We suggest a way of confining quasiparticles by an external potential in a small region of a graphene strip. Transversal electron motion plays a crucial role in this confinement. Properties of thus obtained graphene quantum dots are investigated theoretically for different types of the boundary conditions at the edges of the strip. The (quasi)bound states exist in all systems considered. At the same time, the dependence of the conductance on the gate voltage carries information about the shape of the edges.

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For a quantum dot (QD) in the intermediate regime between integrable and fully chaotic, the widths of single-particle levels naturally differ by orders of magnitude. In particular, the width of one strongly coupled level may be larger than the spacing between other, very narrow, levels. In this case many consecutive Coulomb blockade peaks are due to occupation of the same broad level. Between the peaks the electron jumps from this level to one of the narrow levels, and the transmission through the dot at the next resonance essentially repeats that at the previous one. This offers a natural explanation to the recently observed behavior of the transmission phase in an interferometer with a QD.

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 // ;

We present an analytical solution of the delocalization transition that is induced by an imaginary vector potential in a disordered chain [N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996), cond-mat/9603165]. We compute the relation between the real and imaginary parts of the energy in the thermodynamic limit, as well as finite-size effects. The results are in good agreement with numerical simulations for weak disorder (mean free path large compared to the wavelength).Comment: 3 pages, RevTeX, 2 figure

Distribution of charge induced by a gate voltage in a graphene strip is investigated. We calculate analytically the charge profile and demonstrate a strong(macroscopic) charge accumulation along the boundaries of a micrometers-wide strip. This charge inhomogeneity is especially important in the quantum Hall regime where we predict the doubling of the number of edge states and coexistence of two different types of such states. Applications to graphene-based nanoelectronics are discussed. The new material graphene, a monolayer of carbon atoms with honeycomb lattice structure, is attracting a lot of interest since 2005 when the first transport measurements in this material have been reported [1,2,3]. The interest in 2D electron gases in graphene originates from the Dirac-like spectrum of the low-energy quasiparticles [4]. Several prominent phenomena have been investigated in this "relativistic" system both experimentally and theoretically, including quantum Hall effect (QH) [5,6], weak localization and other effects of disorder [7,8], superconducting proximity effects [9,10,11,12], etc.In the experiments [1, 2, 3], mechanically exfoliated graphene samples were separated from the metallic gate by a b ≈ 0.3µm wide insulating layer (SiO 2 ). The width of the insulator is dictated by the necessity to identify optically the single-layer graphene. In the undoped graphene (half filling), the charge of the conduction electrons is compensated by the charge of the carbon ions forming the lattice. By applying a large (V g 100V) voltage V g to the lower gate one induces a considerable (n e /V g ≈ 7.2 10 10 cm −2 /V [1]) uncompensated charge e×n e in the graphene plane. This extra charge is screened by "image charges" induced in the metallic gate. However, since the images are located 0.6µm below graphene, such a screening becomes effective only in the central region of several microns large graphene samples. As a result, the charge distribution cannot be homogeneous.In this paper, we calculate analytically the charge distribution in the graphene strip and demonstrate a strong increase of the charge density near the strip edges (numerically, a charge accumulation near the edges has been seen in Ref. [13]). For a gate voltage of ≈ 10V the distance between the excess electrons in the sheet is of order ∼ 10nm. This means that for the 0.1 ÷ 1µm wide strips one may speak of a continuous charge distribution and determine the latter minimizing the electrostatic energy of the electrons. In semiconductor heterostructures the electron redistribution has been discussed in the context of compressible/incompressible QH stripes formation [14]. However, in that case electrons were confined by a smooth potential, which resulted in a continuous charge density profile at the edge. As we will see, at the

Shol noise in a chaotic cavity (Lyapunov exponent λ, level spacing δ, lineai dimension L), coupled by two jV-mode point contacts to election reseivons, is studied äs a mcasuie of the ciossovei fiom slochastic quantum tiansport to deteimmistic classical transport The tiansition pioceeds through the formation ofßilly transmitted or leflected scattering states, which we construci exphcitly The fully Uansmitted states contnbute to the rnean cunent 7, but not to the shot-noise powei S We find that these noiseless tiansmission channels do not exist for NS ^k F Shot noise can distmguish deteimmistic scatteimg, chaiactenstic of paiticles, fiom stochastic scattenng, chaiactenstic of waves Particle dynamics is deteimmistic A given initial position and momentum fix the entne tiajectoiy In particulai, they fix whethei the paiticle will be tiansmitted or leflected, so the scattenng is noiseless Wave dynamics is stochastic The quantum uncertamty in position and momentum introduces a probabihstic element mto the dynamics, so it is noisyThe suppiession of shot noise in a conductoi with determimstic scattenng was predicted many yeats ago fiom this qualitative aigument l A bettei undeistandmg, and a quantitative description, of how shot noise measuies the tiansition fiom paiticle to wave dynamics m a chaotic quantum dot was put foiward by Agam, Aleinei, and Laikm," and developed fuithei in Ref 3 The key concept is the Ehienfest time T E , which is the charactenstic time scale of quantum chaos 4 The noise powei S^exp(-T E /T D ) was piedicted to vamsh exponentially with the latio of T E and the mean dwell time T D =TrhlNS m the quantum dot (with δ the level spacing and N the numbei of modes in each of the two pomt contacts thiough which the cunent is passed) A lecent measuiement of the N dependence of S is consistent with this piediction foi T E

We identify the time T between Andreev reflections as a classical adiabatic invariant in a ballistic chaotic cavity (Lyapunov exponent λ), coupled to a superconductor by an N -mode constriction. Quantization of the adiabatically invariant torus in phase space gives a discrete set of periods Tn, which in turn generate a ladder of excited states εnm = (m + 1/2)πh/Tn. The largest quantized period is the Ehrenfest time T0 = λ −1 ln N . Projection of the invariant torus onto the coordinate plane shows that the wave functions inside the cavity are squeezed to a transverse dimension W/ √ N , much below the width W of the constriction.PACS numbers: 05.45. Mt, 73.63.Kv, 74.50.+r, 74.80.Fp The notion that quantized energy levels may be associated with classical adiabatic invariants goes back to Ehrenfest and the birth of quantum mechanics [1]. It was successful in providing a semiclassical quantization scheme for special integrable dynamical systems, but failed to describe the generic nonintegrable case. Adiabatic invariants play an interesting but minor role in the quantization of chaotic systems [2,3].Since the existence of an adiabatic invariant is the exception rather than the rule, the emergence of a new one quite often teaches us something useful about the system. An example from condensed matter physics is the quantum Hall effect, in which the semiclassical theory is based on two adiabatic invariants: The flux through a cyclotron orbit and the flux enclosed by the orbit center as it slowly drifts along an equipotential [4]. The strong magnetic field suppresses chaotic dynamics in a smooth potential landscape, rendering the motion quasiintegrable.Some time ago it was realized that Andreev reflection has a similar effect on the chaotic motion in an electron billiard coupled to a superconductor [5]. An electron trajectory is retraced by the hole that is produced upon absorption of a Cooper pair by the superconductor. At the Fermi energy E F the dynamics of the hole is precisely the time reverse of the electron dynamics, so that the motion is strictly periodic. The period from electron to hole and back to electron is twice the time T between Andreev reflections. For finite excitation energy ε the electron (at energy E F + ε) and the hole (at energy E F − ε) follow slightly different trajectories, so the orbit does not quite close and drifts around in phase space. This drift has been studied in a variety of contexts [5,6,7,8,9], but not in connection with adiabatic invariants and the associated quantization conditions. It is the purpose of this paper to make that connection and point out a striking physical consequence: The wave functions of Andreev levels fill the cavity in a highly nonuniform "squeezed" way, which has no counterpart in normal state chaotic or regular billiards. In particular the squeezing is distinct from periodic orbit scarring [10] and entirely different from the random superposition of plane waves expected for a fully chaotic billiard [11].Adiabatic quantization breaks down near the excitation g...