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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: dirk.poelman@ugent.be
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: dirk.poelman@ugent.be
Received 15 January 2003, in ﬁnal form 3 April 2003Published 16 July 2003Online at stacks.iop.org/JPhysD/36/1850
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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