Graphene is a monolayer thick material consisting only of carbon atoms, which are ar-ranged in a honeycomb lattice. In this geometry, carbon issp2hybridized. 2s, 2px and 2pyorbitals form theσ-bonds between each atom, whilst the 2pzorbitals give rise to the πandπ∗bands in the crystal [8]. Theπandπ∗bands touch at the K and K’ points in re-ciprocal space and form a zero gap semiconductor with linear dispersion relation close to those special points. Theπandπ∗bands are responsible for the phenomenally large interest in graphene’s electronic properties.
Using the tight binding approximation, one can derive the band structure of graphene in the complete energy range [8]. An expansion around the special K and K’ points gives rise to the linear dispersion relation and allows for a more compact and convenient low-energy description using the Dirac equation [33, 34]. Indeed, the behavior of low low-energy electrons in graphene around the K and K’ points mimic the behavior of massless Dirac Fermions, the reason why the linear dispersion around the K and K’ points is often re-ferred to as Dirac cone. A brief summary of all relevant properties is given in the fol-lowing. Comprehensive reviews on the electronic structure of graphene can be found in [35, 36, 37, 38].
2.1.1 Crystal structure
The honeycomb lattice of graphene is not a Bravais lattice, and consequently it has to be described using a triangular lattice with two equivalent atoms in the unit cell [35]. The lattice is depicted in Figure 2.1 (a). The two equivalent sublattices are represented by red and black atoms, respectively, where one atom of each sublattice is surrounded by three atoms of the neighboring sublattice with a distance ofa=1.42 Å [35]. The lattice pa-rameter of the triangular lattice isaG=p
3a=2.46 Å. The lattice vectors are constructed 5
y
Figure 2.1 | Graphene crystal structure. (a) Real space lattice including unit cell vectors. The two differently colored atoms mark the A (black) and B (red) carbon sublattices.(b)Reciprocal lattice including unit cell and Brillouin zone.
by The reciprocal lattice [Figure 2.1 (b)] likewise exhibits a triangular lattice and the vectors are constructed such that~ai·~bj=2πδi j: High symmetry points K, K’ and M of the Brillouin zone are described by
Γ# –K= 2π
The high symmetry points K and K’ differ due to the bipartite lattice and are of special interest for the electronic properties of graphene.
2.1.2 Tight binding dispersion relation
The band dispersion of single layer graphene was first derived by P. Wallace in 1947 [8]
using the tight binding approximation. The tight binding Hamiltonian ˆHfor graphene pzelectrons considering hopping between nearest neighbors with hopping energytand hopping between next-nearest neighbors with hopping energyt0can be written as [35]
Hˆ = −t X
with annihilation and creation operatorsai anda†i, respectively, acting on A sublattice sites andbi andbi†analogous for B sublattice sites. The solution obtained from tight binding with next nearest neighbor consideration yields an energy dispersion [35]
E(~k) = ±t q
3+f(~k)−t0f(~k) (2.5)
2.1 Structure and electronic properties of graphene 7
Figure 2.2 | Tight binding dispersion relation. (a) πandπ∗ bands of graphene. (b)Constant energy contours of theπ-band. (c)Graphene bands along high symmetry directions in reciprocal space. (a-c) Graphene dispersion plotted according to equation2.5with parameterst= 3.5 eV and t’ = 0.08tin order to fit the experimental dispersion of the grapheneπband on Au metal surfaces as reported in [39].
The dispersion relation is depicted in Figure 2.2 (a). As evident from the equation, the latter result differs from the solution which takes into account nearest neighbors only by the additional partt0f(~k). Without next-nearest neighbor hopping, the solution yields two bandsπandπ∗which are symmetric and touch at zero energy. Theπband is com-pletely filled and theπ∗band completely empty. Next-nearest neighbor hopping leads to an asymmetry in theπandπ∗band and an energy shift (doping).
The constant energy contours (CEC) of theπband are depicted in Figure 2.2 (b).
The touching point of the two bands is located at the Brillouin zone corner at K and K’
points, whereas at the M point a van Hove singularity forms. The CEC around the K and K’ points are circular for energies close to the touching point ofπandπ∗and start to
show trigonal warping for larger energies. This behavior is also reflected in the dispersion relation in Figure 2.2 (c), inherent in the different slopes of theπandπ∗bands in# –
ΓK and
# –
KM directions.
2.1.3 Low energy expansion and Dirac cone
Restricting the above discussed tight binding solution in equation 2.5 to nearest neighbor hopping only (i.e.t0=0) and expanding the dispersion around the two special K and K’-points with~k=# –
ΓK+~kGand~k0=# –
ΓK0+~kG0 allows to describe the system with an effective Hamiltonian of the form1[37]
HˆK/K’= ħvF
à 0 kG,x∓i kG,y
kG,x±i kG,y 0
!
(2.7)
close to K and K’ points, i.e. for energies close to the touching point ofπandπ∗bands.
Here, the different signs refer to the K (top) and K’ (bottom) point, with identical energy eigenvalues
The energy dispersion features two solutions, whereλ= ±1 refers to conduction and valence band, respectively andvF=3t a/(2ħ)'106m/s is the Fermi velocity [37]. The spectrum is electron-hole symmetric and linear in~kG, giving rise to a dispersion rela-tion which resembles a cone at the K and K’ points. The unusual constant group velocity vFis fundamentally different from semiconductors, where energy bands are usually ap-proximated by parabolic bands. The linear dispersion relation for low energy graphene physics is depicted in Figure 2.3. In the this low energy approximation the density of states (DOS) per unit cell including spin and valley degeneracy [37]
N(E)=2
is linear in energy with zero states at the Fermi level and shows again that graphene qual-ifies as a zero band gap semiconductor.
Dirac cone
Starting from the Hamiltonian in 2.7 one can derive a description which is formally equiv-alent to the Dirac Hamiltonian for massless particles [33, 34]. Substitution of
¡kG,x,kG,y¢
→ −i¡
∂x,∂y¢
(2.10)
1In this work the~k-vector with respect to K and K’ points is referred to as~kGinstead of~qas common in literature in order to avoid any possible confusion with the scattering vector~qused in chapters 7 and 8.
2.1 Structure and electronic properties of graphene 9
Figure 2.3 | Dirac cone and pseudospin. The image shows the low energy dispersion re-lation. Arrows mark the mo-mentum vectors~kG in black and the pseudospin direction in or-ange. Chirality implies a change of the pseudospin direction with respect to momentum between the electron- and hole like parts of the cone.
leads to a very compact form of the effective Hamiltonian at the K and K’-points [37]:
HˆK= −iħvF~σ·~∇ and HˆK’= −iħvF~σ∗·~∇ (2.12) The two sets of Hamiltonians at K and K’ points lead to identical energy eigenvaluesE:
HˆK/K’ψ±K/K’=Eψ±K/K’; E= ±ħvFkG (2.13) One can see that the Hamiltonian corresponds to the Dirac Hamiltonian for massless particles with reduced speedvF'c/300 compared to the speed of lightc. Consequently the low energy physics of graphene is described formally by Dirac Fermions with renor-malized speed and allows for the verification of principles in QED in the framework of a condensed matter experiment. A prominent example is the Klein paradox [16, 17]. The K and K’ points are often referred to as Dirac points due to this equivalence.
The total wave functionsψKandψK0(spinors) consist of two components each:ψKA, ψKB,ψK’AandψK’B. Withφ=arctan(kG,y/kG,x) the eigenspinors read [37]
In QED, the components represent the real spin. In graphene, the two components ex-press the amplitude of the wave function on the A and B sublattice sites, respectively and are hence referred to aspseudospin.
Withp~= −iħ∇, the Hamiltonian in equation 2.12 can be rewritten to ˆHK =vF~σ·~p resembling the chirality operator, which is formally the projection of pseudospin~σon
momentum~p:
~σ·~p
¯
¯p~¯
¯
ψ±K=η ψ±K (2.15)
The chirality operator has eigenvalues η= ±1 and has important implications for the pseudospin, since it shows that the pseudospin is always parallel or antiparallel to the momentum~p= ħ~kG. In graphene, electrons (holes) have positive (negative) chirality.
The orientation of the pseudospin within the Dirac cone for electrons and holes is de-picted in Figure 2.3.
In addition to the degree of freedom coming from the A and B sublattice label, de-scribed by the pseudospin, a second degree of freedom is related to the K and K’ points
— the valleys — and is occasionally referred to asisospin[37]. In ideal graphene the two valleys are independent, however atomically sharp defects and edges will mix the two valleys and necessitate a combined two-valley 4×4 Hamiltonian description [37]. Taking into account the real electron spin finally leads to the most general low-energy descrip-tion of graphene combining the three internal degrees of freedom: the pseudospin, the isospin and the real spin are represented by a 8×8 Hamiltonian [37].
An important result of the two equivalent sublattices leads to a symmetry protection of the linear crossing points ofπandπ∗bands. In particular if time reversal and inversion symmetry hold, no band gap is opened. However in the presence of a substrate, where A and B sublattice sites feel a different potential, the sublattices are no longer equivalent and the inversion symmetry is broken, leading to a band gap opening [36, 37].
2.1.4 Landau Levels and Anomalous Quantum Hall Effect
Applying a magnetic field B~perpendicular to the graphene plane gives rise to a Lan-dau quantization which shows distinct differences from two-dimensional electron gases.
This results from the completely different behavior of charge carriers in monolayer graphene mimicking massless chiral particles in the low energy regime. In the case of two-dimensional electron gases the Landau levels obey an energy quantization of
En2DEG=E0+ ħeB m∗
µ n+1
2
¶
; n=1, 2, 3, ... (2.16)
giving rise to an equal spacing of Landau Levels, with electron charge e and effective massm∗. In monolayer graphene, the Landau Level sequence has a different form due to the zero mass behavior [40]:
EnMLG= ±vF
p2eħB|n|; n=0,±1,±2, ... (2.17)
In particular, the existence of electron like and hole like states gives rise to Landau Levels both above and below the Fermi level, including a zero energy Landau Level and shows
2.1 Structure and electronic properties of graphene 11
a square root dependence from magnetic fieldB and Landau Level indexn. The spe-cial Landau Level sequence of graphene is a hallmark of the Dirac physics of graphene, discriminating monolayer graphene not only from standard two-dimensional electron gases, but also from bilayer graphene [40]. A further peculiarity of Landau Levels in graphene is the fourfold degeneracy. Besides the two-fold spin degeneracy, the special band structure with inequivalent K and K’ points gives rise to an additional twofold valley degeneracy [41].
Based on this quantization of Landau Levels in graphene, peculiarities are also ob-served for the quantum Hall effect. In contrast to the integer quantum Hall effect in two-dimensional electron gases, the Landau Level atn=0 in graphene is responsible for the absence of a plateau in hall conductivity atn=0. Owing to these peculiarities the quantum Hall effect in graphene is referred to asanomalous quantum Hall effect[7].