Figure 1

Structure series investigated in this paper.Reuse & Permissions

Figure 2

(Color online) (a) The piezoelectric potential isosurfaces at $\pm 50\mathrm{meV}$ of a pyramidal InAs QD with $17\mathrm{nm}$ base length shown for the linear and quadratic parts and for the sum of both, using theoretical values for the piezoelectric constants. Results obtained from experimental values are shown in the last column. (b) Contour plots of the piezoelectric potential $2\mathrm{nm}$ above the wetting layer.Reuse & Permissions

Figure 3

Impact of strain on the local band edges for the cases of a full and a truncated pyramid. [(a) and (c)] The dotted lines indicate the energy shift of the VB imposed by biaxial strain: $\delta E_{\mathrm{HH}∕\mathrm{LH}}=(b∕2){ϵ}_B\left(\mathbf{r}\right)$. The solid line marks the contribution of the hydrostatic strain $\delta E_{\mathrm{CB}}=a_c{ϵ}_H\left(\mathbf{r}\right)$. The resulting local band edge positions are shown in (b) and (d). While the HH band inside the truncated QD is always above the LH band, a crossing between HH and LH band edge occurs in the full pyramid (d).Reuse & Permissions

Figure 4

(Color online) Electronic states classified according to the irreducible representations of the symmetry groups $C_{\infty v}$, $C_{4v}$, and $C_{2v}$. The symmetry lowering from $C_{\infty v}$ down to $C_{4v}$ and $C_{2v}$ can arise from a change of geometry, from a piezoelectric field, or from the ASA effect.Reuse & Permissions

Figure 5

The light-hole fractions of the first five hole states shown for series $A$ (pyramids of different sizes) as function of the QD base length.Reuse & Permissions

Figure 6

(Color online) Lateral scans through the piezoelectric potential: linear part (left), quadratic part (middle), and the sum of both (right), shown for different variations of the QD morphology. (a) A lens-shaped QD (from series $C$ vertical aspect ratio, ${\mathrm{ar}}_V=0.21$) and (b) a full InAs pyramid (from series $A$) with a base length of $17.2\mathrm{nm}$. In the first case, the base edges are oriented along [100]; in the second, the pyramid is rotated by 45°. Hence, the edges are oriented along [110]. (c) The In fraction of the pyramid from (b) is decreased down to 70% (from series $E$). (d) An isotropic diffusion procedure is applied to QD (b) (from series $G$). (e) Here, we show results for a nonisotropic internal InAs composition profile taken from series $F$.Reuse & Permissions

Figure 7

(Color online) Single-particle electron and hole energies for pyramidal QDs with different sizes (series $A$). In (a), only the first-order piezoelectric effect is accounted for, using the experimental value of ${ϵ}_{14}$. In (b), also the second-order piezoelectric effect is taken into account, using the piezoelectric constants from Ref. 11.Reuse & Permissions

Figure 8

(Color online) Single-particle electron and hole energies for lens-shaped QDs (series $C$) versus the vertical aspect ratio ${\mathrm{ar}}_V$. In (a) only the first-order piezoelectric effect is accounted for, using the experimental value of ${ϵ}_{14}$. In (b), also the second order piezoelectric effect is taken into account, using the piezoelectric constants from Ref. 11.Reuse & Permissions

Figure 9

(Color online) Comparison of the electron wave function shapes and state ordering for first- and second-order piezoelectric effects. On the left hand side, a truncated pyramid from series $B$ and, on the right hand side, a flat lens-shaped QD from series $C$ are considered. Results for the electron states in the absence of a piezoelectric field are shown in Fig. 4. The use of first-order experimental constants lead to the same symmetry properties and state ordering as for the calculated values. Energy values in meV are given with respect to the unstrained VB edge.Reuse & Permissions

Figure 10

(Color online) Single-particle electron and hole energies for elongated QDs (series $D$) versus the lateral aspect ratio ${\mathrm{ar}}_L$. No piezoelectricity is included for the results in (a). In (b), only the first-order piezoelectric effect is accounted for, using the experimental value of ${ϵ}_{14}$. In (c), also the second-order piezoelectric effect is taken into account, using the piezoelectric constants from Ref. 11.Reuse & Permissions

Figure 11

(Color online) Single-particle electron and hole energies for series $E$ versus the Ga fraction inside the InGaAs QD. In (a), the first order piezoelectric effect is accounted for, using the experimental value of ${ϵ}_{14}$ only, whereas in (b), also the second-order piezoelectric effect is taken into account, using the piezoelectric constants from Ref. 11.Reuse & Permissions

Figure 12

(Color online) Single-particle electron and hole energies for series $F$ versus the vertical aspect ratio ${\mathrm{ar}}_V$. In (a), the first-order piezoelectric effect is accounted for, using the experimental value of ${ϵ}_{14}$ only, whereas in (b), also the second-order piezoelectric effect is taken into account, using the piezoelectric constants from Ref. 11.Reuse & Permissions

Figure 13

(Color online) (Upper panel) Single-particle electron and hole energies for series $G$ versus the number of annealing steps. In (a), the first-order piezoelectric effect is accounted for, using the experimental value of ${ϵ}_{14}$ only, whereas in (b), also the second-order piezoelectric effect is taken into account, using the piezoelectric constants in Ref. 11. (Lower panel) Probability density (isosurface at 65%) shown for the first three bound electron and hole orbitals as a function of annealing steps. An electron $p$-state reordering occurs between two annealing steps. Only a small degree of annealing is necessary to significantly change the hole wave function shape. Energy values are given with respect to the unstrained VB edge.Reuse & Permissions

Figure 14

(Color online) A possible QD elongation or the effect of the ASA has a qualitatively different impact on the level ordering than a piezoelectric field (here, schematically shown for first-order terms only). In the latter case, the first electron and hole $p$ states are oriented in orthogonal direction. (b) schematically shows the resulting interband absorption spectra. In (b1) the $p$-channel splitting is the sum of the respective electron and hole $p$-state splittings $(\delta p_e+\delta p_h)$, whereas in case (b2), the splitting is determined by the difference $(\delta p_e-\delta p_h)$. Since the polarization (not shown here) of these peaks is rather weak, a distinction between the two $p$-channel peaks might be difficult in experiment. An additional hindsight can provide intraband transition spectra (c). These peaks are nearly 100% polarized and allow a clear assignment of the transition type.Reuse & Permissions

Figure 15

(Color online) (Left panel) Excitonic absorption spectra and (right panel) CB intraband transition spectra shown for a full InAs pyramid and a flat truncated pyramid, both taken from series $B$. In (a) and (b), the results are contrasted for the two approaches for calculating the piezoelectric field.Reuse & Permissions

Figure 16

(Color online) Single-particle electron and hole energies for truncated pyramids (series $B$) as function of the vertical aspect ratio ${\mathrm{ar}}_V$. In (a) only the first-order piezoelectric effects are accounted for, using the experimental value of ${ϵ}_{14}$. In (b), also the second-order is piezoelectric effect are taken into account, using the piezoelectric constants from Ref. 11.Reuse & Permissions

Figure 17

(Color online) (Left panel) Excitonic absorption spectra and (right panel) CB intraband transition spectra shown for parallelepipedal elongated QDs from series $D$ having different lateral orientations, with ${\mathrm{ar}}_L=5∕7$ and ${\mathrm{ar}}_L=7∕5$, respectively. First- and second-order piezoelectric effects are included.Reuse & Permissions

Figure 18

(Color online) (Left panel) Excitonic absorption spectra and (right panel) CB intraband transition spectra shown for two QDs having a circular base. The first one is taken from series $C$ and the other from series $F$, both having the same total amount of InAs inside the QD structure and a similar vertical aspect ratio of (a) ${\mathrm{ar}}_V=0.21$ and (b) ${\mathrm{ar}}_V=0.2$, respectively. First- and second-order piezoelectric effects are taken into account.Reuse & Permissions

Figure 19

(Color online) (Left panel) Excitonic absorption spectra and (right panel) CB intraband transition spectra shown for the unannealed and the strongest annealed QD of series $G$. First- and second-order piezoelectric effects are included.Reuse & Permissions