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Title: Analytical study of bound states in graphene nanoribbons and carbon nanotubes: The variable phase method and the relativistic Levinson theorem

Abstract

The problem of localized states in 1D systems with a relativistic spectrum, namely, graphene stripes and carbon nanotubes, is studied analytically. The bound state as a superposition of two chiral states is completely described by their relative phase, which is the foundation of the variable phase method (VPM) developed herein. Based on our VPM, we formulate and prove the relativistic Levinson theorem. The problem of bound states can be reduced to the analysis of closed trajectories of some vector field. Remarkably, the Levinson theorem appears as the Poincaré index theorem for these closed trajectories. The VPM equation is also reduced to the nonrelativistic and semiclassical limits. The limit of a small momentum p{sub y} of transverse quantization is applicable to an arbitrary integrable potential. In this case, a single confined mode is predicted.

Authors:
 [1]
  1. University of New South Wales, School of Physics (Australia)
Publication Date:
OSTI Identifier:
22617242
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 122; Journal Issue: 6; Other Information: Copyright (c) 2016 Pleiades Publishing, Inc.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 77 NANOSCIENCE AND NANOTECHNOLOGY; BOUND STATE; CARBON NANOTUBES; CHIRALITY; EQUATIONS; GRAPHENE; LEVINSON THEOREM; NANOSTRUCTURES; QUANTIZATION; RELATIVISTIC RANGE; SEMICLASSICAL APPROXIMATION; SPECTRA; VECTOR FIELDS

Citation Formats

Miserev, D. S., E-mail: d.miserev@student.unsw.edu.au, E-mail: erazorheader@gmail.com. Analytical study of bound states in graphene nanoribbons and carbon nanotubes: The variable phase method and the relativistic Levinson theorem. United States: N. p., 2016. Web. doi:10.1134/S1063776116060066.
Miserev, D. S., E-mail: d.miserev@student.unsw.edu.au, E-mail: erazorheader@gmail.com. Analytical study of bound states in graphene nanoribbons and carbon nanotubes: The variable phase method and the relativistic Levinson theorem. United States. doi:10.1134/S1063776116060066.
Miserev, D. S., E-mail: d.miserev@student.unsw.edu.au, E-mail: erazorheader@gmail.com. 2016. "Analytical study of bound states in graphene nanoribbons and carbon nanotubes: The variable phase method and the relativistic Levinson theorem". United States. doi:10.1134/S1063776116060066.
@article{osti_22617242,
title = {Analytical study of bound states in graphene nanoribbons and carbon nanotubes: The variable phase method and the relativistic Levinson theorem},
author = {Miserev, D. S., E-mail: d.miserev@student.unsw.edu.au, E-mail: erazorheader@gmail.com},
abstractNote = {The problem of localized states in 1D systems with a relativistic spectrum, namely, graphene stripes and carbon nanotubes, is studied analytically. The bound state as a superposition of two chiral states is completely described by their relative phase, which is the foundation of the variable phase method (VPM) developed herein. Based on our VPM, we formulate and prove the relativistic Levinson theorem. The problem of bound states can be reduced to the analysis of closed trajectories of some vector field. Remarkably, the Levinson theorem appears as the Poincaré index theorem for these closed trajectories. The VPM equation is also reduced to the nonrelativistic and semiclassical limits. The limit of a small momentum p{sub y} of transverse quantization is applicable to an arbitrary integrable potential. In this case, a single confined mode is predicted.},
doi = {10.1134/S1063776116060066},
journal = {Journal of Experimental and Theoretical Physics},
number = 6,
volume = 122,
place = {United States},
year = 2016,
month = 6
}
  • Graphene nanoribbons are of great interest for pure and applied sciences due to their unique properties which depend on the nanoribbon edges, as, for example, energy gap and antiferromagnetic coupling. Nevertheless, the synthesis of nanoribbons with well-defined edges remains a challenge. To collaborate with this subject, here we propose a new route for the production of graphene nanoribbons from flat carbon nanotubes filled with a one-dimensional chain of Fe atoms by first principles calculations based on density functional theory. Our results show that Fe-filled flat carbon nanotubes are energetically more stable than non flattened geometries. Also we find that bymore » hydrogenation or oxygenation of the most curved region of the Fe-filled flat armchair carbon nanotube, it occurred a spontaneous production of zigzag graphene nanoribbons which have metallic or semiconducting behavior depending on the edge and size of the graphene nanoribbon. Such findings can be used to create a new method of synthesis of regular-edge carbon nanoribbons.« less
  • Highlights: • MWCNTs synthesized and electrochemically oxidized to study the formation of GNR • HRTEM, Raman and XPS confirmed no successful unzipping occurred after oxidation • Electrochemical oxidation very unlikely facilitate formation of intercalated MWCNTs - Abstract: Multiwalled carbon nanotubes (MWCNTs) with different geometrical characteristics and chemical doping have been synthesized and electrochemically oxidized to study the possibility of unzipping, and creating graphene nanoribbon (GNR) nanostructures. Modified glassy carbon electrodes of the MWCNTs have been tested in an aqueous electrolyte via anodic scans in a wide range of potentials, followed by keeping at the maximum potential for different times. Themore » microstructural features, structural defects, and functional groups and their elements have been then studied using high resolution transmission electron microscopy (HRTEM), Raman spectroscopy and X-ray photoelectron spectroscopy (XPS), respectively. All results have confirmed that no successful unzipping occurs in the MWCNTs after electrochemical oxidation, even for the nitrogen-doped MWCNTs (CN{sub x}-MWCNTs) with reactive nitrogen groups and defective bamboo structures. In contrast to the report by Shinde et al. (J. Am. Chem. Soc. 2011, 133, 4168–4171), it has been concluded that the electrochemical oxidation in aqueous electrolytes is very unlikely to facilitate sufficient incorporation of the intercalated molecules among the walls of the MWCNTs. These molecules are, however, responsible for unzipping of MWCNTs.« less
  • Levinson{close_quote}s theorem relates the zero-energy phase shift {delta} for potential scattering in a given partial wave {ital l}, by a spherically symmetric potential that falls off sufficiently rapidly, to the number of bound states of that {ital l} supported by the potential. An extension of this theorem is presented that applies to single-channel scattering by a compound system initially in its ground state. As suggested by Swan [Proc. R. Soc. London Ser. A {bold 228}, 10 (1955)], the extended theorem differs from that derived for potential scattering; even in the absence of composite bound states {delta} may differ from zeromore » as a consequence of the Pauli principle. The derivation given here is based on the introduction of a continuous auxiliary {open_quote}{open_quote}length phase{close_quote}{close_quote} {eta}, defined modulo {pi} for {ital l}=0 by expressing the scattering length as {ital A}={ital a}cot{eta}, where {ital a} is a characteristic length of the target. Application of the minimum principle for the scattering length determines the branch of the cotangent curve on which {eta} lies and, by relating {eta} to {delta}, an absolute determination of {delta} is made. The theorem is applicable, in principle, to single-channel scattering in any partial wave for {ital e}{sup {plus_minus}}-atom and nucleon-nucleus systems. In addition to a knowledge of the number of composite bound states, information (which can be rather incomplete) concerning the structure of the target ground-state wave function is required for an explicit, absolute, determination of the phase shift {delta}. As for Levinson{close_quote}s original theorem for potential scattering, {ital no} {ital additional} {ital information} {ital concerning} {ital the} {ital scattering} {ital wave} {ital function} {ital or} {ital scattering} {ital dynamics} {ital is} {ital required}. {copyright} {ital 1996 The American Physical Society.}« less