Electronic Structure of Helically Coiled Carbon Nanotubes
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Electronic Structure of Helically Coiled Carbon Nanotubes Gian G. Guzman-Verri¹, Lok C. Lew Yan Voon¹, Morten Willatzen ², Jens Gravesen³ ¹Department of Physics, Wright State University, Dayton, OH 45435, U.S.A. ²Mads Clausen Institute for Product Innovation, University of Southern Denmark, Grundvigs Alle 150, DK-6400 Sønderborg, Denmark. ³Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303, DK-2800 Kgs. Lyngby, Denmark ABSTRACT In the present work we calculate the electronic band structure of single-wall helical carbon nanotubes following an effective-mass approach. We include curvature effects and strain due to bending in the band structure. The curvature energy ∆E , and the change in the electronic energy ∆E S due to strain, depend upon the coil pitch and coil diameter of the tube. We find 0.003 ≤ ∆E ≤ 1.3 eV and considered here.
0 ≤ ∆E S ≤ 4.0 eV for the single-wall helical carbon nanotubes
INTRODUCTION During the early 1990s, Dunlap [1] and Itoh et al. [2] predicted the structure of helical carbon nanotubes (HCNTs) and the experimental observation of them was made by Ivanov et al. [3] and Zhang et al. [4] in 1994. Even though the single-wall helical carbon nanotube (SWHCNT) growth process is still challenging, recent interest has been shown due to its potential applications. For instance, HCNTs could lead to devices such as nano-transformers, nanoswitches and self-sensing mechanical resonators due to their coiling [5, 6]. From a theoretical point of view, despite the availability of atomistic models of nanotubes, there has been an interest in continuum models because of their simplicity when dealing with mesoscopic structures [7]. Helical carbon nanotubes are classified by their coil pitch and coil diameter. Coil diameter varies from 10-100 nm and coil pitch varies from 10-200 nm [8]. Akagi et al. [9] calculated the band structure of SWHCNTs using a tight-binding model. Moreover, Tamura and coworkers developed the k.p theory of the same nanostructures, clarifying the origin of the phason line [10]. However, none of these previous studies include the effect of curvature into their calculations; hence, it is the purpose of this work to estimate how the band structure is affected by curvature. SECTION I – REVIEW OF EFFECTIVE MASS THEORY (EMT) Tamura et al. [16, 17] derived an effective-mass theory for SWHCNT from the graphene nearest-neighbor tight-binding Hamiltonian 3
Eψ A (R A ) = −γ ∑ψ B (R A − τ l ), l =1 3
Eψ B (R B ) = −γ ∑ψ A (R B + τ l ). l =1
(1)
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As shown in figure 1, R A and R B are the positions of carbon atoms A and B in the graphene unit cell and τ l (l = 1,2,3) and γ are the separation and the transfer integral between nearestneighbors (2.7 eV) , respectively.
Figure.1 Graphene sheet showing carbon atoms at sites A and B. Vector τ l is the separation between nearest-neighbors. If the wave function amplitudes at atomic sites A and B are written as a superposition of the envelope functions F (r ) , at the Bri
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