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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 maskglass
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)
0 0180
< 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 graphmatching 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
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 5.96
# critical conflicts 1402 0 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