Benchmarking of Mask Fracturing Heuristics

Abstract

Aggressive resolution enhancement techniques such as inverse lithography (ILT) often lead to complex, non-rectilinear 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 correct 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 work, we propose a new method to generate benchmarks with known optimal solutions that can be used to evaluate the suboptimality of mask fracturing heuristics. We also 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. Our results show that 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 3.7× for generated benchmarks, and by as much as 3.6× for actual ILT shapes.

Effective model-based mask fracturing for mask cost reduction

“…All heuristics are run on a single thread and we report the wall time for all the methods. We demonstrate results for CD tolerance γ = 2nm, σ = 6.25nm and ∆ p = 1nm as done in the previous work on benchmarking [16]. We shall report results for ten real ILT mask shapes and ten generated benchmark shapes with known optimal solution, all of which are available at the benchmarking website [17] 4 .…”

“…However, since a mask contains billions of polygons, a practical fracturing method must be extremely fast for any given polygon. In this work, we focus on solving the fixed dose model-based fracturing problem with rectangular shots, as done in the benchmarking work [16]. Although variable-dose shots [18] and non-rectangular shots [19,20] have been proposed, Elayat et al have concluded that fixed-dose fracturing with rectangular shots is the most viable approach to reduce shot count without significant changes to the mask writing tools [21].…”

“…To find shot corner points, we exploit the fact that non-rectilinear boundary segments can be created by using the corner rounding due to e-beam proximity effect. Based on the proximity model, we define a threshold distance L th that corresponds to the longest 45 o line that we can create at the shot corner in a manner similar to [16], as shown in Figure 2. A horizontal or vertical boundary segment, on the other hand, can be written easily by placing the horizontal or vertical edge of a rectangular shot at the segment location.…”

“…We thank Emile Sahouria of Mentor Graphics for preliminary discussions of this idea. We deeply thank Prof. Andrew Kahng, Tuck Boon-Chan and Kwangsoo Han from UCSD for the collaborative benchmarking work [16] that helped us tremendously in developing this method.…”

“…Jiang and Zakhor propose matching pursuit for complex SRAF shapes [13] and a greedy approximation covering method for simpler OPC shapes [14]. Lin et [16]. The authors have shown that even a state-of-the-art prototype [version of] capability within a commercial EDA tool for e-beam mask shot decomposition (PROTO-EDA) can be suboptimal by up to 3.6×.…”