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Recent progress in experimental techniques has made the quantum regime in plasmonics accessible. Since plasmons correspond to collective electron excitations, the electron-electron interaction plays an essential role in their theoretical description. Within the Random Phase Approximation, this interaction is incorporated through a system of equations of motion, which has to be solved self-consistently. For homogeneous media, an analytical solution can be found using the Fourier transform, giving rise to Lindhard theory. When the medium is spatially inhomogeneous, this is no longer possible and one often uses numerical approaches, which are however limited to smaller systems. In this paper, we present a novel semi-analytical approach for bulk plasmons in inhomogeneous media based on the semiclassical (or WKB) approximation, which is applicable when the charge density varies smoothly. By solving the equations of motion self-consistently, we obtain the expressions of Lindhard theory with a spatially varying Fermi wavevector. The derivation involves passing from the operators to their symbols, which can be thought of as classical observables on phase space. In this way we obtain effective (Hamiltonian) equations of motion for plasmons. We then find the quantized energy levels and the plasmon spectrum using Einstein-Brilllouin-Keller quantization. Our results provide a theoretical basis to describe different setups in quantum plasmonics, such as nanoparticles, quantum dots and waveguides., Comment: 97 pages, 21 figures

We employ the tight-binding propagation method to study Klein tunneling and quantum interference in large graphene systems. With this efficient numerical scheme, we model the propagation of a wave packet through a potential barrier and determine the tunneling probability for different incidence angles. We consider both sharp and smooth potential barriers in n-p-n and n-n' junctions and find good agreement with analytical and semiclassical predictions. When we go outside the Dirac regime, we observe that sharp n-p junctions no longer show Klein tunneling because of intervalley scattering. However, this effect can be suppressed by considering a smooth potential. Klein tunneling holds for potentials changing on the scale much larger than the interatomic distance. When the energies of both the electrons and holes are above the Van Hove singularity, we observe total reflection for both sharp and smooth potential barriers. Furthermore, we consider caustic formation by a two-dimensional Gaussian potential. For sufficiently broad potentials we find a good agreement between the simulated wave density and the classical electron trajectories., Comment: 14 pages, 12 figures

We report experimental data and theoretical analysis of Coulomb drag between two closely positioned graphene monolayers in weak magnetic field. Close enough to the neutrality point, coexistence of electrons and holes in each layer leads to a dramatic increase of the drag resistivity. Away from charge neutrality, we observe non-zero Hall drag. The observed phenomena are explained by decoupling of electric and quasiparticle currents which are orthogonal at charge neutrality. The sign of magneto-drag depends on the energy relaxation rate and geometry of the sample., Comment: 12 pages, 7 figures including supplemental information

We present temperature-dependent magneto-transport experiments around the charge neutrality point in graphene and determine the amplitude of potential fluctuations $s$ responsible for the formation of electron-hole puddles. The experimental value $s \approx 20$ meV is considerably larger than in conventional semiconductors which implies a strong localization of charge carriers observable up to room temperature. Surprisingly, in the quantized regime, the Hall resistivity overshoots the highest plateau values at high temperatures. We demonstrate by model calculations that such a peculiar behavior is expected in any system with coexisting electrons and holes when the energy spectrum is quantized and the carriers are partially localized., Comment: 5 pages, 5 figures, corrected typo in author's name

A realisation of a periodically driven microwave system is presented. The principal element of the scheme is a variable capacity, i.e. a varicap, introduced as an element of the resonant circuit. Sideband structures corresponding to different driving signals, have been measured experimentally. In the linear regime we observed sideband structures with specific shapes. The main peculiarities of these shapes can be explained within a semiclassical approximation. A good agreement between experimental data and theoretical expectations has been found., Comment: 19 pages, 11 figures

Spatially separated electron systems remain strongly coupled by electron-electron interactions even when they cannot exchange particles, provided that the layer separation d is comparable to a characteristic distance l between charge carriers within layers. One of the consequences of this remote coupling is a phenomenon called Coulomb drag, in which an electric current passed through one of the layers causes frictional charge flow in the other layer. Previously, only the regime of weak (d>>l) to intermediate (d ~ l) coupling could be studied experimentally. Here we use graphene-BN heterostructures with d down to 1 nm to probe interlayer interactions and Coulomb drag in the limit d<

In this paper we study scattering of two-dimensional massless Dirac fermions by a potential that depends on a single Cartesian variable. Depending on the energy of the incoming particle and its angle of incidence, there are three different regimes of scattering. To find the reflection and transmission coefficients in these regimes, we apply the Wentzel-Kramers-Brillouin (WKB), also called semiclassical, approximation. We use the method of comparison equations to extend our prediction to nearly normal incidence, where the conventional WKB method should be modified due to the degeneracy of turning points. We compare our results to numerical calculations and find good agreement., Comment: Minor revision; several references have been added

Coulomb drag between two unhybridized graphene sheets separated by a dielectric spacer has recently attracted considerable theoretical interest. We first review, for the sake of completeness, the main analytical results which have been obtained by other authors. We then illustrate pedagogically the minimal theory of Coulomb drag between two spatially-separated two-dimensional systems of massless Dirac fermions which are both away from the charge-neutrality point. This relies on second-order perturbation theory in the screened interlayer interaction and on Boltzmann transport theory. In this theoretical framework and in the low-temperature limit, we demonstrate that, to leading (i.e. quadratic) order in temperature, the drag transresistivity is completely insensitive to the precise intralayer momentum-relaxation mechanism (i.e. to the functional dependence of the scattering time on energy). We also provide analytical results for the low-temperature drag transresistivity for both cases of "thick" and "thin" spacers and for arbitrary values of the dielectric constants of the media surrounding the two Dirac-fermion layers. Finally, we present numerical results for the low-temperature drag transresistivity in the case in which one of the media surrounding the Dirac-fermion layers has a frequency-dependent dielectric constant. We conclude by suggesting an experiment that can potentially allow for the observation of departures from the canonical Fermi-liquid quadratic-in-temperature behavior of the transresistivity., Comment: 20 pages, 4 figures

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. 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 review chiral (Klein) tunneling in single-layer and bilayer graphene and present its semiclassical theory, including the Berry phase and the Maslov index. Peculiarities of the chiral tunneling are naturally explained in terms of classical phase space. In a one-dimensional geometry we reduced the original Dirac equation, describing the dynamics of charge carriers in the single layer graphene, to an effective Schr\"odinger equation with a complex potential. This allowed us to study tunneling in details and obtain analytic formulas. Our predictions are compared with numerical results. We have also demonstrated that, for the case of asymmetric n-p-n junction in single layer graphene, there is total transmission for normal incidence only, side resonances are suppressed., Comment: submitted to Proceedings of Nobel Symposium on graphene, May 2010

We consider an application of a general theory for cavities with point-like perturbations for a rectangular shape. Hereby we concentrate on experimental wave patterns obtained for nearly degenerate states. The nodal lines in these patterns may be broken, which is an effect coming only from the experimental determination of the patterns. These findings are explained within a framework of the developed theory., Comment: 14 pages, 3 figures

We present a microwave realization of finite tight-binding graphene-like structures. The structures are realized using discs with a high index of refraction. The discs are placed on a metallic surface while a second surface is adjusted atop the discs, such that the waves coupling the discs in the air are evanescent, leading to the tight-binding behavior. In reflection measurements the Dirac point and a linear increase close to the Dirac point is observed, if the measurement is performed inside the sample. Resonances due to edge states are found close to the Dirac point if the measurements are performed at the zigzag-edge or at the corner in case of a broken benzene ring., Comment: 4 pages, 6 figures

The theoretical interpretation of measurements of "wavefunctions" and spectra in electromagnetic cavities excited by antennas is considered. Assuming that the characteristic wavelength of the field inside the cavity is much larger than the radius of the antenna, we describe antennas as "point-like perturbations". This approach strongly simplifies the problem reducing the whole information on the antenna to four effective constants. In the framework of this approach we overcame the divergency of series of the phenomenological scattering theory and justify assumptions lying at the heart of "wavefunction measurements". This selfconsistent approach allowed us to go beyond the one-pole approximation, in particular, to treat the experiments with degenerated states. The central idea of the approach is to introduce ``renormalized'' Green function, which contains the information on boundary reflections and has no singularity inside the cavity., Comment: 23 pages, 6 figures

We study an interaction of 2D quasiparticles with linear dispersion (graphene) with impurity potentials. It is shown that in 1D potential well (quantum wire) there are discrete levels, corresponding to localized states, whereas in 2D well (quantum dot) there are no such states. Scattering cross-section of electrons (holes) of graphene by an axially symmetric potential well is found and it is shown that for infinetily large energy of incoming particles the cross-section tends to a constant. The effective Hamiltonian for a curved quantum wire of graphene is derived and it is shown that the corresponding geometric potential cannot form 1D bound states., Comment: 2 figures. The paper was sent to JETP Letters. Reasons for replacing previous versions: added references [4-6], corrected typos

We study linear problems of mathematical physics in which the adiabatic approximation is used in the wide sense. Using the idea that all these problems can be treated as problems with operator-valued symbol, we propose a general regular scheme of adiabatic approximation based on operator methods. This scheme is a generalization of the Born-Oppenheimer and Maslov methods, the Peierls substitution, etc. The approach proposed in this paper allows one to obtain "effective" reduced equations for a wide class of states inside terms (i.e., inside modes, subregions of dimensional quantization, etc.) with the possible degeneration taken into account., Comment: 57 pages