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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

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automated layout and phase assignment for dark field psm

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
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
outline1
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
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

Features

Conflict areas (<B)

< B

> B

0

180

0

Phase Assignment Problem

Assign phases 0, 180 to all features s.t. pairs

with separation < B have opposite phases

 b

b  minimum separation

B  minimum separation between same-phase features

conflict graph
Conflict Graph

Vertices: features

Edges: conflicts

(feature pairs with separation< B )

< B

odd cycles in conflict graph
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

outline2
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
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
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
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 compaction1
Iterative Phase Assignment and Compaction

conflict graph

find minimum #

edges to be deleted

for 2-colorobility

already

2-colorable

yes

phase assignment

no

PSM constraints

compaction

one shot 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 assignment1
One-Shot Phase Assignment

conflict graph

find minimum #

edges to be deleted

for 2-colorobility

phase assignment

compaction

conflict edge weight
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
Feature Splitting
  • Splitting features may eliminate odd cycle
  • Green areas: phase shift between 0, 180 degrees
vertical horizontal spacing
Vertical / Horizontal Spacing
  • Introducing a vertical or horizontal gap eliminates all conflict edges that cross gap
  • Optimal algorithm to find min # gaps
outline3
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
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 elimination1
Optimal Odd Cycle Elimination

blue features/red conflicts

conflict graph

matching of odd degree nodes

dual graph

optimal odd cycle elimination2
Optimal Odd Cycle Elimination

blue features/red conflicts

delete green conflicts

matching of odd degree nodes

conflict graph

fast algorithm

3

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
outline4
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
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
Data Flow
  • GDSII  CIF
  • CIF  internal layout representation
  • New layer with phase shift  CIF
outline5
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