Nonmetallization and band inversion in beryllium dicarbide at high pressure

Carbides have attracted much attention owing to their interesting physical and chemical properties. Here, we systematically investigated global energetically stable structures of BeC2 in the pressure range of 0–100 GPa using a first-principles structural search. A transition from the ambient-pressure α-phase to the high-pressure β-phase was theoretically predicted. Chemical bonding analysis revealed that the predicted phase transition is associated with the transformation from sp2 to sp3 C-C hybridization. The electrical conductivity of the high-pressure phase changed from a metal (α-phase) to a narrow bandgap semiconductor (β-phase), and the β-phase had an inverted band structure with positive pressure dependence. Interestingly, the β-phase was a topological insulator with the metallic surface states protected by the time-reversal symmetry of the crystal. The results indicate that pressure modulates the electronic band structure of BeC2, which is an important finding for fundamental physics and for a wide range of potential applications in electronic devices.

that the α-structure is in the thermodynamic ground state. Over the whole pressure range, a new low-enthalpy structure denotes as the β -structure/phase is explored which has the same space group as the α-structure. The enthalpy-pressure relationship for the candidate structures is shown in Fig. 1. It has been seen that, for compressed BeC 2 , the ambient-pressure α-phase is the most stable structure below 56.7 GPa, but the β -phase becomes more favorable from 56.7 to 100 GPa. The pressure evolution of the unite cell volume of BeC 2 in the α -and β -structure is depicted in inset of Fig. 1. The volume collapses about 9.4% around 56.7 GPa, indicating the first-order nature of the phase transition in BeC 2 . No imaginary frequency is observed throughout the whole Brillouin zone, which indicates the two novel phases are dynamically stable in the pressure region from this study.
The atomic arrangements of the competing structures are shown in Fig. 2. The calculated lattice constants of the ambient-pressure are calculated. The α -phase has a monoclinic symmetry, with a = 6.223 Å, b = 2.535 Å and c = 5.904 Å, β = 111.5° at ambient pressure. The Be and C atoms occupy Wyckoff 4i position in the unit cell with Be at (0.672, 0.5, 0.11), C at (0.431, 0.5, 0.227) and (0.467, 1.000, 0.374). The α -phase is similar to that the recently  predicted for MgC 2 of C2/m phase with carbon atoms polymerized into ribbon with a six-membered ring. Planes of Be atoms separate the hexagonal honeycomb layers of carbon atoms, which have the nearest distance with the outer C atom of the ribbon.
The β -phase has a monoclinic symmetry, with lattice parameters of a = 7.879 Å , b = 2.404 Å, c = 3.415 Å, and β = 84.12° at 56.7 GPa. The Be and C atoms occupy Wyckoff 4i positions in the unit cell with Be at (0.212, 0.5, 0.758), C at (0.420, 0.5, 0.9) and (0.418, 1.000, 0.637). It is obvious that the β -phase consists of bulky honeycomb-layer of carbon atoms separated by planes of Be atoms. Figure 3 shows the calculated lattice parameter versus pressure curves for the α -and β -phase. In α -phase, the slope of a is much more steeper than that of b and c. The results show that the inter-atomic forces among the interlayer molecules are weaker than the forces among in the intralayer molecules. For the β -phase, the slopes in different axes are nearly identical. In the inset of Fig. 3, the change of β -phase induced by the pressure is attributable to the elongating of the interlayer (a axis) and shortening of the intralayer (c axis) in α -phase, which also caused the well-organized six-ring carbon rings collapse into the puckered quasi 2-dimension graphite sheet. Due to the occurrence of graphite sheet between neighboring Be atomic layers, the β -phase can be regarded as one of the puckered graphite intercalation compounds.
Nonmetallization of BeC 2 . The calculated band structure and projected density of states (DOS) are shown in Fig. 4. The predicted overlap between the conduction and the valence bands shows the α -phase with sp 2 hybridization is metallic, along with Be-and C-p electrons near the Fermi level. At 56.7 GPa, the β -phase characterized by sp 3 hybridization becomes an insulator with a direct band gap of 0.03 eV (without spin-orbit coupling). Generally, metals under pressure exhibit decreasing interatomic distance and increasing metallic behavior, whereas BeC 2 undergoes an unexpected and often counterintuitive pressure-induced metal-to-insulator transition. This surprising behavior can also be seen in Sodium which shows an optically transparent phase at 200 GPa 16 .
To examine the nature of these bonding states, we plotted the electron localization function (ELF) isosurfaces of BeC 2 (Fig. 5) 17 . The α -phase contains a ribbon of six-member carbon rings, which have sp 2 -like properties. Each inner C atom forms three C-C covalent bonds and each outer C atom has one electron lone pair along with two C-C covalent bonds. The remaining 2p z electrons of the C 6 ring form a delocalized π system. In the β -phase, the puckered honeycomb-layered structure lacks a system of delocalized π bonds with mobile electrons and has more localized electrons resulting in sp 3 hybridization. Each C atom has one electron lone pair and three covalent C-C bonds, resembling diamond structure, accompanied by the appearance of an insulator state. The significant evolution of electronic structures from the sp 2 to sp 3 hybridization of the carbon atoms between the α -and β -phases is similar to the transformation of graphite to diamond under sufficient compression.
Band inversion and topological insulator state in β-phase. Remarkably, the size of the band gap of the β -phase depends positively on pressure in Fig. 6. This peculiar behavior stems from the requirement for symmetry thus forbidding the crossing inverted conduction minimum and valence maximum at Γ point; consequently, the conduction minimum and valence maximum will repel each other more strongly under higher pressure. It is important to check whether the β -phase is a topological insulator at high pressure. Due to the presence of inversion symmetry, the parity criterion can be used to justify the topological properties of the β -phase [18][19][20] . Following the method addressed in ref. 18, we calculated the parities δ 1 at eight time-reversal invariant momenta (TRIMS) (i = 1-8) in three-dimensional Brillouin zone in Table 1. At 40 GPa, only the Γ point has negative parity, whereas the other TRIMS have positive parity. At 50 and 60 GPa, all TRIMS have positive parity. This can also be observed from the band structures of the β -phase around the critical pressure of the phase transition, as shown in Fig. 6. One can clearly observe that the parities of band edges are exchanged when above 40 GPa. The Z 2 topological invariants can then be evaluated from the parities using the following equations, The Z 2 numbers are 1; (111) for 40 GPa and 0; (000) for 50 and 60 GPa. Together with the evolution of band energy as a function of pressure (Fig. 6), we can confirm that the β -phase is a strong topological insulator when below 40 GPa, whereas it is a normal insulator when above 50 GPa.

Conclusion
In summary, we have determined the high-pressure structural evolution of BeC 2 by using the particle swarm optimization technique with first-principles electronic structure calculation. We predicted a pressure-induced phase transition from the α -to β -phase at 56.7 GPa. The structural, dynamical, and electronic properties of BeC 2 under pressure were systemically investigated. The electrical calculation shows that the β -phase has an inverted band structure with a positive pressure dependence. Interestingly, the β -phase is a topological insulator with metallic surface states protected by the time-reversal symmetry of the crystal. These results show that pressure has a strong effect on the fundamental crystal and electronic structure of BeC 2 , and that pressure tuning of the electronic properties offers an effective tool to modulate a wide range of physical properties for its potential applications.

Method
We performed structure predictions through a global minimization of free energy surfaces merging ab initio total-energy calculations implemented in CALYPSO (crystal structure analysis by particle swarm optimization) code [21][22][23][24][25] . The method has successfully predicted the high-pressure structures of various systems, ranging from elements to binary and ternary compounds 26,27 . The first-principles energetic calculations were carried out using density functional theory (DFT) with the Perdew-Burke-Ernzerhof of exchange-correlation as implemented in the Vienna ab initio simulation package (VASP) 28 . The projector augmented wave method has been adopted, with 2s 2 and 2s 2 2p 2 treated as valence electrons for Be and C atoms, respectively 29,30 . For Brillouin zone integration, we used the Monkhorst-Pack scheme and checked convergence of ground state calculations with uniformly