Diameter Dependence of the Dielectric Constant for the Excitonic Transition Energy of Single-Wall Carbon Nanotubes

Abstract

The measured optical transition energies E ii of single-wall carbon nanotubes are compared with bright exciton energy calculations. The E ii differences between experiment and theory are minimized by considering a diameter-dependent dielectric constant , which comprises the screening from the tube and from the environment. Different dependencies are obtained for (EA changing environment changes the diameter dependence for (E S 11 , E S 22 , E M 11 ), but for (E S 33 , E S 44 ) the environmental effects are minimal. The resulting calculated exciton energies reproduce experimental E ii values within AE70 meV for a diameter range (0:7 < d t < 3:8 nm) and 1:2 < E ii < 2:7 eV, thus providing a theoretical justification for E ii , environmental effects and important insights on the dielectric screening in one-dimensional structures. DOI: 10.1103/PhysRevLett.103.146802 PACS numbers: 73.22.Àf, 61.48.De, 78.20.-e, 78.67.Ch The last decade has been marked by an impressive development in understanding the nature of the optical transition energies in quasi-one-dimensional single-wall carbon nanotubes (SWNTs) [1], called E ii , where i ¼ 1; 2; 3; . . . denotes the intersubband transitions between the ith valence and the ith conduction band for a given SWNT. While the interest in excitons and dielectric screening in one-dimensional structures dates from research in -conjugated polymers, in carbon nanotubes the large attention started in 2003 with the so-called ''ratio problem'' [2]. Strong debate still exists about the strength of the exciton binding energy, mostly related to the complex dielectric screening in one-dimensional materials. In 2007, Araujo et al. [3] and Michel et al. [4] showed that the scaling law for the exciton energies explaining the ratio problem [2] breaks down for transitions higher than E M 11 . These results lead to the discussion of the exciton nature for higher energy levels, where quantum-chemistry calculations and solid-state physics (tight binding and firstprinciples) calculations give contradictory pictures [3,5]. Now, the accumulated knowledge in SWNTs, both theoretical and experimental [1], makes it possible to evaluate in detail the dielectric screening in one-dimensional systems.The E ii values are now understood in terms of the bright exciton energy in a framework of a tight binding calculation which includes curvature optimization [1] and manybody effects [1,5,6]. The assignment of E ii for SWNTs over a large region of both diameter (0:7 < d t < 3:8 nm) and E ii (1.2-2.7 eV) values and for a variety of surrounding materials are now available [1], thus making it possible to accurately determine the effect of the general dielectric constant on E ii . By ''general'' we mean comprises the screening from the tube and from the environment. In this work we show a d t -dependent effective values for the exciton calculation that are needed to reproduce the experimental E ii values consistently. The results thus obtained are important for the physics of quasi and truly one-dimensional materials ...

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“…In ref 162 a single parameter (Ce) was used to account for the diameter-dependent environmental effects. 169 For all excitation energies and samples analyzed, we found a maximum discrepancy of 0.03 nm on the tube diameter.…”

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