The Analysis of Diffusion Data by a Method of Moments

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We develop a method for extracting diffusivities from concentration profiles that uses the moments of these profiles. We show that these moments are coefficients of polynomial equations in the fitting parameter Dt; these equations are easily solved. The method does not rely on the estimation of slopes or of functional values; it is therefore both accurate and simple to use. We apply this method to the case of P redistribution by rapid thermal annealing of ion-implanted Si. We have developed a method of data analysis that is based on the computation of moments of the concentration distribution. We have shown that these quantities form the coefficients of polynomials in the fitting parameter Dt. Since the moments are easily evaluated numerically from experimental data, it follows that data fitting reduces to finding roots of polynomials. Not only is this procedure much simpler than the usual minimization schemes, but also it must be remembered that the estimation of moments introduces a measure of smoothing. Nowhere is it necessary to estimate slopes or functional values. One can therefore expect increased accuracy in the measurement of diffusivities. It should also be appreciated that the initial profile can be arbitrary, a condition in line with most experimental situations. We have also demonstrated the method's applicability in the practical case of impurity redistribution in ion-implanted material, but it can be applied equally well to other physical cases that satisfy essentially the same boundary conditions. Although the method requires that the diffusivity be concentration independent and that the boundary conditions be quite simple (given flux, i.e., a Neumann condition), one can anticipate its extension to more complex cases. © 1985, The Electrochemical Society, Inc. All rights reserved.


07 Dec 2019