3.6.2 Electron-Phonon Interaction

where , and are the annihilation and creation operators for phonons with wave-vector , polarization , and energy , and 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 which is zero since the factors and are zero [190]. Similarly, all the odd terms vanish because their time-ordered bracket for phonons contains an odd number of factors. Applying the WICK theorem (Section 3.4.1), only the even terms contribute to the perturbation expansion for the electron-phonon interaction

where the expansion of time-ordered products of electron operators ( ) has been calculated before, see (3.37). Notice that, due to the properties of the annihilation and creation operators for Bosons [190], unless , therefore, one obtains

where 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 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 . The analytical expression regarding the contribution of the self-consistent FOCK self-energy (Fig. 3.9) is given by [112]