• 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 Rule 3: No internal concave points

• Fracture = select ray segments obeying the rules

Fast Yield-Driven Fracture for Variable Shaped-Beam Mask WritingAndrew B. Kahng1, Xu Xu1, and Alex Z. Zelikovsky2

1. CSE Dept. University of California, San Diego 2. CS Department, Georgia State University

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.

Fracture: Decompose a list of polygons into trapezoids (shots)

ABSTRACTFracture in Mask Data Process

• Sliver : : A shot whose minimum dimension <

• Sliver number

Mask CD variation Mask yield

Sliver Minimization Challenge

Gain Based Selection Heuristics 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

• 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

Experimental Results

CONCLUSIONS

BIBLIOGRAPHY

• 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 by 28.9% - 60x speedup• Future work: fracture-friendly OPC

Yield Driven Fracture

Yield Driven Fracture ProblemGiven: • List of rectilinear polygons P • Slivering size Partition: P into non-overlapping trapezoidal shotsTo minimize: Number of shots and number of slivers

Layout ExtractionRET

Circuit Design

Tape OutJob Decomposition

Mask Data Preparation

Mask Making

Writing

Inspection

Metrology

Tonality

PEC Fracture

Job Finishing

Fracture

<

2 shots

Ray-Segment Selection Formulation

concave point

raysray segments

No sliver with good fracture

Conflict pair concave points convex points

0

1 -10

-110

-11

1

In SChosen

Auxiliary Ray Segments

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 3, and no rays partition the sliver in the middle, add one auxiliary ray in the middle.

MethodDesign A Design B

shots slivers CPU shots slivers CPU

Tool A 10754 6111 0 17335 11572 0

Tool B 10455 4451 0 17130 10797 0

Tool C 9755 786 2 17195 6502 3

ILP 9750 417 134 17684 2750 222

Proposed 9786 183 1 17656 2691 4

0 sliversliver

>3