Electronic structure

The electronic band structure of a material allows for describing its phys-ical properties, e.g., electrphys-ical resistivity and optphys-ical absorption. Further-more, it is crucial for understanding the operation principles of related solid-state devices, such as transistors, light emitting diodes and solar cells.

The hybridisation of the two dimensional network of carbon atoms in graphene is key in its resulting electronic properties. A single carbon atom contains a total of six electrons and exhibits the electronic configuration 1s², 2s², 2p². The electrons in the 1s orbitals are the core-level electrons that are not involved in bond formation, while the ones in the 2s and 2p or-bitals are the valence electrons and are involved in bond formation. These valence electrons can hybridise (share electrons) in three ways, such that the carbon atom can have two, three or four adjacent atoms available for bond formation, with the electrons forming the hybrid orbitals sp, sp²and sp³, respectively.

Figure 2.1: Schematic illustration showing the pz orbitals perpendicular to the plane of graphene and theσbonds between neighbouring sp²carbon atoms. Fig-ure adapted from [23].

The carbon atoms in graphene are sp² hybridised, i.e., they have three in-plane neighbours that each form a σ bond, as shown in Fig. 2.1. These bonds are of covalent character and the resulting crystal structure is planar hexagonal, as shown in Fig. 2.2a. The formation of theσ bonds leaves out one 2p orbital (2pz) perpendicular to the graphene plane, which does not participate in the bond formation. The adjacent 2pz orbitals weakly inter-act to form the π band of graphene that contains the delocalised valence electrons. As it follows from theoretical calculations, the electrons in this band mimic relativistic particles, behaving as massless Dirac fermions and are responsible for the excellent conductivity of graphene [2].

Figure 2.2: (a) The hexagonal honeycomb lattice of graphene in real space, formed by the carbon atoms in the two sublattices A and B, with the blue atom in sublattice A and orange atom in sublattice B.di represent the nearest neighbour carbon atoms, with the distance being 1.42 ˚A. The unit cell is formed by the lattice vectorsa₁anda₂. (b) Reciprocal lattice of graphene with the two reciprocal lattice vectorsbi.Γ, K, M are high symmetry points. Figure adapted from [24].

The nearest distance between carbon atoms in graphene is 1.42 ˚A, with the lattice vectors, as shown in Fig. 2.2a:

a1 = a

The lattice parameter is given by a=|a1|=|a2|=1.42√

3 A = 2.46 ˚˚ A and the lattice vectors in reciprocal space are then given by:

b₁ = 2π

The points K and K’, or Dirac points, in Fig. 2.2b, that are located at the corners of the Brillouin zone are very important for the specific physical properties arising in graphene. They are at the positions:

K =

The three nearest-neighbour vectors in real space are:

d₁ = a The tight binding Hamiltonian for electrons in graphene, wherein only electrons that can hop to nearest-neighbour and next-nearest-neighbour atoms are taken into account [24,25], yields the following energy disper-sion relation as a function of the wavevector k:

E±(k) =±~tp The positive and negative signs describe theπ and π^∗ bands that crespond to the dispersion of the bonding and anti-bonding molecular or-bitals, constructed from the pz atomic orbitals on the carbon atoms. The value t is the nearest-neighbour hopping energy (i.e., hopping between carbon atoms at different sublattices) andt⁰ is the next-nearest hoping en-ergy [24]. When the value of t’ is 0, then electron-hole symmetry exists and theπandπ^∗ bands become symmetric.

Close to the Dirac points, the energy dispersion is obtained by expand-ing eq. (2.5). Usexpand-ing k=K+q, withqthe momentum measured relative to the Dirac points and|q| |K|:

E_±(q) =±~u_F|q|, (2.7) when considering only the first term in the expansion.

The termu_F represents the Fermi velocity that is determined theoreti-cally byu_F = 3ta/2[24]. Taking the nearest hopping energytas≈2.8 eV, as found by calculations, then the Fermi velocity becomes u_F ≈ 10⁶ms⁻¹ [25].

This result for the energy dispersion in eq. 2.7 is very different to the usual case as in e.g., inorganic semiconductors, where E(q) = q²/(2m) with m being the electron mass. The discrepancy lies in the fact that uF

does not depend on the energy or the momentum as in the usual case where u = k/m =p

2E/mand the velocity changes significantly with en-ergy. The energy dispersion as given in eq. (2.7) imitates the one that ultrarelativistic particles follow, which is described by the Dirac equation that is also used to describe the energy dispersion of photons [24].

The density of states close to the Dirac point per unit cell (ρ(E)) is given by [24]:

ρ(E) = 2|E|

π~²u²_F . (2.8)

In Fig. 2.3a the resulting band structure of graphene is plotted in 3D for the first Brillouin zone. The occupiedπand unoccupiedπ^∗bands intersect at the K and K’ points, which are the corners of the unit cell of graphene in reciprocal space and the energy dispersion close to these points is linear, as shown in the zoom close to the K point, in Fig. 2.3b.

In summary, the linear energy dispersion ofπ and π^∗ bands near the Dirac points make the electrons in graphene behave as ultrarelativistic particles travelling withuF through the graphene sheet and the transport in graphene is mainly by hopping of electrons from one sublattice to the other [26]. The vanishing density of states near the K points leads to mo-bility of charge carriers surpassing the mobilities observed in silicon [1,2]

and graphene exhibits ballistic charge carrier transport with a high mean free path [2,27].

Figure 2.3: (a) The graphene 3D Brillouin zone showing the π andπ^∗ bands.

The plot was generated using the software MATHEMATICA. (b) Zoom in the K-points, showing the linear intersection ofπandπ∗bands.