Abstract:
A method of determining a defocus direction of a pattern image formed on a reticle, which is projected by an exposure apparatus onto a resist on a substrate as a resist pattern, the exposure apparatus exposing the resist to light via the pattern image on the reticle to form the resist pattern. The method includes an image capturing step of capturing a resist image of the resist pattern that is formed on the substrate by the exposure apparatus, to obtain image data, an extracting step of extracting a feature of the image data to obtain feature data, and a determining step of determining the defocus direction of the pattern image based on the extracted feature data. The resist pattern includes a dual tone line end shortening target having a hollow grating mark and a solid grating mark, and the feature data includes an edge sharpness of a waveform of the image data.

Description:
This application claims the benefit of Japanese Patent Application No. 2005-029829, filed on Feb. 4, 2005, which is hereby incorporated by reference herein in its entirety. 
     FIELD OF THE INVENTION 
     The present invention relates to a technique applied to lithography and, more particularly, to a technique of determining the defocus direction of an image formed on a resist by an exposure apparatus for exposing a resist on a substrate to light via a pattern of a reticle. 
     BACKGROUND OF THE INVENTION 
     In semiconductor manufacture lithography, in which a reticle pattern is projected and transferred onto a wafer by exposing the pattern using exposure light, an apparatus for inspecting the exposure and defocus amount is used. 
     The conventional flow of lithography will be described. 
     A resist pattern is formed by applying a resist, serving as a photosensitizer to a substrate, such as a semiconductor wafer, projecting a mask pattern on the resist by exposure using an exposure apparatus, and then, developing the resist. 
     The dimensions of the formed resist pattern are checked by a scanning electron microscope (measuring SEM or CD-SEM) with a measuring function. The conventional processing contents of the measuring SEM include, e.g., acquiring an electron beam image of an area containing parts with strict dimensional accuracy (step 1), measuring the dimensions (step 2), determining whether the dimensions meet the standards (step 3), and, if not, changing the exposure of the exposure apparatus (step 4, the exposure correction amount is ΔE). For, e.g., a positive resist, if the resist dimensions are too large, the exposure is increased. If the resist width is too small, the exposure is decreased. 
     The relationship between a resist pattern and a film pattern after etching will be described next. 
     The shape of a resist pattern and the shape of a film pattern have a predetermined relationship if the etching conditions are the same. To obtain a film pattern having a predetermined shape, the resist pattern must have a predetermined shape, too. In, e.g., starting a new process, a wafer called an FEM (Focus Exposure Matrix) is prepared by exposing a pattern while changing the focus and exposure in each shot (one exposure cycle). A focus and exposure to obtain a predetermined resist pattern shape are found by measuring the dimensions of the resist pattern in each shot, and cutting the wafer to check its sectional shape. That is, a so-called “condition determination” is performed. 
     With this operation, the exposure (E0) and focus value (F0) to widen the margin are determined. Product wafers are exposed on the basis of these conditions. However, it may be impossible sometimes to obtain a resist pattern with a predetermined shape under the conditions (E0, F0) determined by the “condition determination,” because of various process variations (e.g., a change in resist sensitivity, a variation in thickness of an antireflection film under the resist, or a drift of various kinds of sensors of the exposure apparatus). This is detected in dimension measurement (step 2). In the conventional technique, the change in resist shape caused by process variations is compensated for by correcting the exposure. 
     KLA-Tencor in the United States announced, on Jun. 24, 2003, an “MPX” that enables in-line focus/exposure monitoring as a new option of an overlay measuring apparatus in their “Archer” series. The “MPX” can monitor the focus and exposure by analyzing a unique dual tone line end shortening (LES) target and accurately separating the focus and the exposure. On the basis of the data, management of the defocus and exposure of the exposure apparatus, grasping the apparatus variation, and specifying the cause can be done quickly. KLA-Tencor states that users can suppress any decrease in yield associated with focus, and save the cost of millions of dollars a year by using that option. Techniques related to such a measuring apparatus are disclosed in U.S. Pat. No. 5,629,772, U.S. Pat. No. 5,757,507, U.S. Pat. No. 5,790,254, U.S. Pat. No. 6,137,578, and U.S. Pat. No. 6,577,406. 
     In the prior art, to detect and to cope with process variations, dimensional values, such as a line width are checked by using measuring SEM. If the dimensional values do not meet the standards, the exposure is corrected. However, this method has the problem of focal depth degradation described below. 
     When the exposure changes, the line width changes. On the other hand, the line width rarely changes, even when the focus changes. However, when the focus changes, the sectional shape of the resist changes, although the line width does not change. As described above, the change in sectional shape influences the shape of the film pattern after etching. For this reason, since the prior art is incapable of detecting a focus variation, poor film pattern shapes may be produced in large quantities after etching. 
     As described above, since displacement caused by defocus cannot be corrected by correcting only the exposure, the resist can have no normal sectional shape. In addition, since exposure is not executed at the center of the depth of focus, the depth may be insufficient, and poor film pattern shapes may be produced in large quantities after etching. 
     In “MPX”, the focus amount and exposure are estimated by analyzing a dual tone line end shortening target (to be referred to as a mark, hereinafter). The mark includes a hollow grating mark and a solid grating mark, as shown in  FIG. 1 . Conventionally, the focus amount and exposure are estimated by measuring intervals CD 1  and CD 2  shown in  FIG. 1 . The defocus amount is estimated by using the behavior of the interval CD 1  or CD 2 , which changes with respect to the focus amount, as shown in  FIG. 2 . 
       FIG. 2  shows the relationship between the interval CD 1  or CD 2  and the focus amount at an exposure E=E0. The interval CD 1  or CD 2  is plotted along the ordinate, and the focus amount is plotted along the abscissa. Z0 is the optimum focus amount. The interval CD 1  or CD 2  is minimized when the focus amount is Z0. However, the interval CD 1  or CD 2  changes as an even function with respect to the defocus amount. For this reason, the interval is usable in estimating the defocus amount, but not usable in estimating the defocus direction (i.e., whether the defocus occurs in the positive direction or the negative direction). 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a novel technique of determining the focus direction of an image formed on a resist by an exposure apparatus based on a resist pattern formed through an exposure of the resist to the image. 
     In order to achieve the object, according to a first aspect of the present invention, a method of determining a defocus direction of a pattern image formed on a resist by an exposure apparatus for exposing a resist on a substrate to light via a pattern of a reticle, comprises an image capturing step of capturing a resist image of a resist pattern formed through exposure of a resist to the pattern image by the exposure apparatus, to obtain image data, an extracting step of extracting a feature of the image data to obtain feature data, and a determining step of determining the defocus direction based on the feature data. 
     According to the above aspect, the extracting step includes a differentiating step of differentiating the image data to obtain differential image data. 
     According to the above aspect, the extracting step includes a calculating step of calculating a power spectrum of the differential image data as the feature data. 
     According to the above aspect, in the determining step, the determining step determines the defocus direction based on the feature data and feature data obtained in advance from a resist pattern whose defocus direction is known. 
     According to the above aspect, the determining step determines the defocus direction by projecting the feature data of the image data to probability space data, which has been obtained by projecting the feature data obtained in advance, to one of a Mahalanobis space and a Bayes space. 
     According to the above aspect, the probability space data is prepared with respect to each of two directions of defocus. 
     According to the above aspect, the determining step determines the defocus direction based on two probability data obtained by projecting the feature data of the image data to each of the probability space data. 
     According to the above aspect, the feature data obtained in advance are obtained with respect to a plurality of resist patterns, the resist patterns being obtained with respect to a plurality of values of at least one of a defocus amount and a dose of the exposure. 
     According to a second aspect of the present invention, an apparatus, for determining a defocus direction of a pattern image formed on a resist by an exposure apparatus for exposing a resist on a substrate to light via a pattern of a reticle, comprises a camera configured to capture a resist image of a resist pattern formed through exposure of a resist to the pattern image by the exposure apparatus to obtain image data, and a processor configured to extract a feature of the image data and to obtain feature data, and to determine the defocus direction based on the feature data. 
     According to the above aspect, the processor is configured to differentiate the image data to obtain differential image data. 
     According to the above aspect, the processor is configured to calculate a power spectrum of the differential image data as the feature data. 
     According to the above aspect, the processor is configured to determine the defocus direction based on the feature data, and feature data obtained in advance, from a resist pattern whose defocus direction is known. 
     According to the above aspect, the processor is configured to determine the defocus direction by projecting the feature data of the image data to probability space data, which has been obtained in advance, to one of a Mahalanobis space and a Bayes space. 
     According to the above aspect, the processor is configured to store the probability space data with respect to each of two directions of defocus. 
     According to the above aspect, the processor is configured to determine the defocus direction based on two probability data obtained by projecting the feature data of the image data to each of the probability space data. 
     According to the above aspect, the feature data obtained in advance are obtained with respect to a plurality of resist patterns, the resist patterns being obtained with respect to a plurality of values of at least one of a defocus amount and a dose of the exposure. 
     An exposure apparatus for exposing a resist on a substrate to light via a pattern of a reticle comprises one of the above-described determining apparatuses. 
     A method of manufacturing a device comprises steps of exposing a resist on a substrate to light via a pattern of a reticle using an exposure apparatus as defined above, developing the exposed substrate, and processing the developed substrate to manufacture the device. 
     A device manufacturing method of the present invention comprises a step of determining a defocus direction of a pattern image formed on a resist by an exposure apparatus for exposing a resist on a substrate to light via a pattern of a reticle, using one of the above-described determining apparatuses. 
     According to the present invention, the defocus direction of the exposure apparatus can be determined from the resist pattern. 
     Other objects and advantages, besides those discussed above, shall be apparent to those skilled in the art, from the description of a preferred embodiment of the invention, which follows. In the description, reference is made to the accompanying drawings, which form a part hereof, and which illustrate an example of the invention. Such an example, however, is not exhaustive of the various embodiments of the invention, and, therefore, reference is made to the claims which follow the description for determining the scope of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a view showing a mark (dual tone line end shortening target); 
         FIG. 2  is a graph showing the relationship between the focus amount and CD 1  or CD 2  shown in  FIG. 1 ; 
         FIG. 3  is a schematic view of a waveform of negative focus; 
         FIG. 4  is a schematic view of a waveform of positive focus; 
         FIG. 5  is a view showing marks used in an overlay inspection apparatus; 
         FIG. 6  is a view showing the optical system of the overlay inspection apparatus; 
         FIG. 7  is a functional block diagram for implementing a defocus direction determining method according to an embodiment; 
         FIG. 8  is a flowchart showing the procedure of a feature extracting unit; 
         FIG. 9  is a flowchart showing the procedure of a learning data calculating unit; 
         FIG. 10  is a flowchart showing the procedure of a determining unit; 
         FIG. 11  is a table showing the minimum line width, required alignment accuracy, and required accuracy of the overlay inspection apparatus in each device node; 
         FIG. 12  is a schematic view of a semiconductor exposure apparatus including the overlay inspection apparatus shown in  FIG. 6 ; 
         FIG. 13  is a flowchart showing a device manufacturing method; and 
         FIG. 14  is a flowchart showing a wafer process. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     [Outline of the Present Invention] 
     The outline of the present invention will be described with reference to the accompanying drawings. 
     In the present invention, the defocus direction is estimated by using not only a mark edge interval, such as an interval CD 1  or CD 2  shown in  FIG. 1 , but also, new feature information obtained from a waveform L shown in  FIG. 1 . According to “Yield Management Solutions” Fall 2001 and Winter 2002, issued by KLA-Tencor in the United States, the resist waveform changes between positive focus and negative focus. This tendency becomes noticeable as the defocus amount increases.  FIGS. 3 and 4  show the resist waveform of positive focus and the resist waveform of negative focus, respectively, which have the same absolute value of defocus amount.  FIG. 3  shows a negative focus.  FIG. 4  shows a positive focus. That is, the shape or height changes. 
     An input/output (image capturing) means or step (input/output unit S 701  in  FIG. 7 ) receives unknown waveform data, which is obtained by capturing a mark at an unknown focus amount and exposure, positive focus waveform data obtained by exposure in a predetermined focus amount and exposure, and negative focus waveform data obtained by exposure in a predetermined focus amount and exposure. The input/output means or step also outputs the defocus direction of the unknown waveform determined by a determining means or step (to be described later). 
     A feature extracting means or step (feature extracting unit S 703  in  FIG. 7 ) extracts the difference in resist waveform between positive focus and negative focus. As a feature of the mark shown in  FIG. 1 , the positive focus waveform contains a larger number of sharp edges than the negative focus waveform. More specifically, the difference between positive focus and negative focus includes the edge sharpness and the interval between edges. 
     Edge components are extracted by differentiating the waveform (image data) (differentiated image data is obtained). Additionally, to extract the difference in edge sharpness and edge interval between the positive focus waveform and negative focus waveform, the calculated differentiated waveform is Fourier-transformed to calculate the power spectrum. 
     A learning data calculating means or step (learning data calculating unit S 706  in  FIG. 7 ) removes feature data of a region where the feature data group of positive focus and that of negative focus overlap, to increase the generality of defocus direction determination, thereby, improving the degree of separation of the two feature data groups. For example, the feature data of a waveform having a focus amount near the absolute value of the optimum focus amount is removed from learning data. 
     Next, feature data corresponding to the outliers of each data group is removed, thereby preventing the outliers from influencing the statistic of the two feature data groups. For example, the feature data of a waveform having an exposure largely different from the optimum exposure is removed from learning data. Alternatively, the feature data of a waveform having a focus amount largely different from the optimum focus amount is removed from learning data. 
     A determining means or step (determining unit S 702  in  FIG. 7 ) determines the defocus direction on the basis of the magnitude relationship between the probability that the unknown waveform is a positive focus waveform and the probability that the unknown waveform is a negative focus waveform. To implement this, the positive focus feature data group and negative focus feature data group calculated by the learning data calculating means or step are projected to a Mahalanobis space or Bayes space. With this processing, a positive focus probability space and negative focus probability space are formed. The Mahalanobis space and Bayes space assume a multidimensional normal distribution. The Bayes space is a complete multidimensional normal distribution probability spaced based on the Bayes conditional probability. On the other hand, the Mahalanobis space is a normal distribution, but not a multidimensional normal distribution, in the strict sense. This is described in detail in Ishii, Ueda, Maeda, and Murase, “Easy Pattern Recognition”, Ohm-Sha Ltd., pages 51, 80, 180 and 181. 
     The feature data of the unknown waveform is projected to the positive focus probability space and negative focus probability space. The probability that the unknown waveform is a positive focus waveform and the probability that the unknown waveform is a negative focus waveform are calculated. On the basis of the magnitude relationship between them, the defocus direction is determined. 
     The embodiments of the present invention will be described below in detail with reference to the accompanying drawings. 
     The embodiments to be described below are mere examples of a means for implementing the present invention, and should appropriately be corrected or changed in accordance with various kinds of conditions or the arrangement of the apparatus to which the present invention is applied. The same reference numerals as those in the above-described prior art arrangements denote the same elements, in the following description. 
     First Embodiment 
     An overlay inspection apparatus, which is indispensable in practicing the present invention, will be described first. 
     The overlay inspection apparatus measures the alignment accuracy or distortion of a semiconductor exposure apparatus called a stepper or scanner. As shown in  FIG. 5 , the overlay inspection apparatus measures the relative positional relationship between existing mark  1  and new mark  2  overlaid on it. 
     The overlay inspection apparatus was developed during the 1980s to meet the requirements of high accuracy and mass production, along with size reduction of semiconductor devices, and was introduced in device makers on a full scale in the 1990s. Before the introduction of the overlay inspection apparatus, inspection was done by visually reading a device called a vernier based on the same principle as that of calipers, by using a microscope. Currently, overlay inspection apparatuses are available from makers in Japan and other countries.  FIG. 11  shows the required alignment accuracy, and the like, of overlay inspection. 
       FIG. 11  is quoted from the lithography roadmap in “International Technology Roadmap for Semiconductor: ITRS 99” in 1999. A value of 10 nm or less is already required. It is to be noted that an accuracy up to 0.5 nm is required as the smallest unit. Hence, an accuracy on the sub-nanometer order must be taken into consideration. 
     The arrangement of the overlay inspection apparatus will be described next, with reference to  FIG. 6 . 
     Referring to  FIG. 6 , a halogen lamp is used as a light source  1 . A desired waveform band is selected by various kinds of optical filters  2  and  3 . The light is guided to an optical system  13  by a fiber  4  to Koehler-illuminate marks  6  and  7  on a wafer  5 . An image of the light reflected by the wafer  5  is formed on an image capturing element, such as a CCD camera  12 , through optical systems  8  to  11 . The image is photoelectrically converted. Various kinds of image processing are executed for the image signal to detect the relative positional relationship between the two marks  6  and  7 . 
     The principle of all commercially available currently overlay inspection apparatuses employs “bright field illumination+image processing”, as in the alignment optical system of an exposure apparatus, as shown in  FIG. 12 . 
     The resolving power of the alignment detection system is estimated (because the makers have not disclosed the optical specifications). When the magnification between the wafer and the CCD camera is ×100, and the pixel pitch of the CCD camera is 10 μm, the resolving power is 100 nm/pixel on the wafer surface. Hence, the above-described accuracy of 10 nm or less is achieved here, probably, by various image processing techniques. 
     A defocus direction determining method according to the first of embodiment of the present invention will be described next. 
       FIG. 7  is a functional block diagram for implementing the defocus direction determining method according to the first embodiment of the present invention. 
     Each block may be a device including a dedicated program or processor to implement the function to be below, or may be implemented by executing a control program to control specific hardware corresponding to the function. 
     The outline of this embodiment will be described first with reference to  FIG. 7 . 
     An input/output unit S 701  receives unknown waveform data, which is obtained at an unknown focus amount and exposure by the image capturing element  12  of the overlay inspection apparatus shown in  FIG. 6 , positive focus learning waveform information containing positive focus waveform data, and a focus amount and exposure obtained from an FEM wafer prepared by exposure at a predetermined focus amount and exposure, and negative focus learning waveform information containing negative focus waveform data and a focus amount and exposure obtained from an FEM wafer prepared by exposure, at a predetermined focus amount and exposure. The input/output unit S 701  outputs the defocus direction of the unknown waveform calculated by a determining unit S 702 . 
     The positive focus learning waveform information and negative focus learning waveform information input from the input/output unit S 701  in advance are learned as teaching data. The learning result is held by a learning data storage unit. The learning data can be updated by teaching data input at appropriate times. When the unknown waveform is input, then, the defocus direction is determined on the basis of the learning result. The determination result is output from the input/output unit S 701 . 
     A feature extracting unit S 703  receives the unknown waveform, positive focus learning waveform information, or negative focus learning waveform information from the input/output unit S 701 . Upon receiving the unknown waveform, the feature extracting unit S 703  outputs the feature data to the determining unit S 702 . Upon receiving the positive focus learning waveform information, the feature extracting unit S 703  registers the feature data in a positive focus feature data storage unit S 704  together with the focus amount and exposure. Upon receiving the negative focus learning waveform information, the feature extracting unit S 703  registers the feature data in a negative focus feature data storage unit S 705 , together with the focus amount and exposure. 
     A learning data calculating unit S 706  removes target data from the positive focus feature data (S 704 ) by using the focus amount rage and exposure range, which are registered in a specific data removal information storage unit S 707 , and outputs obtained feature data to a positive focus learning data storage unit S 708 . 
     The learning data calculating unit S 706  also removes target data from the negative focus feature data (S 705 ) by using the focus amount range and exposure range, which are registered in the specific data removal information storage unit S 707 , and outputs the obtained feature data to a negative focus learning data storage unit S 709 . 
     The determining unit S 702  receives the feature data of the unknown waveform calculated by the feature extracting unit S 703 , the feature data of the positive focus learning data (S 704 ), and the feature data of the negative focus learning data (S 705 ), and outputs the defocus direction of the unknown waveform to the input/output unit S 701 . 
     Each functional block will be described below in detail. 
     &lt;Feature Extracting Unit S 703 &gt; 
     As shown in  FIG. 8 , in step S 801 , the feature extracting unit S 703  calculates data by differentiating the unknown waveform or the waveform data of the positive focus learning waveform information or negative focus learning waveform information. In step S 802 , the data calculated in step S 801  is Fourier-transformed to calculate the power spectrum. 
     &lt;Learning Data Calculating Unit S 706 &gt; 
     The learning data calculating unit S 706  is provided to remove the feature data of a waveform having a focus amount near the optimum focus amount and to prevent overlap between the positive focus feature data group and negative focus feature data group, thereby improving the degree of separation of the two feature data groups. 
     More specifically, as shown in  FIG. 9 , in step S 901 , feature data at all exposures corresponding to the focus amount range, including the optimum focus amount registered in the specific data removal information storage unit S 707 , is removed from the positive focus feature data (S 704 ), and the obtained feature data is output to the positive focus learning data storage unit S 708 . 
     Also, in step S 901 , feature data at all exposures corresponding to the focus amount range, including the optimum focus amount registered in the specific data removal information storage unit S 707 , is removed from the negative focus feature data (S 705 ), and the obtained feature data is output to the negative focus learning data storage unit S 709 . 
     &lt;Determining Unit S 702 &gt; 
     As shown in  FIG. 10 , in step S 1001 , the determining unit S 702  receives the positive focus feature data group of the positive focus learning data (S 708 ) and generates a Mahalanobis space as a positive focus probability space. 
     In step S 1002 , the determining unit S 702  receives the negative focus feature data group of the negative focus learning data (S 709 ) and generates a Mahalanobis space as a negative focus probability space. 
     When the Mahalanobis space is used, the distance from the center of the data group can be assumed to be an occurrence probability belonging to the data group. This assumption does not hold in a Euclidean space. In addition, the calculation amount using a Mahalanobis space is smaller than the calculation amount using a Bayes space. 
     In step S 1003 , the feature data of the unknown waveform is projected to the positive focus probability space calculated in step S 1001 , thereby calculating the probability that the unknown waveform is a positive focus waveform. In step S 1003 , the feature data of the unknown waveform is also projected to the negative focus probability space calculated in step S 1002 , thereby calculating the probability that the unknown waveform is a negative focus waveform. 
     If the probability that the unknown waveform is a positive focus waveform, which is calculated in step S 1003 , is higher than the probability that the unknown waveform is a negative focus waveform in step S 1004 , information representing that the defocus direction of the unknown waveform is the positive direction is output to the input/output unit S 701  (S 1006 ). Otherwise, the processing advances to step S 1005 . 
     If the probability that the unknown waveform is a positive focus waveform, which is calculated in step S 1003 , equals the probability that the unknown waveform is a negative focus waveform in step S 1005 , information representing that the defocus direction of the unknown waveform is zero (optimum focus) is output to the input/output unit S 701  (S 1007 ). Otherwise, information representing that the defocus direction of the unknown waveform is the negative direction is output to the input/output unit S 701  (S 1008 ). 
     Second Embodiment 
     In the second embodiment, a Bayes space is used in place of the Mahalanobis space in steps S 1001  and S 1002  of the determining unit S 702  of the first embodiment. 
     In the Bayes space, the distance from the center of the data group can be assumed to be an occurrence probability belonging to the data group, as in the Mahalanobis space. However, the Bayes space is a complete multidimensional normal distribution probability space based on the Bayes conditional probability. On the other hand, the Mahalanobis space is no multidimensional normal distribution in the strict sense, although each dimension is a normal distribution. For this reason, the occurrence probability when the Mahalanobis space is used contains a large error as compared to the occurrence probability when the Bayes space is used. Hence, the robustness of determination can be improved as compared to the first embodiment. 
     Third Embodiment 
     In the third embodiment, the learning data calculating unit S 706  of the first embodiment has the following arrangement. 
     The feature data of a waveform having a focus amount largely different from the optimum focus amount corresponds to the outliers. An object of this embodiment is to remove the data from learning data to prevent the outliers from influencing the statistic of the positive focus feature data group and the statistic of the negative focus feature data group. 
     More specifically, in step S 901  shown in  FIG. 9 , feature data at all exposures corresponding to the focus amount range, including the maximum focus amount registered in a specific data removal information storage unit S 707 , is removed from positive focus feature data (S 704 ), and the obtained feature data is output to a positive focus learning data storage unit S 708 . 
     Also, in step S 901 , feature data at all exposures corresponding to the focus amount range including the maximum focus amount registered in the specific data removal information storage unit S 707  is removed from negative focus feature data (S 705 ), and the obtained feature data is output to a negative focus learning data storage unit S 709 . 
     Fourth Embodiment 
     In the fourth embodiment, a Bayes space is used in place of the Mahalanobis space in steps S 1001  and S 1002  of the determining unit S 702  of the third embodiment. Even in this case, the robustness of the determination can be improved, as compared to the third embodiment, due to the reason described in the second embodiment. 
     Fifth Embodiment 
     In the fifth embodiment, the learning data calculating unit S 706  of the first embodiment has the following arrangement. 
     The feature data of a waveform having an exposure largely different from the optimum exposure corresponds to the outliers. An object of this embodiment is to remove the data from learning data to prevent the outliers from influencing the statistic of the positive focus feature data group and the statistic of the negative focus feature data group. 
     More specifically, in step S 901  shown in  FIG. 9 , feature data at all focus amounts corresponding to the exposure range, including the maximum exposure registered in a specific data removal information storage unit S 707 , is removed from positive focus feature data (S 704 ), and the obtained feature data is output to a positive focus learning data storage unit S 708 . 
     Also, in step S 901 , feature data at all focus amounts, corresponding to the exposure range, including the maximum exposure registered in the specific data removal information storage unit S 707 , is removed from negative focus feature data (S 705 ), and the obtained feature data is output to a negative focus learning data storage unit S 709 . 
     Sixth Embodiment 
     In the sixth embodiment, a Bayes space is used in place of the Mahalanobis space in steps S 1001  and S 1002  of the determining unit S 702  of the fifth embodiment. Even in this case, the robustness of the determination can be improved, as compared to the fifth embodiment, due to the reason described in the second embodiment. 
     Seventh Embodiment 
     In the seventh embodiment, the learning data calculating unit S 706  of the first embodiment has the following arrangement. 
     This embodiment has two objects. 
     The first object is to remove, from learning data, the feature data of a waveform having a focus amount near the absolute value of the optimum focus amount and to prevent an overlap between the positive focus feature data group and negative focus feature data group, thereby improving the degree of separation of the two feature data groups. 
     The second object is to remove, from learning data, the feature data of a waveform having a focus amount largely different from the optimum focus amount, to prevent the outliers from influencing the statistic of the positive focus feature data group and the statistic of the negative focus feature data group, and to remove, from learning data, the feature data of a waveform having an exposure largely different from the optimum exposure, to prevent the outliers from influencing the statistic of the positive focus feature data group and the statistic of the negative focus feature data group. 
     More specifically, in step S 901  shown in  FIG. 9 , feature data at all exposures corresponding to the focus amount range, including the optimum focus amount registered in a specific data removal information storage unit S 707 , feature data at all exposures corresponding to the focus amount range, including the maximum defocus amount, and feature data at all focus amounts corresponding to the exposure range, including the maximum exposure, are removed from positive focus feature data (S 704 ), and the obtained feature data is output to a positive focus learning data storage unit S 708 . 
     Also, in step S 901 , feature data at all exposures corresponding to the focus amount range, including the optimum focus amount registered in the specific data removal information storage unit S 707 , feature data at all exposures corresponding to the focus amount range, including the maximum defocus amount, and feature data at all focus amounts corresponding to the exposure range, including the maximum exposure, are removed from negative focus feature data (S 705 ), and the obtained feature data is output to a negative focus learning data storage unit S 709 . 
     Eighth Embodiment 
     In the eighth embodiment, a Bayes space is used in place of the Mahalanobis space, in steps S 1001  and S 1002  of the determining unit S 702  of the seventh embodiment. Even in this case, the robustness of the determination can be improved as compared to the fifth embodiment due to the reason described in the second embodiment. 
     As an effect common to the above-described embodiments, a conventional overlay inspection apparatus can be used without changing its hardware configuration. 
     When the above-described functions are added to the conventional overlay inspection apparatus, an overlay error, an exposure error, and a focus error can be measured as important performance of an exposure apparatus. 
     Hence, according to the above-described embodiments, exposure by the exposure apparatus can be done in an optimum focus state. More specifically, the conventional overlay inspection apparatus cannot determine the direction of the focus error. However, according to the above-described embodiments, the direction can be determined. For this reason, the focus error can be reduced by, e.g., setting, in the exposure apparatus, an offset value corresponding to the focus error, including the direction measured by the overlay inspection apparatus. 
     [Schematic Arrangement of Exposure Apparatus] 
       FIG. 12  is a schematic view of a semiconductor exposure apparatus including the overlay inspection apparatus shown in  FIG. 6 . 
     Referring to  FIG. 12 , a semiconductor exposure apparatus  21  comprises a reducing projection optical system  23 , which reduces and projects a reticle  22  with a predetermined circuit pattern, a wafer chuck  25 , which holds a wafer  24  on which an underlying pattern and an alignment mark are formed in the preprocess, a wafer stage  26 , which aligns the wafer  24  to a predetermined position (alignment position), and an alignment optical system (alignment scope)  27  used to detect the position of the alignment mark formed on the wafer  24  and to inspect the overlay, the exposure error, and the focus error. The alignment optical system  27  can have the same arrangement as in, e.g.,  FIG. 6 . 
     The alignment optical system  27  sends, to a signal processing unit  27 , an image signal obtained by photoelectrically converting an optical image of the mark on the wafer  24  by the CCD camera  12 . The signal processing unit  27  calculates mark position information on the basis of the image signal. On the basis of the position information calculated by the signal processing unit  27 , a central processing unit  28  positions the wafer stage  26  to correct the misalignment of the wafer. The central processing unit  28  also acquires, through the alignment optical system  27 , the images of various marks to measure the overlay error, the exposure error, and the focus error, and applies the method described in the above embodiments. In this way, an overlay error, an exposure error, and a focus error, including a direction, are measured. 
     [Device Manufacturing Method] 
     A semiconductor device manufacturing process using the exposure apparatus of the embodiment will be described next.  FIG. 13  is a flowchart showing the entire flow of the semiconductor device manufacturing process. In step S 1  (circuit design), the circuit of a semiconductor device is designed. In step S 2  (mask preparation), a mask is prepared on the basis of the designed circuit pattern. 
     In step S 3  (wafer manufacture), a wafer is manufactured using a material such as silicon. In step S 4  (wafer process), called a preprocess, an actual circuit is formed on the wafer by the exposure apparatus, by lithography, using the mask and wafer. In step S 5  (assembly), called a post-process, a semiconductor chip is formed from the wafer prepared in step S 4 . The step includes assembly processes, such as assembly (dicing and bonding) and packaging (chip encapsulation). In step S 6  (inspection), inspections, including an operation check test and a durability test of the semiconductor device manufactured in step S 5 , are performed. A semiconductor device is completed with these processes and shipped, in step S 7 . 
     The wafer process in step S 4  has the following steps ( FIG. 14 ): an oxidation step of oxidizing the surface of the wafer, a CVD step of forming an insulating film on the wafer surface, an electrode formation step of forming an electrode on the wafer by deposition, an ion implantation step of implanting ions into the wafer, a resist process step of applying a photosensitizer to the wafer, an exposure step of transferring the circuit pattern to the wafer after the resist process step by the above-described exposure apparatus, a development step of developing the wafer exposed in the exposure step, an etching step of etching portions other than the resist image developed in the development step, and a resist removal step of removing any unnecessary resist remaining after etching. By repeating these steps, a multilayered structure of circuit patterns is formed on the wafer. 
     As many apparently widely different embodiments of the present invention can be made without departing from the spirit and scope thereof, it is to be understood that the invention is not limited to the specific embodiments thereof, except as defined in the claims.