Quantum dot asymmetry and the nature of excited hole states probed by the doubly positively charged exciton $X^{2+}$

In this experimental and theoretical study, it was found that the emission pattern of the doubly positively charged exciton complex $X^{2+}$ strongly depends on the nature of the involved excited hole states as well as the quantum dot symmetry. The two-hole system in the final state of the $X^{2+}$ recombination for the investigated high-symmetry pyramidal InGaAs quantum dots does not exhibit the singlet-tripletlike arrangement previously observed for the two-electron counterpart. Instead, the final states exhibit two true doublets, in accordance with group-theoretical predictions. Asymmetry is manifested in the photoluminescence spectra of $X^{2+}$ by a significant splitting of one doublet, which is a spectral feature exhibited to some degree by all of the measured quantum dots. The analysis demonstrates that an external magnetic field elevates the symmetry of the quantum dots. This work highlights the exciton complex $X^{2+}$ as a very sensitive probe of the quantum dot shape as well as the nature of the involved quantum states. Thus, its spectral features are very suitable for an efficient uninvasive postgrowth symmetry characterization of quantum dots.

Figure 1

Excitation-power-dependent spectra measured for one QD, QD1. The spectra are normalized and shifted horizontally for clarity. Previously identified excitonic species are labeled according to conventions. Arrows in the top spectra indicate previously unidentified emission lines. The inset shows schematic top and side views of the pyramidal InGaAs/AlGaAs QD structure used in the study.

Figure 2

Excitation-power dependence measured for (a) QD1 and (b) QD2. The spectra are normalized and shifted horizontally for clarity. The weak dotted lines serve as a guide to the eye and mark the positions of the three components in the optical decay of the doubly charged exciton, $X^{2+}$, and the single component in the optical decay of the triply charged exciton, $X^{3+}$.

Figure 3

Derived decay schemes for the doubly positively charged exciton $X^{2+}$, for the different point-group symmetries $C_{3v}$ and the isomorphic $D_{3h}$ and $C_{6v}$. The decay scheme is based on different symmetries of the excited hole wave function $h_n$, according to the irreducible representations $E_{1/2}$ or $E_{3/2}$ of $C_{3v}$, as well as $E_{1/2}$, $E_{5/2}$, or $E_{3/2}$ of $D_{3h}$ and $C_{6v}$. The electron and hole ground levels, $e_1$ and $h_1$, have in all cases been labeled $E_{1/2}$ and $E_{3/2}$, respectively. The $x$- and $y$-polarized emissions are labeled $E$, $E^{\prime \prime }$, and $E_1$ in $C_{3v}$, $D_{3h}$, and $C_{6v}$, respectively. The insets in the middle of (a) show the corresponding single-particle configuration for the initial and final states of $X^{2+}$. The bottom portion of (a) shows one possible spectrum that is compatible with the corresponding decay scheme above, where the electron-hole exchange splitting of the initial states is neglected.

Figure 4

Calculated probability density functions for the ground hole level $h_1$ (top panel) and for the excited hole levels $h_2$ to $h_6$ (subsequent panels) of a cylindrical QD with the symmetry axis along [111] (shown for both top and side view). The valence-band energy (with respect to the valence-band edge of GaAs), the atomiclike orbital of the envelope function, the heavy-hole content with respect to [111], and the labels of $C_{3v}$ irreducible representations for each hole level are also indicated in the panels. Idealized spectra of the optical transitions computed for $X^{2+}$ with two $h_1$ holes and the third hole, $h_n$, in a single excited hole state ($n>1$) are shown to the right. A spectrum was also computed by including both $h_2$ and $h_3$ as possible levels for the third hole (bottom panel). The $x$ axis spans over 12 meV in each spectrum for easy comparison. The small electron-hole exchange interaction was neglected in the computations.

Figure 5

Measured micro-photoluminescence excitation spectrum ($\mu$PLE) with detection on the neutral exciton $X$ is shown together with measured microphotoluminescence ($\mu$PL) spectra for two different excitation powers. Luminescence from exciton complexes with the recombining hole in an excited state are observed in the energy range 1445–1455 meV (enhanced by a factor of 10 or 100). The insets next to the some $\mu$PL and $\mu$PLE peaks schematically illustrate the involved transition.

Figure 6

Computed optical spectra of $X^{2+}$ for a symmetric QD (a) without and (b) with a parabolic curvature. The inset in (a) shows a side view of the QD curvature with the In concentration used for the computations. The inset of (b) shows the energy splitting between the hole levels $h_2$ and $h_3$ as a function of the parabolic curvature, along with a side view of the QD curvature for the particular QD.

Figure 7

Computed optical spectra of $X^{2+}$ in a curved and slightly asymmetric QD (a) without and (b) with a magnetic field of strength $B$ $=$ 1 T. The magnetic field spectra are presented for the two circular polarizations ${\sigma }^{+}$ and ${\sigma }^{-}$.

Figure 8

The dominant initial- and final-state single-particle configurations of the doubly positively charged exciton $X^{2+}$ are shown together with product states and the polarized radiative decay paths. In addition, the corresponding predicted optical recombination spectra connected to each of the symmetries $C_{6v}$, $C_6$, $C_2$, and $C_{2v}$, respectively, are shown underneath each decay scheme. The $z$-polarized transitions are not included. The labels of the $x$- and $y$-polarized and the circularly polarized emission ${\sigma }^{+}$ and ${\sigma }^{-}$ are $E_1$:$x$,$y$ in $C_{6v}$, $B_1$:$x$ and $B_2$:$y$ in $C_{2v}$, ${\sigma }^{+}$:${}^2{E}_1$ and ${\sigma }^{-}$:${}^1{E}_1$ in $C_{6v}$, and ${\sigma }^{+}$, ${\sigma }^{-}$: both $B$ in $C_2$. The dotted lines in the decay scheme correspond to peaks for which the computed intensity is significantly lower compared to the other peaks. The decay schemes are created in consistency with the computed as well as the experimental results. The exact order of the two initial product states $A_1$ and $A_2$ is not determined. The bottom panel shows one possible spectrum that is compatible with the corresponding decay scheme above, where the electron-hole exchange splitting of the initial states is neglected.

Figure 9

(a) Measured normalized polarization-resolved magnetic field spectra, from the same QD (QD2) as in Fig. 2a, at three different field strengths using the same excitation power (power 1). (b) A magnification of the marked area in (a) is shown (bottom) together with a spectrum achieved at the same magnetic field strength ($B=1$ T), but with an increased excitation power. In both panels, the position for the components of the doubly positively charged exciton $X^{2+}$ is indicated by arrows.

Figure 10

The measured energy for the components of the doubly charged exciton $X^{2+}$ plotted as a function of the magnetic field strength for the same QD (QD2) as in Figs. 2a and 4.