A Polynomial Time Exact
Algorithm for Self-Aligned Double
Patterning Layout Decomposition
Z. Xiao, Y. Du, H. Zhang and M. D. F. Wong
Department of Electrical and Computer
Engineering
University of Illinois at Urbana-Champaign
ISPD 2012
Outline
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Introduction
Preliminaries
A polynomial-time exact algorithm for SADP
decomposition
Experimental results
Conclusion
Introduction
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Double patterning lithography (DPL) has become
more and more important for the current sub-32nm
nodes.
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LELE (litho-etch-litho-etch) splits the intended
patterns into two exposures. It suffers from the
inevitable overlay problem during the process.
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The features printed between two masks may be
misaligned.
Introduction
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SADP (Self-aligned double patterning) is a
promising alternative double patterning lithography
that can significantly avoid overlay.
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As a DPL, it also contains two mask steps, namely
core mask and trim mask.
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The problem of generating the core and trim mask
from a 2D designed layout is called SADP
decomposition.
Introduction
Preliminaries
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Overlay in SADP Process
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Overlay control in SADP is achieved by providing sidewall
protection to feature boundaries.
The sidewall must have thickness ws >= 2w0 to fully
tolerate the overlay.
Preliminaries
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Process Rules
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The width of a core pattern is at least wc.
The width of a trim pattern is at least wt.
The distance between core patterns is at least dc.
The distance between trim patterns is at least dt.
The width of sidewall is ws.
The trim overlay is w0.
wc and wt are usually the same, we refer them as wmin.
dc and dt are usually the same, we refer them as dmin.
ws >= 2w0
Auxiliary Cores
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Auxiliary Cores
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An auxiliary core is a core pattern that is placed outside of
a feature to generate parts of sidewall required for the
feature.
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We call a core pattern as a main core when it is placed at
the same location as a feature and generate the sidewall
directly.
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Sometimes we cannot use main cores due to the process
rules. An alternative way is to use auxiliary cores.
Auxiliary Cores
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Alternative layout decompositions
Auxiliary Cores
A graph formulation for SADP
decomposition
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A graph formulation for SADP decomposition
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dmin = wmin = 40nm and ws = 30nm.
The distance between the two features is 35nm.
The sidewall generated by the left feature cannot touch
the right feature since ws = 30nm.
The right feature will be widened 5nm.
A graph formulation for SADP
decomposition
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Given two features, to decompose the layout, we
have the following cases depending on the value of
d:
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If d >= 2ws + wmin, the features are far away from each
other. Either method can produce the features.
If wmin <= d < 2ws + wmin, we cannot use aux-method for
both features. But we can still use core-method for each
feature.
If ws < d < wmin, this layout cannot be decomposed.
If d = ws, we can use core-method for one feature while
use aux-method for another.
If d < ws, this layout cannot be decomposed.
A graph formulation for SADP
decomposition
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Given a layout that consists of a set of target
features, we construct Gs = (V, E) as follows:
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For each feature in the layout, include a vertex in V.
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If the distance d between two features u and v is exactly
ws, insert an edge e = (u, v) in E. If d < ws or ws < d <
wmin, the layout is not decomposable.
A polynomial-time exact algorithm
for SADP decomposition
A polynomial-time exact algorithm
for SADP decomposition
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The ring of core may interact with other cores and
features.
A polynomial-time exact algorithm
for SADP decomposition
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Two two-coloring solutions for each component.
A polynomial-time exact algorithm
for SADP decomposition
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Combine the decompositions and form a complete
final decomposition.
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Two decompositions from two different components
are compatible if they can be merged as a
consistent overlay-free decomposition. Otherwise,
they are incompatible.
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Include exactly one decomposition from each
component to form a final decomposition and all
decompositions selected are mutually compatible.
A polynomial-time exact algorithm
for SADP decomposition
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Given decompositions of all components of Gs, the
final decomposition can be found in polynomial-time.
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Assume that there are k components in Gs, each with
at most two decompositions.
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Construct a boolean formula f.
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For component i, denote its two decompositions as x i and xi
If xi does not exist, include it in f as a disjunction.
Disjunct a clause (xi Λ xj) in f if xi is incompatible with xj
where i≠j.
A polynomial-time exact algorithm
for SADP decomposition
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Let g = f .
g is in 2-conjunctive normal form.
g is satisfiable if and only if there exists a final
decomposition we want.
2SAT problem can be solved in polynomial time.
Experimental results
Experimental results
Conclusion
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This paper proposed a graph formulation that
correctly models the SADP problem.
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They proposed the first polynomial-time exact
algorithm that solves the SADP problem.