## Abstract

We have proposed a modified method to improve the measurement uncertainty of the geometrical thickness and refractive index of a silicon wafer. Because measurement resolution based on Fourier domain analysis depends on the spectral bandwidth of a light source directly, a femtosecond pulse laser having the broad spectral bandwidth of about 100 nm was adopted as a new light source. A phase detection algorithm in Fourier domain was also modified to minimize the effect related to environmental disturbance. Since the wide spectral bandwidth may cause a dispersion effect in the optical parts of the proposed interferometer, it was considered carefully through numerical simulations. In conclusion, the measurement uncertainty of geometrical thickness was estimated to be 48 nm for a double-polished silicon wafer having the geometrical thickness of 320.7 μm, which was an improvement of about 20 times that obtained by the previous method.

© 2012 OSA

## 1. Introduction

A silicon wafer is one of the key elements of semiconductor integrated circuits (ICs). According to recent trends in compact sized electronic devices, thinner wafers have been necessary to improve the degree of integration and functionality of ICs. Moreover, 3D semiconductor packaging has been considered the promising next generation manufacturing process of semiconductors. It can realize a much higher degree of integration and multi-functions in a single packaged device. To build up a 3D semiconductor packaged device, the geometrical thickness of each wafer should be measured precisely before stacking the wafers vertically.

A well-known method to measure the geometrical thickness of a silicon wafer is to use two separate contact or noncontact sensors on both sides of the wafer. The difference between the two readings gives the geometrical thickness of the wafer [1,2]. For higher measurement precision, several research works based on optical interferometry have been proposed and realized [3–5]. In 2010, KRISS also proposed and demonstrated a novel measurement method for extracting the geometrical thickness from the optical thickness using the optical comb of the mode-locked pulse laser [6]. This method had advantages, including high-speed measurement, separate determination of the refractive index and the geometrical thickness in a single operation, and traceability to the length and time standards. Measurement uncertainty was estimated to be about 1 μm, which can restrict potential applications requiring higher accuracy.

In this paper, we have suggested and realized an improved method to achieve higher measurement accuracy, about 20 times better than that obtained by the previous research work [6]. According to the uncertainty evaluation of our previous work, the most dominant factor was uncertainty related to the determination of the optical path difference based on the discrete Fourier transform. That difference theoretically depends on the spectral bandwidth of the light source. To improve the resolution in the Fourier domain, therefore, a femtosecond pulse laser having the spectral bandwidth 10 times broader than that used in the previous work was adopted. Also, a phase detection algorithm was modified to minimize the effect of environmental disturbances. In addition, the dispersion effect was studied through numerical simulations because it may arise due to the wide spectral bandwidth of the light source. Finally, the combined uncertainty of the geometrical thickness of the double polished silicon wafer was improved more than 20 times than that obtained by the previous method.

## 2. Basic principle

A spectrum of interference signal corresponding to an optical path difference, *L* can be expressed as Eq. (1),

*I*(

*f*,

*L*) is the intensity of interference spectrum,

*I*

_{0}(

*f*) is the spectrum of the light source,

*f*is the optical frequency,

*c*is the speed of light, and

*φ*(

*f*,

*L*) is the phase to be measured. After taking the DFT (discrete Fourier Transform) of

*I*(

*f*,

*L*), one peak in positive time domain can be found at the position of

*L*/

*c*, which represents the period of the interference signal in the spectral domain. The time resolution in Fourier domain, Δ

*t,*depends on the spectral bandwidth of the light source,

*N∙*Δ

*f*, which is given bywhere Δ

*f*is the sampling interval of the interference spectrum,

*N*is the sampling number. A 10 times larger spectral bandwidth makes the time resolution 10 times higher, which leads to improvement of the combined uncertainty. To get the phase information corresponding to the optical path difference

*L*, only the peak located in positive time domain is selected with a sampling window, and then inverse-Fourier transformed. The phase

*φ*(

*f*,

*L*) can be extracted by taking the imaginary part of the logarithmic function of the inverse Fourier transform result,

*I'*(

*f*,

*L*) like Eq. (3).

Finally, the optical path difference, *L* can be determined through the linear fit of *φ*(*f*, *L*) as shown in Eq. (4).

## 3. Experiments and uncertainty evaluation

Figure 1
shows the optical layout of the proposed system for measuring the geometrical thickness, *T*, and refractive index, *N*, of a silicon wafer using a femtosecond pulse laser having a spectral bandwidth of about 100 nm at the central wavelength of 1550 nm. The mode spacing was extended from 250 MHz to 50 GHz through a Fabry-Perot filter for efficient detection of the individual comb modes using a conventional optical spectrum analyzer (OSA). Figure 2(a)
shows a spectrum of the light source in the full range, which is 10 times larger than the spectral bandwidth of a light source, ranging from 1535 nm to 1545 nm, of the previous work [6]. Figure 2(b) shows the mode spacing of the optical comb after the Fabry-Perot filter in Fig. 1. The interferometer part was installed as Michelson’s type. It has two paths, of Ray 1 and Ray 2. Similar to the procedure of the previous work, the optical path difference of *L*_{B} + *T* + *L*_{C} - *L*_{A} (≡*L*_{1}) was measured in Ray 1 before inserting the silicon wafer. Then two optical path differences, of *N*·*T* (≡*L*_{2}) and *L*_{B} + *N*·*T* + *L*_{C} - *L*_{A} (≡*L*_{3}), were obtained in Ray 2 with the silicon wafer in a measurement arm. In this work, the optical thickness of the silicon wafer (*L*_{2}) was measured, instead of two additional optical path differences used in the previous work, *L*_{B} - *L*_{A} and *L*_{B} + *N*·*T* - *L*_{A}. Unlike *L*_{1} and *L*_{3}, *L*_{2} is very sensitive to wafer alignment, as in the previous work. But, once the wafer surface is aligned at the right angle to the beam direction, the *L*_{2} measurement can be more stable than the individual measurements of *L*_{B} - *L*_{A} and *L*_{B} + *N*·*T* - *L*_{A} of the previous work because of a common path configuration. As a result, the geometrical thickness, *T*, and the refractive index, *N*, of the silicon wafer can be determined from the obtained three optical path differences *L*_{1}, *L*_{2}, and *L*_{3}, which are given as

Figure 3
shows the Fourier transform results of the two interference spectra of Ray 1 and Ray 2 in Fig. 1, which were obtained by using the OSA with 2^{15} sampling points and a sampling resolution of 0.003 nm in the spectral range of 1500 nm to 1600 nm in wavelength. As mentioned before, to make the interference of *L*_{2} dominant, the optical intensity of the reference arm was adjusted by tilting the reference mirror, M_{1} in Fig. 1 before inserting the silicon wafer. By reducing the optical intensity of the reference arm, an interference term having information of *L*_{2} became dominant to almost 60% of *L*_{3} in terms of peak amplitude, while *L*_{B} - *L*_{A} and *L*_{B} + *N*·*T* - *L*_{A} became relatively weak. The optical intensity of the reference arm was reduced to about 10% of the maximum intensity, which was measured by a conventional optical power meter. In addition, the peak for *L*_{2} also could be recognized practically by blocking the reference path and the *L*_{C} path in Fig. 1.

In Fig. 3(a), a single peak, which contains the information about optical path difference *L*_{1}, was clearly detected at the position of about 5.9 ps. Also, two peaks, which are related to optical path differences *L*_{2} and *L*_{3}, can be identified at the position of about 7.8 ps and 8.8 ps respectively in Fig. 3(b). For data selection through a sampling window for each peak before inverse Fourier transform, 5 data points were symmetrically selected at the center of the peak data position. Then every other data except these sampled data were converted to zero. According to Eq. (3) and Eq. (4), *L*_{1}, *L*_{2}, and *L*_{3} were determined by fitting phase *φ* according to wave vector *k* linearly, as shown in Fig. 4
. The slopes for *L*_{1}, *L*_{2}, and *L*_{3} were 1.777 × 10^{-3} rad·m, 2.323 × 10^{-3} rad·m, and 3.459 × 10^{-3} rad·m, respectively. From these values, the geometrical thickness, *T*, and the refractive index, *N*, of the silicon wafer were determined by using Eq. (5) and Eq. (6).

Table 1
shows the measurement results of 10 repeated measurements. The averaged *T* and *N* of the silicon wafer were measured to be 320.699 μm and 3.621, respectively. Also, the standard deviation of *T* was calculated to be 26.7 nm, which is an improved result by about one order of magnitude in comparison with the precedent task.

The measurement uncertainty of the geometrical thickness, *T*, of a silicon wafer was evaluated according to ISO/IEC 98-3 (GUM) as shown in Table 2
[7]. The dominant uncertainty factor was related to the DFT algorithm. To estimate the uncertainty, first of all, spectra having path differences of 0.1 mm to 2 mm with a step of 0.1 mm were generated, ideally with the same conditions in each of the experiments, which are 2^{15} sampling points and a sampling resolution of 0.003 nm, in the spectral range of 1500 nm to 1600 nm in wavelength. The ideal values of OPD in estimation of uncertainty related to the DFT algorithm were selected to fully cover the actual measurement values of three OPDs shown in Table 1. Then, the differences between the ideal values and the calculated values obtained by the DFT algorithm in use were obtained, which were almost 0.012% for the whole path differences of 0.1 mm to 2 mm. Because the uncertainty depends on OPD, the uncertainty related to the DFT algorithm for *L*_{3} was almost double that for *L*_{1}.

Uncertainty for measurement repeatability came from standard deviations of each of the OPD obtained by 10 repeated measurements. From Table 1, standard deviations of *L*_{1}, *L*_{2}, and *L*_{3}, were 35 nm, 50 nm, and 36 nm, respectively. By dividing the square root of the measurement number into the standard deviations, the uncertainties related to measurement repeatability for *L*_{1}, *L*_{2}, and *L*_{3} could be estimated as 11 nm, 16 nm, and 11 nm, respectively.

Uncertainty for the refractive index of air was roughly about 10^{-6} under general laboratory conditions, which was a minor factor because of the short OPDs. Moreover, for *L*_{2}, the uncertainty was not considered because of common path interferences between front and back surfaces of the silicon wafer. Wavelength uncertainty came from the wavelength accuracy of the OSA in use. According to the technical performance, the wavelength uncertainty of the OSA was given as 0.01 nm in the wavelength range of our use.

Since the spectral bandwidth of the femtosecond pulse laser was wider than that of the mode-locked pulse laser in our previous work, a numerical simulation for studying the dispersion effect of the silicon wafer was performed. The simulated interference spectra for the no dispersion case were generated with *T* of 320.7 μm and *N* of the silicon wafer being a constant value at a center wavelength. In the case of dispersion, the *N* of the silicon wafer was determined by an experimental equation [8]. The spectral bandwidth and sampling resolution were chosen as the same values used in the experiments. According to the simulation results, the dispersion effect caused the shift of peaks for *L*_{2} and *L*_{3} in Fourier domain as shown in Fig. 5
. The peaks for *L*_{2} and *L*_{3} were shifted the same amount, about 2.8 × 10^{-13} s, which corresponds to 42 μm in the optical path. It was too large to ignore in general cases. However, in this work, the dispersion effect could be canceled out according to Eq. (5). Therefore, it was proved that the proposed method was significantly unaffected by the dispersion effect of the silicon wafer. Finally, the uncertainties for *L*_{1}, *L*_{2}, and *L*_{3}, *u*(*L*_{1}), *u*(*L*_{2}), and *u*(*L*_{3}), were estimated to be 108 nm, 140 nm, and 208 nm, respectively.

The combined uncertainty of the *T*, *u*(*T*) can be expressed as

*i*and

*j*are integer numbers and

*u*(

*L*,

_{i}*L*) is a covariance between

_{j}*L*and

_{i}*L*, which can be estimated by use of Eq. (8) using a correlation coefficient,

_{j}*r*(

*L*,

_{i}*L*).

_{j}The correlation term, the second term of Eq. (7) was estimated to be 269 nm with the correlation coefficient of 0.99. Therefore, the combined uncertainty of the *T* of the silicon wafer, *u*(*T*) was determined to be 48 nm (*k* = 1) when measuring a silicon wafer having a *T* of 320.7 μm. The uncertainty of refractive index *N* is easily calculated from that of *T* using Eq. (6).

## 4. Conclusion

A modified spectral resolved interferometer for measuring the geometrical thickness and refractive index of a silicon wafer simultaneously was suggested and demonstrated. It was realized by using a femtosecond pulse laser having spectral bandwidth of about 100 nm. A phase detection algorithm in Fourier domain was also modified to minimize the effect related to environmental disturbance. Since the wide spectral bandwidth may cause a dispersion effect in the optical parts of the proposed interferometer, it was considered carefully through numerical simulations. In conclusion, the measurement uncertainty of geometrical thickness was estimated to be 48 nm (*k* = 1) for a double-polished silicon wafer having the geometrical thickness of 320.7 μm, which was an improvement of about 20 times that obtained by the previous method. It is expected to be used as an alternative standard metrological method for precision calibration of the geometrical thickness of a silicon wafer.

## Acknowledgement

This work was supported in part by the National Program: Development of Application Technologies of Physical Measurement Standards (2012), KRISS.

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