Mean-field theory of nucleation and growth on strained surfaces

Mean-field nucleation and growth modeling is important for understanding various adsorbate-substrate systems, particularly in the context of epitaxial growth. Conventional mean-field theory does not take into account nonlocal interactions, but adparticles may interact with strained islands via long range elastic interactions mediated by the substrate. We show that recent extensions of mean-field theory to deal with nonlocal interactions do not describe such processes faithfully. Here, we derive a generally applicable mean-field theory of adparticle dynamics on strained surfaces, when interdiffusion is neglected. This approach enables us to determine the transport coefficients from the microscopic physics; in particular, we find explicit expressions for the diffusion coefficient and drift velocity at all positions relative to an arbitrarily strained island. We demonstrate the role of strain on island growth, using island strain fields that are dynamically updated, for $\mathrm{Ge}∕\mathrm{Si}\left(001\right)$ parameters. This approach has important applications in the modeling of nucleation and growth of many nanostructures, such as metal nanoclusters, semiconductor hut clusters, and silicide nanowires.

Figure 1

(Color online) [(a) and (b)] Diffusion coefficient, (c) drift velocity, and (d) strain as a function of distance from the island center. The dotted and dash-dotted lines are for ${\alpha }_2={\alpha }_1=-1.75\mathrm{eV}$. The solid (red/gray, black, and green/light gray) lines are for ${\alpha }_2-{\alpha }_1=0.75\mathrm{eV}$, with ${\alpha }_1=-1.75\mathrm{eV}$.

Figure 2

(Color online) Final concentration profiles around an island; the initial island $\text{width}=11$, as shown by the shaded region. The parameter ${\alpha }_2-{\alpha }_1$ equals (a) $0\mathrm{eV}$ (dash blue line), (b) $0.75\mathrm{eV}$ (full black line), and (c) $1.5\mathrm{eV}$ (dashed-dotted red line). In all cases ${\alpha }_1=-1.75\mathrm{eV}$. See text for discussion.