Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation

The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the “diffusion” coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.

Figure 1

Different processes contributing to the diffusion transport of monomers in the zone of influence of a selected island. Both currents 1 and 2 contribute to monomer absorption at the island edge, but only the “net” absorption current 1 is used for the evaluation of kinetic coefficients. Similarly, current 3 defines the “net” desorption.

Figure 2

The concentration profiles of monomers, participating in the separated absorption ($C_1$) and desorption ($C_2$) processes. Their sum $C_1+C_2$ (dashed curve) provides a diffusional profile in the vicinity of a selected island, where both processes are taken into account.

Figure 3

The island nucleation barrier function $\phi \left(R\right)$ corresponding to Eq. (37). The dashed curve illustrates the estimation using the kinetic coefficient expressions suggested in earlier papers [22, 23] with $v=0.25, k=0.01{\lambda }^{-1}$.

Figure 4

Monte Carlo simulation scheme (a) and the probabilities of attachment and detachment depending on the island size $n$ (b).

Figure 5

Distribution function of the supercritical island formation time: Monte Carlo calculations for different kinetic regimes.