Topological semimetal in an $s{p}^2\text{−}s{p}^3$ hybridized carbon network with nodal rings

There have been several proposals that three-dimensional allotropes of carbon formed by graphene networks can support two types of topological nodal line semimetals: type A has closed nodal rings inside the first Brillouin zone (BZ), and type B has nodal lines traversing the whole BZ to be periodically connected. Among these proposals, it has been found that nearly all-$s{p}^2$ hybridized allotropes tend to hold A-type closed nodal rings, while the $s{p}^2\text{−}s{p}^3$ hybridized allotropes tend to hold B-type periodic nodal lines. Here we identify by ab initio calculations an $s{p}^2\text{−}s{p}^3$ hybridized carbon allotrope in $Pcca$ ($D_{2h}^8$) symmetry as a topological nodal line semimetal with two A-type closed nodal rings on the $k_y=0$ and $k_y=\pi$ mirror planes, respectively. The projections of these two nodal rings onto (010) surfaces form concentric ellipses with different size. The drumhead surface states appear in the region enclosed by the bigger ring but disappear inside of the smaller ring. This is consistent with the distribution of the Berry phase calculated along the [010] periodic reciprocal lattice. An effective $k·p$ model has been proposed, and the parameters have been obtained through fitting the ab initio results. The model can reproduce the nodal rings in the shape of an ellipse, satisfying the twofold rotational symmetry. Meanwhile, the simulated x-ray diffraction spectrum matches well with the recently reported distinct diffraction peaks found in the diamond-rich coatings on stainless steel substrate. These results establish a carbon phase with intriguing structural and electronic properties and expand our understandings about topological nodal lines in carbon networks.

Figure 1

The top (a) and side (b) view of oP16 carbon. It has a 16-atom orthorhombic primitive-unit cell in $Pcca$ ($D_{2h}^8$) symmetry with the lattice parameters $a=4.6464$ Å, $b=4.2766$ Å, and $c=5.1325$ Å, occupying the two $8f$ Wyckoff positions of (0.3799, 0.5812, 0.5311) and (0.0936, 0.9189, 0.1087), denoted by ${\mathrm{C}}_1$ (red) and ${\mathrm{C}}_2$ (black), while ${\mathrm{C}}_1$ atoms are in $s{p}^2$ sites and ${\mathrm{C}}_2$ atoms are in $s{p}^3$ sites.

Figure 2

The calculated total energy as a function of volume per atom for diamond, graphite, BC8, bct-${\mathrm{C}}_{16}$, R16, oP16, bco-${\mathrm{C}}_{16}$, ors-${\mathrm{C}}_{16}$, so-${\mathrm{C}}_{12}$, ign-${\mathrm{C}}_6$, oC24, and bct-${\mathrm{C}}_{40}$ carbon.

Figure 3

Phonon band structures and density of states (PDOS) of oP16 carbon. The peak at around 1419 ${\mathrm{cm}}^{-1}$ is related to ${\mathrm{C}}_1$ carbon atoms in $s{p}^2$ bonding, while the peak at around 1044 ${\mathrm{cm}}^{-1}$ is related to ${\mathrm{C}}_2$ carbon atoms in $s{p}^3$ bonding.

Figure 4

(a) The calculated energy bands of oP16 carbon. The blue solid lines represent the band structures obtained from DFT methods, while the red dashed lines represent the band structures obtained from the TB model. (b) Two closed A-type nodal rings on the $k_y=0$ and $k_y=\pi$ mirror planes in the 3D BZ of oP16 carbon. (c) The band-decomposed charge density isosurfaces (0.01 e/${\AA }^3$) for the occupied state around the $E_F$. (d) The partial density of states (DOS) for oP16 carbon. (e) The calculated (010) surface state obtained using 30-layer thick slab geometry along the [010] crystalline direction based on the TB model. (f) The projection of two nodal rings onto the (010) projected plane. The Berry phases inside the small nodal ring (zone I) and outside the large ring (zone III) are 0, while the Berry phase between the small and large nodal rings (zone II) is $\pi$.

Figure 5

(a) Simulated XRD patterns for diamond, graphite, bct-${\mathrm{C}}_{16}$, bco-${\mathrm{C}}_{16}$, so-${\mathrm{C}}_{12}$, ign-${\mathrm{C}}_6$, R16, and oP16 carbon phases. (b) Experimental XRD patterns for the diamond-rich coatings on stainless steel substrate [45]. The x-ray wavelength is 1.5406 Å with a copper source.