1. Introduction

White-light interferometry allows us to produce spectrally-resolved interference for a variety of applications including the 3-D surface profile measurement as concerned in this investigation. A long-established way of implementing white-light interferometry is to use a mechanical scanner so as to vary the optical path difference between the measurement and reference arms while monitoring the resulting interferograms. [2] This technique is termed the z-scanning; z represents the optical path difference. The other way, which may be called the k-scanning with k being the wave number, involves sweeping the wavelength of illumination light progressively by use of a tunable laser, [3] acousto-optical modulator, [4] and liquid crystal cavity. [5] This k-scanning offers the advantage of fast measurement with no need of a mechanical scanner in comparison to the z- scanning, but its performance is restricted by the narrowly-limited wavelength range of tunable sources available in practice. The k-scanning can be performed in the dispersive way using a spectrometer consisting of a diffraction grating and a photo-detector array.[6,7] This dispersive k- scanning permits utilizing the wide spectral range of white light, being considered a useful means of implementing white-light interferometry especially for vibration-desensitized surface profile measurements in harsh environment.

Recently, beyond the well-established capacity of 3-D surface profile metrology, whitelight interferometry began to find new applications in the realm of thickness profile measurement of thin-film layers. Multi-reflection occurring within a thin-film layer causes the resultant phase of white-light interferogram to yield a nonlinear variation with wavelength. Thus, least squares fitting of the measured phase into an appropriate analytical phase model of multi-reflection permits identifying the thickness profile of film layer. Feasibility of this new application was successfully demonstrated by way of z-scanning, [8] k-scanning, [9] and dispersive k-scanning. [10] Now, in this paper an improved version of dispersive k-scanning is proposed with the intention of replacing the time-consuming, iterative least squares fitting by a fast, deterministic method of Fourier transform analysis. Consequently a simple signal processing procedure of spectrally-resolved interference data allows not only the top surface height profile but also the thickness profile to be determined simultaneously, leading to the complete reconstruction of tomographical thickness profile of a thin-film layer.

2. Principles

 

Fig. 1 Optical configuration of dispersive white-light interferometry; LP: linear polarizer, BS: beam splitter, PBS: polarizing beam splitter, ES: entrance slit, DG: dispersive grating, CCD: charge coupled device.

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Figure 1 shows the dispersive white-light interferometer configured in this investigation. A collimated beam of white light is linearly polarized and divided by a polarizing beam splitter so that the thin-film layer on the specimen is illuminated exclusively by p-polarization light, while the reference mirror only by s-polarization light. Two separate spectrometers, respectively designated S and T, are employed in parallel, each comprising an entrance slit, a diffractive grating, and a 2-D photodetector array of charge coupled devices. Spectrometer S is dedicated to only the self-interference occurring among the multi-reflected beams from the specimen, which can be done simply by blocking the reference beam of s-polarization using a linear polarizer. On the other hand, Spectrometer T receives the multi-reflected specimen beams as well as the reference beam by recombining all the beams through a 45°-rotated linear polarizer, which allows us to collect not only the self-interference of the specimen beams but also their individual cross-interferences with the reference beam. The source of white light in use is a tungsten halogen lamp emitting a Gaussian-shaped spectrum of 420 to 780 nm in wavelength, and Spectrometer S and T provide an identical wavelength resolution of 3 nm.

 

Fig. 2 Unfolded diagram depicting four major groups of spectrally-resolved interference signals between the reference and specimen. The reference beam is shown in blue and the multireflected beams from the specimen are in red. The parameter h denotes the distance between the reference mirror and the top surface, and d is the thickness of the thin-film layer.

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The spectrally-resolved interference of white light is described using the spectral density G(ν) that represents the intensity output from a spectrometer as a function of the frequency ν. The interference observed by Spectrometer T encompasses all the contributions that are conveniently grouped into four separate terms as

Fig. 2 depicts how the above four terms are generated individually. The first term G_rr(ν) is the spectral intensity of the beam directly reflected from the reference mirror, while G_nn(ν) in the second term represents the spectral intensity of the n-th reflected beam from the specimen. The third term concerns the cross-interference of the reference beam with the n-th multireflected specimen beams. Lastly, the fourth term relates to the self-interference among the specimen beams, in which the subscripts n and m correspond to the n-th and m-th reflected beams, respectively.

The third and fourth terms, G_nr(ν) and G_nm(ν), involve two time delays, τ_h and τ_d; the first τ_h is associated with the top surface height h of the film layer that is measured with respect to the reference mirror, i.e., τ_h=2h/c, c being the speed of light in vacuum. The refractive index of air is assumed unity. The second time delay τ_d relates to the thickness d of the film layer, which is given as τ_d=2Nd/c with N being the group refractive index of the film. It is obvious that identifying τ_h and τ_d from the sampled G(ν) leads to measurement of the three important characterizing parameters of a thin film; h, d, and N. To that end, Eq. (1) is Fouriertransformed into the form of

The overall result of FG(ν) is symmetrical about τ=0, since the measured signal of G(ν) is real, and appears as a series of peaks of varying amplitudes. Each peak in the positive τ domain is described as Γ[τ-τ*] with τ* being the peak center location. The peak width is affected by the spectral bandwidth of the source, which turns out to be 3.95 fs (FWHM) in our experiment. Each peak can be isolated by use of an appropriate filter such as the Heaviside function and inverse-Fourier-transformed to recover the phase ϕ(ν), following the double Fourier-transform theorem of F^-1FG(ν)=|G(ν)| exp[iϕ(ν)]. Then, using the relation of ϕ(ν)=2πτ*ν, the delay is readily calculated in the deterministic way of τ*=(1/2π)(dϕ/dν). In doing that, overlap between neighboring peaks causes failure in obtaining the true value of τ*, so the minimum measurable limit of τ* is limited by the peak width (FWHM). Note that the maximum extent of τ* is also restricted; in this case by the Nyquist sampling limit that is described as τ* ≤1/(2p) in which p is the spectral resolution of the spectrometer in use.[11]

Now Fig. 3 illustrates how τ_h and τ_d are determined by a two-step procedure using the two spectrometer signals. First, a typical pattern of G(ν) obtained from Spectrometer T is shown in Fig. 3(a) along with its Fourier-transform Γ(τ) in Fig. 3(b). The specimen was a SiO₂ thin-film layer deposited on Si substrate. Many peaks are seen in the resulting Γ(τ), as Eq. (2) implies, at every location satisfying the condition of τ*=τ_h+nτ_d or τ*=(n-m)τ_d for n, m=0,1,2, … (m≠n). Besides, those peaks are densely populated, so it is not easy to identify which peak is related to which time delay. On the other hand, Figs. 3(c) and 3(d) illustrate the results obtained from Spectrometer S, of which Γ(τ) yields evenly-spaced peaks at an interval of τ_d since only the self- interference of the specimen beams is observed in this particular spectrometer. Consequently, the first peak in the positive τ domain can be isolated without the problem of peak identification, which enables to determine the delay τ_d related only to the film thickness d. Second, the self- interference signal G(ν) of Spectrometer S is subtracted from that of Spectrometer T, which enables to separate the cross-interference signal as shown in Fig. 3(e) with its subsequent Γ(τ) in Fig. 3(f). In doing the subtraction, the measured signals of Figs. 3(a) and 3(c) need to be scaled in consideration of the sensitivities of Spectrometer S and T, so that both signals are in the same level in amplitude. In the resulting crossinterference signal of Fig. 3(f), the first peak is obviously concerned only with τ_h and the second peak with τ_h+τ_d. Note that due to exceedingly high reflectance of the bottom surface of the SiO₂ film layer, the second peak shows higher amplitude than the first peak that is related to the top surface of the film layer. Therefore the second peak is selected, of which delay τ_h+τ_d allows τ_h to be obtained more reliably by subtracting the previously determined τ_d. Accurate calculation of the film thickness d from the measured τ_d demands that the group refractive index N of the film be precisely known.

 

Fig. 3 Interference signals G(ν) vs. Fourier-transforms Γ(τ): (a) interference signal from Spectrometer T, (b) Fourier-Transform of (a), (c) interference signal from Spectrometer S, (d) Fourier-Transform of (c), (e) subtracted interference signal, and (f) Fourier-Transform of(e).

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3. Experimental results

Figure 4 depicts how the group refractive index N of the thin film can be decided using a specimen particularly prepared with a stepped layer of thin film. Two sample points, S1 and S2 are selected; one located on the thick side and the other on the thin side. As the relative location of the reference mirror maintains a constant distance with regards to the bottom surface of the film layer deposited right on the flat substrate, the following condition holds; (h+d)₁=(h+d)₂ where subscripts 1 and 2 indicate the sampling points S1 and S2, respectively. This condition is rewritten in terms of the two time delays of τ_d and τ_h as (τ_h+τ_d/N)₁=(τ_h+τ_d/N)₂.

 

Fig. 4 Group refractive index measurement using a stepped film layer: (a) top view of a stepped SiO₂ thin-film layer and (b) its cross sectional view of line A-A’.

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This relation permits N to be decided as N=-(τ_d1-τ_d2)/(τ_h1-τ_h2), which is worked out to be 1.459 for the SiO₂ layer used in this experiment. Dispersion occurring in the film layer of the specimen may cause variation of N with dependence on the film thickness, but its effect is found not significant when the thickness is less than a few micrometers.

Finally, two representative measurement results are shown in Fig. 5, which were taken from SiO₂ thin-film layers deposited on silicon substrate with thickness ranging from 0.5 to 4.0 µm. One of the specimens shown in Fig. 5(a) has a ‘0’-patterned groove etched on the top surface of the SiO₂ film of which bottom surface is flat. The other specimen shown in Fig. 5(c) has a flat top surface profile of a SiO₂ layer that was polished by CMP(Chemical-Mechanical Polishing) Process after being deposited on a check-patterned silicon substrate. These examples demonstrate that the proposed dispersive method is capable of measuring the top surface profile and the film thickness at the same time with resolutions in the nanometer range. The minimum film thickness measurable is found ~0.5 µm, being limited by peak overlap in Γ(τ). The use of 2-D spectrometers enables us to measure one line at a time, so a lateral scanning stage was used for volumetric measurements. The measurement time for a single point is worked out to be ~3 ms when a standard PC equipped with an Intel Pentium 4 CPU run at 2.93 GHz is used.

 

Fig. 5 Exemplary measurement results: (a) 3-D thickness profile of ‘0” patterned SiO₂ film layer on flat Si substrate, (b) cross sectional profile of (a), (c) 3-D thickness profile on patterned Si substrate, and (d) cross sectional profile of (c).

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4. Conclusion

To conclude, the dual-branch detection scheme of dispersive white-light interferometry proposed in this investigation allows reconstructing the tomographical view of film layers with depth resolutions in the nanometer range. The refractive index of a film layer can also be determined to an accuracy of 10^-3 without prior knowledge of geometrical thickness. The Fourier-transform signal processing procedure devised along with the proposed hardware scheme provides advantages of fast measurement speed, no need of mechanical depth scanning, and high immunity to external vibration. This method is currently restricted to single layer measurements, but extension to multiple layers is being under investigation. Finally, it is anticipated that the proposed method of dispersive white-light interferometry will be a powerful metrological tool for thin-film measurements with improved capabilities in terms of the measuring range, speed and accuracy for practical use especially for the semiconductor and flat panel display industry.