Alignment method

A substrate has a plurality of areas, from which several specific areas are selected. The coordinate positions of the specific areas on a static coordinate system are measured and the coordinate positions of the specific areas on the static coordinate system are calculated by the statistic calculation. For the respective specific areas, the calculated coordinate positions are subtracted from the measured coordinate positions to obtain the respective nonlinear position errors of the specific areas. When there is a peculiar area where the nonlinear position error exceeds an allowed value, the coordinate position of at least one area around the peculiar area is measured to obtain the nonlinear position error thereof. Prior to calculating the coordinate positions of the areas on the substrate on the static coordinate system by the use of the coordinate positions of the specific areas, it is judged based on the nonlinear position errors of the peculiar area and the area around the peculiar area whether the coordinate position of the peculiar area is used.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to an alignment method preferable for an 
exposure apparatus, a repair apparatus, an inspecting apparatus, etc., to 
be used in manufacturing, e.g., a semiconductor device, a liquid crystal 
display device, a thin film magnetic head. 
2. Related Background Art 
A step-and-repeat type projection exposure apparatus (stepper) is used in a 
manufacturing process of a semiconductor, etc. and especially in a 
photo-lithography process thereof. In the stepper, a pattern on a mask or 
a reticle (hereinafter referred to as the reticle) is transferred via a 
projection optical system to each of shot areas on a substrate (a 
semiconductor wafer, a glass plate, etc.) with a photosensitive material 
(photoresist) applied thereon in a step-and-repeat manner. The 
semiconductor device is formed by superimposing a plurality of circuit 
patterns on the wafer one over another. Therefore, when projecting and 
exposing the second circuit pattern on the wafer and thereafter, it is 
necessary to align the projected image of the reticle pattern and the 
circuit pattern on the wafer, i.e., the reticle and the wafer accurately. 
Presently, in the stepper, the enhanced-global-alignment (EGA) method is 
the mainstream, which is disclosed in U.S. Pat. Nos. 4,780,617 and 
4,833,621. 
The wafer has a plurality of shot areas each of which is formed with 
alignment marks. The shot areas are regularly arranged on the wafer based 
on arrangement coordinates preliminary set on the wafer. However, even 
though the wafer is shifted by the stepping based on the arrangement 
coordinate values (shot arrangement) upon the design of the plurality of 
shot areas on the wafer, the wafer is not necessarily aligned accurately 
due to the following factors: 
(1) the residual rotation error .theta. of the wafer 
(2) the rectangular degree error w of the stage coordinate system (or shot 
arrangement) 
(3) the linear expansion or contraction (scaling) Rx, Ry of the wafer 
(4) the offset (parallel movement) Ox, Oy of the wafer (center position) 
Arrangement coordinates on the wafer based on these four error amounts (six 
parameters) can be expressed by the linear transformation. Then, the 
linear transformation model for transforming the coordinate system (x, y) 
on the wafer to the stage coordinate system (X, Y) with respect to the 
wafer with the plurality of shot areas arranged regularly thereon can be 
expressed by the use of six transformation parameters a to f as follows: 
##EQU1## 
The six transformation parameters a to f in this equation can be obtained 
by, e.g., the least squares approximation method. Namely, among the 
plurality of shot areas (hereinafter called exposure shots) on the wafer, 
n exposure shots are selected and each of the n exposure shots 
(hereinafter called sample shots) is positioned (alignment) in the 
predetermined positions in accordance with the arrangement coordinates 
upon the design (x1, y1), (x2, y2), . . . , (xn, yn). And, the respective 
coordinate values (XM1, YM1), (XM2, YM2), . . . , (XMn, YMn) of the n 
sample shots on the stage coordinate system (X, Y) are measured. 
Next, the difference (.DELTA.x, .DELTA.y) between the above measured value 
(XMi, YMi) and the arrangement coordinates upon the calculation (Xi, Yi) 
obtained by substituting the arrangement coordinates upon the design (xi, 
yi) (i=1, . . . , n) of the respective sample shots in the above linear 
transformation model (equation (1)) is considered as the alignment error. 
The alignment error .DELTA.x is expressed by .SIGMA.(Xi-XMi).sup.2 and the 
alignment error .DELTA.y is expressed by .SIGMA.(Yi-YMi).sup.2 Further, 
the alignment errors .DELTA.x and .DELTA.y are partially differentiated 
with the six transformation parameters sequentially to obtain six 
expressions and six equations are set up such that the values of the six 
expressions become zero. Then, when the six simultaneous equations are 
solved, the six transformation parameters can be obtained. Thereafter, 
arrangement coordinates of the entire exposure shots on the wafer are 
calculated by using the linear transformation model (equation (1)) 
including the obtained transformation parameters a to f. As a result, when 
the wafer is positioned in accordance with the calculated arrangement 
coordinates, the alignment of the entire exposure shots can be performed 
accurately. 
Also, another method has been proposed in U.S. Pat. Ser. No. 011,697 filed 
Feb. 1, 1993 (abandoned), superseded by co-pending Ser. No. 08/538,467 
filed Oct. 3, 1995, now U.S. Pat. No. 5,561,606 issued Oct. 1, 1996, 
wherein when the preferable approximation accuracy cannot be obtained by 
the above linear equation (equation (1)), the equation of the second 
degree or much higher degree is used. As the linear approximation is 
performed in the above-mentioned EGA method, there is an inconvenience 
that the alignment accuracy is lowered when there is nonlinear distortion 
on the wafer. Then, a so-called weighting EGA method has been proposed in 
the U.S. Pat. No. 005,146 filed Jan. 15, 1993 (abandoned), superseded by 
co-pending Ser. No. 08/360,028 filed Dec. 20, 1994, now U.S. Pat. No. 
5,525,808 issued Jun. 11, 1996. In the weighting EGA method, by using Wi 
corresponding to the distances between an exposure shot and the n sample 
shots, the alignment errors .DELTA.x and .DELTA.y are expressed by 
.SIGMA.Wi (Xi-XMi).sup.2 and .SIGMA.Wi (Yi-YMi).sup.2 respectively. Then, 
the transformation parameters a to f are obtained by the least squares 
method. 
In the above-mentioned alignment methods, there is a case that the 
arrangement coordinates of the sample shots are deviated largely from the 
regularity of the shot arrangement on the wafer owing to the positioning 
error of the wafer at the time of the exposure for the previous layer, the 
partial distortion of the wafer, the damaged or scratched alignment marks, 
or the like. Such a sample shot is called "peculiar shot". The existence 
of only one such abnormal shot causes the reduction of the alignment 
accuracy. Then, the EGA method is applied to a wafer in a lot to calculate 
arrangement coordinates of all exposure shots and arrangement coordinates 
of all the exposure shots are measured by an alignment sensor. Then, in 
all the wafers of the lot, exposure shots where the difference between the 
calculated arrangement coordinates and the measured arrangement 
coordinates is equal to or more than a predetermined value are regarded as 
peculiar shots and are not selected, which method has been proposed in the 
aforesaid U.S. Pat. Ser. No. 011,697 filed Feb. 1, 1993 (abandoned), 
superseded by co-pending Ser. No. 08/361,158 filed Dec. 21, 1994, now U.S. 
Pat. No. 5,561,606 issued Oct. 1, 1996. 
In the EGA method, when the peculiar shot is caused by the measurement 
error, the measurement result needs to be excluded. On the other hand, 
when the peculiar shot is caused by the partial distortion, the exposure 
shots located in the area having the partial distortion need to be 
subjected to the alignment different from that of the other exposure 
shots. However, in the above methods, it is considered indiscriminately 
that all the peculiar shots are caused by the measurement errors and the 
measurement results are not used, whereby the desired alignment accuracy 
cannot be obtained. This problem occurs in the weighting EGA method also. 
Further, in the weighting EGA method, when the measurement error of the 
coordinate positions of the sample shot is large, the alignment accuracy 
of the exposure shots around the sample shot is affected by the 
measurement error largely. As above, when the sample shot has the 
nonlinear arrangement error in the conventional methods, it cannot be 
judged whether the nonlinear arrangement error is caused by the distortion 
of the wafer or the measurement error (the damage or scratch of the 
alignment marks), which reduces the alignment accuracy considerably. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide an alignment method of 
aligning each of shot areas on a substrate and a predetermined position 
with high accuracy and at high speed even though a sample shot has a 
nonlinear arrangement error. 
In the present invention, prior to aligning each of a plurality of areas to 
be processed on a substrate and a predetermined position in a static 
coordinate system for defining a moving position of the substrate, at 
least three areas of the areas to be processed are selected as sample 
areas and coordinate positions of the sample shots on the static 
coordinate system are measured. Then, the measured coordinate positions 
are subjected to statistic calculation to obtain coordinate positions of 
the plurality of areas on the substrate on the static coordinate system. 
Thereafter, in accordance with the calculated coordinate positions, the 
moving position of the substrate is controlled to align each of the 
plurality of areas to be processed and the predetermined position. 
The first alignment method comprises the first step of measuring the 
coordinate positions of the plurality of sample areas on the static 
coordinate system, the second step of obtaining nonlinear error amounts of 
the coordinate positions of the respective sample areas based on the 
measured coordinate positions thereof, the third step of determining as a 
peculiar area, the area of the sample areas where the nonlinear error 
amount exceeds a predetermined allowed value and measuring coordinate 
positions of areas around the peculiar area and the fourth step of 
obtaining nonlinear error amounts of the areas around the peculiar area 
based on the coordinate positions measured in the steps 1 and 3. When the 
nonlinear error amounts of the areas around the peculiar area and the 
nonlinear error amount of the peculiar area have the same trend, 
coordinate positions of the peculiar area and the areas around the 
peculiar area on the static coordinate system are calculated by subjecting 
the coordinate position of the peculiar area measured in the step 1 and 
the plurality of coordinate positions measured in the step 3 to the 
statistic calculation. On the other hand, when the nonlinear error amount 
of the peculiar area and the nonlinear error amounts of the areas around 
the peculiar area have different trends, the plurality of coordinate 
positions measured in the steps 1 and 3 excluding the coordinate position 
of the peculiar area measured in the step 1 are subjected to the statistic 
calculation, thereby to calculate arrangement coordinates of the plurality 
of areas to be processed on the substrate. 
Namely, according to the first alignment method, when there is the peculiar 
area having the large nonlinear error amount among the plurality of sample 
areas on the substrate, the coordinates of the areas around the peculiar 
area on the static coordinate system are measured to obtain the nonlinear 
error amounts thereof. And, when the nonlinear error amount of the 
peculiar area and the nonlinear error amounts of the areas around the 
peculiar area have the same trend, it is deemed that the nonlinear error 
amount of the peculiar area is caused by a partial nonlinear distortion of 
the substrate and the alignment of the areas (the peculiar area and the 
areas around the peculiar area) within the partially distorted region is 
performed based on the measurement results (coordinates). On the other 
hand, when the nonlinear error amount of the peculiar area and the 
nonlinear error amounts of the areas around the peculiar area have the 
different trends, it is deemed that the nonlinear error amount of the 
peculiar area is caused by the measurement error and the measurement 
result of the peculiar area is excluded and the alignment is performed 
based on the measurement results of the others. The alignment of the 
peculiar area is performed by the die-by-die method or based on the 
measurement results of the sample areas other than the peculiar area. 
The second alignment method comprises the first step of selecting at least 
three sample areas and measuring coordinate positions of the sample areas 
on a static coordinate system, the second step of performing statistic 
calculation to the measured coordinate positions to calculate coordinate 
positions of the respective sample areas and calculating the differences 
between the respective calculated coordinate positions of the sample areas 
and the coordinate positions measured in the step 1, i.e., nonlinear error 
amounts of the sample areas, the third step of determining as a peculiar 
area, the sample area of the sample areas where the nonlinear error amount 
exceeds an allowed value, determining as an alternative area, at least one 
area around the peculiar area and measuring a coordinate position of the 
alternative area, the fourth step of calculating a coordinate position of 
the alternative area by performing the statistic calculation to the 
plurality of coordinate positions measured in the steps 1 and 3 and 
obtaining the difference between the calculated coordinate position of the 
alternative area and the measured coordinate position in the third step, 
i.e., nonlinear error amount of the alternative area and the fifth step of 
comparing the nonlinear error amount of the peculiar area and that of the 
alternative area and determining the coordinate position to be used to 
calculate coordinate positions of the plurality of area to be processed on 
the substrate. Based on the coordinate position determined in the fifth 
step and the coordinate positions of the sample areas measured in the 
first step excluding the coordinate position of the peculiar area, 
arrangement coordinates of the plurality of areas to be processed on the 
substrate on the static coordinate system are calculated. 
The present invention is aiming at the point that the arrangement changes 
of the areas to be processed due to the partial distortion of the 
substrate are continuous to a certain degree but the measurement error of 
a sample area is peculiar to its sample area and has no relation to the 
areas around the sample area, i.e., the arrangement changes are 
discontinuous (random). Then, by using the fact that the change of the 
distortion of the areas to be processed on the substrate is different 
between in the case of nonlinear distortion and in the case of the 
measurement error, the present invention specifies as to whether the cause 
of the arrangement changes of the areas to be processed is due to the 
nonlinear distortion or the measurement error. That is, when the nonlinear 
error amount of the peculiar area and the nonlinear error amounts of the 
areas (alternative area) around the peculiar area have the same trend, 
i.e., the nonlinear distortion of the substrate is due to the occurrence 
of the nonlinear error, the measurement results of the peculiar area and 
the areas (alternative area) around the peculiar area are used together 
with the measurement results of the other sample areas to perform the 
alignment. On the other hand, when the nonlinear error amount of the 
peculiar area and the nonlinear error amounts of the areas (alternative 
area) around the peculiar area have different trends, i.e., the occurrence 
of the nonlinear error is due to the measurement error of a sample area, 
the measurement result of the peculiar area is excluded and the alignment 
is performed by the use of the left measurement results. 
According to the present invention as above, regarding the peculiar area 
having the especially large nonlinear error amount, it is judged as to 
whether the nonlinear error amount is caused by the measurement error due 
to the damaged or scratched alignment marks or by the partial distortion. 
When it is caused by the measurement error, the measurement result of the 
peculiar area is excluded to perform the alignment, while when it is 
caused by the partial distortion, the measurement results of the peculiar 
area and the areas (alternative area) around the peculiar area and the 
measurement results of the left sample areas are used to perform the 
alignment. As a result, even though there is the peculiar shot among the 
substrate, the alignment can be performed on the entire surface of the 
substrate with high accuracy.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 2 shows a projection exposure apparatus preferable for applying an 
alignment method of the present invention. In FIG. 2, illumination light 
IL emitted from an extra-high pressure mercury emission lamp 1 is 
reflected by an elliptical mirror 2 and condensed at a second focal point. 
Thereafter, the illumination light IL enters an illumination optical 
system 3 including a collimator lens, an interference filter (wavelength 
selection device), an optical integrator (fly eye lens), an aperture stop 
(.sigma. stop), or the like. The fly eye lens (not shown) is disposed in a 
plane perpendicular to an optical axis AX such that the focal surface of 
the fly eye lens on the side of a reticle R coincides with a Fourier 
transform surface with respect to a pattern formed surface of the reticle 
R. Also, in the vicinity of the second focal point of the elliptical 
mirror 2 is disposed a shutter (rotary shutter with four blades) 37 for 
opening or closing the optical light-path of the illumination light IL by 
a motor 38. In addition to the emission lines from the extra-high pressure 
mercury lamp, the illumination light IL may be a laser light such as an 
excimer laser (KrF excimer laser, ArF excimer laser, etc.) or a higher 
hermonic such as a metal vapor laser, a YAG laser. 
After the illumination light IL (i-lines, etc.) of the wavelength region 
for exposing a resist layer is emitted from the illumination optical 
system 3, approximately all the illumination light IL is reflected by a 
beam splitter 4 and thereafter reaches a mirror 8 via a first relay lens 
5, a variable field stop (reticle blind) 6 and a second relay lens 7. 
Further, the illumination light IL reflected by the mirror 8 illuminates a 
pattern area PA of the reticle R with uniform illuminance via a condenser 
lens 9. The reticle blind 6 is disposed in a plane which has a conjugate 
relationship (image forming relationship) with the pattern formed surface 
of the reticle R. Accordingly, the illumination field of the reticle R can 
be changed at will by changing the largeness and the shape of the opening 
portion of the reticle blind 6 by separately driving a plurality of 
movable blades of the reticle blind 6 by means of a driving system 36. 
The reticle R of FIG. 2 is provided with alignment marks at approximately 
center points of four sides of the rectangular pattern area PA surrounded 
by a light-shielding zone with a predetermined width. When the alignment 
marks (hereinafter called simply the reticle marks) on the reticle R are 
projected to the resist layer of a wafer W via a projection optical system 
13, latent images of the reticle marks are formed on the resist layer. In 
this embodiment, those reticle marks are also used for aligning each of 
the shot areas of the wafer W and the reticle R. Two opposite reticle 
marks on the reticle are multi-marks each of which consists of, e.g., five 
diffraction grating marks arranged in the X direction at regular 
intervals. The diffraction grating mark consists of seven dot marks 
arranged in the Y direction. The other two opposite reticle marks are 
formed such that those multi-marks are rotated at 90.degree.. These 
reticle marks are formed by light-shielding portions such as chrome in 
transparent windows provided in the light-shielding zone of the reticle R. 
Further, in the vicinity of the outer periphery of the reticle R is formed 
two cross marks with light-shielding property so as to be opposite to each 
other. These cross marks are used for alignment (with respect to the 
optical axis of the projection optical system 13) of the reticle R. 
The reticle R is disposed on a reticle stage RS which can be slightly moved 
in the direction of the optical axis AX of the projection optical system 
13, inclined with respect to the plane perpendicular to the optical axis 
AX and moved two-dimensionally and slightly rotated in the plane 
perpendicular to the optical axis AX by a motor 12. A movable mirror 11m 
is fixed to an end of the reticle stage RS so as to reflect a laser beam 
from a laser interferometric measuring machine (laser interferometer) 11. 
The two-dimensional position of the reticle stage RS is constantly 
detected by the laser interferometer 11 with the resolving power of, e.g., 
0.01 .mu.m. Also, two sets of reticle alignment systems (RA systems) 10A 
and 10B are disposed over the reticle R to detect the two cross marks 
formed in the vicinity of the periphery of the reticle R. By shifting the 
reticle stage RS slightly in the X, Y and .theta. directions based on 
measurement signals from the RA systems 10A and 10B, the reticle R is 
positioned such that the center point of the pattern area PA coincides 
with the optical axis of the projection optical axis 13. 
The illumination light IL passed through the pattern area PA of the reticle 
R enters the projection optical system 13 which is telecentric on both 
sides. The projection optical system 13 projects the circuit pattern of 
the reticle R at e.g., 1/5 magnification to the surface of the resist 
layer on the wafer W which coincides with the optimum image forming 
surface of the projection optical system 13. 
The wafer W is attached vacuously to a slightly rotatable wafer holder (not 
shown) and supported by the wafer stage WS via the wafer holder. The wafer 
stage WS is movable two-dimensionally by a motor 16 in a step-and-repeat 
manner. When the pattern of the reticle R has been transferred and exposed 
on a shot area of the wafer W, the wafer stage WS is shifted to set the 
following shot area on the wafer W. A movable mirror 15m is fixed to an 
end of the wafer stage WS to reflect a laser beam from a laser 
interferometer 15. The two-dimensional position of the wafer stage WS is 
constantly detected by the laser interferometer 15 with the resolving 
power of, e.g., 0.01 .mu.m. Namely, a static coordinate system 
(rectangular coordinate system) XY for defining movement of the wafer 
stage WS in a plane perpendicular to the optical axis AX is determined by 
the laser interferometer 15. 
Also, a reference member (glass base plate) 14 is provided on the wafer 
stage WS to measure base lines. The height of the reference member 14 is 
the same as the wafer W. The reference member 14 is provided with a slit 
pattern consisting of five sets of L-shaped patterns of a light 
transmittance type and two sets of reference patterns of a light 
reflective type (duty ratio is 1:1). One of the two sets of reference 
patterns consists of a first mark and a second mark. The first mark has 
three diffraction grating marks arranged in the X direction. Each of the 
diffraction grating marks is formed of seven dot marks arranged in the Y 
direction. The second mark has twelve bar marks extending in the Y 
direction which are arranged in the X direction. The other one of the two 
sets of reference patterns are formed by rotating the above reference 
pattern at 90.degree.. 
An illumination light (exposure light) propagated under the reference 
member 14 by an optical fiber and a mirror, etc. illuminates the slit 
pattern of the reference member 14 from under (in the wafer stage). The 
illumination light transmitted through the slit pattern of the reference 
member 14 passes the projection optical system 13 and forms a projected 
image of the slit pattern on the rear surface (the pattern formed surface) 
of the reticle R. Further, the illumination light transmitted through one 
of the four reticle patterns on the reticle R reaches the beam splitter 4 
via the condenser lens 9 and the relay lenses 7 and 5. Then, the 
illumination light transmitted through the beam splitter 4 is incident on 
a photoelectric detecting device 35 whose light receiving surface is 
disposed in a plane approximately conjugate to a pupil surface (a Fourier 
transform surface with respect to the pattern formed surface of the 
reticle R) of the projection optical system 13. The photoelectric 
detecting device 35 outputs a photoelectric signal SS in accordance with 
the intensity of the illumination light to a main control system 18. 
Hereinafter, the optical fiber, the mirror, the reference member 14 and 
the photoelectric detecting device 35 are called ISS (Imaging Slit Sensor) 
system as a whole. The detailed structure of the ISS system is disclosed 
in, e.g., U.S. Pat. Nos. 4,780,616 and 4,853,745. 
Also, an imaging optical characteristics correcting section 19 is provided 
to adjust the imaging optical characteristics of the projection optical 
system 13. The imaging optical characteristics correcting section 19 
corrects the imaging optical characteristics of the projection optical 
system 13 such as magnification, distortion, field curvature or the like 
by separately driving some lens elements constituting the projection 
optical system 13, especially a plurality of lens elements closer to the 
reticle R. The detailed structure of the imaging characteristics 
correcting section 19 is disclosed in, e.g., U.S. Pat. No. 5,117,225. 
An off-axis type alignment sensor (hereinafter called Field Image Alignment 
(FIA) system) is provided on a side of the projection optical system 13. 
In the FIA system, light of a wide band region emitted from a halogen lamp 
20 is led via a condenser lens 21, an optical fiber 22 to an interference 
filter 23, wherein lights of the wavelength region exposing the resist 
layer and the infrared region are cut. The light transmitted through the 
interference filter 23 is incident on a telecentric objective lens 27 via 
a lens system 24, a beam splitter 25, a mirror 26 and a field stop BR. The 
light emitted from the objective lens 27 is reflected by a prism (or a 
mirror) 28 secured to a lower portion of the lens tube of the projection 
optical system 13 so as not to shield the illumination field of the 
projection optical system 13 and illuminates the wafer W approximately 
perpendicularly. 
The light from the objective lens 27 illuminates a portion on the wafer W 
including an alignment mark (hereinafter simply called the wafer mark). 
The light reflected from the portion is led via the prism 28, the 
objective lens 27, the field stop BR, the mirror 26, the beam splitter 25 
and a lens system 29 to an index plate 30. The index plate 30 is disposed 
in a plane conjugate to the wafer W with respect to the objective lens 27 
and the lens system 29. The image of the wafer mark is formed in a 
transparent window of the index plate 30. As index marks, the index plate 
30 is formed with two linear marks which extend in the Y direction and are 
disposed so as to be apart from each other at a predetermined distance in 
the X direction. The light passed through the index plate 30 is led to an 
image pick-up device (CCD camera or the like) 34 via a first relay lens 
system 31, a mirror 32 and a second relay lens system 33. Then, the image 
of the wafer mark and the image of the index marks are formed on a light 
receiving surface of the image pick-up device 34. An image pick-up signal 
SV from the image pick-up device 34 is supplied to the main control system 
18, in which the position (coordinate value) of the wafer mark in the X 
direction is calculated. The structure of the FIA system is disclosed in, 
e.g., U.S. Pat. No. 4,962,318. Although not shown in FIG. 2, one more FIA 
system for detecting the position of a wafer mark in the Y direction is 
provided. 
Further, a TTL (Through the Lens) type alignment sensor 17 is provided on a 
side of the projection optical system 13. An illumination light, e.g., a 
He-Ne laser beam with the wavelength of 633 nm is directed to a wafer mark 
of the wafer W via mirrors M1, M2 and the projection optical system 13. 
The reflected light from the wafer mark returns to the alignment sensor 17 
via the projection optical system 13 and the mirrors M2 and M1. The 
alignment sensor 17 obtains the position of the wafer mark based on a 
signal obtained by converting the reflected light photoelectrically. 
FIG. 3 shows the structure of the alignment sensor 17 in FIG. 2 in detail. 
The alignment sensor 17 is constituted of a double beam interference type 
alignment system (hereinafter called Laser Interferomatric Alignment (LIA) 
system) disclosed in, e.g., U.S. Pat. No. 5,118,953, and a diffraction 
type alignment system (hereinafter called Laser Step Alignment (LSA) 
system) disclosed in, e.g., U.S. Pat. No. 4,699,515, wherein the same 
optical members are shared maximumly. Its structure will be simply 
described below, but its detailed structure is disclosed in U.S. Pat. No. 
5,151,750. 
In FIG. 3, a laser beam emitted from a He-Ne laser light source 40 is 
divided by a beam splitter 41. The laser beam reflected by the beam 
splitter 41 enters a first beam shaping optical system (LIA optical 
system) 45 via a shutter 42. On the other hand, the laser beam transmitted 
through the beam splitter 41 enters a second beam shaping optical system 
(LSA optical system) 46 via a shutter 43 and a mirror 44. Consequently, 
either the LIA system or the LSA system can be used switchably by 
separately driving the shutters 42 and 43. 
The LIA optical system 45 includes two sets of acoustooptic modulators 
(AOM) and emits two laser beams with a predetermined frequency difference 
.DELTA.f given approximately symmetrically with respect to the optical 
axis of the LIA optical system 45. The two laser beams emitted from the 
LIA optical system 45 reaches a beam splitter 49 via a mirror 47 and a 
beam splitter 48. The two laser beams transmitted through the beam 
splitter 49 are incident on a referential diffraction grating 55 (via lens 
53 and mirror 54) from two different directions at a predetermined 
crossing angle to form an image (intersect). A photoelectric detecting 
device 56 receives the interference light of the diffraction lights 
emanated from the referential diffraction grating 55 approximately in the 
same direction and outputs a sine-wave-like photoelectric signal SR in 
accordance with the intensity of the diffraction lights to the main 
control system 18 (LIA calculating unit in FIG. 4). 
On the other hand, the two laser beams reflected by the beam splitter 49 
are once intersected in a field stop 51 by an objective lens 50 and 
thereafter enter the projection optical system 13 via the mirror M2 (the 
mirror M1 is not shown in FIG. 3). The two laser beams entering the 
projection optical system 13 are once condensed like a spot approximately 
symmetrically with respect to the optical axis AX on the pupil surface of 
the projection optical system 13. Thereafter, the two laser beams become 
each parallel luminous flux inclined at the same angle with respect to the 
optical axis AX in the pitch direction (Y direction) of a wafer mark on 
the wafer W and are incident on the wafer mark from the two different 
directions at a predetermined crossing angle. As a result, the 
one-dimensional interference pattern shifting at a speed corresponding to 
the frequency difference .DELTA.f is formed on the wafer mark. The two 
diffraction lights being generated in the same direction from the wafer 
mark which are .+-.primary diffraction lights (interference light) being 
generated in the direction of the optical axis here are incident on a 
photoelectric detecting device 52 via the projection optical system 13, 
the objective lens 50 and the like. The photoelectric detecting device 52 
outputs a sine-wave-like photoelectric signal SDw corresponding to the 
periods of the change of the light and darkness of the interference 
pattern to a LIA calculating unit 58 (FIG. 4). The LIA calculating unit 58 
calculates the deviation of the position of the wafer mark from the phase 
difference of the waveforms of the two photoelectric signals SR and SDw. 
Also, by using a position signal PDs from the laser interferometer 15, the 
LIA calculating unit 58 obtains the coordinate position of the wafer stage 
WS when the deviation of the position of the wafer mark becomes zero. The 
obtained data is output to an alignment data storing section 61 (FIG. 4). 
The LSA optical system 46 includes a beam expander, a cylindrical lens and 
the like. The laser beam emitted from the LSA optical system 46 enters the 
objective lens 50 via the beam splitters 48 and 49. The laser beam emitted 
from the objective lens 50 is converged in the field stop 51 like a slit 
and thereafter enters the projection optical system 13 via the mirror M2. 
Then, after the laser beam passes approximately the center of the pupil 
surface of the projection optical system 13, it is expanded in the X 
direction in the projection field of the projection optical system 13 and 
illuminates the wafer W as the elongated strip-like spot light so as to be 
directed to the optical axis AX. 
Further, when moving the spot light and the wafer mark (diffraction grating 
mark) on the wafer relatively to each other, the light reflected from the 
wafer mark is received by the photoelectric detecting device 52 via the 
objective lens 50. The photoelectric detecting device 52 only converts 
.+-.primary to .+-.third diffraction lights among the light from the wafer 
mark and outputs a photoelectric signal SDi corresponding to the intensity 
of those diffraction lights to the main control system 18 (LSA calculating 
unit 57 in FIG. 4). A position signal PDs from the laser interferometer 15 
is also input in the LSA calculating unit 57, which then samples the 
photoelectric signal SDi in synchronism with an up-and-down pulse 
generated for the moving amount of the wafer stage WS per unit. Further, 
the LSA calculating unit 57 converts the respective sampling values into 
digital values and stores the digital values in a memory in the order of 
addresses. Thereafter, the LSA calculating unit 57 calculates the position 
of the wafer mark in the Y direction by a predetermined calculation 
process and outputs the calculated data to the alignment data storing 
section 61 (FIG. 4). 
Next, the control system of the projection exposure apparatus in FIG. 2 
will be described with reference to FIG. 4. FIG. 4 is a block diagram 
showing the control system of this embodiment. The main control system 18 
in FIG. 2 is constituted of the LSA calculating unit 57, the LIA 
calculating unit 58, the FIA calculating unit 59, the alignment data 
storing section 61, the EGA calculating unit 62, a memory 63, a shot map 
data section 64, a system controller 65, a wafer stage controller 66 and a 
reticle stage controller 67. 
In FIG. 4, the LSA calculating unit 57, the LIA calculating unit 58 and the 
FIA calculating unit 59 obtain the positions of wafer marks (coordinate 
positions on the rectangular coordinate system XY defined by the 
interferometer 15) based on photoelectric signals from the alignment 
sensors and supply the coordinate positions to the alignment data storing 
section 61. The position data stored in the storing section 61 are 
supplied to the EGA calculating unit 62 as required. The shot map data 
storing section 64 stores the arrangement coordinate value upon the design 
for each shot area on the wafer W. These position data are also supplied 
to the EGA calculating unit 62. The EGA calculating unit 62 calculates six 
transformation parameters a to f of the model function (equation (1)) for 
calculating coordinate positions of all the shot areas on the wafer W 
based on the respective coordinate values from the storing sections 61 and 
64 by the use of the statistic technique (e.g., method of least squares). 
The obtained transformation parameters a to f are sent to the memory 63. 
Further, the EGA calculating unit 62 calculates respective coordinate 
positions of all the shot areas on the wafer W by the use of the model 
function (equation (1)) including the previously obtained transformation 
parameters a to f. The calculated coordination positions are sent to the 
system controller 65. The system controller 65 determines target positions 
by adding the coordinate positions from the EGA calculating unit 62 and 
the base line amount and supplies the determined target positions to the 
stage controller 66. The stage controller 66 drives the wafer stage WS by 
the motor 16 such that the measured value of the laser interferometer 15 
coincides with the target position, whereby each shot area on the wafer W 
is positioned on a predetermined exposure position (projection position of 
the reticle pattern and usually the position of the optical axis of the 
projection optical system 13) in the rectangular coordinate system XY. The 
base line amount which is the distance between the detecting center of the 
FIA system, the LIA system, or the LSA system and the optical axis AX of 
the projection optical system 13 is measured in advance by means of the 
ISS system. The system controller 65 gives instructions to the stage 
controller 67 in accordance with the measured results of the two sets of 
RA systems 10A, 10B or the ISS system. While monitoring the measured value 
of the laser interferometer 11, the stage controller 67 drives the reticle 
stage RS by the motor 12 to effect the alignment of the reticle R. 
Next, an exposure sequence of the alignment method of this embodiment will 
be described with reference to FIG. 1. First, the main control system 18 
loads the wafer W to be exposed onto the wafer stage WS. FIG. 5A shows the 
arrangement of shots on the wafer W and N exposure shots ES1 to ESN are 
formed on the wafer W regularly. Each exposure shot has a chip pattern 
formed by the exposure processes for the previous layers. Also, the 
exposure shots are partitioned with street lines extending in the X, Y 
directions and each provided with two sets of wafer marks Mxi and Myi. 
However, in FIG. 5A, only several exposure shots (e.g., ES3) are provided 
with wafer marks. The wafer mark Mxi of the exposure shot ESi is formed on 
the middle portion of a street line extending in the X direction while the 
wafer mark Myi thereof is formed on the middle portion of a street line 
extending in the Y direction. The wafer mark Mxi has three bar marks which 
extends in the Y direction and are arranged in the X direction at 
predetermined intervals. The wafer mark Myi is formed such that the wafer 
mark Mxi is rotated at 90.degree.. 
In the step 101 in FIG. 1, the system controller 65 detects wafer marks of 
nine exposure shots (hereinafter called sample shots) selected from the N 
exposure shots on the wafer W by using the two sets of FIA systems. The 
FIA calculating unit 59 obtains the coordinate position for each wafer 
mark by processing the waveform of the image pick-up signal SV from the 
FIA system. 
FIG. 6 shows the condition of the wafer mark Mx1 detected by the X-FIA 
system in FIG. 2. The image pick-up device 34 supplies its image pick-up 
signal SV to the FIA calculating unit 59. As shown in FIG. 6, in the image 
pick-up field VSA of the image pick-up device 34, there are the wafer mark 
Mx1 and the index marks FM1 and FM2 on the index plate 30 sandwiching the 
wafer mark Mx1 therebetween. The image pick-up device 34 scans the images 
of the wafer mark Mx1 and the index marks FM1 and FM2 electrically along a 
horizontal scanning line VL. At this time, it is disadvantageous to use 
one scanning line in terms of SN ratio. Therefore, a plurality of 
horizontal scanning lines are provided in the image pick-up field VS and 
levels of image pick-up signals obtained from the respective horizontal 
scanning lines are added and averaged for respective pixels in the 
horizontal direction. After the FIA calculating unit 59 has subjected 
these image pick-up signals to waveform processing and has detected the 
deviation amount between the wafer mark Mx1 and the index marks FM1, FM2, 
it uses the position signal PDs from the interferometer 15 to obtain the 
coordinate position of the wafer mark Mx1 in the X direction when the 
deviation amount becomes zero. Also, the system controller 65 detects the 
wafer mark My1 of the sample shot SA1 by the use of the Y-FIA system. 
Then, the FIA calculating unit 59 subjects the image pick-up signals to 
waveform processing to obtain the coordinate position of the wafer mark 
My1 in the Y direction. The coordinate positions in the X and Y directions 
with respect to the respective sample shots SA2 to SA9 are obtained in the 
similar manner to the above-mentioned operation. These coordinate 
positions are stored in the memory 61. 
FIG. 5B shows an example of a wafer mark preferable for the LIA system. In 
FIG. 5B, a wafer mark MAx is like a diffraction grating arranged with 
predetermined pitches along the X direction. When detecting the wafer mark 
MAx, two laser beams BM.sub.1 and BM.sub.2 are emitted from the LIA 
optical system 45 (FIG. 3) constituting the alignment sensor 17 thereby to 
illuminate the wafer mark MAx at a predetermined crossing angle. The 
crossing angle of the two laser beams BM.sub.1 and BM.sub.2 and the 
pitches of the wafer mark MAx in the X direction are set such that the 
-primary diffraction light B.sub.1 (-1) from the wafer mark MAx by the 
laser beam BM.sub.1 and the +primary diffraction light B.sub.2 (+1) from 
the wafer mark MAx by the laser beam BM2 are directed in the same 
direction (optical axis direction of the projection optical system 13 in 
this embodiment). The interference light of the -primary diffraction light 
B.sub.1 (-1) and the +primary diffraction light B.sub.2 (+1) from the 
wafer mark MAx are received by the photoelectric detecting device 52. 
Then, a photoelectric signal SDw from the photoelectric detecting device 
52 is supplied to the LIA calculating unit 58. The LIA calculating unit 58 
calculates the deviation of the position of the wafer mark MAx in the X 
direction from the phase difference of the reference signal SR and the 
photoelectric signal SDw. And, by using the position signal PDs from the 
interferometer 15, the LIA obtains the coordinate position of the wafer 
stage WS when the deviation becomes zero. 
FIG. 5C shows an example of a wafer mark preferable for the LSA system. In 
FIG. 5C, a wafer mark MBx consists of six dot marks arranged at 
predetermined pitches in the Y direction. When detecting the wafer mark 
MBx, a laser beam is-emitted from the LSA optical system 46 (FIG. 3) 
constituting the alignment sensor 17 to illuminate the wafer W with a spot 
light LXS elongated in the Y direction like a strip. And, the wafer stage 
WS is driven in the X direction to effect relative scanning of the wafer 
mark MBx and the spot light LXS. The diffraction light occurring from the 
wafer mark MBs is received by the photoelectric device 52. A photoelectric 
signal SDi from the photoelectric detecting device 52 is supplied to the 
LSA calculating unit 57, which obtains the coordinate position of the 
wafer mark MBx in the X direction by a predetermined calculation process. 
Next, in the step 102, nonlinear error amounts of the respective sample 
shots SA1 to SA9 are obtained. Therefor, based on the coordinate positions 
upon the design of the wafer marks and the coordinate positions measured 
in the step 101, the EGA calculating unit 62 obtains six transformation 
parameters a to f satisfying the equation (1) by using, e.g., the method 
of least squares. That is, if the coordinate position of the n-th sample 
shot SAn measured in the step 101 is (XM.sub.n, YM.sub.n) and the 
coordinate position obtained by substituting the coordinate position upon 
the design in the equation (1) is (X.sub.n, Y.sub.n), the residual error 
element is expressed by the following equation. It is to be noted that the 
value of m is 9. 
##EQU2## 
And, the values of the transformation parameters a to f of the equation (1) 
are obtained such that the residual error element becomes minimum. This is 
the so-called EGA calculation. 
Next, the EGA calculating unit 62 calculates respective coordinate 
positions of the nine sample shots SA1 to SA9 by the use of the model 
function (equation (1)) including the obtained transformation parameters a 
to f. Further, the EGA calculating unit 62 subtracts the coordinate 
position obtained by the EGA calculation from the coordinate position 
(XM.sub.n, YM.sub.n) measured in the step 101 to obtain the nonlinear 
error amount (Pi, Qi) for each sample shot. The obtained nine nonlinear 
error amounts are stored in the memory 63. 
FIG. 7A shows an example of the respective nonlinear error amounts of the 
sample shots SA1 to SA9 obtained in the step 102 exaggeratedly. In FIG. 
7A, the nonlinear error amount of the sample shot SA1 is indicated by the 
nonlinear error vector &lt;D1&gt;. The start point P1 of the nonlinear vector 
&lt;D1&gt; represents the coordinate position (the coordinate position including 
the linear error amount) calculated in the step 102 while the end point P2 
thereof represents the coordinate position measured in the step 101. 
Therefore, regarding the vector &lt;D1&gt;, &lt;D1&gt;=(P1, Q1) holds. In FIG. 7A, the 
nonlinear error amounts of the other sample shots are also indicated by 
the respective nonlinear error vectors. 
Next, in the step 103, it is checked whether there is a "peculiar shot" in 
the nine sample shots. Then, the X elements of the nonlinear error amounts 
(Pi, Qi) of the sample shots SAi (i=1 to 9) are averaged to obtain the 
average value Pi.sub.0 and the Y elements thereof are averaged to obtain 
the average value Qi.sub.0. Further, the deviation (Pi-Pi.sub.0) of the X 
element of the nonlinear error amount of each sample shot and the 
deviation (Qi-Qi.sub.0) of the Y element thereof are obtained. Next, when 
there is a sample shot in which the value of {(Pi-Pi.sub.0).sup.2 
+(Qi-Qi.sub.0).sup.2 }.sup.1/2 (the absolute value of the deviation 
vector) exceeds a predetermined allowed value, the sample shot is 
specified as the "peculiar shot". 
When there is no peculiar shot among the sample shots SA1 to SA9, the 
alignment is executed by the conventional EGA method in the step 112. 
Namely, the EGA calculating unit 62 calculates coordinate positions of all 
the exposure shots ESi on the wafer W from the equation (1) by using the 
transformation parameters a to f obtained in the step 102 and the 
arrangement coordinate values upon the design. The calculated coordinate 
positions are sent to the system controller 65. Further, the system 
controller 65 adds the base line amount of the FIA system and each of the 
coordinate positions calculated by the EGA calculating unit 62 to correct 
the coordinate positions for the respective exposure shots. The corrected 
coordinate positions (target positions) are output to the stage controller 
66. Then, the stage controller 66 positions the wafer stage WS 
successively in accordance with the input coordinate positions and the 
image of the reticle pattern is projected and exposed on each of the 
exposure shot areas. Thereafter, in the step 113, the exposure operation 
for the following wafer is carried out. 
On the other hand, when among the nine sample shots in FIG. 7A, e.g., the 
absolute value of the deviation vector of the nonlinear error vector &lt;D7&gt; 
of the sample shot SA7 exceeds the allowed value, i.e., the sample shot 
SA7 is the peculiar shot, the operation goes from the step 103 to the step 
104. Then, as shown in FIG. 7B, it may be at least one exposure shot, but 
eight exposure shots located in the peripheral area of the sample shot SA7 
are selected newly as sample shots SA71 to SA78 and the respective 
coordinate values of the sample shots SA71 to SA78 on the stage coordinate 
system XY are measured. 
Next, in the step 105, the correlation degree between the nonlinear error 
amount of the sample shot SA7 and the nonlinear error amounts of the eight 
sample shots SA71 to SA78 is checked. Then, by using the coordinate 
positions of the sample shot SA7 and the sample shots SA71 to SA78 
measured in the steps 101 and 104, the transformation parameters a to f of 
the model function (equation (1)) are calculated by means of the least 
squares method. And, respective coordinate positions of the nine sample 
shots SA7 and SA71 to SA78 are calculated by using the model function 
(equation (1)) including the calculated parameters a to f. Further, the 
respective calculated coordinate positions of the sample shots SA7 and 
SA71 to SA78 are subtracted from the coordinate positions thereof measured 
in the steps 101 and 104 to obtain the nonlinear error amounts (P'7, Q'7) 
and (P71, Q71) to (P78, Q78). The nonlinear error amount (P'7, Q'7) of the 
sample shot SA7 is slightly different from the nonlinear error amount (P7, 
Q7) thereof obtained in the step 102 and therefore recalculated here. 
Also, when calculating the transformation parameters a to f in the step 
105, the respective coordinate positions of the sample shots SA1 to SA6, 
SA8 and SA9 measured in the step 101 may be used together. 
Next, the average value P.sub.0 of the X elements of the nine nonlinear 
error amounts of the sample shot SA7 and the sample shots SA71 to SA78 as 
well as the average value Q.sub.0 of the Y elements thereof are obtained. 
Then, the average value (P.sub.0, Q.sub.0) is subtracted from each of the 
nonlinear error amounts of the sample shots SA7 and SA71 to SA78 to obtain 
the deviation vectors (P'7-P.sub.0, Q'7-Q.sub.0), (P71-P.sub.0, 
Q71-Q.sub.0) to (P78-P.sub.0, Q78-Q.sub.0). The absolute values of the 
nine deviation vectors are called "noncorrelation degrees". When the 
respective nine noncorrelations exceed the predetermined allowed value, it 
is deemed that there is no correlation between the sample shot SA7 and the 
sample shots SA71 to SA78. 
Then, in the step 106, the respective nine noncorrelations and the allowed 
value are compared. When at least one of the nine noncorrelation degrees 
is larger than the allowed value as shown in FIG. 8A, it is judged that 
the trend of the nonlinear error of the sample shot SA7 is different from 
the trends of the nonlinear errors of the sample shots SA71 to SA78 around 
the sample shot SA7, i.e., the result of the measurement of the sample 
shot SA7 is a measurement error caused by the damaged or scratched wafer 
mark. In FIG. 8A, the trend of the nonlinear error vector &lt;D7&gt; of the 
sample shot SA7 is different from the trends of the nonlinear error 
vectors &lt;D71&gt; to &lt;D78&gt; around the nonlinear error vector &lt;D7&gt;. 
Next, in the step 110, the transformation parameters a to f of the model 
function (equation (1)) are calculated by the least squares method by 
using the coordinate positions of the eight sample shots SA1 to SA6, SA8 
and SA9 excluding the peculiar shot SA7. Namely, the values of the 
transformation parameters a to f are obtained such that the residual error 
element obtained by subtracting the residual error of the sample shot SA7 
from the residual error element of the above equation (2) becomes minimum. 
Then, coordinate positions of a plurality of exposure shots on the wafer 
are calculated from the equation (1) based on the transformation 
parameters a to f and the coordinate positions upon the design. As a 
result, the wafer stage WS is positioned successively in accordance with 
the calculated coordinate positions and the image of the reticle pattern 
is exposed on each of the plurality of exposure shots. Thereafter, in the 
step 111, the exposure process for the following wafer is executed. 
When the process condition of the wafer W is preferable and it is deemed 
that the occurrence of the peculiar shot is not necessarily the 
measurement error caused by the damaged or scratched wafer mark, the 
measurement result of the peculiar shot SA7 may be assumed to be right. At 
this time, when exposing the peculiar shot SA7, alignment may be performed 
based on the result of measuring the coordinate position independently of 
the others by the die-by-die method. Besides, when exposing the peculiar 
shot SA7, the weighting EGA method to be described below may be adopted. 
In that case, respective weights given to the coordinate positions of the 
sample shots SA71 to SA78 around the sample shot SA7 are made larger than 
respective weights given to the coordinate positions of the sample shots 
SA1 to SA6, SA8 and SA9. Then, the coordinate position of the peculiar 
shot SA7 is calculated and the wafer stage WS is positioned in accordance 
with the calculated coordinate position. 
On the other hand, in the step 106, when all the nine noncorrelation 
degrees are equal to or less than the allowed value, the trends of the 
nonlinear errors of the sample shot SA7 and the sample shots SA71 to SA78 
around the sample shot SA7 are the same. In this case, it is judged that 
there is a partial nonlinear distortion in an area SA70 (refer to FIG. 5A) 
including the sample shots SA7 and SA71 to SA78 and the operation goes to 
the step 107. In FIG. 8B, the trend of the nonlinear error vector &lt;D7&gt; of 
the sample shot SA7 is approximately equal to the trends of the nonlinear 
error vectors &lt;D71&gt; and &lt;D78&gt;. 
When there is the partial distortion in the area SA70 on the wafer W, the 
respective coordinate positions of all the exposure shots ES1 to ESN on 
the wafer W are calculated by the use of the weighting EGA method. In the 
weighting EGA method, when exposing the exposure shot ESi on the wafer as 
shown in FIG. 9, the weight W.sub.in is given to the respective 17 sample 
shots SA1 to SA9 and SA71 to SA78. And, by using the coordinate positions 
(XM.sub.n, YM.sub.n) of the 17 ample shots measured in the steps 101 and 
104, the coordinate positions (X.sub.n, Y.sub.n) obtained by substituting 
the coordinate positions upon the design of the sample shots in the 
equation (1) and the weight W.sub.in, the residual error element Ei of the 
exposure shot ESi is defined by the following equation. The value of m is 
17 in the equation. 
##EQU3## 
Then, the value of the transformation parameters a to f of the equation (1) 
are obtained such that the residual error element Ei becomes minimum. This 
is called the weighting EGA calculation. Next, the coordinate position of 
the exposure shot ESi on the wafer is calculated by the use of the model 
function (equation (1)) including the obtained transformation parameters a 
to f. As above, in the weighting EGA method, the coordinate positions are 
calculated by obtaining the transformation parameters a to f for each 
exposure shot such that the residual error element Ei of the equation (3) 
becomes minimum. Thereafter, the wafer stage WS is successively positioned 
in accordance with the calculated coordinate positions to expose the image 
of the pattern of the reticle R on each exposure shot (steps 107 and 108). 
In this embodiment, when calculating the coordinate positions of exposure 
shots in the area SA70 having the nonlinear distortion by the weighting 
EGA method in the step 107, the respective weights given to the nine 
sample shots SA7 and SA71 to SA78 in the area SA70 are made larger than 
the respective weights given to the eight sample shots SA1 to SA6, SA8 and 
SA9 located outside the area 70. On the other hand, when calculating the 
coordinate positions of the exposure shots located outside the area SA70 
by the weighting EGA method in the step 108, the respective weights given 
to the nine sample shots SA7 and SA71 to SA78 in the area SA70 are made 
smaller than the respective weights given to the eight sample shots SA1 to 
SA6, SA8 and SA9 located outside the area SA70. Therefore, it is possible 
to position each of the entire exposure shots on the wafer W to the 
exposure position accurately. 
Then, the method of making the above-mentioned weight W.sub.in optimum will 
be described. Here, the weight W.sub.in given to the n-th sample shot is 
defined by the distance LKn between the exposure shot ESi and the n-th 
sample shot as follows: 
##EQU4## 
As is apparent from the equation, the shorter the distance LKn becomes, the 
larger the weight W.sub.in becomes. In the equation (4), when the value of 
the parameter Si is increased, the result approaches that of the 
conventional EGA method. On the other hand, when the value of the 
parameter Si is decreased, the result approaches that of the die-by-die 
method. 
Further, in this embodiment, the parameter Si is set such that a ratio r 
defined below becomes equal to or less than a predetermined value r.sub.0 
for each exposure shot. The ratio r is the value determined by dividing 
"the sum of the weight W.sub.in given to the sample shots outside the area 
SA70 when the coordinate positions of the exposure shots in the area SA70 
are calculated" by "the sum of the weight W.sub.in given to the sample 
shots in the area SA70 when the coordinate positions of the exposure shots 
in the area SA70 are calculated". 
Accordingly, in the step 107, when transferring the reticle pattern to each 
of the exposure shots in the area SA70 on the wafer W of FIG. 5A, the 
comparatively smaller weight W.sub.in is given to the respective eight 
sample shots located outside the area SA70 to obtain those coordinate 
positions. At this time, the comparatively larger weight W.sub.in is given 
to the nine sample shots in the area SA70. On the other hand, in the step 
108, when transferring the reticle pattern to each of the exposure shots 
located outside the area SA70, the weights given to the sample shots in 
the area SA70 are smaller than the weights given to the sample shots 
outside the area SA70 to obtain those coordinate positions. 
After the image of the reticle pattern is transferred to each of the 
exposure shots on the wafer W by the weighting EGA method in the steps 107 
and 108 as above, the exposure process for the following wafer is 
conducted in the step 109. 
Further, as another method of making the weight W.sub.in optimum, when it 
is judged that there is correlation between the peculiar shot SA7 and the 
sample shots SA71 to SA78 surrounding the peculiar shot SA7 in the step 
106, the coordinate positions of all or the greater part of the exposure 
shots on the wafer W of FIG. 5A may be measured to obtain those nonlinear 
error amounts. In this case, the weight W.sub.in is determined in 
accordance with the distribution of those nonlinear error amounts. Also, 
the coordinate positions may be obtained by the weighting EGA method by 
changing the distribution of the weight W.sub.in variously and the weight 
W.sub.in when the residual error element is minimized may be used. 
Also, in the step 105 in FIG. 1, in order to check the correlation between 
the sample shot SA7 and the sample shots SA71 to SA78 around the sample 
shot SA7, the coordinate positions of the nine sample shots SA7 and SA71 
to SA78 are calculated by the use of the transformation parameters a to f 
recalculated by the EGA calculation to obtain the nine nonlinear-error 
amounts (P'7, Q'7) and (P71, Q71) to (P78, Q78). However, in the step 105, 
the above correlation can be checked without re-executing the EGA 
calculation (calculation of the new transformation parameters a to f). 
Namely, the transformation parameters obtained in the step 102 may be 
used. In this case, the coordinate positions of the eight sample shots 
SA71 to SA78 are calculated by the use of the model function (equation 
(1)) including the transformation parameters a to f obtained in the step 
102. Then, in each of the eight sample shots SA71 to SA78, the calculated 
coordinate position is subtracted from the coordinate position measured in 
the step 104 to obtain its nonlinear error amount. As the nonlinear error 
amount of the sample shot SA7 has been obtained in the step 102, the eight 
nonlinear error amounts obtained here and the nonlinear error amount 
obtained in the step 102 are used to obtain the average value (P.sub.0, 
Q.sub.0). Thereafter, if the deviation vectors, i.e., the noncorrelation 
degrees are obtained by the use of the average value (P.sub.0, Q.sub.0), 
it is possible to check the correlation between the sample shot SA7 and 
the sample shots SA71 to SA78. 
Further, when there is no peculiar shot in the plurality of sample shots as 
in the step 112 of FIG. 1 but there is nonlinear distortion in the wafer 
W, the weighting EGA method may be adopted. Also, the weighting EGA method 
may be adopted in the step 110. At this time, the parameter Si is set to 
be the common value for the entire sample shots, e.g., the Si.sub.0 of the 
following equation. In the equation (5), D is the weight parameter. When 
the operator sets the weight parameter D to a predetermined value, the 
parameter Si.sub.0 and the resultant W.sub.in are determined automatically 
. 
EQU Si.sub.0 =D.sup.2 /(8.multidot.log.sub.e.sup.10) (5) 
The physical meaning of the weight parameter D is the range (hereinafter 
simply called the zone) of the sample shots effective for calculating the 
coordinate positions of the exposure shots on the wafer. That is, when the 
zone is large, the number of effective sample shots is increased, so that 
the result approaches that of the conventional EGA method. On the other 
hand, when the zone is small, the number of effective sample shots is 
decreased, so that the result approaches that of the die-by-die method. 
Also, the equation for determining the parameter Si.sub.0 is not limited to 
the equation (5) and the following equation can be used wherein the area 
of the wafer is A mm.sup.2 !, the number of sample shots is m and the 
correction coefficient (positive real number) is C. 
EQU Si.sub.0 =A/(m.multidot.C) (6) 
In the equation (6), the changes of the wafer size (area) and the number of 
the sample shots are reflected to determine the parameter Si.sub.0 thereby 
to prevent the optimum value of the correction coefficient C to be used 
for the determination from being varied excessively. When the correction 
coefficient C is small, the value of the parameters Si.sub.0 is increased, 
approaching the result obtained by the conventional EGA method. On the 
other hand, when the correction coefficient C is large, the value of the 
parameter Si.sub.0 is decreased, approaching the result obtained by the 
die-by-die method. 
Although the value of the parameter Si is changed for each exposure shot in 
accordance with the nonlinear distortion amounts on the wafer in the steps 
107 and 108 of this embodiment, the value of the parameter Si may be set 
based on the value of the parameter Si.sub.0 of the equation (5) or (6), 
thereby increasing or decreasing this reference value. 
Also, although the weight W.sub.in is determined from the equation (4) in 
accordance with the parameter Si, the weight W.sub.in obtained from the 
following equation in accordance with the parameter Si may be used. In 
this case, as shown in FIG. 10, the distance (radius) between the center 
point of the distortion of the wafer (e.g., the center of the point 
symmetry of the nonlinear distortion), e.g., the wafer center and the 
exposure shot ESi on the wafer is made to be LEi. And, the distances 
between the wafer center and m (m=17 in FIG. 10) sample shots are made-to 
be LWl to LWm. Then, the weight W.sub.in defined by the following equation 
is given to respective measurement results of the m sample shots in 
accordance with the distance LEi and the distances LWl to LWm. 
##EQU5## 
As is apparent from the equation (7), the closer the distance LWn between 
the sample shot and the center point of the distortion (wafer center) 
becomes with respect to the distance between the wafer center and the i-th 
exposure shot ESi, the larger the weight W.sub.in given to the alignment 
data becomes. Alternately, the largest weight W.sub.in, is given to the 
alignment data of the sample shots positioned on a circle drawn by the 
radius LEi with the wafer center as its center while the weight W.sub.in 
given to the alignment data is decreased as the sample shots become far 
away from the circle in the radius direction. 
Next, a second embodiment of the present invention will be described with 
reference to FIG. 11. FIG. 11 is a flowchart showing the exposure sequence 
including the alignment method of this embodiment. First, the main control 
system 18 loads a wafer W to be exposed on the wafer stage WS. FIG. 12 
shows the arrangement of shots of the wafer W to be used in this 
embodiment which is basically the same as that of the wafer W in FIG. 5. 
In the step 201 in FIG. 11, the system controller 65 detects respective 
wafer marks of 19 sample shots selected from N exposure shots on the wafer 
W of FIG. 12 by using the two sets of FIA systems. Further, the FIA 
calculating unit 59 processes the waveforms of the image pick-up signals 
SV from the FIA systems for each of the wafer marks to detect the 
deviation of the positions of the respective wafer marks thereby to obtain 
the coordinate position &lt;Am(i)&gt;(i=1 to 19) of each sample shot. The 
operation of the step 201 is the same as that of the step 101 (FIG. 1) of 
the first embodiment. In the step 202, the EGA calculating unit 62 obtains 
the transformation parameters a to f of the equation (1) by using the 
alignment data (coordinate positions) of the sample shots SA1 to SA19 by 
means of the least squares method such that the residual error element of 
the equation (2) is minimized. Next, in the step 203, the variable i is 
set to be 1. Then, in the step.204, the EGA calculating unit 62 calculates 
the coordinate position (Xi, Yi) of the i-th (here i=1) sample shot SAi by 
using the transformation parameters a to f obtained in the step 202. 
Namely, the coordinate position (Xi, Yi) is calculated by substituting the 
coordinate position upon the design of the sample shot SAi in the model 
function (equation (1)) including the transformation parameters a to f. 
The calculated coordinate position (Xi, Yi) is expressed by the vector 
form coordinates &lt;Ae(i)&gt;. 
Next, in the step 205, the coordinate &lt;Am(i)&gt; measured in the step 201 is 
subtracted from the coordinates &lt;Ae(i)&gt; obtained in the step 204 to obtain 
the nonlinear error vector &lt;NLa(i)&gt; of the sample shot SAi. Here, as the 
linear error is eliminated from the arrangement error of the sample shot 
SAi by this subtraction, its nonlinear error vector &lt;NLa(i)&gt; can be 
obtained. Further, in the step 206, it is judged whether the absolute 
value of the nonlinear error vector &lt;NLa(i)&gt; is larger than a 
predetermined allowed value NLc. When the absolute value of the nonlinear 
error vector is equal to or less than the allowed value NLc, i.e., 
.vertline.&lt;NLa(i)&gt;.vertline..ltoreq.NLc holds, it is judged that the 
nonlinear error amount of the sample shot SAi is small and it is not a 
peculiar shot. Accordingly, when .vertline.&lt;NLa(i)&gt;.vertline..ltoreq.NLc 
holds, the operation goes to the step 214, wherein it is determined that 
the alignment data (coordinate position) of the sample shot SAi is 
effective, i.e., the alignment data is to be used in the following step 
217 (weighting EGA method). And, the operation goes to the step 213. 
On the other hand, when .vertline.&lt;NLa(i)&gt;.vertline.&gt;NLc holds in the step 
206, the operation moves to the step 207. In the step 207, the system 
controller 65 selects an exposure shot adjacent to the sample shot SAi as 
an alternative shot SB and detects the wafer marks of the alternative shot 
SBi by the use of the FIA systems. Then, the FIA calculating unit 59 
processes the waveforms of the image pick-up signals SV from the FIA 
systems to obtain the coordinates &lt;BM(i)&gt; of the alternative shot SBi on 
the stage coordinate system XY. 
FIG. 13A shows an example of the respective nonlinear error amounts of the 
sample shots SA1 to SA19 obtained in the step 205 exaggeratedly. In FIG. 
13A, the nonlinear error amount in the sample shot SAi is represented by 
the vector &lt;NLa(i)&gt;. The start point Pi of the vector &lt;NLa(i)&gt; represents 
the coordinate position (including the linear error amount) of the sample 
shot SAi calculated in the step 204 while the end point Qi thereof 
represents the coordinate position of the sample shot SAi measured in the 
step 201. Also, the nonlinear error amounts of the other sample shots are 
represented by the vectors. In FIG. 13A, the respective nonlinear error 
vectors of the three sample shots SA1, SA7 and SA15 are larger than the 
allowed value. Then, e.g., the exposure shots SB1, SB7 and SB15 are 
selected from the exposure shots adjacent to the sample shots SA1, SA7 and 
SA15 as the alternative shots and the respective coordinate positions of 
the selected three alternative shots on the stage coordinate system XY are 
obtained. 
After obtaining the coordinates &lt;Bm(i)&gt; of the alternative shot SBi in the 
step 207, the EGA calculation is executed by the use of the respective 
coordinate positions of the alternative shots SBi and the sample shots 
excluding the i-th sample shot SAi in the step 208. Namely, the 
transformation parameters a to f of the equation (1) are obtained such 
that the residual error element of the equation (2) is minimized. Further, 
the coordinate position &lt;Be(i)&gt; of the alternative shot SBi is calculated 
by the use of these transformation parameters a to f from the equation 
(1). 
Next, in the step 209, the coordinates &lt;Bm(i)&gt; measured in the step 207 are 
subtracted from the coordinates &lt;Be(i)&gt; calculated in the step 208 to 
obtain the nonlinear error vector &lt;NLb(i)&gt; of the alternative shot SBi. 
Then, in the step 210, it is judged whether the absolute value of the 
nonlinear error vector &lt;NLb(i)&gt; is larger than the allowed value NLc. When 
.vertline.&lt;NLb(i)&gt;.vertline.&gt;NLc holds, the operation goes to the step 
211. On the other hand, when .vertline.&lt;NLb(i)&gt;.vertline..ltoreq.NLc 
holds, the nonlinear error amount of the alternative shot SBi is deemed to 
be small. Then, it is judged that the measured nonlinear error amount of 
the sample shot SA is caused by the measurement error and the operation 
goes to the step 215. In the step 215, it is determined that the 
coordinate position of the alternative shot SBi is effective, i.e., its 
coordinate position is to be used in the following step 217 (weighting EGA 
method) and the operation goes to the step 213. The coordinate position of 
the sample shot SAi is not used in the step 217. 
In the step 211, the nonlinear error vector &lt;NLa(i)&gt; is compared to the 
nonlinear error vector &lt;NLb(i)&gt;. That is, the absolute value 
.vertline.&lt;NLb(i)&gt;-&lt;NLa(i)&gt;.vertline. of the difference (the change amount 
of the nonlinear error) of both vectors is compared to the allowed change 
amount .DELTA.NL.sub.0 of the nonlinear error amount. When the absolute 
value is larger than the allowed change amount .DELTA.NL.sub.0, it is 
judged that the cause of the difference between the nonlinear error amount 
of the sample shot SAi and that of the alternative shot SBi is not the 
nonlinear distortion of the wafer but the larger random nonlinear error 
amount and the operation goes to the step 212. In the step 212, it is 
determined that both data (coordinate positions) of the sample shot SAi 
and the alternative SBi are effective, i.e., both data are to be used in 
the step 217 and the operation goes to the step 213. 
On the other hand, when .vertline.&lt;NLb(i)&gt;-&lt;NLa(i)&gt;.vertline. representing 
the change amount of the nonlinear error is equal to or less than the 
allowed change amount .DELTA.NL.sub.0, it is judged that the cause of the 
difference of the nonlinear errors of both shots is the nonlinear 
distortion of the wafer and the operation goes to the step 214. In the 
step 214, the alignment data of the sample shot SAi is made effective, 
i.e., it is determined that the data is to be used in the step 217. 
Next, in the step 213, it is judged whether the variable i has reached the 
total number m (m=19) of sample shots. When the variable i is i&lt;m, the 
operation goes to the step 216. After 1 is added to the value of the 
variable i in the step 216, the operation goes to the step 204 and the 
above-mentioned operation is repeated. On the other hand, when the 
variable i is i=m, the operation goes to the step 217. 
In the step 217, the weighting EGA method is adopted and the transformation 
parameters a to f of the equation (1) are obtained for each exposure shot 
by using a plurality of alignment data of the respective shots made 
effective in the steps 212, 214 and 215 such that the residual error 
element Ei of the equation (3) is minimized. At this time, the weight 
W.sub.in of the equation (3) may be determined by using either the weight 
W.sub.in of the equation (4) or that of the equation (7). Although the 
plurality of alignment data (coordinate positions) used for each of the 
exposure shots are the same in the weighting EGA method, the distances 
between each exposure shot and the sample shots or the alternative shots 
are different. Namely, the weight W.sub.in given to the sample shots and 
the alternative shots is varied for each exposure shot. Then, the 
transformation parameters a to f are calculated for each exposure shot 
thereby to calculate the coordinate positions (shot arrangement) of the 
entire exposure shots on the wafer W. Thereafter, the base line amount is 
added to the calculated coordinate positions and in accordance with the 
obtained coordinate positions, the wafer stage WS is successively 
positioned to expose the reticle pattern on each of the exposure shots on 
the wafer W. After all the exposure shots on the wafer W have been 
exposed, the wafer W is replaced with the following wafer and the steps 
201 to 217 of FIG. 11 are repeatedly performed therefor. 
FIG. 13B shows the residual error elements (nonlinear error amounts) of the 
respective exposure shots when the alignment method of this embodiment is 
applied to the wafer W in FIG. 13A. In FIG. 13A, the 19 sample shots SA1 
to SA19 are selected and there are the three sample shots SA1, SA7 and 
SA15 with the larger nonlinear error amounts. In FIG. 13B, the alternative 
shots SB1, SB7 and SB15 are located inside the respective sample shots 
SA1, SA7 and SA15, i.e., the exposure shots closer to the wafer center 
than the sample shots are selected. Further, the weight parameter D in the 
equation (5) is set to be 100 mm, the allowed value NLc of the nonlinear 
error is set to be 0.1 .mu.m and the allowed change amount .DELTA.NL.sub.0 
is set to be 0.04 .mu.m. Although only the sample shot SA15 has the larger 
nonlinear error amount, such a foreign sample shot may be subjected to the 
alignment by the die-by-die method. Also, when the absolute values of the 
nonlinear error vectors of the entire sample shots are equal to or less 
than the allowed value (allowed value in EGA) NL.sub.0, the alignment may 
be performed by the conventional EGA method. The allowed value NL.sub.0 
is, e.g., 0.1 .mu.m. 
FIG. 14A shows nonlinear error vectors of the respective exposure shots 
when the conventional EGA method is applied to the wafer W in FIG. 13A 
while FIG. 14B shows nonlinear error vectors of the respective exposure 
shots when the conventional weighting EGA method is applied thereto. As is 
apparent from the comparison of FIGS. 13B, 14A and 14B, the nonlinear 
error amounts become small on the entire surface of the wafer W according 
to this embodiment. 
The present invention is not limited to the above-described embodiments and 
is applicable to a scan type exposure apparatus disclosed in, e.g., U.S. 
Pat. No. 5,194,893, a proximity type or contact type exposure apparatus, a 
repair apparatus for repairing a chip pattern in an exposure shot, an 
inspecting apparatus for inspecting a chip pattern or the like. Also, the 
shot arrangement on the wafer, the number of sample shots, the positions 
of the sample shots, the method of determining the alternative shots, the 
largeness of the allowed values may be different from the above ones.