Semimetallic carbon allotrope with topological nodal line in mixed $sp^2$-$sp^3$ bonding networks

Graphene is known as a two-dimensional Dirac semimetal, in which electron states are described by the Dirac equation of relativistic quantum mechanics. Three-dimensional analogues of graphene are characterized by Dirac points or lines in momentum space, which are protected by symmetry. Here, we report a novel 3D carbon allotrope belonging to a class of topological nodal line semimetals, discovered by using an evolutionary structure search method. The new carbon phase in monoclinic $C$2$/m$ space group, termed $m$-$C_8$, consists of five-membered rings with $sp^3$ bonding interconnected by $sp^2$-bonded carbon networks. Enthalpy calculations reveal that $m$-$C_8$ is more favorable over recently reported topological semimetallic carbon allotropes, and the dynamical stability of $m$-$C_8$ is verified by phonon spectra and molecular dynamics simulations. Simulated x-ray diffraction spectra propose that $m$-$C_8$ would be one of the unidentified carbon phases observed in detonation shoot. The analysis of electronic properties indicates that $m$-$C_8$ exhibits the nodal line protected by both inversion and time-reversal symmetries in the absence of spin-orbit coupling and the surface band connecting the projected nodal points. Our results may help design new carbon allotropes with exotic electronic properties.


properties.
Carbon, which is one of the most abundant elements in nature, has a rich variety of structural allotropes due to its ability to form sp, sp 2 , and sp 3 hybridized bonds. Graphene, a single layer of carbon in the honeycomb lattice, consists of all-sp 2 bonds and exhibits the semimetallic band structure with Dirac points. Recently, topological materials including topological insulators (TIs) and topological semimetals (TSMs) have received much attention because of their intriguing physical phenomena and potential applications. The prediction of the TI phase in graphene with spin-orbit coupling (SOC) 1 has stimulated a tremendous amount of theoretical and experimental works to explore new topological materials. The TI state has been evidenced for two-dimensional (2D) HgTe/CdTe quantum wells 2 and three-dimensional (3D) Bi-based chalcogenides [3][4][5] . The band structure of TIs is characterized by the existence of a bulk band gap as well as gapless boundary states that are protected by the nontrivial topology of bulk electronic states. On the other hand, in TSMs, the valence and conduction bands cross each other at discrete points (Dirac and Weyl semimetals) or along curves (nodal line semimetals) in momentum space. In Dirac semimetals, which have been realized for Na3Bi [6][7][8][9] and Cd3As2 10-14 , the band crossing points at the Fermi energy have fourfold degeneracy. By breaking either inversion or time-reversal symmetry in Dirac semimetals, one can obtain Weyl semimetals in which each Dirac point splits into a pair of doubly degenerate Weyl points with opposite chirality. Weyl semimetals have been proposed for compounds containing heavy elements, such as pyrochlore iridates 15 , HgCr2Se4 16 , and transition metal dichalcogenides 17,18 , and the prediction of the Weyl semimetal state in the TaAs family 19,20  In this work, we report a new carbon allotrope belonging to a class of topological nodal line semimetals, based on global optimization and first-principles density functional calculations.
The new carbon phase, termed m-C8, consists of five-membered rings with sp 3 hybridized bonds and sp 2 -bonded carbon networks, and the enthalpy of m-C8 is lower than those of recently proposed carbon allotropes with topological nodal lines. The dynamical stability of m-C8 is verified by phonon spectra and molecular dynamics simulations. Based on the analysis of x-ray diffraction spectra and enthalpy-pressure curves, we propose that m-C8 can be present in detonation soot 47 and a phase transition from graphite to m-C8 can occur under pressure.

Results and Discussion
Structure and stability of a new carbon allotrope. We explored metallic carbon allotropes with sp 2 -sp 3 hybridized bonds by using a computational search method (Methods). Among many low-energy allotropes, we obtained a very distinctive crystal structure in the C2/m space group (Fig. 1a), especially for the N = 8 system. For the N = 16 system, the same low-energy allotrope was also found by the computational search method, while both the number of atoms and C2 atoms, which connect graphitic sheets to five-membered rings, is lying in between those of graphite (1.420 Å) and diamond (1.544 Å).
In eV/atom compared with the diamond phase. On the other hand, m-C8 is more stable by 0.13-0.29 eV/atom than T6 carbon, oC8 carbon, and bco-C16. It is interesting to note that, although the total energy of m-C8 at the equilibrium volume is higher by 0.05 eV/atom than that of IGN, its enthalpy is lower for pressures above 10 GPa, as shown in Fig. 2. From the enthalpy vs pressure curve, a possible synthesis of m-C8 is expected under compression of graphite. It was suggested that graphite may transform to oC8 carbon, which is a denser form of bco-C16, above 65 GPa 39,48 . However, our calculations indicate that m-C8 is lower in enthalpy than bco-C16 up to 77 GPa, and a transition from graphite to m-C8 is more likely to occur at a lower pressure of 60 GPa.
We examined the stability of m-C8 by calculating the full phonon spectra and found no imaginary phonon modes over the entire Brillouin zone (BZ) (Fig. 1b), indicating that m-C8 is dynamically stable. In addition, we carried out first-principles molecular dynamics (MD) simulations at a high temperature of 1500 K. For a 3 2 2   supercell containing 96 atoms, we confirmed that the m-C8 allotrope is stable up to 100 ps (Fig. 1c). Owing to the thermal stability, the synthesis of m-C8 is highly expected under high pressure as well as high temperature. We also calculated the elastic constants of m-C8, and confirmed that the elastic constants meet the criteria for mechanical stability in monoclinic structure 49 . However, the low angle peak at 13.4° does not match any previously reported carbon phases.
This peak was also observed in different detonation soot 47 , providing that an unknown carbon phase should be produced. Our simulated XRD results show that the main (001) (Fig. 5a), the products of the parity eigenvalues (  ) for the occupied bands are listed in Table 2. We find that m-C8 is characterized by the weak Z2 indices (0;111) due to the value of 1    at the Y and Z points.
Since m-C8 has no mirror reflection symmetry, the bulk nodal line does not appear in a mirrorinvariant plane. To visualize the formation of topological surface states, we calculated the surface band structure for a slab geometry composed of 20 graphitic layers, where the (110) surface is exposed to vacuum. As the bulk BZ is projected onto the (110) surface BZ, one can expect that the nodal line is located near the X and S points (Fig. 5a). In fact, the projected band structure clearly shows the formation of the nearly flat surface state connecting the projected nodal points around the X point (Fig. 5b).
In summary, we have predicted a novel carbon allotrope m-C8 in mixed sp 2