Interpenetrating graphene networks: Three-dimensional node-line semimetals with massive negative linear compressibilities

We investigated the stability and mechanical and electronic properties of 15 metastable mixed $s{p}^2\text{−}s{p}^3$ carbon allotropes in the family of interpenetrating graphene networks (IGNs) using density functional theory (DFT). IGN allotropes exhibit nonmonotonic bulk and linear compressibilities before their structures irreversibly transform into new configurations under large hydrostatic compression. The maximum bulk compressibilities vary widely between structures and range from 3.6 to 306 ${\mathrm{TPa}}^{-1}$. We find all the IGN allotropes have negative linear compressibilities with maximum values varying from –0.74 to $–133{\mathrm{TPa}}^{-1}$. The maximal negative linear compressibility of Z33 ($–133{\mathrm{TPa}}^{-1}$ at 3.4 GPa) exceeds previously reported values at pressures higher than 1.0 GPa. IGN allotropes can be classified as either armchair or zigzag type, and these two types of IGNs exhibit different electronic properties. Zigzag-type IGNs are node-line semimetals, while armchair-type IGNs are either semiconductors or node-loop or node-line semimetals. Experimental synthesis of these IGN allotropes might be realized since their formation enthalpies relative to graphite are only 0.1–0.5 eV/atom (that of ${\mathrm{C}}_{60}$ fullerene is about 0.4 eV/atom), and energetically feasible binary compound pathways are possible.

Figure 1

Illustration of design principle used to construct IGNs with larger pores. The building units are armchair chains and zigzag chains in A-type and Z-type allotropes, respectively. Nonstandard primitive cells are used so that $c$ axes are along the pore and chain directions. The black spheres indicate $s{p}^3$ hybridized carbon atoms. The red and blue spheres indicate $s{p}^2$ hybridized carbon atoms along $a$ and $b$ axes, respectively. The same representations are used in the next figures, except that we only use blue spheres to represent all $s{p}^2$ hybridized carbon atoms.

Figure 2

The crystal structures of (a) A-type and (b) Z-type IGN allotropes.

Figure 3

Pressure-dependent enthalpies of IGN carbon allotropes and diamond relative to graphite.

Figure 4

Pressure-dependent (a) relative volumes and (b) bulk compressibilities of carbon allotropes.

Figure 5

Pressure-dependent lattice distances in 13 crystal directions of (a) A13 and (b) Z13. In this work, lattice distance indicates the distance between the two closest lattice points in the corresponding direction.

Figure 6

Pressure-dependent linear compressibilities of IGN allotropes in the most positive and most negative directions. The directions with most positive and negative linear compressibilities are conjugated with each other and in the same surface are perpendicular to the pore direction. In the primitive cell, if $c$ ([001]) represents the pore direction, and $a$ ([100]) and $b$ ([010]) indicate two directions parallel to pore sides, as shown in Fig. 5, then [110] and $\left[1\overline{1}0\right]$ are the directions with the most negative and positive linear compressibilities.

Figure 7

Electronic band dispersion relations and densities of states for (a) Z13, (b) A13, and (c) A33.

Figure 8

Fermi surface (a,b) and isoenergy difference surface (c) in A13. (b) A $\mathrm{\Gamma }$-point-centered local representation of (a) (indicated by the dashed rectangle), with a $k$-point density 2500 times larger than in (a). No Fermi surface can be found in (a) other than the space shown in (b). (c) An isoenergy difference surface (0.01 eV) between the highest-energy valance band and the lowest-energy conduction band. $b_1,b_2$ and $b_3$ in this and following figures indicate the reciprocal lattice directions corresponding to $a,b$, and $c$ of the Bravais lattices.

Figure 9

Fermi surface (a,b,c) and isoenergy difference surface (d) for Z13. (b) The local space enclosed by the dashed rectangle in (a) that includes the Fermi surface, and (c) represents the lower portion of (b), but from different lattice directions. The $k$-point density in (c) is 8 times larger than in (b) and 250 times larger than in (a). (d) The isoenergy difference surface (0.05 eV) between the highest-energy valance band and the lowest-energy conduction band.

Figure 10

Fermi surface (a), isoenergy difference surface (b), and two-dimensional electronic band dispersion in Z11. (a) The Fermi surface using a $k$-point grid at intervals of 0.001, 0.001, and 0.000 5 ${\text{Bohr}}^{-1}$ in the $b_1+b_2$, $b_2-b_1$, and $b_3$ directions, respectively. (b) An isoenergy difference surface (0.05 eV) between the highest-energy valance band and the lowest-energy conduction band. (c, d) Two-dimensional electronic band dispersion viewed from different projections in a plane using a fixed value in the $b_2-b_1$ direction. (e) The band-energy difference within the same plane of reciprocal space used in (c) and (d). (f) The two-dimensional electronic band dispersion in a plane with fixed value in the $b_1+b_2$ direction.