WIAS Preprint No. 1411, (2009)

Profile reconstruction in EUV scatterometry: Modeling and uncertainty estimates


  • Gross, Hermann
  • Rathsfeld, Andreas
  • Scholze, Frank
  • Bär, Markus

2010 Mathematics Subject Classification

  • 35R30 74J20 35J05

2008 Physics and Astronomy Classification Scheme

  • 42.30.Tz


  • EUV scatterometry, inverse scattering, lithography masks, uncertainty estimates




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. The inverse problem can be formulated as a non-linear operator equation in Euclidean spaces. The operator maps the sought mask parameters to the efficiencies of diffracted plane wave modes. We employ a Gauß-Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the calculated ones. We apply our reconstruction algorithm for the measurement of a typical EUV mask composed of TaN absorber lines of about 80 nm height, a period of 420 nm resp. 720 nm, and with an underlying MoSi-multilayer stack of 300 nm thickness. Clearly, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We apply a Monte Carlo procedure and an approximative covariance method to evaluate the reconstruction algorithm. Finally, we analyze the influence of uncertainties in the widths of the multilayer stack by the Monte Carlo method.

Appeared in

  • Meas. Sci. Technol., 20 (2009) pp. 105102 (11 pp).

Download Documents