## Efficient Models and Algorithms for Lithographic Projection Imaging and Related Optical Proximity Effects

Lithography simulation has become a standard tool for the development and optimization of semiconductor fabrication using optical projection lithography. Nowadays, optical projection lithography is used to create 32 nm wide features using a wavelength of 193 nm. The accurate modelling of the relevant effects requires numerical solutions of the Maxwell equations. Appropriate numerical methods such as the finite-difference time-domain method (FDTD) or rigorous couplel wave analysis (RCWA) are applicable to areas of few square micrometers only. On the other hand the design of lithographic masks considers areas with typical sizes in the order of hundreds of micrometers. To bridge this gap, very efficient and accurate compact models have to be developed to describe the resulting light scattering and optical proximity effects.

Mask model optimization in spatial and frequency domains: Enhancement of the scalar model to retrieve electromagnetic field (EMF) effects

A major concern of mask and imaging modeling in lithographic simulations is to obtain a good compromise between accurate models, e.g. fully rigorous mask diffraction and vector imaging computation without the so-called Hopkins approach (shift invariance of the mask diffraction for variation of illumination direction), and fast models like Kirchhoff (scalar) approach. Compact models such as decomposition techniques and boundary layer models provide an alternative to speed up calculations while yielding results with reasonable accuracy.

In this study, correction techniques in the spatial and frequency domains are applied to improve the accuracy of less rigorous but more efficient mask models. In this way it is possible to reproduce the EMF effects predicted by the rigorous model preserving the simplicity and efficiency of the Kirchhoff model.

In the frequency domain, two approaches are considered. First, a Jones pupil function is introduced in the projector pupil plane to describe amplitude, phase and polarization effects which are introduced by the mask (Figures 1 and 2). Second, a correction process performed directly on the scalar spectrum is used to tune the diffraction orders that get into the pupil of the optical system. In this case, since a vector description is needed to include the polarization phenomena, the spectrum for the different polarization components is constructed from the scalar spectrum transformed with the corresponding computed filter (Figure 3). The Jones pupil and correction functions are obtained by calibration with a rigorous model.

Second, a correction process performed directly on the scalar spectrum is used to tune the diffraction orders that get into the pupil of the optical system. In this case, since a vector description is needed to include the polarization phenomena, the spectrum for the different polarization components is constructed from the scalar spectrum transformed with the corresponding computed filter (Figure 3). The Jones pupil and correction functions are obtained by calibration with a rigorous model.

In the spatial domain the well-known boundary layer method is considered. In this case, the thick mask is replaced by a modified thin mask. The bright features of the mask are surrounded with a semi-transparent region with a certain width, transmission and phase (Figure 4). Alternatively, the bright mask features of the Kirchhoff model are modified by adding delta functions to the edges of the absorber. The amplitude and values of the boundary layer and delta function, respectively, are obtained by a calibration with a rigorous model.

In summary, the proposed mask models can be used to predict the impact of sub-wavelength mask diffraction effects by proper corrections of the simple Kirchhoff model. Compared to fully rigorous electromagnetic mask diffraction simulations, the proposed models reduce the simulation time by a factor of more than 1000.

Publications

- V. Agudelo, P. Evanschitzky, A. Erdmann, T. Fühner, F. Shao, S. Limmer and D. Fey, “Accuracy and performance of 3D mask models in optical projection lithography”, Proc. SPIE 7973, 79730O (2011); doi:10.1117/12.879053.
- A. Erdmann, F. Shao, V.Agudelo, T. Fühner, P. Evanschitzky, “Modeling of mask diffraction and projection imaging for advanced optical and EUV lithography”, DOI: 10.1080/09500340.2010.515752, ISSN: 1362-3044 (electronic) 0950-0340 (paper), Journal of Modern Optics, 2010.
- V. Agudelo, P. Evanschitzky, A. Erdmann, T. Fühner, F. Shao, S. Limmer and D. Fey, “Mask Models for the Imaging of Contact Holes in Optical Projection Lithography”, 3th EOS Topical Meeting on Advanced Imaging Techniques, Engelberg, Switzerland, (2010). (poster presentation).
- V. Agudelo, “Evaluation of Mask Models for Contact Holes in Optical Lithography”, 8th IISB Fraunhofer lithography simulation workshop, (2010). (talk)
- V. Agudelo, “Mask model optimization in spatial and frequency domains: Enhancement of the scalar model to retrieve EMF effects”, 9th IISB Fraunhofer lithography simulation workshop, (2010). (talk)

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