Thickness and optical constants

Although the above relationships are not very transparent, they are readily handled by a computer. They allow one to calculate the MSEFS using measurable quantities such as the wavelength and the angle of incidence of incoming radiation, thicknesses, and optical constants of the materials that constitute strata in a multilayer system. These equations offer a tremendous help when the optimization of experimental conditions is undertaken. This point will be illustrated by the material presented in the next section. 

Similar to the surface susceptibility method [503, 507], DR fitting allows one to exclude the film thickness and optical constants as input parameters in spectral simulations, a very important advantage in the case of adlayers when there is no way to determine the film thickness accurately [622], As opposed to spectrum fitting, DR fitting is applicable to the cases when (1) properties of the film depend on the substrate (the general case), (2) preliminary information about the film is poor or absent, and (3) when the film can not be perfectly reproduced in a series of experiments. Moreover, as will be shown below, DR fitting is more sensitive to MO. 

For layered samples, the thickness and optical constants of a particular layer wiU always be correlated parameters. That means that it is not possible to accurately determine both, the thickness and optical constants for a given layer, with just one ellipsometric spectrum. To be able to resolve this correlation it is necessary to collect data for a set of samples with different thicknesses for the layer in question and identical otherwise. Then, by fitting the set of ellipsometric spectra generated, the correlation between the two parameters can be reduced. 

Application Example Thickness and Optical Constants of a Thin Ti02 Film on Si... 

Ellipsometry is a powerful tool to gain the optical properties of materials though measuring the change of polarization state of the probe light after interaction with the sample. It offers a sensitive, nondestructive and comprehensive way to accurately determine film thickness and optical constants of extensive materials, such as metals, ceramics, glasses, semiconductors, and its compounds and composites. These materials can be liquid phase or even gaseous phase, can be isotropic or anisotropic, and can be bulk materials or multi-layer thin films. 

Figure 3.20. Simulated s- and p-polarized IRRAS spectra and optical constants ks and ns of 1-nm thick water layer on quartz as function of angle of incidence.

The inversion of the Drude equations, that is the estimation of unknown thicknesses or optical constants from ellipsometric measurements, relies upon the application of computer-intensive search and optimization methods, which are well within the capabilities of personal computers. The software for solving a wide variety of film problems is now available as part of the instrumentation package from a good number of ellipsometer manufacturers. This has resulted in the fast-widening scope of ellipsometry as reflected in the number of publications in which the technique is dominant. 

Optical techniques can be used to monitor optical thickness and dielectric constant parameters. This includes ellipsometry, multiple reflection interferometry (74), evanescent wave (75), and surface plasmon resonance spectroscopy techniques (43). Ellipsometry has been used widely and routinely to investigate film thickness of pol3mier brush films (76). For optical properties of films, it is important that the average film roughness and imiformity is specified. Often, sampling is localized by the spot size, such that it is necessary to probe and average different areas of a sample. 

Applications for ellipsometry range from the simple determination of thin films thicknesses or optical constants, monitoring reactions and film growth [37], study of plasmonic effects in meta-materials [38], to structural and optical analysis of bio-and nano-materials [39, 40]. 

Usually the samples are multilayered thin films, and the purpose of ellipsometry measurement is to get the film structure and optical constants from the measured ellipsometry parameters, such as the refractive index, extinction coefficient, and thickness of each layer, etc. for the sample. 

Where, the superscripts "meas" and "calc" represent the measured and calculated ellipsometry parameters respectively. In Eq. (3), n and m are the numbers of the measured data points and the analytical parameters, respectively. The unknown parameters in the optical model, such as film thickness or optical constants, are varied and try to produce a "best fit" to the experimental data. The best fitting results will lead to minimum of the RMSE value or a value small enough, and then the physical parameters are obtained once a good fit is achieved. 

X = 633nm, An = 0.18, Anp = 0.17, and Jp = 0.3pm. The elastic constants were given by literature values (Yeh and Gu 1999) and the thicknesses and optical anisotropy were determined by fitting the measured and calculated diffraction... 

Early work in ellipsometry focused on improving the technique, whereas attention now emphasizes applications to materials analysis. New uses continue to be found however, ellipsometry traditionally has been used to determine film thicknesses (in the rang 1-1000 nm), as well as optical constants. " Common systems are oxide and nitride films on silicon v ers, dielectric films deposited on optical sur ces, and multilayer semiconductor strucmres. 

Ellipsometry is a very powerfiil, simple, and totally nondestructive technique for determining optical constants, film thicknesses in multilayered systems, sur ce and... 

The unknown parameters of the model, such as film thicknesses, optical constants, or constituent material fractions, are varied until a best fit between the measured P and A and the calculated P/ and A/ is found, where m signifies a quantity that is measured. A mathematical function called the mean squared error (MSE) is used as a measure of the goodness of the fit ... 

Determination of the optical constants and the thickness is affected by the problem of calculating three results from two ellipsometric values. This problem can be solved by use of the oscillator fit in a suitable wavenumber range or by using the fact that ranges free from absorption always occur in the infrared. In these circumstances the thickness and the refractive index outside the resonances can be determined - by the algorithm of Reinberg [4.317], for example. With this result only two data have to be calculated. 

Manifacier JC, Gasiot J, Fillard JP (1976) A simple method for the determination of the optical constants n, k and the thickness of a weakly ahsorhine thin film. J Phys E 9 1002-1004... 

Fig. 4. Reflectivity of parallel- and perpendicular-polarized light at 1600 cm for the system Ge/Pt/ vacuum determined for platinum films of various thicknesses. In the calculations, the optical constants of bulk platinum were used nce = 4.0, npt = 5.71 + Z23.35. The angle of incidence was 45°. Calculations were performed according to the matrix formalism described in the text.

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