Energy spectrum and optical absorption spectra of carbon nanotubes with chiralities of (10, 10), (11, 9), and (12, 8)
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ONIC PROPERTIES OF SOLID
Energy Spectrum and Optical Absorption Spectra of Carbon Nanotubes with Chiralities of (10, 10), (11, 9), and (12, 8) A. I. Murzashev* and E. O. Shadrin MariEl State University, YoshkarOla, 424000 Russia *email: [email protected] Received August 12, 2013
Abstract—The energy spectra of singlewall carbon nanotubes (CNTs) with chiralities of (10, 10), (11, 9), and (12, 8) are calculated in the static fluctuation approximation for the Hubbard model. The choice of these systems for investigation was dictated by the fact that these systems are the most typical of heterogeneous samples obtained by synthesis. It is shown that in the chosen model and approximation, the type of conduc tion is independent of chirality, which contradicts the generally accepted opinion concerning the critical dependence of the CNT conduction current on the chirality indices. The optical absorption spectra of the CNTs under investigation are calculated using the resultant energy spectra. The shape of the optical absorp tion spectrum averaged over the known weight composition is in good agreement with experimental data. The results of investigation suggest that the rule according to which CNTs exhibit metaltype conductivity when the difference in the chirality indices is a multiple of three; otherwise, they have the semiconductortype con ductivity, is in all probability not general, but has applicability limits that can be established by rigorous cal culations based on the Hubbard model. DOI: 10.1134/S1063776114050148
1. INTRODUCTION In carbon nanotubes (CNTs), carbon is in the sp2 hybridized state. Three hybrid electron orbitals form the σ bands lying deep (~ 20 eV) under the Fermi level [1]. Electrons in these states form the backbone of the system; the fourth nonhybrid orbital forms socalled wandering π bonds, constituting the πelectron sub system. These electrons are partly localized and deter mine the conducting and optical properties of CNTs in view of their proximity to the Fermi level. The CNT energy spectrum was calculated for the first time by the Dresselhaus group [2–4]. It was shown in these publi cations, based on the Wallace computations [5] per formed in 1947 for the carbon plane, that a gap Δ ~ 1 eV exists in the energy spectrum of the πelectron subsystem of the CNT with the (n, m) chirality in the case when difference n – m is not a multiple of three, otherwise it is absent. This statement can be called the multipleofthree rule. Due to the elegance and sim plicity of the results, all subsequent investigations of the electroconducting and optical properties of CNTs were carried out keeping this rule in view. In [2–5], calculations were performed in the oneelectron approximation taking into account only hopping of π electrons from site to site. All effects associated with any correlation of electrons were disregarded. It is well known that the Coulomb interactions in systems with electrons partially localized at sites can be strong. This may lead to a number of interesting effects (i
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