Modeling the dynamics of trapped electrons in quantum dots

TL;DR

The paper addresses how electron-electron and electron-phonon interactions govern trap-assisted dynamics in quantum dots by formulating a minimal Anderson-Holstein impurity model with two dot levels (valence and conduction bands) and a mid-gap trap coupled to a local phonon. It solves the time-dependent Schrödinger equation exactly for two and three electrons to track occupancies ($n_{VB}$, $n_{CB}$, $n_T$) and to quantify emission statistics for exciton and green lines under varying parameters, including the effective interaction $U_{eff}=U-2rac{ ext{lambda}^2}{ obreak obreak }$ and phonon coupling. The key findings are that short transients, set by hopping, lead to quasi-stationary dynamics where occupancies oscillate or stabilize; electron-phonon coupling generates trap sublevels that enable substantial VB/CB participation, while negative $U_{eff}$ produces correlated motion evident in occupancies but not in light-emission factors. The results offer a qualitative mechanism for observed ZnO nanoparticle luminescence trends under UV illumination and highlight that emission channels remain broadly available except in specific VB-initial states, deepening understanding of trap-mediated photophysics in nanoscale emitters.

Abstract

We analyze the effects of electron-electron and electron-phonon interactions in the dynamics of a system of two or three electrons that can be trapped to a localized state and detrapped to ab extended band states of a quantum dot using a simple model. In spite of its simplicity the time dependent problem has no analytical solution but a numerically exact one can be found at a relatively low computational cost. Within this model, we study the time evolution of the electron occupancies of conduction and valence bands and the trap state, as well as the statistical factors influencing light emission of different energies. In most of the analyzed cases, the system dynamics has a very short transient determined by the hopping parameters, that can be of tens of femptoseconds,followed by a quasi-stationary regime in which the electron occupancies either oscillate periodically around their time-averaged values or remain nearly constant. We find signatures of strong electronic correlations in the electronic motion for negative values of the effective electron-electron Coulomb interaction that are not translated to the statistical factors for light emission. Our calculations show that light emission of different energies is always possible except in the especial cases in which the valence band is initially filled with two electrons. In these cases the valence band can lose and recover electrons periodically but exciton emission is negligible at any time. We use this fact to attempt to give a possible explanation for the increase in the intensity of exciton emission with the concomitant decrease in the intensity of the green emission lines upon continuous illumination with ultraviolet radiation, experimentally observed for ZnO nanoparticles suspended in an alcohol.

Figures (15)

• Figure 1: Scheme of all the possible singlet configurations of two electrons in a QD connected to the trap T and $n$ phonons. The left columm depicts the three singlets having both electrons in the dot. The right column depicts the two singlets having one electron in the dot and the other in the trap (top and middle) and the state with the two electrons in the trap (bottom). The states are named according to Eq.(7).
• Figure 2: Scheme of all the possible states of the two-electron system yielding to light emission of different energies represented by the wavy lines. Only one half of the singlet states $DD1,n$ and $D1T,n$ is drawn for simplicity. First row: exciton emission, second row: green1 emission and third row: green0 emission.
• Figure 3: Scheme of all the possible configurations of three electrons in a QD connected to the trap T. The states are named according to Eq. (25).
• Figure 4: Time averaged occupancies for a system of two electrons in a QD with $E_{g}=20 \omega_0$, as a function of $U_{eff}$ and three values of $\lambda$. The initial state has two electrons in the CB ($DD2,n=0$ in Fig. \ref{['esq1']}). Right panel is for a perfectly symmetric case with $\tilde{\epsilon}_{T}=10 \omega_0$, $V_{CB}= V_{VB}=\omega_0$. Left panel is for an asymmetric case with $\tilde{\epsilon}_{T}=15 \omega_0$, $V_{CB}= \omega_0$ and $V_{VB}=0.5 \omega_0$. Black symbols: $n_{VB}$, red symbols: $n_{CB}$ and blue symbols: $n_{T}$. Dashed lines without symbols: $\lambda=0$, dots: $\lambda=1.5\omega_0$ and squares: $\lambda=3 \omega_0$ .
• Figure 5: Time dependent occupancies $n_{CB}(t)$ (red lines), $n_{VB}(t)$ (black lines) and $n_{T}(t)$ (blue lines) for the asymmetric case, $E_{g}= 20 \omega_0$, $\tilde{\epsilon}_{T}=15 \omega_0$, $V_{CB}= \omega_0$ and $V_{VB}=0.5 \omega_0$, two values of $\lambda$ and two values of $U_{eff}$. The initial state has two electrons in the CB ($DD2,n=0$ in Fig. \ref{['esq1']}). The two upper panels are for $\lambda=1.5 \omega_0$ and two lower panels are for $\lambda=3 \omega_0$. The two left panels are for $U_{eff}= -2.5 \omega_0$ and the two right panels are for $U_{eff}= 2.5 \omega_0$. The sign of $U_{eff}$ strongly affects the electron dynamics .