Impact of substrate curvature on grazing-incidence small-angle X-ray scattering signal: theory and example of Ag thin-film growth

Grazing-incidence small-angle X-ray scattering (GISAXS) is a technique of choice for providing information about the morphology of nano- and micro-structures at surfaces and interfaces, also in real time. The geometry of the sample, in particular its curvature, has an impact on the observed X-ray scattering signal. There are a multitude of systems with sophisticated geometries (including curvature), ranging from electronic devices on flexible substrates to biological membranes, for which GISAXS could provide valuable information. Therefore, in this work the effect of the sample geometry on the GISAXS signal is addressed. More specifically the influence of the substrate curvature and extent along the X-ray beam is considered. The analytical expressions accounting for the effects of those two geometrical parameters are provided, and the way to include them in the analysis of GISAXS patterns is described. The calculations reveal that no corrections are needed for small samples (length over distance to the detector ratio smaller than 1%) and radius of curvature |<i>R</i>| > 50 m. These results allow for a combination of GISAXS with substrate curvature measurements. The latter technique is a non-destructive <i>in situ</i> and real-time method providing information about the intrinsic stress in a thin film during its growth. Morphological information from GISAXS is supposed to complement this stress information. Herein this methodology is applied to the growth of Ag thin films deposited by magnetron sputtering with N₂ plasma additive. The analysis of the GISAXS pattern obtained from the sample, which bends during the deposition, provided morphological parameters of the growing film. This methodology can be useful for understanding of the mechanisms at the nanoscale leading to the observed stress state. The ability to perform GISAXS on curved substrates enables its application to more complex systems.