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SEMI E141-0705 © SEMI 2005 6 7 Definitions for Modeling the Reflection from a Sample Surface 7.1 Analysis of Elli psometric Measurements 7.1.1 For the analysis of ellipsometric measurements an optical model d escribing t…
SEMI E141-0705 © SEMI 2005 5
6.4.2.4 electric field vector (E
is
) — Electric field strength of the incident beam perpendicular to the plane of
incidence and perpendicular to the wave vector of the incident beam.
6.4.2.5 electric field vector (E
rp
) — Electric field strength of the reflected beam parallel to the plane of incidence
and perpendicular to the wave vector of the reflected beam.
6.4.2.6 electric field vector (E
rs
) — Electric field strength of the reflected beam perpendicular to the plane of
incidence and perpendicular to the wave vector of the reflected beam.
6.4.2.7 handedness of the coordinate system — The vectors E
ip
, E
is
, and the wave vector k
i
of the incident beam as
well as the vectors E
rp
, E
rs
, and the wave vector k
r
of the reflected beam span a right-handed coordinate system (see
Figure 1).
6.4.2.8 description of the component position — The positions of the ellipsometer components are described by the
following angles, which are specified counterclockwise relative to E
ip
and E
rp
, respectively.
6.4.2.8.1 polarizer azimuth (
P
) — Angle between the plane of incidence and the polarization axis of the light
emerging the polarizer (see Figure 1).
6.4.2.8.2 compensator azimuth (
C
) — Angle between the plane of incidence and the fast axis of the compensator
crystal. If a compensator consists of multiple anisotropic crystals, the axis is defined as the effective axis when the
output is modeled by a single anisotropic crystal (see Figure 1).
6.4.2.8.3 PEM azimuth (
PEM
)
— The PEM azimuth denotes the same angle as
C
for an electronically phase
modulated compensator.
6.4.2.8.4 analyzer azimuth (
A
)
— The analyzer azimuth denotes the angle between the plane of incidence and the
polarization axis of the light emerging the analyzer (see Figure 1).
6.4.3 description of the angle of incidence — The third item to be described for the definition of ellipsometer
equipment is the number of angles of incidence.
6.4.3.1 single-angle ellipsometer (SAE) — With the single-angle ellipsometer, the ellipsometric measurement is
performed at a single angle of incidence.
6.4.3.2 multiple-angle ellipsometer (MAE) — With the multiple-angle ellipsometer, the ellipsometric measurement
is performed at different angles of incidence.
6.4.4 description of measurement wavelength (
— The fourth item to be described for the definition of
ellipsometer equipment is the number of wavelengths used for measurement.
6.4.4.1 single-wavelength ellipsometer (SWE) — With the single-wavelength ellipsometer, one discrete wavelength
is used in the measurement.
6.4.4.2 multiple-wavelengths ellipsometer (MWE) — With the multiple-wavelengths ellipsometer, several discrete
wavelengths are used in the measurement.
6.4.4.3 spectroscopic ellipsometer (SE) — With the spectroscopic ellipsometer many (at least 10) different
wavelengths are used in the measurement.
6.4.5 description of the data acquisition method — The fifth item to be described for the specification of
ellipsometer equipment is the data acquisition method.
6.4.5.1 scanning data acquisition — In scanning data acquisition, the state of polarization is measured wavelength
by wavelength for different positions of one or more optical components.
6.4.5.2 parallel data acquisition — In parallel data acquisition, the state of polarization is measured by
simultaneously varying one or more parameters of the ellipsometer equipment for a defined position of one or more
optical components, e.g. simultaneously measuring the state of polarization for different wavelengths or at different
angles of incidence.
NOTE 2: In the literature, the terms TM (transverse magnetic) and TE (transverse electric) are also used to denote p and s
polarizations, respectively. The TM polarization denotes that the magnetic field vector is perpendicular to the plane of incidence,
while the TE polarization denotes that the electric field vector is perpendicular to the plane of incidence.
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.