Dirac Fermions in Graphene and Graphiteâa view from angle ...

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graphene is obtained from low energy electron microscopy (LEEM) data for each sample. The corresponding angle resolved photoemission spectroscopy (ARPES) data at the K point for each sample are shown in panels d-f. The direct comparison between LEEM and ARPES data allows one to readily correlate the width of the photoemission features and the size of the excitation gap to the morphology of the films and, in particular, to the mean size of the single layer graphene terraces. The main result of this figure is summarized in panel g. Although we see a slight increase in the magnitude of the gap for the smallest terrace size, it is clear that as the terrace size exceeds 50 nm the gap plateaus at 180 meV. In contrast, the predicted and observed quantum confinement gap for exfoliated samples is shown as the dashed line. Clearly, for most of the terrace sizes we studied, the vanishingly small gap induced by quantum confinement can not account for what is observed in our data. These results show that while quantum confinement may have some contribution for small terrace sizes, the gap observed in samples with larger terraces is an intrinsic property of single layer epitaxial graphene and cannot be simply explained by quantum confinement. 4.5.3 Breaking of the sublattice symmetry due to graphene-substrate interaction In our opinion, the more likely scenario is the breaking of the A, B sublattice symmetry. This leads to the rehybridization of the valence and conduction band states associated with the same Dirac point (Fig. 4.10(a)), resulting in a gap at all the K and K ′ points. A necessary prediction of this scenario is the breaking of the six fold rotational symmetry of graphene near the Dirac point energy. For energy well above and/or below E D , the symmetry is restored. For bilayer and trilayer graphene, the breaking of the A, B sublattice equivalence can be a direct consequence of the the AB stacking between different layers. Indeed, topographic Scanning Tunneling Microscopy (STM) images for bilayer graphene have clearly shown inequivalent A and B sublattices 80,84 , similar to what has been observed for graphite 85 . This simply derives from the fact that one sublattice has carbon atoms directly below it while the other does not. Naively it seems that this explanation will not work for single layer graphene. However, it is known that for epitaxially grown graphene, a buffer layer exists 50,49 (see Fig. 4.10(b)). ARPES study of the buffer layer has shown practically the same σ bands as graphene while very different π bands 49 . This is because the π orbitals have hybridized with the dangling bonds from the substrate. The fact that the σ bands are unchanged suggests that, like graphene, the carbon atoms in the buffer layer have also the honeycomb arrangement with similar bond length. Consequently, although the buffer layer is electronically inactive (absence of π orbitals) 49 , structurally it can break the A, B sublattice symmetry when a single layer of graphene grows upon it. This is particularly so in view of the small layer separation of ≈ 2 Å 50 and the AB stacking usually expected for very thin graphene samples. For the single and bilayer graphene, we use a tight binding model with symmetry breaking on the A and B sublattices to fit the symmetry breaking parameters to the observed energy gap. By fitting the dispersion, the symmetry breaking parameter in single layer graphene, defined as half of the difference between the substrate potentials on the A and B sublattices, is determined to be m ≈ 0.13 eV. In bilayer graphene, the symmetry breaking parameters in the top and bottom layers are m 1 ≈ 0.49 eV, m 2 ≈ −0.21 eV respectively. The magnitude of the symmetry breaking parameter is much bigger in the bottom graphene layer than that in single layer graphene, because it is sandwiched between the buffer layer and the top graphene layer. The reason for m 2 to have the opposite sign is because of the AB stacking. This cancels part of the effect in the bilayer graphene and decreases the gap. Therefore, for AB stacking graphene, the eigen-functions average out for many layers, and the gap decreases rapidly. Fig. 4.11 shows additional support for the A, B sublattice symmetry breaking. Panels (a-d) show intensity maps taken on single layer graphene as a function of k x and k y at E F , -0.4, -0.8 eV and -1.0 eV respectively. The dominant features in these panels are the small pockets centered at the six corners of the Brillouin zone. Interestingly, around each corner, there are six faint replicas forming a smaller hexagon. The intensity associated with them is ≈ 4% of the main intensity. Closer inspection shows that the vectors connecting the center of the small hexagon to its six corners are nearly the same as the second shortest 33
Figure 4.10. Proposed mechanism for the gap opening and the structure of epitaxial graphene on SiC. Figure 4.11. Breaking of the six fold symmetry in the intensity map near E D . (a-d) ARPES intensity maps taken on single layer graphene at E F , E D , -0.8 eV and -1.0 eV respectively. Near E D (panel b), the intensity of the six replicas near K shows breaking of six fold symmetry. Note that to enhance the additional feature around E D , the color scale is saturated for the dominant features near K and the replicas. (e) ARPES intensity map of the calculated spectral function at E D in the presence of symmetry breaking on the two carbon sublattices. 34
Page 1 and 2: Dirac Fermions in Graphene and Grap
Page 3 and 4: Dirac Fermions in Graphene and Grap
Page 5 and 6: Contents Abstract 1 Contents Acknow
Page 7 and 8: 7.6 Conclusions and implication of
Page 9 and 10: Curriculum Vitæ Shuyun Zhou Educat
Page 11 and 12: • Selected as Science Highlight f
Page 13 and 14: The unit cell of graphene contains
Page 15 and 16: Projecting this equation into φ(r
Page 17 and 18: Figure 1.5. Schematic drawing of th
Page 19 and 20: Chapter 2 Experimental techniques 2
Page 21 and 22: dispersion E(k z ). Thus k z values
Page 23 and 24: where ɛ is infinitely small. Accor
Page 25 and 26: Langmuirs (1L=10 −6 torr·s) to c
Page 27 and 28: mean free path, E kin is the kineti
Page 29 and 30: a b c d e f graphite SiC f graphite
Page 31 and 32: Figure 3.3. XPS results showing dat
Page 33 and 34: Figure 3.5. Intensity map at -1 eV
Page 35 and 36: Chapter 4 Gap opening in epitaxial
Page 37 and 38: Figure 4.2. Observation of the gap
Page 39 and 40: the EDCs as well as the region of e
Page 41 and 42: Figure 4.7. Dispersions measured in
Page 43: Figure 4.9. (a-c) LEEM images taken
Page 47 and 48: Chapter 5 Metal-insulator transitio
Page 49 and 50: Figure 5.2. (a) Dispersions through
Page 51 and 52: Figure 5.3. (a-f) Data taken throug
Page 53 and 54: Chapter 6 Dirac fermions and effect
Page 55 and 56: Figure 6.2. Linear Λ-shaped disper
Page 57 and 58: Figure 6.4. Detailed low energy dis
Page 59 and 60: Figure 6.6. Detailed dispersion nea
Page 61 and 62: Chapter 7 ARPES study of partially
Page 63 and 64: Figure 7.2. (a) Fermi energy intens
Page 65 and 66: direction, i.e. perpendicular to gr
Page 67 and 68: Figure 7.5. ARPES intensity map mea
Page 69 and 70: Figure 7.7. (a) ARPES intensity map
Page 71 and 72: Figure 7.9. (a) ARPES intensity map
Page 73 and 74: 19. Hüfner, S. Photoelectron Spect
Page 75 and 76: 59. Jiang, Y. J., Yao, D.-Y., Carls
Page 77 and 78: 99. Luk’yanchuk, I. A. & Kopelevi
Page 79 and 80: List of Figures 1.1 (a) Sp 2 bondin
Page 81 and 82: 4.8 Thickness dependence of E D and
Page 83 and 84: 7.2 (a) Fermi energy intensity map.

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