## Abstract

We propose a novel *in situ* aberration measurement technique for lithographic projection lens by use of aerial image based on principal component analysis (AMAI-PCA). The aerial image space, principal component space and Zernike space are introduced to create a transformation model between aerial images and Zernike coefficients. First the aberration-induced aerial images of measurement marks are simulated to form an aerial image space with a statistical Box–Behnken design pattern. The aerial image space is then represented by their principal components based on principal component analysis. The principal component coefficients of the aerial images are finally connected with Zernike coefficients by a regression matrix through regression analysis. Therefore *in situ* aberration measurement can be achieved based on the regression matrix and the principal component coefficients of the detected aerial images. The measurement performance of the proposed AMAI-PCA technique is demonstrated superior compared to that of the conventional TAMIS technique by using a lithographic simulator tool (Prolith). We also tested the actual performance of AMAI-PCA technique on a prototype wafer exposure tool. The testing results show our proposed technique can rapidly measure the aberrations up to high-order Zernike polynomial term with 1σ repeatability of 0.5nm to 2.3nm depending on the aberration type and range.

©2011 Optical Society of America

## 1. Introduction

Projection lens is one of the most important systems in a lithographic tool [1], which is the key contributor to the critical dimension shrinking of semiconductor industry. The aberrations of projection lens can lead to the deterioration of lithographic imaging performance, such as defocus and pattern shift and asymmetry, which reduce the lithographic process latitude [2,3].These adverse lithographic impacts of lens aberrations will become increasingly severe with the critical dimension shrinking and the use of resolution enhancement techniques. Consequently, rapid and accurate *in situ* lens aberration measurement is required to minimize these adverse impacts of lens aberrations during lithographic process [4].

In recent years, several techniques have been reported for *in situ* measurement of projection lens aberration [5–12]. These techniques are similar in way of extracting aberrations by measuring special characteristics of aerial images or exposed resist images of grating marks, such as linewidth or best focus. Transmission image sensor at multiple illumination settings (TAMIS) is a representative aerial image-based technique developed by ASML [12], which is straightforward and fast due to excluding the use of resist compared to the resist-based techniques. In TAMIS technique, the imaging positions of isolated space patterns in 2 directions are measured by an aerial image sensor built into the wafer stage (TIS) at multiple numerical aperture (NA) and partial coherence settings. From lateral (x/y) and axial (z) aerial image position shifts, Zernike coefficients corresponding to astigmatism (Z5), spherical aberration (Z9, Z16) and coma (Z7, Z8, Z14, Z15) can be calculated. This set of aberrations has been found to be most critical for the typically Manhattan-geometric features found in IC layouts. As feature size decreases, the residual wavefront error of projection lens has reached the level of 10mλ or less [13]. The methods with higher accuracy are required to satisfy the measurement requirements of current lithography tools.

In this paper, we propose a novel technique, called AMAI-PCA, for *in situ* aberration measurement (AM) of projection lens in a lithographic tool by use of aerial image (AI) based on principal component analysis (PCA). Principal component analysis of aerial images is utilized to build a regression matrix that links the principal component coefficients of aerial images to the Zernike coefficients of lens aberrations [14]. Once the regression matrix is established by regression analysis, Zernike coefficients can be quickly calculated. The measurement repeatability and accuracy of the proposed AMAI-PCA technique is compared with that of conventional TAMIS technique using a lithographic simulator tool (Prolith) [15] and demonstrated on a prototype wafer exposure tool.

## 2. Theory

#### 2.1 Aerial imaging model in optical lithography

Figure 1 shows the general layout of a lithographic projection system. The system comprises a narrow bandwidth light source, a condenser lens, a mask with the circuit patterns, the

projection lens and the wafer stage. The calculation of optical lithographic aerial image is performed based on Hopkins theory of partially coherent imaging [16]. The aerial image intensity of the mask in the wafer plane can be expressed as

where $O(f\text{'},g\text{'})$ is the diffraction spectrum of a mask and TCC is the transmission cross coefficient. TCC is in turn defined as

where$J({f}_{c},{g}_{c})$ is the effective source function. The source function can be expressed as

Here *σ* is the partial coherence factor of illumination, which denotes the amount of filling of the entrance pupil and equal to the quotient between the numerical apertures of the condenser and projection lenses. In Eq. (2), *H*(*f,g*) is the spatial transfer function and is given by the equation

where *H*
_{0}(*f*,*g*) is the pupil function in absence of aberrations. *W*(*f,g*) represents the lens aberration, defined as optical path difference between the actual wavefront and ideal spherical wavefront. The wavefront aberration is expressed in terms of Fringe Zernike polynomials as

where ${Z}_{n}$is the Zernike coefficient and ${R}_{n}(f,g)$ is the Zernike polynomial.

According to Hopkins theory of partially coherent imaging, the image intensity distributions on wafer plane depend on the wavefront aberration of the projection lens. Image intensity changes uniquely with different aberrations, which allows for the determination of the aberration. Aberration will lead to the deviation of aerial image shape as shown in Fig. 2 . Figure 2(a) shows the ideal aerial image of isolated line formed by a perfect lens without wavefront aberration, and Fig. 2(b) shows the aerial image impacted by coma. As a consequence of the aberration, left-right asymmetry of the “banana” shape appears in the aerial image.

The Zernike wavefront aberration can be calculated from aerial images by a linear transformation if Zernike coefficient values are within a certain range. The principle of the conventional TAMIS technique is to extract special characteristics (lateral (x/y) and axial (z) position shifts) from the aerial image as shown in Fig. 2. However, the accuracy of TAMIS is limited because most information in aerial image is neglected. Only low order coma and spherical aberration can be calculated by TAMIS. In order to improve the Zernike wavefront measurement accuracy and increase the number of Zernike terms that can be measured, it is necessary to fully use the information in the aerial images. However, using the intensity distribution in an aerial image directly is not practical. There are numerous pixels in the measured aerial images and the adjacent pixels are correlated, thus it is unnecessary to use all of the pixels in an aerial image. Establishing a relationship between aberration coefficients and individual measurement pixels is thus to be avoided. Instead, principal component analysis is applied to extract the most critical information from a set of aerial images. Three kinds of parameter spaces are proposed to elucidate the procedure of the proposed AMAI-PCA technique.

- 1. Zernike space: The space constituted by Zernike coefficients.
- 2. Aerial image space: The image intensity distribution corresponding to certain Zernike coefficients combinations.
- 3. Principal component space: The statistical space constructed by principal component analysis of aerial images [17]. The transformation from principal component space to Zernike space relies on the linear transformation built by regression analysis.

Figure 3
illustrates the definition of three kinds of space in AMAI-PCA technique and the transformation relation among them, where *V _{i}* is a principal component coefficient and

*Z*is a Zernike coefficient.

_{i}#### 2.2 Building the aerial image space

An important feature of aberrations' impact on the aerial image is the interactions of individual Zernike coefficients. Care needs to be taken to estimate their strength, as well as the magnitude of pure quadratic terms. Although we expect to establish a linear relationship

for a small number of low-order Zernikes, we need to perform this estimation for a fuller set of up to 33 terms. Thus, a design of experiment based approach is used, with the creation of Zernike treatment combinations handled by a Box–Behnken design [18]. In this method, two parameters are varied at the same time, and are set to their maximum or minimum value. Systematically repeating this process for all pairs of Zernike combinations, the number of treatments is controlled to a manageable $t=4\left(\begin{array}{c}n\\ 2\end{array}\right)+1$, or 2113 runs for *n =* 33 Zernike coefficients. This type of design supports analysis regression using a full quadratic model, including all interactions. An example of complete set treatment combination values for a design with 3 variables is given in Fig. 4
.

The response of aerial image to Zernike coefficients is linear if aberration variation range are small, thus only three values (−20nm, 0, 20nm) is used to represent the variation range of Zernike coefficients. The magnitude of interaction and quadratic terms is tested during the regression step.

#### 2.3 Principal component analysis

Aerial images are calculated for each of the *t* Zernike combinations according to Hopkins imaging model, each image with a total number of pixels *p*. The images are then reshaped into vertical vectors and arranged in an array *AI* of size *p × t*.

The first step in the analysis of created images is separation of Zernike dependence and position dependence. This is performed using Principal Component analysis of image intensity. We use raw data for the analysis instead of the covariance matrix. The goal is to ensure full orthogonality of all image decomposition components to support experimental data fitting in the following steps. Using no scaling and no centering of raw intensity data

Here, *PC _{α}* are the principal components of the aerial images, mutually orthogonal over the space domain. The first principal component,

*PC*

_{1}, is very similar to the mean aerial image, and is not used for the regression analysis; however, its inclusion guarantees that all other principal components are orthogonal to it. Correspondingly,

*V*are normalized vectors of coefficients, representing weights of each principal component in a particular image. The dependence of

_{α}*V*on Zernike coefficients is not analyzed in this step. The advantage of this analysis is the reduction in the number of variables with truncation of the principal component number. A shorter representation is possible if the upper limit in the sum in Eq. (6) is replaced with

_{α}*m*. The generated principal components are arranged by the magnitude of their eigenvalues. The eigenvalues drop off rapidly, thus the first several principal components are most significant. It is then reasonable to represent the full set of images using only the first few principal components with acceptable accuracy. The orthogonal character of the principal components enables to express any aerial image as a unique principal coefficients combination. If first

*m*principal components are used in the model, the relationship between principal component coefficients, principal component and aerial image is expressed as

Here, *AI* is an array of aerial image intensity values of size *p × t*, *PC _{i}* is an array of size

*p ×*1 representing a principal component,

*V*is a corresponding vector of coefficients of size 1

_{i}*× t*. Finally,

*E*is the truncation error array of size

_{T}*p × t*that accounts for the residual when representing the total error in the aerial image set. As the number of utilized principal components

*m*grows, the error is reduced, as illustrated in Fig. 5 .

As seen from Fig. 5, the residual of fitting aerial image using its principal components can be reduced to a small number. When the PC number *m* is 5, the amplitude of residual is already on the level of 10^{−3} shown in Fig. 5(b). Figure 5(c) shows the residual drop rapidly as the PC number increases. Therefore, it is valid to represent the aerial image approximately by using first several PCs. The image sensor is expected to introduce uncorrelated random noise with RMS value of approximately 0.01, therefore the use of the first 5 PC was accepted. Based on this noise assumption, the expected error in *V* estimation was calculated to be less than 10^{−4} for the first 5 coefficients.

#### 2.4 Regression matrix and Zernike aberration retrieval

After the principal component analysis of the aerial image set, a linear regression is applied to obtain the regression matrix between Zernike coefficients and principal component coefficients *V _{i}*. The linear regression equation is given by

where each *RM _{i}* is an array of size 1

*× n*calculated by regression,

*Z*is the matrix of treatment combinations of size

*n × t,*and

*E*is the error of regression of size

_{R}*m × t.*The error of regression includes all terms of order higher than linear.

Equation (8) is critical for the overall performance of the Zernike estimation process. The first four principal components (excluding *PC*
_{1}) obtained a correlation number above 0.99 and thus were selected for future analysis. Standard error of regression for the first five PC coefficients was lower than 0.002. While this is higher than expected error in *V* estimation, it proved to be lower than experimental standard deviation of measured *V* values.

Aberration retrieval is handled by applying Eq. (8) in order to solve for unknown *Z* values, when the regression matrix is known, and *V* coefficients are calculated from an experimentally measured image. During the process of model building, a set of aerial images from simulation were used to get principle components and obtain the regression matrix. In extraction process, regression matrix is applied into principal components to individual image to calculate Zernike value from one aerial image.

## 3. Simulation

The measurement performance of the proposed AMAI-PCA technique was compared to that of TAMIS technique through a standard lithographic simulator tool Prolith, which is widely used in semiconductor industry developed by KLA-Tencor Co. Ltd [19,20]. TAMIS is usually used to measure the low order astigmatism, coma and spherical aberrations. Here the measurement results using AMAI-PCA and TAMIS to measure Z_{5}, Z_{7}, Z_{8}, Z_{9}, Z_{14}, Z_{15}, Z_{16} is simulated by Prolith. The simulation settings are shown in Table 1
. The same measurement mark was used in simulation to guarantee the validity of the comparison. Figure 6
shows the structure pattern of the mark, which is composed of the openings having a width of 250 nm spaced 3000 nm (at wafer level) oriented in the 0°/90° direction.

In TAMIS technique, measurement was carried out under multiple illumination settings. In the simulation, the variation range of NA was from 0.45 to 0.75 with a step of 0.1. The variation range of partial coherence factors was from 0.3 to 0.8 with a step of 0.1. The total number of illumination settings was 24. For AMAI-PCA method, the model was built using the method above. Then, thirty groups of normally distributed random Zernike coefficients were generated to mimic the aberration distribution in lithographic projection optics. Aerial images were simulated and the Zernike aberrations were calculated from one aerial image with AMAI-PCA technique. The absolute deviations are calculated to evaluate the differences between 30 nominal and calculated values of Zernike aberrations. In the following figures, max error, standard error, mean error and RMS error means the maximum value, standard error, average value and root mean square of 30 deviations, respectively. Figure 7 shows the measurement performance comparison between AMAI-PCA and TAMIS with the aberration varying within the range of −10nm to 10nm. Figure 8 shows the measurement performance comparison between AMAI-PCA and TAMIS, when the aberration variation range is within −20nm to 20nm.

The large aberration variation range will decrease the measurement accuracy for the both techniques. From the above simulation results, we could conclude that AMAI-PCA significantly enhances the measurement accuracy compared with TAMIS. The advantage of AMAI-PCA is especially evident when the aberration variation range is large.

## 4. Experiment

#### 4.1 Measurement Method

An aerial image sensor for aberration measurement was integrated into SSA600/10, an ArF lithography tool developed by SMEE [21]. Figure 9 shows the schematic of the sensor. A mask with measurement marks as shown in Fig. 6 is illuminated via the illumination system, and they are projected by means of the projection lens onto the image plane. The aerial image sensor on the wafer stage is covered by the same mark, which is adjusted for magnification. The aerial image sensor converts the light intensity to electrical signal. The sensor carried by wafer stage can move to different positions to measure the aerial images in the exposure field.

In order to measure the aberration of overall field, 9 measurement marks are arranged along x axis of the mask (non-scanning direction). To capture the aerial images of the mark, the sensor measure the light intensity at different position and height values. Through adequate sampling, the aerial image features in the order of tens of nanometers can be extracted. During the experiment, all of the measurements were implemented under conventional illumination with NA equal to 0.75 and partial coherence factor equal to 0.88.

#### 4.2 Experimental Results

The measurement repeatability and accuracy of the Zernike coefficients with AMAI-PCA technique were investigated on SSA600/10.

When we examine the repeatability of AMAI-PCA, the aerial images in 9 locations across the exposure field were measured. Z_{5}, Z_{7}, Z_{8}, Z_{9}, Z_{14}, Z_{15}, and Z_{16} were extracted from each aerial image. The aberrations for each location have been measured for twenty times to evaluate measurement repeatability, that’s to say, there are total 180 aerial images were captured to calculate the aberrations. Figure 10
shows the 1 standard deviation repeatability of the aberration measurement across the exposure field. The actual repeatability is from 0.5nm to 2.3nm, which are larger than simulation results. The main cause of this is that wafer stage positioning accuracy, interferometer accuracy and light intensity noise are worse than expected. In addition, the actual aberration level of projection lens in the prototype machine is larger than expected.

Uncertainty analysis was performed using repeatability of the *V* parameters extracted from the image and the regression matrix. The experimental *V* estimation standard deviation was found to be between 0.005 and 0.020, higher than the 0.002 standard error of regression from Eq. (8), thus making the linear model acceptable for this set of data. The increase of repeatability over the value expected from PCA (less than 10^{−4}) was due to the above mentioned reasons causing centering error of the image. As no absolute image center can be obtained from the measurement, it was estimated using iterative algorithm, and the resulting center error contributed to the error of *V* estimation. From this data, standard error of Zernike estimation was calculated between 0.7 nm to 2.5nm. Compare to the repeatability results, they are matching very well.

In the absence of phase measurement interferometer, we cannot compare the measured aberrations with their absolute value, so an indirect method is used to investigate the accuracy of AMAI-PCA. In 193nm lithographic tool, the linear coma Z_{7} across exposure field changes with respect to the shift of the wavelength. We measured the dependencies of Z_{7} on wavelength and compared them to the dependency derived from the lens design data, which was provided by the lens manufacturer. The accuracy can be evaluated by the correlation between the nominal dependency and the measured dependencies. In the experiment, Z_{7} was measured with AMAI-PCA at nominal wavelength as well as at one negative and two positive wavelength shifts. The measurement results showed wavelength shifts led to large amounts of linear coma. Figure 11
presents a good consistency between the nominal and measured dependencies, which demonstrates the ability of AMAI-PCA to measure the amount of linear coma.

The aberrations can be measured at one illumination setting for AMAI- PCA method, while TAMIS have to do that at multiple numerical aperture (NA) and partial coherence settings. This advantage make the proposed method can reduce the measurement time greatly. Compare to TAMIS measurement time 1 minute per field position [12], AMAI-PCA can finish that in 9 seconds including aerial image acquisitions and data processing.

## 5. Conclusion

In conclusion, a novel AMAI-PCA technique has been proposed to measure aberrations in lithographic projection optics. By using multi-variable statistical design to build the aerial images space and principal component analysis to extract the most critical information of the aerial images, this technique is capable of measuring the aberration accurately and fast. The measurement performance of the proposed AMAI-PCA technique has been analyzed by simulation. It is shown that the proposed technique can measure low-order astigmatism, coma, and spherical terms more accurately compared to the conventional TAMIS technique. These terms are still the dominant aberrations in lithographic tools and are of primary importance. The actual measurement results in an ArF lithography tool SSA600/10 demonstrate that AMAI-PCA technique is capable of determining Zernike coefficients at least up to Z_{16} with a 1 standard deviation repeatability of 0.5nm to 2.3nm depending on the aberration type and range. The measurement time is about 9 seconds per field point, which is faster than TAMIS. By optimizing the test marks, AMAI-PCA technique has great potential to measure Zernike aberrations up to Z_{37} and is expected to be used as a fast method for *in situ* monitoring lens aberration.

## Acknowledgments

The authors would like to thank the technical support of Chinese National Engineering Research Center for Lithographic Equipment. This work was supported by a grant from the National Science Foundation of China (NSFC) (No. 60938003)

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