2.6 Electron-Phonon Interaction

where and are the electron creation and annihilation operators, respectively, and are the phonon creation and annihilation operators, respectively, and is the electron-phonon matrix element

where is the mass density of the CNT, is the normalization length, and is the reduced electron-phonon matrix element of the transition from the initial electronic state to the final state , where is the wave-vector and is the azimuthal quantum number of electrons. Because of energy conservation for a scattering event it holds , where is the wave-vector, is the azimuthal quantum number, and is the polarization of the phonon. The matrix element obeys selection rules arising from wave-vector and azimuthal quantum number conservation, and .

Because in the CNT two degrees of freedom are confined, an electron can only be
scattered forward or backward in the axial direction, preserving or changing
the sign of the band-velocity, respectively. The scattering processes invoke
either *intra-subband* or *inter-subband* transitions. The
intra-subband processes are important for the electrical and the heat transport
in CNTs and for the relaxation of an excited electron or hole in the same
subband. The inter-subband processes contribute to the radiation-less
relaxation of electrons (holes) from a given subband to a subband with a lower
(higher) energy [56].
The scattering of electrons can take place
within a given valley or between two valleys. The two possibilities being
termed *intra-valley* and *inter-valley* scattering processes.