thickness-profile measurement of transparent thin-film layers by white-light scanning interferometry

6
Thickness-profile measurement of transparent thin-film layers by white-light scanning interferometry Seung-Woo Kim and Gee-Hong Kim White-light scanning interferometry is increasingly used for precision profile metrology of engineering surfaces, but its current applications are limited primarily to opaque surfaces with relatively simple optical reflection behavior. A new attempt is made to extend the interferometric method to the thickness-profile measurement of transparent thin-film layers. An extensive frequency-domain analy- sis of multiple reflection is performed to allow both the top and the bottom interfaces of a thin-film layer to be measured independently at the same time by the nonlinear least-squares technique. This rigorous approach provides not only point-by-point thickness probing but also complete volumetric film profiles digitized in three dimensions. © 1999 Optical Society of America OCIS codes: 120.3180, 120.3940, 120.6650, 240.0310, 310.6860. 1. Introduction White-light scanning interferometry produces a short coherence interferogram that permits the optical path difference between test and reference beams to be measured without 2p ambiguity. During past two decades much attention has been paid to three- dimensional surface mapping by use of white-light scanning interferometry; as a result, quite a few tech- niques for such mapping are now available. Kino and Chim 1,2 used Fourier and later Hilbert trans- forms to extract the visibility envelope function in the frequency domain and then to inversely transform the result back into the spatial domain to determine the fringe peak. Caber 3 and Cohen et al. 4 proposed a hardware demodulation scheme that isolates the visibility envelope function by signal filtering. Later, de Groot and Deck 5–7 revisited the Fourier- transform technique to locate the envelope peak di- rectly in the frequency domain by using phase information. There were some attempts to modify the well-established phase-measuring algorithms of laser interferometry to make them suitable for white- light interferometry; Larkin 8 proposed an adaptive nonlinear algorithm of five buckets, and Sandoz 9 sug- gested a seven-bucket phase-measuring algorithm. In addition, Sandoz et al. 10 investigated the possibil- ity of adopting wavelet transform as a new alterna- tive method for fringe peak detection. Although the aforementioned surface-mapping techniques differ from one another in the way in which they accomplish fringe data processing, most of them are well capable of providing nanometer reso- lutions in precision profile measurement of opaque surfaces. 11 In this paper we make an attempt to expand the interferometric principles into the new area of thin-film thickness measurement. Figure 1 shows a representative target surface of interest upon which a transparent thin-film layer has been deposited. Our intention is not simply to measure film thickness, as is usually done in ellipsometry and other thickness-measurement techniques, 12–14 but to construct a three-dimensional volumetric map of the film layer. This type of thickness-profile measure- ment is especially important for quality evaluation of the chemical mechanical polishing process that is in- creasingly adopted in microchip manufacturing for planarization of dielectric layers deposited upon pat- terned wafers. For this purpose we make an ex- tended frequency-domain analysis of white-light interferograms that takes into consideration the mul- tiple reflection that occurs from the thin-film layer. This extensive analysis allows both the top and the bottom interfaces of the film layer to be measured independently at the same time, providing a complete thickness profile that is digitized in three dimensions with a single scan of the surface. The authors are with the Department of Mechanical Engineer- ing, Korea Advanced Institute of Science and Technology, Science Town, Taejon 305-701, Korea. S.-W. Kim’s e-mail address is [email protected]. Received 4 January 1999; revised manuscript received 29 June 1999. 0003-6935y99y285968-06$15.00y0 © 1999 Optical Society of America 5968 APPLIED OPTICS y Vol. 38, No. 28 y 1 October 1999

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Page 1: Thickness-Profile Measurement of Transparent Thin-Film Layers by White-Light Scanning Interferometry

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Thickness-profile measurement of transparentthin-film layers by white-light scanning interferometry

Seung-Woo Kim and Gee-Hong Kim

White-light scanning interferometry is increasingly used for precision profile metrology of engineeringsurfaces, but its current applications are limited primarily to opaque surfaces with relatively simpleoptical reflection behavior. A new attempt is made to extend the interferometric method to thethickness-profile measurement of transparent thin-film layers. An extensive frequency-domain analy-sis of multiple reflection is performed to allow both the top and the bottom interfaces of a thin-film layerto be measured independently at the same time by the nonlinear least-squares technique. This rigorousapproach provides not only point-by-point thickness probing but also complete volumetric film profilesdigitized in three dimensions. © 1999 Optical Society of America

OCIS codes: 120.3180, 120.3940, 120.6650, 240.0310, 310.6860.

n 9

1. Introduction

White-light scanning interferometry produces a shortcoherence interferogram that permits the opticalpath difference between test and reference beams tobe measured without 2p ambiguity. During pastwo decades much attention has been paid to three-imensional surface mapping by use of white-lightcanning interferometry; as a result, quite a few tech-iques for such mapping are now available. Kinond Chim1,2 used Fourier and later Hilbert trans-orms to extract the visibility envelope function in therequency domain and then to inversely transformhe result back into the spatial domain to determinehe fringe peak. Caber3 and Cohen et al.4 proposed

a hardware demodulation scheme that isolates thevisibility envelope function by signal filtering.Later, de Groot and Deck5–7 revisited the Fourier-transform technique to locate the envelope peak di-rectly in the frequency domain by using phaseinformation. There were some attempts to modifythe well-established phase-measuring algorithms oflaser interferometry to make them suitable for white-light interferometry; Larkin8 proposed an adaptive

The authors are with the Department of Mechanical Engineer-ing, Korea Advanced Institute of Science and Technology, ScienceTown, Taejon 305-701, Korea. S.-W. Kim’s e-mail address [email protected].

Received 4 January 1999; revised manuscript received 29 June1999.

0003-6935y99y285968-06$15.00y0© 1999 Optical Society of America

5968 APPLIED OPTICS y Vol. 38, No. 28 y 1 October 1999

onlinear algorithm of five buckets, and Sandoz sug-gested a seven-bucket phase-measuring algorithm.In addition, Sandoz et al.10 investigated the possibil-ity of adopting wavelet transform as a new alterna-tive method for fringe peak detection.

Although the aforementioned surface-mappingtechniques differ from one another in the way inwhich they accomplish fringe data processing, most ofthem are well capable of providing nanometer reso-lutions in precision profile measurement of opaquesurfaces.11 In this paper we make an attempt toexpand the interferometric principles into the newarea of thin-film thickness measurement. Figure 1shows a representative target surface of interestupon which a transparent thin-film layer has beendeposited. Our intention is not simply to measurefilm thickness, as is usually done in ellipsometry andother thickness-measurement techniques,12–14 but toconstruct a three-dimensional volumetric map of thefilm layer. This type of thickness-profile measure-ment is especially important for quality evaluation ofthe chemical mechanical polishing process that is in-creasingly adopted in microchip manufacturing forplanarization of dielectric layers deposited upon pat-terned wafers. For this purpose we make an ex-tended frequency-domain analysis of white-lightinterferograms that takes into consideration the mul-tiple reflection that occurs from the thin-film layer.This extensive analysis allows both the top and thebottom interfaces of the film layer to be measuredindependently at the same time, providing a completethickness profile that is digitized in three dimensionswith a single scan of the surface.

Page 2: Thickness-Profile Measurement of Transparent Thin-Film Layers by White-Light Scanning Interferometry

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2. Multiple Reflection

When light is incident upon a surface coated with atransparent thin-film layer, multiple reflection takesplace, as illustrated in Fig. 2. If the electrical am-plitude of the incident beam is Ei, the total sum ofelectric amplitude of multiply reflected beams can beexpressed as Er 5 REi. The parameter R is definedas the total reflection coefficient and is derived as15

R 5r01 1 r12 exp~2j2b!

1 1 r01r12 exp~2j2b!, (1)

where r01 and r12 represent the Fresnel reflectioncoefficients of the top and the bottom boundaries ofthe film, respectively, and b denotes the phasechange that the reflected wave experiences as ittraverses the thin-film layer once from one boundaryto the other. b may be expressed explicitly as

b 5 kdN1 cos u1, (2)

where k and u1 are the wave number and the propa-gation angle of the incident beam and d and N1 arethe thickness and the refractive index of the thin-filmlayer, respectively.

In fact, N1 varies with k; thus the well-knownCauchy equation that provides an accurate nonlinear

Fig. 1. Measurement of the thickness profile for reconstruction ofa complete three-dimensional map of transparent thin film layers:~a! sectional view of a patterned wafer, ~b! measured thicknessprofile of the film layer.

relationship between N1 and k in the visible region isadopted16:

N1 5 A 1Bl2 1

Cl4 , (3)

where A, B, and C are constants that are character-istic of the film material. When Eqs. ~3! and ~2! aresubstituted into Eq. ~1!, the total reflection coefficientR becomes a complex number that can be arranged inthe standard form R 5 a 1 jb, where a and b are realnumbers. Then the total phase difference of the re-flected beam relative to the incident beam is deter-mined as

C 5 /R 5 tan21~bya!, R 5 a 1 jb. (4)

Note that the phase difference C includes not onlyhe phase change that is due to the optical pathength of the thin-film thickness but also the phasehange on reflection at each boundary. The lattererm is defined by the Fresnel reflection coefficients01 and r12 as described in Eq. ~1!. The reflection

coefficients turn out to be not real but complex num-bers with complicated behavior in practical measure-ments.17,18 Thus, for accurate modeling of lightreflection, careful experiments are necessary to de-termine the precise values of the complex reflectioncoefficients for target materials.19–21 Once the re-

ection coefficients and the refractive indices andropagation angles of all the involved media arenown precisely, the total phase difference C can beegarded as a function that depends merely on waveumber k and film thickness d.

Fig. 2. Multiple reflection of light within a transparent thin-filmlayer.

1 October 1999 y Vol. 38, No. 28 y APPLIED OPTICS 5969

Page 3: Thickness-Profile Measurement of Transparent Thin-Film Layers by White-Light Scanning Interferometry

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3. Frequency-Domain Analysis

Figure 3 is a basic schematic of the white-light scan-ning interferometry considered in this investigation,which includes Michelson, Mirau, and Linnik inter-ferometers. A piezoelectric actuator moves the ob-jective along the z direction toward the test surfacelocated in the xy plane. The resultant white-lightnterferogram is considered an incoherent superposi-ion of individual interferograms of all the monochro-atic waves that constitute the source light.5 Thus

he interference intensity sampled from a surfaceoint ~x,y! of height h is generally described in the

form of double integral1:

Iw~z! 5 g *kc2Dky2

kc1Dky2

F~k! *0

u0

cos@2k~h 2 z!cos u

1 c~k, d, u!#sin u cos ududk, (5)

where g denotes the modulation amplitude and F~k!,kc, and Dk are, respectively, the spectral distributionfunction, the central wave number, and the band-width of the white-light source. In Eq. ~5!, the innerintegral is performed over the full aperture of theobjective whose maximum ray inclination angle is u0,and the outer integral is related to the broad spectraldistribution of the source light.

For simplicity of analysis, the numerical aperture~N.A.! of the objective is assumed to be small enoughthat the approximations cos u ' 1 and sin u ' u apply.Thus Eq. ~5! is simplified as

Iw~z! 512

gu02 *

kc2Dky2

kc1Dky2

F~k!cos@2k~h 2 z! 1 c~k, d!#dk.

(6)

970 APPLIED OPTICS y Vol. 38, No. 28 y 1 October 1999

In addition, with the same assumption of small u, thereflection coefficient of Eq. ~1! is also approximatedas18

rij 5Ni 2 Nj

Ni 1 Nj, rij 5 r01, r12, (7)

which is valid regardless of polarization. Now, if theintensity of Eq. ~6! is Fourier transformed, it is ap-parent that the phase distribution along the spectralwave number is obtained as

Fc~k; h, d! 5 2hk 1 c~k, d!. (8)

It is worthwhile to note that the parameters g andF~k! make no contribution to the phase distributionbecause their values are all real. Thus the phasedistribution Fc~k! is affected mainly by two combinederms, as depicted in Eq. ~8!. The first term, 2hk, isn linear proportion to k and is induced only by theeflection of light from the top boundary of the thin-lm layer. The second term, c~k, d!, represents thehase variation that is due to multiple reflection fromhe thin-film layer. If the thickness of the thin-filmayer is negligible, i.e., if d ' 0, the phase c~k, d!ecomes constant, as it is determined simply by thehase change of light on reflection on the top bound-ry. This case is identical to that in the previoustudy by de Groot and Deck.5–7 When thickness d isonsidered, c~k, d! turns out to be nonlinear withegard to k in a sinusoidal fashion, as implied by Eq.1!. This fact is well demonstrated in the two actualhase distributions shown in Fig. 4, one measuredrom a solid mirror surface and the other from aiO2-coated Si-substrate surface. In the former

case, the nonlinear phase variation is negligible andthe overall phase distribution follows the linear rela-tionship of 2kh. In contrast, in the latter case a

onlinear trend becomes significant because of theultiple reflection behavior explained so far.

4. Nonlinear Least-Squares Fitting

Now we turn our attention to determining surfaceheight h and thickness d. For this, we let Fm~k! bethe actual data of phase distribution computed froma sampled white-light interferogram. An error func-tion is then defined in the integral form of squareddeviations:

x~h, d! 5 *kc2Dky2

kc1Dky2

@Fm~k! 2 Fc~k; h, d!#2dk. (9)

The unknowns h and d are regarded as independentariables and are determined such as to minimizehat error function. The phase component c~k, d! in

Fc turns out to be highly nonlinear with respect to theunknowns; thus an appropriate multidimensionalnonlinear least-squares technique with fast conver-gence capability is necessary. In this investigationwe use the Levenberg–Marquardt nonlinear least-squares algorithm provided by the commercially

Fig. 3. Schematic of white-light scanning interferometry.

Page 4: Thickness-Profile Measurement of Transparent Thin-Film Layers by White-Light Scanning Interferometry

available software MATLAB22 to perform a feasibilitytest.

In searching for the true values of h and d we havefound that many local minima exist, so it is necessaryto make good initial guesses at these values to reachthe global minimum quickly. Figure 5 shows a se-ries of computer simulations that we performed toverify the convergence of the least-squares technique.In the simulation the necessary interferometric in-tensity data were generated from Eq. ~5! for a SiO2layer 0.001–2.000 mm thick deposited upon a Si sub-strate. The wavelength of the white-light source

Fig. 4. Actual phase distributions measured from two representat

Fig. 5. Simulation results for comparison of true and measuredthickness values.

was set to be in the range 452–732 nm, and the N.A.of the objective was taken as 0.3. The dotted–dashed line in the figure shows the true thicknessvalues input to the simulation, and the filled squaresdenote the output thickness values that were finallyfound. The simulation result reveals that 1023 frac-tional convergence is attained within 20 iterations inmost cases. However, when the thickness is lessthan 0.05 mm, correct convergence is not guaranteed;it depends a great deal on the initial guesses of h andd. This is so because the phase change within thethin-film layer becomes either 0 or p regardless ofwave number when thickness d is small; conse-quently the sinusoidal nonlinear variation of thephase term of c~k, d! reduces to a negligible level atwhich the effect of thickness d is relatively insignif-icant.

5. Experiments and Discussion

Independent determination of d and h facilitatesvolumetric mapping of a thin-film layer with both thetop and the bottom taken separately and digitized inthree dimensions. Figure 6 displays an exemplaryresult measured from a test sample: The bottomhas the numerals 52 patterned with tungsten upon aSi substrate; the top is a SiO2 film layer planarized bythe chemical mechanical polishing process. The lat-eral dimensions of the target area are 40 mm 3 25mm, with a pixel resolution of 0.5 mm. The objectiveused was of the Mirau type with a N.A. of 0.4 at 203magnification. The measured film thickness lies inthe range 0.1–0.7 mm.

The measurement accuracy of the proposed tech-

rfaces: ~a! an uncoated mirror and ~b! a SiO2-coated Si substrate.

ive su

1 October 1999 y Vol. 38, No. 28 y APPLIED OPTICS 5971

Page 5: Thickness-Profile Measurement of Transparent Thin-Film Layers by White-Light Scanning Interferometry

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Table 1. Comparison of Sample Step Heights Measured

5

nique was estimated through an indirect comparisonmade with measurements obtained with a stylus-type profiler widely used in the semiconductor indus-try. First, a uniform layer of SiO2 thin film of.2-mm thickness was deposited upon a Si substratend etched into various step heights in the range of0–200 nm. Then their thicknesses were measuredoth with the stylus profiler and with our apparatus.

point-by-point comparison of the measured dataas not feasible because of practical difficulties inatching correspondence points between the two in-

truments. Instead, the average step heights wereomputed from the measured data and were com-ared as summarized in Table 1. The results revealhat the maximum difference between measure-ents made with the two techniques is ;10 nm.We speculate that the main source of error in our

method arises from the first-order approximation ofthe incident angle, which was inevitably made in Eq.~6! for simplicity of analysis. In fact, the reflectionbehavior of light varies with the incident angle andthe target materials in complex ways that were re-ported previously.17–21 Our simplification conse-uently gives rise to measurement errors, especiallyhen high-N.A.-value objectives are used. This

peculation is well verified in the measurement re-ults of Table 2, which we obtained by using threebjectives with different N.A. values. The test re-

Fig. 6. Exemplary measurement result: ~a! microscopic top viewof the test sample, ~b! measured three-dimensional thickness pro-file.

972 APPLIED OPTICS y Vol. 38, No. 28 y 1 October 1999

sult indicates that the maximum deviation reaches;3.0% when the N.A. value is in the range 0.3–0.5,which is that of most commercially available objec-tives.

Figure 7 illustrates another interesting observa-tion that the phase distribution varies in the spectraldomain when it is measured by objectives with dif-ferent N.A. values, although it is sampled from anidentical surface. We believe that the reason for thisvariation is that the equivalent wavelength of lightwithin the objective is modified by the actual angle ofinclination over the aperture. This conclusion re-quires that further studies be made to improve theaccuracy of measurement of thin films by more-accurate determination of the phase shift of light onreflection with the real equivalent wavelength mod-ified by the actual optical path within the objective.

Fig. 7. Experimental results, showing movement of the phasespectrum with various objective magnifications.

by Two Methods

SampleNumber

Measurement Method

Step Height Measured byStylus Profiler ~nm!

Step Height Measured byWhite-Light ScanningInterferometry ~nm!

1 201.0 194.52 173.5 172.23 130.0 135.54 97.5 89.35 25.3 27.2

aThe stylus profiler used in this comparison was an a-step modelmanufactured by the Tencor Company.

Table 2. Comparison of Sample Thicknesses Measuredwith Three Objectives

Magnification ofObjective ~N.A.!

Measured Thickness ~nm!

Sample 1 Sample 2 Sample 3

103 ~0.3! 358.7 511.3 1164.5203 ~0.4! 361.2 506.6 1137.7503 ~0.5! 357.8 520.5 1197.9

Page 6: Thickness-Profile Measurement of Transparent Thin-Film Layers by White-Light Scanning Interferometry

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tection in white light interferometry,” J. Opt. Soc. Am. A 13,

6. Conclusions

We have proposed a new technique for measuring thethickness of transparent thin films through an ex-tended frequency-domain analysis that combineswhite-light scanning interferometry with nonlinearleast-squares fitting. This technique is capable notonly of probing thickness point by point but also ofproviding complete volumetric film profiles digitizedin three dimensions. At present this measurementtechnique provides nanometer resolutions of verticalheight when the film thickness exceeds 0.05 mm.

urther study is consequently necessary for betteronlinear fitting convergence, especially for filmshat are less than 0.05 mm thick. In addition, aore rigorous reflection model is needed to improve

he measurement accuracy from the current level of10 nm by more-accurate determination of phase

hift and equivalent wavelength.

References and Notes1. G. S. Kino and S. S. C. Chim, “The Mirau correlation micro-

scope,” Appl. Opt. 29, 3775–3783 ~1990!.2. S. S. C. Chim and G. S. Kino, “Three-dimensional image real-

ization in interference microscopy,” Appl. Opt. 31, 2550–2553~1992!.

3. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl.Opt. 32, 3438–3441 ~1993!.

4. D. Cohen, P. J. Caber, and C. Brophy, “Rough surface profilerand method,” U.S. patent 5,133,601 ~28 July 1992!.

5. P. de Groot and L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18,1462–1464 ~1993!.

6. L. Deck and P. de Groot, “High-speed noncontact profiler basedon scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 ~1994!.

7. P. de Groot and L. Deck, “Surface profiling by analysis ofwhite-light interferograms in the spatial frequency,” J. Mod.Opt. 42, 389–401 ~1995!.

8. K. G. Larkin, “Efficient nonlinear algorithm for envelope de-

832–843 ~1996!.9. P. Sandoz, R. Devillers, and A. Plata, “Unambiguous profilom-

etry by fringe-order identification in white-light phase-shiftinginterferometry,” J. Mod. Opt. 44, 519–534 ~1997!.

10. P. Sandoz, “Wavelet transform as a processing tool in white-light interferometry,” Opt. Lett. 22, 1065–1067 ~1997!.

11. K. Creath, “Sampling requirements for white light interferom-etry,” presented at the Third International Workshop on Au-tomatic Processing of Fringe Patterns, Bremen Institute ofApplied Beam Technology, Bremen, Germany, 15–17 Septem-ber 1997.

12. L. J. Fried and H. A. Froot, “Thickness measurements of sili-con dioxide films over small geometries,” J. Appl. Phys. 39,5732–5735 ~1968!.

13. T. M. Merklein, “High resolution measurement of multilayerstructures,” Appl. Opt. 29, 505–511 ~1990!.

14. S. Diddams and J. C. Diels, “Dispersion measurements withwhite-light interferometry,” J. Opt. Soc. Am. 13, 1120–1129~1996!.

15. R. Azzam and N. Bashara, Ellipsometry and Polarized Light~North-Holland, Amsterdam, 1987!, p. 283.

16. F. F. Jenkins and H. E. White, Fundamentals of Optics, 4th ed.~McGraw-Hill, New York, 1976!, p. 479.

17. J. F. Biegen, “Determination of the phase change on reflectionfrom two-beam interference,” Opt. Lett. 19, 1690–1692 ~1994!.

18. T. Doi, K. Toyoda, and Y. Tanimura, “Effects of phase changeson reflection and their wavelength dependence in optical pro-filometry,” Appl. Opt. 36, 7157–7161 ~1997!.

19. G. D. Feke, D. P. Snow, R. D. Grober, P. de Groot, and L. Deck,“Interferometric back focal plane microellipsometry,” Appl.Opt. 37, 1796–1802 ~1998!.

20. K. Leonhardt, U. Droste, and H. J. Tiziani, “Topometry forlocally changing materials,” Opt. Lett. 23, 1772–1174 ~1998!.

21. E. W. Rogala and H. H. Barrett, “Phase-shifting interferome-teryellipsometer capable of measuring the complex index ofrefraction and the surface profile of a test surface,” J. Opt. Soc.Am 15, 538–548 ~1998!.

22. The Levenberg-Marquardt function is available as LEASTSQ inthe MATLAB software. For details, refer to Optimization Tool-box for Use with MATLAB ~MathWorks, Natick, Mass., 1992!.

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