Electrons in graphene can travel for several microns without scattering at low temperatures, and their motion becomes ballistic, following classical trajectories. When a magnetic field B is applied perpendicular to the plane, electrons follow cyclotron orbits.Magnetic focusing occurs when electrons injected from one narrow contact focus onto a second contact located an integer number of cyclotron diameters away. By tuning the magnetic field B and electron density n in the graphene layer, we observe magnetic focusing peaks. We use a cooled scanning gate microscope to image cyclotron trajectories in graphene at 4.2 K. The tip creates a local change in density that casts a shadow by deflecting electrons flowing nearby; an image of flow can be obtained by measuring the transmission between contacts as the tip is raster scanned across the sample. On the first magnetic focusing peak, we image a cyclotron orbit that extends from one contact to the other. In addition, we study the geometry of orbits deflected into the second point contact by the tip. KEYWORDS:Graphene, scanning gate microscope, image cyclotron orbits, magnetic focusing. 2The unusual properties of graphene offer new approaches to electronics based on the ballistic motion of electrons. 14,16,17 Electrons that enter the graphene sheet at different angles all travel in a circle -as a consequence of this geometry, the electron flux peaks at a distance d c where circles overlap. As B is increased from zero, the electron transmission from one contact reaches the first magnetic focusing peak when d c = L. As the field continues to increase additional magnetic focusing peaks can occur when L is an integer multiple of d c , if the electron orbit bounces off the edge of the sample specularly. 16,[18][19][20] In this paper, we present images of the cyclotron orbits in graphene associated with the first magnetic focusing peak, recorded using a cooled SPM at 4.2 K with a tip that acts as a movable gate. [13][14][15][16][17] The sample is a high-mobility hBN-graphene-hBN sandwich patterned into a hall bar geometry using reactive ion etching with a mixture of CHF 3 and O (Fig. 1b) The first magnetic focusing peak is clearly shown in in Fig. 2a which presents experimental measurements of R m vs. B and n 1/2 at 4.2K with no tip present. As the density and magnetic field are increased, the transresistance peaks (red) along a track with B 1 ∝ n 1/2 as predicted by theory.At magnetic fields B along either side of the magnetic focusing peak, the transmission between the two contacts is reduced (blue), because cyclotron orbit trajectories are focused away from the receiving contact. Evidence for the second magnetic focusing peak with one bounce off the edge between contacts is seen (black) at magnetic fields B ~ 2B 1 . The intensity of the second peak is reduced by diffuse boundary scattering, which reduces the probability of specular reflection to 5 0.3 to 0.4 in magnetic focusing measurements 20 in graphene, and to almost zero in 1.0 µm wide ballistic graphene w...
We discover and characterise strong quantum scars, or quantum eigenstates resembling classical periodic orbits, in two-dimensional quantum wells perturbed by local impurities. These scars are not explained by ordinary scar theory, which would require the existence of short, moderately unstable periodic orbits in the perturbed system. Instead, they are supported by classical resonances in the unperturbed system and the resulting quantum near-degeneracy. Even in the case of a large number of randomly scattered impurities, the scars prefer distinct orientations that extremise the overlap with the impurities. We demonstrate that these preferred orientations can be used for highly efficient transport of quantum wave packets across the perturbed potential landscape. Assisted by the scars, wave-packet recurrences are significantly stronger than in the unperturbed system. Together with the controllability of the preferred orientations, this property may be very useful for quantum transport applications.
We present a hydrodynamic lattice gas model for two-dimensional flows on curved surfaces with dynamical geometry. This model is an extension to two dimensions of the dynamical geometry lattice gas model previously studied in one dimension. We expand upon a variation of the two-dimensional flat space FrischHasslacher-Pomeau ͑FHP͒ model created by Frisch et al. ͓Phys. Rev. Lett. 56, 1505 ͑1986͔͒ and independently by Wolfram, and modified by Boghosian et al. ͓Philos. Trans. R. Soc. London, Ser. A 360, 333 ͑2002͔͒. We define a hydrodynamic lattice gas model on an arbitrary triangulation whose flat space limit is the FHP model. Rules that change the geometry are constructed using the Pachner moves, which alter the triangulation but not the topology. We present results on the growth of the number of triangles as a function of time. Simulations show that the number of triangles grows with time as t 1/3 , in agreement with a mean-field prediction. We also present preliminary results on the distribution of curvature for a typical triangulation in these simulations.
We present numerical results obtained using a lattice gas model with dynamical geometry. The (irreversible) macroscopic behaviour of the geometry (size) of the lattice is discussed in terms of a simple scaling theory and obtained numerically. The emergence of irreversible behaviour from the reversible microscopic lattice gas rules is discussed in terms of the constraint that the macroscopic evolution be reproducible. The average size of the lattice exhibits power-law growth with exponent at late times. The deviation of the macroscopic behaviour from reproducibility for particular initial conditions ('rogue states') is investigated as a function of system size. The number of such 'rogue states' is observed to decrease with increasing system size. Two mean-field analyses of the macroscopic behaviour are also presented.
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