## Abstract

We describe a method to simultaneously measure thickness variation and refractive index homogeneity of 300 mm diameter silicon wafers using a wavelength-shifting Fizeau interferometer operating at 1550 nm. Only three measurements are required, corresponding to three different cavity configurations. A customized phase shifting algorithm is used to suppress several high order harmonics and minimize intensity sampling errors. The new method was tested with both silicon and fused silica wafers and measurement results proved to be highly repeatable. The reliability of the method was further verified by comparing the measured thickness variation of a 150 mm diameter wafer to a measurement of the wafer flatness after bonding the wafer to an optical flat.

©2012 Optical Society of America

## 1. Introduction

The past decades have seen an unceasing demand for smaller line widths of integrated semiconductor circuits to increase capabilities and processing speed. A decrease of the minimum line width can be achieved by increasing the numerical aperture of a lithographic imaging system or by decreasing the source wavelength. In both cases, a decreased depth of focus of the imaging system is an unavoidable result.

A successful lithography process requires that the exposure site on a silicon wafer remains within the depth of focus during the exposure. Consequently, the decreasing depth of focus leads to increasingly tight specifications for the wafer flatness. This is reflected in the most recent 2011 edition of the International Technology Roadmap for Semiconductors (ITRS-2011) [1] which projects that the site flatness requirement (SFQR for a 26 mm × 8 mm site area) of 300 mm and 450 mm diameter silicon wafers will gradually decrease from 45 nm in 2010 to less than 23 nm in 2015, with an anticipated conversion to 450 mm wafer diameter beginning in 2014. Like the site flatness trend, the nano-topography (peak-to-valley on a 2 mm diameter analysis area) will keep evolving from 11 nm in 2010 to less than 4 nm in 2017.

Over the last several years, the once widely used capacitance gauging methods have been replaced by optical interferometric methods with lower measurement uncertainty and higher spatial resolution [2,3]. Optical interferometers based on an infrared source have the additional advantage that the light can pass through silicon wafers (unless they are highly doped), which makes it possible to measure the refractive index homogeneity of wafers as well as their surface flatness. One important problem which must be considered in interferometric methods for measuring parallel, transparent plates are the multiple reflections which result in intensity distortions due to superposition of the interferograms from all possible interferometer cavities. Transparent plates have been measured by least-squares fitting of the first-order terms in a Twyman-Green interferometer [4,5] and direct filtering of spurious reflection fringes in the frequency domain [6]. Also, Deck [7] described the simultaneous measurement of a significant linear gradient in the refractive index of an 8 mm thick parallel plate, the absolute length of an optical cavity by considering chromatic dispersion [8], and the surface profiles including index homogeneity during a single wavelength scan [9] using Fourier Transform Phase-Shifting Interferometry (FTPSI). In FTPSI, the modulation frequency shift caused by index dispersion and nonlinear wavelength scanning deteriorates the measurement accuracy. Consequently, other research has focused on algorithm design with optimal sampling strategies to compensate for the modulation frequency shift [10–12]. In essence, the FTPSI approach measures phase locked to a specific fringe modulation frequency during a single wavelength scan, which corresponds to a certain optical cavity length among multiple interference fringes generated from many cavities. The air gap distances between surfaces must be determined on the basis of frequency ratios to isolate the first-order modulation frequency from the other high-order terms, which inevitably requires a very complex data-analysis algorithm with large amounts of fringe data for frequency analysis. Besides wavelength scanning methods, several alternatives which mostly target absolute measurement of both geometrical thickness and refractive index of silicon wafers are available, including rotation angle scanning of a wafer [13] and optical comb generation [14]. But, most of this research is limited to a point measurement of a small area, not the whole surface. Other research was directed at measuring the total thickness variation (TTV) over the entire area of a silicon wafer using Haidinger fringes [15] or an infrared Fizeau interferometer [16–18], with the assumption that the index variation is negligible.

In this paper we describe measurements of both thickness variation and index homogeneity of 300 mm wafers using a widely tunable infrared laser Fizeau interferometer called the Improved Infrared Interferometer (IR^{3}). The measurement method is derived from a well-known method for measuring the refractive index homogeneity of a large-aperture parallel plate of glass [19–21]. The original method was further developed by Roberts and Langenbeck [22], and later refined by Schwider et al. [23]. In our method, two of the measurements in the Roberts-Langenbeck method involving the front and back surfaces of a sample, are replaced with a single measurement between front and back. This helps to simplify the measurement process and prevents multiple interference fringes without the need for any coating procedures. In addition, a customized phase-measurement algorithm for our experimental conditions, based on characteristic polynomial theory suggested by Y. Surrel [24,25], is designed [26] to suppress several high-order harmonics and minimize intensity sampling errors.

## 2. Description of the measurement algorithm

The original Roberts-Langenbeck method was developed to measure both thickness variation and index homogeneity of optical-quality glass windows. As shown in Fig. 1(a) , the glass window must be coated initially on both surfaces. The coating must be opaque and highly reflective, but needs to be uniform only on the front face since reflection at the back face is internal. The coatings are successively removed between measurements, as shown in Figs. 1(b) and 1(c). At each step of the measurement process, a single interferometer cavity is measured. When wavelength-shifting interferometry is available, the measurement procedure shown in Fig. 1 can be simplified as shown in Fig. 2 . The measurements of Figs. 1(a) and 1(b) can be combined into a single measurement in which the wafer is the interferometer cavity as is shown in Fig. 2(a). The remaining measurements in Figs. 2(b) and 2(c) are the same as in the Roberts-Langenbeck method. In practice, for the wafer as cavity measurement of Fig. 2(a), the wafer was removed from the cavity formed by the reference and return flats, and installed immediately behind the collimator (see Fig. 3 ). The empty cavity measurement of Fig. 2(c) does not depend on the wafer and does not have to be repeated when more than one wafer is measured.

The spatial variations in the optical path difference (OPD) between the two interfering rays at any point (*x*,*y*) of the surfaces in Fig. 2 are:

*W*

_{1},

*W*

_{2}, and

*W*

_{3}are the wavefronts measured in the three configurations of Fig. 2,

*n*

_{0}is the average refractive index of the sample,

*t*

_{0}is the mean value of the sample thickness,

*z*

_{1}and

*z*

_{2}are the deviations of the two wafer planes taken as positive in the direction of the outer normal to the planes, and

*z*

_{3}and

*z*

_{4}are the deviations from a plane of the return flat and reference flat respectively. Equation (1) and Eq. (2) can be solved for the thickness variation ∆

*t = t-t*and the refractive index variation ∆

_{0}*n = n-n*as follows:

_{0}*x*,

*y*) was omitted to simplify both expressions, and terms proportional to Δ

*n*·Δ

*t*are neglected in the derivation of Eq. (3), because they are insignificant. The most remarkable feature of the solution for the physical thickness variation Δ

*t*is that it is solely dependent on the three measurements

*W*,

_{1}*W*, and

_{2}*W*. Calculating the refractive index variation Δ

_{3}*n*, on the other hand, requires prior knowledge of average thickness

*t*

_{0}and average index

*n*

_{0}.

## 3. Experimental results and discussion

Figure 3 shows the experimental system which is set up for a 300 mm wafer measurement based on the IR^{3} interferometer in Fizeau mode to perform the three measurements shown in Fig. 2. The infrared light source, which has a very wide tuning range from 1520 nm to 1630 nm, delivers the light to the IR^{3} interferometer through a polarization preserving single-mode fiber (SMF). The wavelength shift for phase change is controlled by direct wavelength input through a GPIB communication interface. The interference fringes are captured by an InGaAs charge coupled device (CCD) camera with 640 by 480 pixels resolution and 12-bit digitization.

Two different types of 300 mm diameter wafers, a silicon wafer (p-type, B-doped, annealed) and a silica glass wafer, were used in developing the measurement method. The first step was to determine the wavelength range needed to cover the desired phase change corresponding to a certain cavity length. It was found that a wavelength range from *λ*_{1} = 1550.02 nm to *λ*_{2} = 1550.91 nm generates a 4π phase change for the silicon wafer cavity as depicted in Fig. 4(a)
. Figure 4(b) shows 13 phase-shifted intensity measurements (red circles) for one pixel at nominally equal wavelength increments between *λ*_{1} and *λ*_{2}. The wavelength change corresponding to a phase change of 4π is related to the thickness of the wafer through the following equation:

*t*

_{0}is the physical thickness of the silicon wafer, ∆

*φ*is a total phase change,

*λ*

_{0}is the center wavelength, ∆

*λ*is an overall wavelength change, and

*n*

_{0}is the mean refractive index at the specified center wavelength

*λ*

_{0}. The uncertainty of

*t*

_{0}is 8.7 μm. It was calculated from the uncertainties of the variables in Eq. (4) using Gauss' formula for uncertainty propagation. The wafer thickness calculated from the phase change agrees well with the manufacturing specification for the wafer thickness of 775 ± 25 µm. The refractive index used in Eq. (4) [27,28] and thickness data calculated in Eq. (4) can be applied to determine the refractive index variation in Eq. (3).

The phase-shifting algorithm used in this work requires 13 frames with π/3 phase step, which results in 4π overall phase change. Details of the phase shifting algorithm were described by Chu *et al.* [29]. Figure 5
shows the three OPD measurement results (*W*_{1}, *W*_{2}, and *W*_{3}) in Eq. (1) for both the 300 mm glass wafer and silicon wafer. Tilt terms were not removed from all the OPD maps in Fig. 5, because those tilt terms might represent an actual thickness or refractive index variation. The three OPDs in Fig. 5 were combined according to Eq. (3) to calculate maps of thickness variation ∆*t* and refractive index variation ∆*n*, which are shown in Fig. 6
. The thickness and index variation maps, which show about 93% area of full size wafer by trimming the outer steep edge to emphasize the variation in the central area, represent maps of which tilt terms were removed from ∆*t* and ∆*n* in Eq. (3) to highlight the deviations from the best-fit uniform gradient. The total thickness variation (TTV) excluding constant gradient, was measured to be 5 µm for the glass wafer and 0.5 µm for the silicon wafer. The thickness variation maps are shown in Figs. 6(a) and 6(c). In the case of the glass wafer, the thickness variation is dominated by a quadratic shape. The refractive index variation of the glass wafer is about 1.5 × 10^{−4} peak-to-valley. This value is higher than the known index homogeneity of medium quality fused silica of about 5 × 10^{−6}, but only about 50% larger than our estimate of the measurement uncertainty. The cause of much of the measurement noise in Figs. 5 and 6 is vibration of optics mounts driven by acoustic noise in the clean room. In particular, the measurements involving the large flats were susceptible to vibration that resulted in characteristic “ghost fringes” in the measurement results at twice the fringe spatial frequency. To calculate the uncertainties of ∆*t* and ∆*n* for both glass and silicon wafer, Eq. (5) and (6) were derived from Eq. (3).

*u*(

*W*

_{i = 1,2,3}) is the uncertainty of OPD

*W*

_{i}, which can be estimated as the mean over all pixels of the standard deviation map of OPD

*W*

_{i}calculated from 10 consecutive measurements. According to the references used for the refractive index values of fused silica and silicon [27,28], the uncertainty

*u*(

*n*

_{0}) was estimated as 4 × 10

^{−5}for fused silica and 10

^{−4}for silicon. The uncertainty of

*u*(

*t*

_{0}) for the glass wafer is 41.6 μm. The measurement uncertainties and the resulting uncertainties for the thickness variation

*u*(Δ

*t*) and the refractive index variation

*u*(Δ

*n*) are summarized in Table 1 . The dominant contributors to the uncertainty

*u*(Δ

*n*) appear to be the uncertainty in the mean wafer thickness

*u*(

*t*

_{0}) and mean refractive index

*u*(

*n*

_{0}). Better knowledge of

*t*

_{0}and

*n*

_{0}would result in improved uncertainty of Δ

*n*. In this paper,

*u*(Δ

*n*) of glass wafer were calculated to be 1.9 × 10

^{−5}. The uncertainties

*u*(Δ

*t*) of both glass and silicon wafers have almost the same value of about 16 nm because they do not depend on the uncertainty of

*t*

_{0}and

*n*

_{0}, but only on various environmental influences on the measurements that do not depend on the wafer type.

The effect of neglecting the refractive index variation when measuring the thickness variation can be revealed through simple comparison with thickness variation results equal to half of *W*_{1} divided by *n*_{0}. As shown in Fig. 7
, the TTV of the horizontal center cross-section profile of the glass wafer thickness variation decreased by 30% from 7.5 µm to 5 µm after consideration of refractive index variation. The thickness variation of the silicon wafer decreased by 56% from 1 μm to 0.4 μm. We can conclude that the refractive index variation has more influence on the estimated thickness variation for the silicon wafer.

## 4. Verification of the thickness variation result

We sought to validate the measurement method with an alternative method for measuring the TTV of a silicon wafer by bonding, or optically contacting, the wafer to a thick and very flat glass substrate as shown in Fig. 8 [30–33]. The bonded wafer conforms to the substrate and a measurement of the flatness error of its font surface is a good measure of the TTV. The bonding process was performed in a cleanroom because even small dust particles between the two surfaces prevent the formation of a bond over a large area. The wafer and the substrate were cleaned using a standard cleaning process that leaves the surfaces hydrophilic due to the presence of terminal OH groups. After cleaning, the silicon wafer was placed on the glass substrate and pressure was applied at the wafer center until a bond formed and could be observed to spread from the center to the edge of the wafer. The quality of the bond was easily inspected by viewing the wafer through the glass substrate. Failed bonds were visible as areas with poor contact, most likely caused by trapped dust particles.

Figure 9 shows two thickness variation results of a 150 mm silicon wafer measured using the modified Roberts-Langenbeck method, and measured after bonding the wafer to a fused silica substrate with 60 nm flatness error using a commercial Fizeau interferometer at 633 nm. Figure 9 represents almost 80% (116 mm in diameter) of the full wafer size (150 mm) from the center, the maximum limit which was achievable with no dust inside the gap. The waviness of the results obtained using the modified Roberts-Langenbeck method is due to vibrations that can be reduced through e.g. improved mounts for the flats in the Roberts-Langenbeck test setup. As shown in Fig. 9(c), the two results match closely.

## 5. Conclusion

A simple and straightforward method derived from the Roberts-Langenbeck test was demonstrated to measure the thickness variation and the refractive index homogeneity of 300 mm double-sided polished wafers. The method uses a wavelength-shifting Fizeau interferometer in the infrared to perform three required measurements. We have tested wafers made from fused silica glass and from silicon to identify the different characteristics of thickness variation and index homogeneity. The reliability of the proposed method for thickness variation measurement was verified by comparing the measured thickness variation of a 150 mm diameter wafer to a measurement of the wafer flatness after bonding the wafer to an optical flat.

## Acknowledgments

We gratefully acknowledge financial support by NIST’s Office of Microelectronics Programs (OMP) and unfailing encouragement by its director Dr. Joaquin V. Martinez de Piñillos. Robert E. Parks of the College of Optical Sciences at the University of Arizona introduced us to the art of wafer bonding by demonstrating a wet bonding process.

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