double patterning (4/20 update)

State Key Lab of ASIC & Systems, Fudan University

Double Patterning

Wai-Shing Luk


Background

At the past, chips were continuously getting smaller and smaller, and hence less power consumption.

However, we’re fast approaching the end of the road where optical lithography(光刻) cannot take us where we need to go next.


光刻过程

1. Photo-resist coating

2. Illumination

3. Exposure

4. Etching

5. Impurities Doping

6. Metal connection


Sub-wavelength Lithograph

Feature size <<

lithograph wavelength

45nm vs. 193nm

What you see in the

mask/layout is not what

you get in the chip:

图形失真

成品率下降


What is Double Patterning?

Instead of exposing the photo-resist layer once

under one mask, as in conventional optical

lithography, expose it twice, by splitting the mask

into two, each with features half as dense.


How Difficult is Layout Splitting?

Basically layout splitting problem is NP-hard.


Key Techniques

Novel polygon fracturing algorithm to reduce the

number of rectangles and the total cut-length.

Novel dynamic priority search tree for plane-

sweeping.

Graph-theoretical approach

Convert the coloring problem to a T-join problem and

is then by solved by Hadlock’s algorithm

Decompose the underlying conflict graph into its

tri-connected components using SPQR-tree


New Polygon Fracturing Algorithm

Allow minimal overlapping to reduce the number of rectangles, and hence to reduce the number of conflicts.


Dynamic Priority Search Tree

In plane sweeping, events are frequently

“inserted” and “deleted” to the scan line.

In our PST, all data are stored at the leaf

nodes of PST, making “insert” and “delete”

operations very fast (O(1) time for each tree

rotation). The payoff is that the “query”

operation will be little slower than the

traditional PST.


Conflict Detection

Rule 1: Two rectangles are NOT conflict if their distance is > b.

Rule 2: Two overlapped/touching rectangles are NOT conflict.

Rule 3: Define: a polygon is said to be

rectilinearly convex if it is both x-monotone and y-monotone.

Two rectangles X and Y are conflict if their distance is < b, and there is a path from X to Y that reconstructs a “concave” polygon.

Conflict: (A,C), (B,D), but not (A,B), (A,D) (B,C)!

A

B

C

DE

F

b


Conflict Graph

Blue edge: positive weight (opposite color preferred)

Green edge: negative weight, (same color preferred)


Layout Splitting Problem Formulation

INSTANCE: Graph G = (V,E) and a weight function

w : E N

SOLUTION: Disjoint vertex subsets V0 and V1 where

V = V0 ∪ V1

MINIMIZE: the total cost of edges whose end

vertices in same color.

Note: the problem is linear-time solvable for bipartite

graphs, polynomial-time solvable for planar graphs,

but NP-hard in general.


Graph Theoretic Approach

Q: How to produce a high quality result when the

problem is NP-hard?

A: Observe that G is a nearly planar graph: we can

find a maximum weighted planar subgraph of G (say

G’) using a heuristic method, then obtain the optimal

solution of G’ using Hadlock’s algorithm.

However, the time complexity of this method is still

high.

Solution: Graph partition/decomposition methods


Bi-connected Graph

A vertex is called a cut-vertex of G if removing it will disconnect G.

If no cut-vertex can be found in G, then the graph is called a bi-connected graph.

For example below, a and b are cut-vertices.


Bi-connected Components

A connected graph can be decomposed into

its bi-connected components in linear-time.

Each bi-connected component can be solved

independently without affecting the final sol’n.

Question: Is it possible to further decompose

the graph?


Tri-connected Graph

A pair of vertices is called a separation pair of a bi-connected graph G if removing it will disconnect G.

If no separation pair can be found, then the graph is called a tri-connected graph.

Eg below, {a,b}, {c,d}, {c,e}, {c,f} are separation pairs.


Tri-connected Component


SPQR-Tree

A bi-connected graph can be decomposed into its

tri-connected components in linear-time using a data

structure named SPQR-tree


Four Three Types of Skeleton

A skeleton is classified into four three types:

Trivial (Q): the skeleton contains only two vertices

s and t, and two parallel edges between s and t,

one is virtual and the other one is real.

Parallel (P): the skeleton contains only two

vertices s and t, and multiple parallel edges

between s and t.

Series (S): the skeleton is a cycle graph.

Rigid (R): the skeleton is a tri-connected graph

other than the above types.


Divide-and-Conquer Method

Three basic steps:

Divide a graph into its tri-connected components.

Solve each tri-connected components in a

bottom-up fashion.

Merge the solutions into a complete one in a top-

down fashion.

We calculate two possible solutions for each

components, namely {s, t} in same color and {s, t}

in opposite colors.


Example

s

P

s R

b

h

e e0

S2

2

3

S1

a

b

h

1

5

a

b

c d

e f

g

h

12

3

3 4 4 3

2

5

5

4

4 4

b

hP

2e0' e1'

b

c d

f

g

h

3 4 4 3

5

44

R 4

e1

e2

e2'


Splitting and Stitching

Node splitting (additional rectangle splitting) for

resolving conflicts.

To reduce the number of “cut”, we apply the node

splitting after one color assignment and then perform

the re-coloring

without node splitting with node splitting


45nm SDFFRS_X2 Layer 9, 11


45nm Example


Random, 4K rectangles


fft_all.gds


fft_all.gds, 320K polygons


Current Status of Our SW

fft_all: 320K polygons, 1.3M rectangles

Conflict graph construction within 1 minute

Color assignment within 9 minutes (no

minimization of stitching)

Compare: 26 minutes for just displaying the result

using “eog”

Note: Only g++ 3.4.5 was used, no advanced

compiler optimization has been done yet.


Conclusions

Experiment results show that our method can

achieve 3-10X speedup

We believe that it is a key to the success of

22nm process

Unfortunately we didn’t have chance to try a

realistic 32/22nm layout yet

because nearly everything is confidential

under 90nm

Foundries may move to EUV if DPL fails.

double patterning (4/20 update)

Documents

fudan university

biconnected graph

triconnected graph

bi connected graph g

cycle graph

biconnected components

triconnected components

conflict graph blue