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.