Abbe-PCA:使用主成分分析於Abbe’s kernel 所產生的全新快速微影成像模擬方法

The microlithography simulation is an important part of the VLSI manufacturing process. The aerial image atop the resist surface has long been used as a good first order approximation to the final etched features produced in microlithography. The fundamental ideal to simulate is using the Hopkins’ approach. In general, the Hopkins’ imaging equation is expanded by eigenfunctions, these expansion can be performed numerically using the Singular Value Decomposition (SVD) algorithm. However, there is a huge matrix need to perform SVD, which takes long time. Based on Abbe’s theory, the aerial image is then obtained by an incoherent superposition of all the contributions. Compared two methods, the Abbe’s approach only need fewer matrixes rather than Hopkins’ one. Besides, since the source mask optimization (SMO), the information of the light source embedded in TCC of Hopkins’s approach makes it hard to do SMO, conversely the Abbe’s one has to do source discretization which reserves the information. Therefore, the Abbe’s approach is adopted in our approach. In this thesis, we found that by combing Abbe's theory with the Principle Component Analysis (PCA) method, which is specific to the covariance eigen-decomposition method rather than SVD, we can achieve an extremely efficient computational algorithm to generate the essential kernels for aerial image simulation. The major reason for this speed up is from our discovery that the covariance matrix of Abbe's kernels, which is in the dimension of number of discretized condenser sources, can be easily constructed analytically as well as decomposed to a basis set. As a result, an analytical form of compact and correlated Abbe's kernels can be obtained even without explicit formation of the kernel images. Furthermore, the asymptotic eigenvalues and eigenvectors of the covariance matrix can also be obtained without much computational effort. This discovery also creates a new way to analyze the relationship between sources and final images which can be easily utilized to optimize source shape for lithography process development. Several imaging phenomenon have been readily explained by this method. Extensive experimental results demonstrate that Abbe-PCA is 91X-189X faster than the state-of-the-art algorithm Abbe-SVD. The key contributions to the OPC filed made in this thesis work include: (1) an algorithm for aerial image simulation using PCA, (2) advance source discretization for light source, which forms a special correlation matrix, and (3) the improved vertex based polygon convolution. A major contribution of this thesis is the development of a fast aerial image simulator which is tailored to the subroutine of OPC. OPC structure can be well-designed by computer tools using rule based or OPC-friendly designs, but those are actually inaccurate and full-chip correction is impossible because of the complicated layouts. The best way for OPC is that correcting the structure by pre-lithography simulation which can simulate images of mask patterns and then redesign patterns by simulation result. The usual procedure is that distorting the mask pattern edges and generating simulation to check whether the obtained imaging is better or not and this will repeat several times until the final imaging is good enough. However, imaging simulation by the existing commercial and academic softwares are still too slow that one chip might take at least two to three days with hundreds of computers running simultaneously for computation. Therefore, we propose in this thesis a compact Abbe’s kernel for imaging equation which might simulate faster than before. Another contribution of this thesis is the development of efficient eigen-decomposition. Traditionally, the Hopkins’ approach requires to perform SVD on a huge matrix, where n is the pixel dimension. However, if we discretize the source into concentric source, its correlation matrix is then a relative small matrix. Furthermore, we can discretize the source into squares as a regular mesh, and then the correlation matrix is a Block-Toeplitz-Toeplitz-Block (BTTB) matrix. Since we do the concentric discretization, the toeplitz matrix will be in special form-circulant matrix. According to the circulant theory, the computation time for eigenvalues and eigenvectors are only around the FFT time for a block which is around . We can decompose one eigen-decomposition operation on a big matrix problem into m eigen-decomposition operation on matrix, which complexity reduced from to . The speed up is significant. At last, we improve the look up table of the convolution. The previous approach is first to decompose a polygon into several rectangles, and then look up the result for each rectangle. We introduce the original vertex based rectilinear polygon convolution. For the weakness of this approach, we mention our modification, which prevent the waste of memory. It is more reasonable to implement for real cases. We combine those techniques and establish the simulation flow. Since the illumination system and SMO algorithm are separated, it allows the SMO to be independent of the simulation. Therefore, our simulator can be inserted into any existing SMO frameworks. Experimental results are presented which demonstrate advantages of Abbe-PCA. Results indicate the imaging can decrease computation time, enhance the imaging resolution and reduce memory usage.

並列關鍵字

Abbe’s approach ； aerial image simulation ； optical microlithography ； source mask optimization (SMO), singular-value decomposition (SVD) ； Principle Component Analysis (PCA) ； Circulant matrix

參考文獻

[1] Joseph W. Goodman. Introduction to Fourier Optics. McGraw Hill, 2 edition, 1996.

[3] Heinrich Kirchauer. Photolithography Simulation. PhD thesis, TU Vienna, 1998.

[5] B. J. Lin. Microlithography theory and practice. Lectures in NTU, 2006.

[6] S Hsu, L Chen, Z Li, S Park, K Gronlund, HY Liu, N “An innovative Source-Mask co-Optimization (SMO) method for extending low k1 imaging” In Proc. SPIE, 2008

[7] A. K. Wong. Resolution Enhancement Technique in Optical Lithography. SPIE Press, 2001.

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Abbe-PCA:使用主成分分析於Abbe’s kernel 所產生的全新快速微影成像模擬方法

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