Friedrich-Alexander-Universität Erlangen-Nürnberg

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

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.

Figure 1: Pupil filters amplitude and phase to reproduce phase and polarization effects introduced by a mask composed of lines and spaces of 45 nm size: Jxx and Jyy component of the Jones matrix. Jones formalism is used to introduce aberrations to the projection system to grasp EMF effects.
Figure 2: Process windows for lines and spaces of 45 nm size to evaluate the agreement between the rigorous model and the pupil filter extended model: Green lines correspond to scalar model, blue lines correspond to rigorous model and red lines correspond to the scalar model enhanced with the pupil filters in Figure 1.

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.

Figure 3. Schematic representation of the vector spectrum construction from the scalar spectrum: In this extended model the vector spectrum is constructed via a calibration process with the most rigorous model. Correction functions in amplitude and phase are applied to the scalar spectrum (frequency domain) to obtain the spectra components for the different polarization states.

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.

Figure 4. Process windows for lines and spaces 45 nm size: Green lines represent scalar mask model, blue lines represent rigorous model and red lines represent the scalar extended method. Matching of process windows for the extended scalar model and rigorous model shows a good agreement through pitches from 90 nm to 325 nm. In this case the modification of the scalar model is carried out in the spatial domain. The thick mask model is replaced by a thin mask modified by adding layers of different transmission, phase and width at every edge of the absorber of the mask.

Publications
  

  1. 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.
  2. 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.
  3. 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).
  4. V. Agudelo, “Evaluation of  Mask Models for Contact Holes in Optical Lithography”, 8th IISB Fraunhofer lithography simulation workshop, (2010). (talk)
  5. 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)

 

Mission

SAOT provides an interdisciplinary research and education program of excellence within a broad international network of distinguished experts to promote innovation and leadership in the areas

Optical Metrology
Optical Material Processing
Optics in Medicine
Optics in Communication and Information Technology
Optical Materials and Systems
and Computational Optics.