Objective comparison of concentration profiles in solids by means of mathematical statistics

Abstract

To compare concentration profiles in solids in an objective manner, a multi-dimensional generalization of the comparison of two means by Student's t-test is proposed. The approach is based on piecewise modelling the profiles using rough a priori information, estimating the model parameters and their covariance matrix and comparing the estimated parameters by use of multidimensional test variables. The piecewise modelling is treated in the most important cases in such a way that the partial functions are straight lines or cubic polynoms (splines) with continuity at all knots in each case. It enables us to compare whole profiles as well as the most relevant parts of them. Two ways of estimating the covariance matrix as a generalization of the variance estimation are discussed. The approach is useful for any kind of line profiles if rough information on the profile type is available. As an example SIMS depth-profiles of aluminum obtained from thin-layer systems after different technological heat treatments are considered. These profiles are modeled by three straight lines.Profiling is one of the most significant challenges to solid-state analysis. Therefore, the problem arises to evaluate and to compare multi-dimensional analytical results by means of mathematical statistics analogous to one-dimensional results of classical chemical analysis. The most typical case--treated in the following--is the transition from comparing two means to the comparison of two profiles by generalization of Student's t-test.The test value of the classical t-test 1-1] for comparing two means, 2-1 and 2-2, is calculated according to f = 12-1 --2-2[ ./ nl "n2 qn 1 -t-n 2 (l)

Thickness determination from the GDOS depth profiles using piece-wise linear model functions

“…Of course, by the piece-wise linear modelling, it is possible to describe much more complicated depth profiles reflecting general concentration-vs-depth distributions. Piece-wise linear modelling was firstly proposed as a suitable tool for evaluation of the SIMS depth profiles [10] and this approach has been discussed recently in a more general form [11]. Piece-wise linear model functions appeared also in Auger depth profile data evaluation [13].…”