2.6 Electron-Phonon Interaction

The electron-phonon interaction HAMILTONian for CNTs can be written as [56]

$$\hat{H}_\mathrm{el-ph} \ = \ \sum_{k^\prime,\nu^\prime,k,\nu,\lam... ...me} c_{k^\prime,\nu^\prime} (b_{q,\mu,\lambda}+b^\dagger_{-q,-\mu,\lambda}) \ ,$$ (2.14)

where $c^\dagger_{k^\prime,\nu^\prime}$ and $c_{k,\nu}$ are the electron creation and annihilation operators, respectively, $b^\dagger_{-q,-\mu,\lambda}$ and $b_{q,\mu,\lambda}$ are the phonon creation and annihilation operators, respectively, and ${M}_{k^\prime,\nu^\prime,k,\nu,\lambda}$ is the electron-phonon matrix element

$${M}_{k^\prime,\nu^\prime,k,\nu,\lambda} \ = \sqrt{\frac{\hbar}{2\... ...L\omega_{q,\mu,\lambda}}} \widetilde{M}_{k^\prime,\nu^\prime,k,\nu,\lambda} \ ,$$ (2.15)

where $\rho_\mathrm{CNT}$ is the mass density of the CNT, $L$ is the normalization length, and $\widetilde{M}_{k^\prime,\nu^\prime,k,\nu,\lambda}$ is the reduced electron-phonon matrix element of the transition from the initial electronic state $k,\nu$ to the final state $k^\prime,\nu^\prime$, where $k$ is the wave-vector and $\nu$ is the azimuthal quantum number of electrons. Because of energy conservation for a scattering event it holds $E_{k^\prime,\nu^\prime}-E_{k,\nu}=\pm\hbar\omega_{\pm q,\pm \mu,\lambda}$, where $q$ is the wave-vector, $\mu$ is the azimuthal quantum number, and $\lambda$ is the polarization of the phonon. The matrix element obeys selection rules arising from wave-vector and azimuthal quantum number conservation, $q=k^\prime-k$ and $\mu=\nu-\nu^\prime$.

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.

Subsections

• 2.6.1 Electron-Phonon Matrix Elements

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors