Method for determining arrangement of multiple shot areas on a substrate, exposure method, exposure apparatus, article manufacturing method, program, and information processing apparatus

Abstract

To provide a technique which is advantageous for accurately determining the arrangement of areas on a substrate.SOLUTION: There is provided a method for determining the arrangement of a plurality of areas on a substrate, the method comprising: a first step for using an objective function including regression models for estimating the arrangement, and a regularization term for regulating values of coefficients of the regression models, and position measurement data of a mark allocated to a sample area out of the plurality of areas on a first substrate, for calculating respective values of the plurality of coefficients included in the regression models for setting the objective function to a reference value or lower value; a second step for, on the basis of the values calculated in the first step, extracting coefficients having a value equal to or greater than a threshold, out of the plurality of coefficients; and a third step for, using the regression model including only the coefficients extracted in the second step, and position measurement data of the mark allocated to the sample area out of the plurality of areas on a second substrate, determining the arrangement of the plurality of areas on the second substrate.SELECTED DRAWING: Figure 5

Description

[Technical Field]

[0001] The present invention relates to a method for determining an arrangement of a plurality of shot areas on a substrate, an exposure method, an exposure apparatus, an article manufacturing method, a program, and an information processing apparatus. [Background technology]

[0002] The exposure equipment transfers 10 or more layers of patterns (circuit patterns) onto the substrate in layers, but if the pattern alignment accuracy between layers is poor, it can cause problems with the circuit characteristics. In such cases, the chip will not meet the desired characteristics, resulting in a defective chip and reduced yield. Therefore, precise alignment is required between each of the multiple areas to be exposed on the substrate and the original pattern.

[0003] In an exposure apparatus, alignment marks arranged in each region on a substrate are detected, and each region on the substrate is aligned with the pattern on the original based on the position information of the alignment marks and the position information of the pattern on the original. Ideally, the most accurate alignment would be possible by detecting alignment marks in all regions on the substrate, but this is not realistic from the viewpoint of productivity. Therefore, global alignment methods are currently the mainstream method for aligning a substrate and an original (see Patent Documents 1 and 2).

[0004] In the global alignment method, it is assumed that the relative position of each region on a substrate can be expressed by a function model of the region's position coordinates, and the positions of alignment marks arranged only in multiple (4 to 16) sample regions on the substrate are measured. Next, parameters of the function model are estimated using regression analysis-like statistical calculation processing from the assumed function model and the measurement results of the alignment mark positions. Then, using the parameters and function model, the position coordinates of each region in the stage coordinate system (the arrangement of regions on the substrate) are calculated to perform alignment. In the global alignment method, a polynomial model with stage coordinates as variables is generally used, and scaling, rotation, uniform offset, and other linear polynomials of the stage coordinates are mainly used (see Patent Document 3). A technology using a regression model that also considers higher-order components of the arrangement of regions on the substrate as parameters has also been proposed (see Patent Document 4). [Prior art documents] [Patent documents]

[0005] [Patent Document 1] Japanese Unexamined Patent Publication No. 61-44429 [Patent Document 2] Japanese Patent Application Publication No. 62-84516 [Patent Document 3] Japanese Patent Application Publication No. 6-349705 [Patent Document 4] Patent No. 3230271 Summary of the Invention [Problem to be solved by the invention]

[0006] As device miniaturization and integration become more advanced, improved alignment accuracy is required. Therefore, it is necessary to increase the degrees of freedom of the polynomial order of the function model by using higher-order components. However, if the number of measurement points for measuring the positions of alignment marks on a substrate is small relative to the degrees of freedom of the function model, overfitting occurs, resulting in increased correction errors in unmeasured areas. On the other hand, increasing the number of measurement points for measuring the positions of alignment marks to prevent overfitting increases the measurement time and reduces productivity. Because of this trade-off, a technology is needed that can accurately predict the arrangement of regions on a substrate, including high-order components, using a function model with a small number of measurement points and a high degree of freedom.

[0007] The present invention has been made in view of the above problems of the conventional technology, and has an exemplary object to provide a technology that is advantageous for determining the arrangement of regions on a substrate with high precision. [Means for solving the problem]

[0008] In order to achieve the above object, one aspect of the present invention is a method for determining the arrangement of a plurality of regions on a substrate, the method comprising: A loss function for calculating the coefficients, which includes the coefficients included in and the above multiple and a regularization term for limiting the value of the coefficient, and position measurement data of a mark assigned to a sample region among a plurality of regions on a first substrate, to calculate values ​​of each of a plurality of coefficients included in the regression model so that the objective function is equal to or less than a reference value; a second step of extracting coefficients having values ​​equal to or greater than a threshold from the plurality of coefficients based on the values ​​calculated in the first step; and a third step of determining the arrangement of a plurality of regions on the second substrate using a regression model including only the coefficients extracted in the second step and position measurement data of a mark assigned to a sample region among a plurality of regions on the second substrate.

[0009] Further objects and other aspects of the present invention will become apparent from the following description of the embodiments with reference to the accompanying drawings. [Effects of the Invention]

[0010] According to the present invention, for example, it is possible to provide a technique that is advantageous for determining the arrangement of regions on a substrate with high precision. [Brief explanation of the drawings]

[0011] [Figure 1] 1 is a schematic diagram showing the configuration of an exposure apparatus according to one aspect of the present invention. [Figure 2] FIG. 2 is a schematic diagram showing the configuration of an alignment optical system of the exposure apparatus shown in FIG. [Figure 3] 2 is a flowchart for explaining exposure processing in the exposure apparatus shown in FIG. [Figure 4] FIG. 2 is a diagram showing an arrangement of shot areas on a substrate. [Figure 5] 10 is a flowchart for explaining a process of finding a shot arrangement by optimizing a regression model that estimates a shot arrangement on a substrate. [Figure 6] This is a diagram to explain why the L1 norm tends to produce sparse results. [Figure 7] FIG. 10 is a diagram for explaining the effect of a regularized regression model including an L1 norm. [Figure 8] 10 is a flowchart illustrating a process of determining a shot arrangement using an objective function including a regression model and a regularization term. DETAILED DESCRIPTION OF THE INVENTION

[0012] Hereinafter, embodiments will be described in detail with reference to the accompanying drawings. Note that the following embodiments do not limit the invention according to the claims. Although multiple features are described in the embodiments, not all of these multiple features are necessarily essential to the invention, and multiple features may be combined arbitrarily. Furthermore, in the accompanying drawings, the same reference numerals are used to designate the same or similar components, and redundant explanations will be omitted.

[0013] 1 is a schematic diagram showing the configuration of an exposure apparatus 1 according to one aspect of the present invention. The exposure apparatus 1 is a lithography apparatus used in the manufacturing process of devices such as semiconductor elements. In this embodiment, the exposure apparatus 1 projects a pattern of an original 2 (reticle or mask) onto a substrate 4 via a projection optical system 3, thereby exposing the substrate 4 to light.

[0014] 1, the exposure apparatus 1 has a projection optical system 3 that projects (reduces projection) a pattern formed on an original 2, and a chuck 5 that holds a substrate 4 on which a base pattern and alignment marks have been formed in a previous process. The exposure apparatus 1 also has a substrate stage 6 that holds the chuck 5 and positions the substrate 4 at a predetermined position, an alignment optical system 7 that measures the position of the alignment marks provided on the substrate 4, a control unit CN, and a memory unit SU.

[0015] The control unit CN is configured, for example, by a computer (information processing device) including a CPU, memory, etc., and performs overall control of each unit of the exposure apparatus 1 in accordance with programs stored in the storage unit SU, etc. In this embodiment, the control unit CN not only controls the exposure process in which the substrate 4 is exposed via the original 2, but also functions as a processing unit that determines the arrangement (shot arrangement, area arrangement) of multiple shot areas (multiple areas on the substrate) on the substrate.

[0016] The storage unit SU stores programs and various information (data) required to control each part of the exposure apparatus 1 and perform exposure processing to expose the substrate 4. The storage unit SU also stores programs and various information (data) required for the control unit CN to determine the shot arrangement.

[0017] 2 is a schematic diagram showing the configuration of the alignment optical system 7. The alignment optical system 7 has a function of optically detecting marks assigned to each shot area of ​​the substrate 4 and acquiring position measurement data, and in this embodiment includes a light source 8, a beam splitter 9, lenses 10 and 13, and a sensor 14.

[0018] Light from light source 8 is reflected by beam splitter 9 and passes through lens 10 to illuminate alignment mark 11 or 12 provided on substrate 4. Light diffracted by alignment mark 11 or 12 passes through lens 10, beam splitter 9, and lens 13 and is received by sensor 14.

[0019] The exposure process in the exposure apparatus 1 will be described with reference to FIG. 3. Here, an overview of the steps from aligning the substrate 4 to exposing it will be described. In S101, the substrate 4 is loaded into the exposure apparatus 1. In S102, pre-alignment is performed. Specifically, alignment marks 11 for pre-alignment provided on the substrate 4 are detected by the alignment optical system 7, and the position of the substrate 4 is roughly determined. At this time, detection of the alignment marks 11 is performed for multiple shot areas on the substrate 4, and the overall shift and first-order linear components (magnification and rotation) of the substrate 4 are determined.

[0020] In S103, fine alignment is performed. Specifically, first, based on the results of the pre-alignment, the substrate stage 6 is driven to a position where the alignment marks 12 for fine alignment provided on the substrate 4 can be detected by the alignment optical system 7. Then, the alignment marks 12 provided in each of the multiple shot areas on the substrate 4 are detected by the alignment optical system 7, and the overall shift and first-order linear components (magnification and rotation) of the substrate 4 are precisely determined. At this time, by determining the positions of multiple shot areas, it is also possible to precisely determine higher-order deformation components of the substrate 4. This makes it possible to determine the precise position of each shot area on the substrate 4, i.e., the shot arrangement.

[0021] In S104, the substrate 4 is exposed. Specifically, after fine alignment is performed, the pattern of the original 2 is transferred to each shot area of ​​the substrate 4 via the projection optical system 3. In S105, the substrate 4 is unloaded from the exposure apparatus 1.

[0022] In this embodiment, if distortion occurs in the substrate 4, high-order deformation components are corrected in the fine alignment of S103. Here, a three-dimensional polynomial model is used as an example of a regression model for estimating the shot arrangement, but the present invention is not limited to this. For example, a regression model of any order may be used, or a model other than a polynomial (a trigonometric function model or a logarithmic model) may be used.

[0023] When the deformation of the substrate 4 is expressed by a third-order polynomial model, the positional shift (ShiftX, ShiftY) of each shot area is expressed by the following equation (1). Note that the positional shift of each shot area can also be considered as a correction value for correcting such positional shift.

[0024]

number

[0025] In equation (1), x and y indicate the position of the shot area on the substrate 4. From the actual position measurement data of each shot area on the substrate 4, k1 to k 20 Then, the positional deviation of each shot area is calculated based on the equation (1) for which the coefficients have been determined.

[0026] To obtain position measurement data, the alignment optical system 7 detects alignment marks 12 assigned to some shot areas, so-called sample shot areas (sample areas), of the multiple shot areas on the substrate, as shown in FIG. 4, for example. In FIG. 4, the number of sample shot areas is set to 14. In order to correct high-order deformation components of the substrate 4, many shot areas need to be set as sample shot areas. However, since an increase in the number of sample shot areas is in a trade-off relationship with the measurement time (alignment time), in practice the number of sample shot areas is determined taking into consideration the productivity of the device.

[0027] Hereinafter, a process of optimizing a regression model that estimates a shot arrangement on a substrate and determining the shot arrangement using the regression model in this embodiment will be described with reference to Fig. 5. Note that in this process, optimization of the regression model that estimates a shot arrangement on a substrate is assumed to be performed in the control unit CN in this embodiment, but may also be performed in an external information processing device.

[0028] In S502, first position measurement data is acquired using a test substrate (first substrate). Specifically, the number of sample shot areas on the test substrate is set to a number that does not overlearn with respect to the degrees of freedom of the model, and is sufficient for the degrees of freedom of the model. Then, alignment marks assigned to each sample shot area on the test substrate are detected by the alignment optical system 7 to acquire the first position measurement data.

[0029] In S504, each coefficient of the regression model for estimating the shot arrangement is calculated (first step). Specifically, using the first position measurement data acquired in S502 and a regularized regression model, each value of a plurality of coefficients included in the regression model for estimating the shot arrangement is calculated so that the regularized regression model is equal to or less than a reference value, for example, minimized. Here, the regularized regression model is a function (objective function) including the regression model for estimating the shot arrangement and a regularization term for limiting the value of the coefficient of the regression model.

[0030] In S506, based on the values ​​of the coefficients of the regression model calculated in S504, coefficients having values ​​equal to or greater than a preset threshold are extracted (selected) from the multiple coefficients included in the regression model (second step). Note that with regard to the threshold, the same (common) threshold may be set for all coefficients included in the regression model, or a threshold may be set for each of the multiple coefficients included in the regression model.

[0031] In S508, second position measurement data is acquired using substrate 4 (second substrate), which is the substrate to be actually exposed. Specifically, the number of sample shot areas for substrate 4 is set to a number smaller than the number of sample shots set for the test substrate. Then, alignment marks 12 assigned to each sample shot area of ​​substrate 4 are detected by alignment optical system 7 to acquire second position measurement data.

[0032] In S510, the shot arrangement (arrangement of multiple shot areas) of the substrate 4 is determined (third step). Specifically, the shot arrangement of the substrate 4 (i.e., the positional deviation of each shot area of ​​the substrate 4) is determined using a regression model including only the coefficients extracted in S506 and the second position measurement data acquired in S508. Here, the regression model including only the coefficients extracted in S506 has unnecessary coefficients (models corresponding to them) removed by regularized regression, and only includes important coefficients (models corresponding to them). Therefore, overlearning can be suppressed even if the number of sample shot areas (measurement points) set for the substrate 4 is small.

[0033] As described above, according to this embodiment, even while using a small number of measurement points and a function model with a high degree of freedom, overlearning can be suppressed and the shot arrangement of substrate 4 can be determined with high accuracy, making it possible to correct high-order deformation components. Note that in this embodiment, a test substrate is used in S502, but instead of the test substrate, substrate 4, which is the substrate to be actually exposed, may be used. In other words, instead of using different substrates in S502 and S508, the same substrate (the substrate to be actually exposed) may be used in S502 and S508.

[0034] The regularized regression model in this embodiment will be specifically described below. First, as a comparative example, a general regression model that does not include a regularization term, i.e., has no regularization, and its problems will be described.

[0035] As a simple example, consider a regression model up to second order in the substrate coordinates. The optimal coefficients (k1, k3, k5, k7, k9, k 11), first, each position measurement data Y i and the predicted value T from the regression model i The difference between these values ​​is calculated by adding up the differences between these values ​​for all data points. In equation (2), i is an integer from 1 to n, and n is the number of data points.

[0036]

number

[0037] Equation (2) is generally called the loss function. The optimization of each coefficient of a general regression model without regularization is performed by the above-mentioned coefficients k1 to k 11 This can be replaced with an optimization problem in which a loss function with variables is minimized as an objective function. However, since there is no limit to the size of the variables in this type of optimization, when minimizing the objective function, the coefficients, which are variables (parameters), can become as large as desired. As variables become larger, the likelihood of overfitting generally increases, which increases the correction error in unmeasured areas.

[0038] Therefore, in this embodiment, a regularized regression model is used. Specifically, as the regularized regression model, a regularization term in which the model variable is a function is added to a loss function, which is a general regression model without regularization, as shown in the following formula (3). In formula (3), the regularization term includes a monotonically increasing function of the variable.

[0039]

number

[0040] In this way, by providing a regularization term in the objective function, an increase in the variable is penalized, and therefore it is possible to suppress (prevent) the variable from becoming excessively large. In other words, in an objective function that includes a loss function and a regularization term, it is possible to suppress overfitting.

[0041] The regularization term can be set arbitrarily as long as it is a monotonically increasing function of the variables. In this embodiment, the regularization term is described as including the Lp norm of the variable vector, and in particular, including the L1 norm where p is 1. However, the regularization term is not limited to the L1 norm where p is 1, and for example, the regularization term can include the L2 norm where p is 2. Furthermore, the regularization term can include the Lp norm where p is an arbitrary number. The following equation (4) represents the Lp norm, L1 norm, and L2 norm. In equation (4), D represents the number of variables used, and k represents the value of each variable.

[0042]

number

[0043] The effect of regularizing variables varies depending on the type of norm. The L1 norm, by its nature, tends to make unnecessary coefficients fall to zero, but it can also result in sparse coefficients.

[0044] With reference to Figures 6(a) and 6(b), we explain why the L1 norm tends to produce sparse results, comparing it with the L2 norm. The L1 norm uses the sum of the absolute values ​​of the model coefficients as a penalty. The L2 norm uses the sum of the squares of the model coefficients as a penalty. Both the L1 norm and the L2 norm are regularized regression models in which the penalty decreases as the coefficient decreases. However, the minimization problem of an objective function that includes a norm can be replaced with the problem of searching for a solution that minimizes the loss function within a range defined by the norm. Because the defined range differs between the L1 norm and the L2 norm, the optimal solution also differs between the L1 norm and the L2 norm. As shown in Figure 6(b), in the L2 norm, the defined range for each model variable (two variables) is circular, so the optimal solution within this range tends to be a proportional reduction of the optimal solution without a regularization term (when only the loss function is used). On the other hand, with the L1 norm, as shown in FIG. 6(a), the range defined for each model variable has an edge shape, so the optimal solution tends to have one of the coefficients set to zero. Therefore, by using a regression model and an objective function that includes the L1 norm as a regularization term, it becomes possible to efficiently thin out unnecessary coefficients. Note that in FIGS. 6(a) and 6(b), w1 and w2 correspond to coefficients (e.g., k1 and k3) included in the regression model.

[0045] The effect of a regularized regression model including the L1 norm will be described with reference to Figures 7(a) and 7(b). Figure 7(a) shows the relationship between the number of measurement points and the correction error (correction residual) in the unmeasured region when all coefficients assumed in the regression model (models corresponding to them) are used. Figure 7(b) shows the relationship between the number of measurement points and the correction error (correction residual) in the unmeasured region when the coefficients of the regression model are extracted using the L1 norm as the regularization term (this embodiment).

[0046] When all the coefficients assumed in the regression model are used, as shown in Fig. 7(a), as the number of measurement points is reduced, overfitting begins at a certain number, and the correction error increases rapidly. On the other hand, in this embodiment, as shown in Fig. 7(b), even when the number of measurement points is reduced, the increase in correction error is limited to a gradual change. This is because unnecessary coefficients (or models corresponding to them) are removed by regularized regression and only important coefficients are extracted, so overfitting is suppressed even with a small number of measurement points.

[0047] In this embodiment, the L1 norm is used as the regularization term, but the present invention is not limited to this. For example, the Lp norm or a linear combination of the Lp norms may be used as the regularization term.

[0048] 8, a process for determining a shot arrangement using an objective function including a regression model and a regularization term, i.e., a regularized regression model, without optimizing the regression model for estimating the shot arrangement on the substrate (without extracting coefficients) also constitutes one aspect of the present invention. In this case, in S802, position measurement data is acquired using a substrate 4, which is the substrate to be actually exposed. Specifically, the number of sample shot areas for the substrate 4 is set to a relatively small number. Then, alignment marks 12 assigned to each sample shot area on the substrate 4 are detected by the alignment optical system 7 to obtain position measurement data. Then, in S804, the position measurement data acquired in S802 is applied to the regularized regression model, and the shot arrangement is determined by calculating the values ​​of each of the multiple coefficients included in the regression model so that the regularized regression model is equal to or less than a reference value, for example, minimized. In this way, even when the regression model is not optimized, unnecessary coefficients (or models corresponding to them) can be reduced by including a regularization term. This makes it possible to suppress overlearning even with a small number of measurement points.

[0049] The method for manufacturing an article according to an embodiment of the present invention is suitable for manufacturing articles such as liquid crystal display elements, semiconductor elements, flat panel displays, and MEMS. This manufacturing method includes the steps of exposing a substrate coated with a photosensitive agent using the exposure apparatus 1 or exposure method described above, and developing the exposed photosensitive agent. The developed photosensitive agent pattern is then used as a mask to perform etching and ion implantation processes on the substrate, forming a circuit pattern on the substrate. These exposure, development, etching, and other processes are repeated to form a circuit pattern consisting of multiple layers on the substrate. In subsequent processes, the substrate on which the circuit pattern has been formed is diced, followed by chip mounting, bonding, and inspection. This manufacturing method may also include other well-known processes (e.g., oxidation, film formation, vapor deposition, doping, planarization, resist stripping, etc.). The method for manufacturing an article according to this embodiment is advantageous over conventional methods in at least one of the performance, quality, productivity, and production cost of the article.

[0050] The present invention can also be realized by supplying a program that implements one or more of the functions of the above-described embodiments to a system or device via a network or a storage medium, and having one or more processors in the computer of the system or device read and execute the program.The present invention can also be realized by a circuit (e.g., ASIC) that implements one or more of the functions.

[0051] The invention is not limited to the above-described embodiments, and various changes and modifications can be made without departing from the spirit and scope of the invention. Accordingly, the following claims are appended to apprise the public of the scope of the invention. [Explanation of symbols]

[0052] 1: Exposure device 2: Original 3: Projection optical system 4: Substrate 5: Substrate chuck 6: Substrate stage 7: Alignment optical system CN: Control unit SU: Memory unit

Claims

1. 1. A method for determining the alignment of a plurality of regions on a substrate, comprising: a first step of calculating values ​​of a plurality of coefficients included in a regression model for estimating the arrangement, the values ​​being equal to or less than a reference value, using an objective function including a loss function for determining the plurality of coefficients, the loss function including variables that are included in the regression model for estimating the arrangement, and a regularization term for limiting values ​​of the plurality of coefficients, and position measurement data of marks assigned to a sample region among a plurality of regions on a first substrate; a second step of extracting coefficients having values ​​equal to or greater than a threshold value from the plurality of coefficients based on the values ​​calculated in the first step; a third step of determining an arrangement of the plurality of regions on the second substrate using a regression model including only the coefficients extracted in the second step and position measurement data of marks assigned to sample regions among the plurality of regions on the second substrate; A method comprising:

2. 2. The method according to claim 1, wherein the first step calculates values ​​of a plurality of coefficients included in the regression model that minimize the objective function.

3. 3. The method of claim 1, wherein the number of sample areas on the second substrate is smaller than the number of sample areas on the first substrate.

4. 4. The method of claim 1, wherein the first substrate and the second substrate are different substrates.

5. 4. The method according to claim 1, wherein the first substrate and the second substrate are the same substrate.

6. 6. The method of claim 1, wherein the regularization term comprises an Lp norm.

7. 7. The method of claim 6, wherein the Lp norm comprises an L1 norm or an L2 norm.

8. 8. A method according to any one of claims 1 to 7, wherein the regression model comprises a polynomial model that includes the position of the substrate as a variable.

9. An exposure method for exposing a substrate through an original, comprising: Determining the arrangement of a plurality of regions on a substrate using the method of any one of claims 1 to 8; transferring the pattern of the original onto each of the plurality of regions while positioning the substrate based on the arrangement determined in the step; An exposure method comprising:

10. An exposure apparatus that exposes a substrate through an original, a processing unit for determining an arrangement of a plurality of regions on a substrate; a stage that positions the substrate based on the arrangement determined by the processing unit, The processing unit calculating values ​​of the plurality of coefficients included in the regression model that make the objective function equal to or less than a reference value, using an objective function that includes, as variables, a plurality of coefficients included in a regression model for estimating the arrangement, a loss function for calculating the plurality of coefficients, and a regularization term for limiting values ​​of the plurality of coefficients, and position measurement data of marks assigned to sample regions among a plurality of regions on a first substrate; extracting coefficients having values ​​equal to or greater than a threshold value from the plurality of coefficients based on the calculated values; determining an arrangement of the plurality of regions on the second substrate using a regression model including only the extracted coefficients and position measurement data of marks assigned to sample regions among the plurality of regions on the second substrate; An exposure apparatus characterized by:

11. exposing a substrate using the exposure method according to claim 9; developing the exposed substrate; manufacturing an article from the developed substrate; A method for manufacturing an article, comprising:

12. A program for causing a computer to execute a method for determining the arrangement of a plurality of regions on a substrate, The computer, a first step of calculating values ​​of a plurality of coefficients included in a regression model for estimating the arrangement, the values ​​being equal to or less than a reference value, using an objective function including a loss function for determining the plurality of coefficients, the loss function including variables that are included in the regression model for estimating the arrangement, and a regularization term for limiting values ​​of the plurality of coefficients, and position measurement data of marks assigned to a sample region among a plurality of regions on a first substrate; a second step of extracting coefficients having values ​​equal to or greater than a threshold value from the plurality of coefficients based on the values ​​calculated in the first step; a third step of determining an arrangement of the plurality of regions on the second substrate using a regression model including only the coefficients extracted in the second step and position measurement data of marks assigned to sample regions among the plurality of regions on the second substrate; A program characterized by executing the following.

13. An information processing apparatus that executes the method according to any one of claims 1 to 8.

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