Co-axial spectroscopic snap-shot ellipsometry for real-time thickness measurements with a small spot size

Seung Woo Lee; Sin Yong Lee; Garam Choi; Heui Jae Pahk

doi:10.1364/OE.399777

1. Introduction

Recently, the thin-film process is taking up a large portion of the semiconductor and display process. In the thin-film process, the measurement of the thickness and refractive index plays an essential role in the process control and yield improvement. Since the importance of the thin-film process increases, a real-time thickness measurement technique is required. Also, as the size of the measurement target becomes smaller, the measurement technique having a smaller spot size is required.

Spectroscopic ellipsometry [1–4] is a well-known method that can characterize the thin-film with high accuracy and precision by measuring spectroscopic ellipsometry parameters ($\Delta$, $\psi$). Conventional spectroscopic ellipsometry is configured with mechanically or electrically modulated polarization components (rotating polarizer, liquid crystal, etc). It requires multi-shot measurement, which hinders real-time measurement, while modulating the polarization state of input and output light for ellipsometry measurement. Due to such limited performances in measurement speed, there always has been a demand for faster solutions. Also, conventional ellipsometry has an oblique incidence structure with a large incident angle. Because of this, the light is also obliquely projected and shows a large spot size. Therefore, despite many advantages, both the large spot size due to the oblique incidence optical structure and the long measurement time due to mechanically rotated or electrically activated polarization modulation are obstacles to the introduction of ellipsometry into the in-line manufacturing process [5–7].

There have been various studies to reduce spot size such as adopting an objective lens [8,9] in front of the sample. Those researches can effectively reduce the spot size. Imaging ellipsometry has also been proposed as a solution to the spot size problem [10–14]. Imaging ellipsometry has adopted the objective lens and replaced the detector with the CCD camera. Accordingly, the spot size has been reduced to the value of the CCD$'$s spatial resolution divided by the objective lens magnification. However, both approaches include a polarization modulator like the conventional dual rotating ellipsometer, they require a multi-shot sequence when measuring. Therefore, long measurement time remains to be solved. Even in imaging ellipsometry, the difficulty of obtaining spectroscopic signals, which have many advantages in ellipsometry analysis [15–17], is a major weakness of the imaging ellipsometry. Attempts to acquire spectral data of area have been made such as using a number of band-pass filters or variable spectral filters [18,19] to acquire a spectroscopic signal. Most of these methods are simply repeating the single wavelength ellipsometry measurement. Therefore, the measurement time and step increase in proportion to the resolution of wavelength. Other researches that reduce the spot size by configuring in co-axial hardware and measuring in snap-shot without polarization modulation called micro-ellipsometry [20–22] have been proposed. However this system needs additional hardware components (band-pass filters) and measurement steps to obtain spectral signals which are valuable in ellipsometry analysis.

In the early days of research to reduce the measurement tact time of ellipsometry, speeding up the rotation of synchronized compensator [23] was once regarded as the solution. This method has the advantage of using the conventional dual rotating compensator ellipsometer hardware structure but still has multi-shot measurement sequence. For the one-shot based spectroscopic ellipsometry, researches using multiple spectrometers were proposed [24,25]. This seems to be an intuitive solution to reduce the measurement tact time. But the problem of increased hardware cost due to the use of multiple spectrometers, as well as errors from alignment and calibration differences between the various spectrometers, remain. Next, spectroscopic snap-shot ellipsometry using multi-order retarding components such as thick retarder [26,27] or interferometer structure [28] was proposed. Those methods prove the possibility to measure the spectroscopic ellipsometry signal and thickness in one-shot by high-frequency modulation of spectral signals. But, in our knowledge limit, none of the one-shot or snap-shot based spectroscopic ellipsometry researches tried to solve the spot size issue.

As noted above, many types of research have been conducted to reduce the spot size and the measurement tact time of spectroscopic ellipsometry but none of them solved it simultaneously. In this paper, we propose spectroscopic snap-shot ellipsometry in a co-axial optical structure. Co-axial optical structure makes it possible to apply a high magnification objective lens, which significantly reduces the measurement spot size. For the snap-shot measurement, the spectroscopic ellipsometry signal is modulated into high-frequency in spectral-domain by the multi-order retarder. Specific incidence and azimuth angle are made in the co-axial optical system by measuring the modulated spectroscopic ellipsometry signal at the point of the back focal plane (BFP). Therefore, the spot on the sample is measured at a time, which can achieve the real-time measurement. Detailed analysis, calibration, and optimization process of the proposed method are explained in the following sections. The visible wavelength (420 $\sim$ 750 nm) was selected in the experiment for the verification of the method. Through the proposed method, the spectroscopic ellipsometric parameters and film thickness are measured in a single shot with the spot size diameter about 61.4 $\mu$m.

2. Method

2.1 Hardware configuration

The proposed spectroscopic snap-shot ellipsometer illustrated in Fig. 1 has a co-axial configuration with no moving parts and can measure the spectral signal by snap-shot using a single spectrometer. White LED lights ($I_{in}(\sigma )$) are collimated and pass through polarization state generator (PSG) stage consisted of a polarizer and multi-order retarder. Light passing through the PSG reflects on the beam splitter and incidents on the sample through the objective lens. The light reflected from the sample passes through the beam splitter and to the polarization state analyzer (PSA) stage, which is a linear polarizer. The light passing through the PSA goes through the relay lens, which forms the image of the back focal plane (BFP) of the objective lens at the color camera CCD plane. Before entering the color camera, the image of the BFP is split and formed at the spectrometer fiber plane by the second beam splitter. Here, due to the diameter of the spectrometer fiber, it receives only the specific point of the BFP, which corresponds to the spectral signal ($I_{out}(\sigma )$) of the specific incident angle and azimuth angle.

Fig. 1. Hardware Configuration

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2.2 Incidence and azimuth angle calibration

The relation between the position at the BFP of the objective lens and the incidence angle and the azimuth angle is described in Fig. 2. As illustrated, the light passing through the same point of the BFP proceeds with the same incident angle in the object plane. Let the coordinate of the light passing point at the back focal plane as ($\rho$, $\omega$) in polar coordinate representation, then the incident angle ($\theta$) and the azimuth angle of the plane of incidence ($\Omega$) are calculated as Eq. (1). (Here, maximum pupil radius $\rho _{max} = 2fNA$, $f$ is focal length, and $NA$ is the numerical aperture of the objective lens.)

(1)$$\theta = \sin^{-1}\left({\frac{\rho}{\rho_{max}} }\sin(\theta_{max}) \right) = \sin^{-1}\left({\frac{\rho}{\rho_{max}} } NA \right)\quad,\quad\Omega = \omega .$$

Fig. 2. (a) Relationship between the point at the BFP and the angle of incidence at the object plane. (b) Polarization and plane of incidence change along the point in BFP

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For the calibration of the angle of incidence and azimuth angle of the incidence light, the position of the spectrometer fiber in BFP needs to be determined. To this, additional LED light is illuminated from the spectrometer fiber so the position of it could be imaged as a point of BFP as Fig. 3(a) shows. The image of the illuminated BFP is captured by the review color camera. The radius and the center coordinate of fully illuminated BFP (Fig. 3(b)) and point illuminated by spectrometer fiber (Fig. 3(c)) are calculated by Hough transform circle detection and precise surface fitting edge detection algorithm [29,30]. Referring to Eq. (1), $\theta$ and $\Omega$ are calculated to 64.79$^\circ$ and 1.89$^\circ$.

Fig. 3. (a) The light path of point illumination by spectrometer fiber, (b) BFP in full illumination, (c) BFP in point illumination by spectrometer fiber

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Using this hardware structure, it is possible to acquire the oblique incidence light required for the ellipsometry analysis even in the coaxial optical system configuration. The incident angle and azimuth angle can be adjusted by moving the position of the spectrometer fiber using the micro stage while checking the location of the spectrometer fiber with the review camera in real-time. The incidence and azimuth angle calibration needs to be performed only once when setting the hardware.

2.3 Analysis method

The spectral signal ($I_{out}(\sigma )$) measured by the proposed hardware structure can be analyzed using the Mueller matrix of the PSG ($\mathbf {M}_{PSG}$), sample ($\mathbf {M}_{s}$), and the PSA($\mathbf {M}_{PSA}$) as Eq. (2). The superscript $T$ is a matrix transpose operator, $R(\Omega )$ is the rotation matrix, and the Mueller matrix of the sample ($\mathbf {M}_{s}$) is considered as an isotropic sample. The details of each Mueller matrix element are in the Appendix Eq. (17).

(2)$$I_{out}(\sigma) = \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix} \mathbf{M}_{PSA}\mathbf{R}(-(\Omega+\frac{\pi}{2}))\mathbf{T}_{m}\mathbf{M}_{s}\mathbf{R}(\Omega)\mathbf{M}_{PSG} \begin{bmatrix} I_{in}(\sigma) & 0 & 0 & 0 \end{bmatrix}^{T} .$$

Based on the phase of the multi-order retarder ($\phi$) and using the angle of polarization component ($P, C, A$), Eq. (2) is re-arranged as Eqs. (3) and (4). Considering the axial symmetry, the angle of the polarizer ($P$) and azimuthal angle of the plane of incidence ($\Omega$) are set to zero. The Mueller matrix component of the sample is represented as $M_{11} \sim M_{44}$. The Mueller matrix component of the sample can be calculated in matrix form by using the relations of $DC$, $\alpha _1$, $\beta _1$ in Eq. (4), which has rank 3 (3x4) matrix ($\mathbf {W}_{3\times 4}$). Here, the angle of polarization components ($P, C, A$) are selected as $P = 0, C = \pi /4$, and $A = \pi /4$ by optimization process detailed in the Appendix. C. Also, Eq. (3) can be rewritten as Eq. (5) using $\Delta$ , $\psi$ form.

(3)$$I_{out}(\sigma) = DC + \alpha_1\cos(\phi) + \beta_1\sin(\phi) .$$

(4)$$\begin{bmatrix} DC\\ \alpha_1\\ \beta_1 \end{bmatrix} = \frac{I_{in}(\sigma)}{8} \mathbf{W}_{3\times4} \begin{bmatrix} M_{11}\\ M_{12}\\ M_{33}\\ M_{34} \end{bmatrix}\quad, \mathbf{W}_{3\times4}= \begin{bmatrix} 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 0 & -2 \end{bmatrix} .$$

(5)$$I_{out}(\sigma) = \frac{I_{in}(\sigma)}{4}\left(1 - \cos(2\psi)\cos(\phi) - \sin(2\psi)\sin(\Delta)\sin(\phi) \right) .$$

Here, we use the channeled spectrum technique proposed at the snap-shot ellipsometry [26,27]. Assuming the spectral retardance ($\phi (\sigma )$) of the multi-order retarder as a linear function about the wavenumber ($\sigma = \frac {1}{\lambda }$), the Fourier transform ($\mathfrak {F}\left [ \right ]$) of the measured spectrum $I_{out}(\sigma )$ about wavenumber can be expressed as follows. Here the superscript $*$ is complex conjugate and the detailed equation derivation is given in the Appendix B.

(6)$$\phi(\sigma) = 2\pi L \sigma + \gamma(\sigma) \quad, \quad L = \frac{1}{2\pi}\frac{\partial\phi}{\partial\sigma} \quad, \quad \gamma(\sigma) \approx 0 .$$

(7)$$\begin{aligned} \mathfrak{F}\left[DC(\sigma) \right] = A_0(h)\quad,\quad\mathfrak{F}\left[\alpha_1(\sigma) \right] = & A_1(h)\quad, \quad \mathfrak{F}\left[\beta_1(\sigma) \right] = B_1(h)\quad . \end{aligned}$$

(8)$$\begin{aligned} \mathfrak{F}\left[I_{out}(\sigma) \right] = & A_0(h) + \frac{1}{2}\left(A_1(h-L) + A^*_1(-h-L)\right) + \frac{1}{2j}\left(B_1(h-L) - B^*_1(-h-L)\right) . \end{aligned}$$

Therefore, the $DC$, $\alpha _1$, and $\beta _1$ signals can be separated on the Fourier domain ($h$) by shifting the Fourier transformed signal to the slope ($L$) of the retardance of the multi-order retarder. Even if the spectral retardance of the multi-order retarder is non-linear ($\gamma (\sigma )\neq 0$), the DC signal can be separated from $\alpha _1$ and $\beta _1$ by multiplying the output intensity ($I_{out}(\sigma )$) by $e^{-j \phi }$ and then Fourier transforming it. For the separation of $A_0(h)$, $A_1(h)$, and $B_1(h)$ in the Fourier domain, we use the window function (apodization function) as in Eq. (9). Considering the spectral frequency shifting, the window size $H$ is set to $L/2$. Due to the spectral leakage phenomenon in the Fourier domain, signal distortion may occur at both ends of the calculated ellipsometric signals. To eliminate this, data near both ends of the calculated signal was removed [27].

(9)$$w(h) = \left\{\begin{matrix} 1-\frac{h^4}{H^4} & (|h| \leq H) \\ 0 & (|h| > H) \end{matrix}\right.\quad .$$

The analysis process of calculating the ellipsometric parameters ($\Delta$, $\psi$) from the measured spectrum signal ($I_{out}(\sigma )$) is summarized in Fig. 4. First, each of the measured spectrum $I_{out}(\sigma )$ and shifted spectrum $I_{out}(\sigma )e^{-j\phi (\sigma )}$ is transformed into the Fourier domain (h domain) through Fourier transform. Then, the Fourier coefficients $A_0(h)$, $A_1(h)$, and $B_1(h)$ are calculated by the windowing of the Fourier transform results. The calculated Fourier coefficients ($A_0(h)$, $A_1(h)$, $B_1(h)$) are transformed into spectral domain signals $DC$, $\alpha _1(\sigma )$, and $\beta _1(\sigma )$ by inverse Fourier transform. Finally, using the $W_{(3 \times 4)}$ matrix relation in Eq. (4), ellipsometric parameters of the sample ($\cos (2\psi )$ and $\sin (2\psi )\sin (\Delta )$) are calculated. After calculating the ellipsometric parameters of the sample, the thickness is searched by Levenberg-Marquardt non-linear multivariate regression algorithm to minimize the mean square error (MSE) in Eq. (10). Here, subscript $m$ and $t$ represent measured values and theoretically calculated values, respectively, $d$ is layer thickness, and $i$ is a wavenumber index with a total number $N$. The theoretical ellipsometry values are calculated using the well known Fresnel equation and transfer matrix method. The specifications (spectral reflectance, spectral transmittance, and spectral Mueller matrix, and spectral responses) of optical components are also considered when calculating the theoretical ellipsometry values.

(10)$$\chi^2(d) = \frac{1}{N-1} \sum_{i=1}^{N} \begin{bmatrix} \left \{ \cos(2\psi_{m}(\sigma_i)) - \cos(2\psi_{t}(\sigma_i,d))\right \}^2 + \\ \left \{ \sin(2\psi_{m}(\sigma_i))\sin(\Delta_{m}(\sigma_i)) - \sin(2\psi_{t}(\sigma_i,d))\sin(\Delta_{t}(\sigma_i,d)) \right \}^2 \end{bmatrix} .$$

Fig. 4. Analysis flowchart, simulation for 900 nm thickness SiO$_{2}$/Si layer

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3. Experiment and result

3.1 Hardware specification and experiment process

The detailed hardware component specification for the verification experiment of the proposed method is as follows. The optical system for the experiment was designed for the visible region, which is the easiest way to implement. A 5W white LED which has a visible spectrum (420 $\sim$ 750 nm, 1.33 $\sim$ 2.38 $\times$ 10$^{-3}$ nm$^{-1}$) as shown in Fig. 5(a) is used for the light source. Achromatic wire grid polarizer (WP25M-VIS from Thorlabs) was used for polarizer and analyzer, which is applicable in the wavelength range of 420 $\sim$ 700nm (1.33 $\sim$ 2.38 $\times$ 10$^{-3}$ nm$^{-1}$). For the multi-order retarder, 4092.3 $\mu$m-thick crystal quartz was used and it has a linear spectral retardance as shown in Fig. 5(b). The wavelength unit of Fig. 5 was indicated by wavenumber ($\sigma = 1/\lambda$, [nm$^{-1}$]), so it was possible to confirm the linearity (Eq. (6)) of the spectral retardance $\phi (\sigma )$ about wavenumber. This can be approximated as a linear relationship ($\phi (\sigma )\approx 25482\sigma - 34.452$). The alignment angle of the polarizer and the retarder was adjusted within 0.5 degrees. The characteristics of polarizing components (polarization axis of the polarizer, optical axis, and the spectral retardance of the multi-order retarder) were measured using a transmission type ellipsometer [5,31,32]. For an objective lens, a Nikon CF IC EPI Plan Apo 100x lens with NA 0.95 was used. By adopting this high magnification objective lens, the measurement spot size becomes approximately 61.4 $\mu$m. Here, the diffraction limited spot size can be achieved in the order of 0.71 $\mu$m at a central wavelength of 550 nm according to the Rayleigh criterion [33,34]. For a review camera, Basler’s ace 1300-200uc was adopted. For the measurement, Maya 2000 pro spectrometer was used and the fiber by Avantes corporation which has a 50 $\mu$m diameter was used. The experimental measurement time depends on the integration time of the spectrometer, which was set as 60 milli-seconds. In order to confirm that the method proposed in this study does not have any problem in measuring a wide range of various thicknesses, a total of 13 different samples of SiO$_{2}$ thin-film on the Si substrate having a thickness from 10 nm to 1500 nm were used for the measurement. To evaluate the thickness measurement performance of the proposed method, a wide range of thickness values are tested. The thickness of each sample was compared with the measurement result by a commercially available ellipsometer (Horiba UVISEL). Each sample was measured 30 times repeatedly by the proposed method.

Fig. 5. (a) Spectral intensity of light source $I_{in}(\sigma )$, (b) Spectral retardance $\phi (\sigma )$ of multi-order retarder (Quartz)

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3.2 Measurement result

To verify the accuracy of the ellipsometric parameters calculated in the proposed thickness measurement process, the spectrum signals, and the calculated spectral ellipsometric parameters ($\cos (2\psi )$ , $\sin (2\psi )\sin (\Delta )$) are shown in Fig. 6 and Fig. 7. By plotting the Fig. 6 about wavenumber unit ($\sigma$, [nm$^{-1}$]), the spectral modulation made by retarder ($\cos (\phi ), \sin (\phi )$) appears to be the equal interval periodicity in the measured spectrum. In order to eliminate the effect of the spectral leakage phenomenon, the ellipsometric parameter used in the thickness search was in the range of 500 $\sim$ 700 nm (1.42 $\sim$ 2.0 $\times$ 10$^{-3}$ nm$^{-1}$). In the figure, the samples of nominal thickness 100, 500, 900, and 1500 nm out of 13 different thickness samples are represented. The thickness was calculated by the Levenberg-Marquardt method and the least-squares error between the theory measurement signal was less than 0.003. Converting the ellipsometric parameters ($\cos (2\psi ), \sin (2\psi )\sin (\Delta )$)) into the $\psi (\sigma )$ and $\Delta (\sigma )$ form (degree unit), the maximum mean squared error between the theory and the measured one was 7.795 degrees. The precision of the ellipsometric parameters, which was calculated as three times to the maximum value of standard deviation for each wavelength of the $\psi (\sigma )$ and $\Delta (\sigma )$ signals that is measured repeatedly for 30 times, was 0.2397 degrees. The average, standard deviation (3STD) of 30 times repeated measurements, the absolute difference, and the relative difference ($\%$) of thickness measurement result compared to the reference ellipsometer is illustrated in Table 1. According to the measurement result for the thickness range from 13nm to 1.5$\mu$m, the maximum difference was 2.12 nm in all thickness ranges, and the minimum difference was 0.1 nm compared to the reference ellipsometry measurement result. The relative difference was up to 3.65$\%$. In terms of repeatability, all of the measurements showed repeatability of less than 0.1 nm. Through this, it was confirmed that the spectral ellipsometric parameters and the thickness can be measured using the proposed method.

Fig. 6. Measured spectral intensity $I_{out}(\sigma )$ of each sample

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Fig. 7. Measured and theoretical $\cos (2\psi )$ , $\sin (2\psi )\sin (\Delta )$ of each sample

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Table 1. Thickness measurement result by reference ellipsometer (commercial ellipsometer) and the proposed method. Difference term represents the absolute (relative $\%$) thickness differences between commercial ellipsometer and the proposed method.

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4. Discussion

Through the measurement results (Fig. 6, Fig. 7, and Table 1), the proposed snap-shot method is confirmed to measure spectral ellipsometric parameters and can measure the thickness accurately and precisely. Though it has a small least-squares error (at 0.003 level in $\cos (2\psi ), \sin (2\psi )\sin (\Delta )$ and 7.795 degrees in $\psi (\sigma )$ and $\Delta (\sigma )$) between the theory and the measured values, the following should be considered as the cause of the difference of ellipsometric measurement.

As the light receiving part of the spectrometer fiber is not in a single point but an area of 50 $\mu$m diameter, an overlap of multiple incident angle signals occurs. In this study, the theoretical ellipsometric parameters were modeled by setting the angle of incidence ($\theta$) and azimuth angle of the plane of incidence ($\Omega$) based only on the center coordinates of the region shown in Fig. 3. (c). Considering the spot size of spectrometer fiber to 50$\mu$m, the actual incidence angle ranges from 63.11$^\circ$ to 66.57$^\circ$, and the actual azimuthal angle ranges from 1.07$^\circ$ to 2.70$^\circ$. Therefore, the error can be reduced by averaging the multi-angle incidence signal of the corresponding area for the theoretical intensity signal $I_{out}(\sigma )$ by integrating the intensity signal $I_{out}(\sigma ,\rho ,\Omega )$ in the actual spot area as explained in Eq. (11)

(11)$$I_{out}(\sigma) = \int_{\rho,\Omega \in spot } I_{out}(\sigma,\rho,\Omega)\rho d\rho d\Omega \quad.$$

Non-ideal characteristic of optical components also generates the difference in spectroscopic ellipsometry signals. Especially, the objective lens can make birefringence effect induced from mount stress and intricate combination of lens [35,36]. These birefringence effect at the peripheral of the back focal plane is greater than at the center. Since the peripheral area is used in the proposed method, the lens can affect the ellipsometric parameters by 0.1 for the $\cos (2\psi )$ and by 0.2 for the $\sin (2\psi )\sin (\Delta )$ at wavelength 488 nm according to the measurement results of previous research [36]. And these effects have different values for each wavelength. Therefore, if the lens birefringence is calibrated beforehand, more accurate measurement results will be obtained.

Next, the spectral retardance change of the multi-order retarder with temperature change can be a source of the erroneous signal. This error has been mentioned in previous studies [27,37–39]. The effects of this retardance error are as follows. The difference in spectral retardance ($\phi (\sigma )$) due to temperature change can be expressed as follows. Here, the prime symbol ($'$) denotes that it contains a retardance error.

(12)$$\phi'(\sigma) = \phi(\sigma) + \delta\phi(\sigma) .$$

By Eq. (5), the output intensity considering the error of spectral retardance is expressed as follows.

(13)$$\begin{aligned} I'_{out}(\sigma) & = \frac{I_{in}(\sigma)}{4}\left(1 - \cos(2\psi)\cos(\phi') - \sin(2\psi)\sin(\Delta)\sin(\phi') \right) \\\\ & = \frac{I_{in}(\sigma)}{4}\left(1 - \cos(2\psi)\cos(\phi+\delta\phi) - \sin(2\psi)\sin(\Delta)\sin(\phi+\delta\phi) \right) \\\\ & = \frac{I_{in}(\sigma)}{4}( 1 - \left( \cos(2\psi)\cos(\delta\phi) + \sin(2\psi)\sin(\Delta)\sin(\delta\phi)\right) \cos(\phi) \\ & \quad \quad \quad \quad \quad -(\sin(2\psi)\sin(\Delta)\cos(\delta\phi) - \cos(2\psi)\sin(\delta\phi))\sin(\phi) ). \end{aligned}$$

Therefore, the ellipsometric parameters ($\cos (2\psi '), \sin (2\psi ')\sin (\Delta ')$) calculated using the erroneous light ($I'_{out}(\sigma )$) has the following relationship with the correct ellipsometric parameter ($\cos (2\psi ), \sin (2\psi )\sin (\Delta )$).

(14)$$\begin{aligned} \cos(2\psi') & = \cos(2\psi)\cos(\delta\phi) + \sin(2\psi)\sin(\Delta)\sin(\delta\phi) \\\\ \sin(2\psi')\sin(\Delta') & = \sin(2\psi)\sin(\Delta)\cos(\delta\phi) - \cos(2\psi)\sin(\delta\phi) . \end{aligned}$$

Assuming that the error of spectral retardance is small ($\delta \phi \ll 1, \cos (\delta \phi ) \approx 1, \sin (\delta \phi )\approx \delta \phi$), the above Eq. (14) can be approximated as follows.

(15)$$\begin{aligned} \cos(2\psi') & \approx \cos(2\psi) + \sin(2\psi)\sin(\Delta)\delta\phi \\\\ \sin(2\psi')\sin(\Delta') & \approx \sin(2\psi)\sin(\Delta) - \cos(2\psi)\delta\phi . \end{aligned}$$

This means that the two ellipsometric parameters linearly affect each other, if there is a small error in the spectral retardance. In addition, using this relationship, if it is possible to measure the spectral retardance error, the correct ellipsometric parameters can be calculated as follows.

(16)$$\begin{bmatrix} \cos(2\psi) \\ \sin(2\psi)\sin(\Delta) \end{bmatrix} = \frac{1}{1+(\delta\phi)^2} \begin{bmatrix} 1 & -\delta\phi \\ \delta\phi & 1 \end{bmatrix} \begin{bmatrix} \cos(2\psi') \\ \sin(2\psi')\sin(\Delta') \end{bmatrix} .$$

The spectral retardance error is hard to be quantified in our hardware configuration. However, if it is measured by modifying the hardware according to the method mentioned in the previous studies [27,37,38], ellipsometric parameters can be calibrated through the above relationship.

However, despite these errors, the measured thickness value and the reference value (Fig. 8) shows linearity and $R^2$ as 0.9999, indicating the high linearity. As a result, the possibility of spectroscopic ellipsometry measurement for the thickness and the accuracy of the thickness measurement was verified.

Fig. 8. Comparison of measured values between the reference (commercial ellipsometer) and the proposed methods

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5. Conclusion

In this study, spectroscopic snap-shot ellipsometry of co-axial structure for small spot size and real-time thickness measurement was proposed. The proposed scheme is configured in a co-axial structure without moving parts or modulation devices. By the co-axial optical structure, the high magnification (100x) objective lens can be applied, and therefore, has an advantage that the spot size can be reduced to a diameter of about 61.4 $\mu$m. Only by the one-shot spectrum measurement without hardware drive or extra movement, the spectroscopic ellipsometry parameters ($\cos (2\psi ), \sin (2\psi )\sin (\Delta )$) are measured, which significantly reduce the measurement tact time. The verification of the proposed method was done in the visible wavelength region (420 $\sim$ 750 nm, 1.33 $\sim$ 2.38 $\times$ 10$^{-3}$ nm$^{-1}$). This snap-shot measurement method achieves the real-time measurement for the spot of the sample. The calculated spectroscopic ellipsometry parameters have a difference from the theoretical value less than 0.003. In $\psi (\sigma )$ and $\Delta (\sigma )$ form (degree unit), the maximum mean squared error between the theory and the measured one was 7.795 degrees and the precision was 0.2397 degrees. For the wide range of various thickness samples, the thickness values measured by the proposed method show the error less than 2.12 nm, maximum repeatability less than 0.1 nm, and the linearity $R^2$ value as 0.9999 when compared to the commercial ellipsometer result. These experimental results imply that the method proposed in this study has accuracy and precision performance comparable to conventional ellipsometry. In conclusion, the proposed method was able to solve the issues of large spot size and long measurement time of conventional ellipsometry while maintaining the accuracy and precision level.

Appendix A: Mueller matrix

The detailed Mueller matrices of hardware components are as follows. $\tilde {r}_p$ and $\tilde {r}_s$ represents the reflectance of the P wave component and the S wave component, respectively.

(17)$$\begin{aligned} \mathbf{ R}(\Omega) = & \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(2\Omega) & \sin(2\Omega) & 0 \\ 0 & -\sin(2\Omega) & \cos(2\Omega) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\quad,\quad \mathbf{T}_m = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}\quad,\\ \mathbf{M}_{s} = & \frac{\tilde{r}_p \tilde{r}^*_p + \tilde{r}_s \tilde{r}^*_s}{2} \begin{bmatrix} 1 & -\cos(2\psi) & 0 & 0 \\ -\cos(2\psi) & 1 & 0 & 0 \\ 0 & 0 & \sin(2\psi)\cos(\Delta) & \sin(2\psi)\sin(\Delta) \\ 0 & 0 & -\sin(2\psi)\sin(\Delta) & \sin(2\psi)\cos(\Delta) \end{bmatrix}\quad,\\ \mathbf{M}_{PSG} = & \mathbf{R}(-C) \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos(\phi) & \sin(\phi) \\ 0 & 0 & -\sin(\phi) & \cos(\phi) \end{bmatrix} \mathbf{R}(C) \mathbf{R}(-P) \frac{1}{2} \begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \mathbf{R}(P)\quad,\\ \mathbf{M}_{PSA} = & \mathbf{R}(-A) \frac{1}{2} \begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \mathbf{R}(A) . \end{aligned}$$

Appendix B: Fourier transform relation

The derivation of the Fourier coefficient through the Fourier transform of the measured spectrum is as follows. The latter $j$ represents a complex number $\sqrt {-1}$.

(18)$$\phi(\sigma) = 2\pi L \sigma + \gamma(\sigma) \quad, \quad L = \frac{1}{2\pi}\frac{\partial\phi}{\partial\sigma} \quad, \quad \gamma(\sigma) \approx 0 .$$

(19)$$\begin{aligned} \mathfrak{F}\left[DC(\sigma) \right] = & \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}DC(\sigma)e^{-2\pi j h \sigma}d\sigma = A_0(h)\quad,\\ \mathfrak{F}\left[\alpha_1(\sigma) \right] = & A_1(h)\quad, \quad \mathfrak{F}\left[\beta_1(\sigma) \right] = B_1(h)\quad, \\\\ \mathfrak{F}\left[\alpha_1(\sigma)\cos(\phi(\sigma)) \right] = & \frac{1}{2}\left(\mathfrak{F}\left[\alpha_1(\sigma)e^{2\pi j L \sigma}\right] + \mathfrak{F}\left[\alpha_1(\sigma)e^{-2\pi j L \sigma}\right]\right) \\ = & \frac{1}{2\sqrt{2\pi}}\left( \int_{-\infty}^{\infty}\alpha_1(\sigma)e^{-2\pi j (h - L)\sigma}d\sigma + \int_{-\infty}^{\infty}\alpha_1(\sigma)e^{-2\pi j (h + L)\sigma}d\sigma \right) \\ = & \frac{1}{2}\left(A_1(h-L) + A_1(h+L)\right) =\frac{1}{2}\left(A_1(h-L) + A^*_1(-h-L)\right)\quad, \\\\ \mathfrak{F}\left[\beta_1(\sigma)\sin(\phi(\sigma)) \right] = & \frac{1}{2j}\left(\mathfrak{F}\left[\beta_1(\sigma)e^{2\pi j L \sigma}\right] - \mathfrak{F}\left[\beta_1(\sigma)e^{-2\pi j L \sigma}\right]\right) \\ = & \frac{1}{2j\sqrt{2\pi}}\left( \int_{-\infty}^{\infty}\beta_1(\sigma)e^{-2\pi j (h - L)\sigma}d\sigma - \int_{-\infty}^{\infty}\beta_1(\sigma)e^{-2\pi j (h + L)\sigma}d\sigma \right) \\ = & \frac{1}{2j}\left(B_1(h-L) - B_1(h+L)\right) =\frac{1}{2j}\left(B_1(h-L) - B^*_1(-h-L)\right)\quad, \\\\ \therefore \mathfrak{F}\left[I_{out}(\sigma) \right] = & A_0(h) + \frac{1}{2}\left(A_1(h-L) + A^*_1(-h-L)\right) + \frac{1}{2j}\left(B_1(h-L) - B^*_1(-h-L)\right) . \end{aligned}$$

Appendix C: Polarization hardware optimization

(20)$$\mathbf{W}_{3\times4}= \begin{bmatrix} 2+\cos(2A)(1+\cos(4C)) & 1+2\cos(2A)+\cos(4C) & -\sin(2A)\sin(4C) & 0\\ \cos(2A)(1-\cos(4C)) & 1-\cos(4C) & \cos(2A)\cos(4C) & 0\\ 0 & 0 & 0 & -2\sin(2A)\sin(2C) \end{bmatrix} .$$

From Eq. (3), the matrix $\mathbf {W}_{3\times 4}$ is written as Eq. (20) using the angle of polarization components ($P, C, A$). Since the rank of matrix $\mathbf {W}_{3\times 4}$ is three, only three of four unknowns of the sample Mueller matrix ($M_{11}$, $M_{12}$, $M_{33}$, $M_{34}$) can be obtained. Here, the only possible angle combinations for the rank becomes three, and that is $P = 0, C = \pi /4, 3\pi /4,\ldots$. In this condition, only $M_{11}$, $M_{12}$, and $M_{34}$ can be calculated. The optimal analyzer angle ($A$) is calculated to minimize the condition number (COND) and equally weighted variance (EWV) [40,41] of matrix $\mathbf {W}_{3\times 4}$. The condition number and equally weighted variance values are calculated for every analyzer angle ($A$) as in Fig. 9, and the optimum angle of the analyzer is calculated as $A = \pi /4 , 3\pi /4 ,\ldots$. In conclusion, a set of $P = 0, C = \pi /4$, and $A = \pi /4$ is found as an optimal condition for this research.

Fig. 9. EWV and condition number of matrix $\mathbf {W}_{3\times 4}$ along the analyzer angle

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Acknowledgments

This work was supported in part by the Brain Korea 21 Plus, the Institute of Engineering Research, Institute of Advanced Machines and Design at Seoul National University.

Disclosures

The authors declare no conflicts of interest.

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Sample No.	Nominal Thickness [nm]	Reference Thickness [nm]	Proposed Method [nm]
Sample No.	Nominal Thickness [nm]	Reference Thickness [nm]	Thickness	Difference	3 STD
1	10	13.96	14.46	0.50 ( 3.58 )	0.03
2	30	35.87	37.18	1.31 ( 3.65 )	0.04
3	50	56.62	57.18	0.56 ( 0.99 )	0.03
4	70	78.95	80.94	1.99 ( 2.52 )	0.02
5	90	102.04	101.44	0.60 ( 0.59 )	0.01
6	100	107.94	106.71	1.23 ( 1.14 )	0.01
7	110	123.17	123.27	0.10 ( 0.08 )	0.01
8	130	145.10	144.80	0.30 ( 0.21 )	0.02
9	190	210.81	209.34	1.47 ( 0.70 )	0.03
10	500	494.09	495.97	1.88 ( 0.38 )	0.03
11	600	622.36	624.48	2.12 ( 0.34 )	0.05
12	900	918.50	917.53	0.97 ( 0.11 )	0.02
13	1500	1541.91	1540.23	1.68 ( 0.11 )	0.04

Co-axial spectroscopic snap-shot ellipsometry for real-time thickness measurements with a small spot size

Abstract

1. Introduction

2. Method

2.1 Hardware configuration

2.2 Incidence and azimuth angle calibration

2.3 Analysis method

3. Experiment and result

3.1 Hardware specification and experiment process

3.2 Measurement result

4. Discussion

5. Conclusion

Appendix A: Mueller matrix

Appendix B: Fourier transform relation

Appendix C: Polarization hardware optimization

Acknowledgments

Disclosures

References

Cited By

Figures (9)

Tables (1)

Equations (20)

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