2 and AEI 2 ae, and l can be dominated by N( F ) as well. As shown in Figure 6, N( F ) in C2/m is much higher than the ones in other two structures. Based on an overall analysis of contributions from parameters in two formulas including the contribution from v log , N( F ) play an important role to the much higher T c than other phases.
It is consistent with the calculated three-dimensional Fermi surface which is shown in Figure 5. There are four electronic bands across the Fermi surface for all the three phases, and identified by different colors. As shown in Figure 5, the C2/m phase is filling with much more Fermi surfaces in the brillouin zone than the two other phases, which is corresponding with the high level of N( F ). Therefore, the Fermi surface also support the conclusion that the high values of N( F ) is significant to sustain high T c .

Conclusion
In summary, we have extensively investigated crystal structures and superconductivity of silane. The P2 1 /c, and C2/m structures are found which are thermodynamically, mechanically, and dynamically stable. The superconducting critical temperature T c of the C2/c phase at 300 GPa and P2 1 /c at 400 GPa are 29.65 K and 31.57 K. The superconductivity of the C2/m phase with a transition temperature 106.31 K is found to be mainly attributed to the strong electronphonon coupling due to the high electronic density of states at the Fermi level.

Methods
The most stable structures of silane under high pressures were performed using evolutionary algorithm, as implemented in the USPEX code 28 . This approach has been successfully used in the study of many materials at high pressures 29,30 . The structural relaxations have been performed within the framework of the generalized gradient approximation (GGA) 31 with the Perdew-Burke-Ernzerhof parameterization for the exchange-correlation functional to DFT by using the projector augmented-wave(PAW) method 32 , as implemented in ab initio VASP code 33 . The structures were relaxed at a high cutoff energy of 1000 eV, and a Brillouin zone sampling grid of spacing 2p 3 0.03 Å 21 . The lattice dynamics and electron-phonon coupling have been computed with QUANTUM-ESPRESSO 34 . We used Vanderbilttype ultrasoft pseudopotentials 35 with a cutoff energy of 40 Ry. Phonon frequencies were calculated based on the density functional linear-response method 36 . A Monkhorst-Pack(MP) 37 Brillouin zone sampling grid of spacing 2p 3 0.025 Å 21 with Gaussian smearing of 0.02 Ry were used for the phonon calculations, at 4 3 4 3 4, 5 3 4 3 3 and 3 3 3 3 2 q-point mesh for C2/c, P2 1 /c and C2/m for the electronphonon interaction matrix element, respectively. All the convergences of the planewave basis set and MP sampling are carefully examined by employing higher kinetic energy cutoffs and denser grids sets. The tests of cutoff energy and the validity of potential functions are shown in supplementary information.