U.S. patent application number 09/931258 was filed with the patent office on 2002-02-07 for position detection method and position detector, exposure method and exposure apparatus, and device and device manufacturing method.
This patent application is currently assigned to Nikon Corporation. Invention is credited to Yoshida, Kouji.
Application Number | 20020014601 09/931258 |
Document ID | / |
Family ID | 26377419 |
Filed Date | 2002-02-07 |
United States Patent
Application |
20020014601 |
Kind Code |
A1 |
Yoshida, Kouji |
February 7, 2002 |
Position detection method and position detector, exposure method
and exposure apparatus, and device and device manufacturing
method
Abstract
A parameter calculation unit statistically calculates the
estimations and their certainty of a predetermined number of
parameters, which uniquely specify any position on an object, for
each of a plurality of measured sample sets on the basis of
position information of marks composing the sample set. A valid
value calculation unit calculates statistically valid values of the
predetermined number of parameters on the basis of groups of the
estimations and their certainty of the predetermined number of
parameters for the respective sample sets. Furthermore, an
evaluation unit statistically evaluates if the number of marks
composing a sample set can be reduced. If it is determined that the
number of marks can be reduced, the parameter calculation unit
calculates values of the predetermined number of parameters by
using a new sample set having reduced number of marks. Using the
values of the predetermined number of parameters calculated in this
manner, the position of any area on the object can be accurately
detected.
Inventors: |
Yoshida, Kouji; (Adachi-ku,
JP) |
Correspondence
Address: |
OBLON SPIVAK MCCLELLAND MAIER & NEUSTADT PC
FOURTH FLOOR
1755 JEFFERSON DAVIS HIGHWAY
ARLINGTON
VA
22202
US
|
Assignee: |
Nikon Corporation
2-3, Marunouchi 3-chome
Chiyoda-ku
JP
100-8331
|
Family ID: |
26377419 |
Appl. No.: |
09/931258 |
Filed: |
August 17, 2001 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
09931258 |
Aug 17, 2001 |
|
|
|
PCT/JP00/00855 |
Feb 16, 2000 |
|
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Current U.S.
Class: |
250/548 |
Current CPC
Class: |
G03F 9/7092 20130101;
G03F 9/7046 20130101; G03F 9/7003 20130101 |
Class at
Publication: |
250/548 |
International
Class: |
G01N 021/86 |
Foreign Application Data
Date |
Code |
Application Number |
Feb 17, 1999 |
JP |
11-038220 |
Aug 31, 1999 |
JP |
11-245047 |
Claims
What is claimed is:
1. A position detection method for detecting position information
of any area on an object provided with a plurality of
position-measurement-point- s, the position detection method
comprising: selecting more position-measurement-points than a
minimum number of measurements required to calculate values of a
predetermined number of parameters, which uniquely specify position
information of any area on the object, from the plurality of
position-measurement-points and measuring pieces of position
information of the respective selected position-measurement-poin-
ts; calculating respective positions of the selected
position-measurement-points, based on the measurement results of
the pieces of position information, and estimating probability
density functions which each represent occurrence probability of
the calculated position for respective one of the selected
position-measurement-points; calculating probability density of the
calculated position of each of the position-measurement-points,
based on respective one of the probability density functions; and
evaluating an error of each calculated position relative to
respective reference position while using the respective calculated
probability density's value as a piece of weight information and
calculating values of the predetermined number of parameters, based
on the evaluated errors.
2. The position detection method according to claim 1, wherein the
reference positions are determined in advance based on design
information.
3. The position detection method according to claim 1, wherein the
error evaluation is performed by multiplying the error of each
calculated position relative to the respective reference position
by the respective probability density.
4. The position detection method according to claim 1, wherein the
probability density functions are probability density functions of
normal distribution.
5. The position detection method according to claim 1, wherein
position measurement marks are formed at the
position-measurement-points.
6. The position detection method according to claim 5, wherein a
plurality of position detection marks formed at the plurality of
position-measurement-points include a first number of first marks
of which surface states change in a first direction, and wherein
the piece of position information of each first mark measured in
the step of selecting and measuring comprises pieces of position
information at a plurality of feature portions of each first mark
in the first direction.
7. The position detection method according to claim 6, wherein for
each of the selected first marks, the step of selecting and
measuring measures the piece of position information at a plurality
of positions lined in a direction crossing the first direction.
8. The position detection method according to claim 6, wherein
surface state of each of the first marks changes also in a second
direction different from the first direction, and wherein the piece
of position information of each first mark measured in the step of
selecting and measuring consists of position information, in the
first direction, of a plurality of feature portions lined in the
first direction and position information, in the second direction,
of a plurality of feature portions lined in the second
direction.
9. The position detection method according to claim 8, wherein for
each selected first mark, the step of selecting and measuring
measures at least one of the position information, in the first
direction, of a plurality of positions lined in a direction
crossing the first direction and the position information, in the
second direction, of a plurality of positions lined in a direction
crossing the second direction.
10. The position detection method according to claim 6, wherein the
plurality of marks further include a second number of second marks,
of which surface states change in a second direction different from
the first direction, and wherein for each second mark, the position
information measured in the step of selecting and measuring is
position information, in the second direction, of a plurality of
feature portions.
11. The position detection method according to claim 10, wherein
for each selected second mark, the step of selecting and measuring
measures the position information at a plurality of positions in a
direction crossing the second direction.
12. The position detection method according to claim 5, wherein a
plurality of divided areas are arranged on the object, each of the
divided areas being provided with the position measurement
marks.
13. The position detection method according to claim 12, wherein
parameters associated with respective, representative points of the
plurality of divided areas are included in the predetermined number
of parameters.
14. The position detection method according to claim 13, wherein
parameters associated with other points than the respective,
representative points of the plurality of divided areas are further
included in the predetermined number of parameters.
15. A position detection method for detecting position information
of any area on an object provided with a first number of
position-measurement-po- ints, the position detection method
comprising: selecting a plurality of measurement point subsets
which each consist of a third number of position-measurement-points
and are different from one another, the third number being larger
than a second number and smaller than the first number, the second
number being a minimum number of measurement points required to
calculate a predetermined number of parameters that uniquely
specify position information of any area on the object; and
statistically calculating, for each of the plurality of measurement
point subsets, estimations of the predetermined number of
parameters and certainty of the estimations, based on measurement
results of the third number of position-measurement-points.
16. The position detection method according to claim 15, further
comprising: obtaining statistically valid values of the
predetermined number of parameters, based on the respective groups
of estimations and certainty for the plurality of measurement point
subsets calculated in the step of calculating.
17. The position detection method according to claim 16, wherein
the statistically valid value of each of the predetermined number
of parameters is obtained by calculating average of the
corresponding estimations weighted with the respective certainties,
each of the certainties representing a piece of weight information
for the respective estimation.
18. The position detection method according to claim 15, wherein in
the step of calculating, certainties of position measurement
results of the position-measurement-points are taken in account to
calculate the estimations and certainty thereof.
19. The position detection method according to claim 18, wherein
the step of calculating comprises: calculating, for each of the
plurality of measurement point subsets, respective positions of the
third number of position-measurement-points based on measurement
results of the third number of position-measurement-points and
estimating probability density functions that each represent
occurrence probability of the calculated position of the
respective, selected position-measurement-point; calculating
respective probability densities of the calculated positions of the
position-measurement-points, based on the probability density
functions; and evaluating an error of each of the calculated
positions relative to respective reference position using the
respective calculated probability density's value as a piece of
weight information and calculating estimations of the predetermined
number of parameters, based on the evaluated errors.
20. The position detection method according to claim 15, wherein
position measurement marks are formed at the
position-measurement-points.
21. The position detection method according to claim 20, wherein a
plurality of divided areas, each of which is provided with the
position measurement marks, are arranged on the object.
22. A position detection method for detecting position information
of any area on an object provided with a first number of
position-measurement-po- ints, the position detection method
comprising: selecting a first measurement point subset which
consists of a third number of position-measurement-points, the
third number being larger than a second number and smaller than the
first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object; selecting a plurality of second measurement point
subsets which each consist of a fourth number of
position-measurement-points and are different from one another, the
fourth number being larger than the second number and smaller than
the third number; and statistically evaluating possibility of
replacing the first measurement point subset by one of the
plurality of second measurement point subsets, based on measurement
results of the third number of position-measurement-points
composing the first measurement point subset and measurement
results of sets of the fourth number of position-measurement-points
each composing one of the second measurement point subsets, the
first measurement point subset being used to calculate the
predetermined number of parameters.
23. The position detection method according to claim 22, wherein
the step of evaluating comprises: statistically calculating
estimations of the predetermined number of parameters and certainty
of the estimations, based on measurement results of the third
number of position-measurement-points composing the first
measurement point subset; statistically calculating estimations of
the predetermined number of parameters and certainty of the
estimations for each of the plurality of second measurement point
subsets, based on measurement results of the fourth number of
position-measurement-points; and comparing the estimations and
certainty of the first measurement point subset with the
estimations and certainty for each of the plurality of second
measurement point subsets and evaluating possibility of replacing
the first measurement point subset by one of the plurality of
second measurement point subsets, the first measurement point
subset being used to calculate the predetermined number of
parameters.
24. The position detection method according to claim 23, wherein in
the step of calculating estimations for the first measurement point
subset, certainties of position measurement results of the
position-measurement-points are taken into account upon calculating
the estimations and certainty thereof.
25. The position detection method according to claim 24, wherein
the step of calculating estimations for the first measurement point
subset comprises: calculating respective positions of the third
number of position-measurement-points, which compose the first
measurement point subset, based on measurement results of the third
number of position-measurement-points and estimating probability
density functions that each represent occurrence probability of the
calculated position of a respective point of the third number of
position-measurement-points; calculating respective probability
densities of the calculated positions of the
position-measurement-points, based on the probability density
functions; and evaluating an error of each of the calculated
positions relative to respective reference position using the
respective calculated probability density's value as a piece of
weight information and calculating estimations of the predetermined
number of parameters, based on the evaluated errors.
26. The position detection method according to claim 23, wherein in
the step of calculating estimations for each second measurement
point subset, certainties of position measurement results of the
position-measurement-points are taken into account upon calculating
the estimations and certainty thereof.
27. The position detection method according to claim 26, wherein
the step of calculating estimations for each second measurement
point subset comprises: calculating respective positions of the
fourth number of position-measurement-points, for each of the
plurality of second measurement point subsets, based on measurement
results of the fourth number of position-measurement-points and
estimating probability density functions that each represent
occurrence probability of the calculated position of a respective
point of the fourth number of position-measurement-points;
calculating respective probability densities of the calculated
positions of the position-measurement-points, based on the
probability density functions; and evaluating an error of each of
the calculated positions relative to respective reference position
using the respective calculated probability density's value as a
piece of weight information and calculating estimations of the
predetermined number of parameters, based on the evaluated
errors.
28. The position detection method according to claim 22, wherein
the step of evaluating comprises: statistically calculating, for
each of the second measurement point subsets, estimations of the
predetermined number of parameters and certainty of the
estimations, based on measurement results of the fourth number of
position-measurement-points; statistically calculating position
errors of the position-measurement-poi- nts of the first
measurement point subset through use of the estimations of the
predetermined number of parameters calculated in the step of
calculating estimations and evaluating possibility of replacing the
first measurement point subset by one of the plurality of second
measurement point subsets.
29. The position detection method according to claim 28, wherein in
the step of calculating estimations, certainties of position
measurement results of the position-measurement-points are taken
into account upon calculating the estimations and certainty
thereof.
30. The position detection method according to claim 29, wherein
the step of calculating estimations comprises: calculating
respective positions of the fourth number of
position-measurement-points, for each of the plurality of second
measurement point subsets, based on measurement results of the
fourth number of position-measurement-points and estimating
probability density functions that each represent occurrence
probability of the calculated position of a respective point of the
fourth number of position-measurement-points; calculating
respective probability densities of the calculated positions of the
position-measurement-points, based on the probability density
functions; and evaluating an error of each of the calculated
positions relative to respective reference position using the
respective calculated probability density's value as a piece of
weight information and calculating estimations of the predetermined
number of parameters, based on the evaluated errors.
31. The position detection method according to claim 22, further
comprising a step which, when the step of evaluating finds second
measurement point subsets that can replace the first measurement
point subset, selects a most valid subset from the second
measurement point subsets for replacement and adopts estimations of
the predetermined number of parameters, calculated based on
measurement results of the fourth number of
position-measurement-points of the selected second measurement
point subset, as values thereof, and which, when the step of
evaluating finds no second measurement point subsets that can
replace the first measurement point subset, adopts estimations of
the predetermined number of parameters, calculated based on
measurement results of the second number of
position-measurement-points of the first measurement point subset,
as values thereof.
32. The position detection method according to claim 22, wherein
position measurement marks are formed at the
position-measurement-points.
33. The position detection method according to claim 32, wherein a
plurality of divided areas, each of which is provided with the
position measurement marks, are arranged on the object.
34. A position detection method for detecting position information
of any area on an object provided with a first number of
position-measurement-po- ints, the position detection method
comprising: selecting a plurality of first measurement point
subsets which each consist of a third number of
position-measurement-points and are different from one another, the
third number being larger than a second number and smaller than the
first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object; selecting a plurality of second measurement point
subsets which each consist of a fourth number of
position-measurement-points and are different from one another, the
fourth number being larger than the second number and smaller than
the third number; and statistically evaluating possibility of
replacing the plurality of first measurement point subsets by one
of the plurality of second measurement point subsets, as a
measurement point set used to calculate the predetermined number of
parameters.
35. The position detection method according to claim 34, wherein
the step of evaluating comprises: statistically calculating, for
each of the plurality of first measurement point subsets,
estimations of the predetermined number of parameters and certainty
of the estimations, based on measurement results of the third
number of position-measurement-points; calculating statistically
valid estimations of the predetermined number of parameters and
certainty of the estimations, based on groups of the estimations
and certainty thereof for the plurality of first measurement point
subsets, calculated in the step of calculating estimations for each
first measurement point subset; statistically calculating
estimations of the predetermined number of parameters and certainty
of the estimations for each of the plurality of second measurement
point subsets, based on measurement results of the fourth number of
position-measurement-points; and comparing the statistically valid
estimations and certainty with the estimations and certainty for
each of the plurality of second measurement point subsets and
evaluating possibility of adopting one of the plurality of second
measurement point subsets as a measurement point set used to
calculate the predetermined number of parameters.
36. The position detection method according to claim 35, wherein in
the step of calculating estimations for each first measurement
point subset, certainties of position measurement results of the
position-measurement-points are taken into account upon calculating
the estimations and certainty thereof.
37. The position detection method according to claim 36, wherein
the step of calculating estimations for each first measurement
point subset comprises: calculating respective positions of the
third number of position-measurement-points, for each of the
plurality of first measurement point subsets, based on measurement
results of the third number of position-measurement-points and
estimating probability density functions that each represent
occurrence probability of the calculated position of a respective
point of the third number of position-measurement-points;
calculating respective probability densities of the calculated
positions of the position-measurement-points, based on the
probability density functions; and evaluating an error of each of
the calculated positions relative to respective reference position
using the respective calculated probability density's value as a
piece of weight information and calculating estimations of the
predetermined number of parameters, based on the evaluated
errors.
38. The position detection method according to claim 35, wherein in
the step of calculating estimations for each second measurement
point subset, certainties of position measurement results of the
position-measurement-points are taken into account upon calculating
the estimations and certainty thereof.
39. The position detection method according to claim 38, wherein
the step of calculating estimations for each second measurement
point subset comprises: calculating respective positions of the
fourth number of position-measurement-points, for each of the
plurality of second measurement point subsets, based on measurement
results of the fourth number of position-measurement-points and
estimating probability density functions that each represent
occurrence probability of the calculated position of a respective
point of the fourth number of position-measurement-points;
calculating respective probability densities of the calculated
positions of the position-measurement-points, based on the
probability density functions; and evaluating an error of each of
the calculated positions relative to respective reference position
using the respective calculated probability density's value as a
piece of weight information and calculating estimations of the
predetermined number of parameters, based on the evaluated
errors.
40. The position detection method according to claim 35, further
comprising a step which, when the step of evaluating finds second
measurement point subsets that can replace the first measurement
point subset, selects a most valid subset from the second
measurement point subsets for replacement and adopts estimations of
the predetermined number of parameters, calculated based on
measurement results of the fourth number of
position-measurement-points of the selected second measurement
point subset, as values thereof, and which, when the step of
evaluating finds no second measurement point subsets that can
replace the first measurement point subset, adopts the
statistically valid estimations as values of the predetermined
number of parameters.
41. The position detection method according to claim 34, wherein
the step of evaluating comprises: statistically calculating, for
each of the second measurement point subsets, estimations of the
predetermined number of parameters and certainty of the
estimations, based on measurement results of the fourth number of
position-measurement-points; calculating position errors of all the
position-measurement-points of the plurality of first measurement
point subsets through use of the estimations of the predetermined
number of parameters calculated for each of the second measurement
point subsets and evaluating possibility of replacing the plurality
of first measurement point subsets by one of the plurality of
second measurement point subsets.
42. The position detection method according to claim 41, wherein in
the step of calculating estimations for each second measurement
point subset, certainties of position measurement results of the
position-measurement-points are taken into account upon calculating
the estimations and certainty thereof.
43. The position detection method according to claim 42, wherein
the step of calculating estimations for each second measurement
point subset comprises: calculating respective positions of the
fourth number of position-measurement-points, for each of the
plurality of second measurement point subsets, based on measurement
results of the fourth number of position-measurement-points and
estimating probability density functions that each represent
occurrence probability of the calculated position of a respective
point of the fourth number of position-measurement-points;
calculating respective probability densities of the calculated
positions of the position-measurement-points, based on the
probability density functions; and evaluating an error of each of
the calculated positions relative to respective reference position
using the respective calculated probability density's value as a
piece of weight information and calculating estimations of the
predetermined number of parameters, based on the evaluated
errors.
44. The position detection method according to claim 34, further
comprising a step which, when the step of evaluating finds second
measurement point subsets that can replace the first measurement
point subset, selects a most valid one, for replacement, of the
second measurement point subsets and adopts estimations of the
predetermined number of parameters, calculated based on measurement
results of the fourth number of position-measurement-points of the
selected second measurement point subset, as values thereof, and
which, when the step of evaluating finds no second measurement
point subsets that can replace the first measurement point subset,
statistically calculates estimations of the predetermined number of
parameters and certainty of the estimations, based on measurement
results of the third number of position-measurement-points for each
of the plurality of first measurement point subsets, and adopts as
values of the predetermined number of parameters statistically
valid estimations thereof calculated based on groups of the
estimations and certainty thereof for the plurality of first
measurement point subsets.
45. The position detection method according to claim 44, wherein
the statistically valid value of each of the predetermined number
of parameters is obtained by calculating average of the
corresponding estimations weighted with the respective certainties,
each of the certainties representing a piece of weight information
for the respective estimation.
46. The position detection method according to claim 34, wherein
position measurement marks are formed at the
position-measurement-points.
47. The position detection method according to claim 46, wherein a
plurality of divided areas each of which is provided with the
position measurement marks are arranged on the object.
48. A position detector that detects position information of any
area on an object provided with a plurality of
position-measurement-points, the position detector comprising: a
measurement unit that measures pieces of position information of
more position-measurement-points than a minimum number of
measurements required to calculate values of a predetermined number
of parameters, which uniquely specify position information of any
area on the object, the position-measurement-points being selected
from the plurality of position-measurement-points; an estimation
unit that is electrically connected to the measurement unit and
that detects respective positions of the selected
position-measurement-points, based on the measurement results of
the pieces of position information, estimates probability density
functions which each represent occurrence probability of the
detected position for respective one of the selected
position-measurement-points, and calculates probability density of
the detected position of each of the position-measurement-points;
and a parameter calculation unit that is electrically connected to
the measurement unit and the estimation unit and that evaluates
detection error of each of the detected positions while using the
respective calculated probability density's value as a piece of
weight information and calculates such values of the predetermined
number of parameters that the detection errors become statistically
minimum as a whole, based on the detection errors.
49. The position detector according to claim 48, wherein said
measurement unit comprises an image pickup unit that picks up
images of marks formed on the object.
50. A position detector that detects position information of any
area on an object provided with a first number of
position-measurement-points, the position detector comprising: a
measurement unit that measures positions of the
position-measurement-points; a set-selection unit that selects a
plurality of measurement point subsets which each consist of a
third number of position-measurement-points and are different from
one another, the third number being larger than a second number and
smaller than the first number, the second number being a minimum
number of measurement points required to calculate a predetermined
number of parameters that uniquely specify position information of
any area on the object; and an estimation computing unit that is
electrically connected to the measurement unit and the
set-selection unit and that statistically calculates, for each of
the plurality of measurement point subsets, estimations of the
predetermined number of parameters and certainty of the
estimations, based on measurement results of the third number of
position-measurement-points.
51. The position detector according to claim 50, wherein the
estimation computing unit comprising: an estimation unit that, for
each of the plurality of measurement point subsets, detects
respective positions of the third number of
position-measurement-points, based on the measurement results of
the pieces of position information of the third number of
position-measurement-points, estimates probability density
functions which each represent occurrence probability of the
detected position for the respective point of the third number of
position-measurement-points, and calculates probability density of
the detected position of each of the position-measurement-points;
and a parameter calculation unit that is electrically connected to
the estimation unit and that evaluates detection error of each of
the detected positions while using the respective calculated
probability density's value as a piece of weight information and
calculates such values of the predetermined number of parameters
that the detection errors become statistically minimum as a whole,
based on the detection errors.
52. The position detector according to claim 50, further
comprising: a parameter value determining unit that is electrically
connected to the estimation computing unit and that calculates
statistically valid estimations of the predetermined number of
parameters based on groups of the estimations and certainty
thereof, calculated by the estimation computing unit, for the
plurality of measurement point subsets.
53. A position detector that detects position information of any
area on an object provided with a first number of
position-measurement-points, the position detector comprising: a
measurement unit that measures positions of the
position-measurement-points; a set-selection unit that selects a
first measurement point subsets, which each consist of a third
number of position-measurement-points, and a plurality of second
measurement point subsets which each consist of a fourth number of
position-measurement-points and are different from one another, the
third number being larger than a second number and smaller than the
first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object, the fourth number being larger than the second
number and smaller than the third number; and an evaluation
computing unit that is electrically connected to the set-selection
unit and that evaluates possibility of replacing the first
measurement point subset by one of the plurality of second
measurement point subsets, the first measurement point subset being
used to calculate the predetermined number of parameters.
54. The position detector according to claim 53, wherein the
evaluation computing unit comprises: an estimation calculation unit
that is electrically connected to the measurement unit and that
statistically calculates, for the specific measurement point
subset, estimations of the predetermined number of parameters and
certainty of the estimations, based on measurement results of
position information of position-measurement-points composing the
specific measurement point subset which is selected from the first
measurement point subset and the plurality of second measurement
point subsets; and an evaluation unit that is electrically
connected to the estimation calculation unit and that compares the
estimations and certainty of the first measurement point subset
with the estimations and certainty for each of the plurality of
second measurement point subsets and evaluates possibility of
replacing the first measurement point subset by one of the
plurality of second measurement point subsets, the first
measurement point subset being used to calculate the predetermined
number of parameters.
55. The position detector according to claim 54, wherein the
estimation calculation unit comprises: an estimation unit that
detects respective positions of position-measurement-points
composing the specific measurement point subset, based on the
measurement results of position information of
position-measurement-points composing the specific measurement
point subset, estimates probability density functions which each
represent occurrence probability of the detected position for
respective one of the position-measurement-points of the specific
measurement point subset, and calculates probability density of the
detected position of each of the position-measurement-points; and a
parameter calculation unit that is electrically connected to the
estimation unit and that evaluates detection error of each of the
detected positions while using the respective calculated
probability density's value as a piece of weight information and
calculates such values of the predetermined number of parameters
that the detection errors become statistically minimum as a whole,
based on the detection errors.
56. The position detector according to claim 53, wherein the
evaluation computing unit comprises: an estimation calculation unit
that is electrically connected to the measurement unit and that
statistically calculates, for the specific measurement point
subset, estimations of the predetermined number of parameters and
certainty of the estimations, based on measurement results of
position information of position-measurement-points composing the
specific measurement point subset, which is selected from the
plurality of second measurement point subsets; an evaluation unit
that is electrically connected to the estimation calculation unit
and that calculates position errors of the
position-measurement-points, composing the first measurement point
subset, through use of estimations of the predetermined number of
parameters for each of the polarity of second measurement point
subsets and evaluates possibility of replacing the first
measurement point subset by one of the plurality of second
measurement point subsets.
57. The position detector according to claim 56, wherein the
estimation calculation unit comprises: an estimation unit that
detects respective positions of position-measurement-points
composing the specific measurement point subset, based on the
measurement results of position information of
position-measurement-points composing the specific measurement
point subset, estimates probability density functions which each
represent occurrence probability of the detected position for
respective one of the position-measurement-points of the specific
measurement point subset, and calculates probability density of the
detected position of each of the position-measurement-points; and a
parameter calculation unit that is electrically connected to the
estimation unit and that evaluates detection error of each of the
detected positions while using the respective calculated
probability density's value as a piece of weight information and
calculates such values of the predetermined number of parameters
that the detection errors become statistically minimum as a whole,
based on the detection errors.
58. The position detector according to claim 53, further comprising
a parameter value determining unit that is electrically connected
to the evaluation computing unit and that calculates values of the
predetermined number of parameters, based on evaluation results of
the evaluation computing unit.
59. A position detector that detects position information of any
area on an object provided with a first number of
position-measurement-points, the position detector comprising: a
measurement unit that measures positions of the
position-measurement-points; a set-selection unit that selects a
plurality of first measurement point subsets, which each consist of
a third number of position-measurement-points and are different
from one another, and a plurality of second measurement point
subsets which each consist of a fourth number of
position-measurement-poi- nts and are different from one another,
the third number being larger than a second number and smaller than
the first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object, the fourth number being larger than the second
number and smaller than the third number; and an evaluation
computing unit that is electrically connected to the set-selection
unit and that evaluates possibility of adopting one of the
plurality of second measurement point subsets as a measurement
point subset to calculate the predetermined number of
parameters.
60. The position detector according to claim 59, wherein the
evaluation computing unit comprises: an estimation calculation unit
that is electrically connected to the measurement unit and that
statistically calculates, for the specific measurement point
subset, estimations of the predetermined number of parameters and
certainty of the estimations, based on measurement results of
position information of position-measurement-points composing the
specific measurement point subset which is selected from the
plurality of first measurement point subset and the plurality of
second measurement point subsets, and calculates statistically
valid estimations of the predetermined number of parameters and
certainty of the estimations, based on groups of estimations of the
predetermined number of parameters and certainty of the estimations
for the plurality of first measurement point subsets; an evaluation
computing unit that is electrically connected to the estimation
calculation unit and that compares the statistically valid
estimations and certainty of the first measurement point subset
with the estimations and certainty for each of the plurality of
second measurement point subsets, and evaluates possibility of
adopting one of the plurality of second measurement point subsets
as a measurement point subset to calculate the predetermined number
of parameters.
61. The position detector according to claim 60, wherein the
estimation calculation unit comprises: an estimation unit that is
electrically connected to the measurement unit and that detects
respective positions of position-measurement-points composing the
specific measurement point subset, based on the measurement results
of position information of position-measurement-points composing
the specific measurement point subset, estimates probability
density functions which each represent occurrence probability of
the detected position for respective one of the
position-measurement-points of the specific measurement point
subset, and calculates probability density of the detected position
of each of the position-measurement-points; and a parameter
calculation unit that is electrically connected to the estimation
unit and that evaluates detection error of each of the detected
positions while using the respective calculated probability
density's value as a piece of weight information and calculates
such values of the predetermined number of parameters that the
detection errors become statistically minimum as a whole, based on
the detection errors.
62. The position detector according to claim 59, wherein the
evaluation computing unit comprises: an estimation calculation unit
that is electrically connected to the measurement unit and that
statistically calculates, for the specific measurement point
subset, estimations of the predetermined number of parameters and
certainty of the estimations, based on measurement results of
position information of position-measurement-points composing the
specific measurement point subset which is selected from the
plurality of second measurement point subsets; and an evaluation
unit that is electrically connected to the estimation calculation
unit and that calculates errors of all the
position-measurement-points of the plurality of first measurement
point subsets through use of the estimations of the predetermined
number of parameters calculated for each of the second measurement
point subsets, and evaluates possibility of replacing the plurality
of first measurement point subsets by one of the plurality of
second measurement point subsets.
63. The position detector according to claim 62, wherein the
estimation calculation unit comprises: an estimation unit that
detects respective positions of position-measurement-points
composing the specific measurement point subset, based on the
measurement results of position information of
position-measurement-points composing the specific measurement
point subset, estimates probability density functions which each
represent occurrence probability of the detected position for
respective one of the position-measurement-points of the specific
measurement point subset, and calculates probability density of the
detected position of each of the position-measurement-points; and a
parameter calculation unit that is electrically connected to the
estimation unit and that evaluates detection error of each of the
detected positions while using the respective calculated
probability density's value as a piece of weight information and
calculates such values of the predetermined number of parameters
that the detection errors become statistically minimum as a whole,
based on the detection errors.
64. The position detector according to claim 59, further comprising
a parameter value determining unit that is electrically connected
to the parameter calculation unit and that calculates values of the
predetermined number of parameters, based on evaluation results of
the evaluation computing unit.
65. An exposure method for transferring a predetermined pattern
onto divided areas on a substrate, comprising: calculating a
predetermined number of parameters that pertain to positions of the
divided areas by a position detection method according to claim 1
and calculating arrangement information of the divided areas on the
substrate; and transferring the pattern onto the divided areas
while aligning the substrate based on the arrangement information
of the divided areas calculated in the step of calculating
arrangement information.
66. An exposure apparatus that transfers a predetermined pattern
onto divided areas on a substrate, comprising: a stage unit that
moves the substrate along a movement plane; and a position detector
according to claim 48 that calculates arrangement information of
the divided areas on the substrate mounted on the stage unit.
67. A device manufacturing method including a lithography process,
wherein the lithography process transfers a predetermined pattern
onto divided areas on a substrate using an exposure method
according to claim 65.
68. An exposure method for transferring a predetermined pattern
onto divided areas on a substrate, comprising: calculating a
predetermined number of parameters that pertain to positions of the
divided areas by a position detection method according to claim 15
and calculating arrangement information of the divided areas on the
substrate; and transferring the pattern onto the divided areas
while aligning the substrate based on the arrangement information
of the divided areas calculated in the step of calculating
arrangement information.
69. A device manufacturing method including a lithography process,
wherein the lithography process transfers a predetermined pattern
onto divided areas on a substrate using an exposure method
according to claim 68.
70. An exposure method for transferring a predetermined pattern
onto divided areas on a substrate, comprising: calculating a
predetermined number of parameters that pertain to positions of the
divided areas by a position detection method according to claim 22
and calculating arrangement information of the divided areas on the
substrate; and transferring the pattern onto the divided areas
while aligning the substrate based on the arrangement information
of the divided areas calculated in the step of calculating
arrangement information.
71. A device manufacturing method including a lithography process,
wherein the lithography process transfers a predetermined pattern
onto divided areas on a substrate using an exposure method
according to claim 70.
72. An exposure method for transferring a predetermined pattern
onto divided areas on a substrate, comprising: calculating a
predetermined number of parameters that pertain to positions of the
divided areas by a position detection method according to claim 34
and calculating arrangement information of the divided areas on the
substrate; and transferring the pattern onto the divided areas
while aligning the substrate based on the arrangement information
of the divided areas calculated in the step of calculating
arrangement information.
73. A device manufacturing method including a lithography process,
wherein the lithography process transfers a predetermined pattern
onto divided areas on a substrate using an exposure method
according to claim 72.
74. An exposure apparatus that transfers a predetermined pattern
onto divided areas on a substrate, comprising: a stage unit that
moves the substrate along a movement plane; and a position detector
according to claim 50 that calculates arrangement information of
the divided areas on the substrate mounted on the stage unit.
75. An exposure apparatus that transfers a predetermined pattern
onto divided areas on a substrate, comprising: a stage unit that
moves the substrate along a movement plane; and a position detector
according to claim 53 that calculates arrangement information of
the divided areas on the substrate mounted on the stage unit.
76. An exposure apparatus that transfers a predetermined pattern
onto divided areas on a substrate, comprising: a stage unit that
moves the substrate along a movement plane; and a position detector
according to claim 59 that calculates arrangement information of
the divided areas on the substrate mounted on the stage unit.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This is a continuation of International Application
PCT/JP00/00855, with an international filing date of Feb. 16, 2000,
the entire content of which being hereby incorporated herein by
reference.
BACKGROUND OF THE INVENTION
[0002] 1. Field of the Invention
[0003] The present invention relates to a position detection method
and position detector, an exposure method and exposure apparatus,
and a device and device manufacturing method and, more
particularly, to a position detection method and position detector
for obtaining arrangement information of divided areas on an
object, an exposure method and exposure apparatus using the
position detection method, and a device manufactured using the
exposure method and a manufacturing method thereof.
[0004] 2. Description of the Related Art
[0005] In a lithography process for making semiconductor devices,
liquid crystal display devices, and the like, an exposure apparatus
which transfers a pattern formed on a mask or reticle (to be
generally referred to as a "reticle" hereinafter) onto a substrate
(to be referred to as a "sensitive substrate or wafer" as needed
hereinafter) such as a wafer, glass plate, or the like coated with
a resist or the like via a projection optical system is used. As
such an exposure apparatus, stationary exposure type projection
exposure apparatus such as a so-called stepper, or a scanning
exposure type projection exposure apparatus such as a so-called
scanning stepper is mainly used.
[0006] In these exposure apparatuses, position adjustment
(alignment) between a reticle and wafer must be accurately
performed prior to exposure. To achieve this alignment, position
detection marks (alignment marks) are formed (transferred by
exposure) in the previous lithography process on the wafer in each
of shot regions, and the position of the wafer (or a circuit
pattern on the wafer) can be detected by detecting the positions of
the alignment marks. Alignment is performed on the basis of the
detection results of the positions of the wafer (or a circuit
pattern on the wafer).
[0007] Such alignment schemes include a die-by-die scheme for
performing alignment by detecting alignment marks in each shot, and
an enhanced global alignment (to be abbreviated as "EGA"
hereinafter) scheme for, after measuring alignment marks (position
adjustment marks transferred together with a circuit pattern) at
several positions in a wafer, computing arrangement coordinate
positions of each shot area by a statistical scheme such as a least
square approximation and, upon exposure, stepping a wafer using the
accuracy of a wafer stage and the computation result. This EGA
scheme is disclosed in, e.g., Japanese Patent Laid-Open No.
61-44429 and corresponding U.S. Pat. No. 4,780,617. Of these
schemes, the EGA scheme is prevalently used nowadays in terms of
the throughput of the apparatus.
[0008] In this EGA scheme, in order to determine a plurality of
parameters that uniquely specifies the actual arrangement
coordinate positions of shot areas relative to ideal arrangement
coordinate positions in terms of design, more positions of
alignment marks than the minimum number of those required to obtain
the plurality of parameters are measured. Then, statistically valid
parameter values are determined using a statistical scheme such as
a least square approximation.
[0009] Upon applying such statistical scheme, error analysis is
performed under the premise that "all position measurement results
of alignment marks have the same reliability" (hereafter, this case
is referred to as "prior art 1").
[0010] Also, as disclosed in Japanese Patent Publication No.
7-120621, a technique for determining the predetermined parameters
by executing a fuzzy process based on fuzzy inference using
statistical values such as the average value and variance of the
measured positions of the alignment marks and obtaining the
arrangement coordinate positions of the shot areas has also been
proposed (hereafter, this case is referred to as "prior art
2").
[0011] When all alignment marks are equally formed, the premise of
prior art 1 that "all position measurement results of alignment
marks have the same reliability" is true, but not true when the
shapes of alignment marks differ depending on their positions on a
substrate. Therefore, when the shapes of alignment marks differ
depending on their positions on a substrate, all position
measurement results, whether their reliabilities are high or low,
equally contribute to determination of the arrangement coordinate
positions of shot areas their different reliabilities.
[0012] The accuracy of the arrangement coordinate position
determination of shot areas determined under the above premise
suffices to achieve conventionally required exposure accuracy, but
does not suffice for the increase of integration degree in recent
years.
[0013] Prior art 2 is free from any problem of the accuracy of the
arrangement coordinate position determination of shot areas unlike
in prior art 1. However, since prior art 2 requires a huge
computation volume for fuzzy inference, a long period of time is
required to determine the arrangement coordinate positions of shot
regions, and this makes it difficult to improve the throughput of
exposure. In order to prevent such low throughput, a large-scale
computation resource is needed. However, the use thereof causes the
whole exposure apparatus to be large and complicated.
[0014] Upon applying the conventional statistical scheme, the
positions of alignment marks to be measured on a wafer are
determined empirically or on a trial-and-error basis that after
transferring a pattern onto a wafer while aligning the wafer using
a temporarily selected sample set, the same patterns on the wafer
are measured, and that if desired results are not obtained, another
sample set is selected.
[0015] As described above, in the conventional method, a sample set
of alignment marks as a subset of a set of all alignment marks is
determined by an aleatory method, and the validity of determination
of that sample set is not quantitatively evaluated. Therefore, it
is not guaranteed that the error distribution of the positions of a
plurality of alignment marks as elements of a sample set determined
by the conventional method appropriately reflects the error
distribution of positions of all alignment marks.
[0016] For trying to solve this problem, there is the following
method: the position control using a plurality of parameters which
are obtained using a provisional sample set determined empirically
or arbitrarily and uniquely specify the arrangement coordinate
positions of shot regions, when the sample set includes shot areas
(so-called "isolated shots") having much larger alignment errors
than those of other shot regions, alignment marks contained in such
isolated shots are excluded from the sample set. This method
presupposes that only few isolated shots exist, and that those
alignment marks cause the decrease of alignment accuracy for all
shot regions.
[0017] However, if alignment marks of two isolated shot regions, of
which pattern shift directions are almost opposite to each other
(i.e., the two shot areas having negative correlation), are
selected, high-accuracy alignment is possible. Therefore, exclusion
of the measured position information of alignment marks contained
in an isolated shot area may result in an alignment accuracy
decrease.
[0018] In the case where a wafer is aligned on the basis of the
position measurement results of alignment marks included in a
subset (sample set) selected from a large number of alignment
marks, upon examining the validity of a method for selecting the
desired sample set, evaluating separately individual alignment
marks in the sample set does not have much sense. This is because
it is ideal that the sample set broadly reflects the entire set and
because it is preferable that alignment marks in the sample set
preferably have a position distribution that corresponds to that of
alignment marks in the entire set. For example, in the case where
one of five alignment marks in a sample set is an alignment mark
contained in an isolated shot region, if one fifth of all the
alignment marks are contained in isolated shot regions, the sample
set is more valid than a sample set excluding alignment marks of
isolated shot regions. That is, position errors of measured
alignment marks reflect the position distribution of all the
alignment marks somehow, and should not be carelessly ignored.
However, there have been no proposals concerning a method for
selecting a sample set from the entire set of alignment marks
formed on a substrate so as to perform statistically valid position
control and alignment on the basis of the position measurement
results of alignment marks in the sample set.
[0019] Furthermore, although it is not appropriate to exclude
alignment marks in descending order of position error amounts in
order to reduce the number of sample alignment marks and alignment
measurement time, there have been no proposals concerning a method
for reducing the number of sample alignment marks while maintaining
alignment accuracy.
[0020] That is, an alignment technology that can meet recent
requirements of improved exposure accuracy and throughput is
needed.
[0021] The present invention has been made considering the above
situation, and a first object of the present invention is to
provide a position detection method and position detector which can
accurately and efficiently detect arrangement information of
divided areas on an object.
[0022] A second object of the present invention is to provide an
exposure method and exposure apparatus that can transfer a
predetermined pattern onto a substrate with high accuracy.
[0023] A third object of the present invention is to provide a
device on which fine patterns are accurately formed.
[0024] A fourth object of the present invention is to provide a
manufacturing method of manufacturing a device on which fine
patterns are accurately formed.
SUMMARY OF THE INVENTION
[0025] According to the first aspect, there is provided a first
position detection method for detecting position information of any
area on an object provided with a plurality of
position-measurement-points, the position detection method
comprising a measurement step of selecting more
position-measurement-points than a minimum number of measurements
required to calculate values of a predetermined number of
parameters, which uniquely specify position information of any area
on the object, from the plurality of position-measurement-points
and measuring pieces of position information of the respective
selected position-measurement-poin- ts; an estimation step of
calculating respective positions of the selected
position-measurement-points, based on the measurement results of
the pieces of position information, and estimating probability
density functions which each represent occurrence probability of
the calculated position for respective one of the selected
position-measurement-points; a probability density calculation step
of calculating probability density of the calculated position of
each of the position-measurement-points, based on respective one of
the probability density functions; and a parameter calculation step
of evaluating an error of each calculated position relative to
respective reference position while using the respective calculated
probability density's value as a piece of weight information and
calculating values of the predetermined number of parameters, based
on the evaluated errors.
[0026] According to this method, pieces of position information of
selected position-measurement-points are measured, and the
positions and probability densities of the selected
position-measurement-points are calculated on the basis of the
measurement results. Upon calculating the statistically most valid
values of a predetermined number of parameters, which uniquely
specify position information of any area on an object, while using
the calculated probability densities as pieces of information each
representing the certainty of the position of a respective
position-measurement-point, the error between each calculated
position and a respective reference position is weighted in
accordance with the certainty of the calculated position of the
respective position-measurement-point, i.e., the probability
density at the calculated position of the respective
position-measurement-point. That is, if the probability density is
large, the weight is large, and if the probability density is
small, the weight is small. As a result, when the calculated
position of a position-measurement-point has high certainty, the
degree of influence of the error of the calculated position of the
position-measurement-point relative to its reference position is
high; when the calculated position of a position-measurement-point
has low certainty, the degree of influence of the error of the
calculated position of the position-measurement-point relative to
its reference position is low. Therefore, since statistically valid
values of a predetermined number of parameters which uniquely
specify position information of any area on an object can be
calculated while rationally reflecting respective certainties of
the calculated positions of position-measurement-points, the
position of a area of interest on the object can be accurately
detected.
[0027] In the first position detection method of the present
invention, the reference positions can be determined in advance on
the basis of design information.
[0028] The probability density of each calculated mark (a
position-measurement-point) position directly reflects the
certainty of a respective mark position. Therefore, in the first
position detection method of the present invention, the errors are
evaluated by multiplying the errors of the calculated positions
relative to the respective reference positions by the respective
probability densities of the calculated positions.
[0029] In the first position detection method of the present
invention, normal distributions can be adopted as the probability
density functions. In this way, it is particularly valid to presume
the occurrence probability distributions to be a normal
distribution when variations of errors of the calculated mark
positions relative to the respective reference positions are
expected to be random like normal random numbers. When the
occurrence probability distribution is known, a probability density
function of that probability distribution can be used instead of a
normal distribution. On the other hand, when the occurrence
probability distribution is unknown, it is rational to presume the
occurrence probability distribution to be a normal distribution,
which is the most general probability distribution.
[0030] In the first position detection method of the present
invention, position measurement marks can be formed at the
position-measurement-poin- ts. In such case, the position of each
position-measurement-point can be measured by detecting a
respective position measurement mark. Note that the position
measurement mark can be, e.g., a line-and-space mark, box-in-box
mark, and the like.
[0031] In this case, a plurality of position detection marks formed
at the plurality of position-measurement-points can include a first
number of first marks, of which surface states change in a first
direction, and the position information of each first mark measured
in the measurement step can be position information of a plurality
of feature portions in the first direction of each first mark. In
such a case, the position of a first mark in the first direction
can be calculated by measuring and processing the position
information of the first mark. Also, a probability density function
that represents the occurrence probability of the calculated
position can be estimated based on its design reference position
and measured position information. When the first mark periodically
changes in the first direction like a line-and-space mark, the
average value of positions of a plurality of feature portions in
the first direction such as the boundaries between lines and
spaces, which represents the central position of the first mark,
can be used as the position of the first mark, and the probability
density function that represents the occurrence probability of the
central position can be estimated.
[0032] The position information of each selected first mark can be
measured in the measurement step at a plurality of positions in a
direction perpendicular to the first direction. In this case, since
the number of pieces of position information to be processed
increases, the position of the first mark in the first direction
can be accurately calculated, and the probability density function
that represents the occurrence probability of the calculated
position can be accurately estimated.
[0033] The surface state of each first mark can also change in a
second direction different from the first direction, and the
position information of the first mark measured in the measurement
step can include position information, in the first direction, of a
plurality of feature portions lined in the first direction of the
first mark, and position information, in the second direction, of a
plurality of feature portions lined in the second direction of the
first mark. In this case, the two-dimensional position of the first
mark can be calculated based on the measurement result of the
position information of the first mark, and the probability density
function that represents the occurrence probability of the
calculated position can be estimated based on the design reference
position and measured position information. That is, information
that pertains to the two-dimensional position of the object can be
calculated.
[0034] In the measurement step, for each selected first mark, at
least one of position information in the first direction of a
plurality of feature portions in the first direction and position
information in the second direction of a plurality of feature
portions in the second direction can be measured. In this case,
since the number of pieces of position information to be processed
increases for at least one of the first and second directions, the
position of the first mark in a direction in which the number of
pieces of position information to be processes has increased can be
accurately calculated, and the probability density function that
represents the occurrence probability of the calculated position
can be accurately estimated.
[0035] The plurality of marks can further include a second number
of second marks, of which surface states change in a second
direction different from the first direction, and the position
information of each second mark measured in the measurement step
can be position information of a plurality of feature portions in
the second direction of the second mark. In this case, the position
of the second mark in the second direction can be calculated by
measuring and processing position information of the second mark in
the same manner as for the first mark, and a probability density
function that represents the occurrence probability of the
calculated position can be estimated based on its design reference
position and measured position information. That is, information
that pertains to a two-dimensional position of the object can be
calculated.
[0036] In the measurement step, the position information of each
selected second mark can be measured at a plurality of positions in
a direction perpendicular to the second direction. In this case,
since the number of pieces of position information to be processed
increases, the position of the second mark in the second direction
can be accurately calculated, and the probability density function
that represents the occurrence probability of the calculated
position can be accurately estimated.
[0037] Furthermore, in the first position detection method of the
present invention in which position measurement marks are formed at
position-measurement-points, a plurality of divided areas can be
arranged on an object, and position measurement marks can be
contained in each of the plurality of divided areas. In this case,
the arrangement coordinate position of each divided area on the
object can be accurately detected.
[0038] In addition, the predetermined number of parameters can
include parameters associated with representative points of the
plurality of divided areas. In this case, the arrangement of the
representative points, e.g. the central points, of the plurality of
divided areas on the object can be calculated, the arrangement
being referred to as an arrangement coordinate system.
[0039] Note that the predetermined number of parameters can further
include parameters associated with points other than the
representative points of the plurality of divided areas. In this
case, in addition to the arrangement coordinate system of
representative points of the plurality of divided areas on the
object, a divided area coordinate system that specifies the
direction of pattern transfer, scale, and the like on the divided
areas can be calculated.
[0040] According to the second aspect, there is provided a second
position detection method for detecting position information of any
area on an object provided with a first number of
position-measurement-points, the position detection method
comprising a first step of selecting a plurality of measurement
point subsets which each consist of a third number of
position-measurement-points and are different from one another, the
third number being larger than a second number and smaller than the
first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object; and a second step of statistically calculating, for
each of the plurality of measurement point subsets, estimations of
the predetermined number of parameters and certainty of the
estimations, based on measurement results of the third number of
position-measurement-points. For each measurement point subset, the
certainty of the estimations of the predetermined number of
parameters is calculated using the calculated estimations, and is
determined in accordance with position errors of
position-measurement-points, which are used to calculate the
estimations, relative to respective expected positions. If the
deviation of position errors of the position-measurement-points
used to calculate the estimations is large, the certainty is low;
if the deviation of the position errors is small, the certainty is
high.
[0041] According to this method, a plurality of different
measurement point subsets are selected, and for each of the
measurement point subsets, estimations and their certainty of the
predetermined number of parameters which uniquely specify position
information of any area on an object are statistically calculated.
The estimations and their certainty of the predetermined number of
parameters (to be also referred to as "position parameters"
hereinafter) of each measurement point subset reflect the position
distribution of all position-measurement-points. Therefore, the
position distribution of all position-measurement-points can be
accurately estimated based on respective groups of the estimations
and their certainty of the predetermined number of parameters for
the plurality of measurement point subsets that are selected
empirically or arbitrarily.
[0042] The second position detection method can further comprise
the third step of obtaining statistically valid values of the
predetermined number of parameters, based on the respective groups
of estimations and certainty for the plurality of measurement point
subsets calculated in the second step. In this case, since
statistically valid position parameter values are calculated on the
basis of respective groups of the estimations and their certainty
of position parameters for the measurement point subsets, and the
groups of the estimations and certainty each statistically reflect
the predetermined number of parameters that are statistically
determined by broadly sampling from all
position-measurement-points, statistically valid values of the
predetermined number of parameters for all the
position-measurement-point- s can be accurately calculated.
[0043] Furthermore, the statistically valid value of each of the
predetermined number of parameters is obtained by calculating
average of the corresponding estimations weighted with respective
certainties, each of the certainties representing a piece of weight
information for the respective estimation. In this case, since the
weighted mean of estimations is calculated using the respective
certainties of the estimations as respective weights of the
estimations, the rational evaluation of the estimations can be
performed in which estimations with a low certainty contribute
less, and in which other estimations with a high certainty
contribute more. And statistically valid values of the
predetermined number of parameters for all the
position-measurement-point- s can be accurately and easily
calculated.
[0044] In the second position detection method of the present
invention, in the second step, certainties of position measurement
results of the position-measurement-points can be taken in account
for calculating the estimations and certainty thereof. In this
case, since the estimations and their certainty of the
predetermined number of parameters are calculated considering the
certainties of the position measurement results at the
position-measurement-points, statistically more valid values of the
predetermined number of parameters can be calculated.
[0045] In addition, the second step can comprise an estimation step
of calculating, for each of the plurality of measurement point
subsets, respective positions of the third number of
position-measurement-points based on measurement results of the
third number of position-measurement-points and estimating
probability density functions that each represent occurrence
probability of the calculated position of the respective, selected
position-measurement-point; a probability density calculation step
of calculating respective probability densities of the calculated
positions of the position-measurement-points, based on the
probability density functions; and a parameter calculation step of
evaluating an error of each of the calculated positions relative to
respective reference position using the respective calculated
probability density's value as a piece of weight information and
calculating estimations of the predetermined number of parameters,
based on the evaluated errors. In this case, since errors between
the calculated positions and respective reference positions are
weighted in accordance with the information of the certainties of
the calculated positions of the position-measurement-points, i.e.
the probability densities at the calculated positions of the
position-measurement-points, statistically valid estimations of the
predetermined number of parameters can be calculated which uniquely
specify position information of any area on an object and
rationally reflect the certainties of the calculated positions of
the position-measurement-points.
[0046] In the second position detection method of the present
invention, position measurement marks can be formed at the
position-measurement-poin- ts as in the first position detection
method of the present invention. And, a plurality of divided areas
can be arranged on the object, and position measurement marks can
be contained in each of the plurality of divided areas.
[0047] According to the third aspect, there is provided a third
position detection method for detecting position information of any
area on an object provided with a first number of
position-measurement-points, the position detection method
comprising a first step of selecting a first measurement point
subset which consists of a third number of
position-measurement-points, the third number being larger than a
second number and smaller than the first number, the second number
being a minimum number of measurement points required to calculate
a predetermined number of parameters that uniquely specify position
information of any area on the object; a second step of selecting a
plurality of second measurement point subsets which each consist of
a fourth number of position-measurement-points and are different
from one another, the fourth number being larger than the second
number and smaller than the third number; and a third step of
statistically evaluating possibility of replacing the first
measurement point subset by one of the plurality of second
measurement point subsets, based on measurement results of the
third number of position-measurement-points composing the first
measurement point subset and measurement results of sets of the
fourth number of position-measurement-points each composing one of
the second measurement point subsets, the first measurement point
subset being used to calculate the predetermined number of
parameters.
[0048] According to this, for each of the plurality of second
measurement point subsets, it is evaluated whether or not it is
possible to replace the initially selected sample set (first
measurement point subset) by a sample set including a smaller
number of elements. That is, to determine whether or not it is
possible to reduce the number of position-measurement-points used
to calculate the predetermined number of parameters, it is
evaluated, based on the position measurement results at the
position-measurement-points of the first measurement point subset
and those of each second measurement point subset, whether or not
the position error distribution of position-measurement-points in
the second measurement point subset is similar to that of
position-measurement-point- s of the first measurement point
subset, in other words, whether or not the second measurement point
subset and the first measurement point subset equally reflect the
entire set of all position-measurement-points. Therefore, upon
reducing the number of position-measurement-points as elements of a
sample set, statistical validity of the calculated values of the
predetermined number of parameters can be maintained.
[0049] In the third position detection method of the present
invention, the third step can comprise a fourth step of
statistically calculating estimations of the predetermined number
of parameters and certainty of the estimations, based on
measurement results of the third number of
position-measurement-points composing the first measurement point
subset; a fifth step of statistically calculating estimations of
the predetermined number of parameters and certainty of the
estimations for each of the plurality of second measurement point
subsets, based on measurement results of the fourth number of
position-measurement-points; and a sixth step of comparing the
estimations and certainty of the first measurement point subset
with the estimations and certainty for each of the plurality of
second measurement point subsets and evaluating possibility of
replacing the first measurement point subset by one of the
plurality of second measurement point subsets, the first
measurement point subset being used to calculate the predetermined
number of parameters.
[0050] In this case, the estimations and their certainty of the
predetermined number of parameters calculated based on the position
measurement results at the position-measurement-points of the first
measurement point subset are compared with those calculated based
on the position measurement results at the
position-measurement-points of each second measurement point
subset. In this comparison, the certainties of respective groups of
the estimations of the two measurement point subsets are compared
as well as the groups of the estimations, the certainties each
reflecting deviation of the position error distribution of
position-measurement-points of the respective measurement point
subset. And by examining the two comparison results, the position
error distribution of position-measurement-points of the first
measurement point subset is compared with that of
position-measurement-points of the second measurement point subset.
Therefore, it can be determined whether or not one of the plurality
of second measurement point subsets and the first measurement point
subset equally reflect the entire set of all
position-measurement-points.
[0051] Furthermore, in the fourth step, certainties of position
measurement results of the position-measurement-points can be taken
in account for calculating the estimations and certainty thereof.
In this case, since the estimations and their certainty of the
predetermined number of parameters are calculated considering the
certainties of the position measurement results at the
position-measurement-points, statistically more valid values of the
predetermined number of parameters can be calculated.
[0052] Additionally, the fourth step can comprise an estimation
step of calculating respective positions of the third number of
position-measurement-points, which compose the first measurement
point subset, based on measurement results of the third number of
position-measurement-points and estimating probability density
functions that each represent occurrence probability of the
calculated position of a respective point of the third number of
position-measurement-points; a probability density calculation step
of calculating respective probability densities of the calculated
positions of the position-measurement-points, based on the
probability density functions; and a parameter calculation step of
evaluating an error of each of the calculated positions relative to
respective reference position using the respective calculated
probability density's value as a piece of weight information and
calculating estimations of the predetermined number of parameters,
based on the evaluated errors. In this case, since, for the third
number of position-measurement-points composing the first
measurement point subset, errors between the calculated positions
and their reference positions are weighted in accordance with the
information of the certainties of the calculated positions of the
position-measurement-points, i.e. the probability densities at the
calculated positions of the position-measurement-points,
statistically valid estimations of the predetermined number of
parameters can be calculated which uniquely specify position
information of any area on an object and rationally reflect the
certainties of the calculated positions of the
position-measurement-points.
[0053] In addition, in the fifth step, certainties of position
measurement results of the position-measurement-points can be taken
into account upon calculating the estimations and certainty
thereof. Moreover, the fifth step can comprise an estimation step
of calculating respective positions of the fourth number of
position-measurement-points, for each of the plurality of second
measurement point subsets, based on measurement results of the
fourth number of position-measurement-points and estimating
probability density functions that each represent occurrence
probability of the calculated position of a respective point of the
fourth number of position-measurement-points; a probability density
calculation step of calculating respective probability densities of
the calculated positions of the position-measurement-points, based
on the probability density functions; and a parameter calculation
step of evaluating an error of each of the calculated positions
relative to respective reference position using the respective
calculated probability density's value as a piece of weight
information and calculating estimations of the predetermined number
of parameters, based on the evaluated errors.
[0054] In addition, in the third position detection method
according to this invention, the third step can comprise a fourth
step of statistically calculating, for each of the second
measurement point subsets, estimations of the predetermined number
of parameters and certainty of the estimations, based on
measurement results of the fourth number of
position-measurement-points; a fifth step of statistically
calculating position errors of the position-measurement-points of
the first measurement point subset through use of the estimations
of the predetermined number of parameters calculated in the fourth
step and evaluating possibility of replacing the first measurement
point subset by one of the plurality of second measurement point
subsets.
[0055] In this case, by calculating position errors of
position-measurement-points composing the first measurement point
subset by using the estimations, of the predetermined number of
parameters, calculated on the basis of the position measurement
results at the position-measurement-points of each second
measurement point subset, the position error distribution of
position-measurement-points in the first measurement point subset
can be obtained. Therefore, without calculating the estimations and
their certainty of the predetermined number of parameters on the
basis of the position measurement results at the
position-measurement-points of the first measurement point subset,
it can be determined whether or not one of the plurality of second
measurement point subsets and the first measurement point subset
equally reflect the entire set of all
position-measurement-points.
[0056] Especially, in the case where a second measurement point
subset to be compared is a subset of the first measurement point
subset, by calculating position errors of
position-measurement-points which are included in the first
measurement point subset but not included in the second measurement
point subset, the estimations and their certainty of the
predetermined number of parameters can be calculated for the case
where the first measurement point subset is used as a sample set.
Therefore, it can be quickly determined whether or not one of the
plurality of second measurement point subsets and the first
measurement point subset equally reflect the entire set of all
position-measurement-points.
[0057] Furthermore, in the fourth step, certainties of position
measurement results of the position-measurement-points can be taken
in account for calculating the estimations and certainty thereof.
In this case, since the estimations and their certainty of the
predetermined number of parameters are calculated considering the
certainties of the position measurement results at the
position-measurement-points, statistically more valid values of the
predetermined number of parameters can be calculated.
[0058] In addition, the fourth step can comprise an estimation step
of calculating respective positions of the fourth number of
position-measurement-points, for each of the plurality of second
measurement point subsets, based on measurement results of the
fourth number of position-measurement-points and estimating
probability density functions that each represent occurrence
probability of the calculated position of a respective point of the
fourth number of position-measurement-points; a probability density
calculation step of calculating respective probability densities of
the calculated positions of the position-measurement-points, based
on the probability density functions; and a parameter calculation
step of evaluating an error of each of the calculated positions
relative to respective reference position using the respective
calculated probability density's value as a piece of weight
information and calculating estimations of the predetermined number
of parameters, based on the evaluated errors. In this case, since,
for each of the second measurement point subsets, errors between
the calculated positions and their reference positions are weighted
in accordance with the information of the certainties of the
calculated positions of the position-measurement-points, i.e. the
probability densities at the calculated positions of the
position-measurement-points, statistically valid estimations of the
predetermined number of parameters can be calculated which uniquely
specify position information of any area on an object and
rationally reflect the certainties of the calculated positions of
the position-measurement-points.
[0059] Furthermore, the third position detection method according
to this invention can further comprise the fourth step which, if
the third step finds second measurement point subsets that can
replace the first measurement point subset, selects the most valid,
second measurement point subset for replacement and adopts
estimations of the predetermined number of parameters, calculated
based on measurement results of the fourth number of
position-measurement-points of the selected second measurement
point subset, as values thereof, and which, if the third step finds
no second measurement point subsets that can replace the first
measurement point subset, adopts estimations of the predetermined
number of parameters, calculated based on measurement results of
the second number of position-measurement-points of the first
measurement point subset, as values thereof.
[0060] In this case, if the third step finds second measurement
point subsets that can replace the first measurement point subset,
i.e. if the number of position-measurement-points can be reduced
maintaining the statistical validity, the most valid, second
measurement point subset for replacement is adopted as the sample
set. On the other hand, if the third step finds no second
measurement point subsets that can replace the first measurement
point subset, i.e. if the number of position-measurement-poin- ts
can not be reduced maintaining the statistical validity, the first
measurement point subset is adopted as the sample set. Then, the
estimations of the predetermined number of parameters calculated
based on the position measurement results at the
position-measurement-points of the sample set are adopted as the
values of the predetermined number of parameters. Therefore, the
number of position-measurement-points used to calculate the values
of the predetermined number of parameters can be reduced while
maintaining the statistical validity, and improvement of the
position detection speed can be achieved maintaining the
accuracy.
[0061] In the third position detection method according to this
invention, position measurement marks can be formed at the
position-measurement-poin- ts in the same manner as in the first
position detection method, and a plurality of divided areas, each
of which is provided with the position measurement marks, can be
arranged on the object.
[0062] According to the fourth aspect of this invention, there is
provided a fourth position detection method for detecting position
information of any area on an object provided with a first number
of position-measurement-points, the position detection method
comprising a first step of selecting a plurality of first
measurement point subsets which each consist of a third number of
position-measurement-points and are different from one another, the
third number being larger than a second number and smaller than the
first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object; a second step of selecting a plurality of second
measurement point subsets which each consist of a fourth number of
position-measurement-points and are different from one another, the
fourth number being larger than the second number and smaller than
the third number; and a third step of statistically evaluating
possibility of replacing the plurality of first measurement point
subsets by one of the plurality of second measurement point
subsets, as a measurement point set used to calculate the
predetermined number of parameters.
[0063] According to this method, for each of the plurality of
second measurement point subsets, it is evaluated whether or not it
is possible to replace the plurality of initially selected sample
sets (the plurality of first measurement point subsets) by one
sample set composed of a fewer number of elements. That is, to
determine whether or not the number of position-measurement-points
used to calculate values of the predetermined number of parameters
and the processing volume of the position measurement results can
be reduced, it is evaluated whether or not the position error
distribution of position-measurement-points in one of the plurality
of second measurement point subsets is similar to a position error
distribution, for all position-measurement-points, estimated based
on the position measurement results at the
position-measurement-points of the plurality of first measurement
point subsets. Therefore, upon reducing the number of
position-measurement-points as elements of a sample set and
reducing the processing volume of the position measurement results,
statistical validity of the calculated values of the predetermined
number of parameters can be maintained.
[0064] In the fourth position detection method according to this
invention, the third step can comprise a fourth step of
statistically calculating, for each of the plurality of first
measurement point subsets, estimations of the predetermined number
of parameters and certainty of the estimations, based on
measurement results of the third number of
position-measurement-points; a fifth step of calculating
statistically valid estimations of the predetermined number of
parameters and certainty of the estimations, based on groups of the
estimations and certainty thereof for the plurality of first
measurement point subsets, calculated in the fourth step; a sixth
step of statistically calculating estimations of the predetermined
number of parameters and certainty of the estimations for each of
the plurality of second measurement point subsets, based on
measurement results of the fourth number of
position-measurement-points; and a seventh step of comparing the
statistically valid estimations and certainty with the estimations
and certainty for each of the plurality of second measurement point
subsets and evaluating possibility of adopting one of the plurality
of second measurement point subsets as a measurement point set used
to calculate the predetermined number of parameters.
[0065] In this case, the statistically valid estimations and their
certainty of the predetermined number of parameters calculated
based on the position measurement results at the
position-measurement-points of the plurality of first measurement
point subsets are compared with the estimations and their certainty
of the predetermined number of parameters calculated based on the
position measurement results at the position-measurement-points of
each second measurement point subset. In this comparison, the
certainties of the two groups of the estimations are compared as
well as the groups of the estimations, the certainties each
reflecting deviation of the position error distribution of
position-measurement-points of the respective measurement point
subset. And by examining the two comparison results, the two
position error distributions are compared. Therefore, it can be
determined whether or not one of the plurality of second
measurement point subsets and the first measurement point subset
equally reflect the entire set of all
position-measurement-points.
[0066] Furthermore, in the fourth step, certainties of position
measurement results of the position-measurement-points can be taken
in account for calculating the estimations and certainty thereof.
In this case, since the estimations and their certainty of the
predetermined number of parameters are calculated considering the
certainties of the position measurement results at the
position-measurement-points, statistically more valid values of the
predetermined number of parameters can be calculated.
[0067] Furthermore, the fourth step can comprise an estimation step
of calculating respective positions of the third number of
position-measurement-points, for each of the plurality of first
measurement point subsets, based on measurement results of the
third number of position-measurement-points and estimating
probability density functions that each represent occurrence
probability of the calculated position of a respective point of the
third number of position-measurement-points; a probability density
calculation step of calculating respective probability densities of
the calculated positions of the position-measurement-points, based
on the probability density functions; and a parameter calculation
step of evaluating an error of each of the calculated positions
relative to respective reference position using the respective
calculated probability density's value as a piece of weight
information and calculating estimations of the predetermined number
of parameters, based on the evaluated errors. In this case, since,
for each of the first measurement point subsets, errors between the
calculated positions and their reference positions are weighted in
accordance with the information of the certainties of the
calculated positions of the position-measurement-points, i.e. the
probability densities at the calculated positions of the
position-measurement-points, statistically valid estimations of the
predetermined number of parameters can be calculated which uniquely
specify position information of any area on an object and
rationally reflect the certainties of the calculated positions of
the position-measurement-points.
[0068] In addition, in the sixth step, certainties of position
measurement results of the position-measurement-points can be taken
into account upon calculating the estimations and certainty
thereof. And the sixth step can comprise an estimation step of
calculating respective positions of the fourth number of
position-measurement-points, for each of the plurality of second
measurement point subsets, based on measurement results of the
fourth number of position-measurement-points and estimating
probability density functions that each represent occurrence
probability of the calculated position of a respective point of the
fourth number of position-measurement-points; a probability density
calculation step of calculating respective probability densities of
the calculated positions of the position-measurement-points, based
on the probability density functions; and a parameter calculation
step of evaluating an error of each of the calculated positions
relative to respective reference position using the respective
calculated probability density's value as a piece of weight
information and calculating estimations of the predetermined number
of parameters, based on the evaluated errors.
[0069] Furthermore, the fourth position detection method can
further comprise the eighth step which, if the third step finds
second measurement point subsets that can replace the first
measurement point subset, selects the most valid, second
measurement point subset for replacement and adopts estimations of
the predetermined number of parameters, calculated based on
measurement results of the fourth number of
position-measurement-points of the selected second measurement
point subset, as values thereof, and which, if the third step finds
no second measurement point subsets that can replace the first
measurement point subset, adopts the statistically valid
estimations as values of the predetermined number of parameters.
Therefore, the number of position-measurement-points used to
calculate the values of the predetermined number of parameters can
be reduced while maintaining the statistical validity, and
improvement of the position detection speed can be achieved
maintaining the accuracy.
[0070] In addition, in the fourth position detection method, the
third step can comprise a fourth step of statistically calculating,
for each of the second measurement point subsets, estimations of
the predetermined number of parameters and certainty of the
estimations, based on measurement results of the fourth number of
position-measurement-points; a fifth step of calculating position
errors of all the position-measurement-points of the plurality of
first measurement point subsets through use of the estimations of
the predetermined number of parameters calculated for each of the
second measurement point subsets and evaluating possibility of
replacing the plurality of first measurement point subsets by one
of the plurality of second measurement point subsets.
[0071] In this case, by calculating position errors of
position-measurement-points of the plurality of first measurement
point subsets by using the estimations, of the predetermined number
of parameters, calculated based on the position measurement results
at the position-measurement-points of each second measurement point
subset, the position error distribution for all
position-measurement-points, which will be estimated if the
plurality of first measurement point subsets serve as the sample
set, can be obtained. Therefore, without calculating groups of the
estimations and their certainty of the predetermined number of
parameters on the basis of the position measurement results at the
position-measurement-points of the plurality of first measurement
point subsets and thus the statistically valid estimations and
their certainty of the predetermined number of parameters, it can
be determined whether or not one of the plurality of second
measurement point subsets reflects the entire set of all
position-measurement-points.
[0072] Furthermore, in the fourth step, certainties of position
measurement results of the position-measurement-points can be taken
in account for calculating the estimations and certainty thereof.
In this case, since the estimations and their certainty of the
predetermined number of parameters are calculated considering the
certainties of the position measurement results at the
position-measurement-points, statistically more valid values of the
predetermined number of parameters can be calculated.
[0073] Furthermore, the fourth step can comprise an estimation step
of calculating respective positions of the fourth number of
position-measurement-points, for each of the plurality of second
measurement point subsets, based on measurement results of the
fourth number of position-measurement-points and estimating
probability density functions that each represent occurrence
probability of the calculated position of a respective point of the
fourth number of position-measurement-points; a probability density
calculation step of calculating respective probability densities of
the calculated positions of the position-measurement-points, based
on the probability density functions; and a parameter calculation
step of evaluating an error of each of the calculated positions
relative to respective reference position using the respective
calculated probability density's value as a piece of weight
information and calculating estimations of the predetermined number
of parameters, based on the evaluated errors. In this case, since,
for each of the second measurement point subsets, errors between
the calculated positions and their reference positions are weighted
in accordance with the information of the certainties of the
calculated positions of the position-measurement-points, i.e. the
probability densities at the calculated positions of the
position-measurement-points, statistically valid estimations of the
predetermined number of parameters can be calculated which uniquely
specify position information of any area on an object and
rationally reflect the certainties of the calculated positions of
the position-measurement-points.
[0074] In addition, the fourth position detection method according
to this invention can further comprise the fourth step which, if
the third step finds second measurement point subsets that can
replace the first measurement point subset, selects the most valid
one, for replacement, of the second measurement point subsets and
adopts estimations of the predetermined number of parameters,
calculated based on measurement results of the fourth number of
position-measurement-points of the selected second measurement
point subset, as values thereof, and which, if the third step finds
no second measurement point subsets that can replace the first
measurement point subset, statistically calculates estimations of
the predetermined number of parameters and certainty of the
estimations, based on measurement results of the third number of
position-measurement-points for each of the plurality of first
measurement point subsets, and adopts as values of the
predetermined number of parameters statistically valid estimations
thereof calculated based on groups of the estimations and certainty
thereof for the plurality of first measurement point subsets.
Therefore, the number of position-measurement-points used to
calculate the values of the predetermined number of parameters can
be reduced while maintaining the statistical validity, and
improvement of the position detection speed can be achieved
maintaining the accuracy.
[0075] Note that in the fourth position detection method of this
invention, in the same manner as in the second position detection
method, the statistically valid value of each of the predetermined
number of parameters can be obtained by calculating average of the
corresponding estimations weighted with the respective certainties,
each of the certainties representing a piece of weight information
for the respective estimation.
[0076] Furthermore, in the fourth position detection method of this
invention, position measurement marks can be formed at the
position-measurement-points in the same manner as in the first
position detection method. And a plurality of divided areas each of
which is provided with the position measurement marks can be
arranged on the object.
[0077] According to the fifth aspect of this invention, there is
provided a first position detector that detects position
information of any area on an object provided with a plurality of
position-measurement-points, the position detector comprising a
measurement unit that measures pieces of position information of
more position-measurement-points than a minimum number of
measurements required to calculate values of a predetermined number
of parameters, which uniquely specify position information of any
area on the object, the position-measurement-points being selected
from the plurality of position-measurement-points; an estimation
unit that is electrically connected to the measurement unit and
that detects respective positions of the selected
position-measurement-points, based on the measurement results of
the pieces of position information, estimates probability density
functions which each represent occurrence probability of the
detected position for respective one of the selected
position-measurement-points, and calculates probability density of
the detected position of each of the position-measurement-points;
and a parameter calculation unit that is electrically connected to
the measurement unit and the estimation unit and that evaluates
detection error of each of the detected positions while using the
respective calculated probability density's value as a piece of
weight information and calculates such values of the predetermined
number of parameters that the detection errors become statistically
minimum as a whole, based on the detection errors.
[0078] In this detector, according to the first position detection
method of the present invention, the estimation unit calculates the
positions of the selected marks (position-measurement-points) and
respective probability densities at the calculated mark positions
on the basis of position information of the marks measured by the
measurement unit. And the parameter calculation unit calculates the
values of the predetermined number of parameters that uniquely
specify position information of any area on an object. Therefore,
the predetermined number of parameters can be accurately
calculated, and the position information of any area on an object
can be accurately detected.
[0079] In the first position detector of the present invention, the
measurement unit can comprise an image pickup unit for picking up
images of marks formed on the object. In this case, the position
information of a selected mark can be measured on the basis of
changes in light intensity according to position in the picked-up
mark image.
[0080] According to the sixth aspect, there is provided a second
position detector that detects position information of any area on
an object provided with a first number of
position-measurement-points, the position detector comprising a
measurement unit that measures positions of the
position-measurement-points; a set-selection unit that selects a
plurality of measurement point subsets which each consist of a
third number of position-measurement-points and are different from
one another, the third number being larger than a second number and
smaller than the first number, the second number being a minimum
number of measurement points required to calculate a predetermined
number of parameters that uniquely specify position information of
any area on the object; and an estimation computing unit that is
electrically connected to the measurement unit and the
set-selection unit and that statistically calculates, for each of
the plurality of measurement point subsets, estimations of the
predetermined number of parameters and certainty of the
estimations, based on measurement results of the third number of
position-measurement-points.
[0081] In this detector, for each measurement point subset selected
by the set selection unit, according to the second position
detection method of the present invention, the estimation computing
unit statistically estimates values of the predetermined number of
parameters which uniquely specify position information of any area
on an object and calculates the certainty of the estimations on the
basis of the positions of position-measurement-points measured by
the measurement unit. Therefore, the position distribution of all
position-measurement-points can be accurately estimated based on
respective groups of the estimations and their certainty of the
predetermined number of parameters for the plurality of measurement
point subsets that are selected empirically or arbitrarily.
[0082] In the second position detector according to this invention,
the estimation computing unit can comprise an estimation unit that,
for each of the plurality of measurement point subsets, detects
respective positions of the third number of
position-measurement-points, based on the measurement results of
the pieces of position information of the third number of
position-measurement-points, estimates probability density
functions which each represent occurrence probability of the
detected position for the respective point of the third number of
position-measurement-points, and calculates probability density of
the detected position of each of the position-measurement-points;
and a parameter calculation unit that is electrically connected to
the estimation unit and that evaluates detection error of each of
the detected positions while using the respective calculated
probability density's value as a piece of weight information and
calculates such values of the predetermined number of parameters
that the detection errors become statistically minimum as a whole,
based on the detection errors. In this case, for each of the
plurality of measurement point subset, the parameter calculation
unit weights errors between the calculated positions and their
reference positions in accordance with the information of the
certainties of the calculated positions of the
position-measurement-points calculated by the estimation unit, i.e.
the probability densities at the calculated positions of the
position-measurement-points, and calculates estimations of the
predetermined number of parameters which uniquely specify position
information of any area on an object and rationally reflect the
certainties of the calculated positions of the
position-measurement-point- s. Therefore, statistically valid
estimations of the predetermined number of parameters can be
obtained.
[0083] In addition, the second position detector according to this
invention can further comprise a parameter value determining unit
that is electrically connected to the estimation computing unit and
that calculates statistically valid estimations of the
predetermined number of parameters based on groups of the
estimations and certainty thereof, calculated by the estimation
computing unit, for the plurality of measurement point subsets. In
this case, the statistically valid values of the predetermined
number of parameters for all position-measurement-po- ints can be
accurately obtained.
[0084] According to the seventh aspect of this invention, there is
provided a third position detector that detects position
information of any area on an object provided with a first number
of position-measurement-points, the position detector comprising a
measurement unit that measures positions of the
position-measurement-poin- ts; a set-selection unit that selects a
first measurement point subsets, which each consist of a third
number of position-measurement-points, and a plurality of second
measurement point subsets which each consist of a fourth number of
position-measurement-points and are different from one another, the
third number being larger than a second number and smaller than the
first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object, the fourth number being larger than the second
number and smaller than the third number; and an evaluation
computing unit that is electrically connected to the set-selection
unit and that evaluates possibility of replacing the first
measurement point subset by one of the plurality of second
measurement point subsets, the first measurement point subset being
used to calculate the predetermined number of parameters.
[0085] In this detector, according to the third position detection
method of this invention, the evaluation computing unit
statistically evaluates based on positions of
position-measurement-points measured by the measurement unit
whether or not it is possible to replace the first measurement
point subset as a sample set composed of
position-measurement-points to be measured to calculate values of
the predetermined number of parameters by one of the plurality of
second measurement point subsets each of which is composed of a
fewer number of elements than the first measurement point subset.
Therefore, upon reducing the number of position-measurement-points
as elements of a sample set, statistical validity of the calculated
values of the predetermined number of parameters can be
maintained.
[0086] In the position detector according to this invention, the
evaluation computing unit can comprise an estimation calculation
unit that is electrically connected to the measurement unit and
that statistically calculates, for the specific measurement point
subset, estimations of the predetermined number of parameters and
certainty of the estimations, based on measurement results of
position information of position-measurement-points composing the
specific measurement point subset which is selected from the first
measurement point subset and the plurality of second measurement
point subsets; and an evaluation unit that is electrically
connected to the estimation calculation unit and that compares the
estimations and certainty of the first measurement point subset
with the estimations and certainty for each of the plurality of
second measurement point subsets and evaluates possibility of
replacing the first measurement point subset by one of the
plurality of second measurement point subsets, the first
measurement point subset being used to calculate the predetermined
number of parameters.
[0087] In this case, the evaluation unit compares the estimations
and their certainty of the predetermined number of parameters
calculated by the estimation computing unit for the first
measurement point subset with those calculated for each second
measurement point subset. In this comparison, the certainties of
respective groups of the estimations of the two measurement point
subsets are compared as well as the groups of the estimations, the
certainties each reflecting deviation of the position error
distribution of position-measurement-points of the respective
measurement point subset. And by examining the two comparison
results, the position error distribution of
position-measurement-points of the first measurement point subset
is compared with that of position-measurement-points of the second
measurement point subset. Therefore, it can be determined whether
or not one of the plurality of second measurement point subsets and
the first measurement point subset equally reflect the entire set
of all position-measurement-points.
[0088] Furthermore, the estimation calculation unit can comprise an
estimation unit that detects respective positions of
position-measurement-points composing the specific measurement
point subset, based on the measurement results of position
information of position-measurement-points composing the specific
measurement point subset, estimates probability density functions
which each represent occurrence probability of the detected
position for respective one of the position-measurement-points of
the specific measurement point subset, and calculates probability
density of the detected position of each of the
position-measurement-points; and a parameter calculation unit that
is electrically connected to the estimation unit and that evaluates
detection error of each of the detected positions while using the
respective calculated probability density's value as a piece of
weight information and calculates such values of the predetermined
number of parameters that the detection errors become statistically
minimum as a whole, based on the detection errors. In this case,
since, for the specific measurement point subset, the parameter
calculation unit weights errors between the calculated positions
and their reference positions in accordance with the information of
the certainties of the calculated positions of the
position-measurement-points, i.e. the probability densities at the
calculated positions of the position-measurement-points, and
calculates statistically valid estimations of the predetermined
number of parameters which uniquely specify position information of
any area on an object. Therefore, statistically valid estimations
of the predetermined number of parameters that rationally reflect
the certainties of the calculated positions of the
position-measurement-point- s can be obtained.
[0089] In the third position detector according to this invention,
the evaluation computing unit can comprise an estimation
calculation unit that is electrically connected to the measurement
unit and that statistically calculates, for the specific
measurement point subset, estimations of the predetermined number
of parameters and certainty of the estimations, based on
measurement results of position information of
position-measurement-points composing the specific measurement
point subset, which is selected from the plurality of second
measurement point subsets; an evaluation unit that is electrically
connected to the estimation calculation unit and that calculates
position errors of the position-measurement-points, composing the
first measurement point subset, through use of estimations of the
predetermined number of parameters for each of the polarity of
second measurement point subsets and evaluates possibility of
replacing the first measurement point subset by one of the
plurality of second measurement point subsets.
[0090] In this case, by the evaluation unit calculating position
errors of position-measurement-points composing in the first
measurement point subset by using the estimations, calculated by
the estimation computing unit, of the predetermined number of
parameters for the second measurement point subset, the position
error distribution of position-measurement-points in the first
measurement point subset can be obtained. Therefore, without
calculating the estimations and their certainty of the
predetermined number of parameters on the basis of the position
measurement results at the position-measurement-points of the first
measurement point subset, it can be determined whether or not one
of the plurality of second measurement point subsets and the first
measurement point subset equally reflect the entire set of all
position-measurement-points.
[0091] In addition, the estimation calculation unit can comprise an
estimation unit that detects respective positions of
position-measurement-points composing the specific measurement
point subset, based on the measurement results of position
information of position-measurement-points composing the specific
measurement point subset, estimates probability density functions
which each represent occurrence probability of the detected
position for respective one of the position-measurement-points of
the specific measurement point subset, and calculates probability
density of the detected position of each of the
position-measurement-points; and a parameter calculation unit that
is electrically connected to the estimation unit and that evaluates
detection error of each of the detected positions while using the
respective calculated probability density's value as a piece of
weight information and calculates such values of the predetermined
number of parameters that the detection errors become statistically
minimum as a whole, based on the detection errors. In this case,
since, for the specific measurement point subset, the parameter
calculation unit weights errors between the calculated positions
and their reference positions in accordance with the information of
the certainties of the calculated positions of the
position-measurement-points, i.e. the probability densities at the
calculated positions of the position-measurement-points, and
calculates statistically valid estimations of the predetermined
number of parameters which uniquely specify position information of
any area on an object. Therefore, statistically valid estimations
of the predetermined number of parameters that rationally reflect
the certainties of the calculated positions of the
position-measurement-point- s can be obtained.
[0092] Furthermore, the third position detector according to this
invention can further comprise a parameter value determining unit
that is electrically connected to the evaluation computing unit and
that calculates values of the predetermined number of parameters,
based on evaluation results of the evaluation computing unit.
Therefore, the number of position-measurement-points used to
calculate the values of the predetermined number of parameters can
be reduced while maintaining the statistical validity, and
improvement of the position detection speed can be achieved
maintaining the accuracy.
[0093] According to the eighth aspect of this invention, there is
provided a fourth position detector that detects position
information of any area on an object provided with a first number
of position-measurement-points, the position detector comprising a
measurement unit that measures positions of the
position-measurement-points; a set-selection unit that selects a
plurality of first measurement point subsets, which each consist of
a third number of position-measurement-points and are different
from one another, and a plurality of second measurement point
subsets which each consist of a fourth number of
position-measurement-poi- nts and are different from one another,
the third number being larger than a second number and smaller than
the first number, the second number being a minimum number of
measurement points required to calculate a predetermined number of
parameters that uniquely specify position information of any area
on the object, the fourth number being larger than the second
number and smaller than the third number; and an evaluation
computing unit that is electrically connected to the set-selection
unit and that evaluates possibility of adopting one of the
plurality of second measurement point subsets as a measurement
point subset to calculate the predetermined number of
parameters.
[0094] In this detector, according to the fourth position detection
method of this invention, the evaluation computing unit
statistically evaluates based on positions of
position-measurement-points measured by the measurement unit
whether or not it is possible to replace the plurality of first
measurement point subset as a sample set composed of
position-measurement-points to be measured to calculate values of
the predetermined number of parameters by one of the plurality of
second measurement point subsets each of which is composed of a
fewer number of elements than the first measurement point subset.
Therefore, upon reducing the number of position-measurement-points
as elements of a sample set, statistical validity of the calculated
values of the predetermined number of parameters can be
maintained.
[0095] In the fourth position detector according to this invention,
the evaluation computing unit can comprise an estimation
calculation unit that is electrically connected to the measurement
unit and that statistically calculates, for the specific
measurement point subset, estimations of the predetermined number
of parameters and certainty of the estimations, based on
measurement results of position information of
position-measurement-points composing the specific measurement
point subset which is selected from the plurality of first
measurement point subset and the plurality of second measurement
point subsets, and calculates statistically valid estimations of
the predetermined number of parameters and certainty of the
estimations, based on groups of estimations of the predetermined
number of parameters and certainty of the estimations for the
plurality of first measurement point subsets; an evaluation
computing unit that is electrically connected to the estimation
calculation unit and that compares the statistically valid
estimations and certainty of the first measurement point subset
with the estimations and certainty for each of the plurality of
second measurement point subsets, and evaluates possibility of
adopting one of the plurality of second measurement point subsets
as a measurement point subset to calculate the predetermined number
of parameters.
[0096] In this case, the evaluation unit calculates the
statistically valid estimations and their certainty of the
predetermined number of parameters based on sets of the
predetermined number of parameters, for the plurality of first
measurement point subsets, calculated by the estimation calculation
unit, and compares the statistically valid estimations and their
certainty of the predetermined number of parameters, for each
second measurement point subset, calculated by the estimation
calculation unit with the statistically valid estimations and their
certainty of the predetermined number of parameters. In this
comparison, the certainties of the two groups of the estimations
are compared as well as the groups of the estimations, the
certainties each reflecting deviation of the position error
distribution of position-measurement-points of the respective
measurement point subset. And by examining the two comparison
results, the two position error distributions are compared.
Therefore, it can be determined whether or not one of the plurality
of second measurement point subsets and the first measurement point
subset equally reflect the entire set of all
position-measurement-points.
[0097] Furthermore, the estimation calculation unit can comprise an
estimation unit that is electrically connected to the measurement
unit and that detects respective positions of
position-measurement-points composing the specific measurement
point subset, based on the measurement results of position
information of position-measurement-points composing the specific
measurement point subset, estimates probability density functions
which each represent occurrence probability of the detected
position for respective one of the position-measurement-points of
the specific measurement point subset, and calculates probability
density of the detected position of each of the
position-measurement-points; and a parameter calculation unit that
is electrically connected to the estimation unit and that evaluates
detection error of each of the detected positions while using the
respective calculated probability density's value as a piece of
weight information and calculates such values of the predetermined
number of parameters that the detection errors become statistically
minimum as a whole, based on the detection errors. In this case,
since, for the specific measurement point subset, the parameter
calculation unit weights errors between the calculated positions
and their reference positions in accordance with the information of
the certainties of the calculated positions of the
position-measurement-points, i.e. the probability densities at the
calculated positions of the position-measurement-points, and
calculates statistically valid estimations of the predetermined
number of parameters which uniquely specify position information of
any area on an object. Therefore, statistically valid estimations
of the predetermined number of parameters that rationally reflect
the certainties of the calculated positions of the
position-measurement-points can be obtained.
[0098] In the fourth position detector according to this invention,
the evaluation computing unit can comprise an estimation
calculation unit that is electrically connected to the measurement
unit and that statistically calculates, for the specific
measurement point subset, estimations of the predetermined number
of parameters and certainty of the estimations, based on
measurement results of position information of
position-measurement-points composing the specific measurement
point subset which is selected from the plurality of second
measurement point subsets; and an evaluation unit that is
electrically connected to the estimation calculation unit and that
calculates errors of all the position-measurement-points of the
plurality of first measurement point subsets through use of the
estimations of the predetermined number of parameters calculated
for each of the second measurement point subsets, and evaluates
possibility of replacing the plurality of first measurement point
subsets by one of the plurality of second measurement point
subsets.
[0099] In this case, by the evaluation unit calculating position
errors of position-measurement-points of the plurality of first
measurement point subsets by using the estimations, of the
predetermined number of parameters for each second measurement
point subset, calculated by the estimation calculation unit, the
position error distribution for all position-measurement-points,
which will be estimated if the plurality of first measurement point
subsets serve as the sample set, can be obtained. Therefore,
without calculating groups of the estimations and their certainty
of the predetermined number of parameters on the basis of the
position measurement results at the position-measurement-points of
the plurality of first measurement point subsets and thus the
statistically valid estimations and their certainty of the
predetermined number of parameters, it can be determined whether or
not one of the plurality of second measurement point subsets
reflects the entire set of all position-measurement-points.
[0100] In addition, the estimation calculation unit can comprise an
estimation unit that detects respective positions of
position-measurement-points composing the specific measurement
point subset, based on the measurement results of position
information of position-measurement-points composing the specific
measurement point subset, estimates probability density functions
which each represent occurrence probability of the detected
position for respective one of the position-measurement-points of
the specific measurement point subset, and calculates probability
density of the detected position of each of the
position-measurement-points; and a parameter calculation unit that
is electrically connected to the estimation unit and that evaluates
detection error of each of the detected positions while using the
respective calculated probability density's value as a piece of
weight information and calculates such values of the predetermined
number of parameters that the detection errors become statistically
minimum as a whole, based on the detection errors. In this case,
since, for the specific measurement point subset, the parameter
calculation unit weights errors between the calculated positions
and their reference positions in accordance with the information of
the certainties of the calculated positions of the
position-measurement-points, i.e. the probability densities at the
calculated positions of the position-measurement-points, and
calculates statistically valid estimations of the predetermined
number of parameters which uniquely specify position information of
any area on an object. Therefore, statistically valid estimations
of the predetermined number of parameters that rationally reflect
the certainties of the calculated positions of the
position-measurement-point- s can be obtained.
[0101] The fourth position detector according to this invention can
further comprise a parameter value determining unit that is
electrically connected to the parameter calculation unit and that
calculates values of the predetermined number of parameters, based
on evaluation results of the evaluation computing unit. Therefore,
the number of position-measurement-points used to calculate the
values of the predetermined number of parameters can be reduced
while maintaining the statistical validity, and improvement of the
position detection speed can be achieved maintaining the
accuracy.
[0102] According to the ninth aspect of this invention, there is
provided an exposure method for transferring a predetermined
pattern onto divided areas on a substrate, comprising an
arrangement information calculation step of calculating a
predetermined number of parameters that pertain to positions of the
divided areas by a position detection method according to this
invention and calculating arrangement information of the divided
areas on the substrate; and a transfer step of transferring the
pattern onto the divided areas while aligning the substrate based
on the arrangement information of the divided areas calculated in
the arrangement information calculation step.
[0103] According to this method, a pattern is transferred onto
divided areas while accurately detecting the arrangement of the
divided areas on a substrate using a detection method of the
present invention and aligning the substrate on the basis of the
detection results. Therefore, a pattern can be accurately
transferred onto the divided areas.
[0104] According to the tenth aspect of this invention, there is
provided an exposure apparatus that transfers a predetermined
pattern onto divided areas on a substrate, comprising a stage unit
that moves the substrate along a movement plane; and a position
detector according to this invention that calculates arrangement
information of the divided areas on the substrate mounted on the
stage unit.
[0105] This apparatus transfers a pattern onto divided areas while
moving and aligning the substrate through the stage unit on the
basis of arrangement of the divided areas detected by the position
detection unit of this invention. Therefore, a pattern can be
accurately transferred onto the divided areas.
[0106] In addition, in a lithography process, by performing
exposure using an exposure apparatus according to the present
invention, it is possible to form a multi-layer pattern on a
substrate with high accuracy of superposition, and therefore it is
possible to manufacture a more highly integrated micro device with
high yield, and the productivity can be improved. Accordingly,
another aspect of the present invention is a device manufactured by
using an exposure apparatus of the present invention and a method
of manufacturing a device using an exposure method of the present
invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0107] FIG. 1 is a schematic view showing the arrangement of an
exposure apparatus according to an embodiment;
[0108] FIGS. 2A and 2B are views for explaining exemplary alignment
marks;
[0109] FIG. 3 is a schematic block diagram showing the arrangement
of the main control system in the apparatus shown in FIG. 1;
[0110] FIG. 4 is a view for explaining a design value of a
transferred mark;
[0111] FIG. 5 is a flow chart (part 1) for explaining a position
detection operation;
[0112] FIG. 6 is a flow chart (part 2) for explaining the position
detection operation;
[0113] FIG. 7 is a view for explaining a measurement result of a
mark;
[0114] FIGS. 8A and 8B are views showing an example of a mark
measurement result and probability density function;
[0115] FIGS. 9A and 9B are views showing another example of a mark
measurement result and probability density function;
[0116] FIG. 10 is a flow chart (part 3) for explaining the position
detection operation;
[0117] FIG. 11 is a flow chart for explaining a method of
manufacturing devices using the exposure apparatus shown in FIG. 1;
and
[0118] FIG. 12 is a flow chart showing a process of the wafer
process step of FIG. 11;
[0119] FIG. 13 is a view for explaining a modification (No.1);
[0120] FIG. 14 is a view for explaining another modification
(No.2); and
[0121] FIGS. 15A to 15D are views showing exemplary patterns of
two-dimensional marks used in the modification (No.2).
DESCRIPTION OF THE PREFERRED EMBODIMETNS
[0122] An embodiment of the present invention will be described
hereinafter with reference to FIGS. 1 to 12.
[0123] FIG. 1 shows a schematic arrangement of an exposure
apparatus 100 according to an embodiment of the present invention.
This exposure apparatus 100 is a step-and-scan projection exposure
apparatus. The exposure apparatus 100 comprises an illumination
system 10, a reticle stage RST for holding a reticle R as a mask, a
projection optical system PL, a wafer stage WST on which a wafer W
as a substrate (object) is placed, an alignment microscope AS as an
image pickup unit, a main control system 20 for systematically
controlling the overall apparatus, and the like.
[0124] The illumination system 10 includes a light source, an
illuminance uniforming optical system comprising a fly-eye lens as
an optical integrator, a relay lens, a variable ND filter, a
reticle blind, a dichroic mirror, and the like (none of them are
shown). The arrangement of such illumination system is disclosed
in, e.g., Japanese Patent Laid-Open No. 10-112433. Note that the
light source unit uses a KrF excimer laser light source
(oscillation wavelength=248 nm), an ArF excimer laser light source
(oscillation wavelength=193 nm), a harmonic generator such as an
F.sub.2 laser light source (oscillation wavelength=157 nm),
Kr.sub.2 (krypton dimer) laser light source (oscillation
wavelength=146 nm), Ar.sub.2 (argon dimer) laser light source
(oscillation wavelength=126 nm), copper vapor laser light source,
or YAG laser, an ultra-high-pressure mercury lamp (g-line, i-line,
or the like), or the like. Note that a charged particle beam such
as X-rays, electron rays, or the like may be used in place of light
sent out from the aforementioned light source unit.
[0125] The operation of the illumination system 10 with this
arrangement will be briefly explained below. Illumination light
emitted by the light source unit enters the illuminance uniforming
optical system when the shutter is open. In this way, a large
number of secondary light sources are formed at the exit end of the
illuminance uniforming optical system, and illumination light
components sent out from a large number of secondary light sources
reach the reticle blind. Illumination light transmitted through the
reticle blind is output via an imaging lens system. This
illumination system 10 illuminates slit-like illumination area
portions, which are defined by the reticle blind, on the reticle R
having circuit patterns and the like formed thereon with
illumination light IL with nearly uniform illuminance.
[0126] The reticle R is fixed on the reticle stage RST by, e.g.,
vacuum chucking. The reticle stage RST can be finely driven in the
X-Y plane perpendicular to the optical axis (which coincides with
an optical axis AX of the projection optical system PL, described
later) of the illumination system and can be driven at a designated
scan velocity in a predetermined scan direction (Y-direction in
this case) by a reticle stage driver (not shown) comprising a
magnetic float type two-dimensional linear actuator so as to align
the reticle R. Furthermore, in this embodiment, since the magnetic
float type two-dimensional linear actuator includes a Z-drive coil
in addition to an X-drive coil, Y-drive coil, and the like, the
reticle stage can also be finely driven in the Z-direction.
[0127] The position of the reticle stage RST within a stage moving
surface is always detected by a reticle laser interferometer (to be
referred to as a "reticle interferometer" hereinafter) 16 via a
movable mirror 15 at a resolution of around 0.5 to 1 nm. The
position information of the reticle stage RST from the reticle
interferometer 16 is sent to a stage control system 19, which
drives the reticle stage RST via the reticle stage driver (not
shown) on the basis of the position information of the reticle
stage RST.
[0128] The projection optical system PL is disposed below the
reticle stage RST in FIG. 1, and the direction of its optical axis
AX is defined as the Z-axis direction. As the projection optical
system PL, for example, a refraction optical system which is
both-side telecentric, and has a predetermined reduction ratio
(e.g., 1/5, 1/4, or {fraction (1/6)}) is used. For this reason,
when the illumination area of the reticle R is illuminated with
illumination light IL coming from the illumination optical system,
a reduced-scale image (partial inverted image) of the circuit
pattern on the reticle R within that illumination area is formed on
the wafer W, the surface of which is applied with a resist
(photosensitive agent), via the projection optical system PL.
[0129] The wafer stage WST is disposed on a base BS below the
projection optical system PL in FIG. 1, and a wafer holder 25 is
mounted on the wafer stage WST. The wafer W is fixed by, e.g.,
vacuum chucking or the like on the wafer holder 25. The wafer
holder 25 can tilt in any direction with respect to a plane
perpendicular to the optical axis of the projection optical system
PL by a driver (not shown), and can also be finely movable in the
direction of the optical axis (Z-direction) of the projection
optical system PL. Also, the wafer holder 25 is finely rotatable
about the optical axis AX.
[0130] The wafer stage WST is movable not only in the scan
direction (Y-direction) but also in a direction (X-direction)
perpendicular to the scan direction so as to locate a plurality of
shot areas on the wafer W on an exposure area conjugated with the
illumination region, and performs a step-and-scan operation in
which an operation of scanning and exposing each shot area on the
wafer W and an operation of moving the wafer to the exposure start
position of the next shot are repeated. The wafer stage WST is
two-dimensionally driven by a wafer stage driver 24 including a
motor and the like.
[0131] The position of the wafer stage WST in the X-Y plane is
always detected by a wafer laser interferometer 18 via a movable
mirror 17 at a resolution of around 0.5 to 1 nm. Position
information (or velocity information) WPV of the wafer stage WST is
sent to the stage control system 19, which controls the wafer stage
WST on the basis of this position information (or velocity
information) WPV.
[0132] The alignment microscope AS is an offaxis alignment detector
disposed on the side surface of the projection optical system PL.
This alignment microscope AS outputs image pickup results of
alignment marks (wafer marks) contained in each shot area on the
wafer W. As the alignment marks, an X-position detection mark MX(i,
j) and Y-position detection mark MY(i, j), which are formed on
street lines around a shot area SA(i, j) on the wafer W, as shown
in, e.g., FIG. 2A, are used. As the marks MX(i, j) and MY(i, j), a
line-and-space mark having a periodic structure in the detecting
direction can be used, as represented by, e.g., the mark MX(i, j)
shown on a larger scale in FIG. 2B. Note that FIG. 2B illustrates a
line-and-space mark having three lines. However, the number of
lines in a line-and-space mark used as the mark MX(i, j) (or mark
MY(i, j)) can be three or more. The alignment microscope AS outputs
picked-up image data IMD as its image pickup result to the main
control system 20 (see FIG. 1).
[0133] As shown in FIG. 3, the main control system 20 comprises a
main control unit 30 and storage unit 40. The main control unit 30
comprises a control unit 39 which controls the operation of the
exposure apparatus 100 by, e.g., supplying stage control data SCD
to the stage control system 19 and serves as a set selection unit
for selecting a sample set and replacement candidate set, and a
position arithmetic unit 37. The position arithmetic unit 37
comprises a picked-up image data acquisition unit 31, a mark
information calculation unit 32 for calculating position
information of picked-up marks MX and MY on the basis of the
picked-up image data acquired by the picked-up image data
acquisition unit 31, a parameter calculation unit 33 for
calculating estimations of position parameters which uniquely
determine the arrangement of shot areas SA, a valid value
calculation unit 34 for calculating statistically valid position
parameter values, and an evaluation unit 35 for evaluating
possibility of replacing a sample set by another one containing a
fewer number of elements. The storage unit 40 has picked-up image
data stored area 41, sample set information storage area 42, mark
information storage area 43, estimation storage area 45, and valid
value storage area 44 therein.
[0134] Note that the aforementioned alignment microscope AS,
control unit 39, and position arithmetic unit 37 constitute a
position detector. Also, the mark information arithmetic unit 32,
parameter calculation unit 33, and valid value calculation unit 34
constitute an estimation calculation unit, and the estimation
calculation unit and evaluation unit 35 constitute an evaluation
arithmetic unit. Furthermore, the alignment microscope AS and
picked-up image data acquisition unit 31 constitute a measurement
unit. Moreover, the parameter calculation unit 33 and valid value
calculation unit 34 constitute a parameter determination unit. In
FIG. 3, the flow of data is indicated by the solid arrows, and the
flow of control is indicated by the dotted arrow. The respective
operations of the units in the main control system 20 will be
described later.
[0135] In this embodiment, the main control system 20 is
constituted by combining various units. Alternatively, the main
control system 20 may be constituted as a computer system, and the
respective functions of the units that constitute the main control
unit 30 may be implemented by programs installed in the
computer.
[0136] Referring back to FIG. 1, in the exposure apparatus 100, an
oblique-incident-type multi-point focus detection system is fixed
to a support portion (not shown) that supports the projection
optical system PL, and comprises an illumination optical system 13
which directs imaging light beams used to form a plurality of slit
images in an oblique direction with respect to the direction of the
optical axis AX toward the best imaging surface of the projection
optical system PL, and a light-receiving optical system 14 for
receiving these imaging light beams reflected by the surface of the
wafer W via slits. The stage control system 19 drives the wafer
holder 25 in the Z-direction and an oblique direction on the basis
of wafer position information from this multi-point focus detection
system (13, 14). The detailed arrangement and the like of this
multi-point focus position detection system are disclosed in, e.g.,
Japanese Patent Laid-Open No. 6-283403, its corresponding U.S. Pat.
No. 5,448,332, and the like. The disclosures in the above Japanese
Patent Laid-Open and U.S. patent are incorporated herein by
reference as long as the national laws in designated states or
elected states, to which this international application is applied,
permit.
[0137] In the exposure apparatus 100 with the above arrangement,
the arrangement coordinate position of each shot area on the wafer
W is detected as follows. As a precondition for detecting the
arrangement coordinate position of each shot region, assume that
marks MX(i, j) and MY(i, j) are already formed on the wafer W in
wafer processing up to the previous layer (e.g., in a process for
the first layer).
[0138] Also, X-positions {DX.sub.1(i, j), DX.sub.2(i, j),
DX.sub.3(i, j), DX.sub.4(i, j), DX.sub.5(i, j), DX.sub.6(i, j)} of
boundaries (to be referred to as "edges" hereinafter) between lines
and spaces in a mark DMX(i, j), ideal in terms of design,
corresponding to the mark MX(i, j) are known, as shown in FIG. 4.
In FIG. 4, edge positions DX.sub.k(i, j) (k=1 to 6) are expressed
by DX.sub.k. That is, assume
DX.sub.k+1(i, j)-DX.sub.k(i, j)=.DELTA.X (1)
[0139] for the edge positions DX.sub.k(i, j) (k=1 to 6). Also, the
X-position DX.sub.X of the mark DMX(i, j) is defined by 1 DX X ( i
, j ) = { k = 1 6 DX k ( i , j ) } / 6 ( 2 )
[0140] and is known. Furthermore, a Y-position DY.sub.X(i, j) of
the mark DMX(i, j) is determined upon design, and is known.
[0141] Likewise, Y-positions {DY.sub.1(i, j), DY.sub.2(i, j),
DY.sub.3(i, j), DY.sub.4(i, j), DY.sub.5(i, j), DY.sub.6(i, j)} of
the edges in a mark DMY(i, j), ideal in terms of design,
corresponding to the mark MY(i, j) are known as in the mark DMX(i,
j) That is, assume
DY.sub.k+1(i, j)-DY.sub.k(i, j)=.DELTA.Y (3)
[0142] for edge positions DY.sub.k(i, j) (k=1 to 6). Also, the
Y-position DY.sub.x of the mark DMY(i, j) is defined by 2 DY Y ( i
, j ) = { k = 1 6 DY k ( i , j ) } / 6 ( 4 )
[0143] and is known. Furthermore, an X-position DX.sub.Y(i, j) of
the mark DMY(i, j) is determined upon design, and is known.
[0144] Detection of arrangement coordinate positions of shot areas
on a plurality of wafers W (e.g., wafers for one lot) on which
similar patterns (including the first number of marks) are formed
by wafer processing up to the previous layer will be described
below based on the flow chart shown in FIG. 5 with reference to
other drawings as needed.
[0145] In step 201 in FIG. 5, the control unit 39 selects P (>1)
sample sets S.sub.p{MX(i.sub.pm, j.sub.pm), MY (i.sub.ps,
j.sub.ps)} (p=1 to P, m=1 to M (third number), s=1 to M, M>4),
and also Q (>1) replacement candidate sets R.sub.q{MX(i.sub.qn,
j.sub.qn), MY(i.sub.qt, i.sub.qt)} (q=1 to Q, n=1 to N (fourth
number), t=1 to N, 4.ltoreq.N.ltoreq.M), and stores element
information of the sample sets S.sub.p and replacement candidate
sets R.sub.q in the sample set information storage area 42 in FIG.
3.
[0146] Note that M marks MX(i.sub.pm, j.sub.pm) and M marks
MY(i.sub.ps, j.sub.ps) as elements of each sample set S.sub.p are
respectively selected not to line up upon design. Also, when
comparing any two of the sample sets S.sub.p, each set includes at
least one element that the other does not include.
[0147] N marks MX(i.sub.qn, j.sub.qn) and N marks MY(i.sub.qt,
j.sub.qt) as elements of each replacement candidate set R.sub.q are
also respectively selected not to line up upon design. Furthermore,
when comparing any two of the replacement candidate sets R.sub.q,
each set includes at least one element that the other does not
include.
[0148] In this embodiment, the numbers of marks MX(i.sub.pm,
j.sub.pm) and marks MY (i.sub.ps, j.sub.ps) as elements of the
sample set S.sub.p are equal to each other (M), but may be
different from each other. In such case, each of the numbers of
marks MX(i.sub.pm, j.sub.pm) and marks MY(i.sub.ps, j.sub.ps) as
elements of the sample set S.sub.p must be three or more, and the
total of them must be larger than 6. Also, in this embodiment, the
numbers of marks MX(i.sub.qn, j.sub.qn) and marks MY(i.sub.qt,
j.sub.qt) as elements of the replacement candidate set R.sub.q are
equal to each other (N), but may be different from each other. In
such case as well, each of the numbers of marks MX(i.sub.qn,
j.sub.qn) and marks MY(i.sub.qt, j.sub.qt) as elements of the
replacement candidate set R.sub.q must be three or more, and the
total of them must be larger than 6.
[0149] The first wafer W is loaded onto the wafer holder 25 by a
wafer loader (not shown), and alignment with coarse accuracy
(prealignment) is done by the main control system 20 moving the
wafer via the stage control system 19 so that marks MX(i, j) and
MY(i, j) are placed within the observation field of view of the
alignment microscope AS. Such prealignment is done by the main
control system 20 (more specifically, control unit 39) via the
stage control system 19 on the basis of observation of the outer
shape of the wafer W, the observation result of marks MX(i, j) and
MY(i, j) in a broader field of view, and position information (or
velocity information) from the wafer interferometer 18.
[0150] Referring back to FIG. 5, the positions of the marks
MX(i.sub.pm, j.sub.pm), MY(i.sub.ps, j.sub.ps), MX(i.sub.qn,
j.sub.qn), and MY(i.sub.qt, j.sub.qt) as elements of the sample
sets S.sub.p or replacement candidate sets R.sub.q are measured in
subroutine 202.
[0151] In subroutine 202, the wafer W is moved to locate the first
mark (X-position detection mark MX(i.sub.11, j.sub.11)) at the
image pickup position of the alignment microscope AS in step 211 in
FIG. 6. Such movement is done under the control of the main control
system 20 via the stage control system 19.
[0152] Subsequently, the alignment microscope AS picks up an image
of the mark MX(i.sub.11, j.sub.11) in step 212. Then, the picked-up
image data acquisition unit 31 stores inputted picked-up image data
IMD in the picked-up image data storage area 41 in accordance with
an instruction from the control unit 39, thus acquiring picked-up
image data IMD.
[0153] In step 213, the mark information calculation unit 32 reads
out picked-up image data associated with the mark MX(i.sub.11,
j.sub.11) from the picked-up image data storage area 41, and
extracts the X-positions of six edges as position information in
the mark MX(i.sub.11, j.sub.11) on the basis of the picked-up image
data and position information (or velocity information) WPV from
the wafer interferometer 18 in accordance with an instruction from
the control unit 39. In this manner, (FX.sub.1(i.sub.11, j.sub.11),
FX.sub.2(i.sub.11, j.sub.11), FX.sub.3(i.sub.11, j.sub.11),
FX.sub.4(i.sub.11, j.sub.11), FX.sub.5(i.sub.11, j.sub.11),
FX.sub.6(i.sub.11, j.sub.11)) shown in FIG. 7 are extracted as the
X-positions of the six edges.
[0154] Such extraction of the X-positions of the edge can be
implemented by analyzing a waveform obtained by scanning the
picked-up image data along an XS(i.sub.11, j.sub.11) axis which
passes all the three line portions of the mark MX(i.sub.11,
j.sub.11) and is parallel to the X-axis, as shown in FIG. 7, or by
analyzing a waveform obtained by integrating the picked-up image
data in the Y-direction. Note that the latter method requires a
larger arithmetic volume but can accurately extract the X-positions
of the edges.
[0155] Referring back to FIG. 6, in step 214 the mark information
calculation unit 32 calculates a position FX(i.sub.11, j.sub.11) of
the mark MX(i.sub.11, j.sub.11) and a probability density
PFX(i.sub.11, j.sub.11) of that position as follows on the basis of
the edge positions FX.sub.k(i.sub.11, j.sub.11) (k=1 to 6) of the
mark MX(i.sub.11, j.sub.11) extracted in step 213, and edge
positions DX.sub.k(i.sub.11, j.sub.11) of a ideal mark
DMX(i.sub.11, j.sub.11) corresponding to the mark MX(i.sub.11,
j.sub.11).
[0156] The mark information calculation unit 32 calculates an
X-position FX(i.sub.11, j.sub.11) of the mark MX(i.sub.11,
j.sub.11) by 3 FX ( i pm , j pm ) = { k = 1 6 FX k ( i pm , j pm )
} / 6 ( 5 )
[0157] More specifically, the average value of the edge positions
FX.sub.k(i.sub.11, j.sub.11) (k=1 to 6) of the mark MX(i.sub.11,
j.sub.11) is calculated as the X-position FX(i.sub.11, j.sub.11) of
the mark MX(i.sub.11, j.sub.11).
[0158] The mark information calculation unit 32 then calculates
errors dFX.sub.k(i.sub.11, j.sub.11) of the measured edge positions
FX.sub.k(i.sub.11, j.sub.11) corresponding to ideal edge positions
DX.sub.k(i.sub.11, j.sub.11) by
dFX.sub.k(i.sub.11, j.sub.11)=FX.sub.k(i.sub.11,
j.sub.11)-DX.sub.k(i.sub.- 11, j.sub.11) (6)
[0159] and then calculates an average value dFX(i.sub.11, j.sub.11)
and standard deviation .sigma.X(i.sub.11, j.sub.11) of the errors
dFX.sub.k(i.sub.11, j.sub.11) by 4 dFX ( i pm , j pm ) = { k = 1 6
dFX k ( i pm , j pm ) } / 6 ( 7 ) 5 X ( i pm , j pm ) = [ { k = 1 6
( dFX k ( i pm , j pm ) - dFX ( i pm , j pm ) ) 2 } / 5 ] 1 / 2 ( 8
)
[0160] Note that the X-position FX(i.sub.11, j.sub.11) of the mark
MX(i.sub.11, j.sub.11) given by equation (5), the X-position
DX.sub.X(i.sub.11, J.sub.11) of the aforementioned mark
DMX(i.sub.11, j.sub.11), and the average value dFX(i.sub.11,
j.sub.11) of the errors dFX.sub.k(i.sub.11, j.sub.11) satisfy:
FX(i.sub.11, j.sub.11)=DX.sub.x(i.sub.11, j.sub.11)+dFX(i.sub.11,
j.sub.11) (6A)
[0161] Hence, upon calculating the X-position FX(i.sub.11,
j.sub.11) of the mark MX(i.sub.11, j.sub.11), the value
dFX(i.sub.11, j.sub.11) may be calculated by equation (7), and the
X-position may be calculated by equation (6A) using this value in
place of a calculation given by equation (5).
[0162] Upon calculating the value dFX(i.sub.11, j.sub.11), in place
of a calculation given by equation (7), after the value
FX(i.sub.11, j.sub.11) has been calculated by equation (5), the
value dFX(i.sub.11, j.sub.11) can be calculated by
dFX(i.sub.11, j.sub.11)=FX(i.sub.11, j.sub.11)-DX.sub.x(i.sub.11,
j.sub.11) (6B)
[0163] Since the generation factors of the errors
dFX.sub.k(i.sub.11, j.sub.11) are considered to be random, the mark
information calculation device 32 assumes that their distribution
is a normal distribution, and that a probability density function
f.sub.X11(dx) of the errors dFX.sub.k(i.sub.11, j.sub.11) is given
by 6 f X11 ( dx ) = 1 2 X ( i 11 , j 11 ) exp [ { dx - dFX ( i 11 ,
j 11 ) } 2 2 ( X ( i 11 , j 11 ) ) 2 ] ( 9 )
[0164] Based on such estimation, the mark information calculation
unit 32 calculates a probability density that the value of a
variable dx is dFX(i.sub.11, j.sub.11), i.e., an occurrence
probability pFX(i.sub.11, j.sub.11) that the X-position of the mark
MX(i.sub.11, j.sub.11) takes a value FX(i.sub.11, j.sub.11) by
pFX(i.sub.11, j.sub.11)=f.sub.X11(dFX(i.sub.11,
j.sub.11))={(2.pi.).sup.1/- 2.multidot..pi.X(i.sub.11,
j.sub.11)}.sup.-1 (10)
[0165] The mark information calculation unit 32 stores the position
FX(i.sub.11, j.sub.11) of the mark MX(i.sub.11, j.sub.11) and its
occurrence probability pFX(i.sub.11, j.sub.11) calculated in this
way in the mark information storage area 43. In this manner,
calculations of the mark information that pertains to the first
mark MX(i.sub.11, j.sub.11) are completed.
[0166] FIGS. 8A and 8B show an example wherein the X-position
FX(i.sub.11, j.sub.11) of the mark MX(i.sub.11, j.sub.11) and its
occurrence probability pFX(i.sub.11, j.sub.11) are calculated on
the basis of the measured edge positions FX.sub.k(i.sub.11,
j.sub.11), and FIGS. 9A and 9B show another example.
[0167] In the example shown in FIGS. 8A and 8B, the errors
dFX.sub.k(i.sub.11, j.sub.11) of the measured edge positions
FX.sub.k(i.sub.11, j.sub.11) corresponding to the ideal edge
positions DX.sub.k(i.sub.11, j.sub.11) are relatively uniform, as
shown in FIG. 8A. Note that FIG. 8A illustrates the edge positions
DX.sub.k(i.sub.11, j.sub.11) as elements on the X-axis in the mark
DMX(i.sub.11, j.sub.11), and has symbols DX.sub.k attached onto
those elements. Also, FIG. 8A illustrates the edge positions
FX.sub.k(i.sub.11, j.sub.11) as elements on the X-axis in the mark
MX(i.sub.11, j.sub.11), and has symbols FX.sub.k attached onto
those elements.
[0168] In the mark MX(i.sub.11, j.sub.11) for which the edge
positions FX.sub.k(i.sub.11, j.sub.11) shown in FIG. 8A are
measured, the probability density function f.sub.X11(dx) of the
error distribution is steep, as shown in FIG. 8B. That is, the
standard deviation .sigma.X(i.sub.11, j.sub.11) is small. As a
result, a probability density pFX(i.sub.11, j.sub.11) that an error
takes a value dFX(i.sub.11, j.sub.11), i.e., the X-position of the
mark MX(i.sub.11, j.sub.11) takes a value FX(i.sub.11, .sub.11)
becomes larger than that in FIG. 9B to be described below.
[0169] Meanwhile, in the example shown in FIGS. 9A and 9B, the
errors dFX.sub.k(i.sub.11, j.sub.11) of the measured edge positions
FX.sub.k(i.sub.11, j.sub.11) corresponding to the ideal edge
positions DX.sub.k(i.sub.11, j.sub.11) greatly vary, as shown in
FIG. 9A. Note that FIG. 9A uses the same expression method as FIG.
8A.
[0170] In the mark MX(i.sub.11, j.sub.11) for which the edge
positions FX.sub.k(i.sub.11, j.sub.11) shown in FIG. 9A are
measured, the probability density function f.sub.X11(dx) of the
error distribution is gradual, as shown in FIG. 9B. That is, the
standard deviation .sigma.X(i.sub.11, j.sub.11) of the error
distribution takes a large value. As a result, the probability
density pFX(i.sub.11, j.sub.11) that an error takes the value
dFX(i.sub.11, j.sub.11), i.e., the X-position of the mark
XM(i.sub.11, j.sub.11) assumes the value FX(i.sub.11, j.sub.11)
becomes smaller than that in FIG. 8B mentioned above.
[0171] Referring back to FIG. 6, it is checked in step 215 if mark
information calculations for all the selected marks are complete.
In the aforementioned process, since the calculations of mark
information of only one mark MX(i.sub.11, j.sub.11), i.e., the mark
position FX(i.sub.11, j.sub.11) of the mark MX(i.sub.11, j.sub.11)
and its probability density pFX(i.sub.11, j.sub.11) are complete,
the answer in step 215 is NO, and the sequence advances to step
216.
[0172] In step 216, the control unit 39 moves the wafer W to a
position where the next mark falls within the image pickup field of
view of the alignment microscope AS. Such movement of the wafer W
is done by moving the wafer stage WST when the control unit 39
controls the wafer drive unit 24 via the stage control system 19 on
the basis of the prealignment result.
[0173] After that, the X-positions FX(i.sub.pm, j.sub.pm) of other
marks MX(i.sub.pm, j.sub.pm) and their probability densities
pFX(i.sub.pm, j.sub.pm) the Y-positions FY(i.sub.ps, j.sub.ps) of
marks MY(i.sub.ps, j.sub.ps) and their probability densities
pFY(i.sub.ps, j.sub.ps), the X-positions FX(i.sub.qn, j.sub.qn) of
marks MX(i.sub.qn, j.sub.qn) and their probability densities
pFX(i.sub.qn, j.sub.qn), and the Y-positions FY(i.sub.qt, j.sub.qt)
Of marks MY(i.sub.qt, j.sub.qt) and their probability densities
pFY(i.sub.qt, j.sub.qt) are computed in the same manner as in the
case of the aforementioned mark MX(i.sub.11, j.sub.11) until it is
determined in step 215 that the mark information (mark positions
and probability densities) of all the selected marks has been
calculated. If the mark information of all the selected marks has
been calculated, and the answer in step 215 is YES, subroutine 202
ends. And the sequence advances to step 203 in FIG. 5.
[0174] In step 203, the parameter calculation unit 33 reads out the
X-positions (i.sub.pm, j.sub.pm) (m=1 to M) of the marks
MX(i.sub.pm, j.sub.pm) and their probability densities
pFX(i.sub.pm, j.sub.pm), and the Y-positions FY(i.sub.ps, j.sub.ps)
(s=1 to M) of the marks MY(i.sub.ps, j.sub.ps) and their
probability densities pFY(i.sub.ps, j.sub.ps) from the mark
information storage area 43 for each sample set S.sub.p in
accordance with an instruction from the control unit 39. The
parameter calculation unit 33 then calculates the estimations of
parameters which uniquely specify the arrangement of shot areas
SA(i, j).
[0175] The marks MX(i, j) and MY(i, j) formed on the wafer W
deviate from their ideal positions due to a mismatch between a
stage coordinate system (X, Y) which specifies the position of the
wafer stage WST, and the arrangement coordinate system of shot
areas as a design coordinate system, i.e., a wafer coordinate
system (.alpha., .beta.), and such a mismatch occurs due to the
following four main factors.
[0176] {circle over (1)} Rotation of wafer: This is expressed by a
residual rotation error .theta. of the wafer coordinate system
(.alpha., .beta.) with respect to the stage coordinate system (X,
Y).
[0177] {circle over (2)} Orthogonality of the stage coordinate
system (X, Y): This occurs when the X-axis and Y-axis feed
directions of the wafer stage WST are not accurately orthogonal to
each other, and is expressed by an orthogonality error w.
[0178] {circle over (3)} Linear expansion/shrinkage (wafer scaling
values) in the .alpha.- and .beta.-directions of the wafer
coordinate system (.alpha., .beta.): This occurs when the wafer W
entirely expands/contracts due to wafer processing or the like.
This expansion/shrinkage amount is expressed by wafer scaling
values R.sub.X and R.sub.Y in the .alpha.- and .beta.-directions.
Note that the wafer scaling value R.sub.X in the .alpha.-direction
is represented by the ratio between the actually measured value and
design value of the distance between two points in the
.alpha.-direction on the wafer W, and the wafer scaling value
R.sub.Y in the .beta.-direction is represented by the ratio between
the actually measured value and design value between two points in
the .beta.-direction.
[0179] {circle over (4)} Offset of the wafer coordinate system
(.alpha., .beta.) with respect to the stage coordinate system (X,
Y): This occurs when the wafer W has entirely deviated by an
infinitesimal amount with respect to the wafer stage WST and is
expressed by offset amounts O.sub.X and O.sub.Y.
[0180] When the aforementioned error factors {circle over (1)} to
{circle over (4)} are added, a pattern to be transferred to a
target position (DX, DY), in terms of design, on the wafer
coordinate system (.alpha., .beta.) is transferred to a position
(EX, EY) on the stage coordinate system (X, Y), of which the
position is given by 7 ( EX EY ) = ( R X , 0 0 , R Y ) ( cos , -
sin sin , cos ) ( 1 , - tan w 0 , 1 ) ( DX DY ) + ( O X O Y ) ( 11
)
[0181] Note that various other error factors are also present in
addition to the aforementioned ones upon actual transfer, and the
position (EX, EY) is considered as an expected transfer
position.
[0182] In general, since the orthogonality error w and residual
rotation error .theta. can be considered as infinitesimal amounts,
the target transfer position (DX, DY) and expected transfer
position (EX, EY) are related by 8 ( EX EY ) = ( R X , - R X ( w +
) R Y , R Y ) ( DX DY ) + ( O X O Y ) ( 12 )
[0183] which expresses a first-order approximation of the
trigonometric function in equation (11).
[0184] In the following description, equation (12) can also be
expressed by 9 ( EX EY ) = ( A 11 , A 12 A 21 , A 22 ) ( DX DY ) +
( O X O Y ) ( 13 )
[0185] where
A.sub.11=R.sub.X (14)
A.sub.12=-R.sub.X.multidot.(w+.theta.) (15)
A.sub.21=R.sub.Y.multidot..theta. (16)
A.sub.22=R.sub.Y (17)
[0186] That is, parameters which uniquely specify the arrangement
of shot areas SA(i, j) are six parameters A.sub.11, A.sub.12,
A.sub.21, A.sub.22, O.sub.X, and O.sub.Y. In step 203, the
parameter calculation unit 33 calculates the estimations of these
six parameters of each sample set S.sub.p as follows.
[0187] The parameter calculation unit 33 calculates an expected
transfer X-position EX(i.sub.pm, j.sub.pm) of each mark
MX(i.sub.pm, j.sub.pm) from the ideal transfer position
(DX.sub.X(i.sub.pm, j.sub.pm), DY.sub.X(i.sub.pm, j.sub.pm)) of the
mark MX(i.sub.pm, j.sub.pm) using equation (13), the EX(i.sub.pm,
j.sub.pm) containing the parameters. Subsequently, the parameter
calculation unit 33 calculates an expected transfer Y-position
EY(i.sub.ps, j.sub.ps) of each mark MY(i.sub.ps, j.sub.ps) from the
ideal transfer position (DX.sub.Y(i.sub.ps, j.sub.ps),
DY.sub.Y(i.sub.ps, j.sub.ps)) of the mark MY(i.sub.ps, j.sub.ps)
using equation (13), the EY(i.sub.ps, j.sub.ps) containing the
parameters.
[0188] Then, the parameter calculation unit 33 calculates an error
.sigma.X.sub.pm of the X-position FX(i.sub.pm, j.sub.pm),
calculated based on the measurement result, relative to the
expected transfer X-position EX(i.sub.pm, j.sub.pm) for each mark
MX(i.sub.pm, j.sub.pm) by
.sigma.X.sub.pm=FX(i.sub.pm, j.sub.pm)-EX(i.sub.pm, j.sub.pm)
(18)
[0189] The parameter calculation unit 33 also calculates an error
.sigma.Y.sub.ps of the Y-position FY(i.sub.ps, j.sub.ps),
calculated based on the measurement result, relative to the
expected transfer Y-position EY(i.sub.ps, j.sub.ps) for each mark
MY(i.sub.ps, j.sub.ps) by
.sigma.Y.sub.ps=FY(i.sub.ps, j.sub.ps)-EY(i.sub.ps, j.sub.ps)
(19)
[0190] Normally, it is said that the more identical the ideal mark
shape and actually transferred mark shape are, the more accurate
mark transfer onto the wafer W has been done (except for transfer
position accuracy). Therefore, the mark position is more reliable
as it is calculated from the measurement results of the edge
positions of the mark having a shape more identical to the ideal
mark's shape. Also, the reliability level of the calculated mark
position depends on the occurrence probability of the mark
position, i.e., the probability density.
[0191] Hence, the parameter calculation unit 33 evaluates the
errors .sigma.X.sub.pm and .sigma.Y.sub.ps calculated by equations
(18) and (19) using the probability densities pFX(i.sub.pm,
j.sub.pm) and pFY(i.sub.ps, j.sub.ps), and calculates evaluated
errors .epsilon.X.sub.pm and .epsilon.Y.sub.ps by
.epsilon.X.sub.pm=pFX(i.sub.pm, j.sub.pm).multidot..delta.X.sub.pm
(20)
.epsilon.Y.sub.ps=pFY(i.sub.ps, j.sub.ps).multidot..delta.Y.sub.ps
(21)
[0192] The parameter calculation unit 33 then calculates the
estimations of the six parameters A.sub.11, A.sub.12, A.sub.21,
A.sub.22, O.sub.X, and O.sub.Y which minimize a variation S.sub.p
given by 10 S p = m = 1 M X pm 2 + s = 1 M Y p s 2 ( 22 )
[0193] by applying the method of least squares on the basis of the
evaluated errors .epsilon.X.sub.pm and .epsilon.Y.sub.ps. More
specifically, the parameter calculation unit 33 calculates
estimations A.sub.11p, A.sub.12p, A.sub.21p, A.sub.22p, O.sub.Xp,
and O.sub.Yp of the six parameters A.sub.11, A.sub.12, A.sub.21,
A.sub.22, O.sub.X, and O.sub.Y by solving simultaneous equations
made up of six equations obtained by setting partial differentials
of the variation S.sub.p given by equation (22) by the six
parameters A.sub.11, A.sub.12, A.sub.21, A.sub.22, O.sub.X, and
O.sub.Y to zero.
[0194] The parameter calculation unit 33 stores P groups of
estimations (A.sub.11p, A.sub.12p, A.sub.21p, A.sub.22p, O.sub.Xp,
O.sub.Yp) calculated in this way in the estimation storage area
45.
[0195] The valid value calculation unit 34 reads out the P sets of
calculated estimations (A.sub.11p, A.sub.12p, A.sub.21p, A.sub.22p,
O.sub.Xp, O.sub.Yp) from the estimation storage area 45, and
calculates an X-position deviation .sigma..sub.pX, Y-position
deviation .sigma..sub.pY, and covariance .sigma..sub.pXY associated
with each sample set S.sub.p upon adopting each group of
estimations as follows.
[0196] The valid value calculation unit 34 calculates the expected
transfer X-position EX.sub.p(i.sub.pm, j.sub.pm) of each mark
MX(i.sub.pm, j.sub.pm) from the ideal transfer position
(DX.sub.X(i.sub.pm, j.sub.pm), DY.sub.Y(i.sub.pm, j.sub.pm)) of the
mark MX(i.sub.pm, j.sub.pm) using the estimations (A.sub.11p,
A.sub.12p, A.sub.21p, A.sub.22p, O.sub.Xp, O.sub.Yp) as the values
of the parameters (A.sub.11, A.sub.12, A.sub.21, A.sub.22, O.sub.X,
O.sub.Y) in equation (13). Subsequently, the valid value
calculation unit 34 calculates the expected transfer Y-position
EY.sub.p(i.sub.ps, j.sub.ps) of each mark MY(i.sub.ps, j.sub.ps)
from the ideal transfer position (DX.sub.Y(i.sub.ps, j.sub.ps),
DY.sub.Y(i.sub.ps, j.sub.ps)) of the mark MY(i.sub.ps,
j.sub.ps).
[0197] The valid value calculation unit 34 then calculates an error
.sigma..sub.pX.sub.pm of the X-position FX(i.sub.pm, j.sub.pm),
calculated based on the measurement result, relative to the
expected transfer X-position EX.sub.p(i.sub.pm, j.sub.pm) for each
mark MX(i.sub.pm, j.sub.pm) by
.delta..sub.pX.sub.pm=FX(i.sub.pm, j.sub.pm)-EX.sub.p(i.sub.pm,
j.sub.pm) (23)
[0198] Subsequently, the valid value calculation unit 34 calculates
an error .delta..sub.pY.sub.ps of the Y-position FY(i.sub.ps,
j.sub.ps) calculated based on the measurement result from the
expected transfer Y-position EY.sub.p(i.sub.ps, j.sub.ps) for each
mark MY(i.sub.ps, j.sub.ps) by
.delta..sub.pY.sub.ps=FY(i.sub.ps, j.sub.ps)-EX.sub.p(i.sub.pm,
j.sub.pm) (24)
[0199] The valid value calculation unit 34 then evaluates the
errors .delta..sub.pX.sub.pm and .delta..sub.pY.sub.ps calculated
by equations (23) and (24) using the probability densities
pFX(i.sub.pm, j.sub.pm) and pFY(i.sub.ps, j.sub.ps), and calculates
evaluated errors .epsilon..sub.pX.sub.pm and
.epsilon..sub.pY.sub.ps by
.epsilon..sub.pX.sub.pm=pFX(i.sub.pm,
j.sub.pm).multidot..sigma..sub.pX.su- b.pm (25)
.epsilon..sub.pY.sub.ps=pFY(i.sub.ps,
j.sub.ps).multidot..delta..sub.pY.su- b.ps (26)
[0200] The valid value calculation unit 34 calculates an X-position
deviation .sigma..sub.pX, Y-position deviation .sigma..sub.pY, and
covariance .sigma..sub.pXY respectively by 11 p X = ( [ { m = 1 M (
X pm ) 2 } / ( M - 1 ) ] ) 1 / 2 ( 27 ) p Y = ( [ { s = 1 M ( X p s
) 2 } / ( M - 1 ) ] ) 1 / 2 ( 28 ) p XY = [ { m = 1 M ( X pm Y pm )
} / ( M - 1 ) ] 1 / 2 ( 29 )
[0201] The valid value calculation unit 34 calculates a deviation
.sigma..sub.p upon adopting the estimations (A.sub.11p, A.sub.12p,
A.sub.21p, A.sub.22p, O.sub.Xp, O.sub.Yp) by
.sigma..sub.p={(.sigma..sub.pX).sup.2.multidot.(.sigma..sub.pY).sup.2-(.si-
gma..sub.pXY).sup.2}.sup.1/2 (30)
[0202] The calculated deviations .sigma..sub.p indicate the
certainties of the respective estimations (A.sub.11p, A.sub.12p,
A.sub.21p, A.sub.22p, O.sub.Xp, O.sub.Yp) i.e., the degrees by
which the respective sample sets S.sub.p represent the entire marks
MX and MY.
[0203] The parameter calculation unit 33 checks in step 204 if
there are a plurality of sample sets S.sub.p. In the above, since
the number of sample sets S.sub.p is P (>1), the answer in step
204 is YES, and the sequence advances to step 205.
[0204] In step 205, on the basis of the P groups of estimations
(A.sub.11p, A.sub.12p, A.sub.21p, A.sub.22p, O.sub.Xp, O.sub.Yp)
and deviations .sigma..sub.p that reflect the certainty, the valid
value calculation unit 34 calculates weighted mean of each
parameter of the estimations (A.sub.11p, A.sub.12p, A.sub.21p,
A.sub.22p, O.sub.Xp, O.sub.Yp) using the values (1/.sigma..sub.p)
as respective weight coefficients of the estimations, and obtains a
set of statistically valid parameter values (A.sub.11O, A.sub.12O,
A.sub.21O, A.sub.22O, O.sub.XO, O.sub.YO) That is, the valid value
calculation unit 34 computes 12 A 110 = { p = 1 P ( A 11 p / p ) }
/ { p = 1 P ( 1 / p ) } ( 31 ) A 120 = { p = 1 P ( A 12 p / p ) } /
{ p = 1 P ( 1 / p ) } ( 32 ) A 210 = { p = 1 P ( A 21 p / p ) } / {
p = 1 P ( 1 / p ) } ( 33 ) A 220 = { p = 1 P ( A 22 p / p ) } / { p
= 1 P ( 1 / p ) } ( 34 ) O X0 = { p = 1 P ( O Xp / p ) } / { p = 1
P ( 1 / p ) } ( 35 ) O Y0 = { p = 1 P ( O Yp / p ) } / { p = 1 P (
1 / p ) } ( 36 )
[0205] Also, the valid value calculation unit 34 calculates
deviation .sigma..sub.O upon adopting the statistically valid
parameter values (A.sub.11O, A.sub.12O, A.sub.21O, A.sub.22O,
O.sub.XO, O.sub.YO) by 13 0 = P / p = 1 P ( 1 / p ) ( 37 )
[0206] The valid value calculation unit 34 stores the statistically
valid parameter values (A.sub.11O, A.sub.12O, A.sub.21O, A.sub.22O,
O.sub.XO, O.sub.YO) and deviation .sigma..sub.O calculated in this
way in the valid value storage area 44 as parameter values
(AU.sub.11, AU.sub.12, AU.sub.21, AU.sub.22, OU.sub.X, OU.sub.Y)
and deviation .sigma.U used upon exposure of the wafer W.
[0207] In subroutine 207, the evaluation unit 35 evaluates in
accordance with an instruction from the control unit 39 possibility
of replacing the sample set S.sub.p by any of replacement candidate
sets R.sub.q.
[0208] In subroutine 207, the evaluation unit 35 reads out
X-positions FX(i.sub.pn, j.sub.qn) (n=1 to N) of marks MX(i.sub.qn,
j.sub.qn) and their probability densities pFX(i.sub.qn, j.sub.qn),
and Y-positions FY(i.sub.qt, j.sub.qt) (t=1 to N) of marks
MY(i.sub.qt, j.sub.qt) and their probability densities
pFY(i.sub.qt, j.sub.qt) from the mark information storage area 43
in step 221 of FIG. 10 in the same manner as in the aforementioned
case of the sample sets S.sub.p. The evaluation unit 35 calculates
a estimation (A.sub.11q, A.sub.12q, A.sub.21q, A.sub.22q, O.sub.Xq,
and O.sub.Yq) for parameters A.sub.11, A.sub.12, A.sub.21,
A.sub.22, O.sub.X, and O.sub.Y which uniquely specify the
arrangement of shot areas SA(i, j) in the same manner as in the
case of the sample sets S.sub.p.
[0209] The evaluation device 35 then calculates X-position
deviations .sigma..sub.qX, Y-position deviations .sigma..sub.qY,
and covariances .sigma..sub.qXY of respective replacement candidate
sets R.sub.q upon adopting the Q sets of calculated estimations
(A.sub.11q, A.sub.12q, A.sub.21q, A.sub.22q, O.sub.Xq, O.sub.Yq),
and also deviations .sigma..sub.q upon adopting the Q sets of
calculated estimations (A.sub.11q, A.sub.12q, A.sub.21q, A.sub.22q,
O.sub.Xq, O.sub.Yq) in the same manner as in the case of the sample
sets S.sub.p. The calculated deviations .sigma..sub.q indicate the
certainties of the respective estimations (A.sub.11q, A.sub.12q,
A.sub.21q, A.sub.22q, O.sub.Xq, O.sub.Yq), i.e., the degrees by
which the respective replacement candidate sets R.sub.q represent
the entire marks MX and MY.
[0210] In step 222, the evaluation unit 35 reads out the
statistically valid parameter values (AU.sub.11, AU.sub.12,
AU.sub.21, AU.sub.22, OU.sub.X, OU.sub.Y) and deviation .sigma.U
from the valid value storage area 44, and evaluates similarities
with the estimations (A.sub.11q, A.sub.12q, A.sub.21q, A.sub.22q,
O.sub.Xq, O.sub.Yq) and deviations .sigma..sub.q associated with
the replacement candidate sets R.sub.q. Such evaluation is done by
comprehensively considering a similarity F of parameter values
given by 14 F q = A 11 q - AU 11 + A 12 q - AU 12 + A 21 q - AU 21
+ A 22 q - AU 22 + O Xq - OU X + O Yq - OU Y ( 38 )
[0211] and a similarity G of deviation values given by
G.sub.q=.vertline..sigma..sub.q-.sigma.U.vertline. (39)
[0212] For example, if
C1<F.sub.q.times.G.sub.q (40)
[0213] for a predetermined value C1, it may be determined that the
two sets are similar. In such a case, the similarity F of the
parameter values and the similarity G of deviation values are
equally handled.
[0214] On the other hand, if
C2<F.sub.q+G.sub.q (41)
[0215] for a predetermined value C2, it may be determined that the
two sets are similar. In such a case as well, the similarity F of
the parameter values and the similarity G of deviation values are
equally handled.
[0216] Also, if
C3<F.sub.q+3G.sub.q (42)
[0217] for a predetermined value C3, it is evaluated that the two
sets are similar. In such case, the similarity F of the parameter
values and the triple of the similarity G of deviation values are
equally handled.
[0218] The evaluation unit 35 checks in step 223 if there is a
replacement candidate set R.sub.q that has been found to be similar
as a result of evaluation in step 222. If NO in step 223, the
evaluation unit 35 sends a report indicating this to the control
unit 39. The control unit 39 receives the report and selects new Q
replacement candidate sets, and stores them in the sample set
information storage area 42 in step 225. In this fashion, the
process of subroutine 207 executed if NO in step 223 ends.
[0219] On the other hand, if YES in step 223, the evaluation unit
35 replaces the replacement candidate set information in the sample
set information storage area 42 by information that pertains to
only the replacement candidate set or sets found to be similar.
[0220] The evaluation unit 35 checks in step 226 if there is a
replacement candidate set that is successively found to be similar
a predetermined number of times. For example, when the
predetermined number of times is three, since similarity evaluation
has been done only once for only one wafer, the answer in step 226
is NO, and then the process of subroutine 207 ends.
[0221] If the predetermined number of times is one, the answer in
step 226 is YES, and the evaluation unit 35 selects a replacement
candidate set having the highest similarity from those found to be
similar as a new sample set, and replaces the sample set
information in the sample set information storage area 42 by
information that pertains to the new sample set in step S227, thus
ending the process of subroutine 207. In such a case, the number of
sample sets is one from here on.
[0222] Parallel to the aforementioned process of subroutine 207,
the control unit 39 calculates the arrangement coordinate positions
of shot areas SA(i, j) using the parameter values (AU.sub.11,
AU.sub.12, AU.sub.21, AU.sub.22, OU.sub.X, OU.sub.Y) calculated in
step 205 in FIG. 5. Then, the reticle R and wafer W are aligned on
the basis of the calculated shot area arrangement under the control
of the control unit 39, and the wafer W and reticle R are
synchronously moved at a velocity ratio corresponding to the
projection magnification in opposite directions along the scan
direction (Y-direction) while a slit-like illumination area (the
center of which nearly coincides with the optical axis AX) is
illuminated with illumination light IL, thus transferring a pattern
on a pattern area of the reticle R onto each shot area SA(i, j) in
the reduced scale. Upon completion of pattern transfer onto all the
shot areas SA(i, j), the wafer is unloaded under the control of the
control unit 39.
[0223] If exposure of the first wafer W is complete, and the
process of the subroutine has ended in this way, the control unit
39 checks in step 208 in FIG. 5 if exposure of all wafers (e.g.,
wafers for one lot) is complete. In the above process, since
exposure of only the first wafer is complete, the answer in step
208 is NO. The next wafer is loaded onto the wafer holder 25 by the
wafer loader (not shown) in the same manner as the first wafer, and
alignment with coarse accuracy (prealignment) is done by the main
control system 20 moving the wafer W via the stage control system
19 so that marks MX(i, j) and MY(i, j) can be placed within the
observation field of view of the alignment microscope AS. After
that, steps 202 to 207 in FIG. 5 are repeated for each wafer W in
the same manner as the first wafer until YES is determined n step
208.
[0224] If it is determined in step 208 in FIG. 5 that exposure of
all wafers is complete, the exposure process ends.
[0225] If replacement candidate sets which are successively
determined first to be similar to the sample set a predetermined
number of times are found in step 226 in FIG. 10, since the most
similar one of those sets is used as the subsequent sample set, the
number of sample sets becomes one. As a result, NO is determined in
step 204 in FIG. 5 executed thereafter, and step 206 is executed in
place of step 205 mentioned above. That is, the estimations of
position parameters calculated based on the position measurement
results of marks MX and MY contained in the single sample set are
directly adopted as parameter values (AU.sub.11, AU.sub.12,
AU.sub.21, AU.sub.22, OU.sub.X, OU.sub.Y) used upon exposure of the
wafer W.
[0226] According to the exposure apparatus of this embodiment with
the aforementioned arrangement and operations, a plurality of
sample sets are selected initially, and position parameters are
calculated based on the result of estimating the position
distribution of the entire marks MX and MY according to respective
groups of the estimations and their certainties of position
parameters calculated for the sample sets. Accordingly,
statistically valid position parameter values can be obtained.
Therefore, the wafer W can be aligned very accurately, and the
pattern transfer accuracy can be improved.
[0227] Furthermore, if a set appropriate to replace the sample sets
is found when searching for a replacement candidate set that
reflects the position distribution of the entire marks MX and MY as
much as the sample sets used to calculate the statistically valid
position parameter values and contains fewer elements than any
sample set, the appropriate set is used as a new sample set.
Therefore, since the time required for aligning a wafer W can be
shortened while maintaining statistical validity of the obtained
position parameters, the throughput can be improved.
[0228] In addition, a mark position is calculated on the basis of
the position information of each position measurement mark
(alignment mark) obtained through measuring the alignment marks
associated with each sample set, i.e., the measurement results of a
plurality of edge positions of each alignment mark; the certainty
of that mark position is calculated from the design values of the
edge positions and errors, and parameter values (estimations) that
uniquely specify the arrangement of shot areas (i, j) on the wafer
W are calculated for each sample set using the certainty as the
weight. Therefore, because the arrangement of the shot areas (i, j)
on the wafer W is calculated using the finally obtained accurate
parameter values to align the wafer W, accurate alignment can be
performed, and pattern transfer with high overlap accuracy can be
achieved.
[0229] The manufacture of a device using the exposure apparatus and
method of this embodiment will be described below.
[0230] FIG. 11 is a flow chart of production of devices
(semiconductor chips such as IC or LSI, liquid crystal panels,
CCD's, thin-film magnetic heads, micro machines, or the like) in
this embodiment. As shown in FIG. 11, in step 301 (design step)
function/performance design for the devices (e.g., circuit design
for semiconductor devices) is performed, and also pattern design is
performed. In step 302 (mask fabrication step), a mask formed with
the designed circuit pattern is fabricated. On the other hand, in
step 303 (wafer preparation step) a wafer is prepared using
material such as silicon and the like.
[0231] In step 304 (wafer process step), an actual circuit and the
like are formed on the wafer by lithography, as will be described
later, using the mask and wafer prepared in steps 301 to 303. In
step 305 (device assembly step), devices are assembled using the
wafer processed in step 304. This step 305 includes processes such
as an assembly process (dicing, bonding), packaging step (chip
encapsulation), and the like.
[0232] Finally, in step 306 (inspection step) inspections such as
an operation confirmation test, durability test, and the like of
the devices manufactured in step 305 are performed. After these
steps, the process is complete, and the devices are shipped
out.
[0233] FIG. 12 shows a detailed, exemplary flow of step 304 for
manufacturing semiconductor devices. As shown in FIG. 12, the wafer
surface is oxidized in step 311 (oxidation step). In step 312 (CVD
step), an insulation film is formed on the wafer surface. In step
313 (electrode formation step), electrodes are formed on the wafer
by deposition. In step 314 (ion implantation step), ions are
implanted into the wafer. Steps 311 to 314 mentioned above
constitute a pre-process of each step in the wafer process, and is
selectively executed in accordance with the process required in
each step.
[0234] Upon completion of the pre-process in each step of the wafer
process, a post-process steps is performed as follows. In this
post-process, the wafer is coated with a photosensitive agent in
step 315 (resist formation step), and the above exposure apparatus
transfers the circuit pattern on the mask onto the wafer aligned
using the aforementioned scheme, in step 316 (exposure step). The
exposed wafer is developed in step 317 (development step), and an
exposing member of portions other than portions where the resist
remains is removed by etching in step 318 (etching step). Then, the
resist that has become unnecessary after etching is removed in step
319 (resist removal step).
[0235] By repeating the pre- and post-process steps, multi-layer
circuit patterns are formed on the wafer.
[0236] In this way, devices having a micro-pattern accurately
formed thereon are manufactured with high mass-productivity.
[0237] In the above embodiment, upon detecting the edge positions
of the marks MX(i.sub.pm, j.sub.pm) and MY(i.sub.ps, j.sub.ps), one
edge position is extracted for each edge using the picked-up image
data on a single axis parallel to the direction of the mark pattern
change or the integrated data in the direction perpendicular to the
direction of the mark pattern change, as described above.
Alternatively, as shown in FIG. 13 that representatively shows a
mark MX(i.sub.pm, j.sub.pm), edge positions can be extracted along
each of a plurality of (=H) axes XS.sub.h(i.sub.pm, j.sub.pm) (h=1
to H) parallel to the direction in which the pattern of the mark
MX(i.sub.pm, j.sub.pm) changes. In such a case, edge positions
FX.sub.kh(i.sub.pm, j.sub.pm) (k=1 to 6, h=1 to H) are
extracted.
[0238] Then, an X-position FX(i.sub.pm, j.sub.pm) of the mask
MX(i.sub.pm, j.sub.pm) is calculated by 15 FX ( i pm , j pm ) = { k
= 1 6 h = 1 H FX kh ( i pm , j pm ) } / ( 6 H ) ( 43 )
[0239] Also, an error dFX.sub.kh(i.sub.pm, j.sub.pm) of each edge
position FX.sub.kh(i.sub.pm, j.sub.pm) from the ideal edge position
DX.sub.k(i.sub.pm, j.sub.pm) is calculated by
dFX.sub.kh(i.sub.pm, j.sub.pm)=FX.sub.kh(i.sub.pm,
j.sub.pm)-DX.sub.k(i.su- b.pm, j.sub.pm) (44)
[0240] After that, an average value dFX(i.sub.pm, j.sub.pm) and
standard deviation .sigma.X(i.sub.pm, j.sub.pm) of the errors
dFX.sub.kh(i.sub.pm, j.sub.pm) are respectively calculated by 16
dFX ( i pm , j pm ) = { k = 1 6 h = 1 H dFX kh ( i pm , j pm ) } /
( 6 H ) ( 45 ) X ( i pm , j pm ) = [ { k = 1 6 h = 1 H ( dFX kh ( i
pm , j pm ) - dFX ( i pm , j pm ) ) 2 } / ( 6 H - 1 ) ] 1 / 2 ( 46
)
[0241] The X-position FX(i.sub.pm, j.sub.pm) and standard deviation
.sigma.X(i.sub.pm, j.sub.pm) of the mark MX(i.sub.pm, j.sub.pm)
calculated in this way are statistically more valid than those in
the aforementioned embodiment, since the number of extracted edge
positions is larger than that in the above embodiment.
[0242] Note that the Y-position FY(i.sub.ps, j.sub.ps) of each mark
MY(i.sub.ps, j.sub.ps) can be calculated in the same manner as the
mark MX(i.sub.pm, j.sub.pm). In such case as well, the calculated Y
position FY(i.sub.ps, j.sub.ps) and standard deviation
.sigma.Y(i.sub.ps, j.sub.ps) of the mark MY(i.sub.ps, j.sub.ps) are
statistically more valid than those in the above embodiment, since
the number of extracted edge positions is larger than that in the
above embodiment.
[0243] After that, optimal values of six parameters that uniquely
specify the arrangement of shot areas SA(i, j) on the wafer W are
calculated in the same manner as in the above embodiment. Since the
optimal values of the six parameters are calculated based on
statistically more valid mark positions and their certainties than
the above case, higher accuracy than in the above embodiment can be
assured.
[0244] In the above embodiment, the aforementioned six parameters
are used as those for uniquely specifying the arrangement of shot
areas SA(i, j) on the wafer W. Alternatively, even when the
arrangement of shot areas SA(i, j) are uniquely specified by more
parameters than the above embodiment, their optimal values can be
accurately calculated as in the above embodiment.
[0245] More specifically, in the above embodiment, upon aligning
shot areas SA(i, j) on the wafer, even when appropriate values of
parameters that specify the arrangement of representative points,
e.g. central points of the shot areas SA(i, j), are calculated, and
the central point of a given shot area SA(i, j) is aligned using
those parameters, sufficiently high accuracy of superposition
cannot always be obtained. Such problem is caused by the following
three major factors, as disclosed in, e.g., Japanese Patent
Laid-Open Nos. 6-275496 and 6-349705.
[0246] {circle over (5)} Rotation of shot region: This is caused
when a reticle R has rotated with respect to the stage coordinate
system (X, Y) upon transferring a pattern formed on the reticle R
onto a wafer W or when yawing is accidentally mixed in the movement
of a wafer stage WST upon scanning exposure, and is expressed by a
rotation error .phi. of the wafer coordinate system (.alpha.,
.beta.) with respect to a shot coordinate system having coordinate
axes parallel to the .alpha.- and .beta.-axes.
[0247] {circle over (6)} Orthogonality of shot region: This is
caused by distortion of a pattern formed on a reticle R, distortion
(distortion error) of a projection optical system PL, and the like,
and is expressed by an orthogonality error .chi..
[0248] {circle over (7)} Linear expansion/shrinkage of shot region:
This is caused by an error of the projection magnification upon
projecting a pattern formed on a reticle R onto a wafer W or by the
wafer entirely or partially expanding or shrinking due to a
formation process or the like. This expansion/shrinkage amount is
expressed by wafer scaling values r.sub.X and r.sub.Y in the
coordinate axis directions (i.e., .alpha.- and .beta.-directions)
of the shot coordinate system. Note that the wafer scaling value
r.sub.X in the .alpha.-direction is represented by the ratio
between the actually measured value and design value of the
distance between two points in the .alpha.-direction on the wafer
W, and that the wafer scaling value r.sub.Y in the .beta.-direction
is represented by the ratio between the actually measured value and
design value of the distance between two points in the
.beta.-direction.
[0249] The ideal position (DX, DY) in a shot area is expressed
by
DX=CX+SX (47)
DY=CY+SY (48)
[0250] using a central coordinate position (CX, CY) of that shot
region, and a coordinate position (SX, SY) of the position (DX, DY)
relative to that coordinate position (CX, CY) on the shot
coordinate system. Therefore, when error factors {circle over (5)}
to {circle over (7)} are added to the aforementioned error factors
{circle over (1)} to {circle over (4)}, a pattern to be transferred
at the ideal position (DX, DY) on the wafer coordinate system
(.alpha., .beta.) is transferred to a position (EX, EY) on the
stage coordinate system (X, Y) given by 17 ( EX EY ) = ( R X , 0 0
, R Y ) ( cos , - sin sin , cos ) ( 1 , - tan w 0 , 1 ) ( CX CY ) +
( r X , 0 0 , r Y ) ( cos , - sin sin , cos ) ( 1 , - tan 0 , 1 ) (
SX SY ) + ( O X O Y ) ( 49 )
[0251] Note that various other error factors are also present in
addition to the aforementioned ones upon actual transfer, and that
the position (EX, EY) is considered as an expected transfer
position.
[0252] In general, since the orthogonality errors w and .chi., and
residual rotation errors .theta. and .phi. can be considered as
infinitesimal amounts, the designed transfer position (DX, DY) and
expected transfer position (EX, EY) are related by 18 ( EX EY ) = (
R X , - R X ( w + ) R Y , R Y ) ( CX CY ) + ( r X , - r X ( + ) r Y
, r X ) ( SX SY ) + ( O X O Y ) ( 50 )
[0253] which expresses a first-order approximation of the
trigonometric function in equation (49).
[0254] In the following description, equation (50) is also
expressed by 19 ( EX EY ) = ( A 11 , A 12 A 21 , A 22 ) ( CX CY ) +
( B 11 , B 12 B 21 , B 22 ) ( SX SY ) + ( O X O Y ) ( 51 )
[0255] where
B.sub.11=r.sub.X (52)
B.sub.12=-r.sub.X(.chi.+.phi.) (53)
B.sub.21=r.sub.Y.multidot..phi. (54)
B.sub.22=r.sub.Y (55)
[0256] That is, parameters which uniquely specify the arrangement
of shot areas SA(i, j) are ten parameters All, A.sub.12, A.sub.21,
A.sub.22, B.sub.11, B.sub.12, B.sub.21, B.sub.22, O.sub.X, and
O.sub.Y. In order to obtain optimal values of the ten parameters,
four two-dimensional marks WMp(i, j) (p=1 to 4), which do not line
up upon design and the patterns of which change in the X- and
Y-directions, are formed on each shot area SA(i, j), as shown in,
e.g., FIG. 14. Note that the number of wafer marks is not limited
to four, and can be three or more. As the two-dimensional marks
WMp, marks having patterns shown in, e.g., FIGS. 15A to 15D, can be
used.
[0257] Then, more than five two-dimensional marks WMp(i.sub.m,
j.sub.m) (m>5) including three out of four two-dimensional marks
WMp(i, j) on any shot area are measured. After that, the X- and
Y-positions of the two-dimensional marks WMp(i.sub.m, j.sub.m), and
the probability densities of those X- and Y-positions are
calculated in the same manner as in the above embodiment, thus
calculating optimal values of the ten parameters A.sub.11,
A.sub.12, A.sub.21, A.sub.22, B.sub.11, B.sub.12, B.sub.21,
B.sub.22, O.sub.X, and O.sub.Y.
[0258] In this way, the arrangement of shot areas can be obtained
that takes into account the rotation, orthogonality and
expansion/shrinkage of each shot region. Furthermore, optimal
values of the error factors (r.sub.X, r.sub.Y, .chi., .phi.) are
calculated using equations (32) to (35) for the aforementioned
controllable error factors (r.sub.X, r.sub.Y, .chi., .phi.) and
four parameters (B.sub.11, B.sub.12, B.sub.21, B.sub.22) of the ten
parameters, the optimal values of which have been calculated, and
in accordance with those error factors, corrected are the
magnification of the projection optical system PL, the synchronous
velocity ratio upon scanning exposure, synchronous moving
directions upon scanning exposure, and the like. Thus, the accuracy
of superposition is further improved.
[0259] In the above embodiment, a plurality of initial sample sets
are used, but if one sample set from which accurate position
parameters can be calculated is known, that sample set may be used
as only one initial sample set. Even in such a case, the processing
flow of the above embodiment can be applied. Since the number of
sample sets is one from the beginning, in the processing flow of
the above embodiment, the answer in step 204 is always NO.
[0260] In the above embodiment, whether or not it is possible to
reduce the number of elements of a sample set is evaluated by
comparing the position parameter values and their certainties
calculated from the sample set, and estimations of position
parameters and their certainties calculated from each of a
plurality of replacement candidate sets. Alternatively, such
evaluation can be performed by so-called cross-verification.
[0261] More specifically, the position errors of marks MX and MY
included in a sample set are calculated using the estimations of
the position parameters calculated for each replacement candidate
set, and the position error distribution of the marks MX and MY
included in the sample set is calculated based on that calculation
result, thus evaluating if each replacement candidate set has
statistical characteristics equivalent to that of the sample set.
In this case, replacement possibility can be evaluated without
calculating the position parameter values and their certainties
from the sample set. Such cross-verification is particularly
effective when the replacement candidate set is a subset of the
sample set.
[0262] In the above embodiment, upon reducing the number of
elements of a sample set, a plurality of initial sample sets are
replaced by one new sample set. Alternatively, the number of
elements of each of the plurality of initial sample sets may be
reduced. In this case, for each sample set, a new sample set is
searched for that has characteristics equivalent thereto in the
statistical characteristics of position errors of marks and
consists of fewer elements.
[0263] In the above embodiment, one-dimensional marks MX and MY are
used. Alternatively, two-dimensional marks shown in FIGS. 15A to
15D may be used. Also, upon calculating the ten parameters,
one-dimensional marks MX and MY may be used. As the two-dimensional
mark, for example, a box-in-box mark may be used in addition to
those shown in FIGS. 15A to 15D. Upon detecting the two-dimensional
position of such a two-dimensional mark, the aforementioned
one-dimensional position detection process may be performed twice,
or two-dimensional template matching into which the above
one-dimensional template matching for one-dimensional signals is
extended may be performed on two-dimensional signal waveforms of
two-dimensional marks.
[0264] In the above embodiment, the off-axis alignment method that
measures the positions of alignment marks on a wafer without the
intervention of a projection optical system is adopted.
Alternatively, a TTL (through-the-lens) scheme that measures the
positions of alignment marks on a wafer via a projection optical
system, or a TTR (through-the-reticle) that simultaneously observes
a wafer and reticle via a projection optical system may be adopted.
In the case of the TTR scheme, upon observation, sample alignment
senses the position of a wafer mark where the deviation between a
reticle mark formed on the reticle and a wafer mark formed on the
wafer is zero.
[0265] In the above embodiment, the coordinate positions of the
shot areas are calculated. Alternatively, the step pitch of each
shot may be calculated.
[0266] In the above embodiment, the fly-eye lens is used as an
optical integrator (homogenizer). In place of the fly-eye lens, a
rod integrator may be used. In an illumination optical system using
the rod integrator, the rod integrator is disposed such that its
exit surface is nearly conjugated with the pattern surface of a
reticle R. Such an illumination optical system using the rod
integrator is disclosed in, e.g., U.S. Pat. No. 5,675,401, and the
disclosures in the above U.S. patent is incorporated herein by
reference as long as the national laws in designated states or
elected states, to which this international application is applied,
permit. Also, the fly-eye lens and rod integrator may be combined,
or two fly-eye lenses or rod integrators may be disposed in series
to build double optical integrators.
[0267] In the above embodiment, the present invention is applied to
the step-and-scan scanning exposure apparatus. However, the
application range of the present invention is not limited to such
specific apparatus, and the present invention can be suitably
applied to a stationary exposure apparatus such as a stepper or the
like.
[0268] Even an exposure apparatus using, e.g., ultraviolet rays may
adopt as a projection optical system a reflection system consisting
of reflective optical elements alone or a reflection/refraction
system (a catadioptric system) having both reflective and
refractive optical elements. As the catadioptric projection optical
system, a reflection/refraction system which is disclosed in, e.g.,
Japanese Patent Laid-Open No. 8-171054 and corresponding U.S. Pat.
No. 5,668,672, Japanese Patent Laid-Open No. 10-20195 and
corresponding U.S. Pat. No. 5,835,275, and the like, and has a beam
splitter and concave mirror as reflection optical elements, or a
reflection/refraction system which is disclosed in Japanese Patent
Laid-Open No. 8-334695 and corresponding U.S. Pat. No. 5,689,377,
Japanese Patent Laid-Open No. 10-3039 and corresponding U.S. patent
application Ser. No. 873,605 (application date: Jun. 12, 1997), and
the like, and has a concave mirror and the like as reflective
optical elements without using any beam splitter can be used. The
disclosures in the above Japanese Patent Laid-Opens and U.S.
patents are incorporated herein by reference as long as the
national laws in designated states or elected states, to which this
international application is applied, permit.
[0269] In addition, a reflection/refraction system can be employed
which comprises a plurality of refraction optical elements and two
mirrors (a main mirror being a concave mirror and a sub-mirror that
is a back surface mirror of which the reflection surface is formed
on the opposite side to the incident surface of a refraction
element or plane parallel plate) that are disposed along one axis,
and has the intermediate image, formed by those refraction optical
elements, of a reticle pattern again imaged on a wafer using the
main mirror and sub-mirror, the reflection/refraction system being
disclosed in Japanese Patent Laid-Open No. 10-104513 and U.S. Pat.
No. 5,488,229 corresponding thereto. In this reflection/refraction
system, the main mirror and sub-mirror are disposed in series with
the plurality of refraction optical elements, and an illumination
light passes through a portion of the main mirror, is reflected by
the sub-mirror and the main mirror in turn, passes through a
portion of the sub-mirror and reaches the wafer. The disclosures in
the above Japanese Patent Laid-Open and U.S. patent are
incorporated herein by reference as long as the national laws in
designated states or elected states, to which this international
application is applied, permit.
[0270] Furthermore, as the reflection/refraction-type projection
optical system, a reduction system may be employed which has, e.g.,
a circular image field, is telecentric on both the object plane
side and image plane side, and has a reduction ratio of, e.g.,
{fraction (1/4)} or {fraction (1/5)}. Also, in a scanning exposure
apparatus comprising this reflection/refraction-type projection
optical system, the illumination area of the illumination light may
be a rectangular-slit-shaped area whose center almost coincides
with the optical axis of the projection optical system and which
extends along a direction almost perpendicular to the scanning
direction of a reticle or wafer. By using a scanning exposure
apparatus comprising such a reflection/refraction-type projection
optical system, it is possible to accurately transfer a fine
pattern of about 100 nm L/S pattern onto wafers even with F.sub.2
laser light having, for example, the wavelength of 157 nm as
exposure light.
[0271] Furthermore, as a vacuum ultraviolet light, ArF excimer
laser light or F.sub.2 laser light is used. However, in a case of
containing only a beam-monitor mechanism and reference wavelength
light source in the same environment--controlled chamber as the
projection optical system, a higher harmonic wave may be used which
is obtained with wavelength conversion into ultraviolet by using
nonlinear optical crystal after having amplified a single
wavelength laser light, infrared or visible, emitted from a DFB
semiconductor laser device or a fiber laser by a fiber amplifier
having, for example, erbium (or erbium and ytterbium) doped.
[0272] For example, considering that the oscillation wavelength of
a single wavelength laser is in the range of 1.51 to 1.59 um, an
eight-time-higher harmonic wave of which the wavelength is in the
range of 189 to 199 nm or a ten-time-higher harmonic wave of which
the wavelength is in the range of 151 to 159 nm is emitted.
Especially, when the oscillation wavelength is in the range of
1.544 to 1.553um, an eight-time-higher harmonic wave of which the
wavelength is in the range of 193 to 194 nm, that is, almost the
same as ArF excimer laser light (ultraviolet light) is obtained,
and when the oscillation wavelength is in the range of 1.57 to 1.58
um, a ten-time-higher harmonic wave of which the wavelength is in
the range of 157 to 158 nm, that is, almost the same as F.sub.2
laser light (ultraviolet light) is obtained.
[0273] Furthermore, when the oscillation wavelength is in the range
of 1.03 to 1.12 um, a seven-time-higher harmonic wave of which the
wavelength is in the range of 147 to 160 nm is emitted, and,
especially, when the oscillation wavelength is in the range of
1.099 to 1.106 um, a seven-time-higher harmonic wave of which the
wavelength is in the range of 157 to 158 nm, that is, almost the
same as F.sub.2 laser light (ultraviolet light) is obtained. In
this case, for example, ytterbium-doped fiber laser can be employed
as the single wavelength laser.
[0274] Moreover, the present invention can be applied not only to
an exposure apparatus for producing micro devices such as
semiconductor devices but also to an exposure apparatus that
transfers a circuit pattern onto a glass substrate or silicon wafer
so as to produce reticles or masks used by a light exposure
apparatus, EUV (Extreme Ultraviolet) exposure apparatus, X-ray
exposure apparatus, electron beam exposure apparatus, etc.
Incidentally, in an exposure apparatus using DUV (far ultraviolet)
light or VUV (vacuum ultraviolet) light, a transmission-type
reticle is employed in general. And as the substrate of the
reticle, quartz glass, quartz glass with fluorine doped, fluorite,
magnesium fluoride, or quartz crystal is employed. And an X-ray
exposure apparatus of a proximity method or electron beam exposure
apparatus employs a transmission-type mask (stencil-mask,
membrane-mask); an EUV exposure apparatus employs a reflection-type
mask, and as the substrate of the mask, silicon wafer or the like
is employed.
[0275] Note that the present invention can be applied not only to a
wafer exposure apparatus used in the production of semiconductor
devices but also to an exposure apparatus that transfers a device
pattern onto a glass plate and is used in the production of
displays such as liquid crystal display devices and plasma
displays, an exposure apparatus that transfers a device pattern
onto a ceramic plate and is used in the production of thin magnetic
heads, and an exposure apparatus used in the production of pick-up
devices (CCD, etc.). In addition, in the above embodiment,
positional detection of the alignment marks on a wafer and
alignment of the wafer have been described. However, positional
detection and alignment according to the present invention can be
applied to positional detection of the alignment marks on a reticle
and alignment of the reticle, and also to other units than exposure
apparatuses such as a unit to observe objects and a unit that is
used to detect positions of objects and align them in an assembly
line, process line or inspection line.
[0276] As has been described in detail above, according to the
position detection method and position detection apparatus of the
present invention, since a statistical process is executed on the
basis of position information of position detection points obtained
by measurement of the position detection points on an object,
position information of any area on the object is accurately and
efficiently detected. Therefore, the position detection method and
position detection apparatus of the present invention are suitable
for detecting the position of any area on an object.
[0277] Also, according to the exposing method and exposure
apparatus of the present invention, the positions of a
predetermined number of alignment marks formed on a substrate are
detected using the position detection method of the present
invention, and a predetermined pattern is transferred onto divided
areas while aligning the substrate on the basis of the detection
result. Therefore, the exposing method and exposure apparatus of
the present invention are suitable to perform multi-exposure for
forming a multi-layer pattern with improved accuracy of
superposition between layers. For this reason, the exposing method
and exposure apparatus of the present invention are suitable for
mass-production of devices having a fine pattern.
[0278] While the above-described embodiment of the present
invention is the presently preferred embodiment thereof, those
skilled in the art of lithography systems will readily recognize
that numerous additions, modifications, and substitutions may be
made to the above-described embodiment without departing from the
spirit and scope thereof. It is intended that all such
modifications, additions, and substitutions fall within the scope
of the present invention, which is best defined by the claims
appended below.
* * * * *