Lateral superlattices have attracted major interest as this may allow one to modify spectra of two dimensional (2D) electron systems and, ultimately, create materials with tailored electronic properties 1-8 . Previously, it proved difficult to realize superlattices with sufficiently short periodicity and weak disorder, and most of the observed features could be explained in terms of commensurate cyclotron orbits 1-4 . Evidence for the formation of superlattice minibands (so called Hofstadter's butterfly 9 ) has been limited to the observation of new low-field oscillations 5 and an internal structure within Landau levels 6-8 . Here we report transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions. The substrate's moiré potential 10-12 leads to profound changes in graphene's electronic spectrum. Second-generation Dirac points 13-22 appear as pronounced peaks in resistivity accompanied by reversal of the Hall effect. The latter indicates that the sign of the effective mass changes within graphene's conduction and valence bands. Quantizing magnetic fields lead to Zak-type cloning 23 of the third generation of Dirac points that are observed as numerous neutrality points in fields where a unit fraction of the flux quantum pierces the superlattice unit cell. Graphene superlattices open a venue to study the rich physics expected for incommensurable quantum systems 7-9,22-24 and illustrate the possibility to controllably modify electronic spectra of 2D atomic crystals by using their crystallographic alignment within van der Waals heterostuctures 25 .Since the first observation of Weiss oscillations 1,2 , 2D electronic systems subjected to a periodic potential have been studied in great detail [3][4][5][6][7][8] . The advent of graphene has rapidly sparked interest in its superlattices, too [13][14][15][16][17][18][19][20][21][22] . The principal novelty in this case is the Dirac-like spectrum and the fact that charge carriers are not buried deep under the surface, allowing a relatively strong superlattice potential on a true nanometer scale. One promising avenue for making nanoscale graphene superlattices is the use of a potential induced by another crystal. For example, graphene placed on top of graphite or hexagonal boron nitride (hBN) exhibits a moiré pattern [10][11][12]26 , and graphene's tunneling density of states becomes strongly modified 12,26 indicating the formation of superlattice minibands. The spectral reconstruction occurs near the edges of superlattice's Brillouin zone (SBZ) that are characterized by wavevector G =4/ D and energy E S =v F G/2 (D is the superlattice period and v F graphene's Fermi velocity) 12,22 .To observe moiré minibands in transport properties, graphene has to be doped so that the Fermi energy reaches the reconstructed part of the spectrum. This imposes severe constraints on the misalignment angle  of graphene relatively to hBN. Indeed, D is given by  and the 1.8% difference between the two lattice constants ...

Using a general symmetry-based approach, we provide a classification of generic miniband structures for electrons in graphene placed on substrates with the hexagonal Bravais symmetry. In particular, we identify conditions at which the first moiré miniband is separated from the rest of the spectrum by either one or a group of three isolated mini Dirac points and is not obscured by dispersion surfaces coming from other minibands. In such cases the Hall coefficient exhibits two distinct alternations of its sign as a function of charge carrier density.PACS numbers: 73.22.Pr,73.21.Cd, Recently, it has been demonstrated that the electronic quality of graphene-based devices can be dramatically improved by placing graphene on an atomically flat crystal surface, such as hexagonal boron nitride (hBN) [1][2][3][4][5][6][7]. At the same time, graphene's electronic spectrum also becomes modified, acquiring a complex, energydependent form caused by incommensurability between the graphene and substrate crystal lattices [8][9][10]. In particular, for graphene placed on hBN, the difference between their lattice constants and crystallographic misalignment generate a hexagonal periodic structure known as a moiré pattern [2,3,[8][9][10]. The resulting periodic perturbation, usually referred to as a superlattice, acts on graphene's charge carriers and leads to multiple minibands and the generation of secondary Dirac-like spectra. The resulting new Dirac fermions present yet another case where graphene allows mimicking of QED phenomena under conditions that cannot be achieved in particle physics experiments. In contrast to relativistic particles in free space, the properties of secondary Dirac fermions in graphene can be affected by a periodic sublattice symmetry breaking and modulation of carbon-carbon hopping amplitudes, in addition to a simple potential modulation. The combination of different features in the modulation results in a multiplicity of possible outcomes for the moiré miniband spectrum in graphene which we systematically investigate in this article.To describe the effect of a substrate on electrons in graphene at a distance, d, much larger than the spacing, a, between carbon atoms in graphene's honeycomb lattice, we use the earlier observation [8][9][10][11][12][13][14] that, at d a the lateral variation of the wavefunctions of the p z carbon orbitals is smooth on the scale of a. This is manifested in the comparable sizes of the skew and vertical hopping in graphite and permits an elegant continuummodel description [11][12][13][14] of the interlayer coupling in twisted bilayers and the resulting band structure. A similar idea applied to graphene on a hBN substrate [8][9][10] suggests that a substrate perturbation for Dirac electrons in graphene can be described in terms of simple harmonic functions corresponding to the six smallest reciprocal lattice vectors of the moiré superlattice.Below, we shall use a similar approach to analyse the generic properties of moiré minibands for electrons in graphene subjected to a subs...

The nematic phase transition in electronic liquids, driven by Coulomb interactions, represents a new class of strongly correlated electronic ground states. We studied suspended samples of bilayer graphene, annealed so that it achieves very high quasiparticle mobilities (greater than 10(6) square centimers per volt-second). Bilayer graphene is a truly two-dimensional material with complex chiral electronic spectra, and the high quality of our samples allowed us to observe strong spectrum reconstructions and electron topological transitions that can be attributed to a nematic phase transition and a decrease in rotational symmetry. These results are especially surprising because no interaction effects have been observed so far in bilayer graphene in the absence of an applied magnetic field.

We show theoretically how constant-energy maps of the angle-resolved photoemission intensity can be used to test wave function symmetry in graphene. For monolayer graphene, we demonstrate that the observed anisotropy of ARPES spectra is a manifestation of what has been recently branded as electronic chirality. For bilayer graphene, we show that the anisotropy of the constant-energy maps may be used to extract information about the magnitude and sign of interlayer coupling parameters and about symmetry breaking inflicted on a bilayer by the underlying substrate.

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 ...