Benchmarking of Mask Fracturing Heuristics

Aggressive resolution enhancement techniques such as inverse lithography (ILT) often lead to complex, nonrectilinear mask shapes which make mask writing extremely slow and expensive. To reduce shot count of complex mask shapes, mask writers allow overlapping shots, due to which the problem of fracturing mask shapes with minimum shot count is NP-hard. The need to account for e-beam proximity effect makes mask fracturing even more challenging. Although a number of fracturing heuristics have been proposed, there has been no systematic study to analyze the quality of their solutions. In this paper, we first propose a method to generate tight upper and lower bounds for actual ILT mask shapes by formulating mask fracturing as an integer linear program and solving it using branch and price. Since the integer program requires significant computational resources to compute reasonable bounds, we propose a new method to generate benchmarks with known optimal solutions, that can be used to evaluate the suboptimality of mask fracturing heuristics. To make the generated benchmark shapes realistic, we further propose a novel automated benchmark generation method that takes any ILT shape as input and returns a benchmark shape which looks similar to the input shape and for which the optimal fracturing solution is known. Using these methods, we compare the suboptimality of four mask fracturing heuristics. Our results show that even a state-of-the-art prototype (version of) capability within a commercial EDA tool for e-beam mask shot decomposition can be suboptimal by as much as 2.6× for real ILT shapes and by 6.0× for generated benchmarks.