1. Constrained Pattern Assignment for Standard Cell Based Triple Patterning Lithography H. Tian, Y. Du, H. Zhang, Z. Xiao, M. D.F. Wong Department of ECE, University of Illinois at Urbana Champaign, USA ICCAD 2013

2. Outline Introduction Preliminaries Problem Definition A Hybrid Approach Approach for Local Color Balancing Experimental Results Conclusions

3. Introduction Triple patterning lithography uses three masks to accommodate all the features in a layout. With one more mask than DPL, TPL is able to resolve most of the coloring conflicts and serves as one of the most promising techniques for future lithography solutions.

4. Introduction For standard cell based designs, the same type of standard cells are preferred to be colored in the same way. It is preferred to balance the amount of different color usage.

5. Preliminaries Standard Cell Based Designs  All the standard cells in the cell library has the same height.  Power and ground rails going from the left most to the right most of it.  The same type of cell may corresponds to many instances in a layout.

6. Problem Definition Constrained Pattern Assignment Problem  Given a standard cell based row structure layout, the objective is to find a legal TPL decomposition.  The same type of standard cells has exactly the same coloring solution.  Features in different masks are locally balanced with each other.

7. A Hybrid Approach The algorithm can be divided into two steps:  Fixing the cell boundaries and computing a solution graph for each standard cell.  Utilizing the sliding window approach to local color balancing.

8. A Hybrid Approach Variable Notations  Given a feature, three binary variables are used to represent its mask assignment.  For a feature x i, variables x i1, x i2, x i3 are used to denote its coloring solutions.  If x i is assigned to mask 1, we have x i1 =1, x i2 =0 and x i3 =0

9. A Hybrid Approach Boundary Polygons  A polygon within a standard cell that conflicts or connects with another polygon in any other standard cell.

10. A Hybrid Approach Capturing Boundary Constraints  Boundary conflict: Assume x 1 and x 2 conflict with each other. If x 11 is true, x 21 cannot be true.

11. A Hybrid Approach Capturing Boundary Constraints  Boundary connection: As x 1 connects with x 3, they have to be assigned to the same mask. If x 11 is true, x 31 has to be true.

12. A Hybrid Approach Capturing Boundary Constraints  Native constraint: At any time, exactly one of the three variables for a polygon has to be true. For x 1, if x 11 is true, then both x 12 and x 13 have to be false.

13. A Hybrid Approach Capturing Boundary Constraints  Native constraint: A trivial solution would be setting all variables to be 0. Need one more clause to ensure that for each polygon, at least one of its three binary variable is true.

14. A Hybrid Approach Capturing Cell Inner Constraints Constraint graph Solution graph x2x2 x3x3 x5x5 x6x6 {2} {5,6} {3} Polygon x 2 is assigned to mask 2 Polygon x 3 is assigned to mask 3

15. A Hybrid Approach Capturing Cell Inner Constraints If x 21 is true, x 32 cannot be true.

16. A Hybrid Approach Computing the Solution Graph 2

17. A Hybrid Approach An Extended Partial Max SAT Approach  Constraint of enforcing the same color for the same type of cells: Polygon x 1 is a boundary polygon in cell A 1 and x 2 is a boundary polygon in cell A 2. x 1 and x 2 correspond to the same polygon x in cell A If x 11 is true, x 21 has to be true

18. An Extended Partial Max SAT Approach When no solution exists for the SAT formulation, it means that not all the same type of cells can be colored in the same way. Convert the constrained pattern assignment problem into a partial Max-SAT problem. Hard clause and Soft clause The objective is to find a feasible assignment that satisfies all the hard clauses together with the maximum number of soft ones.

19. Approach for Local Color Balancing A sliding window scheme which targets on locally balancing different masks. Three variables, a 1, a 2 and a 3 with each sliding window. Variable a 1 represents the total area of the polygons assigned to mask 1 covered by the sliding window. The mask with the smallest area is given the highest priority.

20. Experimental Results 3 solutions for cell A 2 solutions for cell B SPC = (3+2)/2 = 2.5

21. Experimental Results

22. Conclusions This paper proposes a novel hybrid approach to solve the constrained pattern assignment problem for standard cell based TPL decompositions. Experimental results show that the proposed algorithm solves all the benchmarks in a very short runtime.