Extracting Crystal Chemistry from Amorphous Carbon Structures

Abstract Carbon allotropes have been explored intensively by ab initio crystal structure prediction, but such methods are limited by the large computational cost of the underlying density functional theory (DFT). Here we show that a novel class of machine‐learning‐based interatomic potentials can be used for random structure searching and readily predicts several hitherto unknown carbon allotropes. Remarkably, our model draws structural information from liquid and amorphous carbon exclusively, and so does not have any prior knowledge of crystalline phases: it therefore demonstrates true transferability, which is a crucial prerequisite for applications in chemistry. The method is orders of magnitude faster than DFT and can, in principle, be coupled with any algorithm for structure prediction. Machine‐learning models therefore seem promising to enable large‐scale structure searches in the future.

: znp and its structural relatives. The znp structure is the lowest-energy network (besides dia/lon and their polytypes) found in our search. The structure contains three types of cages, and they are linked as shown above (yellow, orange, and red, respectively); similar to what we discuss in the main text, these fragments can be combined with dia or lon spacers to form various related structures. The topological information for all of them is provided below: the transitivity is given in green, and the tiling symbols in blue (see also Tables S1-S3). This structure is the lowest-energy network newly found here that does not contain any dia or lon cages. The tiling pattern is colourcoded as in Figure S2; the same tiling but with a different linkage is found in cbn. can be linked to energetic stability, as discussed in L. Öhrström, M. O'Keeffe, Z. Kristallogr., 2013, 228, 343-346 (Ref. [3d] in the main article). Labels are given for some representative structures (cf. Tables S1-S3).
Below a value of ≈ 1.3 (indicated by a vertical dashed line), no low-energy structures are found. In fact, this ratio is largest overall for the diamond structure (dia), where the optimized covalent bond length is r C-C = 1.546 Å, and the next-nearest neighbour distance amounts to r C···C = 2.525 Å, leading to a ratio of 1.63. Figure S6: Structural order parameters and their relation to energetic stability. We examine two parameters, both of which are defined such that they approach 1.0 for perfect tetrahedral coordination:
• Energy above diamond (eV/at.) Percent dia+lon cages in structure S8 Figure S8: No direct correlation between structural simplicity and energetic stability.
We here expand upon the question whether "simpler" carbon allotropes are generally more stable. A measure for structural simplicity can be obtained from the transitivity symbol [N n N e N r N c ], giving the number of independent nodes, edges, rings, and cages (see above). The highest-order measure is thereby N n , followed by N e , and so on. Hence, reading the transitivity symbol as a four-digit number directly enables a qualitative ranking of the allotropes' simplicity (this is straightforward for all N ≤ 9, to which we restrict our analysis here).
For example, dia is the simplest net (transitivity symbol [1111] → rank 1111), followed by unj (rank 1221), lon (rank 1222), and crb, uni, and unc (all rank 1232: note how different nets can have the same transitivity). These examples alone prove that no correlation exists between structural simplicity and energetic stability: dia is the most stable carbon allotrope with fourfold coordination, whereas unc (rank 1232) already lies more than 1 eV/at. above it; similarly, some more complex nets such as cbn (rank 4863) have very low energy (0.16 eV/at. above diamond). The plot above provides a more comprehensive perspective on this. 1 1 Furthermore, there are restrictions on the accessible transitivity symbols: First, for N n nodes there must be at least (N n -1) edges, because these nodes need to be connected in the primitive cell. Furthermore, in all-fourfold coordinated structures, there is also a limit on the maximum number of edges given by N e ≤ (3N n + 1). Both restrictions are indicated in the plot by gray shading. Table S1. Structures found in our search that are known from SACADA. The leftmost column contains a running index (assigned in the order that structures were found; the index is therefore more or less arbitrary). For each structure, the table provides the number of independent nodes N, the topology, transitivity, and tiling symbols. [a] cfe (9-layered SiC polytype, consisting purely of dia and lon cages) [b] known from zeolites (see references in the main text) [c] known in RCSR (http://rcsr.net), but not yet described for carbon allotropes [d] known from metal-organic frameworks (MOFs; in the TTD ToposPro database) §