Few-particle energies versus geometry and composition of ${\text{In}}_x{\text{Ga}}_{1-x}\text{As}/\text{GaAs}$ self-organized quantum dots

Energies of exciton, biexciton, and charged excitons for a large variety of realistic quantum dots are calculated using the configuration-interaction model in conjunction with eight-band $\mathbf{k}\cdot \mathbf{p}$ theory. The interrelation between quantum dot (QD) geometry and composition, the resulting shape and position of electron and hole wave functions, the direct Coulomb energies, and the changes introduced by correlation effects are analyzed in detail. The QD size, the base shape—being either circular, square, or rhombohedral—and the vertical and lateral aspect ratio are varied. Different average compositions and composition profiles, such as the “trumpet shape,” or an isotropic In gradient resulting from postgrowth annealing processes are studied. The resulting spectroscopic signatures turn out to be very sensitive to all these structural and chemical parameters. We analyze their interrelation to address the band-structure inversion problem, gaining information on the QD morphology from its spectroscopic signature. The results are compared to available experimental data.

Figure 1

(Color online) (Upper panel) The evolution of the multiparticle transition energies. (a) No Coulomb interaction is taken into account; the transition energies $\omega \left({\chi }^q\right)$ are degenerate. (b) Coulomb interaction lifts the degeneracy of the transition energies. Four different combinations in the order of $\omega \left({\chi }^q\right)$ can occur (see Fig. 2). (c) Adding correlation can result in a binding biexciton. In principle 24 different sequences of $\omega \left({\chi }^q\right)$ are possible now. (d) Exchange splits the exciton ground state in two nondegenerate dark and two nondegenerate bright states. Since both bright states act as final states for the biexciton decay, $\omega \left(XX\right)$ is split as well. (Lower panel) Energetics of the direct Coulomb interaction of the four many-particle states. The appearance of the negative trion $X^{-}$ decay on the higher- or lower-energy side of the exciton line depends on the relative strength of the additional terms $J_{00}^{\left(ee\right)}$ and $J_{00}^{\left(eh\right)}$ (additional forces are marked as dashed arrows). If $J_{00}^{\left(ee\right)}>\left|J_{00}^{\left(eh\right)}\right|$ the $X^{-}$ decay line appears on the high-energy side and otherwise on the low-energy side. The same rationale applies to $X^{+}$. For the biexciton $XX$ decay the additional forces are not sufficient to create a binding biexciton since $J_{00}^{\left(ee\right)}+J_{00}^{\left(hh\right)}\ge 2\left|J_{00}^{\left(eh\right)}\right|$ always holds.

Figure 2

(Color online) The recombination energy $\omega \left({\chi }^q\right)$ of $X$, $X^{+}$, and $X^{-}$ relative to the exciton $X$ (spectroscopic shift) depends in first order on the Coulomb energies $J_{00}^{\left(eh\right)}$, $J_{00}^{\left(ee\right)}$, and $J_{00}^{\left(hh\right)}$. These quantities in turn depend on the electron and hole wave-function sizes and their mutual positions creating four different possibilities labeled with (a)–(d).

Figure 3

(Color online) The hierarchy of Coulomb effects and their impact on the electron-hole correlation. (a) Electron and hole as calculated by the single-particle model. The first-order corrections are the direct Coulomb integrals. (b) By employing the Hartree method, electron and hole wave functions shrink to account for the mean-field potential imposed by the respective other carrier. The conditional probability of finding an electron for fixed hole positions, ${\left|\Psi \left({\mathbf{r}}_e,{\mathbf{r}}_h\right)\right|}^2$, however, remains unaffected. Hence, electron and hole are still uncorrelated. (c) Both carriers can become correlated by their spin: for opposite spin orientations (for electron-electron interaction: parallel spin orientations) the conditional electron density must vanish at the hole position. Thus, the electron density is anticorrelated with the hole position. This is also known as Fermi correlation and covered by the Hartree-Fock method. (d) For parallel spins, the electron can save Coulomb energy by following the position of the hole. This is called Coulomb correlation and accounted for, e.g., by the CI or the quantum Monte Carlo (QMC) method.

Figure 4

(Color online) The biexciton correlation energy as a function of the used CI configuration $\left(i,j\right)$, where $i$ is the number of electron states and $j$ is the number of hole states being included to build up the CI basis. Results for full CI are compared to that of single-double CI. For the configurations (2,2), (10,2), and (2,10), FCI and SDCI are identical.

Figure 5

Structure series investigated in this paper.

Figure 6

(Color online) Single-particle electron and hole energies shown for series H and A.

Figure 7

(Color online) Probability density at 65% of electron and hole ground-state wave functions for QDs of series H having a height of 5 and 11 ML, respectively.

Figure 8

(Color online) Probability density at 65% of electron and hole ground-state wave functions for pyramidal QDs of series A. Due to the increase in the piezoelectric field with larger QD size the hole wave function tends to elongate along $\left[1\overline{1}0\right]$. As a result the electron-hole overlap decreases and $\left|J_{00}^{\left(eh\right)}\right|$ becomes smaller than $J_{00}^{\left(hh\right)}$ and $J_{00}^{\left(ee\right)}$.

Figure 9

(Color online) (a) The direct Coulomb energies $J_{00}^{\left(ee\right)}$, $J_{00}^{\left(hh\right)}$, and $J_{00}^{\left(eh\right)}$, (b) the difference of the binding energies with respect to the exciton energy calculated for a configuration $\left(10e,10h\right)$, and (c) the correlation energies ${\delta }_{\text{corr}}\left({\chi }^q\right)$ are shown for series A and H. First- and second-order piezoelectric effects are included.

Figure 10

(Color online) (a) The direct Coulomb energies $J_{00}^{\left(ee\right)}$, $J_{00}^{\left(hh\right)}$, and $J_{00}^{\left(eh\right)}$, (b) the spectroscopic shifts calculated for a configuration $\left(10e,10h\right)$, and (c) the degree of correlation shown for series B, C, and F. First- and second-order piezoelectric effects are included.

Figure 11

(Color online) (a) The Coulomb energies $J_{00}^{\left(ee\right)}$, $J_{00}^{\left(hh\right)}$, and $J_{00}^{\left(eh\right)}$, (b) the spectroscopic shifts calculated for a configuration $\left(10e,10h\right)$, and (c) the degree of correlation shown for series of elongated QDs (series D) in the cases of excluded piezoelectric field (left panel) and if first- and second-order effects are taken into account (right panel).

Figure 12

(Color online) (a) the direct Coulomb energies $J_{00}^{\left(ee\right)}$, $J_{00}^{\left(hh\right)}$, and $J_{00}^{\left(eh\right)}$, (b) the spectroscopic shifts calculated for a configuration $\left(10e,10h\right)$, and (c) the degree of correlation shown for series E and G. First- and second-order piezoelectric effects are included.

Figure 13

(Color online) Compilation of measured binding energies, ${\Delta }_{\text{bind}}(\cdot )$ (see Table ) found in literature. The capital letters link the data point to the reference given in Table (first column). Letters X, Y, and Z refer to series of data points, where only representative values are shown in this graph.

Figure 14

(Color online) The difference $\left[E\left(X_p\right)-E\left(X_s\right)\right]$ plotted versus $E\left(X_s\right)$. With $E\left(X_p\right)$ we denote the average value of the $p$-channel transitions taken from the excitonic absorption spectra. The data are shown for the series A–C and E–H. The symbols are used to denote the order of the calculated few-particle resonances, $X$, $XX$, and $X^{\pm }$. The small arrows indicate the order of structures within each series as shown in the structure figures attached to the electronic data (Figs. 6, 10, 12). The numbers in parentheses are related to the ordering index introduced in Table .