ISO 23131-2

ISO/FDIS 23131-2:2025(en)
ISO /TC 107/N3038
Date: 2025-06-19
Secretariat: KATS
Date:
Ellipsometry —
Part 2:
Bulk material model
Ellipsométrie —
Partie 2: Modèle matériel volumique
FDIS stage
ISO/DIS FDIS 23131-2:2025(en)
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All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication
may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying,
or posting on the internet or an intranet, without prior written permission. Permission can be requested from either ISO
at the address below or ISO’s member body in the country of the requester.
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Phone: + 41 22 749 01 11
E-mail: copyright@iso.org
Website: www.iso.org
Published in Switzerland
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ii © ISO 2024 2025 – All rights reserved
ii
ISO/FDIS 23131-2:2025(en)
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Contents
Foreword . v
Introduction . vi
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 . 5
4.3 Special characteristics of the bulk material model . 7
4.4 Validation . 8
4.5 Measurement uncertainty . 9
5 Test report . 11
Annex A (informative) Additions to the bulk material model . 13
Bibliography . 24

Foreword . iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms, definitions, symbols and abbreviations . 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 . 4
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 . 5
4.2.7 Deviations from model assumption M6 . 5
4.2.8 Deviations from model assumption S1 . 5
4.2.9 Deviations from model assumption S2 . 5
4.3 Special characteristics of the bulk material model . 5
4.4 Validation . 6
4.5 Measurement uncertainty . 8
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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 (ε1, ε2) constants . 8
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5 Test report . 10 Formatted: Font: 10 pt
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Annex A (informative) Additions to the bulk material model . 11
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A.1 Fused silica used as dielectric volume reference material . 11
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A.2 Silicon used as semiconducting volume reference material . 11
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iii
ISO/DIS FDIS 23131-2:2025(en)
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A.3 Determination of measurement uncertainties . 13
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A.4 Recommendations for measuring practice . 16
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Bibliography . 17

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ISO/FDIS 23131-2:2025(en)
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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
ISO technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely 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)
patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed patent rights
in respect thereof. As of the date of publication of this document, ISO had not received notice of (a) patent(s)
which may be required to implement this document. However, implementers are cautioned that this may not
represent the latest information, which may be obtained from the patent database available at
www.iso.org/patents. ISO shall not be held responsible for identifying any or all such patent rights.
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and expressions
related to conformity assessment, as well as information about ISO's adherence to the World Trade
Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www.iso.org/iso/foreword.html.
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.
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v
ISO/DIS FDIS 23131-2:2025(en)
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Introduction
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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.
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FINAL DRAFT International Standard ISO/FDIS 23131-2:2025(en)

Ellipsometry— —
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Part 2:
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Bulk material model
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1 Scope
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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 are
1 2
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.
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2 Normative references
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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,
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the latest edition of the referenced document (including any amendments) applies.
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stops: Not at 0.7 cm + 1.4 cm + 2.1 cm + 2.8 cm +
ISO 23131, Ellipsometry — Principles
3.5 cm + 4.2 cm + 4.9 cm + 5.6 cm + 6.3 cm + 7 cm
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ISO/IEC Guide 98--3:2008-09, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in
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measurement (GUM:1995)
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3 Terms, definitions and symbols
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3.1 Terms and definitions
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No terms and definitions are listed in this document.
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ISO and IEC maintain terminology databases for use in standardization at the following addresses:
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stops: Not at 0.71 cm
— IEC Electropedia: available at https://www.electropedia.org/
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3.2 Symbols and abbreviated terms
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For the purpose of this document, the symbols and abbreviated terms given in ISO 23131 and the following
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apply:
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ISO/DIS FDIS 23131-2:2025(en)
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Table 1 — Symbols
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Symbol Description
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R arithmetic roughness average (profile roughness)
a
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Sa arithmetic roughness average (surface roughness)
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Na complex refractive index of the ambient space
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Ns complex refractive index of the substrate
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n refractive index of the ambient space (real part of the complex refractive index N )
a a
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no ordinary refractive index Formatted
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ne extra-ordinary refractive index Formatted
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ns refractive index of the substrate (real part of the complex refractive index Ns) Formatted
...
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extinction coefficient of the ambient space (imaginary part of the complex refractive
...
ka
index Na)
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extinction coefficient of the substrate (imaginary part of the complex refractive index
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k .
s
Ns)
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ko ordinary extinction coefficient
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ke extra-ordinary extinction coefficient
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d penetration depth
p
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φ angle of incidence (AOI) between the incident light wave and the normal to the surface
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Brewster angle; angle of incidence at which the p-polarization for dielectric materials
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φB
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will disappear in the reflected beam (material property)
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ρ ratio of complex amplitude reflection coefficients of p- to s-polarized light
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uncertainty of Ψ in the applied measuring system (being the systematic component of
u 𝑢
sys, 𝛹
sys,
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the measurement uncertainty) .
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standard uncertainty of Ψ from m repeated measurements (being the random
u
𝑢
rnd, rnd, 𝛹
component of the measurement uncertainty)
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uncertainty of Δ in the applied measuring system (being the systematic component of
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u 𝑢 .
sys, sys, 𝛥
the measurement uncertainty)
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standard uncertainty of Δ from m repeated measurements (being the random
u 𝑢
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rnd, rnd, 𝛥
...
component of the measurement uncertainty)
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u 𝑢 combined uncertainty of Ψ
 𝛹
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u 𝑢 combined uncertainty of Δ Formatted
 𝛥 .
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s
𝑠 experimental standard deviation of Ψ
𝛹

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s
𝑠 experimental standard deviation of Δ
𝛥
 Formatted
...
Rs intensity reflection factor/intensity reflectance for s-polarized light Formatted
...
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R intensity reflection factor/intensity reflectance for p-polarized light
p .
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m number of repeated measurements/number of mean values used .
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u 𝑢 combined uncertainty of n
n 𝑛
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u 𝑢 combined uncertainty of k
k 𝑘
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2 © ISO 2024 2025 – All rights reserved
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ISO/FDIS 23131-2:2025(en)
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Symbol Description
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u 𝑢
 𝜀 combined uncertainty of ε1
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u 𝑢 Formatted
combined uncertainty of ε
 𝜀 2 .
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n 𝑛¯ arithmetic mean value of n
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¯
k 𝑘 arithmetic mean value of k
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 arithmetic mean value of ε
𝜀¯ 1
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 𝜀¯ arithmetic mean value of ε
2 .
2 2
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u 𝑢
 𝜀 uncertainty contribution of Ψ to the uncertainty of ε1
1,𝛹
1,
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u 𝑢
𝜀
 1,𝛥 uncertainty contribution of Δ to the uncertainty of ε1
1, Formatted
...
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u 𝑢
𝜀
 2,𝛹 uncertainty contribution of Ψ to the uncertainty of ε2
2,
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u
𝑢
 𝜀 uncertainty contribution of Δ to the uncertainty of ε
2,𝛥 2
2,
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u 𝑢 Formatted
uncertainty contribution of Ψ to the uncertainty of n .
n 𝑛
𝛹

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u 𝑢
𝑛 uncertainty contribution of Δ to the uncertainty of n
n 𝛥

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u
𝑢
k 𝑘 uncertainty contribution of Ψ to the uncertainty of k
𝛹 Formatted
 .
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u 𝑢
k 𝑘 uncertainty contribution of Δ to the uncertainty of k
𝛥

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a
coverage factor of sample (index: samp) for expression of measurement uncertainty
c
samp
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(expanded combined uncertainty)
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a In ISO/IEC Guide 98--3, the coverage factor is designated by “k”.
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4 Bulk material model Formatted
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4.1 Optical path
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Figure 1Figure 1 shows the optical path at a certain angle of incidence φ in the bulk material model. Two half
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spaces are assumed. One half-space is represented by the ambient space (index “a”) that is assigned with the
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complex refractive index N . Usually, air serves as the ambient space and the actual refractive index is in close
a
approximation with n = 1,000 and k = 0,000. The other half-space is represented by the substrate, which is Formatted
a a .
described by the bulk material model and assigned with the complex refractive index Ns. Clause A.1Clause A.1
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describes the optical constants for fused silica, which closely meets the assumptions of the bulk material
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model.
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© ISO 2025 – All rights reserved
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ISO/DIS FDIS 23131-2:2025(en)
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Key
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φ angle of incidence between the incident light wave and the normal to the surface
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Na complex refractive index of the ambient space (for air real, Na = na = 1,000) stops: Not at 0.7 cm + 1.4 cm + 2.1 cm + 2.8 cm +
3.5 cm + 4.2 cm + 4.9 cm + 5.6 cm + 6.3 cm + 7 cm
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
εs complex dielectric function of the substrate (εs = ε1s (ω) + i · ε2s (ω))
φ angle of incidence between the incident light wave and the normal to the surface
Na complex refractive index of the ambient space (for air real, Na = na = 1,000)
εa complex dielectric function of the ambient space (for air real, εa = ε1a = 1,000)
N complex refractive index of the substrate (N = n (λ) + i · k (λ))
s s s s
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ε complex dielectric function of the substrate (ε = ε (ω) + i · ε (ω))
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numbers
Figure 1 — Optical path in the bulk material model
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As a result, it is assumed in the bulk material model that apart from the angle of incidence φ and the
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measurement wavelength λ, only the optical or dielectric constants of the ambient/substrate interface
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determine the measured ellipsometric transfer quantities Ψ and Δ.
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ISO/FDIS 23131-2:2025(en)
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4.2 Assumptions
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4.2.1 General
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The following model assumptions given in Table 2Table 2 are made for the bulk material model.
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Table 2 — Assumptions for the bulk material model
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Assumption Description
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no layer/film is present on the substrate's surface (no native oxide layers, no films of moisture
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M1
and dirt)
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negligible roughness of the substrate relative to the measurement wavelength λ (as a guide
M2
value, Ra or Sa should be at least ≤ λ/250)
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homogeneity of the substrate in the field of analysis, i.e. no lateral (x-‑y) or vertical (z) local
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M3
dependence (gradients) of the optical or dielectric constants
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isotropy or known anisotropy of the substrate, i.e. no unknown direction dependence (e.g.
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M4 resulting from the crystalline structure or due to stress birefringence) of the optical or
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dielectric constants
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chemical purity (one element or defined stoichiometry of a compound) and monophasic
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negligible backside reflections for transparent substrates, i.e. the use of substrates of sufficient
M6
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thickness, typically more than 5 mm
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planarity of the substrate relative to the geometry of the measurement and the plane of
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S1 incidence, i.e. conformity of the substrate’s surface with the ellipsometer reference plane to
obtain a symmetrical optical path
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consistency between the angle of incidence φ applied during measurement and the one used in
S2
the model
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Assumptions M1 to M6 describe the influencing parameters related to the substrate material. Assumptions S1
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and S2 describe measurement-related influencing parameters, which are relevant for the validity and quality
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of the bulk material model. Fused silica, the optical constants of which are described in Clause A.1,Clause A.1,
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conforms with model assumptions M1 to M6 in close approximation. For other materials, such as metals or
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semiconductors, native oxide layers shall be taken into account. For silicon with a native SiO layer, which is
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often used as a reference material just like fused silica, the corresponding information is given in
Clause A.2.Clause A.2. Furthermore, the optical and dielectric constants often depend on the manufacturing Formatted
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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 Formatted
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prove the validity of the bulk model, as shown in the example of the native oxide layer on silicon (see
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Figure A.3).Figure A.3).
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When measuring actual materials and surfaces, it is possible that several assumptions of the bulk material
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model are not met. The occurring model deviations are described in 4.2.2 to 4.2.9.4.2.2 to 4.2.9.
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Native oxide layers —, with a thickness of up to a few nanometres —, are practically unavoidable, especially
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in the case of metals (e.g. Al) and semiconductors (e.g. Si) (see Clause A.2).Clause A.2). These surface layers
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ISO/DIS FDIS 23131-2:2025(en)
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indicate a non-fulfilment of the model assumption because the optical constants of an oxide (SiO or Al O )
2 2 3
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differ significantly from those of the semiconductor or the metal.
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4.2.3 Deviations from model assumption M2
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Physical surfaces show some type of roughness that cause losses due to scattering and depolarization effects.
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For arithmetic roughness average values of highly polished glass and silicon wafers of less than 1 nm, the
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influence of roughness on the measurement in the visible spectral range can be neglected.
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4.2.4 Deviations from model assumption M3
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Materials can show lateral and vertical inhomogeneities. Lateral inhomogeneities can be generally excluded
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when fused silica is used as the reference material (see Clause A.1)Clause A.1) whereas vertical
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inhomogeneities such as surface layers or dispersion layers from polishing processes shall be taken into
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account for this type of reference material, if necessary.
4.2.5 Deviations from model assumption M4
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Materials can be anisotropic. As a consequence, the refractive index is directionally dependent. Birefringent
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materials are described with two refractive indices, no and ne, which is called ordinary or extra-‑ordinary
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refraction and result in two (slightly) differently refracted light beams. Similarly, the directional dependence
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of the extinction is called dichroism and is described by the extinction coefficients k and k . Mechanical stress
o e
states can also cause birefringence (stress birefringence). For the reference materials, fused silica and silicon
(see Clause A.1Clause A.1 and Clause A.2),Clause A.2), material-caused anisotropy, birefringence and
dichroism can be excluded.
4.2.6 Deviations from model assumption M5
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The reference materials conform in good approximation with the requirement for chemical purity and show
stops: Not at 0.71 cm + 0.99 cm + 1.27 cm
a nearly monophasic microstructure (see Clause A.1 and Clause A.2).Clause A.1 and Clause A.2). External
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influences, especially due to temperature variations, can cause changes both in stoichiometry and in the
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microstructure (phase transition).
4.2.7 Deviations from model assumption M6
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For extremely thin transparent substrates or in the presence of internal/buried interfaces (e.g. surface or
stops: Not at 0.71 cm + 0.99 cm + 1.27 cm
dispersion layers), reflections occur at the backside of the substrate or at the internal interfaces. These
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reflections superimpose with the light beam that is reflected at the upper surface and can consequently
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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,
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coating with black paint or by using suitable device-specific appliances. Adjust space between Asian text and numbers, Tab
stops: Not at 0.71 cm + 0.99 cm + 1.27 cm
4.2.8 Deviations from model assumption S1
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The correct orientation of the surface to be measured with respect to the ellipsometer reference plane is only
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provided after alignment of the sample surface (sample orientation along x/y tilt and z height relative to the
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plane and z position of incidence). For corrugated or macroscopically curved surfaces, this can be complicated
stops: Not at 0.71 cm + 0.99 cm + 1.27 cm
or even impossible.
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4.2.9 Deviations from model assumption S2
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The angle of incidence that is actually used during measurement can deviate from the specified angle,
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especially in conjunction with model assumption S1 regarding planarity. If applicable, an angle fit procedure
shall be carried out in order to determine the actually used angle of incidence, which shall then be applied in
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the model. Even with ideally plane-parallel samples, the accuracy of the angle setting, which shall be taken
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into account in the measurement uncertainty analysis according to 4.5,4.5, has to be known.
spacing: single
6 © ISO 2024 2025 – All rights reserved
ISO/FDIS 23131-2:2025(en)
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4.3 Special characteristics of the bulk material model
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The bulk material model is the only model that can be analytically described by Formula (1)Error! Reference
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source not found. and that does not require a fitting procedure. This is because in ellipsometry, a formula
stops: Not at 0.71 cm
system with two independent formulae for p- and s-polarization is used, where two independent parameters
(n and k or ε1 and ε2, respectively) can be determined with one measurement (at one wavelength and one Formatted: Adjust space between Latin and Asian text,
angle of incidence). However, since the optical and dielectric material constants n and k or ε and ε are Adjust space between Asian text and numbers
1 2
wavelength-dependent (show dispersion), these are the two parameters that shall be determined by
Formula (1),Error! Reference source not found., possibly wavelength-dependent, if applicable.

1−
2 2 2
=+n sin 1 tan 
(1)
a 

1+


1−𝜌
2 2 2 2
〈𝜀〉=𝑛 sin 𝜑(1+tan 𝜑( ) ) (1)
a
1+𝜌
where ρ describes the ratio of complex amplitude reflection coefficient of p- and s-polarised light that shall be
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calculated according to the ellipsometric basic Formula (2)Error! Reference source not found. using the
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ellipsometric transfer quantities Ψ and Δ.
−i −𝑖𝛥
= tan e 𝜌=|tan𝛹|𝑒 (2) Formatted: label
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This results in the relationship between the measured ellipsometric transfer quantities Ψ and Δ and the optical
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or dielectric constants that shall be calculated according to Formula (3) and Formula (4).Error! Reference
stops: Not at 0.7 cm + 1.4 cm + 2.1 cm + 2.8 cm +
source not found. and Error! Reference source not found. 3.5 cm + 4.2 cm + 4.9 cm + 5.6 cm + 6.3 cm + 7 cm
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2 2 2 2

tan  cos 2 − sin 2 sin  Adjust space between Asian text and numbers
( )

= n − k =(n sin ) 1+
(3)
1s s s a

1+ sin2cos
( )

(n sin tan) sin4 sin
a
==2nk (4)
2s s s
1+ sin2cos
( )
2 2 2 2
tan 𝜑(cos 2𝛹−sin 2𝛹sin 𝛥)
2 2 2
𝜀 =𝑛 −𝑘 =(𝑛 sin𝜑) (1+ ) (3)
1s s s a
(1+sin2𝛹cos𝛥)
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(𝑛 sin𝜑tan𝜑) sin4𝛹sin𝛥
a
𝜀 =2𝑛 𝑘 = (4)
2s s s
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(1+sin2𝛹cos𝛥)
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The real part n of the complex refractive index N describes the ordinary refraction at the
s s
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ambient/substrate interface according to the law of refraction. The imaginary part k describes the extinction,
s
stops: Not at 0.7 cm + 1.4 cm + 2.1 cm + 2.8 cm +
where a distinction between losses due to absorption and losses due to scattering cannot be made for the
3.5 cm + 4.2 cm + 4.9 cm + 5.6 cm + 6.3 cm + 7 cm
purposes of ellipsometry. For reference materials, losses due to scattering at the ambient/substrate interface
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should be neglectable. If it is consequently assumed that extinction is primarily based on absorption losses,
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then the extinction coefficient k that has been determined by ellipsometry shall also be used for the estimation
s
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of the wavelength-‑dependent penetration depth dp in Formula (5).Error! Reference source not found.
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𝜆

d  = 𝑑 (𝜆)= (5)
( ) p Formatted: Font: 10 pt
p
4𝜋𝑘
s
4k
s
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For a particular spectral range, Formula (5)0 shall also be used for the optical classification of different
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material groups such as dielectric materials (e.g. fused silica, see Clause A.1),Clause A.1), semiconductors (e.g.
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silicon, see Clause A.2)Clause A.2) and metals (e.g. gold).
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© ISO 2025 – All rights reserved
ISO/DIS FDIS 23131-2:2025(en)
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4.4 Validation
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If the bulk material model is strictly met, then the optical or dielectric constants (pseudo constants) that have
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been calculated based on that bulk material model shall be consistent in the measurements at different angles
of incidence. The degree of non-consistency is therefore a measure for the validity or quality of the bulk
material model and can be an indication for the accuracy of the adjustment of the angle of incidence. One
special angle of incidence is the Brewster angle φB, which is closely related to the refractive index ns for ideal
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...
dielectric materials (k = 0 in the considered spectral range) according to Formula (6):Error! Reference
source not found.: Formatted
...
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...
tan = n
tan𝜑 =𝑛 (6)
B s
Bs
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...
Formatted
For dielectric materials, Rp = 0 at the Brewster angle, i.e. there is no reflected p component. For other materials, .
the difference between R and R is at its maximum at the Brewster angle.
p s
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...
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...
Validation of the bulk material model should be performed at three substantially different angles of incidence
with an angular pitch of at least 5 °, if technically possible, where the Brewster angle should not be clearly Formatted
...
outside the range of the measurement angle used.
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...
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...
As anFor example, using fused silica as the reference material (see Table 3)Table 3) with a refractive index of
n = 1,457 at a wavelength of 633 nm in the bulk material model results in a Brewster angle φ of 55,54 °
s B Formatted
...
according to Formula (6).0. Validation of the bulk material model can be carried out in this particular case, for
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...
example, at angles of 50 °, 60 ° and 70 °. If the three calculated n values are almost perfectly consistent (i.e. are
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compliant to the second decimal place), it can be assumed that the bulk material model is in a very good .
approximation and, moreover, that the adjustment of the angle of incidence corresponds to the assumed value
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...
while showing an adequate accuracy.
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...
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Table 3 — Refractive index, extinction coefficient and Brewster angle at 633 nm
...
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...
Silicon
Fused silica Water
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...
(no native oxide)
Formatted Table
...
n 1,457 3,870 to 3,881 1,331 to 1,332
s
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...
ks ≈ 0 0,016 to 0,018 ≈ 0
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...
a
φB 55,54 ° 75,51 ° to 75,55 ° 53,08 ° to 53,10 °
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...
a  Assuming that ks = 0.
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...
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...
Table 4 and Table 5Table 4 and Table 5 show the mutual influence of deviations for fused silica at the
refractive index and the angle of incidence, respectively. Formatted
...
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...
Table 4 — Influence of deviations of the measured angle of incidence on the refractive index ns = ns
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...
(φ + ∆φ) for fused silica at 633 nm
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...
𝛥𝜑
φ  n n
𝑛 𝛥𝑛
s s s s
Formatted Table
...
65 ° — 1,457 —
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...
65,01 ° 0,01 ° 1,457 0,000 Formatted
...
Formatted
65,1 ° 0,1 ° 1,462 0,005
...
Formatted
66 ° 1 ° 1,512 0,055 .
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...
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8 © ISO 2024 2025 – All rights reserved
ISO/FDIS 23131-2:2025(en)
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