Constrained Pattern Assignment for Standard Cell Based Triple Patterning Lithography H. Tian, Y. Du,...
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Transcript of Constrained Pattern Assignment for Standard Cell Based Triple Patterning Lithography H. Tian, Y. Du,...
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
Outline
Introduction Preliminaries Problem Definition A Hybrid Approach Approach for Local Color Balancing Experimental Results Conclusions
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.
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.
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.
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.
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.
A Hybrid Approach
Variable Notations Given a feature, three binary variables are used
to represent its mask assignment. For a feature xi, variables xi1, xi2, xi3 are used to
denote its coloring solutions. If xi is assigned to mask 1, we have xi1=1, xi2=0
and xi3=0
A Hybrid Approach
Boundary Polygons A polygon within a standard cell that conflicts or
connects with another polygon in any other standard cell.
A Hybrid Approach
Capturing Boundary Constraints Boundary conflict:
Assume x1 and x2 conflict with each other.
If x11 is true, x21 cannot be true.
A Hybrid Approach
Capturing Boundary Constraints Boundary connection:
As x1 connects with x3, they have to be
assigned to the same mask. If x11 is true, x31 has to be true.
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 x1, if x11 is true, then both x12 and x13 have
to be false.
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.
A Hybrid Approach
Capturing Cell Inner Constraints
Constraint graph
Solution graph
x2 x3
x5
x6
{2} {5,6} {3}
Polygon x2 is assigned to mask 2
Polygon x3 is assigned to mask 3
A Hybrid Approach
Capturing Cell Inner Constraints
If x21 is true, x32 cannot be true.
A Hybrid Approach
Computing the Solution Graph
2
A Hybrid Approach
An Extended Partial Max SAT Approach Constraint of enforcing the same color for the
same type of cells:
Polygon x1 is a boundary polygon in cell A1 and x2 is a boundary polygon in cell A2.
x1 and x2 correspond to the same polygon x in cell A
If x11 is true, x21 has to be true
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.
Approach for Local Color Balancing
A sliding window scheme which targets on locally balancing different masks.
Three variables, a1, a2 and a3 with each sliding window. Variable a1 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.
Experimental Results
3 solutions for cell A
2 solutions for cell B
SPC = (3+2)/2 = 2.5
Experimental Results
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.