Equilibrium Model of Bimodal Distributions of Epitaxial Island Growth

We present a nanostructure diagram for use in designing heteroepitaxial systems of quantum dots. The nanostructure diagram is computed using a new equilibrium statistical physics model and predicts the island size and shape distributions for a range of combinations of growth temperature and amount of deposited material. The model is applied to Ge on Si(001), the archetype for bimodal island growth, and the results compare well with data from atomic force microscopy of $\mathrm{G}\mathrm{e}/\mathrm{S}\mathrm{i}$ islands grown by chemical vapor deposition.

Figure 1

The nanostructure diagram for $\mathrm{G}\mathrm{e}/\mathrm{S}\mathrm{i}(001)$. The inset scanning tunneling micrograph shows pyramids and domes of different sizes coexisting in equilibrium next to each other, grown by Ge molecular beam epitaxy on Si(001) at $3\mathrm{M}\mathrm{L}/\min $ at $600°\mathrm{C}$ to give a coverage $\theta =8\mathrm{M}\mathrm{L}$. The rainbow color scale indicates the relative numbers of pyramids (4%–96%). The solid yellow line is a contour of relative dome population at the maximum in the island energy and gives an indication of the onset of unstable ripening. The contours indicate relative distribution widths [normalized to the mean island size: pyramids (solid lines); domes (dashed lines)], illustrating the wealth of information that is contained in the nanostructure diagram.

Figure 2

Island size distributions illustrating the information that is available at each point on the nanostructure diagram. The curves give the numbers of islands of each type from Eq. (7), and the total number at $T=600°\mathrm{C}$ and (a) $\theta =4.8\mathrm{M}\mathrm{L}$ $\mathrm{G}\mathrm{e}/\mathrm{S}\mathrm{i}(001)$; (b) $\theta =11.9\mathrm{M}\mathrm{L}$ $\mathrm{G}\mathrm{e}/\mathrm{S}\mathrm{i}(001)$. The vertical scales are different. The bimodal size distribution is characteristic of two island types. Experimental data are also plotted. The broken vertical line indicates the turning point in the pyramid energy at which the distribution was truncated; in both cases the equivalent point for domes is beyond $100000{\mathrm{n}\mathrm{m}}^3$.