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Spectroscopic Ellipsometry: Basic Principles

1.1 Spectroscopic Ellipsometry

Spectroscopic ellipsometry measures the change of light polarization upon its reflection from the sample. A schematic diagram of the experimental setup is displayed in Figure 1.1. The detector of spectroscopic ellipsometry measures the quantities ? and ? at each corresponding wavelength/photon energy. Parameter ? denotes the ratio of the amplitude of p- to s-polarized reflected light, while ? their phase difference. Specifically, p-polarized light has the electric field vector parallel to the plane of incidence, while s-polarized light consists of the electric field vector perpendicular to the incident plane (Figure 1.2).

Typically, the energy range that is commonly used for spectroscopic ellipsometry measurement is the ultraviolet-visible (UV-vis) regime (~0.5-6?eV). In this range, sample properties such as the optical band structures and bandgaps can be investigated. Nevertheless, other regions of the electromagnetic spectrum have also been used in spectroscopic ellipsometry measurements. For instance, the use of mid-to-near-infrared range spectroscopic ellipsometry in the study of low-energy structures in 1T´-phase two-dimensional transition metal dichalcogenides (2D-TMDs), such as their fundamental gap and the anisotropic plasmons, will be discussed in Section 3.4 of Chapter 3.

While spectroscopic ellipsometry is a fast, nondestructive, and surface-sensitive (down to a few angstroms) optical characterization technique, the mathematical analysis involved in extracting the optical parameters from the raw (?, ?) data is not a straightforward process (see Section 1.5 and Figure 1.7). Generally, to analytically elucidate the optical parameters from the raw (?, ?) data, the sample in consideration must be homogenous, isotropic, and of sufficient thickness. In more general cases, complications will arise and optical models with associated numerical approximation techniques are required for the proper elucidation of meaningful optical results.

Figure 1.1 Schematic diagram of spectroscopic ellipsometry with the rotating-analyzer configuration.

Figure 1.2 Electric and magnetic fields for (a) p-polarized and (b) s-polarized waves [1].

1.1.1 p- and s-Polarized Lights and Fresnel Coefficients

The electromagnetic wave features of light can be expressed in terms of its electric, E, and magnetic field, B, components [1]:

where denotes the wave vector, ? denotes the angular frequency, and d denotes the initial phase.

When light is reflected or transmitted through a sample/medium via an oblique angle, the electromagnetic wave can be resolved into two components - p-polarized (in-plane incidence) and s-polarized (perpendicular to incident plane) E-field components, respectively.

For a medium with refractive index n, based on Maxwell's equations and boundary conditions, the amplitude of the reflection coefficient for the p-polarized light is expressed as

Likewise, the amplitude of the transmission coefficient for the p-polarized light can be expressed as

whereas the s-polarized counterparts are expressed as

These equations are known as the Fresnel equations. When the refractive indices are complex, , the Fresnel equations still hold. The complex dielectric function can be obtained via the expression

Based on Snell's law, the Fresnel equations for reflection can be further generalized as

where denotes the complex refractive index and

The reflectances of the p- and s-polarized lights are expressed by

where the light intensity I =?n|E|2. Since the difference between rp and rs is maximized at the Brewster angle [2], ellipsometric measurements are usually performed at incident angles, ?i, typically in the range of 70-80° for the optical characterization of semiconducting systems [3].

In multilayered systems, the resultant amplitude of the reflection coefficients is expressed as the sum of individual components of the reflection and transmission coefficients at each interface. The phase differences of each wave are considered in the analysis.

1.1.2 Representation of Polarized Lights

Electromagnetic waves traversing along the z-direction can be expressed by superimposing two waves that are oscillating parallel to the x- and y-axes. The vector sum of the respective E-fields, Ex and Ey, is given by

where and denote the unit vectors along the respective axes. Ultimately, the phase difference, dy-dx, is the most important quantity that determines the state of the polarization of the resultant wave.

To mathematically represent the polarization states and analyze the effects of the optical components in a neat and elegant manner, they are expressed in the form of Jones vectors and Jones matrices [4].

A complete representation of the polarization of a wave can be expressed in the form of the Jones vector as

which can be further simplified as

where and .

Relative changes to the amplitude and phase are important in spectroscopic ellipsometry. Jones vectors are therefore expressed in terms of normalized intensities. Linearly polarized waves along the x- and y-axes are expressed, respectively, as

When light is linearly polarized at an orientation of 45°,

In the formalism where optical components are expressed in the form of 2?×?2 matrices, they are known as Jones matrices. Based on this formalism, the operation performed on the light by each component in spectroscopic ellipsometry, such as the polarizer, analyzer, and compensator, can be represented as a 2?×?2 matrix operator.

For instance, in the case of a linear polarizer with the azimuthal angle, a, relative to the x-y coordinates of a linearly polarized light, Ei, the process of linear polarization can be expressed as

Transformations by a series of optical components can be represented by the corresponding series of matrix operations.

While the Jones vector is a concise way for describing polarized light, it is unable to express unpolarized light and light that is partially polarized. Therefore, the Stokes parameters (vectors) are used for the description of lights with different polarization [4].

The components of the Stokes vector are

where Ix and Iy denote the intensities of the linearly polarization light along the x- and y-axes, respectively. Likewise, I±45° represents light polarization ±45° to the x-axis, while IL/IR represent intensities of left-/right-circularly polarized light. Finally, ? = dx-dy.

The Stokes vector can also be expressed as

Transformation of a Stokes vector can be expressed via a 4?×?4 matrix representation, also known as a Mueller matrix. The calculation is performed in a fashion similar to the Jones matrix. For instance, when linear polarization oriented at 45° passes a polarizer with transmission axis along the x-direction, the resultant light that emerges from the polarizer is transformed via the following:

1.2 Principles of Ellipsometric Measurements

When light is reflected/transmitted from a sample, the p- and s-polarized components of the incident light undergo changes to their amplitude and phase. Hence, spectroscopic ellipsometry is a technique that capitalizes on these changes where the essential optical parameters are derived. As mentioned, the raw quantities measured using ellipsometry are ? and ?, representing the amplitude ratio and phase difference between reflected or transmitted p- and s-polarized waves, respectively. These two quantities are related complex reflection coefficients via the expression

with rp and rs defined as ratios...