Directed growth of hydrogen lines on graphene: High-throughput simulations powered by evolutionary algorithm

We set up an evolutionary algorithm combined with density functional tight-binding calculations to investigate hydrogen adsorption on flat graphene and graphene monolayers curved over substrate steps. During the evolution, candidates for the new generations are created by adsorption of an additional hydrogen atom to the stable configurations of the previous generation, where a mutation mechanism is also incorporated. Afterwards a two-stage selection procedure is employed. Selected candidates act as the parents of the next generation. The evolutionary algorithm predicts formation of lines of hydrogen atoms on flat graphene. In curved graphene, the evolution follows a similar path except for a new mechanism, which aligns hydrogen atoms on the line of minimum curvature. The mechanism is due to the increased chemical reactivity of graphene along the minimum radius of curvature line (MRCL) and to ${\mathrm{sp}}^3$ bond angles being commensurate with the kinked geometry of hydrogenated graphene at the substrate edge. As a result, the reaction barrier is reduced considerably along the MRCL and hydrogenation continues like a mechanical chain reaction. This growth mechanism enables lines of hydrogen atoms along the MRCL, which has the potential to overcome substrate or rippling effects and could make it possible to define edges or nanoribbons without actually cutting the material.

Figure 1

Illustration of the directed growth of hydrogen atoms on the minimum radius of curvature line. Propagation of the hydrogen atoms along the first and the second minimum radius of curvature lines leads to formation of a kink line.

Figure 2

Flat and curved graphene structures are shown and the adsorption sites are labeled as ortho (O)-, para (P)-, and meta (M)-configurations with respect to the occupied site.

Figure 3

Flow chart of the evolutionary algorithm. The algorithm starts with generation of candidate configurations (red circles) from the parent configurations (blue squares). After DFTB based total energy calculation of the relaxed geometries, preselection is performed among the sister configurations, where the threshold energy is set to 0.2 eV. Preselected configurations are collected at a pool and a second selection procedure is carried out taking total energy and symmetry properties into consideration. Successful candidates (green diamonds) are considered as the stable configurations of the current generation and they are used as parents for the next generation. Candidates that are eliminated at the pool-selection stage are marked with black triangles. (Blue squares, green diamonds, and black triangles are used correspondingly in Figs. 4, 6, and 7, whereas red circles are used in Figs. S2–S8 [48].)

Figure 4

Evolution of generations on flat graphene. Parent atoms (blue squares), stable configurations (green diamonds), and configurations that are eliminated at the pool selection stage (black triangles) are shown throughout the evolution (upper left). Generations are separated with dotted lines and denoted as D, Tr, Te, P, and H for dimers, trimers, tetramers, pentamers, and hexamers, respectively. Each panel corresponds to a family; the family tree is denoted at the upper left corner of each panel. The evolution of generations can also be tracked at the lower panel, where each row stands for a generation and each box for a candidate with the same enumeration with the geometry plots. The stable, eliminated, and mutant configurations are shown with green, black, and pink boxes, respectively. Relative total energies of simulated configurations are shown on the right, where the green zone indicates the 200 meV threshold value. The complete list of simulated geometries can be found in Figs. S2–S6 and their corresponding energies at Table SI [48]. See also the Supplemental Material [56].

Figure 5

Curvature and binding to curved graphene. Energetics of single hydrogen adsorption on curved graphene with bending angles ${52}^{\circ }$ and ${90}^{\circ }$ [(a) and (c), respectively]. In the insets top view and side view of the configurations are illustrated. Pink dots denote tested configurations and blue dots show most favorable positions. Configurations 7 and 6 ensure the positions of the minimum radius of curvature lines (MRCL) for ${52}^{\circ }$ and ${90}^{\circ }$ bending. The MRCLs are depicted in the insets. In (b) and (d) red circles indicate graphene atoms, the blue lines show the radius of curvature (ROC), and the green lines show the inverse radius of curvature (IROC).

Figure 6

Evolution of generations during the hydrogenation of curved graphene (${52}^{\circ }$). The same color codes and notation are applied as in Fig. 4. The complete list simulated structures can be found in Fig. S7 [48].

Figure 7

Evolution of generations during the hydrogenation of curved graphene (${90}^{\circ }$). The same color codes and notation are applied as in Fig. 4. The complete list of simulated structures can be found in Fig. S8 [48].