two-step regression procedure for the optical characterization of thin films

Two-step regression procedure for the optical

characterization of thin films

S. V. Babu, Moses David, and Ramesh C. Patel

A diode array rapid scan spectrometer is used for measuring the intensity of polychromatic light in the 300-

420-nm range reflected from a diamondlike carbon film as a function of wavelength. With a fixed grating

setting, the wavelength range of 120 nm can be covered in 23 ms. From the reflected intensity, a new two-step

regression procedure is utilized to determine refractive index, bandgap, slope of the absorption edge, and film

thickness. The calculated parameters are independent of the starting set and the sequence of parameterestimation. The accuracy of the regression procedure is verified by comparison to the envelope method. It is

shown using simulated data that, for strongly absorbing films, the new regression procedure is more accurate

than the envelope method. The new regression method can handle very noisy reflectance spectra also. Key

words: Diode array reflectometry, regression in reflectometry, two-step regression, envelope method, optical

characterization, optical parameters, refractive index, absorption coefficient, bandgap.

1. Introduction

The accurate determination of optical propertiesand the thickness of thin films is desired in severalindustrial processes and in thin film characterizationinvestigations. Rapid monitoring of changes in opti-cal and geometric parameters of thin films is necessaryin such diverse applications as endpoint detection, insitu surface characterization during deposition, etch-ing, and surface modification. Optical parameterssuch as a refractive index and absorption coefficientare dependent on bonding configurations in the mate-rial composing a thin film and are thus useful indicesfor thin film characterization.

Several techniques are available for this task basedon measurement of the intensity of reflected and/ortransmitted light as a function of wavelength and/orangle of incidence. Reflectance and transmittance oflight are periodic functions of the path length differ-ence between waves reflected from the air-film andfilm-substrate interfaces, respectively. This pathlength difference is related to the angle of incidenceand the wavelength of incident light as well as the filmthickness. Hence, instead of monitoring reflectance/transmittance changes with changing wavelength or

All authors are with Clarkson University, Potsdam, New York13676; R. C. Patel is in the Chemistry Department, the other authorsare in the Department of Chemical Engineering.

Received 5 January 1990.0003-6935/91/070839-08$05.00/0.© 1991 Optical Society of America.

angle of incidence, the variation with changing filmthickness (for growing or dissolving thin films) is alsomeasured.'

Traditional techniques for implementing wave-length or angle of incidence changes rely on the me-chanical movement of optical components. For exam-ple, rotating analyzers are used for the measurement ofangular dependence of reflectance while rotating grat-ings are employed for directing the monochromaticlight to film surface. In such cases, the scan rate islimited by the speed of mechanical movement of thecomponents. Similarly when reflectance is measuredas a function of film thickness (for growing or dissolv-ing thin films), optical parameters are calculated fromchanges that occur over time periods that depend onthe rate of change of film thickness. In cases where thefilm thickness changes slowly but the optical parame-ters change rapidly, rapid monitoring is not possible bysuch means.

A recent development in optical monitoring is theuse of array detectors for wideband monitoring of thetransmittance of multilayer coatings.2-4 In this appli-cation, the deposition of any given layer is terminatedwhen the spectral transmittance profile approachesthe design value. Rapid monitoring is enabled by thearray detector, allowing the accurate termination of adeposition cycle. A film thickness measurement de-vice utilizing polychromatic light and a diode arraydetector is also being marketed commercially by Pro-metrics. The present technique is similar and utilizespolychromatic light incident normal to the film sur-face. An already deposited diamondlike carbon(DLC) film is used here for demonstrating the utility of

1 March 1991 / Vol. 30, No. 7 / APPLIED OPTICS 839

the technique. A simple experimental arrangementutilizing a bifurcated fiber optic cable has been devisedfor the measurement of thin film reflectance. Thereflected light is analyzed by a spectrograph and diodearray detector. Reflectance is measured for a fixedgrating setting by scanning the 300-420-nm wave-length region. On normalization and smoothing of thedetected signal, a two-step regression procedure isused for the determination of the four unknown pa-rameters: refractive index n2; bandgap Eg; slope of theabsorption edge m; and film thickness d.

The time required for a single scan is only 23 is. Inthe results presented in this paper this fast scan capa-bility is not important. However, the technique isbeing applied to characterize thin film growth process-es in situ in our plasma reactor. In this and otherapplications the rapid scan feature is extremely useful.

Many methods have been developed for the com-plete optical analysis of thin films by measurement ofreflectance and/or transmittance measured as a func-tion of wavelength or angle of inoiderice.5-10 For weak-ly absorbing films, several methods are available forthe calculation of optical parameters from the enve-lopes of reflectance and/or transmittance. However,experimentally it is more convenient and simpler tomeasure either the reflectance or the transmittancebut not both. For weakly absorbing films, methodshave been developed for complete optical analysis us-ing reflectance measurements alone.7 10 Explicit for-mulas valid for extrema in reflectance and transmit-tance enable calculation of the spectral dependence ofthe optical constants.

Methods based on nonlinear regression work effi-ciently when the dependent variable varies monotoni-cally, such as when reflectance or transmittance ismeasured as a function of angle of incidence. Regres-sion methods work poorly when the dependent vari-able is oscillatory in nature, such as when reflectanceor transmittance is measured as a function of wave-length. Estimated parameters depend greatly on thestarting solution and their order of determination."These problems are avoided in the new two-step re-gression technique described in this paper. The later-al position of the oscillatory reflectance wave, which isestimated in the initial regression step, is fixed first.The optical parameters are estimated in the secondand final regression analysis. The initial approxima-tion, which is quite accurate for weakly absorbingfilms, gives rise to a final solution, which is found to bevalid even for films having medium to strong absorp-tion. The efficiency of this new technique is comparedwith the solution obtained by the envelope method.7The new regression method is more accurate than theenvelope method when the attenuation parameter adis large.

II. Theory

The theory for the reflectance of light from a thinfilm is well known and is expressed in terms of theFresnel coefficients and the path length difference asfollows1 2 :

R = RR*,

where

r 2tltl exp(-2iW)R = r, +

1 + rjr2 exp(-2i) '

(1)

(2)

are ri and r2 are Fresnel reflection coefficients at theair-film and film-substrate interfaces, respectively, t1and t are Fresnel transmission coefficients for thetransmission of light from air to film and film to air,respectively, and is the normalized path length dif-ference [= (27rn2d)/X for normal incidence]. The Fres-nel coefficients are in turn related to the refractiveindices nl,n2,n3 for air, film, and substrate, respective-ly, and the film thickness d as follows:

nl -n2nl + n2

n 2 -n32 n2 + n3 ;

= 2n,nl + n2

= 2n2nl + n2

(3)

(4)

(5)

(6)

For an absorbing medium with an extinction coeffi-cient k, the corresponding refractive index is replacedby the complex refractive index n - ik. For noncrys-talline materials, k may be related to the bandgap Egand the slope m of the absorption edge by the Taucrelation 3:

ahv = m(hv -Eg)2;

47rkaX

(7)

(8)

When the reflectance is measured experimentally,nonlinear regression can be used for estimating thefour parameters n2 , d, E and m. However, in thiscase, the regression analysis with the four unknowns asindependent parameters is very inefficient due to theoscillatory nature of the reflectance. The curve fittingprocedure is optimized by recognizing that the loca-tions of the extrema in the reflectance curve areuniquely determined in a first approximation by speci-fying the product n2d.12 In this approximation, whichis applicable for weakly absorbing films, extrema occurat wavelengths specified by

Am = (mn2d)- m = 1,2,.... (9)

i.e., the locations of the extrema are uniquely deter-mined by n2d. In this approximation, the other threeparameters influence only the amplitudes of the oscil-lations but not their location.

If m' is the order of an extremum measured from anarbitrary reference extremum (of order mo), then m =m + m' and

1 m0 m

Am 4fl2d 4tl2 d (10)

840 APPLIED OPTICS / Vol. 30, No. 7 / 1 March 1991

0.00400

o 0.00300 / id,

E .

a~~~~~~~~~~~~a

0.00200_ , no0.00200 / wS '¢ § a nd=2948

nd=1865nd=1366

0.00100 1400 4.00 34.00Order of Extrema

Fig. 1. Verification of Eq. (9) for data simulated using different

values of n2d and equation set (1)-(8). The simulated data pointsare represented by symbols, while the straight lines are linear fits of

the data.

Hence, for a constant value of n2d, a plot of 1/Xm vsm' will yield a straight line of slope 1/(4n2d) and inter-cept mo/4n2d). The accuracy of Eq. (10) may be veri-fied for data generated using the exact Eqs. (1)-(8).Figure 1 shows the excellent linearity of plots of 1/Xm vsm for typical reflectance curves obtained for variousvalues of n2d and covers a large number of extrema.The values of n2d calculated from the slopes of theselinear lots are also in excellent agreement with thevalues used for data generation with Eqs. (1)-(8).This result is extremely useful in simplifying the curvefitting algorithm for the experimental reflectancedata.

Finally, if n2 is not a constant but is described well bya Cauchy type relationship,

B2A 2+ C' (11)n2= A + 2

then

n2d = Ad + xd + Cd A' + B + C4 (12)

In such a case, a more involved nonlinear regressionhas to be used for determination of the three parame-ters A', B', and C' from the locations of the extrema inthe reflectance curve.

Determination of the product n2d, which has either aconstant or avariable dependence on wavelength, willpermit the specification of the exact positions of theextrema for a subsequent nonlinear regression analysisto be performed on the reflectance data. Nonlinearregression analysis may be performed by the least-squares technique in which the sum of the squares ofdeviation in reflectance is minimized. For this pur-pose the exact equation set (1)-(8) is used. The Le-venberg-Marquardt method is the technique used inthis paper for specifying the route toward minimiza-tion.1 415 During the final regression analysis, errorminimization is performed by searching from a family

0.30 fl A

C

0.20

0.10 ii ii

60 310 60 410 46Wavelength, nm

Fig. 2. Simulated data show the lack of correlation between the

sum of squares of error (SSE) and the derivation from the truesolution when the extrema are not matched. Here the deviation of

curves A and B is measured from C.

of curves whose extrema are located at wavelengthsidentical to those of the experimental reflectance. Ifthe locations of the extrema are not specified duringregression, the relative magnitudes of error betweentwo different estimates need not necessarily correlateto deviation from the true solution. This is explainedfor two different simulated curves, A and B in Fig. 2,which are compared with curve C. The film parame-ters used for calculation of curve A are very close tothose for C. The summation of the squares of devi-ation of curves A and B from C is performed, and theresults are shown in Fig. 2. In a least-squares sense,curve A deviates much more from curve C than doescurve B, even though the parameters of curve A arevery close to those of curve C. In a least-squares sense,the search for a minimum may, therefore, proceed inthe wrong direction, and convergence is not guaran-teed unless the locations of the extrema are fixed.

111. Determination of Optical Constants and Thickness of

DCL Film

A. Experimental

For the purpose of demonstrating the utility andreliability of the regression method, a rf plasma depos-ited DLC film on silicon substrate from butadienesource gas has been used in the interferometric mea-surements. The details of the deposition process aredescribed elsewhere.16 A bifurcated fiber optic cableis used for directing polychromatic light normal to thefilm surface, as shown in Fig. 3. The same end of thefiber optic cable is used for gathering reflected lightfrom the sample surface and for transporting it to aspectrograph-diode array detector assembly (Prince-ton Instruments model IRY 700) interfaced to a com-puter.

The diode array rapid scan spectrometer (DARSS)is capable of collecting the entire reflectance spectrum


0.40

- Experimental

ExperimentalModel Flt

AlXP I "I Il lJ

310 360Wavelength, nm

Polychromalic Light Sample

Fig. 3. Schematic of the experimental setup used for measuringreflectance.

Fig. 4. Comparison of the experimental DLC film reflectance andthe two-step regression solution.

B. Results for DLC Film

1. Estimation of n2dThe value of n2d is sensitive to the calibration of the

wavelength-diode number relationship for theDARSS. This relationship is linear as shown below:

X = aX+ b,

very rapidly. For a fixed grating setting, the 300-420-nm wavelength range is scanned in 23 ms with the DLCfilm. However, this rapid scan capability is not impor-tant to the measurements discussed here. The simpleexperimental arrangement utilizing a fiber optic cableenables process monitoring in remote locations where,the installation of viewports and other optical compo-nents is not possible or is inconvenient. The accurateestimation of the product n2d requires that a sufficientnumber of extrema exist in the reflectance of the filmin the scanned wavelength range. The number ofextrema that occur in a given wavelength range isdirectly proportional to the film thickness and, there-fore, imposes a limitation on the minimum film thick-ness that can be monitored accurately. Scanning overlonger wavelength ranges or scanning with multiplegrating settings may be necessary for very thin films.

The detected signal is normalized by recording areference reflectance spectrum for quartz-air interfacesince the optical properties for quartz are wellknown.17 The following equation specifies the nor-malization procedure:

R = ( I) X Rq (13)

in which 'm and Iq are the reflection intensities fromthe DLC film and quartz, respectively, Id is the dark'current, and Rq is the reflectance of the quartz-airinterface. Prior to normalization, the recorded spec-tra have been smoothed twice using a Savitsky-Golaysmoothing routine with nine convolution points.' 8

The smoothed reflectance spectrum is shown in Fig. 4.

(14)

in which X is the diode number and a and b are calibra-tion constants. Of the two constants, b is vey sensitiveto the grating setting. One of the side benefits of thisdata fitting technique is the ability to correct thiscalibration by the following procedure:

(1) Convert the diode numbers to wavelengths us-ing an approximate value of b and associate the experi-mentally measured extrema with the resulting wave-lengths.

(2) Estimate n2d from a plot of 1/Xm+l vs m.(3) Generate a reflectance curve using the value of

n2d from step (2). [The approximate values of Eg andm may be used for this purpose since the locations ofthe extrema are independent of them, Eq. (9).]

(4) Compare the experimental and calculated loca-tions of extrema.

(5) Repeat the calculation with incrementalchanges in the value of b until the extrema are accu-rately positioned.

This procedure has been used to correct for thewavelength-diode number calibration for the experi-mental measurements. The locations of the extremaare identified and numbered starting from the maxi-mum wavelength. A plot of 1/Xm vs the order of theextremum m yields a straight line of slope 1/(4n2d).This plot is shown in Fig. 5. From this linear depen-dence, it may be concluded that the refractive index ofthe film exhibits negligible wavelength dependence inthe scanned wavelength range.

2. Estimation of n2, d, Eg, and mThe value of n 2d obtained above is used as an input

for nonlinear regression analysis utilizing an Interna-


0.30

) 0.20

0a4. 6UU,

Z4-U,LK 0.1 0

0-0 410 460i _ . 4 _ :: _. . . . . . . . . . . . . . . . . .

I V

-1-1 . . . . . . . . I . . . . . . . . .

0.0034

o 0.0030

_0E..

0.0026

0.0022 .' . . . . .' 4 ...8 ....... 1 2Order of Extrema

Fig. 5. Linear dependence of 1/X with the order of extrema indi-cates negligible dispersion in the refractive index of the DLC film inthe scanned wavelength range. Experimental data points are repre-sented by symbols, while the straight line is a linear fit of the data.

tional Mathematics Standards Library subroutineRNLIN for performing the error minimization. Theinput parameters for regression include the refractiveindex n1 for air, the complex refractive index n3 - ik3for the substrate, and product n2d, and the normalizedreflectance data at 660 equally spaced intervals. Thefilm parameters calculated by the regression proce-dure are shown in Table I.

Also shown in this table are film parameters calcu-lated by the envelope method. Of the several varia-tions of the envelope method available, a recent meth-od developed by Ohlidal and Navratil for calculation ofthe optical parameters from reflectance measurementsalone (most other methods require a combination ofreflectance and transmittance spectra) is used for thecomparison. 7 In the envelope method chosen, thespectral dependence of reflectance is measured, andexplicit formulas allow the calculation of optical con-stants, their dispersion, and film thickness. Pub-lished literature values19 of these parameters for DLCfilms are also shown in the table and compare favor-ably with the parameters calculated by the regressionprocedure. The fitted reflectance spectrum is com-pared to the experimental curve in Fig. 4 to display theexcellent agreement.

The values of ad for the DLC film in the 300-420-nm

wavelength range lie between 1 and 3. In this range,the optical parameters of the DLC film calculatedusing the envelope method and the new two-step re-gression method are essentially identical. However, itis shown below that there exist a range of optical pa-rameters that include those of thin films of Au, AlN, a-SiH, etc. for which only the two-step regression meth-od yields accurate values. Physically this correspondsto the case of strongly absorbing films in the specificwavelength range of interest. Furthermore, as shownin the next section, the new regression method is alsocapable of yielding the correct set of parameters evenwhen the noise level in the reflectance signal is signifi-cant, as may be the case of films with rough surfaces.

IV. Accuracy of the Two Methods

The following calculations are performed to-test theaccuracy of the two-step regression method. All thetwo-step regression calculations have been performedin double precision arithmetic. First, reflectance dataare generated using equation set (1)-(6) using assumedvalues of parameters n2, d, and k2. A single step non-linear regression analysis with n2, k2, and d as parame-ters is performed initially from the calculated R vs Xdata. The values of the parameters calculated withthis method fluctuate greatly depending on the start-ing values and on their order of estimation. In con-trast, in repeated trials (about forty) the two-step re-gression procedure described in this paper has alwaysconverged to the correct set of values despite the ap-proximation in the first step.

In addition, the sensitivity of the regression methodto the inclusion of noise, always present in experimen-tal data, is investigated for different values of k2 anddifferent degrees of random noise (0-70%) added to thecalculated spectra. The spectra are calculated at 660equally spaced intervals in the chosen wavelengthrange, each interval corresponding to an active ele-ment in the diode array detector used in this study.The random noise is generated by creating a randomnumber between -1 and 1 and scaling it with respect tothe maximum amplitude of the interference spectrum.The calculated spectrum for a noise level of 70% isshown in Fig. 6. The spectrum appears as a thick banddue to the very fine X interval over which noise isadded. The parameters calculated by the two-stepregression method for different levels of noise and k2are shown in Table II. The parameters are accurately

Table I. DLC Film Parameters Estimated by Regression and Published Literature Values19

Regression Method Envelope Method LiteratureValues

Film Thickness, d 1443.1 nm 1437.1 nm

Refractive Index, n2 2.03 2.04 1.8-2.2

Band Gap, Eg 2.6 eV 2.6 eV 0.5-2.8 eV

Slope Abs. Edge, m 3.38x1o-3 eV'nm-' 3.26x 10-3 eV'nm-' 3 x 10-3 eV'nm-


0.13

0.11

0.o02 310 360Wavelength, nm

410 460

Fig. 6. Example of simulated data used for the two-step regressionanalysis. The noise level is measured as a percentage of the maxi-

mum amplitude of the reflectance oscillation.

calculated despite the very high noise level of 70%. Asthe noise level approaches 100%, distinguishing theinterference oscillations from the noise becomes diffi-cult and optical parameters may no longer be estimat-ed accurately.

Finally, the accuracy of the two-step regression

method is compared with that of the envelope methodusing data generated for different values of k2 andassuming negligible dispersion. The results of the twomethods are compared in Tables III and IV for twodifferent film-substrate systems. The data in TableIII are generated for X = 300-420 nm, and the parame-ters are representative of DLC films on silicon. Thedata in Table IV represent a thin silicon film on glass,and the wavelength range covered is from X = 400 to900 nm. This corresponds very closely to the systemstudied by Ohlidal and Navratil7 but with dispersionneglected.

It can be seen from these two tables that both theenvelope method and the two-step regression proce-dure yield similar values for weakly absorbing films.However, with increasing values of k2, the envelopemethod predicts increasingly inaccurate values. Thevalues of k2 beyond which the regression method ismore accurate is seen to be different for the two sets ofconditions. The attenuation parameter ad may bemore useful in characterizing the accuracy of the twomethods. In Table III, at k2 = 0.1 and n2 = 2.0, thevalue of ad changes from 4.2 at X = 300 nm to 3.0 at X =420 nm. In Table IV, at k2 = 0.5 and n2 = 2.0, the valueof ad changes from 3.3 at X = 400 mm to 1.5 at X = 900nm. This appears to indicate that when ad >> 1, theenvelope method yields inaccurate parameters.

For the DLC film chosen above, the value of ad

Table II. Parameters Calculated by the Two-Step Regression Method for Different Values ofk2 and Noise Levelsa

Starting n2 2.0 2.0 2.0 2.0

Starting k2 0.001 0.01 0.1 0.2

Solution with Onoise

n2 2.008 2.0022 1.99990 2.00007

k2 0.00100 0.010019 0.09995 0.19966

Solution with 10%noise

n2 2.004 2.0008 1.99989 2.00008

k2 0.00099 0.009994 0.09992 0.19963

Solution with 50% noise

n2 1.989 1.9951 1.99985 2.00011

k2 0.00097 0.009889 0.09981 0.19945

Solution with 70%noise

n2 1.982 1.9923 1.99983 2.00013

k2 0.00096 0.009846 0.09988 0.19939

a Theoretical spectra are calculated using d = 1000 nm, ni = 1.0, n = 2.0, n 3 = 3.5, k,= k3 = 0.0 in the 300-420-nm wavelength range. All calculations are performed in

double precision arithmetic.


0C

0

I4-U,

- - n=2.0,d=1000 nm,k=0.1, 70% noise

. . . . . . . . I .. .. . . . . . . . . I . . . . . . . . .

Table Iil. Comparison of the Accuracy of Two-Step Regression andEnvelope Methods Using Data Simulated with the Parameters d = 1000

nm, n1 = 1.0, n2 = 2.0, n3 = 3.5, k1 = k3 = 0.0 In the 300-420-nmWavelength Rangea

Starting n2 2.0 2.0 2.0 2.0

Starting k20.001 0.01 0.1 0.2

Envelope Method

n2 2.000 2.000 2.006 2.027

k2 0.00103 0.010004 0.10731 0.11014

Regression Method

n2 2.008 2.002 2.000 2.000

k2 0.00100 0.010019 0.09995 0.1996

a Double precision arithmetic has been used for the two-stepregression calculations.

Table IV. Comparison of the Accuracy of Two-Step Regression andEnvelope Methods Using Data Simulated with the Parameters d = 215 nm,

nj = 1.0, n2 = 4.25, n3 = 1.5, k1 = k3 = 0.0 In the 400-900-nmWavelength Rangea

Starting n2 2.0 2.0 2.0 2.0

Starting k2 0.01 0.1 0.5 1.0

Envelope Method

n2 4.257 4.258 4.325 4.503

k2 0.01019 0.0998 0.4808 0.5683

Regression Method

n2 4.252 4.241 4.247 4.251

k^2 0.01010 0.0992 0.4976 0.9963

I Double precision arithmetic hasregression calculations.

been used for the two-step

Table V. Examples of 1-um Thick Films for Which ad>> 1

Material A,nm a, cm-1 ad

a-SiH[2 0] 300-400 105 - 106 10 - 100

a-C[201 200-300 "- 105 - 10

AlN21 160-200 _ 105 10

Au[22] 300-400 105 - 106 10 - 100

changes from 2.6 at X = 300 nm to 0.4 at X = 420 nm,where both methods are accurate. However, as dis-cussed above, the two-step regression method is supe-rior for systems having larger values of ad. Bothmethods require the occurrence of a sufficient numberof extrema in the scanned reflectance. The number ofextrema is directly proportional to n2 and d and in-versely proportional to X. Hence, when n2 is small, d

has to be large enough to obtain a sufficient number ofextrema in the scanned wavelength range. However,if the film thickness is large, attenuation is also large,ad >> 1, and the envelope method predicts inaccuratevalues. Several examples of 1-Am thick films forwhich ad >> 1 are shown in Table V. Even if d isreduced to 0.5 gim, ad will still be much larger than 1.The new regression method will be used to analyzereflectance data from such films in the near future.

V. Summary and Conclusions

A diode array spectrometer has been used to mea-sure the reflectance spectrum of a DLC film. A newtwo-step regression procedure is utilized to determinethe film thickness, refractive index, bandgap, andslope of the absorption edge of the DLC film. Theoptical properties derived from the regression proce-dure fall within the range of values reported in theliterature 9 and agree with the values calculated by theenvelope method.7 Excellent agreement between thecalculated and experimental reflectance spectra is ob-served in the wavelength range scanned. It is shownusing simulated data that, for strongly absorbing films,the new regression method is more accurate than theenvelope method for calculating optical parameters.The regression method can handle noisy data also.The characterization of thin film growth processesutilizing the rapid scan capability of the DARSS isunder way, and results will be reported in the nearfuture.

This research has been supported in part by a grantfrom the New York State Science and TechnologyFoundation to the Center for Advanced Materials Pro-cessing at Clarkson University and by National Sci-ence Foundation grant ECS-8611298.

References1. F. Rodriguez, P. D. Krasicky, and R. J. Groele, "Dissolution Rate

Measurements," Solid State Technol. 5, 125 (1985).2. B. Vidal, A. Fornier, and E. Pelletier, "Optical Monitoring of

Nonquarterwave Multilayer Filters," Appl. Opt. 18, 3851-3856(1979).

3. F. J. Van Milligen et al., "Development of an Automated Scan-ning Monochromator for Monitoring of Thin Films," Appl. Opt.

24, 1799-1802 (1985).4. L. Li and Y. Yen, "Wideband Monitoring and Measuring Sys-

tem for Optical Coatings," Appl. Opt. 28, 2889-2894 (1989).5. B. Bovard, F. J. Van Milligen, M. J. Messerly, S. G. Saxe, and H.

A. Macleod, "Optical Constants Derivation for an Inhomogene-ous Thin Film from in situ Transmission Measurements," Appl.Opt. 24, 1803-1807 (1985).

6. D. A. Minkov, "Method for Determining the Optical Constantsof a Thin Film on a Transparent Substrate," J. Phys. D 22, 199(1989).

7. I. Ohlidahl and K. Navratil, "Simple Method of SpectroscopicReflectometry for the Complete Optical Analysis of Weakly

Absorbing Thin Films: Application to Silicon Films," Thin

Solid Films 156, 182 (1988).8. J. P. Borgogno, B. Lazarides, and E. Pelletier, "Automatic De-

termination of the Optical Constants, of Inhomogeneous ThinFilms," Appl. Opt. 21, 4020-4029 (1982).

9. L. S. Miller, A. J. Walder, P. Linsell, and A. Blundell, "Reflec-

tometry Measurement of Optical Parameters of Au/SiO2/SiFilms," Thin Solid Films 165, 11 (1988).


10. J. C. Manifacier, J. Gasiot, and J. P. Fillard, "A Simple Methodfor the Determination of the Optical Constants nl, k and theThickness of a Weakly Absorbing Thin Film," J. Phys. E 9,1002(1976).

11. E. Pelletier, P. Roche, and B. Vidal, "Determination Automati-que des Constantes Optiques et de Epaisseur des CouchesMinces: Application aux Couches Dielectriques," Nouv. Rev.Opt. 7, 353 (1976).

12. 0. S. Heavens, Optical Properties of Thin Solid Films (Dover,New York, 1965).

13. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crys-talline Materials (Clarendon, Oxford, 1979).

14. K. Levenberg, "A Method for the Solution of Certain Problemsin Least Squares," Q. Appl. Math. 2, 164 (1944).

15. D. Marquardt, "An Algorithm for Least-Squares Estimation ofNon-Linear Parameters," SIAM J. Appl. Math. 11, 431 (1963).

16. M. David, S. V. Babu, and B. Flint, "Optical Characterization ofDLC, AlN and SiC Thin Films," Appl. Phys. Lett. 57,1093-1095(1990).

17. R. C. Weast, Ed., Handbook of Chemistry and Physics (CRCPress, Boca Raton, FL, 1986).

18. H. H. Madden, "Comments on the Savitzky-Golay ConvolutionMethod for Least-Squares Fit Smoothing and Differentiation ofDigital Data," Anal. Chem. 50, 1383 (1978).

19. V. Natarajan, J. D. Lamb, J. A. Wollam, D. C. Lin, and D. A.Gulino, "Diamondlike Carbon Films: Optical Absorption, Di-electric Properties, and Hardness Dependence on DepositionParameters," J. Vac. Sci. Technol. A 3, No. 3, 681 (1985).

20. M. Hirose, Plasma Deposited Thin Films, J. Mort and F. Jan-sen, Eds. (CRC Press, Boca Raton, FL, 1986).

21. F. Hasegawa, T. Takahashi, K. Kubo, and Y. Nannichi, "PlasmaCVD of Amorphous AlN from Metalorganic Al Source and Prop-erties of the Deposited Films," Jpn. J. Appl. Phys. 26, 1555(1987).


two-step regression procedure for the optical characterization of thin films

Documents