ISO 23131-2:2026

International
Standard
ISO 23131-2
First edition
Ellipsometry —
2026-01
Part 2:
Bulk material model
Ellipsométrie —
Partie 2: Modèle matériel volumique
Reference number
© ISO 2026
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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 Bulk material model . 3
4.1 Optical path .3
4.2 Assumptions .4
4.2.1 General .4
4.2.2 Deviations from model assumption M1 .5
4.2.3 Deviations from model assumption M2 .5
4.2.4 Deviations from model assumption M3 .5
4.2.5 Deviations from model assumption M4 .5
4.2.6 Deviations from model assumption M5 .6
4.2.7 Deviations from model assumption M6 .6
4.2.8 Deviations from model assumption S1 .6
4.2.9 Deviations from model assumption S2 .6
4.3 Special characteristics of the bulk material 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 of the optical (n, k) and dielectric (ε , ε ) constants .9
1 2
5 Test report .10
Annex A (informative) Additions to the bulk material model .12
Bibliography .18

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.
For the bulk material model, a fitting procedure is optional since exactly two independent parameters can
be determined per measurement (per wavelength and at one angle of incidence) using the formula system
consisting of formulae for p- and s-polarization. This, moreover, is the only case where a determination of
target figures (optical or dielectric constants) can be carried out analytically.

v
International Standard ISO 23131-2:2026(en)
Ellipsometry —
Part 2:
Bulk material model
1 Scope
This document specifies the process for determining the optical or dielectric constants by means of
ellipsometric measurements and their analysis based on the bulk material model.
If the assumptions of the bulk material model are strictly met, it is possible to determine the optical constants
(refractive index n and extinction coefficient k) or the dielectric constants (real part ε and imaginary part ε )
1 2
of the material directly. Alternatively, optical ( and ) or dielectric (<ε > and <ε >) pseudo constants
1 2
are determined, which depend on the measurement angle of incidence φ. The degree of consistency of the
pseudo constants in the relevant spectral range, determined from measurements at different angles of
incidence, represents a necessary prerequisite for the validity or quality of the bulk material model.
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/IEC Guide 98-3:2008, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in
me a s ur ement (GUM: 1995)
3 Terms, definitions and symbols
3.1 Terms and definitions
No terms and definitions are listed in this document.
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/
3.2 Symbols and abbreviated terms
For the purpose of this document, the symbols and abbreviated terms given in ISO 23131 and the following
apply:
Table 1 — Symbols
Symbol Description
R arithmetic roughness average (profile roughness)
a
S arithmetic roughness average (surface roughness)
a
N complex refractive index of the ambient space
a
N complex refractive index of the substrate
s
n refractive index of the ambient space (real part of the complex refractive index N )
a a
n ordinary refractive index
o
n extra-ordinary refractive index
e
n refractive index of the substrate (real part of the complex refractive index N )
s s
extinction coefficient of the ambient space (imaginary part of the complex refractive
k
a
index N )
a
extinction coefficient of the substrate (imaginary part of the complex refractive index
k
s
N )
s
k ordinary extinction coefficient
o
k extra-ordinary extinction coefficient
e
d penetration depth
p
φ angle of incidence (AOI) between the incident light wave and the normal to the surface
Brewster angle; angle of incidence at which the p-polarization for dielectric materials
φ
B
will disappear in the reflected beam (material property)
ρ ratio of complex amplitude reflection coefficients of p- to s-polarized light
uncertainty of Ψ in the applied measuring system (being the systematic component of
u
sys,Ψ
the measurement uncertainty)
standard uncertainty of Ψ from m repeated measurements (being the random compo‑
u
rnd,Ψ
nent of the measurement uncertainty)
uncertainty of Δ in the applied measuring system (being the systematic component of
u
sys,∆
the measurement uncertainty)
standard uncertainty of Δ from m repeated measurements (being the random compo‑
u
rnd,∆
nent of the measurement uncertainty)
u combined uncertainty of Ψ
Ψ
u combined uncertainty of Δ
∆
s experimental standard deviation of Ψ
Ψ
s experimental standard deviation of Δ
∆
R intensity reflection factor/intensity reflectance for s-polarized light
s
R intensity reflection factor/intensity reflectance for p-polarized light
p
m number of repeated measurements/number of mean values used
u combined uncertainty of n
n
u combined uncertainty of k
k
u
combined uncertainty of ε
ε
u
combined uncertainty of ε
ε
n
arithmetic mean value of n
arithmetic mean value of k
k
arithmetic mean value of ε
ε
arithmetic mean value of ε
ε
u
uncertainty contribution of Ψ to the uncertainty of ε

1,
a
In ISO/IEC Guide 98-3, the coverage factor is designated by “k”.

TTaabbllee 11 ((ccoonnttiinnueuedd))
Symbol Description
u
uncertainty contribution of Δ to the uncertainty of ε

1,
u
uncertainty contribution of Ψ to the uncertainty of ε

2,
u
uncertainty contribution of Δ to the uncertainty of ε

2,
u
uncertainty contribution of Ψ to the uncertainty of n
n
Ψ
u
uncertainty contribution of Δ to the uncertainty of n
n
∆
u
uncertainty contribution of Ψ to the uncertainty of k
k
Ψ
u
uncertainty contribution of Δ to the uncertainty of k
k
∆
a
coverage factor of sample (index: samp) for expression of measurement uncertainty
c
samp
(expanded combined uncertainty)
a
In ISO/IEC Guide 98-3, the coverage factor is designated by “k”.
4 Bulk material model
4.1 Optical path
Figure 1 shows the optical path at a certain angle of incidence φ in the bulk material model. Two half spaces
are assumed. One half-space is represented by the ambient space (index “a”) that is assigned with the
complex refractive index N . Usually, air serves as the ambient space and the actual refractive index is in
a
close approximation with n = 1,000 and k = 0,000. The other half-space is represented by the substrate,
a a
which is described by the bulk material model and assigned with the complex refractive index N . Clause A.1
s
describes the optical constants for fused silica, which closely meets the assumptions of the bulk material
model.
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
ε complex dielectric function of the ambient space (for air real, ε = ε = 1,000)
a a 1a
N complex refractive index of the substrate (N = n (λ) + i · k (λ))
s s s s
ε complex dielectric function of the substrate (ε = ε (ω) + i · ε (ω))
s s 1s 2s
Figure 1 — Optical path in the bulk material model
As a result, it is assumed in the bulk material model that apart from the angle of incidence φ and the
measurement wavelength λ, only the optical or dielectric constants of the ambient/substrate interface
determine the measured ellipsometric transfer quantities Ψ and Δ.
4.2 Assumptions
4.2.1 General
The model assumptions given in Table 2 are made for the bulk material model.
Table 2 — Assumptions for the bulk material model
Assumption Description
no layer/film is present on the substrate's surface (no native oxide layers, no films of moisture
M1
and dirt)
negligible roughness of the substrate relative to the measurement wavelength λ (as a guide
M2
value, R or S should be at least ≤ λ/250)
a a
homogeneity of the substrate in the field of analysis, i.e. no lateral (x‑y) or vertical (z) local de‑
M3
pendence (gradients) of the optical or dielectric constants
isotropy or known anisotropy of the substrate, i.e. no unknown direction dependence (e.g. re‑
M4 sulting from the crystalline structure or due to stress birefringence) of the optical or dielectric
constants
chemical purity (one element or defined stoichiometry of a compound) and monophasic mi‑
M5 crostructure (no mixed phase as for example in the case of alloys) of the substrate with a high
long-term stability
TTaabbllee 22 ((ccoonnttiinnueuedd))
Assumption Description
negligible backside reflections for transparent substrates, i.e. the use of substrates of sufficient
M6
thickness, typically more than 5 mm
planarity of the substrate relative to the geometry of the measurement and the plane of inci‑
S1 dence, i.e. conformity of the substrate’s surface with the ellipsometer reference plane to obtain
a symmetrical optical path
consistency between the angle of incidence φ applied during measurement and the one used in
S2
the model
Assumptions M1 to M6 describe the influencing parameters related to the substrate material. Assumptions
S1 and S2 describe measurement-related influencing parameters, which are relevant for the validity and
quality of the bulk material model. Fused silica, the optical constants of which are described in Clause A.1,
conforms with model assumptions M1 to M6 in close approximation. For other materials, such as metals or
semiconductors, native oxide layers shall be taken into account. For silicon with a native SiO layer, which is
often used as a reference material just like fused silica, the corresponding information is given in Clause A.2.
Furthermore, the optical and dielectric constants often depend on the manufacturing conditions and the
ageing state of the materials, such that literature and database values of one and the same material can
differ significantly. Therefore, it is recommended to validate the optical/dielectric constants of the materials
used and to approve the validity or quality of the bulk material model in advance by means of multi-angle
measurement. If the sample does not show the expected bulk behaviour in the multi-angle measurement,
the bulk model cannot be used. However, there are cases where this test is not sufficient to prove the validity
of the bulk model, as shown in the example of the native oxide layer on silicon (see Figure A.3).
When measuring actual materials and surfaces, it is possible that several assumptions of the bulk material
model are not met. The occurring model deviations are described in 4.2.2 to 4.2.9.
4.2.2 Deviations from model assumption M1
Native oxide layers, with a thickness of up to a few nanometres, are practically unavoidable, especially
in the case of metals (e.g. Al) and semiconductors (e.g. Si) (see Clause A.2). These surface layers indicate
a non-fulfilment of the model assumption because the optical constants of an oxide (SiO or Al O ) differ
2 2 3
significantly from those of the semiconductor or the metal.
4.2.3 Deviations from model assumption M2
Physical surfaces show some type of roughness that cause losses due to scattering and depolarization effects.
For arithmetic roughness average values of highly polished glass and silicon wafers of less than 1 nm, the
influence of roughness on the measurement in the visible spectral range can be neglected.
4.2.4 Deviations from model assumption M3
Materials can show lateral and vertical inhomogeneities. Lateral inhomogeneities can be generally excluded
when fused silica is used as the reference material (see Clause A.1) whereas vertical inhomogeneities such
as surface layers or dispersion layers from polishing processes shall be taken into account for this type of
reference material, if necessary.
4.2.5 Deviations from model assumption M4
Materials can be anisotropic. As a consequence, the refractive index is directionally dependent. Birefringent
materials are described with two refractive indices, n and n , which is called ordinary or extra-ordinary
o e
refraction and result in two (slightly) differently refracted light beams. Similarly, the directional dependence
of the extinction is called dichroism and is described by the extinction coefficients k and k . Mechanical
o e
stress states can also cause birefringence (stress birefringence). For the reference materials, fused silica
and silicon (see Clause A.1 and Clause A.2), material-caused anisotropy, birefringence and dichroism can be
excluded.
4.2.6 Deviations from model assumption M5
The reference materials conform in good approximation with the requirement for chemical purity and
show a nearly monophasic microstructure (see Clause A.1 and Clause A.2). External influences, especially
due to temperature variations, can cause changes both in stoichiometry and in the microstructure (phase
transition).
4.2.7 Deviations from model assumption M6
For extremely thin transparent substrates or in the presence of internal/buried interfaces (e.g. surface or
dispersion layers), reflections occur at the backside of the substrate or at the internal interfaces. These
reflections superimpose with the light beam that is reflected at the upper surface and can consequently
compromise the ellipsometric transfer quantities Ψ and Δ. Depending on the spectral range 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 appliances.
4.2.8 Deviations from model assumption S1
The correct orientation of the surface to be measured with respect to the ellipsometer reference plane is
only provided after alignment of the sample surface (sample orientation along x/y tilt and z height relative
to the pl
...