17 October 2008 An efficient method for transfer cross coefficient approximation in model based optical proximity correction
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Abstract
Model Based Optical Proximity Correction (MBOPC) is since a decade a widely used technique that permits to achieve resolutions on silicon layout smaller than the wave-length which is used in commercially-available photolithography tools. This is an important point, because masks dimensions are continuously shrinking. As for the current masks, several billions of segments have to be moved, and also, several iterations are needed to reach convergence. Therefore, fast and accurate algorithms are mandatory to perform OPC on a mask in a reasonably short time for industrial purposes. As imaging with an optical lithography system is similar to microscopy, the theory used in MBOPC is drawn from the works originally conducted for the theory of microscopy. Fourier Optics was first developed by Abbe to describe the image formed by a microscope and is often referred to as Abbe formulation. This is one of the best methods for optimizing illumination and is used in most of the commercially available lithography simulation packages. Hopkins method, developed later in 1951, is the best method for mask optimization. Consequently, Hopkins formulation, widely used for partially coherent illumination, and thus for lithography, is present in most of the commercially available OPC tools. This formulation has the advantage of a four-way transmission function independent of the mask layout. The values of this function, called Transfer Cross Coefficients (TCC), describe the illumination and projection pupils. Commonly-used algorithms, involving TCC of Hopkins formulation to compute aerial images during MBOPC treatment, are based on TCC decomposition into its eigenvectors using matricization and the well-known Singular Value Decomposition (SVD) tool. These techniques that use numerical approximation and empirical determination of the number of eigenvectors taken into account, could not match reality and lead to an information loss. They also remain highly runtime consuming. We propose an original technique, inspired from tensor signal processing tools. Our aim is to improve the simulation results and to obtain a faster algorithm runtime. We consider multiway array called tensor data T CC. Then, in order to compute an aerial image, we develop a lower-rank tensor approximation algorithm based on the signal subspaces. For this purpose, we propose to replace SVD by the Higher Order SVD to compute the eigenvectors associated with the different modes of TCC. Finally, we propose a new criterion to estimate the optimal number of leading eigenvectors required to obtain a good approximation while ensuring a low information loss. Numerical results we present show that our proposed approach is a fast and accurate for computing aerial images.
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Romuald Sabatier, Romuald Sabatier, Caroline Fossati, Caroline Fossati, Salah Bourennane, Salah Bourennane, Antonio Di Giacomo, Antonio Di Giacomo, } "An efficient method for transfer cross coefficient approximation in model based optical proximity correction", Proc. SPIE 7122, Photomask Technology 2008, 71221U (17 October 2008); doi: 10.1117/12.801623; https://doi.org/10.1117/12.801623
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