Solving inverse problems of optical microlithography

Abstract

The direct problem of microlithography is to simulate printing features on the wafer under given mask, imaging system, and process characteristics. The goal of inverse problems is to find the best mask and/or imaging system and/or process to print the given wafer features. In this study we will describe and compare solutions of inverse mask problems.Pixel-based inverse problem of mask optimization (or "layout inversion") is harder than inverse source problem, especially for partially-coherent systems. It can be stated as a non-linear constrained minimization problem over complex domain, with large number of variables. We compare method of Nashold projections, variations of Fienap phase-retrieval algorithms, coherent approximation with deconvolution, local variations, and descent searches. We propose electrical field caching technique to substantially speedup the searching algorithms. We demonstrate applications of phase-shifted masks, assist features, and maskless printing.

Double-exposure mask synthesis using inverse lithography

“…The step sizes were calculated using the electrical caching technique discussed in Ref. 5. The double-exposure ILT problem was solved for the target pattern in Fig.…”

“…Hence, there has recently been a revival of interest in pixel-based ILT approaches. [4][5][6][7] In this work, we extend our earlier proposed singleexposure ILT framework 7 to double-exposure lithography ͑DEL͒ systems. Double-exposure lithography has gained a lot of importance recently and is tipped as one of the important technologies for enabling 45 nm and smaller technology nodes.…”

Cascadic multigrid algorithm for robust inverse mask synthesis in optical lithography

“…In order to reduce the computational complexity, iterative methods were proposed to solve the inverse problem via an optimization process, 14,15 and they were further classified as linear, quadratic, and nonlinear optimization problems. 16,17 Meanwhile, Poonawala and Milanfar designed the model-based optical proximity correction system and introduced the steepest descent algorithm for the optimization framework. 18,19 Subsequently, the optimization framework was further generalized for phase-shifting masks 20,21 and partially coherent imaging systems, [22][23][24][25][26] and the optimization algorithm was improved with an active set method 21 and with an augmented Lagrangian method.…”

“…Also, most recently we developed a metric called edge distance error (EDE) to guide mask synthesis. 29,30 Compared to the commonly used metric pattern error, [16][17][18][19][20][21][22][23][24][25][26][27][28] the metric EDE has a dimension of length and is independent of the simulation grid size. The analytical circle-sampling technique and EDE are both independent of grid size.…”

Fast pixel-based optical proximity correction based on nonparametric kernel regression

“…In the past few decades, several PBOPC approaches have been proposed for advanced optical lithography to enhance the imaging performance at nominal settings [19][20][21][22][23][24][25][26][27] or over a range of process variations. [28][29][30][31] Although pixel-based approaches are effective for improving the imaging performance of lithography systems, these approaches are computationally intensive, especially for the current sophisticated large-scale masks.…”