# Mathematical Modelling and Data Analysis

Department 8.4

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### Article

Title: Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry H. Gross, J. Richter, A. Rathsfeld and M. Bär J. Europ. Opt. Soc. Rap. Public. 2010 5 10053 10.2971/jeos.2010.10053 1990-2573 http://www.jeos.org/index.php/jeos\_rp/article/view/10053 Scatterometrie,critical dimensions (CD),inverse problem,profile model,scatterometry 8.41,Scatter-Inv Scatterometry as a non-imaging indirect optical method in wafer metrology is applicable to lithography masks designed for extreme ultraviolet (EUV) lithography , where light with wavelengths of about 13.5 nm is applied. The main goal is to reconstruct the critical dimensions (CD) of the mask, i.e., profile parameters such as line width, line height, and side-wall angle, from the measured diffracted light pattern and to estimate the associated uncertainties. The numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. The inverse problem is expressed as a non-linear operator equation where the operator maps the sought mask parameters to the efficiencies of the diffracted plane wave modes. To solve this operator equation, the deviation of the measured efficiencies from the ones obtained computationally is minimized by a Gauss-Newton type iterative method. In the present paper, the admissibility of rectangular profile models for the evaluations of CD uniformity is studied. More precisely, several sets of typical measurement data are simulated for trapezoidal shaped EUV masks with different mask signatures characterized by various line widths, heights and side-wall angles slightly smaller than 90 degree. Using these sets, but assuming rectangular structures as the basic profiles of the numerical reconstruction algorithm, approximate line height and width parameters are determined as the critical dimensions of the mask. Finally, the model error due to the simplified shapes is analyzed by checking the deviations of the reconstructed parameters from their nominal values.

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