Modelling line edge roughness in periodic line-space structures by Fourier optics to improve scatterometry

Abstract

In the present paper, we propose a 2D-Fourier transform method as a simple and efficient algorithm for stochastical and numerical studies to investigate the systematic impacts of line edge roughness on light diffraction pattern of periodic line-space structures. The key concept is the generation of ensembles of rough apertures composed of many slits, to calculate the irradiance of the illuminated rough apertures far away from the aperture plane, and a comparison of their light intensities to those of the undisturbed, 'non-rough' aperture. We apply the Fraunhofer approximation and interpret the rough apertures as binary 2D-gratings to compute their diffraction patterns very efficiently as the 2D-Fourier transform of the light distribution of the source plane. The rough edges of the aperture slits are generated by means of power spectrum density (PSD) functions, which are often used in metrology of rough geometries. The mean efficiencies of the rough apertures reveal a systematic exponential decrease for higher diffraction orders if compared to the diffraction pattern of the unperturbed aperture. This confirms former results, obtained by rigorous calculations with computational expensive finite element methods (FEM) for a simplified roughness model. The implicated model extension for scatterometry by an exponential damping factor for the calculated efficiencies allows to determine the standard deviation σ r of line edge roughness along with the critical dimensions (CDs), i.e., line widths, heights and other profile properties in the sub-micrometer range. First comparisons with the corresponding roughness value determined by 3D atomic force microscopy (3D AFM) reveal encouraging results.

Bayesian approach to determine critical dimensions from scatterometric measurements

“…This additional information yields more precise estimations of the geometry parameters and to smaller uncertainties. Finally, we compared the results for the approximate Bayesian approach with previous results obtained from the maximum likelihood approach [16,18] and from AFM measurements [27,47]. Estimations of the critical dimension, the side wall angle and the roughness strength of the absorber line are in a very good agreement with AFM measurements.…”

“…The damping strength is directly related to the fluctuation amplitude and the diffraction order. It was shown in recent publications that instead of modeling the full spatially 3D problem of line roughness, a two dimensional line grating including an exponential damping factor for efficiencies can be used [26,27]. In particular, the forward model f j (p) is replaced by the 'damped forward model'…”

Data-driven approaches to optical patterned defect detection

“…Separating the shot noise from the noise due to LER for experimental data is a very challenging task for optical tools, more easily achieved in scatterometry-based critical dimension metrology [47,48] than for imaging defect inspection tools. For this simulation study, by definition we have perfect knowledge of the distribution of the shot noise and removing this would lead to unrealistically good results.…”