Mask blank substrate and method of manufacturing the same

Abstract

SOLUTION: When a rectangular region surrounded by four sides positioned at an inner central portion of four sides parallel to the four sides of the main surfaces is set, the rectangular region having a 5mm at an intersection of diagonal lines of the rectangular main surfaces of 152mm or more and * 152mm or more, the flatness of the rectangular region is 100nm or less, when the smaller one of a difference PVx between the maximum height and the minimum height of Sx and a difference PVy between the maximum height and the minimum height of Sy is defined as PVmin and the larger one is defined as PVmax, PVmin / PVmax is ≤ 1 / 3.EFFECT: It is possible to provide a photomask in which the influence of the flatness of the substrate due to the directionality of the reduction ratio on the IPE is suppressed in the exposure by the high NA generation EUVL using a photomask, particularly in the exposure using an optical system employing an anamorphic lens, and which has a wide exposure area on the main surface and good transfer performance.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to a mask blank substrate used in photolithography and a manufacturing method thereof, and more particularly to a mask blank substrate suitable for mask blanks used to manufacture photomasks for photolithography using EUV (Extreme Ultra Violet) light as exposure light, and a manufacturing method thereof. [Background technology]

[0002] In recent years, interest in AI and IoT has led to the need for computational processing of massive amounts of data, which has led to demands for faster computational processing and lower power consumption. To meet this demand, the performance of IC chips needs to be improved, and miniaturization of electrical wiring is generally adopted as an effective means. The miniaturization of wiring is primarily achieved by increasing the numerical aperture and shortening the wavelength of exposure light, and in recent years, extreme ultraviolet (EUV) lithography (EUVL) using EUV light is beginning to be put into practical use.

[0003] Photomasks are a key element in EUVL, and improving the flatness of glass substrates used as photomask masters (mask blanks) is crucial for achieving accurate exposure. While double-sided simultaneous polishing is the mainstream method for manufacturing conventional glass substrates, this alone does not provide flatness sufficient for EUVL. To achieve high flatness, polishing must be performed on each side individually to adjust the flatness to match the surface shape. This is achieved using local processing techniques such as local etching and local polishing. These techniques remove relatively convex areas on the main surface of the glass substrate, thereby making the entire glass substrate nearly flat. Furthermore, the wavefront correction function of the exposure system can be used to optically correct the surface shape of the photomask, resulting in a photomask with high optical flatness.

[0004] Due to the recent trend towards miniaturization, a high NA is also required for EUVL, and anamorphic lenses are used as the optical system for high-NA generation EUVL. Conventional optical systems reduce the circuit on the mask symmetrically in the vertical and horizontal directions, but optical systems using anamorphic lenses reduce the circuit asymmetrically in the X-axis and Y-axis directions.

[0005] In response to this trend toward miniaturization, the flatness requirements for mask blank substrates are becoming increasingly stringent. The flatness requirements have reached a level that cannot be met by merely pursuing polishing technology, and instead of simply manufacturing mask blank substrates with flatness as an indicator, it is now necessary to selectively manufacture mask blank substrates with shapes that can be used for exposure.

[0006] International Publication No. 2016 / 098452 (Patent Document 1) describes a mask blank substrate that can be suitably used for EUVL, and describes a mask blank substrate having good flatness and a method for manufacturing such a mask blank substrate.

[0007] Japanese Patent Laid-Open Publication No. 2017-116812 (Patent Document 2) describes a mask blank substrate that is applied to EUV lithography using an anamorphic lens. This mask blank substrate specifies local slope angles and their ratios in the X-axis and Y-axis directions, which are different reduction magnifications for the mask blank substrate. While simply reducing the local slope alone cannot sufficiently suppress the effects of misalignment of the imaging position of the substrate to be processed, Japanese Patent Laid-Open Publication No. 2017-116812 (Patent Document 2) claims that in anamorphic exposure, the effects of misalignment of the imaging position of the substrate to be processed, which can occur due to the local slope, can be suppressed relatively easily. [Prior art documents] [Patent documents]

[0008] [Patent Document 1] International Publication No. 2016 / 098452 [Patent Document 2] Japanese Patent Application Publication No. 2017-116812 [Patent Document 3] Japanese Patent Application Laid-Open No. 2010-194705 Summary of the Invention [Problem to be solved by the invention]

[0009] In an optical system that uses an anamorphic lens, the circuit is reduced asymmetrically, but the positional deviation IPE during exposure is calculated using the following formula: IPE=d×tanθ / M (In the formula, d is the flatness of the photomask surface, θ is the angle of incidence of the exposure light, and M is the reduction ratio.) As shown in the figure, the IPE is inversely proportional to the reduction ratio, so the reduction ratio changes depending on the direction of the photomask, and the IPE also changes.

[0010] In other words, the smaller the reduction ratio in the direction, the larger the IPE, so the flatness in that direction has a relatively large effect on the IPE. Therefore, in high-NA generation EUVL optical systems, it can be said that the IPE can be predicted more accurately by dividing the surface shape of the photomask based on that direction.

[0011] The mask blank substrate in WO 2016 / 098452 (Patent Document 1) is a mask blank substrate whose flatness has been isotropically improved, and the flatness of the mask blank substrate does not take into consideration the directionality of the reduction magnification in an optical system that employs an anamorphic lens. Therefore, in an optical system that employs an anamorphic lens, the flatness of the mask blank substrate has a large impact on the IPE.

[0012] On the other hand, in the mask blank substrate of JP 2017-116812 A (Patent Document 2), a local slope is specified, and although the influence of the local surface shape in a predetermined measurement range of the main surface is taken into account, the influence of the surface shape that widely covers the exposure area of ​​the main surface of the mask blank substrate is not taken into account.

[0013] The present invention has been made in view of the above circumstances, and aims to provide a mask blank substrate that provides a photomask that can achieve transfer accuracy that is extremely advantageous for next-generation high-NA EUV exposure in exposure using a photomask, particularly exposure by EUVL, over a wide exposure area of ​​the main surface, particularly over the entire exposure area, and a method for producing such a mask blank substrate with good productivity. [Means for solving the problem]

[0014] As a result of intensive research into solving the above-mentioned problems, the inventors have found that by separating the surface shape of a mask blanks substrate into an X-direction component and a Y-direction component, specifically by polynomial approximating the surface shape of the mask blanks substrate and separating the polynomial-approximated surface shape into an X-direction component and a Y-direction component, and that a mask blanks substrate having different flatness in each of the X-direction component and the Y-direction component of the separated surface shape can provide a photomask that has a small effect on IPE in photomasks used in high NA generation EUVL and has a wide exposure area on the main surface, and in particular, good transfer performance over the entire exposure area, and further that such a mask blanks substrate can be manufactured reliably and with good productivity by local processing under predetermined conditions, thereby completing the present invention.

[0015] Therefore, the present invention provides the following mask blank substrate and method for manufacturing a mask blank substrate. 1. A mask blank substrate having two main surfaces, a first main surface and a second main surface, each of which is rectangular and has dimensions of 152 mm or more × 152 mm or more, When a rectangular region surrounded by four sides is set, the rectangular region has a center at the intersection of the diagonals of the main surface and is located 5 mm inside the four sides of the main surface that are parallel to each of the four sides of the main surface, the flatness of the rectangular region on at least one of the first main surface and the second main surface is 100 nm or less; The surface shape (S) of the rectangular region is separated into an X-direction component (Sx) along one of the four sides of the main surface and a Y-direction component (Sy) along another side perpendicular to the one side, The smaller of the PV value (PVx) which is the difference between the maximum height and minimum height of the X-direction component and the PV value (PVy) which is the difference between the maximum height and minimum height of the Y-direction component is used as the PV. min , the larger one is PV max When PV min / PV max A mask blank substrate characterized in that the ratio (PV ratio) expressed by 2. The mask blank substrate according to 1, characterized in that the surface shape (S) of the rectangular region is polynomial-approximated by the following formula (1), and from formula (1), the X-direction component (Sx) is separated as the following formula (2), and the Y-direction component (Sy) is separated as the following formula (3).

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[0016] The mask blank substrate of the present invention suppresses the effect on IPE during exposure by high NA generation EUVL using a photomask, particularly the effect on IPE of the substrate flatness due to the directionality of the reduction magnification during exposure using an optical system that employs an anamorphic lens, and can provide a photomask that has a wide exposure area on its main surface, and in particular has good transfer performance over the entire exposure area.Furthermore, the mask blank substrate manufacturing method of the present invention allows such mask blank substrates to be manufactured reliably and with high productivity. [Brief explanation of the drawings]

[0017] [Figure 1] FIG. 1 is a conceptual diagram of surface shape separation in the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0018] The present invention will be described in detail below. The mask blank substrate of the present invention has two main surfaces, a first main surface and a second main surface, and a predetermined thickness. The main surfaces of the mask blank substrate of the present invention are preferably rectangular, and may be square or rectangular. The size and thickness of the main surfaces can be appropriately selected according to the size required for the photomask (exposure mask or transfer mask) manufactured using the mask blank substrate and are not particularly limited. However, it is preferable that the size of the main surface is 152 mm or more × 152 mm or more. In other words, it is preferable that the length of any one of the four sides of the main surface is 152 mm or more, and the length of the side perpendicular to this arbitrary side is 152 mm or more. In this case, the lengths of the two perpendicular sides may be the same or different. The upper limit of the size of the main surface is not particularly limited, but is preferably 304 mm or less × 152 mm or less, more preferably 294 mm or less × 152 mm or less, and even more preferably 284 mm or less × 152 mm or less. On the other hand, the thickness of the substrate is usually 6.35 mm.

[0019] Examples of the size of the main surface include 152 mm × 152 mm, 152 mm × 284 mm, 152 mm × 294 mm, and 152 mm × 304 mm. Examples of the thickness of the substrate include 6.35 mm. A substrate with a main surface size of 152 mm × 152 mm and a thickness of 6.35 mm is known as a 6025 substrate, and has two main surfaces, a first main surface and a second main surface, each measuring 6 inches × 6 inches, and a thickness of 0.25 inches.

[0020] The material for the mask blank substrate of the present invention may be a conventionally used material and is not particularly limited, but since the substrate is exposed to high-energy exposure light when drawing a fine pattern, TiO2-doped quartz glass containing 3 to 10 mass% TiO2, which has very high dimensional stability at high temperatures, is preferably used.The raw material substrate for the mask blank substrate can be synthesized, molded, and processed according to conventional methods.

[0021] High flatness is required for mask blank substrates. This is because the higher the flatness, the easier it is to achieve the intended exposure, and it can be said that a substrate with higher flatness is a mask blank substrate that is more suitable for drawing fine patterns. Mask blank substrates have a first main surface on which an exposure pattern (such as a wiring pattern) is formed after film formation, and a second main surface on which no exposure pattern is formed. The second main surface is adsorbed and held by the exposure machine. Of these, reflective masks are the mainstream for cutting-edge applications, so an extremely high level of flatness is required for the first main surface. Therefore, various planarization methods using local processing techniques are being investigated to improve the flatness of the first main surface.

[0022] In recent years, there has been a trend toward higher numerical apertures (NAs) in extreme ultraviolet (EUV) lithography (EUVL) to meet the demand for finer photomask patterns. Specifically, the NA has been increased from the conventional NA of 0.33 to 0.55. This high-NA EUV exposure employs a different anamorphic optical system. While the conventional NA of 0.33 exposed the photomask surface at the same reduction ratio in both the X and Y directions, the high-NA 0.55 photomask has a different reduction ratio in both the X and Y directions. Specifically, in the high-NA 0.55 generation, the exposed pattern shape is not reduced to a similar shape, but is reduced to a non-similar shape that is smaller in one direction than the other, depending on the ratio of the reduction ratios. While this enables further miniaturization of patterns, the impact of the difference in reduction ratio cannot be ignored. At a high NA of 0.55, the influence of positional deviation during exposure (IPE) due to differences in reduction magnification cannot be ignored from the viewpoint of the surface shape of the photomask.

[0023] IPE is a parameter that directly affects exposure quality. EUV light is irradiated at an incident angle θ (for example, 6°) that is inclined relative to the direction perpendicular to the main surface of the photomask substrate, so the positional deviation of the exposure pattern on the wafer occurs depending on the surface shape (height map) of the photomask. IPE is calculated by the following formula: IPE=d×tanθ / M (In the formula, IPE is the positional deviation during exposure, d is the flatness of the photomask surface, θ is the angle of incidence of the exposure light, and M is the reduction ratio.) The smaller the IPE, the better the exposure with less misregistration.

[0024] Regarding the relationship between IPE and reduction magnification M, in a conventional EUV exposure tool with an NA of 0.33, M = 4 regardless of the photomask direction. On the other hand, in an EUV exposure tool with an NA of 0.55, there are directions where M = 4 and directions where M = 8. Therefore, the IPE differs depending on the photomask direction, and the IPE in the direction where M is small has a relatively large impact. For this reason, the IPE of a photomask used in an EUV exposure tool with an NA of 0.55 must be considered separately as the IPE in the X direction (IPEx) and the IPE in the Y direction (IPEy).

[0025] The relationship between the IPE (IPEs) of the photomask surface (entire surface), the IPE in the X direction (IPEx), and the IPE in the Y direction (IPEy) is expressed by the following formula: IPEx=dx×tanθ / Mx IPEy=dy×tanθ / My IPEs=((IPEx) 2 +(IPEy) 2 ) 0.5 (In the formula, dx is the flatness of the photomask surface in the X direction, dy is the flatness of the photomask surface in the Y direction, θ is the angle of incidence of the exposure light, Mx is the reduction ratio in the X direction, and My is the reduction ratio in the Y direction.)

[0026] As expressed by the above formula, in order to consider IPE separately as IPE in the X direction (IPEx) and IPE in the Y direction (IPEy), it is necessary to separate the flatness of the photomask surface into the flatness component in the X direction (dx) and the flatness component in the Y direction (dy).

[0027] The surface shape of a mask blank substrate and the surface shape of a substrate in the state of a mask blank in which a film is formed on the surface of the mask blank substrate can be said to be almost equivalent in practice, and there is substantially no difference in the flatness of the two. Furthermore, the surface shape of a substrate in the state of a mask blank and the surface shape of a photomask can be said to be almost equivalent in practice, and there is substantially no difference in the flatness of the two. In other words, the IPE of a photomask is greatly dependent on the flatness of the mask blank substrate. Therefore, it is important to control the flatness of the mask blank substrate, and it can be said that this can improve the IPE of the photomask.

[0028] Therefore, in the present invention, the surface shape of a mask blanks substrate is separated into an X-direction (X-axis direction) component and a Y-direction (Y-axis direction) component (components in two orthogonal directions), and flatness is evaluated for each of the X-direction component and the Y-direction component of the separated surface shape.The present invention provides a mask blanks substrate that has a difference between the flatness of the X-direction component and the flatness of the Y-direction component (flatness is biased toward either the X-direction or the Y-direction), and is suitable for exposure at a high NA of 0.55 using an anamorphic optical system where there is a difference in reduction magnification between the X-direction and the Y-direction of the photomask, and that can reduce positional deviation during exposure (IPE). If the surface shape of a mask blank substrate is separated into an X-direction component and a Y-direction component and the flatness is evaluated for each of the X-direction component and the Y-direction component, then based on the results, the main surface of the mask blank substrate can be polished more in either the X-direction component or the Y-direction component by a technique such as local processing, thereby making it possible to obtain a main surface whose flatness is biased in either the X-direction or the Y-direction.

[0029] Until now, evaluation of the flatness of the surface shape of a mask blank substrate by separating it into the X-direction component and the Y-direction component has not been performed. Furthermore, the surface shape of a mask blank substrate has not been considered to have flatness biased in either the X-direction or the Y-direction, and polishing to achieve such a surface shape has not been performed.

[0030] To separate the surface shape (S) into the X-direction component (Sx) and the Y-direction component (Sy), polynomial approximation using the least squares method can be applied. For example, a method can be applied in which the surface shape of a mask blank substrate obtained by actual measurement is polynomial-approximated by the following formula (1), the obtained approximate surface shape is used as the surface shape (S), and the X-direction component (Sx) and the Y-direction component (Sy) are separated into the following formulas (2) and (3), respectively. This makes it possible to more accurately separate the flatness of the mask blank substrate into the X-direction component flatness (dx) and the Y-direction component flatness (dy).

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[0031] Fig. 1 is a conceptual diagram of the separation of the surface shape (S) in the present invention. In this case, first, the surface shape of the main surface is measured to obtain the surface shape (Sm) as shown in Fig. 1(A), then the obtained surface shape (Sm) is corrected as necessary and then polynomial-approximated to obtain the approximate surface shape (Sa) as shown in Fig. 1(B), which is used as the surface shape (S) and is separated into the X-direction component (Sx) as shown in Fig. 1(C) and the Y-direction component (Sy) as shown in Fig. 1(D).

[0032] Specifically, the separation and calculation of the X-direction component (Sx) and the Y-direction component (Sy) by polynomial approximation can be performed by the following equations.

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[0033] Sx and Sy can also be expressed as follows: Sx=a1x 1 +a2x 2 +a3x 3 +··· Sy=b1y 1 +b2y 2 +b3y 3 +··· (In the formula, a1, a2, a3... are coefficients of each order, x is the X coordinate, b1, b2, b3... are coefficients of each order, and y is the Y coordinate.)

[0034] When the main surface of a mask blank substrate is a rectangle of 152 mm or more × 152 mm or more, particularly 152 mm × 152 mm or 152 mm × 284 mm, the surface shape of the main surface can be approximated extremely well by approximating it using up to tenth-order terms (n = 10) as in the following formula: Note that the more terms there are, the more accurate the approximation becomes, but there is a trade-off between calculation time and approximation accuracy, and from that perspective, approximation up to tenth-order terms is preferable.

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[0035] From the X-direction component (Sx) and the Y-direction component (Sy) calculated in this way, the PV value (PVx), which is the difference between the maximum height and minimum height of the X-direction component, and the PV value (PVy), which is the difference between the maximum height and minimum height of the Y-direction component, can be calculated.

[0036] The mask blank substrate of the present invention defines a rectangular region surrounded by four sides, centered at the intersection of the diagonals of the main surface, and located 5 mm inside each of the four sides of the main surface that are parallel to each other, and in one or both of the rectangular regions of the first and second main surfaces, particularly the rectangular region of the first main surface, the surface shape (S) of the rectangular region is separated into an X-direction component (Sx) along one of the four sides of the main surface, specifically an X-direction component (Sx) consisting only of X-direction components, and a Y-direction component (Sy) along the other side perpendicular to the one side, specifically a Y-direction component (Sy) consisting only of Y-direction components, and defines the smaller of a PV value (PVx) which is the difference between the maximum height and the minimum height of the X-direction component and a PV value (PVy) which is the difference between the maximum height and the minimum height of the Y-direction component as the PV.min , the larger one is PV max When this is done, PV min / PV max is preferably 1 / 3 (about 0.33) or less, and more preferably 1 / 4 (0.25) or less.

[0037] For example, if the size of the main surface is 152 mm × 152 mm, a 142 mm × 142 mm rectangular region can be set that is centered on the intersection of the diagonals of the main surface and parallel to the four sides of the main surface. Furthermore, if the size of the main surface is 152 mm × 284 mm, a 142 mm × 274 mm rectangular region can be set that is centered on the intersection of the diagonals of the main surface and parallel to the four sides of the main surface. Since it is more effective and efficient to evaluate the center of the main surface when the shape of the main surface is not square, for example, if the size of the main surface is 152 mm × 284 mm, the rectangular region may be substituted with a rectangular region for a 152 mm × 152 mm main surface whose short sides are the same length and whose size is 142 mm × 142 mm and whose center is the intersection of the diagonals of the main surface and parallel to the four sides of the main surface.

[0038] If the PV ratio is within this range, the flatness of the component in one direction will be sufficiently low, and therefore, by determining the direction in which the X and Y directions are set in the exposure tool so that the IPE is more favorable depending on the flatness of the component in one direction and the flatness of the component in the other direction, and then setting the photomask in the exposure tool, it is possible to obtain good transfer accuracy even with EUV exposure with an NA of 0.55, especially exposure using an optical system that employs an anamorphic lens optical system.

[0039] The mask blank substrate of the present invention is a substrate whose main surface has a shape in which there is a difference in flatness between the X direction and the Y direction (two perpendicular directions) on the main surface (flatness is biased toward either the X direction or the Y direction). A photomask manufactured from a mask blank substrate whose main surface has such a surface shape can achieve good transfer accuracy even with EUV exposure at an NA of 0.55. The X direction may be set in the direction of any of the four sides of the rectangular region, and the Y direction is set in a direction perpendicular to the X direction.

[0040] In the mask blank substrate of the present invention, the flatness of one or both of the rectangular regions on the first and second main surfaces, particularly the rectangular region on the first main surface (flatness of the entire surface) is 100 nm or less, more preferably 50 nm or less, and even more preferably 30 nm or less. The flatness is a peak-to-valley (PV) value, which is expressed as the difference between the height at the highest point on the surface and the height at the lowest point. A least-squares plane of the surface can be applied as the reference plane for this height.

[0041] Even if a substrate has a flatness (flatness over the entire surface) exceeding 100 nm, it may have a PV ratio of 1 / 3 or less. However, EUV exposure requires a certain level of flatness over a wide area of ​​the exposed main surface, particularly across the entire exposed area. If the flatness is 100 nm, the PV value (PVx), which is the difference between the maximum and minimum heights of the X-direction component, and the PV value (PVy), which is the difference between the maximum and minimum heights of the Y-direction component, will not exceed 100 nm, and the main surface will not have extremely uneven flatness in either the X-direction or the Y-direction. If the flatness is 100 nm or less, the substrate is suitable for use as a mask blank for EUV exposure. On the other hand, a substrate with a flatness over 100 nm may not be applicable to a photomask used for EUV exposure. In the mask blank substrate of the present invention, the flatness of the rectangular region (flatness over the entire surface) is greater than 0 nm, and the practical lower limit is usually 5 nm or more.

[0042] PV minis preferably 10 nm or less, more preferably 8 nm or less, and even more preferably 5 nm or less. max is preferably more than 0 nm and not more than 30 nm, more preferably more than 0 nm and not more than 24 nm, and even more preferably more than 0 nm and not more than 15 nm. min and PV max is a PV min is 10 nm or less and PV max is preferably more than 0 nm and 30 nm or less, and PV min is 8nm or less and PV max is more preferably more than 0 nm and 24 nm or less, and PV min is 5nm or less and PV max It is more preferable that PV is more than 0 nm and 15 nm or less. min and PV max If is within this range, the flatness is high in both the X direction and the Y direction, the flatness required for EUV exposure is ensured, and higher flatness can be obtained in either the X direction or the Y direction.

[0043] Next, a method for producing a substrate for a mask blank according to the present invention will be described. The method for producing a mask blank substrate of the present invention includes finish polishing of at least one of the first and second main surfaces of the mask blank substrate (raw material substrate for the mask blank substrate). Preferably, the method for producing a mask blank substrate of the present invention further includes local processing of at least one of the first and second main surfaces of the mask blank substrate (raw material substrate for the mask blank substrate), which is carried out before the finish polishing.

[0044] In the manufacture of mask blank substrates, raw material substrates for mask blank substrates are prepared from glass ingots. Raw material substrates for mask blank substrates can usually be prepared by first cutting a glass ingot into a predetermined shape and processing the outer shape, and then polishing the main surfaces and end faces. Polishing can be performed in several stages, such as rough polishing, precision polishing, and ultra-precision polishing. This polishing can be performed using a polishing cloth and abrasive. The abrasive is not particularly limited, but examples that can be used include an aqueous dispersion of cerium oxide having an average primary particle size of 10 to 100 nm, and an aqueous dispersion of silica nanoparticles having an average primary particle size of 10 to 100 nm (aqueous dispersion of colloidal silica).

[0045] The method for producing a mask blank substrate of the present invention includes the steps of: (A-0) a step of measuring the surface shape (S0) of the main surface; (B-0) calculating the surface shape (S) of the main surface by adding the change (ΔS) in the surface shape of the main surface due to finish polishing to the surface shape (S0) of the main surface; (C-0) Separating the surface shape (S) into an x-direction component (Sx) and a y-direction component (Sy); (D-0) Calculate the PV value (PVx), which is the difference between the maximum height and minimum height of the X-direction component (Sx), and the PV value (PVy), which is the difference between the maximum height and minimum height of the Y-direction component (Sy), and determine the smaller of the PV values ​​(PVx and PVy) as the PV. min , the larger one is PV max As a PV min / PV max A process of calculating the ratio (PV ratio) expressed by Includes:

[0046] In step (A-0), the range for measuring the surface shape (S0) of the raw material substrate for the mask blank substrate varies depending on the size of the main surface (substrate size), and the range for determining a predetermined surface shape can be a rectangular area set appropriately depending on the size of the main surface. For example, if the main surface is a rectangle measuring 152 mm x 152 mm, the range for measuring the surface shape (S0) is preferably a rectangular area measuring 142 mm x 142 mm, centered at the intersection of the diagonals of the main surface and parallel to the four sides of the main surface. In the present invention, the surface shape can be measured using, for example, a laser interferometer, and the surface shape can be obtained as a height map of the target surface.

[0047] The shape of the main surface before finish polishing is usually a shape that cancels out the change in shape of the main surface caused by finish polishing (generally a convex shape (a shape in which the central part of the main surface protrudes) or a concave shape (a shape in which the central part of the main surface is recessed)), and even if the flatness of the shape of the main surface before finish polishing is evaluated, the evaluation result will only be different from the evaluation result of the shape of the main surface of the mask blank substrate that is finally obtained.

[0048] The change in the shape of the main surfaces caused by finish polishing is reproducible, and by understanding the change in the shape of the main surfaces after finish polishing under specified polishing conditions and applying this to the shape of the main surfaces before finish polishing, it is possible to predict the shape of the main surfaces after finish polishing performed under specified polishing conditions. This makes it possible to evaluate the shape of the main surfaces obtained after finish polishing at a stage before finish polishing and evaluate the quality of the shape of the main surfaces obtained by finish polishing. The change in the shape of the main surfaces after finish polishing under specified polishing conditions can be obtained by previously processing and evaluating a substrate having a surface shape similar to that of the substrate to be processed.

[0049] In the step (B-0), the change (ΔS) in the surface shape of the main surface due to the finish polishing can be, for example, the change (ΔS) in the surface shape obtained by a prior evaluation using the method described above. Here, the surface shape (S) can be calculated by the following formula using the surface shape (S0) measured in the step (A-0) and the change (ΔS) in the surface shape of the main surface due to the finish polishing: S=S0+ΔS The obtained surface shape (S) is applied to the (C-0) step. This allows the surface shape to be applied taking into account the shape of the main surface obtained after finish polishing.

[0050] In step (C-0), polynomial approximation using the least squares method can be applied to separate the X-direction component (Sx) and the Y-direction component (Sy). Specifically, polynomial approximation as shown in the following formula can be applied to calculate the X-direction component (Sx) and the Y-direction component (Sy).

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[0051] When the main surface of the mask blank substrate is a rectangle of 152 mm or more × 152 mm or more, particularly 152 mm × 152 mm or 152 mm × 284 mm, an extremely good approximation can be achieved by approximating up to the tenth order (n = 10) as in the following formula.

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[0052] In step (D-0), the PV values ​​of the X-direction component (Sx) and the Y-direction component (Sy) obtained in step (C-0) are calculated, and the PV ratio is calculated from the obtained PV values. Specifically, the PV values ​​can be calculated as the PV values ​​(PVx) and PV values ​​(PVy), which are the differences between the maximum and minimum heights, by subtracting the minimum height from the maximum height for each of Sx and Sy.

[0053] When the PV ratio in the (D-0) step is within a predetermined range, finish polishing can be performed without performing local processing. min and PV max However, if both of these are within the predetermined range, finish polishing can be performed without performing local processing. On the other hand, if the PV ratio in the (D-0) step is not within the predetermined range, it is preferable to perform local processing. Also, if the PV ratio in the (D-0) step is not within the predetermined range, it is preferable to perform local processing. min and PV max If one or both of the above are not within the predetermined range, local processing can be performed.

[0054] The method for producing a mask blank substrate of the present invention further comprises: (E) A process of performing local processing to reduce the PV ratio may also include:

[0055] In step (E), specifically, the smaller of the PV values ​​(PVx and PVy) (PV min ) to make the shape of the component in the direction corresponding to the smaller value of PVx or PVy (PV minPerform local processing so as to reduce PVx or PVy corresponding to ). For example, when PVx > PVy, locally process the component in the Y direction (Sy); when PVx < PVy, locally process the component in the X direction (Sx); when PVx = PVy, locally process either the component in the X direction (Sx) or the component in the Y direction (Sy) so as to approach flatness. Thereby, the PV ratio can be lowered. The local processing may process only the component in the direction approaching flatness or may process the components in both directions. From the viewpoint of approaching flatness of either the component in the X direction (Sx) or the component in the Y direction (Sy), it suffices to selectively process the component in the direction approaching flatness. However, in general local processing, when processing a component in one direction, the component in the other direction is also processed. Therefore, local processing may be performed such that the component in the direction approaching flatness is processed more greatly.

[0056] In the present invention, as the local processing, it is possible to perform processing for selectively removing a relatively convex portion with respect to the main surface of the substrate for mask blanks. The local processing can be performed by local polishing, specifically, polishing using a polishing cloth and a polishing agent. The polishing agent is not particularly limited. For example, an aqueous dispersion of cerium oxide having an average primary particle diameter of 10 to 100 nm, an aqueous dispersion of silica nanoparticles having an average primary particle diameter of 10 to 100 nm (colloidal silica aqueous dispersion), etc. can be used. Further, methods such as Magneto Rheological Finishing (MRF) can also be applied to the local processing.

[0057] The surface shape (S) is represented by the following formula based on the surface shape (S1) after local processing and the change (ΔS) in the surface shape of the main surface by finish polishing S = S1 + ΔS and the surface shape (S1) after local processing corresponds to the surface shape (S1) in steps (A - 1) and (B - 1). Ideally, when processing to approach flatness of the component in the X direction (Sx), S ≈ Sy; when processing to approach flatness of the component in the Y direction (Sy), S ≈ Sx.

[0058] The method for producing a mask blank substrate of the present invention further comprises: (A-1) a step of measuring the surface shape (S1) of the main surface after local processing; (B-1) calculating a surface shape (S) of the main surface by adding a change (ΔS) in the surface shape of the main surface due to finish polishing to the surface shape (S1) of the main surface; (C-1) A step of separating the surface shape (S) in the step (B-1) into an x-direction component (Sx) and a y-direction component (Sy); (D-1) PV value (PVx) which is the difference between the maximum height and minimum height of the X-direction component (Sx) in the (C-1) process, and PV value (PVy) which is the difference between the maximum height and minimum height of the Y-direction component (Sy) in the (C-1) process are calculated, and the smaller of the PV values ​​(PVx and PVy) is determined as PV min , the larger one is PV max As a PV min / PV max A process of calculating the ratio (PV ratio) expressed by may also include:

[0059] Steps (A-1), (B-1), (C-1), and (D-1) can be carried out in the same manner as steps (A-0), (B-0), (C-0), and (D-0), respectively, by replacing the surface shape (S0) of the raw material substrate in steps (A-0) and (B-0) with the surface shape (S1) of the main surface after local processing.

[0060] When the PV ratio in the step (D-1) is within a predetermined range, finish polishing can be performed without further local processing. min and PV max However, if the PV ratio in step (D-1) is not within the predetermined range, it is preferable to perform further local processing. min and PV maxIf either or both of the above are not within the predetermined range, further local processing can be carried out. The local processing can be carried out in the same manner as in step (E).

[0061] If the PV ratio in step (D-1) is not within the predetermined range, it is preferable to repeat step (E) and steps (A-1) to (D-1) until the PV ratio falls within the predetermined range. min and PV max If one or both of the above are not within the predetermined range, step (E) and steps (A-1) to (D-1) can be repeatedly performed until the PV ratio falls within the predetermined range.

[0062] The defect level of the main surface obtained by local processing alone may be insufficient, particularly in mask blank substrates for cutting-edge products. Therefore, in the manufacture of mask blank substrates of the present invention, the defect level is reduced by performing finish polishing. In the present invention, finish polishing can be performed by adding a polishing slurry (aqueous colloidal silica dispersion) containing colloidal silica with an average primary particle diameter of 10 to 100 nm to the polishing cloth. Simultaneous double-sided polishing using a double-sided polishing machine is preferably used as the polishing method. The polishing cloth is not particularly limited, but a suede-type soft polishing cloth can be used.

[0063] The method for producing a mask blank substrate of the present invention further comprises: (A-2) measuring the surface shape (S2) of the main surface after finish polishing; (C-2) Separating the surface shape (S2) into an x-direction component (Sx) and a y-direction component (Sy) as the surface shape (S); (D-2) PV value (PVx), which is the difference between the maximum height and minimum height of the X-direction component (Sx) in the (C-2) process, and PV value (PVy), which is the difference between the maximum height and minimum height of the Y-direction component (Sy) in the (C-2) process, are calculated, and the smaller of the PV values ​​(PVx and PVy) is determined as PV min , the larger one is PV max As a PVmin / PV max A process of calculating the ratio (PV ratio) expressed by It is preferred that the composition contains:

[0064] Steps (A-2), (C-2), and (D-2) can be performed in the same manner as steps (A-0), (C-0), and (D-0), respectively, by replacing the surface shape (S0) in step (A-0) with the surface shape (S2) of the main surface after finish polishing. In this case, since the surface shape (S2) is the surface shape after finish polishing, it is not necessary to consider the change (ΔS) in the surface shape of the main surface due to finish polishing, and it is not necessary to perform steps (B-0) or steps corresponding to step (B-1) between steps (A-2) and (C-2).

[0065] In the method for manufacturing a mask blank substrate of the present invention, the PV ratio is preferably 1 / 3 (about 0.33) or less, more preferably 1 / 4 (0.25) or less in the predetermined range (the predetermined range in the evaluation of the value obtained in the step (D-0), the step (D-1), or the step (D-2)). min The predetermined range of is preferably 10 nm or less, more preferably 8 nm or less, and even more preferably 5 nm or less, and PV max The predetermined range is preferably more than 0 nm and not more than 30 nm, more preferably more than 0 nm and not more than 24 nm, and even more preferably more than 0 nm and not more than 15 nm.

[0066] The above-mentioned PV ratio, PV min and PV max The predetermined range is a PV ratio, PV min and PV max For example, when the main surface is a rectangle of 152 mm × 152 mm, the PV ratio and PV min and PV maxIn the method for producing a mask blank substrate of the present invention, the flatness of the rectangular region (flatness of the entire surface) is preferably 100 nm or less, more preferably 50 nm or less, and even more preferably 30 nm or less. The flatness (flatness of the entire surface) is greater than 0 nm, and the practical lower limit is usually 5 nm or more.

[0067] A photomask obtained by forming a film pattern from a mask blank in which a film is formed on the mask blank substrate of the present invention exhibits high flatness during EUV exposure with an NA of 0.55, particularly during exposure using an anamorphic lens optical system, and can be suitably applied to next-generation EUVL. [Example]

[0068] EXAMPLES The present invention will be specifically explained below by showing examples and comparative examples, but the present invention is not limited to the following examples.

[0069] In the examples and comparative examples, the local processing device described in JP 2010-194705 A (Patent Document 3) was used for local processing. Local processing using this device involves controlling the movement speed of a fine polishing tool to polish the entire substrate surface. The polishing tool is moved slowly in relatively convex areas and quickly in relatively concave areas, thereby achieving the desired shape. The processing tool used in the local processing device was a wool felt buff, and the polishing slurry used was a mixture of silica nanoparticles (AJ-3540, manufactured by Nissan Chemical Co., Ltd.) and a trace amount of antifoaming agent (Shin-Etsu Silicone KS-537, manufactured by Shin-Etsu Chemical Co., Ltd.). After polishing, the substrate was washed with a cleaning solution containing KOH to remove the colloidal silica and then dried.

[0070] In the examples and comparative examples, the finish polishing was carried out by placing the locally processed substrate on a carrier of a polishing machine equipped with a suede-type soft abrasive cloth and using a polishing solution containing colloidal silica. After polishing, the substrate was washed with a cleaning solution containing KOH to remove the colloidal silica, and then dried.

[0071] [Example 1] Ten glass substrates (first and second main surfaces: 152 mm × 152 mm square, thickness: 6.35 mm) formed of SiO2 and TiO2 (TiO2 concentration: approximately 7% by mass) were prepared. The edge surfaces of the glass substrates (the four surfaces other than the main surfaces) were chamfered and ground, and the first and second main surfaces were then coarsely and finely polished using a polishing solution containing cerium oxide abrasive grains. The ten glass substrates (Substrates 1-1 to 1-10) were then placed in a carrier of a double-sided polishing machine equipped with a suede-type soft polishing cloth, and the first and second main surfaces were ultra-precision polished using a polishing solution containing colloidal silica abrasive grains. After ultra-precision polishing, the substrates were washed with a cleaning solution containing KOH to remove silica nanoparticles and dried to prepare the raw substrates for mask blanks.

[0072] Next, the surface shape (S0) of the first main surface of the raw material substrate for the mask blank substrate was measured using a surface shape measuring device (UltraFlat, manufactured by Tropel) ((A-0) step). The measurement range of the surface shape of the main surface was a rectangular area of ​​142 mm x 142 mm parallel to the four sides of the main surface, with the intersection of the diagonals of the main surface as the center (the same applies to the following surface shape measurements).

[0073] Next, a substrate having a surface shape similar to that of the substrate to be processed was subjected to finish polishing, and the change in the surface shape (ΔS) due to the finish polishing, which had been measured in advance, was added to the surface shape (S0) to calculate the surface shape (S) (step (B-0)).

[0074] Next, the surface shape (S) was approximated by the following polynomial, taking into account terms up to the tenth order, and separated into an X-direction component (Sx) and a Y-direction component (Sy) (step (C-0)).

number

[0075] Next, for each of the X-direction component (Sx) and Y-direction component (Sy), subtract the minimum height from the maximum height, calculate the difference as the PV value (PVx and PVy), and use the smaller of the PV values ​​(PVx and PVy) as the PV min , the larger PV value is PV max PV when min / PV max The PV values ​​(PVx and PVy) and the PV ratio are shown in Table 1.

[0076] [Table 1]

[0077] In this example, the specified range was set to a PV ratio of 1 / 3 or less, but the PV ratio of none of the substrates was within the specified range, so the component in the direction of the component with the small PV value was determined, and local processing was performed so that the PV value of the component in the direction shown in Table 1 would be small and so that the component in the direction of the component with the small PV value would be polished more (step (E)).

[0078] The surface shape (S1) of the first main surface of the substrate after local processing was measured using a surface shape measuring device (UltraFlat, manufactured by Tropel) (step (A-1)). Furthermore, from the surface shape (S1), the flatness of a 142 mm × 142 mm rectangular area parallel to the four sides of the main surface, centered at the intersection of the diagonals of the main surface (flatness of the entire surface) was calculated. The results are shown in Table 2.

[0079] Next, a substrate having a surface shape similar to that of the substrate to be processed was subjected to finish polishing, and the change in the surface shape (ΔS) due to the finish polishing, which had been measured in advance, was added to the surface shape (S1) to calculate the surface shape (S) (step (B-1)).

[0080] Next, the surface shape (S) was approximated by the same polynomial as in the (C-0) step, and separated into an X-direction component (Sx) and a Y-direction component (Sy) ((C-1) step).

[0081] Next, for each of the X-direction component (Sx) and Y-direction component (Sy), subtract the minimum height from the maximum height, calculate the difference as the PV value (PVx and PVy), and use the smaller of the PV values ​​(PVx and PVy) as the PV min , the larger PV value is PV max PV when min / PV max The PV values ​​(PVx and PVy) and PV ratios after local processing are shown in Table 2.

[0082] [Table 2]

[0083] Since the PV ratio for all of the substrates was within the predetermined range, it was determined that the main surfaces would have the predetermined surface shape after finish polishing, and finish polishing was carried out.

[0084] The surface shape (S2) of the first main surface of the substrate after finish polishing was measured using a surface shape measuring device (UltraFlat, manufactured by Tropel) (step (A-2)). Furthermore, from the surface shape (S2), the flatness of a 142 mm × 142 mm rectangular area parallel to the four sides of the main surface, centered at the intersection of the diagonals of the main surface (flatness of the entire surface) was calculated. The results are shown in Table 3.

[0085] Next, the surface shape (S2) was treated as the surface shape (S) and approximated with the same polynomial as in the (C-0) step, and separated into an X-direction component (Sx) and a Y-direction component (Sy) ((C-2) step).

[0086] Next, for each of the X-direction component (Sx) and Y-direction component (Sy), subtract the minimum height from the maximum height, calculate the difference as the PV value (PVx and PVy), and use the smaller of the PV values ​​(PVx and PVy) as the PV min , the larger PV value is PV max PV when min / PV maxThe PV values ​​(PVx and PVy) and PV ratios of the mask blank substrates after finish polishing are shown in Table 3.

[0087] [Table 3]

[0088] [Example 2] Three glass substrates (first and second main surfaces: 152 mm × 152 mm square, thickness: 6.35 mm) formed of SiO2 and TiO2 (TiO2 concentration: approximately 7% by mass) were prepared. The edge surfaces of the glass substrates (the four surfaces other than the main surfaces) were chamfered and ground, and the first and second main surfaces were then coarsely and finely polished using a polishing solution containing cerium oxide abrasive grains. The three glass substrates (Substrates 2-1 to 2-3) were then placed in a carrier of a double-sided polishing machine equipped with a suede-type soft polishing cloth, and the first and second main surfaces were ultra-precision polished using a polishing solution containing colloidal silica abrasive grains. After ultra-precision polishing, the substrates were washed with a cleaning solution containing KOH to remove silica nanoparticles and dried to prepare the raw substrates for mask blanks.

[0089] Next, the surface shape (S0) of the first main surface of the raw material substrate for the mask blank substrate was measured using a surface shape measuring device (UltraFlat, manufactured by Tropel) ((A-0) step).

[0090] Next, a substrate having a surface shape similar to that of the substrate to be processed was subjected to finish polishing, and the change in the surface shape (ΔS) due to the finish polishing, which had been measured in advance, was added to the surface shape (S0) to calculate the surface shape (S) (step (B-0)).

[0091] Next, the surface shape (S) was approximated by the following polynomial, taking into account terms up to the tenth order, and separated into an X-direction component (Sx) and a Y-direction component (Sy) (step (C-0)).

number

[0092] Next, for each of the X-direction component (Sx) and Y-direction component (Sy), subtract the minimum height from the maximum height, calculate the difference as the PV value (PVx and PVy), and use the smaller of the PV values ​​(PVx and PVy) as the PV min , the larger PV value is PV max PV when min / PV max The PV values ​​(PVx and PVy) and the PV ratio are shown in Table 4.

[0093] [Table 4]

[0094] In this example, the specified range is defined as PV ratio of 1 / 3 or less, PV min is 10nm or less, PV max The PV ratio and PV min and PV max Since none of these values ​​were within the specified range, the component in the direction with the small PV value was determined, and local processing was performed so that the PV value of the component in the direction shown in Table 4 would be small and so that the component in the direction with the small PV value would be polished more (step (E)).

[0095] The surface shape (S1) of the first main surface of the substrate after local processing was measured using a surface shape measuring device (UltraFlat, manufactured by Tropel) (step (A-1)). Furthermore, from the surface shape (S1), the flatness of a 142 mm × 142 mm rectangular area parallel to the four sides of the main surface, centered at the intersection of the diagonals of the main surface (flatness of the entire surface) was calculated. The results are shown in Table 5.

[0096] Next, a substrate having a surface shape similar to that of the substrate to be processed was subjected to finish polishing, and the change in the surface shape (ΔS) due to the finish polishing, which had been measured in advance, was added to the surface shape (S1) to calculate the surface shape (S) (step (B-1)).

[0097] Next, the surface shape (S) was approximated by the same polynomial as in the (C-0) step, and separated into an X-direction component (Sx) and a Y-direction component (Sy) ((C-1) step).

[0098] Next, for each of the X-direction component (Sx) and Y-direction component (Sy), subtract the minimum height from the maximum height, calculate the difference as the PV value (PVx and PVy), and use the smaller of the PV values ​​(PVx and PVy) as the PV min , the larger PV value is PV max PV when min / PV max The ratio (PV ratio) expressed by the following formula was calculated ((D-1) process). In this case, in the first local processing, the PV ratio, PV min and PV max Since either of these did not fall within the specified range, the PV ratio and PV min and PV max Step (E) and steps (A-1) to (D-1) were repeated the number of times shown in Table 5 until all of the values ​​fell within the predetermined range. Table 5 shows the PV values ​​(PVx and PVy) and PV ratios after local processing.

[0099] [Table 5]

[0100] After that, for both substrates, the PV ratio, PV min and PV max Since all of the above were within the predetermined range, it was determined that the main planes would have the predetermined surface shape after finish polishing, and finish polishing was carried out.

[0101] The surface profile (S2) of the first main surface of the substrate after finish polishing was measured using a surface profile measuring device (UltraFlat, manufactured by Tropel) (step (A-2)). Furthermore, from the surface profile (S2), the flatness of a 142 mm × 142 mm rectangular area parallel to the four sides of the main surface, centered at the intersection of the diagonals of the main surface (flatness of the entire surface) was calculated. The results are shown in Table 6.

[0102] Next, the surface shape (S2) was treated as the surface shape (S) and approximated with the same polynomial as in the (C-0) step, and separated into an X-direction component (Sx) and a Y-direction component (Sy) ((C-2) step).

[0103] Next, for each of the X-direction component (Sx) and Y-direction component (Sy), subtract the minimum height from the maximum height, calculate the difference as the PV value (PVx and PVy), and use the smaller of the PV values ​​(PVx and PVy) as the PV min , the larger PV value is PV max PV when min / PV max The PV values ​​(PVx and PVy) and PV ratios of the mask blank substrates after finish polishing are shown in Table 6.

[0104] [Table 6]

[0105] [Comparative Example 1] Three glass substrates (first and second main surfaces: 152 mm × 152 mm square, thickness: 6.35 mm) formed of SiO2 and TiO2 (TiO2 concentration: approximately 7% by mass) were prepared. The edge surfaces of the glass substrates (the four surfaces other than the main surfaces) were chamfered and ground, and the first and second main surfaces were then coarsely and finely polished using a polishing solution containing cerium oxide abrasive grains. The three glass substrates (Substrates 3-1 to 3-3) were then placed in a carrier of a double-sided polishing machine equipped with a suede-type soft polishing cloth, and the first and second main surfaces were ultra-precision polished using a polishing solution containing colloidal silica abrasive grains. After ultra-precision polishing, the substrates were washed with a cleaning solution containing KOH to remove silica nanoparticles and dried to prepare the raw substrates for mask blanks.

[0106] Next, the surface shape (S0) of the first main surface of the raw material substrate for the mask blank substrate was measured using a surface shape measuring device (UltraFlat, manufactured by Tropel).

[0107] Next, a substrate having a surface shape similar to that of the substrate to be processed was subjected to finish polishing, and the change in the surface shape due to finish polishing (ΔS), which had been measured in advance, was added to the surface shape (S0) to calculate the surface shape (S).

[0108] In this example, local processing was performed so that the flatness of the surface shape (S) (flatness of the entire surface) was 100 nm or less, and then finish polishing was performed.

[0109] The surface profile (S2) of the first main surface of the substrate after finish polishing was measured using a surface profile measuring device (UltraFlat, manufactured by Tropel). Furthermore, from the surface profile (S2), the flatness of a 142 mm × 142 mm rectangular area parallel to the four sides of the main surface, centered at the intersection of the diagonals of the main surface (flatness of the entire surface) was calculated. The results are shown in Table 7.

[0110] Next, the surface shape (S2) was approximated by the following polynomial, taking into account terms up to the tenth order, and separated into an X-direction component (Sx) and a Y-direction component (Sy).

number

[0111] Next, for each of the X-direction component (Sx) and Y-direction component (Sy), subtract the minimum height from the maximum height, calculate the difference as the PV value (PVx and PVy), and use the smaller of the PV values ​​(PVx and PVy) as the PV min , the larger PV value is PV max PV when min / PV max The PV values ​​(PVx and PVy) and PV ratios of the mask blank substrates after finish polishing are shown in Table 7.

[0112] [Table 7]

Claims

1. A mask blank substrate having two main surfaces, a first main surface and a second main surface, each of which is rectangular and has dimensions of 152 mm or more × 152 mm or more, When a rectangular region surrounded by four sides is set, the rectangular region has a center at the intersection of the diagonals of the main surface and is located 5 mm inside the four sides of the main surface that are parallel to each of the four sides of the main surface, the flatness of the rectangular region on at least one of the first main surface and the second main surface is 100 nm or less; The surface shape (S) of the rectangular region is separated into an X-direction component (Sx) along one of the four sides of the main surface and a Y-direction component (Sy) along another side perpendicular to the one side, The smaller of the PV value (PVx) which is the difference between the maximum height and the minimum height of the X-direction component and the PV value (PVy) which is the difference between the maximum height and the minimum height of the Y-direction component is determined as the PV min , the larger one is PV max When PV min / PV max A mask blank substrate characterized in that the ratio (PV ratio) represented by

2. 2. The mask blank substrate according to claim 1, wherein the surface shape (S) of the rectangular region is polynomial-approximated by the following formula (1), and the X-direction component (Sx) is separated from formula (1) as the following formula (2), and the Y-direction component (Sy) is separated from formula (1) as the following formula (3). [Equation 1] (In the formula, k and n represent the degree of the polynomial, a and b represent the coefficients, x represents the X coordinate, and y represents the Y coordinate.)

3. The PV min is 10 nm or less, and PV max 2. The mask blank substrate according to claim 1, wherein the thickness is greater than 0 nm and not more than 30 nm.

4. The PV min is 5 nm or less, and PV max 2. The mask blank substrate according to claim 1, wherein the thickness is greater than 0 nm and not more than 15 nm.

5. A method for manufacturing a mask blank substrate, comprising finish polishing of at least one of a first main surface and a second main surface of a raw material substrate for a mask blank substrate, (A-0) measuring the surface shape (S0) of the main surface; (B-0) calculating a surface shape (S) of the main surface by adding a change (ΔS) in the surface shape of the main surface due to the finish polishing to the surface shape (S0) of the main surface; (C-0) Separating the surface shape (S) into an x-direction component (Sx) and a y-direction component (Sy); (D-0) PV value (PVx) which is the difference between the maximum height and the minimum height of the component (Sx) in the X direction, and PV value (PVy) which is the difference between the maximum height and the minimum height of the component (Sy) in the Y direction are calculated, and the smaller of the PV values ​​(PVx and PVy) is determined as PV min , the larger one is PV max As a PV min / PV max A step of calculating a ratio (PV ratio) represented by A method for manufacturing a mask blank substrate, comprising:

6. 6. The manufacturing method according to claim 5, wherein the finish polishing is carried out when the PV ratio in the (D-0) step is within a predetermined range.

7. Further, local processing performed before the finish polishing is included, If the PV ratio in step (D-0) is not within the predetermined range, (E) A step of performing the local processing so as to reduce the PV ratio. The method of claim 5, comprising:

8. Furthermore, (A-1) measuring the surface shape (S1) of the main surface after local processing; (B-1) calculating a surface shape (S) of the main surface by adding a change (ΔS) in the surface shape of the main surface due to the finish polishing to the surface shape (S1) of the main surface; (C-1) A step of separating the surface shape (S) in the step (B-1) into an x-direction component (Sx) and a y-direction component (Sy); (D-1) PV value (PVx) which is the difference between the maximum height and the minimum height of the component (Sx) in the X direction in the (C-1) step, and PV value (PVy) which is the difference between the maximum height and the minimum height of the component (Sy) in the Y direction in the (C-1) step are calculated, and the smaller of the PV values ​​(PVx and PVy) is determined as PV min , the larger one is PV max As a PV min / PV max A step of calculating a ratio (PV ratio) represented by The method of claim 7, comprising:

9. 9. The manufacturing method according to claim 8, wherein the finish polishing is carried out when the PV ratio in the step (D-1) is within a predetermined range.

10. The method according to claim 8, wherein, when the PV ratio in step (D-1) is not within a predetermined range, step (E) and steps (A-1) to (D-1) are repeatedly performed until the PV ratio falls within the predetermined range.

11. Furthermore, (A-2) measuring the surface shape (S2) of the main surface after finish polishing; (C-2) separating the surface shape (S2) into an x-direction component (Sx) and a y-direction component (Sy) as a surface shape (S); (D-2) The PV value (PVx) which is the difference between the maximum height and the minimum height of the X-direction component (Sx) in the (C-2) step, and the PV value (PVy) which is the difference between the maximum height and the minimum height of the Y-direction component (Sy) in the (C-2) step are calculated, and the smaller of the PV values ​​(PVx and PVy) is determined as the PV min , the larger one is PV max As a PV min / PV max A step of calculating a ratio (PV ratio) represented by The method according to claim 6 or 9, comprising:

12. The predetermined range of the PV ratio is 1 / 3 or less, The main surface of the substrate is a rectangle of 152 mm or more × 152 mm or more, The surface shape is a surface shape of a rectangular region surrounded by four sides, with the center at the intersection of diagonals of the main surface and located 5 mm inside the four sides of the main surface that are parallel to each other, and The flatness of the rectangular region is 100 nm or less. The method of claim 11, wherein a substrate is produced.

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