We have proposed and demonstrated a novel method that can determine both the geometrical thickness and refractive index of a silicon wafer at the same time using an optical comb. The geometrical thickness and refractive index of a silicon wafer was determined from the optical thickness using phase information obtained in the spectral domain. In a feasibility test, the geometrical thickness and refractive index of a wafer were measured to be 334.85 μm and 3.50, respectively. The measurement uncertainty for the geometrical thickness was evaluated as 0.95 μm (k = 1) using a preliminary setup.
©2010 Optical Society of America
Silicon is an indispensable element of, and a foundation layer in, semiconductor wafer manufacturing. It is essential to know the geometrical thickness of a silicon wafer accurately for quality control and process management of the manufacturing process. In the emerging field of semiconductor packaging, which is realized by stacking multiple interconnected wafers, the geometrical thickness of a silicon wafer needs to be controlled as one of the most important parameters, because a variation in the thicknesses of each wafer layer can lead to serious problems, such as disconnections or undesired connections between chips, circuits, and wafers.
In general, the geometrical thickness can be measured using two types of metrological technique: contact and noncontact. Contact-type measurements use a stylus, and this is the most widely used method because of its ease of operation. However, it can damage the polished surface of a silicon wafer, and can scratch the surface. Noncontact-type measurements have been employed as an alternative method to avoid damage caused by mechanical contact. One of the noncontact methods uses dual focused laser lights on the front and back surfaces, based on the confocal principle. The geometrical thickness is measured by detecting the focusing positions on both surfaces. The repeatability of such measurements is reported to be < 1 μm in the range 30 μm to 10 mm . Similarly, optical interferometers have also been adopted instead of dual focusing lenses for high precision measurements of wafer thickness . Although interferometry is a sensitive displacement measuring technique, rigorous alignment is required for developing its strong point, its high precision. In the same manner, capacitive sensors installed on the front and rear sides of a wafer can be utilized by monitoring the variation in an induced current. Such methods using dual detectors have the advantages of a simple setup and easy handling. However, they need careful alignment of the dual detectors. They need to be isolated from both electromagnetic and background noise, because the geometrical thickness is derived from other physical properties, such as the current, voltage, and intensity distribution in the case of noncontact methods, except for interferometry. Ultrasonic wave inspection and X-ray imaging have also been used for this application, but it has many practical difficulties because of the harsh handling conditions required [3,4]. A propagation medium, usually water, is needed for ultrasonic waves, because ultrasonic waves are absorbed easily in air. Therefore, a wafer needs to be soaked in water to be measured using ultrasonic waves. Since hard X-rays with a wavelength of 0.1 nm to 0.01 nm can penetrate solid objects, they are employed for nondestructive inspections, such as in diagnostic radiography and crystallography, and because of the high photon energy of X-rays, shielding has to be considered for safety.
Optical interferometry has been widely used to carry out high precision geometrical thickness measurements on transparent media. However, it cannot be exploited widely because it measures the optical thickness instead of the geometrical thickness. To extract the geometrical thickness from the optical thickness, either the refractive index of the medium needs to be known, or additional measurement steps, such as the use of rotating samples or blocking lights need to be employed [5–11]. These may produce undesired errors related to mechanical errors as well as environment errors or vibrations.
In this paper, we demonstrate a novel method that can measure both the geometrical thickness and the refractive index of a silicon wafer at the same time. To measure these properties of a silicon wafer, infrared light with a wavelength near to 1.5 μm was selected as the light source, because this wavelength could penetrate the silicon wafer and also be reflected from the surface of the silicon wafer. The optical comb generated by the phase modulation of a seed laser was exploited for the spectral domain analysis. Using the phase values obtained in the spectral domain, which is based on spectral interferometry [11,12], the geometrical thickness was determined over a time period without using any mechanical moving parts or knowledge on the refractive index of the wafer. This led to high-speed measurements, and it did not require either soaking the wafer or the need to shield the measurement system. Since the optical thickness is determined using the optical wavelength, it can be traceable to the definition of meter. A single operation can minimize mechanical errors, environment errors, and vibration. A double-sided polished wafer having a nominal geometrical thickness of 335 μm was measured or the feasibility test was performed.
2. Thickness and refractive index measurements
The intensity of the interference signal having an optical path difference (OPD) of Δ can be expressed by12,13]
In our work, an optical comb was adopted as a wide-spectrum light source having a frequency stability of 10−8–10−9 to realize high-speed measurements. The wavelength of the light source was chosen in the infrared regime near to 1550 nm, so that the light could partially pass through a silicon wafer for noncontact measurements.
Figure 1 shows the optical layout of the measurement system. The light source consisted of a distributed feed-back (DFB) laser (Toptica, DL DFB), two erbium-doped fiber amplifiers (Luxpert, LXI2000), and a comb generator (OptoComb, WTAS-01). The DFB laser was a seed laser with a typical linewidth of 500 kHz to 4 MHz at 5 μs, which corresponds to 10−8~10−9 in terms of frequency stability. The optical comb was generated by the modulating phase of an amplified seed laser having an optical power of 20 mW. The spectral bandwidth of the light source was ~20 nm, with the center wavelength at 1540.2 nm, shown as Fig. 2(a) . The pulse train emitted from the light source had a repetition rate of 25 GHz with a pulse duration of 0.5 ps. Figure 2(b) shows the spectrum of the optical comb in the wavelength range of 1535 to 1545 nm. The mode spacing was equal to the repetition rate of 25 GHz, which corresponded to ~0.2 nm in terms of the wavelength. The repetition rate was locked to a Rb-reference clock (Novatech, 2975AR) having a frequency stability of 10−11 at 10 s. The pulses emitted from the light source were divided by a beam splitter (BS), and then reflected from the reference and the front and rear sides of a silicon wafer and a mirror. The reflected light was recombined and formed the interference signal composed of the reference and the front side, the reference (M1) and the rear side, the reference and the mirror (M2), and other interference, including the interference between the front and the rear sides of the wafer. The interference signals were detected using an optical spectrum analyzer (OSA) for the spectral domain analysis.
The interference signal traveling along Ray 1 in Fig. 1 contains information about the optical path difference, which is expressed as L1 + T + L2 – L0 (≡ A). The light traveling along Ray 2 passes through the silicon wafer and a part of the light is reflected from the front and rear surfaces of the silicon wafer. The interference signal traveling along Ray 2 in Fig. 1 also contains information on several optical path differences, which correspond to L1 – L0 (≡ B), L1 + N·T – L0 (≡ C), where N is the refractive index of the silicon wafer, L1 + N·T + L2 – L0 (≡ D), and other interference, which is too weak to be observed. Finally, the thickness and refractive index, T and N, can be determined from Eq. (3) and Eq. (4).
To extract optical path differences, the values of A, B, C, and D from the interference signals traveling along Rays 1 and 2 in a single operation, and the two optical spectra of the interference signals traveling along Rays 1 and 2 were measured using an OSA. Since the value of the OPD could be determined from Eq. (2), the obtained interference spectra were Fourier-transformed to obtain their periods.
Each interference spectrum was sampled from in the wavelength range 1535 to 1555 nm within 8196 points, which led to a wavelength resolution of ~0.004 nm. Within the wavelength range used, the wavelength uncertainty of the OSA (Agilent 86142B) was about 0.01 nm after calibration using stabilized light sources.
Figures 3(a) and 3(b) show the interference spectra of Rays 1 and 2, respectively. Figure 3(c) shows their discrete Fourier transform results, which are plotted using an offset for easy comparison. The Fourier spectrum of Ray 1 contains two major peaks: the first peak represents the background produced from the intensity distribution of the light source, and the most dominant peak occurring at a period of ~0.87 × 10−11 s of the interference spectrum contains the information related to the optical path difference, A. Background peaks were commonly observed with Rays 1 and 2. Ray 2 showed three peaks, which correspond to the optical path differences of B, C, and D. These peaks could be distinguished easily by checking their amplitudes. The intensity ratio of Peaks B, C, and D was 0.87: 0.47: 1, since the refractivity of the silicon wafer was measured to be 0.26 in our experiments. Accordingly, the peaks occurred at 0.31 × 10−11 s, 0.83 × 10−11 s, and 1.61 × 10−11 s and represent the optical path differences of D, B, and C, respectively. In addition, the peaks could also be distinguished by blocking off the light at the front and rear of the silicon wafer. To determine the geometrical thickness, it is important to determine the differences (D – A) and (C – B) rather than the absolute values of A, B, C and D. The geometrical thickness of a silicon wafer is usually in the region of several hundred micrometers, which is thick enough to avoid the range in ambiguity of 6 mm. Even if the absolute values of A, B, C, and D had an ambiguity, then any problems from the ambiguity can be avoided by determining the differences, (D–A) and (C–B) mathematically. To determine the peak position precisely, a second-order curve-fitting routine was performed on the upper part of each peak. For one-shot measurements, the optical path difference, A, was measured in advance before inserting the silicon wafer. Under stable environmental conditions, the deviation of A was within 0.1 μm after 30 repeated measurements. One of the practical methods used to realize one-shot measurements more strictly is to use a fast optical switch before the detector, or to use a couple of OSAs to obtain two interference spectra at the same time. The optics were aligned well by obtaining the null fringe using a visible light with a wavelength of 633 nm to avoid an unwanted path difference between Rays 1 and 2 caused by any tilting. Although it is not shown in Fig. 3(c), a peak occurring at 4 × 10−11 s and its harmonics were also observed. The peak occurring at 4 × 10−11 s arises from pulse-to-pulse interference, which corresponds to the repetition rate of the light source, 25 GHz.
Table 1 shows measurement results obtained from a double-sided polished wafer. The thickness and the refractive index of the silicon wafer were measured to be 334.85 μm and 3.50, respectively. The standard deviation of the geometrical thickness and refractive index of the silicon wafer from 10 repeated measurements were 0.49 μm and 0.004, respectively. Even if the measurement resolution was not the best among the other techniques mentioned in Section 1, our technique has the advantage of measuring both the geometrical thickness and the refractive index of the silicon wafer at the same time in a single operation. In the near future, we plan to improve the performance of the measurement system by extending the spectral bandwidth of the light source. This can be achieved using a femtosecond pulse laser with a photonic crystal fiber.
3. Discussion and summary
The measurement resolution of the geometrical thickness was achieved by determining the period of the interference signal in the Fourier domain. Since the periods were obtained using a discrete Fourier transform (DFT), the measurement resolution depended on the spectral bandwidth of the interference spectrum. In this work, the resolution of the DFT was 2.64 × 10−13 s, the reciprocal of the spectral bandwidth of the light source used was 3.80 THz, which could resolve the optical thickness to 40 μm. For better resolution, the indices of the peak locations in the Fourier domain were calculated to the third decimal place using a second-order curve fitting of the interference spectra within the full-width half maximum values of the peaks, which corresponded to a length of 0.05 μm. In addition, 8196 zeros were added to the raw spectrum data to improve the peak detection resolution in the Fourier domain. We expect to improve this by exploiting a femtosecond pulse laser with a spectral bandwidth that can be extended to several hundred nanometers using the nonlinear self-phase modulation effect. For example, a measurement resolution of 5 nm can be achieved by simply replacing the light source with a femtosecond laser having a spectral bandwidth of 300 nm.
The measurement uncertainty of the geometrical thickness was evaluated according to ISO/IEC 98-3 (GUM). The uncertainty of the refractive index of air is usually 10−6 under laboratory conditions. The uncertainty of the wavelengths of the OSA was 0.01 nm in the wavelength range 1480 to 1570 nm. The contribution to the uncertainty from these two uncertainty sources was around 10 nm in length for T = 334.85 μm. The uncertainty of the optical path differences of A, B, C, and D was mostly contributed by the uncertainty of the DFT algorithm and the measurement repeatability. The uncertainty of the DFT algorithm was evaluated using computer-generated ideal interference spectra. The ideal spectra had optical path differences of 0.5 to 3.5 mm with a step of 0.1 mm, whose errors after the DFT calculations were < 0.10% when optical path difference was > 1.0 mm. The errors increased to 0.70% for shorter optical path differences below 1.0 mm, because the interference spectrum had only a few periods within the spectral bandwidth of the light source, 20 nm. By adjusting the position of the reference, the mirror, and the silicon wafer, the optical path differences of A, B, C, and D could be > 1.0 mm to minimize the error caused by the DFT algorithm. In this work, the values of A, B, C, and D were selected to be 1.30 mm, 2.32 mm, 1.15 mm, and 0.47 mm, respectively.
The uncertainty attributed to random effects was evaluated by repeated measurements of A, B, C, and D. Therefore, the uncertainty of the optical path differences of A, B, C, and D was estimated to be 1.06 μm, 0.97 μm, 0.73 μm, and 1.17 μm, respectively. Since A, B, C, and D are partially correlated, the correlation terms were taken into account in evaluating the uncertainty of T. Because the uncertainties of the optical path differences were almost the same, the correlation coefficient, r, can be approximated by 
The correlation coefficient was calculated to be 0.80. From Eq. (1), with the correlation coefficient, the uncertainty of the coupling between A, B, C, and D was determined to be –1.75 μm. Finally, the uncertainty of the geometrical thickness was evaluated to be 0.95 μm (k = 1) for a 334.85 μm silicon wafer. In this study, the most dominant uncertainty factor was the DFT algorithm used. The uncertainty can be decreased by increasing the sampling number and the length of the raw data through adopting a mode-locked femtosecond pulsed laser that has a wide spectral bandwidth. The measurement repeatability can be improved by stabilizing the environmental conditions.
In this paper, a new method for measuring both the geometrical thickness and the refractive index of a silicon wafer at the same time was proposed and realized. It required no additional measurement steps, such as rotating or translating the sample, which led to high-speed measurements. In general, optical path differences should be close to zero for a light source having a wide spectrum due to its short coherence length. However, by adopting a mode-locked laser instead of a wide spectral light source, we could observe periodic interference signals because of pulse-to-pulse interference, which meant that there was interference between the different pulses. This makes a flexible measuring system equivalent to a nonequal path interferometer. Moreover, in the spectral domain, the optical path differences could be obtained, even if the pulses do not meet in the time domain. The optical comb can be considered as a combination of monochromatic light sources having well-defined wavelengths. It is similar to an extension of multiwavelength interferometry. Based on this principle, the geometrical thickness could be determined from the four optical path differences in the interferometer, which are measured in the spectral domain analysis. The geometrical thickness of a double side polished wafer was measured to be 334.85 μm, with a measurement uncertainty of 0.95 μm (k = 1). The uncertainty of the DFT algorithm, which is the most dominant uncertainty factor, can be reduced by adopting a mode-locked femtosecond pulse laser having a spectral bandwidth of several hundred nanometers.
This work was supported in part by the National Program: 3-dimensional Profile Measurement Technology on Microscopic Integrated Structures, KRISS.
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