Maximum entropy: A new approach to non‐destructive deconvolution of depth profiles from angle‐dependent XPS

Abstract

The application of the maximum entropy method to nondestructive depth profiling by angle-dependent XPS is described. The algorithm gives the set of depth profiles that has maximum SkillingJaynes entropy, subject to the condition that the calculated data agree with the measured data within the experimental precision. The method does not require an inverse transform, is robust to experimental noise and is not restricted to small numbers of components. The programme can determine which of a set of prior estimates for the depth profiles is most probable; however, the reconstruction near the surface is virtually independent of this choice. Further, the method also estimates the accuracy of the reconstruction from a single data set. It is illustrated using model data and by a re-analysis of angleresolved XPS data sets available in the literature.

Nondestructive in‐depth composition profile of oxy‐hydroxide nanolayers on iron surfaces from ARXPS measurement

“…This can lead to artificial variations in the intensities and to meaningless results, as can be seen from Eqns (8) and (9). In addition, as the measured sample area changes with the emission angle, the ARXPS intensities have to be normalized before using the data for MEM calculations.…”

“…The apparent composition X A (ϑ) of elements as a function of emission angle is then obtained by substituting Eqns (8) and (9) in Eqn (3). The constants of proportionality for each element cancel in each term, and therefore, for a sample containing N elements:…”

“…Note, however, that if the data contain enough information about the sample structure, then the reconstruction is virtually unaffected by the choice of the initial estimate or the model. 9 The functional Q (Eqn 13) was then maximized iteratively. This procedure involves the random generation of n j,i values, to calculate the value of Q at the nth iteration and to compare it with the value of Q at the (n 1)th iteration.…”

“…Livesey and Smith first applied this method to ARXPS studies. 9,12 Other examples of MEM applications can be found in Opila et al, 5,13 where the method was applied to ARXPS data from gate dielectric layers and from metal layers on SAM films, and in Champaneria et al, 14 where the method was applied to ARXPS data from high-k dielectric layers on silicon. Splinter et al 15 applied the MEM to the curve fitting of the XPS spectra.…”

Formation of Al/Au Bimetallic Interface Studied by Angle-Resolved X-Ray Photoelectron Spectroscopy (ARXPS)

“…5). The in-depth concentration profiles were also extracted after two stages of the sample XPS signal in the binary system [26,27]. The effects due to the elastic scattering were not included in the model [28].…”