Chapter 11 Rotated graphene moiré superstructures on various transition metal surfaces
11.4 Formation mechanisms of graphene moiré superstructures on the Ru(0001) surface
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Figure 11.7 Atomic configurations of 1- and 2-layer Pt clusters on various graphene/Ru(0001) templates, with the corresponding formation energy of Pt clusters on graphene/Ru(0001) templates (EPt) showing at the bottom.
11.4 Formation mechanisms of graphene moiré superstructures on the Ru(0001) surface
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Figure 11.8 Two models of graphene on the Ru(0001) surface with grey representing the graphene film and blue representing the Ru(0001) substrate. In model I, h denotes the height oscillation degree and r denotes the radius of the hump area.
In the following, the binding energies and curvature energies of graphene/Ru(0001) systems are investigated first. Figure 11.9 shows the atomic configurations of graphene on the four high- symmetric sites of the Ru(0001) surface, i.e., ATOP, FCC, HCP and Bridge as referred above. Because of the fluctuations of graphene layers on the Ru(0001) surface with rotation angles less than 20º are always site-dependent, the binding energies of graphene at the four high-symmetric sites of the Ru(0001) surface are calculated separately based on the definition:
𝐸𝐵 = (𝐸𝑇−𝐸𝑆𝑢𝑏− 𝐸𝐺𝑟𝑎)/𝑁𝐶 (11.8)
where 𝐸𝑇 is the total energy of a structure with graphene lattice on a high-symmetric site of Ru(0001) lattice, as shown in Figure 11.9(a), 𝐸𝑆𝑢𝑏 and 𝐸𝐺𝑟𝑎 are the energies of freestanding Ru(0001) and freestanding graphene in the cell, respectively. 𝑁𝐶 denotes the number of C atoms in the graphene lattice.
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Figure 11.9 (a) Lattice relationships of graphene on the four high-symmetric sites of the Ru(0001) surface. (b) Binding energies of graphene with the four sites of the Ru(0001) surface as a function of the distance. (c) Illustration of a graphene/Ru(0001) common cell with different lattice relationships distinguished by colors. (d) Linear fitting of the curvature energies of corrugated graphene moiré superstructures. (e) Fitting of the height of graphene layers on the Ru(0001) surface versus the length of graphene/Ru(0001) common cell (L). (f) Curvature energies of graphene layers on the Ru(0001) surface as a function of the rotation angle and the dashed line denotes the binding energy increase of graphene/Ru(0001) with graphene changing from an ultra-flat to a corrugated structure.
From Figure 11.9(b), we see that the binding energies of graphene on the FCC, HCP and Bridge sites are quite strong with optimized distances of ~2.1Å, in contrast, that on the ATOP site becomes much weaker with an optimized distance of ~3.6 Å, which are in agreement with the height oscillation degree of the 0º graphene moiré superstructure on the Ru(0001) surface, implying that the corrugated graphene layers are cooperative phenomena by distinct interactions between graphene and different sites of the Ru(0001) surface. Therefore, the binding energy of graphene on the Ru(0001) surface can be estimated by the summation of that on the four high-symmetric sites of the Ru(0001) surface. In a graphene/Ru(0001) common cell, as shown in Figure 11.9(c), areas that correspond to graphene on different sites are marked by different colors. There are 1/3, 1/8 and 1/8 graphene lattice locating on the ATOP, FCC and HCP sites with the rest locating on the Bridge site, based on this, the binding energy is estimated to be ~ -208.7 eV. Using the binding energy of graphene on the ATOP site as the reference
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of vdW interaction, the binding energy increase of the graphene/Ru(0001) superstructure with graphene changing from an ultra-flat to a corrugated structures is ~ 136.9 eV.
The curvature energy of a graphene/Ru(0001) superstructure is mainly from hump part of the graphene layer, and for graphene, its curvature energy (𝐸𝐶) and curvature radius (R) satisfy a relationship of 𝐸𝐶 ∝ 1
𝑅2 .421,422 From the model I of Figure 11.8(a), the curvature radius of a corrugated graphene layer on a TM substrate can be obtained by the formula of (𝑅 − ℎ)2+ 𝑟2= 𝑅2, where h and r denote the height and the radius of the hump section, respectively. Due to ℎ ≪ 𝑟, 𝑅 =ℎ2+𝑟2
2ℎ ≈𝑟2
ℎ , and therefore 𝐸𝐶 ∝ (ℎ
𝑟2)2. By fitting curvature energies of the corrugated graphene layers on the Ru(0001) surface in Figure 11.9(d), we get:
𝐸𝐶 = 50539.98 × (ℎ 𝑟2)
2
(11.9) Besides, it is found that the height of graphene layer (h) on the Ru(0001) surface and the length (L) of graphene/Ru(0001) common cell satisfy a relationship of ℎ = ℎ0× (𝛽 ∙ 𝐿 + 𝛾 ∙ 𝐿2), as shown in Figure 11.9(e). Because of ℎ = ℎ0 happens at 𝐿 = 𝐿0, where ℎ0 and 𝐿0 correspond to the 0º graphene/Ru(0001) superstructure, we get
ℎ = ℎ0× ((1
𝐿0− 𝛾 ∙ 𝐿0) ∙ 𝐿 + 𝛾 ∙ 𝐿2) (11.10)
and 𝛾 =-1.25e-3 by fitting with the corrugated graphene/Ru(0001) superstructures.
Assuming graphene layers on the Ru(0001) are always corrugated, Figure 11.9(f) gives curvature energies of graphene layers obtained from Eqs. (11.8) and (11.9), with the binding energy increase value of ~136.9 is marked by a dash line. It turns out the demarcation of the corrugated and ultra-flat graphene structures is 19.5º, which perfectly agrees with our DFT results, confirming the proposal that graphene layer on a TM surface is determined by the competition of binding energy and curvature energy of a graphene/Ru(0001) superstructure. Besides, the ultra-flat and corrugated graphene/Ru(0001) superstructures with large rotation angles are compared, as displayed in Figure 11.10, it is very clear that the ultra-flat ones always show advantages in total energy when the rotation angle is larger than 20º, further confirming the prominent roles of the corrugated graphene/Ru(0001) superstructures at large rotation angles.
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Figure 11.10 Comparison the total energies of the ultra-flat and corrugated graphene layers on the Ru(0001) substrate with larger rotation angles, and the unit of the total energy here is eV.