Energy transfer of excitons between quantum wells separated by a wide barrier

We present a microscopic theory of the excitonic Stokes and anti-Stokes energy-transfer mechanisms between two widely separated unequal quantum wells with a large energy mismatch (Δ) at low temperatures (T). Several important intrinsic energy-transfer mechanisms have been examined, including dipolar coupling, real and virtual photon-exchange coupling, and over-barrier ionization of the excitons via exciton-exciton Auger processes. The transfer rate is calculated as a function of T and the center-to-center distance d between the wells. The rates depend sensitively on T for plane-wave excitons. For localized excitons, the rates depend on T only through the T dependence of the exciton localization radius. For Stokes energy transfer, the dominant energy transfer occurs through a photon-exchange interaction, which enables the excitons from the higher-energy wells to decay into free electrons and holes in the lower-energy wells. The rate has a slow dependence on d, yielding reasonable agreement with recent data from ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ quantum wells. The dipolar rate is about an order of magnitude smaller for large d (e.g., $d=175\mathrm{}\AA )$ with a stronger range dependence proportional to $d^{-4}.$ However, the latter can be comparable to the radiative rate for small d (e.g., $d<~80\AA ).\mathrm{}$ For anti-Stokes transfer through exchange-type (e.g., dipolar and photon-exchange) interactions, we show that thermal activation proportional to $\exp \left(-\Delta {/k}_{B}T\right)$ is essential for the transfer, contradicting a recent nonactivated result based on the Förster-Dexter’s spectral-overlap theory. Phonon-assisted transfer yields a negligibly small rate. On the other hand, energy transfer through over-barrier ionization of excitons via Auger processes yields a significantly larger nonactivated rate which is independent of d. The result is compared with recent data.