Chapter 5 Alignment of graphene islands on low symmetric Cu substrates
5.4 Results and Discussions
5.4.1 Alignment of graphene islands on Cu{111}-based low symmetric surfaces
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by PAW.349 Periodic boundary conditions are applied. Along the step edge or graphene edge direction, the length difference between graphene edge and step edge is limited to 3.0%, and to avoid strain in graphene the graphene lattice length is kept unchanged. Along the direction perpendicular to the step direction, the distance between two steps is set to be ~20 Å for all structures. Along the out of plane direction, the vacuum spacing between neighboring images is set to be larger than 12 Å. For structural optimization, the force criteria on each atom is less than 0.01 eV/Å, and an energy convergence of 10-4 eV is used.
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Figure 5.5 (a) Formation energies and configurations of various graphene edges attaching to the two straight steps of Cu{111}-based low symmetric surfaces. (b) Sketch of graphene islands along a curved step varying from -90 ~ 90º. Graphene islands along straight steps all show same orientation.
Above calculations do not consider tilted Cu step edges, but there must be a large amount of tilted step edges between the transition of the two straight steps. Therefore, we further calculated the formation energies of different graphene edges attaching to various tilted step edges, as shown in Table 5.1. Graphene ZZ direction (G<10>) and Cu<110> direction are chosen as the references of graphene orientaion and Cu step edge orientations respectively, along the diagonal of the table, the relative angle of the grpahene edge to its reference and that of Cu step edge to its reference are same, and the formation energy is smallest along both row and column directions. It is worth noting that, except the matched interfaces (iv) of Figure 5.3(d) where tilted graphene edges match the tilted Cu step edges, the interfaces (iii) with graphene ZZ edge attaching to tilted Cu step edges are stable than other unmatched interfaces with tilted garphene edges attaching to tilted Cu step edges. The corresponding atomic configurations with lowest formation energies are shown in Figure 5.6 (a), graphene edges and Cu step edges all match very well at the interfaces, satisfying all requirements of interface (type (iv) of Figure 5.3(d)). Moreover, in consistent with the alignment of graphene on the straight Cu step edges, ZZ segments of graphene tilted edges prefer to align along <110> step segments of the SE<211>𝑖×<110> step edges, and AC segments of graphene tilted edges prefer to align along <211> step segments of the SE<110>𝑖×<211>step edges.
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Table 5.1 Formation energies of graphene edges attaching to different Cu step edges of Cu{111}-based low symmetric surfaces.
Ef (eV/Å) G<10> G<14> G<13> G<12> G<23> G<34> G<11>
SE0<110> (Cu<110>) 0.35 0.44 0.46 0.49 0.50 0.47 0.45
SE<211>3×<110> (Cu<145>) 0.49 0.35 0.51
SE<211>2×<110> (Cu<134>) 0.48 0.37 0.51 0.53
SE<211><110> (Cu<123>) 0.46 0.49 0.39 0.54 0.52
SE<110>2×<211> (Cu<235>) 0.47 0.49 0.40 0.53
SE<110>3×<211> (Cu<347>) 0.49 0.40 0.52
SE0<211> (Cu<112>) 0.46 0.50 0.46 0.47 0.55 0.53 0.39
Figure 5.6 (a) Matched interfaces of graphene edges attaching to different kinked Cu step edges. (b) The intact evolution of graphene orientation (dark lines) versus different step edges. Light lines showing kink density of the step edges.
Choosing the Cu[110] step direction in Figure 5.5(b) as the reference of 0º, the graphene ZZ direction with the reference is used to represent the orientation of graphene island. Figure 5.6(b) shows the calculated graphene orientation profile (solid dark line) as a function of the orientation of the Cu
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step edges on the Cu{111}-based high-index substrates. The small hexagons in Figure 5.6(b) denote the corresponding orientations graphene islands. On Cu substrates with the straight SE0<110> (60º× 𝑛, n is an integer), graphene ZZ edge aligns along the step edges and the graphene orientation is 0º. On Cu substrates with the straight SE0<211> (30º+60º× 𝑛), graphene AC edge is parallel to the step, meanwhile, graphene ZZ direction is parallel to Cu<110> direction and therefore the graphene orientation is also 0º. On Cu substrates with tilted step edges, as inferred in the interface (type (iv) in Figure 5.3(d)), the kink height difference between the tilted graphene edge and the tilted Cu step edge would induce a small misorientation angle between their segments. As shown in Figure 5.6 (b), on Cu substrates with SE<211>𝑖×<110> step edges, graphene ZZ direction deviates from Cu<110> direction with a small misorientation angle varying in a range of 0º ± 0.74º (solid black line). Similarly, on Cu substrates with SE<110>𝑖×<211> step edges, graphene AC (ZZ) direction deviates from Cu<211> (Cu<110>) direction with a small misangle ranging in 0º ± 0.41º. Although segments have more contributions and should be the dominant factors in determining the orientation of graphene islands, the effects of kinks cannot be ignored especially for the step edges with a high kink density. Thus, the dotted dark lines representing graphene orientation determined by kinks are also plotted. On Cu substrates with step edges with a high kink density, graphene islands may have a large chance to show two orientations.
However, on all Cu{111}-based high-index low symmetric surfaces, the two orientations are quite close, with a difference less than 1º, which is difficult be identified in experiments.
In summary, graphene orientation is not sensitive to the step edge variation on Cu{111}-based high-index low symmetric substrate. Therefore, unidirectionally aligned graphene islands can be easily achieved on these substrates regardless of the directions of the step edges, which means that Cu{111}- based high-index low symmetric substrates show a very large tolerance for the synthesis of unidirectionally aligned graphene.
5.4.2 Alignment of graphene islands on Cu{100}-based high-index low symmetric substrates