In this section, the crystal structures of single and bilayer graphene in real and momentum (reciprocal space) space are described and a comparison between these two is presented as well.
1.4.1 Single layer graphene
Graphene is an arrangement of carbon atoms in a honeycomb pattern. Each carbon atom has f our valance electrons in its outer orbit three of which, one 2s and two 2p electrons (2px and 2py), are used to form three sigma (σ) bonds with the neighbouring atoms. Theseσ bonds lie in the plane of the graphene sheet with the bond angle of 120◦ between two neighbouringσbonds. Therefore, each carbon atom in graphene honeycomb lattice is sp2 hybridized. This gives rise the trigonal structure of hybridized state. The remaining electron in pz orbital is aligned perpendicular to the plane of the graphene sheet forming theπbond. The electrons inπorbitals are free to move in the plane of the graphene sheet and responsible for the electronic and transport properties of graphene. sp2 hybridization of carbon atoms in graphene leads to the trigonal structure. Thus graphene sheet can also be thought of as made up of two interpenetrating triangular sublattices named A and B. Therefore each of the carbon atoms in one sublattice of graphene sheet has three nearest neighbours from the second sublattice and six next-nearest neighbour from same sublattice, Fig. 1.6. The position of A andB atoms are not
equivalent because it is not possible to connect them with a lattice vector of the form R~ =n1~a1 +n2~a2, where n1 and n2 are integers. The lattice unit vectors of hexagonal lattice in position space can be written as [42]
~a1 = 3
2 ac−c xˆ+
√3
2 ac−c y, ~aˆ 2 = 3
2 ac−c xˆ−
√3
2 ac−c yˆ (1.1) where ac−c ≈1.42˚A is the carbon-carbon bond length and |~a1|=|~a2|=√
3ac−c is the lattice constant. The three nearest-neighbor vectors in real space are given by
~δ1 = 1
2 ac−c xˆ+
√3
2 ac−c y, ~ˆ δ2 = 1
2 ac−c xˆ−
√3
2 ac−c y, ~ˆ δ3 =−ac−c xˆ while the six second-nearest neighbors are located at
~δ10 =±~a1, ~δ20 =±~a2, ~δ30 =±(~a2−~a1)
Figure 1.6: Left: honeycomb crystal structure of graphene, made out of two interpenetrating sublattices denoted by ‘A’ and ‘B’. Carbon atoms on two sub- lattices are marked by ‘green’ and ‘purple’ color. The lattice unit vectors and nearest-neighbour vectors are denoted by~a1,~a2 and~δ1,~δ2 and~δ3, respectively.
It has two atoms per unit cell. ac−c is the carbon-carbon bond length. Right:
direct lattice of honeycomb lattice.
The reciprocal lattice of direct graphene lattice can be constructed as shown in Fig. 1.7. The reciprocal lattice can also be thought of as the rotation of 90◦ of direct graphene lattice. The reciprocal-lattice vectors are given by
~b1 = 2π 3 ac−c
ˆ
x+ 2π
√3ac−c
ˆ
y, ~b2 = 2π 3ac−c
ˆ
x− 2π
√3 ac−c
ˆ
y (1.2)
The reciprocal lattice vectors~b1 and~b2can be obtained from the following relation
~ai·~bj = 2π δi,j
The first Brillouin zone is the smallest volume in the reciprocal space bounded by the planes bisecting the vectors to the nearest reciprocal lattice points as shown in Fig. 1.7. The three points in the first Brillouin zone Γ,K and K0 are of special interest because graphene shows interesting physics near these points. Γ point is located at the center of the first Brillouin zone while K and K0 are at the corners of the first Brillouin zone where the valance band and conduction band touches one another. These two points are connected by time-reversal symmetry. The position of these two points in the momentum space is given by
K~ = 2π
3 ac−c xˆ+ 2π 3√
3 ac−c y, ~ˆ K0 = 2π
3ac−c xˆ− 2π 3 √
3 ac−c yˆ (1.3)
First Brillouin zone 𝒃𝟐 𝒃𝟏
Figure 1.7: Left: graphene honeycomb lattice in reciprocal lattice space. ~b1 and~b2 are the reciprocal lattice vectors. The points K and K0 are called the Dirac points and Dirac cones are located at these points. These two points are connected by the time reversal symmetry. Γ point denotes the center of the Brillouin zone. Right: It shows the construction of reciprocal lattice from direct graphene lattice. Green hexagon shows the first Brillouin zone of graphene
lattice in reciprocal space.
1.4.2 Bilayer graphene
Monolayer graphene is a building block of other graphitic materials [12]. Graphite structure can be obtained from the stacking of the graphene sheets one on the top of the other. The simplest generalization is bilayer graphene. Bilayer graphene can be thought of as made from two coupled monolayers. These two layers are separated by 3.35A◦, the interplanar distance, and coupled by the weak Van der Waals forces.
Bilayer graphene can be obtained from the stacking of two monolayers of carbon atoms in a Bernal or Rhombohedral stacking [11]. The crystal lattice structure of bilayer graphene [16–18] is given in Fig. 1.8. These two layers are arranged in a way such that one type of atoms in the lower layer, sayB1, are directly below to an atom on the top layer, say A2. These two atomic sites are referred as ‘dimer’
sites. The atomic orbitals on dimer sites are strongly coupled by an inter-layer couplingγ1. The atomic sites which are not directly above or below to one another called ‘non-dimer’ sites. The dimer sites in bilayer graphene are responsible for the formation of high energy bands [43–45] in the electronic spectrum. All four atomic sites A1, B1 in bottom layer and A2, B2 in top layer are inequivalent.
Bilayer graphene has four atoms per unit cell. The reciprocal lattice of bilayer
Figure 1.8: Left: schematic honeycomb crystal lattice of bilayer graphene, side view, containing four sites in the unit cell: A1 (black circles) and B1 (green circles) in the bottom layer, and A2 (green circles) and B2 (purple circles) in the top layer. Right: Reciprocal lattice of bilayer graphene. It is also a honeycomb lattice similar to single layer graphene.~b1 and~b2are primitive reciprocal lattice vectors. The shaded hexagon is the first Brillouin zone with Γ indicating the
center,K+ and K− showing two non-equivalent corners.
graphene is shown in Fig. 1.8 [46]. The reciprocal lattice of bilayer graphene is
also a hexagonal lattice similar to single layer graphene. Figure 1.9 shows the
Figure 1.9: Schematic of the plane view of the honeycomb crystal lattice of bilayer graphene. ~a1, ~a2 are the primitive lattice vectors. Bottom layer is denoted by solid hexagons while top layer denoted by dashed hexagons. The shaded rhombus indicates the conventional unit cell. ais the lattice constant.
plane view of the graphene bilayer. The bottom and top layers are denoted by solid and dashed hexagons, respectively. The dimer sites, B1−A2, are shown by green circles whereas non-dimer sites, A1 and B2, are shown by black and purple circles, respectively. Shaded region shows the conventional (non-primitive) unit cell and a=|a1|=|a2| is the lattice constant, the distance between adjacent unit cells, a= 2.46˚A [18,47]. The lattice constant is different from the carbon-carbon bond lengthac−c =a/√
3 = 1.42˚A, which is the distance between adjacent carbon atoms. The primitive lattice vectors~a1 and~a2 may be defined as
~a1 = a 2 xˆ+
√3 a
2 y, ~aˆ 2 = a 2 xˆ−
√3 a
2 yˆ (1.4)
The primitive reciprocal lattice vectors~b1 and~b2 can be calculated from the fol- lowing relation between primitive direct lattice vectors and primitive reciprocal lattice vectors.
~ai·~bj = 2 π δi,j
where δi,j = 0 f or i 6= j and δi,j = 0f or i = j. The primitive reciprocal lattice vectors are given by
~b1 = 2 π
a xˆ+ 2 π
√3 a y, ~bˆ 2 = 2π
a xˆ− 2 π
√3 a yˆ (1.5)
As shown in Fig. 1.8, the reciprocal lattice is hexagonal lattice, and the first Brillouin zone is a hexagonal. Two of the six points, one at each corner, in the first Brillouin zone are inequivalent denoted by K+ and K−, also denoted by K and K0. These two points are connected by time reversal symmetry. The physics of bilayer graphene is important at these points.These two points are called the Dirac points where the valance band and conduction band touch one another. The position vectors of Dirac points, also referred to as valleys, in first Brillouin zone are
K~ = 4π
3 ax, ~ˆ K0 =−4π 3 axˆ
In the following section, we discuss the band structures of single and bilayer graphene. We also show how to apply the tight binding method to calculate the energy momentum dispersion for both graphene systems.