Bilayer graphene (BLG) is composed of two monolayer graphenes (MLGs) in Bernal or AB stacking, meaning that the A sublattice of one layer is on top of the B sublattice of the other layer, as shown in Fig. 1.7. This is the typical stacking pattern of 3D graphite as well. Its charge carriers, as we show below, reveal non-relativistic, ”Schr¨odinger” (quadratic dispersion) and relativistic ”Dirac” (chiral symmetry, unusual Berry phase) features.
Due to their peculiar nature, as discussed below, BLG holds the promise of revolutionizing electronics, since its band gap is directly controllable by a perpendicular electric field over a wide range of parameters [17, 18, 19, 20, 21]
(up to 250 meV [22]), unlike existing semiconductor technology. Moreover, unlike monolayer graphene, whose effective model, namely the Dirac equation was thoroughly investigated in QED and relativistic quantum mechanics, understanding the low energy properties of BLG represents a new challenge.
The band structure of BLG also follows from tight-binding calculations.
In addition to the intralayer hopping of the graphene layers, an interlayer hopping, t⊥ ≈ 0.3-0.4 eV should be taken into account, since the typical distance between the layers is d = 3.3 ˚A. Additional hopping processes are also present, but these are neglected for the sake of simplicity. A tight
A2 B2
Figure 1.7: The lattice structure of BLG with the relevant hopping processes, the vertical green arrow denotes a perpendicular electric field.
binding calculation for the kinetic energy gives
Hgraphene =X particles on layer 1 or 2, sublatticeA orB with momentum kand spinσ, as visualized in Fig. 1.7, and f(k) has already been defined for MLG. Here, ∆ represents a layer dependent chemical potential, which arises upon switching on a perpendicular electric field. Due to the 4 atoms in the unit cell, BLG possesses 4 band, two of them touching each other at zero energy for ∆ = 0, and two others separated by±t⊥, as shown in Fig. 1.8
The low energy part of the spectrum in BLG is obtained by integrating out the high energy modes, leading to an effective 2×2 description as[18, 23]
H = the presence of a finite electric field, the band touching disappears and a finite bandgap appears, whose size is easily tunable by the electric field. However, screening due to electron interactions becomes relevant in this case, and the induced gap is related to the external potential, Uext, created by the electric
-4
Figure 1.8: The energy spectrum of BLG in the Brillouin zone together with the low energy part of the spectrum around zero energy for ∆ = 0.
field as [18, 24]
2∆ =Uext+ e2dδn
2Acεrε0, (1.14)
where δn = P
p(n1p −n2p) is the dimensionless density imbalance between the two layers with nip the particle density of state p on the ith layer. The induced gap to a good approximation is given by [18, 17]
∆ =
and the density imbalance reads
δn= 4ρ0∆ ln (|∆|/2t⊥), (1.16) with λ = e2dρ0/Acεrε0 ∼ 0.1−0.5 the dimensionless screening strength, ε0
the permittivity of free space and ρ0 = Acm/2π~2 the density of states per valley and spin in the limit ∆→0. For SiO2/air interface,εr≈2.5 (εr = 25 for NH3,εr = 80 for H2O), which reduces the effects of screening. This extra tunability of its bandgap makes bilayer graphene a promising candidate as well for future electronic devices. By applying a dual-gate structure [19, 22, 20, 25, 21] with top and back-gate, the size of the gap together with the total number of charge carriers can be tuned independently.
The physical properties of BLG with ∆ = 0 are as surprising as those of MLG, and vaguely speaking, in spite of the different topology of the low energy Hamiltonians in Eqs. (1.3) and (1.13), it is a ”factor of 2 times mono-layer graphene”. It exhibits a universal minimal conductivity at the charge neutrality point, which is twice as large as that of MLG. Its wavefunction in momentum space is
whereα =± corresponds to positive and negative energy states asEα(p) = α|p|2/2m and ϕp = arctan(py/px). The Berry phase is calculated to be
±2π [26] and in spite of its massive quasiparticles, it exhibits chiral symme-try unlike particles obeying the standard Schr¨odinger equation. The Klein tunneling is also peculiar in BLG: no perfect transmission occurs for perpen-dicular incidence, in contrast to MLG, but perfect reflection. This perfect reflection (instead of the perfect transmission) is viewed as another manifes-tation of Klein tunneling, because the effect is again due to the chirality of the quasiparticles (fermions in MLG and BLG exhibit chiralities that resem-ble those associated with spin 1/2 and 1, respectively, as follows from the π and 2π Berry phases). For MLG, an electron wavefunction at the barrier interface matches perfectly the corresponding wavefunction for a hole with the same direction of pseudospin due to chiral symmetry, yieldingT = 1. In contrast, for BLG, the chiral symmetry requires a propagating electron with wavevector k to transform into a hole with wavevector ik(rather than−k), which is an evanescent wave inside a barrier.
Additionally, BLG is also characterized by a featureless optical conductiv-ity which is twice that of MLG, as shown in Fig. 1.6. One essential difference, however, with respect to MLG is the finite density of states around half fill-ing. Due to this, a short range electron-electron interaction is usually found to be marginally relevant, leading to all sorts of phase transition in BLG, at least on a theoretical level [27, 28, 29, 30]. The experimental verification of such phase transitions still remains to be seen.
The Landau level structure in a perpendicular magnetic field also shows Dirac like (chiral) and Schr¨odinger like features. In the absence of a gap, the Landau levels per spin and valley read as
Enα =αp
n(n+ 1)eB⊥/m (1.18)
with α = ± and n non-negative integer and B⊥ is the perpendicular com-ponent of the magnetic field to the BLG sheet. First of all, it contains two degenerate zero modes (in contrast to the single zero mode in graphene),
−4 0
−3
−2