U.S. patent application number 13/472598 was filed with the patent office on 2016-03-24 for method for designing topographic patterns for directing the formation of self-assembled domains at specified locations on substrates.
This patent application is currently assigned to INTERNATIONAL BUSINESS MACHINES CORPORATION. The applicant listed for this patent is Joy Cheng, Kafai Lai, Chi-Chun Liu, Jed W. Pitera, Charles T. Rettner. Invention is credited to Joy Cheng, Kafai Lai, Chi-Chun Liu, Jed W. Pitera, Charles T. Rettner.
Application Number | 20160085896 13/472598 |
Document ID | / |
Family ID | 49650548 |
Filed Date | 2016-03-24 |
United States Patent
Application |
20160085896 |
Kind Code |
A1 |
Cheng; Joy ; et al. |
March 24, 2016 |
METHOD FOR DESIGNING TOPOGRAPHIC PATTERNS FOR DIRECTING THE
FORMATION OF SELF-ASSEMBLED DOMAINS AT SPECIFIED LOCATIONS ON
SUBSTRATES
Abstract
Methods and computer program products for designing topographic
patterns for directing the formation of self-assembled domains at
specified locations on substrates. The methods include generating
mathematical models that operate on mathematical descriptions of
the number and locations of cylindrical self-assembled domains in a
mathematical description of a guiding pattern.
Inventors: |
Cheng; Joy; (San Jose,
CA) ; Lai; Kafai; (Poughkeepsie, NY) ; Liu;
Chi-Chun; (San Jose, CA) ; Pitera; Jed W.;
(Portola Valley, CA) ; Rettner; Charles T.; (San
Jose, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Cheng; Joy
Lai; Kafai
Liu; Chi-Chun
Pitera; Jed W.
Rettner; Charles T. |
San Jose
Poughkeepsie
San Jose
Portola Valley
San Jose |
CA
NY
CA
CA
CA |
US
US
US
US
US |
|
|
Assignee: |
INTERNATIONAL BUSINESS MACHINES
CORPORATION
Armonk
NY
|
Family ID: |
49650548 |
Appl. No.: |
13/472598 |
Filed: |
May 16, 2012 |
Current U.S.
Class: |
716/55 |
Current CPC
Class: |
G03F 7/0002 20130101;
G06F 2111/06 20200101; G06F 30/39 20200101 |
International
Class: |
G06F 17/50 20060101
G06F017/50 |
Goverment Interests
[0001] This invention was made with government support under
FA8650-10-C-7038 awarded by the Defense Advanced Research Projects
Agency (DARPA). The government has certain rights in the invention.
Claims
1. A method of designing a guiding pattern opening in a layer on a
substrate, the guiding pattern opening yielding a set of
self-assembled domains at specified locations within the guiding
pattern opening when the guided pattern opening is filled with a
self-assembly material that undergoes directed self-assembly, the
method comprising: (a) specifying a number and corresponding
locations of self-assembled domains; (b) generating a mathematical
description of an initial guiding pattern opening based on said
specified number and locations of self-assembled domains and
designating said initial guiding pattern opening as a current
guiding pattern opening; (c) using a computer, computing a
mathematical model to generate calculated numbers and calculated
high-probability locations of self-assembled domains within said
current guiding pattern opening, wherein said function represents
the relative probability that said self-assembled domains will form
at said specified locations within said current guiding pattern
opening; (d) comparing the calculated number of high-probability
locations of said mathematical model with said specified number of
self-assembled domains and comparing the calculated locations of
said high-probability locations with said specified locations of
self-assembled domains; (e) adjusting the current guiding pattern
opening based on said comparing of step (d); and (f) repeating
steps (c) to (e) until both (i) said calculated number of
self-assembled domains and said specified number of self-assembled
domains is the same and (ii) said high-probability locations of
self-assembled domains and said specified locations of
self-assembled domains agree within a specified range.
2. The method of claim 1, wherein said mathematical model is
approximated by a function comprising a first contribution that
represents the interaction between a self-assembled domain formed
at a particular location and said guiding pattern opening, and a
second contribution that represents the interaction between a
self-assembled domain formed at said particular location and all
other self-assembled domains within said current guiding pattern
opening.
3. The method of claim 2, including: dividing said current guiding
pattern opening into a set of geometric elements.
4. The method of claim 3, wherein said guiding pattern opening is
represented by a set of two-dimensional points, line segments,
curve segments or combinations thereof.
5. The method of claim 3, wherein said first interaction is a sum
of contributions corresponding to said geometric elements.
6. The method of claim 3, wherein said first contribution does not
include interactions of those geometric elements of the guiding
pattern opening that are separated from said self-assembled domain
location by another geometric element.
7. The method of claim 3, wherein said second contribution does not
include interactions of other self-assembled domains that are
separated from said self-assembled domain location by a geometric
element of the guiding pattern opening.
8. The method of claim 2, wherein said predicted self-assembled
domains are (i) modeled as cylinders oriented perpendicular to a
plane and (ii) represented within said mathematical model as a
point given by the intersection of the axis of each cylinder with
said plane.
9. The method of claim 2 wherein said second interaction is a sum
of the interaction of all different pairs of said self-assembled
domains.
10. The method of claim 2, wherein said mathematical model is
further approximated by a third contribution that represents
locations where self-assembled domains are specified not to form
within said guiding pattern opening.
11. The method of claim 10, wherein said third contribution
interaction is a sum of the interactions between (i) locations
where self-assembled domains are specified not to form and (ii)
said guiding pattern opening.
12. The method of claim 1, wherein said mathematical model is
represented by: F ( { x i , y i } , { x l , y l } ) .apprxeq. i = 1
N l = 1 M g ( ( x i - x l ) 2 + ( y i - y l ) 2 ) + i = 1 N - 1 j =
i + 1 N h ( ( x i - x j ) 2 + ( y i - y j ) 2 ) ##EQU00010## where:
F is the relative probability, expressed as an effective potential
acting on a self-assembled domain, of the formation of a set of
self-assembled domains within a guiding pattern shape, {x.sub.i,
y.sub.i} are locations of self-assembled domains, {x.sub.l,
y.sub.l} are the locations of the walls of a guiding pattern
opening, g is a function that describes the interaction of a
self-assembled domain modeled as a cylinder and the wall of the
guiding pattern opening, h is a function that describes the
interaction between pairs of self-assembled domains modeled as
cylinders, N is the number of self-assembled domains within the
guiding pattern opening, and M is the number of points defining the
wall locations of the guiding pattern opening.
13. The method of claim 1, wherein said mathematical model is
represented by: F ' ( { x i , y i } , { x l , y l } ) .apprxeq. i =
1 N l = 1 M g ( ( x i - x l ) 2 + ( y i - y l ) 2 ) + i = 1 N - 1 l
= i + 1 N h ( ( x i - x j ) 2 + ( y i - y j ) 2 ) - u = 1 L l = 1 M
g ( ( x u - x l ) 2 + ( y u - y l ) 2 ) ##EQU00011## where: F' is
the relative probability, expressed as an effective potential
acting on a self-assembled domain, of the formation of a set of
self-assembled domains within a guiding pattern shape, {x.sub.i,
y.sub.i} are locations of self-assembled domains, {x.sub.l,
y.sub.l} are the locations of the walls of a guiding pattern
opening, {x.sub.u, y.sub.u} are locations where self-assembled
domains are not to be formed, g is a function that describes the
interaction of a self-assembled domain modeled as a cylinder and
the wall of the guiding pattern opening, h is a function that
describes the interaction between pairs of self-assembled domains
modeled as cylinders, N is the number of self-assembled domains
within the guiding pattern opening, M is the number of points
defining the wall locations of the guiding pattern opening; and L
is the number of locations where self-assembled domains are not to
be formed.
14. A method of designing a guiding pattern opening in a layer on a
substrate, the guiding pattern opening yielding a set of
self-assembled domains at specified locations within the guiding
pattern opening when the guided pattern opening is filled with a
self-assembly material that undergoes directed self-assembly, the
method comprising: (a) specifying a number and corresponding
locations of self-assembled domains; (b) generating a mathematical
description of a guiding pattern opening based on said specified
number and locations of self-assembled domains; (c) defining a
mathematical model of the probability of the formation of
self-assembled domains based on said mathematical description of
said guiding pattern opening; (d) using a computer, using said
mathematical model to compute the probability that said number of
specified self-assembled domains will form at said specified
locations within the guiding pattern opening; (e) determining
derivatives of said mathematical model with respect to parameters
of said mathematical description of said guiding pattern opening;
and (f) maximizing the probability for formation of self-assembled
domains at said specified locations within said initial guiding
pattern opening using said derivatives to adjust said locations
defining the walls of said guiding pattern opening using an
optimization algorithm.
15. The method of claim 14, wherein said optimization algorithm is
an iterative optimization algorithm repeating steps (d) through (f)
until a probability of self-assembled domains forming at said
specified locations reaches a specified value.
16. The method of claim 14, herein said optimization algorithm
performs gradient descent iterations to reduce said derivatives to
specified values, said specified self-assembled domain locations
kept fixed during the optimization and wherein only (i) the
parameters of said mathematical description of said guiding pattern
opening and (ii) a step length of the gradient descent are allowed
to change.
17. The method of claim 16, wherein said derivatives of said
mathematical model used in said gradient descent iterations are
represented by: x l ( n + 1 ) = x l ( n ) - .gamma. .differential.
F ( n ) .differential. x l ##EQU00012## y l ( n + 1 ) = y l ( n ) -
.gamma. .differential. F ( n ) .differential. y l ##EQU00012.2##
where: n is the iteration step of the optimization, .gamma. is a
positive constant controlling the step length of the descent, F(n)
represents F({(x.sub.i,y.sub.i)}, {(x.sub.l(n),y.sub.l(n))}),
{x.sub.i, y.sub.i} are locations of self-assembled domains, and
{x.sub.l, y.sub.l} are the location of the walls of a guiding
pattern opening.
18. The method of claim 14, wherein said mathematical model is
approximated by a function comprising a first contribution that
represents the interaction between a self-assembled domain formed
at a particular location and said guiding pattern opening and a
second contribution that represents the interaction between a
self-assembled domain formed at said particular location and all
other self-assembled domains within said current guiding pattern
opening.
19. The method of claim 18, including: dividing said current
guiding pattern opening into a set of geometric elements.
20. The method of claim 19, wherein said guiding pattern opening is
represented by a set of two-dimensional points, line segments,
curve segments or combinations thereof.
21. The method of claim 19, wherein said first interaction is a sum
of contributions corresponding to said geometric elements.
22. The method of claim of claim 18, wherein said mathematical
model is further approximated by a third contribution that
represents magnitudes of derivatives of said first and said second
contributions with respect to said specified locations of said
self-assembled domains.
23. The method of claim 18, wherein said predicted self-assembled
domains are modeled as cylinders oriented perpendicular to a plane,
said mathematical model operating on a point given by the
intersection of the axis of each cylinder with said plane.
24. The method of claim 18, wherein said second interaction is a
sum of the interaction of all different pairs of said
self-assembled domains.
25. The method of claim 18, wherein said first contribution does
not include interactions with those geometric elements of the
guiding pattern opening that are separated from said self-assembled
domain location by at least one other geometric element.
26. The method of claim 18, wherein said second contribution does
not include other self-assembled domains that are separated from
said self-assembled domain location by a geometric element of the
guiding pattern opening.
27. The method of claim 18, wherein said mathematical model
includes a third contribution wherein: said third contribution
represents locations where self-assembled domains specified not to
form within said guiding pattern opening.
28. The method of claim 27, wherein said third contribution
interaction is a sum of the interactions between (i) locations
where self-assembled domains are specified not to form and (ii)
said guiding pattern opening.
29. The method of claim 14, wherein said mathematical model is
represented by: F ( { x i , y i } , { x l , y l } ) .apprxeq. i = 1
N l = 1 M g ( ( x i - x l ) 2 + ( y i - y l ) 2 ) + i = 1 N - 1 j =
i + 1 N h ( ( x i - x j ) 2 + ( y i - y j ) 2 ) ##EQU00013## where:
F is the relative probability, expressed as an effective potential
acting on a self-assembled domain, of the formation of a set of
self-assembled domains within a guiding pattern shape, {x.sub.i,
y.sub.i} are locations of self-assembled domains, {x.sub.l,
y.sub.l} are the locations of the walls of a guiding pattern
opening, g is a function that describes the interaction of a
self-assembled domain modeled as a cylinder and the wall of the
guiding pattern opening, h is a function that describes the
interaction between pairs of self-assembled domains modeled as
cylinders, N is the number of self-assembled domains within the
guiding pattern opening, and M is the number of points defining the
wall locations of the guiding pattern opening.
30. The method of claim 14, wherein said mathematical model is
represented by: F ' ( { x i , y i } , { x l , y l } , { x u , y u }
) .apprxeq. i = 1 N l = 1 M g ( ( x i - x l ) 2 + ( y i - y l ) 2 )
+ i = 1 N - 1 j = i + 1 N h ( ( x i - x j ) 2 + ( y i - y j ) 2 ) -
u = 1 L l = 1 M g ( ( x u - x l ) 2 + ( y u - y l ) 2 )
##EQU00014## where: F' is the relative probability, expressed as an
effective potential acting on a self-assembled domain, of the
formation of a set of self-assembled domains within a guiding
pattern shape, {x.sub.i, y.sub.i} are locations of self-assembled
domains, {x.sub.l, y.sub.l} are the locations of the walls of a
guiding pattern opening, {x.sub.u, y.sub.u} are locations where
self-assembled domains are not to be formed, g is a function that
describes the interaction of a self-assembled domain modeled as a
cylinder and the wall of the guiding pattern opening, h is a
function that describes the interaction between pairs of
self-assembled domains modeled as cylinders, N is the number of
self-assembled domains within the guiding pattern opening, M is the
number of points defining the wall locations of the guiding pattern
opening, and L is the number of locations where self-assembled
domains are not to be formed.
31. A non-transitory computer readable storage device having a
computer readable program code embodied therein, said computer
readable program code comprising an algorithm adapted to implement
a method for designing a guiding pattern opening in a layer on a
substrate, the guiding pattern opening yielding a set of
self-assembled domains at specified locations within the guiding
pattern opening when the guided pattern opening is filled with a
self-assembly material that undergoes directed self-assembly, said
method comprising the steps of: (a) generating a mathematical
description of an initial guiding pattern opening based on a
user-specified number of, and corresponding locations of,
self-assembled domains; (b) designating said initial guiding
pattern opening as a current guiding pattern opening; (c) computing
a mathematical model to generate calculated numbers and calculated
high-probability locations of self-assembled domains within said
current guiding pattern opening, wherein said function represents
the relative probability that said self-assembled domains will form
at said specified locations within said current guiding pattern
opening; (d) comparing the calculated number of high-probability
locations of said mathematical model with said specified number of
self-assembled domains and comparing the calculated locations of
said high-probability locations with said specified locations of
self-assembled domains; (e) adjusting the current guiding pattern
opening based on said comparing of step (d); and (f) repeating
steps (c) to (e) until both (i) said calculated number of
self-assembled domains and said specified number of self-assembled
domains is the same and (ii) said high-probability locations of
self-assembled domains and said specified locations of
self-assembled domains agree within a specified range.
32. A non-transitory computer readable storage device having a
computer readable program code embodied therein, said computer
readable program code comprising an algorithm adapted to implement
a method for designing a guiding pattern opening in a layer on a
substrate, the guiding pattern opening yielding a set of
self-assembled domains at specified locations within the guiding
pattern opening when the guided pattern opening is filled with a
self-assembly material that undergoes directed self-assembly, said
method comprising the steps of: (a) generating a mathematical
description of a guiding pattern opening based on a user-specified
number of, and corresponding locations of, self-assembled domains;
(b) defining a mathematical model of the probability of the
formation of self-assembled domains based on said mathematical
description of said guiding pattern opening; (c) using said
mathematical model to compute the probability that said number of
specified self-assembled domains will form at said specified
locations within the guiding pattern opening; (d) determining
derivatives of said mathematical model with respect to parameters
of said mathematical description of said guiding pattern opening;
and (e) maximizing the probability for formation of self-assembled
domains at said specified locations within said initial guiding
pattern opening using said derivatives to adjust said locations
defining the walls of said guiding pattern opening using an
optimization algorithm.
Description
FIELD OF THE INVENTION
[0002] The present invention relates to the field of integrated
circuit fabrication; more specifically, it relates to a method and
computer system for designing topographic patterns for directing
the formation of self-assembled domains at specified locations on
substrates.
BACKGROUND
[0003] Directed self-assembly (DSA), which combines self-assembling
materials and a lithographically defined prepattern on a photomask,
is a potential candidate to extend optical lithography. A
lithographically-defined guiding pattern serves to direct the
self-assembly process and the pattern formed by the self-assembling
materials. The resolution enhancement and self-healing effects of
DSA are particularly useful for extending the resolution of optical
lithography and to rectify the ill-defined patterns printed by
optical lithography. To take full advantage of DSA for extending
optical lithography requires modeling of the guiding pattern to be
used in the photomask. Current models and methods produce guiding
patterns that often cause the number and locations of DSA domains
to vary from the specified number and/or locations. Accordingly,
there exists a need in the art to mitigate the deficiencies and
limitations described hereinabove.
SUMMARY
[0004] A first aspect of the present invention is a method of
designing a guiding pattern opening in a layer on a substrate, the
guiding pattern opening yielding a set of self-assembled domains at
specified locations within the guiding pattern opening when the
guided pattern opening is filled with a self-assembly material that
undergoes directed self-assembly, the method comprising: (a)
specifying a number and corresponding locations of self-assembled
domains; (b) generating a mathematical description of an initial
guiding pattern opening based on the specified number and locations
of self-assembled domains and designating the initial guiding
pattern opening as a current guiding pattern opening; (c) using a
computer, computing a mathematical model to generate calculated
numbers and calculated high-probability locations of self-assembled
domains within the current guiding pattern opening, wherein the
function represents the relative probability that the
self-assembled domains will form at the specified locations within
the current guiding pattern opening; (d) comparing the calculated
number of high-probability locations of the mathematical model with
the specified number of self-assembled domains and comparing the
calculated locations of the high-probability locations with the
specified locations of self-assembled domains; (e) adjusting the
current guiding pattern opening based on the comparing of step (d);
and (f) repeating steps (c) to (e) until both (i) the calculated
number of self-assembled domains and the specified number of
self-assembled domains is the same and (ii) the high-probability
locations of self-assembled domains and the specified locations of
self-assembled domains agree within a specified range.
[0005] A second aspect of the present invention is a method of
designing a guiding pattern opening in a layer on a substrate, the
guiding pattern opening yielding a set of self-assembled domains at
specified locations within the guiding pattern opening when the
guided pattern opening is filled with a self-assembly material that
undergoes directed self-assembly, the method comprising: (a)
specifying a number and corresponding locations of self-assembled
domains; (b) generating a mathematical description of a guiding
pattern opening based on the specified number and locations of
self-assembled domains; (c) defining a mathematical model of the
probability of the formation of self-assembled domains based on the
mathematical description of the guiding pattern opening; (d) using
a computer, using the mathematical model to compute the probability
that the number of specified self-assembled domains will form at
the specified locations within the guiding pattern opening; (e)
determining derivatives of the mathematical model with respect to
parameters of the mathematical description of the guiding pattern
opening; and (f) maximizing the probability for formation of
self-assembled domains at the specified locations within the
initial guiding pattern opening using the derivatives to adjust the
locations defining the walls of the guiding pattern opening using
an optimization algorithm.
[0006] A third aspect of the present invention is a non-transitory
computer readable storage device having a computer readable program
code embodied therein, the computer readable program code
comprising an algorithm adapted to implement a method for designing
a guiding pattern opening in a layer on a substrate, the guiding
pattern opening yielding a set of self-assembled domains at
specified locations within the guiding pattern opening when the
guided pattern opening is filled with a self-assembly material that
undergoes directed self-assembly, the method comprising the steps
of: (a) generating a mathematical description of an initial guiding
pattern opening based on a user-specified number of, and
corresponding locations of, self-assembled domains; (b) designating
the initial guiding pattern opening as a current guiding pattern
opening; (c) computing a mathematical model to generate calculated
numbers and calculated high-probability locations of self-assembled
domains within the current guiding pattern opening, wherein the
function represents the relative probability that the
self-assembled domains will form at the specified locations within
the current guiding pattern opening; (d) comparing the calculated
number of high-probability locations of the mathematical model with
the specified number of self-assembled domains and comparing the
calculated locations of the high-probability locations with the
specified locations of self-assembled domains; (e) adjusting the
current guiding pattern opening based on the comparing of step (d);
and (f) repeating steps (c) to (e) until both (i) the calculated
number of self-assembled domains and the specified number of
self-assembled domains is the same and (ii) the high-probability
locations of self-assembled domains and the specified locations of
self-assembled domains agree within a specified range.
[0007] A fourth aspect of the present invention is a non-transitory
computer readable storage device having a computer readable program
code embodied therein, the computer readable program code
comprising an algorithm adapted to implement a method for designing
a guiding pattern opening in a layer on a substrate, the guiding
pattern opening yielding a set of self-assembled domains at
specified locations within the guiding pattern opening when the
guided pattern opening is filled with a self-assembly material that
undergoes directed self-assembly, the method comprising the steps
of: (a) generating a mathematical description of a guiding pattern
opening based on a user-specified number of, and corresponding
locations of, self-assembled domains; (b) defining a mathematical
model of the probability of the formation of self-assembled domains
based on the mathematical description of the guiding pattern
opening; (c) using the mathematical model to compute the
probability that the number of specified self-assembled domains
will form at the specified locations within the guiding pattern
opening; (d) determining derivatives of the mathematical model with
respect to parameters of the mathematical description of the
guiding pattern opening; and (e) maximizing the probability for
formation of self-assembled domains at the specified locations
within the initial guiding pattern opening using the derivatives to
adjust the locations defining the walls of the guiding pattern
opening using an optimization algorithm.
[0008] These and other aspects of the invention are described
below.
BRIEF DESCRIPTION OF THE DRAWINGS
[0009] The features of the invention are set forth in the appended
claims. The invention itself, however, will be best understood by
reference to the following detailed description of illustrative
embodiments when read in conjunction with the accompanying
drawings, wherein:
[0010] FIGS. 1A-1F are cross-sectional views illustrating an
exemplary method of forming a pattern in a substrate using directed
self-assembly;
[0011] FIGS. 2A-2D illustrate domain formation of a self-assembly
material in a guiding pattern;
[0012] FIGS. 3A and 3B illustrate Monte Carlo simulation results of
directed self-assembly in two different guiding patterns;
[0013] FIGS. 4A through 4D illustrate the directed self-assembly
domain probability function P(x,y) in two different guiding
patterns according to embodiments of the present invention;
[0014] FIGS. 5A and 5B illustrate the features of domain formation
of a self-assembly material in a guiding pattern;
[0015] FIG. 6 illustrates the cylinder indicator function
W(x,y,{r}) according to embodiments of the present invention;
[0016] FIG. 7 illustrates the relationship between the domain
probability function P(x,y) and the corresponding potential of mean
force F(x,y) according to embodiments of the present invention;
[0017] FIG. 8 illustrates the behavior of a self-assembled cylinder
location under the influence of the potential of mean force F(x,y)
according to embodiments of the present invention;
[0018] FIG. 9 is an abstract representation of directed
self-assembly in terms of cylinder locations and the guiding
pattern walls according to embodiments of the present
invention;
[0019] FIGS. 10A-10C illustrate the terms of equation (8) according
to embodiments of the present invention;
[0020] FIG. 11 is a flowchart of a forward guiding pattern design
method according to embodiments of the present invention;
[0021] FIGS. 12A-12D illustrate a gradient based optimization
algorithm based on derivatives of equations (9) and (10) according
to embodiments of the present invention;
[0022] FIGS. 13A-13C illustrate the differences between
optimization of the derivatives to a minima and optimization of the
derivatives to a zero;
[0023] FIG. 14 is a flowchart of an inverse guiding pattern design
method according to embodiments of the present invention; and
[0024] FIG. 15 is a schematic block diagram of a computer that may
be used in implementing preferred methods disclosed herein.
DETAILED DESCRIPTION
[0025] The semiconductor industry is constantly attempting to
manufacture smaller and smaller transistors in order to increase
the performance and decrease the cost of integrated circuits.
Current transistor feature sizes range from 65 nanometers (nm) down
to 22 nm. These sizes are at the limit of what can be patterned
directly by optical lithography.
[0026] The embodiments of the present invention relate to a
patterning technology known as directed self-assembly (DSA) which
can produce these smaller sized images needed in advanced
integrated circuit fabrication. Directed self-assembly is a hybrid
approach that uses optical, imprint or electron beam lithography to
produce a "guiding pattern" on a substrate which is then coated
with a self-assembling material. The self-assembling material then
undergoes self-assembly, the formation of regular well-defined
structures or domains whose size, shape, and arrangement are
defined by the chemical structure of the self-assembly material and
the geometry of the guiding pattern. The guiding patterns serve to
direct the self-assembled domains to form in particular locations
or orientations. The chemical difference between the domains in the
self-assembling material can then be exploited to transfer a
pattern down into the substrate by selectively dissolving one of
the domains and using the remaining domain(s) as an etch mask.
Preferred aspects of the present invention are methods of designing
guiding patterns for DSA.
[0027] FIGS. 1A-1F are cross-sectional views illustrating an
exemplary method of forming a pattern in a substrate using directed
self-assembly. In FIG. 1A, formed on substrate 100 is a patterned
layer 105 having an opening 107 in which a top surface 108 of
substrate 100 is exposed. In one example, patterned layer 105 is
formed using a photolithographic process and may comprise a
patterned photoresist layer. In one example, patterned layer 105 is
a hard mask formed using a patterned photoresist layer as a
template. In one example, patterned layer 105 may be formed using
imprint technology in which a three-dimensional pattern is pressed
into a polymer layer and the thinner regions are then removed to
expose the underlying substrate. Opening 107 is a guiding pattern
for DSA. Opening 107 has a width W1. In FIG. 1B, a DSA material 110
is formed in the opening 107. In FIG. 1C, DSA material 110 has
self-assembled into outer domains 110A and inner domains 110B. In
one example, heating the DSA material initiates the DSA process.
DSA materials and the self-assembly process are described in more
detail infra. In FIG. 1D, inner domains 110B (see FIG. 1C) are
removed, exposing substrate 100. In one example, the inner domains
are removed by dissolution in a solvent. In one example, the inner
domains are removed by plasma etching or reactive ion etching
(RIE). In FIG. 1E trenches 115 are etched into substrate 100. In
one example, trenches 115 are formed using a wet etch. In one
example, trenches 115 are formed using RIE. In FIG. 1F patterned
layer 105 and outer domains 110A (see FIG. 1E) are removed leaving
trenches 115 in substrate 100. Trenches 115 have a width W2, where
W2 is less than W1. In one example, substrate 100 is a
semiconductor substrate. In one example, substrate 100 represents a
layer (e.g., of oxide, nitride, polysilicon, other dielectric
material or metal) to be patterned on a semiconductor substrate or
on a non-semiconductor substrate.
[0028] FIGS. 2A-2D illustrate domain formation of a self-assembly
material in a guiding pattern. In FIGS. 2A-2D the inner DSA domains
are approximately cylindrical in shape while the outer domains take
the shape of the guiding pattern. FIG. 2A is an isometric view and
FIG. 2B is a top view of a guiding pattern. In FIGS. 2A and 2B, a
guiding pattern 120 is formed in a layer 125 formed on a substrate
130. A physical guiding pattern is an opening in a layer and is
three-dimensional. A design for a guiding pattern is
two-dimensional and defines the perimeter of the guiding pattern at
the surface (top or bottom) of layer 125. In the example of FIGS.
2A and 2B, guiding pattern 130 is in the form of two intersecting
cylinders. FIG. 2C is an isometric view and FIG. 2D is a top view
of DSA of a domain formed in a cylinder-forming DSA material. In
FIGS. 2C and 2D, two cylindrical inner domains 135A and 135B are
surrounded by an outer domain 140.
[0029] The hybrid character of DSA poses a challenge for the
computational tools currently used to design lithographic masks.
The structure to be formed by the first patterning step (e.g.,
opening 107 of FIG. 1A) is now the guiding pattern, not the final
on-chip structure (e.g., trenches 115 of FIG. 1F). The guiding
pattern has to be appropriately shaped in order for the
self-assembled domains produced in the second patterning step
(e.g., domains 110A and 110B of FIGS. 1C and 1D) to form the final
structure.
[0030] A key issue with the use of DSA in lithographic mask design
is the need for a model to predict the location and structure of
the self-assembled domains, since it is these domains that form the
final pattern of interest. Since self-assembly is a partially
stochastic process, with some degree of randomness, a model can
only predict the probable locations and structures of the
self-assembled domains. Some guiding pattern shapes will reliably
yield the same type of self-assembled domains at the same locations
within the guiding pattern. Other guiding pattern shapes can cause
the self-assembling material to behave in an unreliable fashion,
varying the number, shape, and location of the self-assembled
domains in a stochastic manner. An example of this stochastic
behavior is shown in FIG. 3.
[0031] FIGS. 3A and 3B illustrate Monte Carlo simulation results of
directed self-assembly in two different guiding patterns. In FIG.
3A, formation of two inner domains 145A and 145B in a "good"
guiding pattern 150 has been simulated eight times. Guiding pattern
150 is "good" because the domains 145A and 145B reliably produced
in the desired locations all eight times. In FIG. 3B, formation of
two inner domains 155A and 155B in a "bad" guiding pattern 160 has
been simulated eight times. Guiding pattern 160 is "bad" because
only in four of the eight simulations have the domains 155A and
155B reliably produced in the desired locations. Guiding pattern
160 has also generated a third domain 155C in the other four
simulations with roughly equal probability as domains 155A and
155B. While Monte Carlo simulations are accurate, they are orders
of magnitude too slow to be used in mask design where thousands of
guiding patterns would need to be simulated.
[0032] In order to accurately predict number and locations of inner
domains, it is necessary to know the probability of self-assembled
domain formation at each point (x,y) on the surface of the
substrate. This "DSA domain probability function" P(x,y) is
schematically illustrated in FIG. 4. If the high-probability
locations coincide with designed final pattern locations, we have a
"good" guiding pattern design. If the high-probability locations
deviate from designed final pattern (e.g., they occur in the wrong
positions, or there are more or fewer high-probability locations
than desired), then the guiding pattern is "bad."
[0033] FIGS. 4A through 4D illustrate the directed-self-assembly
domain probability function P(x,y) in two different guiding
patterns according to embodiments of the present invention. FIG. 4A
is a top view of a "good" guiding pattern 165. FIG. 4B is a top
view of the probability distribution for domain formation
represented by four probability regions 170A, 170B, 170C, and 170D
progressing from highest probability in region 170A to lowest
(non-zero) probability in region 170D. There are two
high-probability regions 170A. Within guiding pattern 165
P(x,y)>0. Without guiding pattern 165 P(x,y)=0. FIG. 4C is a top
view of a "bad" guiding pattern 175. FIG. 4D is a top view of the
probability distribution for domain formation represented by four
probability regions 180A, 180B, 180C and 180D progressing from
highest probability in region 180A to lowest (non-zero) probability
in region 180D. Within guiding pattern 175 P(x,y)>0. Without
guiding pattern 175 P(x,y)=0. There are three high-probability
regions 180A, when only two are wanted.
[0034] The most common self-assembling materials used in DSA are
block copolymers. Block copolymers are linear polymer chains whose
chemical composition changes along their length, with "blocks" of
different types of monomer. A common material made up of two
different monomers (a "di-block copolymer") is
polystyrene-block-polymethylmethacrylate (PS-b-PMMA). The chemical
differences between the monomers making up each block provide the
thermodynamic driving force for self-assembly. A small preference
for interactions between similar monomers (styrene with styrene,
methacrylate with methacrylate) is amplified by the large number of
monomers in the polymer to cause microphase separation into regions
composed almost entirely of one monomer or the other. The volume
fractions of each block in the final polymer control the shapes of
the self-assembled domains. A di-block copolymer made up of equal
volume fractions of its two monomers will adopt a lamellar
morphology, while decreasing the fraction of one block away from
0.5 produces a morphology with inner cylindrical domains of the
minority block surrounded by an outer domain of the majority block.
Further reductions in the volume fraction of the minority block
(below 0.33) produce a morphology of minority block spheres
embedded in the majority matrix. The overall molecular weight of
the polymer defines the size of each domain, with larger polymers
producing larger domains. With PS-b-PMMA (PMMA is the minority
block), heating to about 200.degree. C. initiates the self-assembly
process.
[0035] FIGS. 5A and 5B illustrate the features of domain formation
of a self-assembly material in a guiding pattern. In FIG. 5A, an
A-b-B polymer is placed in a guiding pattern 190 formed in a
patterned layer 192 on a substrate 195. There are three
interactions that describe the domain formation in FIG. 5B. The
first interaction is within the A-b-B polymer itself (i.e., the
attraction of A monomers to A monomers and B monomers to B
monomers). The second interaction is between the A-b-B polymer and
the material of the patterned layer 192. The third interaction is
between the A-b-B polymer and substrate 195.
[0036] To model the DSA process, the three interactions described
supra must be considered. During the self-assembly process, only
the self-assembling material moves or rearranges, so the substrate
and guiding pattern can be treated as rigid objects. If there is a
sufficient model for the interactions of all the components,
simulation techniques such as molecular dynamics (MD), Langevin
dynamics (LD), or Monte Carlo (MC) or other optimization techniques
such as self-consistent field theory (SCFT) can be used to find the
stable self-assembled structure adopted by the self-assembly
material in the presence of a particular guiding pattern shape and
substrate. All of these methods involve significant computational
expense. For example, a typical DSA scenario might involve a 50 nm
thick layer of PS-b-PMMA with a molecular weight of 100 kD
assembling in a guiding pattern of area 200.times.100 nm.sup.2.
This total volume of 10.sup.6 nm.sup.3 is occupied by
6.5.times.10.sup.3 polymers made up of .about.10.sup.8 atoms. Using
one of the above simulation techniques in this scenario would
require hundreds to thousands of computer processing unit
(CPU)-hours to produce an accurate estimate of the self-assembled
structure. Scaling this approach up to consider the .about.10.sup.9
transistors on a modern microprocessor would require an intractable
10.sup.11 CPU-hours.
[0037] To avoid intractable CPU time and still generate accurate
guiding patterns, the embodiments of the present invention disclose
a simplified model of the directed self-assembly process for the
case of cylinder-forming self-assembling polymers confined in a
relatively small guiding patterns of about 500 nm by about 500 nm.
While the simplified model is described for the case where the
self-assembled cylinders extend the full thickness of the
self-assembled material, it is also applicable to model the
behavior of partial cylindrical domains that do not extend through
the full thickness of the material.
[0038] An indicator function is a function whose value is 1 if a
condition is fulfilled and 0 if it is not. The indicator function
W(x,y,{r}) considers a cylindrical region of a specified radius
whose axis is aligned normal to the top surface of the substrate
and passes through the point (x,y). Mathematically, it is a
function of both the location (x,y) as well as the set of positions
of all the atoms in the self-assembling polymers ({r}). In
practice, the value of W is only influenced by the subset of atoms
of the self-assembling polymers that are within the cylindrical
region centered at (x,y). The cylinder radius is approximately the
same radius as the cylinders formed by the self-assembling polymer.
W(x,y,{r}) takes a value of 1 if this cylindrical region is
entirely filled with the minority component of the polymer (e.g., a
cylindrical self-assembled domain is present and centered on (x,y))
and 0 otherwise. The shape of the indicator function and the values
it takes on in different locations are illustrated in FIG. 6. An
alternative, but more complex indicator function, could include an
outer region or cylindrical shell which needs to be occupied by the
majority component in addition to the requirement for the inner
cylinder to be occupied by the minority component.
[0039] FIG. 6 illustrates the cylinder indication function
W(x,y,{r}) according to embodiments of the present invention. In
FIG. 6, a cylindrical inner domain 200 of radius "r" is located at
the position x,y and surrounded by an outer domain 205.
[0040] Using the framework of statistical mechanics, the
expectation value <W(x,y,{r})> can be calculated by
integrating over all possible configurations of the atoms of the
self-assembling material:
W ( x , y , { r } ) = .intg. V { r } W ( x , y , { r .fwdarw. } ) -
U ( { r } ) / kT .intg. V { r } - U ( { r } ) / kT ( 1 )
##EQU00001##
Where:
[0041] <W(x,y,{r})> is the probability for finding a
self-assembled cylinder centered at (x,y), averaged over all
possible values of {r},
[0042] {r} is the set of coordinates of all the atoms of the
self-assembling material, and
[0043] U({r}) is the energy associated with a particular
configuration of coordinates, and each coordinate is allowed to
vary throughout the full volume V.
[0044] This probability goes to zero for points (x,y) that lie
within the guiding pattern material rather than within the open
area of the guiding pattern. It will be maximized at positions
(x,y) that correspond to favorable environments for a
self-assembled cylinder. It is a probability rather than a
certainty because of the stochastic nature of the self-assembly
process. In some guiding patterns, it is possible for two or more
different arrangements of self-assembled cylinders to form. This
situation was illustrated in FIG. 3B.
[0045] Importantly, the function <W(x,y,{r})> is exactly the
"DSA domain probability function" P(x,y) we need from a predictive
DSA model so:
P(x,y)=W(x,y,{r}) (2)
[0046] The corresponding "potential of mean force" F(x,y) in
reduced units is defined as:
F(x,y)=-ln W(x,y,{r}) (3)
W(x,y,{r})=e.sup.-F(x,y) (4)
[0047] Or, in terms of P(x,y), making use of Equation (2):
F(x,y)=-ln P(x,y) (5)
P(x,y)=e.sup.-F(x,y) (6)
[0048] These relationships between P(x,y) and F(x,y) are
illustrated in FIG. 7. A "potential of mean force" is a term of art
in statistical mechanics defined as an effective potential energy
that is a function of one or more coordinates (x and y in this
case) and that reproduces the observed probability distribution(s)
in those coordinates when simulations are carried out using the
effective potential energy.
[0049] FIG. 7 illustrates the relationship between the domain
probability function P(x,y) and the corresponding potential of mean
force F(x,y) according to embodiments of the present invention. In
FIG. 7, there are two high-probability regions indicated by the
pairs of upper and lower cones. The axes of the P(x,y) function are
x, y and P where P is the probability of domain formation at
coordinates x and y. The axes of the F(x,y) function are x, y and F
where F is the free energy of domain formation at coordinates x and
y. P(x,y) and F(x,y) are related by equations (5) and (6).
[0050] F(x,y) is an effective potential acting on a self-assembled
cylinder centered at (x,y). A cylinder that happens to form at a
location with a large value of F will tend to move from that
position to one with a lower value of F. This movement could occur
by the gradual, coordinated movement of all of the polymer chains
forming the cylinder, or by the dissolution of the cylindrical
structure and its reformation in another location. Stable locations
for cylinders correspond to local minima in F, and the most
probable locations correspond to low-lying (or global) minima in F.
These properties are illustrated in FIG. 8.
[0051] FIG. 8 illustrates the behavior of a self-assembled cylinder
location under the influence of the potential of mean force F(x,y)
according to embodiments of the present invention. In FIG. 8, a
domain 210 formed between the two minima of F(x,y) where
df/dx.noteq.0 will migrate to the closest minima where df/dx=0 (to
the left in FIG. 8). Note F(x,y) is bounded by infinity where
f(x)=.infin., implying P(x,y)=0 so domain 210 cannot form in the
location marked by the "X."
[0052] While it is possible to use full three-dimensional
simulations as described earlier to calculate F(x,y) (or
<W(x,y,{r})>), such an approach would be very expensive since
different simulations would likely be required for each (x,y) value
of interest. Instead, an approximate two-dimensional form for F
that is simple and quick to calculate is used as illustrated in
FIG. 9. FIG. 9 is an abstract representation of directed
self-assembly in terms of cylinder locations and the guiding
pattern walls according to embodiments of the present invention.
Any given guiding pattern (e.g., guiding pattern 215), in top view,
can be abstracted as a set of open regions (places that will be
filled by self-assembly material) bounded by walls 220A and 220B of
the material composing the guiding pattern. The combined system of
guiding pattern and self-assembly material can be abstracted to a
set of N (i=1 . . . N) cylinder locations (x.sub.i,y.sub.i) (N=4 at
locations 225A, 225B, 225C and 225D in FIG. 9) within guiding
pattern 215 whose walls are defined by a set of M (l=1 . . . M)
points, the locations (x.sub.l,y.sub.l). This two-dimensional
abstraction of the DSA problem is the first key approximation DSA
model according to embodiments of the present invention. The
abstraction reduces the number of variables in the problem by many
orders of magnitude, greatly speeding any calculations needed.
[0053] Working with this abstraction, the fundamental approximation
is that F can be approximated as a sum of pairwise (or "two-body")
terms. By "pairwise" it is meant that each term in the sum is
dependent on the positions of only two entities in the abstraction
(i.e., two cylinders, or a cylinder and a wall point, but not a
cylinder and two wall points at the same time). This pairwise
approximation is the second key approximation the DSA model
according to embodiments of the present invention. The complete
equation for F, even in the context of the abstraction, contains
contributions from one-body, two-body, three-body and higher terms.
This "many-body expansion" is a standard tool for representing
energy functions in statistical mechanics; here it is applied to an
effective potential F instead. Part of the motivation for working
in terms of F rather than P is that it is straightforward to
decompose energy-like terms (i.e., F) in a many-body expansion, but
probabilities cannot be similarly decomposed into sums. One-body
terms can be ignored since they only contribute an additive
constant to F. Three-body and higher terms need to be discarded
since the computational cost scales as n.sup.m (where n is the
number of entities in the problem and m is the order of the term,
m=3 for a three-body term). The pairwise approximation of retaining
only the two-body terms strikes a balance between computational
time and accuracy. Equation 7 expresses F given the two key
approximations discussed above.
F ( { x i , y i } , { x l , y l } ) .apprxeq. i = 1 N l = 1 M g ( x
i , y i , x l , y l ) + i = 1 N - 1 j = i + 1 N h ( x i , y i , x j
, y j ) ( 7 ) ##EQU00002##
Where:
[0054] F is the relative probability, expressed as an effective
potential acting on a self-assembled domain, of the formation of a
set of self-assembled domains within a guiding pattern shape,
[0055] {x.sub.i, y.sub.i} are locations of self-assembled
domains,
[0056] {x.sub.l, y.sub.l} are the locations of the walls of a
guiding pattern opening,
[0057] g is a function that describes the interaction of a
self-assembled domain modeled as a cylinder and the wall of the
guiding pattern opening,
[0058] h is a function that describes the interaction between pairs
of self-assembled domains modeled as cylinders,
[0059] N is the number of self-assembled domains within the guiding
pattern opening, and
[0060] M is the number of points defining the wall locations of the
guiding pattern opening.
[0061] The first double sum in Equation (7) represents
contributions to F from interactions of the material composing each
cylinder with the material forming the walls of the guiding
pattern, and the second double sum represents interactions between
the material composing pairs of cylinders. These interactions are
illustrated schematically in FIG. 10A. In practice, the summed
functions g (the "cylinder-wall interaction") and h (the
"cylinder-cylinder" interaction) are only functions of the
distances between the interacting components:
F ( { x i , y i } , { x j , y j } ) .apprxeq. i = 1 N l = 1 M g ( (
x i - x l ) 2 + ( y i - y l ) 2 ) + i = 1 N - 1 j = i + 1 N h ( ( x
i - x j ) 2 + ( y i - y j ) 2 ) ( 8 ) ##EQU00003##
Where again:
[0062] F is the relative probability, expressed as an effective
potential acting on a self-assembled domain, of the formation of a
set of self-assembled domains within a guiding pattern shape,
[0063] {x.sub.i, y.sub.i} are locations of self-assembled
domains,
[0064] {x.sub.l, y.sub.l} are the locations of the walls of a
guiding pattern opening,
[0065] g is a function that describes the interaction of a
self-assembled domain modeled as a cylinder and the wall of the
guiding pattern opening,
[0066] h is a function that describes the interaction between pairs
of self-assembled domains modeled as cylinders,
[0067] N is the number of self-assembled domains within the guiding
pattern opening, and
[0068] M is the number of points defining the wall locations of the
guiding pattern opening.
[0069] Knowledge of the physics of self-assembling materials gives
some guidance to the functional forms of g and h. The dense,
disordered nature of the self-assembly material means that all
interactions are screened, decaying to zero at long distances. For
g, we know that cylinders cannot form close to wall elements, and
form readily at the center of circular guiding patterns that are
.about.10-20 times the cylinder diameter. In larger circular
guiding patterns, multiple cylinders form. For h, we know that
cylinders cannot overlap one another, and have a characteristic
separation distance d in the bulk. Examples of representative
functions for g and h are shown in FIG. 10 in graphical form. They
can be described mathematically as well. An important feature of
the approximation of F is that it is an explicit, and at least a
once (but ideally a twice), differentiable function of both the
cylinder locations {(x.sub.i,y.sub.i)} and the locations of the
walls of the guiding pattern {(x.sub.l,y.sub.l)}.
[0070] Another physical consideration that can be included in the
model is that the material forming the walls of the guiding pattern
is solid, and can therefore occlude or block interactions. This
means that the self-assembling domain at a particular location only
interacts with a wall element at another location if no other wall
elements lie in between them. For example, in FIG. 9,
self-assembled domain 225A can only interact with regions of wall
elements 220A not blocked by wall element 220B and can interact
only with regions of wall element 220B in direct lines from
self-assembled domain 225A. Similarly, two self-assembling domains
can be prevented from interacting if a wall element(s) lies between
them. For example, self-assembled domain 225A can interact with
self-assembled domains 225B and 225C but will not interact with
self-assembled domain 225D due to the intervening wall material of
walls 220B. This effect can be implemented in practice by
restricting the sums in equation (7) or (8) to only those
domain-wall element or domain-domain pairs that are not separated
by intervening wall elements.
[0071] FIGS. 10A-10C illustrate the terms of equation (8) according
to embodiments of the present invention. In FIG. 10A, domain A can
interact with all regions of wall W of guiding pattern GP that are
not blocked in a direct line by domain B. Likewise, domain B can
interact with all regions of wall W of guiding pattern GP that are
not blocked in a direct line by domain A. Domain A can also
interact with domain B. In FIG. 10B, the domain to guiding pattern
wall interaction g(r) is plotted as energy (where T is temperature
and k.sub.B is Boltzmann's constant) versus domain to wall
distance. Since FIG. 10A is symmetrical, FIG. 10B is the same for
both domain A and domain B. In FIG. 10C, the domain to domain
interaction h(r) is plotted as energy versus domain to domain
distance.
[0072] Equation (8) can be used to obtain a fast and accurate
approximation to F for use in designing guiding patterns in two
modes, a forward mode and an inverse mode described infra. In the
forward mode, the model is used to predict self-assembled domain
locations within a given guiding pattern shape. These locations can
then be compared to a desired target design, and the guiding
pattern shape iteratively modified and re-evaluated with the model
until the desired design is achieved.
[0073] FIG. 11 is a flowchart of a forward guiding pattern design
method according to embodiments of the present invention. In step
230, the number and locations of cylindrical domains are specified
(e.g., the inner domains 135A and 135B of FIG. 2C). In step 235, an
initial guiding pattern is generated. This becomes the current
guiding pattern. In step 240, the self-assembly domain probability
model (e.g., equation 8 or 12) is defined and in step 245, the
probability model is used to model the number and locations (minima
of F) of cylindrical domains using the current guiding pattern. In
step 250, the model results with respect to number and locations of
cylindrical domains are compared to the specified number and
locations of cylindrical domains. If the number of modeled domains
is the same as the specified number of domains and the locations
are within a specified range from the specified locations, the
method terminates, otherwise the method proceeds to step 260. In
step 260, the current guiding pattern is modified and the method
loops back to step 245. Optionally, in step 255, additional filters
based on the properties of F at each minimum can also be used. For
example, a threshold value of F may be required, ensuring a high
probability of self-assembled cylinder formation at each minimum. F
can be required to have high curvature at each minimum, implying
that the location is well-specified. Minima with low curvature in F
correspond to broad troughs with many possible cylinder locations
of nearly equal probability. Testing of this approach by comparison
of the cylinder locations predicted by F with those observed
experimentally in a given guiding pattern shape shows a predictive
root-mean-square accuracy of 1-2 nm in domain location.
[0074] If equation (8) is differentiable (the derivative of F with
regards to cylinder or wall locations can be calculated) it is
possible to instead operate in an inverse mode. In an inverse mode,
the number and locations of cylindrical domains are specified, and
the guiding pattern shape is optimized directly based on values of
the derivatives of F. This produces the ideal guiding pattern shape
for a given pattern without the iterative cycles of comparing and
modifying of the forward model.
[0075] In an inverse mode, an initial guiding pattern shape
{(x.sub.l(0),y.sub.l(0))} (0 indicates the initial or 0.sup.th step
of optimization) is constructed from specified domain locations
{(x.sub.i,y.sub.i)} and the guiding pattern shape is optimized to
minimize F({(x.sub.i,y.sub.i)}, {(x.sub.l,y.sub.l)}). By minimizing
F({(x.sub.i,y.sub.i)}, {(x.sub.l,y.sub.l)}), the probability of
finding self-assembled cylinders at the specified domain locations
is maximized. The partial derivatives of F in terms of the x.sub.l
and y.sub.l variables describing the guiding pattern shape, i.e.,
the sets {.differential.F/.differential.x.sub.l} and
{.differential.F/.differential.y.sub.l}, provide information about
how F will change with changes in the guiding pattern shape. If the
partial derivative .differential.F/.differential.x.sub.0 is
positive, F will increase if x.sub.0 is increased and decrease if
it is decreased. If a particular partial derivative is zero, F will
be unaffected by small changes in that variable. At a local
extremum (minimum or maximum) of F all partial derivatives are
zero. Given that the partial derivatives of F are available, a
simple gradient-based optimization (or more sophisticated
optimization algorithms) to find a guiding pattern shape that
minimizes F can be used. Gradient-based optimization is an
iterative algorithm that updates the optimization variables by
small increments in the direction opposite of the corresponding
partial derivative:
x l ( n + 1 ) = x l ( n ) - .gamma. .differential. F ( n )
.differential. x l ( 9 ) y l ( n + 1 ) = y l ( n ) - .gamma.
.differential. F ( n ) .differential. y l ( 10 ) ##EQU00004##
Where:
[0076] n is the iteration step of the optimization,
[0077] .gamma. is a positive constant controlling the step length
of the descent,
[0078] F(n) represents F({(x.sub.i,y.sub.i)},
{(x.sub.l(n),y.sub.l(n))}),
[0079] {x.sub.i, y.sub.i} are locations of self-assembled domains,
and
[0080] {x.sub.l, y.sub.l} are the location of the walls of a
guiding pattern openings.
[0081] The step length .gamma. has an initial large value but if at
any point F(n+1)>F(n), iteration n+1 is rejected, .gamma. is
decreased and a new set of {(x.sub.l(n+1),y.sub.l(n+1))} guiding
pattern coordinates is generated. Gradient descent iterations are
continued until a minimum in F is found. It is important to note
that the cylinder locations {(x.sub.i,y.sub.i)} are kept fixed
during this optimization; only the points describing the guiding
pattern shape {(x.sub.l,y.sub.l)} are allowed to vary. The use of
this approach to optimize guiding pattern shapes is shown
schematically in FIG. 12. Alternative optimization schemes could be
used similarly.
[0082] FIGS. 12A-12D illustrate a gradient based optimization
algorithm based on derivatives of equation (8) or (12) according to
embodiments of the present invention. In FIG. 12A, an initial
guiding pattern 265 and specified domain locations 270A and 270B
are specified. In FIG. 12B, F(0) is, for example, -20 and the
direction and length of the arrows indicates the direction and
magnitude of adjustments to be made to guiding pattern 265 using
equations (9) and (10) based on
.differential.F/.differential.x.sub.i and
.differential.F/.differential.y.sub.i. In FIG. 12C, after the
optimization called for in FIG. 12B, there are still some
adjustments to be made as indicated by the arrows, but F(1) has
been reduced, for example, to -50. In FIG. 12D, after the
optimization called for in FIG. 12C, there are no more adjustments
to be made as indicated by the lack of arrows, and F(2) has been
reduced, for example, to -80.
[0083] Simply using F as the objective function will find the
guiding pattern shape most likely to yield self-assembled cylinders
at the desired locations, but does not guarantee that these desired
locations correspond to local minima of F. In mathematical
language, the optimization ensures that the partial derivatives of
F in all of the guiding pattern shape coordinates are zero, but
does not guarantee that the partial derivatives of F in the
self-assembled cylinder locations are also zero. This means that it
may be possible to move the self-assembled cylinder locations
within the optimized guiding shape to achieve an even lower value
of F, at the cost of no longer satisfying the target design. To
simultaneously optimize for this second condition as well, the
method of Lagrange multipliers can be used. The method of Lagrange
multipliers is a mathematical technique for optimizing a function
(F, in this case) subject to additional constraints. A new function
is established consisting of the original function plus one term
for each constraint which is the product of a new variable, the
Lagrange multiplier, and a term that is zero when the constraint is
satisfied.
[0084] To simultaneously minimize F and ensure that the partial
derivatives of F in terms of the self-assembled cylinder locations
{(x.sub.i,y.sub.i)} are zero, we introduce a new objective function
G:
G = F + i = 1 N ( .lamda. i ( .differential. F .differential. x i )
2 + .mu. i ( .differential. F .differential. y i ) 2 ) ( 11 )
##EQU00005##
Where:
[0085] .lamda..sub.i and .mu..sub.i are the Lagrange multipliers
(>0) for the x- and y-components of the derivative of F at
cylinder location i.
[0086] Minimizing G requires both minimizing F and satisfying the
condition that these derivatives go to zero. Since G now contains
terms in {.differential.F/.differential.x.sub.i} and
{.differential.F/.differential.y.sub.i)}, the partial derivatives
of G in terms of the guiding pattern shape
{.differential.G/.differential.x.sub.l} and
{.differential.G/.differential.y.sub.l} will themselves contain
contributions of the form
.differential.({.differential.F/.differential.x.sub.i})/.differential.x.s-
ub.l. These second derivative contributions reflect how changing
the guiding shape at a particular point changes the derivative of F
at a particular cylinder location, and are the reason why ideal
forms of F are twice differentiable. Other forms of constraint
(e.g., constraints on higher derivatives of F or constraints
reflecting lithographic design rules) can be introduced similarly.
The difference between optimizing F and optimizing G is illustrated
in FIG. 13.
[0087] FIGS. 13A-13C illustrate the differences between
optimization of the derivatives to a minima and optimization of the
derivatives to a zero. In FIG. 13A, a guiding pattern 275 and two
specified domain locations 280A and 280B are illustrated. The
partial derivatives of F in terms of these domain locations
(.differential.F/.differential.x.sub.i,
.differential.F/.differential.y.sub.i) are shown as arrows. In FIG.
13B, optimizing for F only (so that
-.differential.F/.differential.x.sub.i and
-.differential.F/.differential.y.sub.i are not both zero) results
in the actual domain locations 285A and 285B which are offset from
respective specified locations 280A and 280B. In FIG. 13C G is
optimized instead of F (-.differential.F/.differential.x.sub.i and
-.differential.F/.differential.y.sub.i are both zero) so actual
domains 290A and 290B are in the same locations as respective
specified locations 280A and 280B. Optimizing G finds the minimum
of F that also ensures the self-assembled cylinder locations are
stable and do not move during the DSA process.
[0088] Most mask designs, while they explicitly specify desired
self-assembled cylinder domain locations, also implicitly specify
undesired self-assembled cylinder locations (i.e., anywhere a
self-assembled cylinder is not requested). This can be treated in
an inverse model by adding a new set of L "undesired locations"
{(x.sub.u,y.sub.u)} and optimizing a modified F';
F ' ( { x i , y i } , { x l , y l } , { x u , y u } ) .apprxeq. i =
1 N l = 1 M g ( x i , y i , x l , y l ) + i = 1 N - 1 j = i + 1 N h
( x i , y i , x j , y l ) - u = 1 L l = 1 M g ( x u , y u , x l , y
l ) ( 12 ) ##EQU00006##
Where:
[0089] F' is the relative probability, expressed as an effective
potential acting on a self-assembled domain, of the formation of a
set of self-assembled domains within a guiding pattern shape,
[0090] {x.sub.i, y.sub.i} are locations of self-assembled
domains,
[0091] {x.sub.l, y.sub.l} are the locations of the walls of a
guiding pattern opening,
[0092] {x.sub.u, y.sub.u} are locations where self-assembled
domains are not to be formed,
[0093] g is a function that describes the interaction of a
self-assembled domain modeled as a cylinder and the wall of the
guiding pattern opening,
[0094] h is a function that describes the interaction between pairs
of self-assembled domains modeled as cylinders,
[0095] N is the number of self-assembled domains within the guiding
pattern opening,
[0096] M is the number of points defining the wall locations of the
guiding pattern opening; and
[0097] L is the number of locations where self-assembled domains
are not to be formed.
[0098] The corresponding form of equation (12) that depends only on
distances rather than locations is:
F ' ( { x i , y i } , { x j , y j } ) .apprxeq. i = 1 N l = 1 M g (
( x i - x l ) 2 + ( y i - y l ) 2 ) + i = 1 N - 1 j = i + 1 N h ( (
x i - x j ) 2 + ( y i - y j ) 2 ) - u = 1 L l = 1 M g ( ( x u - x l
) 2 + ( y u - y l ) 2 ) ( 13 ) ##EQU00007##
[0099] Again, F' can be directly optimized by gradient-based
optimization in analogy to Equations 9 and 10:
x l ( n + 1 ) = x l ( n ) - .gamma. .differential. F ' ( n )
.differential. x l ( 14 ) y l ( n + 1 ) = y l ( n ) - .gamma.
.differential. F ' ( n ) .differential. y l ( 15 ) ##EQU00008##
[0100] If desired, it is also possible to use the Lagrange
multiplier method to additionally ensure that the optimization of
F' also ensures that the target via locations correspond to local
minima of F', by constructing a new target function G' in the same
manner as equation 12:
G ' = F ' + i = 1 N ( .lamda. i ( .differential. F ' .differential.
x i ) 2 + .mu. i ( .differential. F ' .differential. y i ) 2 ) ( 16
) ##EQU00009##
[0101] Optimizing G' by numerical approaches will yield a guiding
pattern shape that: (i) maximizes the probabilities of
self-assembled domains form at the desired locations; (ii)
minimizes the probabilities of self-assembled domains forming at
the undesired locations; and (iii) ensures that the target via
locations correspond to local minima in F' (local maxima in the
probability). If there are other known properties of the desired
solution (e.g., smoothness or size of the guiding pattern shape)
they can be included in the optimization by adding additional terms
to F' or G' in a similar fashion.
[0102] FIG. 14 is a flowchart of the inverse guiding pattern design
method according to embodiments of the present invention. In step
300, the number and locations of cylindrical domains is specified
(e.g., the inner domains 135A and 135B of FIG. 2C). In step 305, a
guiding pattern is generated. In step 310, the self-assembly domain
probability model (e.g., equation 8 or 12) is defined and in step
315, the probability model is used to model the number and
locations of cylindrical domains using the guiding pattern. In step
320, the partial derivatives of the model are determined and in
step 325, the probability of formation of the specified number of
cylindrical domains at specified locations is maximized using an
optimization algorithm based on the partial derivatives of the
model. In one example, the optimization algorithm is a gradient
descent algorithm as described supra.
[0103] As will be appreciated by one skilled in the art, aspects of
the present invention may be embodied as a system, method or
computer program product. Accordingly, aspects of the present
invention may take the form of an entirely hardware embodiment, an
entirely software embodiment (including firmware, resident
software, micro-code, etc.) or an embodiment combining software and
hardware aspects that may all generally be referred to herein as a
"circuit," "module" or "system." Furthermore, aspects of the
present invention may take the form of a computer program product
embodied in one or more computer readable medium(s) having computer
readable program code embodied thereon.
[0104] Any combination of one or more computer readable medium(s)
may be utilized. In one example, the computer readable medium is a
computer readable storage medium. A computer readable storage
medium may be, for example, but not limited to, an electronic,
magnetic, optical, electromagnetic, infrared, or semiconductor
system, apparatus, or device, or any suitable combination of the
foregoing. More specific examples (a non-exhaustive list) of the
computer readable storage medium would include the following: an
electrical connection having one or more wires, a portable computer
diskette, a hard disk, a random access memory (RAM), a read-only
memory (ROM), an erasable programmable read-only memory (EPROM or
Flash memory), an optical fiber, a portable compact disc read-only
memory (CD-ROM), an optical storage device, a magnetic storage
device, or any suitable combination of the foregoing. In the
context of this document, a computer readable storage medium may be
any tangible medium that can contain, or store a program for use by
or in connection with an instruction execution system, apparatus,
or device.
[0105] In one example, the computer readable medium is a computer
readable signal medium. A computer readable signal medium may
include a propagated data signal with computer readable program
code embodied therein, for example, in baseband or as part of a
carrier wave. Such a propagated signal may take any of a variety of
forms, including, but not limited to, electro-magnetic, optical, or
any suitable combination thereof. A computer readable signal medium
may be any computer readable medium that is not a computer readable
storage medium and that can communicate, propagate, or transport a
program for use by or in connection with an instruction execution
system, apparatus, or device.
[0106] In one example, program code embodied on a computer readable
medium may be transmitted using any appropriate medium, including
but not limited to wireless, wireline, optical fiber cable, RF,
etc., or any suitable combination of the foregoing.
[0107] Computer program code for carrying out operations for
aspects of the present invention may be written in any combination
of one or more programming languages, including an object oriented
programming language such as Java, Smalltalk, C++ or the like and
conventional procedural programming languages, such as the "C"
programming language or similar programming languages. The program
code may execute entirely on the user's computer, partly on the
user's computer, as a stand-alone software package, partly on the
user's computer and partly on a remote computer or entirely on the
remote computer or server. In the latter scenario, the remote
computer may be connected to the user's computer through any type
of network, including a local area network (LAN) or a wide area
network (WAN), or the connection may be made to an external
computer (for example, through the Internet using an Internet
Service Provider).
[0108] Aspects of the present invention are described below with
reference to flowchart illustrations and/or block diagrams of
methods, apparatus (systems) and computer program products
according to embodiments of the invention. It will be understood
that each block of the flowchart illustrations and/or block
diagrams, and combinations of blocks in the flowchart illustrations
and/or block diagrams, can be implemented by computer program
instructions. These computer program instructions may be provided
to a processor of a general purpose computer, special purpose
computer, or other programmable data processing apparatus to
produce a machine, such that the instructions, which execute via
the processor of the computer or other programmable data processing
apparatus, create means for implementing the functions/acts
specified in the flowchart and/or block diagram block or
blocks.
[0109] These computer program instructions may also be stored in a
computer readable medium that can direct a computer, other
programmable data processing apparatus, or other devices to
function in a particular manner, such that the instructions stored
in the computer readable medium produce an article of manufacture
including instructions which implement the function/act specified
in the flowchart and/or block diagram block or blocks.
[0110] The computer program instructions may also be loaded onto a
computer, other programmable data processing apparatus, or other
devices to cause a series of operational steps to be performed on
the computer, other programmable apparatus or other devices to
produce a computer implemented process such that the instructions
which execute on the computer or other programmable apparatus
provide processes for implementing the functions/acts specified in
the flowchart and/or block diagram block or blocks.
[0111] Generally, the method described herein with respect to
methods for designing topographic patterns for directing the
formation of self-assembly domains at specified locations on
substrates is practiced with a general-purpose computer and the
methods described supra in the flow diagrams of FIGS. 11 and 14 may
be coded as a set of instructions on removable or hard media for
use by the general-purpose computer.
[0112] FIG. 15 is a schematic block diagram of a computer that may
be used in implementing preferred methods disclosed herein. In FIG.
15, computer system 400 has at least one microprocessor or central
processing unit (CPU) 405. CPU 405 is interconnected via a system
bus 410 to a random access memory (RAM) 415, a read-only memory
(ROM) 420, an input/output (I/O) adapter 425 for connecting a
removable data and/or program storage device 430 and a mass data
and/or program storage device 435, a user interface adapter 440 for
connecting a keyboard 445 and a mouse 450, a port adapter 455 for
connecting a data port 460 and a display adapter 465 for connecting
a display device 470.
[0113] ROM 420 contains the basic operating system for computer
system 400. The operating system may alternatively reside in RAM
415 or elsewhere as is known in the art. Examples of removable data
and/or program storage device 430 include magnetic media such as
floppy drives and tape drives and optical media such as CD ROM
drives. Examples of mass data and/or program storage device 435
include electronic, magnetic, optical, electromagnetic, infrared,
and semiconductor devices. Examples of a computer-readable medium
include a semiconductor or solid state memory, magnetic tape, a
removable computer diskette, a random access memory (RAM), a
read-only memory (ROM), a rigid magnetic disk and an optical disk.
Current examples of optical disks include compact disk-read only
memory (CD-ROM), compact disk-read/write (CD-R/W) and DVD. In
addition to keyboard 445 and mouse 450, other user input devices
such as trackballs, writing tablets, pressure pads, microphones,
light pens and position-sensing screen displays may be connected to
user interface 440. Examples of display devices include cathode-ray
tubes (CRT) and liquid crystal displays (LCD).
[0114] A computer program with an appropriate application interface
may be created by one of skill in the art and stored on the system
or a data and/or program storage device to simplify the practicing
of this invention. In operation, information for the computer
program created to run the present invention is loaded on the
appropriate removable data and/or program storage device 430, fed
through data port 460 or typed in using keyboard 445.
[0115] Thus the embodiments of the present invention provide
methods and computer program products for designing topographic
patterns (i.e., guiding patterns) for directing the formation of
self-assembly domains at specified locations on substrates which a
very high probability that that the number of actual domains will
be the same as the specified number of domains and that they will
form in the specified locations.
[0116] The descriptions of the various embodiments of the present
invention have been presented for purposes of illustration, but are
not intended to be exhaustive or limited to the embodiments
disclosed. Many modifications and variations will be apparent to
those of ordinary skill in the art without departing from the scope
and spirit of the described embodiments. The terminology used
herein was chosen to best explain the principles of the
embodiments, the practical application or technical improvement
over technologies found in the marketplace, or to enable others of
ordinary skill in the art to understand the embodiments disclosed
herein.
* * * * *