ISO 23131-3:2026

International
Standard
ISO 23131-3
First edition
Ellipsometry —
2026-01
Part 3:
Transparent single layer model
Ellipsométrie —
Partie 3: Modèle de couche unique transparente
Reference number
© ISO 2026
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Published in Switzerland
ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 Terms and definitions .1
3.2 Symbols and abbreviated terms .1
4 Transparent single layer model . 2
4.1 Optical path .2
4.2 Assumptions .3
4.2.1 General .3
4.2.2 Deviations from model assumption M1 .4
4.2.3 Deviations from model assumption M2 .4
4.2.4 Deviations from model assumption M3 .4
4.2.5 Deviations from model assumption M4 .5
4.2.6 Deviations from model assumption M5 .5
4.2.7 Deviations from model assumption M6 .5
4.2.8 Deviations from model assumption M7 .5
4.2.9 Deviations from model assumption S1 .5
4.2.10 Deviations from model assumption S2 .6
4.3 Special characteristics of the transparent single layer model .6
4.4 Validation .7
4.5 Measurement uncertainty .8
4.5.1 Measurement uncertainty of the ellipsometric transfer quantities Ψ and Δ .8
4.5.2 Measurement uncertainty u of the layer thickness d .9
d
5 Test report .12
Annex A (informative) Additions to the transparent single layer model .13
Bibliography .29

iii
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out through
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with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are described
in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the different types
of ISO document should be noted. This document was drafted in accordance with the editorial rules of the
ISO/IEC Directives, Part 2 (see www.iso.org/directives).
ISO draws attention to the possibility that the implementation of this document may involve the use of (a)
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This document was prepared by Technical Committee ISO/TC 107, Metallic and other inorganic coatings, in
collaboration with ISO/TC 35, Paints and varnishes, SC 9, General test methods for paints and varnishes.
A list of all parts in the ISO 23131 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.

iv
Introduction
The ellipsometry measuring method is a phase-sensitive reflection technique using polarized light in
the optical far-field. Ellipsometry has been established as a non-invasive measuring method in the field
of semiconductor technology, in particular the field of integrated production. The method was originally
conceived as a single-wavelength measuring method, then as a multiple-wavelength and later as a
spectroscopic measuring method.
Ellipsometry can be used to determine optical or dielectric constants of any material as well as the layer
thicknesses of at least semi-transparent layers or layer systems. Ellipsometry is an indirect measuring
method, the analysis of which is based on model optimization. The measurands, which differ according to the
procedural principle, are converted into the ellipsometric transfer quantities Ψ (psi, amplitude information)
and Δ (delta, phase information). The physical target quantities of interest (optical or dielectric constants,
layer thicknesses) are determined based on these measurands by means of a parameterized fit.
Ellipsometry shows a high precision regarding the ellipsometric transfer quantities Ψ and Δ, which can
be equivalent to a layer thickness sensitivity of 0,1 nm for ideal layer substrate systems. As a result, the
measuring method can detect even the slightest discrepancies in surface characteristics. This is closely linked
to the homogeneity and the isotropy of the material surface. In order to achieve high precision, carrying out
measurements at the exact same measuring point is a prerequisite for inhomogeneous materials. The same
applies to the orientation of the incident plane relative to the material surface for anisotropic materials.
In the transparent single layer model, a fitting procedure is used to determine the layer thickness d and the
refractive index n of the layer while the optical constants of the substrate are known. For ideal transparent
materials, it is assumed that k = 0. Therefore, fitting only refers to the layer thickness d, which is by definition
independent of the wavelength and of the angle of incidence, and the wavelength-dependent refractive
index n, which does not depend on the angle of incidence. Consequently, the transparent single layer model
allows the determination of the layer thickness and of the refractive index.

v
International Standard ISO 23131-3:2026(en)
Ellipsometry —
Part 3:
Transparent single layer model
1 Scope
This document uses ellipsometric measurements and their analysis to specify the method for the
determination of the layer thickness d of a transparent layer and the optical (refractive index n) or dielectric
(real part ε ) constants/functions based on the transparent single layer model within a spectral region, for
which k = 0 applies.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content constitutes
requirements of this document. For dated references, only the edition cited applies. For undated references,
the latest edition of the referenced document (including any amendments) applies.
ISO 23131, Ellipsometry — Principles
ISO 23131-2:2026, Ellipsometry — Part 2: Bulk material model
ISO/IEC Guide 98-3, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in
measurement (GUM:1995)
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document, the following term and definition applies.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https://www.iso.org/obp
— IEC Electropedia: available at https://www.electropedia.org/
3.1.1
transparent single layer
layer of which the light extinction can be neglected within the considered spectral region
3.2 Symbols and abbreviated terms
For the purpose of this document, the symbols and abbreviated terms given in ISO 23131 and ISO 23131-2
and the following apply:
Table 1 — Symbols
Symbols Description
n refractive index (real part of the complex refractive index N)
k extinction coefficient (imaginary part of the complex refractive index N)
N complex refractive index
ε real part of the complex dielectric function
ε imaginary part of the complex dielectric function
ε complex dielectric function
φ angle of incidence between the incident light wave and the normal to the surface
N complex refractive index of the ambient space
a
N complex refractive index of the layer
l
N complex refractive index of the substrate
s
R maximum roughness depth
t
R arithmetic roughness average (profile roughness)
a
root-mean-square deviation (in ISO 23131 and some evaluation software, root-mean-square
D
RMS
error is still used)
D minimum of the root-mean-square deviation
RMS,min
d fitting result for the layer thickness of the transparent single layer (sample)
fit
calculated layer thickness of the transparent single layer for single-wavelength ellipsometry
d
calc
and a single angle of incidence (sample)
u uncertainty of the fit on the layer thickness model parameter (sample)
fit
a half width of the rectangular distribution
uncertainty of modelling in the transparent single layer model regarding the layer thickness
u
mod
(sample)
uncertainty of the fit on the layer thickness model parameter (layer thickness standard/ref-
u
fit,ref
erence layer)
expanded combined uncertainty (measurement uncertainty) of the reference materials
U
ref
(layer thickness standard/reference layer)
a
coverage factor that has been used to indicate the measurement uncertainty (expanded
c
ref
combined uncertainty) of the reference materials (layer thickness standard/reference layer)
u combined uncertainty of the layer thickness (layer thickness standard/reference layer)
ref
combined uncertainty of the layer thickness d considering all uncertainty components (sam-
u
d
ple)
minimum uncertainty when no other uncertainty determination is possible; defined as equal
u
min
to 0,5 nm
a
coverage factor for expression of measurement uncertainty (expanded combined uncertain-
c
samp
ty) of sample
a
The coverage factor is denoted “k” in ISO/IEC Guide 98-3.
4 Transparent single layer model
4.1 Optical path
Figure 1 shows the optical path at a certain angle of incidence φ in the transparent single layer model. The
ambient space (index “a”) is assigned with the complex refractive index N . Usually, air serves as the ambient
a
space and the refractive index is in close approximation with n = 1,000 and k = 0,000. The transparent
a a
single layer is assigned with the layer thickness d and the complex refractive index N (index “l” for layer).
l
In the transparent single layer model, it is assumed that k = 0. The substrate is assigned with the complex
l
refractive index N .
s
Incident light from the ambient space strikes the interface between the ambient space and the transparent
single layer; part of it is reflected. Another part is refracted and experiences multiple reflections in the
transparent single layer at the layer/substrate and the layer/ambient interfaces. The measured signal thus
represents the superposition of the multiple reflections (interference) in the transparent single layer.
Key
φ angle of incidence between the incident light wave and the normal to the surface
N complex refractive index of the ambient space (for air real, N = n = 1,000)
a a a
d layer thickness of the transparent single layer
N complex refractive index of the layer (for transparent layers real, N = n (λ), k (λ) = 0)
l l l l
N complex refractive index of the substrate (N = n (λ) + i · k (λ))
s s s s
Figure 1 — Optical path in the transparent single layer model
In the transparent single layer model, it is assumed that, in addition to the angle of incidence φ, only the
layer thickness d and the refractive index n or the real part of the dielectric function ε of the layer as well
l 1l
as the optical (n , k ) or dielectric (ε , ε ) constants of the substrate determine the measured ellipsometric
s s 1s 2s
transfer quantities Ψ and Δ.
Clause A.1 gives the refractive indices of specified transparent layer materials. Clause A.2 and Clause A.3
describe the reference layers for transparent single layers on opaque or transparent substrates.
4.2 Assumptions
4.2.1 General
The following model assumptions given in Table 2 are made for the transparent single layer model.

Table 2 — Assumptions for the transparent single layer model
Assumption Description
negligible extinction (i.e. neither absorption nor scattering losses) of the layer (k (λ) = 0) in the
l
M1
considered spectral region
no layers in addition to the transparent single layer, neither on the single layer (e.g. moisture or
M2
dirt films) nor between the layer and the substrate (e.g. interfacial layers)
negligible roughness of the layer (and of the substrate) relative to the measurement wavelength λ:
typically, R or S ≤ λ/250
a a
M3
negligible maximum roughness depth relative to the layer thickness d: typically,
R ≤ 1 nm + d/1 000
t
homogeneity of the substrate and the layer, especially of the layer thickness d in the field of analy-
M4 sis (FOA), i.e. no lateral (x-y) or vertical (z) local dependence (gradients) of the optical or dielectric
constants
isotropy or known anisotropy of the layer (and of the substrate), i.e. no unknown directional
M5
dependence of the optical or dielectric constants
defined chemical composition, crystallographic microstructure and atomic packing factor of the
M6
layer (and of the substrate) with sufficient stability (relative to the measuring time)
only the layer surface and the layer/substrate interface but not the backside reflections of trans-
M7 parent substrates contribute to the reflected signal, i.e. the use of substrates of sufficient thick-
ness, typically of more than 5 mm
planarity of the sample’s surface in relation to the orientation of the beam path during the meas-
S1 urement, i.e. conformity of the layer’s surface with the ellipsometer reference plane to obtain a
symmetric optical path
consistency between the angle of incidence φ applied during measurement and the one used in
S2
the model
Assumptions M1 to M7 specify the influencing parameters related to the layer and substrate material.
Assumptions S1 and S2 describe measurement-related influencing parameters which are relevant for the
validity and quality of the transparent single layer model.
The transparent reference layers described in Clause A.2 conform with model assumptions M1 to M7 in
very close approximation. When measuring actual materials and surfaces, it is possible that several
assumptions of the transparent single layer model are not met. The fundamental model-specific requirement
is assumption M1. Possible deviations from the assumptions M2 to M7 and S1 to S2 are specified for the
bulk material model in ISO 23131-2 and shall apply accordingly to the transparent single layer model in this
document.
4.2.2 Deviations from model assumption M1
Absorption or scattering losses occurring in the considered spectral region cause non-conformance with
assumption M1. It is nearly impossible to capture extinction coefficients k (λ) ≤ 0,1 by using ellipsometry as
l
a method which measures in reflection.
4.2.3 Deviations from model assumption M2
As a result of the manufacturing process of the transparent single layer, interfacial layers can develop, which
then also need to be taken into account in the model.
4.2.4 Deviations from model assumption M3
Physical substrate surfaces show some type of roughness that cause losses due to scattering as well as
depolarization effects. If model assumption M3 is not met, the roughness can be described as the interfacial
layer between the substrate and the transparent single layer (as well as between the transparent single
[1]
layer and the ambient space in case of roughness of the layer surface) using the effective medium theory .
The maximum roughness depth R , which is a multiple of the arithmetic mean roughness value R , is usually
t a
applied as the thickness of this interfacial roughness layer (see Figure 2).

Figure 2 — Arithmetic mean roughness value Ra and maximum roughness depth R of an interfacial
t
roughness layer using the example of the layer/substrate interface
4.2.5 Deviations from model assumption M4
Materials can show lateral and vertical inhomogeneities. Lateral inhomogeneities of the layer thickness
can be identified by measurements carried out on different measuring points (mapping with a sufficiently
small FOA or high-resolution imaging). These lateral inhomogeneities of the layer thickness are also called
thickness non-uniformities. Vertical gradients of the optical/dielectric constants shall be taken into account
in the model. SiO reference layers on silicon and borosilicate glass show a high degree of lateral homogeneity
(see Clause A.2 and Clause A.3).
4.2.6 Deviations from model assumption M5
The layer and the substrate can be anisotropic, e.g. as a result of the coating process or as a result of
interfacial surface tension between the layer and the substrate. As a consequence, the refractive index is
directionally dependent. Layer materials can also be birefringent and are therefore described similar to bulk
materials using two refractive indices for the ordinary and the extraordinary refraction. Anisotropies of the
layers (and of the substrate) shall be taken into account in the model. For SiO reference layers on silicon and
borosilicate glass, anisotropy and birefringence can be mostly excluded (see Clause A.2 and Clause A.3).
4.2.7 Deviations from model assumption M6
Modifications of the chemical composition, crystallographic microstructure or atomic packing factor are
deviations from the model assumptions. For SiO reference layers on silicon and borosilicate glass, these
parameters are well-defined and have long-term stability (see Clause A.2 and Clause A.3).
4.2.8 Deviations from model assumption M7
For transparent layers on ultra-thin transparent substrates, reflections occurs on the backside of the
substrate, which overlap with the light beam reflected on the layer surface and thus compromise the
ellipsometric transfer quantities Ψ and Δ. Depending on the spectral region that is considered, backside
reflections can be reduced experimentally to a value that can be neglected by roughening, taping, coating
with black paint or by using suitable device-specific fixtures.
4.2.9 Deviations from model assumption S1
The correct alignment of the surface to be measured with respect to the ellipsometer reference plane is only
given after sample adjustment of the focus in the z direction and the alignment of the x-y sample surface to
the ellipsometer reference plane. For corrugated or macroscopically curved surfaces, this can be complicated
or even impossible.
4.2.10 Deviations from model assumption S2
The angle of incidence actually used during measurement can deviate from the specified angle, especially
in conjunction with model assumption S1 regarding planarity. If necessary, an angle fit procedure shall be
carried out in order to determine the actually used angle of incidence, which shall then be applied in the
model. Even with ideally plane-parallel samples, the accuracy of the angle setting, which shall be taken into
account in the measurement uncertainty analysis according to 4.5, has to be known.
4.3 Special characteristics of the transparent single layer model
Due to the interference of the light waves reflected by the ambient/layer interface and the layer/substrate
interface, the spectra of the ellipsometric transfer quantities Ψ and Δ for transparent layers can show
oscillations whose occurrence and magnitude or number depend on the layer thickness d, the refractive
indices of the layer, the substrate and the ambient space as well as on the range of the measurement
wavelength. Corresponding spectra of SiO reference layers of different layer thicknesses d are described in
Clause A.2 (on silicon being the opaque substrate) and Clause A.3 (on borosilicate glass being the transparent
substrate). The amplitude of the oscillation (Figure A.4 and Figure A.7) depends on the refractive index
contrast of the interfaces ambient/layer and the layer/substrate, in particular for Ψ. The number of
oscillations in Ψ and Δ increases with the given contrast of the layer to the ambient space or to the substrate
with increasing layer thickness d. At a fixed wavelength, the maxima/minima are reproduced for multiples
of the so-called thickness period [see Formula (A.1) and Figure A.5 and Figure A.9].
The layer thickness d in the transparent single layer model is a parameter, which is independent of the
wavelength and the angle of incidence. This allows a multi-wavelength and a multi-angle of incidence fit
to the layer thickness, where the root-mean-square deviation (D ) between the ellipsometric transfer
RMS
quantities that have been determined by experiment and those that have been determined by model
simulation is minimised by using Formula (1).
2 2
U  exp. expp. 
mod. mod.
   
 
1 1
i i i i 2
 
D      
(1)
RMS

    
21UuV uU21V
sys, sys,
i1  i   i 
 
 
where
U is the number of measuring points;
V is the number of free parameters (2U ≥ V).
NOTE Formula (1) can be found in ISO 23131. In that document and in some evaluation programmes, u and
sys,Ψ
exp. exp. exp. exp.
u are still designated  and  . In addition, the designation of  and  as the experimental
sys,Δ
   
i i i i
standard deviation can lead to confusion. They actually designate the systematic uncertainties of the ellipsometric
measuring system that are usually determined by means of statistics.
Formula (1) describes the fit quality for the determination of the fit parameters, which can be achieved
based on the specified model. In software products of different software developers, other parameters can
also be used for the description of the fit quality and for the D calculation (and thus also for the fit). For
RMS
example, N, C, S parameters (see ISO 23131) or ρ, the ratio of complex amplitude reflection coefficients of p-
to s-polarized light. Normalization of these measurands [for Ψ and Δ in Formula (1), it is the division by u
sys,Ψ
and u ] can also be achieved by other means, leading to a different weighting of several measuring points.
sys,Δ
For this reason, the D values are only comparable under identical measurement conditions.
RMS
If the optical/dielectric constants of the layer material are known, the layer thickness is the only parameter
that can be determined directly from the fit based on the transparent single layer model. In that case, either
the optical/dielectric constants that have been experimentally determined according to ISO 23131-2 or
database values are used for the substrate or layer material.
If the optical/dielectric constants of the layer are unknown or if the model-based simulated ellipsometric
transfer quantities do not sufficiently match the experimentally determined ellipsometric transfer quantities
(high value of the D minimum), an additional fit to the parameters of the specified dispersion formula for
RMS
the material constants n and ε can be made, and the layer thickness d and the material constants mentioned
can be determined simultaneously.

For thin layers (n · d < 30 nm), the simultaneous determination of the layer thickness d and the refractive
index n can be difficult due to a possible correlation of d and n.
4.4 Validation
For the determination of the layer thickness d according to 4.3, a first validation of the applied optical/
dielectric constants is carried out. This validation and, if necessary, the validation of the layer architecture,
is carried out according to the principles shown in Figure 3. In addition to the D value, plausibility of the
RMS
results shall also be analysed. Especially for measurements at a single wavelength, the initial layer thickness
for the fit shall be in the correct interference order (see Clause A.8), since the ellipsometric transfer
quantities of different interference orders are identical (Figure A.5 and Figure A.9).
A fit for measurements at a single-wavelength is only possible if measurements at different angles are
available. The absolute value of the D minima may only be compared under identical conditions (sample,
RMS
model, measurement), as demonstrated in Table A.1 to Table A.4 for standard or reference layers.
Figure 3 — Validation of the transparent single layer model
Based on the validation according to Figure 3, it is ensured that the obtained D value is the smallest
RMS
possible value under the given conditions and that the model-simulated ellipsometric transfer quantities
match the experimentally determined ellipsometric transfer quantities in the best possible way.
The initial values of the fit parameter for the layer thickness d and the initially used optical/dielectric
constants (for the substrate and the transparent single layer), either determined from measurement or by
using database values, shall be sufficiently close to the layer thickness d that is to be determined by fitting,
or sufficiently close to the actual optical/dielectric constants of the layer/substrate. Otherwise, there is a
risk that the D minimum is a locally defined minimum value and that the fitting result does not represent
RMS
reality. For non-ideal samples it can be necessary to take into account the roughness and the gradients of the
layer thickness or refractive indices. This shall be implemented in the model in order to obtain the best fit
for the layer thickness d.
If the transparent single layer model is strictly met, a pronounced minimum in the D function according
RMS
to Formula (1) is achieved for each fit at different angles of incidence for the same layer thickness.
Figure 4 illustrates the D function for the layer thickness fit of an SiO layer thickness standard with
RMS 2
d = (388,4 ± 2,1) nm as specified. Despite the different D minimum values at different angles of incidence,
RMS
the fits in the spectral region from 192 nm to 1 700 nm lead to nearly identical fitting results for the layer
thickness of 389,63 nm at the angles of incidence of 65 ° and 70 °, and for the layer thickness of 389,68 nm
at an angle of incidence of 75 °. The fitting result for the layer thickness resulting from the fitting procedure
at all three angles simultaneously is 389,65 nm. In addition, the results of spectral fits at different angles of
incidence are compared to the results from single-wavelength measurements in Table A.1 to Table A.4.

Key
D root-mean-square deviation
RMS
d layer thickness, in nm
1 layer thickness fit at the angel of incidence (AOI) of 65 °
2 layer thickness fit at AOI of 70 °
3 layer thickness fit at AOI of 75 °
4 layer thickness fit at AOI of 65 °, 70 ° and 75 °
Figure 4 — Measurement of a layer thickness standard with d = (388,4 ± 2,1) nm as specified — root-
mean-square deviation of the layer thickness fit of SiO on silicon in the wavelength range from
192 nm to 1 700 nm from fits at different angles of incidence
The root-mean-square deviation (D ) is a measure for the quality of the fit, whereas small values of the
RMS
D minimum represent a better correspondence between those ellipsometric quantities that have been
RMS
simulated based on the model (Ψ , Δ ) and those that have been determined experimentally from the
mod mod
measurement (Ψ , Δ ) under otherwise identical boundary conditions (sample, spectral region, angle of
exp exp
incidence, fit parameter, number of measurement points). The minimum in the D function corresponds
RMS
to the best fit for the layer thickness d or the refractive index n or the parameters of a dispersion function
under the given boundary conditions (see examples in Clause A.5).
4.5 Measurement uncertainty
4.5.1 Measurement uncertainty of the ellipsometric transfer quantities Ψ and Δ
The measurement uncertainties of the ellipsometric transfer quantities Ψ and Δ shall be calculated in
accordance with ISO 23131-2:2026, 4.5.1.
For correlated quantities, the combined uncertainties of Ψ or Δ shall be obtained from the systematic and
random uncertainties of Ψ and Δ according to ISO/IEC Guide 98-3 using Formula (2) and Formula (3):
uu u (2)
sysr,,nd
or
uu u (3)
sysr,,nd 
The systematic uncertainty component can be specific to the equipment and method used. It can also be
dependent on the wavelength and angle applied. In practice, the values of the uncertainties u are therefore
sys
provided by the measuring system, while the values of u can be calculated from repeated measurements.
rnd
Under repeatability conditions (after a single sample adjustment without sample repositioning), m
measurements are carried out at a single angle of incidence in the same wavelength range on either
reference, standard or application samples in the field of analysis (often to evaluate the long-term stability
of the measuring system). The results shall be used to calculate the standard uncertainty u according to
rnd
Formula (4) and Formula (5) where s is the experimental standard deviation of Ψ or Δ at a given wavelength
λ. These values can also be provided after the (possibly automated) measurement by the measuring system.
s

u   (4)

rnd,
m
or
s

u   (5)

rnd,
m
Clause A.6 gives the experimental standard deviations for Ψ and Δ for SiO layers on silicon of different layer
thicknesses. Figure A.10 and Figure A.11 illustrate that s and s are independent of the layer thickness and
Ψ Δ
are hence essentially dependent of the measuring system. In the visible spectral range, s is typically below
Ψ
0,1 ° and s is up to 0,5 °.
Δ
Due to the high precision of ellipsometry, the following usually applies: uu and uu
sysr,,ΨΨnd sysr,,∆∆nd
[see Formula (2) and Formula (3)]. This is the reason why repeated measurements (m ≥ 5) are normally
omitted. It is common practice to carry out only a small number of single measurements (after repositioning
and rotating the sample) in order to check, for example, homogeneity and isotropy of the sample.
4.5.2 Measurement uncertainty u of the layer thickness d
d
4.5.2.1 General
In order to determine the layer thickness d and, if necessary, the optical/dielectric constants n or ε of the
single layer, while the optical/dielectric constants of the substrate are known, a parameterized fitting
procedure is applied and used to determine the parameter d from the condition that D approaches
fit RMS
D [see Formula (1)].
RMS,min
This procedure is applied in the example given in 4.4. Here, the optical constants of the material are either
used from a database or determined by fit from a different measurement. In the example, a certified layer
thickness standard with d = (388,4 ± 2,1) nm and c = 2 is used. After the fit procedure on the basis of the
ref
transparent single layer model, the result of the thickness determination is d = 389,65 nm. The software
fit
provides the uncertainty of the layer thickness fit with u = 0,01 nm.
fit
When using an appropriate model (layer architecture, applied optical/dielectric constants of the transparent
single layer and the substrate), the uncertainty u basically indicates the sensitivity of the algorithm for the
fit
determination of the layer thickness fit parameter and is usually negligible (see Clause A.4 and Table A.6).
However, it is also correlated with the experimental conditions. Higher values of u can also indicate a
fit
correlation between the fit parameters.
The model-correlated uncertainty u usually contributes essentially to the uncertainty (see Clause A.7).
mod
It describes, for example, deviations from the transparent single layer model with regard to the layer
architecture (e.g. interfacial roughness layers) as well as the actual optical/dielectric constants of the
transparent single layer and the substrate as a function of the specified fit parameters.
4.5.2.2 Determination of u , Option 1 using the D function and traceable layer thickness
d RMS
standards or reference layers
From the metrological perspective, the availability of the D function is bound to spectral measurements
RMS
and the capability of the applied software program to calculate D . The D function normalized to
RMS RMS
the minimum and related to the variation of the layer thickness can be used to estimate the modelling
uncertainty u (see Figure 5).
mod
Key
D /D root-mean-square deviation normalized to the minimum value
RMS RMS, min
d layer thickness, in nm
Figure 5 — Measurement of a layer thickness standard with d = (388,4 ± 2,1) nm as specified
– Normalized root-mean-square deviation of the layer thickness fit of SiO on silicon in the
wavelength range from 192 nm to 1 700 nm at an AOI of 65 °, 70 ° and 75 °
A layer thickness range from (d − a) to (d + a) is determined from the normalized D curve, for which
fit fit RMS
D
RMS
12≤≤ applies. Within this range, half width a of the rectangular distribution is assumed for the
D
RMSm, in
layer thickness values. The modelling uncertainty in the transparent single layer model u results from
mod
the assumption of a rectangular distribution according to Formula (6) as in ISO/IEC Guide 98-3.
a
u = (6)
mod
In Figure 5, the half width determined from the curve of the normalized D function is a = 0,79 nm.
RMS
Consequently, the modelling uncertainty is u = 0,46 nm with uu . The uncertainty contributions
mod fitmod
u and u are correlated quantities and shall be added up. For the ideal case of the sample being a layer
fit mod
thickness standard with a nearly perfect layer architecture and very well-known optical/dielectric
constants, this results in u + u = 0,47 nm.
fit mod
Traceable layer thickness standards (based on the transparent single layer model) are often used for
instrument verification, without instrument calibration in the classic sense. The layer thickness of the used
layer thickness standard should therefore be as close as possible to the layer thickness to be measured.
The uncertainty of the layer thickness fit u for the verification measurement using the layer thickness
fit,ref
standard is also provided by the evaluation programme. Based on the expanded combined uncertainty
(measurement uncertainty) U of the layer thickness standard as specified, the combined uncertainty of
ref
the layer thickness of the standard shall first be determined according to Formula (7).
U
ref
u = (7)
ref
c
ref
c is the coverage factor as specified, which is typically 2 (95 % confidence level). Consequently, there are
ref
two contributions u and u by the layer thickness standard/reference layer, which shall be taken into
ref fit,ref
account for the expanded uncertainty of the layer thickness of the sample.
The correlated quantities u and u of the sample and the non-correlated quantities u and u of
fit mod ref fit,ref
the layer thickness standard shall be used in Formula (8) according to ISO/IEC Guide 98-3 to obtain the
combined uncertainty of the layer thickness.
2 22
uuuuu (8)

d fitmod fitr, ef ref
In the example shown in Formula (8), using a layer thickness standard as a sample, u = 0,46 nm,
mod
u = u = 0,01 nm and u = 1,05 nm. The measurement result shall be calculated with Formula (9) using
fit,ref fit ref
the coverage factor c = 2 as follows:
samp
ddcu (9)
fitsamp d
or numerically expressed d = (389,65 ± 2,3) nm.
The numerical value of the expanded combined uncertainty (measurement uncertainty) according to
Formula (9) has been derived assuming that the optical/dielectric material constants of the ideal sample
used (layer thickness standard) are very well-known and therefore only layer thickness fitting is required.
In addition to the layer thickness, fitting is usually also required for the parameters of the dispersion
function of the optical/dielectric constants in case of real samples, which raises the number of fit parameters
and can result in a significant broadening of the D function in comparison to Figure 5. In the result of
RMS
a multi-parameter fit, the uncertainty of modelling u and hence the uncertainty of the layer thickness
mod
u significantly increases as the parameter a significantly increases. This applies to the layer thickness
d
standard that is used as the sample in Table A.6 and Figure A.10.
4.5.2.3 Determination of u , Option 2 using an estimation correlated with the layer thickness
d
As an alternative to option 1, especially in the case where no D function or layer thickness standards or
RMS
reference layers are available, an estimation correlated with the layer thickness can be used to determine
the uncertainty of the layer thickness u according to Formula (10).
d
uu05,( 0,)005d (10)
dmin fit
In Formula (10), u is the minimum uncertainty when using this method. It shall be defined with a
min
constant value of 0,5 nm. If no fitting procedure has been possible at all (e.g. for measurements at a certain
wavelength), the layer thickness calculated by the evaluation programme is used instead of d .
fit
This estimation takes into account that the accuracy of the optical/dielectric constants is usually limited
to the second decimal place. A residual uncertainty, i.e. the contribution of 0,5 nm in Formula (10), due to
the interface roughness and other non-ideal characteristics, shall always be taken into account. This is
particularly relevant for ultra-thin layers below 10 nm. Using the estimation according to Formula (10),
the uncertainty of the layer thickness in the given example is u = 1,23 nm. Similar to Formula (9), the
d
measurement result shall be calculated from Formula (11) using c = 2 as follows:
samp
ddcu (11)
fitsamp d
or numerically expressed d = (389,65 ± 2,46) nm.
For all layers (ideal and non-ideal layers) that correspond to the transparent single layer model, a first
fit is generally carried out on the layer thickness and subsequently a second fit is carried out on the layer
thickness and the optical/dielectric constants.
The fitting result of the second fit for the layer thickness and for the newly determined optical/dielectric
constants shall be used in case of the following scenarios:
...