U.S. patent number RE44,792 [Application Number 12/587,415] was granted by the patent office on 2014-03-04 for rapid scattering simulation of objects in imaging using edge domain decomposition.
This patent grant is currently assigned to The Regents of the University of California. The grantee listed for this patent is Konstantinos Adam. Invention is credited to Konstantinos Adam.
United States Patent |
RE44,792 |
Adam |
March 4, 2014 |
Rapid scattering simulation of objects in imaging using edge domain
decomposition
Abstract
A complex two-dimensional layout of a photomask or other
three-dimensional object is systematically decomposed into a finite
number of elementary two-dimensional objects with the ability to
cause one-dimensional changes in light transmission properties. An
algorithmic implementation of this can take the form of creation of
a look-up table that stores all the scattering information of all
two-dimensional objects needed for the synthesis of the
electromagnetic scattered field from the original three-dimensional
object. The domain is decomposed into edges, where pre-calculated
electromagnetic field from the diffraction of isolated edges is
recycled in the synthesis of the near diffracted field from
arbitrary two-dimensional diffracting geometries. The invention has
particular applicability in die-to-database inspection where an
actual image of a mask is compared with a synthesized image that
takes imaging artifacts of comers, edges and proximity into
account. Another application is optical proximity correction which
consists of evaluating the image of every feature on a mask and
improving it by introducing edge shifts and iteratively adjusting
the amounts of these shifts.
Inventors: |
Adam; Konstantinos (Belmont,
CA) |
Applicant: |
Name |
City |
State |
Country |
Type |
Adam; Konstantinos |
Belmont |
CA |
US |
|
|
Assignee: |
The Regents of the University of
California (Oakland, CA)
|
Family
ID: |
32599928 |
Appl.
No.: |
12/587,415 |
Filed: |
September 3, 2009 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
|
|
60415510 |
Oct 1, 2002 |
|
|
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Reissue of: |
10677136 |
Sep 30, 2003 |
7266480 |
Sep 4, 2007 |
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Current U.S.
Class: |
703/2; 703/17;
703/6; 703/19; 430/5; 716/53 |
Current CPC
Class: |
G03F
7/70591 (20130101); G03F 7/705 (20130101); G03F
1/84 (20130101); G03F 1/36 (20130101); G03F
7/70941 (20130101); G03F 1/26 (20130101) |
Current International
Class: |
G06F
17/50 (20060101); G03F 1/36 (20120101) |
Field of
Search: |
;703/2,6,17,19
;716/2,4,52,53 ;430/5,40 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Adam et al., "Methodology for Accurate and Rapid Simulation of
Large Arbitrary 2D Layouts of Advanced Photomasks", 21st Annual
Bacus Symposium on Photomask Technology, Oct. 3, 2001. cited by
applicant .
Konstantinos Adam, "Domain Decomposition Methods for the
Electromagnetic Simulation of Scattering From ThreeDimensional
Structures With Applications in Lithography", A dissertation
submitted in Fall 2001 to the University of California, Berkeley.
cited by applicant .
Adam et al., "Efficient Phase Defect Modeling Using Domain
Decomposition Methods", EIPBN Conference, May 2002. cited by
applicant .
Adam et al., "Simplified Models for Edge Transitions in Rigorous
Mask Modeling", 26th Annual International Symposium on
Microlithography, Mar. 1, 2001. cited by applicant .
Stolowitz Ford Cowger LLP, "Listing of Related Cases", Jul. 31,
2013, 1 page. cited by applicant .
Adam et al., "Efficient Phase Modeling using Domain Decomposition
Methods", EIPBN Conference, May 2002. cited by applicant.
|
Primary Examiner: Phan; Thai
Attorney, Agent or Firm: Stolowitz Ford Cowger LLP
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATIONS
.[.This patent application.]. .Iadd.This application is a Reissue
Application of its parent application Ser. No. 10/677,136, filed on
Sep. 30, 2003, now U.S. Pat. No. 7,266,480 which .Iaddend.claims
priority from Provisional Patent Application No. 60/415,510 filed
Oct. 1, 2002 for "METHOD FOR RAPID SCATTERING SIMULATION OF OBJECTS
IN IMAGING VIA DOMAIN DECOMPOSITION AND SPECTRAL MATCHING," and is
related to application Ser. No. 10/241,242 filed Sep. 10, 2002 for
"CHARACTERIZING ABERRATIONS IN AN IMAGING LENS AND APPLICATIONS TO
VISUAL TESTING AND INTEGRATED CIRCUIT MASK ANALYSIS," .Iadd.all of
.Iaddend.which are incorporated herein by reference for all
purposes.
Claims
What is claimed is:
1. A method .[.for simulation of light scattering in large three
dimensional objects in a optical imaging system.]. comprising
.[.the steps of.].: .[.a) decomposing a three dimensional object
into a plurality of elementary two dimensional objects, b) defining
the two dimensional objects by edges of the objects, c).]. applying
edge domain decomposition .[.for the edges of the.]. .Iadd.to
.Iaddend.two dimensional objects .Iadd.corresponding to a three
dimensional object .Iaddend.to identify light scattering by
.[.the.]. .Iadd.associated .Iaddend.one dimensional edges, and
.[.d).]. summing scattered light from all edges to simulate light
scattering in .[.two.]. .Iadd.three .Iaddend.dimensions.
2. The method of claim 1 and further including: .[.e).]. spectrally
mapping the edge domain decomposition.
3. The method of claim 2 wherein the three dimensional object
comprises a photo mask with multiple and interconnected
openings.
4. The method of claim 3 wherein .[.step c).]. .Iadd.applying edge
domain decomposition .Iaddend.includes computational methods which
speed-up calculations through use of table look-up of previously
calculated objects and filtering that sequentially invokes more
rigorous simulation as needed.
5. The method of claim 4 wherein .Iadd.an .Iaddend.electromagnetic
field from .[.the.]. diffraction of isolated edges is recycled in
.[.the.]. synthesis of .Iadd.a .Iaddend.near diffracted field for
arbitrary two dimensional diffracting geometries.
6. The method of claim 5 wherein .[.the.]. .Iadd.a
.Iaddend.scattered field from all edges are summed to simulate
light scattering .[.in.]. .Iadd.by .Iaddend.the .[.two.].
.Iadd.three .Iaddend.dimensional .[.objects.].
.Iadd.object.Iaddend..
7. The method of claim 6 wherein scattered light is polarized and
.[.step c).]. .Iadd.applying edge domain decomposition
.Iaddend.includes separate decomposition for light parallel to
edges (TE) and for light perpendicular (TM) to edges.
8. The method of claim 7 wherein .[.step c).]. .Iadd.applying edge
domain decomposition .Iaddend.includes pre-simulating all distinct
edges in the object with all possible illumination directions and
field polarizations.
9. The method of claim 8 .Iadd.further comprising decomposing the
three dimensional object into a plurality of the elementary two
dimensional objects, .Iaddend.wherein .[.in step a).]. the two
dimensional objects comprise rectangles.
10. The method of claim 9 wherein the object is a phase shifting
mask and .[.step c) of.]. pre-simulating light scattering from
edges includes phase effects of etch depths in defining the phase
shifting mask.
11. The method of claim 10 wherein the simulation of light
scattering is based on .[.through the.]. .Iadd.a .Iaddend.lens
optical spectra.
12. The method of claim 10 wherein the simulation of light
scattering is based on a raised cosine function for .[.the.].
optical spectra.
13. The method of claim 10 wherein the simulation of light
scattering is based on a Gaussian function for .[.the.]. optical
spectra.
14. The method of claim 10 wherein the simulation of light
scattering is based on a rect-function for .[.the.]. optical
spectra.
15. The method of claim 3 wherein the object is a phase shifting
mask and .[.step c).]. .Iadd.applying edge domain decomposition
.Iaddend.includes pre-simulating light scattering from edges
.Iadd.and .Iaddend.includes phase effects of etch depths in
defining the phase shifting mask.
16. The method of claim 15 wherein scattered light is polarized and
.[.step c).]. includes separate decomposition for light parallel to
edges .[.(TB).]. .Iadd.(TE) .Iaddend.and for light perpendicular
(TM) to edges.
17. The method of claim 16 wherein .[.step c).]. .Iadd.applying
edge domain decomposition .Iaddend.includes pre-simulating all
distinct edges in the object with all possible illumination
directions and field polarizations.
18. The method of claim 1 wherein .[.step c).]. .Iadd.applying edge
domain decomposition .Iaddend.includes pre-simulating all distinct
edges in the object with all possible illumination directions and
field polarizations.
.Iadd.19. A method comprising: applying edge-domain decomposition
to the a plurality of one-dimensional edges derived from a
three-dimensional object to identify light scattering by the
one-dimensional edges; and using a computer, summing scattered
light from the one-dimensional edges to simulate the light
scattering by the three-dimensional object..Iaddend.
.Iadd.20. The method of claim 19, further comprising spectrally
mapping the edge domain decomposition..Iaddend.
.Iadd.21. The method of claim 19, wherein the three dimensional
object comprises a photo mask with multiple and interconnected
openings..Iaddend.
.Iadd.22. The method of claim 19, wherein the applying edge-domain
decomposition comprises using computational methods that speed up
calculations by using lookup tables of previously calculated
objects and filtering that sequentially invokes more rigorous
simulation as needed..Iaddend.
.Iadd.23. The method of claim 19, wherein an electromagnetic field
from diffraction of isolated edges is recycled in synthesis of a
near diffracted field for arbitrary two-dimensional diffracting
geometries..Iaddend.
.Iadd.24. The method of claim 19, wherein one or more scattered
fields from the one-dimensional edges are summed to simulate the
light scattering by the three-dimensional object..Iaddend.
.Iadd.25. The method of claim 19, wherein the scattered light is
polarized and the applying edge-domain decomposition includes
performing separate decompositions for light parallel to the
one-dimensional edges (TE) and for light perpendicular (TM) to the
one-dimensional edges..Iaddend.
.Iadd.26. The method of claim 19, wherein the applying edge-domain
decomposition includes distinct edges in the three-dimensional
object with one or more possible illumination directions and field
polarizations..Iaddend.
.Iadd.27. The method of claim 26, wherein the three-dimensional
object is a phase-shifting mask and the pre-simulating accounts for
phase effects of etch depths in the phase shifting
mask..Iaddend.
.Iadd.28. The method of claim 19, wherein two-dimensional objects
defining the three-dimensional object comprise
rectangles..Iaddend.
.Iadd.29. The method of claim 19, wherein the applying edge-domain
decomposition is based on through-the-lens optical
spectra..Iaddend.
.Iadd.30. The method of claim 19, wherein the applying edge-domain
decomposition is based on a raised cosine function for optical
spectra..Iaddend.
.Iadd.31. The method of claim 19, wherein the applying edge-domain
decomposition is based on a Gaussian function for optical
spectra..Iaddend.
.Iadd.32. The method of claim 19, wherein the applying edge-domain
decomposition is based on a rect-function for optical
spectra..Iaddend.
.Iadd.33. The method of claim 19, wherein: the three-dimensional
object is a phase shifting mask, and the applying edge-domain
decomposition includes pre-simulating light scattering from the
one-dimensional edges and phase effects of etch depths in the phase
shifting mask..Iaddend.
.Iadd.34. The method of claim 33, wherein: the scattered light is
polarized; and the applying edge-domain decomposition includes
performing separate decompositions for light parallel to the
one-dimensional edges (TE) and for light perpendicular (TM) to the
one-dimensional edges..Iaddend.
.Iadd.35. The method of claim 34, wherein the applying edge-domain
decomposition includes pre-simulating distinct edges in the
three-dimensional object with a plurality of possible illumination
directions and field polarizations..Iaddend.
.Iadd.36. The method of claim 19, wherein the applying edge-domain
decomposition includes pre-simulating distinct one-dimensional
edges in the three-dimensional object with a plurality of
pre-selected possible illumination directions and field
polarizations..Iaddend.
.Iadd.37. A non-transitory computer-readable storage medium storing
computer-executable instructions for causing a computer to perform
a method comprising: applying edge-domain decomposition to a
plurality of one-dimensional edges composing a three-dimensional
object to identify light scattering by the one-dimensional edges;
and summing scattered light from the one-dimensional edges to
simulate the light scattering by the three-dimensional
object..Iaddend.
.Iadd.38. The non-transitory computer-readable storage medium of
claim 37, wherein the method further comprises spectrally mapping
the edge domain decomposition..Iaddend.
.Iadd.39. The non-transitory computer-readable storage medium of
claim 37, wherein the three dimensional object comprises a photo
mask with multiple and interconnected openings..Iaddend.
.Iadd.40. The non-transitory computer-readable storage medium of
claim 37, wherein the applying edge-domain decomposition comprises
using computational methods that speed up calculations by using
look-up tables of previously calculated objects and filtering that
sequentially invokes more rigorous simulation as
needed..Iaddend.
.Iadd.41. The non-transitory computer-readable storage medium of
claim 37, wherein an electromagnetic field from diffraction of
isolated edges is recycled in synthesis of a near diffracted field
for arbitrary two-dimensional diffracting geometries..Iaddend.
.Iadd.42. The non-transitory computer-readable storage medium of
claim 37, wherein one or more scattered fields from the
one-dimensional edges are summed to simulate the light scattering
by the three-dimensional object..Iaddend.
.Iadd.43. The non-transitory computer-readable storage medium of
claim 37, wherein the scattered light is polarized and the applying
edge-domain decomposition includes performing separate
decompositions for light parallel to the one-dimensional edges (TE)
and for light perpendicular (TM) to the one-dimensional
edges..Iaddend.
.Iadd.44. The non-transitory computer-readable storage medium of
claim 37, wherein the applying edge-domain decomposition includes
pre-simulating distinct edges in the three-dimensional object with
one or more possible illumination directions and field
polarizations..Iaddend.
.Iadd.45. The non-transitory computer-readable storage medium of
claim 37, wherein two-dimensional objects defining the
three-dimensional object comprise rectangles..Iaddend.
.Iadd.46. The non-transitory computer-readable storage medium of
claim 44, wherein the three-dimensional object is a phase-shifting
mask and the pre-simulating accounts for phase effects of etch
depths in the phase shifting mask..Iaddend.
.Iadd.47. The non-transitory computer-readable storage medium of
claim 37, wherein the applying edge-domain decomposition is based
on through-the-lens optical spectra..Iaddend.
.Iadd.48. The non-transitory computer-readable storage medium of
claim 37, wherein the applying edge-domain decomposition is based
on a raised cosine function for optical spectra..Iaddend.
.Iadd.49. The non-transitory computer-readable storage medium of
claim 37, wherein the applying edge-domain decomposition is based
on a Gaussian function for optical spectra..Iaddend.
.Iadd.50. The non-transitory computer-readable storage medium of
claim 37, wherein the applying edge-domain decomposition is based
on a rect-function for optical spectra..Iaddend.
.Iadd.51. The non-transitory computer-readable storage medium of
claim 37, wherein: the three-dimensional object is a phase shifting
mask, and the applying edge-domain decomposition includes
pre-simulating light scattering from the one-dimensional edges and
phase effects of etch depths in the phase shifting
mask..Iaddend.
.Iadd.52. The non-transitory computer-readable storage medium of
claim 51, wherein: the scattered light is polarized, and the
applying edge-domain decomposition includes performing separate
decompositions for light parallel to the one-dimensional edges (TE)
and for light perpendicular (TM) to the one-dimensional
edges..Iaddend.
.Iadd.53. The non-transitory computer-readable storage medium of
claim 52, wherein the applying edge-domain decomposition includes
pre-simulating distinct edges in the three-dimensional object with
a plurality of possible illumination directions and field
polarizations..Iaddend.
.Iadd.54. The non-transitory computer-readable storage medium of
claim 37, wherein the applying edge-domain decomposition includes
pre-simulating distinct one-dimensional edges in the
three-dimensional object with a plurality of pre-selected possible
illumination directions and field polarizations..Iaddend.
.Iadd.55. A method of simulating an aerial image projected by a
multiple-opening mask having a plane defined by an X-axis and a
Y-axis, comprising: receiving layout data representing physical
dimensions of one or more openings in the mask; producing first
scattered field data for a selected one of the openings using a
first two-dimensional model oriented along the X-axis; producing
second scattered field data for the selected opening using a second
two-dimensional model oriented along the Y-axis; producing third
scattered field data for the selected opening using a Kirchoff mask
model; and using a computer, producing composite scattered field
data for the selected opening from the first scattered field data,
the second scattered field data, and the third scattered field
data..Iaddend.
.Iadd.56. The method of claim 55, wherein the layout data includes
data concerning phase shift properties of the one or more
openings..Iaddend.
.Iadd.57. The method of claim 55, wherein the layout data includes
data concerning binary mask properties of the one or more
openings..Iaddend.
.Iadd.58. The method of claim 55, wherein the X-axis is oriented in
a TM direction for a polarized electromagnetic wave to be
transmitted through the selected opening and the Y-axis is oriented
in a TE direction for the polarized electromagnetic
wave..Iaddend.
.Iadd.59. The method of claim 55, wherein the method further
comprises decomposing the data representing the physical dimensions
of the one or more openings into data defining the one or more
openings as rectangles..Iaddend.
.Iadd.60. The method of claim 59, wherein the decomposing produces
mutually disjoint rectangles..Iaddend.
.Iadd.61. The method of claim 55, further comprising performing
optical proximity correction using the composite scattered field
data..Iaddend.
.Iadd.62. The method of claim 55, further comprising performing
die-to-database inspection using the composite scattered field
data..Iaddend.
.Iadd.63. The method of claim 55, wherein the producing the
composite scattered field data is performed by linearly combining
the first scattered field data, the second scattered field data,
and the third scattered field data..Iaddend.
.Iadd.64. The method of claim 55, wherein the first scattered field
data and the second scattered field data are produced using
pre-calculated values stored in one or more look-up
tables..Iaddend.
.Iadd.65. A non-transitory computer-readable storage medium storing
computer-executable instructions for causing a computer to perform
a method of simulating an aerial image projected by a
multiple-opening mask having a plane defined by an X-axis and a
Y-axis, the method comprising: receiving layout data representing
physical dimensions of one or more openings in the mask; producing
first scattered field data for a selected one of the openings using
a first two-dimensional model oriented along the X-axis; producing
second scattered field data for the selected opening using a second
two-dimensional model oriented along the Y-axis; producing third
scattered field data for the selected opening using a Kirchoff mask
model; and producing composite scattered field data for the
selected opening from the first scattered field data, the second
scattered field data, and the third scattered field
data..Iaddend.
.Iadd.66. The non-transitory computer-readable storage medium of
claim 65, wherein the layout data includes data concerning phase
shift properties of the one or more openings..Iaddend.
.Iadd.67. The non-transitory computer-readable storage medium of
claim 65, wherein the layout data includes data concerning binary
mask properties of the one or more openings..Iaddend.
.Iadd.68. The non-transitory computer-readable storage medium of
claim 65, wherein the X-axis is oriented in a TM direction for a
polarized electromagnetic wave to be transmitted through the
selected opening and the Y-axis is oriented in a TE direction for
the polarized electromagnetic wave..Iaddend.
.Iadd.69. The non-transitory computer-readable storage medium of
claim 65, wherein the method further comprises decomposing the data
representing the physical dimensions of the one or more openings
into data defining the one or more openings as
rectangles..Iaddend.
.Iadd.70. The non-transitory computer-readable storage medium of
claim 69, wherein the decomposing produces mutually disjoint
rectangles..Iaddend.
.Iadd.71. The non-transitory computer-readable storage medium of
claim 65, wherein the method further comprises performing optical
proximity correction using the composite scattered field
data..Iaddend.
.Iadd.72. The non-transitory computer-readable storage medium of
claim 65, wherein the method further comprises performing
die-to-database inspection using the composite scattered field
data..Iaddend.
.Iadd.73. The non-transitory computer-readable storage medium of
claim 65, wherein the producing the composite scattered field data
is performed by linearly combining the first scattered field data,
the second scattered field data, and the third scattered field
data..Iaddend.
.Iadd.74. The non-transitory computer-readable storage medium of
claim 65, wherein the first scattered field data and the second
scattered field data are produced using pre-calculated values
stored in one or more look-up tables..Iaddend.
.Iadd.75. A method of performing optical proximity correction for a
mask, comprising: receiving layout data representing the mask, the
layout data defining polygons and phase assignments; assigning edge
types to edges of one or more of the polygons, wherein the edge
type for a respective edge is based at least in part on phase
assignment data for an area adjacent to the respective edge;
simulating an aerial image produced by the mask using
two-dimensional models for the edges of the one or more polygons,
wherein the two-dimensional models are selected based at least in
part on the respective edge types of the edges and wherein the
simulating produces edge image intensities; and using a computer,
modifying the layout data by applying an optical proximity
correction method to the layout data using the aerial
image..Iaddend.
.Iadd.76. The method of claim 75, wherein the edge type is one of a
Cr-layer/0.degree. edge, a Cr-layer/90.degree. edge, a
Cr-layer/270.degree. edge, a 0.degree./90.degree. edge, or a
0.degree./270.degree. edge..Iaddend.
.Iadd.77. The method of claim 75, further comprising decomposing
one or more of the polygons in the layout data into mutually
disjoint rectangles..Iaddend.
.Iadd.78. The method of claim 75, wherein the polygons comprise
non-Manhattan geometries..Iaddend.
.Iadd.79. The method of claim 75, wherein the aerial image is
simulated as being produced by quadrupole
illumination..Iaddend.
.Iadd.80. The method of claim 75, wherein the modifying includes
adding one or more of an edge shift or a sub-resolution assist
feature to the layout data..Iaddend.
.Iadd.81. The method of claim 75, wherein the simulating ignores
corner effects..Iaddend.
.Iadd.82. The method of claim 75, wherein the simulating includes
using one or more look-up tables to determine the aerial
image..Iaddend.
.Iadd.83. The method of claim 82, wherein the one or more look-up
tables are generated by calculating electromagnetic field
solutions..Iaddend.
.Iadd.84. A non-transitory computer-readable storage medium storing
computer-executable instructions for causing a computer to perform
an optical proximity correction method for a mask, the method
comprising: receiving layout data representing the mask, the layout
data defining polygons and phase assignments; assigning edge types
to edges of one or more of the polygons, wherein the edge type for
a respective edge is based at least in part on phase assignment
data for an area adjacent to the respective edge; simulating an
aerial image produced by the mask using two-dimensional models for
the edges of the one or more polygons, wherein the two-dimensional
models are selected based at least in part on the respective edge
types of the edges and wherein the simulating produces edge image
intensities; and modifying the layout data by applying an optical
proximity correction method to the layout data using the aerial
image..Iaddend.
.Iadd.85. The non-transitory computer-readable storage medium of
claim 84, wherein the edge type is one of a Cr-layer/0.degree.
edge, a Cr-layer/90.degree. edge, a Cr-layer/270.degree. edge, a
0.degree./90.degree. edge, or a 0.degree./270.degree.
edge..Iaddend.
.Iadd.86. The non-transitory computer-readable storage medium of
claim 84, wherein the method further comprises decomposing one or
more of the polygons in the layout data into mutually disjoint
rectangles..Iaddend.
.Iadd.87. The non-transitory computer-readable storage medium of
claim 84, wherein the polygons comprise non-Manhattan
geometries..Iaddend.
.Iadd.88. The non-transitory computer-readable storage medium of
claim 84, wherein the aerial image is simulated as being produced
by quadrupole illumination..Iaddend.
.Iadd.89. The non-transitory computer-readable storage medium of
claim 84, wherein the modifying includes adding one or more of an
edge shift or a sub-resolution assist feature to the layout
data..Iaddend.
.Iadd.90. The non-transitory computer-readable storage medium of
claim 84, wherein the simulating ignores corner
effects..Iaddend.
.Iadd.91. The non-transitory computer-readable storage medium of
claim 84, wherein the simulating includes using one or more look-up
tables to determine the aerial image..Iaddend.
.Iadd.92. The non-transitory computer-readable storage medium of
claim 91, wherein the one or more look-up tables are generated by
calculating electromagnetic field..Iaddend.
.Iadd.93. A method of inspecting a mask, comprising: receiving data
produced by scanning a beam across a surface of the mask and
imaging scattered electromagnetic fields that result from the
scanning; and comparing the received data to benchmark data,
wherein the benchmark data is produced by applying an edge domain
decomposition method (EDDM) to layout data representing the
mask..Iaddend.
.Iadd.94. The method of claim 93, wherein the benchmark data is
produced at substantially a same time as the scanning..Iaddend.
.Iadd.95. The method of claim 93, further comprising generating a
database storing the benchmark data..Iaddend.
.Iadd.96. The method of claim 93, wherein the benchmark data
represents a defect-free version of the mask..Iaddend.
.Iadd.97. The method of claim 93, further comprising outputting a
signal indicating whether the mask is defective based on the
comparing..Iaddend.
.Iadd.98. The method of claim 93, wherein the benchmark data
includes data produced by: defining a defect-free area of the mask;
simulating the defect-free area of the mask using the EDDM; and
simulating a defect within the defect-free area of the
mask..Iaddend.
.Iadd.99. The method of claim 98, wherein the simulating a defect
includes evaluating an aerial critical dimension (CD)..Iaddend.
.Iadd.100. A non-transitory computer-readable storage medium
storing computer-executable instructions for causing a computer to
perform a method, the method comprising comparing received data to
benchmark data, wherein the received data is produced at least in
part by scanning a beam across a surface of a mask to produce a
scattered electromagnetic field, and wherein the benchmark data is
produced at least in part by applying an edge domain decomposition
method (EDDM) to layout data representing the mask..Iaddend.
.Iadd.101. The non-transitory computer-readable storage medium of
claim 100, wherein the benchmark data is produced at substantially
a same time as the scanning..Iaddend.
.Iadd.102. The non-transitory computer-readable storage medium of
claim 100, wherein the benchmark data is stored in a
database..Iaddend.
.Iadd.103. The non-transitory computer-readable storage medium of
claim 100, wherein the benchmark data represents a defect-free
version of the mask..Iaddend.
.Iadd.104. The non-transitory computer-readable storage medium of
claim 100, wherein the method further comprises, based on the
comparing, outputting a signal indicating whether the mask is
defective..Iaddend.
.Iadd.105. The non-transitory computer-readable storage medium of
claim 100, wherein the benchmark data includes data produced by:
defining a defect-free area of the mask; simulating the defect-free
area of the mask using the EDDM; and simulating a defect within the
defect-free area of the mask..Iaddend.
.Iadd.106. The non-transitory computer-readable storage medium of
claim 105, wherein the simulating a defect includes evaluating an
aerial critical dimension (CD)..Iaddend.
.Iadd.107. A method for simulating light scattering by a
three-dimensional object in an optical imaging system, the method
comprising: applying edge-domain decomposition to the
three-dimensional object to identify light scattering by
one-dimensional edges that define the three-dimensional object; and
using a computer, summing scattered light from all the edges to
simulate light scattering in two dimensions..Iaddend.
Description
BACKGROUND OF THE INVENTION
This invention relates generally to optical imaging systems, and
more particularly the invention relates to simulation and modeling
of electromagnetic scattering of light in such imaging systems. The
invention has applicability to optical proximity correction in
photo masks and mask image inspection, but the invention is not
limited thereto.
The general components of an optical lithography tool, shown
schematically in the diagram of FIG. 1 are the illumination system,
the projection system, the photomask (also called reticle), and the
photoresist, spun on top of a semiconductor wafer. The operation
principle of the system is based on the ability of the resist to
record an image of the pattern to be printed. The mask, already
carrying this pattern, is flooded with light and the projector
forms an image of all mask patterns simultaneously onto (and into)
the resist. The inherent parallelism of this process is the main
reason why optical photolithography is favored over any other
lithography, since it facilitates a very high throughput of 30-120
wafers per hour. The light intensity distribution on top of the
resist surface is commonly referred to as aerial image. The resist
itself is a photosensitive material whose chemical composition
changes during light exposure. The pattern is thereby stored in
form of a latent (bulk) image within the resist. After exposure has
occurred, the resist is developed by means of a chemical process
that resembles the process of developing photographic film. After
development, the exposed parts of the resist remain or dissolve
depending on its polarity (negative or positive, respectively). The
end-result of the lithography process is a more or less exact
(scaled or not) replica of the mask pattern on the wafer surface
that will play the role of a local protective layer (mask) for
subsequent processing steps (etching, deposition,
implantation).
The role of the illumination system is to deliver a light beam that
uniformly trans-illuminates the entire reticle. See FIG. 2(a). It
typically consists of various optical elements, such as lenses,
apertures, filters and mirrors. The light source is responsible for
generating very powerful and monochromatic radiation. Power is
necessary because it is directly related to throughput.
Monochromaticity is important because high quality refractive (or
reflective, in the case of EUV lithography) optics can only be
fabricated for a very narrow illumination bandwidth.
State-of-the-art optical lithography tools employ excimer lasers as
their light source. Deep Ultra Violet (DUV) lithography is the term
used for lithography systems with illumination wavelengths
.lamda.=248 nm (excimer laser with KrF), .lamda.=193 nm (excimer
laser with ArF) and .lamda.=157 nm (excimer laser with F.sub.2).
The successful development of current and future optical
photolithography technologies is hinged upon research advances in
both excimer laser technology and novel materials that possess the
required properties (high optical transmission at DUV wavelengths,
thermal properties, stability after heavy DUV radiation exposure)
by which the optical elements of the system will be made.
All illumination systems in optical projection printing tools are
designed to provide what is known as Kohler illumination. By
placing the source or an image of the source in the front focal
plane of the condenser column, the rays originating from each
source point illuminate the mask as a parallel beam, as seen in
FIG. 2. Here, each source point emits a spherical wave that is
converted by the illumination system into a plane wave incident on
the object (photomask). The angle of incidence of the plane wave
depends on the location of the source point (.alpha.,.beta.) with
respect to the optical axis (0,0). Each parallel beam is a plane
wave whose direction of propagation depends on the relative
position of the source point with respect to the optical axis.
Nonuniformity in the brightness of the source points is averaged
out so that every location on the reticle receives the same amount
of illumination energy. As we will see in subsequent Sections
Kohler illumination can be modeled in a concise mathematical
way.
In addition to dose uniformity, the lithography process should also
maintain directional uniformity such that the same features are
replicated identically regardless of their orientations. The shape
of the light source is therefore circular (or rotationally
symmetric) in traditional optical lithography, although this is not
true for certain advanced illumination schemes such as quadrupole
illumination, where directional uniformity is sacrificed in order
to maximize the resolution of features with certain
orientations.
The coherence of the light source is another important attribute.
Temporal coherence is usually not a big concern, since the narrow
bandwidth of excimer lasers implies high temporal coherence.
Spatial coherence (or just plain coherence) on the other hand is
always carefully engineered and, in most cases, adjustable. Using
special scrambling techniques, the light emitted from any point of
the source is made completely uncorrelated (incoherent) to the
light emitted from every other point. However, light gathers
coherence as it propagates away from its source. The frequently
quoted partial coherence factor a is a characteristic of the
illumination system and is a measure of the physical extent and
shape of the light source. The larger the light source, the larger
the partial coherence factor, and the light source has a lower
degree of coherence. In the limit of an infinite source, imaging is
incoherent and .sigma.=.infin.. On the other hand, the smaller the
light source, the smaller the partial coherence factor, and the
higher the degree of coherence. Imaging with a point source is
fully coherent and .sigma.=0. Note that a point source in a Kohler
illumination will result in a single plane wave illuminating the
mask and the angle of incidence of this wave depends on the
relative position of the point source with respect to the optical
axis. For partial coherence factors between zero and infinity,
imaging is partially coherent. Typical partial coherence factors in
optical lithography range from 0.3 to 0.9.
The projection system typically consists of a multi-element lens
column (up to 30-40 lenses) that may also have apertures, filters,
or other optical elements, and it is a marvel of engineering
precision in order to be able to reliably project images with
minimum dimensions on the order of 100 nm for state-of-the-art
systems. One of the main reasons for the required high precision is
control of the aberrations or deviations of the wavefront from its
ideal shape. Two relevant parameters of the projection system are
the numerical aperture, NA, and the reduction factor, R. The
numerical aperture is, by definition, the sine of the half-angle of
the acceptance cone of light-rays as seen from the image side of
the system. The ratio of image-height to object-height is, by
definition, the magnification factor M of the system. The inverse
of the magnification factor is the reduction factor, R. Since a
typical system in photolithography projects at the image plane a
scaled-down version of the object (mask), M is less than 1 and R is
greater than 1. State-of-the-art systems currently have reduction
factors of R=4 or 5. Note that two numerical apertures exist in the
projection system, namely NA.sub.i (or simply NA) and NA.sub.o,
which refer to the half-angle of the acceptance cone as seen from
the image side and from the object (mask) side, respectively. They
are related through the reduction factor as follows:
.times..times. ##EQU00001##
For a circularly shaped light source, the partial coherence factor
.sigma. mentioned above is related to the numerical apertures of
both the projection system and the illumination system.
Specifically, .sigma. is given by:
.sigma..times..times. ##EQU00002## where NA.sub.c is the numerical
aperture of the condenser lens (illumination system) and NA.sub.p
is the numerical aperture of the projector lens. Some confusion
arises from the fact that, in the above equation, the reduction
factor of the imaging system is implicitly taken into account. FIG.
2(b) clarifies the situation by showing simplified diagrams of two
optical systems with parameters NA=0.5, .sigma.=0.5 and R=5 or
R=1.
The photomask, also called reticle, carries the pattern to be
printed at a given lithography processing step. The masks of
integrated circuits having large die-sizes or footprints (that is,
occupying large areas on the semiconductor wafer), typically carry
just one copy of the chip pattern. A matrix of several chip
patterns is contained in one mask whenever the chip size permits.
Note that the mask is drawn R times the actual size on the
semiconductor wafer, since the dimensions of the circuit will be
scaled down by the reduction factor, R. For this reason it is not
sufficient to just provide feature sizes, since it may not be
immediately obvious from the context whether these are photomask
(object) or resist (image) sizes. A typical convention for
distinguishing photomask feature sizes from resist feature sizes is
to include in parenthesis the reduction factor R. For example, a
600 nm (4.times.) line has a size of 600 nm on the mask, and would
produce a 600 nm/4=150 nm line if used in a 4.times.imaging system.
Similarly, a 130 nm (1.times.) line refers to the size of a line at
the image (wafer) plane and would result from the printing of a 130
nm line on the mask for a system with R=1, or a 520 nm (4.times.130
nm) line on the mask for a system with R=4, or a 1.3 .mu.m
(10.times.130 nm) line on the mask for a system with R=10.
Depending on their operation principle, photomasks can be divided
into two broad categories: conventional binary or chrome-on-glass
(COG) masks and advanced phase-shifting masks (PSM).
A binary or COG mask consists of a transparent substrate (mask
blank), covered with a thin opaque film that bears the desired
pattern. Light can either pass unobstructed through an area not
covered by the opaque film or be completely blocked if it is
incident on an area that is protected by the film. This binary
behavior of the transmission characteristic of the mask is
responsible for its name. The mask blank for DUV lithography
typically consists of fused silica glass that has excellent
transmission at .lamda.=248 nm and somewhat poorer but acceptable
transmission at .lamda.=193 nm and .lamda.=157 nm. The opaque film
is typically on the order of 100 nm thick and has a chromium (Cr)
composition.
Adding phase modulation to the photomask can profoundly increase
the attainable resolution. This is the principle followed by
phase-shifting masks (PSM), which employ discrete transmission and
discrete phase modulation. There are many different flavors of PSMs
depending on the way that the phase modulation is achieved. One of
the most promising PSM technologies is what is known as alternating
phase-shifting mask (alt. PSM, or APSM). Here are cut-planes of
geometry of a binary (COG) mask (a) and an alternating phase-shift
mask (b). The ideal electric field distribution for the binary mask
(c) leads to a poor image intensity distribution (e) at the image
plane, whereas the ideal electric field distribution for the alt.
PSM (d), because of destructive interference, leads to a robust
image. The principle of an alt. PSM is compared with that of a
binary mask in FIG. 3. The center line is bordered by transmitting
regions with 180.degree. phase difference on an alt. PSM and by
clear areas of the same phase on a binary mask. The phase
difference on the alt. PSM leads to destructive interference,
resulting in a sharp dark image. The binary mask image is not as
sharp because of the lack of phase interaction. The 180.degree.
phase difference is created by etching trenches, also called
phase-wells, into the fused silica substrate during the alt. PSM
fabrication process, which is now more complex than the COG
fabrication process. The difference in the amount of material
removed d.sub.etch is such that the path length difference between
light passing through the different phase regions is half of the
wavelength in air.
Sub-wavelength lithography, where the size of printed features is
smaller than the exposure wavelength, places a tremendous burden on
the lithographic process. Distortions of the intended images
inevitably arise, primarily because of the nonlinearities of the
imaging process and the nonlinear response of the photoresist. Two
of the most prominent types of distortions are the wide variation
in the linewidths of identically drawn features in dense and
isolated environments (dense-iso bias) and the line-end pullback or
line-end shortening (LES) from drawn positions. The former type of
distortion can cause variations in circuit timing and yield,
whereas the latter can lead to poor current tolerances and higher
probabilities of electrical failure.
Optical proximity correction or optical proximity compensation
(OPC) is the technology used to compensate for these types of
distortions. OPC is loosely defined as the procedure of
compensating (pre-distorting) the mask layout of the critical IC
layers for the lithographic process distortions to follow. This is
done with specialized OPC software. In the heart of the OPC
software is a mathematical description of the process distortions.
This description can either be in the form of simple shape
manipulation rules, in which case the OPC is referred to as
"rule-based OPC," or a more detailed and intricate process model
for a "model-based OPC." The OPC software automatically changes the
mask layout by moving segments of line edges and adding extra
features that (pre-) compensate the layout for the distortions to
come. Although after OPC has been performed the mask layout may be
quite different than the original (before OPC) mask, the net result
of this procedure is a printed pattern on the wafer that is closest
to the IC designer's original intent.
In the early 1990's the problem of OPC for large mask layouts was
addressed formally as an optimization problem. The size of such a
problem is formidable, but through introduction of appropriate
constraints--induced primarily from mask fabrication constraints,
local optimal mask "points" were successfully demonstrated. This
research has led to commercially available software tools that
perform OPC on a full-chip scale. All three approaches rely heavily
on speedy calculations of the image intensity at selected points of
the image field. Although the methods by which they achieve this
appear seemingly different, they are nevertheless the same, in the
sense that they all rely on a decomposition of the kernel of the
imaging equation for partially coherent light.
Domain decomposition techniques, where a large electromagnetic
problem is broken up into smaller pieces and the final solution is
arrived at by synthesizing (field-stitching) the elemental
solutions, have been proposed for the study of one-dimensional
binary (phase only) diffractive optical elements. Others working on
the same problem have demonstrated how to create and use a
perturbation model for binary edge-transitions based on the product
of the ideal, sharp transition and the continuous field variations
in the vicinity of the edge. Independently, a similar technique to
field-stitching has been proposed for the simulation of
two-dimensional layouts of advanced photomasks (alternating PSM,
masks with OPC) that properly models interactions from neighboring
apertures and furthermore takes advantage of the spectral
properties of the diffracted fields to come up with a compact model
for the edge-transitions.
The present invention is directed to an improved methodology for
OPC on optical masks.
SUMMARY OF THE INVENTION
In accordance with the invention, a complex two-dimensional layout
of a photomask or other three-dimensional object is systematically
decomposed into a finite number of elementary two-dimensional
objects with the ability to cause one-dimensional changes in light
transmission properties. An algorithmic implementation of this can
take the form of creation of a look-up table that stores all the
scattering information of all two-dimensional objects needed for
the synthesis of the electromagnetic scattered field from the
original three-dimensional object.
More particularly, the domain is decomposed into edges, where
pre-calculated electromagnetic field from the diffraction of
isolated edges is recycled in the synthesis of the near diffracted
field from arbitrary two-dimensional diffracting geometries. In
this invention, the sum of the complex fields is taken rather than
the product due to the linearity of the Kirchoff-Fresnel
diffraction integral, which makes the decomposition method
possible.
The invention has particular applicability in die-to-database
inspection where an actual image of a mask is compared with a
synthesized image that takes imaging artifacts of comers, edges and
proximity into account. A very important current challenge in
synthesizing images is the correct evaluation of electromagnetic
effects at edges of phase-shifted mask features. These
electromagnetic effects are a strong function of the edge
topography of the nonplanar features introduced during the mask
writing process. The inspection of every feature on a 50M
transistor chip must be accomplished in an inspection time on the
order of 5 hours for feature sizes of 100 nm. Future inspection
rates are expected to hit 200M inspection pixels per hour, which
would allow a mask with 50 nm features and 200M transistors to be
inspected at a pixel size of 25 nm in about 5 hours.
Another application is optical proximity correction which consists
of evaluating the image of every feature on a mask and improving it
by introducing edge shifts and iteratively adjusting the amounts of
these shifts. A typical OPC correction process on a 50M transistor
mask takes about a full day. Since the number of iterations in the
OPC shifts is on the order of 5 the full mask image is computed on
average roughly at the same speed of the die-to-database
inspection, or 50M transistors every 5 hours for the 100 mn
generation and 200M transistors per hour for the 50 nm
generation.
The invention and objects, features and applications thereof will
be more apparent from the following detailed description and
appended claims when taken with the drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a general diagram of an optical photolithography
tool.
FIG. 2(a) illustrates the illumination system of FIG. 1 with
Kohler's method.
FIG. 2(b) illustrates numerical apertures of illumination and
projection lenses in the tool of FIG. 1.
FIGS. 3(a)-3(f) are a comparison of operational principles of an
alternating phase shift mask (alt. PSM) and a binary (COG) mask as
used in the invention.
FIG. 4 illustrates light decomposition of an opening into two
edges.
FIGS. 5(a)-5(f) illustrate an edge domain decomposition method in
accordance with the invention.
FIGS. 6(a)-6(f) illustrate an edge domain decomposition method
applied on an isolated line.
FIG. 7 illustrates an edge domain decomposition method applied on
an arbitrary polygon.
FIGS. 8(a)-8(d) illustrate dependence of edge scattering on edge
profile and polarization.
FIGS. 9(a)-9(d) illustrate complex differences of true
edge-scattering from ideal and sharp transition.
FIGS. 10(a)-10(d) illustrate the spectra of complex differences
shown in FIGS. 9A-9D.
FIG. 11 illustrates edge domain decomposition on a large mask
layout.
FIGS. 12(a)-12(c) illustrate edge domain decomposition applied on
an isolated square hole (scattered field).
FIGS. 13(a)-13(c) illustrate edge domain decomposition applied on
an isolated square hole (spectrum).
FIGS. 14(a)-14(c) illustrate edge domain decomposition applied on
an isolated square island (scattered field).
FIGS. 15(a)-15(c) illustrate edge domain decomposition applied on
an isolated square island (spectrum).
FIG. 16 illustrates edge domain decomposition on a large mask
layout (images).
FIGS. 17(a), 17(b) illustrate matched bandwidth approximation to
edge diffraction.
FIGS. 18(a), 18(b) illustrate matched bandwidth approximation
applied to the hole of FIG. 12.
FIGS. 19(a), 19(b) illustrate defect assessment using prior art
techniques.
FIG. 20 illustrates domain decomposition in accordance with an
embodiment of the invention when a defect is present.
FIGS. 21(a), 21(b) illustrate layout and amplitude of scattered
field below the mask at the observation plane.
FIGS. 22(a)-22(d) illustrate a 90 degree phase defect present in
the center of the bottom-left hole of a mask.
FIGS. 23(a)-23(d) illustrate a defect present in the center of the
bottom-right hole of a mask.
FIGS. 24(a)-24(b) are a comparison of the decomposition method with
the rigorous determined scattering.
FIGS. 25(a)-25(b) are plan views of defective masks of FIGS. 22 and
23.
FIGS. 26(a)-26(b) are spectra plots (magnitude) of errors of FIGS.
22 and 23.
DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS
Adam and Neureuther, "SIMPLIFIED MODELS FOR EDGE TRANSITIONS IN
RIGOROUS MASK MODELING," 26.sup.th Annual International Symposium
on Microlithography, Mar. 1, 2001, discloses domain decomposition
methods (DDM) that allow the division of a larger mask diffraction
problem into a set of constituent parts or single openings in the
mask.
The present invention provides an extension of the disclosed domain
decomposition methods (DDM). A domain decomposition method based on
edges uses pre-calculated electromagnetic field from the
diffraction of isolated edges which is recycled in the synthesis of
the near diffracted field from arbitrary two-dimensional
diffracting geometries. Although at first glance it might be
tempting to discard edge-decomposition techniques as inaccurate for
masks with large vertical topography and features with small
lateral dimensions, it will be shown that for a large set of
practical situations, they are accurate. A key difference in this
method from the prior art is that the sum of the complex fields is
taken instead of the product. The reason for this is in the
linearity of the Kirchhoff-Fresnel diffraction integral, and this
linearity makes possible the decomposition method.
A natural extension of the DDM that is also based on the linearity
of the Kirchhoff-Fresnel diffraction integral is shown in FIG. 4.
An opening with transmission T and phase .phi. in an infinitely
thin, opaque screen can be decomposed by way of the linearity of
the Kirchhoff-Fresnel diffraction integral into two edges. The
uniform plane wave illumination is subtracted to restore the light
level everywhere. The 1D mask layout is first decomposed into
single opening masks, as noted above, and subsequently each opening
is decomposed into two edges and a uniform illumination field as in
FIG. 4. Such a decomposition can be proven to be exact for
infinitely thin and perfectly conducting screens through the use of
the electromagnetic form of Babinet's principle, but it should
break down if applied to photomasks with "thick" vertical
structures (compared to .lamda.) and not perfectly absorbing screen
materials. Also, because of the reduced ability of light to
penetrate through small (compared to .lamda.) openings, it is
expected to break down for small mask features. This decomposition
method will be hereafter referred to as the edge domain
decomposition method or edge-DDM. The application of the edge-DDM
to a single isolated space and an isolated line are described next,
followed by a systematic evaluation of the limits of applicability
of the method.
Consider the isolated mask space (opening) shown in FIGS.
5(a)-5(f). The amplitude of the complex Ey-field under Ey (TE)
illumination is shown in FIG. 5(c) and the scattered field across
the observation plane is shown in FIG. 5E. According to FIG. 4, the
field across the observation plane can be alternatively obtained by
using the scattered field from an edge that has the same vertical
profile as the space, as shown in FIGS. 5(b) and 5(d). The
resulting field from this edge-DDM is overlaid on the plot of FIG.
5E along with the field from the edge-scattering. The normalized
mean square error 1 (NMSE) between the r-mask and the edge-DDM is
0.19%. In FIG. 5(f) the spectra of the r-mask and edge-DDM are
compared and it is seen that they differ by 0.13% in a NMSE sense
if the whole spectrum of propagating waves is considered and by
merely 0.007% within the collection ability of a projection system
with NA=0.8 and R=4. Since only the portion of the spectrum that is
collected by the projection system contributes to the image
formation, the exceptional accuracy of the edge-DDM is this regime
is more than adequate for accurate image simulations.
Similar steps can be followed for the isolated line shown in FIG.
6(a) illuminated with Ex (TM) polarized light. Note that it is not
typical to have a line as shown here, where in both sides the glass
has been etched. Normally only one side of the line is
phaseshifted, but this example simply aims to introduce the
principle of the edge-DDM. A line with one side phase-shifted and
the other left intact can still be decomposed via the edge-DDM
using the diffraction fields from the two different edges. The
amplitude of the complex Ex field everywhere in the domain of FIG.
6(a) is shown in FIG. 6(c) and for the edge of FIG. 6(b) in FIG.
6(d) respectively. The true (r-mask) scattered field is compared
with the synthesized field with edge-DDM in FIG. 6(e) and the
spectra are compared in FIG. 6(f). The error (NMSE) is seen to be
small in both the near field and spectra plots and specifically the
error of the through-the-lens (TTL) spectrum is just 1.5E-5.
In the examples of FIG. 5 and FIG. 6 the edge-DDM is seen to be
accurate compared with the simulation of the exact mask structure.
Although the vertical mask topography (180.degree..apprxeq.170
nm.apprxeq.1.lamda.) is comparable to the wavelength, the fact that
the lateral dimensions of the space and the line are relatively
large compared with the wavelength (CD=400 nm>2.times.193
nm=2.times..lamda.) renders the edge-DDM accurate. One expects that
as the lateral sizes become smaller and the vertical mask
topography even larger the edge-DDM will eventually break down. The
limits of the edge-DDM were investigated through exhaustive
simulation of isolated spaces and lines on the mask with various
etched depths and amounts of undercut. The sidewall angles were
always kept vertical and no rounding at the bottom of the etched
wells was included. The mask CD was varied in the set {2 .mu.m, 1
.mu.m, 0.6 .mu.m, 0.4 .mu.m, 0.2 .mu.m(.about.1.lamda.), 0.1
.mu.m}, the etched depth in {0.degree., 90.degree., 180.degree.,
270.degree., 360.degree.} and the amount of underetch in {0 nm, 25
nm, 50 nm, 75 nm, 100 nm}. All 2 (space/line) by 2 (TE/TM
illumination) by 6 (CD sizes) by 5 (etched depths) by 5
(underetch)=600 cases were run and the rigorous (r-mask) scattered
field was automatically compared with that obtained from edge-DDM.
The results of the investigation were spectacular. The edge-DDM can
readily achieve better than 1% accuracy (in a NMSE sense) in the
near field for CD=400 nm (4.times.) or larger regardless of depth,
underetch and polarization. The accuracy in the TTL-spectrum is
sufficient in those cases even for inspection (R=1) simulations.
For lines, the edge-DDM maintains excellent accuracy in the
TTL-spectrum with R=4 down to a mask CD of 100 nm
(.about.0.5.lamda.) with 90.degree. of etched depth and 0 nm of
undercut, or mask CD of 200 nm (.about.1.lamda.) with 270.degree.
of etched depth and 25 nm of undercut. For spaces, the accuracy of
the method is acceptable at least down to mask CD of 200 nm
(.about.1.lamda.) with 270.degree. of etched depth and 50 nm of
underetch.
Current state-of-the-art photomasks and processes utilize
sub-resolution assist features that are never less than one
wavelength in size on the mask. This clearly implies that the
edge-DDM has the potential to accurately simulate the most advanced
photomask technologies (OPC, alt. PSM, OPC on alt. PSM). Again, the
true power of the method lies on its application in 2D mask
layouts. However, one key difference with the DDM of mask openings
is the following: only one rigorous 2D simulation of an edge
suffices for the scattered field reconstruction of any arbitrary
size opening on the mask. The DDM of openings requires a separate
simulation of every different mask size. Next, the application of
the edge-DDM in 2D layouts is developed.
The application of the edge-DDM in 2D layouts is straightforward. A
mask layout comprises of a large number of edges positioned at
different locations and having different orientations. No matter
how complicated the mask technology, there are usually only a small
number of different types of edges present in the layout. For
example, a single exposure 0.degree./90.degree./270.degree. alt.
PSM has five types of edges: i) Cr-layer/0.degree. edge, ii)
Cr-layer/90.degree. edge, iii) Cr-layer/270.degree. edge, iv)
0.degree./90.degree. edge and 0.degree./270.degree. edge. Depending
on the orientation of each edge in the layout it "sees" and
responds to the incident field differently. If the illuminating
field is a TE (E.sub.y) normally incident plane wave and the edge
is oriented along the y-axis it "sees" TE illumination, but if it
is oriented along the x-axis it "sees" TM illumination. This is
illustrated in FIG. 7. Depending upon the orientation of each edge
in the layout it "sees" and responds to the incident field
differently. Here the illuminating field is a TE (E.sub.y) normally
incident plane wave and the edges that are oriented along the
y-axis "see" TE illumination (parallel to the edge), whereas the
edges that are oriented along the x-axis "see" TM illumination
(perpendicular to the edge). A separate 2D edge simulation is
required to capture the different response of the edge to different
field polarizations. The response of each edge to its respective
illumination is taken into account rigorously from a pre-stored 2D
edge-diffraction simulation. However, corner effects are
effectively ignored, since the finite extent of each edge is not
rigorously taken into account. Instead, the scattered field at the
end points of every edge is abruptly terminated (truncated) in a
perfect square-wave fashion to the field value of the kmask model.
Comer effects in typical imaging situations are mapped at the
extremities of the spectra and do not contribute to the image
formation. Owing to this observation, the edge-DDM applied in 2D
layouts is successful in accurately capturing the true
electromagnetic behavior of edges in a rapid manner, while the
insignificant comer effects are safely ignored.
Slightly different embodiments of the edge-DDM applied in 2D
layouts can be devised. One that is powerful and general in its
application is described next. Then, scattering results from
various edge profiles that would be encountered in alt. PSMs are
presented for completeness. Next, application of the edge-DDM in
simple 2D mask layouts is presented and the results of the method
are compared with fully rigorous r-mask models in order to
establish the validity of this approach. Finally, a large portion
of a real layout from a single exposure
0.degree./90.degree./270.degree. alt. PSM that is too big for 3D
rigorous simulation is simulated via the edge-DDM and the aerial
image is compared with the simple k-mask model.
First, every distinct edge that is present on the layout is
pre-simulated with all possible illumination directions (field
polarizations) that are required, based on the orientations of the
edge encountered within the layout. For simplicity only
Manhattan-type layouts are considered, where the edges are oriented
along either the x- or the y-axis and require electromagnetic
simulation under both parallel and perpendicular to the edge
polarizations. The extension to non-Manhattan layouts although much
more computationally laborious is straightforward. Subsequently,
each distinct mask layer is broken up into a set of mutually
disjoint (non-overlapping) rectangles. For example, a plain binary
mask consists of two layers, namely the absorption layer (typically
Cr-based) and the clear (uncovered) layer, whereas a more advanced
0.degree./90.degree./270.degree. three-phase alt. PSM consists of
four layers, namely the absorption layer and the 0.degree. clear,
90.degree. clear and 270.degree. clear layers. Next, looping
through all rectangles of each layer, the type of each one of the
four edges of the rectangle is determined based on what its
neighboring layer is. Trivial edges (when the neighboring rectangle
is of the same type) are discarded. It is also possible that only
part of a single edge from a rectangle is neighboring with one
layer and the rest is neighboring with a different layer (or
different layers). In that case the edge is broken up in multiple
edges, in a way that tracks the neighboring layer. What is left at
the end of this process is a set of all non-trivial edges that are
present in the layout (location, size and orientation) and their
type, i.e. which are the two mask layers on each side of the edge.
Given the mask layout, the k-mask (Kirchhoff-mask) model is readily
available. Finally, the respective (complex) difference of the true
edge-scattering from the ideal and sharp k-mask model is added to
all non-trivial edges accordingly. This revised mask model that
results from the edge-DDM includes accurate information about the
electromagnetic scattering from the edges and it is a
quasi-rigorous mask model (qr-mask) in the same sense that the term
was used where only 2D mask simulations were used to approximate a
3D electromagnetic problem.
The above algorithm was implemented in a MATLAB environment. The
part that deals with the polygon and edge extraction can probably
be judged as rudimentary and most likely cannot compete in speed
with more sophisticated implementations in production caliber CAD
software, but it suffices for the purposes of this work, that is,
to demonstrate the proof of concept of the edge-DDM and the
feasibility of including accurate edgescattering information in
rapid aerial image simulations.
Examining the departure of the scattered field from an edge from
its ideal, sharp edge-transition and how it depends on the profile
of the edge and the incident field polarization can provide
intuition of the physical mechanisms involved.
FIG. 8 shows the amplitude of the (near) scattered field from an
edge with various profiles. The ideal, sharp edge-transition is
also shown on every graph for comparison. In FIG. 8(a) and (b) the
edge-scattering for different etched depths (0.degree., 180.degree.
and 360.degree.) and vertical sidewalls with no undercut is shown
for TE (E.sub.y) and TM (E.sub.x) illumination respectively. The
effect of undercut is depicted in FIG. 8(c) and (d), where the
edgescattering for different amounts of undercut (0 nm, 25 nm, 50
nm and 75 nm) and vertical sidewalls with 180.degree. etched depth
is shown for TE and TM illumination respectively. It is interesting
to note that increasing the etched depth is "making the edge
response slower", meaning that the near scattered field from the
edge requires more distance to reach its "on" value from its "off"
value, although this effect is more pronounced for the first
increment from 0.degree. to 180.degree. rather than from
180.degree. to 360.degree.. Similarly, increasing the amount of
undercut appears to "make the edge response faster" and furthermore
this behavior appears to be linear, since for every increment of
the undercut the edge-transition gains a constant distance in its
"off-on" behavior.
It is instructive to isolate the difference of the true
edge-scattering from the ideal, sharp edge-transition. This is done
in FIG. 9, which depicts the amplitude of this (complex) difference
for all respective plots of FIG. 8. Here the amplitude of the
complex difference of the true electromagnetic field scattered from
an edge from the ideal, sharp transition (step-function) is shown
in FIGS. 9(a)-(d) for all edge profiles and illumination
polarizations of FIG. 8 respectively. FIGS. 9(a) and (c) are for TE
polarization and FIGS. 9(b) and (d) are for TM polarization. The
spatial extent of the difference grows with increasing etched
depths, as seen in FIGS. 9(a) and (b), and shrinks with increasing
undercuts, as seen in FIG. 9(d) for TM polarization. The behavior
in FIG. 9(c) appears somewhat erratic, but if the phase is also
taken into account, it is shown in FIG. 10 that this is also the
case for TE polarization. Observe that, indeed, by increasing the
etched depth the difference has a larger spatial extent, whereas
increasing the amount of undercut confines the spatial extent of
the difference. This is crucial, because it is exactly the spatial
extent of the difference that affects its spectral distribution. If
the difference is well confined, its spectral distribution is
spread out to higher spatial frequencies and the lower frequency
content is small. In such a case the complex difference can be
neglected and the approximation of the ideal, sharp transition is
adequate.
However, when the difference has a larger spatial extent, then its
lower frequency content becomes significant and it cannot be
neglected. This is shown in FIG. 10, which depicts the magnitude of
the spectra of all respective difference distributions of FIG. 9.
Here the spectra of the complex differences shown in FIG. 9 are
depicted here respectively. FIGS. 10(a) and (c) are for TE
polarization and FIGS. 10(b) and (d) are for TM polarization.
Observe that the through-the-lens (TTL) parts of these spectra,
which for typical lithographic imaging situations (R-4,
NA.about.0.7-0.8) extent up to .about..+-.0.008 (nm).sup.-1 are
smaller for smaller etched depths and larger undercuts, regardless
of polarization, as expected from the observations on the plots of
FIG. 8 and FIG. 9. Observe that the TTL spectrum of the difference
attains larger values when the etched depth increases and smaller
values when the amount of undercut increases, as expected from FIG.
8 and FIG. 9. Remember that the normalization used in spectra plots
is that a constant (DC) value of 1 (clear field) would produce a
Dirac .delta.-function at zero frequency with amplitude of 19.4 (
{square root over (.eta.)}= {square root over (
.mu..sub.0/.epsilon..sub.0)}.apprxeq.19.4 .OMEGA.).
The application of the edge-DDM on an isolated square hole and an
isolated square island (post) and comparison of the results with
fully rigorous 3D mask simulations (rmasks) are presented here. The
edge profile has a glass etched depth of 180.degree. and an
undercut of 50 nm. The dimensions of both the hole and the island
are 400 nm.times.400 nm (4.times.), corresponding to approximately
2.times.2 wavelengths, for .lamda.=193 nm.
FIG. 12(a) and (b) depict the amplitude of the scattered field
across the observation plane below the hole obtained via the
edge-DDM and via fully rigorous 3D simulation, respectively. The
amplitude of their difference (error of the edge-DDM) is shown in
FIG. 12(c). Locally the amplitude of the error is seen to reach a
discouraging level of almost 50% of the clear field value. The
normalized mean square error is however more contained and is only
about 4.9% if calculated within a 3 .mu.m by 3 .mu.m area or 3.6%
if calculated within the 0.4 .mu.m by 0.4 .mu.m open area of the
hole, as shown in FIG. 12. From the work presented so far, it
should be obvious that neither of these error levels is immediately
relevant in the simulation of the image formation. What is rather
more relevant is the error incurred on the diffraction orders that
are collected by the imaging optics. If the spatial frequencies of
the error depicted in FIG. 12(c) are sufficiently high, this error
will be discarded by the projection lens. This concept is shown in
FIG. 13, where the spectra (magnitude) of the scattered fields from
edge-DDM and from rigorous 3D simulation are shown in FIG. 13(a)
and (b) respectively. The marginal circles indicating the
propagating plane waves (with |{right arrow over
(k)}|.ltoreq.2.pi./.lamda.) and the collected by the imaging system
plane waves (with |{right arrow over
(k)}|.ltoreq.2.pi.NA(1+.sigma.)/.lamda.R), for NA=0.75, R=4 and
.sigma.=0.3, are indicated on these plots. The magnitude of the
error in the spectrum of the field obtained from edge-DDM is shown
in FIG. 13. Observe that most of the error is concentrated in the
higher spatial frequencies for |k.sub.x| approaching 0.03 and only
small levels of error exist in the collected orders. The normalized
mean square errors of all propagated and collected orders are 2.2%
and 2E-4 respectively.
Similarly, the amplitude of the scattered fields across the
observation plane below the isolated island obtained with the
edge-DDM and with fully rigorous 3D simulation and the amplitude of
the error are shown in FIG. 14. Again, a locally significant error
amplitude near the comers is evident. However, the energy of this
error is concentrated mostly at the higher spatial frequencies, for
|k.sub.x| approaching 0.03 and the error in the collected orders is
seen to be small, only 0.16% in a NMSE sense. Note that the large
DC component of the isolated island mask layout was subtracted in
order to make meaningful comparisons of the edge-DDM with fully
rigorous simulations. Otherwise this large DC component would
artificially show an even better accuracy, since the error level in
comparison with a much stronger signal (containing the DC term)
would be smaller.
Each fully rigorous mask simulation of the patterns of FIG. 12 and
FIG. 14 that was performed with TEMPEST used approximately 1.8
Gbytes of memory and required almost 2 days on a 550 Mhz CPU. On
the other hand, once all of the required 2D edge simulation results
are available the scattered field can be obtained with the edge-DDM
in less than 1 sec. The simulation of the isolated
edge-diffractions is an off-line procedure. It only needs to be
performed once and the edge-diffraction results are recycled as
many times as are necessary for the reconstruction of the scattered
fields of arbitrary layouts. In the above examples, it took
approximately 5 min for each rigorous, isolated-edge 2D
simulation.
The edge-DDM is used here to generate the scattered field from a
three level (0.degree./90.degree./270.degree.) alt. PSM. A portion
of the arbitrary target layout is shown in FIG. 11 before and after
the phase assignment. Rectangle and edge extraction is performed
within the 3 .mu.m by 4 .mu.m (1.times.) coherence window shown and
the scattered field obtained with the edge-DDM inside the 12 .mu.m
by 16 .mu.m (4.times.) coherence window is also shown in FIG. 11
for TE and TM illuminations at .lamda.=193 nm. Note that a fully
rigorous 3D mask simulation of a 12 .mu.m by 16 .mu.m (4.times.)
layout would require close to 40 Gbytes of memory and would only be
feasible on a multi-CPU architecture. The scattered field is
obtained with the edge-DDM implementation in less than 1 min. This
time includes the polygon and edge extraction and the field
synthesis from pre-calculated edge-diffractions. Therefore, the
results of the edge-DDM cannot be immediately compared with a fully
rigorous solution, but the accuracy of the edge-DDM within the
imaging system's collection ability can reveal the inaccuracies of
the ideal k-mask, where all transitions are assumed perfectly
sharp. This is done in FIG. 16, where the images (at the 30%
intensity level) obtained with the k-mask and the edge-DDM for two
different undercuts (0 nm and 50 nm) are compared. The system
parameters are .lamda.=193 nm, NA=0.7, .sigma.=0.3 and R=4. The
inaccuracies of the k-mask model are revealed at various locations
in the image as seen in FIG. 16. The effect on the image of
changing the undercut from 0 nm to 50 nm can be quickly and
accurately evaluated. Observe the subtle differences evident in the
intensity contours of the 0 nm and 50 nm masks of FIG. 16.
Although the edge-DDM presented earlier achieves excellent accuracy
and tremendous speed-ups as compared to a fully rigorous mask
simulation, there is still one more parameter to exploit in order
to further speed-up the method. There is no need to carry all the
details of the true field scattering as far as the imaging process
is concerned (with larger than 1 reduction factors), since the
higher spatial frequencies are not used. If a lumped parameter
model can be devised that closely matches the TTL spectrum of the
scattered field then the task of an accurate field representation
is accomplished. A rect-function, a raised cosine function and a
Gaussian function can successfully mimic the TTL spectra of
scattered fields from various size and profile openings. A similar
principle can be applied here with the purpose of matching the TTL
spectra of edgediffracted fields. Assuming a restriction to
piecewise constant models, the scattered field from an isolated
edge under TE or TM illumination can be approximated using the
multistep functions shown in FIG. 17. Although not shown in FIG.
17, the amplitude, phase and size of all steps are optimized to
achieve the best spectral matching of the collected (TTL) orders
with the orders resulting from the continuous edge-diffracted
field. Note that the key factor for a good spectral match is to
capture the finite extent of the "off-on" edge-transition. A
piecewise constant function is probably not the best choice to
achieve this task, but on the other hand a piecewise constant
function offers a good lumped parameter approach and it lends
itself nicely as an extension of the unmatched Kirchhoff approach,
where all edge-transitions are assumed perfectly sharp. Instead of
approximating the true edge-diffraction by a single step function
(geometrical or Kirchhoff approximation) a multi-step piecewise
constant function is employed. The amplitudes, phases and sizes of
all steps are optimized to achieve the best spectral matching of
the collected orders with the orders resulting from the continuous
edge-diffracted field. The matched bandwidth approximations to the
field diffracted by a 180.degree. edge with 50 nm undercut under TE
and TM illuminations are shown in (a) and (b) respectively.
If the matched bandwidth edge-DDM is applied on the 400 nm by 400
nm (4.times.) isolated hole of FIG. 12 the scattered field across
the observation plane can be represented in the compact way shown
in FIG. 18(a). Recall that the edges parallel to the y-axis "see"
TE polarization and the edges parallel to the x-axis TM
polarization. The multi-step matched BW approximations to the
edge-diffraction shown in FIG. 17 for TE and TM illuminations
intermingle in the x- and y-directions to produce the multi-color
mask of FIG. 18(a), where each color represents a different
amplitude and phase region of the mask. The spectrum of the error
of this approach is depicted in FIG. 18(b). Note that compared to
the error shown in FIG. 13(c) this error has increased by almost
four-fold. Nevertheless, the error in the TTL spectrum still
remains at low enough levels for accurate imaging simulations. A
better approximating function, other than the piecewise constant
step function, that is more capable of capturing the true
edge-diffraction profile (such as linear or higher order
polynomial) should result in better spectral matching and
consequently more accurate imaging simulations.
Consider now a domain decomposition strategy appropriate for
handling wavelength size phase defects. The bright future for alt.
PSMs as a major resolution enhancement technique (RET) has been
plagued primarily by the difficulty to reliably inspect them, flag
locations that phase defects are present and subsequently repair
the defective locations. State-of-the-art inspection systems
utilize a focused laser beam that can be either at-wavelength (up
to .lamda.=248 nm), meaning that the wavelength of the inspection
beam is the same as the wavelength that the alt. PSM was designed
for, or emit light of a larger wavelength. This beam scans the
entire mask and the reflected or transmitted scattered fields from
the particular mask location under inspection are imaged with a
very high resolution imaging system that typically has a reduction
factor of R=1 and NA close to 1. But here is the caveat: After the
signal from the inspection tool for a particular mask location is
available it needs to be compared to something in order to
determine whether or not a phase-defect (or other defect) is
present within that location. Such a comparison is in general very
difficult, because not only does it need to be accurate in order to
reliably flag defective locations, but it also needs to be rapid
enough for the entire reticle to be inspected in a reasonable
amount of time.
The most promising direction that has been adopted by researchers
working on the problem has been what is known as a die-to-database
comparison. By that, it is meant that the local signal is compared
to the signal that the same mask location would produce if it were
defect-free. The problem now shifts to building the database of
signals from defect-free masks. Note that all possible mask
geometries that will be encountered in every inspected reticle need
to exist in the database. One way to building the database would be
to produce a (set of) test reticle(s) that includes every single
geometry situation that is anticipated to exist in all designed
layouts to come. This (set of) test reticle(s) is then meticulously
inspected with a reliable tool (maybe an atomic force
microscope--AFM) to guarantee that it is defect-free and finally
all signals from the different locations are collected and placed
in the vault (database). Clearly, the immensity of this task has to
do not only with the a priori precise anticipation of what layout
situations to take provision for, not only with manufacturing the
test reticle(s), characterizing it and building the database, but
also with the sheer volume of data that such a database would
contain. Moreover, a precise diagnosis of the type, size and
location of a phase defect requires an even larger database of
benchmark signals, where every possible combination of defect type,
size and location in every possible layout configuration needs to
exist in the database. Performing such a task in a way similar to
the one outlined is impractical.
Simulation can again come to the rescue, at least in principle. If
a simulation tool that can rapidly and accurately predict the
expected signal resulting from a defect-free or defective location
on the mask is available, then the task of building the database is
simplified tremendously. Moreover, if such a tool is really fast
(and accurate) then a database is not even needed. The required
benchmark signal from the defect-free layout is generated in-situ,
while the inspection system gathers measurements. The problem with
this solution is that to date there exists no simulation tool that
possesses both the rapidity and accuracy properties. The family of
simulation tools that rigorously solve Maxwell's equations around
the reticle (such as TEMPEST) and subsequently use a vectorial
formulation for the image formation to calculate the expected
optical signal from the inspection system are impractical because
of large memory and time requirements. On the other hand the
accuracy of speedier simulation programs that circumvent the
solution of Maxwell's equations around the mask is
unacceptable.
Yet another important consideration of an inspection system for
alt. PSMs should be the following: Phase-defects that are not
critically affecting the image that a projection printing tool will
produce on the wafer surface should be discarded, i.e. not flagged
and not repaired. But this is also a difficult task, since the only
bullet-proof way of achieving this would be to expose wafers with
the suspect defect present and examine if the printed resist images
have intolerable artifacts. Clearly, such a process now involves
the close cooperation of the inspection system with the projection
printing tool. However, an accurate and properly calibrated
simulation tool can alternatively assess the tendency of the defect
to print or cause other image artifacts. Consequently, the costly
step of exposing wafers can be avoided and a decision of repairing
or not the defect rests fully upon the simulation result. Again, to
date, results from no simulation tool, that is fast enough in
calculating expected images with phase defects present, can be
trusted to base repair decisions on.
The domain decomposition methods described above are appropriate
for solving the problem of the rapid and accurate evaluation of the
benchmark signal from nondefective masks for the inspection system.
A domain decomposition method is presented that addresses the
problem of the defect printability assessment, after a defect is
previously found. The generation of benchmark signals from
defective mask locations can also be based on the domain
decomposition technique, although more work will be needed to adapt
it and test it for the high-NA inspection optics with R=1. As will
soon be obvious, the speed and efficiency of the method are
inherited by virtue of the domain decomposition methods presented
above, although for simplicity edge-DDM is not explicitly employed
in the following. In the simulation examples, the illumination
wavelength is 193 nm and the phase-wells are designed accordingly,
so as to provide the required phase shifts for that wavelength.
Consider the example depicted in FIG. 19. In FIG. 19(a) the near
scattered field below a small layout of a 0.degree./180.degree.
alt. PSM is shown when a square 120 nm.times.120 nm (4.times.),
180.degree. post phase defect is present at various locations
within the layout. Note that although neither the layout nor the
phase defect geometry change, a different simulation altogether is
needed because of the different relative positions of the defect
within the layout. Each one of the nine 3D simulations in the
matrix shown in FIG. 19(a) takes approximately 10 hrs on a 450 MHz
CPU and utilize 300 Mb of memory. Subsequently, the aerial images
for all nine situations are calculated for an imaging system with
R=4, NA=0.75 and .sigma.=0.3. Then, the fractional change of
CD.sub.aerial at the middle and at the end of the line because of
the defect is found for all nine defect locations and is shown in
FIG. 19(b). From the plot in FIG. 19(b) it can be deduced that when
the defect is located in the middle of the phase wells or is mostly
"tucked" underneath the absorbing layer (columns 3 and 1 of FIG.
19(a) respectively) the CD.sub.aerial is not critically affected,
but locations such as those in column 2 of FIG. 19(a) cause larger
CD.sub.aerial variations. In the former cases the defects could
probably be discarded whereas in the latter repair seems necessary.
Clearly, the luxury of the 10 hr-long simulations is only available
as a proof of concept and such an approach is not viable for
full-chip characterizations.
Now, consider the paradigm shown in FIG. 20. A domain decomposition
method is again invoked, where instead of a one-step simulation of
the defective layout two separate rigorous simulations are
performed, one with just the defect-free layout and one with just
the defect in clear surroundings. The scattered field below the
defect is then shifted accordingly, such that the defect is
effectively placed at the location it appears in the defective
mask, and the uniform background light is subtracted, so as to
retrieve the (complex) signal of the local perturbation that the
defect causes. This perturbation is added to the defect-free layout
signal for an approximation to the signal from the rigorous
simulation of the defective mask. The non-defective mask simulation
can be performed rapidly using the edge-DDM decomposition methods
described above.
A strategy can now be devised, where a database of the near
electromagnetic field scattered from isolated phasedefects in clear
surroundings is created for all possible potential defects that are
critical. The volume of such a database can be reduced based on the
earlier observations that defects with similar footprints (lateral
dimensions) and small shape perturbations are equivalent from an
electromagnetic point-of-view.
The suggested decomposition method will be introduced through the
following example: Suppose a 150 nm (1.times.), 1:1.5 semi-dense,
90.degree./270.degree. alt. PSM contact mask. The layout and the
scattered field across the observation plane under normal incidence
Ey (TE) polarization are shown in FIG. 21.
Now, suppose that a 200 nm by 200 nm (4.times.) 90.degree. post
defect is present in the center of the bottom left hole. According
to the paradigm of FIG. 20, the mask scattering simulation can be
broken up into two constituent parts. This approach is shown in
FIG. 22. In FIG. 22(a) the phase defect is simulated in an isolated
configuration at the location that it appears in the layout. In
FIG. 22(b) and (c) the near field across the observation plane
below the defective alt. PSM is shown, using the domain
decomposition method and the complete mask respectively. The
differences are not discernible from the plots of FIGS. 22(b) and
(c) and one has to look at the amplitude of the error in FIG.
22(d). Again, not only is the error level of the decomposition low,
but the frequency content is very high, so that it will be filtered
out by the imaging system.
Next, suppose that the same 90.degree. post defect is present in
the center of the bottom right hole. The important observation here
is that no new simulation is necessary for the edge decomposition
method. The scattering of the isolated phase defect can be recycled
from the database, i.e., from FIG. 22(a). However, the scattered
field is now properly shifted in the lateral x-direction and also
(a key step) it is propagated in the z-direction. This results in
the plot of FIG. 23(a), where a larger diffraction spreading is
evident. In FIGS. 23(b) and (c) the near field across the
observation plane below the defective alt. PSM is shown, using the
domain decomposition method and the complete mask respectively. The
amplitude of the error is shown in FIGS. 23(d) and similar
observations apply in FIG. 22(d).
The near fields across cut-lines passing through the center of the
bottom two holes (90.degree., 270.degree.) are shown for the two
defective masks in FIG. 24, comparing the rigorous (r-mask) and the
decomposition methods (DDM). The non-defective case is also plotted
for comparison.
Next, the images of both rigorous and DDM approaches, for both
defective masks are compared in FIG. 25, (a) for the 90.degree.
hole and (b) for the 270.degree. hole. The optical system
parameters are .lamda.=193 nm, R=4, NA=0.75 and .sigma.=0.3. The
agreement between the r-mask and the DDM is good, a normalized mean
square error of less than 0.3% is incurred (measured only in the
area of the defective hole). The k-mask model is insufficient for
capturing these effects and is only shown for comparison.
The reason of the excellent agreement in the predictions of the
aerial image can be traced back to the plots of FIG. 22(d) and FIG.
23(d). The high spatial frequency variation of the error means that
it is mapped at the extremities of the spectrum of propagating
plane waves and is not collected by the optical system. The
normalized spectra (magnitude) of the near field errors are shown
in FIG. 26. Observe that the implied accuracy from these plots
should be enough even for R=1 and NA.about.0.8-0.9, which would be
sufficient for inspection simulations.
Thus the invention includes a suitable extension of the domain
decomposition framework for the simulation of alternating phase
shift mask with phase defects. The value of this approach is
believed to be significant in the problem of rapid assessment of
defect printability once a defect has been found, or in the
die-to-database comparisons of inspection systems when deciding
about the existence or not of a phase-defect.
The invention is an extension of known domain decomposition methods
that introduces tremendous versatility, since only a small number
of isolated edge-diffraction simulations is shown to contain all
the necessary information for the synthesis of the scattered field
from arbitrary 2D mask layouts and subsequent accurate imaging
simulations. The limits of this method are reached when the mask
features are smaller than a wavelength in size and the vertical
mask topography is large. Through a systematic process it was
determined that features as small as at least 200 nm bearing
270.degree. of phase-wells can be accurately decomposed via the
edge-DDM, at .lamda.=193 nm. The method was tested for simple 2D
layouts, where rigorous mask simulations are possible. Excellent
accuracy accompanied by speed-up factors of 172,800 (1 sec vs. 2
days) have been demonstrated. The accuracy of the method is
attributed to the fact that on one hand the edge-diffraction
phenomena are modeled rigorously through 2D edge-diffraction
simulations and that on the other hand the errors incurred during
the synthesis (primarily near the comers) are mapped at the
extremities of the spectrum of propagating plane waves and do not
contribute to the image formation. It was emphasized that all
necessary rigorous 2D simulations of the diffraction from isolated
edge profiles is performed off-line and recycled for the
diffraction calculation of arbitrary layouts. If needed, the
accuracy of the edge profile diffraction simulations can be pushed
to extreme limits, since it is performed only once. This can aid
for example in the correct simulation of unusual edge profiles
whose geometrical details would require excessive discretization of
the domain. It was also shown that the exact details of the
edge-diffraction are not necessary for accurate imaging simulations
(with larger than 1 reduction factors) and a piecewise constant,
multistep edge-transition model that matches the TTL spectrum was
introduced. The range of validity of the edge-DDM is expected to be
appropriate for rapid and accurate evaluation of aerial images
whenever speed is critical, as for example in full-chip OPC
software and die-to-database comparisons in the inspection of alt.
PSMs.
While the invention has been described with reference to specific
embodiments, the description is illustrative of the invention and
is not to be construed as limiting the invention. Various
modifications and applications may occur to those skilled in the
art without departing from the true scope and spirit of the
invention as defined by the appended claims.
* * * * *