Ellipsometry

Michael Hofmann, summersemester 2021

Ellipsometry is an optical technique which investigates surfaces covered with thin films. This method is based on the change of polarization of light upon reflection at a surface and works on partly reflecting and non opaque surfaces. It is a non-destructive measurement technique and can be used in situ, which is primarily applied to measure thickness of thin films. As a result of its very high sensitivity, it is used for layers from 0,1 nm up to multiple micrometers of thickness. Due to the non-destructive character of the technique, it can even be used to observe growth or disappearance of thin films.

Basic principle

The theoretical basis of ellipsometry goes back to the works of P. Drude in the 19^th century. To understand the manner of function it is necessary to know what polarized light is and how it works.

Polarized light

Light can be described as a transverse electromagnetic wave, where the electric field $\begin{array}{l}\vec{E}\end{array}$ and the magnetic field $\begin{array}{l}\vec{B}\end{array}$ are perpendicular to each other and to the direction of propagation as shown in figure 1.


Figure 1: A linear-polarized sinusoidal electromagnetic wave, the electric field (blue arrows) oscillates in the ±x-direction, and the orthogonal magnetic field (red arrows) oscillates in phase with the electric field, but in the ±y-direction. By Emmanuel Boutet; Onde electromagnetique.svg; CC BY-SA 3.0, Creative Commons — Attribution-ShareAlike 3.0 Unported — CC BY-SA 3.0

The direction of polarization describes the direction in which the electric vector $\begin{array}{l}\vec{E}\end{array}$ points. The electric vector can be expressed with following equation:

$\begin{array}{l}\displaystyle \vec{E}(z,t)=\vec{E}_{0} \; cos\left( \omega t-\vec{k}z+\delta \right)\end{array}$

in which $\begin{array}{l}\delta\end{array}$ describes the phase shift.
The vector $\begin{array}{l}\vec{k}\end{array}$ is the wave vector, which represents the direction of wave propagation. It is perpendicular to the electric vector and its magnitude is given by the wavelength:

$\begin{array}{l}\displaystyle \vec{E} \cdot \vec{k}=0\qquad \qquad \qquad \qquad k=\frac{2\pi}{\lambda}\end{array}$

The angular frequency $\begin{array}{l}\omega\end{array}$ is linked with the frequency $\begin{array}{l}v\end{array}$ of the wave:

$\begin{array}{l}\displaystyle \omega=2\pi v\end{array}$

The electric vector $\begin{array}{l}\vec{E}\end{array}$ can be represented in two components (the x and y component). In complex notation the following is received:

$\begin{array}{l}\displaystyle E_j=E_{0 j} \; e^{i(\omega t-kz+\delta_j)}\end{array}$

with $\begin{array}{l}j= x,y\end{array}$

To assess the polarization of an electromagnetic wave only the amplitude ratio of the two components and the phase shift is important. Therefore, the JONES-vector describes just this:

$\begin{array}{l}\begin{pmatrix} E_{0x} \: e^{i\delta_x}\\ E_{0y} \: e^{i\delta_y} \end{pmatrix}\end{array}$

There are 3 different polarizations of an electromagnetic wave: linear, circular, and elliptic. The differences are explained in the paragraphs below and visualized in Figure 2.

Linear-polarized light:
In the case of linear-polarized electromagnetic radiation, there is only one direction of polarization, which is constant. This means that if you consider an electromagnetic wave, which propagates along the z-direction, the electric vector $\begin{array}{l}\vec{E}\end{array}$ is always parallel to the x-y-plane and is always oriented along the same constant direction. Therefore, the phase shift $\begin{array}{l}\delta_x-\delta_y\end{array}$ has either to be 0 or $\begin{array}{l}\pi\end{array}$ . Of course, the magnitude of the electric field vector changes both as a function of the z-coordinate and as a function of time.

Circular-polarized:
If a wave linear-polarized in the x-direction and a wave linear-polarized in the y-direction with the same amplitude, but with a phase difference of a quarter wavelength ( $\begin{array}{l}\delta_x-\delta_y=\frac \pi 2\end{array}$ or $\begin{array}{l}\delta_x-\delta_y=\frac 3 2\:\pi\end{array}$ ) interfere, an electromagnetic wave is created, whose electric vector $\begin{array}{l}\vec{E}\end{array}$ rotates around the direction of propagation as a function of time and as a function of the z-coordinate. The electric vector $\begin{array}{l}\vec{E}\end{array}$ no longer changes its length as a function of time or the z-coordinate but rotates around the z-axis.

Elliptical-polarized:
If a wave linear-polarized in the x-direction and a wave linear-polarized in the y-direction with different amplitudes and a phase difference other than zero or a half wavelength ( $\begin{array}{l}\delta_x-\delta_y \neq 0\end{array}$ and $\begin{array}{l}\delta_x-\delta_y \neq \pi\end{array}$ ) is present, an electromagnetic wave is created whose electric vector $\begin{array}{l}\vec{E}\end{array}$ rotates as a function of time parallel to the the x-y-plane and at the same time changes its length. The tip of the field vector describes an ellipse in the x-y plane.
Thus, the electric vector changes both its length and its orientation as a function of time and the z-coordinate.


Figure 2: Three basic types of polarization: linear, circular, and elliptical https://www.microwaves101.com/encyclopedias/polarization; last visited 28.06.2021.

Fundamental equation of ellipsometry

The following section is based on coordinate system as shown in figure 3, in which the z-axis is the direction of the wave propagation, the x-axis is parallel and the y-axis is perpendicular to the plane of the incident and reflected light. The coordinate axis x and y are the same as the coordinate axis p and s, as used in other literature.


Figure 3: Coordinate system to describe polarized light Uni Münster, Apparative Methoden in der Physikalischen Chemie: Versuch Ellipsometrie

When linear-polarized light is reflected at a surface, it changes its polarization, because the x- and y-component of the electric vector $\begin{array}{l}\vec{E}\end{array}$ are attenuated differently. Additionally, the reflection also induces a phase shift $\begin{array}{l}\Delta\end{array}$ between the incident and the reflected light beam. The shift is different for the x- and y-component of the light. These changes depend on the angle of incidence $\begin{array}{l}\phi\end{array}$ (defined as in Figure 4), the wavelength $\begin{array}{l}\lambda\end{array}$ and the refractive index $\begin{array}{l}n\end{array}$ of the two interacting mediums. In general, linear-polarized light changes to elliptical-polarized light. The technique to measure these changes is called ellipsometry.

There are two parameters that are measured.
The first one is the phase shift $\begin{array}{l}\Delta\end{array}$ between the incident light and the reflected light. Thereby the parameter $\begin{array}{l}\Delta\end{array}$ does not describe the phase shift between the x- and y-component of the incident light beam but the change of it when the light gets reflected:

$\begin{array}{l}\displaystyle \Delta=\left(\delta_x^r-\delta_y^r\right)-\left(\delta_x^i-\delta_y^i\right)\end{array}$

On this, the upper indexes $\begin{array}{l}r\end{array}$ and $\begin{array}{l}i\end{array}$ stand for the reflected or the incident light.

The second parameter which gets measured is $\begin{array}{l}tan \, \Psi\end{array}$ . It describes the relative amplitude attenuation of the reflection.

$\begin{array}{l}\displaystyle tan\, \Psi = \frac{ \;\; \frac{\mid E_{0x}^r\mid} {\mid E_{0x}^i\mid} \;\; } {\;\;\frac{\mid E_{0y}^r\mid} {\mid E_{0y}^i\mid} \;\;}\end{array}$

The parameters $\begin{array}{l}\Delta\end{array}$ and $\begin{array}{l}\Psi\end{array}$ are called the ellipsometric angles and they lead to the fundamental equation of ellipsometry:

$\begin{array}{l}\displaystyle \frac {R_x}{R_y} = tan \Psi \; e^{i\Delta}\end{array}$

$\begin{array}{l}R_x\end{array}$ and $\begin{array}{l}R_y\end{array}$ are the complex reflection coefficients and describe the relation between the incident and reflected light:

$\begin{array}{l}\displaystyle R_j= \frac {\mid E_{0j}^r\mid} {\mid E_{0j}^i\mid} \; e^{i \left( \delta_j^r -\delta_j^i \right)}\end{array}$

with $\begin{array}{l}j=x,y\end{array}$

Determination of refraction index

The reflection coefficients $\begin{array}{l}R_x\end{array}$ and $\begin{array}{l}R_y\end{array}$ at an interface of a semi-infinite body as shown in figure 4 are given by the Fresnel equations.


Figure 4: Reflection and Refraction of light at one Interface Punckt, Christian: Untersuchungen zur optischen Beobachtung musterbildender katalytischer Oberflächenreaktionen, Berlin, 2005

$\begin{array}{l}\displaystyle R_x=\frac {n_1\; cos\phi_0 - n_0\; cos\phi_1 } {n_1\; cos\phi_0 + n_0\; cos\phi_1}\end{array}$

$\begin{array}{l}\displaystyle R_y=\frac {n_0\; cos\phi_0 - n_1\; cos\phi_1 } {n_0\; cos\phi_0 + n_1\; cos\phi_1}\end{array}$

By using Snell's law of refraction, which says

$\begin{array}{l}\displaystyle sin \phi_0 \; n_0 = sin \phi_1 \; n_1\end{array}$

the angle $\begin{array}{l}\phi_1\end{array}$ can be eliminated in the Fresnel equations.
Thus, when the angle of incidence $\begin{array}{l}\phi_0\end{array}$ and the refraction index $\begin{array}{l}n_0\end{array}$ (air: $\begin{array}{l}n_0 = 1\end{array}$ ) are known, the complex refraction index $\begin{array}{l}n_1\end{array}$ of the medium can be determined by measuring the ellipsometric angles and insert the Fresnel equations in the fundamental equation of ellipsometry.
The complex refraction index $\begin{array}{l}n\end{array}$ is also linked by the Kramers-Kronig-relation with the complex permittivity. Thus, the permittivity of a material can be determined by an ellipsometric measurement.

Measuring of thin films

Ellipsometry is most widely used to measure the thickness of thin films. In the following paragraph a semi-infinite medium coated with a single thin film as shown in figure 5 is considered.


Figure 5: Reflection and refraction of light on two interfaces Punckt, Christian: Untersuchungen zur optischen Beobachtung musterbildender katalytischer Oberflächenreaktionen, Berlin, 2005

When there is a thin film layer on a substrate, multiple reflections, and transmissions in between the first layer take place. Thus, the light beams, which are reflected at different points, interfere and change the reflection coefficients $\begin{array}{l}R_x\end{array}$ and $\begin{array}{l}R_y\end{array}$ compared to a reflection at a medium without a thin film layer on it. This changes the polarization of the reflected light, which can be measured by a change of the ellipsometric angles $\begin{array}{l}\Psi\end{array}$ and $\begin{array}{l}\Delta\end{array}$ . The deviation is directly dependent on the thickness $\begin{array}{l}d\end{array}$ and the refractive index $\begin{array}{l}n_1\end{array}$ of the layer.

The reflection coefficients can be expressed by means of the Fresnel equations. The reflection coefficients between medium 0 (air) and medium 1 (thin film layer) stay the same as in the chapter before:

$\begin{array}{l}\displaystyle R_{01x}=\frac {n_1\; cos\phi_0 - n_0\; cos\phi_1 } {n_1\; cos\phi_0 + n_0\; cos\phi_1}\end{array}$

$\begin{array}{l}\displaystyle R_{01y}=\frac {n_0\; cos\phi_0 - n_1\; cos\phi_1 } {n_0\; cos\phi_0 + n_1\; cos\phi_1}\end{array}$

The reflection coefficients between medium 1 (thin film layer) and medium 2 (substrate) are:

$\begin{array}{l}\displaystyle R_{12x}=\frac {n_2\; cos\phi_1 - n_1\; cos\phi_2 } {n_2\; cos\phi_1 + n_1\; cos\phi_2}\end{array}$

$\begin{array}{l}\displaystyle R_{12y}=\frac {n_1\; cos\phi_1 - n_2\; cos\phi_2 } {n_1\; cos\phi_1 + n_2\; cos\phi_2}\end{array}$

As in determination of the refraction index Snell's law can be used to eliminate the angles $\begin{array}{l}\phi_1\end{array}$ and $\begin{array}{l}\phi_2\end{array}$ .
After the interference of the different reflections, it results in the following reflection coefficients for a single layer system:

$\begin{array}{l}\displaystyle R_x = \frac {R_{01x} + R_{12x} \; e^{-i \: 2 \beta}} {1 + R_{01x} \; R_{12x} \; e^{-i \: 2 \beta}}\end{array}$

$\begin{array}{l}\displaystyle R_y = \frac {R_{01y} + R_{12y} \; e^{-i \: 2 \beta}} {1 + R_{01y} \; R_{12y} \; e^{-i \: 2 \beta}}\end{array}$

with $\begin{array}{l}\beta = 2 \pi \: \frac{d}{\lambda} \: \sqrt{n_1^2 - n_0^2 \; sin^2 \phi_0}\end{array}$

in which $\begin{array}{l}d\end{array}$ is the thickness of the thin film layer.

Thus, when the angle of incidence $\begin{array}{l}\phi_0\end{array}$ and the refraction indices $\begin{array}{l}n_0\end{array}$ and $\begin{array}{l}n_1\end{array}$ are known, the thickness of the film can be determined by measuring the ellipsometric angles and insert the Fresnel equations in the fundamental equation of ellipsometry. Due to the trigonometric functions the thickness $\begin{array}{l}d\end{array}$ can not be determined absolutely but just periodic values of it. Hence, the approximate range of the film thickness must be known.

Data analysis

The analytical calculation of the refraction index and thickness parameters of a sample is only possible for very simple cases (isotropic, homogenous material, semi-finite body). General conditions make it much more difficult to determine the wanted quantities. Therefore, a layer model of the sample is built, and an iterative process tries to find the best fit of the model with respect to the measured data by varying unknown optical constants and thickness parameters. The parameters, which provide the best fit, are issued as the final result.

Instrumentation


Figure 6: PCSA configuration of an ellipsometer H. G. Tompkins and E. A. Irene (Editors), Handbook of Ellipsometry William Andrews Publications, Norwich, NY (2005)

There are different configurations for an ellipsometric measurement. The basic PCSA configuration is shown in the figure above. PCSA stands for the different components and their order. It consists of a light source, the linear polarizer (P), a compensator (C), the sample (S), an analyzer (A), which is also a linear polarizer, and the detector. The light source is typically a Hg lamp ( $\begin{array}{l}\lambda\end{array}$ =546,1 nm) or a He-Ne laser ( $\begin{array}{l}\lambda\end{array}$ =632,8 nm). The left side with the linear polarizer and the compensator generates a known and defined polarization state of the incident light. The compensator is normally a $\begin{array}{l}\frac{\lambda}{4}\end{array}$ -waveplate. The right side with analyzer and detector is used to detect the change in polarization state. Hence, the ellipsometric angles $\begin{array}{l}\Delta\end{array}$ and $\begin{array}{l}\Psi\end{array}$ can be established and unknown quantities, such as layer thickness or refractive index, can be calculated.

Literature

Visscher, W. H. M.: Ellipsometry and other reflection techniques. Materialwissenschaft und Werkstofftechnik, September 1973, Vol.4(6), pp.302-306
Tompkins, Harland G.: Handbook of ellipsometry. Andrew [u.a.], Norwich, NY 2005
Punckt, Christian: Untersuchungen zur optischen Beobachtung musterbildender katalytischer Oberflächenreaktionen (Dissertation, Naturwissenschaften), Researchgate, Berlin 2005
Daniel Faltermeier: Ellipsometrie an organischen Dünnfilmen und Einkristallen zur Bestimmung der optischen und strukturellen Eigenschaften (Dissertation, Naturwissenschaften), online Publikationen der Universität Stuttgart 2007
Universität Münster: Apparative Methoden in der Physikalischen Chemie, Versuch Ellipsometrie, 2009

[1] Visscher, W. H. M.: Ellipsometry and other reflection techniques. Materialwissenschaft und Werkstofftechnik, September 1973, Vol.4(6), pp.302-306

[2] Tompkins, Harland G.: Handbook of ellipsometry. Andrew [u.a.], Norwich, NY 2005

[3] Punckt, Christian: Untersuchungen zur optischen Beobachtung musterbildender katalytischer Oberflächenreaktionen (Dissertation, Naturwissenschaften), Researchgate, Berlin 2005

[4] Daniel Faltermeier: Ellipsometrie an organischen Dünnfilmen und Einkristallen zur Bestimmung der optischen und strukturellen Eigenschaften (Dissertation, Naturwissenschaften), online Publikationen der Universität Stuttgart 2007

[5] Universität Münster: Apparative Methoden in der Physikalischen Chemie, Versuch Ellipsometrie, 2009

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