Cellular automaton model of phase separation during annealing of nonstoichiometric silicon oxide layers

Abstract

A cellular automaton (CA) model is proposed that simulates the structural evolution of a nonstoichiometric SiO _m (m < 2) layer with a thickness of 3–30 nm, involving the formation of silicon nanoclusters during thermal annealing at temperatures within 900–1200°C. The model does not take into account the crystalline or amorphous structure of the nanoclusters. The three-dimensional CA model implemented on a cubic 0.54-nm mesh grid by the SoftCAM software is synchronous, does not use the Margolus block neighborhood, and is open to the incorporation of ab initio calculations for Si _x O _y clusters. The state of the CA cell is determined by three variables (x, y, z), which take discrete values 0, 1, 2, …, and 255 (corresponding to the numbers of silicon and oxygen atoms and conditional vacancies in the cell), and the fourth variable δ, which takes values 0, 1, and 2, indicating that the given cell belongs to a silicon nanocluster, SiO _x matrix, or their interface. The local transition rules are defined based on the following considerations: (i) each cell is characterized by a scalar “free energy” dependent only on the cell state (analogous to thermodynamic potentials); (ii) this free energy is defined as the sum of three components: internal energy U(x, y), elastic energy G(z), and surface energy E(δ); and (iii) the exchange of matter between the cells is determined by probabilities dependent on the difference of the free energies according to the Fermi–Dirac relation. The model traces the evolution of the total number of clusters, their average size, and the average distance between them. The results of the numerical simulation are consistent with the published experimental data.