Patent Publication Number: US-8532366-B2

Title: Position detecting method

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. patent application Ser. No. 11/244,558, filed Oct. 5, 2005, which claims priority from Japanese Patent Application No. 2004-293026, filed Oct. 5, 2004. The contents of the documents cited in this section are hereby incorporated herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to a position detecting method, and more particularly, though not exclusively, to a method for detecting a position of an alignment mark provided on a wafer. The present invention is suitable, for example, for manufactures of various devices including semiconductor chips such as ICs and LSIs, display devices such as liquid crystal panels, sensing devices such as magnetic heads, and image-pickup devices such as CCDs. 
     Recent high-performance and inexpensive electronic apparatuses need more economic and precisely manufactured semiconductors installed in them, and require an exposure apparatus that exposes a semiconductor circuit pattern to have precision and efficiency in the transfer of a circuit pattern of a reticle or a mask (collectively referred to as a “reticle” hereinafter) onto a wafer and a glass plate (collective referred to as a “wafer” hereinafter), onto which a photosensitive material (referred to as “resist” hereinafter) is applied. In general, precise exposure of a circuit pattern requires a precise alignment between the reticle and the wafer. 
     A conventional alignment method exposes alignment marks on a wafer at the same time when the circuit pattern of the reticle is exposed, and sequentially measures positions of plural preset alignment marks among the alignment marks formed on the wafer after the exposure process, using an alignment detection optical system. After the position measurement result is statistically processed to calculate the entire shot arrangement, and the wafer is positioned in relation to the reticle based on the calculation result. 
     The alignment marks are indexes to align the reticle and the wafer with high precision. Recent fine processing to circuit patterns requires a precise alignment mark. The special semiconductor manufacturing technology, such as a chemical mechanical polishing (“CMP”) process, has been recently introduced. Along with this, a wafer induced shift (“WIS”) as a positional detection error caused by a wafer process occurs, scatters shapes of the alignment marks among wafers and among shots, and deteriorates the alignment accuracy. One solution for this problem is an offset correction to correct the WIS, as disclosed in Japanese Patent Application, Publication No. 2004-117030. The offset correction previously calculates a true position of an alignment mark and a shift amount from the alignment mark detected by a detection system, and uses the offset value to correct the position of the alignment mark detected by the detection system. The position of the alignment mark actually detected by the detection system is also referred to as an “actual position” hereinafter. Conventionally, as in Japanese Patent Application No. 2004-117030, a linear function is used to calculate the shift amount. 
     Other prior art relating to the position detecting method include, for example, Japanese Patent Applications Nos. 6-151274 and 8-94315. 
     However, the WIS occurs due to a single cause, an interaction among plural causes, and an apparatus cause, such as an exposure apparatus and an alignment optical system. Therefore, the linear function has a difficulty to predict a shift amount, or to detect a position of an alignment mark precisely. The apparatus cause is also referred to as a tool induced shift (“TIS”). An interaction between the process error and the apparatus error is referred to as a TIS-WIS interaction. 
     BRIEF SUMMARY OF THE INVENTION 
     With the foregoing in mind, it is an exemplified object of the present invention to detect a position precisely. 
     A method according to one aspect of the present invention of detecting a position of a mark based on an image signal of the mark includes steps of obtaining a first position of the mark by performing a first process for the image signal, extracting plural feature values from the image signal based on the first position, and detecting the position of the mark by obtaining an offset value for the first position based on the plural feature values. 
     An exposure apparatus according to another aspect of the present invention to detect a position of a mark formed on an object to be exposed to light based on an image signal of the mark, and to expose the object to the light based on the detected position includes a detector configured to output the image signal, and a processor configured to obtain a first position of the mark by performing a first process for the image signal, to extract plural feature values from the image signal based on the first position, and to detect the position of the mark by obtaining an offset value for the first position based on the plural feature values. 
     A method according to still another aspect of the present invention of detecting a position of a mark based on an image signal of the mark includes steps of obtaining a first position of the mark by performing a first process for the image signal, extracting plural feature values from the image signal, and detecting the position of the mark by obtaining an offset value for the first position based on a function, of which an order is higher than two, having the plural feature values as variables. 
     An exposure apparatus according to another aspect of the present invention to detect a position of a mark formed on an object to be exposed to light based on an image signal of the mark, and to expose the object to the light based on the detected position includes a detector configured to output the image signal, and a processor configured to obtain a first position of the mark by performing a first process for the image signal, to extract plural feature values from the image signal, and to detect the position of the mark by obtaining an offset value for the first position based on a function, of which an order is higher than two, having the plural feature values as variables. 
     A method of manufacturing a device according to another aspect of the present invention includes steps of exposing an object to light using the above exposure apparatus, developing the exposed object, and processing the developed object to manufacture the device. 
     Other objects and further features of the present invention will become readily apparent from the following description of the embodiments with reference to accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a flowchart for explaining an inventive position detecting method. 
         FIGS. 2A and 2B  are waveforms for explaining a detection method of an alignment signal waveform in the position detecting method shown in  FIG. 1 . 
         FIG. 3  is a waveform for explaining a calculation method of a waveform evaluation value from an alignment signal waveform in the position detecting method shown in  FIG. 1 . 
         FIG. 4  is a schematic scatter diagram among a shift ΔP 1  between a true value and a signal process P 1 , waveform evaluation values C 1 , C 2 , and C 3 , and a difference P 2 −P 1  between signal processes P 1  and P 2 . 
         FIG. 5  is a view showing a relationship among the waveform evaluation values C 1 , C 2  and C 3  and signal processes P 1  and P 2  in relation to one signal waveform. 
         FIG. 6  is a graph showing a relationship between an illusion amount due to the wafer process and a residue after a correction. 
         FIG. 7  is a schematic block diagram of an exposure apparatus according to one embodiment of the present invention. 
         FIG. 8  is a schematic optical-path view showing principal components in an alignment optical system shown in  FIG. 7 . 
         FIGS. 9A and 9B  are schematic plane views of an alignment mark shown in  FIG. 7 . 
         FIGS. 10A and 10B  are schematic plane views of another alignment mark shown in  FIG. 7 . 
         FIG. 11  is a graph of typical detection results when the alignment marks shown in  FIGS. 9A ,  9 B,  10 A and  10 B are optically detected. 
         FIG. 12  is a schematic block diagram showing main functional modules in an alignment signal processor shown in  FIG. 7 . 
         FIG. 13  is a flowchart for explaining a method for manufacturing devices (semiconductor chips such as ICs, LSIs, and the like, LCDs, CCDs, etc.) using the exposure apparatus shown in  FIG. 1 . 
         FIG. 14  is a detailed flowchart for Step  4  of wafer process shown in  FIG. 13 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A description will now be given of a position detecting method according to one aspect of the present invention, with reference to the accompanying drawings. The same element in each figure is designated by the same reference numeral, thus a description thereof will be omitted. 
       FIG. 7  is a block diagram of an exposure apparatus  100 . The exposure apparatus  100  is a projection exposure apparatus that exposes onto a wafer a circuit pattern of a reticle, e.g., in a step-and-repeat or a step-and-scan manner. Such an exposure apparatus is suitable for a submicron or quarter-micron lithography process, and this embodiment exemplarily describes a step-and-scan exposure apparatus (which is also called “a scanner”). The “step-and-scan manner”, as used herein, is an exposure method that exposes a mask pattern onto a wafer by continuously scanning the wafer relative to the mask, and by moving, after a shot of exposure, the wafer stepwise to the next exposure area to be shot. The “step-and-repeat manner” is another mode of an exposure method that moves a wafer stepwise to an exposure area for the next shot every shot of cell projection onto a wafer. 
     Referring to  FIG. 7 , the exposure apparatus  100  includes a projection optical system  120 , a wafer chuck  145 , a wafer stage  140 , an alignment optical system (or alignment scope)  150 , an alignment signal processor  160 , and a controller  170 . The projection optical system  120  projects a reduced size of a reticle  110  that has a desired pattern, such as a circuit pattern. The wafer chuck  145  holds a wafer  130 , onto which a primary coat pattern and alignment marks  180  have been formed in a pretreatment step. The wafer stage  140  positions the wafer  130  in place. The alignment optical system  150  measures a position of the alignment mark  180  on the wafer  130 .  FIG. 7  omits a light source, and an illumination optical system that illuminates the reticle  110  using light from the light source. 
     The controller  170  includes a CPU and a memory (not shown), and controls actions of the exposure apparatus  100 . The controller  170  is electrically connected to an illumination apparatus (not shown), a reticle stage (not shown), the wafer stage  140 , and the alignment signal processor  160 . The controller  170  positions the wafer  130  through the wafer stage  140  based on positional information of the alignment mark from the alignment signal processor  160 . A position detecting method  1000  of this embodiment to be executed by the controller  170  will be described later. 
     A description will now be given of a detection principle of an alignment mark  180 .  FIG. 8  shows the schematic optical path of the main elements in the alignment optical system  150 . Referring to  FIG. 8 , the illumination light from an alignment light source  151  passes through a lens  153  after being reflected on a beam splitter  152 , and illuminates an alignment mark  180  on the wafer  130 . The (reflected or diffracted) light from the alignment mark  180  passes through the lens  153 , the beam splitter  152 , and a lens  154 , and is split by a beam splitter  155 , and received by CCD sensors  156  and  157 . 
     The alignment mark  180  is magnified by an imaging magnification of about 100 times by lenses  153  and  154 , and imaged on the CCD sensors  156  and  157 . The CCD sensors  156  and  157  are used to measure the offsets of the alignment mark  180  in the X and Y directions, respectively, and are arranged at a rotational angle of 90° relative to the optical axis. The CCD sensor may use a line sensor, which preferably uses, in this case, a cylindrical lens having power in a direction perpendicular to the measurement direction to condense the light in the direction perpendicular to the measurement direction, to execute an optical integration, and to average the information in the perpendicular direction. A description will be given of a positional measurement in the x direction, because the measurement principle is the same between the x and y directions. 
     The alignment marks  180  are arranged on a scribe line for each shot and may use, for example, alignment marks  180 A and  180 B shown in  FIGS. 9A ,  9 B,  10 A and  10 B. The alignment mark  180  generalizes alignment marks  180 A and  180 B. Here,  FIGS. 19A and 19B  are plane sectional views of the alignment mark  180 A.  FIGS. 10A and 10B  are plane sectional views of the alignment mark  180 B. In  FIGS. 9A ,  9 B,  10 A and  10 B, each of the alignment marks  180 A and  180 B include four mark elements  182 A and  182 B arranged at regular intervals.  FIGS. 9A ,  9 B,  10 A and  10 B omit the resist, which is actually applied to the alignment marks  180 A and  180 B. 
     The alignment mark  180 A arranges, as shown in  FIG. 9A , four rectangular mark elements  182 A at a pitch of 20 μm in the x direction, each of which has a size of 4 μm in a measurement x direction and 20 μm in a non-measurement y direction. The mark element  182 A has a concave sectional shape as shown in  FIG. 9B . The alignment mark  180 B arranges four mark elements  182 B, each of which replaces, as shown in  FIGS. 10A and 10B , a contour of the mark element  182 A in  FIGS. 9A and 9B  with a line width (critical dimension) of 0.6 μm. 
     Whichever is used, the alignment mark  180 A or  180 B in  FIGS. 9A and 9B  or  10 A and  10 B, the CCD sensor  156  generally takes an image as shown in  FIG. 11 , due to a generation and interference of scattered rays at a lens&#39; edge outside a NA of the lenses  153  and  154  in the alignment optical system  150 . The alignment mark  180 A has a dark contour, while the alignment mark  180 B has a dark or bright concave part, as characterized and often observed by a bright field. Here,  FIG. 11  is a graph showing a typical detection result when the alignment marks  180 A and  180 B shown in  FIGS. 9A ,  9 B,  10 A and  10 B are optically detected. 
     The alignment signal processor  160  provides an alignment signal processing to an image of the alignment mark  180 , which has been thus taken, as follows:  FIG. 12  is a schematic block diagram showing main functional modules in the alignment signal processor  160 . 
     Referring to  FIG. 12 , an A/D converter  161  digitalizes the alignment signals from the CCD sensors  156  and  157 . Various signal processors in a storage unit  162  remove noise from the digitized alignment signal, and store the resultant signal in memory. A mark center detector  163  performs a digital signal processing for a stored digitized alignment signal. The mark center detector  163  executes a position detecting method  1000 , which will be described later, by an operational element for the digitalized alignment signal, and detects a center position of the alignment signal. The CPU  164  is connected to an A/D converter  161 , the storage unit  162 , the mark center detector  163 , and outputs control signals to them so as to control their actions. A communication part  165  communicates with the controller  170  shown in  FIG. 7  for communications of necessary data, control commands, etc. 
     Various methods have been proposed to the digital signal processing by the mark center detector  163 , which includes a method for detecting edge parts of the alignment signal and for calculating the distance between the edges, and a template or pattern matching that uses a normalized function. The present invention proposes a position detecting method  1000  for detecting a center position in the alignment signal as one example of a digital signal processing. The signal source may be a two-dimensional or one-dimensional signal. A two-dimensional image is voting-processed after its horizontal pixels are arranged in a perpendicular direction to create a histogram, averaged with respect to the main components, and converted into a one-dimensional image. The proposed, inventive digital signal processing independently performs X and Y measurements, and the base signal processing for positioning is a one-dimensional signal processing. For example, the digital signals representative of the two-dimensional images on the CCD sensors  156  and  157  are integrated and averaged, and converted into a one-dimensional line signal. 
     Referring now to  FIG. 1 , a description will be given of a position detecting method  1000  of a first embodiment. Here,  FIG. 1  is a flowchart of the position detecting method  1000 . 
     Initially, an extraction method is set for a signal waveform of the mark element  182 A or  182 B based on a waveform feature value to be obtained (step S 1002 ). The “waveform feature value,” as used herein, means a parameter that characterizes a signal waveform of the mark element  182 A or  182 B. Step S 1002  may apply various methods to the alignment signal waveform extracting method. However, this embodiment uses a method for extracting a signal waveform from the reflectance at the mark element  182 A or  182 B (or for obtaining a center position of the mark element based on the symmetry of the signal waveform), and a method for first-order-differentiating a signal waveform and for extracting the signal waveform (or for obtaining the center position of the mark element from the first order differentiated value of the signal waveform). 
     Next, the signal waveform of the mark element  182 A or  182 B is obtained with the above extracting method (step S 1004 ). This step obtains the waveform from the mark element  182 A or  182 B, and the controller  160  calculates the signal waveform of the mark element  182 A or  182 B. 
     Next, a true position or value of the mark element  182 A or  182 B and an actual position of the mark element  182 A or  182 B are obtained (step S 1006 ). The “true position,” as used herein, is a value of an ideal position of the mark element  182 A or  182 B, and calculated, for example, by simulation. Since a shift amount between the actual position and the true position serves as learning data for a predicting model, which will be described later, this embodiment refers to the alignment signal data having a known true position as teacher data. 
     Referring now to  FIGS. 2A and 2B , a description will be given of a method for detecting a position of a mark element using various signal processes. Here,  FIGS. 2A and 2B  are waveforms for explaining a detection method of an alignment signal waveform. 
     Initially, for example, as shown in  FIG. 2A , S(x) in a measuring direction x relative to an alignment signal row y is defined as Equation 1: Equation 1 corresponds to Equation 24 in Japanese Patent Application, Publication No. 8-94315 in which a is set to WC−WW/2 and b is set to WC+WW/2. 
     
       
         
           
             
               
                 
                   
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     As shown in  FIG. 2A , S(x) at a certain point x is calculated from Equation 1, and S(x) is obtained by varying x. The minimum of S(x) or the maximum of 1/S(x) shown in  FIG. 2A  is set to a mark position. The minimum value of S(x) is calculated with precision of sub-pixel by approximating a discrete signal S(x) with a function. This specification refers to this detection processing method as P 1 . 
     Another mark element position detecting method sets sections L and R for an alignment signal y as shown in  FIG. 2B . After the signal is filtered to remove random noises in each section, the mark position is calculated by using a midpoint between an x coordinate indicative of a minimum value in the section L and an x coordinate indicative of a maximum value in the section R for the first-order-differentiated signal. Since the first-order-differentiated signal is a discrete signal, the minimum and maximum values are calculated with precision of sub-pixel, for example, by approximating the first-order-differentiated signal with a function. This specification refers to this detection processing method as P 2 . 
     A shift amount is calculated between the true value and the actual position obtained in the above detecting method (step S 1008 ). In Step S 1008 , the true value is known when the alignment waveform is a simulated waveform, and the shift amount is obtained as an offset between the true value and the actual position (or signal processing result), which will be described later, when the wafer process error occurs. The wafer process error can be calculated by the simulation, and occurs, for example, in accordance with a step amount of the alignment mark, a resist&#39;s thickness, and coverage on the wafer. 
     The shift amount can be calculated without a true value calculated by the simulation. For example, a position of the alignment mark is detected based on the alignment waveform, the wafer is positioned based on the detection result, and the exposure follows. Thereafter, the wafer is inspected by using an overlay inspector. The shift amount between the actual position and the true value is calculated from the inspection result. In this case, the actual position corresponds to a position of the alignment mark obtained from the alignment waveform, and the true value corresponds to a position at which the shift amount detected by the overlay inspector is zero. 
     Next, two or more waveform feature values are extracted from the obtained signal waveform (step S 1010 ). 
     A description will be given of an extraction of the waveform feature value. This embodiment refers to not only a waveform evaluation value that specifies a waveform shape but also an arbitrary signal processing result as a waveform feature value. In this embodiment, the waveform feature value includes 1) a difference between the signal waveform of the mark element  182 A or  1828  and another signal waveform different from that of the signal waveform of the mark element  182 A or  1828 , 2) a waveform evaluation value as a gradient difference of a signal waveform of the mark element  182 A or  1828 , 3) a waveform evaluation value as a contrast of the signal waveform of the mark element  182 A or  1828 , and 4) a waveform evaluation value as a difference between left and right heights in the signal waveform of the mark element  182 A or  1828 . The results of the detection processes P 1  and P 2  are originally given in the coordinate, and the coordinate values do not provide a feature amount. Therefore, this embodiment defines a waveform feature value, for example, as a difference P 2 -P 1  between the signal processing P 2  and the signal processing P 1 . 
     Referring now to  FIG. 3 , a description will be given of the waveform evaluation value. Here,  FIG. 3  is a waveform for explaining a method for calculating the waveform evaluation value from the alignment signal. 
       FIG. 3  sets the left and right sections of the waveform for the waveform from which one element is extracted out of the alignment signal, and quantifies a waveform shape in each section. 
     A first waveform evaluation value C 1  is defined as Equation 2, where a L  and b L  are maximum and minimum values of the signal row y in the section L, a R  and b R  are maximum and minimum values of the signal row y in the section R. 
     
       
         
           
             
               
                 
                   
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     A waveform evaluation value C 2  is defined as Equation 3 where L L  is a length of a box when the signal row y in the section L is expressed by a boxplot used for the statistic approach, L R  is a length of a box when the signal row y in the section R is expressed by a boxplot used for the statistic approach. See R. Becker, J. Chambers, and A. Wilks, “S Language I”, Kyoritsu Publishing, for the boxplot. 
     
       
         
           
             
               
                 
                   
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     The waveform evaluation value C 3  is defined as Equation 4, where A L  is an absolute value of the minimum value of the first order differentials in the section L shown in  FIG. 2B , A R  is an absolute value of the maximum value of the first order differentials in the section R.
 
 C 3 =A   L   −A   R   [EQUATION 4]
 
     Since the waveform evaluation value is obtained for each alignment mark element, an average of four waveform evaluation values is used to obtain a waveform value for each alignment mark when four mark elements exist as in this embodiment. 
     Next, a function is generated to correct a shift between the true value and the actual position of the mark element  182 A or  182 B (step S 1012 ). Characteristically, step S 1012  generates a function using a waveform feature value. Thereby, an offset amount can be precisely predicted even when there is an error due to an interaction of WIS caused by plural causes, and a WIS-TIS interaction. As a result, the position detecting method  1000  can precisely detect a position. The above function is available for each semiconductor process. 
     First Embodiment 
     A description will now be given of a method for predicting a shift amount between the true value and the detection processing result that uses a N-th order polynomial that includes a term of an interaction, plural waveform evaluation values, and a multivariate analysis that uses a result of the detection processes P 1  and P 2 . 
       FIG. 4  is a scatter diagram that is formed by an optical simulation among a shift ΔP 1  between the true value and the signal process P 1  when a wafer process error is generated, and a difference P 2 −P 1  between the signal processes P 1  and P 2  and waveform evaluation values C 1 , C 2  and C 3 . 
     A target variable is set to ΔP 1  between the true value and the signal process P 1 , and is predicted by using an equation model defined as Equation 5, P 2 −P 1 , C 1 , C 2  and C 3  as predictor variables, and multivariate analysis: 
     
       
         
           
             
               
                 
                   
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                           ( 
                           
                             
                               P 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             - 
                             
                               P 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ) 
                         
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         1 
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     + 
                     
                       
                         
                           c 
                           3 
                         
                         · 
                         
                           ( 
                           
                             
                               P 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             - 
                             
                               P 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ) 
                         
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     + 
                     
                       
                         
                           c 
                           4 
                         
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         1 
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     + 
                     
                       
                         
                           d 
                           1 
                         
                         · 
                         
                           ( 
                           
                             
                               P 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             - 
                             
                               P 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ) 
                         
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         1 
                         · 
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                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
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                       ⁢ 
                       
                           
                       
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                       3 
                     
                   
                 
               
               
                 
                   [ 
                   
                     EQUATION 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ] 
                 
               
             
           
         
       
     
     The second and subsequent terms in Equation 5 are terms indicative of an interaction. Here, the “interaction” is a term expressed by a product of predictor variables, such as a product C 1 ×C 2 . In other words, it corresponds to an error of WIS caused by an interaction of plural causes. This embodiment optimizes a coefficient parameter by using Akaike Information Criterion (“AIC”) as a regression predicting equation where N=3. 
     Next, a shift between the true value and the actual position of the mark element  182 A or  182 B is corrected based on the above function (step S 1014 ). Accordingly, an offset amount can be predicted with precision even when there is an error due to an interaction of WIS of plural causes and a WIS-TIS interaction. As a result, the position detecting method  1000  can detect a position with precision. 
     Step S 1014  uses the model predicted by this embodiment to obtain the waveform evaluation values C 1 , C 2  and C 3  and signal processing results P 1  and P 2  for the actual waveform. The shift amount ΔP 1  is predicted from the obtained value, and corrected with respect to the first signal processing result P 1 . 
       FIG. 6  plots the predicted residue of the model in relation to the teacher data in this embodiment, in which R-squared, an index of application easiness of the model is 0.9756. The illusion amount (API) due to the wafer process error improves from 200 nm to 30 nm in the range, and provides a validity of this embodiment. Here,  FIG. 6  is a graph showing a relationship between the illusion amount due to the wafer process and the residue after correction. 
     While this embodiment uses AIC as an evaluation approach that adds a penalty term to a square sum of the predicted residue to prevent over-fitting of the regression model, the inventive approach to improve the model generalization performance is not limited to this embodiment and may use a cross validation, a bootstrap method, etc. to determine the N-th order. After the N-th order is determined, each coefficient parameter can be determined through a neural network. 
     Second Embodiment 
     While the first embodiment includes a term of an interaction in the predicting model equation, the present invention is not limited to this embodiment and may predict by using a N-th order polynomial that does not include an interaction. Equation 6 below is a model equation in the second embodiment: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
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                     1 
                   
                   = 
                   
                     
                       
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                           i 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             a 
                             
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                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           a 
                           
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                             ⁢ 
                             
                                 
                             
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                         ⁢ 
                         
                             
                         
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                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           a 
                           
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                             ⁢ 
                             
                                 
                             
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                   [ 
                   
                     EQUATION 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ] 
                 
               
             
           
         
       
     
     Third Embodiment 
     The present invention is applicable when the waveform feature value includes only a waveform evaluation value. The third embodiment uses a model equation defined as Equation 7, in which the waveform feature values are C 1 , C 2  and C 3 : 
     
       
         
           
             
               
                 
                   
                     Δ 
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                         · 
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                           2 
                         
                         · 
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                         · 
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                     + 
                     
                       
                         
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                         · 
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                         · 
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                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                   
                 
               
               
                 
                   [ 
                   
                     EQUATION 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ] 
                 
               
             
           
         
       
     
     Fourth Embodiment 
     The present invention is applicable when the waveform feature value is merely a signal processing result. The fourth embodiment uses a model equation defined as Equation 8, in which the signal processing results are P 1 , P 2  and P 3 : It is enough that P 3  originates from one of other various signal processing methods different from that for P 1  and P 2 , such as a method for setting a slice level and detecting a mark position, and a method for detecting a mark position using template matching. 
     
       
         
           
             
               
                 
                   
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                     + 
                     
                       
                         b 
                         1 
                       
                       · 
                       
                         ( 
                         
                           
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                             ⁢ 
                             
                                 
                             
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                             2 
                           
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                         ) 
                       
                       · 
                       
                         ( 
                         
                           
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                             ⁢ 
                             
                                 
                             
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                             ⁢ 
                             
                                 
                             
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                             1 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     EQUATION 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ] 
                 
               
             
           
         
       
     
     In exposure, the light from the illumination apparatus illuminates the reticle  110 , and the projection optical system  120  images a pattern of the reticle  110  onto the wafer  130 . This embodiment provides an arc-shaped or ring-shaped image surface, and the overall pattern of the reticle  110  is transferred to the wafer  130  when the reticle  110  and the wafer  130  are scanned at a reduction speed ratio. The position detecting method  1000  of this embodiment detects a position of the wafer  130  with high precision. Even when the detection results of the alignment marks scatter, this method can maintain both the necessary alignment accuracy and throughput. As a result, the exposure apparatus  100  provides fine processing with an improved throughput and productivity. 
     Referring now to  FIGS. 13 and 14 , a description will be given of an embodiment of a device manufacturing method using the above exposure apparatus  100 .  FIG. 13  is a flowchart for explaining the fabrication of devices (i.e., semiconductor chips such as IC and LSI, LCDs, CCDs, etc.). Here, as an example, a description will be given of a fabrication of a semiconductor chip. Step  1  (circuit design) designs a semiconductor device circuit. Step  2  (mask fabrication) forms a mask having a designed circuit pattern. Step  3  (wafer preparation) manufactures a wafer using materials such as silicon. Step  4  (wafer process), which is referred to as a pretreatment, forms actual circuitry on the wafer through photolithography using the mask and wafer. Step  5  (assembly), which is also referred to as a posttreatment, forms into a semiconductor chip the wafer formed in Step  4  and includes an assembly step (e.g., dicing, bonding), a packaging step (chip sealing), and the like. Step  6  (inspection) performs various tests on the semiconductor device made in Step  5 , such as a validity test and a durability test. Through these steps, a semiconductor device is finished and shipped (Step  7 ). 
       FIG. 14  is a detailed flowchart of the wafer process in Step  4 . Step  11  (oxidation) oxidizes the wafer&#39;s surface. Step  12  (CVD) forms an insulating film on the wafer&#39;s surface. Step  13  (electrode formation) forms electrodes on the wafer by vapor disposition and the like. Step  14  (ion implantation) implants ions into the wafer. Step  15  (resist process) applies a photosensitive material onto the wafer. Step  16  (exposure) uses the exposure apparatus  200  to expose the circuit pattern on the mask onto the wafer. Step  17  (development) develops the exposed wafer. Step  18  (etching) etches parts other than the developed resist image. Step  19  (resist stripping) removes the disused resist after etching. These steps are repeated and multi-layer circuit patterns are formed on the wafer. The device fabrication method of this embodiment may manufacture higher quality devices than the conventional one. 
     Further, the present invention is not limited to these preferred embodiments, and various variations and modifications may be made without departing from the scope of the present invention. 
     This application claims a benefit of foreign priority based on Japanese Patent Application No. 2004-293026, filed on Oct. 5, 2004, which is hereby incorporated by reference herein in its entirety as if fully set forth herein.