Double patterning (4/20 update)

Double Patterning

Wai-Shing Luk

State Key Lab of ASIC & Systems, Fudan University

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
b
 Rule 1: Two rectangles are NOT
conflict if their distance is > b.
 Rule 2: Two overlapped/touching
rectangles are NOT conflict.
 Rule 3: A C
 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 F
if their distance is < b, and there is
a path from X to Y that reconstructs
a “concave” polygon. E B D
 Conflict: (A,C), (B,D), but not (A,B),
(A,D) (B,C)!


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
a
a 5
5 b
1 b
s
e2'
4 4 h
2
c d S1
1
2
f
e 3 4 4 3 P
5
3 b
g
4
s R
e2 e0' e1'
2
h

P h

b

4 4
b
c d
2
f
e0 e1 3 3
e 4 4
h 5
S2 3 g

R 4

h


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)

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