Phonon-Assisted Recombination of Free Excitons in Compound Semiconductors

In this paper we study the band-edge optical transitions in "direct" semiconductors involving the interacting exciton-phonon system, especially the longitudinal-optical (LO)-phonon-assisted recombination of free excitons. By introducing an extended Green's-function approach to include propagation in and between the various bands in the exciton spectrum, we obtain rather directly the transition rate to lowest order in the exciton-photon coupling which contains the effects of the interactions between the exciton and lattice vibrations to all orders, and thus includes line broadening, shift, asymmetry, and all renormalizations. Under the appropriate conditions (sufficiently separated bands), this result is equivalent to an expression previously derived by Toyozawa, but has the virtue that all terms appearing in it (in particular, the asymmetry term) are given in precise and general expressions. The relation of the general result to that obtained by conventional perturbation theory is discussed. The radiative decay of free excitons by the one- LO- and two-LO-phonon-assisted processes are formulated in the framework of perturbation theory. The limitation of this approach and its relationship to that in which the exciton-photon interaction is treated more accurately are briefly discussed. It is also shown that the second-order perturbation-theory result for the one-phonon-assisted processes (both for absorption and emission), including the contributions from the full intermediate state (hydrogenic) spectrum, can be evaluated exactly in closed form. The one- and two-LO-phonon-assisted emission spectra for CdS are calculated for several temperatures up to 77°K using only experimentally determined parameters and taking the anisotropy of the valence band into account. Except for the one-LO peak at $T=77\mathrm{}°$K, where the polariton effects are important, the calculated line shapes and widths, as well as the intensity ratio of peaks, are found to be in good accord with the observed spectra. The corresponding calculations for ZnO at 77°K are in good agreement with experimental spectrum. The zero-LO exciton peak due to the one-acoustic-phonon-assisted process is calculated in weak exciton-photon coupling approximations for both the deformation potential and piezoelectric couplings. The widths of the calculated lines are orders of magnitude smaller than the observed widths, and we conclude that the difficulty lies in the use of the weak exciton-photon coupling approach.