Electron-phonon interaction in the dynamics of trap filling in quantum dots

We analyze theoretically the effects of electron-phonon interaction in the dynamics of an electron that can be trapped to a localized state and detrapped to an extended band state of a small quantum dot (QD) using a simple model system. It consists of a one-dimensional tight-binding linear chain of a few sites, having a discrete set of energy levels mimicking the discrete levels of the conduction band of the QD that is connected at its end to another site, the trap, having a single energy level well below the conduction band, where the electron is allowed to interact with a local phonon of a single frequency. In spite of its simplicity, the time-dependent model has no analytical solution but a numerically exact one can be found, producing a rich dynamics. The electronic motion is quasiperiodic in time, with oscillations around a mean value that are basic characteristics of the weak and strong coupling regimes of electron-phonon interaction and set the timescales of the system. Using values of the parameters appropriate for defects in semiconductor QDs, we find these timescales to range typically from tenths of picoseconds to a few picoseconds. The values of the time-averaged trap occupancy strongly depend on the the strength of the electron-phonon interaction and can be as large as $40%$ when the coupling is most efficient, independently of other parameters. An interesting result of the present paper is the formation of resonances at specific values of the electron-phonon coupling parameter that only exist when several levels are allowed to coherently cooperate in the filling of the trap. They are characterized by a trap occupancy that is a periodic function of time with large amplitude and period picturing an electron that is periodically trapped and detrapped. We conclude that the formation of these resonances is a robust consequence of electron-phonon interaction in small systems. Electron-phonon interaction is an efficient mechanism that can provide ca. $50%$ filling of a deep trap state on a subpicosecond to picosecond timescale, much faster than radiative decay occurring in timescales of tens of picoseconds to nanoseconds, while the occupancy of this state will be smaller than ca. $1%$ in the absence of electron-phonon coupling.