Line-shape analysis of ellipsometric spectraon thin organic films

Hans Arwin, Jan Mrtensson, and Roger Jansson

A methodology for the line-shape analysis of ellipsometric spectra on thin (<200-A) organic films ispresented. Four different line shapes are employed: Gaussian, Lorentzian, phase-relaxed Lorentzian,and a critical-point line shape. An analysis of analytic data addresses the problem of the modeling ofunsymmetric absorption bands. The method is exemplified by an analysis of thin films of phthalocya-nine and poly(3-hexylthiophene), and we show that the number and type of resonances in an absorptionband can be obtained. The possibility of resolving the cause of a shift in the peak position of anabsorption band is also demonstrated. In the case being studied the shift is due to the redistribution ofthe oscillator strengths between the individual resonances in the band and not to shifts in the energies ofthe resonances.

Key words: Line-shape analysis, spectroscopic ellipsometry, thin films.

1. Introduction

Ellipsometry has now evolved to a stage where spec-troscopic data can be measured routinely in manyresearch laboratories. Ellipsometers with wave-length-scanning facilities are also commercially avail-able. However, most reports involving spectroscopicdata taken by ellipsometry deal with bulk opticalproperties of semiconductors and metals.' Relativelyfew studies involve thin films, but there is no doubtthat thin-film spectroscopic data would be valuable inmany surface problems.2 A plausible cause of therare appearance of spectroscopic applications on thinfilms is the classical ellipsometric problem of theseparation of film thickness from film optical proper-ties. On a wavelength-by-wavelength basis this sep-aration is in principle impossible on opaque filmssince there are two measured data but three un-knowns: the film thickness and the real and imagi-nary part of the film refractive index. On thintransparent films the separation is virtually impossi-ble because of the high correlation between the filmthickness and the film refractive index. This can beshown mathematically in thin-film linear expansionsof the ellipsometric equations.3 Experimentally this

The authors are with the Laboratory of Applied Physics, Depart-ment of Physics and Measurement Technology, Link6ping Insti-tute of Technology, S-581 83 Link6ping, Sweden.

Received 28 August 1991.0003-6935/92/316707-09$05.00/0.c 1992 Optical Society of America.

effect can be seen in kinetic data on thin-film growth,e.g., in the report by Cuypers et al.,4 where the noiselevel in the thickness and refractive index is consider-ably higher than when they are combined by calculat-ing the mass per unit area with the de Feijtersformula.5 Another limiting factor is that the de-tailed microstructure is not always known for thinfilms. Interface roughness and film nucleation can,e.g., make the interpretation of thin-film ellipsomet-ric data quite complicated.

However, access to spectroscopic ellipsometric datacan provide additional information, which opens newpossibilities for analysis. Besides the obvious factthat more information is available for interpretationand redundancy checks in a spectroscopic measure-ment, there are also interpretation strategies thatmake use of the wavelength dependence of the opticalproperties of the substrates or film material. Incombination with multiple-angle-of-incidence mea-surements, spectroscopic ellipsometry has proved tobe a powerful surface analytic tool.7-9

In our laboratory spectroscopic ellipsometric stud-ies on thin organic films adsorbed or deposited onmetal and semiconductor surfaces are in focus. Thefilms are either of biological interest, e.g., proteins,10or of electronic interest, e.g., conducting polymers"and phthalocyanines.12 The thicknesses of the filmsare typically in the range of 1-30 nm. Once theoptical properties of a film have been determined, onemay proceed one step further and analyze and inter-pret the optical structure in the film spectrum.Composition, anisotropy, or energy position and line-

1 November 1992 / Vol. 31, No. 31 / APPLIED OPTICS 6707

widths of absorption bands or individual resonancesmay be of interest.

In the present study absorption band studiesthrough line-shape analysis are addressed. This canbe seen as an extension of the pioneering study ofAspnes on the analysis of critical points in semiconduc-tors. 3"14 In this paper, procedures for the derivativeanalysis of ellipsometrically determined spectra onthin organic films are presented. The purpose of theanalysis is to break down an absorption band intoindividual resonances. In Section 2 the fitting rou-tines and the line shapes employed are presented.A comparison between the interpretation of analyticdata with different line shapes is presented, and thetechnique is applied to experimentally determinedspectra on thin organic films. Finally various ap-proaches in terms of the accuracy of the fits and therelevance of confidence intervals are discussed.

where k is the order of the derivatives; k = 2 is used inthis study. If 2 is analyzed, the imaginary partinstead of the real part of the expression in thebrackets on the right-hand side of Eq. (2) is used.The line shapes are called Gaussian, Lorentzian,relaxed Lorentzian, and critical-point line shape.They are described in detail below by superscripts G,L, R, and C, respectively.

The Gaussian line shape is a real-valued functionand has the character of an absorption line, but it canbe used only to model e2. The expression is

LG(E) = A exp{- [(E - Eg)F]21, (3)

where A is the amplitude, Eg is the resonance energy,and r the phenomenological broadening.

The Lorentzian line shape in its complex form isgiven by

2. Experimental

A. Ellipsometric Data

Thin organic films were deposited by spinning tech-niques on gold substrates prepared by vacuum evapo-ration.' 2 Spectroscopic ellipsometric data, the com-plex reflection ratio' 5 p(Em) = tan Tm exp(iAm), weremeasured with a rotating-analyzer ellipsometer' 6 at256 equidistant photon energies Em (m = 1, . . . , 256)in the photon-energy range of 1.5-5 eV. The com-plex dielectric function, e = el + i 2, of the films werecalculated in the three-phase (one-layer) model.'5Observe that in an ellipsometric measurement, we canobtain both El and E2 without performing a Kramers-Kronig transformation. Three-point smoothing (ap-plied five times) where a data point ym is replaced byym = (ym-, + ym + ym+,)/4 is used. After smooth-ing and numerical derivation were accomplished,curve fitting was done by means of least-squaresregression. The signal-to-noise level in the currentfiltered data permits line-shape analysis on secondderivatives.

B. Model Line Shapes

Dielectric function spectra versus photon energy Ewas calculated by using the different line shapes L(E)described below. For derivatives the exact analyticexpressions were used. The general form of thecomplex dielectric functions used is

•4

E(E) = El(E) + i 2(E) = 1 + I L(E) + B(E), (1)j=l

where E indicates that several lines (up to four) canbe fitted simultaneously. The background functionB(E) is assumed to vary slowly with E and is ne-glected in derivatives of e(E). The expressions thatwe actually used for the derivatives of E are therefore

akEl(E) 54

j=l

rakL (E)Re DE",

LL( = EAL(E) = E -E il (4)

The imaginary part of Eq. (4) is the well-knownCauchy function:

Im[LL(E)] = AF

1(5)1 + [(E - Eg)lr]2

which corresponds to a peak of height A/F at E = Egwith a half-width of 2F.

The relaxed Lorentzian line shape is obtained byintroducing mathematically a phase factor 13 in theLorentzian line shape:

L R(E) = A exp(ip)°LRE-Eg + iF (6)

The critical-point line shape is quantum mechanicallyderived and is based on a parabolic-band approxima-tion in reciprocal space.'3 It has been used exten-sively in the determination of critical points from theoptical spectra of semiconductors. Its general formis

Lc(E)= A exp(i3)(E - Eg + irF)P (7)

where is a phase factor and p. is the order of theresonance; and [i are parameters that containinformation about the type and dimension of thecritical point. Here is used as a free parameter toaccount for unsymmetric absorption bands and p =- 1/2, which corresponds to a three-dimensional par-abolic-band approximation. A prefactor E- 2 in Eq.(7) is neglected since it varies slowly over the narrowenergy ranges studied. L(E) should be consideredonly as an approximation for obtaining the derivatives.It should be used only in the vicinity of a resonance,and it is not applicable to describing the full dielectricfunction over a wider photon-energy range becausethe parabolic-band approximation introduces errorinto the low-order derivatives.

6708 APPLIED OPTICS / Vol. 31, No. 31 / 1 November 1992

Figure 1 illustrates the complex dielectric functionfor a Lorentzian-type optical absorption band calcu-lated from Eqs. (1) and (4) with A = 1 eV, Eg = 3 eV,F = 0.2 eV, and B(E) = 0. Observe that 2 issymmetric and that El becomes negative over part ofthe energy spectrum; e1 and 2 are not independentbut are interrelated through the Kramers-Kronigrelations.' 7 For comparison Fig. 1 also shows aGaussian line ( 2 only) with the same parameters asfor the Lorentzian line, except for the amplitude(A = 5). Notice that a Gaussian line falls off muchfaster than a Lorentzian line does. For the Lorentz-ian line Fig. 1 also shows the absorption coefficient

= 4k/X = 2E 2 /nX, (8)

where n and are the real and imaginary parts,respectively, of the complex refractive index N = n +ik = vE and X is the wavelength of the light. Observethe great difference in the energy position of thepeaks in E2 and a. This is discussed further below.

The four models differ markedly in shape, whichcan be seen in Fig. 2(a) where the second derivativesof 2 corresponding to Gaussian, Lorentzian, andcritical-point line shapes are shown. The Gaussianfalls fastest and has pronounced side peaks, while thecritical-point line has small side peaks. Figure 2(b)shows the second derivatives of critical-point lineshapes for different values of the phase 13. Thesecurves show that it is possible to fit models tounsymmetric absorption bands by using the phase 3as a free parameter. The curve for P = Tr/4 corre-sponds to a symmetric band and is identical to thecritical-point line shape in Fig. 2(a). Unsymmetriclines may in a similar way be generated in the relaxedLorentzian model.

C. Least-Squares Regression

We determined the optimal parameters in the line-shape functions by least-squares regression by mini-

G)w

wCO

i

PHOTON ENERGY (eV)

(a)

w

T/

2 3 4PHOTON ENERGY eV)

(b)

Fig. 2. (a) Second derivatives of the imaginary part of thedielectric functions of Gaussian, Lorentzian, and critical-pointtypes with resonances at 3 eV and amplitudes of 1, 0.2, and 18 eV,respectively. The line broadenings are 0.2 eV, and the phase inthe critical-point model is ,B = 7r/4. (b) Second derivatives of theimaginary part of the dielectric function for different values of thephase 3 of a critical-point-type resonance at 3 eV with an amplitudeof 18 eV and line broadening of 0.2 eV.

mizing

-2 2 3 4 5

PHOTON ENERGY (eV)

Fig. 1. Complex dielectric function e = El + iE2 and the absorptioncoefficient a of Lorentzian (superscripts L) and Gaussian (super-script G, E2 only) types of optical absorption line with a resonance at3 eV. In both cases the line broadenings are 0.2 eV. Theamplitude of the Gaussian line is normalized to the Lorentzian.

M2 a2 0El(Epm) _X = I ~~~~ak I

M=M1 E .lexp

IEkel(Em)1 12

aEk calcJ(9)

where the first term is the measured data and thesecond is obtained from Eq. (2); ml and m2 are the lowand high summation limits, which in this analysis arein the range of 1-256. If the imaginary part of thecomplex dielectric function is studied, 2 replaces e1.The range of ml-m 2 should be chosen to be largeenough that the essential features of an absorptionband lie within the corresponding photon-energyrange without necessarily including information aboutthe background and other absorption bands.

We found the minimum of the nonlinear functionX2 by making a least-squares fit with a parabolicexpansion' 8 of x2. The parameters used in the anal-ysis are the amplitudes Aj, the broadenings Fj, theresonance energies Egi, and in the case of unsymmet-

1 November 1992 / Vol. 31, No. 31 / APPLIED OPTICS 6709

rical lines the phase factors by. Since up to four linescan be fit, the number of parameters P is a minimumof three for a single symmetric line and a maximum of16 for four unsymmetrical lines.

D. Confidence Intervals

The computation of the confidence intervals involvesthe Student-t statistics' 9 for the degree of freedomNfree and a specified level of confidence. Here Nfree =M2 - - 1 - P, where P is the number of adjustablemodel parameters. With a specified confidence of90% the true value of a parameter will be with 90%probability

Pntrue = Pn + 8 (Pn), (10)

where Pn is obtained in the least-squares fittingprocedure and

8(Pn) = t(Nfree, 0.9) Is 2(pn)]1/2. (11)

Here t(Nfree, 0.9) is the tabulated value for the Stu-dent-t statistics, and the parameter variances S2(Pn)are obtained from

s 2 (Pn) Nfree82 X2 /5Pn2 (12)

However, the confidence intervals are mathematicalquantities providing information about how well theparameters separate in the models and give onlylimited information about absolute precision. Largeintervals are signs of parameter correlation. Valuesof the intervals depend on the data modeled, noiselevel, number of lines, etc. Typically the 90% confi-dence interval is less than a few milli electron volts forthe resonance energies in the spectra analyzed here.

3. Model Calculations on Analytic Line-Shape Data

Results obtained on ideal spectra are presented here.This serves as a background to the analysis of theexperimental data in Section 4. Our purpose is toshow some problems and pitfalls that one can encoun-ter when performing this type of analysis of opticalspectra. First, symmetric absorption bands are dis-cussed. Then there follows a discussion of the analy-sis of the absorption coefficient a as an example of anunsymmetric spectrum derived from a symmetricspectrum. Finally a general unsymmetric analysisis discussed.

A. Symmetric Bands

The second derivatives of the Gaussian, Lorentzian,and critical-point line shapes were calculated with thefollowing parameters: A = 1 eV (dimensionless forGaussian), Eg = 3 eV, r = 0.2 eV. For the critical-point line shape we use p = - 1/2 and = IT!4, whichcorrespond to a symmetric absorption line. Table 1summarizes the results when derivatives of the imag-inary part ( 2e2/E 2) of the three line shapes arefitted to each other. The correct value of the reso-nance energy is obtained in all cases, which is naturalsince all the data are symmetric. The obtainedvalues on the amplitudes and broadening parametersvary between the models, which is due to the fact thatthey are defined differently in the different models.The general conclusion is that as far as the resonanceenergy is concerned, independent of the fitting func-tion used, we obtained the correct value for symmet-ric absorption lines.

B. Absorption Coefficient (

In transmission spectroscopy the absorption coeffi-cient a is the experimentally determined quantity.It is therefore of interest to discuss the analysis of a,especially since the peak of a resonance apparentlylies at different energies in E2 and a (see Fig. 1). Thereason for this shift is the weighting with and therefractive index n as can be seen in Eq. (8). Thiseffect is less pronounced if the background [B(E) inEq. (1)] is high but can be significant in real data.12Because of the wavelength weighting, the absorptioncoefficient is an intrinsically unsymmetric functionaround the resonance energy. The results will bewrong therefore if symmetric line shapes are used inthe analysis of a. This is clearly seen in Table 2,where the results of fitting Gaussian and Lorentzianlines to a in Fig. 1 as well as to its second derivativeare presented. It is seen that the errors are large ifthe analysis is carried out on x directly. However,even on the second-derivative level, the errors aregreat if symmetric line shapes are used. In contrastthe unsymmetric critical-point line shape gives acorrect value of the resonance energy. The 90%confidence intervals for the resonance energies arealso greater (5 meV) with the Gaussian and Lorentz-ian models compared with the relaxed Lorentzianmodel (2 meV). The conclusion is that, if the absorp-tion band being analyzed is unsymmetric for funda-

Table 1. Parameter Values Obtained by Fitting Second Derivatives of Gaussian, Lorentzian, and Critical-Point Line Shapes to the Second Derivativesof c 2 of Calculated Line Shapes by Using the Three Models Above with A = 1 eV, Eg = 3 eV, and r = 0.2 eV

Model in Interpretation

Gauss Lorentz Critical Point

Model for A Eg r A Eg r A Eg rData - (eV) (e) (eV) (eV) (eV) (eV-1/2 ) (eW) (eV)

Gauss 1.000 3.000 0.200 0.383 3.000 0.239 7.389 3.000 0.097Lorentz 2.970 3.000 0.168 1.000 3.000 0.200 26.135 3.000 0.080Critical point 0.146 3.000 0.408 0.143 3.000 0.509 1.000 3.000 0.200

6710 APPLIED OPTICS / Vol. 31, No. 31 / 1 November 1992

Table 2. Parameter Values Obtained by Fitting Model Functions to a in Fig. 1 and Its Second Derivativea

Model in Interpretation

Gauss Lorentz Relaxed Lorentz

A Eg r A Eg I A Eg F 13Function - (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV)

a 4.540 3.228 0.504 1.709 3.205 0.344 - - - -

a2 aI/E 2 2.141 3.074 0.212 0.948 3.071 0.256 1.012 3.001 0.256 -0.84

aThe parameter values used to calculate a were A = 1 eV, Eg = 3 eV, and F = 0.2 eV.

mental reasons, it is necessary to use a phase-relaxedapproach to obtain the correct resonance energy.

C. Fitting to Unsymmetric Critical-Point Bands

There is no reason to assume a priori that absorptionbands are symmetric. Therefore the general ap-proach to an analysis is to start with unsymmetricmodels. Here some results from the analysis of thefive critical-point bands in Fig. 2(b) are presented.These analytic data were interpreted with the Gauss-ian, the Lorentzian, as well as the relaxed Lorentzianmodel. Our purpose is to demonstrate the magni-tude of the errors that can result if symmetric linesare used for fitting to unsymmetric lines. The re-sults are presented in Table 3. The worst-case er-rors in the determination of resonance energies withsymmetric models are almost 0.3 eV, which indeed isunacceptable. However, if the 13 phase is included asa free parameter as in the relaxed Lorentzian model,the correct energies are obtained independently of theasymmetry. The value of 13 is a measure of theasymmetry of a line. With 3 defined in the relaxedLorentzian model, the extremes are = 0 and 13 = iTr,which corresponds to a symmetric line and an upside-down line, respectively. The latter represents a dipor a hole in the spectrum, which occurs in materialswhere spectral hole burning is possible.20 However,these phenomena are not present at room tempera-ture, and if becomes too large one should be careful.If 11I becomes larger than !r/2, the hole characterdominates and other models should be considered.However, recall that the main emphasis is to deter-mine the resonance energies as correctly as possible,and in this respect a phase-relaxed approach appar-ently gives superior results for unsymmetrical bands.

The confidence intervals are typically in the range of10-20 meV for the resonance energies when theGaussian and Lorentzian models are used, while thephase-relaxed Lorentzian models result in intervalsof < 3 meV.

4. Applications to Experimental Data

Two applications of derivative line-shape analysis toreal data are presented in this section. The objectiveis to point out the type of information that can beobtained. The physical significance of the resultsare discussed in the more materials-oriented refer-ences given below.

A. Phthalocyanines

Ellipsometrically determined optical spectra of thinfilms of tetrasulfonated phthalocyanines (TSPc) withZn, Ni, and Cu as the substituted metal ion have beenreported recently.12 Figure 3 shows the ellipsometri-cally determined dielectric function of a 76-A film ofNi-substituted TSPc (NiTSPc) prepared on a goldsurface by spinning. The fabrication and details ofthe analysis are given in Ref. 12. With derivativeline-shape analysis, as described in Section 3, the Qband around 2 eV was resolved in three resonances.The energy positions of these resonances are shownby the arrows marked L1, L2, and L3 in Fig. 3. It wasalso shown that the energy position of one of theresonances for Ni-containing TSPc depends on thethickness of the film. Here a further analysis of aspectrum of NiTSPc is done. The questions ad-dressed are the following: (1) Which line shape bestrepresents the shape of the actual absorption lines?(2) How does one decide that there are three and nottwo or four resonances in the Q band? The objective

Table 3. Parameter Values Obtained by Fitting Gaussian, Lorentzian, and Relaxed Lorentzian Line Shapes to the Critical-PointCalculated Data in Fig. 2(b)

Model in Interpretation

Gauss Lorentz Relaxed Lorentz

Eg r A Eg F A Eg `Phase A (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) 13

-7r/4 0.128 3.276 0.423 0.103 3.255 0.500 0.139 2.999 0.504 -1.580 0.143 3.135 0.415 0.126 3.123 0.498 0.136 2.998 0.500 -0.807r/4 0.149 3.000 0.413 0.135 3.000 0.498 0.135 3.000 0.498 -0.017r/2 0.143 2.865 0.415 0.129 2.876 0.502 0.138 3.000 0.503 0.79

37r/4 0.128 2.724 0.423 0.110 2.740 0.513 0.139 2.999 0.504 1.56

1 November 1992 Vol. 31, No. 31 APPLIED OPTICS 6711

C,

2-

I tL, tL 32 3 4

PHOTON ENERGY leVI

Fig. 3. Complex dielectric function, e = El + iE2, versus photonenergy for a 76-A film of NiTSPc on a gold substrate. The arrowsshow the energies for the three resonance lines found in theline-shape analysis.

quantity that the conclusions are based on is theexactness of a fit in terms of x2 defined in Eq. (9).

Question (1) is answered by curve fitting to thesecond derivative of E2 of the spectrum in Fig. 3 in theenergy range of 1.72-2.43 eV (ml = 20 andm 2 = 80).We found that for the four models used, that is,Gauss, Lorentz, relaxed Lorentz, and critical point,the values of x2 were 6079, 613, 224, and 828,respectively. Fitting to el with Lorentzian, relaxedLorentzian, and critical-point models results in 1623,486, and 867, respectively, for x2. The values of themodel parameters differ slightly between the fits to eland E2. If the criterion that the lowest value of x2

represents the best fit is used, the relaxed Lorentzianmodel used on E2 would give the best values on theline-shape parameters (X2 = 224). These values aregiven in Table 4. In Fig. 4(a) the second derivative ofE2 and the best fit are shown.

Let us now turn to how to decide the number oflines to be used in the fitting. By visual inspection ofFig. 4(a) it is obvious that the fit to a model with threelines is quite good. Adding a fourth line does notimprove the fit since the computer program puts twoof the lines at the same energy. On the other hand,if only one or two lines is used, the fit becomes poorwith x2 = 29,000 and 10,570, respectively, to becompared with x2 = 224 for three lines. The fitswith two lines and one line are shown in Figs. 4(b)and 4(c), respectively. It can be concluded that threelines satisfactorily model the Q band.

Table 4. Parameter Values Obtained by Fiting with the RelaxedLorentzian Model to the Second Derivative of e2 of the Spectrum in Fig. 3

Line-Shape Parameter Values

A Eg rLine (eV) (eV) (eV) 13

L, 0.062 1.852 0.118 -0.20L2 0.158 2.018 0.130 -0.21L3 0.162 2.138 0.136 0.56

50

-50

-100

1.8 2.2 1.8 2.2 1.8 2.2PHOTON ENERGY (eV)

Fig. 4. Best fits with the relaxed Lorentzian model (dashed curve)to the second derivative (solid curve) of the imaginary part of thedielectric function of the NiTSPc film shown in Fig. 3: (a) modelwith three resonances, (b) model with two resonances, (c) modelwith one resonance.

The analysis above demonstrates that curve fittingcan give valuable information about the details inoptical properties for materials with relatively sharpfeatures in their spectrum. Examples of such detailsare the number of resonances in an absorption bandand values of resonance energies as shown above.Additional information can be obtained, e.g., shifts inresonance energies caused by gas absorption, anddifferences in resonance energies for different substi-tuted metal ions. The analysis of TSPc and thephysical significance of the results are presented inmore detail elsewhere. 2'2 '

B. Conducting Polymers

The optical properties of the thin films of a conduct-ing polymer, poly(3-hexylthiophene) (P3HT), andsome results on the stability of this polymer havebeen reported." A further line-shape analysis of aseries of spectra of a 16.3-nm film of doped P3HT,which was stored in air for a period of 27 days, ispresented here. Figure 5 shows the imaginary partof the dielectric function measured at different timesduring storage. It is clear that the polymer is notstable in the actual storage conditions since the shapeand magnitude of the broad absorption band changewith time. Also here one of the questions addressedis how to deconvolute the absorption band in reso-nance lines. A quantification of the kinetic changesin the values of the parameters of the absorption linesconstituting the absorption band is also performed.

From Fig. 5 it is clear that the peak energy of theband as a whole is shifting toward higher energiesand that its magnitude is decreasing. However, themechanisms are not obvious from visual inspection.By numerical line-shape fitting to the second deriva-tives, we found that the data for the as-prepared filmcan be resolved into three resonance lines. Also thebands for films stored for 4, 7, and 12 days can befitted to three lines, while only two lines can beresolved in the bands for the films stored for 15 and

6712 APPLIED OPTICS / Vol. 31, No. 31 1 November 1992

2.6

4,

0wcczuJW

Wuszz0ciW

2.41

2.2

PHOTON ENERGY (eV)

Fig. 5. Time variation in the imaginary part of the dielectricfunction of a 16.3-nm film of P3HT upon a gold substrate. Thenumber of days that the film has been stored in air is given on eachcurve.

27 days since the third line becomes quite weak (seebelow). The values up to day 15 on the parametersobtained with the relaxed Lorentzian model are shownin Fig. 6. It is interesting to notice that the energiesare relatively constant and that the main changes arein the amplitudes and broadenings of the individuallines. The conclusion is that the apparent shift inthe absorption band is due mainly to a decrease in theamplitude of the resonance line with the lowestenergy and broadening and an increase in the ampli-tude of the central line. The resonance energies arefairly constant.

5. Discussion

A line-shape analysis of an optical spectrum of amaterial may be carried out for different reasons.It may be of fundamental interest to learn about theenergy levels in the material by determining theenergy position of the individual absorption linesconstituting an absorption band. In those cases it isimportant to have models that give absolute values ofthe energies that are as correct as possible. In othercases it may be of interest to follow the changes inpeak positions resulting from physical or chemicalchanges in the material induced, e.g., by light, temper-ature, gas exposure, pressure, or electric fields. Inthe latter case it is more important that we determinerelative peak positions with high precision.

The basic idea presented here is to take a spectrumwith an unknown number of absorption lines, with anunknown type of symmetry, broadening and magni-tude, and model such a spectrum with a few lines witha limited number of parameters. It is obvious thatthe data can never be mimicked exactly, and all thefits will be approximations. In this paper the exact-ness of a fit is used as a criterion for obtaining the bestparameter values. One must also bear in mind thatline shapes, e.g., the critical-point type, are applicableonly to derivatives and have limited meaning forunderivated spectra. Also the relaxed Lorentzianmodel has limitations because of the mathematically

a.4

155 10STORAGE TIME (DAYS)

(a)

STORAGE TIME (DAYS)(b)

5)

0z

z

0W2

z

STORAGE TIME (DAYS)(C)

Fig. 6. Time variations in the line-shape parameters of the threeresonance lines C labeled 1, 2, and 3 obtained by fitting the relaxedLorentzian model to the second derivative of the data in Fig. 5: (a)resonance energies, (b) amplitudes, (c) line broadenings. Curve 1,resonance L1; curve 2, resonance L2 ; curve 3, resonance L3.

induced phase. Therefore we should use criticallythe values of the parameters obtained in the analysiswithout forgetting the actual model used in theevaluation.

However, if one goes back to basics, the purpose ofthe analysis is to determine poles in the dielectricfunction in the negative complex half-plane. The

1 November 1992 Vol. 31, No. 31 / APPLIED OPTICS 6713

_ 3 . .__ ___

2 ~ ~ ~ ~ *I , .~~~~~l

,3n

positions of these poles represent the energy levels inthe material under investigation. As pointed out byAspnes,'4 we can determine the energy positions of asingle pole by a phase analysis of the Fourier trans-form of the dielectric function without assuming aline-shape model. Thus it is plausible that the un-symmetric but artificial line shapes that mathemati-cally give a superior fit to the data in fact are better touse than the symmetric line shapes. The results offitting to the analytic data on unsymmetric bands inTables 2 and 3 support this hypothesis.

It is appropriate here to include a short comparisonbetween the different line shapes. The Gaussianline is applicable when absorption lines are symmet-ric, and the broadening can be attributed to statisticalphenomena. Instrumentally induced broadening issuch an example. Gaussian lines are often used inphotoemission studies.22 The Lorentzian model hasthe advantage that it generates a full complex dielec-tric function. It is also easily understood since it canbe derived from the harmonic oscillator model forlight interactionwith matter.17 However, the Lorent-zian model also gives a symmetric absorption line.We introduced the relaxed Lorentzian model to over-come the problem with unsymmetric absorption lines.The critical-point model can be justified from a morerealistic theoretical point of view. The disadvantageis that it is valid only when it is close to a resonanceand can be used only to model derivatives of thedielectric function in a narrow energy range. Alsothe critical-point line shape can handle unsymmetricabsorption lines. This is an important feature, whichwas demonstrated with the model calculations inSection 4. The model also has an additional freedomin the parameter p, which contains the dimension ofthe resonance, which can be important for thin films,especially if they are anisotropic.

The results from the conducting polymers and thephthalocyanines show that a line-shape analysis pro-vides information about details in optical spectra.The difference between shifts in resonance energiesand other changes such as variations in line broaden-ings or magnitudes can be distinguished. This fea-ture provides valuable information about the physicsbehind the observed changes. A possible extensionof the analysis would allow for anisotropic resonances.Thin films are ordered in many cases and havedifferent optical properties that are parallel or perpen-dicular to a surface.

The rest of this discussion describes some details inthe fitting procedures, such as optimizing, how toselect the appropriate number of derivatives andsmoothing level, and their influence on precision.We also include a comment on the finding thatslightly different results are obtained when fitting isdone to the real or the imaginary part of e.

Second derivatives are used in the analysis. In-creasing the order of derivatives would enhance thesharp features in the spectra, which would increasethe precision in the determination of resonance en-ergy. Less interference from near-lying bands also

would be an advantage. However, the noise levelalso increases, which counteracts the increased reso-lution caused by sharpening of the spectrum. Inbulk analysis third derivatives are often used,23 butwe must point out that in the bulk case we obtained aspectrum by analytic inversion of ellipsometric data,while we deduced thin-film optical spectra from nu-merical inversion by using two measured spectra thatdiffer only slightly. Empirically it is found that thesecond derivative is optimal for the applications pre-sented here. By being fitted to the first and thirdderivative, x2 becomes larger as a result of modelmismatch (a first derivative) and more noise (a thirdderivative). The values of the parameters obtainedgenerally differ little between fits to the second andthird derivatives, while the first derivative resultsdiffer considerably.

The noise level can be reduced by more filtering.However, this will distort the data and introduceerrors in the parameter values. Empirically it wasfound that the three-point smoothing used here isadequate. With less filtering the fitting algorithmbecomes trapped too easily in false minima, andincreased filtering will introduce unwanted distor-tion, e.g., in the line broadening. We performed afurther test by using unfiltered data, and as startingvalues we used the parameter values obtained onfiltered data. Few or no changes in the parametervalues were obtained, but the confidence intervalsand x2 increased. Distortion caused by filtering andderivation was addressed recently by Garland et al.24

They suggested that both data and the model shouldbe differentiated to obtain the best result. Suchprocedures will be implemented in our programs,although it will not improve the results of the presentdata. In principle a confidence interval may bespecified in the parameter of main interest, and thelevel of filtering can then be selected. The filteringas described in Section 2 gives 90% confidence inter-vals of < 1 meV in resonance energies and linebroadenings in the analysis of the phthalocyaninedata in Fig. 3. Without filtering the confidenceintervals become 10 times larger.

In a more detailed analysis of the data on TSPc onefinds that the values on the parameters becomeslightly different if fitting is done to the real instead ofthe imaginary part of the dielectric function. Thedifferences depend on the model but were <0.01 eVin the relaxed Lorentzian model and < 0.002 eV in theLorentzian model. We suggest that the magnitudeof these differences be taken as a measure of theprecision.2' A possible improvement therefore seemsto be to fit to the complex dielectric function insteadof to the real and imaginary parts separately. How-ever, this has not been implemented because gener-ally a fit is much better (lower x2) to the imaginarypart. The parameter values obtained with 2 aretherefore the most representative. A possible expla-nation for these differences may be that, for a givenresonance, 2 fundamentally has a variation over anarrower photon energy than cl and thus is less

6714 APPLIED OPTICS / Vol. 31, No. 31 / 1 November 1992

sensitive to background effects and to near-lyingresonances.

6. Conclusions

A line-shape analysis of ellipsometrically determinedspectra on thin films provides new information abouttheir physics.

Large errors in line-shape parameters are obtainedif unsymmetric absorption bands are modeled withsimple Lorentzian or Gaussian models.

These errors can be decreased significantly if phase-relaxed models are used.

With a line-shape analysis it is possible to resolvethe cause of shifts in the energy position and thechanges in shape of the complex absorption bands.

This work was supported by grants from the Na-tional Swedish Board for Technical Development.Ingemar Lundstr6m is acknowledged for valuablecomments.

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