Diagrams viewing in the directions of z-x and z-y planes of Si-terminated 6H-SiC substrate. For illustration, the unit cell of 12 atoms is depicted in (a) with yellow color balls for Si atoms and cyan for C atoms, and is repeated 4 times first along the x-direction shown in (b) and then the set of four unit cells is repeated also 4 times along the y ′ path that subtends at an angle 60o north of x-axis (see (c) and Fig. 2 ). The grey balls in (a)–(c) are Si atoms which repeat periodically the Si atoms on top. The distance between Si–Si (C–C) is 3.08 Å in both (b) and (c). In the z-direction, the distance between Si–Si (C–C) is 2.52 Å, whereas between Si–C it is 0.624 Å and 1.896 Å for the intra- and inter-bilayer, respectively.
Substrate of Si-terminated 6H-SiC(0001) taken as the initial configuration input in LAMMPS software for performing simulated annealing. Standard periodic conditions are applied along the x-axis and along a path in y ′ -direction of the substrate, which has an orthorhombic structure with lattice parameters a = 3.08 Å in x-axis direction, b = 3.08 Å in the direction y ′ along a path that subtends an angle of 60o north of x-axis, and c = 15.12 Å in z-axis direction. The orthorhombic substrate has angles α = β = 90o, γ = 120o subtended between z- and x-axis, z- and y ′ -axis, and x- and y ′ -axis, respectively. Same color notations for Si and C atoms are used as in Fig. 1 .
The binding energy (per atom) E b for an infinite free graphene layer calculated with the Tersoff (solid circles or dashed line) and TEA (open circles or full line) potentials at different values of lattice constant a 0. The lowest values of E b are a 0 = 2.53 and 2.56 Å for Tersoff and TEA potentials, respectively. These values may be compared with the tight-binding calculation of Reich et al. 51 (a 0 = 2.468 Å) and DFT calculation of Gan and Srolovitz 52 (a 0 = 2.472 Å). Notice that the E b corresponding to Tersoff and TEA potentials crosses at a 0 = 2.584 Å and for a 0 > 2.6 Å the E b for TEA potential is relatively lower in energy.
One-layer graphene overlaid on Si-terminated 6H-SiC(0001) obtained by the simulated annealing method for (a) Tersoff (second column) and (b) TEA (third column) potentials. In the second and third columns at the bottom corner on the right, the integer is the hexagon number. The average distance of separation between the graphene buffer layer and surface is about 2.43 Å for TEA potential.
The variation of the binding energy (per atom) E b plotted against the equilibrium annealing time steps (Δt = 0.5 fs) at (a) 1200 K for Tersoff potential, (b) 1100, and (c) 1200 K for TEA potential.
Comparison of the average bond-length ℓ (Å) versus annealing temperature T (in units Kelvin) between results calculated using TEA (open circle) and Tersoff (solid circle) potentials. Our criterion of a bond-length is ℓ ≤ 1.6 Å for the formation of graphene. This criterion yields T tr = 1200 and 1500 K for TEA and Tersoff potentials, respectively (see text).
Same as Fig. 6 except for the binding energy (per atom) E b .
(a) Pair correlation function g(r) of carbon atoms obtained using TEA potential at different annealing temperature T (in units of Kelvin) for the one-layer graphene which emerges for T ≥ 1200 K. At T < 1200 K, it displays typical crystalline structure. (b) Same as Fig. 8(a) except that the MD simulations were done using Tersoff potential. The one-layer graphene emerges at T ≥ 1500 K. At T < 1300 K, it displays typical crystalline structure. Only few hexogons are seen at T = 1400 K.
Two-layer graphene overlaid on 6H-SiC(0001) obtained by simulated annealing method with TEA potential. In the second and third columns at the bottom corner on the right, the integer is the hexagon number. The first graphene “buffer” layer refers to one closest to the top surface of substrate and has an average distance of separation about 2.37 Å, and the second layer corresponds to one next to the first graphene layer and these graphene layers are separated by an average distance about 3.13 Å.
(a) The average bond-length ℓ (Å) versus annealing temperature T (in units of Kelvin) obtained by the simulated annealing using TEA potential for two-layer graphene. Notations used are: first-layer graphene, open circle; second-layer, solid circle. (b) The binding energy (per atom) E b (in units of eV) versus annealing temperature T (in units of Kelvin) obtained by simulated annealing using TEA potential for two-layer graphene. Same notations as Fig. 10(a).
Parameters used for carbon, silicon, and silicon carbide in MD simulation for Tersoff  and TEA  potentials. Explicit definition of these parameters are described in text. The bond order parameters γ ik , c ik , d ik, and h ik in the three-body interactions (Eq. (9) ) are crucial in the present simulated annealing method. They are more quantitatively fitted to experimentally observed values for the TEA potential (see text).
Distance parameters before and after minimization of 6H-SiC(0001) substrate before simulated annealing process for growing one- and two-layer graphene. The entities cc refers to distance of separation between two C-rich sheets and cs between first C-rich sheet (see text) and substrate for the case of studying one-layer graphene. For studying two-layer graphene, cc(upper) and cc(lower) have the same meaning as cc in the one-layer case of graphene except that they refer to upper and lower sets of C-rich layers, respectively. The cs in the case of two-layer graphene corresponds to separation between the first C-rich sheet and substrate and cc′ between the second- and third-layer of C-rich sheets (see text).
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