4.1 Starting from graphite-2H

We did a preliminary test at 20 GPa starting from the graphite-2H structure, and successfully found diamond as the ground state, and M-carbon and bct-C$_4$ as metastable states. In the calculation we set $d_{\rm max}$=2.5 , $W$=4000 kbar$\cdot $$^3$ and $\delta {h}$=0.6 .

Figure 8.6: (a) Enthalpy evolution during the compression of graphite-2H at 20 GPa; (b) Enthalpy evolution during the compression of graphite-3R at 20 GPa (black line: enthalpies for best structures with constant cell matrix; red line: enthalpies for best structures after full relaxation).

Fig. 8.6 shows the enthalpy evolution. Graphene layers (Fig. 8.7a), persisted until the 15th generation. Then, upon sufficient cell deformation, the layers began to buckle, and the planar structure transformed into 3D-networks of sp$^3$-hybridized carbon atoms. Lonsdaleite with 6-membered rings (Fig. 8.7b) appeared as the best structure in the 16th generation. We observed in the same generation the bct-C$_4$ structure with 4+8 membered rings (Fig. 8.7c), and M/W-carbon structures containing 5+7 membered rings (Fig. 8.7e,f), appeared shortly after. Lonsdaleite survived for a few generations until it transformed into a hybrid structure made of alternating layers of M-carbon and diamond (Fig. 8.7g, we refer to it as M+D type), followed by the transition to another hybrid structure made of bct-C$_4$ and diamond (Fig. 8.7d, similarly, we refer to it as B+D type). Diamond was dominant in the following generations. At the 51st generation, the system reverted to graphite.

Figure 8.7: Structures observed during compression of graphite-2H at 20 GPa. (a) graphite-2H (2 $\times $ 2 $\times $ 2 supercell of the calculation model); (b) lonsdaleite; (c) bct-C$_4$ with 4+8 membered rings; (d) Z-carbon (belonging to B+D type); (e) M-carbon with 5+7 membered rings; (f) W-carbon with 5+7 membered rings; (g) M+D type carbon. Polygons are highlighted by different colors (quadrangle: turquoise; pentagon: green; hexagon: blue).

The power of the evolutionary metadynamics method lies in that it is highly suitable for harvesting low-energy metastable structures in addition to the ground state. Those previously proposed candidate structures for the product of cold compression of graphite, bct-C$_4$, M, and W-carbon are all easily recovered in a single simulation. More interestingly, we also observed many low-energy structures based on 5+7 or 4+8 topology. The B+D type structure (Fig. 8.7d) observed in the simulation is actually the oC16 structure (sometimes called Z carbon), recently suggested as a candidate for superhard graphite. Since oC16 inherits layers of bct-C$_4$ and diamond, there is no surprise that its thermodynamic properties are intermediate between these two structures. The 5+7 class of structures shows a larger diversity. In some of these structures, 5-membered rings form pairs, while in others these they are single. The difference of the 5-membered ring pairs’ orientation leads to two allotropes: M-carbon (Fig. 8.7e) and W-carbon (Fig. 8.7f). Some structures can be thought of as combinations of layers of the M-carbon and diamond structures (M+D carbon, as shown in Fig. 8.7g)