Correlation of structural and few-particle properties of self-organized $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots

Charged $(X^{+},X^{-},X{X}^{+})$ and neutral $(X,XX)$ exciton complexes in single $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots (QDs) are investigated by cathodoluminescence spectroscopy. The relative spectral positions of the few-particle transition energies compared to the $X$ transition are shown to be strongly correlated to the QD size. Starting from an unprecedented detailed knowledge about the size, shape, and composition of the investigated quantum dots these energies are calculated using an eight-band $\mathbf{k}\bullet \mathbf{p}$ theory for the single-particle states and the configuration interaction method for the few-particle states. The observed strong variation of the few-particle energy positions is found to originate from a depletion of the number of excited states in the QDs when they become smaller. Then the degree of correlation is reduced. From a detailed comparison of the numerical results with the experimental data we identify the number of hole states bound in the QD to be the key parameter for size and sign variations of the relative few-particle energies.

Figure 1

(Color online) (a) A typical spectrum as measured through a $100\text{−}\mathrm{nm}$ aperture. (b) Temporal evolution of (a) consisting of 500 spectra, each being integrated for $80\mathrm{ms}$. The spectral jitter allows us to identify two overlapping single-QD spectra marked by dashed $\left(Q{D}_1\right)$ and full $\left(Q{D}_2\right)$ lines in (a). The different recombinations are marked as follows. $X$: exciton; $XX$: biexciton; $X^{+∕-}$: positively/negatively charged exciton (trion); $X{X}^{+}$: positively charged biexciton.

Figure 2

Two single-QD spectra as identified by the spectral jitter. $X$: exciton; $XX$: biexciton; $X^{+∕-}$: charged excitons; $X{X}^{+}$: charged biexciton. The unmarked lines do belong to different QDs. The upper spectrum demonstrates a predominantly positively charged QD with no $X^{-}$ emission, whereas the lower spectrum shows a QD with emission from $X^{-}$ also.

Figure 3

(Color online) Energy distances between the recombination energy of the few-particle states as a function of and relative to the neutral-exciton recombination energy. The straight lines are linear fits as guides for the eyes.

Figure 4

(Color online) (a) Absolute Coulomb charging and exciton binding energies for the truncated pyramidal $\left(a1\right)$ and the pyramidal-shaped $\left(a2\right)$ QDs. (b) and (c) The predicted few-particle energies (relative to the exciton transition energy) for these structures. While in graph (b) the correlation effects are removed in that a ground state configuration (2,2) is applied, in graph (c) the configuration (6,6) corresponds to six bound electron and hole states (i.e., three electron and hole levels), thus accounting for a fixed degree of correlation.

Figure 5

(Color online) Comparison between measured (a) and calculated (b) and (c) spectroscopic shifts. The calculation is done for different CI configurations. The number of electron states the CI accounts for is fixed at two in panel (b) and at six in (c), whereas the number of hole states is being varied in the range between eight and two.