Transmission resonances and supercritical states in a one-dimensional cusp potential

Villalba, Vı́ctor M.; Greiner, Walter

doi:10.1103/physreva.67.052707

Cited by 48 publications

(53 citation statements)

References 12 publications

(16 reference statements)

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“…For simplicity, we assume that v(x) is a square well of width d and depth u although more realistic potentials [15][16][17][18] can also be considered. In the present case the eigenfunctions Ψ are combinations of plane waves and/or exponentials that have to be matched at x = ±d /2, see Appendix A.…”

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confidence: 99%

Tunable Plasmonic Reflection by Bound 1D Electron States in a 2D Dirac Metal

Jiang

Pan

et al. 2016

Phys. Rev. Lett.

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We show that surface plasmons of a two-dimensional Dirac metal such as graphene can be reflected by line-like perturbations hosting one-dimensional electron states. The reflection originates from a strong enhancement of the local optical conductivity caused by optical transitions involving these bound states. We propose that the bound states can be systematically created, controlled, and liquidated by an ultranarrow electrostatic gate. Using infrared nanoimaging, we obtain experimental evidence for the locally enhanced conductivity of graphene induced by a carbon nanotube gate, which supports this theoretical concept.Plasmon scattering and plasmon losses in Dirac materials, such as graphene and topological insulators, are problems of interest to both fundamental and applied research. It is an outstanding challenge to understand various kinds of interaction (electron-electron, electron-phonon, electron-photon, electron-disorder) responsible for these complex phenomena [1][2][3][4][5] . At the same time, control of plasmon scattering is critical if this class of materials is to become a new platform for nanophotonics [6][7][8][9] .One source of plasmon scattering is long-range inhomogeneity of the electron density, which causes local fluctuations in the plasmon wavelength λ p . If the inhomogeneities are weak, those of size comparable to the average λ p are expected to be the dominant scatterers 10,11 Surprisingly, recent experiments have revealed that one-dimensional (1D) defects of nominally atomic width can act as effective reflectors for plasmons with wavelengths as large as a few hundred nm. Strong plasmon reflection was observed near grain boundaries 12,13 , topological stacking faults 14 , as well as nanometerscale wrinkles and cracks 11,12 in graphene. If this anomalous reflection is indeed an ubiquitous effect largely unrelated to the specific nature of a defect, it calls for a universal explanation. In this Letter we attribute its origin to electron bound states commonly occurring near 1D defects. We show that optical transitions involving the bound states can produce strong dissipation at small distances x from the defect and therefore, alter plasmon dynamics. To support this idea we present a theoretical analysis of an exactly solvable model, which illustrates qualitative and quantitative characteristics of the bound states and predicts how their optical response depends on the tunable parameters of a 1D potential well. We also report an attempt to probe the predicted effects experimentally. Our approach is to employ an ultranarrow electric gate in the form of a carbon nanotube (CNT) to create a precisely tunable 1D barrier in graphene. This device enables a systematic investigation and control of plasmon propagation, including, in principle, an implementation of a plasmon on-off switch (Fig. 1). What we find is that the measured real-space profile of the plasmon amplitude (Fig. 4) cannot be accounted for by a local change in λ p alone. Instead, the data are consistent with the presence of an enhanced ...

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confidence: 99%

Tunable Plasmonic Reflection by Bound 1D Electron States in a 2D Dirac Metal

Jiang

Pan

et al. 2016

Phys. Rev. Lett.

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“…In two dimensions, the Dirac equation in the presence of the spatially dependent electric field (1) can be written as [13] …”

Section: Solution Of the Dirac Equation And Energy Resonancesmentioning

confidence: 99%

“…Here the Dirac equation presents halfbound states with the same asymptotic behavior of those obtained with the potential barrier. Supercritical states with E = −m have been also studied in a one-dimensional cusp potential [13,14] and in a class of short-range potentials [15].…”

Section: Introductionmentioning

confidence: 99%

Resonant states in an attractive one-dimensional cusp potential

Villalba¹,

González-Díaz

2007

Phys. Scr.

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Abstract. We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetric attractive cusp potential. The components of the spinor solution are expressed in terms of Whittaker functions. We compute the bound states solutions and show that, as the potential amplitude increases, the lowest energy state sinks into the Dirac sea becoming a resonance. We characterize and compute the lifetime of the resonant state with the help of the phase shift and the Breit-Wigner relation. We discuss the limit when the cusp potential reduces to a delta point interaction.

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“…Analytic solutions have been obtained for the the square well [2,3], Woods-Saxon potential [4], cusp potential [5], and Hulthén potential [6] as well as asymmetric barriers [7,8], multiple barriers [9], and a class of short-range potentials [10]. The successful isolation of graphene [11] has led to renewed interest in the transmission-reflection problem for the one-dimensional Dirac equation.…”

Section: Introductionmentioning

confidence: 99%

“…Transmission resonances and supercriticality [1] (bound states occurring at E = −m, where E is the particles energy and m the particles mass) of relativistic particles in onedimensional potential wells have been studied extensively [1][2][3][4][5][6][7][8][9][10]. Analytic solutions have been obtained for the the square well [2,3], Woods-Saxon potential [4], cusp potential [5], and Hulthén potential [6] as well as asymmetric barriers [7,8], multiple barriers [9], and a class of short-range potentials [10].…”

Section: Introductionmentioning

confidence: 99%

Quasi-exact solution to the Dirac equation for the hyperbolic-secant potential

Hartmann

Portnoi

2014

Phys. Rev. A

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We analyze bound modes of two-dimensional massless Dirac fermions confined within a hyperbolic secant potential, which provides a good fit for potential profiles of existing top-gated graphene structures. We show that bound states of both positive and negative energies exist in the energy spectrum and that there is a threshold value of the characteristic potential strength for which the first mode appears. Analytical solutions are presented in several limited cases and supercriticality is discussed.

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Transmission resonances and supercritical states in a one-dimensional cusp potential

Abstract: We solve the two-component Dirac equation in the presence of a spatially one dimensional symmetric cusp potential. We compute the scattering and bound states solutions and we derive the conditions for transmission resonances as well as for supercriticality.

Cited by 48 publications

References 12 publications

Tunable Plasmonic Reflection by Bound 1D Electron States in a 2D Dirac Metal

Tunable Plasmonic Reflection by Bound 1D Electron States in a 2D Dirac Metal

Resonant states in an attractive one-dimensional cusp potential

Quasi-exact solution to the Dirac equation for the hyperbolic-secant potential

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