Abstract
The dependence of the frequency and broadening of optical phonons on carrier concentration is studied in carbon nanotubes within an effective-mass approximation. In metallic nanotubes, the frequency shift exhibits a logarithmic divergence and the broadening vanishes discontinuously when the Fermi level reaches the half of the optical-phonon frequency for the longitudinal mode with displacement in the axis direction, while the transverse mode is not affected. In semiconducting nanotubes, the frequency is raised for both transverse and longitudinal modes, exhibiting a behavior similar to level crossing.
Highlights
Optical phonons are strongly influenced by electron–phonon interactions.1–3) In a previous work,4) the interaction effect was shown to cause characteristic diameter dependence of the frequency and broadening of phonons in metallic and semiconducting nanotubes within the continuum model for both electrons and phonons
Large change in the phonon spectra was experimentally observed for varying electron concentration by the application of a gate voltage.5,6) The purpose of this paper is to study the Fermi-level dependence of optical phonons
In the Raman spectra of single-wall nanotubes the socalled G band is usually fit with two components Gþ and G arising from transverse and longitudinal optical phonons at the Brillouin-zone center.27,28) Semiconducting tubes have sharp Gþ and G, while metallic tubes have a broad down-shifted G and a sharp Gþ,29–37) some experiments seem to show that the G peak appears only in nanotube bundles.38) The G band shows a strong diameter dependence, being lower in frequency for smaller diameters
Summary
Optical phonons are strongly influenced by electron–phonon interactions.1–3) In a previous work,4) the interaction effect was shown to cause characteristic diameter dependence of the frequency and broadening of phonons in metallic and semiconducting nanotubes within the continuum model for both electrons and phonons. The characteristic properties, first predicted in a tight-binding model, can be described well in a kp scheme or an effective-mass approximation.7) The longwavelength phonons and their interaction with electrons are described quite well in a continuum model.8) In this paper, we calculate the frequency shift and broadening of phonons in the lowest order approximation as a function of the Fermi level, using the continuous phonon model within the kp scheme.
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