Chapter 5 Alignment of graphene islands on low symmetric Cu substrates
5.3 Modeling and simulation methods
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connected and thus shows the highest density of kinks. Figures 5.2 (b) and (e) show the structure of Cu{100}-based high-index low symmetric substrates and the kink density profile as a function of the Cu step edge orientation, respectively. Due to C4V symmetry of the Cu(100) surface, the non-repeating range of the step edge orientation for the kink density profile is 0°~45°, which varies from one Cu<110>
direction to its neighboring Cu<100> direction. The highest kink density appears at SE<100><110> or SE<110><100> at a step edge direction of 26.57°. Due to the C2V symmetry of the Cu(110) surface, the non- repeating range of step edge orientation on Cu{110}-based high-index low symmetric substrates is 0°~90°, which varies from one Cu<110> direction to the neighboring Cu<112> direction and then to the neighboring Cu<001> direction. In the 0°~35.26°, 35.26°~70.53°, and 70.53°~90° ranges, step edges are dominated by <110> segments (SE<211>𝑖×<110>), <211> segments (SE<110>𝑖×<211> and SE<100>𝑖×<211>), and <100> segments (SE<211>𝑖×<100>), respectively.
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edge and step edge are parallel (𝑙1 // 𝑙2) and the unit lengths of the two edges are close to each other, and using 𝑙 to denote the unit length of tilted graphene edge or tilted Cu step edge, i.e.,𝑙1≈ 𝑙2=𝑙. However, due to the lattice mismatch between graphene and Cu substrate, the kink heights of tilted graphene edge and that of tilted step edge are different, which induces a small misorientation angle (∆𝛾) between the graphene edge segment and the step edge segment.
Figure 5.3 Four interfacial configurations of different types of graphene edges attaching to different types of step edges. In (d), 𝑙1 and 𝑙2 represent the unit length of tilted graphene edge or tilted Cu step edge; 𝑑1 and 𝑑2 are the kink heights of graphene edge and Cu step edge, 𝛾1 denotes the angle between the tilted graphene edge and the edge segment, and 𝛾2 denotes the angle between the tilted Cu step edge and its segment.
According to the geometrical relationship of interface (iv), the misangle can be derived as followed:
∆𝛾 = 𝛾2− 𝛾1= asin𝑑2
𝑙 − asin𝑑1
𝑙 (5.1)
where 𝑑1 and 𝑑2 are the kink heights of graphene edge and Cu step edge, respectively; 𝛾1 is the angle between the tilted graphene edge and the edge segment, and similarly 𝛾2 is the angle between the tilted Cu step edge and its segment. Obviously, the misorientation angle, ∆𝛾, will change the orientation of graphene islands gradually when the Cu step edge deviates from the straight step edge direction.
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Moreover, the larger kink height difference, the larger misorientation angle and therefore the larger orientation change.
Figure 5.4 (a) Atomic model for calculating the formation energy of a graphene edge attaching to a step edge. (b-c) Atomic models for calculating graphene edge energy.
In order to compare the stabilities of different interfaces quantitatively, we build atomic models with a graphene nanoribbon attaching to a step edge of Cu substrate to mimic the interface between a graphene edge and a Cu step edge (Figure 5.4). The other edge of the graphene nanoribbon is terminated by hydrogen to avoid its interaction with the Cu terrace. Therefore, the formation energies of a graphene edge attaching to a Cu step edge can be defined as: 𝐸𝑓 = 𝜀𝑒𝑑𝑔𝑒− 𝐸𝐵, where 𝜀𝑒𝑑𝑔𝑒 represents the edge energy of a pristine graphene edge and 𝐸𝐵is the binding energy between the edge of the graphene nanoribbon and the Cu substrate step edges.
The binding energy (𝐸𝐵) is thus calculated by: 𝐸𝐵 = (𝐸𝐶𝑢+ 𝐸𝐺𝑟𝑎− 𝐸𝑇𝑜𝑡− 𝐸𝑣𝑑𝑊)/𝐿, where 𝐸Cu, 𝐸Gra, 𝐸Tot and 𝐸vdW are the energies of the Cu substrate slab, graphene ribbon, graphene ribbon attaching to the Cu substrate and the vdW interaction between graphene ribbon and the Cu substrate, respectively. L is the length of the super cell along the step edge direction.
The models used to calculate the formation energy of a pristine graphene edge are shown in Figure 5.4 (b-c). Structure II in Figure 5.4(b) is a graphene nanoribbon with its two edges passivated by hydrogen, and the structure I in Figure 5.4(c) is half of the structure II and only one of its edges is passivated by hydrogen. Thus, 𝜀𝑒𝑑𝑔𝑒 can be obtained by: 𝜀𝑒𝑑𝑔𝑒 = (2𝐸𝐼− 𝐸𝐼𝐼)/𝐿′, where 𝐸𝐼𝐼 and 𝐸𝐼 are the total energies of the wo graphene ribbons with same length, 𝐿′.
5.3.2 Simulation methods
The calculations are performed by DFT-D3 method with VASP.346,347 The exchange-correlation functional is treated by GGA,348 and the interaction between valence electrons and ion cores is treated
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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.