The Electronic Structure of Graphene

The electronic properties of graphene originate from its very special crystal structure. The graphene honeycomb lattice can be seen as two triangu-lar sublattices lattice A and lattice B with one C atom in each. The two sublattices are bonded by sp² hybridisation of the 2s, the 2px and 2py C orbitals forming strong in-planeσbonds. The remaining 2pzorbital is ori-ented perpedicular to the graphene x-y-plane. The neighbouring 2pzorbital form delocalised states across the graphene plane, so-calledπ-bonds. Elec-trons move easily in theseπstates resulting in the exceptionally electrical conductivity of graphene.

Figure6.6a shows the hexagonal graphene Brillouin zone (shaded in grey).

The graphene reciprocal lattice vectorsb₁,b₂connect theΓ-point of the first Brillouin zone to the neighbouringΓ-points (indicated as blue arrows). The graphene Brillouin zone contains three high symmetry points:

Γ-point in the zone center,

M-point at the mid point of the Brillouin zone edge, K-point at the corners of the Brillouin zone.

The graphene band structure of pritstine graphene on the level ofPBEis shown in Fig. 6.6 b. We used the PBE+vdW lattice parameter given in Tab.6.1. The graphene unit cell contains twoπorbitals forming two bands.

These bands are referred to asπ band, the bonding lower energy valence band (indicated by an arrow in Fig.6.6b), andπ^∗band, the anti-bonding higher energy conduction band. The π bands touch at the K-point and close the band gap which is why graphene is often referred to as a zero-gap

Figure 6.6.: Subfigure (a) shows the graphene Brillouin zone (shaded in grey) and the high symmetry pointsΓ,MandK(marked by blue points). (b) The band dispersion of graphene on the level ofPBE. Thek-path is chosen along the high symmetry lines of the graphene Brillouin zone as indicated by arrows in Subfigure (a).

semiconductor. The energy-momentum dispersion of theπandπ^∗bands around theK-point is approximately linear. The energy where the twoπ bands touch is called the Dirac point (ED) (see Fig6.6b). Theπbands of an undoped graphene sheet are half-filled,E_D equals the Fermi energy.

The doping of the graphene layer shifts the Dirac point with respect to the Fermi level. In a neutral graphene layerE_D equals the Fermi energy. The position of the Dirac point is very sensitive to small doping concentrations.

This is a direct consequence of the linear dispersion of the bands in the close vicinity of the K-point in the grapheneBZ. In reciprocal space the electronic structure of graphene in the vicinity of the Dirac point is formed like a cone. The amount of dopants is proportional to the area enclosed by the Fermi surface which can be approximated by a circle with the area k_F²π. In this simple model we find that the position of the Dirac point (ED) is proportional to the amount of dopants (δq) in the graphene layer as ED∝ p

|δq|.

However, as can be seen in Fig6.6b theπ bands are not fully symmetric in the vicinity of the k-point [35, 50,79]. In the following, we examine the shift of the Dirac point with respect to the Fermi level for different doping concentrations. The position of the Dirac point with respect to the Fermi level is influenced by hole and electron doping of the graphene layer. We simulate the doping of the graphene layer using the virtual-crystal approximation (VCA) [336,278,261]. TheVCAensures that the simulation cell remains neutral and therefore allows to calculate a doped crystal using periodic boundary conditions. This is achieved by removing a fraction of

the positive nuclear charge δq of all carbon atoms and adding the same amount of negative charge to the conduction electrons. This means that a fractional electron of the charge−_δqis transferred to the conduction band, which gives the Fermi level in theVCA.

Figure 6.7.: The position of the graphene Dirac point with respect to the Fermi level for doping in the range of±0.005e⁻/atom on the level ofPBE.PBE+vdWlattice para-meter were used (Tab.6.1).

We used the two atomic graphene cell introduced in Sec. 6.1 with the lattice parameter optimised on the level ofPBE+vdWas given in Tab.6.1.

The Fermi surface for different doping changes only in a small region around theK-point in the graphene Brillouin zone. In order to accurately account for these small changes, we chose a very dense off-Γk-grid with 48 points in the in-plane x and y direction. A full band structure similar to Fig6.6b was calculated for 28 different doping concentrations using PBE. We chose a doping range of ±0.005 e⁻/atom, which corresponds to a doping con-centrations of 0.08%. Within theVCA higher doping concentrations could be included. However, for doping con-centrations much higher than the ones considered here at the order of a few tenth of the atomic charge local effects in the electronic structure may not be captured by theVCA[261].

Figure6.7shows the Dirac energy (E_D)

with respect to the Fermi level. In the range between−0.005e⁻/atom and zero the graphene layer is hole doped, so that the Dirac point lies above the Fermi level. In this case the Dirac cone is not fully occupied resulting in a p-type doped graphene layer (see diagram in Fig. 6.7). In the range between zero and 0.005e⁻/atom the graphene layer is n-type doped and the Dirac point is shifted below the Fermi level. In Chapter12, we will use the here presented analysis of doping in pristine graphene to aid a better understanding of the doping of epitaxially grown graphene.