Theoretical derivation of the bi- and tri-molecular trion formation coefficients

A theoretical investigation of the trion formation process from free carriers in a single GaAs/Al_{1-x}Ga_{x}As quantum well is presented. The mechanism for the formation process is provided by the interaction of the electrons and holes with phonons. The contributions from both the acoustic and optical phonons are considered. The dependence of both bi-molecular and tri-molecular formation rates on temperature is calculated. We demonstrate that they are equivalent for negatively and positively charged excitons.