Electronic Transistions in Incommensurate Double Walled Carbon Nantoubes and Analysis of Band Structure

Abstract

The study of intertube electronic transitions and band structure of incommensurate double walled carbon nanotubes were made. It was possible in the case of double walled carbon nanotubes with super imposed extended Brillouin zones corresponding to constituent of single walled carbon nanotubes. The selection reules allowed the intertube transitions which were geometric and permittd to determine the transitions and band structure of double walled carbon nanotubes. This was powerful tool with Rayleigh spectroscopy used for structural identification. All existing methods for optical spectra of double walled carbon nanotubes used the number of transitions as an input parameter which was equal to the number of transitions in pristine single walled carbon nanotubes in the measured spectral range. It was found that the intertube transition was doubled due to symmetry breaking with respect to the electron hole permutation. The analysis predicted the splitting and values of dipole matrix elements for possible intertube transitions. We have used the nearest neighbor tight binding approximation for analyzing band structure of double walled carbon nanotubes. It was found that unconventional electronic transitions in double walled carbon nanotubes with much greater structural diversity. The expressions found for energy dispersions contained additional terms which corresponded to cross band interaction and increased the accuracy of calculations. We have calculated more compact and more accurate expressions for the energies of optical transitions. We have also used effective Hamiltonian for analyzing band structure of incommensurate double walled carbon nanotubes. The obtained results were found in good agreement with previously obtained results.