Ellipsometry errors

In the case of Langmuir monolayers, film thickness and index of refraction have not been given much attention. While several groups have measured A versus a, [143-145], calculations by Knoll and co-workers [146] call into question the ability of ellipsometry to unambiguously determine thickness and refractive index of a Langmuir monolayer. A small error in the chosen index of refraction produces a large error in thickness. A new microscopic imaging technique described in section IV-3E uses ellipsometric contrast but does not require absolute determination of thickness and refractive index. Ellipsometry is routinely used to successfully characterize thin films on solid supports as described in Sections X-7, XI-2, and XV-7. 

Dielectric constants of metals, semiconductors and insulators can be detennined from ellipsometry measurements [38, 39]. Since the dielectric constant can vary depending on the way in which a fihn is grown, the measurement of accurate film thicknesses relies on having accurate values of the dielectric constant. One connnon procedure for detennining dielectric constants is by using a Kramers-Kronig analysis of spectroscopic reflectance data [39]. This method suffers from the series-tennination error as well as the difficulty of making corrections for the presence of overlayer contaminants. The ellipsometry method is for the most part free of both these sources of error and thus yields the most accurate values to date [39]. 

One of the drawbacks of ellipsometry is that the raw data cannot be directly converted from the reciprocal space into the direct space. Rather, in order to obtain an accurate ellipsometric thickness measurement, one needs to guess a reasonable dielectric constant profile inside the sample, calculate A and and compare them to the experimentally measured A and values (note that the dielectric profile is related to the index of refraction profile, which in turn bears information about the concentration of the present species). This procedure is repeated until satisfactory agreement between the modeled and the experimental values is found. However, this trial-and-error process is complicated by an ambiguity in determining the true dielectric constant profiles that mimic the experimentally measured values. In what follows we will analyze the data qualitatively and point out trends that can be observed from the experimental measurements. We will demonstrate that this... 

It is easy to figure out why this is. The theory of ellipsometry assumes that the surface is atomically flat. It is possible to model roughness as a series of declivities in the surface. These are taken as being full of solution. Thus, the ellipsometer sees pools of solution where it assumes the electrode surface should be. Especially in the determination of submonolayers, the result can contain significant errors in n and K that have been calculated on the assumption of a completely smooth surface (Brusic and Cahan, 1969). 

From the data reported in Fig. 2.2 and from spectroscopic ellipsometry measurements, the anisotropic complex optical constants of oriented PPV have been determined [32,69]. Several different data analyses, described previously, were carried out on the 7Z and T spectra in order to extract n, both below and above the HOMO-LUMO transition (transparent free-standing film and bulk material, respectively). In order to evaluate n below 1.6 eV, where the sum of 1Z and T is equal to 1 within experimental error, a numerical inversion of the 7Z and T spectra was performed by assuming k = 0... 

NiFe)S0jj 6H20 by xray diffraction. The two types of measurement can be compared by using published density and refractive index. We believe that the discrepancy between the two may be an error in the ellipsometry due to the serious roughening of the surface. In our experience atmospheric corrosion almost always leads to sufficient roughening to affect ellipsometric data. This roughening must be taken into account in any model of the corrosion process. 

Figure 29 Activation plot of the mean relaxation rate of PS as measured by different experimental techniques BDS with evaporated electrodes ( capped ) and with nanostructured electrodes ( uncapped ), AC and DC calorimetry, ellipsometry, and X-ray reflectometty. Parameters to be varied are the molecular weight and the layer thicknesses of the sample as indicated. Within the error margins, the measurements of all applied techniques deliver coinciding results. Additionally, a VF fit to the dataof a 37 nm thick sample (/14,=58900gmol" ) is displayed (dashed line). Data taken from Tress, M. Erber, M. Mapesa, E. etal. Macromolecules2010, 43,9937. ...

Figure 30 Glass transition temperature fg of PMMA as measured by ellipsometry (filled symbols) and characteristic temperature r (at 120.74 Hz) measured by BDS (open symbols) plotted vs. layer thickness d. The measurements were carried out on different substrates having hydrophilic (a, Si/SiOjj to hydrophobic (b, gold coating c, silanized with hexamethyidisilazane (HMDS)) surfaces, (d) Additionally, covalently bonded PMMA brushes on Si/SiOx with strong attractive interfacial interactions were studied. The calorimetrically determined Tg of the bulk is shown by dashed lines, and the experimental error of 0.5 K (shaded region) is plotted for comparison. Reproduced with permission from Erber, M. Tress, M. Mapesa, E. U. etal. Macomolecules 2010, 43, 7729.2 ...

Neuschaefer-Rube U., Hozapfel W., Wirth F. Surface measurement applying focusing reflection ellipsometry configurations and error treatment. Measurement 2003 33 163-171... 

However, recently the fundamentals of ellipsometry have been questioned [77, 78]. The assumptions made in the four-layer model are maybe not close enough to the real situation. In the applied fitting procedure, the interface is considered as a uniform layer with a uniform refractive index. In reality, there is certainly a concentration gradient at the interface. This is a weak point of the method, but it is difficult to assess the error induced by this approximation. 

Three-parameter ellipsometry is based on the fact that there can be three independent equations and there are three unknown variables pertaining to a surface film. All three equations are related to actual quantities measured with a fixed set of wavelength and angle of incidence. Gottesfeld et al measured and R which are and respectively, and A. In another of their works, measurements of R and R in addition to A and were made. This amounts to a redundant use of the equations because the ij/ function is not independent of the R and R functions, but these four measurements may serve to enhance the accuracy of determination by suppressing measurement errors. 

Many earlier attempts to utilize reflectance data in combination with ellipsometry data seem to have failed because of improper experimental conditions which resulted in poor resolution. In particular, the proper choice of an angle of incidence has been found to be crucially important for achieving high resolution, as will be discussed in connection with error analysis in Section III.6. The often-used angle of incidence near the principal angle of incidence for many metals) turns out to be an extremely... 

The combined ellipsometry-reflectance method—three-parameter ellipsometry—has been subject to error analysis. Cahan showed that, while it is possible in principle to obtain an unambiguous solution for the optical constants and thickness of a film by three-parameter ellipsometry, the method does not guarantee that a solution can be obtained in practice. He also pointed out, by working with sample data from electrochemically produced films, that the numbers obtained as the solution are not necessarily physically real when the three-layer model is inadequate for the particular system. Chung, Lee, and Paik" studied the forward and reverse sensitivity analyses for three-parameter ellipsometry to obtain the forward sensitivity coefficients (dMldoi) > and the reverse sensitivity coefficients for a passive film on nickel (here... 

The optical principles and equations used in the technique are concisely summarized. The combined reflectance-ellipsometry (three-parameter ellipsometry) method and spectroscopic ellip-sometry are expected to be applied to an increasing number of studies in interfacial electrochemistry. The importance of proper experimental conditions, especially the proper choice of incidence angle is emphasized. Instrumentation, experimental methods, and error and sensitivity problems are dealt with. Some typical and recent applications in electrochemistry are reviewed. 

The sum in Equation [13] is taken over N wavelength points (A ), z is a vector of the fitted parameters (in this case, d nterface df and A), and m is the dimensionality of z (i.e. 3). The quantities in the denominator (8Pexp( i)) re the errors in the experimental data. The is a good metric for the goodness of fit to ellipsometry data. If is near 1 the fit is a good fit, but if it is much greater than 1 the model does not fit the data. After the parameters are determined, it is also necessary to calculate the error limits as well as the correlation coefficients of all the fitted parameters. 

Ellipsometry provided accurate and reproducible measurement of film thickness to correlate with the cooperativity of the polymer. The film thickness for each of the polymer film systems prepared from 2%, 5%, and 10% solutions was recorded in Table 4.3-1, Table 4.3-2, and Table 4.3-3 respectively. Each row represents one sample giving a total of twenty samples. The error was determined by the difference in the experimental and model curves for each sample overall, the error was a small percentage of the film thickness. 

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