Shape transitions of metastable surface nanostructures

A shape transition between surface nanostructures which, as a function of island size, are associated with minima in formation energy per atom is modeled using a Fokker-Planck equation. We find that metastable states, associated with positive gradients in island chemical potential, can dominate the dynamics of the transition. The resulting bimodal island size distribution function is metastable to Ostwald ripening which has important implications for the self-organization of quantum dots.

Figure 1

(a) Energy per atom ${ϵ}_s\left(N\right)$ and (b) chemical potential ${\mu }_s\left(N\right)$ plotted as a function of $N∕N_1^E$ for island shapes 1 (solid line) and 2 (dashed line). Minima in ${ϵ}_s\left(N\right)$ occur at $N_1^E,N_2^E$ for shapes 1 and 2, respectively. Shape 1 will transform into shape 2 at $N_T$. Chemical potentials of islands 1 and 2 have minima at $N_1^C$ and $N_2^C$, respectively.

Figure 2

Island size distribution function $f(N∕N_1^E)$ evolving with time $t$ (scaled units). Shape 1 transforms into shape 2 at $N_T$.

Figure 3

Construction linking the mean-field chemical potential ${\overline{\mu }}^M$ with the chemical potentials of the two shapes, ${\mu }_1\left(N_1^M\right)$ and ${\mu }_2\left(N_2^M\right)$. In the linear approximation, the chemical potential variation of the two shapes is approximated by straight lines of different slope.