Mask blanks substrate and method for manufacturing the same

A method for manufacturing a mask blank substrate having two main surfaces of a first main surface and a second main surface of 152 mm×152 mm square and having a thickness of 6.35 mm, the method comprising:

• a local processing step for one or both of the first main surface and the second main surface;
• and
• a finish polishing step subsequent to the local processing step,
• wherein the local processing step includes:
• (A) a step of estimation of a change in the shapes of the surfaces before and after the finish polishing step of the first main surface and the second main surface by simulation or measurement of the change in the shapes of the surfaces before and after the finish polishing step of the first main surface and the second main surface on another, previously processed substrate;
• (B) a step of locally processing one or both of the first main surface and the second main surface;
• (C) a step of measuring the shape of the first main surface and the shape of the second main surface after the step (B) as the shape of the surfaces before the finish polishing step;
• (D) a step of predicting the shapes of the first main surface and the second main surface after the finish polishing step based upon the estimation of the change in the shapes of the surfaces before and after the finish polishing step of the first main surface and the second main surface of the step (A) to the shapes of the surfaces before the finish polishing step obtained in the step (C);
• and
• (E) a step of determining if the process is complete by determining if the shapes of the first main surface and the second main surface predicted in the step (D) have a flatness (TIR) of 50 nm or less, and
• wherein the shapes of the first main surface and the second main surface in the step (E) are determined to have the flatness (TIR) of the calculation surface of 50 nm or less by the following steps:
• (1) arranging the first main surface and the second main surface of the substrate along a substantially vertical direction, and setting a calculation region of a square shape extending in a horizontal direction through four sides of a 138 mm×138 mm square along four sides of the first main surface and the second main surface in a central portion of the first main surface and the second main surface;
• (2) cutting out a portion in the calculation region from the first main surface in a state of facing the first main surface to obtain a first region surface;
• (3) setting a vertical plane passing through a reference point that is an arbitrary point on a central axis of a square shape of the calculation region and orthogonal to the central axis as a reference plane, setting a rotation axis passing through the reference point and parallel to any one side of four sides of a regular square that is an intersection line between the square shape of the calculation region and the vertical plane, and cutting out a portion in the calculation region is from the second main surface in a state of facing the second main surface by rotating the substrate 180 degrees along the rotation axis from a state of facing the first main surface, to obtain a second region surface;
• (4) calculating a least square plane in each of the first region surface and the second region surface;
• (5) converting the first region surface and the second region surface into a height map of the first region surface and a height map of the second region surface with respect to respective positions of the first region surface and the second region surface on the respective least square planes, where the height map of the second region surface is obtained by fitting with terms up to a 15th order of a Legendre polynomial to separate a wavelength component that contributes to the shape of the first main surface and a wavelength component that does not contribute to the shape of the first main surface from each other;
• (6) setting the height map of the second region surface as a reverse height map of the second region surface by symmetrically moving the height map with respect to a perpendicular plane passing through the rotation axis and along a 90 degree direction of the rotation;
• and
• (7) creating a map of a calculated height (Z coordinate) by adding a height of the height map of the first region surface and a height of the reverse height map of the second region surface at each position (X coordinate and Y coordinate) on the reference plane, and defining the map of the calculated height as the calculation surface.