Theory of angle-resolved photoemission from the bulk bands of solids. III. Effects of intra-atomic and interatomic electron-phonon interactions

In this third paper of a series, the influence of phonons upon the angle-resolved spectra of electrons photoemitted from the bulk bands of solids is considered. When a tight-binding formalism is used to describe the electronic states of solids, it is possible to divide the electron-phonon interaction into terms that mix orbitals located on the same site (intra-atomic) and on different sites (interatomic). The intra-atomic interactions can be treated simply by using a basis set in which the atomic orbitals follow the displacement of the atoms. The effects of this intra-atomic interaction are to weaken the usual $k$-selection rules in a manner discussed in previous works. When the initial states only are perturbed by the interatomic electron-phonon interaction, the primary modification of the angle-resolved photoelectron spectra is a broadening and shifting of the features observed at low temperatures while the conservation of $k$ and polarization selection rules appear to remain essentially intact. When the final states are modified by the interatomic electron-phonon interaction, the direct $k$-conserving component of the spectra decreases as the temperature increases, while an angle-averaged-like component increases. The electron-phonon interaction also decreases the mean free path of the photoelectron, thereby increasing the tendency of the spectra to resemble one-dimensional density of states. The form of the spectra expected in various cases is discussed.