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.

Rectangle blanket problem: Binary integer linear programming formulation and solution algorithms

“…It basically asks to maximize the covered area while using a given number of primitive shapes. Chan et al (2014) tackle a similar problem in the context of integrated circuit manufacturing where the rectangles are allowed to overlap. After transforming the binary input image appropriately, RBP can also be considered as a generalization of Bentley's classic maximum sum subsequence problem to two dimensions for multiple subsequences (Bentley, 1984).…”

“…Demiröz et al (2014) determine rectangle blankets for crude human silhouettes at different locations offline before using them in a generative model to compute occupancy probability at a location. In a slightly different work, Chan et al (2014) find the minimum number of rectangles that best fit an integrated circuit layout off-line prior to the minimization of the mass production time.…”

Sparse nonlinear inverse imaging for shot count reduction in inverse lithography

“…is the position of the center point, and W and L are the width and height of the rectangle. Similar to Chan et al's algorithm, a library can be formed by the potential shots that can be used to recover the mask pattern [27]. These shots defined by the rectangular functions can include those shots with different sizes and center positions in the region of the target mask.…”

“…It is represented by a 125 × 125 pixel image, and the pixel size is 0.5 nm. It should be noted that the number of potential rectangles to recover the mask pattern can be huge even for a very small region, and thus the size of the library formed can be extremely large [27]. To make the algorithm tractable, we limit the shots to be squares with fixed sizes.…”

“…An algorithm to fracture the curvilinear patterns is developed, though it still relies on the approximation to edge representation [26]. Another formulation based on integer linear programming method is established by Chan et al, and the problem is solved with the branch and price algorithm [27]. This formulation demonstrates a significant shot count reduction over the traditional methods.…”