## Abstract

An interferometric method using an optical comb is proposed and realized to measure the total physical thickness of a multi-layered wafer even if the refractive index of each layer is not given. For a feasibility test, two-layered and three-layered silicon-on-glass wafers were chosen as samples and were measured. An uncertainty evaluation was conducted to estimate the performance capabilities of the proposed method. To verify the measured values, the wafers were also measured by a contact-type standard instrument. For the three-layered wafer, the total physical thickness distribution was determined in a selected area.

© 2017 Optical Society of America

## 1. Introduction

A silicon-on-insulator (SOI) wafer has a vertically stacked structure in which a thin silicon layer is bonded onto an insulating substrate. This structure can reduce the parasitic device capacitance and therefore reduce power consumption levels and improve the performance capabilities of semiconductor devices [1]. For these reasons, the demand for SOI wafers for microelectronics, image sensors, and MEMS devices has increased with the development and commercialization of wearable devices, smart devices, and Internet of Things technology [2]. Various materials can be selected for use as the insulating layer for SOI wafers, depending on the purpose of use, and SOI wafers can have various types of stacked structures depending on the bonding and manufacturing processes used.

The most important aspect during semiconductor manufacturing processing based on optical lithography is nonuniformity of the substrate thickness [3]. The wafer thickness and thickness distribution measurements can contribute to reduce the defect rate and improve the throughput. Thickness measurement techniques are classified into the two types of contact-type and non-contact-type methods. Contact-type methods measure the thickness by a simple means [4]. However, this method can physically damage sample surfaces by, for instance, scratching them, and they can also distort the thickness value due to elastic deformation when the stylus physically comes into contact with a specimen, especially when thin wafers are involved. Moreover, contact-type methods are associated with practical difficulties related to real-time inspections of the thickness and its distribution owing to the slow measurement speed of these methods. On the other hand, non-contact methods can rapidly determine the optical thickness without causing any damage. Among non-contact methods, the use of an optical interferometer is common because it allows the user to determine thickness values accurately. Because the physical thickness equals the optical thickness divided by the refractive index of the specimen, the pre-determined refractive index must be accurate enough to prevent degradation of the measurement precision in optical interferometry. More importantly, the exact determination of the refractive index is never a simple task owing to the dispersion of the specimen. Even if the refractive index of the specimen is easily obtained at various wavelengths from numerous look-up tables, it can differ according to the purity of the specimen and spectral distribution of the light source used. To solve these problems, several types of interferometric methods operating with an optical comb [5–14] or a wide spectral bandwidth source [15–19] for simultaneous measurements of the physical thickness, thickness profile, and group refractive index of a single-layered specimen were studied in the precedent researches.

In this paper, an optical interferometric means of measuring the total physical thickness of a multi-layered SOI wafer is proposed and demonstrated. The difference here is that the proposed method measures not a single-layered but a multi-layered object. Even if the refractive index value of each layer is not accurately known, the proposed method can determine the total physical thickness. Because SOI wafers are always polished after the bonding process, their total thickness and thickness distribution instead of the thickness values of each layer must be monitored to prevent defects and failures. For a feasibility test, two types of samples are chosen; one is a two-layered silicon-on-glass (SOG) wafer fabricated by an anodic bonding process and the other is a three-layered SOG wafer bonded by an adhesive layer. The total physical thickness values of the samples were measured by the proposed method. To estimate the measurement performance, an uncertainty evaluation was also conducted. More importantly, to cross-check the measurement results, the obtained values were compared with those measured by a contact-type calibrated standard instrument. In addition, the three-layered SOG wafer was measured in an area of 20 mm × 2 mm to ascertain the total thickness distribution. Through this work, the authors would like to demonstrate the technical possibility of measuring the total physical thickness of a multi-layered wafer composed of different materials whose optical properties are not accurately known.

## 2. Basic principle

A spectral-domain interferometer can measure optical path differences (OPDs) by calculating the periods of the interference spectrum. The interference spectrum is generated by a wide-spectral light source and is sampled by a spectrum analyzer. In the frequency domain, the interference spectrum, I(*f,l*), can be expressed by Eq. (1) in terms of the optical frequency (*f*), OPD (*l*), the spectral distribution of a light source (I_{0}(*f*)), and speed of light in a vacuum (*c*):

*c/l*in the frequency domain [6–13]. When the interference spectrum undergoes Fourier transformation, a peak corresponding to the period of

*c/l*is observed. More specifically, the peak is located at

*l/c*because the Fourier-transformed signal is shown in the time domain, which is the inverse of the frequency domain. Therefore, the OPD (

*l*) can be calculated from the location of the peak. To determine the location of the peak precisely, several well-known methods can be used, such as the phase-slope detection method, the center-of-gravity method, and a fitting method.

Figure 1 shows a schematic diagram of the total physical thickness measurement of the multi-layered specimen. The specimen has an n-layered structure, and each layer has a physical thickness of T_{i} and a refractive index of N_{i} (i = 1, 2,…, n). Here, L_{1} and L_{2} are the optical path lengths passing through air along the measurement path and the reference path, respectively. In Fig. 1, W_{1} is a beam transmitted directly through the specimen, W_{1}´ is a beam travelling along the measurement path in air, W_{2} is a beam transmitted after one-time reflection at the first and last surfaces of the specimen, and W_{3} is a beam travelling along the reference path. To determine the total physical thickness, the three OPDs of OPD_{1}, OPD_{2}, and OPD_{3}, as correspondingly shown in Eqs. (2)-(4), are required. Here, OPD_{1} and OPD_{3} are the OPDs between the reference path and the measurement path without and with the specimen, respectively. OPD_{2} is defined as the double-optical thickness of the specimen, which is created by W_{1} and W_{2} in Fig. 1. With OPD_{1}, OPD_{2}, and OPD_{3}, the total physical thickness (T* _{total}*) can be calculated easily by Eq. (5):

## 3. Experiments and results

Figure 2(a) shows the optical configuration of the measurement system for measuring the total physical thickness of a SOG wafer. The light emitted from the light source is delivered to the two paths of a reference path and a measurement path through an optical fiber, a collimation lens (CL), and a beam splitter (BS1) in sequence. The diameter of the collimated beam was 0.76 mm in terms of 1/e^{2} width. The reference and the measurement paths are defined as counter-clockwise and clockwise travelling paths from beam splitter 1 (BS1) to beam splitter 2 (BS2) through a mirror (M2) and another mirror (M1) in Fig. 2(a), respectively. An interference signal is created by combining two light beams travelling along the reference and measurement paths with detection using a spectrometer. The interferometer installed was a Mach-Zehnder type to suppress sensitivity on alignment or the parallelism of the specimen during movable measurements. All optical components were fixed on a ‘U’-shaped plate as one body, which stood vertically in this case. A two-axis scanning stage driven by two stepping motors was utilized to move the specimen to measure the thickness distribution or for positioning at a certain location. Figure 2(b) shows a photographical view of the experimental setup.

Figures 2(c) and 2(d) show cross-sectional images of a two-layered SOG wafer and a three-layered SOG wafer. Both cross-sectional images of two SOG wafer samples diced by using a wafer dicing machine were acquired through a commercial microscope system using 5x magnification. For the two-layered SOG wafer (Plan Optik AG) with a diameter of 150 mm, a thin silicon wafer with a nominal thickness of 50 μm is attached onto a thick glass substrate with a nominal thickness of 500 μm by an anodic bonding process. For the three-layered SOG wafer, silicon urethane fluoric acrylic adhesive in a gel state was spread onto a glass substrate (Corning Eagle XG) and a silicon wafer was then placed on the adhesive and cured by a UV light and heat. The nominal thickness values of the silicon wafer, the adhesive layer, and the glass substrate were 775 μm, 20 μm, and 700 μm, respectively. The three-layered SOG wafer was diced in dimensions of 30 mm × 3 mm for easy handling in our experiments.

For our experiments, a femtosecond pulse laser with a spectral bandwidth of 80 nm at a center wavelength of 1550 nm (M-Comb, Menlo Systems) was used as a light source. The interference spectra in a wavelength range of 1510 nm to 1590 nm were then obtained with 16384 sampling points using an optical spectrum analyzer (AQ6370C, Yokogawa). Because the period of the interference spectrum in the frequency domain contains an OPD according to Eq. (1), the spectrum in the wavelength domain was converted to that in the frequency domain. Due to the inverse relationship between the wavelength and the frequency, the irregular interval of the converted spectrum in the frequency domain was adjusted to an equal interval with the help of linear interpolation. For clear detection of the peak signal in the Fourier domain, the Gaussian window function was applied to the interpolated spectrum. The spectrum was zero-padded to enhance the measurement resolution of the OPD and then was Fourier transformed.

Figures 3(a)-3(c) present the interference spectra in the case of no sample, a two-layered SOG wafer, and a three-layered SOG wafer in the measurement arm, respectively. Similarly, Figs. 3(d)-3(f) show the amplitude signals in the Fourier domain in the case of no sample, a two-layered SOG wafer, and a three-layered SOG wafer, respectively. The OPD_{1} can easily be determined by peak A_{1} in Fig. 3(d). For the two-layered SOG wafer and the three-layered SOG wafer, the peaks for OPD_{2} and OPD_{3} in the Fourier domain were carefully selected with nominal values of the thickness and refractive index (N_{1} = 3.6, T_{1} = 50 μm, N_{2} = 1.5, and T_{2} = 500 μm for the two-layered SOG wafer and N_{1} = 3.6, T_{1} = 775 μm, N_{2} = 1.47, T_{2} = 30 μm, N_{3} = 1.5, and T_{3} = 700 μm for the three-layered SOG wafer), as several peaks were observed, as shown in Figs. 3(e) and 3(f), owing to multiple reflections between the surfaces of the layered structure. OPD_{2} and OPD_{3} for the two-layered SOG wafer and the three-layered SOG wafer correspond to peak B_{3} and peak B_{4} in Fig. 3(e) and peak C_{4} and peak C_{2} in Fig. 3(f), respectively. To determine the locations of the peaks precisely, the center-of-gravity method was applied to the sampling points only over the half-maximum of the selected peaks. Here, the other peaks of B_{1} and B_{2} in Fig. 3(e) represent for the OPDs of 2∙N_{1}∙T_{1} and 2∙N_{2}∙T_{2} for the two-layered SOG wafer, respectively. Similarly, the peaks of C_{1} and C_{3} in Fig. 3(f) correspond to the OPDs of 2∙N_{2}∙T_{2} + 2∙N_{3}∙T_{3} and 2∙N_{1}∙T_{1} for the three-layered SOG wafer.

Figure 4(a) shows the total physical thickness values of the two-layered SOG wafer for 20 repeated measurements at a stationary point. The average total physical thickness was 551.729 μm with a standard deviation of 57 nm. Because OPD_{1} was determined during an open path of the measurement arm, it was measured twice, before and after obtaining OPD_{2} and OPD_{3}, with the sample positioned in the measurement path. The average value of OPD_{1} was 1850.345 μm, which was used as a representative value for the calculation of the total physical thickness. Similarly, Fig. 4(b) shows the total physical thickness values of the three-layered SOG wafer for 20 repeated measurements at a stationary point. The average total physical thickness was 1497.461 μm with a standard deviation of 37 nm. OPD_{1} was also measured in the same manner used for the two-layered SOG wafer, and the average value of OPD_{1} was 1850.305 μm. Moreover, the total physical thickness distribution for the three-layered SOG wafer was measured using a two-axis scanning stage with an interval of 0.1 mm in a range of 20 mm (*x*) × 2 mm (*y*), as shown in Fig. 4(c).

An uncertainty analysis was performed based on Eq. (5) to estimate the measurement performance. The uncertainty when determining the OPD is derived from the uncertainty of the discrete Fourier transform (DFT) algorithm and the wavelength accuracy of the optical spectrum analyzer in use, the uncertainty of the measurement repeatability, and the uncertainty of the refractive index of air. The uncertainty of the DFT algorithm and the wavelength accuracy was estimated by a numerical simulation. The interference spectra were theoretically generated under sampling conditions exactly identical with the actual measurement conditions at a set of OPDs ranging from 1500 μm to 8000 μm with an interval of 100 μm, which covers all the OPD_{1}, OPD_{2}, and OPD_{3} of the two- and the three-layered SOG wafers used in our experiments. For the wavelength accuracy of 0.01 nm, disturbance of less than 0.01 nm was added randomly onto each wavelength of the theoretical interference spectrum. At each OPD, the difference between the ideally given OPD and the OPD determined using the DFT algorithm taking into consideration disturbances of the wavelength was calculated; this is defined as the OPD error, ΔOPD. According to the simulation results, the uncertainty for the DFT algorithm and the wavelength accuracy was estimated to be less than 1.24 × 10^{−5} for all OPDs used in terms of the relative error of ΔOPD/OPD. The uncertainty associated with measurement repeatability can be estimated by dividing the standard deviation of the repeatedly measured values by the square root of the number of measurement. The uncertainty of the refractive index of air can be estimated using the variations of the temperature, pressure, and relative humidity under laboratory conditions. The uncertainty value of the phase change on reflection in case of OPD_{2} was calculated to be about a few pm by applying Fresnel reflection coefficient to glass-air and silicon-air interfaces, which is extremely small enough to be negligible in the uncertainty budget in comparison with other uncertainty components. The corresponding expanded uncertainties for the two-layered SOG wafer and the three-layered SOG wafer were estimated to be 34 nm (*k* = 2) and 41 nm (*k* = 2). Tables 1 and 2 summarize the uncertainty components of the two wafers [20]. The coupling between OPD_{1}, OPD_{2}, and OPD_{3} was considered as the correlation term according to our previous work [8].

To verify the measurement results obtained by the proposed method, the total physical thicknesses of the identical samples were measured at a pre-determined point by a contact-type standard instrument. This instrument has traceability to a length standard with measurement uncertainty of 0.1 μm (*k* = 2) through reference gauge blocks which are precisely calibrated by a laser interferometer. According to the results of 20 consecutive measurements, the average total physical thicknesses of the two-layered SOG wafer and the three-layered SOG wafer were 551.714 μm and 1497.460 μm, respectively. Figure 5 shows comparative measurement results of the total physical thickness values of the two SOG wafers with their measurement uncertainties.

## 4. Summary

In this paper, we proposed and implemented an interferometer that can measure the total physical thickness of a multi-layered specimen at a high speed without refractive index information, unlike conventional interferometers. The proposed interferometer works based on spectral-domain interferometry using an optical comb of a femtosecond pulse laser and an optical spectrum analyzer. A two-axis precision stage driven by stepping motors used to scan the sample to measure the thickness distribution. To suppress sensitivity upon alignment or parallelism of the sample, the interferometer was installed in a transmission-type configuration.

An uncertainty evaluation of the proposed method was conducted to estimate its measurement performance. The uncertainty for determining the OPD was derived from the uncertainty of the DFT algorithm and the wavelength accuracy of the optical spectrum analyzer in use, the uncertainty of the measurement repeatability, and the uncertainty of the refractive index of air. Among the uncertainty components, the most influential value was found to be the uncertainty of the DFT algorithm and the wavelength accuracy of the optical spectrum analyzer. The total average thicknesses of the two-layered and the three-layered SOG wafers were measured and found to be 551.729 μm with measurement uncertainty of 34 nm (*k* = 2) and 1497.461 μm with measurement uncertainty of 41 nm (*k* = 2), respectively. For the three-layered SOG wafer, the thickness distribution was also measured in an area of 20 mm × 2 mm. To cross-check, the same SOG wafers were also measured at a pre-determined point using a contact-type standard instrument. The comparative measurement results show that the measured values obtained by the proposed method and the standard instrument were well within their measurement uncertainties with regard to each other. Therefore, the proposed method can measure the total physical thickness and thickness distribution with uncertainty of less than 50 nm without refractive index information. This method is also expected to be utilized for measuring the total physical thickness of laminated structures for high-quality display panels as well as SOG wafers for the next-generation semiconductors.

## Funding

Development of Application Technologies of Physical Measurement Standards (17011023); Development of Next Generation Converging Measurement Standards and Technology for Dimensional Metrology (17011024).

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