Modeling optical transition energies in semiconducting single wall carbon nanotubes and assignment of their chirality

Abstract

An empirical model of nearest neighbor hopping parameter (γ0) in tight binding (TB) model of single wall carbon nanotubes is proposed in order to calculate first and second optical transition energies of semiconducting SWCNTs. A highly systematic and nearly linear pattern is observed when the γ0, as calculated from experimental optical transition energies of semiconducting SWCNTs, were scaled by a chirality combination term (2n-m) and plotted against tube diameters. Based on this observation, two empirical expressions of γ0 are formulated for mod 1 and mod 2 type semiconducting SWCNTs. In this model of γ0, observations from various optical spectroscopic experiments are incorporated. First and second optical transition energies (E11 and E22) for all semiconducting SWCNTs within minimum and maximum diameter range of 0.4 to 3 nm are calculated using this empirical γ0. Calculated values showed excellent agreement with experimental values for all type of chiralities over the full diameter range and precisely reflected the chirality effect on transition energies. The so called ‘ratio problem’ of first two optical transition energies are also adreseed and formulated through another empirical equation to give proper E22 to E11 ratio for any chirality. The proposed empirical γ0 highly improved the calculation from simplest tight binding model and enables it to give almost accurate qualitative and quantitative prediction of transition energies of semiconducting SWCNTs. A new technique for chirality assignment of semiconducting SWCNT is also proposed which is independent from any prior graphical plot or tabulated data. The technique is based on solving a set of empirical equations for unknown chirality (n, m), using values of radial breathing mode (RBM) frequency and first or second optical transition energies (E11 or E22) of semiconducting SWCNTs from available resonant Raman scattering data. The proposed technique can determine chiral index (n, m) of unknown semiconducting tube’s unambiguously in most of the cases. Chiral index (n, m) of a number of semiconducting tubes are successfully determined using this technique. The way of detecting and correcting any ambiguous assignments are also presented, which gives completeness to this technique. The technique is especially useful for determining chirality of isolated semiconducting Single wall carbon nanotubes.