Body-Centered Orthorhombic ${\mathrm{C}}_{16}$: A Novel Topological Node-Line Semimetal

We identify by ab initio calculations a novel topological semimetal carbon phase in all-$s{p}^2$ bonding networks with a 16-atom body-centered orthorhombic unit cell, termed bco-${\mathrm{C}}_{16}$. Total-energy calculations show that bco-${\mathrm{C}}_{16}$ is comparable to solid fcc-${\mathrm{C}}_{60}$ in energetic stability, and phonon and molecular dynamics simulations confirm its dynamical stability. This all-$s{p}^2$ carbon allotrope can be regarded as a three-dimensional modification of graphite, and its simulated x-ray diffraction (XRD) pattern matches well a previously unexplained diffraction peak in measured XRD spectra of detonation and chimney soot, indicating its presence in the specimen. Electronic band structure calculations reveal that bco-${\mathrm{C}}_{16}$ is a topological node-line semimetal with a single nodal ring. These findings establish a novel carbon phase with intriguing structural and electronic properties of fundamental significance and practical interest.

Figure 1

Structure of bco-${\mathrm{C}}_{16}$ and its phonon dispersion. (a) Top and side view of the all-$s{p}^2$ bco-${\mathrm{C}}_{16}$ in $Imma$ ($D_{2h}^{28}$) symmetry with one-third double ($d_2$, $d_3$) and two-thirds single ($d_1$) carbon-carbon bonds. It has a 16-atom body-centered orthorhombic structure with lattice parameters $a=7.8061\AA$, $b=4.8772\AA$, $c=3.2372\AA$, occupying the $8i$ (0.3231, 0.25, 0.1258) and $8f$ (0.0885, 0.5, 0.5) Wyckoff positions, denoted by ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$, respectively. (b) Calculated phonon dispersion curves of bco-${\mathrm{C}}_{16}$ at zero pressure.

Figure 2

Calculated energy versus volume per atom for bco-${\mathrm{C}}_{16}$ compared to graphite, fcc-${\mathrm{C}}_{60}$, cR6, cT8, and rh6 carbon in all-$s{p}^2$, oC8 in mixed $s{p}^2\text{−}s{p}^3$, and diamond in all-$s{p}^3$ bonding networks. The dashed line indicates the energy level of carbyne chain.

Figure 3

X-ray diffraction (XRD) patterns. (a) Simulated XRD patterns for graphite, diamond, rh6, cT8, cR6, fcc-${\mathrm{C}}_{60}$, and bco-${\mathrm{C}}_{16}$. (b) Experimental XRD patterns for detonation soot of TNT and diesel oil [36] and chimney soot [37]. X-ray wavelength is 1.5406 Å with a copper source.

Figure 4

Calculated bulk and surface band structures of bco-${\mathrm{C}}_{16}$ at equilibrium lattice parameters. (a) The bulk band structure along several high-symmetry directions. $G_1$ and $G_2$ indicate the irreducible representation of the two crossing bands, respectively. (b) The Brillouin zone with several high-symmetry momenta indicated. The nodal ring (red circle), formed by the band crossing points, is in the shaded mirror plane. (c) and (d) show the surface states for different zigzaglike (c) and beardlike (d) terminations of the (100) surface. The surface flat band (red line) can be outside or inside the surface projected nodal ring.