Strained bilayer graphene: Band structure topology and Landau level spectrum

Abstract

We show that topology of the low-energy band structure in bilayer graphene critically depends on mechanical deformations of the crystal which may easily develop in suspended graphene flakes. We describe the Lifshitz transition that takes place in strained bilayers upon splitting the parabollic bands at intermediate energies into several Dirac cones at the energy scale of few meV. Then, we show how this affects the electron Landau level spectra and the quantum Hall effect. PACS numbers: 73.22.Pr,71.70.Di Electrons in bilayer graphene exhibit quite unusual properties: they can be viewed as 'massive chiral fermions' with parabolic dispersion at intermediate energies and Berry phase 2π [1,2], in contrast to monolayer graphene, where charge carriers are Berry-phase-π quasiparticles with linear dispersion [3,4]. Here, we show that topology of the low-energy band structure of electrons in bilayer graphene critically depends on mechanical deformations of the crystal. Strain determines the number of Dirac mini-cones in the low-energy part of the spectrum, below the saddle point in the electron dispersion: two with the Berry phases π in a strongly strained crystal instead of four (three with Berry phase π and one with −π) in an unperturbed crystal [1]. These spectral features are tracked down to the evolution of the Landau levels for electrons in a magnetic field, and we predict their manifestation in the quantum Hall effect in strained bilayers.Bilayer graphene (BLG) consists of two honeycombs of carbons, A 1(2) , B 1(2) in the bottom (top) layer and its band structure can be understood using the Slonczewski-0 0 0 0 0 0 r FIG. 1: Top: Top view of the unperturbed (left) and deformed (right) bilayer graphene lattice. The top and bottom graphene layer is shown in yellow (light) and blue (dark), respectively. Bottom: Electronic band structure of unperturbed bilayer graphene in the vicinity of the Brillouin zone corners K and K'.Weiss tight-binding model for graphite [5]. Within layers, each A site is surrounded by three B sites (and vice versa), with relative positions given by vectors e 1 = (0, 1)r AB , e 2 = ( √ 3 2 , − 1 2 )r AB and e 3 = (− √ 3 2 , − 1 2 )r AB , where r AB is a distance between carbon sites (in equilibrium, r AB ≈ 1.4A), and coupling γ 0 ∼ 3eV between π-orbitals of the closest carbons. The two layers are arranged according to Bernal stacking shown in Fig. 1: sites A 2 appear on top of B 1 , with the interlayer coupling γ 1 ∼ 0.4eV, whereas sites A 1 and B 2 are positioned over the hexagons in the other layer and are coupled by 'skew' coupling γ 3 ∼ 0.3eV. All this gives rise [1] to a pair of 'low-energy' bands touching each other in the Brillouin zone corners K and K' which correspond to the electron states located on the sites A 1 and B 2 , and two bands split by the energy ±γ 1 which correspond to electron states located on the sites A 2 and B 1 (see Fig. 1). The degeneracy of the low-energy bands is the result of cancellation of the contributions from the intra-and interlayer hops in directions e 1 ...

Strain and curvature induced evolution of electronic band structures in twisted graphene bilayer

“…Compared with single-layer graphene, graphene bilayers, including AA stacked, AB stacked (Bernal) and twisted graphene bilayer, display even more complex electronic band structures and intriguing properties because of the interplay of quasiparticles between the Dirac cones on each layer [16][17][18][19][20][21][22][23][24][25][26][27][28] . Recently, several groups addressed the physics of the strained graphene bilayer (either AA-or AB-stacked graphene bilayer) theoretically and obtained many interesting results [29][30][31][32][33][34] . Despite many suggestive findings and potential applications, there have unfortunately been no experimental studies of the effect of strain on the electronic band structures of the graphene bilayer.…”

Strong interaction effects at a Fermi surface in a model for voltage-biased bilayer graphene

“…For a perfect bilayer formed from two monolayers stacked in AB configuration with interlayer coupling strength t ∼ O(0.1)t, where t is the hopping parameter in the original tight-binding Hamiltonian, the electron dispersion relation in the vicinity of the Dirac point is quadratic, only becoming approximately relativistic (i.e., linear) for ka t /t [10]. Theoretical studies suggest however that the presence of a "skew" interlayer coupling with strength t 3 breaking the trigonal symmetry of the crystal, the parabolic bands split to form separate Dirac cones so that N f = 4 is a reasonable approximation for ka t 3 t /t 2 [3]. A similar effect results from mechanical deformation.…”

“…1 shows results from μa t = 0.1 on 48 3 , for two values of the exciton source j . In the normal channel the time-asymmetric form of C N is manifest, and changes in character as k increases.…”

“…On the assumption that a smooth curve may be drawn through the admittedly noisy data, there is no evidence for any significant finite volume artifacts, or systematic difference between the two channels. Figure 3 plots E(k) for various j at μa t = 0.1 on 48 3 . The nonmonotonicity observed above becomes more pronounced as j → 0, and since there is little shift in the minimum with j , we adopt the pragmatic procedure of identifying the Fermi momentum k F with the value of k where E is minimum.…”

“…First, the description in terms of N f = 4 relativistic species (i.e., N f = 2 electrons and N f = 2 holes, where N f counts the number of fourcomponent spinors) is only justified by the band structure of the tight-binding model in the presence of an interlayer "skew" coupling breaking the trigonal symmetry of the underlying lattice [3]. Second, the interaction between charge densities is simplified to be a local four-Fermi contact, although a more realistic unscreened Coulomb interaction can be modeled with the introduction of a third spatial lattice direction to capture the electrodynamics [4].…”