阅读说明：本技术 光罩热膨胀校正方法 (Method for correcting thermal expansion of mask ) 是由 任茂华 白源吉 谈文毅 于 2021-08-02 设计创作，主要内容包括：光罩热膨胀校正方法,包括对一组晶圆执行曝光程序并生成配方,执行数据挖掘及数据解析以生成多个迭置参数,由该多个迭置参数取出多个预定参数,对每一预定参数执行线性回归,及生成该每一预定参数的决定系数。(A method for mask thermal expansion calibration includes performing an exposure process on a set of wafers to generate a recipe, performing data mining and data analysis to generate a plurality of overlay parameters, extracting a plurality of predetermined parameters from the plurality of overlay parameters, performing linear regression on each predetermined parameter, and generating a coefficient for determining each predetermined parameter.)

1. A method for thermal expansion correction of a mask, comprising:

performing an exposure process on the first group of wafers and generating a first recipe;

performing data mining and data parsing to generate a plurality of stacking parameters;

extracting a plurality of preset parameters from the plurality of superposed parameters;

performing a linear regression on each of the plurality of predetermined parameters; and

a first coefficient of determination (coefficient) for each of the predetermined parameters is generated.

2. The method of claim 1, further comprising:

if the first determination coefficients of the predetermined parameters are within the acceptable range, the exposure process is performed on other wafers according to the first recipe.

3. The method of claim 1, further comprising:

if at least one first decision coefficient of at least one predetermined parameter in the plurality of predetermined parameters is not in the acceptable range, executing the exposure program on a second group of wafers and generating a second formula;

performing the data mining and the data parsing to generate the plurality of update overlay parameters;

extracting a plurality of updating preset parameters from the plurality of updating superposed parameters;

performing the linear regression on each updated predetermined parameter of the plurality of updated predetermined parameters; and

a second coefficient of determination (coefficient) for each of the updated predetermined parameters is generated.

4. The method of claim 3, further comprising:

if the second determination coefficients of the updated predetermined parameters are all within the acceptable range, the exposure process is performed on other wafers according to the second recipe.

5. The method of claim 3, wherein for each of the updated predetermined parameters, the second determination coefficient is generated according to the following formula:

wherein:

R_j ²is the second determining coefficient;

n is the number of wafers in the second group of wafers, and N is a positive integer greater than 2;

ActY_ithe ith wafer of the second group of wafers corresponds to the actual value of each updated predetermined parameter;

PredY_ithe predicted value of each updated predetermined parameter corresponding to the ith wafer of the second group of wafers; and

avg (ActY) is the average value of the second set of wafers corresponding to each of the updated predetermined parameters.

6. The method of claim 5, wherein the predetermined parameter is updated for each of the plurality of predetermined parameters if 1 ≧ R_j ²>0.9, the second determination factor is within the acceptable range.

7. The method of claim 1, further comprising:

linearly combining a plurality of first decision coefficients of the plurality of predetermined parameters to generate a first weighted decision coefficient; and

if the first weighting determination factor is within an acceptable range, the exposure process is performed on other wafers according to the first recipe.

8. The method of claim 1, further comprising:

linearly combining a plurality of first decision coefficients of the plurality of predetermined parameters to generate a first weighted decision coefficient;

if the first weighting decision coefficient is not in the acceptable range, executing the exposure program on a second group of wafers and generating a second formula;

performing the data mining and the data parsing to generate the plurality of update overlay parameters;

extracting a plurality of updating preset parameters from the plurality of updating superposed parameters;

performing the linear regression on each updated predetermined parameter of the plurality of updated predetermined parameters; and

generating a second decision coefficient for each updated predetermined parameter.

9. The method of claim 8, further comprising:

performing the linear combination on a plurality of second decision coefficients of the plurality of updated predetermined parameters to generate a second weighted decision coefficient; and

if the second weight determination coefficient is within the acceptable range, the exposure process is performed on other wafers according to the second recipe.

10. The method of claim 9, wherein for each of the updated predetermined parameters, the second determination coefficient is generated according to the following formula:

wherein:

R_j ²is the second determining coefficient;

n is the number of wafers in the second group of wafers, and N is a positive integer greater than 2;

ActY_iis the ith wafer of the second group of wafersActual values corresponding to the each updated predetermined parameter;

PredY_ithe predicted value of each updated predetermined parameter corresponding to the ith wafer of the second group of wafers;

avg (ActY) is the average value of the second set of wafers corresponding to each of the predetermined parameters; and

if the second weight determination coefficient is between 0.9 and 1, the second weight determination coefficient is within the acceptable range.

11. A method for thermal expansion correction of a mask, comprising:

performing an exposure process on the first group of wafers and generating a first recipe;

performing data mining and data parsing to generate a plurality of stacking parameters;

performing a linear regression on each of the plurality of stacking parameters; and

a first determination coefficient (coefficient) of each of the stacking parameters is generated.

12. The method of claim 11, further comprising:

if the first determination coefficients of the stacking parameters are within the acceptable range, the exposure process is performed on other wafers according to the first recipe.

13. The method of claim 11, further comprising:

if at least one first decision coefficient of at least one stacking parameter in the stacking parameters is not in the acceptable range, executing the exposure program on a second group of wafers and generating a second formula;

performing the data mining and the data parsing to generate the plurality of update overlay parameters;

performing the linear regression on each updated stacking parameter of the plurality of updated stacking parameters; and

generating a second coefficient of determination (coefficient) for each of the updated stacking parameters.

14. The method of claim 13, further comprising:

if the second determination coefficients of the updated overlay parameters are within the acceptable range, the exposure process is performed on other wafers according to the second recipe.

15. The method of claim 13, wherein for each of the updated stacking parameters, the second determining coefficient is generated according to the following formula:

wherein:

R_j ²is the second determining coefficient;

n is the number of wafers in the second group of wafers, and N is a positive integer greater than 2;

ActY_ithe actual value of each updated stacking parameter corresponding to the ith wafer of the second group of wafers;

PredY_ithe predicted value of each updated stacking parameter corresponding to the ith wafer of the second group of wafers; and

avg (acty) is the average value of the second set of wafers corresponding to each updated stacking parameter.

16. The method of claim 15, wherein for each updated stacking parameter, if 1 ≧ R²>0.9, the second determination factor is within the acceptable range.

17. The method of claim 11, further comprising:

linearly combining a plurality of first decision coefficients of the plurality of stacking parameters to generate a first weighting decision coefficient;

if the first weighting determination factor is within an acceptable range, the exposure process is performed on other wafers according to the first recipe.

18. The method of claim 11, further comprising:

linearly combining a plurality of first decision coefficients of the plurality of stacking parameters to generate a first weighting decision coefficient;

if the first weighting decision coefficient is not in the acceptable range, executing the exposure program on a second group of wafers and generating a second formula;

performing the data mining and the data parsing to generate the plurality of update overlay parameters;

performing the linear regression on each updated stacking parameter of the plurality of updated stacking parameters; and

generating a second decision coefficient for each updated stacking parameter.

19. The method as recited in claim 18, further comprising:

performing the linear combination on a plurality of second decision coefficients of the plurality of updated stacking parameters to generate a second weighting decision coefficient;

if the second weight determination coefficient is within the acceptable range, the exposure process is performed on other wafers according to the second recipe.

20. The method of claim 19, wherein for each of the updated stacking parameters, the second determining coefficient is generated according to the following formula:

wherein:

R_j ²is the second determining coefficient;

n is the number of wafers in the second group of wafers, and N is a positive integer greater than 2;

ActY_ithe actual value of each updated stacking parameter corresponding to the ith wafer of the second group of wafers;

PredY_ithe predicted value of each updated stacking parameter corresponding to the ith wafer of the second group of wafers;

avg (acty) is the average value of the second set of wafers corresponding to each of the updated stacking parameters; and

if the second weight determination coefficient is between 0.9 and 1, the second weight determination coefficient is within the acceptable range.

Technical Field

The present invention relates to a method for correcting thermal expansion of a mask, and more particularly, to a method for correcting thermal expansion of a mask by correcting a linear regression generated curve.

Background

During the exposure of the wafer, the mask is heated by the deep ultraviolet light, which not only causes the mask to deform, but also causes the exposed product to generate the sequelae of improper exposure. To improve the problem of improper exposure, the astml provides a mask Heating Calibration Method. The exposure temperature of the photomask during exposure of a first batch of wafers is used for predicting the influence of the exposure of the next batch of wafers, and exposure alignment is corrected, so that the potential risk of wrong exposure overlay (overlay) caused by thermal expansion of the photomask is reduced. However, in the field of foundry, there is no method for providing any recipe correction when there is an error in the recipe (sub-recipe) for exposing the wafer, and the correction of the exposure alignment alone is not performed according to the symptoms, and the product is easily out of specification and the yield is damaged.

Disclosure of Invention

The embodiment provides a method for correcting thermal expansion of a photomask, which comprises the steps of executing an exposure program on a group of wafers to generate a formula, executing data mining and data analysis to generate a plurality of stacking parameters, extracting a plurality of preset parameters from the stacking parameters, executing linear regression on each preset parameter of the preset parameters, and generating a coefficient of determination (coefficient of determination) of each preset parameter.

Another embodiment provides a method for mask thermal expansion correction, comprising performing an exposure process on a set of wafers and generating a recipe, performing data mining and data parsing to generate a plurality of overlay parameters, performing linear regression on each overlay parameter of the plurality of overlay parameters, and generating a coefficient for determining each overlay parameter.

Drawings

FIG. 1 is a schematic diagram of an embodiment of a thermal expansion correction system for a reticle.

FIG. 2 is a flowchart illustrating a method for thermal expansion correction of a mask according to an embodiment.

FIG. 3 is a diagram illustrating an example of 4 stacking parameters generated by the computing system of FIG. 1.

FIG. 4 is a diagram illustrating an example of 4 update overlay parameters generated by the computing system of FIG. 1.

FIG. 5 is a flowchart illustrating a method for thermal expansion correction of a mask according to another embodiment.

FIG. 6 is a flowchart illustrating a method for thermal expansion correction of a mask according to another embodiment.

FIG. 7 is a flowchart illustrating a method for thermal expansion correction of a mask according to another embodiment.

[ Specification of predetermined symbols ]

10 mask thermal expansion correction system

12 computing system

14 memory

16 scanner

18 first formulation

20 second formulation

22 first group of wafers

24 second group of wafers

Thermal expansion correction method for 30, 100, 200, 300 mask

S32-S60, S102-S134, S202-S226, S302-S330

K1 to K4 parameters

Time T

% offset

Detailed Description

FIG. 1 is a schematic diagram of an embodiment of a system 10 for thermal expansion correction of a reticle. The mask thermal expansion correction system 10 includes a computing system 12, a memory 14, and a scanner 16, the memory 14 and the scanner 16 may be coupled to the computing system 12, and the memory 14 may store a first recipe 18 and a second recipe 20 when the computing system 12 generates the first recipe 18 and the second recipe 20. FIG. 2 is a flowchart illustrating an embodiment of a method 30 for mask thermal expansion calibration, the method 30 comprising:

step S32, the scanner 16 performs an exposure process on the first set of wafers 22 and generates a first recipe 18;

step S34, the computing system 12 performs data mining and data parsing to generate a plurality of overlay parameters;

step S36, the computing system 12 extracts a plurality of predetermined parameters from the plurality of stacking parameters;

step S38, the computing system 12 performs a linear regression on each of a plurality of predetermined parameters;

step S40, the computing system 12 generates a first coefficient of determination (coefficient of determination) for each predetermined parameter;

in step S42, is a plurality of first decision coefficients of a plurality of predetermined parameters within an acceptable range? If yes, go to step S44; otherwise, executing step S48;

in step S44, the scanner 16 performs an exposure process on other wafers according to the first recipe 18.

Step S48, the scanner 16 performs an exposure process on the second group of wafers 24 and generates a second recipe 20;

step S50, the computing system 12 executes data mining and data parsing to generate a plurality of update stacking parameters;

step S52, the computing system 12 extracts a plurality of updating predetermined parameters from the plurality of updating stacking parameters;

step S54, the computing system 12 performs a linear regression on each updated predetermined parameter of the plurality of updated predetermined parameters;

step S56, the computing system 12 generates a second decision coefficient for each updated predetermined parameter;

in step S58, are the second determination coefficients for updating the predetermined parameters within the acceptable range? If yes, go to step S60; otherwise, the third set of wafers replaces the second set of wafers 24 to perform step S48 to update the second recipe 20;

in step S60, the scanner 16 performs an exposure process on other wafers according to the second recipe 20.

In step S32, assuming that the first group of wafers includes 25 wafers, when the scanner 16 performs the exposure process on the first group of wafers 22, the scanner 16 sequentially exposes the 1 st to 25 th wafers of the first group of wafers 22 one hundred times, and the sensor of the scanner 16 collects the temperature variation and distribution accumulated on the mask when the scanner 16 performs one hundred times exposure on each wafer, and the computer of the esmolol technology in the computing system 12 generates the first recipe 18 according to the temperature variation and distribution data accumulated on the mask. When scanner 16 exposes a wafer of first set of wafers 22, a coarse compensation may be performed using data collected from exposing a previous wafer, for example, when scanner 16 exposes a third wafer of first set of wafers 22, a coarse compensation may be performed using data collected from exposing a second wafer of first set of wafers 22. In step S48, the second recipe 20 is generated similarly, and will not be described herein.

In steps S34 and S50, the computing system 12 may perform data mining and data parsing using another computer (not a computer of esomo technology) to generate a plurality of stacking parameters and a plurality of updated stacking parameters. Fig. 3 is a schematic diagram illustrating that the computing system 12 generates 4 stacking parameters K1-K4, in which the number of stacking parameters may be 39 or other numbers, but fig. 3 only illustrates 4 stacking parameters K1-K4 for convenience of illustration. In FIG. 3, the horizontal axis T indicates the time for performing the exposure process on 25 wafers sequentially using the same mask, e.g., T₁To t₂Is the time for performing the exposure process on the 1 st wafer, t₂To t₃Is carried out on the 2 nd waferTime of line exposure procedure, t₂₅To t₂₆Is the time for performing the exposure process on the 25 th wafer, and so on, the vertical axis% refers to the offset of each stacking parameter, which can be positive or negative, and in fig. 3, the vertical axis is marked with 10 to represent the offset as 10%, and the vertical axis is marked with-20 to represent the offset as-20%. In step S36, the computing system 12 extracts predetermined parameters from the overlay parameters, for example, the computing system 12 extracts 20 predetermined parameters from 39 overlay parameters, and the 20 predetermined parameters are parameters of the 39 overlay parameters that are more related to the thermal expansion of the mask, so the parameters can be preset as the predetermined parameters in the computing system 12.

Assuming that, among the 4 overlay parameters K1-K4 in fig. 3, the overlay parameter K1 is the horizontal offset (transition) of the overlay of the exposure patterns on the wafer, the overlay parameter K2 is the vertical offset of the overlay of the exposure patterns on the wafer, the overlay parameter K3 is the horizontal magnification (verification) of the overlay of the exposure patterns on the wafer, and the overlay parameter K4 is the vertical magnification of the overlay of the exposure patterns on the wafer. Assuming that the deviation of the exposure pattern on the wafer to the right is positive and the deviation to the left is negative in the stacking parameter K1, it can be seen from fig. 3 that the exposure pattern on the wafer is deviated to the left; assuming that the shift to the upper side is positive and the shift to the lower side is negative in the stacking parameter K2, the exposure pattern on the wafer is shifted to the lower side as shown in fig. 3; in addition, since the shift amounts of the overlay parameters K3 and K4 are positive numbers, it can be seen that the mask exhibits thermal expansion rather than contraction during exposure.

2 of the 4 stacking parameters K1 to K4, K1 and K2 are not predetermined parameters, and 2 stacking parameters K3 and K4 are predetermined parameters, because the stacking parameters K1 and K2 related to displacement are less related to the thermal expansion of the mask, and the stacking parameters K3 and K4 related to the magnification are more related to the thermal expansion of the mask. Therefore, in step S38, the computing system 12 can perform linear regression on the predetermined parameters K3, K4, and the slope of each line segment after the linear regression is performed on the predetermined parameters K3, K4 changes greatly, for example, when the scanner 16 exposes the 10 th wafer, the offset of the stacking parameters K3, K4 does not increase or decrease, which is different from the ideal state that the offset should be proportional to the number of exposed chips, and thus the predetermined parameters K3, K4 should be corrected.

In steps S40 and S56, the computing system 12 generates a first determination coefficient for each predetermined parameter according to the following equation, which is described by a predetermined parameter j, such as the predetermined parameters K3 and K4:

wherein:

R_j ²a first decision coefficient being a predetermined parameter j;

n is the number of wafers in the first set of wafers 22, and N is a positive integer greater than 2;

ActY_iis the actual value of the predetermined parameter j for the ith wafer of the first set of wafers 22;

PredY _iis the predicted value of the predetermined parameter j corresponding to the ith wafer of the first group of wafers 22; and

avg (acty) is the average value of the first set of wafers 22 corresponding to the predetermined parameter j.

If 1 ≧ R_j ²>0.9, the first determination factor is within the acceptable range, and if the first determination factor of each of the predetermined parameters generated in step S40 is within the acceptable range, which means that the offset of each of the predetermined parameters K3 and K4 is almost proportional to the number of exposed chips, the computing system 12 controls the scanner 16 to perform the subsequent exposure process on other wafers according to the first recipe 18.

If the first determination coefficient of at least one of the plurality of predetermined parameters generated in step S40 is not in the acceptable range, i.e., 0.9 ≧ R_j ²If the offsets representing not all of the predetermined parameters K3, K4 are substantially proportional to the number of exposed dies, the computing system 12 performs step S48 to enable the scanner 16 to perform the exposure process on the second set of wafers 24 and generate the second recipe 20, wherein the second set of wafers 24 is different from the first set of wafers 22. The computing system 12 performs data mining and data parsing to generate a plurality of update stack parameters in step S50, and following the embodiment of fig. 3, the update stackThe stacking parameters are still K1-K4, but there will be a different offset from FIG. 3. Fig. 4 is a schematic diagram of the computing system 12 generating 4 updated overlay parameters K1-K4, as shown in fig. 3, in fig. 4, the horizontal axis T indicates the time for performing the exposure process on 25 wafers sequentially using the same mask, and the vertical axis% indicates the offset of each overlay parameter, which may be positive or negative. Since the predetermined parameters are preset in the computing system 12, the computing system 12 retrieves the updated predetermined parameters from the updated stacking parameters K1 to K4 in step S52, which are still K3 and K4.

In step S54, the computing system 12 performs linear regression on the updated predetermined parameters K3 and K4, and the slope of each line segment after the updated predetermined parameters K3 and K4 perform linear regression is limited, for example, when the scanner 16 exposes the mask and the 1 st to 25 th wafers, the offset of the overlay parameters K3 and K4 is steadily increased, which is close to the ideal condition that the offset should be proportional to the number of exposed wafers, so the predetermined parameters K3 and K4 should no longer need to be corrected.

In step S56, the computing system 12 generates a second decision coefficient for updating the predetermined parameters K3 and K4, and the equation generated by the second decision coefficient is the same as that in step S40, and thus is not repeated herein. If the second determination coefficients generated in step S56 for each updated predetermined parameter K3, K4 are all within the acceptable range, i.e. 1 ≧ R_j ²>0.9, the computing system 12 may control the scanner 16 to perform a subsequent exposure process on the other wafers according to the second recipe 20. If at least one of the two second determination coefficients of the two updated predetermined parameters K3 and K4 generated in step S56 is not in the acceptable range, i.e., 0.9 ≧ R_j ²If the offset representing at least one of the two updated predetermined parameters K3, K4 is not substantially proportional to the number of exposed dies, the scanner 16 may be used to perform the exposure process on the third set of wafers and update the second recipe 20 in step S48, and the computing system 12 may perform the following steps until it is determined in step S58 that the updated second determination coefficients are within the acceptable range, and the third set of wafers is different from the first set of wafers 22 and the second set of wafers 24.

FIG. 5 is a flowchart illustrating another embodiment of a method 100 for mask thermal expansion correction, the method 100 comprising:

step S102, the scanner 16 performs an exposure process on the first set of wafers 22 and generates a first recipe 18;

step S104, the computing system 12 executes data mining and data analysis to generate a plurality of superposition parameters;

step S106, the computing system 12 extracts a plurality of preset parameters from the plurality of superposed parameters;

step S108, the computing system 12 performs a linear regression on each of the plurality of predetermined parameters;

step S110, the computing system 12 generates a first coefficient of determination (coefficient) for each predetermined parameter;

step S112, the computing system 12 linearly combines a plurality of first decision coefficients of a plurality of predetermined parameters to generate a first weighting decision coefficient;

in step S114, is the first weight determination coefficient within the acceptable range? If yes, go to step S116; otherwise, executing step S120;

in step S116, the scanner 16 performs an exposure process on other wafers according to the first recipe 18.

Step S120, the scanner 16 performs an exposure process on the second group of wafers 24 and generates a second recipe 20;

step S122, the computing system 12 performs data mining and data parsing to generate a plurality of update stacking parameters;

step S124, the computing system 12 extracts a plurality of updating preset parameters from the plurality of updating superposition parameters;

step S126, the computing system 12 performs a linear regression on each updated predetermined parameter of the plurality of updated predetermined parameters;

step S128, the computing system 12 generates a second decision coefficient for each updated predetermined parameter;

step S130, the computing system 12 performs the linear combination on the second decision coefficients of the updated predetermined parameters to generate second weighting decision coefficients;

step S132, is the second weight decision coefficient within the acceptable range? If yes, go to step S134; otherwise, replacing the second group of wafers 24 with the third group of wafers to execute step S120 to update the second recipe 20;

in step S134, the scanner 16 performs an exposure process on other wafers according to the second recipe 20.

Compared to the mask thermal expansion correction method 30, the computing system 12 of the mask thermal expansion correction method 100 performs a linear combination of a plurality of first determination coefficients of a plurality of predetermined parameters to generate a first weighting determination coefficient in step S112. The equation for the linear combination is as follows:

wherein:

R²is a first weight determining coefficient;

R_j ²is the jth first decision coefficient;

w_jis R_j ²The weight of (c); and

m is the total number of first decision coefficients.

In step S114, if the first weighting determination coefficient is within the acceptable range, i.e., 1 ≧ R²>0.9, the computing system 12 controls the scanner 16 to perform an exposure process on other subsequent wafers according to the first recipe 18 in step S116. In step S114, if the first weighting determination coefficient is not in the acceptable range, i.e., 0.9 ≧ R²The scanner 16 performs an exposure process on the second set of wafers 24 and generates a second recipe 20 in step S120. In addition, in step S130, the computing system 12 linearly combines a plurality of second decision coefficients of a plurality of predetermined parameters to generate a second weighting decision coefficient, which is generated in the same manner as the first weighting decision coefficient, and thus, is not described again. In step S132, if the second weighting determination coefficient is within the acceptable range, i.e., 1 ≧ R²>0.9, the computing system 12 controls the scanner 16 to perform an exposure process on other subsequent wafers according to the second recipe 20 in step S134. In step S132, if the second weighting determination coefficient is not in the acceptable range, i.e., 0.9 ≧ R²The scanner 16 executes the exposure process on the third group of wafers and updates the second group of wafers in step S120Recipe 20, computing system 12 then performs subsequent steps until it is determined in step S132 that the updated second weighting determination coefficients are within the acceptable range, and the third set of wafers is different from the first set of wafers 22 and the second set of wafers 24.

FIG. 6 is a flowchart illustrating an embodiment of a method 200 for mask thermal expansion correction, the method 200 comprising:

step S202, the scanner 16 performs an exposure process on the first set of wafers 22 and generates a first recipe 18;

step S204, the computing system 12 executes data mining and data analysis to generate a plurality of stacking parameters;

step S206, the computing system 12 performs linear regression on each stacking parameter of the plurality of stacking parameters;

step S208, the computing system 12 generates a first determination coefficient (coefficient of determination) for each stacking parameter;

in step S210, are the first determination coefficients of the stacking parameters within the acceptable range? If yes, go to step S212; otherwise, executing step S216;

in step S212, the scanner 16 performs an exposure process on other wafers according to the first recipe 18.

Step S216, the scanner 16 performs an exposure process on the second group of wafers 24 and generates a second recipe 20;

step S218, the computing system 12 performs data mining and data parsing to generate a plurality of update stack parameters;

step S220, the computing system 12 performs a linear regression on each of the plurality of updated stacking parameters;

step S222, the computing system 12 generates a second decision coefficient for each updated stacking parameter;

in step S224, are the second determination coefficients of the updated stacking parameters within the acceptable range? If yes, go to step S226; otherwise, the third set of wafers replaces the second set of wafers 24 to perform step S216 to update the second recipe 20;

in step S226, the scanner 16 performs an exposure process on other wafers according to the second recipe 20.

In comparison with the mask thermal expansion correction method 30, the mask thermal expansion correction method 200 does not extract the predetermined parameters from the stacking parameters as in step S36, but performs linear regression on each stacking parameter of the stacking parameters in step S206 to generate the first determination coefficient. The method 200 does not extract the predetermined updated parameters from the updated stacking parameters as shown in step 52, but performs linear regression on each of the updated stacking parameters in step S220 to generate the second determination coefficient. The mask thermal expansion correction method 200 generates the first and second determining coefficients in the same manner as the mask thermal expansion correction method 30, except that the mask thermal expansion correction method 200 generates the first and second determining coefficients for all of the overlay parameters, rather than the first and second determining coefficients for only the predetermined parameters.

FIG. 7 is a flowchart illustrating another embodiment of a method 300 for mask thermal expansion correction, the method 300 comprising:

step S302, the scanner 16 performs an exposure process on the first set of wafers 22 and generates a first recipe 18;

step S304, the computing system 12 executes data mining and data analysis to generate a plurality of stacking parameters;

step S306, the computing system 12 performs linear regression on each of the plurality of stacking parameters;

step S308, the computing system 12 generates a first determination coefficient (coefficient of determination) for each stacking parameter;

step S310, the computing system 12 linearly combines a plurality of first decision coefficients of a plurality of superposition parameters to generate a first weighting decision coefficient;

in step S312, is the first weight determination coefficient within the acceptable range? If yes, go to step S314; otherwise, executing step S318;

in step S314, the scanner 16 performs an exposure process on other wafers according to the first recipe 18.

Step S318, the scanner 16 performs an exposure process on the second group of wafers 24 and generates a second recipe 20;

step S320, the computing system 12 executes data mining and data analysis to generate a plurality of updating and stacking parameters;

step S322, the computing system 12 performs a linear regression on each of the plurality of updated stacking parameters;

step S324, the computing system 12 generates a second determination coefficient for each updated stacking parameter;

step S326, the computing system 12 performs the linear combination on the second determination coefficients of the updated stacking parameters to generate second weighting determination coefficients;

in step S328, is the second weight determination coefficient within the acceptable range? If yes, go to step S330; otherwise, the third set of wafers replaces the second set of wafers 24 to perform step S318 to update the second recipe 20;

in step S330, the scanner 16 performs an exposure process on other wafers according to the second recipe 20.

Compared to the mask thermal expansion calibration method 100, the mask thermal expansion calibration method 300 does not extract the predetermined parameters from the stacking parameters in step S106, but performs a linear regression on each stacking parameter of the stacking parameters in step S306 to generate the first determination coefficient. The method 300 does not extract the predetermined updated parameters from the updated stacking parameters as shown in step S124, but performs linear regression on each of the updated stacking parameters in step S322 to generate the second determination coefficient. The mask thermal expansion correction method 300 generates the first and second determining coefficients in the same manner as the mask thermal expansion correction method 100, except that the mask thermal expansion correction method 300 generates the first and second determining coefficients for all the overlay parameters, rather than the first and second determining coefficients for only the predetermined parameters; and the mask thermal expansion correction method 300 may linearly combine all of the first determining coefficients to generate first weighted determining coefficients and may linearly combine all of the second determining coefficients to generate second weighted determining coefficients.

The embodiment provides a method for correcting thermal expansion of a photomask, which updates a formula when a determination coefficient is not in an acceptable range until the updated determination coefficient is in the acceptable range, so that the next batch of wafers can be prevented from being exposed by using an incorrect formula, the product meets the specification and the yield of the product is improved.

The above-mentioned embodiments are only preferred embodiments of the present invention, and all equivalent changes and modifications made by the claims of the present invention should be covered by the scope of the present invention.