The limits of ultrahigh-resolution x-ray mapping: estimating uncertainties in thin-film and interface structures determined by phase retrieval methods

Capturing subtle details at the sub-Angstrom level is key to understanding the structural basis of many intriguing interfacial phenomena in epitaxial thin films and nanostructures. X-ray phase retrieval methods are ideally suited to this task but the usual approaches for determination of uncertainties, based on refining a parametrized model, are not applicable in this case. Here we describe a method to estimate the uncertainties of the system electron density, obtained by phase retrieval, and of parameters of interest obtained from it. The method is based on the bootstrap approach and it can be generally applied to surface x-ray scattering data. Several examples are given which illustrate the method's utility in determining uncertainties arising from random and systematic errors. The approach also provides a quantitative measure of the validity of structural solutions obtained by phase retrieval methods. (paper)