A moiré pattern is formed when two copies of a periodic pattern are overlaid with a relative twist. We address the electronic structure of a twisted two-layer graphene system, showing that in its continuum Dirac model the moiré pattern periodicity leads to moiré Bloch bands. The two layers become more strongly coupled and the Dirac velocity crosses zero several times as the twist angle is reduced. For a discrete set of magic angles the velocity vanishes, the lowest moiré band flattens, and the Dirac-point density-ofstates and the counterflow conductivity are strongly enhanced.L ow-energy electronic properties of few layer graphene (FLG) systems are known (1-8) to be strongly dependent on stacking arrangement. In bulk graphite 0°and 60°relative orientations of the individual layer honeycomb lattices yield rhombohedral and Bernal crystals, but other twist angles also appear in many samples (9). Small twist angles are particularly abundant in epitaxial graphene layers grown on SiC (10, 11), but exfoliated bilayers can also appear with a twist, and arbitrary alignments between adjacent layers can be obtained by folding a single graphene layer (12,13).Recent advances in FLG preparation methods have attracted theoretical attention (14)(15)(16)(17)(18)(19)(20) to the intriguing electronic properties of systems with arbitrary twist angles, usually focusing on the two-layer case. The geometry of the bilayer system is characterized by a twist angle θ and by a translation vector d. Commensurability is determined only by θ. Sliding one layer with respect to the other in a commensurate structure modifies the unit cell but leaves the bilayer crystalline. In this work we find it convenient to regard the AB stacking as the aligned configuration. The positions of the carbon atoms in the two misaligned layers labeled by R and R 0 are then related by R 0 ¼ MðθÞðR − τÞ þ d, where M is a 2-D rotation matrix within the graphene plane, and τ is a vector connecting the two atoms in the unit cell.The problem is mathematically interesting because a bilayer forms a two-dimensional crystal only at a discrete set of commensurate rotation angles; for generic twist angles Bloch's theorem does not apply microscopically and direct electronic structure calculations are not feasible. For twist angles larger than a few degrees the two layers are electronically isolated to a remarkable degree, except at a small set of angles which yield low-order commensurate structures (16,19). As the twist angles become smaller, interlayer coupling strengthens, and the quasiparticle velocity at the Dirac point begins to decrease.Here we focus on the strongly coupled small twist angle regime. We derive a low-energy effective Hamiltonian valid for any value of d and for θ ≲ 10°irrespective of whether or not the bilayer structure is periodic. We show that it is meaningful to describe the electronic structure using Bloch bands even for incommensurate twist angles and study the dependence of these bands on θ. ModelWe construct a low-energy continuum model Hamiltoni...

Because graphene is an atomically two-dimensional gapless semiconductor with nearly identical conduction and valence bands, graphene-based bilayers are attractive candidates for high-temperature electron-hole pair condensation. We present estimates which suggest that the Kosterlitz-Thouless temperatures of these twodimensional counterflow superfluids can approach room temperature.

Energy transfer to acoustic phonons is the dominant low-temperature cooling channel of electrons in a crystal. For cold neutral graphene we find that the weak cooling power of its acoustic modes relative to their heat capacity leads to a power-law decay of the electronic temperature when far from equilibrium. For heavily doped graphene a high electronic temperature is shown to initially decrease linearly with time at a rate proportional to n;{3/2} with n being the electronic density. The temperature at which cooling via optical phonon emission begins to dominate depends on graphene carrier density.

Commensurate-incommensurate transitions are ubiquitous in physics and are often accompanied by intriguing phenomena. In few-layer graphene (FLG) systems, commensurability between honeycomb lattices on adjacent layers is regulated by their relative orientation angle θ, which is in turn dependent on sample preparation procedures. Because incommensurability suppresses inter-layer hybridization, it is often claimed that graphene layers can be electrically isolated by a relative twist, even though they are vertically separated by a fraction of a nanometer. We present a theory of interlayer transport in FLG systems which reveals a richer picture in which the specific conductance depends sensitively on θ, single-layer Bloch state lifetime, in-plane magnetic field, and bias voltage. We find that linear and differential conductances are generally large and negative near commensurate values of θ, and small and positive otherwise.Experimental advances in the fabrication of graphenebased structures [1,2] have now provided researchers with a multitude of systems that have strikingly distinct electronic properties. By engineering the substrate underlying exfoliated samples [3][4][5], identifying exfoliated fragments with folds[6], or controlling epitaxial growth conditions [7,8], the size and shape of the honeycomb lattice arrays [9,10] and the number of graphene layers and their orientations can all be varied. This structural diversity nourishes hopes for a future carbon-based electronics[11] with band-structure and transport characteristics that can be tailored for different types of applications.FLG has advantages over single-layer-graphene because it has a larger current-carrying capacity and because its electronic properties are sensitive to more engineerable system parameters [12]. In nature it appears in a variety of stacking arrangements, the most common being Bernal and rhombohedral sequences which can form three dimensional lattices. It has been understood for some time [13] [16]. The present work is motivated primarily by the need to achieve a more complete understanding of transport in these graphitic nanostructures, which currently appear to provide the most promising platform for applications.In a bilayer system, the relative rotation angle θ can be classified as either commensurate or incommensurate [17]. In the former case the misaligned bilayer system still forms a crystal, albeit one with larger lattice vectors and more than four atoms per unit cell. Commensurability occurs at a countably infinite set of orientations; but the probability that a randomly selected orientation angle is commensurate vanishes. The energy bands of commensurate twisted multilayers disperse approximately linearly with momentum [18][19][20], except at energies very close to the Dirac point. However, the Dirac velocity is reduced FIG. 1: Interlayer (RC) equilibration rate as a function of twist angle θ. These results were calculated for two layers with equal carrier densities (n = 5 × 10 12 cm −2 ) and ǫFτ = 3, where ǫF is the Fermi e...

The Hofstadter butterfly spectral patterns of lattice electrons in an external magnetic field yield some of the most beguiling images in physics. Here we explore the magnetoelectronic spectra of systems with moiré spatial patterns, concentrating on the case of twisted bilayer graphene. Because long-period spatial patterns are accurately formed at small twist angles, fractal butterfly spectra and associated magnetotransport and magnetomechanical anomalies emerge at accessible magnetic field strengths.