Topological Nodal-Net Semimetal in a Graphene Network Structure

Topological semimetals are characterized by the nodal points in their electronic structure near the Fermi level, either discrete or forming a continuous line or ring, which are responsible for exotic properties related to the topology of bulk bands. Here we identify by ab initio calculations a distinct topological semimetal that exhibits nodal nets comprising multiple interconnected nodal lines in bulk and have two coupled drumheadlike flat bands around the Fermi level on its surface. This nodal net semimetal state is proposed to be realized in a graphene network structure that can be constructed by inserting a benzene ring into each $\text{C}─\text{C}$ bond in the bct-${\mathrm{C}}_4$ lattice or by a crystalline modification of the (5,5) carbon nanotube. These results expand the realm of nodal manifolds in topological semimetals, offering a new platform for exploring novel physics in these fascinating materials.

Figure 1

(a) The 40-atom unit cell of bct-${\mathrm{C}}_{40}$ with lattice parameters $\mathrm{a}=12.9012\AA$, $c=2.4599\AA$, occupying the $8h$ (0.7257, 0.7257, 0.5), $16l_1$ (0.7370, 0.6096, 0.5), and $16l_2$ (0.7615, 0.0558, 0.5) Wyckoff positions denoted by ${\mathrm{C}}_1$, ${\mathrm{C}}_2$ and ${\mathrm{C}}_3$, respectively. (b) A 20-atom primitive cell of bct-${\mathrm{C}}_{40}$ viewed as a crystalline modification of the (5,5) CNT.

Figure 2

Calculated energies vs volume per atom for bct-${\mathrm{C}}_{40}$ compared to graphite, fcc-${\mathrm{C}}_{60}$ [60], bco-${\mathrm{C}}_{16}$ [35], (5,5) CNT, bct-${\mathrm{C}}_4$ [17], diamond, and IGN-${\mathrm{C}}_6$ [32].

Figure 3

Calculated bulk electronic band structures of bct-${\mathrm{C}}_{40}$ at equilibrium lattice parameters using the standard GGA-PBE functional. (a) The results along several high-symmetry directions. $G_1$ and $G_2$ indicate the irreducible representations of the two crossing bands, respectively. (b) The bulk Brillouin zone with several high-symmetry points indicated at $G$(0.00, 0.00, 0.00); $X$(0.00, 0.00, 0.50); $P$(0.25, 0.25, 0.25); $N$(0.00, 0.50, 0.00); $Z$($-0.50$, 0.50, 0.50); $L$(0.26, 0.26, $-0.26$), and $A$($-0.26$, 0.74, 0.26). (c) Two symmetric boxed-asterisk shaped nodal nets consisting of six nodal lines of 3-0-2, 7-0-8, 6-7-1, 1-2-5, 6-0-5, and 4-0-1, which link the high-symmetry nodal points 0 (0.192, 0.192, $-0.192$), 1 ($-0.301$, 0.699, 0.301), 2 (0.195, 0.195, 0.305), and others (points 3 through 8) that are generated from points 0, 1, and 2 by symmetry operations.

Figure 4

Calculated surface band structures of bct-${\mathrm{C}}_{40}$ using the standard GGA-PBE functional. (a,b) The (110) surface states obtained using an eight-layer-thick slab geometry along the [110] direction; (c,d) The (100) surface states obtained using a ten-layer-thick slab geometry along the [100] direction. In (b,d) the surface dangling bonds are saturated with hydrogen atoms. The surface bands are marked in red solid and green dot lines. (e,f) The projected nodal lines on the $\overline{X}\text{−}\overline{P}\text{−}\overline{L}\text{−}\overline{G}$ (110) and $\overline{Z}\text{−}\overline{A}\text{−}\overline{L}\text{−}\overline{G}$ (100) planes of the bulk Brillouin zone, corresponding to the projected surface Brillouin zones of the (110) and (100) planes in (a–d).