Optimal phase conflict removal for layout of dark field alternating phase shifting masks

We describe new, efficient algorithms for layout modification and phase assignment for dark field alternating-type phase shifting masks in the single exposure regime. We make the following contributions. First, we suggest new two-coloring and compaction approach that simultaneously optimizes layout and phase assignment which is based on planar embedding of an associated conflict graph. We also describe additional approaches to cooptimization of layout and phase assignment for alternating PSM. Second, we give optimal and fast algorithms to minimize the number of phase conflicts that must be removed to ensure two colorability of the conflict graph. We reduce this problem to the T-join problem which asks for a minimum weight edge set A such that a node u is incident to an odd number of edges of A if u belongs to a given node subset T of a weighted graph. Third, we suggest several practical algorithms for the T-join problem. In sparse graphs, our algorithms are faster than previously known methods. Computational experience with industrial VLSI layout benchmarks shows the advantages of the new algorithms.