## Abstract

In this paper, we propose a novel method for measuring the coma aberrations of lithographic projection optics based on relative image displacements at multiple illumination settings. The measurement accuracy of coma can be improved because the phase-shifting gratings are more sensitive to the aberrations than the binary gratings used in the TAMIS technique, and the impact of distortion on displacements of aerial image can be eliminated when the relative image displacements are measured. The PROLITH simulation results show that, the measurement accuracy of coma increases by more than 25% under conventional illumination, and the measurement accuracy of primary coma increases by more than 20% under annular illumination, compared with the TAMIS technique.

©2007 Optical Society of America

## 1. Introduction

The projection optics system is one of the most important systems in a step-and-scan lithographic tool. Aberrations of the projections optics cause a deterioration of the image quality as well as a significant reduction of process latitude of the lithographic process [1–4]. Coma aberration is one of the most important aberrations because of its striking effects on printed patterns. Coma causes lateral image displacements that depend on pattern features and illumination settings, and then leads to overlay errors. Coma also causes linewidth asymmetry of printed lines, which influences the lithographic resolution and critical dimension uniformity [5–7]. As the critical dimension shrinks, especially with the use of resolution enhancement techniques, impacts of coma aberration on the image quality become more serious. To predict and minimize its adverse effects on printed patterns, fast and accurate in-situ measurement techniques for measuring the coma aberration are necessary [8].

Recently, a number of in-situ methods for measuring coma aberration of lithographic projection optics have been reported, such as three-beam interference [9,10], two-beam interference [11], aberration ring test [12], Litel in-situ interferometer[13], and transmission image sensor (TIS) at multiple illumination settings (TAMIS) [14,15]. Among these methods, the TAMIS is a commonly used sensor-based technique. A binary mark is used as the measurement mark in the TAMIS technique. The image displacements of the measurement mark at multiple numerical aperture (NA) and partial coherence settings are measured by the TIS, which is an aerial image measurement device built into the wafer stage. Using the image displacements, the Zernike coefficients corresponding to coma can be determined. Advantages of the TAMIS include robustness and speed, because it is a straightforward measurement technique that does not involve exposure of resist, and it does not rely on the formation of an intermediate image in resist which is subsequently analyzed by a scanning electron microscope, an overlay inspection tool or an optical microscope. However, distortion, which is a kind of lateral aberrations, also causes lateral image displacements. When coma is measured, the impact of distortion on lateral image displacements can not be eliminated in the TAMIS technique, which influences the measurement accuracy of coma. The impact of distortion on lateral image displacements is also neglected in the coma measurement method that we reported last year [16].

In this paper, a novel in-situ method for measuring the coma aberration of lithographic projection optics is proposed. A new mark which contains two fine-segmented phase-shifting gratings and two sufficiently large binary gratings is used in this method. The Zernike coefficients corresponding to coma aberration are extracted from the relative image displacements between the phase-shifting gratings and the binary gratings at multiple NA and partial coherence settings. The measurement accuracy of coma can be improved because the phase-shifting gratings are more sensitive to coma than the binary gratings used in the TAMIS technique, and the impact of distortion on displacements of aerial image can be eliminated when the relative image displacements are measured. Variation ranges of sensitivities are the key factor which influences the measurement accuracy of coma. Using the lithographic simulator PROLITH, the variation ranges of sensitivities are calculated. Compared with the TAMIS technique, the measurement accuracy of coma obviously increases because the variation ranges of the sensitivities are enlarged in this novel method.

## 2. Theory

In the present method, a novel mark is proposed, as shown in Fig. 1. The mark is composed of two orthogonal alternating phase-shifting gratings which act as measurement marks and two orthogonal binary gratings which act as reference marks. The linewidth of the measurement mark is 250nm, and the pitch is 500nm. The linewidth of the reference mark is 2µm, and the pitch is 4µm. We denote the two measurement marks as mark A and mark C and the two reference marks as mark B and mark D. Marks A and B are used to measure the x-coma, and marks C and D are used to measure the y-coma.

The transmission function of an alternating phase-shifting grating is

where p is the width of a line and a space on the alternating phase-shifting grating. The spectrum of the alternating phase-shifting grating is the Fourier transformation of *t*(*x*),

where *f _{x}*=sin

*θ*/

*λ*is the spatial frequency variables and the sinc function is defined as $\mathrm{sin}c\left(x\right)=\frac{\mathrm{sin}\left(\pi x\right)}{\pi x}$. The transmission function of a binary grating is

The spectrum of the binary grating can be expressed as

From Eqs. (2) and (4), it can be seen that the zero-order diffraction light of the alternating phase-shifting grating and the even-order diffraction light of the binary grating are missing. For a phase-shifting grating, more high-order diffraction light can pass through the pupil, and without the zero-order diffraction light, the intensity of high-order diffraction light can be higher. So the intensity near the pupil edge should be higher than that of a binary grating which has the same linewidth. So the impact of coma aberration on image displacements of the phase-shifting grating becomes more obvious.

Coma aberration can be represented by the Zernike coefficients Z_{7}, Z_{8}, Z_{14}, and Z_{15}, which are the coefficients of the Zernike polynomials. The Zernike polynomials which represent the wavefront aberrations in the projection lenses can be expressed as [17]

$$={Z}_{1}+{Z}_{2}\rho \mathrm{cos}\theta +{Z}_{3}\rho \mathrm{sin}\theta +{Z}_{4}\left(2{\rho}^{2}-1\right)+{Z}_{5}{\rho}^{2}\mathrm{cos}2\theta +$$

$${Z}_{6}{\rho}^{2}\mathrm{sin}2\theta +{Z}_{7}\left(3{\rho}^{2}-2\right)+\rho \mathrm{cos}\theta +{Z}_{8}\left(3{\rho}^{2}-2\right)\rho \mathrm{sin}\theta +\dots +$$

$${Z}_{14}\left(10{\rho}^{4}-12{\rho}^{2}+3\right)\rho \mathrm{cos}\theta +{Z}_{15}\left(10{\rho}^{4}-12{\rho}^{2}+3\right)\rho \mathrm{sin}\theta +\dots $$

where ρ is the normalized radius of the exit pupil, and θ is the azimuthal angle. Z_{7} represents third-order x-coma, Z_{8} represents third-order y-coma, Z_{14} represents fifth-order x-coma, and Z_{15} represents fifth-order y-coma. When high-order coma aberrations are neglected, the aberration functions which influence the image displacements can be written as

Equations (6) and (7) include so-called distortion in the linear terms. The distortion causes image displacements for large patterns as well as for fine ones. However, image displacements caused by coma depend on the pattern size and density. Therefore, to distinguish coma from distortion, it is necessary to notice the relative image displacements of a fine grating pattern to a sufficiently large pattern. In the present technique, the linewidths of marks B and D are sufficiently large, so almost all their spectra are concentrated about the origin [9]. The relative image displacements between marks A and B, marks C and D are measured by an aerial image sensor such as the TIS, and the x-coma and y-coma can be extracted from the relative image displacements, which can be expressed as

where *ΔX _{A}*,

*ΔX*,

_{B}*ΔY*and

_{C}*ΔY*are the image displacements of marks A, B, C and D, respectively. The impact of distortion on image displacements can be eliminated by measuring the relative image displacements which can be expressed as

_{D}From Eqs. (10) and (11), the relative image displacements caused by coma depend on the spectrum distribution in the pupil. The maximum relative image displacement caused by primary coma occurs at *ρ*=1, and the minimum image displacement caused by primary coma occurs at *ρ*=0. The maximum relative image displacement caused by secondary coma occurs at *ρ*=0, and the minimum image displacement caused by secondary coma occurs near *ρ*=0.8. This is different from that of primary coma. As mentioned above, for the mark proposed in the paper which contains phase-shifting gratings, more high-order diffraction light can pass through the pupil. The diffraction spectrum has its orders positioned in the regions where large phase errors are introduced by coma, and consequently the sensitivity to relative image displacements is large.

The relative image displacements caused by coma also depend on the NA of the projection optics and partial coherence of the illumination system, because the intensity distribution of diffraction light varies with the NA and the partial coherence. The maximum and minimum sensitivities can be obtained by proper illumination settings. The relative image displacements are approximately linear with the Zernike coefficients, for marks A and B, their relative image displacement can be written as

where *ΔX*(*NA*,*σ*) is the relative image displacement at the given NA and partial coherence setting, which can be measured by the aerial image sensor. *S*
_{1}(*NA*,*σ*) and *S*
_{2}(*NA*,*σ*) are sensitivities and can be expressed as

The sensitivities vary with the NA and partial coherence settings, and can be calculated by lithographic simulator such as PROLITH.

For each NA and partial coherence setting, a linear equation is obtained. With multiple NA and partial coherence settings, a set of equations can be written as

Equation (15) is over-determined and can be solved by the least square method. Using the aerial image sensor, the relative image displacements of the mark in different field positions are measured at multiple illumination settings. Zernike coefficients Z_{7}, Z_{8}, Z_{14} and Z_{15} in different field positions can be determined. As can be seen from Eq. (15), when the variation ranges of the sensitivities are larger, the span of the data used for the least square fit are larger and the least square fit has higher accuracy, so the variation ranges of the sensitivities are the key factor which influences the measurement accuracy of the Zernike coefficients.

## 3. PROLITH simulation

In the present paper, the relative image displacements of the marks are used to extract the coma aberration. The measurement accuracy of the relative image displacements depends on the overlay accuracy of the lithographic tool. The measurement accuracy of the coma can be estimated as

where MA is the measurement accuracy of Zernike coefficients corresponding to coma, OA is the overlay accuracy of the lithographic tool, S_{max} and S_{min} are the maximum and minimum values of sensitivity S.

Using the lithographic simulator PROLITH, we calculated the sensitivities for the present technique and the TAMIS technique at multiple NA and partial coherence settings, which are commonly used in mainstream lithographic tools. The wavelength used in the simulation is 193nm. The variation range of the NA of projection optics is 0.5~0.8. The variation ranges of the partial coherence under conventional illumination and annular illumination are 0.25~0.85 and 0.3~0.8, respectively. The annular width is 0.3. The mark used in the TAMIS simulation is a binary grating with a linewidth of 250nm.

Figures 2(a) and 2(b) show the sensitivities of Z_{7} for the present technique at multiple NA and partial coherence, under conventional illumination and annular illumination, respectively. Figures 2(c) and 2(d) show the sensitivities of Z_{7} for the TAMIS technique at multiple NA and partial coherence, under conventional illumination and annular illumination, respectively. From the simulation results of sensitivities, the measurement accuracy can be estimated by Eq. (16), and the overlay accuracy of the lithographic tool is assumed to be 2nm. The simulation res ults of the sensitivities of Z_{7} and the estimated measurement accuracy are shown in Table 1. From Fig. 2 and Table 1, the measurement accuracy of Z_{7} increases by 25% and 21.7% under conventional illumination and annular illumination, respectively, compared with the TAMIS technique.

Figures 3(a) and 3(b) show the sensitivities of Z_{14} for the present technique at multiple NA and partial coherence, under conventional illumination and annular illumination, respectively. Figures 3(c) and 3(d) show the sensitivities of Z_{14} for the TAMIS technique at multiple NA and partial coherence, under conventional illumination and annular illumination, respectively. The simulation results of the sensitivities of Z_{14} and the estimated measurement accuracy are shown in Table 2. From Fig. 3 and Table 2, the measurement accuracy of Z_{14} increases by 36.8% and 2.2% under conventional illumination and annular illumination, respectively, compared with the TAMIS technique.

The y-coma is extracted from the relative image displacements between marks C and D at multiple illumination settings. The measurement and calculation principle of y-coma is similar to that of x-coma. Therefore, the variation ranges of the sensitivities of Z_{8} and Z_{15} equal to that of Z_{7} and Z_{14}, respectively.

From the simulation above, it is clear that the measurement accuracy of coma obviously increases in this novel method, because the variation ranges of the sensitivities are enlarged. The measurement accuracy of coma increases by more than 25% under conventional illumination, and the measurement accuracy of primary coma increases by more than 20% under annular illumination, compared with the TAMIS technique.

## 4. Conclusion

A novel method for measuring the coma aberration of lithographic projection optics based on relative image displacements at multiple illumination settings has been proposed. A novel mark, which comprises two fine-segmented phase-shifting gratings and two sufficiently large binary gratings, has been presented. The coma aberration can be calculated from the relative image displacements between the phase-shifting gratings and the binary gratings at multiple illumination settings, using the least square method. From the theoretical analysis, it is obvious that the measurement accuracy of coma can be improved by this method, because the phase-shifting gratings are more sensitive to the aberrations than the binary gratings, and the impact of distortion on displacements of aerial image can be eliminated. Using the lithographic simulator PROLITH, we have calculated the sensitivities corresponding to coma at a series of illumination settings and analyzed the measurement accuracy. According to the simulation results, the measurement accuracy of coma has increased by more than 25% under conventional illumination, and the measurement accuracy of primary coma has increased by more than 20% under annular illumination, compared with the TAMIS technique.

## Acknowledgements

This work was supported by a grant from the Key Basic Research Program of Science and Technology Commission of Shanghai Municipality NO. 07JC14056 and National Natural Science Foundation of China under Grant NO. 60578051. The authors would like to thank NERCLE (National Engineering Research Center for Lithographic Equipment, China) for the support of this work.

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