Origin of Apparent Critical Thickness for Island Formation in Heteroepitaxy

We find that a continuum model of heteroepitaxy exhibits a sharp crossover with increasing coverage, from planar growth to island formation. The “critical thickness” at which this Stranski-Krastanov transition occurs depends sensitively on misfit strain, with a dependence strikingly similar to that seen experimentally. The initial planar growth occurs because of intermixing of deposited material with the substrate. While the transition is strictly kinetic in nature, it depends only weakly on growth rate. The role of surface segregation is also discussed.

Figure 1

Evolution of structure and composition during heteroepitaxy, for nominal ${\mathrm{S}\mathrm{i}}_{0.60}{\mathrm{G}\mathrm{e}}_{0.40}$ on Si(001), at deposition rate of ${10}^4$ (arbitrary units). The onset of nonplanar morphology is more abrupt at lower growth rates. Colors indicate composition, from pure Si substrate (bottom) to ${\mathrm{S}\mathrm{i}}_{0.62}{\mathrm{G}\mathrm{e}}_{0.38}$ (top). The bottom panel is the initial surface (slightly nonplanar), and subsequent panels are at equal time intervals. The figure shows one unit cell of periodic system; the lateral size is $640w_s$. Surface-layer thickness $w_s$ is indicated by a black rectangle in bottom panel; the vertical scale is greatly expanded to show the small perturbation. The rectangle is repeated in the same position in subsequent panels for reference. Surface steps are a graphical artifact.

Figure 2

Apparent critical thickness vs composition $c_{\mathrm{d}\mathrm{e}\mathrm{p}}$ of the deposited material, for growth rates 1, ${10}^2$, and ${10}^4$ (in arbitrary units). The bottom curve corresponds to the slowest growth rate. (The dotted curve is the slowest growth rate, including surface segregation as described in the text.) The inset shows experimental results for ${\mathrm{I}\mathrm{n}}_x{\mathrm{G}\mathrm{a}}_{1-x}\mathrm{A}\mathrm{s}$ on GaAs(001), from Ref. 2 (open circles) and from Ref. 6 (solid squares). The lines in the inset are theoretical curves copied from the main panel, with arbitrary vertical scaling chosen to emphasize similarity to experiment.

Figure 3

Evolution of perturbation amplitude $h_q$ and surface composition $\xi$ with deposition thickness, at fixed growth rate (1 in arbitrary units). The left (blue) curve of each type (solid, dashed, and dotted) is for the deposition of ${\mathrm{G}\mathrm{e}}_{0.40}{\mathrm{S}\mathrm{i}}_{0.60}$; the right (red) curves are for ${\mathrm{G}\mathrm{e}}_{0.25}{\mathrm{S}\mathrm{i}}_{0.75}$. In each case we use the value of $q$ that determines the critical thickness in the case. The solid curves are results without surface segregation. For results with surface segregation, we show separately the composition ${\xi }^{(1)}$ of the surface monolayer (upper dotted curves) and ${\xi }^{(2)}$ of the rest of $w_s$ (dashed curves). The horizontal dashed line illustrates “critical composition.”