Ellipsometric angles

Two measured ellipsometric angles T and d at a fixed wavelength and a fixed angle of incidence enable calculation of a maximum of two other properties, e. g. the film thickness and refractive index of a transparent layer. Multiple angle measurements increase the number of measured quantities and hence the number of properties which can be determined for a specific sample, although even under these condi-... 

Elllpsometry. The automated ellipsometer is essentially equivalent to that described by Hauge and Dill ( 9). Data acquisition and analysis was made through the use of a Digital Equipment Corporation MINC 11/23 computing system. The detailed description of the system was reported previously ( 8). The measured ellipsometric angles and 4 are converted into the... 

Fig. 32 Dependence of cosine of A and tangent of where A and are ellipsometric angles related to the change of amplitude and phase shift of the incident polarized light, on the wavelength of the incident polarized light collected from PDMAEMA- -PAA brushes immersed in solutions of a constant ionic strength 0.001 M) and pH ranging from 3.52 to 9.50. The data in figures a and b (c and d) have been collected at the position 1 (4). For clarity the data for cos(A) and tan(V ) collected at pH > 3.53 were shifted vertically by - 0.2 relative to each previous set...

Figure 9.13 Adsorption of nitrogen to a single basal plane of graphite at a temperature of 46.2 K as determined by ellipsometry. Plotted is the change in the ellipsometric angle A versus pressure. The subsequent adsorption of at least four layers at defined pressures can clearly be distinguished. Redrawn from Ref. [389].

It is useful to define the ellipsometric angles, j/ and A, for reflected and refracted light. These are related to the ratios... 

The results of this model calculation can be used to extract unknown film properties, such as the film refractive index and thickness. Normally, however, an ellipsometer will yield only two independent observables (the two ellipsometric angles defined in equation (3.6), for example). For that reason, only two film properties can be extracted from a single measurement. If the film is nonabsorbing with a real refractive index, an ellipsometric measurement can provide the real part of the refractive index, n, and the thickness. For... 

At a substrate temperature of 340 K, the adsorption of CH3 is observed with ellipsometry. The change of the ellipsometric signal is depicted in Fig. 11.7. The points 1, 2, and 3 label consecutive stages in the evolution of the ellipsometry data points in the plane point 1 represents the C D layer at the beginning of the experiment. With the onset of the CH3 flux, the ellipsometric angles shift at first to larger values of and A (point 1 to 2), followed by a shift to larger values of but smaller values of A (point 2 to 3). 

This variation of the ellipsometric angles is compared with a two-layer optical model the open squares correspond to a hard a C H film with refractive index h = 2.1 — i0.13 and variable film thickness d. At point 1, the film thickness is d = 31.62 nm while d = 31.71 nm at point 1. A polymer-like film with refractive index n = 1.62 — 0 on top of a 31.62 nm thick hard film is represented by open circles. In both model curves, film thickness increases by Sd = 0.01 nm between consecutive model points in the direction of decreasing values for A. 

Let us calculate the small contributions of nanometer layers to ellipsometric parameters. In long-wavelength approximation in the first order with respect to the small parameter djX for<54 = 4 - Lo and M = A-A, where 4, A and 4 o> o te the ellipsometric angles of an ultrathin film and a bare substrate (d, = 0), respectively, we obtain the following approximate formulas ... 

Similar unambiguous expressions for , can be obtained by combining the changes in the ellipsometric angles and the differential reflectance of s- or p-polarized light, i.e., the ratios SAI dJiJ Rj " or It must be emphasized that the... 

Let us consider the effect of dielectric nanometer film on the ellipsometric angles and on the reflectance of linearly polarized light from transparent substrate ( = 0). 

For the contributions of an ultrathin layer to ellipsometric angles we obtain the following approximate formulas ... 

Equation (111), in terms of the ellipsometric angles A and P, then reads ... 

Measurement of a Single Set or Multiple Sets of Ellipsometric Angles A and ... 

An obvious attractive feature of ellipsometry is that two parameters are obtained in a single measurement. If there is only one perfectly homogeneous surface with a complex refractive index, the real and complex part of the refractive index can be calculated from the two ellipsometric angles. The situation becomes more complicated if the assumed system becomes more complicated every thin film has a thickness and a complex refractive index, which means three additional parameters. 

The single-film model is relatively straightforward, especially for the case of a transparent film on a transparent substrate with a known refractive index. In that case the film thickness and refractive index can be obtained by measuring the ellipsometric angles at a single angle of incidence. For a few cases there are analytical solutions that directly relate film thickness and refractive index to the ellipsometric angles. 

The ellipsometric angles can be determined quite accurately and rather independently on absolute intensity. Several reversal and calibration methods are available so the ellipsometric angles can be obtained accurately and bias-free. 

An ellipsometer can be highly sensitive. The sensitivity for a film thickness can be of the order of r/nm for the sensitivity of A for a film thickness or a change in film thickness. As ellipsometric angles can be measured with about 0.01°... 

In principle matrix methods analogous to the ones discussed in the section about neutron reflectivity can be applied to calculate ellipsometric angles for an arbitrary refractive index profile (Lekner 1987) and analytical approximations have also been developed (Charmet and de Gennes 1983). In practice the use of ellipsometry to obtain fine details of the structure of interfaces at the level of tens of angstrom units is likely to be difficult and to require extreme care. 

Since the air bearing surface of the slider is carbon-overcoated, the same carbon overcoat was placed on some of the silicon strips to evaluate the PFOM film thickness and ellipsometric measurement procedure on carbon- and non-carbon-overcoated substrates. A nominally 12.5-nm-thick layer of sputtered carbon was deposited on silicon strips, and the strips were dip coated with PFOM. The ellipsometric angles A and T were measured. The two-layer model (two films on an absorbing substrate) was used with the optical constants for the materials listed in table 4.6 in calculating the PFOM thickness from A and T on carbon-overcoated silicon. The apparent... 

Optical Constants from Ellipsometry and Ellipsometric Angles A and P for the Carbon-Overcoated Silicon Strips... 

Indicates the carbon thickness used in the calculation of the PFOM thickness from ellipsometric angles A and P. 

The refractive index, n, and extinction coefficient, k, values were calculated from the measured values of ellipsometric angles P and A, using equation (2) ... 

This setup allows the determination of the unknown ellipsometric angles and can be operated in various modes. Each optical component modifies the state of polarization. Since any state of polarization can be represented by a complex Jones vector consisting of two columns, the effect of each optical components is described by a complex 2x2 matrix. The Jones formalism provides an elegant means for a quantitative description [4]. 

So far we introduced two quantitites which account for changes in the state of reflection upon reflection. We also discussed how these quantities can be measured. The next chapter deals with the theory of reflection and illustrates means for a calculation of the ellipsometric angles of a given optical layer system. 

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