A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers

Precision Engineering Journal of the International Societies for Precision Engineering and Nanotechnology 26 (2002) 396–404

A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers Wei Gao a,∗ , Peisen S. Huang b,1 , Tomohiko Yamada a , Satoshi Kiyono a a

Department of Mechatronics and Precision Engineering, Tohoku University, Aramaki Aza Aoba 01, Sendai 980-8579, Japan b Department of Mechanical Engineering, SUNY at Stony Brook, Stony Brook, NY 11794-2300, USA Received 14 November 2001; received in revised form 28 January 2002; accepted 27 February 2002

Abstract A two-dimensional (2D) angle probe was developed to realize a new scanning multi-probe instrument employing 2D angle probes for flatness measurement of large silicon wafers. Each probe, which utilizes the principle of autocollimation, detects the 2D local slope components of a point on the wafer surface. The 2D local slopes (angles) are obtained through detecting the corresponding 2D positions of the reflected light spot on the focal plane of the object lens using position-sensing devices (PSDs). To make the probe compact in size, it is more effective to improve the sensitivity of angle detection by selecting proper PSDs than using an objective lens with a larger focal distance. Two kinds of photo devices, linear lateral effect PSDs and quadrant photodiodes (QPD), for sensing 2D positions were discussed theoretically and experimentally. It was shown that a QPD is the best for highly sensitive 2D angle detection. In the experiments, a compact prototype angle probe with dimensions 90(L) mm × 60(W) mm × 30(H) mm employing a QPD as the PSD was confirmed to have a resolution of approximately 0.01 arc-second. © 2002 Elsevier Science Inc. All rights reserved. Keywords: Measurement; Angle probe; Two-dimensional; Slope probe; Flatness; Surface profile; Local slope; Autocollimator; Sensitivity; Error-separation; Silicon wafer; Quadrant photodiode

1. Introduction Flatness is a critical parameter of silicon wafers for making integrated circuits, and measurement of wafer flatness is an essential process for both wafer manufacturers and device manufacturers. Meanwhile, to make devices with improved functionality at a reduced cost, the design rule (width of the device pattern) has been becoming smaller and the silicon wafer size larger. It is predicted that the design rule/wafer size will be 130 nm/∅300 mm, 35 nm/∅450 mm in the near future [1]. This brings a big challenge to the metrology of wafer flatness. The site flatness of a wafer is required to be at the same level as the design rule, and the tolerance for the global flatness over the whole wafer is also very tight. To properly evaluate the flatness of such large wafers, im∗ Corresponding author. Tel.: +81-22-217-6951, fax: +81-22-217-6951. E-mail addresses: [email protected] (W. Gao), [email protected] (P.S. Huang). 1 Tel: +1-631-632-8329; fax: +1-631-632-8544.

proved measuring instruments with high accuracy and high measurement speed must be developed [2]. The scanning probe instruments are promising tools for measurement of large flat surfaces [3]. To achieve the required measurement accuracy and measurement speed, however, it is necessary to separate the flatness profile from motion errors of scanning [4–9]. From this point of view, it is more preferable to use angle probes because only the angular motion errors will affect the measurement [10,11]. Several scanning angle probe instruments using one or two one-dimensional (1D) angle probes for measuring the cross-sectional profile of a flat surface along a line have been developed [12–17]. When these 1D instruments are used to measure the flatness of the entire flat surface, it is necessary to move the probe along X- and Y-directions respectively to scan the surface in a raster scanning pattern [18]. The measurement procedure is quite timeconsuming. It is also difficult to determine the relationship between the sectional profiles along different lines accurately, since the relative heights of the sectional profiles are indeterminate.

0141-6359/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 6 3 5 9 ( 0 2 ) 0 0 1 2 1 - 6

W. Gao et al. / Precision Engineering 26 (2002) 396–404

In this research project, we aim to develop an accurate and fast scanning multi-probe instrument employing two-dimensional (2D) angle probes. As described in the next section, the probes are moved by a linear carriage along the radial direction of the wafer while the wafer is being rotated by a spindle. The 2D local slopes over the entire wafer surface can be obtained in the one spiral scanning. This results in a fast measurement. The flatness profile of the wafer surface can be evaluated from the 2D outputs of the probes without the influence of the scanning motion errors of the linear carriage and the wafer spindle. To realize the proposed error-separation instrument, 2D angle probes with high resolution and high accuracy must be used. However, there are no commercially available angle probes, which can detect 2D surface local slopes. It is also not a good solution to employ two 1D angle probes, which have already been developed by the authors [19–22], to realize the 2D local slope measurement. Development of such 2D angle probes is, thus, of high priority in this project. In this paper, a prototype 2D angle probe was designed and built for exploration of the probe design criteria by focusing on the probe sensitivity and compactness. The probe utilizes the principle of autocollimation [23]. Different position-sensing devices (PSDs; photo-detectors) were compared from the point of view of sensitivity. Theoretical analysis and experimental results are presented after a description of the new error-separation method.

2. The scanning multi-probe instrument for wafer flatness measurement Fig. 1 shows the schematic of the scanning multi-probe instrument. The wafer sample is mounted on a spindle. A sensor unit is moved by a linear sensor carriage along X-direction to scan the wafer surface while the wafer is be-

397

Fig. 2. Sampling positions on the wafer surface.

ing rotated by the spindle. There are two 2D angle probes with a probe distance of D in the sensor unit. Fig. 2 shows coordinates of sampling points on the wafer surface. The scanning in X-direction starts from the center of the wafer, and the sampling positions are numbered as xi (i = 1, 2, . . . , M). For the sake of clarity, the probe distance D is assumed to be equal to the sampling intervals in the following description. At each sampling position xi , the 2D surface local slopes at the points along two concentric circles are detected by the two angle probes simultaneously. The sampling positions along the circle are numbered as θ j (j = 1, 2, . . . , N). The Y-directional outputs µ1y (xi , θ j ), µ2y (xi , θ j ) of the angle probes of Sensor unit A at xi , which correspond to the tilts about X-axis, can be expressed as: µ1y (xi , θj ) = fy (xi , θj ) + eCX (xi ) + eSX (xi , θj ) (1) µ2y (xi , θj ) = fy (xi+1 , θj ) + eCX (xi ) + eSX (xi , θj ), i = 1, 2, . . . , M − 1, j = 1, 2, . . . , N

(2)

where, eCX (xi ) is the roll error of the sensor carriage, eSX (xi , θ j ) is the angular motion component of the wafer spindle about X-axis. fy (x, θ ) is the Y-directional local slope of the wafer surface, which is defined as: ∂f (x, θ ) (3) fy (x, θ ) = ∂y Taking the difference of Eqs. (1) and (2) gives: µy (xi , θj ) = fy (xi+1 , θj ) − fy (xi , θj ), i = 1, 2, . . . , M − 1, j = 1, 2, . . . , N

(4)

For a fixed θ j (j = 1, 2, . . . , N), fy (xi , θ j ) can be obtained from the integration of µy : fy (xi , θj ) = Fig. 1. Schematic of the scanning multi-probe instrument for wafer flatness metrology.

i
k=1

µy (xk , θj ), fy (x1 , θj ) = 0, i = 2, 3, . . . , M − 1

(5)

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For a fixed xi (i = 2, 3, . . . , M), the relative profile height f(xi , θ j ) of the wafer surface along the i-th concentric circle can then be calculated from the integration of fy (xi , θ j ): f (xi , θj ) =

j
t=2

fy (xi , θt )(θt − θt−1 )xi + f (xi , θ1 ), j = 2, 3, . . . , N − 1

(6)

where f(xi , θ 1 ) (i = 2, 3, . . . , M) are the profile heights along X-axis. It should be noted that the angular pitch (θ t −θ t −1 ) should be chosen in such a way that the profile at the outermost circle can be evaluated with enough height resolution. Since f(xi , θ 1 ) (i = 1, 2, . . . , M) are indeterminate, however, the entire profile of the wafer surface can not be correctly expressed from the profiles f(xi , θ j ) along concentric circles obtained in Eq. (6). To determine f(xi , θ 1 ) (i = 1, 2, . . . , M), we can use two kinds of methods. In the first method, we keep the wafer spindle stationary at the position of θ 1 and move the sensor carriage along X-direction so that the wafer surface is scanned by the sensor unit again. The X-directional output of the angle probe 1, µ1x (xi , θ j ), which corresponds to the tilt about Y-axis, can be expressed by: µ1x (xi , θj ) = fy (xi , θj ) + eCY1 (xi )

(7)

where, eCY1 (xi ) is the yaw error of the sensor carriage during the second scanning. To remove the influence of the yaw error eCY1 (xi ), we place an autocollimator outside the sensor carriage to monitor the yaw error during the scanning. The sectional profile f(xi , θ j ) (j = 1) along the radial direction can, thus, be calculated from the integration of fx (xi , θ j ): f (xi , θj ) =

i
t=2

fx (xt , θj )s, f (x1 , θj ) = 0, i = 2, 3, . . . , M − 1, j = 1

(8)

In the second method to determine f(xi , θ 1 ) (i = 1, 2, . . . , M), we use only the X-directional outputs of the angle probes µ1x (xi , θ j ), µ2x (xi , θ j ), without using the external autocollimator. µ1x (xi , θ j ), µ2x (xi , θ j ) are expressed by: µ1x (xi , θj ) = fx (xi , θj ) + eCY (xi ) + eSY (xi , θj ) + α1x

(9)

fx (x, θ ) =

∂f (x, θ ) ∂x

+ eCY (xi ) + eSY (xi , θj ) + α2x , (10)

where, eCY (xi ) is the yaw error of the sensor carriage, eSY (xi , θ j ) is the angular motion component of the wafer spindle about Y-axis, α 1x and α 2x are the offsets of each probe, which are the probe outputs when a perfect flat surface is detected by the probes. fx (x, θ ) is the X-directional local

(11)

The differential output ∆µx (xi , θ j ), in which the motion errors are cancelled, can be expressed as: µx (xi , θj ) = fx (xi+1 , θj ) − fx (xi , θj ) + (α2x − α1x ), i = 2, 3, . . . , M − 1, j = 1

(12)

fx (xi , θ j ) can be obtained from the integration of ∆µx (xi , θ j ): fx (xi , θj ) =

i


µx (xk , θj )

k=1

+(α2x − α1x )i, fx (x1 , θj ) = 0, i = 2, 3, . . . , M − 1, j = 1

(13)

The sectional profile f(xi , θ j ) (j = 1) along the radial direction can, thus, be calculated from the integration of fx (xi , θ j ): f (xi , θj ) =

i
t=2

1 fx (xt , θj )s + αi 2 s, f (x1 , θj ) = 0, 2 i = 2, 3, . . . , M − 1, j = 1

(14)

where α (=α2x − α1x ) is referred to as the zero-adjustment error [24] of Sensor unit A. In this case, it can be seen that a parabolic error term in the calculated profile f(xi , θ j ) is caused by the zero-adjustment error. Since this error is proportional to the square of the wafer radius, it becomes very large for large wafers. To remove the influence of the zero-adjustment error, we use the system shown in Fig. 3. In Fig. 3, the wafer is mounted on a special spindle [25]. Another sensor unit (Sensor unit B) consisting of two 2D angle probes with a probe distance of D was added to scan the other side of the wafer surface (Side 2). Sensor units A and B are mounted on the same linear sensor carriage. The X-directional outputs of Sensor unit B corresponding to those of Sensor unit A shown in Eqs. (1) and (2) are expressed as: τ1x (xi , θj ) = gx (xi , θj ) − eCY (xi ) − eSY (xi , θj ) + β1x

µ2x (xi , θj ) = fx (xi+1 , θj ) i = 1, 2, . . . , M − 1, j = 1

slope of the wafer surface, which is defined as:

(15)

τ2x (xi , θj ) = gx (xi+1 , θj ) − eCY (xi ) − eSY (xi , θj ) + β2x , i = 1, 2, . . . , M − 1, j = 1

(16)

where, gx (xi , θ j ) is the X-directional local slope of the profile g(xi , θ j ) of Side 2. β 1x and β 2x are the offsets of each probe in Sensor unit B.

W. Gao et al. / Precision Engineering 26 (2002) 396–404

399

Fig. 3. Schematic of the scanning multi-probe instrument with two sensor units.

Following the first scanning, the two sensor units scan the wafer surface again after the exchange of their positions (reversal). The sensor outputs in the second scanning are: µr1x (xi , θj ) = −gx (xi+1 , θj ) − erCY (xi ) − erSY (xi , θj ) + α1x

(17)

µr2x (xi , θj ) = −gx (xi , θj ) − erCY (xi ) − erSY (xi , θj ) + α2x

(18)

τr1x (xi , θj ) = −fx (xi+1 , θj ) + erCY (xi ) + erSY (xi , θj ) + β1x

(19) 3. Principle of the 2D angle probe

τr2x (xi , θj ) = −fx (xi , θj ) + erCY (xi ) + erSY (xi , θj ) + β2x , i = 1, 2, . . . , M − 1, j = 1

(20)

From the probe outputs of the two scannings, the zero-adjustment errors of the two sensor units can be obtained from the following equations. M

β −α =

where β (=β2x − β1x ) is referred to as the zero-adjustment error of Sensor unit B. The sectional profile f(xi , θ j ) (j = 1) in Eq. (14) as well as the entire profiles of the two sides of the wafer can be obtained accurately by compensating α and β. It should be pointed out that setting the probe distance D to be equal to the sample interval s was just for simplicity of explanation. In practice, D is set to be larger than so that more data are obtained in the radial direction and the proposed method can be realized more accurately through reducing influences of random errors.

As described earlier, 2D angle probes are necessary for realizing the proposed error-separation method. A 2D angle probe is typically a sensor for detecting the 2D tilt of a surface as shown in Fig. 4. The 2D surface local slopes can also be detected by such a probe with a thin laser beam. The simplest way to construct a 2D angle probe is to utilize the method of optical lever. As can be seen in Fig. 4, a laser

1
{(τr2x (xi , θj ) − τr1x (xi , θj )) 2M i=1

− (µ2x (xi , θj ) − µ1x (xi , θj )) + (τ2x (xi , θj ) − τ1x (xi , θj ))−(µr2x (xi , θj )−µr1x (xi , θj ))} (21) M−1

α+β =−


1 [µ1x (xi+1 , θj ) − µ2x (xi , θj ) 2(M − 1) i=1

+ τ1x (xi+1 , θj ) − τ2x (xi , θj ) + µr2x (xi+1 , θj ) − µr1x (xi , θj ) + τr2x (xi+1 , θj ) − τr1x (xi , θj )] (22)

Fig. 4. The 2D tilt of a surface (the optical source is omitted for clarity).

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beam is projected onto the test surface. The optical spot of the reflected beam on a photo-detector with a distance L from the sample surface will move in the X- and Y-directions if the sample tilts about the Y- and X-axes. The 2D components of the tilt, α and β, can be calculated from the moving distances y and x of the spot on the photo-detector as follows: y α = (23) 2L β =

x 2L

(24)

This method is simple but errors arise when distance L changes. In this study, we employ the technique of autocollimation [23] to solve this problem. As shown in Fig. 5, an objective lens is placed between the sample and the photo-detector. If the photo-detector is placed at the focal position of the lens, the relationship between the tilt and the readout of the photo-detector becomes: y α = 2f

(25)

x 2f

(26)

β =

where f is the focal distance of the objective lens. As can be seen in Eqs. (25) and (26), the distance between the sample surface and the autocollimation unit consisting of the objective lens and the photo-detector does not affect the angle detection. In the case of flatness metrology of silicon wafer, local slopes of the wafer surface are very small and the sensitivity of the angle probe is required to be very high. The sensitivity of the angle probe based on autocollimation can be improved by choosing an objective lens with a long focal distance. However, this will influence the compactness of the angle probe. Here, we discuss how to improve the sensitivity of the angle probe by choosing proper photo-detectors, without increasing the focal distance of the lens.

Fig. 5. Detection of 2D tilt by autocollimation (the incident beam is omitted for clarity).

The 2D linear lateral effect PSD is widely used to detect the 2D position of a light spot [26]. PSDs provide continuous position information and have the advantage of good linearity. Position detection is also not affected by the intensity distribution of the light spot. Let the sensitive length of the PSD be LP in both X- and Y-directions, the 2D positions x and y can be obtained from the 2D output xout PSD and yout PSD of the PSD, which are calculated from the photoelectric currents IX1 , IX2 , IY 1 , and IY 2 (Fig. 5) through the following equations: xout

PSD

=

yout

PSD

=

(IX1 − IX2 ) 4f 2 x = β × 100% = (IX1 + IX2 ) LP LP (27) (IY 1 − IY 2 ) 4f 2 y = α × 100% = (IY 1 + IY 2 ) LP LP (28)

It can be seen that the sensitivities of a 2D PSD, which are defined as xout PSD /x (or xout PSD /β) and yout PSD /y (or yout PSD /α), respectively, are mainly determined by the sensitive length and are not adjustable. Since the sensitivities are inversely proportional to the sensitive length, a PSD with a short sensitive length is preferred for getting high sensitivity. In Eqs. (27) and (28), when an objective lens with a focal distance of 40 mm is used, a 0.01 arc-second angle α (or β) only corresponds to a position change x (or y) of approximately 4 nm. Assume the required resolution of the angle probe is 0.01 arc-second and the dynamic range (measurement range/resolution) is 10,000, the preferred sensitive length of the PSD, which corresponds to the measurement range of the angle probe, is calculated to be approximately 40 ␮m. However, commercially available PSDs typically have sensitive lengths of several millimeters [26–28], which generate unnecessarily large measurement ranges of angle. Considering the fact that the practical signal-to-noise ratios (dynamic ranges) of the current/voltage conversion amplifiers used to pick up the photoelectric currents are not easy to exceed 10,000, it is difficult to achieve the required resolution of angle detection, which is determined by the measurement range and the dynamic range. Another parameter to determine the resolution of a PSD is the noise current. The noise current level of a 2D PSD is several times larger than that of a 1D PSD [26–28]. From this point of view, it is more feasible to use 1D PSDs instead of 2D PSDs. As shown in Fig. 6, however, two 1D PSDs, with sensitive directions aligned perpendicularly, are necessary for detecting the 2D angle information. This results in a more complicated structure. Misalignment of the sensitive axes of each PSD will also reduce the measurement uncertainty. Moreover, just as a 2D PSD, the resolution of a 1D PSD, which is basically dominated by the sensitive length and the dynamic range, cannot be expected high enough. Another possible photo-detector is the quadrant photodiode (QPD). As shown in Fig. 7, a QPD is placed at or slightly

W. Gao et al. / Precision Engineering 26 (2002) 396–404

401

Fig. 6. The 2D tilt detection using two 1D PSDs.

apart from the focal point of the objective lens so that a light spot with a width of DP is generated on the QPD. For simplicity, assume the shape of the light spot is rectangular and the intensity distribution of the light spot is uniform. The 2D position of the light spot can be calculated by [24]: xout

yout

QPD

QPD

=

(I1 + I4 ) − (I2 + I3 ) × 100% (I1 + I2 + I3 + I4 )

=

2 4f x = β DP DP

=

(I1 + I2 ) − (I3 + I4 ) × 100% (I1 + I2 + I3 + I4 )

=

4f 2 y = α DP DP

(29)

Fig. 8. A prototype angle probe.

It should be pointed out that if the shape of the light spot is not rectangular but round, the relationships shown in Eqs. (29) and (30) will become nonlinear [30]. The Gaussian intensity distribution of the laser beam will also influence the linear relationships [30].

(30)

where I1 , I2 , I3 , and I4 are the photoelectric currents from the QPD cells. It can be seen that the sensitivity of the QPD for position and/or angle detection is inversely proportional to the width of the light spot on the sensitive window of the QPD. The width of the light spot is a function of the location of the QPD relative to the focal position of the objective lens along the optical axis of the autocollimation unit [29]. A proper measurement range/sensitivity of position and/or angle detection can, thus, be obtained through adjusting the location of the QPD. Extremely high sensitivity and resolution can be achieved by using this technique.

Fig. 7. The 2D tilt detection using a QPD.

Fig. 9. Calibration result of the 2D angle probe of using a QPD. (a) X-directional output of the 2D angle probe. (b) Y-directional output of the 2D angle probe.

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4. The prototype 2D angle probe A prototype 2D angle probe was designed and built. Fig. 8 shows a schematic view of the probe. A laser diode with a wavelength of 780 nm was used as the optical source. The output light from the laser diode unit was a collimated beam with a diameter of 1 mm. An achromatic lens with a short focal distance of 40 mm was employed as the objective lens for the sake of compactness. A QPD was used as the photo-detector to detect the 2D angle information. Another 1D PSD was also used in the same probe to detect the 1D angle information about Y-axis so that the sensitivities of using PD and PSD can be compared experimentally. The light beam passing through the objective lens was split into two beams, which were received by the QPD and the PSD, respectively. The PSD has a sensitive length of 2.5 mm, which was the shortest we could find in the market. The probe was designed to be within 90(L) mm × 60(W) mm × 30(H) mm in size. Fig. 9 shows a calibration result of the 2D angle probe of using the QPD. The position of the PD on the optical axis was carefully adjusted to get high sensitivity of angle detection. A Nikon photoelectric autocollimator with a resolution of 0.05 arc-second was used as the reference. The test surface in Fig. 8, which was mounted on a manual tilt stage, can be tilted manually about the X- and Y-axis,

Fig. 10. Comparison of using 1D PSD and QPD as the PSD (note that the two horizontal scales are 10 times different).

respectively. The tilt was detected by the angle probe and the Nikon autocollimator simultaneously. In Fig. 9, the horizontal axes show the output of the Nikon autocollimator in arc-seconds, and the vertical axes show the outputs of the angle probe in percentage, which are defined in Eqs. (29) and (30). The X-directional output corresponds to the tilt about Y-axis (β), and the Y-directional output corresponds the tilt about X-axis (α). As can be seen in the figure, the angle probe of using a QPD has the ability to detect the 2D angle information in a range of approximately 200 arc-second.

Fig. 11. Results of resolution test. (a) Result 1 of resolution testing. (b) Result 2 of resolution testing. (c) Result 3 of resolution testing.

W. Gao et al. / Precision Engineering 26 (2002) 396–404

Fig. 10 shows a comparison of the probe output of using the QPD and the 1D PSD. Since the 1D PSD can only detect the tilt about Y-axis, the comparison was made only with the X-directional output of the QPD. Note that the two horizontal scales in the graph are 10 times different. It can be seen that the sensitivity of using the QPD is approximately 30 times higher than that of using the PSD. Fig. 11 shows the experimental results of testing the resolution of the angle probe using the QPD. In the test, since the tilt range was very small, a tilt stage driven by PZTs was employed to tilt the test surface. A sinusoidal voltage signal was applied to the PZT driver of the tilt stage to tilt the test surface periodically about Y-axis. The X-directional outputs of the angle probe and the Nikon autocollimator with respect to time are plotted in the figure. The voltage signals applied to the PZT driver are also shown in Figs. 11b and c. As can be seen in Fig. 11a, both the angle probe and the Nikon autocollimator followed the tilt motion very well when the amplitude of the tilt was approximately 0.7 arc-second. When the PZT voltage was reduced to 30 mV peak-to-valley as shown in Fig. 11b, the output of the Nikon autocollimator showed an amplitude of approximately 0.1 arc-second with some noises corresponding to its resolution level, which was 0.05 arc-second. The angle probe responded to the 0.1 arc-second tilt motion quite well (Fig. 11b). When the amplitude of the tilt motion was reduced to approximately 0.03 arc-second, which was smaller than the resolution of the Nikon autocollimator, the tilt motion cannot be distinguished in the output of the Nikon autocollimator (Fig. 11c). On the other hand, however, the angle probe still showed a good response to the tilt motion. As can be seen in Fig. 11c, the resolution of the angle probe is better than 0.01 arc-second, which is estimated to be high enough for the purpose of wafer metrology.

5. Conclusions A scanning multi-probe instrument has been proposed for flatness measurement of large silicon wafers. A sensor unit consisting of two 2D angle probes was used in the instrument to measure the flatness accurately without being influenced by the motion errors of the linear sensor carriage and the wafer spindle. A prototype 2D angle probe has been designed and made. To improve the sensitivity of angle detection, the method of using two 1D PSDs, and that of using a single QPD as the photo-detectors have been theoretically discussed and experimentally investigated. The angle probe has been confirmed to have a resolution better than 0.01 arc-second. It should be pointed out that we have only verified the sensitivity and resolution of the angle probe and obtained the criteria for designing high-resolution 2D angle probes. Further experiments of testing the probe characteristics, such as

403

the frequency bandwidth, are necessary. To accurately use the probe, a calibration process is also essential. The authors are applying an in situ self-calibration method [31–33] to the calibration of the 2D angle probe when the probe and the wafer are mounted in the scanning system shown in Fig. 1 or Fig. 3. Building multiple 2D angle probes, performing flatness measurement and analyzing flatness measurement uncertainties will also be carried out in our future works.

Acknowledgments This project is conducted under a Japan Society for the Promotion of Science (JSPS) grant-in-aid for scientific research (No. 12555032). It is also supported by a JSPS/National Science Foundation, USA (NSF) International Joint Research Grant and a grant from Mazda Science Foundation. References [1] International SEMATECH: International Technique Roadmap for Semiconductors. [2] Evans CJ, Davies A, Shmitz T, Parks R, Shao L-Z. Interferometric metrology of substrates for VLSI. Proceedings of the euspen Second International Conference; 2001 May 27–31; Turin, Italy. p. 388–91. [3] Tyler Estler W. Calibration and use of optical straightness in the metrology of precision machines. Opt Eng 1985;24(3):372–9. [4] Whitehouse DJ. Some theoretical aspects of error separation techniques in surface metrology. J Phys E: Sci Instrum 1976;9:531–6. [5] Evans CJ, Hocken RJ, Tyler Estler W. Self-calibration: reversal, redundancy, error separation, and ‘absolute testing’. Ann CIRP 1996;45(2):617–34. [6] Kiyono S, Gao W. Profile measurement of machined surface with a new differential method. Prec Eng 1994;16(3):212–8. [7] Gao W, Kiyono S. High accuracy profile measurement of machined surface by the combined method. Measurement 1996;19(1):55–64. [8] Gao W, Kiyono S. On-machine measurement of machined surface using the combined three-point method. JSME Int J Series C 1997;40(2):253–9. [9] Gao W, Kiyono S. Development of an optical probe for profile measurement of mirror surfaces. Opt Eng 1997;36(12):3360–6. [10] Von Bieren K. Pencil beam interferometer for aspherical optical surfaces. Laser diagnostic. SPIE 1982;343:101–8. [11] Takacs PZ. Nanometer precision in large surface profilometry. Proceedings of Ninth International Conference on Precision Engineering; 1999 Aug 29–Sept 1; Osaka, Japan. p. 301–10. [12] Ennos AE, Virdee MS. High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation. Prec Eng 1982;4(1):5–8. [13] Ennos AE, Virdee MS. Precision measurement of surface form by laser autocollimation. Industrial application of laser technology. SPIE 1983;388:252–7. [14] Kiyono S, Asakawa Y, Inamoto M, Kamada O. A differential laser autocollimation probe for on-machine measurement. Prec Eng 1993;15(2):68–76. [15] Takacs PZ, Bresloff CJ. Significant improvements in long trace profile measurement performance. Optics for high-brightness synchrotron radiation beamlines II. SPIE 1996;2856:236–45. [16] Huang PS, Xu XR. Design of an optical probe for surface profile measurement. Opt Eng 1999;38(7):1223–8.

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