Regular papers

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

¹ Department Mathematical Modelling and Data Analysis, Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany
² Department Dimensional Nanometrology, Physikalisch-Technische Bundesanstalt (PTB), Braunschweig, Germany
³ Department Radiometry with Synchrotron Radiation, Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany

^* hermann.gross@ptb.de

Received: 12 December 2013
Revised: 3 January 2014

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.