X-ray reflectivity measurements of nanoscale structures: Limits of the effective medium approximation

Specular X-ray reflectivity (SXR) can be used, in the limit of the effective medium approximation (EMA), as a high-resolution shape metrology for periodic patterns on a smooth substrate. The EMA means that the density of the solid patterns and the spaces separating the periodic patterns are averaged together. In this limit the density profile as a function of pattern height obtained by SXR can be used to extract quantitative pattern profile information. Here we explore the limitations of SXR as a pattern shape metrology by studying a series of linear grating structures with periodicities ranging from 300 nm to 16 mu m and determining at which length scales the EMA breaks down. We also study the angular dependence of the grating orientation with respect to the incident X-ray beam. The gratings systematically are rotated through a series of azimuthal angles with the incident X-ray beams ranging from 0 degrees to 90 degrees. The applicability of the EMA is related to the coherence length of the X-ray source. When the coherence length of beam is larger than the physical dimension of grating periodicities, EMA can be applied for characterizing nanostructures. For our slit-collimated X-ray source, the coherence length in the direction parallel to the long axis of the slit is on the order of 900 nm while the coherence length along the main axis of the beam appears to be in the range of (22 to 26) mu m.