Alignment method

An alignment method for achieving accurate alignment by accurately eliminating an isolated area with a large nonlinear component of an alignment error from sample areas. A conversion parameter is calculated by statistical processing on the basis of a result obtained by measuring the position of each sample area on a substrate to be processed in advance, and each area on the substrate is aligned on the basis of arrangement coordinate values calculated using the conversion parameter. This method relates to a method of aligning each of a plurality of areas to be processed arranged on the substrate on the basis of arrangement coordinates on a first coordinate system (x, y) set on the substrate to a predetermined process position in a second coordinate system (X, Y) for defining the moving position of the substrate.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to an alignment method for an exposure 
apparatus for sequentially exposing a pattern on a reticle onto, e.g., 
shot areas on a wafer and, more particularly, to an alignment method which 
is suitably applied to a case wherein each shot area on a wafer is 
sequentially aligned to the exposure position on the basis of arrangement 
coordinate values calculated based on statistical processing. 
2. Related Background Art 
Upon manufacture of semiconductor elements, liquid crystal display 
elements, and the like by a photolithography process, a projection 
exposure apparatus, which projects a pattern image on a photomask or a 
reticle (a term "reticle" will be used hereinafter as an example) onto 
each of shot areas on a wafer, coated with a photosensitive material, via 
a projection optical system, is used. In recent years, as a projection 
exposure apparatus of this type, a so-called step-and-repeat type exposure 
apparatus, and more particularly, a reduction projection type exposure 
apparatus (stepper) are popularly used. In such an apparatus, a wafer is 
placed on a stage which is two-dimensionally movable, and an operation for 
sequentially exposing a pattern image on a reticle onto shot areas on a 
wafer is repeated by stepping the wafer using the stage. 
For example, since a semiconductor element is formed by repetitively 
exposing a large number of layers of circuit patterns on a wafer, 
alignment between a pattern image on a reticle and each shot area on which 
a circuit pattern has already been formed, i.e., alignment between the 
wafer and the reticle must be accurately performed when the second and 
subsequent circuit patterns are to be projected and exposed onto the 
wafer. The alignment of the wafer in the conventional stepper or the like 
is achieved by a method called enhanced global alignment (to be 
abbreviated as "EGA" hereinafter) (for example, see Japanese Patent 
Laid-Open No. 61-44429). 
SUMMARY OF THE INVENTION 
It is an object of the present invention to achieve accurate alignment by 
accurately eliminating an isolated shot with a large nonlinear component 
of an alignment error from sample shots in an alignment method wherein a 
conversion parameter is calculated by statistical processing on the basis 
of a result obtained by measuring the position of each sample shot on a 
wafer to be processed in advance, and each shot area on a wafer is aligned 
on the basis of arrangement coordinate values calculated using the 
conversion parameter. 
An alignment method according to the present invention relates to a method 
of aligning each of a plurality of areas to be processed arranged on a 
substrate on the basis of arrangement coordinates on a first coordinate 
system (x, y) set on the substrate to a predetermined process position in 
a second coordinate system (X, Y) for defining the moving position of the 
substrate. 
The method of the present invention comprises: the first step of measuring 
coordinate positions, on the second coordinate system (X, Y), of N (N is 
an integer not less than 2) sample areas selected in advance from a 
plurality of areas to be processed; the second step of calculating 
nonlinear components of deviation amounts of the coordinate positions of N 
sample areas from corresponding design positions on the basis of the 
coordinate positions measured in the first step, and calculating a 
variation E(N) of the N nonlinear components; and the third step of 
calculating nonlinear components of deviation amounts of the coordinate 
positions of (N-1) sample areas, obtained by excluding a predetermined 
sample area from the N sample areas, on the basis of the coordinate 
positions measured in the first step, and calculating a variation E(N-1, 
h) of these (N-1) nonlinear components. 
Furthermore, the method of the present invention comprises: the fourth step 
of selecting the (N-1) sample areas used in the third step as sample areas 
for calculation when the variation E(N-1, h) of the nonlinear components 
calculated in the third step is smaller than the variation E(N) of the 
nonlinear components calculated in the second step, or selecting the N 
sample areas as sample areas for calculation when the variation E(N-1, h) 
of the nonlinear components calculated in the third step is not less than 
the variation E(N) calculated in the second step; and the fifth step of 
calculating the arrangement coordinate values, on the second coordinate 
system (X, Y), of the plurality of areas to be processed on the substrate 
by executing statistical processing of the coordinate positions, measured 
in the first step, of the sample areas for calculation selected in the 
fourth step. 
In this case, if it is determined in the fourth step that the variation 
E(N-1, h) of the nonlinear components calculated in the third step is 
smaller than the variation E(N) of the nonlinear components calculated in 
the second step, it is preferable to repeat the second to fourth steps 
after replacing the N sample areas in the second step by the (N-1) sample 
areas used in the third step. 
In the third step, it is preferable to calculate the nonlinear components 
of the deviation amounts, from corresponding design positions, of the 
coordinate positions of N sets of (N-1) sample areas, obtained by 
sequentially excluding an i-th (i is an integer from 1 to N) sample area 
from the N sample areas, on the basis of the coordinate positions measured 
in the first step, calculate variations of the N sets of (N-1) nonlinear 
components, and obtain a minimum one of the variations of the N sets of 
(N-1) nonlinear components. 
Furthermore, the method of the present invention comprises: the step of 
repeating the second and third steps while sequentially excluding a 
predetermined sample area from the sample areas until the variation of the 
(N-1) nonlinear components calculated in the third step coincides with a 
predetermined expected value E.sub.0 within a predetermined tolerance 
range .epsilon.; and the step of calculating the coordinate positions, on 
the second coordinate system (X, Y), of the plurality of areas to be 
processed on the substrate by executing statistical processing of the 
coordinate positions, measured in the first step, of the sample areas 
which are left in the former step. 
In this case, an example of the predetermined expected value E.sub.0 is 
determined on the basis of a variation of measurement values of the 
coordinate positions, on the second coordinate system (X, Y), of these 
sample areas. 
The present invention will become more fully understood from the detailed 
description given hereinbelow and the accompanying drawings which are 
given by way of illustration only, and thus are not to be considered as 
limiting the present invention. 
Further scope of applicability of the present invention will become 
apparent from the detailed description given hereinafter. However, it 
should be understood that the detailed description and specific examples, 
while indicating preferred embodiments of the invention, are given by way 
of illustration only, since various changes and modifications within the 
spirit and scope of the invention will become apparent to those skilled in 
the art from this detailed description.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
On a wafer, a plurality of shot areas (chip patterns) with alignment marks 
called wafer marks are formed, and these shot areas are regularly arranged 
on the basis of arrangement coordinate values set in advance on the wafer. 
However, the wafer cannot always be accurately aligned due to the 
following factors even when the wafer is stepped on the basis of the 
design arrangement coordinate values (shot arrangement) of the plurality 
of shot areas on the wafer: 
(1) residual rotation error .theta. of wafer; 
(2) orthogonality error w of stage coordinate system (or shot arrangement); 
(3) linear expansion/contraction (scaling) parameters Rx and Ry of wafer; 
and 
(4) offset (translation) parameters Ox and Oy of wafer (central position). 
In this case, coordinate conversion of the wafer based on these four error 
amounts (six parameters) can be described by a first-order conversion 
equation. Thus, a first-order conversion model for converting a coordinate 
system (x, y) on a wafer on which a plurality of shot areas with wafer 
marks are regularly arranged into a coordinate system (X, Y) on a stage as 
a still coordinate system can be expressed as follows using six conversion 
parameters a to f: 
##EQU1## 
The six conversion parameters a to f in this conversion equation can be 
obtained by the EGA method as follows. First, some shot areas are selected 
from shot areas (chip patterns) as a plurality of objects to be exposed on 
a wafer. Design coordinate values, on a coordinate system (x, y), of the 
selected shot areas (to be referred to as "sample shots" hereinafter) are 
respectively represented by (x.sub.1, y.sub.1), (x.sub.2, y.sub.2), . . . 
, (x.sub.n, y.sub.n), and the wafer marks on these sample shots are 
aligned to predetermined reference or fiducial positions. Actual 
coordinate values (XM.sub.1, YM.sub.1), (XM.sub.2, YM.sub.2), . . . , 
(XM.sub.n, YM.sub.n) of the sample shots on a coordinate system (X, Y) on 
a stage at that time are measured. 
The difference (.DELTA.x, .DELTA.y) between a computational arrangement 
coordinate value (X.sub.i, Y.sub.i) (i=1, . . . , n) obtained by 
substituting the design arrangement coordinate value (x.sub.i, y.sub.i) of 
each of the selected wafer marks in the above-mentioned first-order 
conversion model, and a coordinate value (XM.sub.i, YM.sub.i) measured 
upon alignment is considered as an alignment error. One alignment error 
component .DELTA.x is expressed by, e.g., a sum of (X.sub.i 
-XM.sub.i).sup.2 in association with i, and the other alignment error 
component .DELTA.y is expressed by, e.g., a sum of (Y.sub.i 
-YM.sub.i).sup.2 in association with i. 
When equations for sequentially partially differentiating these alignment 
error components .DELTA.x and .DELTA.y by the six conversion parameters a 
to f to respectively yield 0 are set, and these six simultaneous equations 
are solved, the six conversion parameters a to f are obtained. 
Calculations for calculating the six conversion parameters a to f of 
equation (1) by the least square method in this manner are called EGA 
calculations. Thereafter, each shot area on the wafer can be aligned on 
the basis of the arrangement coordinate values calculated using the 
first-order conversion equation using the conversion parameters a to f as 
coefficients. Alternatively, when sufficient approximation accuracy cannot 
be obtained by the first-order conversion equation, wafer alignment may be 
achieved using higher-order (e.g., 2nd-order or higher) equations. 
In the above-mentioned EGA type alignment method, a plurality of sample 
shots often include a so-called isolated shot which has a particularly 
larger nonlinear component, obtained by subtracting a linear component 
from an alignment error, than those of other sample shots. Such an 
isolated shot is generated due to a measurement error caused by, e.g., 
collapse of wafer marks belonging to the sample shot on the wafer, a local 
nonlinear distortion on the wafer, an alignment error of a wafer stage 
upon transfer of a reticle pattern of the first layer onto the wafer, and 
the like. Therefore, when the arrangement coordinate values of other shot 
areas are to be calculated, alignment data (measured coordinate value) of 
such an isolated shot is preferably rejected. 
For this reason, an isolated shot is detected by methods 1 to 3 below, and 
the EGA type alignment is performed by rejecting the detected isolated 
shot. 
1 A shot area which has an alignment error equal to or larger than a 
predetermined reference value is determined to be an isolated shot. For 
example, FIG. 1A exaggeratingly shows an example of alignment errors of 
sample shots distributed on a wafer 41 as an object to be exposed. In FIG. 
1A, the design arrangement coordinate values of shot areas including 
sample shots are determined on a coordinate system (x, y) on the wafer 41. 
On the other hand, the coordinate values of eight sample shots SB.sub.1 to 
SB.sub.8 (more specifically, the coordinate values of wafer marks) on a 
stage coordinate system (X, Y) as a coordinate system of a wafer stage on 
which the wafer 41 is placed are measured. 
The alignment errors of the eight sample shots SB.sub.1 to SB.sub.8 are 
respectively expressed by vectors VB.sub.1 to VB.sub.8. For example, the 
start point of the vector VB.sub.1 represents the design central 
coordinate value, on the stage coordinate system (X, Y), of the sample 
shot SB.sub.1, and the end point of the vector VB.sub.1 represents the 
central coordinate value, measured on the stage coordinate system (X, Y), 
of the sample shot SB.sub.1. In this case, the design central coordinate 
values on the stage coordinate system (X, Y) are calculated by 
substituting estimated values of the six parameters a to f and the design 
values on the coordinate system on the wafer in equation (1). The 
estimated values of the six parameters a to f are obtained by so-called 
global alignment, i.e., by measuring the positions, on the stage 
coordinate system (X, Y), of two two-dimensional alignment marks on the 
wafer 41 while assuming linear expansion/contraction of the six parameters 
to be isotropic (Rx=Ry) and the orthogonality error w to be 0. 
FIG. 1B shows the absolute values .vertline.VB.sub.1 .vertline. to 
.vertline.VB.sub.8 .vertline. of the vectors of the alignment errors of 
the eight sample shots SB.sub.1 to SB.sub.8 shown in FIG. 1A. A sample 
shot which has an absolute value equal to or larger than a predetermined 
reference value VB, i.e., the second sample shot SB.sub.2, is rejected. 
2 The EGA calculations are performed to classify alignment errors into 
linear components and nonlinear components, and a sample shot having a 
nonlinear component equal to or larger than a predetermined reference 
value is rejected. 
FIG. 2A shows another example of the vectors VB.sub.1 to VB.sub.8 of the 
alignment errors of the eight sample shots SB.sub.1 to SB.sub.8 on the 
wafer 41, and FIG. 2B shows the absolute values (the absolute values of 
alignment errors) of the vectors VB.sub.1 to VB.sub.8 of these sample 
shots SB.sub.1 to SB.sub.8. In this case, the values of the six conversion 
parameters a to f which satisfy equation (1) in a least square method 
manner are calculated by the EGA calculations on the basis of the design 
arrangement coordinate values, on the coordinate system on the wafer 41, 
of the sample shots, and the measured coordinate values on the stage 
coordinate system. These six conversion parameters a to f and design 
arrangement coordinate values are substituted in equation (1) to calculate 
computational arrangement coordinate values except for linear errors, on 
the stage coordinate system, of the sample shots SB.sub.1 to SB.sub.8. A 
vector from a first computational arrangement coordinate position to a 
computational arrangement coordinate value except for its linear error is 
the vector of the linear component of the alignment error. 
When the vectors of the linear components are subtracted from the vectors 
of the alignment errors shown in FIG. 2A, vectors VBN.sub.1 to VBN.sub.8 
of nonlinear components are obtained in units of sample shots SB.sub.1 to 
SB.sub.8, as shown in FIG. 3A. FIG. 3B shows the absolute values 
.vertline.VBN.sub.1 .vertline. to .vertline.VBN.sub.8 .vertline. of the 
vectors of the nonlinear components of the alignment errors of the sample 
shots SB.sub.1 to SB.sub.8, and a sample shot which has a nonlinear 
component of an absolute value larger than a predetermined reference 
value, e.g., the eighth sample shot SB.sub.8, is rejected. 
3 Standard deviations of the absolute values of the vectors of alignment 
errors are calculated in units of sample shots on the wafer, and a sample 
shot in which the absolute value of the vector of the alignment error is 
equal to or larger than a predetermined multiple of the standard deviation 
is rejected. 
Of the above-mentioned techniques, in the method of rejecting a sample shot 
in which the absolute value of the vector of the alignment error is equal 
to or larger than a predetermined reference value like in 1, for example, 
in the example shown in FIG. 1A, even the vector VB.sub.8 which has a poor 
direction balance as a whole is not rejected since it has a small absolute 
value. When the rotation, orthogonality, or linear expansion/contraction 
(scaling) of the entire wafer 41 is particularly large, most of sample 
shots are rejected unless the predetermined reference value (a value 
corresponding to the reference value VB in FIG. 1B) is set to be 
considerably large, and accurate alignment is disabled. In addition, 
depending on the direction of the vector of the alignment error to be 
rejected, a nonlinear component may be emphasized. In this manner, a wrong 
sample shot may be rejected. 
In the method of correcting linear components by executing the EGA 
calculations and comparing the absolute values of the obtained nonlinear 
components with a predetermined reference value like in 2, a probability 
that a wrong sample shot is rejected is considerably lowered unlike in 1. 
However, since linear components calculated in method 2 are calculated 
using the coordinate value of an isolated shot to be rejected, it is 
considered that accurate linear components are not obtained. Therefore, 
the finally obtained absolute values (corresponding to the distribution 
shown in FIG. 3B) of the nonlinear components of the alignment error are 
inaccurate, and a wrong sample shot may be rejected, e.g., near the 
predetermined reference value. 
Even when a reference value for rejection is statistically set to be a 
variable value on the basis of the standard deviations of the absolute 
values of the alignment errors like in 3, since linear components are not 
subtracted, a wrong sample shot may be rejected like in 1. Furthermore, 
method 3 may be combined with method 2. Even by this method, since the 
alignment error of a sample shot to be rejected is included as a basis of 
the calculations, a wrong sample shot may be rejected, e.g., near the 
reference value as in method 2. 
An isolated shot to be rejected is reliably rejected by the following 
method. 
First, the coordinate positions, on a second coordinate system (stage 
coordinate system (X, Y)), of N sample areas (sample shots SA.sub.1 to 
SA.sub.N) selected on the wafer (substrate) 41 are measured, a variation 
(e.g., a predetermined multiple of a standard deviation) E(N) of nonlinear 
components of deviation amounts of the measured N coordinate positions 
from their design arrangement coordinate values is calculated. Thereafter, 
when a region which is expected to have a large distortion on the 
substrate 41 is known in, e.g., the manufacturing process completed so 
far, a sample area SA.sub.h in the region which is expected to have a 
large distortion is regarded as an isolated shot and is rejected. Using 
the measured coordinate positions of the remaining (N-1) sample areas, a 
variation E(N-1, h) (e.g., a predetermined multiple of a standard 
deviation) of nonlinear components of deviation amounts of these (N-1) 
measured coordinate positions from their design arrangement coordinate 
positions is calculated. 
Thereafter, when the variation E(N-1, h) of the (N-1) nonlinear components 
is smaller than the variation E(N) of the N nonlinear components, 
alignment is performed by, e.g., the EGA method using the measured 
coordinate positions of the (N-1) sample areas after the sample area 
SA.sub.h is rejected as an isolated shot. In this case, upon calculation 
of the variation E(N-1, h) of the (N-1) nonlinear components, since the 
measurement result of the sample area SA.sub.h as an isolated shot is not 
included, the variation of the nonlinear components can be accurately 
evaluated. Therefore, an isolated shot which has a large linear component 
of the deviation amount (alignment error) of the measured coordinate 
position from the design coordinate position can be accurately rejected, 
and alignment accuracy can be improved. 
In the above-mentioned fourth step, when the variation E(N-1, h) of the 
(N-1) nonlinear components calculated in the third step is smaller than 
the variation E(N) of the N nonlinear components calculated in the second 
step, a sample area SA.sub.f in a region which is expected to have a large 
distortion is further rejected as a second isolated shot from the (N-1) 
sample areas used in the third step. A variation E(N-2, f) of the 
nonlinear components of the deviation amounts of the measured coordinate 
positions of the obtained (N-2) sample areas is calculated. When the 
variation E(N-2, f) of the (N-2) nonlinear components is smaller than the 
variation E(N-1, h) of the (N-1) nonlinear components, the sample area 
SA.sub.f is rejected as an isolated shot. Thereafter, isolated shots are 
similarly rejected. Even when there are a plurality of isolated shots, all 
these isolated shots can be accurately rejected, and accurate alignment 
can be performed. 
In the third step, nonlinear components of deviation amounts of (N-1) 
sample areas obtained by sequentially excluding an i-th (i is an integer 
from 1 to N) sample area from the N sample areas are calculated on the 
basis of the coordinate positions measured in the first step, and 
variations E(N-1, i) of N sets of (N-1) nonlinear components from their 
design coordinate positions are calculated. When a minimum one of these 
variations of the N sets of nonlinear components is to be calculated, a 
sample area SA.sub.h, which has the largest nonlinear component, of these 
N sample areas, can be specified as an isolated shot. In addition, since 
calculations are made on the basis of the measurement results of the 
remaining samples obtained by rejecting an isolated shot, evaluation can 
be accurately achieved. 
Embodiment 1 
The first embodiment of the alignment method according to the present 
invention will be described below with reference to the accompanying 
drawings. In this embodiment, the present invention is applied to the 
alignment operation of an exposure apparatus (stepper) which exposes a 
pattern image on a reticle onto each shot area on a wafer as a 
photosensitive substrate by the step-and-repeat method. Note that the 
present invention can also be applied to a scanning exposure type exposure 
apparatus such as a step-and-scan type apparatus. 
FIG. 4 shows an exposure apparatus of this embodiment. Referring to FIG. 4, 
exposure light IL emerging from an illumination optical system 1 
illuminates a reticle 2 with an almost uniform illuminance. The reticle 2 
is held on a reticle stage 3, and the reticle stage 3 is movable and 
finely rotatable in a two-dimensional plane on a base 4. A main control 
system 6 for controlling the operation of the entire apparatus controls 
the operation of the reticle stage 3 via a driving device 5 on the base 4. 
Under the exposure light IL, a pattern image on the reticle 2 is projected 
onto each shot area on a wafer 8 via a projection optical system 7. The 
wafer 8 is placed on a wafer stage 10 via a wafer holder 9. A Z-axis is 
defined in a direction parallel to the optical axis of the projection 
optical system 7, and an orthogonal coordinate system of a two-dimensional 
plane perpendicular to the Z-axis is taken to define X- and Y-axes. The 
wafer stage 10 is constituted by an X-Y stage for two-dimensionally 
aligning the wafer 8 in a plane perpendicular to the optical axis of the 
projection optical system 7, a Z stage for aligning the wafer 8 in the 
Z-direction parallel to the optical axis of the projection optical system 
7, a stage for finely rotating the wafer 8, and the like. 
A movable mirror 11 is fixed on the upper surface of the wafer stage 10, 
and a laser interferometer 12 is arranged to oppose the movable mirror 11. 
Although simply illustrated in FIG. 4, the movable mirror 11 is 
constituted by a plane mirror which has a reflection surface perpendicular 
to the X-axis, and a plane mirror which has a reflection surface 
perpendicular to the Y-axis. The laser interferometer 12 is constituted by 
two X-axis laser interferometers for irradiating laser beams toward the 
movable mirror 11 along the X-axis, and a single Y-axis laser 
interferometer for irradiating a laser beam toward the movable mirror 11 
along the Y-axis. One X-axis laser interferometer and the Y-axis laser 
interferometer measure the X- and Y-coordinates of the wafer stage 10. The 
coordinate system (X, Y) defined by the X- and Y-coordinates measured in 
this manner will be referred to as a stage coordinate system or a still 
coordinate system hereinafter. 
The rotational angle of the wafer stage 10 is measured on the basis of the 
difference between the measurement values of the two X-axis laser 
interferometers. The X-coordinate information, Y-coordinate information, 
and rotational angle information measured by the laser interferometer 12 
are supplied to a coordinate measurement circuit 12a and the main control 
system 6. The main control system 6 controls the alignment operation of 
the wafer stage 10 via a driving device 13 while monitoring the supplied 
coordinates. Although not shown in FIG. 4, the same three-axis 
interferometer system as that at the wafer side is also provided to the 
reticle side. 
An imaging characteristic controller 14 is attached to the projection 
optical system 7 of this embodiment. The imaging characteristic controller 
14 adjusts the projection magnification and distortion of the projection 
optical system 7 by adjusting the interval between predetermined lens 
groups of those constituting the projection optical system 7 or by 
adjusting the pressure of a gas in a lens chamber between the 
predetermined lens groups. The operation of the imaging characteristic 
controller 14 is also controlled by the main control system 6. 
An off-axis type alignment system 15 which adopts an imaging processing 
method is arranged on the side surface of the projection optical system 7. 
In this alignment system 15, illumination light from a light source 16 is 
irradiated onto a position near an X-axis wafer mark (alignment mark) Mx 
on the wafer 8 via a collimator lens 17, a beam splitter 18, a mirror 19, 
and an objective lens 20. In this case, a base line amount as the interval 
between an optical axis 20a of the objective lens 20 and an optical axis 
7a of the projection optical system 7 is measured in advance. Light 
reflected by the wafer mark Mx is irradiated onto an index plate 22 via 
the objective lens 20, the mirror 19, the beam splitter 18, and a focusing 
lens 21, and an image of the wafer mark Mx is formed on the index plate 
22. 
The light transmitted through the index plate 22 propagates toward a beam 
splitter 24 via a first relay lens 23, and the light transmitted through 
the beam splitter 24 is focused on the imaging surface of an X-axis image 
pickup element 26X comprising a two-dimensional CCD by an X-axis second 
relay lens 25X. On the other hand, the light reflected by the beam 
splitter 24 is focused on the imaging surface of a Y-axis image pickup 
element 26Y comprising a two-dimensional CCD by a Y-axis second relay lens 
25Y. On the imaging surfaces of the image pickup elements 26X and 26Y, the 
image of the wafer mark Mx and the images of index marks on the index 
plate 22 are formed to overlap each other. The image pickup signals from 
the image pickup elements 26X and 26Y are supplied to the coordinate 
measurement circuit 12a. 
FIG. 5 shows a pattern on the index plate 22 shown in FIG. 4. Referring to 
FIG. 5, an image MxP of the wafer mark Mx consisting of three linear 
patterns are formed at the central portion. An XP direction as the pitch 
direction of the image MxP and a YP direction as the longitudinal 
direction of the image MxP are respectively conjugate with the X- and 
Y-directions of the stage coordinate system of the wafer stage 10 shown in 
FIG. 4. Two index marks 31A and 31B are formed to sandwich the wafer mark 
image MxP therebetween along the XP direction, and two index marks 32A and 
32B are formed to sandwich the wafer mark image MxP therebetween along the 
YP direction. 
In this case, images in a detection region 33X which surrounds the index 
marks 31A and 31B and the wafer mark image MxP in the XP direction are 
picked up by the X-axis image pickup element 26X in FIG. 4, and images in 
a detection region 33Y which surrounds the index marks 32A and 32B, and an 
image of a Y-axis wafer mark (not shown; a pattern obtained by rotating 
the X-axis wafer mark Mx through 90.degree.) in the YP direction are 
picked up by the Y-axis image pickup element 26Y in FIG. 4. The scanning 
directions upon reading of photoelectric conversion signals from the 
pixels of the image pickup elements 26X and 26Y are respectively set in 
the XP and YP directions. Thus, by processing the image pickup signals 
from the image pickup elements 26X and 26Y, the positional deviation 
amount, in the XP direction, between the X-axis wafer mark image MxP and 
the index marks 31A and 31B, and the positional deviation amount, in the 
YP direction, between the Y-axis wafer mark image and the index marks 32A 
and 32B can be obtained. Therefore, in FIG. 4, the coordinate measurement 
circuit 12a measures the X-coordinate, on the stage coordinate system (X, 
Y), of the wafer mark Mx on the basis of the positional relationship 
between the image of the wafer mark Mx on the wafer 8 and the index marks 
on the index plate 22a and the measurement results from the laser 
interferometer 12 at that time, and supplies the measured X-coordinate to 
the main control system 6. Similarly, the Y-coordinate, on the stage 
coordinate system (X, Y), of the Y-axis wafer mark is also measured, and 
is supplied to the main control system 6. 
The operation executed when each of shot areas on the wafer to be exposed 
is aligned, and a pattern image on the reticle 2 is exposed on each shot 
area in this embodiment will be described below with reference to the flow 
chart in FIG. 6. The wafer 8 to be exposed is loaded onto the wafer holder 
9 shown in FIG. 4. 
FIG. 7 shows the arrangement of shot areas on the wafer 8. Referring to 
FIG. 7, shot areas ES.sub.1, ES.sub.2, . . . , ES.sub.M (M is an integer 
equal to or larger than 3) are regularly formed on the wafer 8 along the 
coordinate system (x, y) set on this wafer, and chip patterns are 
respectively formed on the respective shot areas ES.sub.i (i=1 to M) in 
the process completed so far. The shot areas ES.sub.i are separated by 
street lines with a predetermined width in the x- and y-directions. X-axis 
wafer marks Mx.sub.i are formed at the central portions of the street 
lines which contact the shot areas ES.sub.i and extend in the x-direction, 
and Y-axis wafer marks My.sub.i are formed at the central portions of the 
street lines which contact the shot areas ES.sub.i and extend in the 
y-direction. 
Each of the X- and Y-axis wafer marks Mx.sub.i and My.sub.i is defined by 
arranging three linear patterns at a predetermined pitch in a 
corresponding one of the x- and y-directions. These patterns are formed as 
recess or projecting patterns on the undercoat of the wafer 8. An 
x-coordinate (design coordinate value) x.sub.i of each wafer mark Mx.sub.i 
on the coordinate system (x, y) on the wafer 8, and a y-coordinate (design 
coordinate value) y.sub.i of each wafer mark My.sub.i are known, and are 
stored in a storage unit in the main control system 6 shown in FIG. 4. In 
this case, the x-coordinate of each wafer mark Mx.sub.i and the 
y-coordinate of each wafer mark My.sub.i are respectively regarded as the 
x- and y-coordinates of a corresponding shot area ES.sub.i. 
Also, two two-dimensional global alignment marks (not shown) for rough 
alignment (global alignment) are formed on the wafer 8, and the coordinate 
values of these two global alignment marks on the coordinate system (x, y) 
on the wafer 8 are known. After the number of unknown conversion 
parameters of equation (1) is reduced to four by assuming linear 
expansion/contraction to be isotropic (Rx=Ry) and the orthogonality error 
w of the stage coordinate system to be 0, the main control system 6 shown 
in FIG. 4 measures the coordinate values, on the stage coordinate system 
(X, Y), of the two two-dimensional global alignment marks on the wafer 8 
via the alignment system 15. Based on the measurement results, the values 
of the simplified four conversion parameters in equation (1) are 
determined. 
Thereafter, the main control system 6 calculates an initial value of a 
computational X-coordinate of each wafer mark Mx.sub.i and an initial 
value of a computational Y-coordinate of each wafer mark My.sub.i on the 
stage coordinate system (X, Y) by substituting these four conversion 
parameters, the design x-coordinate of each wafer mark Mx.sub.i, and the 
design y-coordinate of each wafer mark My.sub.i in equation (1). At the 
same time, the main control system 6 calculates an initial value of a 
computational Y-coordinate of the center of each wafer mark Mx.sub.i and 
an initial value of a computational X-coordinate of the center of each 
wafer mark My.sub.i. By driving the wafer stage 10 on the basis of the 
initial values of the design coordinate values on the stage coordinate 
system (X, Y), the wafer marks Mx.sub.i and My.sub.i are sequentially 
driven into an observation field of the alignment system 15, and their 
accurate coordinate values are measured. 
Finally, in this embodiment as well, coordinate conversion from the 
coordinate system (x, y) on the wafer 8 into the stage coordinate system 
(X, Y) is expressed by equation (1) using the six conversion parameters a 
to f. These six conversion parameters are determined as follows. 
In step 101 in FIG. 6, N (in FIG. 7, N=9) sample shots SA.sub.1 to SA.sub.N 
are selected in an arbitrary arrangement from all shot areas ES.sub.1 to 
ES.sub.M on the wafer 8, as shown in FIG. 7, and the coordinate values, on 
the stage coordinate system (X, Y), of the sample shots SA.sub.i (i=1 to 
N) are measured via the alignment system 15 shown in FIG. 4. Measuring the 
coordinate value of each sample shot SA.sub.i is to measure the coordinate 
values, on the stage coordinate system (X, Y), of the X- and Y-axis wafer 
marks of the sample shot SA.sub.i. The coordinate value of each sample 
shot SA.sub.i measured on the stage coordinate system (X, Y) is 
represented by (XM.sub.i, YM.sub.i). In this case, an initial value 
(design value) of the computational coordinate value on the stage 
coordinate system (X, Y), which initial value is used for driving each 
sample shot SA.sub.i into the observation field of the alignment system 
15, is represented by (X.sub.0i, Y.sub.0i). 
In step 102, nonlinear components of alignment errors in units of N sample 
shots SA.sub.i are obtained. The alignment error is an error between the 
computational coordinate value and the actually measured coordinate value. 
The alignment error of each sample shot SA.sub.i is numerically expressed 
by a vector V.sub.i which has the initial value (X.sub.0i, Y.sub.0i) of 
the computational coordinate value as a start point, and the measured 
coordinate value (XM.sub.i, YM.sub.i) as an end point. 
FIG. 8A exaggeratingly shows vectors V.sub.1 to V.sub.N of the alignment 
errors of the N sample shots SA.sub.1 to SA.sub.N on the wafer 8. 
Referring to FIG. 8A, for example, a start point 34 of the vector V.sub.1 
of the shot area SA.sub.1 corresponds to the initial value of the 
computational coordinate value, and an end point 35 corresponds to the 
measured coordinate value. In this case, each vector V.sub.i is expressed 
by a sum of a linear portion VL.sub.i and a nonlinear portion VN.sub.i. 
Since the nonlinear portion VN.sub.i corresponds to a component generated 
due to the measurement error, a local distortion on the wafer 8, or the 
like, the nonlinear component is obtained as follows. 
First, the main control system 6 calculates the values of the six 
conversion parameters a to f which satisfy equation (1) using a simple 
least square method on the basis of the design coordinate values and 
measured coordinate values of the N sample shots (in other words, their 
wafer marks). Such calculations are called EGA calculations. In the EGA 
calculations, when the coordinate value of an n-th sample shot SA.sub.n 
measured on the stage coordinate system is represented by (XM.sub.n, 
YM.sub.n), and the coordinate value calculated by substituting the design 
coordinate value and the six conversion parameters in equation (1) is 
represented by (X.sub.n, Y.sub.n), the residual error component is given 
by the following formula: 
##EQU2## 
Note that the value of m is 9 in the example shown in FIG. 8A. 
The values of the conversion parameters a to f in equation (1) are 
determined to minimize the residual error component. Then, the main 
control system 6 calculates vectors by subtracting computational 
arrangement coordinate values (X.sub.i, Y.sub.i) calculated using the 
conversion parameters a to f obtained as described above from the 
coordinate values (XM.sub.i, YM.sub.i) of the sample shots SA.sub.i. 
Furthermore, the main control system 6 calculates the absolute values of 
these vectors to obtain nonlinear components of the alignment components 
of the N sample shots SA.sub.1 to SA.sub.N. More specifically, the 
nonlinear component is the absolute value of the nonlinear portion 
VN.sub.i of the vector V.sub.i which represents the alignment error. 
Thereafter, the main control system 6 calculates a value E(N) three times 
the standard .sigma. deviation of the nonlinear components of the 
alignment errors of the N sample shots. The value E(N) is shown in the 
left end portion in FIG. 8B. 
In step 103, the main control system 6 performs the above-mentioned EGA 
calculations of the (N-1) sample shots SA.sub.2 to SA.sub.N obtained by 
rejecting the first sample shot SA.sub.1 from the N sample shots SA.sub.1 
to SA.sub.N to obtain the six conversion parameters of equation (1). More 
specifically, the main control system 6 determines the values of the six 
conversion parameters a to f of equation (1), which minimize the residual 
error component as a square sum of the difference between the coordinate 
values (XM.sub.n, YM.sub.n), measured on the stage coordinate system, of 
the (N-1) sample shots SA.sub.2 to SA.sub.N, and the design coordinate 
values (X.sub.n, Y.sub.n) based on equation (1). Then, the main control 
system 6 calculates the nonlinear components of the alignment errors by 
subtracting the computational coordinate values from the coordinate values 
measured on the stage coordinate system (X, Y) in units of (N-1) sample 
shots SA.sub.2 to S.sub.AN, and calculates a 3.sigma. value E(N-1, 1) of 
these (N-1) nonlinear components. 
Similarly, the main control system 6 performs the EGA calculations of (N-1) 
sample shots obtained by sequentially rejecting the second, third, . . . , 
N-th sample shots from the N sample shots SA.sub.1 to SA.sub.N in FIG. 8A 
to obtain the nonlinear components of the alignment errors, and then 
calculates 3.sigma. values of these (N-1) nonlinear components. The 
3.sigma. value of the nonlinear components of the alignment errors of the 
(N-1) sample shots obtained when the i-th sample shot SA.sub.i (i=1 to N) 
is rejected is represented by E(N-1, i). 
FIG. 8B shows these N 3.sigma. values E(N-1, 1) to E(N-1, N). As shown in 
FIG. 8B, a minimum value of the 3.sigma. values E(N-1, 1) to E(N-1, N) is 
E(N-1, h), i.e., the variation of the nonlinear components of the 
alignment errors obtained when the h-th sample shot SA.sub.h is rejected. 
When the 3.sigma. values E(N-1, i) include a plurality of minimum values, 
a value corresponding to a rejected sample shot having the smallest number 
is adopted. The flow then advances to step 104, and the main control 
system 6 compares the minimum value E(N-1, h) of the 3.sigma. values 
E(N-1, i) with the 3.sigma. value E(N) calculated in step 102. If E(N) is 
larger than E(N-1, h), i.e., E(N)&gt;E(N-1, h) holds, the flow advances to 
step 105 to set an initial value of a variable j to be 1. Thereafter, in 
step 106, the remaining (N-j) sample shots (in this case, (N-1) sample 
shots) after the h-th sample shot SA.sub.h is rejected is defined as a new 
set of sample shots, as shown in FIG. 9A, and the flow advances to step 
107. 
This means to redo the EGA type alignment by rejecting the sample shot 
SA.sub.h with a particularly large nonlinear component of the alignment 
error, i.e., an isolated shot. Thus, alignment can be performed with high 
accuracy by rejecting an isolated shot as a sample shot causing a 
measurement error or a sample shot in a region with a large distortion. In 
addition, the 3.sigma. value E(N-1, h) used for confirming whether or not 
the h-th sample shot is to be rejected is calculated without the 
measurement value of the h-th sample shot. For this reason, a sample shot 
to be rejected can be accurately determined. 
On the other hand, if it is determined in step 104 that the 3.sigma. value 
E(N) is equal to or smaller than the 3.sigma. value E(N-1, h), since the 
nonlinear component does not decrease even when the sample shot is 
rejected, it is considered that a sample shot with a particularly large 
nonlinear component is not present. For this reason, the flow advances to 
step 112, and the EGA type alignment is performed using the initial N 
sample shots SA.sub.1 to SA.sub.N to expose a pattern image on the reticle 
2 onto each of the shot areas ES.sub.1 to ES.sub.M on the wafer 8. More 
specifically, the values of the six conversion parameters a to f of 
equation (1) are determined by the least square method using the 
measurement values of the initial N sample shots SA.sub.1 to SA.sub.N, and 
the arrangement coordinate values, on the stage coordinate system (X, Y), 
of these shot areas are calculated on the basis of these six conversion 
parameters, and the design arrangement coordinate values of the shot areas 
ES.sub.1 to ES.sub.M. Based on the calculated arrangement coordinate 
values, these shot areas are aligned. Thereafter, the flow advances to 
step 113 to perform exposure of the next wafer. 
Reverting to step 107, it is checked in step 107 if the number (N-j) (in 
this case, (N-1)) of the remaining sample shots is larger than a 
predetermined minimum value N.sub.min (N.sub.min is an integer equal to or 
larger than 1). If the number (N-j) is larger than the minimum value 
N.sub.min, the flow advances to step 108. If the number (N-j) is decreased 
to the minimum value N.sub.min, the flow advances to step 112, and the EGA 
type alignment is performed using the remaining (N-j) sample shots to 
achieve exposure. 
In step 108, the 3.sigma. value E(N-j, h) of the nonlinear components after 
the h-th sample shot is rejected, which value is obtained in step 103 or 
109 (to be described later), is defined as a 3.sigma. value E(N-j) of the 
nonlinear components of the remaining (N-j) (in this case, (N-1)) sample 
shots. The value E(N-j) (when j=1) is shown at the left end portion in 
FIG. 9B. Thereafter, the flow advances to step 109, and the EGA 
calculations are performed for (N-j-1) sample shots obtained by 
sequentially rejecting the first, second, . . . , and (N-j)-th sample 
shots from the remaining (N-j) sample shots to obtain the nonlinear 
components of the alignment error, and the 3.sigma. values of these 
(N-j-1) nonlinear components are calculated. Note that the 3.sigma. value 
of the nonlinear components of the alignment errors of the (N-j-1) sample 
shots obtained when an i-th (i=1 to N-j) sample shot is rejected from the 
(N-j) sample shots is represented by E(N-j-1, i). 
FIG. 9B shows the (N-j-1) 3.sigma. values E(N-2, 1) to E(N-2, N) obtained 
when j=1. In FIG. 9B, the order of sample shots to be rejected is 
expressed using that of the initial N sample shots. As shown in FIG. 9B, a 
minimum value of the 3.sigma. values E(N-2, 1) to E(N-2, N) is E(N-2, f), 
i.e., the variation of the nonlinear components of the alignment errors 
obtained when the f-th sample shot SA.sub.f is rejected. In this case as 
well, when these 3.sigma. values E(N-2, i) include a plurality of minimum 
values, a value corresponding to a rejected sample shot having the 
smallest number is adopted. The f-th sample shot is assumed to be the h-th 
sample shot of the (N-j) sample shots. 
The flow then advances to step 110, and the main control system 6 compares 
the minimum value (N-j-1, h) of the 3.sigma. values E(N-j-1, i) (in this 
case, E(N-2, i)) with the 3.sigma. value E(N-j) set in step 108. If E(N-j) 
is larger than E(N-j-1, h), i.e., E(N-j)&gt;E(N-j-1, h) is satisfied, the 
value of the variable j is incremented by one in step 111. Thereafter, the 
flow returns to step 106 to define the remaining (N-j) sample shots 
((N-j-1) sample shots before increment of the variable j) obtained after 
the h-th sample shot is rejected, as a new set of sample shots, and the 
flow advances to step 107. 
On the other hand, if it is determined in step 110 that the 3.sigma. value 
E(N-j) is equal to or smaller than the 3.sigma. value E(N-j-1, h), since 
the nonlinear component does not decrease even by rejecting a sample shot, 
it is considered that a sample shot with a particularly large nonlinear 
component is not present. Thus, the flow advances to step 112, and the EGA 
type alignment is performed using one set of remaining (N-j) sample shots. 
Thereafter, a pattern image on the reticle 2 is exposed onto each of the 
shot areas ES.sub.1 to ES.sub.M on the wafer 8. 
In this embodiment, steps 111 and 106 to 110 are repeated until the number 
of remaining sample shots decreases to N.sub.min in step 107 or until all 
the (N-j) values E(N-j-1, i) as the 3.sigma. values of the nonlinear 
components obtained by rejecting a corresponding one sample shot become 
equal to or larger than E(N-j) in step 110. With this processing, since 
all isolated shots with particularly large nonlinear components of 
alignment errors due to the measurement error of the alignment system 15 
or a local distortion on the wafer 8 are finally rejected from the N 
sample shots SA.sub.1 to SA.sub.N shown in FIG. 7, alignment can be 
performed with higher accuracy. When a plurality of (about 25) wafers in a 
single lot are to be continuously subjected to exposure processing, 
isolated shots may be rejected from only the first wafer or only several 
wafers from the first wafer, and the EGA method may be applied to the 
subsequent wafers in accordance with the sample shot arrangement from 
which the isolated shots are rejected. 
In the above-mentioned embodiment, as shown in FIG. 7, the positions of the 
X- and Y-axis wafer marks of the sample shots SA.sub.1 to SA.sub.N are 
measured. However, the positions of both the X- and Y-axis wafer marks of 
each sample shots need not always be measured. For example, the number of 
sample shots may be doubled, so that the positions of only the X-axis 
wafer marks of odd-numbered sample shots SA.sub.1, SA.sub.3, . . . may be 
measured, and the positions of only the Y-axis wafer marks of the 
even-numbered sample shots SA.sub.2, SA.sub.4, . . . may be measured. In 
this case, the magnitudes of the nonlinear components of the alignment 
errors may be obtained in units of wafer marks, and a wafer mark with a 
large nonlinear component may be rejected as an isolated mark. 
Even when the number of sample shots after isolated shots are rejected is 
equal to or larger than N.sub.min, if it is expected that the EGA accuracy 
may deteriorate due to a small number of sample shots, at least one shot 
area may be designated as a sample shot from shot areas except for the 
previously selected sample shots to increase the number of sample shots. 
Furthermore, step 107 is provided in FIG. 6, and if it is determined in 
step 107 that the number of remaining sample shots has reached N.sub.min, 
the flow advances to step 112 to perform alignment and exposure. However, 
since the number of isolated shots with large nonlinear components is 
relatively smaller than that of the initial number of all sample shots, 
the flow normally advances from step 104 or 110 to step 112 before the 
number of sample shots decreases to N.sub.min in step 107. Therefore, step 
107 need not always be provided. 
The number of initially designated sample shots need only be equal to or 
larger than the minimum value N.sub.min, and be equal to or smaller than 
the number of all shots on the wafer. Furthermore, since the minimum value 
N.sub.min is determined in accordance with the number of parameters 
included in the model equation (corresponding to equation (1) of this 
embodiment) corresponding to the regularity of the shot arrangement on the 
wafer, and this embodiment requires at least each three X and Y marks, 
N.sub.min =3. The model equation is not limited to equation (1), and may 
have any other equations as long as they are set by appropriately 
determining parameters required for expressing the shot arrangement on the 
wafer. Furthermore, all the shot areas on the wafer may be designated as 
sample shots, and after isolated shots are rejected like in this 
embodiment, the sample shot arrangement (the number and positions of 
sample shots) may be determined on the basis of the remaining shot areas 
after all the isolated shots are rejected. In this case, a problem of low 
calculation accuracy of the EGA calculations due to a decrease in the 
number of alignment data (measured coordinate values) of sample shots used 
in the EGA method can be solved. In particular, when the sample shot 
arrangement is optimized in the first wafer of a lot, calculations for 
rejecting isolated shots need not be performed in the second and 
subsequent wafers unlike in this embodiment, and the processing time 
required for one lot can be shortened. 
Although the calculation amount of this embodiment is considerably larger 
than that of the normal EGA method, since the computation performance of 
recent computers has remarkably improved, the wait time from measurement 
of the coordinate values of the wafer marks to exposure can be shortened 
to a negligible time even when the calculations of this embodiment are 
performed. 
Embodiment 2 
The second embodiment of the present invention will be described below with 
reference to FIG. 10. In this embodiment, when a local distortion is 
generated on the wafer 8, alignment is performed by a weighted EGA method 
as an improved EGA method. More specifically, although the basic operation 
of this embodiment is the same as that in the flow chart shown in FIG. 6, 
the operations in steps 102 to 112 in FIG. 6 are executed in units of shot 
areas ES.sub.i (i=1 to M) on the wafer 8 in FIG. 7 to determine a sample 
shot to be rejected, and alignment is performed using the measurement 
results of the remaining sample shots. In place of the EGA calculations in 
steps 102, 103, 109, and 112, the following weighted EGA calculations are 
performed. 
For example, as shown in FIG. 10, when exposure onto an arbitrary shot area 
ES.sub.i on the wafer 8 is to be performed, the coordinate values, on the 
stage coordinate system (X, Y), of nine sample shots SA.sub.1 to SA.sub.9 
are initially measured. At this time, in the step corresponding to step 
102 in FIG. 6, a weight W.sub.in is assigned to an n-th (n=1 to 9) sample 
shot of the nine sample shots. 
Using the measured coordinate values (XM.sub.n, YM.sub.n) of the nine 
sample shots, the computational coordinate values (X.sub.n, Y.sub.n) based 
on equation (1), and the weight W.sub.in, the residual error component Ei 
of the shot area ES.sub.i is defined as follows: 
##EQU3## 
Note that the value of m is 9 in the above equation. 
The values of the conversion parameters a to f are determined, so that the 
residual error component Ei assumes a minimum value. Then, the 
computational arrangement coordinate values of the nine sample shots are 
calculated by substituting these conversion parameters a to f and the 
design arrangement coordinate values of the nine sample shots in equation 
(1). The nonlinear components (strictly speaking, nonlinear components 
other than a distortion around the shot area ES.sub.i) of the alignment 
errors of the sample shots are calculated, and a 3.sigma. value E(N) (N=9) 
of these nonlinear component is calculated. Thereafter, the residual error 
component given by equation (3) is similarly used in place of equation (2) 
unlike in the first embodiment. With this processing, alignment can be 
accurately performed in consideration of the influence of a local 
distortion on the wafer 8, and a sample shot having a nonlinear component 
which is not included in the local distortion (e.g., a measurement error) 
can be accurately rejected as an isolated shot. 
A method of optimizing the weight W.sub.in will be explained below. As an 
example, in this embodiment, when the distance from the shot area ES.sub.i 
to the n-th sample shot SA.sub.n is represented by LK.sub.n, as shown in 
FIG. 10, the weight W.sub.in to be given to the n-th sample shot whose 
measurement result is utilized is determined as follows: 
##EQU4## 
Note that a parameter S.sub.i is a parameter for changing the degree of 
weighting. 
As can be seen from this equation, as the sample shot has a shorter 
distance LK.sub.n to the shot area ES.sub.i, the weight W.sub.in to be 
given to its measurement result becomes larger. In equation (4), when the 
value of the parameter S.sub.i increases, the obtained result becomes 
approximate to that obtained by the normal EGA method; when the value of 
the parameter S.sub.i decreases, the obtained result becomes approximate 
to that obtained by a die by die method. In this embodiment, the parameter 
S.sub.i is set as defined by equation (5) below. In the following 
equation, D is the weighting parameter. When an operator sets the value of 
the weighting parameter D to be a predetermined value, the parameter 
S.sub.i, and hence, the weight W.sub.in are automatically determined. 
EQU S.sub.i =D.sup.2 /(8.multidot.log.sub.e 10) (5) 
The physical meaning of this weighting parameter D is the range of sample 
shots (to be simply referred to as "zone" hereinafter) effective for 
calculating the coordinate position of each shot area on the wafer. More 
specifically, when the zone is wide, since the number of effective sample 
shots is large, the obtained result becomes approximate to that obtained 
by the normal EGA method. On the other hand, when the zone is narrow, 
since the number of effective sample shots is small, the obtained result 
becomes approximate to that obtained by the die by die method. 
In the above-mentioned second embodiment, the weight W.sub.in is determined 
based on equation (4) using the parameter S.sub.i. Alternatively, a weight 
W.sub.in ' calculated by equation (6) below using the parameter S.sub.i 
may be used. In this case, as shown in FIG. 11, the modification central 
point (e.g., the center of point symmetry of a nonlinear distortion) of 
the wafer 8 is defined as a wafer center W.sub.C, the distance (radius) 
between the wafer center W.sub.C and the shot area ES.sub.i on the wafer 8 
is represented by LE.sub.i, and the distances (radii) between the wafer 
center W.sub.C and m (in FIG. 11, m=9) sample shots SA.sub.1 to SA.sub.9, 
are respectively represented by LW.sub.1 to SW.sub.9. The weight W.sub.in 
' defined by equation (6) below using the distance LE.sub.i and the 
distances LW.sub.n is given to the measurement result of the n-th sample 
shot SA.sub.n. 
##EQU5## 
Then, the weight W.sub.in in equation (3) for calculating the residual 
error component is replaced by this weight W.sub.in ', and the conversion 
parameters of equation (1) are calculated by the weighted EGA method. In 
this case, even when the center of an almost point symmetrical distortion 
is present on the wafer 8, only a sample shot having a nonlinear component 
other than the point symmetrical distortion (e.g., a measurement error) 
can be accurately rejected, and alignment can be achieved with high 
accuracy. 
In the above-mentioned embodiment, the present invention is applied to 
alignment of a wafer in the exposure apparatus. However, the present 
invention may be applied to, e.g., so-called vernier evaluation of the 
exposure apparatus. In this vernier evaluation, after a first measurement 
mark is exposed on each of a large number of shot areas on a wafer, a 
second measurement mark is exposed to overlap the first measurement mark 
in the second exposure, and after development, the deviation amount 
between the first and second measurement marks on a selected shot area 
(sample shot) on the wafer is measured. Then, the characteristics such as 
the regularity (registration) of the arrangement of shot areas are 
evaluated on the basis of the deviation amount. In this case, a sample 
shot with a larger nonlinear component is rejected based on the 
measurement results of initially selected sample shots according to the 
present invention, thus allowing accurate evaluation of various 
characteristics. 
On the other hand, when the variation E(N-1, h) of the (N-1) nonlinear 
components is smaller than the variation E(N) of the N nonlinear 
components, the corresponding sample shot SA.sub.h may be rejected as an 
isolated shot, and when the variation E(N-1, h) of the (N-1) nonlinear 
components after the isolated shot is rejected has a value near a 
predetermined expected value E.sub.0, alignment may be performed by, e.g., 
the EGA method using the measured coordinate positions of the remaining 
(N-1) sample areas. 
When the variation E(N-1, h) of the (N-1) nonlinear components calculated 
in the above-mentioned third step does not have a value near the 
predetermined expected value E.sub.0, the remaining (N-1) sample areas may 
be defined as a new set of sample areas, and the same operations as in the 
second and third steps may be repeated. When the variation of the 
nonlinear components of the sample areas after an isolated shot is 
rejected has reached a value near the predetermined expected value 
E.sub.0, alignment is performed by, e.g., the EGA method using the 
remaining sample areas. 
On the other hand, when the second and third steps are repeated within a 
range wherein the number of sample areas other than that rejected in the 
third step is equal to or larger than a predetermined allowable value 
N.sub.min, the number of sample areas can be prevented from decreasing too 
much to destroy an averaging effect. 
Embodiment 3 
The third embodiment of the present invention will be described below with 
reference to FIG. 12. 
In step 116 in FIG. 12, an expected value E.sub.0 (to be described later) 
of nonlinear components and a variation .epsilon. of this expected value 
are determined. In this embodiment, assuming that the repetitive movement 
accuracy (reproducibility of alignment accuracy) of the wafer stage 10 in 
FIG. 4 is represented by .sigma..sub.step and the measurement accuracy 
(measurement reproducibility) of the alignment system 15 is represented by 
.sigma..sub.sens, the expected value E.sub.0 is expressed by the square 
root of a sum of the squares of .sigma..sub.step and .sigma..sub.sens. 
EQU E.sub.0 =(.sigma..sub.step.sup.2 +.sigma..sub.step.sup.2).sup.1/2(7) 
The variation .epsilon. is set to be, e.g., a fraction of the expected 
value E.sub.0. 
Then, steps 101 to 106 are executed. Since these steps are the same as 
those shown in FIG. 6, a detailed description thereof will be omitted. 
In step 117, it is checked for all the (N-j+1) (in this case, N) 3.sigma. 
values E(N-j, i) (i=1 to N-j+1) calculated in the same step 103 or 109 as 
that described above if the following relation is satisfied: 
EQU .vertline.E(N-j, i)-E.sub.0 .vertline.&lt;.epsilon. (8) 
More specifically, it is checked if all the 3.sigma. values E(n-j, i) 
coincide with the expected value E.sub.0 within the range of the variation 
.epsilon.. If inequality (8) is satisfied for all the 3.sigma. values 
E(n-j, i), since it is considered that all isolated shots with larger 
nonlinear components are rejected, the flow advances to step 112 as in the 
above-mentioned operation. Then, alignment is performed by the EGA method 
using the remaining (N-j) sample shots to perform exposure. On the other 
hand, if at least one of the 3.sigma. values E(n-j, i) does not satisfy 
inequality (8), since the remaining sample shots still include an isolated 
shot with a larger nonlinear component, the flow returns to step 107 as in 
the above-mentioned operation. 
Since steps 107 to 113 are the same as those shown in FIG. 6, a detailed 
description thereof will be omitted. 
In this embodiment, steps 111, 106, 117, and 107 to 110 are repeated until 
all the 3.sigma. values E(n-j, i) coincide with the expected value E.sub.0 
with the range of the variation .epsilon. in step 117, until the number of 
remaining sample shots decreases to N.sub.min in step 107, or until all 
the (N-j) values E(N-j-1, i) as the 3.sigma. values of the nonlinear 
components respectively obtained by rejecting one sample shot become equal 
to or larger than E(N-j) in step 110. With this processing, since all 
isolated shots with particularly large nonlinear components of alignment 
errors due to the measurement error of the alignment system 15 or a local 
distortion on the wafer 8 are finally rejected from the N sample shots 
SA.sub.1 to SA.sub.N shown in FIG. 7, alignment can be performed with 
higher accuracy. 
The expected value E.sub.0 is not limited to that given by equation (7). 
For example, .sigma..sub.step and .sigma..sub.sens may be multiplied with 
weights and, then, the products may be averaged. Also, one of 
.sigma..sub.step and .sigma..sub.sens may be used upon determination of 
the expected value E.sub.0, and error factors other than .sigma..sub.step 
and .sigma..sub.sens may be taken into consideration. To summarize, the 
expected value E.sub.0 can be determined based on factors 
(.sigma..sub.step, .sigma..sub.sens, and the like) other than a true 
nonlinear error of an isolated shot included in the above-mentioned 
nonlinear component. In addition, the variation .epsilon. may be 
determined in correspondence with required alignment accuracy or the like. 
In the above-mentioned embodiment, the minimum value N.sub.min as the 
number of finally remaining sample shots is determined. Instead, a maximum 
value N.sub.max of the number of sample shots to be rejected as isolated 
shots may be determined, and when the number of isolated shots has reached 
N.sub.max, alignment may be performed by the EGA method using the 
remaining sample shots. 
As the expected value of the 3.sigma. value of nonlinear components in the 
case of the vernier evaluation described above, the square root of a sum 
of the squares of .sigma.1.sub.step and .sigma.2.sub.step as repetitive 
movement accuracy (reproducibility of alignment precision) of the wafer 
stage of the exposure apparatus used in the first and second exposure 
processes, and .sigma..sub.sens as measurement precision of the sensor for 
measuring the positional deviation amount between the first and second 
measurement marks may be used. 
From the invention thus described, it will be obvious that the invention 
may be varied in many ways. Such variations are not to be regarded as a 
departure from the spirit and scope of the invention, and all such 
modifications as would be obvious to one skilled in the art are intended 
to be included within the scope of the following claims. 
The basic Japanese Application Nos. 16115/1994 filed on Feb. 10, 1994 and 
16116/1994 filed on Feb. 10, 1994 are hereby incorporated by reference.