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

Abstract

1-D gridded design is one of the most promising solutions that can enable the scaling to 10 nm technology node and beyond. Line-end cuts are needed to fabricate 1-D layouts, where two techniques are available to resolve the conflicts between cuts: 1) cut redistribution and 2) cut mask assignment. In this paper, we consider incorporating the two techniques to enable the manufacturing of cut patterns in 1-D 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 $95.0{\times }$ 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.