Clustering and pair correlation function in atom probe tomography

The measurement of the composition of small clusters from 3D maps as provided by atom probe tomography or Monte-Carlo simulations is a very tricky issue. A method based on pair correlation functions was developed. The analytical expression of the pair correlation function as a function of the phase composition, the number density and the size of spherical particles for a two-phase mono-dispersed system has been established. A best-fit procedure applied to experimental pair correlation function is shown to be a simple, fast and elegant way to determine the concentration of clusters and that of the parent phase as well as the radius and the number density of clusters. Application to carbon-doped silicon demonstrates the validity of this approach. Results were found very close to those derived by other means. This method was also applied to boron clustering in implanted silicon where clusters are not visible in 3D images. The advantage of this approach over other methods such as erosion or cluster identification is discussed.