Analysis method, analysis apparatus, measurement method, measurement apparatus, exposure method, and exposure apparatus

Abstract

Provided is an analysis method capable of expressing non-linear deformations that are not proportional to a position within a substrate.　The analysis method comprises: preparing measurement position information, which is measured position information of a plurality of measured parts (MP) formed on a substrate; and fitting, with respect to the measurement position information, a reference function, which is a sum of at least one function obtained by multiplying a proportional coefficient with a standard function represented using a Bessel function of the first kind, and calculating an appropriate value for at least one proportional coefficient.

Description

[Technical field]

[0001] The present invention relates to an analysis method, an analysis apparatus, a measurement method, a measurement apparatus, an exposure method, and an exposure apparatus. This application claims priority based on Japanese Patent Application No. 2021-073425, filed on April 23, 2021, the contents of which are incorporated herein by reference. [Background technology]

[0002] In lithography technology used in the manufacture of semiconductor integrated circuits and the like, a pattern is formed by superimposing another pattern on an existing pattern formed on a substrate. In the lithography process, the substrate may undergo not only linear deformation, which is uniform expansion and contraction proportional to the position in the substrate, but also nonlinear deformation, which is not proportional to the position in the substrate. One method for expressing nonlinear deformation of a substrate is disclosed in, for example, Patent Document 1. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] International Publication No. 2015 / 189001 Summary of the Invention

[0004] According to a first aspect, an analysis method includes: preparing measurement position information, which is measured position information of a plurality of measurement targets formed on a substrate; and fitting a reference function, which is a sum of at least one function obtained by multiplying a base function expressed using a first kind Bessel function by a proportionality coefficient, to the measurement position information to calculate an appropriate value of at least one of the proportionality coefficients. The reference function is expressed as a product of a first kind Bessel function for the distance from the reference position of the substrate to the measurement portion and a trigonometric function based on the deflection angle from the reference position to the measurement portion. . According to a second aspect, an analysis method includes preparing measurement position information, which is measured position information of a plurality of measured portions formed on a substrate, and fitting a reference function, which is the sum of at least one function obtained by multiplying a base function expressed using a first kind of Bessel function by a proportionality coefficient, to the measurement position information to calculate an appropriate value of at least one of the proportionality coefficients, wherein the base function is expressed using a first kind of Bessel function for a distance from a reference position of the substrate to the measured portions, and the first kind of Bessel function is a function that takes an extreme value with respect to the base position information of the measured portions arranged on the periphery of the substrate. No. 3According to this aspect, the analysis device includes an input unit that inputs measurement position information, which is measured position information of a plurality of measurement target parts formed on a substrate, an analysis unit that analyzes the measurement position information of the measurement target parts measured by the position measurement unit, and an output unit that outputs an analysis result including at least the calculated suitable value of the proportionality coefficient to an external device, and the analysis unit: mentioned above By the above analysis method, an optimum value of at least one of the proportionality coefficients is calculated. No. 4 According to this aspect, the measurement method includes measuring positions of a plurality of measurement target portions formed on a substrate, and obtaining measurement position information of the measurement target portions; mentioned above and calculating an optimal value of at least one of the proportionality coefficients by the analysis method. No. 5 According to the aspect of the present invention, the measurement apparatus includes a position measurement unit that measures the position of a measurement target portion formed on a substrate, and an analysis unit that analyzes measurement position information of the measurement target portion measured by the position measurement unit, and the analysis unit: mentioned above By the above analysis method, an optimum value of at least one of the proportionality coefficients is calculated. No. 6 According to the embodiment, the exposure method includes: mentioned above The measurement method includes measuring positions of a plurality of measured portions on the substrate, on which a first pattern is formed in a predetermined positional relationship with respect to the measured portions, acquiring measurement position information of the measured portions, and calculating an appropriate value of at least one of the proportionality coefficients, estimating a position of the first pattern on the substrate based on the acquired measurement position information and the appropriate value of at least one of the proportionality coefficients, and exposing a second pattern onto the substrate based on the estimated position of the first pattern. No. 7 According to the aspect of the present invention, an exposure apparatus includes a position measurement unit that measures the position of a measurement portion formed on a substrate, and an exposure optical system that exposes a desired pattern onto the substrate, mentioned above The exposure method is as follows. No. 8According to the aspect of the present invention, an exposure apparatus includes a stage that moves while placing a substrate on which a measurement portion is formed, and a substrate transport device that transports the substrate and places it on the stage, and exposes a desired pattern onto the substrate, mentioned above Based on the measurement results obtained by the measurement method, an apparatus constant that defines the operation of the substrate transport apparatus is determined. [Brief description of the drawings]

[0005] [Figure 1] 1 is a diagram showing the schematic arrangement of a measurement apparatus and exposure apparatus according to an embodiment; [Figure 2A] FIG. 2 is a diagram showing an example of a substrate. [Figure 2B] FIG. 2 is a diagram showing an example of a substrate. [Figure 3A] FIG. 4 is a diagram showing an example of a first kind Bessel function included in a basis function. [Figure 3B] FIG. 4 is a diagram showing an example of a first kind Bessel function included in a basis function. [Figure 4] FIG. 1 shows approximate values ​​of some criterion functions. [Diagram 5] 11 is a diagram showing an example of a virtual deformation amount in an XY in-plane direction of a substrate, expressed using a basis function. [Figure 6A] FIG. 11 is a diagram showing another example of the first kind Bessel function included in the basis function. [Figure 6B] FIG. 11 is a diagram showing another example of the first kind Bessel function included in the basis function. [Figure 7] FIG. 13 is a diagram showing an example of a vector diagram. [Figure 8] FIG. 1 is a diagram illustrating a schematic configuration of an analysis device according to an embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0006] (Measurement Apparatus and Exposure Apparatus of the Embodiments) Fig. 1 is a diagram showing a schematic configuration of a measurement apparatus 1 and an exposure apparatus 2 according to an embodiment. The exposure apparatus 2 includes the measurement apparatus 1 according to an embodiment, which is shown surrounded by a dashed line in Fig. 1. 1, the measurement apparatus 1 of the embodiment may be part of the configuration of the exposure apparatus 2, or may be an independent apparatus separate from the exposure apparatus 2. In the following, the measurement apparatus 1 will be described as constituting part of the configuration of the exposure apparatus 2.

[0007] The X, Y, and Z directions indicated by arrows in FIG. 1 and the following figures are perpendicular to each other, and the X, Y, and Z directions indicate the same direction in each figure. Hereinafter, the directions indicated by the arrows will be referred to as the +X direction, the +Y direction, and the +Z direction, respectively. Moreover, the position in the X direction is called the X position, and the position in the Y direction is called the Y position.

[0008] The exposure apparatus 2 is an apparatus that exposes and transfers a predetermined pattern onto the surface of the substrate WF. In addition to the measurement apparatus 1, the exposure apparatus 2 includes an exposure optical system 20, which is surrounded by a two-dot chain line in Fig. 1, a stage 30 on which the substrate WF is placed, and a mask stage 24 on which a mask 25, which is an original of a pattern to be exposed onto the substrate WF, is placed. Note that, although Fig. 1 shows two stages 30 whose positional relationship in the X direction is interchangeable with each other, there may be only one stage 30.

[0009] The exposure optical system 20 includes an illumination optical system 21 and a projection optical system 22. The illumination optical system 21 irradiates illumination light onto a mask 25, and the projection optical system 22 exposes the substrate WF with exposure light that has been transmitted through the mask 25 and has been given a light and dark pattern. Note that the mask 25 is not limited to a transmissive mask, and may be a reflective mask.

[0010] The stage 30 is equipped with a driving mechanism such as a linear motor, and is capable of moving in the X and Y directions on a guide 33, as well as moving a small distance (for example, about a few millimeters) in the Z direction. The position of the stage 30 is measured as the position of an encoder plate 32 provided on the stage 30 by an encoder head 23 provided so as to maintain a predetermined positional relationship with the projection optical system 22, and is output as a signal S2a.

[0011] The mask stage 24 is equipped with a driving mechanism such as a linear motor, and is capable of moving in the Y direction on the mask guide 27, as well as moving by small distances (for example, about a few millimeters) in the X and Z directions. The position of the mask stage 24 is measured as the position of an encoder plate 26 provided on the mask stage 24 by an encoder head 28 provided so as to maintain a predetermined positional relationship with the projection optical system 22, and is output as a signal S4.

[0012] The exposure apparatus 2 is, for example, a scanning type exposure apparatus that exposes the substrate WF while scanning a stage 30 on which the substrate WF is placed, and a mask stage 24 on which a mask 25 is placed, relative to the projection optical system 22 in the Y direction. However, the exposure apparatus 2 may be a step-and-repeat type exposure apparatus that performs exposure while the stage 30 on which the substrate WF is placed and the mask stage 24 on which the mask 25 is placed are stationary relative to the projection optical system 22.

[0013] The means for measuring the positions of the stage 30 and the mask stage 24 is not limited to the encoder system described above, but may also use a laser interferometer or a magnetic scale. The stage 30 has a substrate chuck portion 31 at a portion on which the substrate WF is placed. The substrate chuck portion 31 adsorbs the substrate WF by a so-called vacuum chuck or electrostatic chuck.

[0014] The substrate WF exposed by the exposure apparatus 2 contains electronic materials such as semiconductors, and near its surface (the surface on the +Z side) there is an existing pattern formed in a previous exposure process, and a photosensitive material (not shown) is formed on the surface of the substrate WF.

[0015] 2A is a diagram showing an example of the substrate WF, showing a view of the substrate WF as seen from the +Z direction. On the substrate WF, as part of an existing pattern (not shown), a plurality of predetermined measurement parts MP, such as alignment marks for position measurement, are formed at positions that have a predetermined positional relationship with the existing pattern. The measurement parts MP are not limited to a dedicated pattern provided for position measurement, and may be part of a circuit pattern that constitutes an electronic circuit formed on the substrate WF.

[0016] 2A is, for example, a central point such as the center of gravity of a circular substrate WF, the X axis represents an axis that passes through the origin O and extends in the X direction, and the Y axis represents an axis that passes through the origin O and extends in the Y direction. In addition, the distance from the origin O to one measured part MP is r, and the deflection angle from the +X direction to one measured part MP with the origin O as the reference is θ.

[0017] 2A, the intervals between the multiple measured parts MP in the X direction and the Y direction are all equal, but the intervals between the multiple measured parts MP in the X direction and the Y direction may both be unequal. Also, the multiple measured parts MP do not necessarily need to be arranged at lattice points of a lattice arrangement parallel to the X direction and the Y direction, and may be arranged at random positions on the substrate WF.

[0018] The position of each measurement portion MP is generally known from the design data of the pattern including the measurement portion MP that was exposed and transferred in the previous exposure process. However, since the substrate WF has been deformed in the XY in-plane direction by processing applied to the substrate WF after the exposure process that formed the measurement portion MP, the actual exact position of the measurement portion MP is unknown.

[0019] Fig. 2B is an enlarged view showing, as an example, the positions of the measurement parts MP (MP1-MP4) in an arbitrary small area AR of the substrate WF, which is shown in Fig. 2A surrounded by a dashed circle. Due to deformation of the substrate WF, each measurement part MP (MP1-MP4) is disposed at a position shifted by a position represented by ~DV4 in the in-plane direction of the substrate WF from the reference positions MD1-MD4 determined by the above-mentioned design data. In other words, the positional deviation amount of the measurement parts MP (MP1-MP4) from the reference positions MD1-MD4, i.e., the displacement vectors DV1-DV4, represent the deformation amount of the substrate WF in the XY in-plane direction.

[0020] Similarly, other measured parts MP not shown in Figure 2B are assumed to be positioned at positions shifted from reference positions MD1 to MD4 in the in-plane direction of the substrate WF by positions represented by displacement vectors DV, respectively, due to deformation of the substrate WF. The X and Y components of the displacement vector DV are ΔX and ΔY, respectively. The displacement vector DVj of the j-th (j is an arbitrary natural number) measured part MP is DVj=(ΔXj, ΔYj).

[0021] The positions where the measured parts MP (MP1 to MP4) are actually located are positions measured by the position measurement unit 10 described later, and are therefore referred to as the measurement positions MM (MM1 to MM4) of the measured parts MP below. The measurement positions MM (MM1 to MM4) of each measured part MP (MP1 to MP4) are positions obtained by adding a displacement vector DV (DV1 to DV4) to the reference position MD (MD1 to MD4). In other words, the displacement vectors DV1 to DV4 of each measured part MP1 to MP4 are the positional differences between the measurement positions MM1 to MM4 and the reference positions MD1 to MD4.

[0022] As an example, the reference position MD and the measurement position MM are represented by XY coordinate values ​​with the origin O as the center (0,0). Here, the relationship between the position (X, Y) of the reference position MD in the XY coordinates (Cartesian coordinates) and the position (r, θ) of the reference position MD in the polar coordinates, that is, the distance r and the deviation angle θ from the origin O to the reference position MD of the measured part MP described above, is represented by equations (1) to (4).

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[0023] Hereinafter, the reference positions MD of the multiple measured parts MP will be collectively or individually referred to as reference position information IRP. Hereinafter, the displacement vectors DV of the multiple measurement targets MP will be collectively or individually referred to as measurement position information IMP.

[0024] Referring again to FIG. 1, the measurement apparatus 1 and substrate transport apparatus 40 included in the exposure apparatus 2 will be described. The measuring apparatus 1 includes a position measuring unit 10, which is enclosed by a dotted line and which measures the position of a measurement target part MP formed on the substrate WF, and a control unit 15. Here, a position calculation unit 13, which will be described later, is included in both the position measuring unit 10 and the control unit 15. The control unit 15 further includes an analysis unit 16, a display unit 17, and an output unit 18, which will be described later.

[0025] The position measurement unit 10 has a mark detection unit 11 that detects the measured portion MP (see Figure 2A) on the substrate WF, an encoder head 12 that measures the position of the stage 30 when the mark detection unit 11 detects the measured portion MP, and a position calculation unit 13.

[0026] The mark detection unit 11 is, for example, an imaging optical microscope that measures the relative positional deviation in the X and Y directions between a reference index in the mark detection unit 11 and the measured part MP on the substrate WF, and outputs the measurement result as a signal S1. The mark detection unit 11 may be configured to irradiate an irradiation pattern onto the substrate WF using laser light or the like, and measure the position of the measured part MP based on the change in intensity of the returned light.

[0027] Similar to the above-described encoder head 23, the encoder head 12 measures the position of the stage 30 as the position of an encoder plate 32 provided on the stage 30, and outputs the position of the stage 30 as a signal S2. Note that instead of the encoder head 12, a position measurement system using a laser interferometer or a magnetic scale may be used.

[0028] The control unit 15 sends a signal S3 to the stage 30 based on the signal S2 from the encoder head 12 or the signal S2a from the encoder head 23, thereby controlling the position of the stage 30. The control unit 15 sends a signal S5 to the mask stage 24 based on the signal S4 from the encoder head 28, thereby controlling the position of the mask stage 24.

[0029] As an example, the substrate transport device 40 has a load guide 41 and a load arm 42 that moves along the load guide 41. The substrate transport device 40 is controlled by a signal S6 from the control unit 15, receives the substrate WF from an external device such as a track system, and places the substrate WF on the substrate chuck unit 31 on the stage 30. Note that the substrate chuck unit 31 is a member that performs the operation of transferring the substrate WF, and therefore constitutes a part of the stage 30 and is also included in the substrate transport device 40.

[0030] When the substrate WF is placed on the stage 30, the control unit 15 sends a signal S3 to the stage 30 based on the reference position information IRP of the multiple measured parts MP, and sequentially moves the stage 30 to a position where each of the multiple measured parts MP on the substrate WF faces the mark detection unit 11. With the mark detection unit 11 facing each of the multiple measured parts MP, the control unit 15 receives a signal S1 from the mark detection unit 11 and a signal S2 from the encoder head 12.

[0031] Based on these signals, the position calculation unit 13, which is included in the control unit 15 and is part of the position measurement unit 10, calculates the measurement position MM or displacement vector DV of each of the multiple measured parts MP on the substrate WF. As described above, the measurement position MM of the measured part MP is the position obtained by adding the displacement vector DV to the reference position MD of the measured part MP. Conversely, the difference between the measurement position MM of the measured part MP and the reference position MD is the displacement vector DV.

[0032] The analysis unit 16 analyzes the measurement position information IMP of the multiple measurement targets MP measured by the position measurement unit 10, i.e., the displacement vectors DV of the multiple measurement targets MP, and determines a reference function that fits to the measurement position information IMP. That is, it determines a reference function that represents the amount of deformation in the XY in-plane direction of the substrate WF including the multiple measurement targets MP. Then, the analysis unit 16 estimates the position of the existing pattern formed on the substrate WF based on the determined reference function.

[0033] When determining the reference function, the analysis unit 16 prepares in advance a reference function that is the sum of at least one function obtained by multiplying a proportionality coefficient with a base function expressed using a first kind of Bessel function. Then, an appropriate value of the proportionality coefficient is determined so that the reference function fits to the measurement position information IMP of the multiple measured parts MP. As the base function, a function expressed by the first kind of Bessel function for reference position information IRP, which is the reference position MP of the multiple measured parts MP formed on the substrate WF, is used. Details of the base function, the reference function, and the method of determining the reference function, i.e., the method of determining the proportionality coefficient, will be described later.

[0034] The control unit 15 may cause the output unit 18 to output the optimum value of the proportionality coefficient determined by the analysis unit 16 and other analysis results to an external device different from the measurement device 1 via the network line NW. The control unit 15 may cause the display unit 17 to display the optimum value of the proportionality coefficient determined by the analysis unit 16 and other analysis results. It should be noted that the measurement apparatus 1 or the exposure apparatus 2 does not necessarily have to include at least one of the output unit 18 and the display unit 17.

[0035] When measurement of the measurement positions MM or the displacement vectors DV of the multiple measurement portions MP on the substrate WF is completed, the control unit 15 sends a signal S3 to the stage 30 on which the substrate WF is placed, and moves the stage 30 to a position facing the projection optical system 22. Then, the exposure apparatus 2 moves the position of the stage 30 based on the position of the existing pattern on the substrate WF estimated by the analysis unit 16, i.e., aligns it with the position of the existing pattern, and exposes the pattern formed on the mask 25 onto the substrate WF.

[0036] The existing pattern on the substrate WF may also be called a first pattern, and the pattern formed on the mask 25 and newly exposed onto the substrate WF may also be called a second pattern. A plurality of first patterns may be formed at different positions on the substrate WF. In this case, the second patterns are exposed a plurality of times at different positions on the substrate WF in alignment with the plurality of first patterns.

[0037] In the measurement apparatus 1 shown in FIG. 1, the control unit 15 of the measurement apparatus 1 also controls the exposure apparatus 2, but the control unit 15 only controls the measurement apparatus 1, and the control unit that controls the exposure apparatus 2 may be provided separately from the control unit 15.

[0038] (Measuring method of the embodiment) The analysis method of the embodiment analyzes the characteristics of the measurement position information IMP, which is measured position information of a plurality of measurement targets MP formed on the substrate WF, by fitting a reference function, which will be described later, to the measurement position information IMP. As the reference function, a sum of at least one function obtained by multiplying a proportionality coefficient to a base function expressed using a first kind Bessel function is used.

[0039] The optimal value of the proportionality coefficient determined by fitting the reference function to the measurement position information IMP represents the deformation amount of the substrate WF corresponding to a specified deformation mode among the displacement amounts of multiple measured parts MP from the reference position MD (reference position information IRP). A specific example of the analysis method according to the embodiment will be described below.

[0040] In the analysis method of the embodiment, the first kind Bessel function J is used as a reference function constituting a reference function. m A function using μr (μr) is used. Here, m is a subscript representing the order of the first kind of Bessel function, and r is the distance from the origin O (center) of the substrate WF to the measured portion MP, as shown in FIG. 2A. However, to facilitate mathematical treatment, r is not the actual distance, but is treated below as a value normalized to be 1 on the outer periphery of the circular substrate WF. As an example, if the measured portion MP is at the midpoint between the origin O of the substrate WF and any point on the outer periphery of the substrate WF, r of that measured portion MP is 0.5. Moreover, μ is a constant to be described later.

[0041] The reference function is, for example, the following formula (5) or a function expressed by formula (5).

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[0042] 3A and 3B show the first kind Bessel function J included in the basis function. m (μ mk FIG. 3A shows an example of a constant μ determined by the subscripts m and k. mk3A is a table showing some example values ​​of the constant μ mk The value of is the mth order Bessel function of the first kind J with r as argument. m In (r), as r increases from 0, J m (r) is the value of r that takes the kth extreme value (maximum or minimum value) other than 0. Therefore, J m (μ mk r) is a function that has an extreme value at r=1.

[0043] In the table of FIG. 3A, the constants μ range from 0 to 7 for m and from 1 to a predetermined value depending on the value of m for k. mk where the constant μ mk The value of m may be 8 or more, and the value of k may be any value equal to or greater than 1. mk The value of is the m-th order Bessel function of the first kind J m It can be easily calculated from (r). Note that when m=0, the value of r that takes the extreme value of k=1, i.e., the first extreme value, is 0, and is not suitable for use in equations (5) and (6). Therefore, in the table in Figure 3A, the column for m=0, k=1 is lined out, indicating that the values ​​in this column are inappropriate.

[0044] In FIG. 3A, the constant μ mk The value of is displayed with six or more significant digits, but the constant μ mk The value of does not necessarily have to match the value shown in Figure 3A with an accuracy of six or more significant digits. Depending on the usage situation, i.e., the accuracy required for the analysis, it may be sufficient to match with, for example, two significant digits, or it may be sufficient to match with, for example, three significant digits.

[0045] FIG. 3B shows the Bessel function of the first kind J, defined by the subscripts m and k. m (μ mk The graph shows some of the functions J0(μ r) for r ranging from 0 to 1. The curve C02 is a function J0(μ 02 Similarly, the curve C11 shows the function J1(μ11 r), and curve C12 is function J1(μ 12 r), and curve C21 is function J2(μ 21 r), and curve C22 is function J2(μ 22 r), and curve C31 is function J3(μ 31 r), and curve C32 is function J3(μ 32 r) are shown respectively.

[0046] As described above, each function shown in FIG. 3B has an extreme value (maximum or minimum value) at r=1. The function J corresponding to k=1 m (μ m1 In the curves C11, C21, and C31 corresponding to k=2, the first non-zero extreme value is reached at r=1 as r increases. m (μ m2 In the curves C12, C22, and C32 corresponding to r), a second non-zero extreme value is reached at r=1 as r increases.

[0047] 4 is a diagram showing approximate values ​​of some of the criterion functions represented by the subscripts m and k among the criterion functions represented by the formula (5) or (6). The criterion function C11c is J1(μ 11 r) × cos(1θ), and the reference function C11s is J1(μ 11 r) × sin(1θ). The reference function C21c is J2(μ 21 r) × cos(2θ), and the reference function C21s is J2(μ 21 r) × sin(2θ).

[0048] The basis function C31c is J3(μ 31 r) × cos(3θ), the basis function C31s is J3(μ 31 r) × sin(3θ). The reference function C22c is , m=2, k=2 J2(μ 22 r) × cos(2θ), and the basis function C22s is m = 2, k=2 J2(μ22 r) × sin(2θ).

[0049] 4, within a circle centered at the origin O and with a radius of r=1, the values ​​of each criterion function (C11c-C22c, C11s-C22s) are indicated by adding a "+" to the region where they are greater than a predetermined positive value, and by adding a "-" to the region where they are smaller than a predetermined negative value. Regions without a "+" or "-" are regions where the value of the criterion function is greater than a predetermined negative value and smaller than a predetermined positive value, i.e., close to zero.

[0050] 4, by appropriately selecting the values ​​of the subscripts m and k, the basis functions given by equations (5) and (6) become functions whose values ​​increase and decrease in various predetermined shapes within a circle of radius 1. Therefore, by using one of these basis functions (C11c-C22c, C11s-C22s, etc.) or by combining several of the basis functions, it is possible to express, for example, a virtual deformation in the XY in-plane direction of a circular substrate WF.

[0051] Hereinafter, a vector representing a virtual deformation amount of the measured portion MP of the substrate WF expressed by the reference function is called a virtual displacement vector IV, and the X and Y components of the virtual displacement vector IV are Δx and Δy, respectively. The virtual displacement vector IVj of the jth (j is any natural number) measured portion MP is IVj=(Δxj, Δxj). The X component Δxj and the Y component Δyj of the virtual displacement vector IVj are expressed as follows using the reference function: This is expressed by equations (7) and (8).

[0052]

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[0053] As described above, in this specification, the first kind Bessel function J as shown in Equation (7) and Equation (8) is used. m (μ mk The sum of at least one function obtained by multiplying the criterion function expressed using the function (r) by the proportionality coefficient α is called the reference function. Note that when there is only one criterion function, the criterion function itself becomes the reference function.

[0054] 5 is a diagram showing, as an example, the virtual deformation amount in the XY in-plane direction of the circular substrate WF, in which the deformation amount in the X direction is represented by using the reference function C21c and the deformation amount in the Y direction is represented by using the reference function C21s. That is, in the example shown in FIG. 5, IVj=(Δxj, Δxj) is expressed as follows: This is expressed by the following equations (9) and (10).

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[0055] In addition, Δxj, Δxj are expressed as a virtual displacement vector IVj(Δxj, Δxj) Moreover, formula (9) and formula (10) can be expressed as one formula (11).

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[0056] By setting the value of the proportionality coefficient α to an appropriate value (optimum value), the actual deformation amount (displacement vector DV) of the measured portion MP of the substrate WF can be approximately expressed using the reference function. The value of the proportionality coefficient α may be determined, for example, by the least squares method. That is, for N measured portions MPj of the substrate WF, the values ​​of Ex and Ey obtained by substituting Δxj and Δyj of equations (9) and (10), or equation (11) into equations (12) and (13), respectively, are The proportionality coefficient α may be set to an appropriate value so as to be minimum.

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[0057] Alternatively, the appropriate value of the proportionality coefficient α may be set using other known methods. By setting an appropriate proportionality coefficient α, the reference functions shown in equations (7) and (8) can be fitted to the measurement position information IMP, i.e., the displacement vector DV(ΔX, ΔY), which is the measured position information of multiple measured parts MP.

[0058] The value of the proportionality coefficient α set corresponding to one base function is an index indicating to what extent the deformation mode represented by that base function is included in the measurement position information IMP. Therefore, by calculating the proportionality coefficient α corresponding to each base function, it is possible to analyze the amount of various deformation modes included in the deformation of the substrate WF.

[0059] The reference functions expressed by equations (7) and (8) are expressed as vectors with Δxj, Δxj as virtual displacement vectors IVj(Δxj, Δxj) as in equation (11) above, and are expressed as follows: It can also be written as equation (15).

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[0060] Incidentally, even when written as in equation (15), the reference function is still the sum of at least one function obtained by multiplying a proportionality coefficient to a reference function expressed using a first kind Bessel function.

[0061] By setting the value of the proportionality coefficient β in equation (15) to an appropriate value, the actual deformation amount (displacement vector DV) of the measured portion MP of the substrate WF can be approximately expressed by using a function including one or more reference functions expressed by equation (15). The proportionality coefficient β value may be determined, for example, by the above-mentioned least squares method, in the same way as the proportionality coefficient α value described above.

[0062] In addition, when setting the proportionality coefficients α and β of the reference functions expressed by the formulas (7) and (8) or (15), the values ​​of the proportionality coefficients α and β corresponding to the subscripts m and k of the predetermined values ​​may be determined as 0. In other words, when determining the proportionality coefficients α and β, it is not necessary to calculate the proportionality coefficients α and β corresponding to the reference functions corresponding to all combinations of the subscripts m and k. In the extreme, the values ​​of the proportionality coefficients α and β corresponding to only one reference function consisting of one combination of the subscripts m and k may be determined as shown in the formulas (9) to (11).

[0063] Below, we will show examples of some of the reference functions included in the reference function expressed by equation (15) that are important when analyzing the deformation of the actual substrate WF and their corresponding proportionality coefficients. When analyzing the substrate WF, it is advisable to determine the value of the proportionality coefficient β corresponding to at least one of the reference functions shown below.

[0064] The first example is where the subscripts m = 2 and k = 1, and the proportionality coefficient β1 in equation (15) mk Against The corresponding criterion function and the proportionality coefficient β2 mk The two criterion functions are: In order to easily distinguish the correspondence with equation (15), the proportionality coefficient β1 21 or proportionality coefficient β2 21 The results obtained by multiplying these are shown in equations (16) and (17), respectively.

[0065]

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[0066] The second example is when the subscripts m = 2 and k = 2, and the proportionality coefficient β1 in equation (15) mk Against The corresponding criterion function and the proportionality coefficient β2 mk The two criterion functions are: In order to easily distinguish the correspondence with equation (15), the proportionality coefficient β1 22 or β2 22 The results obtained by multiplying these are shown in equations (18) and (18), respectively.

[0067]

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[0068] The third example is where the subscripts m = 1 and k = 2, and the proportionality coefficient β1 in equation (15) mk Against The corresponding basis function, with subscripts m = 1, k = 3, and the proportionality coefficient β1 in equation (15) mk To In order to easily distinguish the correspondence with Equation (15), the proportionality coefficient β1 corresponding to these criterion functions is 12 , or proportionality coefficient β1 13 The results obtained by multiplying these are shown in equations (20) and (21), respectively.

[0069]

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[0070] The fourth example is where the subscripts m = 2 and k = 1, and the proportionality coefficient β3 in equation (15) mk Against The corresponding criterion function and the proportionality coefficient β4 mk The two criterion functions are: In order to easily distinguish the correspondence with equation (15), the proportionality coefficient β3 corresponding to these criterion functions is 21 , and the proportionality coefficient β4 21The results obtained by multiplying are shown in equations (22) and (23), respectively.

[0071]

number

number

[0072] Note that the multiple base functions (C21c, C21s, etc.) for setting the proportionality coefficients α and β may be selected to be functions that are mutually orthogonal within a circle of r≦1 centered on the origin O. For example, base functions C21c and C21s are orthogonal because the integrated product of these functions is 0 within a circle of r≦1 centered on the origin O.

[0073] In addition, the proportionality coefficients α and β are set by using a plurality of Bessel functions of the first kind J m (μ mk At least some or all of the functions (r) having different subscripts m and k may be mutually orthogonal within a circle of r≦1 centered on the origin O. In other words, the reference function may include two or more base functions represented by two or more Bessel functions of the first kind that are mutually orthogonal within the plane of the substrate WF (r≦1). In addition, the two Bessel functions of the first kind J m (μ mk r) is orthogonal if the two Bessel functions of the first kind J m (μ mk r) in a circle with r≦1 centered on the origin O, the integral value becomes 0. In other words, the product of two Bessel functions of the first kind J m (μ mk This means that the inner product of r (r) within a circle with r≦1 centered at the origin O is 0.

[0074] In addition, when determining the appropriate values ​​of the proportionality coefficients α and β, residual position information, which is position information obtained by removing the positional deviation amounts associated with the translational deviation, rotation, and uniform expansion and contraction of the substrate WF from the measured position information (displacement vector DV) of multiple measured portions MP, may be used as the measured position information IMP.

[0075] Here, the amount of positional deviation caused by the translational deviation, rotation, and uniform expansion and contraction of the substrate WF can be calculated using general formulas shown as formulas (24) to (26).

number

number

number

[0076] Here, in equation (24), a is a parameter that represents the uniform expansion and contraction of the substrate WF in the X direction, d is a parameter that represents the uniform expansion and contraction in the Y direction, b and c are the rotation of the substrate WF, e is a translational deviation in the X direction of the substrate WF, and y is a parameter that represents the translational deviation in the X direction. Furthermore, Xj and Yj are the X and Y positions, respectively, of the reference position MD of the jth measured part MP, and N is the number of measured parts MP. Then, εxj and εyj are the amount of deviation of the X position of the jth measured part MPj from the reference position MD due to the translational deviation, rotation, and uniform expansion and contraction of the substrate WF determined by parameters a to f, and Represents the amount of deviation in the Y position.

[0077] εxj and εyj expressed by equation (24) are substituted into equations (25) and (26), respectively, and the least squares method or the like is used to determine the values ​​of parameters a to f so that, for example, the values ​​of Lx and Ly are minimized. Then, the determined values ​​of parameters a to f are substituted into equation (22) to calculate εxj and εyj for each of the N measurement points MP. The residual position information, which is the position information obtained by removing the positional deviation amounts (εxj, εyj) associated with the translational shift, rotation, and uniform expansion and contraction of the substrate WF from the measured displacement vector DVj(ΔXj, ΔYj) of the jth measured part MPj, is calculated as (ΔXj-εxj, ΔYj-εyj).

[0078] As described above, instead of the measured position information (displacement vector DV) of the multiple measured parts MP, the residual position information (ΔXj-ε xj, ΔYj-εyj) may be used for the analysis.

[0079] (Variations of the analysis method) The following describes a modified analysis method. However, since most of the configuration of the modified analysis method is common to the analysis method of the above-mentioned embodiment, the following describes the differences. In the analysis method of the embodiment, all of the first kind Bessel functions constituting the basis function have an extreme value at r=1. In contrast, the analysis method of the modified example is different in that all of the first kind Bessel functions constituting the basis function take 0 at r=1.

[0080] In the analysis method of the modified example, the first kind Bessel function J is used as a reference function constituting the reference function. m (λ mk r), where λ mk is the μ in the analysis method of the embodiment. mk Similarly, the Bessel function of the first kind J m It is a constant with the degree m of and an arbitrary natural number k as subscripts. The reference function used in the analysis method of the modified example is, for example, a function expressed by the following formula (27) or formula (28).

number

number

[0081] 6A and 6B show the first kind Bessel function J included in the basis function. m (λ mk FIG. 6A shows an example of a constant λ determined by the subscripts m and k. mk 6B is a table showing some example values ​​of the constant λ shown in FIG. mk The value of is the mth order Bessel function of the first kind J with r as argument. m In (r), as r increases from 0 (excluding r=0), J m (r) is the value of r that takes the kth 0. Therefore, J included in equations (27) and (28) m (λ mk r) is a function that is 0 at r=1.

[0082] In the table of FIG. 6A, the constants λ range from 0 to 7 for m and from 1 to a predetermined value depending on the value of m. mk where the constant λ mk The value of m may be 8 or more, and the value of k may be any value greater than or equal to 1. mk The value of is the m-th order Bessel function of the first kind J m It can be easily calculated from (r).

[0083] FIG. 6B shows the Bessel function of the first kind J, defined by the subscripts m and k. m (λ mk The graph shows some of the functions J0(λ r) for m=0 and k=1. 01 Similarly, the curve B11 shows the function J1(λ 11 r), and curve B12 is function J1(λ 12 r), and curve B21 is function J2(λ 21 r), and curve B22 is function J2(λ 22 r), and curve B31 is function J3(λ 31 r), and curve B32 is function J3(λ 32 r) are shown respectively.

[0084] As described above, each function shown in FIG. 6B is 0 at r=1. The function J corresponding to k=1 is m (λ m1 In the curves B11, B21, and B31 corresponding to k=2, the function J first reaches 0 at r=1 as r increases (except for r=0). m (λ m2 In the curves B12, B22, and B32 corresponding to r, the (Except for ) take the second zero at r=1.

[0085] In the analysis method of the modified example, the proportionality coefficient μ mk The proportionality coefficient λ mk Then, similarly to the analysis method of the above-mentioned embodiment, the appropriate values ​​of the proportionality coefficients α and β can be determined by using, for example, the least squares method.

[0086] In the analysis method of the modified example, the base function constituting the reference function is 0 at r=1, i.e., at the outer periphery of the substrate WF. In other words, the first kind of Bessel function included in the base function is a function that takes the value 0 for the reference position information IRP of the measured part MP arranged on the periphery of the substrate WF. For this reason, the values ​​of the proportionality coefficients α and β can be determined without being significantly affected by the deformation near the outer periphery of the substrate WF, i.e., without being significantly affected by the displacement vector DV of the measured part MP arranged near the outer periphery of the substrate WF.

[0087] On the other hand, in the analysis method of the above-mentioned embodiment, the first kind of Bessel function included in the base function constituting the reference function takes an extreme value at r=1, that is, at the outer periphery of the substrate WF. In other words, the first kind of Bessel function included in the base function is a function that takes an extreme value with respect to the reference position information IRP of the measured part MP arranged on the periphery of the substrate WF. Therefore, at the outer periphery of the substrate WF, the fluctuation of the value of the base function with respect to the fluctuation of r is small. As a result, the values ​​of the proportionality coefficients α and β can be determined without being significantly affected by the local fluctuation of the displacement vector DV of the measured part MP arranged near the outer periphery of the substrate WF.

[0088] The reference function is the Bessel function of the first kind J m (κr) does not necessarily have to be a function that takes an extreme value or 0 at r=1. In other words, the proportionality coefficient κ is mk , or λ mk It may be a value other than .

[0089] In either of the analysis methods of the above-mentioned embodiment and the analysis methods of the modified examples, the reference function may not only use the Bessel function of the first kind, but may be combined with another reference function. For example, it may be used in combination with a reference function using Zernike polynomials as a reference function. In this case, fitting can be performed for low-order components using a reference function with Zernike polynomials as a reference function, and fitting can be performed for high-order components using a reference function with Bessel function of the first kind as a reference function.

[0090] (Measuring method of the embodiment) The measurement method of the embodiment will be described below. The measurement method of the embodiment measures the positions of multiple measurement targets MP formed on the substrate WF, and obtains measurement position information IMP of the measurement targets MP, i.e., the displacement vector DV of the multiple measurement targets MP. Then, the above-mentioned analysis method is used to calculate appropriate values ​​for the proportionality coefficients α and β included in the reference functions shown in equations (7) and (8), or equation (15).

[0091] The method for measuring the position of the measured part MP has been explained in the description of the measurement apparatus and exposure apparatus of the above-mentioned embodiment, so a repeated explanation will be omitted here. Also, the method for calculating the reference function and the appropriate values ​​of the proportionality coefficients α and β has been explained in the description of the analysis method of the above-mentioned embodiment and modified example, so a repeated explanation will be omitted here.

[0092] (Exposure method of the embodiment) Hereinafter, an exposure method according to an embodiment will be described with reference to Fig. 7. Fig. 7 is a diagram showing a substrate WF on which a plurality of first patterns A1, the outer periphery of which is shown as a dashed rectangular shape, are formed. A plurality of fine patterns (not shown) are formed inside the outer periphery of the first pattern A1, which is shown as a dashed line. A plurality of measurement portions MP are formed on the substrate WF at positions that have a predetermined positional relationship with the first pattern A1.

[0093] The exposure method of the embodiment measures the positions of the multiple measurement targets MP on the substrate WF by the measurement method of the embodiment described above, obtains measurement position information IMP of the measurement targets MP, and calculates an appropriate value for at least one of the proportionality coefficients α, β in the reference function described above. Then, based on the obtained measurement position information IMP and the appropriate value of the at least one proportionality coefficient, the position of the first pattern A1 formed on the substrate WF in a predetermined positional relationship with the multiple measurement targets MP is estimated.

[0094] The position of the first pattern A1 is estimated based on the reference function into which optimal values ​​of the proportional coefficients α and β are substituted, and the design position data (AX, AY) of the first pattern A1. That is, optimal values ​​are substituted into the proportional coefficients α and β of the reference function shown in Equation (7), Equation (8), or Equation (15). The design position data (AX, AY) of the j-th first pattern A1j in the polar coordinates are calculated by substituting rj and θj of this reference function. By inputting the values ​​of the positions (Arj, Aθj), the amount of positional deviation Δxj in the X direction and the amount of positional deviation Δyj in the Y direction from the designed position are calculated (estimated).

[0095] Therefore, the estimated position of the jth first pattern A1j in the XY coordinates is This is the estimated position (AXj+Δxj, AYj+Δyj) obtained by adding the above-mentioned Δxj and Δyj to the designed position (AXj, AYj) in the XY coordinates of turn A1j. Then, based on the estimated position of the first pattern A1, a second pattern (not shown) is exposed onto the substrate WF so as to be aligned with the first pattern A1.

[0096] In addition, when the appropriate values ​​of the proportionality coefficients α and β are determined after removing the positional deviation amounts associated with the translational deviation, rotation, and uniform expansion and contraction of the substrate WF using equation (24) or the like, correction is required for the estimated position (AXj+Δxj, AYj+Δyj) of the first pattern A1 described above.

[0097] This correction is performed by substituting the design position (AXj, AYj) of the jth first pattern A1j in the XY coordinate system into equation (24) into which the parameters a to f determined as described above have been substituted. Then, εxj and εyj are added to the above-mentioned estimated position of the first pattern A1. That is, the estimated position of the first pattern A1 after correction becomes the estimated position (AXj+Δxj+εxj, AYj+Δyj+εyj).

[0098] The exposure method of the embodiment can be performed, for example, by using the exposure apparatus 2 of the above-described embodiment. As an example, the second pattern exposed onto the substrate WF in the exposure method of the embodiment is obtained by exposing a pattern original formed on a mask 25 onto the substrate WF via the projection optical system 22.

[0099] (Additional Description of Measurement Apparatus and Exposure Apparatus of the Embodiments) Additional details regarding the measurement apparatus 1 and exposure apparatus 2 of the embodiment will be described. The control unit 15 may display the fitting result of the measurement position information IMP by the reference function to which the appropriate values ​​of the proportionality coefficients α and β are applied (substituted) as a vector diagram. Here, the vector diagram is a diagram in which the deformation amounts of multiple measurement targets MP on the substrate WF are displayed as vectors with the reference position MD of each measurement target MP as the reference (starting point), as shown in Fig. 7. In this case, the scale representing the vectors may be greatly enlarged compared to the scale representing the position of the measurement target MP.

[0100] The vector diagram displayed on the display unit 17 may be a vector diagram showing the virtual deformation amount of each measured part MP calculated by the fitted reference function, or may be a vector diagram showing the difference between the actual deformation amount (displacement vector DV) of each measured part MP and the virtual deformation amount of each measured part MP calculated by the fitted reference function.

[0101] The control unit 15 may obtain information on the history of the processing of the analyzed substrate WF via the network line NW, and analyze the relationship between the information on the history and the analysis results obtained by the analysis unit 16. For example, a correlation may be obtained between the processing parameters of an external processing device (etching device, deposition device, CMP (Chemical Mechanical Planarization), etc.) different from the measurement device 1 that was used before the substrate WF was carried into the measurement device 1 and the appropriate value of the proportionality coefficient determined by the analysis unit 16, and the processing parameters may be changed and optimized by feeding back the correlation to the processing device. As the processing parameters, for example, the temperature applied to the substrate WF in the process in the processing device, the polishing speed in CMP, etc. may be changed and optimized. Such analysis and feedback of the correlation does not necessarily have to be performed by the control unit 15 or the analysis unit 16, and may be performed by a host computer provided separately from the measurement device 1 or the external processing device.

[0102] Furthermore, the optimum value of the proportionality coefficient determined by the analysis unit 16 may be continuously monitored, and a warning may be issued if the coefficient deviates from a preset threshold value. The warning may detect an abnormality in an external processing device. For example, the deterioration (lifespan) of electrodes in an etching device may be estimated. Also, a flag may be attached to the history of substrates WF that deviate from the threshold value.

[0103] (Additional explanation regarding the exposure apparatus of the embodiment) Additional details regarding the exposure apparatus 2 of the embodiment will be described. The exposure apparatus 2 of the embodiment determines the operating state of the substrate transport device 40 (load guide 41, load arm 42, substrate chuck unit 31) based on the measurement results (proportional coefficients α, β) obtained by the measurement method of the embodiment described above. The operating state of the substrate transport device 40 is, for example, the moving speed of the substrate WF with respect to the substrate chuck unit 31 when the substrate WF is placed on the substrate chuck unit 31 on the stage 30, or the relative inclination of the substrate WF and the substrate chuck unit 31.

[0104] The operating state of the substrate transport device 40 may be the timing of generation of a holding force for holding the substrate WF on the substrate chuck portion 31. The change in the timing of generation of the holding force may be a change in the timing at which the holding force is generated over substantially the entire surface of the substrate chuck portion 31 relative to the timing at which the substrate WF is placed. Alternatively, the timing at which the holding force is generated in a predetermined portion of the substrate chuck portion 31 may be shifted forward or backward relative to the timing at which the holding force is generated in other portions.

[0105] These operating states are changed by changing the values ​​of the apparatus constants stored in the control unit 15 or the like of the exposure apparatus 2. Therefore, it can be said that the exposure apparatus 2 determines the apparatus constants that define the operation of the substrate transport apparatus 40 based on the measurement results (proportionality coefficients α, β) obtained by the measurement method of the embodiment.

[0106] By changing these apparatus constants, it is possible to change the state in which the substrate WF is attracted to the substrate chuck portion 31, and to change the state of deformation of the substrate WF due to suction. As described above, the apparatus constants include any one of the apparatus constants that determine the moving speed of the substrate WF when the substrate WF is placed on the stage 30, the relative inclination between the substrate WF and the stage 30, or the timing of generation of a holding force for holding the substrate WF on the stage 30.

[0107] In addition, the device constants may be changed after the proportional coefficients α and β are determined using the measurement method of the embodiment while the substrate WF is adsorbed to the stage 30 (substrate chuck portion 31), and then the substrate WF is removed from the substrate chuck portion 31 and before the substrate WF is adsorbed to the substrate chuck portion 31 again.

[0108] When exposing one lot of substrates consisting of a predetermined number of substrates WF, the second and subsequent substrates WF may be exposed using the apparatus constant determined as described above for the first substrate WF.

[0109] (Embodiments of the analysis device) The analysis unit 15 included in the exposure apparatus 2 or the measurement apparatus 2 described above may be incorporated in an analysis apparatus separate from the exposure apparatus 2 or the measurement apparatus 2 . 8 is a diagram illustrating a schematic configuration of an analysis device 3 according to an embodiment. The analysis device 3 includes an input unit 19 in addition to the analysis unit 16, display unit 17, output unit 18, and position calculation unit 13 described above. In addition, in analysis apparatus 3 configured separately from exposure apparatus 2 or measurement apparatus 2, at least one of position calculation unit 13 and display unit 17 described above may not be included.

[0110] The analysis device 3 receives measurement position information IMP, which is measured position information of multiple measurement targets formed on the substrate WF, from an input unit 19, and outputs an appropriate value calculated by the analysis unit 16 from an output unit 18 to an external device such as an exposure device, a measurement device, or a display device. The measurement position information IMP may be input to the input unit 19 via a network line NW, or via a dedicated signal line provided between the input unit 19 and the external device. The output from the output unit 18 to the external device may also be via the network line NW, or via a dedicated signal line provided between the output unit 18 and the external device. The input section 19 and the output section 18 may be integrally configured as an input / output section. The analysis device 3 may output not only optimal values ​​as analysis results, but also data corresponding to the various vector diagrams described above and other processing results.

[0111] (Effects of the Analysis Method of the Embodiment and Modifications) (1) The analysis method of the embodiment or the modified example includes preparing measurement position information IMP, which is measured position information of a plurality of measured portions MP formed on a substrate WF, and fitting a reference function, which is the sum of at least one function obtained by multiplying a base function expressed using a first kind of Bessel function by a proportionality coefficient, to the measurement position information IMP to calculate an appropriate value of at least one proportionality coefficient. With this configuration, the deformation of the substrate WF can be easily separated into deformation components corresponding to the base functions, and the amount of deformation in the deformation mode corresponding to the base functions can be analyzed.

[0112] (Effects of the analysis device according to the embodiment) (2) The analysis device 3 of the embodiment includes an input unit 19 that inputs measurement position information IMP, which is measured position information of multiple measurement targets formed on the substrate WF, an analysis unit 16 that analyzes the measurement position information IMP of the measurement targets MP measured by the position measurement unit, and an output unit 18 that outputs an analysis result including at least the calculated optimum value of the proportionality coefficient to an external device. The analysis unit 16 calculates at least one optimum value of the proportionality coefficient by the analysis method of the embodiment or the modified example described above. With this configuration, the deformation of the substrate WF can be easily separated into deformation components corresponding to the base functions, and the amount of deformation in the deformation mode corresponding to the base functions can be analyzed.

[0113] (Effects of the measurement method according to the embodiment) (3) The measurement method of the embodiment includes measuring the positions of multiple measured parts MP formed on the substrate WF to obtain measurement position information IMP of the measured parts MP, and calculating, by an analysis method of the embodiment or the modified example, an appropriate value of at least one proportionality coefficient in a reference function which is the sum of at least one function obtained by multiplying a base function expressed using a first kind Bessel function by a proportionality coefficient. With this configuration, the deformation of the substrate WF can be measured for each deformation component corresponding to the base function.

[0114] (Effects of the Measuring Device of the Embodiment) (4) The measuring apparatus 1 of the embodiment includes a position measurement unit 10 that measures the position of a measured part MP formed on a substrate WF, and an analysis unit 16 that analyzes measurement position information IMP of the measured part MP measured by the position measurement unit 10. The analysis unit 16 calculates, by an analysis method of the embodiment or the modified example, an appropriate value of at least one proportionality coefficient in a reference function which is the sum of at least one function obtained by multiplying a base function expressed using a first kind Bessel function by a proportionality coefficient. With this configuration, the deformation of the substrate WF can be measured for each deformation component corresponding to the base function.

[0115] (Effects of the Exposure Method of the Embodiment) (5) In the exposure method of the embodiment, the positions of the multiple measured portions MP on the substrate WF are measured by the measurement method of the embodiment, measurement position information IMP of the measured portions is obtained, and an appropriate value of at least one proportionality coefficient in a reference function which is a sum of at least one function obtained by multiplying a proportionality coefficient with a base function expressed using a first kind of Bessel function. Then, based on the obtained measurement position information IMP and the appropriate value of the at least one proportionality coefficient, the position of a first pattern A1 formed on the substrate WF in a predetermined positional relationship with the multiple measured portions MP is estimated, and a second pattern is exposed onto the substrate WF based on the estimated position of the first pattern. With this configuration, the second pattern can be exposed by aligning its position with respect to the first pattern A1 formed on the deformed substrate WF.

[0116] (Effects of the Exposure Apparatus of the Embodiment) (6) From one perspective, the exposure apparatus 2 of the embodiment comprises a position measurement unit 10 that measures the position of a measurement portion MP formed on the substrate WF, and an exposure optical system 20 that exposes a desired pattern onto the substrate WF, and performs the exposure method of the embodiment. With this configuration, the second pattern can be exposed by aligning its position with respect to the first pattern A1 that is formed on the deformed substrate WF in a predetermined positional relationship with the measured portion MP. (7) From another perspective, the exposure apparatus 2 of the embodiment includes a stage 30 that moves while placing a substrate WF on which a measured portion MP is formed, and a substrate transport device 40 that transports the substrate WF and places it on the stage 30, and exposes a desired pattern onto the substrate WF. An apparatus constant that specifies the operation of the substrate transport device 40 is determined based on the measurement results obtained by the measurement method of the embodiment. With this configuration, the substrate WF can be placed on the stage 30 with deformation of the substrate WF reduced, and the second pattern can be exposed with a more aligned position relative to the first pattern A1 formed on the substrate WF.

[0117] The present invention is not limited to the above. Other aspects conceivable within the scope of the technical idea of ​​the present invention are also included in the scope of the present invention. This embodiment may combine all or part of the above aspects. [Explanation of symbols]

[0118] 1: measurement device, 2: exposure device, 3: analysis device, 10: position measurement unit, 11: mark detection unit, 12: encoder head, 13: position calculation unit, 15: control unit, 16: analysis unit, 17: display unit, 18: output unit, 20: exposure optical system, 21: illumination optical system, 22: projection optical system, 23: encoder head, 24: mask stage, 25: mask, 40: substrate transport device, WF: substrate, MP: measured part, MM: measurement position, MD: reference position, DV: displacement vector, IRP: reference position information, IMP: measurement position information

Claims

1. preparing measurement position information which is measured position information of a plurality of measurement targets formed on a substrate; fitting a reference function, which is a sum of at least one function obtained by multiplying a base function expressed using a first kind Bessel function by a proportionality coefficient, to the measurement position information to calculate an appropriate value of at least one of the proportionality coefficients; Equipped with The analysis method, wherein the reference function is expressed as a product of a first kind Bessel function for a distance from a reference position on the substrate to the measured portion and a trigonometric function based on an angle from the reference position to the measured portion.

2. 2. The analysis method according to claim 1, an analysis method, wherein the first kind of Bessel function included in the reference function is a function that takes an extreme value with respect to reference position information of the measurement portion disposed on the periphery of the substrate.

3. Preparing measurement position information which is measured position information of a plurality of measurement target portions formed on a substrate; fitting a reference function, which is a sum of at least one function obtained by multiplying a base function expressed using a first kind Bessel function by a proportionality coefficient, to the measurement position information to calculate an appropriate value of at least one of the proportionality coefficients; Equipped with the reference function is expressed using a first kind of Bessel function for a distance from a reference position on the substrate to the measurement portion, The analysis method, wherein the first kind Bessel function is a function that takes an extreme value with respect to reference position information of the measurement portion disposed on the periphery of the substrate.

4. 4. The analysis method according to claim 3, The analysis method, wherein the reference function is expressed using a product of the first kind Bessel function and a trigonometric function based on an angle from the reference position to the measured portion.

5. In the analysis method according to claim 2 or 3, As the reference function, an m-th order first kind Bessel function Jm(μr) (m is a natural number) having a value obtained by multiplying a distance r from the reference position to the measured part by a constant μ as an argument, a cos function having an angle m times the deflection angle θ from the reference position to the measured part as an argument, and a quadratic function consisting of a cos function An analytical method including two functions, each of which is represented by one of the following two equations, as a product of an original vector and [0010] [0025]

6. 6. The analysis method according to claim 5, The m is 2, The constant μ is set such that a second-order Bessel function of the first kind J2(μr) takes a first non-zero extreme value at a distance r corresponding to an edge of the substrate as the distance r increases from 0.

7. 6. The analysis method according to claim 5, The m is 2, The constant μ is set such that a second-order Bessel function of the first kind J2(μr) takes a second non-zero extreme value at a distance r corresponding to an edge of the substrate as the distance r increases from 0.

8. In the analysis method according to claim 2 or 3, The reference functions include two functions obtained by multiplying a first-order Bessel function of the first kind J1 (μ1r) having as an argument a value obtained by multiplying a distance r from the reference position to the measured portion by a constant μ1, and a first-order Bessel function of the first kind J1 (μ2r) having as an argument a value obtained by multiplying a distance r from the reference position to the measured portion by a constant μ2, by a two-dimensional vector consisting of a cos function and a cos function having as an argument an argument an angle θ from the reference position to the measured portion, and each of the functions includes two functions represented by either of the following two expressions: The constant μ1 is set such that a first-order Bessel function of the first kind J1(μ1r) takes a second non-zero extreme value at a distance r corresponding to the edge of the substrate as the distance r increases from 0; The constant μ2 is set such that a first-order Bessel function of the first kind J1(μ2r) takes a third non-zero extreme value at a distance r corresponding to an edge of the substrate as the distance r increases from 0. [0030] [0045]

9. In the analysis method according to claim 2 or 3, The reference function includes two functions represented by either of the following two expressions as a product of a second-order first kind Bessel function J2(μr) having as an argument a value obtained by multiplying a distance r from the reference position to the measured portion by a constant μ, and a two-dimensional vector consisting of a cos function and a cos function having as an argument an angle twice the deflection angle θ from the reference position to the measured portion, The constant μ is set such that a second-order Bessel function of the first kind J2(μr) takes a first non-zero extreme value at a distance r corresponding to an edge of the substrate as the distance r increases from 0. [0050] [006]

10. In the analysis method according to claim 2 or 3, an analysis method, wherein the first kind Bessel function included in the reference function is a function that becomes 0 with respect to the reference position information of the measured portion disposed on the periphery of the substrate.

11. In the analysis method according to claim 1 or 3, The analysis method, wherein the reference function includes two or more basis functions represented by two or more Bessel functions of the first kind that are mutually orthogonal within the plane of the substrate.

12. In the analysis method according to claim 1 or 3, An analysis method using, as the measurement position information, residual position information, which is position information obtained by removing the amount of positional deviation due to translational deviation, rotation, and uniform expansion and contraction of the substrate from the measured position information of the multiple measured portions.

13. an input unit for inputting measurement position information which is measured position information of a plurality of measurement targets formed on a substrate; an analysis unit that analyzes measurement position information of the measured part measured by a position measurement unit; an output unit that outputs an analysis result including at least the calculated optimum value of the proportionality coefficient to an external device; The analysis device according to claim 1 , wherein the analysis unit calculates an optimum value of at least one of the proportionality coefficients by the analysis method according to claim 1 .

14. measuring positions of a plurality of measurement target portions formed on a substrate to obtain measurement position information of the measurement target portions; Calculating an optimal value of at least one of the proportionality coefficients by the analysis method according to claim 1; A measurement method comprising:

15. a position measurement unit that measures the position of a measurement target portion formed on a substrate; an analysis unit that analyzes measurement position information of the measured part measured by the position measurement unit, The analysis unit performs the analysis method according to claim 1 or 3. A metrology device for calculating an optimal value of at least one of said proportionality coefficients.

16. 16. The measurement device according to claim 15, A measurement device comprising an output unit that outputs an analysis result including at least the calculated optimum value of the proportionality coefficient to an external device different from the measurement device.

17. 16. The measurement device according to claim 15, A measurement device comprising a display unit that displays an analysis result including at least the calculated optimum value of the proportionality coefficient.

18. 18. The measurement device according to claim 17, The display unit displays a fitting result by the reference function to which the appropriate value of the proportionality coefficient is applied as a vector diagram.

19. measuring positions of a plurality of the measurement target portions on the substrate by the measurement method according to claim 14, acquiring measurement position information of the measurement target portions, and calculating an appropriate value of at least one of the proportionality coefficients; estimating a position of a first pattern formed on the substrate in a predetermined positional relationship with respect to the plurality of measured portions, based on the acquired measurement position information and the appropriate value of at least one of the proportionality coefficients; exposing a second pattern onto the substrate based on the estimated position of the first pattern; An exposure method comprising:

20. a position measurement unit that measures the position of a measurement target portion formed on a substrate; an exposure optical system that exposes a desired pattern onto the substrate; An exposure apparatus that performs the exposure method according to claim 19.

21. a stage on which a substrate having a measurement portion formed thereon is placed and moved; a substrate transport device that transports the substrate and places it on the stage, 15. An exposure apparatus, comprising: a measuring device for measuring a position of the substrate transport device; a measuring device for measuring a position of the substrate transport device;

22. 22. The exposure apparatus according to claim 21, wherein the apparatus constant includes any one of a moving speed of the substrate when the substrate is placed on the stage, a relative inclination between the substrate and the stage, or a timing of generation of a holding force for holding the substrate on the stage. Exposure equipment.

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