Graphene has demonstrated great promise for future electronics technology as well as fundamental physics applications because of its linear energy-momentum dispersion relations which cross at the Dirac point [1,2]. However, accessing the physics of the low density region at the Dirac point has been difficult because of the presence of disorder which leaves the graphene with local microscopic electron and hole puddles [3][4][5], resulting in a finite density of carriers even at the charge neutrality point. Efforts have been made to reduce the disorder by suspending graphene, leading to fabrication challenges and delicate devices which make local spectroscopic measurements difficult [6,7].Recently, it has been shown that placing graphene on hexagonal boron nitride (hBN) yields improved device performance [8]. In this letter, we use scanning tunneling microscopy to show that graphene conforms to hBN, as evidenced by the presence of Moiré patterns in the topographic images. However, contrary to recent predictions [9,10], this conformation does not lead to a sizable band gap due to the misalignment of the lattices. Moreover, local spectroscopy measurements demonstrate that the electron-hole charge fluctuations are reduced by two orders of magnitude as compared to those on silicon oxide. This leads to charge fluctuations which are as small as in suspended graphene [6], opening up Dirac point physics to more diverse experiments than are possible on freestanding devices.

The Schrödinger equation dictates that the propagation of nearly free electrons through a weak periodic potential results in the opening of bandgaps near points of the reciprocal lattice known as Brillouin zone boundaries 1 . However, in the case of massless Dirac fermions, it has been predicted that the chirality of the charge carriers prevents the opening of a bandgap and instead new Dirac points appear in the electronic structure of the material 2,3 . Graphene on hexagonal boron nitride exhibits a rotation-dependent moiré pattern 4,5 . Here, we show experimentally and theoretically that this moiré pattern acts as a weak periodic potential and thereby leads to the emergence of a new set of Dirac points at an energy determined by its wavelength. The new massless Dirac fermions generated at these superlattice Dirac points are characterized by a significantly reduced Fermi velocity. Furthermore, the local density of states near these Dirac cones exhibits hexagonal modulation due to the influence of the periodic potential.Owing to its hexagonal lattice structure with a diatomic unit cell, graphene has low-energy electronic properties that are governed by the massless Dirac equation 6 . This has a number of consequences, among them Klein tunnelling [7][8][9][10] , which prevents electrostatic confinement of charge carriers and inhibits the fabrication of standard semiconductor devices. This has motivated a number of recent theoretical investigations of graphene in periodic potentials 2,3,[11][12][13][14][15] , which explored ways of controlling the propagation of charge carriers by means of various superlattice potentials. On the analytical side, one-dimensional potentials render particle propagation anisotropic 2,3,11,14 and generate new Dirac points, where the electron and hole bands meet, at energies ±hv F |G|/2 given by the reciprocal superlattice vectors G (refs 2,3), where v F is the Fermi velocity. Numerical approaches have extended several of these results to the case of two-dimensional potentials 2,3,14,15 . Unlike for Schrödinger fermions, the periodic potentials generally induce new Dirac points but do not open bandgaps in graphene, owing to the chiral nature of the Dirac fermions.Recent scanning tunnelling microscope (STM) topography experiments have reported well-developed moiré patterns in graphene on crystalline substrates, which suggests that the latter generate effective periodic potentials 4,5,16,17 . Of particular interest is hexagonal boron nitride (hBN), because it is an insulator which only couples weakly to graphene. Furthermore, graphene on hBN exhibits the highest mobility ever reported for graphene on any substrate 18 , and has strongly suppressed charge inhomogeneities 4,5 . Hexagonal boron nitride is a layered material whose planes have the same atomic structure as graphene, with a 1.8% longer lattice constant. The influence of the weak graphene-substrate interlayer coupling on the electronic transport and spectroscopic properties of graphene is not well understood. In particular, there ...

We study the level spacing statistics P (s) in many-body Fermi systems and determine a critical two-body interaction strength Uc at which a crossover from Poisson to Wigner-Dyson statistics takes place. Near the Fermi level the results allow to find a critical temperature T ch above which quantum chaos and thermalization set in.PACS numbers: 05.45.+b, 05.30.Fk, 24.10.Cn The Random Matrix Theory (RMT) was developed to explain the general properties of complex energy spectra in many-body interacting systems such as heavy nuclei, many electron atoms and molecules [1]. Later, it has found many other successful applications in different physical systems. Among the most recent of them we can quote models of quantum chaos where RMT appears due to the classically chaotic but deterministic underlying dynamics [2]. One of the most direct indications of the emergence of quantum chaos is the transition of the level spacing statistics P (s) from Poisson to WignerDyson (WD) distribution. This property has been widely used to detect the transition from integrability to chaos not only in systems with few degrees of freedom [2] but also in solid-state models with many interacting electrons [3]. It was also applied to determine the delocalization threshold in noninteracting disordered systems [4].While the conditions for the appearance of the WD distribution in noninteracting systems is qualitatively well understood the situation is more intricate in presence of interaction. Indeed, in this case the size of the total Hamiltonian matrix grows exponentially with the number of particles and it becomes very sparse as a result of the two-body nature of the interaction. Due to that it was initially not obvious whether switching on the interaction would lead to the WD statistics. To study this problem a two-body random interaction model (TBRIM) had been proposed [5,6]. This model consists of n fermions which can occupy m unperturbed energy orbitals with mean one-particle level spacing ∆. The multiparticle states are coupled by two-body random transition matrix elements of typical strength U . It was found that a sufficiently strong U leads to a level mixing and appearance of WD statistics. Very recently the interest for this model has been renewed and its statistical properties were investigated in more details [7]. This raise of interest was stimulated by the understanding that many statistical properties of real physical systems such as the rare-earth Ce atom [8] and the 28 Si nucleus [9,10] are well described by the TBRIM. In addition this model is quite similar to the s-d shell model used for a description of complex nuclei [9,10]. Since interaction is generically of two-body nature it is reasonable to assume that this model will be also useful for a description of interacting electrons in clusters [11] and mesoscopic quantum dots [12].While the statistical properties of the TBRIM were studied in some details, surprisingly, the most important question of the critical interaction strength U c at which the WD level spacing statisti...

The overlap of two wave packets evolvmg m time with shghtly different Hamiltomans decays exponentially a e~y', for perturbation strengths U greater than the level spacing Δ We present numencal evidence for a dynamical System that the decay rate γ is given by the smallest of the Lyapunov exponent λ of the classical chaotic dynarrucs and the level broadening ί/ 2 /Δ that follows from the golden rule of quantum mechanics This imphes the ränge of vahdity U> \j\Ä for the perturbation-strength independent decay rate discovered by Jalabert and Pastawski [Phys Rev Lett 86, 2490(2001 Perturbation theory breaks down once a typical matnx element U of H l connectmg different eigenstates of H 0 becomes greater than the level spacing Δ Then the eigenstates of H, decomposed mto the eigenstates of H 0 , contam a laige number of non-neghgible components The distnbution p(E) (local spectral density) of these components over energy has a Lorentzian formwith a spreading width Γ-ί/ 2 /Δ given by the golden rule [6,7] A simple calculation m a landom-matiix model gives an aveiage decay Mccexp(-IY) governed by the same golden uile width This should be contrasted with the exponential decay M^exp(-λί) obtamed by Jalabert and Pastawski [3], which is governed by the Lyapunov exponent λ of the classical chaotic dynamics Smce the landom-matiix model has by construction an infinite Lyapunov exponent, one way to umiy both results would be to have an exponential decay with a late set by the smallest of Γ and λ We will m what follows present nu mencal evidence for this scenano, usmg a dynamical System in which we can vaiy the lelative magnitude of Γ and λ Theie exists a third energy scale, the mverse of the Ehienfest time T E , that is smaller than the Lyapunov exponent by a factor logarithmic m the System's effective Planck constant In om numencs we do not have enough Orders of magnitude between l/r E and λ to distinguish between the two, so that our findings lemam somewhat inconclusive in this respect Because Γ cannot become bigger than the band width B of H 0 (we are interested m the regime // ;

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The crystal structure of a material plays an important role in determining its electronic properties. Changing from one crystal structure to another involves a phase transition which is usually controlled by a state variable such as temperature or pressure. In the case of trilayer graphene, there are two common stacking configurations (Bernal and rhombohedral) which exhibit very different electronic properties [1][2][3][4][5][6][7][8][9][10][11]. In graphene flakes with both stacking configurations, the region between them consists of a localized strain soliton where the carbon atoms of one graphene layer shift by the carbon-carbon bond distance [12][13][14][15][16][17][18]. Here we show the ability to move this strain soliton with a perpendicular electric field and hence control the stacking configuration of trilayer graphene with only an external voltage. Moreover, we find that the free energy difference between the two stacking configurations scales quadratically with electric field, and thus rhombohedral stacking is favored as the electric field increases. This ability to control the stacking order in graphene opens the way to novel devices which combine structural and electrical properties. * leroy@physics.arizona.edu 2 Multilayer graphene has attracted interest in large part due to the ability to induce a sizable band gap with the application of an electric field. The exact nature of the electronic properties of multilayer graphene is controlled both by the number of layers as well as their stacking configuration. The equilibrium in-plane crystal structure of graphene is hexagonal [19], and deviations from this equilibrium require a large amount of energy. Upon stacking multiple graphene sheets, Bernal-stacking -where the A-sublattice of one layer resides above the B-sublattice of the other layer -represents the lowest energy stacking configuration. Thus under normal circumstances, any two graphene layers in a graphite stack will be Bernal-stacked with respect to one another. However, when examining layers more than one apart, there can be multiple nearly-degenerate stacking configurations (2 (n−2) such configurations for n layers) [1]. For example, in the simplest case of trilayer graphene, the top layer may lie directly above the bottom layer (denoted Bernal-or ABA-stacked), or may instead be configured such that one sublattice of the top layer lies above the center of the hexagon of the bottom layer (denoted rhombohedrally-or ABC-stacked). Applying a perpendicular electric field breaks the sublattice symmetry differently depending on the stacking configuration, and thus is capable of re-ordering the energy hierarchy of the stacking configurations [1][2][3][4][5][6][7][8][9][10][11]. As a consequence, multilayer graphene exhibits the rare behavior of crystal structure modification, and hence modification of electronic properties, via the application of an external electric field.To examine this effect, we perform scanning tunneling topography (STM) and scanning tunneling spectroscopy (STS) measur...

We construct a scattering theory of weakly nonlinear thermoelectric transport through sub-micron scale conductors. The theory incorporates the leading nonlinear contributions in temperature and voltage biases to the charge and heat currents. Because of the finite capacitances of sub-micron scale conducting circuits, fundamental conservation laws such as gauge invariance and current conservation require special care to be preserved. We do this by extending the approach of Christen and Büttiker [Europhys. Lett. 35, 523 (1996)] to coupled charge and heat transport. In this way we write relations connecting nonlinear transport coefficients in a manner similar to Mott's relation between the linear thermopower and the linear conductance. We derive sum rules that nonlinear transport coefficients must satisfy to preserve gauge invariance and current conservation. We illustrate our theory by calculating the efficiency of heat engines and the coefficient of performance of thermoelectric refrigerators based on quantum point contacts and resonant tunneling barriers. We identify in particular rectification effects that increase device performance.PACS numbers: 85.50.Fi

We investigate the spin Hall effect in ballistic chaotic quantum dots with spin-orbit coupling. We show that a longitudinal charge current can generate a pure transverse spin current. While this transverse spin current is generically nonzero for a fixed sample, we show that when the spin-orbit coupling time is short compared to the mean dwell time inside the dot, it fluctuates universally from sample to sample or upon variation of the chemical potential with a vanishing average.