CRMA: Incorporating Cut Redistribution with Mask Assignment to Enable the Fabrication of 1D Gridded Design

1D gridded design is one of the most promising solutions that can enable the scaling to 10nm technology node and beyond. Line-end cuts are needed to fabricate 1D layouts, where two techniques are available to resolve the conflicts between cuts: cut redistribution and cut mask assignment. In this paper, we consider incorporating the two techniques to enable the manufacturing of cut patterns in 1D gridded design. We consider both 2-mask case (double patterning is performed on the cuts) and 3-mask case (triple patterning is performed on the cuts). We first present an accurate integer linear programming (ILP) formulation that can solve the co-optimization of cut redistribution and mask assignment optimally. In addition, we propose efficient graph-theoretic approaches based on a novel integrated graph model and a longest-path-based refinement algorithm. Experimental results demonstrate that our graph-theoretic approaches are orders of magnitude faster than the ILP-based method and meanwhile it can obtain very comparable results. For 2-mask case, comparing with the method that solves mask assignment and cut redistribution optimally but separately, our graph-theoretic approach that solves the two tasks simultaneously can achieve 95.0× smaller cost on average. We also extend our graph-theoretic approach to 3-mask case. Comparing with the method that reduces the 3-mask problem to 2-mask problem and solves it indirectly, our innovative approach that solves the problem directly based on a novel framework of identifying and solving 4-cliques can achieve 7.6% smaller cost on average.