2003-review-A1_02 2003 JPhysD DP PFS.pdf

INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 36 (2003) 1850–1857 PII: S0022-3727(03)58303-6 Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review Dirk Poelman and Philippe Frederic Smet Department of Solid State Sciences, Ghent University, Krijgslaan 281 S1, B-9000 Gent, Belgium E-mail: Received 15 January 2003, in final form 3 April 2003 Published 16
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  I NSTITUTE OF  P HYSICS  P UBLISHING  J OURNAL OF  P HYSICS  D: A PPLIED  P HYSICS J. Phys. D: Appl. Phys.  36  (2003) 1850–1857 PII: S0022-3727(03)58303-6 Methods for the determination of theoptical constants of thin films from singletransmission measurements: a criticalreview Dirk Poelman and Philippe Frederic Smet Department of Solid State Sciences, Ghent University, Krijgslaan 281 S1, B-9000 Gent,BelgiumE-mail: Received 15 January 2003, in final form 3 April 2003Published 16 July 2003Online at Abstract Optical transmission measurements are commonly used for the routinedetermination of thin film optical constants. This paper presents anoverview of the different methods of evaluating these transmission data,leading to values for the complex refractive index. Three different groups of methods are distinguished using: (1) at least two different opticalmeasurements; (2) dispersion relations or general physical constraints toapproximate the behaviour of the wavelength-dependent refractive index;and (3) a ‘virtual’ measurement as a second variable. The methods fromgroups (2) and (3) (requiring only a single transmission measurement) aretreated in more detail and are evaluated in terms of their accuracy. 1. Introduction The knowledge of accurate values of the wavelength-dependent complex refractive index of thin solid films isvery important, both from a fundamental and a technologicalviewpoint. It yields fundamental information on the opticalenergy gap (for semiconductors and insulators), defect levels,phonon and plasma frequencies, etc. Moreover, the refractiveindex is necessary for the design and modelling of opticalcomponents and optical coatings such as interference filters.If the model of an isotropic, homogeneous and plane-parallel thin film can be adopted, the real and imaginaryparts of the refractive index at each wavelength completelydetermine the optical properties of the film (in most instances,the film thickness can be determined at the same time).Therefore, two independent measurements are necessary ateach wavelength in order to solve for the unknowns  n(λ) and  k(λ) .If the above-mentioned requirements are not met (forinhomogeneous films [1–3] or rough films [4,5]), it is stillpossible in most cases to model the optical properties of the‘non-ideal’ thin films, albeit with somewhat more complexity.Numerous methods have been devised for the determi-nation of the refractive index of thin films. Advantages anddisadvantagesofthemostimportantmethodswillbediscussed;Three different groups of methods will be distinguishedusing:(1) at least two different optical measurements,(2) dispersion relations or general physical constraints toapproximate the behaviour of the wavelength-dependentrefractive index and(3) a ‘virtual’ measurement as a second variable.Special attention will be given to methods, using only a singletransmissionspectrum(groups(2)and(3)), sincetheserequirethe least experimental effort. The dependence of the results onexperimental errors will not explicitly be discussed here; thishas been done in detail in other studies [6–8]. 2. Results and discussion 2.1. Methods using two independent measurements Since many years, the combination of a normal incidencetransmission measurement and a near-normal incidencereflectance measurement—in short the ( R,T  ) method—hasbeen used for the ( n,k ) determination [9–12]. This method 0022-3727/03/151850+08$30.00 © 2003 IOP Publishing Ltd Printed in the UK  1850  Optical constants of thin films suffers a few disadvantages:(a) It is very difficult to obtain sufficiently accurateabsolute specular reflectance data. In spectrophotometricmeasurements, there are two common methods for measuringspecular reflectance. Using the simplest experimental set-up,one measures the light attenuation upon a single reflectance atthe sample surface, which is then compared to the reflectanceof a calibrated mirror (see figures 1( a ) and ( b )). This methodobviously necessitates such a calibrated mirror, which isboth expensive and, moreover, only has limited shelf life:its reflectance is only guaranteed for a few months due toageing effects. A more elegant way for measuring absolutereflectance is the VW-set-up, introduced by Strong [13]and shown schematically in figures 1( c ) and ( d  ). Here, amoveablemirrorisfirstplacedina‘reference’position,andthecorresponding combined reflectance of the two fixed mirrorsand the moveable mirror is acquired. Then, the moveablemirrorisswitchedtoits‘measurement’positionandthefilmtobeinvestigatedismounted. Now(figure1( d  )), theonlyopticaldifference with the reference measurement (figure 1( c )) is adouble reflectance off the sample under test. Thus, divisionof the ‘measurement’ and ‘reference’ values exactly yields thesquareofthereflectanceneeded,irrespectiveofthereflectanceof the mirrors used. The only obvious disadvantage of theVW-method is that the two reflectances on the sample occuron different spots. Therefore, the sample should be perfectlyhomogeneous over at least the distance between the two spots,including the size of the beams, typically 25mm in total [14].(b) Usually, it is nearly impossible to measuretransmittance and reflectance at exactly the same spot on thefilm; if the film is not perfectly homogeneous (in compositionor in thickness), which is inevitable, this further decreases theoverall accuracy of the method.(c) There is no easy way to measure normal incidencereflectance. In the usual methods (described above), the angleof incidence is 7˚ [14]. For maximum accuracy, this angle (a) (b)(c) (d) Figure 1.  Experimental set-ups for measuring specular reflectance:using a calibrated mirror, in reference position ( a ) and measurementposition ( b ) and using the VW-set-up, in reference position ( c ) andmeasurement position ( d  ). should be taken into account in the subsequent calculations. Italso makes the measurements sensitive to polarization effects.(d) The ( n,k ) determination from ( R,T  ) is based on anumerical inversion. Numerous papers have been publishedon different inversion methods [10] and on the accuracy limitsof the technique [15–17], but it usually remains a source of errors and a possibility for multiple solutions.Remark that if the optical transmission of a thin filmis extremely low due to excessive absorption, reflectionmeasurements may be the only choice. In that case, thecombination of two reflection measurements, using differentincidentanglesand/ordifferentpolarization,canbeemployed.Numerous other methods, using two different measure-ments, have been published. We will only sum up a few of these as examples:(a) Transmission measurements on two films of differentfilm thickness [18]: obviously, this method requires that therefractive index is independent of film thickness. In addition,it cannot be used as a routine analysis method: most often, thethin films will have to be specifically grown in two thicknessesfor this analysis. A related method is based on measurementson films deposited on partly metallized substrates [19], againrequiring additional care upon film deposition.(b) Ellipsometry: this is a very powerful technique(especially spectroscopic ellipsometry) for the determinationof the optical constants of thin films and substrates [20]. Ithas found a widespread use in the microelectronics industry[21], for the analysis of monolayer-thick oxide layers onsemiconductor wafers. The strength of the technique, itsextreme surface sensitivity, is its weakness at the same time:a very thin contamination layer on a thin film will yieldgrossly different optical constants for the film. In recent years,ellipsometric measurements have also been performed intransmission mode [22], rather than the usual reflection mode.While careful modelling of the investigated layer system forthe analysis of ellipsometric measurements can certainly givegood results, we will limit ourselves to spectrophotometrictechniques in the rest of this paper. 2.2. Fitting of dispersion relations A method, which is widely in use, assumes a specific—moreor less empirical—dispersion equation for the wavelength-dependent complex refractive index. This method is beingused in several commercial software packages for thin filmopticaldesignandanalysis(TfCalc[23]andFilmwizard[24]).The equations used most often are:(a) The Cauchy equations [25]. These equations arecompletely empirical and were first proposed by Cauchy(1789–1827). They are well suited to model transparentmaterials like SiO 2 , Al 2 O 3 , Si 3 N 4 , BK7 glass, etc: n(λ) = A n  +  B n λ 2  +  C n λ 4  + ··· k(λ) = A k  +  B k λ 2  +  C k λ 4  + ··· (1)where wavelengths are expressed in microns.  A n ,  B n ,  C n ,  A k , B k  and  C k  are the six fitting parameters. Very often, the series1851  D Poelman and P F Smet expansion is ended after the first two terms and the terms in λ − 4 are not used.(b) The Sellmeier relation [25,26]. This formula was firstderived by Sellmeier (1871). It is applicable to transparentmaterials (like the Cauchy equations) and to semiconductors(Si, Ge, GaAs, etc) in the infrared. The Sellmeier equation is ageneralizationoftheCauchyformulae. TheoriginalSellmeierrelation is used for completely transparent materials ( k  =  0);however, it is sometimes extended to cover the absorbingregime, using an additional formula for  k(λ) : n(λ) =  A n  +  B n λ 2 λ 2 − C 2 n  1 / 2 k(λ) = 0 or  k(λ) =  n(λ)  B 1 λ  +  B 2 λ +  B 3 λ 3  − 1 (2) A n , B n , C n , B 1 , B 2  and B 3  arethefittingparametersinthiscase.(c) The Lorentz classical oscillator model [25,27]: n 2 − k 2 = 1 +  Aλ 2 λ 2 − λ 20  +  gλ 2 /(λ 2 − λ 20 ) 2 nk  = A √  gλ 3 (λ 2 − λ 20 ) 2 +  gλ 2 (3)with  λ 0  the oscillator central wavelength,  A  the oscillatorstrength and  g  the damping factor. In the first of the twoformulae,theoneattheright-handsiderepresentsthedielectricfunction at infinite energy (zero wavelength). For mostpurposes, it is more realistic to replace it by a fitting parameter ε ∞ , representing the dielectric function at wavelengths muchsmaller than measured. The equations (3) are easily solved for n  and  k , but yield rather unwieldy expressions when writtendown explicitly.One of the advantages of these equations over those fromCauchy and Sellmeier, is that they present a set of coupledequations for  n  and  k , consistent with the Kramers–Kronig(KK) relations. The Forouhi–Bloomer set of equations alsotake this into account.(d) The Forouhi–Bloomer dispersion relations have beendeveloped for modelling the complex index of refraction of crystalline semiconductors and dielectrics on a rigorous basis[28–31]. The formula for  n(E)  is deduced explicitly from k(E)  using the KK-relations: k(E) = q  i = 1 A i (E − E g ) 2 E 2 − B i E  +  C i n(E) = n( ∞ )  + q  i = 1 B o i E  +  C o i E 2 − B i E  +  C i (4)with B o i  = A i Q i  − B 2 i 2 +  E g B i  − E 2g  +  C i  C o i  = A i Q i  (E 2g  +  C i )B i 2  − 2 E g C i  Q i  =  12 ( 4 C i  − B 2 i  ) 1 / 2 (5)Not all the parameters in equations (4) are independent;dependences are defined in equations (5). Therefore, only n( ∞ ) ,  A i ,  B i ,  C i  and  E g  remain as independent fittingparameters. The equations have been implemented in a seriesofthinfilmanalysistools, usingreflectancemeasurementsandmainly focused on the semiconductor industry [32].Remark that the Forouhi–Bloomer equations wereessentially introduced to only model the interband region of materials [31], i.e. at photon energies higher than the bandgap energy; however, they also have been applied in the sub-bandgap region [33]; it will be evaluated whether they canprovide the same level of precision as the other dispersionequationsintheregionofnormaldispersionofawidebandgapmaterial.(e) The Drude model: for metals, the dielectric function isgoverned by free carriers. When  ω p  is the plasma frequency( ω 2p  = 4 πne 2 /m)  and  ν  the electron scattering frequency, theDrude dielectric function is given by [34]: ε(ω) = 1 − ω 2p ω(ω  + i ν) (6)Usually, the parameters in the dispersion equations aredetermined using a least squares fitting procedure, comparingthe experimental transmission spectrum against the spectrum,calculated from ( n,k ) and the usual equations for thetransmission of an absorbing thin film, as given in appendix.In most instances, it is straightforward to include the filmthickness as a fitting parameter.Like the Forouhi–Bloomer equations, the Sellmeier andLorentz-oscillator dispersion equations can be extended tomultiple oscillators if needed [35]; for some materials, it isnecessary to combine an oscillator-type dispersion relationwith the Drude model.Any of the dispersion equations can yield very goodresults for a large number of materials and over a quitelarge wavelength region. The applicability of an equationis guaranteed if there is a good fit between the experimentaltransmission spectrum and the one, calculated from thedispersion equations. Indeed, all of these dispersion equationsare slowly varying functions of wavelength, and there is nomeanstoobtainagoodfitoveralargewavelengthrange, usinga wrong set of ( n,k ) data. This is due to the fact that  n  and  k (togetherwiththefilmthickness d  )verydirectlydeterminetheformofatransmissionspectrum(alsoseefigure2):thespacingof the interference fringes follows from the product of thefilm thickness and the refractive index (the optical thickness).Iftherefractiveindexdispersionisnegligible(whichisusuallythe case well away from the absorption edge), the order of interference  m  at the transmission maximum at wavelength λ 1  is: m =   λ 2 λ 1 − λ 2   (7)where  λ 1  and  λ 2  are the wavelengths of two adjacenttransmission maxima ( λ 1  > λ 2 ) and the symbol [ x ] stands fornearest integer. The maximum transmittance is determinedby the product of   d   and the extinction coefficient  k , and theheight of the fringes is determined in first approximation by n  (from equation (A1), with  k  =  0). In this way, with a littleexperience, the thickness and ( n,k ) can often be estimatedfrom a transmission spectrum at first sight. The transmissionspectrum of figure 2 easily allows such a ‘paper and pencil’1852  Optical constants of thin films approach. In this case, the extinction coefficient  k  is so low,that the transmission extrema are found from (see appendix): nd   = mλ 2 nd   = (m  + 1 )λ 4(8)for transmission maxima and minima, respectively. Thedetermination of the wavelengths of the transmission extrema,together with the value of the interference order (formula 7)then allows to determine the product  nd   at each extremumwavelength. The accurate determination of the real part of therefractive index at only one wavelength (e.g. at a point wherethe absorption is negligible) makes it possible to determineboth the film thickness and  n  at each extremum wavelength.The result of this procedure in the case of the spectrum of figure 2 is also shown in table 1 and figure 3.However, one must take care in the case of very thinor strongly absorbing films, since they show no interferencefringesandfitscangivemultiplesolutions. Itis,unfortunately,not easy to quantify the error limits for the most general caseof film thickness,  n  and  k .The dispersion relation method is very powerful and canyield very accurate data, especially for thin films (or even thinfilmstacks)withsomeknownparameters(likefilmthicknessesor material data from previous experiments). The onlylimitation of the method is that one has to make assumptionsabout the type of dispersion relation before starting the fittingprocedure.More general physical constraints on the complex refrac-tiveindex,notimplyingaspecificfunctionalrelationship,have Figure 2.  Experimental transmission spectrum of an electron beamdeposited SrS thin film on fused silica. Table 1.  Results of different analysis methods for single transmission spectra of a SrS thin film.Method Data range (nm) Film thickness (nm) RMS (%)  n  (2000nm)  n  (500nm)Cauchy 300–2000 798.3 0.71 1.997 2.091Sellmeier 300–2000 796.3 0.31 1.995 2.101Lorentz 300–2000 793.9 0.36 2.002 2.107Paper and pencil N.A. 792.6 N.A. 1.994 2.114Forouhi–Bloomer 500–2000 793.0 0.38 1.988 2.096Swanepoel N.A. 794.5 N.A. 1.995 2.107Chambouleyron 300-2000 796.6 0.22 1.988 2.101RMS: deviation between experimental data and fitted results. been explicitly used by Chambouleyron  et al  [36–39] for aconstrained optimization approach to the extraction of   n  and k  from transmittance data. The authors assumed the follow-ing physical constraints, usually valid for semiconductors andinsulators in the region of normal dispersion:(1)  n(λ)  1 and  k(λ)  0 for all  λ .(2)  n(λ)  and  k(λ)  are decreasing functions of   λ .(3)  n(λ)  is convex (which translates into  n  (λ)  0).(4) There is a infliction point  λ infl  such that  k(λ)  is convex if  λ  λ infl  and concave if   λ < λ infl .Then, a pointwise optimization method was developed forthe least squares fitting of experimental transmission spectraagainst the spectra, calculated from  n(λ)  and  k(λ) , explicitlytaking into account these constraints [36]. While thismethod was shown to yield reliable results, it presents aserious mathematical challenge. Therefore, the method wasconverted into a mathematically and computationally easierunconstrained optimization [37]. This was cleverly madepossible by a change to new fitting variables, automaticallyincluding the physical constraints. For example, theconstrained optimization of   n(λ)    1 could be convertedto the unconstrained optimization of a variable  v , with n(λ) = 1 +  v 2 (λ) .It is worth inquiring how well the previously discusseddispersion relations comply with the physical constraints setby Chambouleyron: ã  The Cauchy equations comply with the first threeconstraints, provided all the fitting constants are positive.Ifthecoefficient C k  isnegative,thelastconstraintcanalsobe fulfilled. ã  The Sellmeier relation for  n(λ)  is also consistent withthe first three constraints, at wavelengths well above thecritical wavelength  C n . For  k(λ) , the constraints can allbe fulfilled, dependent on the fitting coefficients. ã  The Lorentz oscillator model is intrinsically consistentwiththeconstraints, againifthewavelengthissufficientlylonger than the oscillator wavelength. ã  The Drude model only applies to metals, not tosemiconductors or insulators in their transparent region,where they show normal dispersion and thus obviously isnot consistent with the constraints. ã  Since the Forouhi–Bloomer equations were introducedto describe the refractive index dispersion of materialsat energies higher than the band gap, the constraints areintrinsically not fulfilled.All the methods discussed so far, require the user tomake certain assumptions about the nature of the refractive1853


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