Development of oxide precipitates in silicon: calculation of the distribution function of the classical theory of nucleation by a nodal-points approximation

The classical theory of nucleation in solids is mathematically expressed by a system of differential equations for temporal development of cluster distribution (sizes and their concentration). Cluster sizes reach hundreds of nanometers during long annealing times, requiring us to deal with up to 107–108 differential equations. The full numerical simulation grows linearly with the number of equations, making the numerical solution extremely time-consuming. In this paper we develop a nodal-points approximation method with a logarithmic efficiency, which allows us to calculate the cluster distribution very quickly. The method is based on modified Becker–Doring equations solved precisely only within a given set of nodal points and approximated in between them. Availability of the method is shown by monitoring the kinetics of oxygen precipitation in Czochralski silicon for the case of a three-stage annealing for 8 h at 600 °C+4 h at 800 °C+8 h at 1000 °C, where the number of monomers in the clusters reaches more than 2 × 107. Examples are discussed, mainly about the development of a concentration gap and concentration wavelet of the cluster distribution and about interstitial oxygen concentration.