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In situ surface topography measurement method of granite base in scanning wafer stage with laser interferometer
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 1–6 www.elsevier.de/ijleo
In situ surface topography measurement method of granite base in scanning wafer stage with laser interferometer Le Hea,b,, Xiangzhao Wanga,b, Weijie Shia,b a Information Optics Laboratory, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China b Graduate School of the Chinese Academy of Sciences, Beijing 100039, China
Received 18 January 2006; accepted 20 March 2006
Abstract Topography of a granite surface has an effect on the vertical positioning of a wafer stage in a lithographic tool, when the wafer stage moves on the granite. The inaccurate measurement of the topography results in a bad leveling and focusing performance. In this paper, an in situ method to measure the topography of a granite surface with high accuracy is present. In this method, a high-order polynomial is set up to express the topography of the granite surface. Two double-frequency laser interferometers are used to measure the tilts of the wafer stage in the X- and Y-directions. From the sampling tilts information, the coefficients of the high-order polynomial can be obtained by a special algorithm. Experiment results shows that the measurement reproducibility of the method is better than 10 nm. r 2006 Elsevier GmbH. All rights reserved. Keywords: Laser interferometer; Topography measurement; Least squares method; Scanning wafer stage; Lithographic tool
1. Introduction Generally, a wafer stage is designed to move on the surface of a granite base in a modern lithographic tool [1,2]. Therefore, the vertical position of the wafer stage is varied with the height of the granite surface when the wafer stage is scanning on the granite surface [3,4]. The vertical position of the wafer stage impacts on the leveling and focusing control in lithographic tools greatly. With the decrease of critical dimension (CD) in current low-k1 and high-NA optical lithography, the depth of focus (DOF) becomes ultra-small. The less DOF needs more accurate leveling and focusing control Corresponding author. Information Optics Laboratory, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China E-mail address: [email protected] (L. He).
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.036
method which monitors and adjusts the vertical position of the wafer stage. In order to keep the surface of the wafer stage in the DOF during the scanning, the topography information of the granite base surface is required for the focusing and leveling system [5]. Thus, fast and accurate measurement methods of granite topography become necessary [6]. Because of the limitation of the common architecture of step-and-scan lithographic tools, the topography of granite base surface is difficult to be measured directly. Moreover, in different conditions, the topography of granite base has some small deformation which is not expected to be introduced between the measurement condition and working condition. Thus, an in situ measurement method is necessary. In the traditional method, some special position sensors are used to measure the relative height between the wafer bottom and the granite surface to obtain the granite topography.
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L. He et al. / Optik 119 (2008) 1–6
Because of the limitation of manufacture technique, the method is impressionable at some conditions, such as the cleanliness level of the wafer or the vibration of the wafer stage, and so on. The measurement accuracy of the method is several hundred nanometers and it is not accurate enough to modern lithographic tools [7–10]. In this paper, a novel in situ measurement method to determine the granite base topography with doublefrequency laser interferometers is presented. In this method, a high-order polynomial is set up to express the topography of granite base firstly. Then two doublefrequency laser interferometers are used to measure the tilts of the wafer stage in the X- and Y-directions with the measurement accuracy better than 0.5 mrad. From the sampling tilts information, the coefficients of the high-order polynomial can be obtained by a special algorithm. Thus, the granite base topography is determined with the defined high-order polynomial.
2. Principle The granite base surface can be polished with several hundred nanometers accurateness by modern process technology. This kind of topography can be expressed as a curve surface and a better approximate expression method is high-order polynomial (at least three orders) as below [11,12]: Zðx; yÞ ¼
N X N X
C ij xi yj ,
in this method. Thus, Eq. (2) is simplified to Eq. (3): N X N 1 X N X X C ij xi yj þ C ij xi yj Zðx; yÞ ¼ C 11 xy þ i¼2 j¼2
þ
Zðx; yÞ ¼ C 00 þ C 01 y þ C 10 x þ C 11 xy þ
N X N X
ð3Þ
As we know, the tilt of a line can be calculated by the trigonometric formula if the vertical and horizontal distances between the two random points in this line are measured. If this line is a part of an object, the tilt of this line is the tilt of this object in a special position. According to this principle, two double-frequency laser interferometers are used to measure the position of wafer stage with several nanometers accuracy. The measurement positions are used to calculate the tilts of the wafer stage. As shown in Fig. 1, two interferometers emit three parallel beams to the side face of the wafer stage, respectively, the relative distances X1, X2, X3 and Y1, Y2, Y3 are measured. The tilts of wafer stage in X- and Y-directions are calculated by ½ Y 1 þ Y 3=2 Y 2 Tx ¼ , dy ½ X 1 þ X 3=2 X 2 Ty ¼ , ð4Þ dx where dx and dy are the vertical distances between the top beam and bottom beam for X and Y interferometers, respectively. Since Tx and Ty are the tilts of wafer stage in X- and Y-directions, they are also calculated by Txðx; yÞ ¼
where Z is the height of the granite surface in position (x, y) and Cij is the coefficient of the polynomial. Because of the limitation of the common architecture of step-and-scan lithographic tools, the local height of granite base surface is difficult to be measured directly. Thus, an indirect measurement method is introduced here. Considering that the inherent shortcomings of an indirect measurement calculation method, some inherent constant components in Eq. (1) such as constant tilts of granite base should be ignored to get the more accurate local topography of the granite base surface. As a result, Eq. (1) is written as
C ij xi yj .
i¼0 j¼2
i¼2 j¼0
(1)
i¼0 j¼0
N X 1 X
N X N 1 X N X qZ X ¼ jC ij xi yj1 þ jC ij xi yj1 qy i¼2 j¼2 i¼0 j¼2
þ
N X
C i1 xi ,
i¼1
Tyðx; yÞ ¼
N X N N X 1 X qZ X ¼ iC ij xi1 yj þ iC ij xi1 yj qx i¼2 j¼0 i¼2 j¼0
þ
N X
C 1j yj .
ð5Þ
j¼1
C ij xi yj
i¼2 j¼2
þ
1 X N X i¼0 j¼2
C ij xi yj þ
N X 1 X
C ij xi yj .
ð2Þ
i¼2 j¼0
The items C00, C01y, C10x in Eq. (2) describe the fixed position planes, which can be considered as an inherent system placement errors of the granite. They will not be considered in the calculation of granite base topography
Fig. 1. Rotation measurement model of laser interferometer.
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From the difference partial differential relationship between the height and tilt of granite base surface shown in Eq. (5), the granite topography can be calculated by the integration of tilts if the measurement data are sufficient. But this method is not accurate because the side face of the wafer stage is not absolutely vertical to the bottom face of the wafer stage. This influence is expressed as Txmeasured ðx; yÞ ¼ Txgranite ðx; yÞ þ Txside ðxÞ, Tymeasured ðx; yÞ ¼ Tygranite ðx; yÞ þ Tyside ðyÞ,
ð6Þ
where Txmeasured, Tymeasured are the gross tilts of wafer stage, Txgranite, Tygranite are the gross grads of granite base topography, and Txside, Tyside are the inherent nonorthogonality noises of wafer stage. In the process of integration, the error from Txside and Tyside will be accumulated to decrease the accuracy of the calculated topography. Considering the principle of the measurement, the measured tilts will not change if the incident points of the beams to the side face of granite base are fixed. Since in the same X-direction line the Txside is equivalent, and in the same Y-direction line the Tyside is equivalent as well, if the measurement points are selected in the same X-direction line or Y-direction line, the difference method is used to eliminate the Txside and Tyside as below: DTxðx; yÞ ¼ Txmeasured ðx; y þ dyÞ Txmeasured ðx; yÞ ¼ Txgranite ðx; y þ dyÞ Txgranite ðx; yÞ ¼
N X N X
jC ij xi ððy þ dyÞj1 yj1 Þ,
3
granite base topography. Eq. (7) is used to determine the coefficients with least square fitting method [13]. In order to get the enough fitting accuracy, the tolerance in the fitting method is set below 1017 m. If the number of measurement points is P P, there are 2P(P1) equations to be built according to Eq. (7). The calculation accuracy depends on the number of measurement data and the number of orders of high polynomial. But too much measurement points will cost too much measurement time, too high-order polynomial will bring too much coefficients and add the complexity of the fitting method. In this method, a three- or four-order polynomial is used commonly. Fig. 2 shows the basic selection principle of measurement points in this method. In this figure, the Y-direction is the scan direction of lithographic tools, and the X-direction is the step direction. There are P P points with equal intervals in X- or Y-directions, respectively. The measurement area covers the whole valid area of granite surface on which the scanning wafer stage moves. A step-and-repeat measurement method is used to get the tilts of wafer stage, as the following steps: (i) In the leftmost column, the wafer stage steps along the positive Y-direction and its position is measured in each sampling point. Then in the same column, the wafer stage steps along the negative Y-direction, its position is remeasured in each sampling point. (ii) The wafer stage steps from the leftmost column to the rightmost column, in each column repeats step i.
i¼0 j¼2
DTyðx; yÞ ¼ Tymeasured ðx þ dx; yÞ Tymeasured ðx; yÞ ¼ Tygranite ðx þ dx; yÞ Tygranite ðx; yÞ ¼
N X N X
iC ij yj ððx þ dxÞi1 xi1 Þ.
ð7Þ
i¼2 j¼0
From Eq. (7), it can be seen that the inherent nonorthogonality noises of wafer stage are eliminated from the measurement tilts and the items including the coefficient C11 have been counteracted. Considering that the item C11xy reflects two fixed directional derivatives and can be handled alone, the elimination of this item will improve the expected calculation accurateness of granite base topography. Thus, the final expression of granite base topography is Zðx; yÞ ¼
N X N X
C ij xi yj þ
1 X N X
i¼2 j¼2
þ
N X 1 X
C ij xi yj
i¼0 j¼2
C ij xi yj .
ð8Þ
i¼2 j¼0
From Eq. (8), there are (N1)(N1)+4(N1) coefficients Cij should be determined to get the expression of
Fig. 2. Layout of measurement positions in the granite surface.
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(iii) The wafer stage steps from the rightmost column to the leftmost column, in each column repeats step i. In this method, all the sampling points will be measured four times. The mean value of these four values will be used to fitting the granite base topography. The inherent nonorthogonality noises of wafer stage can be calculated as follows: ^ Txside ðxÞ ¼ meanðTxðx; yÞ Txðx; yÞÞ, ^ Tyside ðyÞ ¼ meanðTyðx; yÞ Tyðx; yÞÞ,
ð9Þ
where function mean(x) is to calculate the average value ^ ^ of the element x, and Txðx; yÞ and Tyðx; yÞ are the fitting tilts in position (x, y). The calculation residuals are calculated by ^ yÞ Txside ðxÞ, RTxðx; yÞ ¼ Tðx; yÞ Tðx; ^ yÞ Tyside ðyÞ. RTyðx; yÞ ¼ Tðx; yÞ Tðx;
ð10Þ
The residuals are used to estimate the algorithm used in this method.
height offset of the surface is 575.76 nm. Fig. 3(b) is the 3 sigma value of each item of the fitting polynomial. The maximum 3 sigma value of all items is 95.27 nm. Fig. 4(a) is the fitting topography of granite base surface according to the average tilts measurement 10 times. Fig. 4(b) is the standard deviation layout of the 10 times measurements. The maximum deviation is 8.263 nm, which means the reproducibility of this method is better than 8.263 nm. Fig. 5 shows the residuals of Txðx; yÞ and Tyðx; yÞ in the experiment. The maximum residual of Txðx; yÞ is 0.74 mrad and the maximum residual of Tyðx; yÞ is 0.59 mrad. And the 3 sigma values of residuals of Txðx; yÞ and Tyðx; yÞ are 0.99 and 0.56 mrad, respectively, which are very close to the measurement accurateness of the interferometer. Then the square of the multiple correlation coefficients (R_square) is used to evaluate goodness of the fitting algorithm [14]: R_square ¼ 1
3. Experiments and results An experimental scanning wafer stage model with double-frequency interferometer was used to test this method. The valid surface area of granite base was 102 mm 153 mm. A four-order polynomial was used to express the topography of granite and 51 51 sample points were selected to measure. There were 21 coefficients calculated by 5100 equations according to Eqs. (7) and (8). In the least square method, the maximum number of iterations was set to 400 and the tolerance was set to 5 1018 m. Fig. 3(a) is the fitting topography of the granite base surface. The maximum
SSEðn 1Þ , SSTðv 1Þ
(11)
where n is the number of sampling data, v the degree of freedom and defined as n minus the number of fitted coefficients m. SSE and SSR are the sum of squares due to error and the sum of squares of the regression, respectively [15]: SSE ¼
n X ðyi y^ i Þ2 , i¼1
n X SST ¼ ðyi y¯ Þ2 ,
ð12Þ
i¼1
where y^ i , yi and y¯ are the fitting data, measurement data and average data of the measurement data, respectively.
Fig. 3. Fitting topography and 3 sigma value of each item in fitting polynomial. (a) Fitting topography of granite; (b) 3 sigma value of each item in polynomial.
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5
Fig. 4. Fitting topography according to average measurement tilts and its standard deviation layout. (a) Fitting topography according to the average tilts measurement 10 times. (b) Standard deviation of measurements.
Fig. 5. Residuals of Tx(x,y) (a) and Ty(x, y) (b).
In this method, Tx and Ty are the actual measurement data, the R_square is calculated by 2600 R_square ¼ 1 2579 P2550 2 2 ^ ^ ððTxðiÞ TxðiÞÞ þ ðTyðiÞ TyðiÞÞÞ Pi¼1 , 2550 2 2 ¯ ¯ i¼1 ððTxðiÞ TxðiÞÞ þ ðTyðiÞ TyðiÞÞÞ
which proves the validity of this indirect measurement method. In order to increase the accuracy, the inherent planes of all the area of granite surface are not considered in the fitting polynomial. As a result, some special measurements should be designed to calculate these inherent planes.
ð13Þ ^ ^ ¯ ¯ where RxðiÞ, RyðiÞ and RxðiÞ, RyðiÞ are the fitting data according to Eq. (5) and the average measurement data, respectively. The R_square of Fig. 3 is 0.977 and the average R_square of 10 times measurements is 0.977,
4. Conclusion A novel in situ measurement method to determine the granite topography with double-frequency laser interferometers has been presented in this paper. In this
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method, a high-order polynomial is set up to express the topography of granite base firstly. Then two doublefrequency laser interferometers are used to measure the tilts of the wafer stage in the X- and Y-directions with the measurement accuracy better than 0.5 mrad. From the sampling tilts information, the coefficients of the high-order polynomial have been obtained by a special algorithm. Thus, the granite base topography is determined with the defined high-order polynomial. Experiment results show that the measurement reproducibility of the method is better than 10 nm. The R_square of the fitting topography of granite surface is better than 0.977. Compared to the traditional method, the novel indirect measurement method uses the existing instrument and avoids the local architecture modification of current lithographic tools.
Acknowledgment This work was supported by the National High Technology Research and Development 863 Program (2002AA4Z3000) of China.
References [1] M. Watanabe, Y. Oshida, Y. Nakayama, et al., Focusing and leveling based on wafer surface profile detection with interferometry for optical lithography, Proc. SPIE 2197 (1994) 980–989. [2] A. Tran, J. DeWitt, C. Boettcher, Positioning accuracy issues for 0.35 micron critical dimension microlithography, Proc. SPIE 2087 (1994) 87–98.
[3] P.D. Henshaw, D.P. DeGloria, S.A. Kelly, et al., Realtime stage position measurement with nanometer-scale accuracy, Proc. SPIE 3051 (1997) 913–921. [4] G.E. Sommargren, Linear/angular displacement interferometer for wafer stage metrology, Proc. SPIE 1088 (1989) 268–271. [5] M.L. Schattenburg, et al., Sub-100 nm metrology using interferometrically produced fiducials, J. Vac. Sci. Technol. B 17 (1999) 2692. [6] B.M. La Fontaine, J. Hauschild, M.V. Dusa, et al., Study of the influence of substrate topography on the focusing performance of advanced lithography scanners, Proc. SPIE 5040 (2003) 570–581. [7] De Roover, Motion control of a wafer stage, Ph.D. Paper, The Netherlands, 1997, ISBN:90-407-1562-9. [8] M.L. Schattenburg, Nanometrology in nanomanufacturing, in: NASA Tech Briefs, Nanotech 2003 Conference. [9] E.H. Lucy, Y. Pao, T.N. Chang, Input shaper designs for minimizing the expected level of residual vibration in flexible structure, in: Proceedings of the American Control Conference, 1997, pp. 3542–3546. [10] Flemming Tinker, Cheshire: in-situ metrology system and method, United States Patent No. 6301009B1, 2001. [11] M. do Carmo, Differential Geometry of Curve and Surface, Prentice-Hall, Englewood Cliffs, NJ, 1976. [12] A.M. Ancikov, Seventeen Papers on Topography & Differential Geometry, American Mathematical Society, Providence, RI, ISBN 0821817922, 1970. [13] N.R. Draper, H. Smith, Applied Regression Analysis, third ed., Wiley, New York, 1998. [14] Full Product Family Help of Matlab Release 13, /http:// www.mathworks.com/productsS. [15] P.R. Bevington, D.K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, second ed., WCB/McGraw-Hill, Boston, 1992.
Optik
Optics
Optik 119 (2008) 1–6 www.elsevier.de/ijleo
In situ surface topography measurement method of granite base in scanning wafer stage with laser interferometer Le Hea,b,, Xiangzhao Wanga,b, Weijie Shia,b a Information Optics Laboratory, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China b Graduate School of the Chinese Academy of Sciences, Beijing 100039, China
Received 18 January 2006; accepted 20 March 2006
Abstract Topography of a granite surface has an effect on the vertical positioning of a wafer stage in a lithographic tool, when the wafer stage moves on the granite. The inaccurate measurement of the topography results in a bad leveling and focusing performance. In this paper, an in situ method to measure the topography of a granite surface with high accuracy is present. In this method, a high-order polynomial is set up to express the topography of the granite surface. Two double-frequency laser interferometers are used to measure the tilts of the wafer stage in the X- and Y-directions. From the sampling tilts information, the coefficients of the high-order polynomial can be obtained by a special algorithm. Experiment results shows that the measurement reproducibility of the method is better than 10 nm. r 2006 Elsevier GmbH. All rights reserved. Keywords: Laser interferometer; Topography measurement; Least squares method; Scanning wafer stage; Lithographic tool
1. Introduction Generally, a wafer stage is designed to move on the surface of a granite base in a modern lithographic tool [1,2]. Therefore, the vertical position of the wafer stage is varied with the height of the granite surface when the wafer stage is scanning on the granite surface [3,4]. The vertical position of the wafer stage impacts on the leveling and focusing control in lithographic tools greatly. With the decrease of critical dimension (CD) in current low-k1 and high-NA optical lithography, the depth of focus (DOF) becomes ultra-small. The less DOF needs more accurate leveling and focusing control Corresponding author. Information Optics Laboratory, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China E-mail address: [email protected] (L. He).
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.036
method which monitors and adjusts the vertical position of the wafer stage. In order to keep the surface of the wafer stage in the DOF during the scanning, the topography information of the granite base surface is required for the focusing and leveling system [5]. Thus, fast and accurate measurement methods of granite topography become necessary [6]. Because of the limitation of the common architecture of step-and-scan lithographic tools, the topography of granite base surface is difficult to be measured directly. Moreover, in different conditions, the topography of granite base has some small deformation which is not expected to be introduced between the measurement condition and working condition. Thus, an in situ measurement method is necessary. In the traditional method, some special position sensors are used to measure the relative height between the wafer bottom and the granite surface to obtain the granite topography.
ARTICLE IN PRESS 2
L. He et al. / Optik 119 (2008) 1–6
Because of the limitation of manufacture technique, the method is impressionable at some conditions, such as the cleanliness level of the wafer or the vibration of the wafer stage, and so on. The measurement accuracy of the method is several hundred nanometers and it is not accurate enough to modern lithographic tools [7–10]. In this paper, a novel in situ measurement method to determine the granite base topography with doublefrequency laser interferometers is presented. In this method, a high-order polynomial is set up to express the topography of granite base firstly. Then two doublefrequency laser interferometers are used to measure the tilts of the wafer stage in the X- and Y-directions with the measurement accuracy better than 0.5 mrad. From the sampling tilts information, the coefficients of the high-order polynomial can be obtained by a special algorithm. Thus, the granite base topography is determined with the defined high-order polynomial.
2. Principle The granite base surface can be polished with several hundred nanometers accurateness by modern process technology. This kind of topography can be expressed as a curve surface and a better approximate expression method is high-order polynomial (at least three orders) as below [11,12]: Zðx; yÞ ¼
N X N X
C ij xi yj ,
in this method. Thus, Eq. (2) is simplified to Eq. (3): N X N 1 X N X X C ij xi yj þ C ij xi yj Zðx; yÞ ¼ C 11 xy þ i¼2 j¼2
þ
Zðx; yÞ ¼ C 00 þ C 01 y þ C 10 x þ C 11 xy þ
N X N X
ð3Þ
As we know, the tilt of a line can be calculated by the trigonometric formula if the vertical and horizontal distances between the two random points in this line are measured. If this line is a part of an object, the tilt of this line is the tilt of this object in a special position. According to this principle, two double-frequency laser interferometers are used to measure the position of wafer stage with several nanometers accuracy. The measurement positions are used to calculate the tilts of the wafer stage. As shown in Fig. 1, two interferometers emit three parallel beams to the side face of the wafer stage, respectively, the relative distances X1, X2, X3 and Y1, Y2, Y3 are measured. The tilts of wafer stage in X- and Y-directions are calculated by ½ Y 1 þ Y 3=2 Y 2 Tx ¼ , dy ½ X 1 þ X 3=2 X 2 Ty ¼ , ð4Þ dx where dx and dy are the vertical distances between the top beam and bottom beam for X and Y interferometers, respectively. Since Tx and Ty are the tilts of wafer stage in X- and Y-directions, they are also calculated by Txðx; yÞ ¼
where Z is the height of the granite surface in position (x, y) and Cij is the coefficient of the polynomial. Because of the limitation of the common architecture of step-and-scan lithographic tools, the local height of granite base surface is difficult to be measured directly. Thus, an indirect measurement method is introduced here. Considering that the inherent shortcomings of an indirect measurement calculation method, some inherent constant components in Eq. (1) such as constant tilts of granite base should be ignored to get the more accurate local topography of the granite base surface. As a result, Eq. (1) is written as
C ij xi yj .
i¼0 j¼2
i¼2 j¼0
(1)
i¼0 j¼0
N X 1 X
N X N 1 X N X qZ X ¼ jC ij xi yj1 þ jC ij xi yj1 qy i¼2 j¼2 i¼0 j¼2
þ
N X
C i1 xi ,
i¼1
Tyðx; yÞ ¼
N X N N X 1 X qZ X ¼ iC ij xi1 yj þ iC ij xi1 yj qx i¼2 j¼0 i¼2 j¼0
þ
N X
C 1j yj .
ð5Þ
j¼1
C ij xi yj
i¼2 j¼2
þ
1 X N X i¼0 j¼2
C ij xi yj þ
N X 1 X
C ij xi yj .
ð2Þ
i¼2 j¼0
The items C00, C01y, C10x in Eq. (2) describe the fixed position planes, which can be considered as an inherent system placement errors of the granite. They will not be considered in the calculation of granite base topography
Fig. 1. Rotation measurement model of laser interferometer.
ARTICLE IN PRESS L. He et al. / Optik 119 (2008) 1–6
From the difference partial differential relationship between the height and tilt of granite base surface shown in Eq. (5), the granite topography can be calculated by the integration of tilts if the measurement data are sufficient. But this method is not accurate because the side face of the wafer stage is not absolutely vertical to the bottom face of the wafer stage. This influence is expressed as Txmeasured ðx; yÞ ¼ Txgranite ðx; yÞ þ Txside ðxÞ, Tymeasured ðx; yÞ ¼ Tygranite ðx; yÞ þ Tyside ðyÞ,
ð6Þ
where Txmeasured, Tymeasured are the gross tilts of wafer stage, Txgranite, Tygranite are the gross grads of granite base topography, and Txside, Tyside are the inherent nonorthogonality noises of wafer stage. In the process of integration, the error from Txside and Tyside will be accumulated to decrease the accuracy of the calculated topography. Considering the principle of the measurement, the measured tilts will not change if the incident points of the beams to the side face of granite base are fixed. Since in the same X-direction line the Txside is equivalent, and in the same Y-direction line the Tyside is equivalent as well, if the measurement points are selected in the same X-direction line or Y-direction line, the difference method is used to eliminate the Txside and Tyside as below: DTxðx; yÞ ¼ Txmeasured ðx; y þ dyÞ Txmeasured ðx; yÞ ¼ Txgranite ðx; y þ dyÞ Txgranite ðx; yÞ ¼
N X N X
jC ij xi ððy þ dyÞj1 yj1 Þ,
3
granite base topography. Eq. (7) is used to determine the coefficients with least square fitting method [13]. In order to get the enough fitting accuracy, the tolerance in the fitting method is set below 1017 m. If the number of measurement points is P P, there are 2P(P1) equations to be built according to Eq. (7). The calculation accuracy depends on the number of measurement data and the number of orders of high polynomial. But too much measurement points will cost too much measurement time, too high-order polynomial will bring too much coefficients and add the complexity of the fitting method. In this method, a three- or four-order polynomial is used commonly. Fig. 2 shows the basic selection principle of measurement points in this method. In this figure, the Y-direction is the scan direction of lithographic tools, and the X-direction is the step direction. There are P P points with equal intervals in X- or Y-directions, respectively. The measurement area covers the whole valid area of granite surface on which the scanning wafer stage moves. A step-and-repeat measurement method is used to get the tilts of wafer stage, as the following steps: (i) In the leftmost column, the wafer stage steps along the positive Y-direction and its position is measured in each sampling point. Then in the same column, the wafer stage steps along the negative Y-direction, its position is remeasured in each sampling point. (ii) The wafer stage steps from the leftmost column to the rightmost column, in each column repeats step i.
i¼0 j¼2
DTyðx; yÞ ¼ Tymeasured ðx þ dx; yÞ Tymeasured ðx; yÞ ¼ Tygranite ðx þ dx; yÞ Tygranite ðx; yÞ ¼
N X N X
iC ij yj ððx þ dxÞi1 xi1 Þ.
ð7Þ
i¼2 j¼0
From Eq. (7), it can be seen that the inherent nonorthogonality noises of wafer stage are eliminated from the measurement tilts and the items including the coefficient C11 have been counteracted. Considering that the item C11xy reflects two fixed directional derivatives and can be handled alone, the elimination of this item will improve the expected calculation accurateness of granite base topography. Thus, the final expression of granite base topography is Zðx; yÞ ¼
N X N X
C ij xi yj þ
1 X N X
i¼2 j¼2
þ
N X 1 X
C ij xi yj
i¼0 j¼2
C ij xi yj .
ð8Þ
i¼2 j¼0
From Eq. (8), there are (N1)(N1)+4(N1) coefficients Cij should be determined to get the expression of
Fig. 2. Layout of measurement positions in the granite surface.
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(iii) The wafer stage steps from the rightmost column to the leftmost column, in each column repeats step i. In this method, all the sampling points will be measured four times. The mean value of these four values will be used to fitting the granite base topography. The inherent nonorthogonality noises of wafer stage can be calculated as follows: ^ Txside ðxÞ ¼ meanðTxðx; yÞ Txðx; yÞÞ, ^ Tyside ðyÞ ¼ meanðTyðx; yÞ Tyðx; yÞÞ,
ð9Þ
where function mean(x) is to calculate the average value ^ ^ of the element x, and Txðx; yÞ and Tyðx; yÞ are the fitting tilts in position (x, y). The calculation residuals are calculated by ^ yÞ Txside ðxÞ, RTxðx; yÞ ¼ Tðx; yÞ Tðx; ^ yÞ Tyside ðyÞ. RTyðx; yÞ ¼ Tðx; yÞ Tðx;
ð10Þ
The residuals are used to estimate the algorithm used in this method.
height offset of the surface is 575.76 nm. Fig. 3(b) is the 3 sigma value of each item of the fitting polynomial. The maximum 3 sigma value of all items is 95.27 nm. Fig. 4(a) is the fitting topography of granite base surface according to the average tilts measurement 10 times. Fig. 4(b) is the standard deviation layout of the 10 times measurements. The maximum deviation is 8.263 nm, which means the reproducibility of this method is better than 8.263 nm. Fig. 5 shows the residuals of Txðx; yÞ and Tyðx; yÞ in the experiment. The maximum residual of Txðx; yÞ is 0.74 mrad and the maximum residual of Tyðx; yÞ is 0.59 mrad. And the 3 sigma values of residuals of Txðx; yÞ and Tyðx; yÞ are 0.99 and 0.56 mrad, respectively, which are very close to the measurement accurateness of the interferometer. Then the square of the multiple correlation coefficients (R_square) is used to evaluate goodness of the fitting algorithm [14]: R_square ¼ 1
3. Experiments and results An experimental scanning wafer stage model with double-frequency interferometer was used to test this method. The valid surface area of granite base was 102 mm 153 mm. A four-order polynomial was used to express the topography of granite and 51 51 sample points were selected to measure. There were 21 coefficients calculated by 5100 equations according to Eqs. (7) and (8). In the least square method, the maximum number of iterations was set to 400 and the tolerance was set to 5 1018 m. Fig. 3(a) is the fitting topography of the granite base surface. The maximum
SSEðn 1Þ , SSTðv 1Þ
(11)
where n is the number of sampling data, v the degree of freedom and defined as n minus the number of fitted coefficients m. SSE and SSR are the sum of squares due to error and the sum of squares of the regression, respectively [15]: SSE ¼
n X ðyi y^ i Þ2 , i¼1
n X SST ¼ ðyi y¯ Þ2 ,
ð12Þ
i¼1
where y^ i , yi and y¯ are the fitting data, measurement data and average data of the measurement data, respectively.
Fig. 3. Fitting topography and 3 sigma value of each item in fitting polynomial. (a) Fitting topography of granite; (b) 3 sigma value of each item in polynomial.
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Fig. 4. Fitting topography according to average measurement tilts and its standard deviation layout. (a) Fitting topography according to the average tilts measurement 10 times. (b) Standard deviation of measurements.
Fig. 5. Residuals of Tx(x,y) (a) and Ty(x, y) (b).
In this method, Tx and Ty are the actual measurement data, the R_square is calculated by 2600 R_square ¼ 1 2579 P2550 2 2 ^ ^ ððTxðiÞ TxðiÞÞ þ ðTyðiÞ TyðiÞÞÞ Pi¼1 , 2550 2 2 ¯ ¯ i¼1 ððTxðiÞ TxðiÞÞ þ ðTyðiÞ TyðiÞÞÞ
which proves the validity of this indirect measurement method. In order to increase the accuracy, the inherent planes of all the area of granite surface are not considered in the fitting polynomial. As a result, some special measurements should be designed to calculate these inherent planes.
ð13Þ ^ ^ ¯ ¯ where RxðiÞ, RyðiÞ and RxðiÞ, RyðiÞ are the fitting data according to Eq. (5) and the average measurement data, respectively. The R_square of Fig. 3 is 0.977 and the average R_square of 10 times measurements is 0.977,
4. Conclusion A novel in situ measurement method to determine the granite topography with double-frequency laser interferometers has been presented in this paper. In this
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method, a high-order polynomial is set up to express the topography of granite base firstly. Then two doublefrequency laser interferometers are used to measure the tilts of the wafer stage in the X- and Y-directions with the measurement accuracy better than 0.5 mrad. From the sampling tilts information, the coefficients of the high-order polynomial have been obtained by a special algorithm. Thus, the granite base topography is determined with the defined high-order polynomial. Experiment results show that the measurement reproducibility of the method is better than 10 nm. The R_square of the fitting topography of granite surface is better than 0.977. Compared to the traditional method, the novel indirect measurement method uses the existing instrument and avoids the local architecture modification of current lithographic tools.
Acknowledgment This work was supported by the National High Technology Research and Development 863 Program (2002AA4Z3000) of China.
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