U.S. patent application number 15/049838 was filed with the patent office on 2016-06-16 for lithography mask functional optimization and spatial frequency analysis.
The applicant listed for this patent is D2S, Inc.. Invention is credited to P. Jeffrey Ungar.
Application Number | 20160171142 15/049838 |
Document ID | / |
Family ID | 42139485 |
Filed Date | 2016-06-16 |
United States Patent
Application |
20160171142 |
Kind Code |
A1 |
Ungar; P. Jeffrey |
June 16, 2016 |
LITHOGRAPHY MASK FUNCTIONAL OPTIMIZATION AND SPATIAL FREQUENCY
ANALYSIS
Abstract
In an electronic design automation technique for optical
proximity correction, a mask is represented by a function with an
exact analytical form over a mask region. Using the physics of
optical projection, a solution based on a spatial frequency
analysis is determined. Spatial frequencies above a cutoff are
determined by the optical system do not contribute to the projected
image. Spatial frequencies below this cutoff affect the print (and
the mask), while those above the cutoff only affect the mask.
Frequency components in the function below this cutoff frequency
may be removed, which will help to reduce computational
complexity.
Inventors: |
Ungar; P. Jeffrey;
(Sunnyvale, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
D2S, Inc. |
San Jose |
CA |
US |
|
|
Family ID: |
42139485 |
Appl. No.: |
15/049838 |
Filed: |
February 22, 2016 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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14257865 |
Apr 21, 2014 |
9268900 |
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15049838 |
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13686682 |
Nov 27, 2012 |
8707222 |
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14257865 |
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12973686 |
Dec 20, 2010 |
8321819 |
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13686682 |
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11864343 |
Sep 28, 2007 |
7856612 |
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12973686 |
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60827295 |
Sep 28, 2006 |
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Current U.S.
Class: |
716/54 |
Current CPC
Class: |
G06F 30/39 20200101;
G21K 5/00 20130101; G03F 1/00 20130101; G03F 1/70 20130101; G03F
1/28 20130101; G06F 30/398 20200101; G06F 2111/10 20200101; G06F
2119/18 20200101; G03F 1/50 20130101; G06F 17/14 20130101; G06F
30/00 20200101 |
International
Class: |
G06F 17/50 20060101
G06F017/50 |
Claims
1. A system comprising: at least one electronic processor coupled
to a computer-readable medium, wherein the computer-readable medium
comprises computer-executable code that causes the electronic
processor to perform the following: providing a target mask,
wherein the target mask is in a spatial domain; performing a
frequency domain transformation on the target mask to obtain a
first mask, wherein the first mask is in a frequency domain; and
computing a first cost function for a first mask to obtain a first
value.
2. The system of claim 1 wherein the computer-executable code
comprises: altering the first mask in a frequency domain to obtain
a second mask, wherein the second mask is in the frequency domain;
computing the first cost function for the second mask to obtain a
second value; and repeating the altering the first mask and the
computing the first cost function for the second mask until the
second value is less than the first value, wherein the performing a
frequency domain transformation, computing a first cost function,
and altering the first mask are performed using at least one
electronic processor.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This patent application is a continuation of U.S. patent
application Ser. No. 13/686,682, filed Nov. 27, 2012, issued as
U.S. Pat. No. 8,707,222 on Apr. 22, 2014, which is a continuation
of U.S. patent application Ser. No. 12/973,686, filed Dec. 20,
2010, issued as U.S. Pat. No. 8,321,819 on Nov. 27, 2012, which is
a divisional of U.S. patent application Ser. No. 11/864,343, filed
Sep. 28, 2007, issued as U.S. Pat. No. 7,856,612 on Dec. 21, 2010,
which claims the benefit of U.S. provisional patent application
60/827,295, filed Sep. 28, 2006. These applications are
incorporated by reference along with all other references cited in
this application.
BACKGROUND OF THE INVENTION
[0002] This invention relates to the field of electronic design
automation, and in particular, the optimization of masks used for
photolithographic printing of circuit designs, known in the field
as optical proximity correction.
[0003] Production-scale chip manufacturing uses photolithographic
techniques to build up layers of materials on a wafer to create the
transistors, wires, and so on, that realize a chip design. The
sizes of the features to be printed on the wafer are approaching
the limits set by the wavelength of the light, the optical
projection system, and the behavior of the light sensitive
materials used, among many other factors.
[0004] Diffraction effects from the wavelength of the light source
and the limits of the projection optics motivated the development
of optical proximity correction (OPC) techniques to adjust the
shapes on the mask to print more like the desired result on the
wafer. For example, a square may have serifs added to its corners
to compensate for excessive rounding of the corners of the printed
feature, or the ends of a rectangle may have "hammerheads" added to
further ensure the printed feature reaches the desired line
end.
[0005] The first OPC methods were based on simple rules for
modifying the shapes on the mask, but as the technology was pushed
closer to optical resolution limits, model-based optimization,
which adjusts the features on the mask to improve the calculated
printed image, was developed. Two significant advantages of
model-based OPC are the ability to account for proximity effects
(warping a nearby feature will affect how a given feature prints)
and to accommodate photoresist behavior.
[0006] Sometimes, features are found to print with greater fidelity
if extra features are added to the mask that are too small to print
themselves, but nevertheless favorably affect the way nearby main
features print, especially over a range of process conditions.
Introduction of these so-called subresolution assist features
(SRAFs) is still generally done according to preset rules.
Typically they are inserted first and held fixed as OPC is applied
to the main features on the mask.
[0007] There are significant problems in applying these methods as
the industry moves to ever smaller on-wafer dimensions. The rules
used to insert SRAFs are becoming more complex and less reliable.
The standard OPC methods do not have the flexibility needed to
achieve the best results and require post-OPC verification and
manual intervention.
[0008] What is needed is a practical model-based method for mask
design that can automatically determine a mask that both satisfies
mask manufacturing and verification criteria, and produces the
desired print on the wafer over a range of process conditions, such
as exposure and focus variation. Such a method will generally
result in a mask that warps existing layout geometry and adds or
subtracts SRAFs from anywhere, including in ways that split or join
the layout features.
BRIEF SUMMARY OF THE INVENTION
[0009] Model-based OPC and more general "inverse lithography"
methods are both iterative optimization algorithms that adjust
parameters defining the mask until the predicted print is within
acceptable tolerances for a set or range of conditions. They differ
chiefly in how the mask is represented, which is typically as
simple geometry for OPC, and as pixels comprising an image of
transmission values for "inverse lithography".
[0010] Unless the mask representation automatically satisfies all
desired mask constraints and characteristics, such as allowed
transmission values or minimum feature size, the formula measuring
a mask's suitability will introduce terms that add a cost related
to the violation of these constraints. In field of "inverse
problems", introducing these terms is known as "regularization",
and is a means of selecting a solution from a potentially infinite
set of solutions that fits the desired outcome equally or similarly
well but also has other a priori desirable properties.
[0011] This invention advantageously and most generally represents
the mask as a function with an exact analytical form over the mask
region. Furthermore, the present invention uses the physics of
optical projection to design the solution based on a spatial
frequency analysis. From a physics perspective, spatial frequencies
above a cutoff determined by the optical system do not contribute
to the projected image. Spatial frequencies below this cutoff
affect the print (and the mask), while those above the cutoff only
affect the mask.
[0012] In some embodiments of the present invention, the mask
function is expressed in a Fourier basis, which exposes perfectly
the separation between the low frequencies that affect the print
and the high frequencies that only adjust the mask. In other
embodiments, different basis sets, such as wavelets, may be used
that expose this frequency separation to a lesser degree.
[0013] The present invention expresses the desirability of a given
mask function as a functional (a mathematical form that is a
function of a function over a defined region) that calculates the
"cost" with respect to the predicted printed image and possibly
various a priori mask constraints. Since the present invention
expresses the mask with an exact analytical form, the cost and its
functional derivatives can be evaluated analytically and
consistently, which enables the use of efficient cost minimization
algorithms.
[0014] In one embodiment, the invention first determines a
band-limited continuous-tone mask function that is clamped through
regularization to the minimum and maximum allowable mask values,
and then proceeds to determine a mask quantized to the allowable
values that is consistent with this mask through further
regularization of the solution. The frequency cutoff used may be
higher than the optical cutoff to accommodate clamping and
quantization and may change as the optimization proceeds.
[0015] The purpose of the clamping, which may be "soft" in that
small excursions outside the range are allowed, is both to
accommodate the kind of mask being designed and to avoid unphysical
values that would imply amplification. For both clamping and mask
value quantization, regularization terms are incorporated into the
cost functional in a manner familiar to one skilled in the art.
Their effect and their specific form may be adjusted on any
schedule during the course of the optimization.
[0016] In an embodiment, the quantized optimal mask is decomposed
into geometric shapes for further regularization to satisfy various
geometric constraints. The shapes thus determined may appear as
perturbed idealized mask shapes with extra features added or
removed (SRAFs), or more generally as a pattern without a simple
correspondence to the ideal target layout pattern.
[0017] In an embodiment, the spatial locality of the optical
projection process is used in addition to the frequency locality to
enable the decomposition of a full mask into overlapping tiles in
which the cost functional and functional derivatives may be
evaluated in a manner consistent with handling the entire mask at
once.
[0018] In an embodiment, the invention is a method including:
partitioning a layout mask into a plurality of regions, each region
comprising geometric shapes; generating a two-dimensional pixel
representation of a region; generating a analytical function (e.g.,
Fourier basis function) to represent the pixel representation of
the region; performing a transformation on the analytical function
to obtain a frequency space representation of the analytical
function of the region; and based on the frequency space
representation, removing low frequency components of the analytical
function of the region.
[0019] Other objects, features, and advantages of the present
invention will become apparent upon consideration of the following
detailed description and the accompanying drawings, in which like
reference designations represent like features throughout the
figures.
BRIEF DESCRIPTION OF THE DRAWINGS
[0020] FIG. 1 shows a system of the present invention for
performing electronic design automation using a graphics
processor.
[0021] FIG. 2 shows a simplified system block diagram of a computer
system used to execute software of the invention.
[0022] FIG. 3 schematically shows a mask represented by simple
geometric shapes.
[0023] FIG. 4 schematically shows a mask represented as pixels
comprising an image of transmission values.
[0024] FIG. 5 schematically shows a mask represented as a general
function giving the mask transmission over the mask area.
[0025] FIG. 6 schematically shows in one dimension a band-limited
mask transmission function that has been regularized to avoid large
excursions beyond the minimum and maximum allowable mask
values.
[0026] FIG. 7 schematically shows in one dimension a mask
transmission function that has been further regularized to strongly
favor allowable mask values.
DETAILED DESCRIPTION OF THE INVENTION
[0027] FIG. 1 shows a system of the present invention. In an
embodiment, the invention is software that executes on a computer
workstation system, such as shown in FIG. 1. FIG. 1 shows a
computer system 1 that includes a monitor 3, screen 5, cabinet 7,
keyboard 9, and mouse 11. Mouse 11 may have one or more buttons
such as mouse buttons 13. Cabinet 7 houses familiar computer
components, some of which are not shown, such as a processor,
memory, mass storage devices 17, and the like.
[0028] Mass storage devices 17 may include mass disk drives, floppy
disks, magnetic disks, optical disks, magneto-optical disks, fixed
disks, hard disks, CD-ROMs, recordable CDs, DVDs, recordable DVDs
(e.g., DVD-R, DVD+R, DVD-RW, DVD+RW, HD-DVD, or Blu-ray Disc),
flash and other nonvolatile solid-state storage (e.g., USB flash
drive), battery-backed-up volatile memory, tape storage, reader,
and other similar media, and combinations of these.
[0029] A computer-implemented or computer-executable version of the
invention may be embodied using, stored on, or associated with
computer-readable medium. A computer-readable medium may include
any medium that participates in providing instructions to one or
more processors for execution. Such a medium may take many forms
including, but not limited to, nonvolatile, volatile, and
transmission media. Nonvolatile media includes, for example, flash
memory, or optical or magnetic disks. Volatile media includes
static or dynamic memory, such as cache memory or RAM. Transmission
media includes coaxial cables, copper wire, fiber optic lines, and
wires arranged in a bus. Transmission media can also take the form
of electromagnetic, radio frequency, acoustic, or light waves, such
as those generated during radio wave and infrared data
communications.
[0030] For example, a binary, machine-executable version, of the
software of the present invention may be stored or reside in RAM or
cache memory, or on mass storage device 17. The source code of the
software of the present invention may also be stored or reside on
mass storage device 17 (e.g., hard disk, magnetic disk, tape, or
CD-ROM). As a further example, code of the invention may be
transmitted via wires, radio waves, or through a network such as
the Internet.
[0031] FIG. 2 shows a system block diagram of computer system 1
used to execute software of the present invention. As in FIG. 1,
computer system 1 includes monitor 3, keyboard 9, and mass storage
devices 17. Computer system 1 further includes subsystems such as
central processor (CPU) 202, system memory 204, input/output (I/O)
controller 206, display adapter 208, serial or universal serial bus
(USB) port 212, network interface 218, graphics processor (GPU)
220, FPGA 225, and specialized processor 228 (e.g., ASIC, physics
processor). The invention may also be used with computer systems
with additional or fewer subsystems. For example, a computer system
could include more than one processor 202 or 220, or both. Such a
system may be referred to as a multiprocessor system. The system
may include on-chip or external cache memory.
[0032] The computer system may include any number of graphics
processors. The graphics processor may reside on the motherboard
such as being integrated with the motherboard chipset. One or more
graphics processors may reside on external boards connected to the
system through a bus such as an ISA bus, PCI bus, AGP port, PCI
Express, or other system buses. Graphics processors may on separate
boards, each connected to a bus such as the PCI Express bus to each
other and to the rest of the system. Further, there may be a
separate bus or connection (e.g., Nvidia SLI or ATI CrossFire
connection) by which the graphics processors may communicate with
each other. This separate bus or connection may be used in addition
to or in substitution for system bus.
[0033] Each processor, CPU or GPU, or both, may be a dual core or
multicore processor, where there are multiple processor cores on a
single integrated circuit. The system may also be part of a
distributed computing environment. In a distributed computing
environment, individual computing systems are connected to a
network and are available to lend computing resources to another
system in the network as needed. The network may be an internal
Ethernet network, Internet, or other network.
[0034] Arrows such as 222 represent the system bus architecture of
computer system 1. However, these arrows are illustrative of any
interconnection scheme serving to link the subsystems. For example,
speaker 220 could be connected to the other subsystems through a
port or have an internal connection to central processor 202.
Computer system 1 shown in FIG. 1 is but an example of a computer
system suitable for use with the present invention. Other
configurations of subsystems suitable for use with the present
invention will be readily apparent to one of ordinary skill in the
art.
[0035] Computer software products may be written in any of various
suitable programming languages, such as C, C++, C#, Pascal,
Fortran, Perl, MatLab (from MathWorks, Inc.), SAS, SPSS, Java,
JavaScript, and AJAX. The computer software product may be an
independent application with data input and data display modules.
Alternatively, the computer software products may be classes that
may be instantiated as distributed objects. The computer software
products may also be component software such as Java Beans (from
Sun Microsystems) or Enterprise Java Beans (EJB from Sun
Microsystems).
[0036] An operating system for the system may be one of the
Microsoft Windows.RTM. family of operating systems (e.g., Windows
95, 98, Me, Windows NT, Windows 2000, Windows XP, Windows XP
.times.64 Edition, Windows Vista, Windows CE, Windows Mobile),
Linux, HP-UX, UNIX, Sun OS, Solaris, Mac OS X, Alpha OS, AIX,
IRIX32, or IRIX64, or combinations of these. Other operating
systems may be used. A computer in a distributed computing
environment may use a different operating system from other
computers.
[0037] Furthermore, the computer may be connected to a network and
may interface to other computers using this network. For example,
each computer in the network may perform part of the task of the
many series of steps of the invention in parallel. Furthermore, the
network may be an intranet, internet, or the Internet, among
others. The network may be a wired network (e.g., using copper),
telephone network, packet network, an optical network (e.g., using
optical fiber), or a wireless network, or any combination of these.
For example, data and other information may be passed between the
computer and components (or steps) of a system of the invention
using a wireless network using a protocol such as Wi-Fi (IEEE
standards 802.11, 802.11a, 802.11b, 802.11e, 802.11g, 802.11i, and
802.11n, just to name a few examples). For example, signals from a
computer may be transferred, at least in part, wirelessly to
components or other computers.
[0038] Mask Design Through Inverse Lithography with Spatial
Frequency Analysis
[0039] Production-scale chip manufacturing uses photolithographic
techniques to build up layers of materials on a wafer to create the
transistors, wires, and so on, that realize a chip design. The
sizes of the features to be printed on the wafer are approaching
the limits set by the wavelength of the light, the optical
projection system, and the behavior of the light sensitive
materials used, among many other factors.
[0040] Diffraction effects from the wavelength of the light source
and the limits of the projection optics motivated the development
of optical proximity correction (OPC) techniques to adjust the
shapes on the mask to print more like the desired result on the
wafer. For example, a square may have serifs added to its corners
to compensate for excessive rounding of the corners of the printed
feature, or the ends of a rectangle may have "hammerheads" added to
further ensure the printed feature reaches the desired line
end.
[0041] The first OPC methods were based on simple rules for
modifying the shapes on the mask, but as the technology was pushed
closer to optical resolution limits, model-based optimization,
which adjusts the features on the mask to improve the calculated
printed image, was developed. Two significant advantages of
model-based OPC are the ability to account for proximity effects
(warping a nearby feature will affect how a given feature prints)
and to accommodate photoresist behavior.
[0042] Sometimes, features are found to print with greater fidelity
if extra features are added to the mask that are too small to print
themselves, but nevertheless favorably affect the way nearby main
features print. Introduction of these so-called subresolution
assist features (SRAFs) is still generally done according to preset
rules. Typically they are inserted first and held fixed as OPC is
applied to the main features on the mask.
[0043] There are significant problems in applying these methods as
the industry moves to ever smaller on-wafer dimensions. The rules
used to insert SRAFs are becoming more complex and less reliable.
The standard OPC methods do not have the flexibility needed to
achieve the best results and require post-OPC verification and
manual intervention.
[0044] What is needed is a practical model-based inverse
lithography method that can automatically determine a mask that
both satisfies mask manufacturing and verification criteria, and
produces the desired print on the wafer over a range of process
conditions, such as exposure and focus variation. Such a method
will generally result in a mask that warps existing layout geometry
and adds or subtracts SRAFs from anywhere, including in ways that
split or join the layout features.
[0045] Typically, inverse lithography methods have pixelized the
mask transmission values. The mask pixels are then adjusted to
optimize the computed print in the photoresist as well as some
measure of mask design "goodness." For example, the mask pixel
values may be constrained to be fully transmitting (1) or absorbing
(0) to produce a chrome-on-glass binary mask, or if not fully
constrained, the optimization procedure may be tuned to strongly
favor these values. In the field of inverse theory, directing the
solution to have desired a priori characteristics is known as
regularization. Other regularization terms are often employed to
affect the solution, such as measures that penalize complex
structures.
[0046] In contrast to pixel-based methods, this invention
represents the mask as a continuous function and uses the physics
of optical projection to design the inverse solution based on a
spatial frequency analysis. From a physics perspective, spatial
frequencies above a cutoff determined by the optical system do not
contribute to the projected image.
[0047] Frequencies below this cutoff affect the print (and the
mask), while those above the cutoff only affect the mask.
[0048] We represent the mask in a basis set that automatically
exposes the separation between the low frequencies that affect the
print and the high frequencies that only adjust the mask. The plane
wave (Fourier) basis makes this separation perfectly, but other
basis sets such as wavelets may be used.
[0049] The first stage in optimizing the mask function seeks a
continuous tone solution that is clamped to the minimum and maximum
allowable mask values. For a binary mask, this means that a
regularizing cost functional is added to a functional that measures
the goodness of the predicted printed features to prevent the mask
values from straying outside of the interval [0, 1]. We include
spatial frequencies up to some multiple of the cutoff to
accommodate this clamping.
[0050] The output of this stage will have produced an optimal
clamped solution for the various criteria included in the fitting
functional. These may include terms that measure variation with
exposure (dose), depth of focus, or that weight different parts of
the printed image according to the needs of the design.
[0051] The solution obtained needs to be pushed further to be
quantized to the allowable mask values, such as 0 and 1 for a
binary mask. This is accomplished through another idea.
[0052] Mask design through "inverse lithography" using separation
of spatial frequencies to control printed image versus mask
manufacturability.
[0053] Typical inverse lithography methods pixelize the computed
aerial image and the mask values under optimization on a dense
spatial grid. The mask pixel values are adjusted to optimize both
the computed print in the resist and some measure of mask
"goodness" (which is known as regularization in "inverse theory").
For example, the mask transmission values may be constrained to be
0 or 1, or at least strongly favor these values in a solution, or a
measure of the solution's complexity may be minimized to avoid
complex, "snowflake" structures.
[0054] The regularization is used because the inverse problem (what
mask will print a desirable image) is severely
underconstrained--many masks will print essentially the same image,
so other criteria must be used to select appropriate solutions.
From a physics perspective, spatial frequencies above a cutoff
determined by the optical system do not contribute to the projected
image; therefore, frequencies below this cutoff affect the print
(and the mask), while those above only affect the mask.
[0055] A computer-implemented method of the invention includes:
representing the mask values in different basis sets with a view to
exposing automatically the separation between the low frequencies
that affect the print and the high frequencies that only adjust the
mask. The Fourier basis provides the separation perfectly, but
others such as various wavelet representations can be used.
[0056] A given (desirable) printed image must result from a
band-limited aerial image (which is all the optical system can
form), but there may be many such images that would work. The
method regularizes this reduced solution further by favoring
real-space mask pixels with values in (0 . . . 1), wider "process
windows" (solution sensitivity to exposure and defocus), and so
forth.
[0057] Taking the forward view, the system will print from a
band-limited aerial image, so one may preprocess the ideal resist
print (round the corners, and so forth) to be a better match to
what is achievable to begin with, and actually construct an aerial
image to achieve.
[0058] In either case, the high frequency coefficients may then be
adjusted while holding the low frequency values constant or
otherwise forced close to their original values to optimize the
mask manufacturability.
[0059] Quasi-Static Regularization for Mask Manufacturability
[0060] Continuous-tone mask functions that solve the inverse
lithography problem are quantized to allowable values; for example,
0 or 1 for a binary chrome-on-glass mask.
[0061] Starting with an optimized mask function, such as a
continuous tone, clamped solution utilizing another invention, we
slowly mix in a new cost functional to further regularize the
solution. The mixing in may be done "quasi-statically" to bring the
cost functional and the mask from pure goodness-of-fit to printed
layout and process characteristics to pure manufacturability by
keeping the total cost functional minimized step-by-step. The aim
of this gradual mixing in is to do thermodynamically optimal work
on the mask function to bring it to manufacturable values.
[0062] The functional we mix in to push the intermediate mask
values to the allowable values may depend on the mask itself in
some fixed, self-consistent, or otherwise changing way. The idea is
that any value not close to an allowed value needs to move away
from where it is without any particular bias as to which direction
it goes.
[0063] One way to accomplish this is for the new cost functional
term to place an energy maximum locally at intermediate current
mask values to cause them to "roll" downhill towards an allowed
value on either side of the hill. The new functional can also favor
allowed values by having local minima at these values. Intermediate
mask values very close to an allowed value may instead be pushed
toward the nearest allowed value by placing the hill on the far
side.
[0064] By doing multiple steps where the quantizing functional
becomes a larger fraction of the total cost functional and where
the hills placed underneath intermediate values are adjusted to
chase them to any allowed value, the mask function is brought to
quantized values in a controlled and optimal way.
[0065] A computer-implemented method of the invention includes:
Start with optimized mask (perhaps continuous tone and phase,
pixelized, or otherwise parameterized). Mix in a new cost to
"regularize" the solution to be more desirable (e.g., to favor a
binary mask, or others). Conceptually, adjust the mixing
quasi-statically to bring the functional from pure goodness-of-fit
to printed layout to pure manufacturability, keeping the total cost
functional minimized step-by-step.
[0066] An objective is to do thermodynamically optimal (reversible)
"work" on the solution to adjust to manufacturabilty.
[0067] Use spatial frequency separation described above to
advantage.
[0068] Inverse Lithography with Simplified Resist Model to Generate
Assist Features for Traditional Model-Based Optical Proximity
Correction
[0069] Inverse lithography methods generally require substantial
computing resources and may generate smooth, curvy features that
need to be converted to rectilinear mask geometry. Traditional
model-based OPC is fast and can use more sophisticated resist
models, but cannot generate subresolution assist features (SRAFs)
automatically and self-consistently.
[0070] Take a mask function optimized through inverse lithography,
and use it as input to a traditional model-based OPC program. The
inverse lithography process will provide the SRAFs and initial
shapes, and the traditional OPC program can further correct and
validate the results using better models.
[0071] A computer-implemented method of the invention includes: Use
a simplified resist model that has an analytical functional form to
do inverse modeling faster, but take the resulting mask (after
modifying to be a set of shapes as for binary, PSM, EAPSM, or
others, manufacturable masks), and optimize it using a traditional
OPC program with a more complex resist model.
[0072] In this way, the extra mask features needed to print well
are generated by the inversion, but then further corrected and
validated by a program with access to better models (which may take
more computational effort to use).
[0073] Fast Eigenfunction Decomposition of the Real Space Partially
Coherent Transmission Operator using High Resolution Fourier Space
Grids
[0074] Calculating projected images with partially coherent
illumination can be done efficiently using an eigenfunction
expansion of the Hopkins partial coherence operator. This expansion
can be accomplished in real space or Fourier space. A naive
discretization of the illumination source and coherent transfer
function of the optical projection system followed by matrix
diagonalization is likely to be both inaccurate and computationally
intensive, even if iterative, large-scale diagonalization schemes
are used.
[0075] In this invention, we super-sample the illumination source
and the coherent optical transfer functions in Fourier space to
represent slim or circular shapes more accurately. We may also
smooth the transition at the edges of these shapes to further
symmetrize the calculated real space samples.
[0076] To accelerate the diagonalization of the real-space Hopkins
partial coherence operator, we use the translational invariance of
the mutual intensity illumination function to compute the
matrix-vector product via element-wise multiplication and Fast
Fourier Transforms (FFTs). To determine the eigenvectors with the
largest eigenvalues (greatest contribution to the operator), only
matrix-vector products are needed by solvers such as SVDPACK or
ARPACK.
[0077] The operations count (when discretizing space on an n by n
grid) for the matrix-vector product is reduced from O(n.sup.4) to
O(n.sup.2 log(n.sup.2)) .
[0078] Representation and decomposition of the partially coherent
transmission matrix in real-space with the effective source mutual
intensity and the coherent amplitude point-spread transfer function
represented on a high-resolution grid in Fourier space. A
computer-implemented method of the invention includes:
[0079] 1. Represent quantities on a real-space 2D grid (nx by ny).
Transmission matrix will be on a (nx*ny) by (nx*ny) real space
grid, assuming we flatten the 2D space to a single index.
Diagonalize this (Hermitian) matrix to decompose (goal is to keep
largets eigenvalue vectors only). Eigenvectors so obtained are the
real-space "kernels."
[0080] The computational effort to do the diagonalization (or
singular value decomposition) increases quickly with real-space
grid resolution, being O((nx*ny) 3). The resolution needed for a
good representation is not too high, however, with n=nx=ny=64 being
adequate and requiring only minutes to diagonalize.
[0081] 2. The real-space transmission matrix is
[0082] T(r1,r2)=J0minus(r1,r2)*K(r1)*conjugate(K(r2))
[0083] J0minus(,) is mutual intensity just before the mask
[0084] K( )is coherent amplitude transfer function
[0085] r1, r2 coordinates of two real-space points.
[0086] It is typical for J0minus(r1,r2) to be a function of r1-2
only (translational invariance). These quantities are conveniently
specified in Fourier space; e.g., F{K}( ) is generally a (circular)
low pass filter, and F{J0minus}( ) is essentially the illumination
pattern (Kohler, Annular, QUASAR, or others).
[0087] These slim or circular shapes, or both, are represented on a
square grid with enough resolution to capture their essential
characteristics, but not exceed the transmission matrix
diagonalization. One technique is to use a Fourier space grid that
has many times more samples than the target real-space grid (while
retaining the same frequency extents). This effectively just embeds
the result in a real-space grid with larger extents, so the needed
values are obtained by inverting the fine-Fourier-grid samples to
real-space, and then retaining only the values within the target
real-space grid cell.
[0088] If the Fourier-space grid resolution is an integral factor
times greater, then after inversion the real-space values within
the target real-space grid cell will lie on the target real-space
grid points.
[0089] Light Source Optimization
[0090] The characteristics of the light source in a
photolithography projection system have a significant effect on the
quality of the resulting images printed on the wafer. These
parameters may be optimized to print certain layers.
[0091] According to the invention, there are various
approaches:
[0092] (1) Optimize a set of orthonormal functions and eigenvalues
constrained to represent a valid Hopkins partial coherence operator
to optimize the illumination source.
[0093] (2) Instead, optimize the mutual intensity function directly
in Fourier space or real space taking advantage of small
perturbations yielding small variations in any eigenfunction
decomposition.
[0094] (3) Optimize the light source directly using Abbe's method
of statistically independent ray bundles.
[0095] A computer-implemented method of the invention includes:
Optimize the orthonormal kernels and singular values defining a
sparse decomposition of the TCC matrix (in real or Fourier
space).
[0096] Optimize the incident mutual intensity directly. Take
advantage of small variations in J yielding small variations in TCC
or intensity, or both.
[0097] Optimize source light distribution directly and formulate
partial coherence via Abbe's method of statistically independent
ray bundles.
[0098] Use spatial frequency separation described earlier to
advantage.
[0099] Mask Functional Optimization and Spatial Frequency
Analysis
[0100] In the field of optical proximity correction, a mask for
photolithography is typically represented as geometric shapes of a
foreground transmission value set against a backdrop with a
background transmission value. FIG. 3 shows this representation
schematically. The foreground transmission values need not all be
the same, but they will be fixed to values allowed by the mask
technology, and the geometric shapes may be more general than those
illustrated.
[0101] The fundamental limitation is that new shapes cannot be
systematically introduced during the course of mask
optimization.
[0102] More general "inverse" lithography methods have typically
represented a mask as an image of pixels with transmission values,
as shown in FIG. 3. In this case, it is the pixel values that are
adjusted, which will construct a mask with many intermediate
transmission values that must be "regularized" to assume only the
allowed values. Moreover, a consistent calculation of the projected
intensity and any derived quantities requires integrations over
each piecewise constant pixel area, and this is usually neglected
since it may be too computationally intensive.
[0103] The present invention represents a photolithography mask as
a function with a specific mathematical form that evaluates to the
mask transmission value over the mask area (see FIG. 5). Using a
specified mathematical form permits consistent and efficient
evaluation of functionals of the mask that compute the cost of a
mask function with respect to fitting a desired printed target
pattern, or to satisfying a priori criteria a mask function should
meet to be acceptable. (Instead of "cost," "energy," "merit," or
other terms familiar to one skilled in the art may be used to
describe such functionals.) This consistency enables the use of
fast optimization algorithms that rely on derivative
information.
[0104] The physics of the optical projection process shows that
only spatial frequencies in the mask function up to a cutoff can
contribute to the final image. For example, for a system using a
quasi-monochromatic light source with wavelength .lamda..sub.0, an
extended source shape that is a disk (known as Kohler illumination)
with partial coherence factor .sigma., and image-forming optics
with numerical aperture NA, only spatial frequencies present in the
mask function up to
f cut = NA .lamda. 0 ( 1 + .sigma. ) ##EQU00001##
[0105] can contribute to the projected image. (The image itself is
quadratic in the mask function and so contains frequencies up to
2f.sub.cut.)
[0106] The mask function, as well as other functions such as the
image intensity, and so on, may be expressed mathematically in
terms of a basis set. Desirable basis sets may exploit locality in
the frequency or spatial domains depending on the needs of a
specific calculation, but in any case should permit exact results
to be calculated within any other approximations applied. One
useful basis set that may be employed to exploit the frequency
cutoff is a Fourier basis. For example, conceptually the mask
function and the projected intensity may be written as
m ( r ) = f m ~ ( f ) 2 .pi. f r ##EQU00002## I ( r ) = f I ~ ( f )
2 .pi. f r ##EQU00002.2##
[0107] and the band-limited nature of the intensity and of the mask
that is "seen" by the optics may be made explicit by restricting
the sums to |f|.ltoreq.f.sub.max for some selected f.sub.max.
[0108] In an embodiment, the maximum frequency used to represent
the mask may be greater than the optics cutoff. Furthermore, an
embodiment need not parameterize band-limited functions describing
the mask, intensity, or other quantity of interest explicitly in
terms of its Fourier coefficients provided complete consistency is
maintained between real space and Fourier space samples. More
generally, an embodiment that uses other basis sets will ensure
such consistency is maintained.
[0109] To formulate the mask design task as a functional
optimization problem, the present invention defines a "cost" or
"energy" functional that compares the predicted printed pattern to
a target layout over the entire mask area. In a very general way,
the total cost may be expressed as
E=.intg.d.sup.2r .epsilon.[m(r),p(r)]
[0110] where m(r) is the mask function, p(r) is the target pattern,
and s is an energy density that is a functional of the mask and
target pattern functions. In an embodiment, the form of the energy
density may contain terms that are not simple local functions of
the mask and target pattern. These may include but are not limited
to convolutions of the mask or target pattern with filter functions
and various spatial derivatives.
[0111] An optimal mask will have a cost that is stationary with
respect to small variations in the mask function. To first order in
the variation
.delta. E = .intg. 2 r .delta. .delta. m ( r ) .delta. m ( r ) = 0
##EQU00003##
[0112] where .delta..epsilon./.delta.m(r) is the functional
derivative of the energy density with respect to the mask function.
With the mask expressed in terms of a basis set, the derivative of
the cost with respect to a basis set coefficient is conveniently
obtained in a variationally consistent manner, which can then be
used in fast optimization algorithms. For example, if a Fourier
basis is used,
.delta. E .delta. m ~ ( - f ) = .intg. 2 r .delta. [ m ( r ) , p (
r ) ] .delta. m ( r ) - 2.pi. f r ##EQU00004##
[0113] In this case, all the derivatives may be calculated
wholesale via fast Fourier transform (FFT) algorithms.
[0114] The present invention may use different forms of the cost
functional, and these forms may be modified as the optimization
proceeds. An example of a simple form that measures how well a mask
is predicted to print a desired target is
E=.intg.d.sup.2r(R[I(r)]-p(r)).sup.2
[0115] where I(r) is the projected intensity image (which is a
functional of the mask), and R is a functional of the intensity
that predicts the final developed resist pattern. In practice, more
sophisticated cost functionals may be necessary or provide better
results.
[0116] Without further direction, an optimized mask function will
evaluate to transmission values that are not restricted to those
allowed for a particular mask manufacturing process. Furthermore,
some of the values may lie well outside the range allowed for
passive transmission of light; for example, a mask cannot add
energy by having a large region with a transmission coefficient of
2.
[0117] To address the second of these issues, an embodiment of the
present invention may add a regularizing term to the cost
functional that penalizes a mask function that assumes values
outside of the allowed range. As shown schematically in one
dimension in FIG. 4, such a term can prevent large excursions
beyond specified limits. An embodiment that exploits a spatial
frequency analysis may use a larger frequency limit to provide for
this "soft" clamping effect.
[0118] Once a restricted optimized mask function has been found via
cost minimization algorithms familiar to one skilled in the art,
further regularization may be performed to quantize the mask to
allowable values. The cost functional term may be mixed in on any
schedule found to effective, and it's form may adjust as the
regularization optimization proceeds. FIG. 7 shows the result from
FIG. 6 after regularization to assume values of one or zero, as for
a binary chrome-on-glass mask. Note how the area near x=0.75 of
value <0.4 has transformed into an "SRAF" of full transmission
value but width too narrow to print.
[0119] Once quantized, the mask may be transformed into geometry to
be further regularized to satisfy geometric manufacturability
criteria.
[0120] In an embodiment, the spatial locality of the optical
projection process is used in addition to the frequency locality to
enable the decomposition of a full mask into overlapping tiles in
which the cost functional and functional derivatives may be
evaluated in a manner consistent with handling the entire mask at
once. The spatial localization means that there need not be any
cross-tile matching conditions imposed on basis sets used to
represent the mask function within each tile.
[0121] This description of the invention has been presented for the
purposes of illustration and description. It is not intended to be
exhaustive or to limit the invention to the precise form described,
and many modifications and variations are possible in light of the
teaching above. The embodiments were chosen and described in order
to best explain the principles of the invention and its practical
applications. This description will enable others skilled in the
art to best utilize and practice the invention in various
embodiments and with various modifications as are suited to a
particular use. The scope of the invention is defined by the
following claims.
* * * * *