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# Superconducting high-pressure phases of disilane

Contributed by Ho-Kwang Mao, April 20, 2010 (sent for review March 11, 2010)

## Abstract

High-pressure structures of disilane (Si_{2}H_{6}) are investigated extensively by means of first-principles density functional theory and a random structure-searching method. Three metallic structures with *P*-1, *Pm*-3*m*, and *C*2/*c* symmetries are found, which are more stable than those of XY_{3}-type candidates under high pressure. Enthalpy calculations suggest a remarkably wide decomposition (Si and H_{2}) pressure range below 135 GPa, above which three metallic structures are stable. Perturbative linear-response calculations for *Pm*-3*m* disilane at 275 GPa show a large electron-phonon coupling parameter *λ* of 1.397 and the resulting superconducting critical temperature beyond the order of 10^{2} K.

It is known that the highest superconductive critical temperature (*T*_{c}) found for a conventional superconductor is 39 K for MgB_{2} (1) at ambient pressure. Cuprate superconductors have much higher critical temperatures. The cuprate superconductor discovered has a critical temperature of 93 K (2), and mercury-based cuprates have critical temperatures in excess of 130 K. Pressure causes extraordinary changes in materials and modifies their properties. This often provides a path for synthesis of novel materials. Applying BCS theory to hypothetic metallic hydrogen, Ashcroft realized that it is a conventional superconductor with a very high-*T*_{c} (3). A *T*_{c} of the order of 10^{2} K was further proposed under very strong compression by quantitative calculations (4). This value compares favorably with those in cuprate superconductors. However, hydrogen remains insulating up to extremely high pressures, at least up to about 342 GPa (5).

It was recently predicted that group IVa hydrides would also present a high superconducting critical temperature, while becoming metallic at lower pressures due to chemical precompression (6). Theoretical (7–12) and experimental (13–15) studies of silane, and theoretical studies of germane (16) and stannane (17, 18), have investigated possible metallization and superconducting phase transitions at high pressures. Indeed, the theoretical studies on germane and stannane have predicted very high *T*_{c} of 64 K at 220 GPa (16) and 80 K at 120 GPa (17), respectively. These results sufficiently encouraged us to prompt studies on a wider range of hydrides to confirm the prediction (6). Disilane containing a large fraction (3/4) of H atoms is also an important hydrogen-rich compound and leads to interesting properties under high pressure. Furthermore, it is more readily available for experimental studies because of the higher boiling and melting points than silane, germane, and stannane. However, studies on disilane are very scarce.

Here, we have explored the crystal structures of disilane in a wide pressure range from 50 to 400 GPa, and three favored structures, i.e., *P*-1, *Pm*-3*m*, and *C*2/*c*, are found above 135 GPa. Remarkably, the large *T*_{c} of 80 K at 200 GPa for *P*-1 and 139 K at 275 GPa for *Pm*-3*m* are predicted by quantitative calculations. Up to now, the superconductive *T*_{c} of 139 K is much higher than other group IVa hydrides reported and comparable with those in cuprate superconductors.

## Results and Discussion

The XY_{4}-type hydrides in group IVa, i.e., methane (CH_{4}), silane (SiH_{4}), germane (GeH_{4}), and stannane (SnH_{4}), have been investigated extensively to uncover the high-pressure crystal structures and pressure-induced metallization in recent years, as mentioned above. Methane, the first candidate explored, is found to have no clues of metallization up to 400 GPa by both theoretical (26) and experimental (27) work. The disputed metallization of silane suggested that it might not metallize until much higher pressure above 200 GPa (28). A metallic monoclinic structure of germane above 196 GPa (16) and superconducting crystal structures of stannane above 96 GPa (18) have been reported recently. Below these pressures, germane and stannane are unstable with respect to elemental decompositions. In general, high pressure is needed to metallize group IVa hydrides, so 50 GPa is selected as a starting pressure point for investigating the possible superconducting high-pressure phases of disilane.

We carried out unconstrained searches at fixed pressures in the range from 50 to 400 GPa using one, two, three, four, and six SiH_{3} formula units (f.u.) per cell, and found many different relaxed structures. Fig. 1 shows our calculated enthalpies of the candidate structures including random-searching and XY_{3}-type ones, together with the decomposition (2Si + 3H_{2}) level (red dot line). At low pressures below about 135 GPa, *P*2_{1}/*c* and *P*-1 are the most favorable structures with the lowest enthalpies. However, in the range from 50 to 135 GPa, disilane with these two structures is unstable and decomposes into a Si and H_{2} mixture. A strikingly wide decomposition pressure range from 50 to 135 GPa is revealed. The *P*-1 disilane becomes stable above about 135 GPa, which is obviously energetically favored over the others until the pressure close to about 275 GPa. In the vicinity of 275 GPa, the *Pm*-3*m* disilane has the lowest enthalpies among these structures, about 4 meV/disilane lower than the *C*2/*c* one (see Fig. 1 *Inset*). Above 300 GPa, the *C*2/*c* disilane is the most favorable structure and about 9 meV/disilane lower than the *Pm*-3*m* one (see *Inset* of Fig. 1). Obviously, the three structures we suggested, i.e., *P*-1, *Pm*-3*m*, and *C*2/*c*, are the most favored structures of disilane under high pressures.

The favored structures of disilane obtained in this work are shown in Fig. 2, and the corresponding parameters at respective pressures are listed in Table 1. The *P*-1 phase consists of two f.u. (i.e., Si_{4}H_{12}) in a triclinic crystal lattice, as shown in Fig. 2*A*. There are eight unequivalent atoms in the primitive cell. The six H and four Si atoms occupy the crystallographic 2*i* sites. The calculated lattice parameters of this triclinic phase (shown in Table 1) suggest a low symmetry (space group *P*-1) at 175 GPa. The triclinic lattice with space group *P*1 was identified by an X-ray diffraction pattern of phase II of AlH_{3} (30) under high pressures, too. The two minima of Si-H bonds are 1.530 and 1.537 Å at 175 GPa, related to the unequivalent atoms H1, H6, and Si2 as listed in Table 1. As discussed below, these shortest bonds contribute the high-frequency vibrational modes of phonons. The *Pm*-3*m* disilane contains one SiH_{3} unit in each primitive cell as shown in Fig. 2*B*. There are two unequivalent atoms in the primitive cell. The H atom occupies the crystallographic 3*d* site and Si atom locates at the 1*b* site. The lattice parameters at 275 GPa are shown in Table 1. This structure is remarkably simple and completely different from the *P*-1 and *C*2/*c* phases of disilane. One silicon atom occupies the center of cubic cell, and three H atoms locate at the centers of edges of the cubic lattice. This is the well-known *Pm*-3*m* ReO_{3} or UO_{3} type structure. In the lattice of *Pm*-3*m* structure, H atoms are grouped in eight regular triangles in the corners of cube, which is characterized by the same distances (1.59 Å at 275 GPa) of the shortest H-H and Si-H bonds. The structure of *C*2/*c* disilane contains four f.u. in a monoclinic crystal lattice, as shown in Fig. 2*C*. The corresponding lattice parameters of this triclinic crystal at 300 GPa are listed in Table 1. There are four unequivalent atoms, including three H atoms and one Si atom, in the conventional cell occupying the crystallographic 8*f* sites, where the Si atoms have eightfold coordination with the average Si-H bond length of 1.554 Å.

The mechanical stability of structure provides a useful insight into the stability of crystals. The strain energy of a crystal must be positive against any homogeneous elastic deformations, i.e., the matrix of elastic constants *C*_{ij} must be positive definite (31). It should note that negative values are not prohibited for *C*_{ij} (32). To evaluate the mechanical stability of the three phases, elastic constants have been calculated and listed in Table 2. Obviously, the elastic constants of the three structures satisfy the mechanical stability criteria (31, 33), indicating that the three structures are mechanically stable.

The calculated electronic band structure and projected density of states (DOS) for *P*-1 phase at 175 GPa, *Pm*-3*m* phase at 275 GPa, and *C*2/*c* phase at 300 GPa reveal that these phases are metallic. The less dispersed valence and conduction bands near the Fermi level intensify the large electronic DOS at the Fermi level, which are 4.02, 5.96, and 2.80 states/spin/Ry/unit cell (if a unit cell contains a Si_{2}H_{6} f.u.) for *P*-1, *Pm*-3*m*, and *C*2/*c*, respectively. These high DOS values might favor the superconducting behavior. The charts of phonon band structure and the projected DOS of *P*-1, *Pm*-3*m*, and *C*2/*c* phases calculated at selected pressures are shown in Fig. 3. The absence of imaginary frequency modes indicates that these structures are stable dynamically. A direct result from PVDOS 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, as expected. Remarkably, the two branches of high-frequency modes, about 70 THz for *P*-1 disilane, are mainly due to the two minimum Si-H bond stretching vibrations (Fig. 3*A*). In the *C*2/*c* disilane, the variety of Si-H bond lengths contribute nearly continuous phonon frequencies by Si-H/Si-H-Si bond stretching/bending vibrations above 24 THz (Fig. 3*C*).

The analyses of lattice dynamic calculations at the zone-center ( point) phonons are useful for further high-pressure experiments. The phonons at point of the three phases can be classified by the irreducible representations of the point group *C*_{i}(*P*-1), *O*_{h}(*Pm*-3*m*), and *C*_{2h}(*C*2/*c*). The optical modes can be analyzed by the group theory. In *P*-1 phase, , where the *A*_{u} modes are IR active and *A*_{g} modes are Raman active. For the *Pm*-3*m* phase, , thereinto, the *T*_{1u} modes are IR active and *T*_{2u} modes are silent. In the crystal of *C*2/*c*, , where the *A*_{u}, *B*_{u} modes are IR active and the *A*_{g}, *B*_{g} are Raman active. The details of the calculated IR/Raman active frequencies are listed in Table 3.

To explore the superconductivity of these three phases we suggested, the EPC parameter *λ*, the logarithmic average phonon frequency (*ω*_{log}), and the Eliashberg phonon spectral function *α*^{2}*F*(*ω*) have been investigated at high pressures. The resulting *λ* for *P*-1 phase at 175, 200 GPa and *Pm*-3*m* phase at 275 GPa are 0.980, 1.089, and 1.397, respectively, indicating that the EPC is fairly strong. The calculated *λ* for *C*2/*c* phase at 300 GPa is 0.812, a little lower than the other ones. The theoretical spectral function *α*^{2}*F*(*ω*) and the integrated *λ*(*ω*) as a function of frequency at selected pressures are shown in Fig. 4. The superconducting critical temperature can be estimated from the Allen–Dynes modified McMillan equation (34) , which has been found to be highly accurate for many materials with *λ* < 1.5. The *ω*_{log} is calculated directly from the phonon spectrum. The Coulomb pseudopotential *μ*^{∗} is often taken as 0.1 for most metals, an appropriate one proposed by Ashcroft is 0.13 for hydrogen dominant metallic alloys that has been adopted in the works of GeH_{4} and SnH_{4} (16, 17). For *P*-1 phase at 175 and 200 GPa, the calculated *ω*_{log} are 1121.42 K and 1164.28 K. Using *μ*^{∗} of 0.1 and 0.13, the estimated *T*_{c} are 75.63, 64.62 K at 175 GPa, and 91.78, 80.10 K at 200 GPa, respectively. The calculated *ω*_{log} of *Pm*-3*m* at 275 GPa and *C*2/*c* at 300 GPa are 1443.60 and 866.13 K, respectively. Using *μ*^{∗} of 0.1 and 0.13, the estimated *T*_{c} are 153.44, 138.86 K for *Pm*-3*m* phase and 41.83, 33.97 K for *C*2/*c* phase, respectively. Remarkably, the estimated *T*_{c} of *Pm*-3*m* phase reaches a very high value, i.e., a *T*_{c} beyond the order of 10^{2} K. These current studies inevitably stimulate the future high-pressure experiments on the structural and conductivity measurements.

## Conclusions

We have used a random structure-searching method to explore the crystal structures of disilane in a wide pressure range from 50 to 400 GPa. Three metallic structures, i.e., *P*-1, *Pm*-3*m*, and *C*2/*c* are found and energetically much superior to those of XY_{3}-type candidates under high pressure. We have revealed a wide decomposition pressure range from 50 to 135 GPa, above which these three structures are thermodynamically, mechanically, and dynamically stable. Perturbative linear-response calculations for the three structures are performed at selected pressures. *T*_{c} of the *P*-1 phase at 175 and 200 GPa are 65 and 80 K. For *C*2/*c* phase at 300 GPa, the estimated *T*_{c} is 34 K. Remarkably, the estimated *T*_{c} of *Pm*-3*m* phase at 275 GPa reaches a very high value of 139 K, a *T*_{c} beyond the order of 10^{2} K.

## Methods

We have studied high-pressure phases of Si_{2}H_{6} by means of first-principles density functional theory (DFT) and the random structure-searching method (8), which have been used successfully in recent works (19, 20, 21). We design a set of initial structures by choosing random unit cells, renormalizing the volume to a reasonable value, and inserting the desired number and types of atoms at random positions. Each of these structures is then relaxed to a minimum of the enthalpy at a given pressure. Searches are then repeated for different numbers of SiH_{3} formula units per cell. Moreover, XY_{3}-type candidates are considered for comparison. A wide variety of XY3-type candidates have been checked, including X=N, B, Al, Ga, In, Y, Re, Cr, Mn, Fe, Cu, La, Mo, Ti, Co, Ce, Bi, and Y=H, F, Cl, Br, I, etc. We found three structures (shown in Fig. 2) that are more stable than the XY_{3}-type candidates energetically.

The underlying ab initio structure relaxations are performed by means of DFT implemented in the Vienna ab initio simulation package VASP code (22). For the initial search over structures, the *k*-point sets are generated separately for each unit cell, and no symmetry restrictions are applied. Brillouin zone (BZ) sampling using a grid of spacing 2*π* × 0.05 *Å*^{-1} and a plane-wave basis set cutoff of 280 eV are found to be sufficient. But we recalculate the enthalpy curves with higher accuracy using a BZ sampling of 2*π* × 0.025 *Å*^{-1}, a plane-wave basis set with an energy cutoff of 460 eV, and the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) density functional (23). The plane-wave pseudopotential method within the PBE-GGA, through the Quantum-ESPRESSO package (24) is employed to study the lattice dynamics and electron-phonon coupling (EPC) for *P*-1, *Pm*-3*m*, and *C*2/*c* structures at selected pressures, where the three structures are fully reoptimized by VASP calculations. Phonon frequencies are calculated based on the density functional linear-response method (25). The Monkhorst–Pack (MP) grids of 11 × 8 × 8, 16 × 16 × 16, and 10 × 10 × 6 are used for the three structures accordingly. Subsequently, EPC are calculated in the first BZ on the same MP *q*-point meshes using individual EPC matrices obtained with the 14 × 10 × 10, 18 × 18 × 18, and 14 × 14 × 9 *k*-point mesh for the three structures, respectively. All the convergences of the plane-wave basis set and MP sampling are carefully examined by employing higher kinetic energy cutoffs and denser grids sets.

## Acknowledgments

We are thankful for financial support from the National Natural Science Foundation of China (No. 10979001), the National Basic Research Program of China (No. 2005CB724400), the Cheung Kong Scholars Program of China, Changjiang Scholar and Innovative Research Team in University (No. IRT0625), and the National Found for Fostering Talents of Basic Science (No. J0730311). H.K.M. was supported as part of the EFree, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001057.

## Footnotes

^{1}To whom correspondence may be addressed. E-mail: cuitian{at}jlu.edu.cn or h.mao{at}gl.ciw.edu.Author contributions: T.C. designed research; X.J., X.M., Z.H., Y.M., B.L., T.C., G.Z., and H.-K.M. performed research; X.J., T.C., and H.-K.M. analyzed data; and X.J., T.C., and H.-K.M. wrote the paper.

The authors declare no conflict of interest.

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