Stochastic generation of complex crystal structures combining group and graph theory with application to carbon

A method is introduced to stochastically generate crystal structures with defined structural characteristics. Reasonable quotient graphs for symmetric crystals are constructed using a random strategy combined with space group and graph theory. Our algorithm enables the search for large-size and complex crystal structures with a specified connectivity, such as threefold ${\mathrm{sp}}^2$ carbons, fourfold ${\mathrm{sp}}^3$ carbons, as well as mixed ${\mathrm{sp}}^2\text{−}{\mathrm{sp}}^3$ carbons. To demonstrate the method, we randomly construct initial structures adhering to space groups from 75 to 230 and a range of lattice constants, and we identify 281 new ${\mathrm{sp}}^3$ carbon crystals. First-principles optimization of these structures show that most of them are dynamically and mechanically stable and are energetically comparable to those previously proposed. Some of the new structures can be considered as candidates to explain the experimental cold compression of graphite.

Figure 1

The flow chart of our algorithm to generate, n-connected crystalline network.

Figure 2

The total energies per atom of the ${\mathrm{sp}}^3$ carbon crystals belonging to space groups from 75 to 230 relative to diamond (a). The total energies are mapped as radius and the space group numbers are mapped as degrees. The red circles and blue solid five-point stars denote the ${\mathrm{sp}}^3$ carbon crystals discovered in previous work and those newly discovered in our present work, respectively. The scatter plot of the relative average energy (eV/atom) against equilibrium volume (${\AA }^3$/atom) for ${\mathrm{sp}}^3$ carbons with tetragonal, hexagonal, and cubic symmetry (b).

Figure 3

The perspective crystalline views of the clathrate-like 215-10-III-4-001 and the diamond-like 201-3-fff-4-001, 228-3-lll-4-001, and 230-3-jjj-4-001 with only six-member rings.

Figure 4

The perspective crystalline views of 174-10-IID-4-001, 179-3-DDn-4-001, 176-5-IID-4-001, 193-3-iiD-4-001, and 194-3-hhD-4-001 can be structurally translated from graphite.

Figure 5

The simulated XRDs of our ${\mathrm{sp}}^3$ hexagonal carbons, which can be considered as good candidates for explaining the superhard carbon crystals synthesized by cold-compressing graphite.

Figure 6

The perspective crystalline views of the most stable four new tetragonal ${\mathrm{sp}}^3$ carbon allotropes discovered by ${\mathrm{RG}}^2$. They are 137-3-eef-4-001, 109-4-ffj-4-001, 139-3-iic-4-001, and 138-4-ddM-4-001.