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Reflectometry-based wavelength scanning interferometry for thickness measurements of very thin wafers
Optics Express, Vol. 18, Issue 7, pp. 6522-6529 (2010)
http://dx.doi.org/10.1364/OE.18.006522
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– Abstract
1. Introduction
2. Principle
3. Experimental results and simulations
4. Conclusion
– References and links
– Abstract
1. Introduction
2. Principle
3. Experimental results and simulations
4. Conclusion
– References and links
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Abstract
With the development of microelectronics, the demand for silicon wafers is greatly increased for various purposes, especially the use of thin wafers for smart cards, cellular phones and stacked packages. In this paper, we describe an innovative scheme of combining wavelength scanning interferometry (4 nm tuning range centered at 1550 nm) with spectroscopic reflectometry that enables us to measure the thickness profile of thin wafers below 100 μm with high thickness resolution. The performance of this method is compared with that of an existing technique and verified by measuring several thin wafers.
© 2010 OSA
1. Introduction
Silicon wafers are widely used in technologies such as semiconductor integrated circuits, cellular phones and solar cells, and have been getting thinner to meet device performance and packaging requirements. As microelectronics devices continue to shrink, the demand for thin wafers is increasing. Reducing the wafer thickness enables chip makers to enlarge the capacity of integrated circuits by multichip packaging (MCP) and to build smaller packages. Moreover due to the advancement of industrial wafer thinning techniques, it is possible to make wafers thinner than 100 μm easily and cheaply (mass production and cost reduction). This in turn accelerates the electronics applications. Thin wafers of other materials such as fused silica are also finding increasing application in micro-optic devices. In these cases, the wafer thickness can be critical as light propagation through the wafer is common and the optical path length can be a first order parameter that defines performance. To increase the product rate and prevent inferior goods, the test and evaluation of wafers is as critical as their fabrication.
Many approaches have been developed for characterization of wafer thickness and flatness. A micrometer is the most straightforward, but damage is a risk, particularly for thin wafers. Capacitance gauging techniques measure thickness variation by scanning two probes facing each other on the top and bottom surfaces of wafer in a constant gap. It is a single point measurement and requires a precision stage for complete coverage. As a general metrological tool for 3-D surface mapping, optical interferometry is widely used. Optical techniques in reflection using visible light measure simultaneously the front side and back side flatness and obtain wafer thickness variation by calculating distance variation between two sides, but this needs a complex optical system [1]. Another optical approach is to use a tunable infrared light source for which undoped silicon wafers are transparent, and measure the wafer thickness variation directly by analyzing the reflected beams from a wafer using a phase-shifting algorithm [2] or a Fourier-based analysis with wavelength scanning [3]. But, these techniques are not focused on absolute thickness measurements.
M. J. Jansen, H. Haitjema, and P. H. J. Schellekens, “A scanning wafer thickness and flatness interferometer,” Proc. SPIE 5856, 334–345 (2004). [CrossRef]
T. L. Schmitz, A. Davies, C. J. Evans, and R. E. Parks, “Silicon wafer thickness variation measurements using the National Institute of Standards and Technology infrared interferometer,” Opt. Eng. 42(8), 2281–2290 (2003). [CrossRef]
L. L. Deck, “Multiple surface phase shifting interferometry,” Proc. SPIE 4451, 424–431 (2001). [CrossRef]
In this paper we describe an innovative configuration of wavelength scanning interferometry based on spectroscopic reflectometry that allows one to measure three-dimensional absolute thickness of thin wafers and to extend the limit of the measurable thickness of wafers below 100 μm with a small tuning range. The principle of spectroscopic reflectometry has already been successfully applied to thin-film metrology [4, 5]. It is reported that the Fourier-based method of wavelength scanning interferometry can be used to measure absolute thickness [6–8], but the measurement is limited to thicker wafers because the 1st order frequency in the interference frequency spectrum must be resolvable. This leads to a minimum measurable sample thickness with the Fourier-based method that follows the relation of dmin=λ2/(nΔλ) where Δλ is the wavelength tuning range and n is the group refractive index. So the minimum thickness measurable for a silicon wafer with the Fourier-based technique and a 4 nm-tunable (500 GHz) 1550 nm laser, for example, is approximately 170 μm. Thus, the application of reflectometry to wavelength scanning interferometry enables a significant extension of thickness measurements and represents a significant advance.
Y.-S. Ghim and S.-W. Kim, “Fast, precise, tomographic measurements of thin films,” Appl. Phys. Lett. 91(9), 091903 (2007). [CrossRef]
Y.-S. Ghim and S.-W. Kim, “Spectrally resolved white-light interferometry for 3D inspection of a thin-film layer structure,” Appl. Opt. 48(4), 799–803 (2009). [CrossRef] [PubMed]
L. L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” Proc. SPIE 4778, 218–226 (2002). [CrossRef]
A. Suratkar, Y.-S. Ghim, and A. Davies, “Uncertainty analysis on the absolute thickness of a cavity using a commercial wavelength scanning interferometer,” Proc. SPIE 7063, 70630R (2008). [CrossRef]
2. Principle
Figure 1
shows a schematic diagram of a commercial wavelength scanning interferometer (Zygo MST) configured for this investigation [9]. We draw a generic Fizeau configuration inside the MST to provide a complete schematic of what is needed for the measurement. We do not use a transmission flat on the instrument, but rather use the 100 mm collimated beam from the interferometer as the wave front incident on the wafer. This incident light is reflected from the top and bottom surfaces and later imaged on a CCD camera inside the interferometer, yielding the self-interference fringes from the wafer. The light source is a thermally tuned laser in this instrument, and the tuning range is from 1546 nm to 1550 nm. A CCD camera operating at 60 Hz captures a series of digitized intensity images of the wafer interference pattern as the wavelength is tuned. Therefore each frame represents the interferogram at a specific wavelength. We obtained 600 frames during a 500 GHz tune.
This is the Zygo VeriFire MSTTM (http://www.zygo.com/?/met/interferometers/verifire/mst).
Fig. 1 (a) A schematic diagram of wavelength scanning interferometer for measuring of characteristics of a wafer; BS: beam splitter, CL: collimating lens, IL: imaging lens, CCD: charge coupled device, (b) single-side polished wafer as a reference, and (c) double-side polished wafer as a specimen.
We used a double-side polished (DSP) wafer as our test sample to obtain interference fringes corresponding to the light reflected from the two surfaces. On the other hand, for a single-side polished (SSP) wafer, significant specular reflection only occurs at the polished front surface and diffuse scattering occurs at the back side therefor no interference pattern is observed. A SSP wafer is used for a reference measurement and this determines the scale of the intensity versus wavelength for the DSP measurement. Figure 1(b) and (c) show how the incident light on SSP and DSP wafers is reflected from both the front and back surfaces.
Our method is based on an optical measurement, so the measured optical path length must be converted to thickness in nanometers by inversely scaling by the silicon refractive index. The wafer must be transparent over the wavelength range therefore the wafer must be undoped. The refractive index values for the wavelength range of the measurement are taken from the literature [10].
C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998). [CrossRef]
Our procedure consists of two measurement steps; first, measure the reflectance of a SSP wafer as a reference specimen; and second, measure the reflectance of a DSP wafer as the test specimen.
For SSP wafer, the incident light is only reflected at the top air-wafer interface and scatters at the bottom wafer-air interface so the theoretical reflectance ℜT(k)ref is approximately given as |r|2, while for DSP wafers the incident light undergoes multiple reflections at the top and bottom interfaces so the theoretical reflectance ℜT(d;k)sam is expressed in a more complex way as [11]Note that r=(1-n)/(1+n), the refractive index of air is 1, the refractive index of the wafer is n, d is the wafer thickness, and k is the wave-number. The experimental reflectance of the sample ℜE(k)sam can be obtained by the relationship between the sample and reference, and by measuring the spectral density distribution of the sample and reference reflectance, respectively [4, 11]. Once the spectral reflectance ℜE(k)sam is obtained, the wafer thickness d is found by fitting the analytical model of ℜT(d;k)sam to the measured data of ℜE(k)sam so as to minimize the total sum over the entire wave-number range ofThe above equations include (x,y) coordinates, thus we carry out this fitting for every pixel to obtain the thickness estimate over the entire wafer. The values for n are taken from the literature and the values for k are taken from independent measurements of the laser using a wavemeter [12]. The index is taken to be constant over the wafer, but d is not. So the wafer thickness d is the only free parameter that needs to be found numerically using an appropriate one-dimensional nonlinear optimization technique. The well-known Levenberg-Marquardt algorithm was used to search the true value of d that minimizes the merit function of ζ(d) [13]. We have tested Eq. (2) for various values of m=1, 2, 3, 4, and 5, and m=2 was found to yield optimal results. The superscripts ‘T’ and ‘E’ refer to ‘theory’ and ‘experiment’, respectively.
Y.-S. Ghim and S.-W. Kim, “Fast, precise, tomographic measurements of thin films,” Appl. Phys. Lett. 91(9), 091903 (2007). [CrossRef]
Figure 2(a)
shows the representative graphs of the merit function and its lower envelope versus thickness value at a single pixel for a measurement of a 200 μm thick wafer. We can see numerous local minima which ride upon the slowly varying envelope. On the scale of the noise, the minimum of this lower envelope (global minimum) is well defined, as shown in right of Fig. 2 (a). The lower envelope of the merit function curvature is a strong function of the tuning range and, as expected, the minimum is more sharply defined for larger tuning. This can be seen in Fig. 2(b) which shows a plot of the lower envelope of a simulated merit function as a function of the thickness fit parameter d for different tuning ranges. If the tuning range is broad and multiple interference oscillations are present over the measurement range, the merit function shows a unique minimum at the absolute value of d. For small tuning ranges, only a fraction of an interference period is captured and the merit function minimum is now impacted by phase-dependent systematic errors. All interference-based measurements are susceptible to such errors, often called ripple error or fringe bleed-through. The ripple error is present in our measurement, but it can be significantly reduced by unwrapping. This is because the fringe error primarily causes a shift in the envelope of the merit function (Fig. 2(a)) rather than a shift in the local minima; therefore the minimization returns a value of d from an erroneous adjacent local minimum. The local minima are separated by λc /2n (λc: the mean wavelength in the tuning range); therefore the profile of d over the wafer will show discrete thickness jumps by this amount. If we assume such jumps are not physical and that the pixel-by-pixel variations are less than this, we can remove the jumps. This is demonstrated with simulation and the results are shown in Fig. 3
. We simulate the measurement for a parabolic thickness profile ranging from 206 μm to 207 μm using a 4 nm tuning. We see discrete jumps in the simulation in increments of 222.66 nm and this is equivalent to ε = λc /2n. Assuming the wafer is smooth, we unwrap the jumps to retrieve the true thickness profile (part (b) of the figure). The simulation shows that the discrete steps are ± ε about the true thickness, and this will be well defined for wafers with a few fringes across the profile. Thus, the absolute thickness can be determined. We expect this to break down for extremely thin wafers, very narrow tuning ranges, and low signal-to-noise conditions. We will explore this limit in a future publication.
Fig. 2 Plot of real and simulation data using the optimization technique: (a) merit function ζ(d) and its lower envelope vs. thickness value with a 4 nm tuning range (real data), (b) the lower envelope of merit function ζ(d) vs. thickness value according to the tuning range; black: 4 nm tuning range, red: 30 nm tuning range, blue: 100 nm tuning range (simulation).
3. Experimental results and simulations
We have tested the method by measuring the thickness profile of two samples, a nominally 200 μm thick silicon wafer and a 60 μm thick silicon wafer. Our data is unwrapped, as described above. The ~200 μm wafer can be measured with the Zygo MST which uses the Fourier analysis methodology published by L. Deck [14]. This provides a partial intercomparison for validation. This Fourier approach used by the MST determines the spatial phase variation of the dominant spectral peaks in the Fourier spectrum. The analysis of the phase of the peak corresponding to the interference of the top and bottom reflections yields a spatial map of the optical thickness variation of the sample.
L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003). [CrossRef] [PubMed]
Figure 4(a)
shows an interferogram of the ~200 μm wafer viewed with monochromatic light in our experiment and Fig. 4(b) represents the result of the MST Fourier analysis which is the relative thickness map of the wafer (the optical length values have been inversely scaled by the refractive index). The absolute thickness map of the same area, as measured with our analysis, is shown in Fig. 4(c). The root mean square (RMS) and peak to valley (PV) numbers compare well. The MST measurement shows an RMS of 0.310 μm and a PV of 1.93 μm for the relative thickness measurement, and our measurement shows an RMS of 0.309 μm and a PV of 1.91 μm after the average absolute thickness has been removed. We can compare the two results more directly by looking at a difference map where we remove the average thickness from our absolute thickness map and subtract the Fourier-based result in Fig. 4(b), see Fig. 4(d). The difference map has an RMS of 8 nm and a PV of 34 nm, respectively. The spatial structure in the difference map is clearly correlated with the interferogram and is predominantly twice the fringe frequency.
Fig. 4 Comparisons of measurement results: (a) an interferogram of a wafer with ~200 μm thickness viewed in monochromatic light, (b) the relative thickness map of (a) measured with the MST using the Fourier method, (c) the absolute thickness map of (a) measured with the proposed method and (d) the difference between our measurement and the MST Fourier measurement, after removing the average absolute thickness from our measurement (1 pixel = 110 μm).
We have simulated both our method and the Fourier method to assess measurement errors and the origin of the fringe ripple error. We assume a perfect linear wavelength tune over the same wavelength range as our commercial instrument. We simulated the measurement of a silicon wafer with a very slight plano-convex geometry and the thickness ranging from ~206 μm to ~207 μm.
The simulation results are shown in Fig. 5
. Figure 5(b) shows the error map for the Fourier method and shows the strong twice-fringe frequency structure similar to what is observed in the experiment (Fig. 4(d)). This error map has an RMS of 4.79 nm and a PV of 16.66 nm, which corresponds to the phase errors of ~ 0.31% and ~ 1.07% at a 1550 nm wavelength. The error map for our technique is shown in Fig. 5(c), with a relatively small RMS of 0.07 nm and PV of 0.35 nm. The residual ripple error for our technique is significantly smaller. This suggests that the ripple error present in the difference map for the 200 μm wafer measurement (Fig. 4(d)) is primarily present in the Fourier-method measurement. We also investigated the dependence on the tuning range in simulation. Figure 5(d) shows the PV of the measurement error for both methods as a function of tuning range. The error with the Fourier method decreases with tuning range, but that of our algorithm is small and is approximately constant over the tuning range considered. Note that uncertainty/noise sources such as tuning rate non-linearity, temperature, intensity fluctuations, etc…, were not included in the simulation, and these will add uncertainty to our reflectometry-based measurement.
Fig. 5 Simulation results: (a) the interferogram and structure of the virtual wafer with ~200 μm thickness, (b) the error map of the virtual wafer measured with Fourier technique, (c) the error map of the virtual wafer measured with our proposed technique, (d) the simulated PV of the error maps of the virtual wafer using the two methods as the tuning range of source is increased.
Figure 6
shows a measurement of a ~ 60 μm wafer using our technique. There is no independent thickness data for such a thin wafer for comparison, but the results show plausible thickness variation that matches the interferogram structure. Our algorithm has no difficulty extracting small thickness values, even with such a small tuning range, and confirms that the method is capable of measuring absolute wafer thickness below 100 μm. The RMS and PV values of the thickness maps are 0.559 μm and 2.45 μm, respectively.
Fig. 6 An exemplary measurement result: (a) an interferogram of specimen with ~60 μm thickness viewed in monochromatic light and (b) 3-D thickness map profile, 1 pixel=100 μm.
There are many factors that add uncertainty to our measurement. A detailed analysis is currently underway and will be the topic of a future publication. For this paper, we have carried out preliminary uncertainty estimates for individual parameters that we estimate will have significant effects: uncertainty in the refractive index, uncertainty in wavelength, and repeatability. The results are shown in Table 1
for the 200 μm thickness wafer.
4. Conclusion
To conclude, we have proposed and tested a method using a reflectometry-based wavelength scanning interferometer for measurements of the absolute thickness profile of DSP wafers. This technique has extended the measurement range to thickness below 100 μm and improved the precision compared to the Fourier-based technique. This is an important extension that benefits thin-wafer manufacturing, a critical technology for many microelectronics and micro-optic devices. Our initial tests confirm that the method is capable of profiling thin wafers less than 100 μm thick, and thus is well suited for thin wafer inspection.
Acknowledgements
This material is partially based upon work supported by the National Science Foundation under Grant 0348142. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
References and links
M. J. Jansen, H. Haitjema, and P. H. J. Schellekens, “A scanning wafer thickness and flatness interferometer,” Proc. SPIE 5856, 334–345 (2004). [CrossRef] | |
T. L. Schmitz, A. Davies, C. J. Evans, and R. E. Parks, “Silicon wafer thickness variation measurements using the National Institute of Standards and Technology infrared interferometer,” Opt. Eng. 42(8), 2281–2290 (2003). [CrossRef] | |
L. L. Deck, “Multiple surface phase shifting interferometry,” Proc. SPIE 4451, 424–431 (2001). [CrossRef] | |
Y.-S. Ghim and S.-W. Kim, “Fast, precise, tomographic measurements of thin films,” Appl. Phys. Lett. 91(9), 091903 (2007). [CrossRef] | |
Y.-S. Ghim and S.-W. Kim, “Spectrally resolved white-light interferometry for 3D inspection of a thin-film layer structure,” Appl. Opt. 48(4), 799–803 (2009). [CrossRef] [PubMed] | |
L. L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” Proc. SPIE 4778, 218–226 (2002). [CrossRef] | |
L. L. Deck, C. V. Peski, and R. Eandi, “Measurements of hard pellicles for 157 nm lithography using Fourier transform phase-shifting interferometry,” Proc. SPIE 5130, 555–559 (2003). [CrossRef] | |
A. Suratkar, Y.-S. Ghim, and A. Davies, “Uncertainty analysis on the absolute thickness of a cavity using a commercial wavelength scanning interferometer,” Proc. SPIE 7063, 70630R (2008). [CrossRef] | |
This is the Zygo VeriFire MSTTM (http://www.zygo.com/?/met/interferometers/verifire/mst). | |
C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998). [CrossRef] | |
H. G. Tompkins, and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (John Wiley & Sons. Inc, 1999). | |
A. Suratkar, Absolute distance (thickness) metrology using wavelength scanning interferometry (UNC Chralotte, 2009). | |
The Levenberg-Marquardt function is available as LEASTSQ in the MATLAB software. | |
L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003). [CrossRef] [PubMed] |
OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.4290) Instrumentation, measurement, and metrology : Nondestructive testing
(120.4640) Instrumentation, measurement, and metrology : Optical instruments
(160.6000) Materials : Semiconductor materials
ToC Category:
Instrumentation, Measurement, and Metrology
History
Original Manuscript: January 5, 2010
Manuscript Accepted: January 24, 2010
Published: March 15, 2010
Citation
Young-Sik Ghim, Amit Suratkar, and Angela Davies, "Reflectometry-based wavelength scanning interferometry for thickness measurements of very thin wafers," Opt. Express 18, 6522-6529 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6522
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References
- M. J. Jansen, H. Haitjema, and P. H. J. Schellekens, “A scanning wafer thickness and flatness interferometer,” Proc. SPIE 5856, 334–345 (2004). [CrossRef]
- T. L. Schmitz, A. Davies, C. J. Evans, and R. E. Parks, “Silicon wafer thickness variation measurements using the National Institute of Standards and Technology infrared interferometer,” Opt. Eng. 42(8), 2281–2290 (2003). [CrossRef]
- L. L. Deck, “Multiple surface phase shifting interferometry,” Proc. SPIE 4451, 424–431 (2001). [CrossRef]
- Y.-S. Ghim and S.-W. Kim, “Fast, precise, tomographic measurements of thin films,” Appl. Phys. Lett. 91(9), 091903 (2007). [CrossRef]
- Y.-S. Ghim and S.-W. Kim, “Spectrally resolved white-light interferometry for 3D inspection of a thin-film layer structure,” Appl. Opt. 48(4), 799–803 (2009). [CrossRef] [PubMed]
- L. L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” Proc. SPIE 4778, 218–226 (2002). [CrossRef]
- L. L. Deck, C. V. Peski, and R. Eandi, “Measurements of hard pellicles for 157 nm lithography using Fourier transform phase-shifting interferometry,” Proc. SPIE 5130, 555–559 (2003). [CrossRef]
- A. Suratkar, Y.-S. Ghim, and A. Davies, “Uncertainty analysis on the absolute thickness of a cavity using a commercial wavelength scanning interferometer,” Proc. SPIE 7063, 70630R (2008). [CrossRef]
- This is the Zygo VeriFire MSTTM ( http://www.zygo.com/?/met/interferometers/verifire/mst ).
- C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998). [CrossRef]
- H. G. Tompkins, and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (John Wiley & Sons. Inc, 1999).
- A. Suratkar, Absolute distance (thickness) metrology using wavelength scanning interferometry (UNC Chralotte, 2009).
- The Levenberg-Marquardt function is available as LEASTSQ in the MATLAB software.
- L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003). [CrossRef] [PubMed]
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