Theory of Nucleation and Growth of a Thin Film. II. Application of the Path-Probability Method

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Abstract
Nucleation and growth of a thin film on a substrate by vapor deposition are studied theoretically using a statistical mechanical technique called the path-probability method for irreversible cooperative phenomena. The thin film is assumed a monolayer and is represented by a two-dimensional lattice system, each lattice point being either vacant or occupied by an adatom or an ``impurity.'' The atoms are supplied from the gas phase onto the film and may migrate on the film surface. The ``impurity'' acts as the prenucleation centers which attract adatoms; it can be either a defect in the substrate surface or a prenucleated atom and is assumed immobile in the analysis. Kinetic differential equations for three variables which specify the state of the film are derived as the most probable path. The limit of the stationary state leads to the gas—liquid isotherm identical to the one derived from the equilibrium theory, showing the internal consistency of the treatment. The stationary limit gives information on the maximum attainable density, and leads to the existence of a discontinuous ``deposition'' temperature; this temperature is lower for smaller impurity density. The process of film growth is calculated by integrating the kinetic equations for given pertinent parameters such as the rate of impinging atoms, density of impurities and the activation energies for migration and evaporation. Under certain parameters the density of the film stays on the first plateau value for some time before it increases suddenly to the second plateau. The first plateau is identified as the critical state of nucleation. The size of the critical nuclei is deduced; when the impurity density is larger, the film can grow with smaller rate of impingement and hence the critical size is larger.

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