Grid graph : Draw two rays from each concave point

1. Fast Yield-Driven Fracture for Variable Shaped-Beam Mask Writing Andrew B. Kahng1, Xu Xu1, and Alex Z. Zelikovsky2 1. CSE Dept. University of California, San Diego 2. CS Department, Georgia State University Fracture in Mask Data Process ABSTRACT Gain Based Selection Heuristics The aggressive use of RET techniques with each successive process generation have presented new challenges for current fracture tools, which are at the heart of layout data preparation. One main challenge is to reduce the number of small dimension trapezoids (slivers) to improve mask yield. Some commercial tools are available for handling the sliver minimization problem in fracture. The integer linear programming (ILP) method can significantly reduce sliver number at the expense of long runtime. In this work, we propose a new ray-segment selection heuristic which can find a near-optimal fracture solution in practical time while being flexible enough to take into account all specified requirements. We also extend the heuristics with the introduce of auxiliary ray-segments. Compared with state-of-art sliver-driven fracturing tools, the proposed method reduces the number of slivers in the fractures of two industry testcases by 76.7% and 58.6%, respectively, without inflating the runtime and shot count. Similarly, compared with the previous ILP based fracture, the new method reduces the number of slivers by 56.1% and 2.2% respectively,with more than 60X speedup and negligent shot count overhead. • For any ray segment i, weight of i • W(i)= increased sliver number after using i • For any conflict pair (i, j), gain of i • G(i)=W(j)-W(i) = sliver number saved by using I • Initially, the set S = {All ray • segments from concave points} • While (S≠Ø) • - Choose one ray segment i with • the largest gain, delete its • conflict pair from the S • - If there is a ray segment j connected • with i, add j into S • - Update the gains of ray segments • in S Circuit Design Layout Extraction RET Tape Out Fracture Job Decomposition -1 Tonality 1 0 Mask Data Preparation PEC Fracture 0 Job Finishing 0 Writing 1 -1 Mask Making Inspection 1 Metrology -1 Fracture:Decompose a list of polygons into trapezoids (shots) 1 Sliver Minimization Challenge In S Chosen • Sliver : A shot whose minimum dimension < e • Sliver number • Mask CD variation Mask yield Auxiliary Ray Segments <  Yield Driven Fracture Sliver number may be reduced with the introduction of auxiliary ray segments Auxiliary ray segment addition rule: If two rays form a sliver whose length grater than 3e, and no rays partition the sliver in the middle, add one auxiliary ray in the middle. • Yield Driven Fracture Problem • Given: • List of rectilinear polygons P • Slivering size e • Partition:Pinto non-overlapping trapezoidal shots • To minimize: Number of shots and number of slivers 2 shots No sliver with good fracture >3e Ray-Segment Selection Formulation CONCLUSIONS 0 sliver sliver • Grid graph :Draw two rays from each concave point • Rays are divided into non-intersected ray-segments • Conflict pair: two ray segments from the same point • Rule 1: One of ray segments from any concave point must be used • Rule 2: At most one ray segment in each conflict pair can be used • Rule 3: No internal concave points • Fracture = select ray segments obeying the rules • Compared with two commercial fracture tools: • - Reduce sliver number by 76.7% and 58.6% • - No runtime overhead • Compared with previous ILP method: • - Reduce sliver number by28.9% • - 60x speedup • Future work: fracture-friendly OPC rays ray segments concave point Experimental Results BIBLIOGRAPHY • Kahng et al., “Yield- and Cost-Driven Fracturing for Variable • Shaped-Beam Mask Writing”, BACUS 2004 • Nakao et al. “A new figure fracturing algorithm for variable-shaped • EB exposure-data generation” , ECJ 2003 • Cobb et al. “High performance Hierarchical fracturing” SPIE 4754 • Cobb et al. “Hierarchical GDSII based fracturing and job deck • system” SPIE 4562 Conflict pair concave points convex points