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Theory of highly excited semiconductor nanostructures including Auger coupling: exciton-bi-exciton mixing in CdSe nanocrystals

We present a theory of highly excited interacting carriers confined in a semiconductor nanostructure, incorporating Auger coupling between excited states with different number of excitations. The Coulomb matrix elements connecting exciton, bi-exciton and tri-exciton complexes are derived and an intuitive picture of breaking neutral multi-exction complexes into positively and negatively charged multi-exciton complexes is given. The general approach is illustrated by analyzing the coupling of biexciton and exciton in CdSe spherical nanocrystals. The electron and hole states are computed using atomistic $sp^3d^5s^*$ tight binding Hamiltonian including an effective crystal field splitting and surface passivation. For each number of electron-hole pairs the many-body spectrum is computed in the configuration-interaction approach. The low-energy correlated biexciton levels are broken into charged complexes: a hole and a negatively charged trion and an electron and a positively charged trion. Out of a highly excited exciton spectrum a subspace coupled to bi-exciton levels via Auger processes is identified. The interaction between correlated bi-exciton and exciton states is treated using exact diagonalization techniques. This allows to extract the spectral function of the biexciton and relate its characteristic width and amplitude to the characteristic amplitude and timescale of the coherent time evolution of the coupled system. It is shown that this process can be described by the Fermi's Golden Rule only if a fast relaxation of the excitonic subsystem is accounted for.