TECHNICAL NOTE

Numerical algorithm forspectroscopic ellipsometry of thick transparent films

Salvador Bosch, Julio Perez, and Adolf Canillas

We present a numerical method for spectroscopic ellipsometry of thick transparent films. When ananalytical expression for the dispersion of the refractive index ~which contains several unknown coeffi-cients! is assumed, the procedure is based on fitting the coefficients at a fixed thickness. Then thethickness is varied within a range ~according to its approximate value!. The final result given by ourmethod is as follows: The sample thickness is considered to be the one that gives the best fitting. Therefractive index is defined by the coefficients obtained for this thickness. © 1998 Optical Society ofAmerica

OCIS codes: 310.0310, 120.4530, 120.2130.

1. Introduction

Ellipsometry is a standard characterization tech-nique for thin-film manufacture because of its intrin-sic high sensitivity and its nondestructive nature.One of the particular applications in which ellipsom-etry provides the best accuracy is in the study of verythin films. In this case, spectrophotometric methodsare not useful since there are no interference extremain the transmittance or the reflectance spectra.Conversely, for relatively thick films ~several wave-lengths of the considered working range! the use ofellipsometric methods is less widespread becausethey face several drawbacks, making spectrophotom-etry the most common practice.

Recently, an interesting study of the characteriza-tion of thick transparent films by variable-angle spec-troscopic ellipsometry has been published.1 In thisstudy, the specific problems related to the thicknessof the layer have been revised. One of the topicsimplicated in this study is the numerical methodsthat have been used for the inversion of the measuredellipsometric data. In fact, the detailed treatment ofthis problem is not presented in the text, but itssignificance in the characterization of the thick layersis evident.

The authors are with the Universitat de Barcelona, Departa-ment Fısica Aplicada i Electronica, Diagonal 647, Barcelona,E-08028 Spain.

Received 24 June 1997; revised manuscript received 14 October1997.

0003-6935y98y071177-03$15.00y0© 1998 Optical Society of America

Let us analyze Figs. 2 and 3 of Ref. 1. To illustratethe model calculation proposed, the authors considera theoretical spectrum of the D, C angles correspond-ing to a 9-mm layer on a silicon substrate. Theyassume that the optical properties of the layer aregiven by the Cauchy model

n~l! 5 A 1 Byl2 1 Cyl4 (1)

with the values A 5 1.5364, B 5 5.564 3 1013 nm2,and C 5 8.010 3 1014 nm4. The spectrum is calcu-lated for an angle of incidence of 70° in the 500–850wavelength region. It is explained in the text in Ref.1 that the plots of x2 versus thickness were obtainedwith “a series of least squares minimizations to findthe coefficients ~A, B, C!, with the thickness fixed foreach minimization.” In the particular case of Fig. 2,various x2 minima are obtained, and there is no doubtof the identification of the right thickness since thecorrect one has a noticeably lower value for x2 thanthe wrong ones. In the case of working with simu-lated data having a small amount of statistic noiseadded, a plot like Fig. 3 is obtained. The authorsnote here that the minima are now closer togetherand the values of x2 at the valleys are similar. It isevident that the existence of these side minima is animportant drawback for the proposed method, be-cause it leads to the requirement of a previous deter-mination of the thickness range of the layer byprofilometer measurements.

Our aim is to propose a numerical method for theinversion of the ellipsometric data that does not leadto the existence of side spurious minima. Themethod is simple, easy to be implemented numeri-

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cally, and shows good convergence properties whentested with simulated data having statistical noiseadded.

2. Method

In its most basic implementation, the method that wepresent comprises the two following steps:

1. Perform the ellipsometric data inversion ~assum-ing a fixed thickness! by use of a monochromatic al-gorithm to find ~n, k! for each wavelengthindependently.

2. Use a best-fitting algorithm for the data ~n, k!obtained in the previous step to determine the con-stants that define the dispersion model adopted.Compute the x2 that corresponds to the best fitting.

These two steps allow for the computation of x2 forthe thickness assumed in step 1. Repeating the pro-cess for a set of thicknesses within a variation range,one gets a plot x2 versus layer thickness. We willshow that this procedure does not lead to spuriousminima and also that it is very stable against noise inthe data.

Let us explain in more detail the two steps. Step1 is a repeated application of any single-layer mono-chromatic ellipsometry algorithm that determines ~n,k! at a known thickness from the ellipsometric data~D, C!, for example, a two-dimensional simplex on thevariables ~n, k!.2 Approximate values for the un-knowns are needed to start any monochromatic algo-rithm, but it is important to note that these must besupplied only for one starting wavelength l0 since theresult ~n, k! for one wavelength may be the approxi-mate value for the next one ~as it is always quiteclose, say 10 nm apart!. In our particular case wemust supply an approximate value for the refractiveindex napprox at one of the wavelengths ~l0! of therange of measurements. Since our method assumesa fixed thickness for step 2, inverting D~l0! and C~l0!will generate n~l0! and k~l0!. Note that in the caseof computed ellipsometric data corresponding to anideal transparent layer we will have k [ 0, but fornoisy data, k will be small but nonzero ~the valuesmay increase with noise!. For the next wavelength,say l1, we may use n~l0! instead of napprox, repeatingthe monochromatic algorithm. For real ellipsomet-ric data, we expect a behavior for k~l! similar to thenoisy case. Thus at the end of step 2, we have thefull spectrum n~l! and k~l! at a fixed layer thicknessfor the wavelength range considered.

Step 2 consists of applying a best-fitting algorithmto the spectrum obtained in step 1. Thus we need toassume a dispersion formula, fit the coefficients ofthis formula, and calculate the final merit function.This merit function may be the x2 estimator, but itcould also be any other measure of the deviation be-tween data and model. In our implementation of themethod, the dispersion formula is the one shownabove and the best-fitting procedure used is the sin-gular value decomposition.3

Regarding the repetition of the process, to obtain

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the plot of x2 versus layer thickness, it is evident thatthis is the most simple version of a one-dimensionalminimization algorithm. The limits of the thicknessrange, and the step for increments defines the num-ber of times that steps 1 and 2 will be performed andalso the accuracy in the final thickness determina-tion.

3. Results

To illustrate the performances of our numerical pro-cedure, we have applied it to the same theoreticalspectrum of Figs. 2 and 3 in Ref. 1, but with differentlevels of statistical noise added to the ellipsometricdata. We start by computing the spectroscopic ~D,C! values corresponding to an ideal sample consistingof a perfect 9-mm-thick layer on a semi-infinite siliconsubstrate. Three cases have been considered: ex-act ~D, C! data, and data with both s 5 0.3° and 1.0°rms noise added. The previously described methodhas been applied to the three sets of theoretical spec-tra, when thicknesses from 8.7 to 9.3 mm were con-sidered.

Figure 1 shows two sets of refractive-index valuesand their best fittings to the Cauchy model @Eq. ~1!#by singular value decomposition corresponding to twoassumed thicknesses, 9.0 and 9.05 mm, for the datawith s 5 1.0° noise added. As explained in the de-scription of step 1, the use of the result for one wave-length as the starting point for the next one in theSIMPLEX algorithm guarantees the continuity in thevalues of the refractive index and, consequently, thesmoothness of the subsequent fitting. A problemcould appear in the vicinity of the singular pointsrelated to the thickness cycle of transparent layers,i.e., when thickness ~d!, refractive index ~n!, wave-

Fig. 1. Refractive-index values and best fittings to the Cauchymodel corresponding to assumed thicknesses, 9.0 and 9.05 mm, forthe data with s 5 1.0° noise added.

length ~l!, and incidence angle ~f! fulfill the expres-sion

d 5ml

2În2 2 sin2 fm 5 0, 1, 2 (2)

since one may obtain a meaningless value of n fornoisy data. It is unlikely to meet the conditions

Fig. 2. Plots of x2 versus layer thickness for three data sets cor-responding to noise levels s 5 0°, 0.3°, 1.0°.

leading to Eq. ~2! however, but even in this case it iseasy to protect the point-by-point reinitialization ofthe SIMPLEX algorithm against misleading values of nby considering the starting value for the initializationto be more than one wavelength, that is, by looking atthe last two or three n values found and not allowingvariations that are too large.

Figure 2 plots x2 versus layer thickness for threesets of data having s 5 0°, 0.3°, and 1.0° noise added.The interval between thicknesses is 20 nm for the fullrange and 10 nm in the inset for the zone near theminimum.

From the figure we note that the method proposedworks quite well, giving only one minima for all thenoise levels considered. One may note that there isa small but noticeable displacement of the absoluteminimum with respect to the 9-mm value. This isdue to the noise introduced in the data that maygenerate statistic fluctuations that become more im-portant as the noise level increases.

In conclusion, the numerical method proposedeliminates some of the drawbacks of the ellipsometricdata analysis developed in Ref. 1 for thick layers.

This research has been partially supported by theComision Interministerial de Ciencia y Tecnologıaunder contract TIC95-1119-C02-01.

References1. S. Guo, G. Gustafsson, O. J. Hagel, and H. Arwin, “Determina-

tion of refractive index and thickness of thick transparent filmsby variable-angle spectroscopic ellipsometry: application tobeczocyclobutene films,” Appl. Opt. 35, 1693–1699 ~1996!.

2. S. Bosch, F. Monzonıs, and E. Masetti, “Ellipsometric methodsfor absorbing layers: a modified downhill simplex algorithm,”Thin Solid Films 289, 54–58 ~1996!.

3. W. H. Press, S. A. Teukolsky, W. T. Wetterling, and B. P. Flan-nery, Numerical Recipes in C, 2nd ed. ~Cambridge U. Press,Cambridge, England, 1992!, Chaps. 10 and 15.

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