Electronic structure of carbon nanotubes on graphene substrates

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Electronic structure of carbon nanotubes on graphene substrates

Benedetta Flebus and Allan H. MacDonald

Phys. Rev. Research 2, 022041(R) – Published 15 May 2020

Abstract

Allotropes of carbon, including one-dimensional carbon nanotubes and two-dimensional graphene sheets, continue to draw attention as promising platforms for probing the physics of electrons in lower dimensions. Recent research has shown that the electronic properties of graphene multilayers are exquisitely sensitive to the relative orientation between sheets and in the bilayer case exhibit strong electronic correlations when close to a magic twist angle. Here we investigate the electronic properties of a carbon nanotube deposited on a graphene sheet by deriving a low-energy theory that accounts for both rotations and rigid displacements of the nanotube with respect to the underlying graphene layer. We show that this heterostructure is described by a translationally invariant, a periodic, or a quasiperiodic Hamiltonian, depending on the orientation and the chirality of the nanotube. Furthermore, we find that, even for a vanishing twist angle, rigid displacements of a nanotube with respect to a graphene substrate can alter its electronic structure qualitatively. Our results identify a promising direction for strong correlation physics in low dimensions.

Received 6 September 2019
Revised 25 March 2020
Accepted 27 March 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.022041

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Electronic structure

Graphene Nanotubes

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Benedetta Flebus¹ and Allan H. MacDonald ²

¹Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles, California 90095, USA
²Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

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Issue

Vol. 2, Iss. 2 — May - July 2020

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Images

Figure 1
Single-wall carbon nanotube on a graphene substrate. The nanostructure geometry can be characterized by starting from Bernal $A B$ stacking between a graphene sheet and an unrolled nanotube with general chirality and then displacing the nanotube or changing the orientation of its axis. In Bernal stacking, atoms located at the $B$ sites of the carbon nanotubes are above $A$ sites of the graphene layer. (a) Structure of an unrolled carbon nanotube, which is specified by a primitive translation vector $\vec{T}$ along the nanotube axis and a perpendicular chiral vector $\vec{L}$ . Both vectors can be written as linear combinations of the primitive lattice vectors $\vec{a}$ and $\vec{b}$ . (b) ${\vec{G}}_{2}$ and ${\vec{G}}_{3}$ are reciprocal lattice vectors that connect equivalent graphene Brillouin-zone corners. The momentum states near $K$ and its equivalent corners together host the Dirac point of valley $K$ . Momenta near the other three Brillouin-zone corners host valley $K^{'}$ , the time-reversed counterpart of valley $K$ . (c) $A B$ stacking between graphene and the unrolled nanotube applies when the orientation of the nanotube axis matches its chirality angle $η$ .
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Figure 2
Electronic structure of a carbon nanotube on a graphene sheet. Spectral function $A (p_{y}, ω)$ of the carbon nanotube for rigid displacements (a) along the direction transverse and (b) parallel to the nanotube axis. (c) Dependence of the (absolute value of) conduction ( $| 〈 c | T | c 〉 |$ ) and valence ( $| 〈 v | T | v 〉 |$ ) band hybridization matrix elements (7) at the graphene momentum $k_{x} = 0$ on displacement transverse to the nanotube axis. For the dimensionless displacements $x_{⊥} \neq 0, 1 / 2$ , the two matrix elements have different values, explaining the asymmetry of the spectral function (a). (d) For displacements along the nanotube axis, the valence and conduction bands experience the same interaction strength, explaining the symmetry of the spectral function (b). The spectral densities, frequencies, momenta, and tunneling strength are plotted in dimensionless units, i.e., $A \to (v / a) A, p_{y} \to a p_{y}, ω \to ω (a / v)$ , and $t \to t (a / v)$ .
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Figure 3
Four-band model and valley-dependent spectral functions. (a) Energy spectrum of the $4 \times 4$ Hamiltonian (8) for graphene momentum $k_{x} = 0$ and different values of dimensionless displacement $x_{⊥}$ . Far from the Dirac point, the spectrum displays the same features observed in the spectral function [see Fig. 2]. However, in contrast with the spectral function, the four-band spectrum exhibits an energy gap. (b) Energy spectrum of the $4 \times 4$ Hamiltonian (8) for a finite $k_{x}$ and $x_{⊥} = 1 / 4$ . For $k_{x} > 0$ ( $< 0$ ), the gap closes at the left (right) of the Dirac point. The total contribution of electronic states with finite $k_{x}$ gives rise to a gap closing at the Dirac point, which is displayed by the spectral density in Fig. 2. (c) Nanotube spectral density $A (p_{y}, ω)$ at the valleys $K$ and $K^{'}$ , which are related by time-reversal symmetry. A valley-projected time-reversal symmetric spectrum is degenerate for $K$ and $K^{'}$ (top panels). When valley-projected time-reversal symmetry is broken (bottom panels), a right-going state at momentum $p_{y}$ in one valley still has a left-going degenerate Kramers partner at momentum $- p_{y}$ in the other valley. The spectral densities, frequencies, momenta, and tunneling strength are plotted in dimensionless units, i.e., $A \to (v / a) A, p_{y} \to a p_{y}, ω \to ω (a / v)$ , and $t \to t (a / v)$ .
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