Relaxation kinetics of excitons in cuprous oxide

Abstract

Due to a simple band structure, the excitons in cuprous oxide $({\mathrm{Cu}}_{2}\mathrm{O})$ are a model system for kinetic studies. Cuprous oxide appeared to be a host for a Bose-Einstein condensate of excitons, as the excitons showed transient kinetic energy distributions which matched those expected for a Bose gas near the critical density for Bose-Einstein condensation. However, recent absolute measurements of the exciton density made it clear that two-exciton annihilation is limiting the exciton density to far below the quantum density. This paper reconciles the measured exciton density with the observed exciton energy distributions by using a Boltzmann equation approach. We include experimentally determined rates for acoustic- and optical-phonon emission, conversion between exciton spin states, and two-exciton annihiliation, and use recent diffusion-Monte-Carlo estimates of the exciton-exciton elastic scattering cross section. Many experiments intending to produce a dense exciton gas in ${\mathrm{Cu}}_{2}\mathrm{O}$ used surface photoexcitation, and we found it important to include the resulting spatial inhomogeneities in the model by following the exciton occupation numbers as functions of space as well as momentum and time. A detailed but straightforward numerical integration of the resulting Boltzmann equation does in fact match the experimental results, without the assumption of quantum statistics for the excitons.