Handbook of Modern Coating Technologies

Basic principles of ellipsometry

• Basic principles of ellipsometry

The Fresnel reflection and transmission equations for polarized light are the foundation for the analysis of ellipsometric data. As shown in Fig. 2—1, the incident light is often linearly polarized, whose electric-field components orient parallel (p-) and perpendicular (s-) to the plane of incidence. When this linearly polarized light is reflected by a planar—layered speci­men at an oblique incidence, it encounters actually multiple interferences in the specimen, which change the amplitudes and phases of the s- and p-components of the light. The ellips- ometer measures those changes. The change in amplitudes between the s- and p-compo- nents in the reflected light is named as the ellipsometric parameter ф, and the difference in the phases as another ellipsometric parameter A. In brief, these two ellipsometric para­meters (ф, A) are defined by [27]:

FIGURE 2-1 Schematic of ellipsometry [27,47].

where p is complex reflectance ratio, that is the ratio of complex reflection coefficients for the parallel (R_p) and perpendicular (R_s) polarizations according to the following equations [24]:

^Rp

P 5 ¹ Rs

where r_01p, r_{12 p}, r_01s, and r₁₂ are the Fresnel reflection coefficients at the ambient-film (0—1) and film-substrate (1—2) interfaces for p- and s-polarized lights. Here three media (ambient, film, and substrate) are indicated by the subscript of 0, 1, and 2, respectively. And the phase angle в expresses the phase difference between the two boundaries of the film. According to the Snell's law and the Fresnel equation, the following relationships can be deduced [24]:

_ N₁cos6₀ — N₀cos6₁ ^r°^1,p N₁cos0₀ + N₀cos0₁

_ N₀cos#₀ — N₁cos0₁ ^r01,s N₀cos#₀ + N₁cos0₁

N₂cos0₁ — N₁cos0₂ ^r12,p N₂cos0₁ + N₁cos0₂

N₁cosO₁ — N₂cosO₂

Гіо c — ---------------------

^;      NicosOi 1 NcosOo

2nd                    2nd    2 о о л ч і /о

в —        N_icosO_i —         (N? —N2sin²O₀ )^1/2

A                     A

where N₀, N_b and N₂ are the complex optical constants (optical indices or complex refractive indices) of the ambient, film, and substrate, respectively. The complex refractive index is nor­mally written as N — n 1 ik, where n and k are called the refractive index and extinction coef­ficient. In some literature, the dielectric functions є appear in the above equations because of the relationship є — N².

From the abovementioned equations, we can express the fundamental equation of ellipsometry as follows [24]:

According to Eqs. (2.1)—(2.4), it is easy to derive that:

Thus the ellipsometric parameters (ф, A) are functions of the optical properties of the ambient-film-substrate system (since only one film is involved here, this model is called single-layer model), including the complex refractive indices of the ambient (N₀), film (N₁), substrate (N₂), as well as the film thickness (d), the vacuum wavelength (A) of the ellips- ometer light beam and the incidence angle (O₀). The functional dependence of ф and A on these optical properties can be symbolically expressed briefly as:

tanф e^,A — /(N₀, N₁, N₂, d, O₀, A)                                                 (2.13)

If the values of N₀, N₂, O₀, and A are known in advance from experiments or references, the optical constants (N₁) and thickness (d) of the thin film can be determined using the above relationship. Especially for the most common situation, the ellipsometric experiments are carried out in the air, the system becomes air-film-substrate and the complex refractive index of air is always treated as N₀ — 1, which makes data analysis easier.

• Data analysis procedure

The ellipsometric parameters (ф, A ) are collected within given spectral range and incident angles by using an ellipsometer. However, the measured data (ф and Д) do not give direct
information about the properties of the sample. To extract meaningful physical information about the sample, it is necessary to perform a data analysis procedure. Usually the data anal­ysis procedure includes the following steps [9,27,45,48], as outlined in Fig. 2—2.

The first step is to construct a felicitous optical model to describe the sample system. Usually each material in the system, including the substrate, bulk material, ambient and so on, are always looked as a "layer” and the model is built layer by layer. There are also possi­ble interface layers (gradual transition area between two materials) and surface roughness layer (gradual transition area between the top material and ambient) should be considered. In this step, the number of layers and basic structure concerning the contents of each layer must be assigned so that an optical model is built. A reasonable optical model must repre­sent an approximated structure of the real sample system very well. Additionally some essen­tial information about the real sample, such as the compositions and morphologies, is very helpful for building a "good” model. For example, for an unknown sample, various comple­mentary characterization techniques including X-ray photoelectron spectroscopy (XPS),

FIGURE 2-2 Flowchart of the SE data analysis [27]. SE, Spectroscopic ellipsometry.

atomic force microscopy (AFM) and scanning electron microscopy (SEM) could be used to obtain the exact information about the compositions, roughness and morphologies of the sample. On the other hand, it must be remembered that the construction of the optical model follows the "simple-better” rule, which means that the best sample description is often the matched curve produced by the simplest model.

After the optical model is constructed, the optical constants N (or the dielectric functions є, є = N²) of each layer in the model should be assigned. For the simplex situation, the opti­cal constants are a real "constant,” which likes the optical constant of the air is always given as N_air = 1. However, for the most materials, the optical constants are not a "constant” and they always change with the wavelength, that is, the optical constants N are the function of the wavelength A. To describe the function relationship, many models such as Drude [49], Cauchy [50], Sellmeier [51], Lorentz [52], Tauc-Lorentz [53], and so on, have been developed. In addition, to model the mixing layer which contains two or more kinds of materials, effec­tive medium approximation (EMA) [27] model is introduced. A typical example of EMA layer is the surface roughness, which could be looked as the mixture of the top layer material and the air (or void). The optical constants of the EMA layer are the average values of the optical constants of each component. For instance, the optical constants of a classical EMA layer, Bruggeman EMA [54] layer, can be calculated with Eq. (2.14), where є_а and є_ь are the dielec­tric functions of two components a and b, V is the volume fraction of the first component (component a) and є^ is the average dielectric functions of the EMA layer. Choosing a felici­tous N—A model based on the optical properties of every layer of the sample, is the second step for analyzing the ellipsometric data.

V ^{Єь 2 Єе}Я 1 ₍1 _ V) ^Єа ~ ^£ef = 0                                          (2.14)

єь 1 2SeJf                   Єа 1 2SeJf

Generating theoretical data according to the optical model and the optical constants of every layer and comparing them with the measured data are the third step in the SE data anal­ysis process. Because Eqs. (2.3)—(2.12) are so complex that we cannot solve them directly, data fitting is necessary. Data fitting is the process to search the optimal values of some para­meters by adjusting other model parameters, and the Marquardt—Levenberg regression algo­rithm is the most common one for data fitting. In this algorithm, the mean square error (MSE) is introduced to evaluate the fitting effect. MSE is the sum of the squares of the differences between the measured and generated data [9]. Usually a small MSE value (usually is given as less than 3) describes a good fit because it means that the generated data are very close to the measured data. For the ellipsometric data analysis, if the MSE value is great, it means that the data fitting results are not good and the data analysis procedure must be repeated all over again by adjusting model or assigning new optical constants or dielectric functions.