### Introduction

Carbon, which is one of the most abundant elements in nature, has a rich variety of structural allotropes due to its capacity to form *sp*, *sp*^{2} and *sp*^{3} hybridized bonds. Graphene, a single layer of carbon in a honeycomb lattice, consists of all-*sp*^{2} bonds and exhibits a semimetallic band structure with Dirac points. Recently, topological materials, including topological insulators (TIs) and topological semimetals (TSMs), have received a great deal of attention because of the intriguing physical phenomena underlying their behavior, as well as their 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 work to explore new topological materials. The TI state has been demonstrated for two-dimensional (2D) HgTe/CdTe quantum wells^{2} and three-dimensional (3D) Bi-based chalcogenides.^{3} The band structure of a TI 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. By contrast, 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 with Na_{3}Bi (ref. 4) and Cd_{3}As_{2},^{5} the band crossing points at the Fermi energy have four-fold 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,^{6} HgCr_{2}Se_{4},^{7} and transition metal dichalcogenides,^{8} and the prediction of the Weyl semimetal state in the TaAs family^{9} has been verified experimentally.^{10}

In nodal line semimetals, the formation of Dirac nodes along a closed loop or line requires inversion and time-reversal symmetries without SOC in Cu_{3}(Pd, Zn)N (ref. 11), Ca_{3}P_{2},^{12} and alkaline-earth metals (Ca, Sr, Yb)^{13} and compounds *AX*_{2} (A=Ca, Sr, Ba; *X*=Si, Ge, Sn).^{14} When SOC is included, additional non-symmorphic symmetry is necessary to protect the nodal line against gap opening in ZrSiS.^{15} In non-centrosymmetric PbTaSe_{2}, the nodal line is protected by mirror reflection symmetry even in the presence of SOC.^{16} Meanwhile, the TSM phase has also been reported for 3D carbon networks constructed from graphene, such as Mackay-Terrenes crystals,^{17} interpenetrated graphene networks (IGNs)^{18} and bco-C_{16}.^{19} Due to the negligible SOC,^{20} the nodal lines of these semimetallic carbon allotropes are protected by a combination of inversion and time-reversal symmetry. Recent experiments have demonstrated the synthesis of 3D graphene networks with high-electrical conductivities by a chemical vapor deposition technique, although the associated crystal structure has not yet been identified.^{21} Other 3D metallic carbon allotropes have been proposed, including *H*18 carbon^{22} and *T*6 carbon^{23} in mixed *sp*^{2}*-sp*^{3} bonding networks, and ThSi_{2}-type tetragonal bct4 carbon^{24} and *H*6 carbon^{25} with all-*sp*^{2} bonding. However, the band overlap occurs at different crystal momenta in *H*18 and *T*6 carbon,^{22, 23} whereas the metallic nature of bct4 and *H*6 carbon arises from the twisted *π* states that make these allotropes dynamically unstable.^{26}

In this work, we report a new 3D 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*-C_{8}, consists of five-membered rings with *sp*^{3} hybridized bonds and *sp*^{2}-bonded carbon networks, and the enthalpy of *m*-C_{8} is lower than those of recently proposed carbon allotropes with topological nodal lines. The dynamic stability of *m*-C_{8} is verified by phonon spectra and molecular dynamics simulations. On the basis of the analysis of X-ray diffraction patterns and enthalpy-pressure curves, we propose that *m*-C_{8} may be present in detonation soot^{27} and a phase transition from graphite to *m*-C_{8} can occur under pressure.

### Methods

#### Computational structure search and electronic structure calculations

We explored new carbon allotropes with *sp*^{2}*-sp*^{3} hybridized bonds by using a computational search method,^{28} in which the conformational space annealing (CSA) algorithm^{29} for global optimization is combined with first-principles electronic structure calculations. The efficiency of this approach has been demonstrated by successful applications to predict Si and C allotropes with direct band gaps.^{30, 31} For various carbon systems with *N* atoms per unit cell (*N*=8, 12, 16, 20), we optimized the degrees of freedom, such as atomic positions and lattice parameters, with the number of configurations set to 40 in the population size of the CSA.

The goal of computational materials design is to obtain the optimal crystal structure with specific target properties, which can be expressed in terms of an objective function.^{28} In this work, the objective function is composed of two parts: enthalpy and penalty. For all configurations, a local minimization was performed for the enthalpy within the framework of density functional theory. Our work aimed to find low-energy 3D carbon allotropes with exotic electronic properties. Thus, the penalty function was designed to prevent carbon structures from forming all-*sp*^{2} or all-*sp*^{3} hybridized bonds, such as diamond and graphite, and to promote mixed *sp*^{2}-*sp*^{3} bonding networks. In each configuration, *sp*^{2}- and *sp*^{3}-hybridized atoms were determined by using the bond lengths of 1.42 and 1.54 Å. If configurations only consisted of *sp*^{2}- or *sp*^{3}-hybridized atoms, the penalty function, defined as eV, was included in the objective function, so that configurations with high-objective function values were excluded. For the first-principles calculations we used the functional form proposed by Armiento and Mattsson (AM05) (ref. 32) for the exchange-correlation potential and the projector augmented wave potentials,^{33} as implemented in the VASP code.^{34} In graphite, interlayer interactions are described more accurately with the AM05 functional compared to other functional forms of the exchange-correlation potential (Supplementary Table S1). The wave functions were expanded in plane waves up to an energy cutoff of 600 eV. At the final stage of optimization, we used an even higher energy cutoff of 800 eV.

### Results and Discussion

#### Structure and stability of a new carbon allotrope

Among many low-energy allotropes (Supplementary Table S2), we obtained a very distinctive crystal structure in the *C*2*/m* space group (Figure 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, although both the number of atoms per unit cell and the cell volume are doubled. The monoclinic *C*2*/m* structure, denoted as *m*-C_{8}, has the equilibrium lattice parameters, *a*=7.010 Å, *b*=2.480 Å, *c*=6.608 Å and *β*=71.2°, and four inequivalent Wyckoff positions, *4i* (0.272, 0, 0.446), (0.632, 0, 0.79), (0.781, 0, −0.072) and (0.007, 0, 0.878), occupied by the C_{1}, C_{2}, C_{3} and C_{4} atoms, respectively. The *m-*C_{8} allotrope is characterized by five-membered carbon rings interconnected by graphitic carbon networks, similar to penta-graphene ribbons linked with hexa-graphene ribbons.^{37} The graphitic networks are composed of *sp*^{2}-bonded C_{1} atoms, with the bond length of =1.412 Å, whereas the C_{2}, C_{3}, and C_{4} atoms forming five-membered rings are all *sp*^{3}-bonded, with the bond lengths of =1.553 Å, =1.527 Å, =1.540 Å, =1.514 Å and =1.570 Å. Note that the bond length between the C_{1} and C_{2} atoms, which connect graphitic sheets to five-membered rings, is =1.493 Å, between those of graphite (1.420 Å) and diamond (1.544 Å).

##### Figure 1.

(**a**) Side and top views of the atomic structure of *m*-C_{8} in *C*2*/m* space group. The lattice parameters in the monoclinic structure are *a*=7.010 Å, *b*=2.480 Å, *c*=6.608 Å, *β*=71.2° and C_{1}, C_{2}, C_{3} and C_{4} denote the four inequivalent Wyckoff positions. (**b**) Calculated phonon spectra of *m*-C_{8} at zero pressure and (**c**) potential energy fluctuations during MD simulations at 1500 K for a 2 × 4 × 2 supercell.

In Table 1, the calculated equilibrium volume, lattice parameters, bond lengths and total energy of *m*-C_{8} are compared to those of diamond, graphite and several recently reported metallic allotropes: *T*6 carbon^{25} and IGN,^{20} in mixed *sp*^{2}*-sp*^{3} bonding networks, and oC8 carbon^{38} and bco-C_{16},^{19} in all-*sp*^{2} bonding networks. The equilibrium volume of *m*-C_{8} is 6.80 Å^{3} per atom, between those of graphite and diamond. Because of the mixture of *sp*^{2} and *sp*^{3} hybridized bonds, the *m*-C_{8} allotrope has four different bond angles of 93.7, 107.3, 114.1 and 121.3°, which deviate from the ideal bond angles of graphite (120°) and diamond (109.5°). Owing to the induced strain, the *m*-C_{8} structure has an excess energy of 0.22 eV/atom compared to the diamond phase. By contrast, *m*-C_{8} is more stable by 0.13 0.29 eV/atom than *T*6 carbon, oC8 carbon and bco-C_{16}. It is interesting to note that although the total energy of *m*-C_{8} at the equilibrium volume is higher by 0.05 eV/atom than that of IGN, its enthalpy is lower for pressures above 10 GPa (Figures 2a and b). From the enthalpy vs pressure curve, a possible synthesis of *m*-C_{8} is expected under compression of graphite. It has been suggested that graphite may transform to oC8 carbon, which is a denser form of bco-C_{16}, above 65 GPa.^{19, 38} However, our calculations indicate that *m*-C_{8} is lower in enthalpy than bco-C_{16} up to 77 GPa, and a transition from graphite to *m*-C_{8} is more likely to occur at the lower pressure of 60 GPa.

##### Figure 2.

(**a**) Total energy as a function of volume and (**b**) enthalpy as a function of pressure curves for diamond, graphite, *T*6 carbon, IGN, bco-C_{16}, oC8 and *m*-C_{8}. The transition from graphite to *m*-C_{8} occurs at a pressure of 60 GPa.

We examined the stability of *m*-C_{8} by calculating the full phonon spectra and found no imaginary phonon modes over the entire Brillouin zone (BZ) (Figure 1b), indicating that *m*-C_{8} is dynamically stable. In addition, we carried out first-principles molecular dynamics (MD) simulations at a temperature of 1500 K. For a 2 × 4 × 2 supercell containing 128 atoms, we confirmed that the *m*-C_{8} allotrope is stable for up to 100 ps (Figure 1c). Owing to the thermal stability, the synthesis of *m*-C_{8} is expected under high-pressure and high-temperature. We also calculated the elastic constants of *m*-C_{8}, and confirmed that the elastic constants meet the criteria for mechanical stability in monoclinic structure.^{39}

The X-ray diffraction patterns of *m*-C_{8} were simulated and compared to the experimental data from detonation soot (sample Alaska B),^{27} along with those of graphite, diamond, *T*6 carbon, bco-C_{16} and IGN, as shown in Figure 3. In the detonation soot, the prominent peaks ~26.5° and 43.9° are attributed to the graphite (002) and diamond (111) diffractions, respectively. The (101) peak of *T*6 carbon, (101) peak of bco-C_{16}, and (001) peak of IGN match with the experimental X-ray diffraction data located at 37.4°, 30.0° and 21.4°, respectively. 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,^{27} indicating that an unknown carbon phase should be produced. Our simulated X-ray diffraction results show that the main (001) peak of *m*-C_{8} reasonably explains the unidentified peak at 13.4°. Moreover, the (111), (002) and (112) peaks of *m*-C_{8} match those in the experimental X-ray diffraction pattern located at 25.1°, 27.8° and 32.1°, respectively, indicating the presence of *m*-C_{8} in the specimen produced by detonation experiments.^{27}

##### Figure 3.

The simulated X-ray diffraction patterns for graphite, diamond, *T*6 carbon, bco-C_{16}, IGN and *m*-C_{8} are compared to those experimentally observed for detonation soot of TNT (sample Alaska B).^{27} Arrows indicate the X-ray diffraction peaks related to *m*-C_{8}.

#### Band structure of a new carbon allotrope

Finally, we examined the band topology of *m*-C_{8}. In Figure 4a, the band structure exhibits linear dispersions around the Fermi level where the valence and conduction bands touch, similar to graphene. The linear bands are mainly derived from the *sp*^{2} hybridized C atoms in graphitic sheets, as illustrated in the distribution of charge densities (Figure 4b), whereas more dispersive bands far from the Fermi level are associated with the *sp*^{3} hybridized bonds in five-membered rings. From the band structure in the full BZ, we find that the crossing points of the valence and conduction bands form a continuous nodal line piercing the extended BZ, without interfering with *sp*^{3}-hybridized bands (Figures 4 and 5). Thus, the *m*-C_{8} allotrope with symmorphic symmetry belongs to a class of topological nodal line semimetals.^{11, 12, 13, 14, 15, 16, 17, 18, 19} Recently, two types of topological nodal line semimetals were proposed, depending on systems with and without SOC.^{39} In the former, because the SOC may open up gaps at the band crossing points, both inversion and time-reversal symmetries are insufficient to protect the band crossings, whereas additional non-symmorphic symmetry can protect the nodal line.^{40} In carbon systems such as *m*-C_{8}, with a negligible SOC, the topological nodal line survives by a combination of inversion and time-reversal symmetries.

##### Figure 4.

(**a**) Band structure of *m*-C_{8}. The thicknesses of the red and blue colored bands represent the degrees of confinement for the *sp*^{2} and *sp*^{3} hybridized atoms, respectively. (**b**) Distribution of the charge densities for the linear bands (A_{1} and A_{2}) near the Fermi energy, which are mainly derived from the carbon chains in the *sp*^{2} bonding networks.

##### Figure 5.

(**a**) The 3D Brillouin zone (black polyhedron) with several high-symmetry momenta and the nodal lines (red lines) at the Fermi energy in the monoclinic structure of *m*-C_{8}. TRIM points (vertices in green rhombohedral) and their projection onto the (110) surface BZ are indicated. (**b**) Topologically protected (110)-surface band (blue line) nestled inside the bulk nodal line (red dots).

Based on the analysis of parity eigenvalues, Fu and Kane proposed the *Z*_{2} topological invariants to describe the topological nature of TIs.^{41} Similarly, the parities of energy states can be used to assign the *Z*_{2} topological invariants in topological nodal line semimetals. For the eight time-reversal invariant momenta (Figure 5a), the products of the parity eigenvalues (δ) for the occupied bands are listed in Table 2. We found that *m*-C_{8} is characterized by the weak *Z*_{2} indices (0;111) due to the value of *δ*=−1 at the *Y* and *Z* points. As *m*-C_{8} has no mirror reflection symmetry, the bulk nodal line does not appear in a mirror-invariant 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 and points (Figure 5a). In fact, the projected band structure clearly shows the formation of the nearly flat surface state connecting the projected nodal points around the point (Figure 5b).

In summary, we have predicted a novel carbon allotrope *m*-C_{8} with mixed *sp*^{2}*-sp*^{3} bonding networks using the evolutionary structure search method. The monoclinic structure of *m*-C_{8} is composed of five-membered rings connected by graphitic sheets. The stability of *m*-C_{8} is verified by calculating the full phonon spectra and MD simulations at a temperature of 1500 K. From the analysis of the electronic band structure, we have identified that *m*-C_{8} belongs to the class of topological nodal line semimetals, exhibiting the topological nodal line in bulk and the topological surface states at surface boundaries. As the SOC is extremely weak in *m*-C_{8}, the nodal line is protected by the coexistence of inversion and time-reversal symmetries. Although it remains a challenge to synthesize the crystalline form of *m*-C_{8}, our results provide not only a perspective for the novel electronic structure of carbon allotropes but also promote future studies to explore new carbon allotropes with exotic electronic and transport properties.

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### Acknowledgements

This work was supported by Samsung Science and Technology Foundation under Grant No. SSTF-BA1401-08.

Supplementary Information accompanies the paper on the NPG Asia Materials website

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