Many-body dynamics of the decay of excitons of different charges in a quantum dot

We calculate the photoluminescence spectrum of a single semiconductor quantum dot strongly coupled to a continuum as a function of light frequency, gate voltage, and magnetic field. The spectrum is dominated by the recombination of several excitonic states which can be considered as quantum quenches in which the many-body nature of the system is suddenly changed between initial and final states. This is associated with an Anderson orthogonality catastrophe with a power-law singularity at the threshold. We explain the main features observed experimentally in the region of stability of the trion $X^{-}$, the neutral exciton $X^0$, and the gate-voltage-induced transition between them.

Figure 1

PL spectrum as a function of gate voltage. The intensity is normalized by twice the maximum intensity at $V_g=-0.35$ V [31]. Parameters are $T=1.75$ K, $E_e^0=-25$ meV, $E_h^0=1398.5$ meV, $U_{hh}=U_{eh}=U_{ee}+2U_{ee}^{\prime }=20.97$ meV, $U_{ee}^{\prime }=3.52$ meV, $\mathrm{\Delta }=1$ meV, and $\lambda =8$.

Figure 2

Most relevant energy levels of the zero-bandwidth model and the PL transitions (indicated with arrows). $|{\mathrm{\Psi }}_g{\rangle }_i$ and $|{\mathrm{\Psi }}_g{\rangle }_f$ are the initial and final ground states, respectively. $|{\mathrm{\Psi }}_e{\rangle }_f$ is an excited FS. Here $V_{ee}^0=\lambda (U_{ee}+U_{ee}^{\prime })/e$, $V_{ee}^1=\lambda U_{ee}/e$, and $\mathrm{\Delta }V=\lambda (U_{eh}-U_{ee}-U_{ee}^{\prime })/e$. The hopping between the QD level and the reservoir effective level is $V=1$ meV. Other parameters are as in Fig. 1.

Figure 3

(a) Green solid line: Energy gain ${\mathrm{\Delta }}_i$ of the IS before photoemission due to hybridization. Blue dashed line: Energy gain ${\mathrm{\Delta }}_f$ for the FS after photoemission. (b) Change of the emitted photon energy gain ${\mathrm{\Delta }}_i-{\mathrm{\Delta }}_f$ due to hybridization.

Figure 4

PL intensity close to the onset of the $X^0$ plateau (green solid line) and at the center of the $X^0$ plateau (orange solid line). The orange solid and green open circles are the corresponding digitized results of Kleemans et al. [11]. $I_0=I(1351.6$ meV) for $V_g=-0.35$ V.

Figure 5

PL intensity as a function of the frequency ${\omega }^{\prime }={\omega }_e^{*}-\omega$ shift from the threshold (${\omega }_e^{*}$) in a log-log scale for three values of the gate voltage, $-0.144$, $-0.214$, and 0.3 V. The thresholds are 1349.74, 1350.34, and 1352.54 meV, respectively. The intensity is normalized by $I_0=I(k_B{T}_K/\hslash )$ calculated for $V_g=-0.144$ V, where $T_K\sim 0.083$ K.

Figure 6

PL intensity in the presence of an external magnetic field producing a Zeeman splitting 0.57 meV of the electronic level of the QD and an $\sim 1.05$ meV splitting for the hole level at $T=1.75$ K.

Figure 7

PL intensity as a function of the frequency shift from the threshold for different values of the magnetic field. The left side panel is in the exciton decay regime ($V_g=-0.3\mathrm{V}$) and the right side panel is for trion decay ($V_g=-0.144\mathrm{V}$). The PL intensity for the spin up projection $I_{\uparrow }\left(\omega \right)$ (not shown) is exponentially suppressed for $g_h{\mu }_{B}B>k_{B}T$. Other parameters are as in Fig. 5.