Topological nodal line semimetal in an all-sp 2 monoclinic carbon

Topological nodal line semimetal is an exotic class of quantum materials featuring the continuous line of nodes inside the first Brillouin zone. Here we identify by systematical ab initio calculations a new all-sp 2 hybridized carbon allotrope with monoclinic C2/c (C2h6) symmetry which is termed as bcm-C16. Total energy calculations show that our proposed bcm-C16 carbon is energetically comparable to or stable than the previously proposed bco-C16, bct-C16, and oP16 carbon. Its dynamical stability has been confirmed by phonon mode calculations. Detailed analysis of the electronic properties show that bcm-C16 carbon is a topological nodal line semimetal with a single closed nodal ring around the Γ high symmetric point, protected by spatial inversion (P) and time-reversal (T) symmetry. When the nodal ring is projected onto the (001) surface, a topologically protected drumhead-like surface state can be seen inside or outside the nodal ring depending on the different surface terminations. Moreover, we also examined the tensile-strain robustness of the electronic properties of bcm-C16 carbon. The nodal ring is robust under a tensile-strain along the crystalline x- and z-directions up to 20%. In addition, the simulated x-ray diffraction pattern (XRD) of bcm-C16 carbon matches with the experimental pattern found in the detonation and chimney soot experiment. The present proposal has enriched the family of carbon allotropes with topological nodal lines, and pave the way for further theoretical and experimental studies.

Carbon can form a rich variety of allotropes due to its ability to form sp, sp 2 , and sp 3 carbon-carbon bonds [24][25][26]. In recent years, a series of carbon allotropes have been proposed with topological nodal lines [27][28][29][30][31][32][33][34][35][36][37][38][39]. The topological nodal lines in carbon networks can be roughly classified into type-A with close nodal ring such as Mackay-Terrones carbon crystal (MTC) [28], body-centered orthorhombic C 16 (bco-C 16 ) [30], body-centered tetragonal C 16 (bct-C 16 ) [31] and base-centered orthorhombic ors-C 16 [32] carbon in all-sp 2 hybridized network; or type-B with two separate nodal lines such as interpenetrated graphene network C 6 (ign-C 6 ) [33], orthorhombic C 24 (oC 24 ) [34], and simple orthorhombic C 12 (so-C 12 ) [36] carbon in sp 2 -sp 3 hybridized network. Recently, a new hexagonal series of topological nodal line carbon phases was proposed for all-sp 2 hybridized networks but with B-type paired nodal lines on the edge of the first BZ rather than A-type closed nodal rings [38], and soon after an sp 2 -sp 3 hybridized oP16 carbon [39] with two closed A-type nodal rings is proposed and it is pointed out that the type-A/B nodal line behavior is actually affected by the lattice anisotropy while the bonding type is not the essence.
In this paper, we identify through ab initio calculations a new three-dimensional (3D) all-sp 2 hybridized carbon allotrope in C2/c (C 6 2h , space group No. 15) symmetry. This new carbon phase has 16-atom base-centered monoclinic unit cell, thus termed as bcm-C 16 carbon. Total energy calculations show that our proposed bcm-C 16 carbon is energetically comparable to or stable than the previously proposed bco-C 16 [30], bct-C 16 [31], and oP16 [39] carbon. Its dynamical stability has been confirmed with phonon mode calculations. Detailed analysis of the electronic properties show that bcm-C 16 carbon is a topological nodal line semimetal with a single closed nodal ring around the high symmetry Γ point, protected by P and T symmetry. When the nodal ring is projected onto the (001) surface, a topologically protected drumhead-like surface state can be seen inside or outside the nodal ring depending on the different surface terminations. In addition, we also examined the tensile-strain robustness of the electronic properties of bcm-C 16 . The nodal ring is robust under a tensile-strain along the crystalline x-and z-directions up to 20%. Meanwhile, the simulated x-ray diffraction pattern (XRD) of bcm-C 16 carbon matches with the experimental pattern found in the detonation soot [40] and chimney soot [41] experiment, indicating its possible existence and synthesis route. The present proposal has enriched the family of carbon allotropes with topological nodal lines.

Computational method
We have systematically discovered the bcm-C 16 carbon structure with a structural search process based on a Monte Carlo algorithm as reported for R16 [42], O16 [43], and BC14 carbon [44]. Highly efficient Tersoff potential [45] and more accurate first-principles calculations were used to screened and refined the energetics which resulted in the identification of bcm-C 16 carbon. The ab initio calculations are carried out based on density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) [46]. The all-electron projector augmented wave (PAW) [47] method was adopted with 2s 2 2p 2 treated as valence electrons. The generalized gradient approximation (GGA) developed by Armiento-Mattsson (AM05) [48] was adopted as an exchange-correlation potential. A 8 × 8 × 8 Monkhorst-Pack grid of Brillouin zone (BZ) sampling is used and an energy cutoff of 800 eV is set for the plane-wave basis. The structures are fully relaxed until the total energy difference is less then 10 −6 eV and convergence criteria for atomic forces is set to be 10 −3 eV Å −1 . The electronic properties are calculated using the Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) [49] method, and the phonon properties are calculated with phonopy package [50]. To further explore the topological electronic properties, we establish a tight-binding (TB) model using the maximally localized Wannier functions (MLWFs) [51,52] implemented in Wannier90 package [53] and searched the band crossing points in the entire BZ with WannierTools package [54].

Results and discussion
We firstly characterize the crystalline structure of bcm-C 16 carbon. Figure 1 , associated with C 1 (sp 2 ) = C 1 (sp 2 ), C 2 (sp 2 ) = C 2 (sp 2 ), C 1 (sp 2 ) − C 1 (sp 2 ), C 2 (sp 2 ) − C 2 (sp 2 ), and C 1 (sp 2 ) − C 2 (sp 2 ), respectively. There are also three kinds of bond angles are: The primitive cell of bcm-C 16 carbon contains eight atoms as shown in figure 1(b). It has only four symmetry operations including an inversion center and lack of mirror symmetry unlike the reported all-sp 2 topological semimetals bco-C 16 [30] and bct-C 16 [31]. Figure 2 shows the calculated energy versus volume curves of bcm-C 16 carbon comparing with graphite, diamond [55], and the reported bco-C 16 [30], bct-C 16 [31], ors-C 16 [32], and oP16 [39] carbon that also have 16 carbon atoms in unit cell. It is seen that the equilibrium energy of bcm-C 16 carbon is  Calculated total energy per atom as a function of volume for bcm-C 16 carbon compared to graphite, diamond [55] and other reported bco-C 16 [30], bct-C 16 [31], oP16 [39], and ors-C 16 [32] carbon allotropes. −8.659 eV/atom, which is very close to −8.671 eV/atom for bco-C 16 carbon, and lower than −8.570 eV/atom for oP16 carbon and −8.369 eV/atom for bct-C 16 carbon, while slightly higher than −8.810 eV/atom for ors-C 16 carbon, showing its good energetic stability. We have also calculated the bulk modulus (B 0 ) by fitting with Murnaghan's equation of state [57] to the energy-volume curve of bcm-C 16 as 301 GPa, which is smaller than 451 GPa for diamond, 315 GPa for bco-C 16 , and larger than 250 GPa for bct-C 16 carbon and 298 GPa for ors-C 16 carbon. The calculated equilibrium structural parameters, total energy per atom, equilibrium volume per atom and bulk modulus are listed in table 1, comparing with available experimental and reported data [30-32, 37, 39, 55].
To further examine the dynamical stability of bcm-C 16 carbon, we have calculated the phonon band structure and phonon density of states (PDOS) as shown in figure 3. The highest phonon frequency is located at the high-symmetric X point with a value of ∼1561 cm −1 , which is comparable to the highest phonon frequency ∼1571 cm −1 of all-sp 2 bco-C 16 [30] and lower than but close to ∼1600 cm −1 for graphite [58]. There are two main peaks at around 1524 and 1381 cm −1 in the PDOS of bcm-C 16 carbon. The peak at around 1524 cm −1 and 1381 cm −1 are related to C 1 and C 2 carbon atoms, respectively. There is no imaginary phonon frequency through the entire BZ and in PDOS, thus confirming the dynamical stability of bcm-C 16 .
Next we discuss the electronic properties of bcm-C 16 carbon. Figure 4(a) depicts the calculated bulk band structures at equilibrium lattice parameters using HSE06 functional [49] with the Fermi energy (E F )  parameters (a, b, c, and β), volume per atom V 0 , bond lengths d C-C , total energy per atom E tot , electronic band gap E g and bulk modulus B 0 for diamond, bct-C 16 , bcm-C 16 , bco-C 16 , ors-C 16 and oP16 carbon at zero pressure, compared to available experimental and reported data [30-32, 37, 39, 55, 56].  set to zero. It can be seen that the highest occupied band and lowest unoccupied band cross each other along the high-symmetric X-Γ direction and exhibit a linear dispersion due to the band inversion mechanism [59,60] with opposite parity eigenvalues. Further analysis of the electronic band structure in 3D BZ shows that the band crossing nodes form a continuous nodal ring around the Γ point inside the 3D BZ as shown in figure 4(b), which is protected merely by PT symmetry. Unlike bct-C 16 , bco-C 16 , CaP 3 [19] or CaTe [22] nodal line semimetals, whose nodal lines lie in a certain mirror plane, in our case the nodal ring is twisted around the high symmetric Γ point due to the lack of mirror symmetry. It is well known that, when the nodal line is projected into some planes, there is a drumhead like surface state inside or outside the nodal ring depending on the surface terminations. In order to investigate the topological surface states in our system, we have constructed an eight-band tight-binding (TB) model for bcm-C 16 carbon with the MLWFs method based on the p x orbitals of carbon atoms by using the WANNIER90 package [51,52]. The calculated electronic density of states and TB band structures are shown in figure S1 of the supplemental materials (https://stacks.iop.org/NJP/24/043007/mmedia) [61]. The TB band structures match well with the DFT band structures, confirms the effectiveness of our established TB models.
Based on this TB model, we have calculated the surface states along H-Γ-X high-symmetric directions for the (001) surface as shown in figures 4(c)and (d) with the iterative surface Green's function method as implemented in WannierTools package [54]. When the surface termination is a zigzag type, a drumhead-like surface state is seen outside the projected nodal ring as shown in figure 4(c) and when the surface termination is a arm-chair type, the drumhead like surface state seen inside the projected nodal ring as in figure 4(d); while the different types of surface terminations are shown in figure S2 of the supplemental materials [61]. The calculated surface states can be detected by the angle resolved photoemission spectroscopy (ARPES) experiments [62].
Next we investigate the robustness of the nodal ring against the applied tensile strain along the crystalline x-and z-directions (denoted as ε x and ε z , respectively). The change of the band crossing points near the Fermi level is depicted in figure 5(a) with an increase of tensile strain ε x , from 5 to 25%, respectively. It can be seen that the band crossing point around the Fermi level robust within the applied strain limit along x-direction. However, with an increase of strain ε x , the band crossing point slightly move toward X high symmetric point and below to the Fermi level due to the change in BZ. Further analysis of band crossing point in 3D BZ with the applied tensile strain along x-direction demonstrates that the size of the nodal ring becomes larger with an increase of applied strain along x-direction as shown in figures S3(a) and (b) in the supplementals materials [61]. On the other hand, when the tensile strain is applied along z-direction from 5 to 20% the band crossing point move toward Γ point and near to Fermi level as depicted in figure 5(b). While in 3D BZ the size of nodal ring becomes smaller with an increase of applied strain as shown in figures S3(c) and (d) in the supplementals materials [61]. At 25% tensile strain along z-direction, the band crossing point is not robust and the bcm-C 16 nodal line semimetal has been converted into a direct band gap semiconductor. This change in the topological phase is associated with the change of double bond (d 1 ) into a single bond in the crystal structure of bcm-C 16 carbon at 25% strain along z-direction. On the other hand, with the applied strain along x-direction the double bond remain conserve within the applied strain limit, due to which band crossing is robust. The changes in bond lengths with the applied strain along x-and z-directions are given in table S1 in the supplementals materials [61].
Finally, to guide for experimental observations, we simulated XRD pattern of bcm-C 16 carbon with wavelength 1.5406 Å, along with those of diamond, graphite, bco-C 16 [30], bct-C 16 [31], ors-C 16 [32], and compare with experimental data from detonation soot of TNT and diesel oil [40] and chimney soot [41] as shown in figure 6. It can be observed that in XRD pattern of bcm-C 16 carbon, the first dominant peak at 2θ ≈ 30 • matches well with the peaks at 2θ ≈ 30 • of detonation soot of TNT and diesel oil [40] and chimney soot [41]. Simulated XRD patterns of the bcm-C 16 carbon also exhibit a main peak at a position slightly higher than 30 • , similar to bco-C 16 carbon [30]. The high energetic stability and well agreement of XRD pattern between the simulated and experiments XRD patterns suggest the existence of the bcm-C 16 carbon in the detonation and chimney soot.   [30], bct-C 16 [31], ors-C 16 [32], and bcm-C 16 carbon phases. (b) Experimental XRD patterns for detonation soot [40] and chimney soot [41]. The x-ray wavelength is 1.5406 Å with a copper source.

Summary
In conclusion, we have identified based on ab initio calculations a new all-sp 2 hybridized bcm-C 16 carbon. Total energy calculations have confirmed its energetic stability and its dynamical stability has been confirmed with phonon mode analysis. The electronic band structures show that it is a topological nodal line semimetal with a single nodal ring around the high symmetric Γ point, and the nodal line is robust under a large tensile strain ε x and ε z along the crystalline x-and z-directions up to 20%. Simulated XRD patterns show a good match with the experimental findings in the detonation and chimney soot experiments, suggesting that it may be one of the experimental phases. Our findings have enriched the family of carbon allotropes with topological properties and pave the way for further theoretical and experimental researches.

Conflict of interest
There are no conflicts to declare.