Theory of the lifetime of an exciton incoherently created below its resonance frequency by inelastic scattering

When an exciton in semiconductor is scattered and its energy is decreased far below the resonance energy of the bare exciton state, it has been considered that an exciton-polariton is created immediately by the scattering process because there is no exciton level at that energy. However, according to the recent time-resolved measurements of P emission originating from inelastic exciton-exciton scattering, it looks rather natural to consider that the exciton-polariton is created in a finite time scale which is restricted by a coherence volume of the exciton after the scattering. In this interpretation, the exciton remains in this time scale far below its resonance energy as a transient state in a series of processes. We propose an expression of the P-emission lifetime depending on the coherence volume of the scattered excitons through the conversion process from them to the polaritons. The coherence volume of the scattered excitons appears in the calculation of the inelastic scattering process on the assumption of a finite coherence volume of the bottleneck excitons. Time-resolved optical-gain measurements could be a way for investigating the validity of our interpretation.

Figure 1

Sketch of P-emission process depicted in dispersion relations. Excitons created by pumping are relaxed to the bottleneck region (dashed arrows). Then, they are scattered to higher exciton states with $n>1$ and to photonlike polariton states with conserving the energy. The emission with the lower energy is called the P emission.

Figure 2

Schematic diagrams of (a) conventional interpretation and (b) our interpretation of the dynamics toward the P emission. The escape time ${\tau }_{\text{escape}}$ of polariton is estimated to be quite short compared to the observed lifetime of the P emission. We interpret that the lifetime reflects the conversion time ${\tau }_{\text{conv}}$ from scattered excitons to polaritons. If the excitons after the inelastic scattering have a coherence length longer than the radiation wavelength, they can be converted quickly to polaritons as in the conventional interpretation. However, if the coherence length is quite short, it restricts the conversion time ${\tau }_{\text{conv}}$, and our interpretation is rather appropriate.

Figure 3

(a) At each emission frequency, the $k$ distribution of scattered exciton is plotted with gray color. It is calculated from Eq. (58) for ZnO with fitting parameters ${\lambda }_{\text{screen}}=14.4\mathrm{nm}$ and ${\lambda }_{\text{coh}}^0=80\mathrm{nm}$. (b) The coherence length ${\lambda }_{\text{coh}}$ of the scattered exciton is calculated from the width at half maximum of the distribution. At the present conditions, ${\lambda }_{\text{coh}}$ does not strongly depend on the emission frequency.