Morphology Response to Strain Field Interferences in Stacks of Highly Ordered Quantum Dot Arrays

Twofold stacked $\mathrm{I}\mathrm{n}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ quantum dot (QD) layers are grown on GaAs(001) substrates patterned with square arrays of shallow holes. We study the surface morphology of the second InGaAs QD layer as a function of pattern periodicity. Comparing our experimental results with a realistic simulation of the strain energy density $E_{\mathrm{s}\mathrm{t}\mathrm{r}}$ distribution, we find that the second InGaAs QD layer sensitively responds to the lateral strain-field interferences generated by the buried periodic QD array. This response includes the well-known formation of vertically aligned QDs but also the occurrence of QDs on satellite strain energy density minima. Our calculations show that base size and shape as well as lateral orientation of both QD types are predefined by the $E_{\mathrm{s}\mathrm{t}\mathrm{r}}$ distribution on the underlying surface.

Figure 1

InGaAs QDs grown on square hole patterns with (a) 103 nm, (b) 157 nm, (c) 179 nm and (d) 200 nm periodicity, overgrown with a twofold stack of InGaAs QDs. With increasing periodicity, the number of elongated structures between two neighboring QDs increases. The square areas, indicated in (a) and (d), are magnified in Figs. 4b, 4d. (e) The average number per prepatterned site of aligned QDs, satellite QDs, and QDs on other sites as a function of periodicity.

Figure 2

Strain energy density distribution generated by a buried QDM. (a) Cross-sectional view and (b) top view for a QDM capped with 13 nm GaAs. The main minimum directly above the molecule and the saddle points in the $[100]$ and $[010]$ directions are visible. The inset shows an AFM image of a typical QDM.

Figure 3

Simulation of the strain energy density distribution for arrays with (a) 103 nm, (b) 157 nm, (c) 179 nm, and (d) 200 nm periodicity. In (a) only the main minima are visible. If the periodicity is increased, elongated satellite minima develop between main minima.

Figure 4

Comparison of theory and experiment. (a) and (b) compare the strain energy density simulation with the AFM results for the unit cell of a pattern with 103 nm periodicity and in (c) and (d) for 200 nm periodicity. All images are magnifications from Figs. 1, 3. Three different areas are identified: (i) low strain energy density in the main minimum, (ii) an area with higher strain energy density, and (iii) satellite minima. The corresponding areas in (b),(d) are the vertically aligned QD, a depression area around the vertically aligned QD and the satellite QD in (d).

Figure 5

Simulation without anisotropy. (a) The strain energy density simulation of the same QDM as in Fig. 2b. Here the anisotropic contribution is neglected resulting in an elliptical strain energy density distribution. In (b) we calculate the same array as in Fig. 3d. We observe (i) the main strain energy density and (ii) areas with higher strain. (iii) Minima develop between the second nearest neighbors in the $[1\overline{1}0]$ directions.