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Transcript
Materialanalytik
Praktikum
Ellipsometry
B508
Stand: 05.09.2011
Aims:
Measurement of the thickness of different dielectric thin films.
Table of Contents
1.
INTRODUCTION
1
2.
BASIC CONCEPTS
1
2.1.
Polarized light
1
2.2.
Reflection of polarized light
3
2.3.
Principle of Ellipsometry
4
3.
THE EXPERIMENT
7
3.1.
Devices / Samples
7
3.2.
Performing the experiment
7
4.
EVALUATION OF THE EXPERIMENT
7
5.
BIBLIOGRAPHY
8
APPENDIX A – REFRACTIVE INDEX FOR SELECTED MATERIALS
8
B508: Ellipsometry
1. Introduction
Ellipsometry is an optical measurement technique which is used to determine the dielectric
properties (complex index of refraction) and thickness of thin (a few Å – a few µm) transparent films. It utilizes polarized light and its changing properties upon reflection and transmission at or through dielectric media.
Quickness, precision, and the non-destructive nature of this method are strengths of the method. Therefore, ellipsometry is a standard measurement technique used in the industry (e.g.
microelectronics, solar cells…), as well as in basic research throughout natural and engineering sciences.
This lab course is designed to convey the basic knowledge about the ellipsometry measurement technique. It involves a theoretical introduction to the method starting with the theoretical background of polarized light, as well as practical measurements and the required preparations.
2. Basic Concepts
Ellipsometry is based on the use of polarized light. Thus, our theory chapter will start right
there and will later be devoted to the method itself. Hence, we will discuss briefly the
•
•
Basics of polarized light.
Basics of ellipsometry.
2.1. Polarized light
Most generally, light can be treated as a transverse electromagnetic wave which consists of
sinusoidally oscillating electric (E) and magnetic (H) fields perpendicular to each other and to
the propagation direction of the wave. For simplicity the mathematical representation is usually only based on the electric field E. The electric field E of the wave can further be divided
into two components, the so-called polarizations, usually called Ep and Es. These polarizations
are plane waves, which can be expressed by
E S = Eˆ S exp(iωt )
(1)
with the amplitude Ês , the angular frequency ω and the time t and
E P = Eˆ P exp(i (ωt − ∆) )
(2)
where ∆ denotes the phase between both polarizations.
If a light wave impinges on a surface, the area which is perpendicular to the surface and which
contains the wave, is called the plane of incidence. In this context the meaning of the indices s
and p is easy to understand: The p-polarization is the plane wave which is oscillating parallel
to the plane of incidence, whereas the s-polarization is oscillating perpendicular (from Ger-
1
B508: Ellipsometry
man “senkrecht”) to it. Thus, both polarizations are perpendicular to each other and to the
propagation direction of wave.
Figure 1: Polarization of light. (a) The light is linearly polarized, i.e. both waves (polarizations) are in phase. (b) The light is zircularly polarized, i.e. both polarization amplitudes are
the same and both waves are shifted by a phase of 90°. (c) The light is elliptically polarized,
i.e. both polarizations’ amplitudes are different and both waves are not in phase.
To calculate the resulting electric field E, the two components (polarizations) simply have to
be added (as vectors). This is illustrated in Fig. 1 for three cases:
i)
Es and Ep are in phase, Δ = 0
The resulting electrical field E will be a plane wave, the position (angle to Es or Ep) is determined by the ratio of the amplitudes Ês/Êp of the polarizations.
ii)
Es and Ep are shifted by Δ = 90° and Ês = Êp
The resulting wave is circularly polarized, i.e. the wave describes a rotation around the axis of propagation. The amplitude of the wave stays constant, since Ês = Êp.
iii)
Es and Ep are arbitrarily shifted in phase and Ês ≠ Êp
2
B508: Ellipsometry
This case is the most general case for a light wave. The wave is elliptically polarized, i.e.
the electric field vector E is rotating around the axis of propagation. Its amplitude however
is not constant, as in case ii), therefore the projection of the movement describes an ellipse. Cases i) and ii) can be seen as special cases of case iii).
2.2. Reflection of polarized light
The two polarizations are not just a theoretical construct, completely s- or p-polarized light
shows different behavior in nearly all optical properties. One very distinct difference can be
observed in the case of reflection from a material surface, e.g. the reflection at a GaAs sample
measured in air, which is presented in Fig. 2. The measured reflectance is plotted versus the
incidence angle ρ. The slope of the resulting curve for the s-polarized light Rs is monotonously
increasing with increasing angle. In contrast, the curve Rp for p-polarized light decreases to
zero between 0° and an angle close to 75°, steeply increasing to unity between that angle and
90°. The angle where Rp = 0 is called “Brewster Angle” ρB. It can be shown, that this angle
can be determined by
tan ρ B =
n2
,
n0
(3)
where n2 and n0 are the refractive indices of the measured sample and the medium in which
the wave travels, before it impinges on the sample.
If a thin layer with refractive index n1 is added to the system, the Brewster angle will be altered. On the one hand, Rp will depend on the substrate and the layer, therefore the term
“Pseudo-Brewster Angle” is commonly used, on the other hand ρB will shift to
•
•
Lower angles for n1 < n2.
Higher angles for n1 > n2.
3
B508: Ellipsometry
0
20
60
80
1,0
0,9
0,9
0,8
0,8
RS
RP
0,7
Reflactance
40
1,0
0,6
0,7
0,6
0,5
0,5
0,4
0,4
0,3
0,3
0,2
0,2
0,1
0,1
0,0
0,0
0
20
40
60
80
Incidence Angle ρ
Figure 2: Reflectance R vs. incidence angle ρ for an air-GaAs interface. The reflectance R
differs for the two polarizations. The angle characterized by the minimum in reflectance for Rp
is called the “Brewster Angle”.
2.3. Principle of Ellipsometry
In Fig. 3 the basic principle of ellipsometry is illustrated schematically. As light source (here a
He-Ne laser with λ = 632.8nm) is used, which emitts unpolarized light. This light is linearly
polarized by a polarizer (operated at a fixed azimuth angle of 45°) and can optionally be zircularly polarized by a so-called quarterwave plate (compensator). The light beam will hit the
sample (simplest case: substrate + dielectric layer on top) at a defined angle ρ. Angles in the
vicinity of the Brewster angle are best suited to yield good measurement results, since ellipsometry is most sensitive to film parameters in this range.
The light will penetrate into the layer until it reaches the interface between the layer and the
substrate, where it will be reflected parially. Finally the beam leaves the layer under the same
exit angle as the incidence angle ρ. If a wave of light travels in a dielectric medium, it will
exhibit changes in its properties. The amplitude of both polarization directions (Ês and Êp)
will change as will the phase Δ between both polarizations. The beam will then reach a rotating analyzer, which allows to measure the reflectance for all phases Δ. Eventually a photodetector will measure the intensity of the incoming light beam as a function of the angle of the
rotating analyzer.
Measurement results are usually expressed in terms of the ellipsometric parameters Δ and Ψ,
which are defined by
Rp
,
(4)
tanψ e −i∆ =
Rs
where Rs and Rp are the reflectance of the s- and p-polarization. Finally the recorded data is
4
B508: Ellipsometry
transferred to a computer for the calculation of layer thickness(es) and/or refractive indices.
Figure 3: Schematic illustration of the ellipsometer working principle (after [1]).
One question remains: How are the properties of a wave changed when propagating through a
dielectric material? Since answering this question in detail would exceed the time limitations
of this lab course, we will give a qualitative description of the processes to give a better understanding of the experiment and to point out how this gained knowledge should influence
the execution of our experiments.
Fig. 4 shows a schematic view of the light propagation in an ellipsometer experiment. A light
wave traveling in a medium of refractive index n0 is impinging on a dielectric layer (n1) on top
of a substrate (n2). The light will partially be reflected, illustrated here by the reflection coefficient r01, the remaining fraction will be transmitted, represented by the transmission coefficient t01. The transmitted fraction will penetrate through the dielectric layer to the interface
between layer and substrate. There the light will again be partially reflected and transmitted,
represented by the coefficients r12 and t12 respectively. This process will be repeated again and
again, thus multiple beams will leave the sample and contribute to the measurement result. To
calculate the thickness of the layer or its refractive index from the intensities of the reflected
light involves a huge amount of straightforward mathematics and therefore is done by special
ellipsometer software.
For allpying a "simple" theory the substrate should be infinitely thick, in practice this is obviously not possible. If the substrate is a dielectric material (i.e. absorption = 0), e.g. glass, all
the light that is transmitted through the n1 / n2 interface can penetrate through the substrate
and be reflected from the bottom side. Therefore this fraction of the incident light also "finds a
way" through the sample to the detector and can thus influence the measurement. Since this
influence is not desired, the backside of transparent substrates is usually blackened to reduce
the reflection from the backside of the substrate.
5
B508: Ellipsometry
Figure 4: Schematic illustration of the multi-reflection of an incident beam on a thin dielectric layer, as in the case of an ellipsometer measurement. The beam is partially reflected at the
interface of an incident medium n0 and a thin layer of refractive index n1, the rest of the beam
is transmitted to the thin layer. The transmitted beam will undergo further reflections and or
transmissions at different interfaces (n0 / n1 interface and between n1 and the substrate n2),
resulting in multiple beams leaving the sample towards an detector and therefore influencing
the measurement.
6
B508: Ellipsometry
3. The experiment
3.1. Devices / Samples
PLASMOS SD2300 Ellipsometer
We will investigate 4 types of thin films on different substrates:
Sample #
1
2
3
4
Thin Film Layer
SiO2
Porous Si #1
Porous Si #2
TiO2
Substrate
Si
Si
Si
Glass
The complex refractive indices for all layers and substrates can be found in Appendix A.
The refractive index n for the porous silicon samples can be estimated according to the
simplest case of the Brüggemann approximation by adding the refractive indices for silicon
and air weighted by their respective volume fractions. The porosity of the porous silicon
(Psi) sample 1 is ≈ 90 %, for sample 2 it is ≈ 40 %. For k the value of bulk silicon can be
taken.
3.2. Performing the experiment
• Estimate the refractive index n for the porous silicon samples.
• Calculate the Brewster angles for the given sample substrates and estimate the PseudoBrewster angles of the samples.
• Set up the ellipsometer and the ellipsometer software, the polarizer has to be manually
aligned to 45° with the help of the adviser.
• Implement the model for the system(s) to be analyzed in the ellipsometer software, i.e. put
in values for the number of layers and the complex refractive indices of the layer(s) and the
substrate (Important: k-values have to be put in as –k in the software!).
• Measure the film thickness d and the refractive index n of the layers at four different angles
of incidence ρ (40°, 50°, 60°, and 70°). Perform five measurements at each angle and sample.
4. Evaluation of the experiment
• Plot the layer thickness d and the refractive index n as a function of the incidence angle ρ
for the measured samples.
• Calculate the statistic error of a regular measurement from the respective 5 measurements
per sample and angle (Add error bars into the plots).
• Estimate the measurement error induced by changing the incidence angle ρ.
• Discuss which measurement contains the most reliable data for each sample based on the
7
B508: Ellipsometry
results of the error evaluation.
• Discuss possible reasons why the thin layer shows larger errors than the thicker layers.
5. Bibliography
[1] Operating Manual – PLASMOS SD 2300, PLASMOS GmbH, München, 1993
[2] Ellipsometry and polarized light, R.M.A. Azzam and N.M. Bashara, Elsevier Science,
Amsterdam, 1987
[3] J.A. Woollam Co., Inc. / www.jawoollam.com
[4] Optics (3rd edition), E. Hecht, Addison Wesley Longman, Reading, 1998
Appendix A – Refractive Index for selected materials
Material
Si
SiO2
TiO2
Glass
n
3.875
1.455
2.88
1.5
8
k
0.018
0
0
0