Automated Layout and Phase Assignment for Dark Field PSM Andrew B. Kahng, Huijuan Wang, Alex Zelikovsky UCLA Computer Science Department Supported by a grant from Cadence Design Systems, Inc.

Outline Phase assignment for dark field Alt PSM Removing odd cycles from conflict graph –previous work –proposed methods Algorithms for odd cycle elimination Implementation experience Conclusions

Outline Phase assignment for dark field Alt PSM Removing odd cycles from conflict graph –previous work –proposed methods Algorithms for odd cycle elimination Implementation experience Conclusions

Alternating PSM conventional mask glass Chrome phase shifting mask Phase shifter 0 E at mask 0 0 E at wafer 0 0 I at wafer 0

Phase Assignment Problem Features Conflict areas (<B) < B > B Assign phases 0, 180 to all features s.t. pairs with separation < B have opposite phases b  minimum separation B  minimum separation between same-phase features  b

Conflict Graph < B Vertices: features Edges: conflicts (feature pairs with separation < B )

Odd Cycles in Conflict Graph No valid phase assignment exists, because of odd cycle (triangle) in conflict graph Valid assignment  2-colorable  bipartite  no odd cycles

Breaking an Odd Cycle  B

Outline Phase assignment for dark field Alt PSM Removing odd cycles from conflict graph –previous work –proposed methods Algorithms for odd cycle elimination Implementation experience Conclusions

Previous Work Interactive methods (Ooi et al., Moniwa et al.) –detect odd cycles –manually widen spacing for chosen pairs Compaction method (Ooi et al.) –symbolic layout from mask layout –phase assignment in symbolic layout –PSM design rules –compaction of symbolic layout

Proposed Methods Iterative coloring and compaction One-shot phase assignment Conflict edge weight Splitting of features Vertical/horizontal spacing Layer assignment

Iterative Phase Assignment and Compaction Iterate until conflict graph becomes bipartite: Compact the layout and find conflict graph Find minimum set of edges to be deleted from conflict graph for 2-colorability Add new separation constraints: one per deleted edge

Iterative Phase Assignment and Compaction find minimum # edges to be deleted for 2-colorobility conflict graph already 2-colorable PSM constraints compaction phase assignment no yes

One-Shot Phase Assignment Find conflict graph Find minimum set of edges to be deleted from conflict graph for 2-colorability Assign phases such that only chosen conflict edges connect features of the same phase Compact layout with PSM design rules: –B-separation if features have the same phase –b-separation if features have different phase

One-Shot Phase Assignment conflict graph compaction phase assignment find minimum # edges to be deleted for 2-colorobility

Conflict Edge Weight Compaction moves all features left Constraint graph contains arcs between edges Critical path between leftmost, rightmost features Conflict edges not on critical path: break for free critical path

Feature Splitting Splitting features may eliminate odd cycle Green areas: phase shift between 0, 180 degrees

Vertical / Horizontal Spacing Introducing a vertical or horizontal gap eliminates all conflict edges that cross gap Optimal algorithm to find min # gaps

Layer Assignment

Outline Phase assignment for dark field Alt PSM Removing odd cycles from conflict graph –previous work –proposed methods Algorithms for odd cycle elimination Implementation experience Conclusions

Optimal Odd Cycle Elimination Construct conflict graph G Construct dual graph D Find odd-degree vertices ODD in D Find minimum weighted perfect matching of ODD (weights = the length of path) Delete all edges of G which correspond to paths of the minimum matching of ODD

Optimal Odd Cycle Elimination conflict graph dual graph matching of odd degree nodes blue features/red conflicts

Optimal Odd Cycle Elimination conflict graphmatching of odd degree nodes delete green conflictsblue features/red conflicts

Fast Algorithm For each odd degree vertex V in dual graph –Voronoi region  even degree vertices which are closer to V than to any other odd degree vertex Connect two vertices if there is an edge between their Voronoi regions –edge weight  path cost in dual graph Find matching between odd degree nodes in Voronoi graph 3

Outline Phase assignment for dark field alt PSM Removing odd cycles from conflict graph –previous work –proposed methods Algorithms algorithm for odd cycle elimination Implementation experience Conclusions

Compaction Shape constraints Connectivity constraints Spacing constraints (PSM design rules) Bellman-Ford solution for constraint graph for one-dimensional constraint graph in x- direction Flip design and solve in y-direction

Data Flow GDSII  CIF CIF  internal layout representation New layer with phase shift  CIF

Results

Outline Phase assignment for dark field alt PSM Removing odd cycles from conflict graph –previous work –proposed methods Algorithms algorithm for odd cycle elimination Implementation experience Conclusions