Population decay time and distribution of exciton states analyzed by rate equations based on theoretical phononic and electron-collisional rate coefficients

Population distributions and transition fluxes of the $A$ exciton in bulk GaN are theoretically analyzed using rate equations of states of the principal quantum number $n$ up to 5 and the continuum. These rate equations consist of the terms of radiative, electron-collisional, and phononic processes. The dependence of the rate coefficients on temperature is revealed on the basis of the collisional-radiative model of hydrogen plasma for the electron-collisional processes and theoretical formulation using Fermi's ``golden rule'' for the phononic processes. The respective effects of the variations in electron, exciton, and lattice temperatures are exhibited. This analysis is a base of the discussion on nonthermal equilibrium states of carrier-exciton-phonon dynamics. It is found that the exciton dissociation is enhanced even below 150 K mainly by the increase in the lattice temperature. When the thermal-equilibrium temperature increases, the population fluxes between the states of $n>1$ and the continuum become more dominant. Below 20 K, the severe deviation from the Saha-Boltzmann distribution occurs owing to the interband excitation flux being higher than the excitation flux from the $1S$ state. The population decay time of the $1S$ state at 300 K is more than ten times longer than the recombination lifetime of excitons with kinetic energy but without the upper levels $(n>1$ and the continuum). This phenomenon is caused by a shift of population distribution to the upper levels. This phonon-exciton-radiation model gives insights into the limitations of conventional analyses such as the ABC model, the Arrhenius plot, the two-level model $(n=1$ and the continuum), and the neglect of the upper levels.