3.6.2 Electron-Phonon Interaction

The electron-phonon interaction HAMILTONian can be written as [190]

$\displaystyle \hat{H}^\mathrm{el-ph}_\mathrm{I}(t_1) \ = \ \int d{\bf r_1} \
...\bf q_1} \hat{A}_{\bf q_1}(t_1)
 \right)\hat{\psi}_\mathrm{I}({\bf r_1},t_1)\ ,$ (3.47)

where $ \hat{A}_{{\bf q},\lambda}(t)=(b_{{\bf q},\lambda}e^{-i\omega_{{\bf q},\lambda}t} \ + \ b^\dagger_{{\bf -q},\lambda} e^{+i\omega_{{\bf q},\lambda}t})$, $ b_{{\bf q},\lambda}$ and $ b^\dagger_{{\bf q},\lambda}$ are the annihilation and creation operators for phonons with wave-vector $ {\bf q}$, polarization $ \lambda$, and energy $ \hbar\omega_{{\bf q},\lambda}$, and $ M_{{\bf q},\lambda}$ is the electron-phonon interaction matrix element. The zero-order perturbation gives the non-interacting GREEN's function. The first-order term of the perturbation expansion must vanish because it contains the factor $ \langle
\hat{A}_{{\bf q},\lambda}\rangle$ which is zero since the factors $ \langle b_{{\bf
q},\lambda}\rangle$ and $ \langle b^\dagger_{{\bf q},\lambda}\rangle$ are zero [190]. Similarly, all the odd terms vanish because their time-ordered bracket for phonons contains an odd number of $ \hat{A}_{{\bf q},\lambda}$ factors. Applying the WICK theorem (Section 3.4.1), only the even terms contribute to the perturbation expansion for the electron-phonon interaction

 G^{1}_\mathrm{N} & = &\displaystyle \lang...
...t_2)\rangle}_{\displaystyle K^1_\mathrm{N}}\ \ ,
 \end{array}\end{displaymath} (3.48)

where the expansion of time-ordered products of electron operators ( $ F^1_\mathrm{N}$) has been calculated before, see (3.37). Notice that, due to the properties of the annihilation and creation operators for Bosons [190], $ \langle A_{{\bf q_1},\lambda}(t_1) A_{{\bf q_2},\lambda}(t_2)\rangle=0$ unless $ {\bf q_2}={\bf - q_1}$, therefore, one obtains

 K^1_\mathrm{N} \displaystyle = \ \sum_{{\bf...
...bda}\ i\hbar
 D_{\lambda_0}({\bf q_1},t_1,t_2) \ ,
 \end{array}\end{displaymath} (3.49)

where $ D_{\lambda_0}({\bf q_1},t_1,t_2)$ is the non-interacting phonon GREEN's function (see Appendix D). FEYNMAN diagrams for this expansion are similar to Fig. 3.4, but one should only replace the COULOMB interactions with non-interacting phonon GREEN's functions [190]. However, the contributions of the diagrams (a), (b), and (f) are zero. They are non-zero only if the phonon wave-vector $ {\bf q}$ is zero, but such phonon is either a translation of the crystal or a permanent strain, and neither of these meant to be in the HAMILTONian. The lowest order self-energies due to electron-phonon interaction are also referred to as HARTREE and FOCK self-energy by analogy to the treatment of the electron-electron interaction. However, the HARTREE self-energy due to electron-phonon interaction is zero since it corresponds to phonons with $ {\bf q=0}$. The analytical expression regarding the contribution of the self-consistent FOCK self-energy (Fig. 3.9) is given by [112]

 \Sigma_\mathrm{el-ph}({\bf r_1},t_1;{\bf r_...
...;{\bf r_2},t_2) D_{\lambda}({\bf q_1};t_1,t_2) \ .
 \end{array}\end{displaymath} (3.50)

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