Atomic collapse, Lorentz boosts, Klein scattering, and other quantum-relativistic phenomena in graphene

Shytov, A. V.; Rudner, Mark S.; Gu, Nan; Katsnelson, M. I.; Levitov, L. S.

doi:10.1016/j.ssc.2009.02.043

Cited by 112 publications

(83 citation statements)

References 44 publications

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“…However, in single-layer graphene (SLG) the realization of FP interferometers is challenging. The absence of a band gap and the Klein tunneling hamper the efficiency of sharp potential steps between the n-and p-type regions, which play the role of the interferometer mirrors [7][8][9]. Theory suggests that smooth barriers enhance the visibility of interference [10,11] due to Klein collimation [12].…”

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Fabry-Pérot Interference in Gapped Bilayer Graphene with Broken Anti-Klein Tunneling

et al. 2014

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We report the experimental observation of Fabry-Pérot interference in the conductance of a gate-defined cavity in a dual-gated bilayer graphene device. The high quality of the bilayer graphene flake, combined with the device's electrical robustness provided by the encapsulation between two hexagonal boron nitride layers, allows us to observe ballistic phase-coherent transport through a 1-μm-long cavity. We confirm the origin of the observed interference pattern by comparing to tight-binding calculations accounting for the gate-tunable band gap. The good agreement between experiment and theory, free of tuning parameters, further verifies that a gap opens in our device. The gap is shown to destroy the perfect reflection for electrons traversing the barrier with normal incidence (anti-Klein tunneling). The broken anti-Klein tunneling implies that the Berry phase, which is found to vary with the gate voltages, is always involved in the Fabry-Pérot oscillations regardless of the magnetic field, in sharp contrast with single-layer graphene. DOI: 10.1103/PhysRevLett.113.116601 PACS numbers: 72.80.Vp, 73.23.-b Interference of particles is a manifestation of the wave nature of matter. A well-known realization is the double-slit experiment, which cannot be described by the laws of Newtonian mechanics, but requires a full quantum description. This experiment has been performed with photons [1,2], electrons [3], and even molecules [4]. Another setting widely used in optics is the Fabry-Pérot (FP) interferometer, where a photon bounces back and forth between two coplanar semitransparent mirrors. Partial waves transmitted after a distinct number of reflections within this cavity interfere and give rise to an oscillatory intensity of the transmitted beam as the mirror separation or the particle energy is varied.In solid-state physics, graphene has proven to be a suitable material for probing electron interference at cryogenic temperatures [5,6]. However, in single-layer graphene (SLG) the realization of FP interferometers is challenging. The absence of a band gap and the Klein tunneling hamper the efficiency of sharp potential steps between the n-and p-type regions, which play the role of the interferometer mirrors [7][8][9]. Theory suggests that smooth barriers enhance the visibility of interference [10,11] due to Klein collimation [12]. Recently, ultraclean suspended SLG devices have shown FP interference with stunning contrast using cavity sizes of more than 1 μm [13][14][15].In bilayer graphene (BLG) potential steps between n-and p-type regions lead to evanescent interface states resulting in a zero-transmission at normal incidence, known as anti-Klein tunneling [7]. Furthermore, in BLG a band gap can be induced by a transverse electric field [16][17][18][19]. The Berry phase of π in SLG has been predicted [20] and observed [10] to cause a phase jump of π in the FP fringes at weak magnetic field (B). In gapless BLG the Berry phase is known to be 2π, but it is yet to be understood how the Berry phase in gapped BLG influ...

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Fabry-Pérot Interference in Gapped Bilayer Graphene with Broken Anti-Klein Tunneling

et al. 2014

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“…The transition at B = B c can be linked to the classical dynamics of a massless particle, characterized by closed orbits at B > B c and open trajectories at B < B c [16]. A simple but intuitive picture of the spectrum (2) can be obtained from the Bohr-Sommerfeld quantization condition…”

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Collapse of Landau Levels in Gated Graphene Structures

Rudner

Young

et al. 2011

Phys. Rev. Lett.

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We describe a new regime of magnetotransport in two dimensional electron systems in the presence of a narrow potential barrier imposed by external gates. In such systems, the Landau level states, confined to the barrier region in strong magnetic fields, undergo a deconfinement transition as the field is lowered. We present transport measurments showing Shubnikov-de Haas (SdH) oscillations which, in the unipolar regime, abruptly disappear when the strength of the magnetic field is reduced below a certain critical value. This behavior is explained by a semiclassical analysis of the transformation of closed cyclotron orbits into open, deconfined trajectories. Comparison to SdH-type resonances in the local density of states is presented.Electron cyclotron motion constrained by crystal boundaries displays many interesting phenomena, such as skipping orbits and electron focusing, which have yielded a wealth of information on scattering mechanisms in solids [1,2]. Since the 1980s, many groups have investigated electron transport in semiconducting two-dimensional electron systems (2DES), where gateinduced spatially varying electric fields can be used to alter cyclotron motion. A variety of interesting phenomena were explored in these systems, including quenching of the quantum Hall effect [3,4], Weiss oscillations due to commensurability between cyclotron orbits and a periodic grating [5], pinball-like dynamics in 2D arrays of scatterers [6], and coherent electron focusing [7].The experimental realization of graphene [8], a new high-mobility electron system, affords new opportunities to explore effects that were previously inaccessible. In particular, attempts to induce sharp potential barriers in III-V semiconductor quantum well structures have been limited by the depth at which the 2DES is buriedtypically about 100nm below the surface [9]. In contrast, electronic states in graphene, a truly two-dimensional material, are fully exposed and thus allow for potential modulation on ∼ 10 nm length scales using small local gates and thin dielectric layers [10][11][12][13]. Significantly, such length scales can be comparable to the magnetic length ℓ B = c/eB, which characterizes electronic states in quantizing magnetic fields. In this Letter we focus on one such phenomenon, the transformation of the discrete Landau level spectrum to a continuum of extended states in the presence of a static electric field.The behavior which will be of interest for us is illustrated by a toy model involving the Landau levels of a massive charged particle in the presence of an inverted parabolic potential U (x) = −ax 2 . Competition between the repulsive potential and magnetic confinement gives rise to a modified harmonic oscillator spectrum ε n (p y ) = e m B 2 − B 2 c (n + 1/2) − 2ap 2 y e 2 (B 2 − B 2 c ) (1) for B > B c , where m is the particle mass, p y is the y component of momentum, and B c = √ 2ma/e is the critical magnetic field strength. For strong magnetic field, B > B c , the spectrum consists of discrete (but dispersive) energy ban...

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“…(28) demonstrate that the supersymmetric τ 3 -QED theory given in Eq. (27) supplemented by a Φ 4 -term given in Eq. (33) provides a theoretical framework that extends Jackiw-Pi's original chiral gauge theory [34], represented here by Eq.…”

Section: Discussionmentioning

confidence: 99%

“…If we had assumed the parameter m displayed in Eq. (27) to be odd as well, the parity transformation here proposed would not leave the action invariant, as it would also change by a sign. Instead, we take the viewpoint of considering the mass parameter as a fixed "bare" mass, here included for the sake of generality, left unchanged by parity as the physical mass would get contributions from non-trivial minima of the power-counting renormalisable A 6 -potential (mind Eq.…”

Section: The φ -Sectormentioning

confidence: 97%

“…[34]) as a sector of a wider and richer planar gauge field theory: the so-called Supersymmetric τ 3 -QED [45]. With our work, supersymmetric quantum field theory becomes one more item to be added to the list [26,27] of topics originated from high-energy physics which have the potentiality to be realized in table-top experiments with graphene.…”

Section: Introductionmentioning

confidence: 98%

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A supersymmetric model for graphene

Abreu

Andrade

Assis

et al. 2011

J. High Energ. Phys.

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In this work, we focus on the fermionic structure of the low-energy excitations of graphene (a monolayer of carbon atoms) to propose a new supersymmetric field-theoretic model for this physical system. In the current literature, other proposals for describing graphene physics have been contemplated at the level of supersymmetric quantum mechanics. Also, by observing the inhomogeneities between neighbor carbon atoms, Jackiw et al. have set up an interesting chiral Abelian gauge theory. We show in this paper that our formulation encompasses models discussed previously as sectors of an actually richer (supersymmetric) planar gauge model. Possible interpretations for the fields involved in the present graphene model are proposed and the question of supersymmetry breaking is discussed.

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Atomic collapse, Lorentz boosts, Klein scattering, and other quantum-relativistic phenomena in graphene

Cited by 112 publications

References 44 publications

Fabry-Pérot Interference in Gapped Bilayer Graphene with Broken Anti-Klein Tunneling

Fabry-Pérot Interference in Gapped Bilayer Graphene with Broken Anti-Klein Tunneling

Collapse of Landau Levels in Gated Graphene Structures

A supersymmetric model for graphene

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