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Author: Gross, Hermann × WGL Institute: WIAS ×

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Now showing 1 - 4 of 4
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    Mathematical modelling of indirect measurements in periodic diffractive optics and scatterometry
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gross, Hermann; Model, Regine; Bär, Markus; Wurm, Matthias; Bodermann, Bernd; Rathsfeld, Andreas
    In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms to determine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the grating structure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method offer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bottom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonic solution of Maxwell's equations. It determines the incident and scattered electric and magnetic fields, which fully specify the light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction patterns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the elliptic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structure parameters to evaluate the applicability and sensitivity of different scatterometric measurement methods ...
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    Profile reconstruction in EUV scatterometry: modeling and uncertainty estimates
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Gross, Hermann; Rathsfeld, Andreas; Scholze, Frank; Bär, Markus
    Scatterometry as a non-imaging indirect optical method in wafer metrology is also relevant to lithography masks designed for Extreme Ultraviolet Lithography, where light with wavelengths in the range of 13 nm is applied. The solution of the inverse problem, i.e. the determination of periodic surface structures regarding critical dimensions (CD) and other profile properties from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed parameters. With decreasing feature sizes of lithography masks, increasing demands on metrology techniques and their uncertainties arise. 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. For typical EUV masks the ratio period over wave length is so large, that a generalized finite element method has to be used to ensure reliable results with reasonable computational costs ...
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    Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gross, Hermann; Rathsfeld, Andreas
    In this work, we discuss some aspects of numerical algorithms for the determination of periodic surface structures (gratings) from light diffraction patterns. With decreasing structure details of lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method are applied to simple periodic line structures in order to evaluate the quality of the manufacturing process. Using scatterometry, geometrical parameters of periodic structures including period (pitch), side-wall angles, heights, top and bottom widths of trapezoid shaped bridges can be determined. The mathematical model for the scattering is based on the time-harmonic Maxwell's equations and reduces in case of grating structures to the Helmholtz equation. For the numerical simulation, e.g. finite element methods can be applied to solve the corresponding boundary value problems. More challenging is the inverse problem, where the grating geometry is to be reconstructed from the measured diffraction patterns. Restricting the class of gratings and the set of measurements ...
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    Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Gross, Hermann; Henn, Mark-Alexander; Heidenreich, Sebastian; Rathsfeld, Andreas; Bär, Markus
    We investigate the impact of line edge and line width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The 2D rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method (FEM) providing a numerical solution of the two-dimensional Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. ...