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Nucleation
Nucleation is an important step in growth that helps determine the final structure of a thin film. Many growth methods rely on nucleation control such as atomic-layer epitaxy (atomic layer deposition). Nucleation can be modeled by characterizing surface process of adsorption, desorption, and surface diffusion.

Adsorption and Desorption
Adsorption is the interaction of a vapor atom or molecule with a substrate surface. The interaction is characterized the sticking coefficient, the fraction of incoming species thermally equilibrated with the surface. Desorption reverses adsorption where a previously adsorbed molecule overcomes the bounding energy and leaves the substrate surface.

The two types of adsorptions, physisorption and chemisorption, are distinguished by the strength of atomic interactions. Physisorption describes the Van der Waals bonding between a stretched or bent molecule and the surface characterized by adsorption energy $$E_{p}$$. Evaporated molecules rapidly lose kinetic energy and reduces its free energy by bonding with surface atoms. Chemisorption describes the strong electron transfer (ionic or covalent bond) of molecule with substrate atoms characterized by adsorption energy $$E_{c}$$. The process of physi- and chemisorption can be visualized by the potential energy as a function of distance. The equilibrium distance for physisorption is further from the surface than chemisorption. The transition from physisorbed to chemisorbed states are governed by the effective energy barrier $$E_{a}$$.

Crystal surfaces have specific bonding sites with larger Ea values that would preferentially be populated by vapor molecules to reduce the overall free energy. These stable sites are often found on step edges, vacancies and screw dislocations. After the most stable sites become filled, the adatom-adatom (vapor molecule) interaction becomes important.

Nucleation Models
Nucleation kinetics can be modeled considering only adsorption and desorption. First consider case where there are no mutual adatom interactions, no clustering or interaction with step edges.

The rate of change of adatom surface density n, where J is the net flux, $$\tau_{a}$$ is the mean surface lifetime prior to desorption and $$\sigma$$ is the sticking coefficient:

$${dn\over dt}=J \sigma-{n\over \tau_{a}} $$

$$n = J\sigma\tau_{a}[1-exp({-t\over\tau_{a}})] n = J\sigma\tau_{a}[exp({-t\over\tau_{a}})]$$

Adsorption can also be modeled by different isotherms such as Langmuir model and BET model. The Langmuir model derives an equilibrium constant b based on the adsorption reaction of vapor adatom with vacancy on the substrate surface. The BET model expands further and allows adatoms deposition on previously adsorbed adatoms without interaction between adjacent piles of atoms. The resulting derived surface coverage is in terms of the equilibrium vapor pressure and applied pressure.

Langmuir model where $$P_{A}$$ is the vapor pressure of adsorbed adatoms:

$$\theta = {bP_{A}\over (1+bP_{A})}$$

BET model where $$p_{e}$$ is the equilibrium vapor pressure of adsorbed adatoms and $$p$$ is the applied vapor pressure of adsorbed adatoms:

$$\theta ={X p \over (p_{e}-p)[1+(X-1){p\over p_{e}}]}$$

As an important note, surface crystallography and differ from the bulk to minimize the overall free electronic and bond energies due to the broken bonds at the surface. This can result in a new equilibrium position known as “selvedge”, where the parallel bulk lattice symmetry is preserved. This phenomenon can cause deviations from theoretical calculations of nucleation.

Surface Diffusion
Surface diffusion describes the lateral motion of adsorbed atoms moving between energy minima on the substrate surface. Diffusion most readily occurs between positions with lowest intervening potential barriers. Surface diffusion can be measured using glancing-angle ion scattering. The average time between events can be describes by :

$$\tau_{d}=(1/v_{1})\exp(E_{d}/kT_{s})$$

In addition to adatom migration, clusters of adatom can coalesce or deplete. Cluster coalescence through processes, such as Ostwald ripening and sintering, occur in response to reduce the total surface energy of the system. Ostwald repining describes the process in which islands of adatoms with various sizes grow into larger ones at the expense of smaller ones. Sintering is the coalescence mechanism when the islands contact and join.