Analysis of band-gap formation in squashed armchair carbon nanotubes

H. Mehrez, A. Svizhenko, M. P. Anantram, M. Elstner, and T. Frauenheim

Phys. Rev. B 71, 155421 – Published 28 April 2005

Abstract

The electronic properties of deformed armchair carbon nanotubes are modeled using constraint free density functional tight binding molecular dynamics simulations. Independent from CNT diameter, deforming path can be divided into three regimes. In the first regime, the nanotube deforms with negligible force. In the second one, there is significantly more resistance to deforming with the force being $\sim 40 – 100 nN ∕ per$ CNT unit cell. In the last regime, the CNT loses its hexagonal structure resulting in force drop-off followed by substantial force enhancement upon deforming. We compute the change in band gap as a function of deforming and our main results are: (i) A band gap initially opens due to interaction between atoms at the top and bottom sides of CNT. The $π$ -orbital approximation is successful in modeling the band-gap opening at this stage. (ii) In the second regime of deforming, large $π − σ$ interaction at the edges becomes important, which can lead to band-gap oscillation. (iii) Contrary to a common perception, nanotubes with broken mirror symmetry can have zero band gap. (iv) All armchair nanotubes become metallic in the third regime of deforming. Finally, we discuss both differences and similarities obtained from the tight binding and density functional approaches.

Received 6 December 2004

DOI:https://doi.org/10.1103/PhysRevB.71.155421

Authors & Affiliations

H. Mehrez, A. Svizhenko, and M. P. Anantram

Mail Stop: 229-1, Center for NanoTechnology and NASA Advanced Supercomputing Division, NASA Ames Research Center, Moffett Field, California 94035-1000, USA

M. Elstner and T. Frauenheim

Theoretische Physik, Universitat Paderborn, D-33098 Paderborn, Germany

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Vol. 71, Iss. 15 — 15 April 2005

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Images

Figure 1
(Color online) The inset in (b) displays the atomic configuration of a deformed CNT between tip and sample. $d_{Tip − Sub}$ , and $d_{TB}$ are the tip-sample separation distance and the distance between top and bottom sides of CNT, respectively. (a) $d_{TB}$ vs $d_{Tip − Sub}$ ; CNT deforming undergoes three regimes, $d_{TB} > 3.8 Å, 3.8 Å > d_{TB} > 2.3 Å$ , and $d_{TB} < 2.3 Å$ . These regimes are separated by the horizontal dashed lines. In the first regime, the slope of $d_{TB}$ vs $d_{Tip − Sub}$ is $\sim 0.88 \pm 0.07$ while in the second regime the slope is $\sim 0.6 \pm 0.05$ , where the corresponding fits are represented as thick solid lines. At $d_{TB} < 2.3 Å$ , CNT undergoes structural transformation and the honeycomb structure cannot be identified. (b),(c) Force exerted on the tip (top graphene sheet) as a function of $d_{Tip − Sub}$ and $d_{TB}$ , respectively. In all panels, every thin line corresponds to one simulation and results for (6,6) and (12,12) CNT are highlighted as thick dotted lines and dashed lines, respectively.Reuse & Permissions
Figure 2
(Color online) $E_{g}$ of (6,6) CNT at $E_{F}$ as a function of $d_{TB}$ for different simulations. In (a) and (b) only the top graphene layer is displaced, while in (c) and (d) both layers are moved to deform CNT. The insets in each panel show the atomic configuration of the corresponding CNT unit cell for $d_{TB} 2.90 \pm 0.05 Å$ where gray atoms and dark gray atoms indicate top and bottom sides of CNT, respectively.Reuse & Permissions
Figure 3
(Color online) (a) and (b) show the band gap at $E_{F}$ and $2 ∣ V_{π π^{*}} ∣$ of (6,6) CNT for simulation results shown in Figs. 2c, 2d; respectively. Continuous lines and dashed lines correspond to $E_{g}$ using 4 orbitals/atom and a single $π$ -orbital/atom, respectively. Thin lines with triangle down symbols and dashed lines with triangle-up symbols represent $2 ∣ V_{π π^{*}} ∣$ using 4 orbitals/atom and a single $π$ -orbital/atom, respectively.Reuse & Permissions
Figure 4
(Color online) (a),(b) and ( $a^{'}$ ),( $b^{'}$ ) show the band gap at $E_{F}$ and $2 ∣ V_{π π^{*}} ∣$ of (6,6) CNT for simulation results shown in Figs. 2c, 2d due to top-bottom interactions $(H_{TB})$ and edge effects $(H_{ED})$ , respectively. Top and bottom of CNT are shown in the insets of Fig. 2 with gray and dark gray atoms. In these panels, continuous and dashed lines correspond to $E_{g}$ using 4 orbitals/atom and a single $π$ -orbital/atom, respectively. Thin lines with triangle down symbols and dashed lines with triangle-up symbols represent $2 ∣ V_{π π^{*}} ∣$ using 4 orbitals/atom and a single $π$ -orbital/atom, respectively.Reuse & Permissions
Figure 5
(Color online) Band gap of armchair CNTs as a function of deforming. (a), (b), (c), (d), (e), and (f) correspond to (7,7), (8,8), (9,9), (10,10), (11,11), and (12,12) tubes, respectively. Dotted lines with open circles correspond to all interactions, continuous lines are interactions up-to the second nearest neighbor and dashed lines correspond to results when top-bottom interactions are omitted. The insets in each panel show the atomic configuration of the corresponding CNT unit cell for $d_{TB}$ closest to 3.2 Å where top and bottom sides of CNT are highlighted.Reuse & Permissions
Figure 6
(Color online) Energy gap of (6,6) CNT at $E_{F}$ as a function of separation distance between top and bottom sides of CNT $(d_{TB})$ for simulations shown in Figs. 2a, 2b, 2d. Dotted lines with open circles are the original calculations within the DF-TB model. Solid and dashed lines correspond to calculations using SCTB and DFT, respectively.Reuse & Permissions

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