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SEMI E141-0705 © SEMI 2005 7 7.4 material and layer definition — the optical properties of a material d etermine how the complex refractive index N i is given as a funct ion of wavel ength and environm ental parameters (…
SEMI E141-0705 © SEMI 2005 6
7 Definitions for Modeling the Reflection from a Sample Surface
7.1 Analysis of Ellipsometric Measurements
7.1.1 For the analysis of ellipsometric measurements an optical model describing the optical parameters and layer
thickness values of the sample must be provided. The optical model is regarded as part of the necessary substrate
data set
4
. The objective of the ellipsometric measurement is to determine the value of at least one parameter within
the optical model such that the measurand calculated from the optical model is in optimum consistency with the
measurand (raw data value
1
) determined in the measurement. For a correct optical model and in the absence of
measurement errors, the measurand calculated from the optical model coincides with the measurand (raw data
value
1
) determined in the measurement.
7.1.2 The set of adjusted parameter values is the measurement result (i.e. the converted measurement data
1
determined by the ellipsometric measurement). In some special cases the measurement result can be calculated from
the measurands or raw data analytically.
7.1.3 Typically, in ellipsometry the analysis is performed by calculating the expected value of the measurand (i.e.
raw data value from the parameter values provided in the optical model). The parameter values of interest within the
optical model are then intentionally adjusted in a manner such that optimum consistency between the calculated and
measured value of the measurand (i.e. the raw data value is obtained). The adjusted parameter values are the
measurement result (i.e. the converted measurement data).
7.2 Definition of the Optical Parameters — The optical parameters are used as substrate data set and the optimized
parameter values after measurement and analysis denote the measurement result (i.e. the converted measurement
data).
7.2.1 layer index (i) — The layer index i identifies the layer for remote access. The index i = ‘a’ describes the
ambient medium (n
a
, k
a
). The index i = ‘s’ describes the substrate medium (n
s
, k
s
). The intermediate layers are
numbered in the direction from the substrate to the ambient (i.e. i = 1 denotes the first layer on the substrate) (see
Figure 2).
7.2.2 n
i
— Denotes the refractive index of layer i being a positive real number.
7.2.3 k
i
— Denotes the extinction coefficient of layer i. The value of the extinction coefficient k
i
must be set as a
positive number |k
i
| for better readability in programs and printouts, independently of the notation of the complex
refractive index and complex dielectric function (see ¶¶7.2.4, 7.2.6, and 7.2.7).
NOTE 3: The extinction coefficient k describes absorbing media if |k| > 0 and transparent media if |k| = 0. The absorption
coefficient
and the absorption index
are also two common optical parameters used for describing absorbing media. The
extinction coefficient k is related to the absorption coefficient
by
= 4k/
and to the absorption index
by
= k/n.
7.2.4 N
i
— Denotes the complex refractive index of layer i. Depending on the physical notation the complex
refractive index is calculated by N
i
= n
i
– j k
i
or N
i
= n
i
+ j k
i
(see ¶7.2.3).
7.2.5
i
= N
i
2
— Complex dielectric function of layer i.
7.2.6
1i
=
n
i
2
– k
i
2
— Real part of the dielectric function of layer i.
7.2.7
2i
— Imaginary part of the dielectric function of layer i. Depending on the physical notation, the imaginary
part of the dielectric function is calculated by
2i
= −2n
i
k
i
or
2i
= 2n
i
k
i
(see ¶7.2.4, ¶7.2.5). The value of the
imaginary part of the dielectric function
2i
must be set as a positive number |
2i
| for better readability in programs
and printouts, independently of the notation of the complex refractive index and complex dielectric function (see
¶¶7.2.3, 7.2.4, and 7.2.5).
7.2.8 t
i
— Metric thickness of the layer i. The method for counting is identical as for the refractive indices, but no
thickness is provided for the ambient and the substrate.
7.3 ambient medium — The modeling always requires a semi-infinite space (or material) for the incident and the
reflected beam that is not part of the sample. The ambient is the propagation medium immediately before and after
reflection at the sample surface. In many cases the ambient medium is air or vacuum with n
a
= 1 and k
a
= 0.
4 Terminology here is as defined in SEMI E127.
SEMI E141-0705 © SEMI 2005 7
7.4 material and layer definition — the optical properties of a material determine how the complex refractive index
N
i
is given as a function of wavelength and environmental parameters (e.g., temperature). A layer consists of a
single material or a combination of several materials with a metric thickness t
i
. This definition also applies to
complex structures as index gradients, interfaces, and roughness that can be modeled as a series of layers.
7.5 substrate medium — Lowest (see ¶7.2.1) material involved in the reflection with a complex refractive index N
s
.
A sample has only one substrate material, which is treated as semi-infinite.
7.6 Ellipsometric Measurand (raw data set)
5, 6, 7, 8, 9
7.6.1 r
p
= E
rp
/E
ip
— Complex amplitude reflection coefficient parallel to the plane of incidence.
7.6.2 r
s
= E
rs
/E
is
— Complex amplitude reflection coefficient perpendicular to the plane of incidence.
7.6.3
= r
p
/r
s
= tan
e
j
— Ratio of the complex amplitude reflection coefficients.
7.6.4
p
— Phase shift of E
rp
relative to E
ip.
7.6.5
s
— Phase shift of E
rs
relative to E
is.
7.6.6
=
p
-
s
— Phase shift between p and s components of the electric field strength.
7.6.7 tan
= |r
p
|
/|r
s
| — Ratio of the absolute values of the amplitude reflection coefficients.
7.6.8 S
x
— Stokes parameters (x = 0, 1, 2, 3).
NOTE 4: The four Stokes parameters describe the polarization ellipse using the physical dimension of energy. The four Stokes
parameters involve three independent parameters that are necessary to describe the polarization ellipse. For totally polarized light,
the Stokes parameters describe a sphere with the radius S
0
and represent the parameters
and
in a Cartesian coordinate
system: S
0
2
= S
1
2
+ S
2
2
+ S
3
2
. The parameter S
0
is proportional to the energy of the light wave. For elliptically and totally
polarized light, the Stokes parameters are calculated from the parameters
and
as follows: S
1
= – S
0
cos 2
, S
2
= S
0
sin 2
cos
, and S
3
= S
0
sin 2
sin
. The Stokes vector, consists of four vector components, which are the Stokes parameters.
7.6.9 s
x
= S
x
/S
0
— normalized Stokes parameters (x = 1, 2, 3).
NOTE 5: The normalized Stokes parameters are calculated from the Stokes parameters as follows: s
1
= S
1
/S
0
, s
2
= S
2
/S
0
, and s
3
=
S
3
/S
0
.
7.6.10 The results are typically given as [
,
](
0
,
), [tan
, cos
](
0
,
), [s1, s2, s3](
0
,
) dependent
on the measurement angle of incidence
0
, and the wavelength
see ¶3.4) The parameter array
provides all
environmental conditions (influence quantities) relevant for the measurement. All relevant parameters shall be
provided in SI units. At least the sample temperature must be specified, but additional parameters (e.g. ambient
pressure, measurement time, sample orientation, the measurement position
10
, composition, and strain) may have to
be added.
7.7 Qualification of Ellipsometric Data
7.7.1 The qualification of an ellipsometer is verified by measuring certified reference samples and test procedures
11, 12, 13, 14
.
5 Born, M.; Wolf, E.: “Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, Cambridge
University Press, ISBN 0521642221.
6 Azzam, R. M. A., Bashara, N. M.: "Ellipsometry and Polarized Light", Elsevier Science Publishers B. V., ISBN 0444870164.
7 Tompkins, H. G.; McGahan W.A.: "Spectroscopic Ellipsometry and Reflectometry: A User's Guide”, Wiley-Interscience, ISBN 0471181722.
8 Muller, R. H.: "Definitions and Conventions in Ellipsometry", Surface Science Vol. 16 (1969), pp. 14-33.
9 Röseler, A.: Infrared Spectroscopic Ellipsometry, Akademie Verlag Berlin, ISBN 3-05-500623-2.
10 See SEMI M20.
11 See SEMI E89-1104E.
12 See SEMI MF576.
13 Metrology Tool Gauge Study Procedure for the International 300 mm Initiative (I300I), Technology Transfer # 97063295A-XFR,
International 300 mm Initiative, June 15, 1997.
14 Eastman, S. A.: Evaluating Automated Wafer Measurement Instruments. Technology Transfer # 94112638A-XFR; SEMATECH February 28,
1995.
SEMI E141-0705 © SEMI 2005 8
NOTE 6: The reference samples are typically certified thickness standards using reference materials. For thickness
measurements typical samples are thermal oxides on silicon ranging from 4 nm to 800 nm. For refractive index measurements,
the reference sample thickness is typically in the range from 80 nm to 500 nm.
NOTE 7: SEMI MF576-01 was originally published by ASTM International as ASTM F 576-78. It was formally approved by
ASTM balloting procedures and adhered to ASTM patent requirements. Though ownership of this standard has been transferred
to SEMI, it has not been formally approved by SEMI balloting procedures and does not adhere either to SEMI Regulations
dealing with patents or to SEMI Editorial Guidelines. It was available at www.semi.org, last published by ASTM International as
ASTM F 576-01. Hence in SEMI MF576-01, the notation for parameters required in data acquisition and modeling is aligned but
could not retroactively be fully harmonized with the definitions specified in the presented guide. This does not constitute any
constraints since these few parameters may easily be superseded by the newer definitions provided.
7.7.2 Whenever the thickness and refractive index measurement (if performed) is within specified limits for the
reference sample, the ellipsometer is defined as qualified and hence capable of measuring the respective type of
sample.
7.7.3 The qualification status of an ellipsometer is defined and must be specified by providing the following
parameters: (1) the relevant parameters for sample identification, (2) the relevant optical parameters of the sample,
(3) the expected measurement result, (4) the actual measurement result, (5) the tolerated deviation between the
actual and the expected measurement result, (6) all relevant parameters of the ellipsometer system (e.g. all relevant
parameters of the positions of the optical components) the angle of incidence, and (7) all relevant environmental
parameters.
7.7.4 Recommended Units and Symbols — The units as listed in Table 1 are recommended for the description of the
listed parameter.
Table 1 Recommended Units
Physical Quantity Symbol Unit(s)
Metric thickness t µm, nm, Å
Wavelength
µm, nm, Å
Photon energy E eV
Wave number
cm
-1
Angles
,
,
0
,
p
,
c
,
PEM
,
A
° or deg (360° are a full circle)
Electric field vectors E
ip
, E
is
, E
rp
, E
rs
V/m
Optical parameters (refractive indices) N, n, k
no unit
Complex dielectric function
no unit
Pseudo <
>
no unit
#1
See SEMI E30.5 for the preferred thickness unit (Å).
#2
The pseudo dielectric constant <> is calculated from the raw data set using a sample model consisting of only the ambient and substrate
without layers.