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.
