Complexation of Cationic Surfactant and Anionic Polymer at theAir-Water Interface
Atef Asnacios,*,† Dominique Langevin,† and Jean-Francois Argillier‡
Centre de Recherche Paul Pascal, Avenue Albert Schweitzer, 33600 Pessac, France, andInstitut Francais du Petrole, 1 et 4 avenue du Bois-Preau, 92506 Rueil,Malmaison Cedex, France
Received February 13, 1996; Revised Manuscript Received August 15, 1996X
ABSTRACT: Equilibrium surface tension measurements have been carried out on solutions of a non-surface-active anionic polyelectrolyte (poly(acrylamide sulfonate)) mixed with various surfactants. Whilethe surface tension does not change when the polymer is added to nonionic and anionic surfactant solutions,a strong synergistic lowering of the surface tension is found with cationic surfactants. In the latter case,we also find that the surface tension is practically independent of the amount of polymer over the rangeof concentrations studied. Assuming that the polymer stretches out at the interface to form a neutralcomplex with the surfactant, this behavior can be explained by the classical Gibbs adsorption equation.Furthermore, ellipsometry measurements on these solutions are consistent with our adsorption model.
Introduction
The study of interactions between surfactants andpolymers is a rapidly growing field of interest in colloidscience.1 Indeed, many practical systems for industrialapplications contain mixtures of polymers and surfac-tants. These mixtures are widely used as thickenersin water-based formulations such as paints, drillingmuds, etc. Moreover, in biology, membranes are beauti-ful examples of architectures where lipids (surfactants)and proteins (polymers) form important structuredcomplexes. Whereas there is an extensive literature onsurfactant solutions on one hand and polymer solutionson the other hand, less is known for the mixed polymer-surfactant systems. It has recently been suggested thatpolymer-surfactant solutions are similar to polymer-polymer solutions.2 If two polyelectrolytes of oppositecharge are mixed, the two polyions associate, thusreleasing the counterions in the solution and increasingthe entropy of the solution. It is also observed that thebehavior of polyelectrolyte-surfactant solutions is simi-lar in that no association for neutral or similarlycharged surfactants occurs, yet there is a strong as-sociation for oppositely charged surfactant-polymercombinations.2 However, surfactant-polymer mixturesare more complicated because the surfactant aggregates(i.e. micelles, bilayers, etc.) must also be considered.3-6
In this work, our objective is to study the polymer-surfactant interaction at the air-water interface. Theseinteractions are important to many practical applica-tions such as colloidal stabilization, wettability, andadhesion. While complexation in the bulk is generallyrelated,7 only few studies have focused on surfactant-polyelectrolyte interactions at a surface.8-11 Therefore,our work is centered around poly(acrylamide sulfonate)(a model anionic polyelectrolyte for drilling muds) incombination with a variety of common surfactants.First, the bulk polymer-surfactant interactions arecharacterized by viscosity measurements. We then usea combination of surface tension and ellipsometrymeasurements to probe the surface complexes at theair-water interface.
Experimental Section
Materials. Nonionic surfactant penta(ethylene glycol)mono-n-decyl ether (C10E5) from Nikko (Nikkol BD-5SY) andanionic surfactant dioctyl sulfosuccinate (AOT) from Sigma(99%) were used as received. Cationic surfactant, dodecyltri-methylammonium bromide (C12TAB) from Aldrich (99%), wasrecrystallized (2 g of C12TAB:10 mL of ethyl acetate:1 mL ofethyl alcohol) three times before use. For all three surfactants,no minimum in the surface tension versus surfactant concen-tration was observed. Potassium bromide (KBr) was suppliedby Fluka.The polyanion is a statistical copolymer of acrylamide (AM)
and acrylamidomethylpropanesulfonate (AMPS) synthesizedby SNF Floerger. The sample used is composed of 75 mol %of neutral AM and 25 mol % of charged AMPS monomers. Thestructure of the polymer is shown in Figure 1. The polymer’schemical structure has been characterized by titration via abromidation reaction for the amide function and potentiometrictitration for the sulfonate. The molecular weight and poly-dispersity of the polymer were measured by size exclusionchromatography (SEC) coupled with multiangle light scatter-ing, which gave Mw ) 2.8 × 106 and Mw/Mn ) 1.5. In 1 g/LNaCl, the average radius of a polymer chain is Rw ) 110 nm.To eliminate any traces of surfactant molecules and lowmolecular weight impurities, the polymer solutions werepassed through an ultrafiltration unit with a 20 000 cutoffmembrane. Final concentrations were determined using aShimatzu TOC 5050 total carbon analyzer. After this purifi-cation, the polymer displayed no surface activity at concentra-tions below 2000 ppm (see Figure 2). Pure water was takenfrom a Millipore Milli-Q system. Polymer-surfactant mix-tures were obtained by mixing pure surfactant and purepolymer solutions.Surface Tension Measurement. Experiments were per-
formed at room temperature (20 ( 1 °C). Measurements werecarried out in a Teflon trough (6 cm diameter) housed in aPlexiglas box with an opening for the tensiometer. The surfacetension was measured with an open-frame version of theWilhelmy plate (to avoid the wetting problems of a classicalplate12). The rectangular (20 mm× 10 mm) open frame, made
† Centre de Recherche Paul Pascal.‡ Institut Francais du Petrole.X Abstract published in Advance ACS Abstracts, October 15,
experimental results and we can thus use eq 11 tointerpret the surface tension isotherms.Applying eq 11 to the best fit of the AM/AMPS 750
ppm-DTAB surface tension curve below the cac givesan area per surfactant molecule of
This value is much greater than the 45 Å2 found for thesaturated surface of polymer-free C12TAB solutions.Likewise, using eq 8, we obtain the following area perpolymer molecule:
In addition to these area measurements, we also useellipsometry to determine the adsorbed layer thicknessat the air-water interface. Unfortunately, this tech-nique does not give a direct thickness measurement;instead two “ellipsometric angles” ψ and ∆ (related tointensity change and phase shift in reflected light) aremeasured and a model is required to obtain the ad-sorbed layer thickness. We analyze our data using ahomogeneous and isotropic adsorbed layer model. Work-ing at very low bulk concentrations, we consider oursystem as a mixed layer of surfactant, polymer, andwater (with unknown refractive index nd and layerthickness d) adsorbed between two infinite mediums,air and water.Under these conditions, nd and d can be calculated,
in principle, from the changes δψ and δ∆ in themeasured values of ψ and ∆ with and without anadsorbed layer.15,16 Hence, two sets of measurementsare performed, one on pure water and one on mixedpolymer-surfactant solutions. But, we find that δψ iszero within experimental error, which is expected in thecase of thin adsorbed layers.17,18 Thus, we can only relyon one measurement (δ∆) to determine the two un-knowns, nd and d.We overcome this difficulty by two different methods:(1) Although δψ cannot be measured, we know at least
that it must be less than the ellipsometer resolution.Thus, we solve the ellipsometric equation (for a trans-parent layer between air and water16) using the mea-sured δ∆ value and simulated values of δψ within theexperimental error.17 In that case, we determine the“range of possible values of nd and d” instead of the“true” (nd,d) couple. At 20 °C and angle of incidence Φ) 57°, we find for the pure water interface
and, for AM/AMPS 750 ppm-C12TAB solutions near thecac (0.7 mM C12TAB),
Thus
and taking -0.001° < δψ < -0.01° leads to
(2) We replace the “missing” δψ measurement by anavailable information on the adsorbed layer (in this case,AS+). First, we use the assumptions made in the surfacetension discussion to estimate the adsorbed layer com-
position. Then the measured δ∆ is combined with theAS+ value given by eq 11 to get the thickness andrefractive index of the adsorbed layer from the opticalcharacteristics of its components (see Appendix). Withδ∆ ) -1.55 ( 0.02° and AS+ ) 78 ( 5 Å2, we find
Here, we see that the characteristic length of a polymermolecule in the plane of the interface (2(AP-/π)1/2 ≈ 800Å) is, at least, about 15 times its length perpendicularto this interface. This flat polymer conformation is inagreement with the assumptions used when applyingthe Gibbs adsorption equation. In fact, the “ion ex-change” process (between nearly all sodium counterionsand C12TABmolecules) would be energetically favorableonly if the bound molecules of surfactant were nearenough to the air.Furthermore, one can calculate from eq A.2 the water
volume per adsorbed polymer-surfactant complex. Fornd ) 1.44 we find
This amount corresponds to 10% water and then to agel-like adsorbed layer. This gel structure is in agree-ment with the “texture” of the precipitates observed inbulk after the cac (as pointed out by Buckingham et al.,9complexation occurs first at the interfaces where con-centrations are higher than in bulk). Moreover, we havealso observed this gel behavior in foam films made frommixed polymer-surfactant solutions near the cac.21Finally, as the nd and d values found using surface
tension measurements are in good agreement (withinthe experimental errors) with those based only onellipsometric parameters, one can conclude that theassumptions used in our Gibbs adsorption equationanalysis are probably reasonable.
Summary and Conclusion
We have shown a synergistic surface tension loweringdue to coadsorption of a non-surface-active polyanionand oppositely charged surfactant ions. Relative viscos-ity measurements reveal a difference in the polyelec-trolyte-surfactant interaction in the bulk and at theair-solution interface. While there is formation of ahighly surface-active complex at the interface, there isno significant binding between the two species in thebulk at the very low surfactant concentrations wherethe synergistic effect takes place.We use the Gibbs adsorption equation to interpret the
surprising observation that surface tension is indepen-dent of the polymer concentration in the range 75-750ppm. This phenomenon can be explained by an “ionexchange” process where sodium counterions of anadsorbed polyelectrolyte molecule have been replacedby surfactant ions. Moreover, we see that this ionexchange process is accompanied by a relatively flat AM/AMPS conformation that allows all bound surfactanttails to be at the air-water interface. In the case of amore rigid polyelectrolyte, all charged monomers maynot be simultaneously close enough to the air-solutioninterface, counterion exchange would not be so energeti-cally favorable, and one would expect surface tensionto be dependent on polymer concentration. We arecurrently clarifying this issue with a rigid polyelectro-lyte.
AS+ ) 1/ΓS+ ≈ 78 Å2
AP- ) fAS+ ≈ 5 × 105 Å2
∆w ) 0.18 ( 0.04°
∆cac ) -1.37 ( 0.02°
δ∆ ) ∆cac - ∆w ) -1.55 ( 0.02°
1.37 < nd < 1.51 and 10 < d < 60 Å
nd ) 1.44 ( 0.07 and d ) 30 ( 20 Å
Vw ≈ (1 ( 0.8) × 106 Å3
7416 Asnacios et al. Macromolecules, Vol. 29, No. 23, 1996
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Finally, the results may also be dependent on theaffinity between the polymer charged groups and thesurfactant. We plan to compare the interaction betweenC12TAB and a sulfonated or a carboxylated polymer.
Acknowledgment. We are grateful to V. Bergeronfrom the Ecole Normale Superieure and A. Audibert, J.Lecourtier, and C. Noik from the Institut Francais duPetrole for many interesting and useful discussions andsuggestions. We thank S. N. F. Floerger for providingthe polymer samples, G. Muller for the molecular weightmeasurements, and N. Thurston for his help with theviscosity measurements. We are grateful to S. C.Russev from the University of Sofia for providing thepolynomial inversion program used in ellipsometry. Wealso acknowledge I.F.P. for partial financial support ofthis work.
AppendixUsing the Drude approximation for thin layers,18,19
we get
where na, nw, and nd are the refractive indexes of air,water, and adsorbed layer respectively; Φ is the angleof incidence and λ is the laser wavelength. Here wehave only one equation for the two unknowns d and nd;thus we need to establish another relation betweenthose quantities. For this, we rely on our surfacetension results.A second independent relationship between d and nd
can be established if we again assume an adsorbedhomogeneous layer composed of polyelectrolyte, surfac-tant, and water. We then focus our attention on a layervolume element corresponding to one polymer molecule,for which we can write
where AP- is the area polymer molecule (taken from theGibbs equation) and VP-,nP- and VS+,nS+ are the volumesand refractive indexes of the polymer molecule and thef coadsorbed surfactant molecules, respectively. Vw andnw are the volume and refractive index of water in thelayer. Equation A.3 expresses the fact that the molarrefractions of the species in the surface layer areadditive.20 Realizing that VP- ) Mw/(N0FP-) and VS+ )fMS+/(N0FS+) (where N0 is Avogadro’s number) andcombining eqs A.2 and A.3, we can eliminate the
unknown water volume and find the desired relation-ship between d and nd,
where
Using eqs 4 and 8, one can express eq A.4 as a functionof the measured area per surfactant molecule AS+, therate of charged monomer x, and the monomer meanweight Mmono:
Now, we have two equations ((A.1) and (A.5)) for twounknowns (nd and d) that can thus be determined fromthe measured δ∆ and the calculated AS+.
References and Notes
(1) Goddard, E. D.; Ananthapadmanabhan, K. P. Interaction ofSurfactants with Polymers and Proteins; CRC Press: BocaRaton, FL, 1993.
(2) Lindman, B. Adv. Colloid Interface Sci. 1992, 41, 149.(3) Ibragimova, Z. Kh.; Kasaikin, V. A.; Zezin, A. B.; Kabanov,
V. A. Polym. Sci. USSR (Engl. Transl.) 1986, 28 (8), 1826-1833 (Vysokomol. Soedin., Ser. A 1986, 28, 1640).
(4) Bianna-Limbele, W.; Zana, R. Macromolecules 1987, 20,1331-1335.
(5) Chandar, P.; Somasundaran, P.; Turro, N. J.Macromolecules1988, 21, 950-953.
(6) Dubin, P. L.; Rigsbee, D. R.; Gan, L.-M.; Fallon, M. A.Macromolecules 1988, 21, 2555-2559.
(7) Cabane, B.; Duplessix, R. J. Phys. (Fr.) 1982, 43, 1529-1542.(8) Goddard, E. D. Colloid Surf. 1986, 19, 301-329.(9) Buckingham, J. H.; Lucassen, J.; Hollway, F. J. Colloid
Interface Sci. 1978, 67, 423.(10) Shubin, V. Langmuir 1994, 10, 1093-1100.(11) Argillier, J. F.; Ramachandran, R.; Harris, W. C.; Tirrel, M.
J. Colloid Interface Sci. 1991, 146, 242.(12) Mann, E. K. Thesis, Paris, 1992.(13) Meunier, J. J. Phys. (Fr.) 1987, 48, 1818-1831.(14) Radlinska, E. Z.; Gulik-Krzywicki, T.; Lafuma, F.; Langevin,
D.; Urbach, W.; Williams, C. E.; Ober, R. Phys. Rev. Lett.1995, 74, 4237-4240.
(15) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and PolarizedLight (North-Holland Personal Library); North-Holland: Am-sterdam, 1992.
(16) Drolet, J. P.; Russev, S. C.; Ivanov, M. I.; Leblanc, R. M. J.Opt. Soc. Am. A 1994, 11 (12), 3284-3291.
(17) The experimental accuracy on δψ is (0.01°, while one expectsδψ ≈ -0.003° for a homogeneous layer of d ) 20 Å and nd )1.42 at Φ ) 57°. In fact, if λ ()6328 Å) is the laserwavelength, δ∆ ∝ (d/λ) and δψ ∝ (d/λ)2, so δψ is close to zerofor thin layers (d , λ) and approaches more rapidly theinstrument resolution than δ∆.15
(18) Den Engelsen, D.; De Koning, B. J. Chem. Soc., FaradayTrans. 1 1974, 70, 1603.
(19) Drude, P. Ann. Phys. 1889, 36, 532, 865.(20) Handbook of Chemistry and Physics, 66th ed.; CRC Press:
Real time monitoring of the growth of transparent thin filmsby spectroscopic ellipsometry
M. Kildemoa) and B. Drevillonb)Laboratoire de Physique des Interfaces et des Couches Minces (UPR 258 du CNRS), Ecole Polytechnique,91128 Palaiseau, France
~Received 12 October 1995; accepted for publication 19 February 1996!
In situ spectroscopic ellipsometry~SE!, in the near ultra-violet ~UV!-visible range, is extensively used for real timemonitoring of thin film growth.1 Among the various nonin-vasive optical techniques, SE appears as a more powerfultechnique for thin films because of its high sensitivity tolayer thickness (d) and refractive index (n). Moreover, as aconsequence of recent advances in optical instrumentation,real time SE is now compatible with most of the kineticsinvolved in thin film processing.2–4 Thus, this fast and sen-sitive optical technique makes possible the feedback controlof the thickness or/and refractive index in real time.4–8 Inparticular, in the case of transparent materials deposited onabsorbing substrates,n andd can be extracted from the mea-surement of the ellipsometric anglesC andD by direct in-version of the Fresnel equations.9–13 Such a simultaneousdetermination ofn andd cannot be performed using conven-tional reflectometry measurements. However, the processcontrol of the growth of transparent films, by real time singlewavelength ellipsometry, is generally restricted to thin layers~, 2000 Å typically!.6 This can be a severe limitation inmany practical applications. Nevertheless, the latter limita-tion can be overcome by combining real time ellipsometrytrajectories simultaneously recorded at different photonenergies.4,7,8
Furthermore, determinations of the outer-layer dielectricresponse based on virtual-interface methods have recentlybeen described.14,15 They have successfully been applied tothe growth of absorbing semiconductor materials. Neverthe-less, such promising methods cannot simply be extrapolatedto transparent materials because of the presence of interfer-ences in these films. As a matter of fact, when applyingvirtual-interface methods14 to transparent thin films likesilica, a serious decrease of the accuracy in the determination
of the refractive index of the outermost layer is found asfunction of the film thickness.
Before extrapolation to feedback control, real time moni-toring of the growth of transparent thin films by spectro-scopic phase modulated ellipsometry~SPME!2 is evaluatedhere. For such applications, SPME can take advantage of itsfast recording capability. The films consist of silicon alloys~nitride or oxide! deposited on crystalline silicon~c-Si! byplasma enhanced chemical vapor deposition~PECVD!. Twoon-line methods for the simultaneous determination ofn andd are compared and discussed. The first one is based on thefast inversion of the Fresnel equations and can be used evenif the films are grown at high deposition rates~.30 Å s21!.On the other hand, a much better precision can be obtainedusing a slower least square fit procedure based on singlephoton energy SPME. More generally, it is shown that theon-line monitoring of transparent thin films can be achievedwithout the need of an extensive real time spectroscopiccapability.4,7,8
II. EXPERIMENTAL DETAILS
The film growth is studiedin situ using a spectroscopicphase modulated ellipsometer UVISEL~of ISA JobinYvon!.2 The ellipsometer is directly adapted to the plasmachamber by means of optical fibers. Let us briefly recall that,in SPME, the high frequency polarization modulation~50kHz! is provided by a photoelastic device. Two detectionsystems can be used. First, the ellipsometer can record spec-troscopic measurements, continuously ranging from 1.5 up to5 eV, by means of a 10 cm focus monochromator. Moreover,at each photon energy, the modulation amplitudeA of thephotoelastic device fulfills the condition
J0~A!50, ~1!
where J0 is the zero order Bessel function. On the otherhand, a 20 cm focus spectrograph can also be used, allowingthe simultaneous recording of four photon energies. In thepresent study, the kinetic ellipsometry trajectories at 2.0, 2.8,
a!Also at Applied Optics Group, Norwegian Institute of Technology, Trond-heim, Norway.
and 3.5 eV are sequentially recorded at 2 or 3 s intervals.~with 0.3 s integration at each photon energy!. In the lattercase,A is fixed whatever the photon energy, the condition~1! being fulfilled at 4 eV.
Finally, the ellipsometric ratior5tanC exp(iD) is de-termined from the quantitiesI s and I c , which are directlyprovided by the numerical signal processing system:2
I s5sin 2C sin D, I c5sin 2C cosD. ~2!
The conventional calibration procedure of SPME2,16was sys-tematically used, without particular care to the remainingexperimental problems, such as those encountered whenC.45°.17
Silicon oxide~SiO2! and silicon nitride~SiNx! transpar-ent thin films are deposited onc-Si substrates at 70 °C byPECVD using a dual mode microwave/radio frequency sys-tem described elsewhere.18,19 In this study, SiO2 films aregrown with deposition ratesr d ranging from 1.5 to 32 Å s21,the deposition rate of SiNx being estimated at 4.0 Å s21. Theprevious deposition rates were determined from kinetic ellip-sometry measurements. This PECVD process leads to thegrowth of homogeneous films as revealed by the resultsshown in Fig. 1, which displays the cycling real time trajec-tory, recorded at 3.8 eV, during the growth of SiO2. Theexperimental trajectory is compared with a calculation cor-responding to the uniform growth, at 1.5 Å s21, of a materialwith refractive indexn51.45. A very good agreement isobserved in Fig. 1. More generally, such homogeneous be-havior is observed whatever the deposition conditions ofSiO2 and SiNx used in the present study.
The on-line determination ofn andd, using a few realtime trajectories, is described below. After film deposition,n and d can also be determined fromin situ spectro-scopic measurements, using the full spectrum capability. Theagreement between the two determinations ofd remainswithin 1% in all the investigated cases. Thus, the precisionsgiven below correspond to the noise on the on-line determi-nation ofn andd.
III. THEORETICAL BACKGROUND
Let us recall that the ellipsometric ratior is defined fromthe complex reflectances corresponding to the polarizationorientations parallel (r p) and perpendicular (r s) to the planeof incidence, respectively, by
r5r p /r s . ~3!
In the case of an ideal double-layer structure consistingof a homogeneous-isotropic thin film deposited on a semi-infinite absorbing substrate with smooth interfaces, the equa-tions that giver in terms of film refractive index and thick-ness are well known.20 These equations lead always to aquadratic expression as:
a~n,r!X21b~n,r!X1c~n,r!50, ~4!
with X5exp(2 ib) and
b5~4p/l!b854p~dl!n cosF, ~5!
wherel is the wavelength andF the refraction angle for thefilm ~related to the angle of incidenceF0 by the Snell’s law!.Besides, if the film is transparent,n, F, andb are real values,making the modulus ofX equal to 1. As a consequence, Eq.~4! and its complex conjugate share the same solution, pro-viding the condition:
G~n!5~ uau22ucu2!22ua* b2cb* u250. ~6!
It can be shown, from Eq.~6!, that the dielectric function«(«5n2) of the layer satisfies a fifth-degree polynomialequation.12,21 Then one derives:
X05uau22ucu2
cb*2a* b, ~7!
and finally the film thickness:
d5Arg~X0!l
4pAn22 sin2 F0
. ~8!
However, there is a singularity forb52kp or:
d5kl
2An22sin2 F0
. ~9!
The behavior of Eq.~6! in typical operating conditions~weaksubstrate absorption! is shown in Fig. 2. In this case, theprecision on the determination ofn is related to the slope ofG(n) nearG(n)50. In particular, this precision is very poorfor very thin films, according to Eq.~9!, and then increases.The precision is found maximum near
d52~k11/2!l
4An22sin2 F0
, ~10!
and then decreases until the first singularity@see Eq.~9!#; andso on. In the UV range, which corresponds to high substrateabsorption, a more complex behavior can be obtained.
More generally, the features displayed in Fig. 2 illustratethe limitations of the use of inversion methods in the case ofsingle wavelength ellipsometry.6
FIG. 1. Real time examination, at 3.8 eV, of the PECVD growth of SiO2 ona c-Si substrate.
1957Rev. Sci. Instrum., Vol. 67, No. 5, May 1996 Transparent thin film This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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5000–6000 Å thick. Thus, the fitting procedure provides bet-ter precision than the inversion method. Furthermore, as aconsequence of the correlation betweenn andd, the productnd ~orb8! can be determined more precisely, as alreadypointed out. Figure 10 shows thatnd can be determined witha precision of 1%, a 0.8% precision being obtained forb8. Itwas verified that the combination of two or three photonenergies does not induce a significant increase of precision inthe present case~d,6000 Å!.
V. SUMMARY AND CONCLUSION
We have presented a real time monitoring of the growthby PECVD of transparent thin films by SPME. Two on-linemethods have been compared. The direct determination ofnandd, without need of any particular modeling, can be per-formed in 0.2 and 1–2 s, respectively, up to 6000 Å, using aconventional PC 486 computer. However, a decrease of thecomputation time, in both cases, can be expected from im-provements in hardware and software optimization. The firstone is based on the inversion of the Fresnel equations. Com-bining the measurements simultaneously recorded at twophoton energies,n andd can be determined with a relativeprecision of 1.5% in case of thin films~'1500 Å! deposited
at low deposition rates. At high deposition rates, a 5% rela-tive precision is obtained on 5000-Å-thick films. More gen-erally, the precision is found to decrease with the thickness.A number of experimental problems encountered in the in-version method can be overcome by using a least squarefitting procedure. Furthermore, a better precision onnd ~1%or less! can be obtained using this fitting procedure, evenwhen using single photon energy measurements. The lattermethod allows the monitoring of the surface layer and caneasily be extended to absorbing materials.
In conclusion, the present SE study shows that it be-comes possible, from the real time analysis of measurementsrecorded at few photon energies, to control transparent ma-terial growth through a closed-loop adjustment of processvariables. Such variables can be the gas flow ratios that es-tablish alloy composition in the case of PECVD. Thus, it canbe anticipated thatin situ spectroscopic ellipsometry, with afast recording capability as SPME, appears to be a very use-ful tool for real time control of thin film growth.
ACKNOWLEDGMENTS
The authors are grateful to B. Equer, O. Hunderi~fromthe University of Trondheim!, J. C. Deutsch, and J. P. Gail-lard ~from CMO Grenoble! for stimulating discussions. Thiswork was sponsored by De´legation Generale pourl’Armement under Contract No. DRET 94-062.
1See, for instance,Spectroscopic Ellipsometry, Thin Solid Films233, 234~1993!.
3R. W. Collins, I. An, H. V. Nguyen, Y. Li, and Y. Lu,Optical Charac-terization of Real Surfaces and Filmsin Physics of Thin Films, edited byK. Vedam~Academic, San Diego, 1994!, Vol. 19.
4W. M. Duncan and S. A. Henck, Appl. Surf. Sci.63, 9 ~1993!.5D. E. Aspnes, W. E. Quinn, and S. Gregory, Appl. Phys. Lett.57, 2707~1990!.
6I. F. Wu, J. B. Dottelis, and M. Dagenais, J. Vac. Sci. Technol. A11, 2398~1993!.
7S. A. Henck, W. M. Duncan, L. M. Loewenstein, and S. W. Butler, J. Vac.Sci. Technol. A11, 1179~1993!.
8W. M. Duncan, S. A. Henck, J. W. Kuehne, L. M. Loewenstein, and S.Maung, J. Vac. Sci. Technol. B12, 2779~1994!.
9Y. Yoriume, J. Opt. Soc. Am.73, 888 ~1983!.10D. Charlot and A. Maruani, Appl. Opt.24, 3368~1985!.11S. Bosch, Surf. Sci.289, 411 ~1993!.12J. Lekner, Appl. Opt.33, 5159~1994!.13M. Kildemo and B. Dre´villon, Appl. Phys. Lett.67, 918 ~1995!.14D. E. Aspnes, J. Opt. Soc. Am. A10, 974 ~1993!.15S. Kim and R. W. Collins, Appl. Phys. Lett.67, 3010~1995!.16O. Acher, E. Bigan, and B. Dre´villon, Rev. Sci. Instrum.60, 65 ~1989!.17J. Campmany, E. Bertran, A. Canillas, J. L. Andu`jar, and J. Costa, J. Opt.Am. A 10, 713 ~1993!.
18J. C. Rostaing, F. Coeuret, B. Dre´villon, R. Etemadi, C. Godet, J. Huc, J.Y. Parey, and V. Yakovlev, Thin Solid Films236, 58 ~1993!.
19R. Etemadi, C. Godet, M. Kildemo, J. E. Boure´e, R. Brenot, and B.Drevillon, J. Non-Cryst. Solids187, 70 ~1995!.
20R. M. A. Azzam and N. M. Bashara,Ellipsometry and Polarized Light~North-Holland, Amsterdam, 1977!.
21J. P. Drolet, S. C. Russev, M. I. Boyanov, and R. Leblanc, and R. Leblanc,J. Opt. Soc. Am. A11, 3284~1994!.
22W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Velterling,Numerical Recipes~Cambridge University Press, Cambridge, 1986!.
FIG. 9. On-line measurements of the refractive index (n) and thickness ofplasma deposited SiNx using single photon energy SPME. The method isbased on the least square fitting of the outermost 200-Å-thick layer.
FIG. 10. On-line measurements of the optical thickness of plasma depositedSiNx using single photon energy SPME. The method is based on the leastsquare fitting of the outermost 200-Å-thick layer.
1960 Rev. Sci. Instrum., Vol. 67, No. 5, May 1996 Transparent thin film This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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Untersuchung dünner Si3N4 - Schichten mit Ellipsometrie und RBS
von
Tanja Dessauvagie
Diplomarbeit in Physik angefertigt im Institut für Strahlen- und Kernphysik
vorgelegt der
Mathematisch-Naturwissenschaftlichen Fakultät der
Rheinischen Friedrich-Wilhelms-Universität Bonn
im September 1997
12
2.2 Lösung der ellipsometrischen Gleichung
2.2.1 Einleitung Eine andere Möglichkeit die Lösungspaare (n1,d) zu erhalten wurde von Drolev [DRO94] entwickelt. Diese Methode führt die ellipsometrische Gleichung auf ein Polynom 5. Grades zurück. Das Problem ist dann darauf reduziert, die Nullstellen (oder Wurzeln) dieses Polynoms zu finden. Die Koeffizienten dieses Polynoms werden, wie in der oben erwähnten numerischen Methode auch, durch den Einfallswinkel ϕ0, den reellen Brechungsindex des umgebenden Mediums (meistens Luft), den komplexen Brechungsindex des Substrats und das gemessene ellipsometrische Verhältnis ρ bestimmt. Bei den numerischen (klassischen) Methoden hat man das Problem, daß die Lösung nur sehr langsam konvergiert oder überhaupt nicht erreicht wird. Dies hängt vor allem von der Wahl der Anfangswerte der gesuchten Parameter ab. Ein weiteres Problem stellt die Tatsache dar, daß es normalerweise mehr als eine mathematische oder physikalische Lösung für eine Messung des Systems gibt, d.h. es gibt mehrere Lösungen, die die gleichen Werte für Ψ und ∆ ergeben. Man muß alle physikalisch sinnvollen Lösungen kennen, um zwischen den erhaltenen Lösung zu entscheiden, welche für das System die Zutreffende ist. Mit numerischen Methoden ist es schwierig alle Lösungen zu einem Problem zu finden oder im voraus zu wissen, wieviele Lösungen es gibt, da die Anzahl von System zu System verschieden sein kann. Es gibt also keine Garantie, daß alle Lösungen gefunden werden. Bei der analytischen Methode wird ausgenutzt, daß die Lösung in zwei Schritte unterteilt werden kann: im ersten Schritt löst man eine Gleichung, die nur vom Brechungsindex abhängt, im zweiten Schritt wird die Dicke der Schicht aus diesem Brechungsindex berechnet. Man kann zeigen, daß sich die Gleichung für den Brechungsindex, die auf den ersten Blick recht kompliziert aussieht (ein Polynom 24. Grades), auf ein Polynom 5. Grades zurückführen läßt. Obwohl diese Methode nicht vollkommen analytisch ist (mindestens eine Nullstelle dieses Polynoms muß iterativ gefunden werden), ist sie direkt und komplett in dem Sinn, daß man von Anfang an weiß, daß es maximal fünf Nullstellen gibt (Polynom 5. Grades). Außerdem kann man von diesen fünf Lösungen gut erkennen, welche Lösungen sinnvoll sind und welche nicht. Die einzige Vorausetzung, die gemacht werden muß, ist die, daß der relative Brechungsindex n = n1/n0 reell, d.h. die Schicht transparent sein soll.
2.2.2 Herleitung des Polynoms 5. Grades Um das Polynom 5. Grades herzuleiten, beginnt man mit einer Umformung von ρ in Gleichung (2.16). Man führt eine zusätzliche komplexe Variable X ein, alle anderen Variablen bezeichnen die üblichen Werte: Schichtdicke d, Brechungsindizes der Umgebung, Schicht, Substrat n0, n1, n2, Wellenlänge λ, Einfallswinkel ϕ0, und die Fresnel´schen Reflexionskoeffizienten rij für die Reflexion an der ij. Grenzschicht.
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6 Literatur [Ana75] K.V. Anand, S.K. Momodu: Applications of ellipsometry in
North-Holland Publishing Company, New York, (1977) [Deh95] E.Dehan, P.Temple-Boyer, R.Henda, J.J.Pedroviejo, E.Scheid :
Optical and structural properties of SiOx and SiNx materials. Thin Solid Films 266 (1995), p.14-19
[Dro94] J.-P. Drolet, S.C. Russev, M.I. Boyanov, R.M. Leblanc: Polynomial inversion of the single transparent layer problem in ellipsometry. J. Opt. Soc. Am. A/Vol. 11, No12/December 1994, p.3284-91
[Gri96] V.A.Gritsenko, A.D.Milov: Wigner crystallization of electrons and holes in amorphous silicon nitride. Antiferromagnetic ordering of localized electrons and holes as a result of a resonance exchange interaction. JETP Letters, Vol.64, No.7, 10.Oct.1996
[Mar90] G.Marx : Aufbau und Test einer Kurzzeit-Temper-Anlage. Diplomarbeit, Universität Bonn, (1990)
[Men96] M.Mendel : Aufbau und Test eines Ellipsometers zu Bestimmung der optischen Eigenschaften dünner dielektrischer Schichten. Diplomarbeit, Universität Bonn, (1996)
[Mod04] Handbook of Modern Ion Beam Materials Analysis. Materials Research Society, Pittsburgh, Pennsylvania
[Möl92] A.Möller : Aufbau und Test eines elektronischen Temperaturreglers für die RTA-Kurzzeit-Temper-Anlage, Diplomarbeit, Universität Bonn (1992)
[Zha92] S.-L.Zhang, J.-T.Wang, W.Kaplan, M.Östling : Silicon nitride films deposited from SiH2Cl2-NH3 by low pressure chemical vapor deposition: kinetics, thermodynamics, composition and structure. Thin Solid Films, 213 (1992), p.182-191
[Zie77] J.F.Ziegler : The Stopping and Ranges of Ions in Matter, Vol. 4 : Helium, Stopping Powers and Ranges in All Elements, Pergamon Press 1977
Mixed monolayers of cationic surfactants and anionic polymersat the air-water interface: Surface tension and ellipsometrystudies
A. Asnacios1, D. Langevin2,a, and J.F. Argillier3
1 Laboratoire de Physique des Materiaux Divises, Universite de Marne la Vallee (IFI) 2, Cite Descartes,Champs-sur-Marne, 77454 Marne la Vallee, Cedex 2, France
2 Laboratoire de Physique des Solides, Batiment 510, Universite Paris Sud, 91405 Orsay, France3 Institut Francais du Petrole, 1 Avenue de Bois-Preau, 92852, Rueil Malmaison, France
Received: 16 October 1997 / Revised: 14 May 1998 / Accepted: 15 May 1998
Abstract. Equilibrium surface tension measurements have been carried out on mixed solutions of a non-surface active polyelectrolyte (polyacrylamide sulfonate) and cationic surfactants. A strong synergisticlowering of the surface tension is found in the concentration range where no appreciable complexation ofsurfactant and polymer occurs in the bulk solution (as seen from viscosity measurements). The surfacetension decrease does not depend upon the polymer molecular weight, and there is a limited influence ofthe surfactant chain length. The influence of the degree of charge of the polymer is more important: forsmall degree of charge, the complexation is less cooperative, and the structure of the surface complex islooser.
PACS. 68.10.Cr Surface energy (surface tension, interface tension, angle of contact, etc.) –78.66.SqComposite materials – 82.70.Gg Gels and sols
1 Introduction
Interactions between surfactants and polymers is a rapidlygrowing field of interest in colloid science [1]. Many prac-tical systems for industrial applications contain mixturesof polymers and surfactants, which are widely used asthickeners in water based formulations such as paints,drilling muds, etc. In these applications, polyelectrolytesare of particular interest, because of the important roleof polymer charges. Biological membranes are structuredcomplexes of lipids and proteins, which are also chargedpolymers. Whereas there is an extensive literature on sur-factant solutions on one hand, and polymer solutions onthe other hand, much less is known for the mixed solu-tions. Polyelectrolytes solutions are less well understoodthan neutral polymer solutions, but recent work has al-lowed to improve the current knowledge [2]. These poly-mers form much more extended structures than neutralpolymers, with effective persistence lengths much largerthan those of neutral polymers. The polyelectrolyte solu-tions are in the semi-dilute range at very small polymerconcentrations; however, the viscosity of the solutions in-creases less rapidly with concentration than for neutralpolymers, because the polyelectrolyte chains are rod-likeand less strongly entangled. The influence of the counte-rions is subtle: partial counterion condensation frequently
occurs and the concentration of counterions is enhancedclose to the polyion. When two polyelectrolytes of oppo-site charge are mixed, the two polyions associate, thus re-leasing the counterions in the solution, and increasing theentropy of the solution [3]. It was observed that the be-haviour of polyelectrolyte-surfactant solutions is similar tothe behaviour of polyelectrolyte-polymer solutions: no as-sociation when the surfactant or the polymer are nonionicor when the two species have the same charge, and strongassociation for opposite charges. The case of surfactant-polymer mixtures is however less simple, because the sizeand the shape of the surfactant aggregates can vary [4–9].
Polymers and surfactants also form complexes at sur-faces, either solid-fluid or fluid-fluid surfaces. Complex-ation in bulk or at a surface are generally related [10].Surface complexation is also important for practicalapplications such as colloidal stabilization, wettability, ad-hesion, etc. Only few studies have focused on surfactant-polyelectrolyte complexation at a surface [11–16]. In thiswork we present a study of polymer-surfactant complex-ation at the free surface of an aqueous solution. We havestudied a model polymer for drilling muds formulations:polyacrylamide sulfonate. A first study has already beenreported [17]. Here, the study is extended, and the role ofdifferent parameters is reported: surfactant chain length,polymer molecular weight, and polymer degree of charge.We have characterized the bulk complexation by viscosity
906 The European Physical Journal B
measurements, and the surface complexation by surfacetension and ellipsometry measurements.
2 Experimental section
2.1 Materials
The cationic surfactants, dodecyl and hexadecyl trimethylammonium bromides (DTAB and CTAB) from Aldrich(99%), were recrystallized (2 g surfactant, 10 ml ethyl ac-etate, 1 ml ethyl alcohol) 3 times before use. For bothsurfactants, no minimum in the surface tension versus sur-factant concentration was observed. Potassium bromide(KBr) was supplied by Fluka.
The polyanion (synthesized by SNF Floerger) is a sta-tistical copolymer composed of neutral monomers of acry-lamide (AM) and charged monomers of acrylamido methylpropane sulfonate (AMPS). The polymer chemical struc-ture has been characterized by titration via a bromidationreaction for the amide function and potentiometric titra-tion for the sulfonate. The molecular weight and poly-dispersity of the polymer were measured by size exclu-sion chromatography (SEC) coupled with multiangle lightscattering. Two different molecular weights were studied,Mw = 2.2× 106 and Mw = 4× 105. For Mw = 2.2× 106,two degrees of charge were investigated: f = 25 mol%and 10 mol% (f is the fraction of charged monomers;the number of charged monomers per macromolecule isx = fN , where N is the total number of monomers). Thecharacteristics of the three polymer samples studied aresummarized in Table 1. To eliminate any traces of surfac-tant molecules and low molecular weight impurities, thepolymer solutions were passed through an ultrafiltrationunit with a 20 000 cut-off membrane. Final concentrationswere determined using a total carbon analyzer ShimatzuTOC 5050. After this purification, the polymer displays nosurface activity at concentrations below 2000 ppm. Purewater was taken from a Millipore Milli-Q system. Polymer-surfactant mixtures were obtained by mixing pure surfac-tant and pure polymer solutions.
2.2 Surface tension measurement
Experiments were performed at room temperature20±1 ◦C. Measurements were carried out in a Teflontrough (6 cm diameter) housed in a Plexiglas box with anopening for the tensiometer. The surface tension was mea-sured with an open frame version of the Wilhelmy plate(to avoid the wetting problems of the classical plate [18]).The rectangular (20 mm × 10 mm) open frame, madefrom a 0.19 mm diameter platinum wire, was attached toa force transducer (HBM Q11) mounted on a motor allow-ing it to be drawn away from the surface at a controlledconstant rate.
For mixed solutions at low concentrations of surfac-tant and polyelectrolyte, the approach to the equilibriumcould take more than 3 hours and we did not find any reli-able method to get the equilibrium surface tension of such
a system by extrapolation to infinite time. Thus, it was as-sumed arbitrarily that equilibrium had been reached whenthe surface tension variation was less than 0.01 mN/m over10 minutes.
The reproducibility, including long equilibration timeand/or contamination effects, was 0.5 mN/m for mixedsolutions. Surface tensions measured on polymer-free so-lutions of surfactants were in good agreement with theliterature values.
2.3 Viscosity measurement
Relative viscosities of polymer solutions and mixedpolymer-surfactant solutions were measured using a lowshear viscosimeter (Contraves 30) which has a coaxialcylindrical geometry. The shear thinning behaviour ofthe polymer has been characterized and all subsequentmeasurements have been done at low shear rates, below0.5 s−1.
2.4 Ellipsometry
Ellipsometric angles were measured by means of a PLAS-MOS (SD 2300) rotating-analyzer ellipsometer. Measure-ments on free surface of water, ethanol and cyclohexanewere in good agreement with the values given in refer-ence [19]. In our experiments, two sets of measurementsare performed, one on pure water (reference ψ0, ∆0), thesecond on the solution (ψd, ∆d). The thickness d and therefractive index nd of the adsorbed layer are then deducedfrom the ellipsometric angles variations δψ = ψd−ψ0 andδ∆ = ∆d − ∆0. Assuming flat and homogeneous layers,the ellipsometric equations are inverted according to ref-erence [20]. For the layers studied here, the thicknessesare very small and δψ is of the order of the experimentalaccuracy. It is then important to check the consistencyof the values of d and nd determined by this way, withthose obtained from a different analysis based on δ∆ andthe determinations of area per molecules from the Gibbsequation (see Ref. [17] for details).
3 Results and discussion
We have worked with polymer/surfactant solutions offixed polymer concentration and varying surfactant con-centration. We have thus studied the changes induced bythe presence of fixed amount of polymer in the surfactantsolutions.
3.1 Role of the surfactant
Mixed solutions of DTAB and AMPS show a synergisticlowering of surface tension at very low surfactant concen-trations. As seen in Figure 1, the surface tension curveexhibits two break points: the first one, known as the crit-ical aggregation concentration (CAC), corresponds to the
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References
1. E.D. Goddard, K.P. Ananthapadmanabhan, Interactionof Surfactants with Polymers and Proteins (CRC press,1993).
8. I. Iliopoulos, U. Olsson, J. Phys. Chem. 98, 1500-1505(1994).
9. X. Li, Z. Lin, J. Cai, L.E. Scriven, H.T. Davis, J. Phys.Chem. 99, 10865-10878 (1995).
10. B. Cabane, R. Duplessix, J. Phys. 43, 1529-1542 (1982).
11. E.D. Goddard, Coll. Surf. 19, 301-329 (1986).12. J.H. Buckingham, J. Lucassen, F. Hollway, J. Coll. Int.
Sci. 67, 423 (1978).13. V. Shubin, Langmuir 10, 1093-1100 (1994).14. J.F. Argillier, R. Ramachandran, W.C. Harris, M. Tirrel,
J. Coll. Int. Sci. 146, 242 (1991).15. V.G. Babak, M.A. Anchipolovskii, G.A. Vikhoreva, I.G.
Lukina, Coll. J. 2, 145-152 (1996) (translated from russian)16. I. Nahringbauer, Langmuir 13, 2242-2249 (1997).17. A. Asnacios, D. Langevin, J.F. Argillier, Macromol. 29,
7412 (1996).18. E.K. Mann, Ph.D. thesis, Paris, 1992.19. J. Meunier, J. Phys. 48, 1818-1831 (1987).20. J.P. Drolet, S.C. Russev, M.I. Ivanov, R.M. Leblanc, J.
Opt. Soc. Am. A 11, 3284-91 (1994).21. A. Espert, unpublished.22. D. Den Engelsen, B. De Koning, J.C.S., Faraday I, 70,
1603 (1974).23. P. Drude, Ann. Phys. 36, 532, 865 (1989).24. V. Bergeron, D. Langevin, A. Asnacios, Langmuir. 12,
1550 (1996).25. Y. Jayalakshmi, D. Langevin, submitted for publication.
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Analytic methods in the determination of opticalproperties by spectral ellipsometry
1. INTRODUCTIONEllipsometry is a primary technique for determining theoptical properties, including thickness, of bulk materialsand thin films. It is based on the principle that fully po-larized light, on reflection, undergoes a change inpolarization.1 By illuminating the sample with light ofknown polarization, wavelength, and angle of incidence,one may obtain information about the surface properties.On reflection the state of polarization of the reflected lightis historically written, in ellipsometry, as
r 5rp
rs5 tan~c!exp~iD!, (1)
where rp is the complex P polarization reflectance, rs isthe complex S polarization reflectance (the ratio of the in-cident to reflected electric fields), c represents the ratio ofthe intensities of the two polarizations, and D is the rela-tive phase between the two polarizations.
A homogeneous substrate, without any thin films, isgenerally characterized by having uru > 0. This is a con-sequence of ursu . 0. The condition uru 5 0 occurs at theBrewster angle when urpu 5 0. With the addition of thinfilms the condition ursu 5 0 can easily be made to occur.When this happens, uru → `. This is not a consequenceof coordinate choice for rp and rs but is inherent in thedefinition of r. This paper explores one consequence of asimple thin-film system, in which ursu → 0 on ellipsom-etry.
The reflectivity ratio r given in Eq. (1) is readily calcu-lated for monochromatic light for a plane interface, in-cluding a number of thin films. The usual starting pointis with the derivation of the Fresnel reflection equations(see Ref. 1, Chap. 4). For the simplest case of reflectionfrom a planar surface defined as the interface betweentwo homogeneous isotropic media r is found to be a func-tion of at least five parameters,
r 5 r~nsup , nsub , ksub , u, l!, (2)
where it is implicitly assumed that the superstrate istransparent (ksup 5 0). With the addition of homoge-neous isotropic thin films, each layer adds three more pa-rameters: the indices of refraction (n and k) and thick-ness. Implicit in the derivation is the assumption thatthe longitudinal coherence of the light source is greaterthan the thickness of the thin-film stack under investiga-tion.
Given a set of measurements of c and D for each wave-length and angle of incidence, estimation of the remain-ing parameters is made by starting from some initialmodel of the interface. In normal practice some param-eters are held fixed while others are allowed to vary. Themost general ellipsometric measurement for a givensample would be to vary the wavelength, polarization,and angle of incidence over some interesting range. Thisis not always possible, given constraints of in situ thin-film measurement requirements. For example, the angleof incidence is constrained to be fixed on the basis of avail-able view ports on a thin-film deposition chamber. Ellip-someters that run with a fixed angle of incidence whilevarying the wavelength are spectral ellipsometers (SE’s).
With a starting model of the surface, the challenge is toinvert Eq. (2), given a set of measurements of c and D foreach wavelength. Several techniques have been devel-oped for this purpose.2,3 SE’s often employ analyticmethods such as the Levenberg–Marquardt4 algorithmfor this purpose. Essentially what is done is to fit thenonlinear SE equations to the data. Success of analyticschemes in general requires continuity and existence offirst partial derivatives of r. These conditions are easilyviolated when the SE is operated near or beyond certaincritical angles of the material under investigation. Theprincipal sources of trouble are discontinuities in D thatarise from the branch cut of the tangent (and arctan)
2228 J. Opt. Soc. Am. A/Vol. 15, No. 8 /August 1998 David U. Fluckiger
robust10 for finding global minima by means of floating-point arithmetic in contrast to other genetic algorithmsthat rely on bit manipulations.
REFERENCES AND NOTES1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and
Polarized Light (North-Holland, New York, 1996).2. D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno,
C. K. Carniglia, W. E. Case, J. A. Dobrowolski, U. J. Gibson,T. T. Hart, F. C. Ho, V. A. Hodgkin, W. P. Klapp, H. A.Macleod, E. Pelletier, M. K. Purvis, D. M. Quinn, D. H.Strome, R. Swenson, P. A. Temple, and T. F. Thonn, ‘‘Mul-tiple determination of the optical constants of thin-filmcoating materials,’’ Appl. Opt. 23, 3571–3596 (1984).
3. J. P. Drolet, S. C. Russev, M. I. Boyanov, and R. M. Leblanc,‘‘Polynomial inversion of the single transparent layer prob-lem in ellipsometry,’’ J. Opt. Soc. Am. A 11, 3284–3291(1994).
4. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vet-terling, Numerical Recipes in C (Cambridge U. Press, NewYork, 1991).
5. G. E. Jellison, ‘‘The use of the biased estimator in the inter-
pretation of spectroscopic ellipsometry data,’’ Appl. Opt. 30,3354–3360 (1991).
6. C. M. Herzinger, H. Yao, and P. G. Snyder, ‘‘Determinationof AlAs optical constants by variable angle spectroscopic el-lipsometry and a multisample analysis,’’ J. Appl. Phys. 77,4677–4687 (1995).
7. A Verity Instruments Series-10 Spectral Ellipsometer wasused in obtaining data. The wavelength increment is de-termined by the spectrograph optics and the grating disper-sion, which varies from 9.5 nm at the short-wavelength endto 9.1 nm at the long-wavelength end.
8. E. D. Palik, ed., Handbook of Optical Constants of Solids,Volume 1 and 2 (Academic, New York, 1991).
9. K. Price and R. Storn, ‘‘Differential evolution—a simple andefficient adaptive scheme for global optimization over con-tinuous spaces,’’ Tech. Rep. TR-95-012 (International Com-puter Science Institute, University of California at Berke-ley, Berkeley, Calif., 1995) (available throughftp.icsi.berkeley.edu).
10. R. Storn and K. Price, ‘‘Minimizing the real functions of theICEC’96 contest by differential evolution,’’ in Proceedings ofthe 1996 IEEE Conference on Evolutionary Computation(ICEC ’96) (Institute of Electrical and Electronics Engi-neers, New York, 1996), pp. 842–844.
2232 J. Opt. Soc. Am. A/Vol. 15, No. 8 /August 1998 David U. Fluckiger
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Bayesian inference analysis of ellipsometry data
N. P. Barradas*School of Electronic Engineering, Information Technology and Mathematics, University of Surrey, Guildford,
Surrey GU2 5XH, United Kingdom
J. L. Keddie† and R. SackinSchool of Physical Sciences, University of Surrey, Guildford, Surrey GU2 5XH, United Kingdom
~Received 8 September 1998!
Variable angle spectroscopic ellipsometry is a nondestructive technique for accurately determining thethicknesses and refractive indices of thin films. Experimentally, the ellipsometry parametersc and D aremeasured, and the sample structure is then determined by one of a variety of approaches, depending on thenumber of unknown variables. The ellipsometry parameters have been inverted analytically for only a smallnumber of sample types. More general cases require either a model-based numerical technique or a series ofapproximations combined with a sound knowledge of the test sample structure. In this paper, the combinatorialoptimization technique of simulated annealing is used to perform least-squares fits of ellipsometry data~bothsimulated and experimental! from both a single layer and a bilayer on a semi-infinite substrate using what iseffectively a model-free system, in which the thickness and refractive indices of each layer are unknown. Theambiguity inherent in the best-fit solutions is then assessed using Bayesian inference. This is the only way toconsistently treat experimental uncertainties along with prior knowledge. The Markov chain Monte Carloalgorithm is used. Mean values of unknown parameters and standard deviations are determined for each andevery solution. Rutherford backscattering spectrometry is used to assess the accuracy of the solutions deter-mined by these techniques. With our computer analysis of ellipsometry data, we find all possible models thatadequately describe that data. We show that a bilayer consisting of a thin film of poly~styrene! on a thin filmof silicon dioxide on a silicon substrate results in data that are ambiguous; there is more than one acceptabledescription of the sample that will result in the same experimental data.@S1063-651X~99!02905-0#
Ellipsometry is a fast accurate technique for measuringthe optical constants, interfacial roughness, and thicknessesof thin films. The technique has become widespread over thelast 30 years in a diverse range of fields. Two recent reviews@1,2# highlight examples of ellipsometry applications. Theseinclude the determination of glass transition temperatures inpolymer thin films@3#, thin film swelling @4#, adsorption ofsmall molecules at solid/liquid interfaces@5,6#, the character-ization of Langmuir-Blodgett films@7# and the determinationof damage depth profiles from ion implantation in siliconwafers@8#. For a given sample, ellipsometry measures ellip-ticity r, which is written as
r5Rp
Rs5tanceiD, ~1!
whereRp andRs are the Fresnel reflection coefficients, withp denoting the plane of reflection ands denoting the planeperpendicular to it@9#. The Fresnel coefficients are depen-dent on experimental parameters: the angle-of-incidence of
light, w0 ~conventionally measured from the sample surfacenormal!, and the wavelength of radiation,l. The Fresnelcoefficients are also functions of material parameters: com-plex refractive indices of each of the components,N, andeach of the layer thicknesses,d. The parameterD is thechange in phase difference between thep ands componentscaused by reflection, whilec is the ratio of the amplituderatios of thep and s light components before and after re-flection. The ellipticity is measured by the analysis of theelliptically polarized light reflected from a flat, smoothsample surface. The ratio can be described algebraically byan expression derived from the Fresnel coefficients forn lay-ers on a semi-infinite substrate, wheren is any integer num-ber @9#. Starting with~c,D! pairs obtained at knownw0 andl, one can invert this expression, under certain circum-stances outlined below, to find values for the unknown pa-rameters, such asN, d, and the roughness of each layer in asample. Only a few specific cases have as yet been invertedanalytically. Droletet al. @10# summarized these structures as~1! a single layer with unknown complex refractive index ona known substrate;~2! a substrate with two layers and withone layer thickness being unknown;~3! multilayer systemswith any one unknown layer thickness;~4! multilayer sys-tems with unknown substrate complex refractive index;~5! asymmetric system of one layer of unknown thickness andreal refractive index embedded in two identical phases hav-ing a real index; and~6! an optically absorbing layer on asubstrate with a complex refractive index, and with the thick-ness of the layer being unknown.
*Permanent address: Instituto Tecnolo´gico Nuclear, E.N. 10, 2685Sacave´m, Portugal.
†Author to whom correspondence should be addressed. Electronicaddress: [email protected]
Cases~1!–~5! require measurements onone substrate atonewavelength andoneangle. Case~6! is somewhat differ-ent in that it requirestwo different angles of incidence ontwodifferent substrates. It should be noted that inherent correla-tion exists between measurements at two different values ofw0 and l. Hence, increasing the number of measurementsdoes not necessarily increase the amount of informationabout a sample. Moreover, the use of multiplel values po-tentially introduces more unknown material parameters dueto optical dispersion and absorption. In some cases, however,the number of unknowns can be reduced by using analyticalor empirical expressions to describe a material’s opticalproperties@11,12#.
A number of numerical inversion methods have been de-veloped which are suitable for different problems. There areat least three categories of methods, as outlined below.
~1! There are exact numerical methods suitable for whenthe number of unknowns is equal to the number ofc andDmeasurements. The Reinberg method@13# is most often citedfor single-angle, single-wavelength ellipsometry to deter-mine the unknown refractive index and thickness of a thinnonabsorbing film on a reflecting substrate with known op-tical constants. A similar method is the functional-link neuralnetwork approach of Parket al. @14#. Other techniques arecited by Droletet al. @10#.
~2! When the number of unknown variables is greaterthan the number ofc andD data points, an exact solution canobviously not be written. Multiparameter fitting methods, ofwhich the Levenberg-Marquardt@15# algorithm is commonlyused, are suitable for data from variable angle spectroscopicellipsometry~VASE! ~a technique where manyw0 andl areused!. Such an algorithm is used to minimize the differencebetween the experimental~c,D! spectra and the simulatedspectra generated from the Fresnel equations@12,9#. Modelscan be built to include any number of layers, with complexrefractive indices, on any substrate and in any ambient me-dium. Other physical properties such as biaxiality@16# andsurface roughness can also be included in the model. A re-lated technique is the backpropagation neural networkmethod of Fried and Masa@17#, which is trained to recognizecharacteristics of~c,D! spectra.
~3! There exist inversion-after-approximation schemes,such as that described by Charmet and de Gennes@18# formultiple-angle single-wavelength ellipsometry. This tech-nique can determine an arbitrary refractive index profile overdepths much greater thanl/4p.
The preceding methods, although for the most part quickto perform, are all limited when little or nothing is knownabout the test sample. The exact numerical methods are goodonly for specific, very simple samples. The multiparameterfitting methods only globally minimize the difference be-tween the simulated and experimental spectra if the initialguess solution is close to the global minima. Otherwise, localminima solutions are given. The neural network schemespresented in the literature have, so far, only been trainedsuccessfully for a limited set of solutions. The Charmet–deGennes method@18# can only give results for specific sampletypes over certain depth scales. Knowledge of the substrateand ambient refractive indices is also required. There isclearly a need for amodel-independenttechnique when asample is presented in which the structure is not well known.
The use of VASE has become widespread for the study ofcomplex samples, as it is the most powerful ellipsometrytechnique. In any approach to the analysis of VASE data, theaim is to determine a unique physical description, or model,of the unknown sample. Except in the case of exact datainversion, however, there is a lingering question of theuniqueness of the model obtained. Hence, prior to fitting thedata to a model, one should assess the solution space forambiguities. That is, one should find the number of exactsolutions that exist for a given set of~c,D! pairs. A study ofambiguity in solution space has recently been presented byPolovinkin and Svitasheva@19# using a ‘‘step-by-step move-ment’’ numerical method to search solution space for datafrom a single unknown layer deposited on a semi-infinitesubstrate. However, expanding this technique for ann-unknown problem would not be trivial. Unless the incre-ments used in the steps are infinitesimally small, there is afinite possibility that such a search will not find all solutionsto the problem. The search might miss a description of thesample that is the ‘‘true’’ description. We have thereforedeveloped an alternative method for assessing ambiguities.
In this paper, we use the simulated annealing~SA! algo-rithm to perform multiparameter least-square fits to ellipsom-etry data. Simulated annealing is a global optimization algo-rithm designed to find the absolute minimum~or maximum!of any given function@20–22#. It is completely general inthat it entails in principle no restrictions on the function to beminimized. In the case of ellipsometry data analysis, no as-sumptions need to be made about the sample’s physicalproperties. SA has solved previously intractable problemssuch as the traveling salesman problem@23#, and it is widelyapplied in fields ranging from ion beam analysis@24–29# tonatural language processing@30#. We show here that SAfinds solutions that correctly reproduce VASE data. How-ever, since it is a stochastic technique, if more than onesample structure can fit the data, SA will randomly find onlyone of the possible structures. To overcome this limitation,we have also applied the Bayesian inference, which is theonly way to consistently treat incomplete and noisy datawhen additional prior information is known. It is realisedusing the Markov chain Monte Carlo~MCMC! algorithm@31–33#. The MCMC algorithm explores the whole param-eter space, and is therefore able to find each and every solu-tion that is consistent with the data. Beneficially, it providesconfidence limits on the solutions obtained. It has alreadybeen successfully applied to other techniques@34–37#. Weapply SA and MCMC techniques to both theoretically gen-erated and experimental ellipsometry data.
II. THEORY OF ELLIPSOMETRY: DERIVATION OF THEFRESNEL REFLECTION COEFFICIENTS
The central equation of ellipsometry is given by Eq.~1!.In this section, we present expressions for the Fresnel coef-ficients@12#. For anm-layer system on a substrate~in whichm is a positive integer and the substrate is them11th layer!,Snell’s law states
PRE 59 6139BAYESIAN INFERENCE ANALYSIS OF ELLIPSOMETRY DATA
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ACKNOWLEDGMENTS
N.P.B. was partially supported by the U.K. Engineeringand Physical Sciences Research Council~EPSRC! throughGrant No. GR/L78512. We also thank the EPSRC for a grant
~Grant No. GR/L12066! toward the purchase of the ellipsom-eter used in this work and for financial support for R.S. Fi-nally, the comments of Dr. Chris Jeynes on a first draft of thepaper are greatly appreciated.
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F. Krimmel, and K. Bethge, Fresenius J. Anal. Chem.353, 734~1995!.
@9# R. M. A. Azzam and N. M. Bashara,Ellipsometry and Polar-ized Light~North-Holland, Amsterdam, 1977!, Chap. 4.
@10# J.-P. Drolet, S. C. Russev, M. I. Boyanov, and R. M. Leblanc,J. Opt. Soc. Am. A11, 3284~1994!.
@11# B. D. Johs, W. A. McGahan, and J. A. Woollam, Thin SolidFilms 253, 25 ~1994!.
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@13# A. R. Reinberg, Appl. Opt.11, 1273~1972!.@14# G. H. Park, Y. H. Pao, B. Igelnik, K. G. Eyink, and S. R.
Leclair, IEEE Trans. Neural Netw.7, 816 ~1996!.@15# D. W. Marquardt, J. Soc. Ind. Appl. Math.11, 431 ~1963!.@16# A. Zuber, H. Ja¨nchen, and N. Kaiser, Appl. Opt.35, 5553
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314, 128 ~1998!.@20# S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Science220,
671 ~1983!.@21# Emile Aarts and Jan Korst,Simulated Annealing and Boltz-
mann Machines: A Stochastic Approach to Combinatorial Op-timization and Neural Computing~Wiley, Chichester, 1989!.
@22# R. H. J. M. Otten and L. P. P. van Ginneken,The AnnealingAlgorithm ~Kluwer, Norwell, MA, 1989!.
@23# W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery,Numerical Recipes in Fortran, 2nd ed.~CambridgeUniversity Press, Cambridge, 1992!, p. 438.
@24# N. P. Barradas, C. Jeynes, and R. Webb, Appl. Phys. Lett.71,291 ~1997!.
@25# N. P. Barradas, P. K. Marriott, C. Jeynes, and R. P. Webb,Nucl. Instrum. Methods Phys. Res. B136-138, 1157~1998!.
@26# N. P. Barradas, C. Jeynes, R. P. Webb, U. Kreissig, and R.Grotzschel, Nucl. Instrum. Methods Phys. Res. B149, 233~1999!.
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@30# Y. A. Wilks, B. M. Slator, and L. M. Guthrie,Electric Words:Dictionaries, Computers, and Meanings~MIT Press, Cam-bridge, 1996!, p. 203–206.
@31# Maximum Entropy and Bayesian Methods in Inverse Prob-lems, edited by C. R. Smith and W. T. Grandy~Reidel, Dor-drecht, 1985!.
@32# Maximum Entropy in Action, edited by B. Buck and V. A.Macaulay~Clarendon, Oxford, 1994!.
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@37# N. P. Barradas, C. Jeynes, M. Jenkin, and P. K. Marriott, ThinSolid Films ~to be published!.
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PRE 59 6151BAYESIAN INFERENCE ANALYSIS OF ELLIPSOMETRY DATA
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Characterization of inhomogeneous ®lms by multiple-angle ellipsometry
SteÂphane Colard*, Martine Mihailovic
LASMEA, UMR 6602 CNRS, Universite Blaise Pascal de Clermont-Ferrand, Les CeÂzeaux, 63177 AubieÁre Cedex, France
Abstract
High performance devices often require good quality layers, homogeneous composition and crystallinity, then the homogeneity in depth of
the layers is of major interest. Our study investigates the possibility of evaluating the inhomogeneity of the refractive index of transparent
®lms deposited on absorbing substrate by multiple-angle ellipsometry. We propose a method which is based upon the determination of the
best experimental conditions leading to the minimal uncertainty one can expect on the physical parameters to be determined. A theoretical
study of an inhomogeneous Al2O3 layer deposited on an InP substrate is presented as an example. q 1998 Elsevier Science S.A. All rights
1. INTRODUCTIONEllipsometry is a well-known technique used to determinethe optical properties of thin films. It is based on theprinciple that polarized light changes polarization statewhen reflected. Once a sample is irradiated with light ofknown polarization, wavelength, and incidence angle, in-formation regarding the optical properties of the film andits thickness may be collected.
The ellipsometric equation is well known and is pre-sented in many texts. If the measured sample can betreated as a bulk substrate with a perfect surface, the re-flection ratio can be expressed as
r 5 rp /rs 5 tan~ c!exp~iD!, (1)
where rp and rs are the Fresnel reflection ratios for thecomponents parallel (p) and perpendicular (s), respec-tively, to the plane of incidence. Written in polar form,the complex ratio can be expressed with the two param-eters c and D (the ellipsometric angles), where tan c andD describe the amplitude ratio and the phase difference,respectively, between the p and s components. If thesample consists of a substrate with one or two more films,the measured reflection ratio can be written as
Rp /Rs 5 tan c exp~iD!, (2)
where the reflection coefficients Rp and Rs are functionsof the Fresnel reflection coefficients for the interfaces andthe film thicknesses.1
For standard single-wavelength ellipsometry of an ab-sorbing film with a known substrate, there are usuallythree unknowns: n, the real part of the refractive index;k, the imaginary part of the refractive index (the extinc-tion coefficient); and d, the film thickness. Once theangles c and D have been measured, the problem is thento find an algorithm to inverse this data and determinethe three unknowns, since no direct analytical solutioncan be found. The ellipsometric angles are measured atmore than one incidence angle.
Standard inversion algorithms require a starting point,or initial value. This initial value must usually be near
the correct value or the algorithm does not converge andcomputation time may be high, although with the expo-nentially rapid development of computers, the latter re-quirement is less of a concern.
We propose to use a genetic algorithm to inverse ellip-sometric data. Genetic algorithms do not require a start-ing point and in general have a greater range of conver-gence than other inversion techniques. They are readilyadaptable to many different problems and also have lowcomputation times. They also do not require the evalua-tion of derivatives, instead relying on a merit function toimprove performance.
2. BACKGROUNDSome of the first methods used to solve the ellipsometricinversion problem were polynomial inversion techniques.Among the first were those of McCrackin and Colson.2
Their algorithm determined two of the parameters, n andd, when a third, k, was known. The algorithm still re-quired an initial value, and this value had to be close tothe solution. However, uncertainties in the measure-ments and in the method’s precision rendered this algo-rithm impractical. Other researchers, such as Reinberg3
and later Easwarakhanthan et al.,4 used a multidimen-sional Newton algorithm (or a variation) to compute thefilm properties. Computing time was good, but this tech-nique had some convergence problems, especially for filmsless than 40 nm thick.
More recently, Urban5 proposed a new method, usingan algorithm of variably damped least squares. He com-puted the intersection of two solution curves at differentincidence angles. Two intersections are found, one for aplot of n versus d and one for d versus k. These curvesintersect at the correct solution of (n, k, d). If the modelis correct and the data are precise, the two solutionsshould be almost identical. Drolet et al.6 proposed a newapproach to the inversion problem. They considered thecase of nonabsorbing layers (k 5 0) and separated theproblem into two steps: One step implies solving anequation that depends only on the refractive index, and
G. Cormier and R. Boudreau Vol. 17, No. 1 /January 2000 /J. Opt. Soc. Am. A 129
sults of other inversion techniques. The GA was testedon a number of test cases, and results obtained are quitegood. Also, the genetic algorithm does not require astarting value, so evaluation of an unknown sample ispossible. The only problem to be aware of is the period-icity of the sample thickness. If the layer periodicity iswithin the search bounds, the algorithm may converge tothe periodic thickness, and although the answer is math-ematically correct, it does not represent the actual physi-cal thickness. Computation times of the GA vary, but atrial of 200 generations with a population of 100 requiresapproximately 16 s to compute on a P-II 400-MHz per-sonal computer. Some problems with GA’s can be solvedwithout the use of elitism, but in this case elitism wasnecessary to ensure convergence. Without elitism, thealgorithm fluctuated wildly and did not converge to aminimum.
7. CONCLUSIONA new method has been proposed to invert ellipsometricdata. A genetic algorithm has been applied to data to de-termine the optical properties of thin films. The GA iseasy to implement and does not require much computa-tion time. A number of test cases have been done tosimulate real situations. The GA performs well, converg-ing to the best possible values. The GA can be used in anumber of different situations, such as cases with differ-ent numbers of measurements or different errors on themeasurements.
The GA can easily be applied to systems with morethan one layer, since the method does not require evalu-ation of derivatives and can easily be adapted to anynumber of variables.
ACKNOWLEDGMENTSThe authors thank Noyan Turkkan for his help with ge-netic algorithms, the reviewers for their helpful com-ments, and the Natural Sciences and Engineering Re-search Council of Canada for financial support.
All correspondence should be addressed to Roger Bou-dreau, Ecole de genie, Universite de Moncton, Moncton,NB, Canada E1A 3E9.
REFERENCES1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Po-
larized Light (North-Holland, New York, 1996).2. F. L. McCrackin and J. P. Colson, ‘‘Computational tech-
niques for the use of the exact Drude equations in reflectionproblems,’’ in ‘‘Ellipsometry in the measurement of surfacesand films,’’ Natl Bur. Stand. Misc. Publ. 256, 61–82 (1964).
3. A. R. Reinberg, ‘‘Ellipsometer data analysis with a smallprogrammable desk calculator,’’ Appl. Opt. 11, 1273–1274(1972).
4. T. Easwarakhanthan, C. Michel, and S. Ravelet, ‘‘Numeri-cal method for the ellipsometric determination of opticalconstants and thickness of thin films with microcomput-ers,’’ Surf. Sci. 197, 339–345 (1988).
5. F. K. Urban, ‘‘Ellipsometry algorithm for absorbing films,’’Appl. Opt. 32, 2339–2344 (1993).
6. J.-P. Drolet, S. C. Russev, M. I. Boyanov, and R. Leblanc,‘‘Polynomial inversion of the single transparent layer prob-lem in ellipsometry,’’ J. Opt. Soc. Am. A 11, 3284–3291(1994).
7. G. E. Jellison, ‘‘Use of the biased estimator in the interpre-tation of spectroscopic ellipsometry data,’’ Appl. Opt. 30,3354–3360 (1991).
8. S. Bosch, F. Monzonıs, and E. Masetti, ‘‘Ellipsometric meth-ods for absorbing layers: a modified downhill simplex al-gorithm,’’ Thin Solid Films 289, 54–58 (1996).
9. J. H. Holland, Adaptation in Natural and Artificial Systems(University of Michigan Press, Ann Arbor, Mich., 1975).
10. D. E. Goldberg, Genetic Algorithms in Search, Optimizationand Machine Learning (Addison-Wesley, Reading, Mass.,1989).
11. T. Eisenhammer, M. Lazarov, M. Leutbecher, U. Schoffel,and R. Sizmann, ‘‘Optimization of interference filters withgenetic algorithms applied to silver-based heat mirrors,’’Appl. Opt. 32, 6310–6315 (1993).
12. L. Davis, Handbook of Genetic Algorithms (Van NostrandReinhold, New York, 1991).
13. A. H. Wright, ‘‘Genetic algorithms for real parameter opti-mization,’’ in Foundation of Genetic Algorithms, G. J. E.Rawlins, ed. (Morgan Kaufmann, San Mateo, Calif., 1991),pp. 205–218.
14. Z. Michalewicz, Genetic Algorithms (Springer-Verlag, NewYork, 1992).
15. S. Bosch, J. Perez, and A. Canillas, ‘‘Numerical algorithmfor spectroscopic ellipsometry of thick transparent films,’’Appl. Opt. 37, 1177–1179 (1998).
Table 6. Data Generated for a Film of n 5 2.2, k5 0.22, and d 5 10 nm on a Substrate of ns 5 4.05
and ks 5 0.028 with l 5 546.1 nm
IncidenceAngle
f
Generated Data Noise Added
D c Dm cm
50 172.393 31.959 172.39a 31.96b
70 145.506 12.886 145.49a 12.89b
a Added noise of 0.02° standard deviation.b Added noise of 0.01° standard deviation.
Table 7. Data Generated for a Film of n 5 2.2, k5 0.22, and d 5 10 nm on a Substrate of ns 5 4.05
and ks 5 0.028 with l 5 546.1 nm
IncidenceAngle
f
Generated Data Noise Added
D c Dm cm
45 174.267 34.835 174.30a 34.84b
50 172.393 31.959 172.41a 31.96b
70 145.506 12.886 145.48a 12.88b
80 32.836 13.417 32.84a 13.40b
a Added noise of 0.02° standard deviation.b Added noise of 0.01° standard deviation.
134 J. Opt. Soc. Am. A/Vol. 17, No. 1 /January 2000 G. Cormier and R. Boudreau
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Direct numerical inversion method for kineticellipsometric data. I. Presentation of themethod and numerical evaluation
Dmitri Kouznetsov, Alfred Hofrichter, and Bernard Drevillon
There is growing interest in the deposition of opticalfilters for various applications. As the coating de-sign becomes increasingly complicated, increasedprecision in the control of the refractive index and inthe growth rates during the coating deposition is be-coming crucial for high performance and for highlyreproducible processes. Because of its phase sensi-tivity, ellipsometry can detect small changes in thick-ness and refractive index and is therefore consideredto be one of the most sensitive methods for opticalcoating control. It is already a widely used methodfor the controlling the growth process by molecularbeam epitaxy1 or by end-point detection in etchingprocesses.2
In recent years several authors have shown theusefulness of spectroscopic ellipsometry for in situcontrol of optical filters.3,4 In particular, it has beenshown that by monitoring of the trajectory of the
ellipsometric parameters open-loop control �layerend-point detection� can be achieved. When themeasured ellipsometric parameters are compared inreal time with a precalculated target, these trajecto-ries can be used to stop the deposition of individuallayers of multilayer stacks without reliance on depo-sition time. This can be achieved by detection of theminimum distance between the measurement pointand the targeted layer end point3 or by means ofcomparing the length of the measured trajectory withthe length of the precalculated target trajectory.5Another method,6 based on the fact that the ellipso-metric trajectory of a constant-index layer is a closedloop in the plane of the ellipsometric parameters, ishowever limited to the deposition of constant-indexlayers with a comparatively high individual thick-ness. These types of control are sensitive mainlyto changes of the optical thickness and allow com-pensation for process variations leading to growth-rate changes during the deposition. To a certainextent they also allow successful deposition whenrefractive-index changes occur, provided that theserefractive-index changes remain small and the op-tical structures are relatively simple.
Closed-loop control of transparent optical coatings,however, with a real-time adjustment of all relevantprocess parameters, seems at present out of scopewithout knowledge of the optical parameters �thick-ness, refractive index, absorption if any� of the cur-rently deposited layer, because the relationshipbetween the measured ellipsometric parameters and
When this research was performed, the authors were with theLaboratoire de Physique des Interfaces et des Couches Minces,Ecole Polytechnique, Palaiseau, France. A. Hofrichter �[email protected]� is now with the Department of ThinFilms, Central Research Labs�Automotive, Saint-Gobain SekuritDeutschland GmbH and Co. KG, Glasstrasse 1, D-52134 Herzo-genrath, Germany.
Received 5 April 2001; revised manuscript received 15 October2001.
The use of the polynomial method for inversion ofellipsometric data has been previously investigated�see Refs. 19–21, and references therein�. Charlotand Maruani21 discovered that in the well-known for-mulas for the case of a single nonabsorbing layer ona semi-infinite substrate, the thickness of the layercan be decoupled from the unknown dielectric con-stant. This leads to a fifth-degree polynomial in �;among its solutions one finds the solution for theunknown dielectric constant of the layer. Severalauthors have presented methods to eliminate byalgebraic methods as many of the solutions as pos-sible19,20 and have performed extensive error-propagation analysis. These methods, however,remain limited to a simple sample configuration.Our method, in contrast, is applicable to any sampleconfiguration. As long as its Abeles matrices areknown, or have been reconstructed during the growthor by initial modeling, any multilayer sample can beconsidered for the inversion of a layer grown on top ofit.
The method proposed in this study is based on apolynomial expansion for the coefficients of theAbeles transfer matrices of the newly grown layer.As the evaluations in Section 4 show, it is sufficient tokeep only second-order terms to achieve reasonableprecision, although, in principle, expansion to ahigher degree is possible. The degree of this expan-sion gives the degree of the corresponding polynomialequation for �, because the terms before dx in rela-tions �8� and �9� are of the order of � and �1, respec-tively. When we keep only second-order terms of dxduring the subsequent polynomial multiplication anddivision, the highest and lowest degrees of � are oforder 2 and 2, respectively. The dielectric con-stant and the thickness of the newly grown layer aredecoupled by elimination of dx. During this elimi-nation, two of those polynomials are multiplied byeach other so that we finally obtain the eighth-degreepolynomial in �, the master function whose roots yieldthe dielectric constant of the outmost layer. Sincethe coefficients of the � polynomials are determinednumerically, it is impossible to reduce the number ofsolutions by algebraic means. However, becauseonly films with an optical thickness less than ��4 areconsidered, the numerical filtering is considerablysimplified, and the results presented in Section 4 andin Part II of this study11 show that it allows for asuccessful reconstruction of the refractive-index pro-file.
C. Extension to Other Optical Configurations
The polynomial methods mentioned above generallythe ellipsometric angles � and � as input data for theinversion formulas. Unfortunately, most ellipsom-eters do not determine � and � directly �see Section2�, and thus the use of conventional polynomial meth-ods necessitates an initial recalculation procedure.This recalculation has to deal with the optical inde-termination of the ellipsometric angles, because the
relation between them and the ellipsometric signal isnot bijectif. Our method, however, allows us to ex-press directly the experimental measured intensitiesas a function of the refractive index and of the thick-ness of the newly grown layer. This method is evenadapted to the case of the incoherent reflection model,since it is possible to incorporate the products�rs,p*rs,p� averaged over the optical path. It is alsopossible to adapt it to almost any ellipsometric oreven photometric in situ sensor, as long as it is mea-suring two independent combinations of the Fresnelcoefficients. This is true not only in the case of re-flection but can also be done for transmission mea-surements. Because any of these optical sensors aremeasuring intensities and not the electromagneticfield, conjugated products of the Fresnel coefficientsarise in the polynomial expansion of the measuredsignal. In the present development this limits thedirect inversion to the case of nonabsorbing materi-als, since otherwise mixed products of the type�n��m�* arise, which considerably complicates deter-mining the dielectric function by calculation. Itmust be noted that this is not the case if the ellipso-metric angles � and � are used, but as already men-tioned, they are generally not measured directly.
6. Summary and Conclusion
A method for the direct numerical inversion of theellipsometric signal for the real-time monitoring andcontrol of the growth of optical filters has been pre-sented. This method is based on the polynomial ex-pansion of the Abeles matrices of a newly growntransparent layer and the recorded changes of theellipsometric signal. The coefficients of these poly-nomials are determined numerically, and the refrac-tive index is determined by solution of an eighth-degree polynomial in �. The validity of the second-order approximations has been checked by analyticaland numerical means. The method has been ap-plied to the inversion of theoretical data for constant-index layers, linear gradients, and alternating-indexmultilayer filters. For noise-free data and a suffi-ciently small step size the algorithm allows a perfectreconstruction of the refractive-index profile. Thequality of the reconstruction decreases when the stepsize is increased or when experimental noise ispresent. Using multiwavelength information canconsiderably reduce the influence of experimentalnoise. Successful inversion can already be attainedby use of 10–15 wavelengths simultaneously. Inthis case a typical precision of 0.01 for the refractive-index determination and a precision of better than1% for the overall thickness determination is found.
To our knowledge, the most novel result in thisstudy is that a direct inversion method has been de-veloped, which allows the thickness and refractive-index determination of a very thin �20–50-A� layerwithout the use of numerical fitting procedures or theuse of dispersion laws to reduce the number of inde-pendent parameters. It is applicable not only tosingle-wavelength and multiwavelength ellipsometryon semi-infinite substrates but can also be used in the
case of incoherent reflection. Furthermore, themethod is general and can be easily adapted to otheroptical instruments, including other ellipsometerconfigurations such as rotating analyzer ellipsom-etry, reflectance difference spectroscopy, or reflec-tance spectroscopy.
In conclusion, the results of this investigation showthe appearance of a new powerful algorithm for thereal-time inversion of kinetic optical data. Part II ofthis study11 will be devoted to applying this method tothe monitoring of thin-film growth by phase-modulated ellipsometry.
This research was financially supported by JobinYvon SA �group Horiba�. The authors are gratefulfor useful discussions with P. Bulkin and B. Kaplan.
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9. M. Kildemo and B. Drevillon, “Real time monitoring of thegrowth of transparent thin films by spectroscopic ellipsom-etry,” Rev. Sci. Instrum. 67, 1956–1960 �1996�.
10. M. Kildemo, R. Brenot, and B. Drevillon, “Spectroellipsometricmethod for process monitoring semiconductor thin films andinterfaces,” Appl. Opt. 37, 5145–5149 �1998�.
11. A. V. Hofrichter, D. Kouznetsov, P. Bulkin, and B. Drevillon,“Direct numerical inversion method for kinetic ellipsometricdata. II. Implementation and experimental verification,”Appl. Opt. 41, 4519–4525 �2002�.
12. F. Abeles, “Recherches sur la propagation des ondes electro-magnetiques sinusoıdales dans les millieux stratifies. Appli-cation aux couches minces,” Ann. Phys. �Paris� 5, 596–640,706–782 �1950�.
13. B. Drevillon, “Phase modulated ellipsometry from the ultravi-olet to the infrared: in situ applications to the growth ofsemiconductors,” Prog. Cryst. Growth Charact. Mater. 27,1–87 �1993�.
14. Y. H. Yang and J. R. Abelson, “Spectroscopic ellipsometry ofthin films on transparent substrates: a formalism of datainterpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 �1995�.
15. M. Kildemo, R. Ossikovski, and M. Stchakovsky, “Measure-ment of the absorption edge of thick transparent substratesusing the incoherent refelction model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313, 108–113�1998�.
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11 Ellipsometry for optical su rface study appl ications
Y M GEI>REMICIIAEL AND K T V GRATTAN. Cioy Un;".,,;ly. Lor.OOn
11 .1 Introduction
Thi. <n'pter pro" iOn • brief di",u»ioo of the rriocipk> "oO practice of e llipsometrk m<a,urem"n" for Of>Iio. 1 .urf.,,. .,ud i .. , e<peci.lly using "1"",.1 f,I:>re -t.a,.~ <Ie"kos. Wi,h the ~."el"flmen" ,,. ... fee<"' years in t1>e opli"., fibre CO)m",unk .. ions m.rket, • '""go of rotl1l'O""nts ooJ dev"'.' h.ve bee" e .... ''''' and u..u in ,hi, [,eld ",i,h ,he resul' Ih., ,he 'spin orr into <optioal (,bre m<asuremon" .00 >en>or 'y>!om. h •• been ,'cry .i~" if,<.nt. This offer.> the po«ibi lity of the c",,,ioo of. "'"go of n<w dcvice. employing t«hno logi •• ofton aJor'eJ fmm the ,.I""ommu"".,i"". field. Speci.lised oplk.1 fibre .. in p.~ ic"lar "sed ",,(,ide lhe lciec<>m",unie"ioo, ;n""Slt)'. 'R: 00'" ",><lily .v.il,b le .nd .s. , ,,,ull highly biTcfrin~o"' (HiBi) fibre can be used 10 enabk polari"""oo ","e. '0 be m"nla'oOO ... ',h 'n fib,..,. _ . m.jor >d,'.nlago for ,be do"elopmon' of c"mpac!. fibre optic e llip","'e"ic in>l",m<n""i",,_
Thi, oh.p' or "" 'i<w, •• " -,,.1 in",rumon,,1 d.,'olopmcnl> b.>«I on fibre ""'ic 'o<~nol o;:y, "nd fol l""'ing • descrip'ion of ,he "nderlying m.,hem.,'c.1 b<as's, ' ho prioc'ple, . nd ,pplie,'io", of • ran¥o of eliip"'mOlric d<"i«' on: c"".idered. foc"'ing in panicul., uflOn flOl.ri .. ,i"" bascd dlip>Omc'ry .00 fib,.., op' io d.,'k .. in pankul,,_ Rofo",flCe i, mod< '0 • range of 'oc~niqu .. . 00 de,kes rO'J>O"leJ in 'he Ii'o""'",,, 000 oW" .. ,ion, of ....,"', '00 me', "",_O' 'ySlcms of 'his 'ype 'rc consi<k:n:d,
11 ,1 ,1 Polarisatiofl 01 light afld ellipsometry
Ugh' w",'." being ' ,"o,verse <loc'rom'gne'ic "',,'es.,.., <h.ra ... ri>«l by lime_ , '.ry;n~ .I<wi< . 00 m.~ne'i< fi <ld '-0<,"", whi<h ar. perpendi,ulor to e.," o,h<r .nd norma l '0 'hc "'''-0 prop.galion "ircr,ion, The orirnla,ion.1 charnercri"",, of ooe of 'he,"" oarriers of ' he field. obsc,,-oo '" • fixoo poin' io 'I"'<e .00 'i"'e, <k:fio" 'he flO laris-a,ion of ' h< elo<'rom.~ne'ic ,",'OR '-, For "",i<o1 srody PUr'»OS<s, ;, is <"",'en, ;"".l '0 .... in ,h. elwri< f,e ld S(n:"~(~, I:, b.!><d on lire [",., lha l ..-hen li¥h' intera«> with maller, the force excncJ on the
'<SI",:ti" e ly for ' ho .• and P J>Olari>at ion ,tates .nd fJ i. the phase change the n' uit i-refi«teJ """ 'e inside ,he film .,periences as it I' .. · ..... s Ih. film, tile o"e", 11 complex ' mpli'u"" ,d lo<tioo coeflicirnl'. R, .nd R, .. f'" ,he p ,nd. J>OI.ri .. ,ion "",c. c.n be f"""u l.,ed by addin~ thc , ,,,,«,,ive pa" i.1 w",,,. 'hat make up the ,e,u ltan' len""od ,,"",'0 ' Thu, R, .nd R, may be gi" en by'
' or, + ",,~-Ar R,_ V 1 + 'orr '''''' !
R '01, + ''''~ }l.r • - 1 + 'or ,', ,,e r..f
"llere tit< film ph.", Ihic ....... p. i, giWtl by
II = h(0 (N~ - Nt ,in' (oito)lf'
11.12
11.13
' 1.1 4
.nd C. i, tit< .nglo of indden<e , d i, fi lm thick""". ,1,1, (= n _ j t) i. the comp le, ' of"",tiw ind .. of tit< fitm .nd ) is the "'",'e lo"g, h of the Ii!lill,
If P i, .-pre,sed in terms of Np ond N, (i" eqn, I ' 12 and I ' 13 .1:>ow). the ",e"u,«I ell ip<omc:lric .ngles 'i' .nd ~ can be rel"ed 10 the opti<al pl'Op<~i<s of thc 'hin film 'Y"= by eqn 11.15,
I ' ) I;' "", +FI"...-12.' I +'OL" , ,,~-!2,'
tan l." _ p _ ~ ~
I + '. ''''''~ ,' " 'or , + " "~ , . .1
11.15
I, i. """m«l Ihal the fl lro i. optic.lIy isotropic and Oornogen""". " 'ith • u"if(H'fll r<froc1i," iodex N,• throughoot iI, ,hick"" ... J. Th • • mbi.o, .nd ,u!»""te m«l i •• re . Iso considered "'mi -i nf,oile. homo~eo<Oll' .nd optically i><>tropic with ref,ac,ive iodi«. No .nd ,1,1, ""I,.,,:ti ,'ely."
Ellip<om<lry . llow. the m',"'UfOme n' of two , .. I, .. " . '" .nd Ll.. whic h de«,ihe • • h,oge in ,o. polari""t"," stat. c'u""" by tit< ,.mpl •. These two para.,.te .. are funot i"", of ,o. opt ical corrSlanlS Of ,he (,Im th ickness ,h."""en,tio" For "n id<al bulk ,ub>lTlltc. ,he mea,ufoo ellipsomOiric pa"mcte .... '" and .... can be di ''''''i)' inwre«l to give "I"ic.1 coostan" " and t . For mo", comp iicat«l st"",tures. whkh i""lude multi pl. l.y .... notl-ide. 1 io,.rf ..... Of grad ie"" in the film optic.1 ptOJ'I<n i.,. to. , n.I)'tical i"'· .... ioo of "'Itl 1 1.15 to "xt",,1 to. OJIt i<;al Consl'nts .nd the strue,"ral iof(H'fllOl ion is u,ual ly imJ>OSSibl., In such ca",s • .. h .nee" data .naly.i' ,«hniQuos.re ' equire" to build a ch.rocteri"ie model of tit< matcri .1 that i. being mc .. u,«I. to c."".<" .imult.ncously and uniquely the optic.1 coo."n" f",m ' ho m .. ,ured &ta. Tlt<re aro ,'.riou, num<ric.1 tochoique, . nd p'rame"i, moo. I."- " for on. ly. iog .!lip,ome"i, dOl. 00" t he dOiermin .. i"" of the optica l paramc'er> of , .. "mpie,
For . gi,'en .mb icnt_film_,ubst.-." .)'Stem . •• inglc w,,'c!cngth mc",uremcn' of '" and Ll. " one .ngle of incidence, ¢Jo, p""id« enough in formation to
J Au...,. R. M. A. -.I D •• Iw>. N. M .. EWp''''''?' aM pal.,;,,,, !;gior. A""".-dam Noo h flolland. 1971.
4, V.,k<~ , A .• Op,jc, « "i. fll,., . r-:ooh·lloliond l'ubli,i'i Co. Am,tcrdom. 1%0, I )1<",- E. and Zaj><. A .. ap,ie •. Add;,;.,o·W«l<y ruDii>h i"i C"",,,,,.y. 10< .. 1979 6 T"mkin,. II. G .. AU .. ,:. G.'J,- '" Ellif''''''''''?'' ",..!em", I'T<», I.., .. I WJ 7 B",,,,.,,, N. M .. Bock ....... ". B. , ro;t 11. 11, " . C. k ..... n' k>~I"f""""" ;n
,!ilp"""",,·. Nonh )101 1.00 l'uol" hinl Company, AmSl<rdim, I%'!. S 11."&,, P. S .. 'R",,,,,, d<,"k>pn><n'" in in''''' ''''''',",''''' in ' lIif"<'l"<\'Y". S.if Sri
1 '180 96 I (t8
9, GOIf.,d, A. ond B.,ch.). M" I.,rod"",;"" 10 ,"a,,,, "",W, i. opfie •. )"1,, Wi!q ,00 Son., 1975,
10 Loo~h"",'. R S .. G""",,,ric~1 Q..J rh)',,·, ·~! (}pri .... L<>O<loo. l""i<ffi'I •. I97J. II r",,",. R. F"I.,md.,.""" to ,1/odP", Or';", 11<:>1,. ~ ind"" ,,.j Win", ... I""., '963 11 M«:,..,~in. F. L . • ,.j Col,.... J. P. , ·C'""!"'''t .... 1 , ,,,, hn~,,,,, r", tho u>< of tho
I) M<C""'kin. F. L. Eli<> r . ... gh., Robm R. Slrt>mbo.-g -.l lbrol.! L St<inb<'1I. 'Me .... """,", of tho Th"k..,,, ,ro;t ~ ,f"", i v" I","" ofVer)' t",n Film. o,.j Opt",1 F'ror<""" of SU1f""",·. A . p~}.ia ~.,j C",,"i,,ry, 1% I 67A14) 3lil J))
14 Moj W\ CII>n~ and U","l.1 J. Gi"""'. 'Opt",.1 «>0>"'" "'"nI\;",,''''' of ,hi. fi lm. by ......... ><,,,,h """t..t". Arrb..! 0>'ie.,. 19ij 24{() 504-lO7.
I j, Monof,,;"',}, C. G ..... }. -.l Fillard.). P .• 'A ,impl< rr.clllod for til< """",,i"'t"" of ,II< "",.,.1 ,on>l>"" •. k and ,II< t~i<kne .. of. ~'''ldy oboorbi"i th in film', J"""",! of Phpie, [;, Sci""ifie I."",,",",,. 1976 9 1001_ 1001.
16, C»<. Will i. m E .• 'AI",mO; m<tllod for <" ,,,, i'g onin fi lm "",O;.II""m"«,,. f,O"I S{><Ctropho<omc',," 1"<'''.''1''",,,,'. Awli'" (}pri<3, 198) ll(l2) 1 ~J2-1 ~)~.
17 Oro"'. L R""<v. S. c.. Boy ..... v. M. L and t.;,ol....,. R. M. (1994). ·Poly""",i.1 im·"" ... of the 'ingi< ''''' ''I'''"''' loY<" !"''''km in eI hl"""""ry' . """,,,,,,I « Dr'ica! ,5oci.,y of AmPrieQ. 1_ (0) " ill) n!4-J19 L
I g, ldn<T,)" 'In" ""OO of ,,~«,ioo elkif"OOl<tfk do .. ·• Appli..! (}pric,. 1"94 33(22) 5 159_5 165
19 Do.f. M. C and L,k ..... ) .. ·R.n«,"", and ' .... ""i .. "" dhJ"OO><'ry of. ,.,if""" I.y,,'· , """,,,,,,I of (}pri""! So";,/)' of A""";ca (.1, 1987 4{ II) 2(l\I()-2100,
10 I lo l "i>~, A, It M. and SeMI". r . M, L. 0 .. '0p0i<>1 m,,.,,,,,,,,,,,nt of the ",t,,,,;,·. ind<>. l.1yer th"'kne .. and "olw"" ' II>nK" or 'hin f,lms', Awli'" ap,i,·,. 1989 11l(1J) mS-5 1 (l.I
21, lo<s<h<,. D, II" D<try. R. J . • ro;t a.""". M. J .. 'L".,,·SqUJ'" An, ll';'" of to< I' il m· Su"" .. '" I'ToI>km in ~lIf"OI!1<IlY', .Mw1Ia' of'~' (}pri"'! s.x,,,y of A,.<_. 197 1
mboyanov
Oval
Von gewöhnlicher bis hin zu ungewöhnlicher
spektroskopischer ellipsometrischer Invertierung zur
optischen Charakterisierung dünner Schichten
Inauguraldissertation
zur
Erlangung der Würde eines Doktors der Philosophie
vorgelegt der
Philosophisch-Naturwissenschaftlichen Fakultät
der Universität Basel
von
R. Shui-Ching Ho (何瑞清)
aus Hongkong, China
Basel, 2005
33
graphischer Interpolation oder mit interpoliertem Vergleich von tabellarisch aufgezeichneten Daten. Im heutigen Sinn würden solche Methoden Unannehmlichkeiten und Umstände bereiten. Die graphische Darstellung [Arc62] der ellipsometrischen Gleichungen muss beispielsweise für jedes Substrat wiederholt werden [Ghe68]. Die exakten ellipsometrischen Gleichungen zur Bestimmung der optischen Konstanten einer auf einem Substrat abgeschiedenen Schichte waren vor dem Einzug des modernen Computers praktisch nicht brauchbar [So72]. Der Einzug des elektronischen Computers in der zweiten Hälfte des vorigen Jahrhunderts gab der Ellipsometrie neuen Antrieb. Die Ausführung von Rechnungen durch computergesteuerte Programme wurde ermöglicht. Es waren McCrackin et al. [Mcc63], die zum ersten Mal die Programmierung eines numerischen Verfahrens ausführten. Zahlreiche Verfahren [Hol67a, So72, Rei72, Cel76, Yor83] wurden vorgeschlagen und ausprobiert, um die ellipsometrischen Gleichungen effizienter zu invertieren. Holmes [Hol67a] brachte ein modifiziertes Verfahren vor, das die Berechnung der Variablen n und k von derjenigen der Variable d abtrennt. Oldham [Old68] überprüfte ein anderes auf Newton-Iteration basierendes Verfahren zur Berechnung der optischen Konstanten absorbierender Filme mit bekannter Schichtdicke. Dieses Verfahren hätte weniger Rechenschritte als dasjenige von McCrackin et al. [Mcc63] benötigt, allerdings war die Wahrscheinlichkeit des Scheiterns war größer, weil mehrdeutige Lösungen nicht in Betracht gezogen wurden. Ghezzo [Ghe68] arbeitete mit einem neuen Begriff und behauptete, dass alle oben erwähnten Beschränkungen und widrigen Umstände überwunden wurden. Aber seine Methode setzt eine unabhängige Bestimmung des n, der Schichtdicke, voraus, z.B. mittels den Ansäten von Abelès [Abe50a] oder Lewis [Lew64]. Die gleiche ungünstige Vorraussetzung weisen zahlreiche Publikationen [Ghe68, Ved68, Yam75, Han73, So72] auf. Beispielsweise arbeitete die Methode von Vedam et al. [Ved68] mit zwei verschiedenen Medien. Die Suche nach intelligenten Invertierungsansätzen war in dem relativ jungen Computerzeitalter eine Mode. Mit der rasch zunehmenden Rechengeschwindigkeit des modernen Prozessors gegen das Ende des letzten Jahrhunderts wurden mathematisch anspruchsvollere Algorithmen und Überlegungen, wie Genetischer Algorithmus [Cor00], multidimensionaler Newtonscher Algorithmus [Eas88], gedämpfter quadratischer Mittelwert [Urb93], Polynom der fünfter Ordnung [Dro94], CSA (,,classical simulated annealing‘‘) [Kir83, Pol00], FSA (,,fast simulated annealing‘‘) [Szu87], GSA (,,generalized simulated annealing‘‘) [Tsa96, Xia97, Mun98, Gut99, Ell00, Yu03], in der ellipsometrischen Invertierung und spektroskopisch angewendet.
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mboyanov
Oval
Von gewöhnlicher bis hin zu ungewöhnlicher
spektroskopischer ellipsometrischer Invertierung zur
optischen Charakterisierung dünner Schichten
Inauguraldissertation
zur
Erlangung der Würde eines Doktors der Philosophie
vorgelegt der
Philosophisch-Naturwissenschaftlichen Fakultät
der Universität Basel
von
R. Shui-Ching Ho (何瑞清)
aus Hongkong, China
Basel, 2005
33
graphischer Interpolation oder mit interpoliertem Vergleich von tabellarisch aufgezeichneten Daten. Im heutigen Sinn würden solche Methoden Unannehmlichkeiten und Umstände bereiten. Die graphische Darstellung [Arc62] der ellipsometrischen Gleichungen muss beispielsweise für jedes Substrat wiederholt werden [Ghe68]. Die exakten ellipsometrischen Gleichungen zur Bestimmung der optischen Konstanten einer auf einem Substrat abgeschiedenen Schichte waren vor dem Einzug des modernen Computers praktisch nicht brauchbar [So72]. Der Einzug des elektronischen Computers in der zweiten Hälfte des vorigen Jahrhunderts gab der Ellipsometrie neuen Antrieb. Die Ausführung von Rechnungen durch computergesteuerte Programme wurde ermöglicht. Es waren McCrackin et al. [Mcc63], die zum ersten Mal die Programmierung eines numerischen Verfahrens ausführten. Zahlreiche Verfahren [Hol67a, So72, Rei72, Cel76, Yor83] wurden vorgeschlagen und ausprobiert, um die ellipsometrischen Gleichungen effizienter zu invertieren. Holmes [Hol67a] brachte ein modifiziertes Verfahren vor, das die Berechnung der Variablen n und k von derjenigen der Variable d abtrennt. Oldham [Old68] überprüfte ein anderes auf Newton-Iteration basierendes Verfahren zur Berechnung der optischen Konstanten absorbierender Filme mit bekannter Schichtdicke. Dieses Verfahren hätte weniger Rechenschritte als dasjenige von McCrackin et al. [Mcc63] benötigt, allerdings war die Wahrscheinlichkeit des Scheiterns war größer, weil mehrdeutige Lösungen nicht in Betracht gezogen wurden. Ghezzo [Ghe68] arbeitete mit einem neuen Begriff und behauptete, dass alle oben erwähnten Beschränkungen und widrigen Umstände überwunden wurden. Aber seine Methode setzt eine unabhängige Bestimmung des n, der Schichtdicke, voraus, z.B. mittels den Ansäten von Abelès [Abe50a] oder Lewis [Lew64]. Die gleiche ungünstige Vorraussetzung weisen zahlreiche Publikationen [Ghe68, Ved68, Yam75, Han73, So72] auf. Beispielsweise arbeitete die Methode von Vedam et al. [Ved68] mit zwei verschiedenen Medien. Die Suche nach intelligenten Invertierungsansätzen war in dem relativ jungen Computerzeitalter eine Mode. Mit der rasch zunehmenden Rechengeschwindigkeit des modernen Prozessors gegen das Ende des letzten Jahrhunderts wurden mathematisch anspruchsvollere Algorithmen und Überlegungen, wie Genetischer Algorithmus [Cor00], multidimensionaler Newtonscher Algorithmus [Eas88], gedämpfter quadratischer Mittelwert [Urb93], Polynom der fünfter Ordnung [Dro94], CSA (,,classical simulated annealing‘‘) [Kir83, Pol00], FSA (,,fast simulated annealing‘‘) [Szu87], GSA (,,generalized simulated annealing‘‘) [Tsa96, Xia97, Mun98, Gut99, Ell00, Yu03], in der ellipsometrischen Invertierung und spektroskopisch angewendet.
mboyanov
Oval
323
[Cha85] D. Charlot & A. Maruani, Appl. Opt. 24, 3368 (1985).
[Cha91] T. S. Chao, C. L. Lee, T. F. Lee, J. Electrochem. Soc. 138, 1756
(1991).
[Cho90] S. Chongsawangvirod, E. A. Irene, A. Kalnitsky, S. Tay, J. Ellul,
J. Andrews, J. electrochem. Soc. 137 (11), 35536 (1990).
[Cla75] R. A. Clarke, R. L. Tapping, M. A. Hooper, L. Young, J.
Electrochem. Soc. 122, 1347 (1975).
[Cla94] J. P. Clarke, J. Vac. Sci. Tech. A 12 (2) 594 (1994).
[Coh73] R. W. Cohen, G. D. Cody, M. D. Coutts, B. Abeles, Phys. Rev.
B8 3689 (1973).
[Cor00] G. Corner, R. Boudreau, J. Opt. Soc. Am. A 17 129 (2000).
[Cue95] R. Cueff, G. Baud, J. P. Besse, M. Jacquet, Thin solid Films
266, 198 (1995).
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Birkhäuser, Basel, (1994).
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[Dit76] R. W. Ditchburn, Light. 3rd ed. London: Academic Press 1976.
[Dit97] G. Dittmar, U. Richter, U. Wielsch, Manual for Spectraray
software for ellipsometers of SE 800 / SE850 and SE900 / SE950
series. Sentech Instruments GmbH, Berlin, (1997).
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UNIVERSITE PAUL CEZANNE, AIX MARSEILLE III
TITRE :
CONTROLE SPECTROPHOTOMETRIQUE LARGE BANDE DE FILTRES INTERFERENTIELS EN COURS DE DEPOT
THESE
Pour obtenir le grade de
DOCTEUR DE L’UNIVERSITE PAUL CEZANNE, AIX MARSEILLE III
Discipline : Optique, Image et Signal Ecole Doctorale : Physique et Sciences de la matière
Présentée et soutenue publiquement par
Bruno BADOIL Le 8 Novembre 2007
JURY
M. Jean-Marc BLONDY (rapporteur)
M. Jean TABOURY (rapporteur) Mme Marie-Françoise RAVET-KRILL
Mme Catherine GREZES-BESSET M Serge HUARD
M Fabien LEMARCHAND (co-directeur de thèse) M Michel LEQUIME (directeur de thèse)
1 Einleitung In der Arbeitsgruppe werden dünne Halbleiterschichten mit der gestörten Winkelkorrelation (PAC) untersucht. Hierfür implantiert man radioaktive Sonden. Es ist wichtig, möglichst genau die Schichtdicke d zu kennen, da bei einer zu hohen Implantationsenergie diese Sonden auch in das Substrat gelangen und so die anschließenden Messungen verfälschen. Mit dem Ellipsometer sollen Eigenschaften wie Schichtdicke und Brechungsindex der verwendeten Proben bestimmt werden. Der Name „Ellipsometrie“ tauchte erstmals 1945 in einem Artikel von Alexander Rother in der Review of Scientific Instruments auf. Die Polarisation eines Lichtstrahls wird senkrecht (s-Polarisation) beziehungsweise parallel (p-Polarisation) zur Einfallsebene betrachtet. Trifft der Strahl nun auf eine Grenzfläche, so ist die Reflexion für die s- und p- Polarisation resultierend aus den Stetigkeitsbedingungen der Maxwellschen und der Fresnel´schen Gleichungen verschieden. Durch die Änderung des Polarisationszustandes durch die Reflexion des Messlichtes können die dielektrischen Eigenschaften einer Probe bestimmt werden. 1883 beobachtete der deutsche Physiker Paul Drude (1864-1906) eine zeitabhängige Änderung der Phasenverschiebung zwischen der senkrechten und parallelen Komponente von polarisiertem Licht. Vier Jahre nach seiner Entdeckung veröffentlichte er ein Gleichungssystem für p- und s- Polarisation, die theoretische Grundlage der Ellipsometrie. Allerdings erlangte diese Methode erst mit dem Aufkommen der ersten Computer in den 60-er Jahren eine größere Bedeutung. Da man sich dabei den veränderten Polarisationszustand einer elektromagnetischen Welle nach der Reflexion an einer Grenzfläche oder einem dünnen Film zu Nutze macht, kann man die Ellipsometrie als ein „phasenempfindliches Reflexionsexperiment mit polarisiertem Licht an einer Grenzfläche“[Sar04] betrachten. Im allgemeinen Fall ist das reflektierte Messlicht elliptisch polarisiert, was der Methode schließlich den Namen gab. Die wichtigste Anwendung ist die Vermessung dünner Schichten. Mittels der Ellipsometrie kann man zerstörungsfrei Informationen über optischen Konstanten sowie der Schichtdicke einer Probe erhalten. Bereits im Jahre 1996 wurde im Rahmen einer Diplomarbeit von M. Mendel ein Nullellipsometer in der Arbeitsgruppe für nukleare Festkörperphysik am Helmholtz-Institut für Strahlen- und Kernphysik der Universität Bonn konstruiert. Das damalige numerische Ausleseprogramm wurde im Folgejahr verbessert und baute auf einen analytischen Lösungsweg nach dem Vorbild von Jean-Pierre Drolet [Dro94] auf. Die Aufgabe dieser Diplomarbeit besteht darin, dieses Programm von C++ in LabVIEW umzuschreiben. Damit wird eine digitale Datenauslese mittels eines neu angeschlossenen digital-analog Wandlers (DAC) ermöglicht. Anschließend sollen Testmessungen vorgenommen werden, um die Qualität der Messergebnisse für verschiedene Schichtdicken für unterschiedlichen Materialien zu untersuchen.
Abbildung 1.1: Ellipsometer Anfang des 20. Jahrhunderts [Dru02]
1
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2.3 Lösung der ellipsometrischen Gleichung Um aus der ellipsometrischen Gleichung die erwünschten Informationen wie Schichtdicke d und den Brechungsindex der zu untersuchenden Schicht zu erhalten, wurde zu Beginn ein Computerprogramm verwendet, dass die Lösungen iterativ berechnet hatte [Men96]. Es bezog seine Anfangswerte lediglich aus einem
Diagramm und war zudem recht kompliziert in der Handhabung [Des97]. Um diese Probleme zu umgehen, wurde 1997 in Rahmen einer Diplomarbeit das Programm „XEllip“ in C++ geschrieben, das einen analytischen Ansatz wählt und die Nachteile einer iterativen Lösung zu umgehen versucht. „XEllip“ sollte nun in LabVIEW umgeschrieben werden, um eine digitale Datenauslese zu ermöglichen. Das neue Programm „Elli“ basiert auf der gleichen analytischen Methode und d zu bestimmen. Entwickelt wurde dieser Lösungsweg bereits 1994 vom kanadischen Physiker
1n
Δ−Ψ
1n
Jean-Piere Drolet an der Universität Québec [Dro94]. Er führte die ellipsometrische Gleichung auf ein Polynom 5.Grades zurück und reduzierte das Problem auf die Berechnung der Nullstellen dieses Polynoms. 2.3.1 Warum keine iterative Lösung? Der Nachteil der in der Vergangenheit verwendeten numerischen Methoden besteht darin, dass die jeweilige Lösung nur sehr langsam konvergiert, manchmal sogar gar nicht erreicht wird. Sie ist stark abhängig von den gewählten Anfangsparametern der gesuchten Werte. Des Weiteren existiert in der Regel mehr als eine mögliche mathematische oder physikalische Lösung für die Messung eines Systems, d.h. verschiedene Systeme können zu gleichen Werten für Ψ und Δ führen. Man müsste alle sinnvollen physikalischen Lösungen kennen, um die für das gegeben System zutreffende auszuwählen. Dies führt dazu, dass man zum Beispiel eine zweite Messung unter einem anderen Einfallswinkel durchführen muss, um die richtige Lösung „herauszufiltern“. Meistens kennt man im Voraus nicht einmal die Anzahl der möglichen Ergebnisse, da diese für verschiedene Systeme unterschiedlich sein kann. Daher hat man keine Sicherheit, dass alle Lösungen überhaupt gefunden werden [Dro94]. 2.3.2 Lösungsmethode nach Drolet Die von Drolet vorgeschlagene analytische Lösung basiert darauf, dass der Lösungsweg in zwei Schritte unterteilt wird. Als erstes ermittelt man den Brechungsindex der zu untersuchenden Schicht, um daraus anschließend die Schichtdicke d zu berechnen.
1n
Die einzige Voraussetzung ist, dass die dünne Schicht einen reellen relativen Brechungsindex haben muss, d.h. transparent ist. Es gilt:
reellnnn ==
0
1
Für absorbierende Schichten wurde 1962 von Dagman [Dag62] eine analytische Methode entwickelt, um die Schichtdicke und den Brechungsindex zu bestimmen. Leider benötigt man hier vier verschiedene Messungen mit zwei verschiedenen Einfallswinkeln und Umgebungsmedien.
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65
7 Literatur [AEG79] AEG TELEFUNKEN, Halbleiterübersicht 1978/79, Heilbronn (1978) [Azza77] R. M. A. Azzam, N.M Bashara: Ellipsometry and Polarized Light.
North-Holland Publishing Company, New York (1977) [Dag62] E.E Dagman: Analytical solution of the inverse ellipsometry problem in the modeling of a single-layer reflecting system.
Opt. Spectrosc. (USSR) 62, 500-503 (1987) [Des97] T. Dessauvagie: Untersuchung dünner Si3N4 – Schichten mit Ellipsometrie und RBS. Diplomarbeit, Universität Bonn (1997) [Dro94] J.-P. Drolet, S.C. Russev, M.I. Boyanov, R.M. Leblanc: Polynomial Inversion of the single transparent layer problem in Ellipsometry.
J. Opt. Soc. Am. A/Vol. 11, No 12/December 1994, p.3284-91 [Dru02] P. Drude: The Theorie of Optics.
New York, Dover Publications (1902) [Fre69] M. Frenzel: Ellipsometrische Bestimmung der Dicke und Brechzahl
dünner Schichten auf Silizium. Kristall und Technik, 4, 1, S. 149-160 (1969)
[HO94] Handbook of Optics,
Vol. 2, 2nd edition. McGraw-Hill (1994) [Kern92] W. Kern: Handbook of semiconductor Wafer Cleaning Technologie. no. 5, 723-8 (1992) [Lab07] LabVIEW - Versionshinweise, August 2007 [Lab98] Benutzerhandbuch LabVIEW, Auflage Juli 1998 [MeG95] Melles Griot, Optikkatalog, Bensheim (1995) [Men96] M. Mendel: Aufbau und Test eines Ellipsometers zur Bestimmung der
optischen Eigenschaften dünner dielektrischer Schichten. Diplomarbeit, Universität Bonn (1996)
[Muth99] J.F Muth: Absorption coefficient and refractive indexof GaN, AlN and
AlGaN alloys. MRS Internet J. Nitride Semicond. Res. 4S1, G5.2 (1999) [Ni09] Datenblatt USB-6008, National Instruments (2009) [Pohl83] R. W. Pohl: Optik und Atomphysik.
13. Auflage, Springerverlag (1983)
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Extraction of complex refractive index of absorbing filmsfrom ellipsometry measurement
Mickaël GilliotUniversité de Reims Champagne-Ardenne, GReSPI, BP 1039, 51687 Reims Cedex 2, France
a b s t r a c ta r t i c l e i n f o
Article history:Received 11 June 2011Received in revised form 20 January 2012Accepted 16 April 2012Available online 24 April 2012
Numerical extraction of complex refractive index of an unknown absorbing layer inside a multilayer samplefrom ellipsometry measurement is discussed. The approach of point by point extraction considering all pointsof spectroscopic data as independent data points is investigated. This problem has typically multiple solutionsand the standard method consisting in fitting calculated to experimental point is likely to converge to a wrongsolution if a precise guess value is not given. An alternate method is proposed, based on the determination ofcontours of the ellipsometric function, to provide all solutions in an as extended as wanted range of complexrefractive index values. The method is tested through different kinds of sample examples. Errors relative toany of the parameters used in the sample model are calculated and discussed. This method should be helpfulin many practical cases of ellipsometry data interpretation.
Optical characterization techniques and especially spectroscopicellipsometry are favorite techniques for surface and thin film analyses,as attested by its use for decades in semiconductor industry and themore and more extended field of applications for example to flatpanel displays, photovoltaics materials, polymers, organic materialsor biotechnology [1,2]. Although instrumentation has inheritedmuch progress, analysis of data is still a delicate task because of thehighly non-linear character of optical equations. Only a limited numberof cases allowing unambiguous analytical inversion, such as the deter-mination of complex refractive index of the ambient–substrate systemand the determination of the layer thickness of the ambient–layer–substrate system provided that all optical constants are known.
General process of spectroscopic ellipsometry data interpretationconsists in building a parametric representative model of the samplewhere unknownmaterials are represented bydispersion laws andfittinggenerated to experimental data by varying some of the parameters.However materials cannot always be represented by theoretical laws.Fitting procedure furthermore requires the knowledge of estimatesvalues of the parameters as a starting point. Alternatively numerical pro-cedures referred as point by point methods can be used to extract someparameters of interest from each point of the spectrum considered as anindependent data point. Numerical procedures have been developed toextract refractive index and thickness of an unknown transparent layer
on known substrate [3,4] but the problem of direct data inversion ismore complicated when the layer is absorbing.
This paper is focused on point by point inversion processes. Awidespread point by point procedure concerning simultaneous deter-mination of refractive index and extinction coefficient (and possiblythickness) is to consider them as variables to fit ellipsometric functionto experimental data for each point of the spectrum. Interestingadditionalworks have beenpublished to extract point by point refractiveindex, extinction coefficient and possible thickness at the same time bycomputing solution curves when one parameter is varied [5–8] andintersecting such solution curves for multiple measurements suchas multiple angle of incidence data or multiple thicknesses data of agrowing sample [9–11]. Other interesting works use spectroscopic datato extract at the same time thicknesses of different films inside a stackby extracting spectra of unknown complex refractive index whenenergy-independent parameters are varied. The solution is then selectedbased on smoothness criteria of the spectral dielectric function [12,13].Direct fitting processes as well as computation of parametric solutioncurves however often require the knowledge of guess values.
In this paper an analysis of the problem of point by point extractionof complex refractive index of a layer of known thickness in an arbitrarystack of layers from ellipsometry data is performed. This inversion caseis of particular interest because a large number of materials are trans-parent over some spectral range, especially semiconductors for energybelow the band-gap, where thickness can be accurately determinedby other numerical techniques [3,4,14,15]. The difficulty is that thereare typically multiple solution couples of the refractive index andextinction coefficient for a given set of sample parameters and
Relative derivatives of n and k according to any input parametersfor the solution of interest n=2.485, k=0.092 at 5.8 eV for the SiN/Si sample are given in Table 6 using previous formulas. The appropri-ateness of the linearization for measurement error has been moreextensively confirmed by comparison with the exact calculations ona larger number of values (not shown here).
Especially errors according to the principal parameters Ψe, Δe andd can be considered. For instance an error of 1 nm on d would leadto partial error contributions of dn=2.12×10−2(0.9%) and dk=7.43×10−5(0.1%). An error of 0.1° onΨewould lead to partial error con-tributions of dn=2.64×10−4(0.01%) and dk=7.12×10−4(0.8%). Anerror of 0.1° on Δe would lead to partial error contributions of dn=3.04×10−4(0.01%) and dk=1.13×10−4(0.1%). All these errors are lessthan 1%, and are in agreement with example results obtained using erro-neous data.
In fact errors on input parameters can hardly be precisely knownand reasonable margins of errors for these parameters are considered.Corresponding margins of errors for final results depend on the sampleparameters and measured data, and would be different for eachparticular case. The error formulas allowan easy implementation in stan-dard procedures so that accuracy of the obtained solutions can be sys-tematically calculated and checked for each set of new input data.
7. Conclusion
A method to extract point by point n and k of an unknown layer inany stack of thin films from ellipsometric measurement has beenpresented.
The method neither requires the use of a parametric dispersiondielectric function nor requires guess values as starting points of aminimization process. It provides accurately all possible solutions ofthe ellipsometric equations in a given (n,k) range that can be aslarge as desired. This inversion process is of particular interest whenthe materials is completely unknown and when estimate values forcomplex refractive index can not be provided.
Efficiency of the method has been tested by different examples.
Error effects due to the different input parameters and experimentaldata uncertainties have been investigated.
This method should be useful in many practical cases where thick-ness of the considered absorbing layer can be knownby other techniquesor by ellipsometry in the possible range of transparency of thematerials.
References
[1] H. Arwin, U. Beck, M. Schubert (Eds.), Proceedings of the 4th International Con-ference on Spectroscopic Ellipsometry, Wiley-VCH Verlag, Weinheim, 2008.
[2] H.G. Tompkins (Ed.), Proceedings of the 5th International Conference on Spectro-scopic Ellipsometry, volume 519 of Thin Solid Films, Elsevier, Amsterdam, 2011.
[3] F. McCrackin, Natl. Bur. Std. Tech. Note (1969) 479.[4] D. Charlot, A. Maruani, Appl. Opt. 24 (1985) 3368.[5] M. Malin, K. Vedam, Surf. Sci. 56 (1976) 49.[6] B.D. Cahan, Surf. Sci. 56 (1976) 354.[7] F.K. Urban III, Appl. Opt. 32 (1993) 2339.[8] T. Easwarakhanthan, P. Mas, M. Renard, S. Ravelet, Surf. Sci. 216 (1989) 198.[9] D. Barton, F.K. Urban III, Thin Solid Films 516 (2007) 119.
[10] F.K. Urban III, D. Barton, Thin Solid Films 517 (2008) 1063 35th International Con-ference on Metallurgical Coatings and Thin Films (ICMCTF).
[11] F.K. Urban III, D. Barton, T. Tiwald, Thin Solid Films 518 (2009) 1411 Proceedingsof the 36th International Conference on Metallurgical Coatings and Thin Films.
[12] D.E. Aspnes, A.A. Studna, E. Kinsbron, Phys. Rev. B 29 (1984) 768.[13] H. Arwin, D. Aspnes, Thin Solid Films 113 (1984) 101.[14] J. Lekner, Appl. Opt. 33 (1994) 5159.[15] J.-P. Drolet, S.C. Russev, M.I. Boyanov, R.M. Leblanc, J. Opt. Soc. Am. A 11 (1994)
3284.[16] R.M.A. Azzam, N.M. Bashara, Ellipsometry and Polarized Light, North-Holland,
Amsterdam, 1977.[17] H.G. Tompkins, E.A. Irene (Eds.), Handbook of Ellipsometry, William Andrew
Publishing/Noyes, 2005.[18] E. Elizalde, J.M. Frigerio, J. Rivory, Appl. Opt. 25 (1986) 4557.[19] J.C. Comfort, F.K. Urban, Thin Solid Films 270 (1995) 78 22nd International Con-
ference on Metallurgical Coatings and Thin Films.[20] S. Bosch, F. Monzonis, E. Masetti, Thin Solid Films 289 (1996) 54.[21] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in
FORTRAN; The Art of Scientific Computing, 2nd edition Cambridge UniversityPress, New York, NY, USA, 1993.
[22] S.A. Alterovitz, B. Johs, Thin Solid Films 313–314 (1998) 124.[23] G. Cormier, R. Boudreau, J. Opt. Soc. Am. A 17 (2000) 129.[24] T. Ross, G. Cormier, J. Opt. Soc. Am. A 27 (2010) 319.[25] J.N. Hilfiker, N. Singh, T. Tiwald, D. Convey, S.M. Smith, J.H. Baker, H.G. Tompkins,
Thin Solid Films 516 (2008) 7979 Proceedings of the EMRS 2007 Fall MeetingSymposium H: Current trends in optical and X-ray metrology of advanced mate-rials and devices II Warsaw, Poland.
[26] E. Palik, Handbook of Optical Constants of Solids, Academic Press, New York,1985/1991.
Table 5n and k solutions found for the gold/alumina/aluminum sample using exact values (toplines) and erroneous values (bottom lines) for Ψ, Δ, and d input parameters.
Input parameters Output parameters
Ψ(∘) Δ(∘) d (nm) n k
40.1014 70.1856 20.0 0.4485 0.23260.6650 1.9100
40.0693 70.2733 20.2 0.4521 0.23030.6619 1.9109
Table 6Partial derivatives of n and k solutions for the SiN/Si sample at 5.8 eV relative to thedifferent parameters: experimental ellipsometric angles (Ψe and Δe), thickness (d),angle of incidence (θ), and refractive and extinction indices of substrate (ns and ks).
5574 M. Gilliot / Thin Solid Films 520 (2012) 5568–5574
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Real-time combined reflection and transmission ellipsometry for film-substrate systems
M. Elshazly-Zaghloul
Electrical Engineering Department, Faculty of Engineering, Cairo University, Cairo, Egypt
ABSTRACT
Combined reflection and transmission ellipsometry of film-substrate systems provides a wealth of experimental information that proves helpful with extracting the system parameters. In such a technique, both the reflection and transmission ellipsometric functions are simultaneously measured. The technique itself requires a special design of the sample for fixed and for multiple angles of incidence measurements. The data reduction could be done using numerical methods which are tedious and time consuming resulting in an overall slower technique. A closed-form inversion for the system parameters (film thickness, film optical constant, and substrate optical constant) would render a fast technique that is suitable for real-time applications. The sample used is a three-film-thickness sample and measurements are carried out at two angles of incidence. Real simple closed-form equations are derived through successive transformations and algebraic manipulations to obtain the optical constants of the film and the substrate, in addition to the film thickness. We propose the use of obtained several values of the film thickness to provide a measure of the accuracy of the experimental measurements. In addition, the special case of a bare-substrate system is considered, and the use of the several values of the substrate refractive index to provide a measure of the accuracy of the experimental measurements in such a case is also proposed. A simple software program, with a limited number of code lines, is developed and tested yielding perfect results.
Keywords: Ellipsometry, real time, thin films, film-substrate system, optical constants, combined reflection and transmission, closed-form inversion
1. INTRODUCTION
Combined reflection and transmission (CRT) ellipsometry combines two types of ellipsometry; reflection and transmission. The idea was introduced in Reference 1, where we discussed the technique, sample design, and linear approximation of the equations of both reflection and transmission ellipsometry. Also, the information is published in References 2 and 3.
On the other hand, we developed and published closed-form inversion methods for several of the other ellipsometry problems using deductive mathematics, rigorous algebraic manipulations, and repeated transformations. For reflection ellipsometry on uniaxial crystals, we developed and presented closed-form formulas to calculate the crystal optical constants by algebraically solving the involved equations governing the process.4 We derived a closed-form formula to obtain the optical constant of the substrate of a film-substrate system using reflection ellipsometry with no transparency conditions on the ambient, the film, or the substrate.5 In Reference 6, we presented a closed-form formula for transmission ellipsometry on transparent-film transparent-substrate systems to obtain the film refractive index. In Reference 7, closed-form formulas are presented to characterize an unsupported film/pellicle using transmission ellipsometry, where it was previously thought not possible. In Reference 8, we presented closed-form design formulas for non-negative film-substrate transmission polarization devices that are equally applicable to transmission ellipsometry.
In our continued effort to provide closed-form formulas to different types of ellipsometry, we present in this paper closed-form (CF) formulas to obtain the optical constants of the film and the substrate, and the film thickness of any film-substrate system using CRT ellipsometry. The use of closed-form formulas with any ellipsometer instrument would provide for real-time applications, hence real-time CRT ellipsometry. The CF formulas are deductively derived using
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rigorous algebraic manipulations and repeated transformations with no conditions on the optical constants of the ambient, film, and/or substrate with regard to transparency.
The real-time CRT ellipsometer can triple as both real-time reflection and real-time transmission ellipsometers, since the data are already available, combined with the available CF formulas for both ellipsometers. The wealth of experimental and algebraic information available in such cases, and in real-time CRT ellipsometry, provides for more than the parameters of the film-substrate system under testing. It provides also for much-needed definitive information on the accuracy of the measurements and of the experimental system(s) involved.
In principle, CRT ellipsometry should not be any slower than either reflection or transmission ellipsometry by itself. In combining the two techniques into one, they operate in parallel and no speed is lost. Same argument applies to real-time CRT ellipsometers and it should not be any slower than either of the other two ellipsometers; no speed is lost.
In the following sections, we discuss the real-time CRT ellipsometer, the sample considerations, some experimental considerations of the instrument, and the mathematical equations used for both reflection and transmission ellipsometry. We culminate with the presentation of the simple CF formulas for the film and substrate optical constants, and for the film thickness, in addition to the corresponding formulas for the special case of a bare-substrate system.
The inclusion in a software environment of all of the presented CF formulas in this paper is simple and straight forward. The developed software program is short, concise, and includes a limited number of code lines that are executed only once. It swiftly yields perfect answers, since deductive mathematics was used in the derivation of the CF formulas.
2. REAL-TIME COMBINED REFLECTION AND TRANSMISSION ELLIPSOMETRY
As indicated by the name, combined reflection and transmission ellipsometry is a combination of two types of ellipsometry; reflection and transmission. The most widely used type is reflection ellipsometry.9, 10 A simple internet search would return hundreds of thousands of results, soon to approach a million.11 In contrast, transmission ellipsometry is sparingly used. Some experimental considerations are in order in such a case, which we discuss briefly in Section 2.3.
The equations governing reflection from, and refraction through, material interfaces are used in both ellipsometry types, obviously differently.1, 8, 12 Such equations represent the performance of the physical model of the sample, and are used to deduce model parameters from experimental measurements using one of the available data reduction methods.
Data reduction methods available for reflection ellipsometry are mathematical or heuristic. Mathematical methods are numerical,13 analytical,5 or hybrid methods thereof.14 Heuristic methods are either genetic algorithms15 or artificial neural networks,16 up till now.
Data reduction methods available for transmission ellipsometry are also mathematical or heuristic. Mathematical methods are numerical for transparent unsupported films/pellicles,17 analytical for unsupported films/pellicle,7 analytical for transparent-films transparent-substrate systems,6 or hybrid methods.18
It is clear that the combined reflection and transmission (CRT) ellipsometry provides much more information on the physical model, than either reflection ellipsometry or transmission ellipsometry alone. That provides an unparalleled opportunity for data reduction, leading to closed-form inversion. In Section 5, we present such a closed form inversion. We present a closed-form equation to calculate the optical constant of the film, a second for the optical constant of the substrate, and a third (set of equations) for the film thickness.
Mathematically, no conditions on the opacity of the ambient, film, and/or substrate are required or applied. In section 2.3, we discuss some experimental limitations that may or may not be applicable, based on the film-substrate system under consideration.
Clearly, the closed-form equations to directly and fully characterize the film-substrate system are real simple to implement in software. It is equally simple and straightforward to integrate with the CRT ellipsometer hardware, regardless of the actual instrument in use. Any ellipsometer, adopting any ellipsometric technique to measure the ellipsometric parameters ψ and ∆ would equally work perfectly fine.
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[9] Losurdo, M. and Hingerl, k., Editors, [Ellipsometry at the nanoscale], Springer, Berlin, 2013. [10] Fujiwara, H., [Spectroscopic ellipsometry: principles and applications], Wiley, West Sussex, 2007. [11] www.google.com, August 8, 2013 @ 3:58 am. [12] Zaghloul, A. R. M. and Yousef, M. S. A., “Ellipsometric function of a film-substrate system: detailed analysis and
closed-form inversion,” J. Opt. Soc. Am. A 16, 2029 – 2044 (1999). [13] Zaghloul, A. R. M., Azzam, R. M. A., and Bashara, N. M., “Inversion of the nonlinear equations of reflection
ellipsometry on film-substrate systems,” Surf. Sci. 56, 87-96 (1976). [14] Drolet, J.-P, Russive, S. C., Boyanov, M. I., Leblanc, R. M., “Polynomial inversion of the single transparent layer
problem in ellipsometry,” J. Opt. Soc. Am. A 11, 3284-3291 (1994). [15] Zaghloul, Y. A. and Zaghloul, A. R. M., “Single-angle-of-incidence ellipsometry,” Appl. Opt. 47, 4579-4588
(2008). [16] Chen, X., Liu, S., Zhang, C., and Zhu, J., “Improved measurement accuracy in optical scatterometry using fitting
error interpolation based library search,” Measurement 46, 2638-2646 (2013). [17] Azzam, R. M. A., “Transmission ellipsometry on transparent unbacked or embedded thin films with application to
soap films in air,” Appl. Opt. 30, 2801-2806 (1991). [18] Elshazly-Zaghloul, M., Zaghloul, Y. A., and Zaghloul, A. R. M., “Transmission ellipsometry of transparent-film
transparent-substrate systems: polynomial inversion for the substrate optical constant,” Proc. SPIE 6286, 62860G, 1-10 (2006).
[19] Zaghloul, A. R. M., [Ellipsometric function of a film-substrate system: Applications to the design of reflection-type optical devices and to ellipsometry], PhD dissertation, Univ. of Nebraska, 1975.
[20] Zaghloul, A. R. M. and Yousef, M. S. A., “Unified analysis and mathematical representation of film-thickness behavior of film-substrate systems,” Appl. Opt. 45, 235-264 (2006).
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Single-angle-of-incidence single-element rotating-polarizer (Single SERP) ellipsometer for film-substrate systems
A. R. M. Zaghloul
Electrical Engineering Department, Faculty of Engineering, Cairo University, Cairo, Egypt
ABSTRACT
The single-element rotating-polarizer ellipsometer is where a rotating polarizer is inserted into the incident beam and the reflected-signal intensity is detected using a photodetector. The polarizer is either rotated mechanically or electromagnetically. The angle of incidence of the beam is adjusted to detect the angles where the detector signal is dc. The ellipsometric function of the film-substrate system under measurement is of a unity magnitude at those detected angle(s). The number of required measurements (such angles of incidence) is related (directly proportional) to the number of system parameters to be determined: film thickness is one parameter, film optical constant is two parameters, and substrate optical constant is two parameters. The more parameters to be determined, the more the number of measurements required. This creates film-thickness bands, which number and width depend on the system physical properties and the wavelength used for measurement, and where a continuum exists above a certain film-thickness value. Accordingly, full characterization of film-substrate systems is limited to systems with large film thicknesses for the required multiple angles of incidence to exist. In this paper, we use only one detected angle of incidence to fully characterize the film-substrate system. This allows for film-substrate systems with much smaller film thicknesses to be fully characterized. A fast genetic algorithm is used to heuristically obtain all the system parameters: film thickness and optical constants of the film and the substrate, or any subset thereof.
Keywords: Ellipsometer, single element, thin films, film-substrate system, optical constants, single angle of incidence, genetic algorithm
1. INTRODUCTION
We consider any ellipsometer to be composed of two main parts; the instrument and the data reduction method. The instrument, in turn, is composed of the different devices, their position in the instrument, and their sequence of operation and their motion, if any, that lead to the ellipsometrically measured quantities (the technique). On the other hand, the data reduction method takes the measured quantities and renders the parameters of the system under measurement. This view, when properly adopted, can lead to better ellipsometers; better in the sense of faster, simpler, easier to operate, and/or cheaper. For example, reducing the number of instrument components (devices) leads to an improved accuracy by eliminating one or more sources of experimental errors. Also, the reduction of the number of components would, in most cases, lead to a simpler ellipsometer. In addition, such a reduction leads to a cheaper ellipsometer by reducing the cost involved. The different motion of different components, if any, and the sequence of operation dictate the technique of measurement. The smaller the number of components, the less the motion involved. The smaller the number of components is involved in motion, the faster the instrument and accordingly the ellipsometer as a whole. Similarly, the less involved the data reduction method is, the faster is the ellipsometer. This view of an ellipsometer is adopted in this paper. By providing the appropriate data reduction technique, we convert a multi angle-of-incidence ellipsometer to a single angle-of-incidence ellipsometer. In the following section, we discuss the new ellipsometer in some detail, and conclude with testing results that prove the point.
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where, P is the polarization azimuth of the signal measured counter-clockwise looking into the beam. Therefore, if the incident linear polarization is rotated at an angular speedω , the detector signal would become;
( )2 22 2cos sinp sD k R t R tω ω= + . (9)
The polarization rotation can be effected by mechanically rotating a polarizer, or electromagnetically rotating the beam polarization. For the mathematical condition considered here, Eq. (6) holds, and the signal detector is DC. That is clear by the use of one of the fundamental trigonometric equalities ( 2 2sin cos 1α α+ = ).
Operation of the SERP ellipsometer is real simple. If the rotation is provided mechanically or electromagnetically, the angle of incidence is changed and the DC signal condition is detected. At these angle(s) of incidence, the condition of ρ =1 is satisfied. Such information is necessary and sufficient to fully characterize the film-substrate system, as
discussed in the next subsection.
The experimental accuracy of the instrument is high since we only measure the angle of incidence, which is the most accurately-measured experimental angle in ellipsometry. Also, highly accurate digital circuits exist to detect the DC signal condition. In addition, since no other parameters than the angle of incidence are measured, we have eliminated the errors associated with the measurements of any other parameters in other ellipsometers; reducing the number of error sources introduced by the instrument itself. In fact, not measuring the intensity of the reflected beam (as a value as in other rotating-element(s) ellipsometry or as a zero in null ellipsometry) eliminates one more source of instrumental errors. As such, we are only left with one source of experimental errors; an absolute minimum in ellipsometry as we know it.
2.3 Data reduction
In ellipsometry, data reduction is commonly done using numerical methods. Numerical methods are developed using either forward or inverse calculations. In forward-calculations methods, the calculations start with the unknown value which is to be determined to calculate a known measured quantity, or a minimized function thereof.11 The starting value of the unknown which is to determine is then changed and forward calculations repeated until a calculated value of the measured parameter coincides closely with the actual measured value, or its associated minimized function. The starting value of the unknown quantity in this case is the solution chased. Clearly, the speed of such a method depends on the choice of the value of the unknown parameter to start with, and on its associated function. Also, such choices may lead to the failure of the method itself, where no correct answers are obtained. Several numerical techniques are used to reduce the computational effort of such methods. Commercial ellipsometers are commonly delivered with a computer program following such a data reduction strategy.
Numerical inversion methods, using reverse calculations, start with the measured values and obtains the unknown parameter(s) through numerical calculations.12 Such techniques are more accurate, much faster, guarantees reaching a solution, and easier to implement in software.
On the other hand, closed-form inversion methods are algebraically derived to directly calculate the optical constants of the film and substrate, and/or the film thickness, from measured quantities. For reflection ellipsometry, closed-form inversion formulas exist to calculate the optical constant of the substrate, 13 and for the film thickness. 1, 5 In addition, closed-form inversion formulas are derived to calculate the refractive index of the film and film thickness, of transparent-film transparent-substrate systems using transmission ellipsometry.14 For transmission ellipsometry on an unsupported film/pellicle, closed-form formulas to obtain the film optical constant and thickness are reported.15
Polynomial inversion is an inversion method that is a hybrid numerical and algebraic method; partially numerical and partially algebraic. It algebraically provides a higher order polynomial in the unknown parameter that must be solved numerically to obtain the required parameter value. A polynomial inversion method to calculate the film refractive index of a transparent-film absorbing-substrate is available.16 A different polynomial inversion method to obtain the substrate optical constant of transmission ellipsometry on transparent-film transparent-substrate systems is published.17
All data reduction methods discussed above are based on deductive mathematics, regardless of the direction of calculations being forward or backward (inversion), numerical or algebraic. Heuristic methods, which are conceptually different, are also available to do data reduction.18 A heuristic method is based on the idea of assuming a value for the
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[13] Elshazly-Zaghloul, M. and Zaghloul, A. R. M., “Closed-form inversion of the reflection ellipsometric function of a film-substrate system: absorbing substrate optical constant,” J. Opt. Soc. Am. A 22, 1630-1636 (2005).
[14] Zaghloul, A. R. M., Elshazly-Zaghloul, M., and Zaghloul, Y. A., “Transmission ellipsometry of transparent-film transparent-substrate systems: closed-form inversion for the film optical constant,” Proc. SPIE 6240, 62400M, 1-10 (2006).
[15] Zaghloul, A. R. M., Elshazly-Zaghloul, M., and Zaghloul, Y. A., “Transmission ellipsometry on unsupported film/pellicle: closed-form inversion,” Proc. SPIE 6682, 66820J, 1-12 (2007).
[16] Drolet, J.-P, Russive, S. C., Boyanov, M. I., Leblanc, R. M., “Polynomial inversion of the single transparent layer problem in ellipsometry,” J. Opt. Soc. Am. A 11, 3284-3291 (1994).
[17] Elshazly-Zaghloul, M., Zaghloul, Y. A., and Zaghloul, A. R. M., “Transmission ellipsometry of transparent-film transparent-substrate systems: polynomial inversion for the substrate optical constant,” Proc. SPIE 6286, 62860G, 1-10 (2006).
[18] Polya, G., [How to solve it: A new aspect of mathematical method], Princeton U. Press, New Jersey (1945). [19] Zaghloul, A. R. M., El-Bahy, M. M., and Abou-seada, M. S., “Analysis and application of single-element rotating-
polarizer (SERP) ellipsometer: Film-thickness determination,” Opt. Commun. 61, 363-368 (1987). [20] El-Bahy, M. M., [Single-element rotating-polarizer ellipsometer], M. S. Thesis, Univ. of Cairo, 1987. [21] Spears, W. M., [Using neural networks and genetic algorithms as heuristics for NP-complete problems], MS Thesis,
George Mason University, 1989. [22] Haupt, R. L. and Haupt, S. E., [Practical genetic algorithms]. 2nd Ed., John Wiley, New Jersey, 2004. [23] Cormier, G. and Boudreau, R., “Genetic algorithm for ellipsometric data inversion of absorbing layers,” J. Opt. Soc.
Am. A 17, 129–134 (2000). [24] Kudla, A., “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455–456, 804–
808 (2004). [25] Zaghloul, Y. A. and Zaghloul, A. R. M., “Single-angle-of-incidence ellipsometry,” Appl. Opt. 47, 4579-4588
(2008). [26] Fernandes, V. R., Vicente, C. M. S., Pecoraro, E., Karpinsky, D., Kholkin, A. L., Wada, N., André. P. S., and
Ferreira, R. A. S., “Determination of refractive index contrast and surface contraction in waveguide channels using multiobjective genetic algorithm applied to spectroscopic ellipsometry,” J. Lightwave Tech. 29, 2971-2978 (2011).
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Errors in inversion of ellipsometric equations for transparent films
Mickaël Gilliot ⁎Université de Reims Champagne-Ardenne, BP 1039, 51687 Reims Cedex 2, France
a b s t r a c ta r t i c l e i n f o
Article history:Received 24 September 2012Received in revised form 16 June 2013Accepted 27 June 2013Available online 10 July 2013
Keywords:EllipsometryData inversionThin filmsOptical propertiesThicknessRefractive index
The traditional method of single-angle ellipsometric inversion to determine the refractive index and thickness oftransparentfilms is re-examined. Error formulas are derived and analyzed. It is shown that consideration of theseerrors allows the improvement of the determination of thickness and then that extraction of refractive index canbe performed with improved confidence by an additional calculation step. The method is tested through differ-ent sample examples. It should be helpful in many practical cases of ellipsometry data interpretation, especiallywhen standard fitting techniques do not work.
Ellipsometry is a technique of great interest to characterize thinfilms. Its sensitivity has been known for a long time and has beenthe subject of thorough works and progress [1–3]. Information ofdifferent kinds can be extracted from measurements. The techniqueas well as the fields of application have been extensively developed[4,5]. A key issue of ellipsometry still remains in the data interpretationbecause of the indirect nature of the measured quantities. Measuredangles depend on many factors such as multilayer structure of thesamples, interference effects inside layers, optical properties of thematerials, thickness of layers, microstructures, which prevents intuitiveunderstanding of spectroscopic data.
The general method in spectroscopic ellipsometry data interpreta-tion is to represent the sample by a multilayer structure and representunknown materials by dispersion functions and adjust the dispersionparameters and unknown thicknesses to fit generated to experimentaldata. This task may be difficult because of complexity of materialsthat cannot always be represented by theoretical laws. Alternately,methods referred as wavelength-by-wavelength methods can be ofgreat interest. These methods consist in the rigorous resolution ofellipsometric equations for each point of the considered spectrum toextract refractive index and thickness using single-angle measurementfor transparent films, or additionally also extract extinction coefficientfor absorbing films using multiple measurements.
In this paper a method is proposed to improve the results given bythe classical wavelength-by-wavelength inversion method originally
proposed by McCrackin to determine thickness and refractive indexof a transparent layer in a multilayer system from spectroscopicellipsometry data. Derivation of error formulas and careful investigationof errors related to input experimental data is added to improve thethickness determination over the whole considered spectrum. Thenrefractive index can be confidently extracted using this optimizedvalue of thickness by solving the ellipsometric equations as a functionof complex refractive index. The method is tested with different exam-ples. Note that it can also be used to extract thickness, refractive indexand extinction coefficient of an absorbing layer provided that the layerpresents some spectral range of transparency.
2. Theory
Ellipsometry measures the change of polarization state betweenincident and reflected light on a sample resulting on reflection at the dif-ferent interfaces and interferences within the layers. The ellipsometricangles Ψ and Δ are related to Fresnel coefficients of the sample rp andrs, respectively for p-polarized (parallel to the plane of incidence) ands-polarized light (perpendicular to the plane of incidence) by
ρ ¼ rprs
¼ tanΨ eiΔ: ð1Þ
In this paper the ellipsometric ratio and ellipsometric angles aregenerally denoted as ρ, Ψ and Δ, and are also denoted as ρe, Ψe and Δe
when they refer to input data used as experimental data.Assuming the sample under investigation is represented by a stack
of films on a substrate and that only one transparent layer with thick-ness d and refractive index n is unknown in the stack, the inversion
process consists in solving equations for these two parameters makingthe modelled response equal to the experimental response. Theoreticaloptical response is calculated as a function of n and d using standardmatrix formalism which basic principles are reminded in Appendix A,and can be reduced as:
rp;s ¼ap;sX þ bp;sa′p;sX þ b′p;s
; ð2Þ
whereap;s; bp;s; a′p;s; b
′p;s are coefficients determined byoptical properties
of other materials within the sample and that depend on refractiveindex n, and X is a propagation factor that depends on both refractiveindex n and thickness d of the considered unknown transparent layeras:
X ¼ e− j4πdλffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2−N2
0 sin2ϕ0
p; ð3Þ
where N0, ϕ0 and λ are the refractive index of the incidence medium,the angle of incidence and wavelength of light on the sample.
Knowledge of an input ellipsometric data point represented by thecomplex ellipsometric ratio ρe, provides, according to notations ofEq. (2), the following equation:
ρe ¼ repres
¼ apX þ bpa′pX þ b′p
a′sX þ b′sasX þ bs
; ð4Þ
and the ellipsometric equation can be recast as a polynomial in X:
ρea′pas−apa′s
� �X2 þ ρeb′pas þ ρea′pbs−bpa
′s−apb
′s
� �X
þ ρeb′pbs þ bpb′s
� �¼ 0:
ð5Þ
The coefficients of the polynomial depend on the experimentally
determined complex ellipsometric ratio ρe and on the coefficients ap;
bp; a′p; b′p; as; bs; a
′s; b
′s. These coefficients can be calculated using formal-
ism and formulas presented in Appendix A as a function of thicknessesand optical properties of othermaterials that are supposed to be knownwithin the sample and also as a function of the unknown refractiveindex n that is considered as a variable.
From this last equation two X roots functions and subsequently
Note ed ¼ edr þ jedi is a function expressing values of thickness as afunction of refractive index whereas d is the actual value of thicknessof the layer. For a given value of n, these functions generally returncomplex numbers. m is the film thickness multiplicity specifyingthat solution is not unique but has periodic behaviour. As thicknessis a real quantity, the solutions of the problem are particular valuesof n where imaginary part of thickness function is zero, which canbe found by numerically solving:
edi ¼ 0: ð7Þ
The principle of this inversion referred as McCrackin inversion hasbeen proposed for more than forty years and implemented in differentways [6–13], including an original polynomial derivation [14,15]. Thekey point of the method used in this work is the additional consider-ation of errors associated to this inversion process. For error estimationof thickness and refractive index values, thickness functions are
linearized around the solution points yielding the following sensitivitycoefficients:
∂n∂x ¼ −
∂edi∂x∂edi∂n
∂d∂x ¼ ∂edr
∂n∂n∂x þ ∂edr
∂x ;
ð8Þ
where r and i subscripts stand for real and imaginary parts and x is anyof the input parameters such as experimental ellipsometric angles,angle of incidence or optical constants or thicknesses of surroundingmedia and layers. Note the expression of thickness error must alsoinclude the multiplicity related part. Formulas can be algebraicallydeveloped yielding more elaborated expressions which are presentedin Appendix B. Practically in the developed algorithm sensitivity coeffi-cients are calculated directly using numerical derivatives in theseequations.
The inversion procedure yields n and d solutions but in thepresented method, only the thickness information is used. Errors asso-ciated to this inversion are at the same time calculated and thicknessis estimated by considering points of the spectrum presenting thelowest sensitivity to errors. Using this estimated value of thickness, re-fractive index is calculated in a second step by solving the ellipsometricequations as a function of the complex refractive index, which can beperformed using least-squared minimization or a more elaboratedprocedure solving all possibilities of complex refractive index in a de-sired range of exploration [16]. Extracted values of complex refractiveindex should then have imaginary parts very close to zero. By extensioncomplex refractive index values can be extracted for an absorbing filmpresenting a range of transparency where thickness can be evaluated.
3. Numerical procedure
The inversion procedure has been written with the important fol-lowing features:
• The roots of edi are directly found by splitting a given range of nvalues and using a Brent type algorithm [17].
• Both X roots are considered. Whereas in some early implementationsof the inversion process [7,9], the only one root giving the smallestimaginary part of ed for an initial value of n was considered, it is nowknown that both roots are of interest [18].
• For each of these X roots, multiple n solutions are considered. Forexample the case of a 200 nm layer of silicon dioxide (N = 1.466),on silicon (N = 4.673−0.144i) at incident wavelength of 453 nmand angle of 70° has two solutions n = 1.466 and n = 1.562. Itappears to be of importance to consider all possible n solutions sothat the actual solution has no risked to be ignored.
• Film thickness multiplicity m is considered using the wavelengthdependence of the multiplicity factor. As spectral data are in factconsidered, the choice of the order is automatically performed byplotting a few orders (2 or 3 orders are enough) for each point ofthe spectrum and selecting the most regular horizontal line for thethickness as a function of wavelength.
• Errors associated to uncertainties on input parameters are systemat-ically calculated and examined with care. Thickness is evaluatedusing points of the spectrum where erros are the lowest.
• Refractive index is then calculated using the estimated value of thick-ness by solving the ellipsometric equations as a function of complexrefractive index with the help of a method based on the determina-tion of contours of real and imaginary parts of the ellispometricfunction in the n-k plane described elsewhere [16].
301M. Gilliot / Thin Solid Films 542 (2013) 300–305
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Partial derivatives in the preceding formulas can be expanded as:
∂ed∂Ψe ¼ j
λ4π
1þ tan2Ψe� �
eiΔe
KXY
∂ed∂n ¼ j
λ4π
− LKXY
þ j4πnedλY2
!;
ðB:4Þ
with the following notations:
Y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2−N2
0 sin2ϕ0
q;
u ¼ AX2 þ BX þ C;
v ¼ A′X2 þ B′X þ C
K ¼2AX þ Bð Þv− 2A′X þ B′
� �u
v2
L ¼∂A∂nX
2 þ ∂B∂nX þ ∂C
∂n
� �v− ∂A′
∂nX2 þ ∂B′
∂nX þ ∂C′
∂n
� �u
v2
ðB:5Þ
Finally sensitivity coefficients are expressed as:
∂n∂Ψe ¼ 1þ tan2Ψe
� � I j eiΔe
KXY
� �I j L
KXYþ 4πned
λY2
� �
∂d∂Ψe ¼
λ4π
1þ tan2Ψe� �I eiΔ
e
KXYL
KXY− j4πn
edλY2
� �" #
I j LKXY
þ 4πnedλY2
� �ðB:6Þ
Using similar calculations, one also gets:
∂n∂Δe ¼ − tanΨe
IeiΔ
e
KXY
� �I j L
KXYþ 4πned
λY2
� �
∂d∂Δe ¼ − λ
4πtanΨe
IeiΔ
e
KXYj LKXY
þ 4πnedλY2
� �" #
I j LKXY
þ 4πnedλY2
� �ðB:7Þ
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305M. Gilliot / Thin Solid Films 542 (2013) 300–305
