Corrections to the Optical Transition Energies in Single-Wall Carbon Nanotubes of Smaller Diameters
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Corrections to the Optical Transition Energies in Single-Wall Carbon Nanotubes of Smaller Diameters Georgii G. Samsonidze1, Riichiro Saito5, Jie Jiang5, Alexander Grüneis5,6, Naoki Kobayashi5, Ado Jorio7, Shin G. Chou2, Gene Dresselhaus3, and Mildred S. Dresselhaus1,4 1
Department of Electrical Engineering and Computer Science, 2Department of Chemistry, Francis Bitter Magnet Laboratory, and 4Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, U.S.A. 5 Department of Physics, Tohoku University and CREST JST, Aoba, Sendai 980-8578, Japan 6 Leibniz Institute for Solid State and Material Research Dresden, D-01171 Dresden, Germany 7 Depto. de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG 30123-970, Brazil 3
ABSTRACT Optical spectroscopy characterization of carbon nanotube samples requires accurate determination of their band structure and exciton binding energies. In this paper, we present a non-orthogonal density-functional based tight-binding calculation for the electronic transition energies in single-wall carbon nanotubes. We show that the curvature-induced rehybridization of the electronic orbitals, long-range atomic interactions, and geometrical structure relaxation all have a significant impact on the electronic transition energies in the small diameter limit. After including quasiparticle corrections and exciton binding energies, the calculated electronic transition energies show good agreement with the experimental transition energies observed by photoluminescence and resonance Raman spectroscopy. INTRODUCTION AND BACKGROUND Optical spectroscopy techniques, such as resonance Raman spectroscopy (RRS), optical absorption, and photoluminescence (PL), are commonly used for the characterization of singlewall carbon nanotubes (SWNTs). The optical properties of SWNTs are primarily determined by the electronic transitions between Van Hove singularities (VHSs) in the density of states (DOS) arising from the one-dimensional structure of the SWNTs. These transition energies (Eii) for SWNTs of different structural (n,m) indices are commonly summarized in the so-called Kataura plot [1]. The Kataura plot depicts the Eii transition energies as a function of SWNT diameters (dt) or inverse SWNT diameters (1/dt). For each (n,m) SWNT, the diameter is given by dt =
a
π
n 2 + nm + m 2 ,
(1)
where a = 3aCC is the graphene lattice constant, and aCC = 0.142 nm is the interatomic distance S S [2]. The Eii energies in the Kataura plot are arranged in bands ( E11S , E22 , E11M , E33S , E44 , E22M , etc.) for semiconducting (S) and metallic (M) SWNTs, respectively, where the index i enumerates the VHSs in the valence and conduction bands, such that i increases with increasing distance from the Fermi level. Within each Eii band in the Kataura plot, the Eii energies follow “family”
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patterns for SWNTs with 2n + m = 3p + r, where p is an integer and r = 0,1,2 represents metallic, semiconducting type I (S I) and type II (S II) SWTNs, respectively. Recently, PL stud
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