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