  We use the energy analysis as a perturbative method to study the effect of
electron-radial breathing mode (RBM) phonon interaction on the electrical
conductivity of long metallic zigzag carbon nanotubes (CNTs). The band
structure of zigzag CNTs is calculated by exerting zone-folding method on
relations derived by using the nearest neighbor approximation of tight-binding
expression for the $\pi$ valence and conduction bands of graphene. The small
hollow cylinder model, with two different approximations, is used to obtain the
RBM frequency in our calculation. As the result, we have calculated the effects
of electron$ - $RBM phonon interaction on the conductance of zigzag CNTs. It
has been observed that current is a step$ - $like function of bias voltage
because of the absorption or emission by electron injection in the system.
Moreover, the dependence of the conductance to the temperature in low bias and
high bias voltages has been studied. In this paper, we propose a simple and
useful method for phonon spectroscopy. Also, since RBM mode determines the
geometry and structure of CNT, this approach can be used for characterization
of CNTs.
