WIAS Preprint No. 1711, (2012)

Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry


  • Gross, Hermann
  • Henn, Mark-Alexander
  • Heidenreich, Sebastian
  • Rathsfeld, Andreas
  • Bär, Markus

2010 Mathematics Subject Classification

  • 78A46 65N30 62P35


  • diffraction gratings, metrology


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. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using the LER/LWR perturbed efficiency datasets for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values.

Appeared in

  • Appl. Optics, 51 (2012) pp. 7384--7394.

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