Shapes of Two-Phonon Recombination Peaks in Silicon

Abstract

The radiative recombination spectrum of excitons in $n$-type Si exhibits structure which has been analyzed in terms of one- and two-phonon processes. A model for radiative recombination of excitons is presented which predicts one-phonon peak shapes which are in good agreement with experiment and two-phonon peak shapes which are in fair agreement with experiment. Calculations are described with particular attention devoted to phonon decoupling approximations. The observed width of the ${\ensuremath{\Delta}}_{1}(25)$ two-phonon peak is accounted for very well by phonon dispersion effects. Neither the ${\ensuremath{\Sigma}}_{1}(60)$ nor the ${\ensuremath{\Gamma}}_{{25}^{\ensuremath{'}}}(64)$ two-phonon peak alone can span the observed width of the assigned peak. We conclude that this peak is composite. The calculations provide the following best estimates of the magnitudes (meV) of intervalley electron-phonon matrix elements ${\ensuremath{\Delta}}_{1}(25)\ensuremath{-}261$, ${\ensuremath{\Sigma}}_{1}(48)\ensuremath{-}80$, ${\ensuremath{\Sigma}}_{1}(60)\ensuremath{-}178$, and the intravalley (valence band) matrix element ${\ensuremath{\Gamma}}_{{25}^{\ensuremath{'}}}(64)\ensuremath{-}403$ where the phonons are designated by their symmetry and energy (meV). Also included are certain intermediate results which show the effects of the approximations and permit modifications of the model to be tested without extensive calculation.