Abstract
Effects of interactions with electrons on optical phonons are studied in an effective-mass approximation. The longitudinal mode with displacement in the axis direction is lowered in its frequency, while the transverse mode with displacement in the circumference direction is raised, in metallic nanotubes. The shifts are opposite but their absolute values are smaller in semiconducting nanotubes. Only the longitudinal mode has a considerable broadening in metallic nanotubes. In the presence of an Aharonov–Bohm magnetic flux, the broadening appears for the transverse mode and diverges when the induced gap becomes the same as the frequency of the optical phonon.
Highlights
Carbon nanotubes are quasi-one-dimensional materials made of sp2 -hybridized carbon networks.1) The electronic states change from metallic to semiconducting depending on the tubular circumferential vector characterizing a nanotube.The characteristic properties were predicted by calculations in tight-binding models2–11) and in a kp scheme or an effective-mass approximation.12–14) Electron–phonon interactions can play an important role in the behavior of optical phonons in carbon nanotubes
An effective Hamiltonian has been derived for describing the interaction between long-wavelength optical phonons and electrons in carbon nanotubes
It has been used for the evaluation of the frequency shift
Summary
Carbon nanotubes are quasi-one-dimensional materials made of sp2 -hybridized carbon networks.1) The electronic states change from metallic to semiconducting depending on the tubular circumferential vector characterizing a nanotube. The characteristic properties were predicted by calculations in tight-binding models2–11) and in a kp scheme or an effective-mass approximation.12–14) Electron–phonon interactions can play an important role in the behavior of optical phonons in carbon nanotubes. Recent first-principles calculations reported lowering of phonon energy compared to that in the two-dimensional graphite.15–17) The purpose of this paper is to study effects of electron–phonon interactions on long-wavelength optical phonons based on the kp scheme. Electronic properties have been understood by those of the graphite plane using a periodic boundary condition, the phonon modes of nanotubes are not given by the zone-folded modes of planes because they fail to give breathing modes.7) In a previous work, a continuum model suitable for a correct description of long-wavelength acoustic phonons was constructed.18) In this paper we shall introduce a similar continuum model of optical phonons and derive the Hamiltonian for electron–phonon interactions.
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