9 Strain in Epitaxial Graphene
9.2. Interface Models for Epitaxial Graphene
It would be computationally more efficient to use smaller-cell approximant phases to the true (6√
3×6√
3)-R30◦supercell. Indeed, different smaller-cell approximant phases have been used in literature [214,215,334,335,289, 243]. However, the residual artificial strain and incorrect bonding in those phases hinder a meaningful surface energies comparison. Sclauzero and Pasquarello [288] compared three different structural models to the ZLG phase, a SiC-(√
3×√
3)-R30◦supercell covered by an (2×2) graphene cell, an unrotated (4×4) SiC cell covered by a (5×5) graphene cell and the experimentally observed SiC-6√
3×6√
3-R30◦ZLGstructure. They com-pared the energy of the three SiC-graphene interface models and identified the SiC-6√
3×6√
3-R30◦ZLGstructure to be the most stable one.
We constructed commensurate graphene-3C-SiC(111) interface structures from rotated hexagonal supercells of 3C-SiC(111) and graphene. We ad-dress artificially induced strain, dangling bond saturation, corrugation in the carbon nanomesh and the surface energy for different coincidence structures.
Figure 9.2.: Comparison of the surface energies for six different smaller-cell approximant models of the3C-SiC(111)ZLGphase, relative to the bulk-terminated (1×1) phase, as a function of the C chemical potential within the allowed ranges using PBE+vdW. The Si rich (3×3) and (√
3×√
3) reconstructions as well as the (6√ 3×6√
3)-R30◦ZLGphase from Fig.10.1are included for comparison. The shaded areas indicate chemical potential values outside the strict thermodynamic stability limits of Eq.4.15.
By rotating a graphene layer on a SiC substrate Sclauzero and Pasquarello [290] found several model interfaces which give almost perfect commen-surability between the graphene and SiC supercells. For a selection of such interface models, they calculated the change in energy of a plane free-standing graphene layer due to the strain necessary to built a commensurate SiC-graphene interface model [290]. They found a change in energy to be smaller than 40 meV for all structures except the (√
3×√
3)-R30◦ model.
We include six different smaller-cell approximant models of the3C-SiC(111) ZLGphase in our analysis:
To our knowledge, structure (a) and (e) have not yet been discussed in literature. We calculated a slightly rotated (5×5) approximant to theZLG phase [242] (structure (b)), a periodicity sometimes seen in experiment [210, 264,307]. Structure (b) has also been discussed in Ref. [290] and Ref. [289].
Structure (c) was first introduced by Sclauzero and Pasquarello [290]. Hass et al. [131] used basic trigonomy to show that for the (6√
3×6√ 3) SiC supercell, three rotational angle relative to SiC substrate can be found the experimentally observed 30◦ (ZLG) and two additional rotational angle
±2.204◦. The graphene layer rotated by 2.204◦can be constructed with the supercell matrix (N), introduced in Sec.11
N =
In addition to theZLGstructure, we include the SiC-(6√
3×6√
3) supercell with a graphene layer rotated by 2.204◦ (structure (d)). In Figure 9.2, we compare the surface energies of the six different smaller-cell approximant models listed above. We also included the Si rich (3×3) and (√
3×√ 3)
SiC cell ZLG cell Angleα NsurfSiC NZLG NDBSi δC[Å] Strain [%] γ[eV/SiC(1x1)]
Table 9.1.: Parameters of the simulatedZLG-3C-SiC(111) interface structures in different commensurate supercells: the3C-SiCandZLGperiodicity, the rotation angle (α), the num-ber of surface Si atoms (NsurfSiC) and C atoms in theZLG(NZLG), the number of unsaturated Si bonds (NDBSi ) (see AppendixE), the corrugation of theZLGlayer (δC), the strain in the carbon layer, and the relative surface energy at the carbon rich graphite-limit (γ|µ
C=Ebulkgraphite).
reconstructions as well as the (6√
3×6√
3)-R30◦ ZLGphase from Ch. 8.
All structural models consist of six3C-SiCbilayer covered by a graphene lattice, the bottom C atoms of the SiC substrate have been saturated by hydrogen. The top three SiC-bilayers and the carbon layer above are fully relaxed usingPBE+vdW(residual energy gradients: 8·10−3eV/Å or below).
The surface energies (γ) were calculated using Eq.4.9. All surface energies are given relative to the bulk-terminated (1×1) surface. The range of the Si and C chemical potential are given in Sec.4.2. For instance, the popular (√
3×√
3)-R30◦[334,214] approximant intersects the graphite stability line at a relative surface energy in the carbon rich limit (ref. Graphite) of 0.15 eV, far above the actually stable phases.
In Table 9.1 we listed the parameters of the simulated ZLG-3C-SiC(111) interface structures for the different smaller-cell approximant models shown in Fig. 9.2: the 3C-SiC and ZLG periodicity, the rotation angle (α), the number of surface Si atoms (NsurfSiC) and C atoms in theZLG(NZLG). We also list the number of unsaturated Si bonds (NDBSi ). The Si dangling bonds were identified on the basis of geometric position and bond length differences (for a detailed discussion see AppendixE). Table9.1further includes the corrugation of the ZLGlayer (δC), the strain in the carbon layer, and the relative surface energy at the carbon rich graphite-limit (γ|
µC=Ebulkgraphite).
In structure (b) the strain in the carbon ad-layer is small with 0.27% and the Si dangling bond saturation is improved if compared to the experimentally observed 6√
3×6√
3-R30◦. However, the relative surface energy intersects
with the chemical potential limit for graphite at −0.35 eV, still higher by 0.06 eV than even the closest competing Si-rich phase, the (√
3×√
3)-R30◦ Si adatom phase. The (5×5) phase is either a nonequilibrium phase, or its structure is not the same as that assumed in Ref. [242].
So far different mechanisms were assumed to stabilise the SiC-(6√
3×6√ 3)-R30◦ & (13×13)-ZLGstructure. First, the surface energy is reduced by minimising the strain between the SiC substrate and theZLGlayer [290].
Sclauzero and Pasquarello [288] pointed out a correlation between the bind-ing energy and the corrugation of theZLG-layer. In their study, the binding energy was reduced with a reduction of the corrugation. It also has been discussed that the structural properties are driven by saturation of dangling Si bonds of the top SiC-bilayer through bond formation with theZLGC atoms [83,290]. Structure (d) and theZLGexperience the same lattice mis-match. Both phases saturate 89 Si atoms at the SiC-ZLGinterface, leaving 19 Si dangling bonds (Tab9.1). The corrugation of theZLG-phase is higher by 0.09 Å. If we follow the line of argument and compare the two structure models, the ZLG and structure (d), we would assume that the binding energy of strucutre (d) should be lower, because the lattice mismatch and the Si dangling bond saturation is the same, but the corrugation of structure (d) is lower. However, this is not observed. TheZLGphase is 0.15 eV lower in surface energy Tab9.1and inLEEDmeasurement show a 30◦ rotation between the substrate and theZLG-layer, e.g. in Ref. [263].
Certainly, the lattice mismatch, the corrugation and the dangling bond saturation play a crucial role in the stabilisation of theZLGphase, but these three factors cannot explain why the 30◦ rotation is preferred energetically over the 2.2◦ (Tab.9.1, and Fig.9.2). To shed some light on the origin of the difference in surface energy between the two different rotation angles, we compare the bond length distribution of these two phases.
Figure9.3shows the bond length distribution in the (6√
3×6√
3) & (13× 13)-ZLGlayer for two different rotational angles. The bond length distri-bution of the 30◦rotatedZLGlayer is taken from Fig.9.1(shown in grey in Fig.9.3). Structure (d), rotated by 2.204◦, is shown in blue in Fig.9.3.
We applied a Gaussian broadening of 0.002 Å for visualisation purposes only. The average bond length of structure (d) is 1.434 Å, which is 0.012 Å smaller than for theZLG-layer. In Section9.1, we demonstrated that the two peak structure can be understood as a result of a mixture of regions with asp2-like andsp3-like hybridisation in theZLGlayer. The bond length of thesp3-like hybridised C atoms in theZLG-layer range from 1.468 Å to 1.504 Å. If we compare the two curves in Fig.9.3, we see that in the bond length range of thesp3-hybridised C atoms of theZLGlayer are only very
Figure 9.3.: Bond length distribution in the (6√ 3×6√
3) & (13×13)-ZLGlayer for two different rotational angles. The bond length distribution for theZLGlayer in the 30◦ rotated structure from Fig.9.1is shown in grey and the ZLG structure rotated by 2.204◦ in blue. A Gaussian broadening of 0.002 Å was applied for visualisation purposes only.
The two vertical lines indicate the bond length of free-standing graphene in the optimised structure on aPBE+vdWlevel and the average bond length in theZLGlayer rotated by 2.204◦.
few C-C bonds of structure (d). Indeed, the maximum C-C bond length in structure (d) is 1.472 Å, indicating that the C-C bonds withsp3characteristic are compressed. Most bonds of structure (d) are between the two peaks of theZLGstructure. From the analysis shown in Fig.9.3, we conclude that the surface energy of structure (d) is increased by the compression of the sp3-like hybridised C-C bonds and the stretching of thesp2-like hybridised C-C bonds. The difference in the bond length distribution originates in the detailed bonding situation determined by the position of the Si dangling bonds. However, also the electronic structure of the specific arrangement will play a role in the stabilisation of one structure over the other.
Finally, we discuss the influence of artificially strained approximant phases on electronically relevant properties, such as the formation energies of defects. To calculate the defect formation energy, we substract from the surface energy (Eq.4.9) of the system including the defect (γdef) the surface energy of the defect free surface (γZLG) and account for additional C atoms or C vacancies by the carbon chemical potentialµGraphiteC (Eq.4.15)
∆G = 1 Ndef
γdef−γZLG−NCµGraphiteC
, (9.2)
whereNCis the number of substituted or vacant C atoms andNdefthe num-ber of defects in the slab. For negative defect formation energies (∆G) the defect is stable and for positive∆Gthe formation of the defect is unlikely.
As an example, we consider a specific class of C-rich hexagon-pentagon-heptagon (H5,6,7) defects suggested as an equilibrium feature of theZLG phase in Ref. [256]. The defects consist of three carbon heptagons and pentagons surrounding one carbon hexagon. The central hexagon is rotated by 30◦ with respect to the graphene lattice. This defect incoporates two additional carbon atoms (NC=2) , lowering the average C-C bond length in theZLG. Qiet al.suggested two different defect positions, “hollow” and
“top”, shown in Fig.9.4.
In Figure9.4the H(5,6,7) defect is shown for both positions in the approxim-ated 3×3 and in the (6√
3×6√
3)-R30◦ ZLGphase. The 3×3 is a supercell of the approximated (√
3×√
3)-R30◦ZLGphase with a massively strained carbon layer (see Tab.9.1). All four phases were fully relaxed using the same procedure as described above. Indeed, both defects would be more stable than the hypothetical (√
3×√
3)-R30◦ ZLGapproximant when in-cluded in a(3×3)arrangement as done in Ref. [256]: Using Eq.9.2gives
∆G = −1.75 eV per defect for the hollow and ∆G = −2.93 eV per de-fect for the top position. However, the same dede-fects are unstable when included into and compared to the correct (6√
3×6√
3)-R30◦ ZLGphase:
∆G = +5.28 eV per defect for the hollow and∆G = +5.27 eV for the top site, again at the chemical potential limit for graphite. The formation energy of the H(5,6,7) defect in the (6√
3×6√
3)-R30◦ ZLGphase is too high to make the formation of these defects likely. The formation of a H(5,6,7) defect introduces two additional C atoms, lowering the overall strain of the C-C bonds in the approximated (√
3×√
3)-R30◦ ZLG phase, which leads to a stabilisation of the defect. This example illustrates the importance of a careful interpretation of any results obtained by using an approximated interface structure.
Figure 9.4.: The hexagon-pentagon-heptagon (H5,6,7) defect in the zero-layer graphene shown in the approximated 3×3 cell (insert a and b) and in the (6√
3×6√
3)-R30◦ZLG phase. The defect was placed in two different positions. In inset a and c the defect is placed at the “hollow” position with a silicon atom of the underlying SiC bilayer in the middle of the central hexagon and at the “top” position.