ppt - UCSD VLSI CAD Laboratory

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Automated Layout and Phase
Assignment for Dark Field PSM
Andrew B. Kahng, Huijuan Wang, Alex Zelikovsky
UCLA Computer Science Department
http://vlsicad.cs.ucla.edu
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
phase shifting mask
glass
Chrome
Phase shifter
0 E at mask 0
0 E at wafer 0
0 I at wafer 0
Phase Assignment Problem
Assign phases 0, 180 to all features s.t. pairs
with separation < B have opposite phases
Features
b
0
Conflict areas (<B)
<B
>B
180
0
b  minimum separation
B  minimum separation between same-phase features
Conflict Graph
Vertices: features
Edges: conflicts
(feature pairs with separation < B )
<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
conflict graph
find minimum #
edges to be deleted
for 2-colorobility
already
2-colorable
yes
no
PSM constraints
compaction
phase assignment
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
find minimum #
edges to be deleted
for 2-colorobility
phase assignment
compaction
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
blue features/red conflicts
matching of odd degree nodes
conflict graph
dual graph
Optimal Odd Cycle Elimination
blue features/red conflicts
delete green conflicts
matching of odd degree nodes
conflict graph
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 xdirection
• Flip design and solve in y-direction
Data Flow
• GDSII  CIF
• CIF  internal layout representation
• New layer with phase shift  CIF
Results
TEST
Layout1 Layout2 Layout3
# polygons
3769
6914
36227
# rectangles
4549
8691
36227
Conflict graph runtime
1.88
1.40
19.99
Dual graph runtime
4.45
0.23
42.63
Voronoi graph runtime
0.06
0
0.18
Matching runtime
1.1
0.26
# critical conflicts
1402
0
5.96
5672
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
Conclusions
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