Selected Examples in X-ray Reflectometry

Lukáš Horák

Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic

lukas.horak@matfyz.cuni.cz

X-ray reflectivity (XRR) is a well-established and, in principle, straightforward technique for determining the depth profile of electron density in thin films [1]. The presence of interference fringes (Kiessig fringes) in the reflectivity curve typically allows for a direct and accurate estimation of layer thickness. However, in many practical cases, these fringes are weak, distorted, or even completely absent (Figure 1); yet meaningful structural information can still be extracted.

Figure 1. XRR curves measured on Hydrogenated U layers. As it is visible in the inset with the zoomed part, no thickness fringes are present for the sample SO3. Is it still possible to determine/estimate the layer thickness by XRR?.

Even when sample imperfections such as surface curvature or thickness inhomogeneity suppress or smear the interference fringes, the absorption contrast can still be exploited to reliably estimate the average layer thickness.

For each sample, one must carefully consider the appropriate level of model complexity for data fitting. Should the model be kept as simple as possible, even if it fails to adequately reproduce the measured data? In cases where the fit is as poor as for some samples shown in Figure 2, how trustworthy are the extracted parameters? Furthermore, is it feasible to fit the electron density depth-profile directly, and if so, can such an approach yield a unique and physically meaningful solution?

Figure 2. (left) XRR curves of BIO/DTO bilayers on YSZ substrate fitted by the most simple model of two layers with rough interfaces; (right) density depth-profiles visualized for the refined parameters of the model. How reliable are the fitting results when the experimental data are evidently not fully fitted?

This presentation will explore several such cases, where the absence of clear fringes or the use of oversimplified models did not prevent the retrieval of reliable density profiles. Surprisingly, even models that poorly fit the experimental data can yield results comparable to those obtained using significantly more complex and computationally demanding approaches. This raises important questions about the trade-off between model fidelity and interpretability, especially in the context of peer review and publication.

1. L. G. Parratt, Phys. Rev. 95, 359–369 (1954). DOI: 10.1103/PhysRev.95.359

The work was supported by the project GA ČR, reg. No. 24-12710S