Ellipsometric parameter

RgureS Three-dimensional plot of predicted ellipsometric parameter data versus angle of incidence and wavelength. 

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. WooUam, and S. A. Alterovitz. /. ofAppL Phys. 60, 3293, 1986. First use of computer drawn three-dimensional surfaces (in wavelength and angle of incidence space) for ellipsometric parameters r and A and their sensitivities. 

In addition to its handedness, a vibration ellipse is characterized by its ellipticity, the ratio of the length of its semiminor axis to that of its semimajor axis, and its azimuth, the angle between the semimajor axis and an arbitrary reference direction (Fig. 2.13). Handedness, ellipticity, and azimuth, together with irradiance, are the ellipsometric parameters of a plane wave. 

Although the ellipsometric parameters completely specify a monochromatic wave of given frequency and are readily visualized, they are not particularly conducive to understanding the transformations of polarized light. Moreover, they are difficult to measure directly (with the exception of irradiance, which can easily be measured with a suitable detector) and are not adaptable to a... 

The Stokes parameters are related to the ellipsometric parameters as follows ... 

Thus, the Stokes parameters are equivalent to the ellipsometric parameters although less easily visualized, they are operationally defined in terms of measurable quantities (irradiances). Additional advantages of the Stokes parameters will become evident as we proceed. Note that Q and U depend on the choice of horizontal and vertical directions. If the basis vectors 0 ( and , ... 

A In connection with another symbol it is a difference, ellipsometric parameter... 

Figure 6.4-4 Elliptical polarization The path of the tip of the electric vector as projected into a plane perpendicular to the direction of propagation. The ellipse is generated by mutually perpendicular oscillations with amplitudes and rp, resp., which are phase shifted by an angle A. It is fully defined by the ellipsometric parameters A and tp = tan ( rj / r ).

Figure 6.4-5 Simulated spectra of a strong oscillator (strength/ = 200 10 cm with resonance at t> real and imaginary part of the dielectric function = s + k", refractive index n and absorption index k, ellipsometric parameters A and ip, as well as reflectance R for the angles of incidence and the polarization states stated.

With ellipsometry the polarization state of reflected radiation rather than just its intensity, is experimentally determined. Ellipsometry is not so much another experimental technique but a more thorough variety of the traditional ones, whether external or internal reflection. Two results per resolution element, namely the ellipsometric parameters (cf. Eq. 6.4-17) and A, are derived independently from the measurements. These can further be evaluated for the two optical functions of the medium behind the reflecting surface or other two data of a more complex sample. In any case there is no information necessary from other spectral ranges as it is for Kramers-Kronig relations. In comparison to the conventional reflection experiment, ellipsometry grants more information with a more reliable basis, e.g. since no standards are needed. 

Now the state of polarization of the reflected radiation can be determined completely and specified by the coordinates on a Poincar6 sphere or by the full set of four Stokes parameters s (cf. Sec. 3.2) the latter are related to the ellipsometric parameters by... 

The ellipsometric parameters xl> and A experimentally determined with a homogeneous thick sample, are algebraically related with the components of the dielectric function, which in turn define the refractive index and the absorption index (Bom and Wolf, 1980). The parameters for the strong oscillator used to simulate the spectra shown in Fig. 6.4-5 were chosen to resemble the strong infrared resonance of quartz glass. Radiation reflected from such a sample was measured ellipsometrically the evaluation led to the results presented In Fig. 6.4-14. For weaker absorbers such as many molecular compounds. 

The analysis is carried out using the Drude equations this leads to a combination of the ellipsometric thickness and the refractive index Increment. These characteristics of the adsorbate cannot be unambiguously separated. Conversion of the refractive index increment into the composition of the adsorbate layer is usually done by assuming drt/dx to be the same as in a fluid of composition x for 0 not too high this is usually allowed, but problems may arise when the adsorbate differs substantially from the solution, for Instance because of alignment of adsorbed chain molecules. The result obtained is not unique, in the sense that different profiles may lead to the same pair of ellipsometric parameters. Therefore, normally totally adsorbed amounts are presented. For accurate measurements a good optical contrast between adsorbate and solution is mandatory. 

Figure 4a illustrates the spectral dependence of ellipsometric parameters and A for the hybrid sample Ag/APTES/Si. Experimental spectra were fitted by the optical response of one effective layer. According to model calculations for this sample, the thickness of the effective layer and APTES film was 5.3 and 11.5 nm, respectively. The spectral dependence of the dielectric function for the effective layer (Fig. 4b) possesses two features. The low-energy peak at 2.2 eV can be attributed to the residual material of the solution containing the P VP-coated Ag nanoparticles. The peak can be also contributed by the interparticle dipole-dipole couplings of nanoparticles on solid substrates. The peak at the 3.4 eV is related to the surface plasmon resonance of metal nanoparticles and corresponds to the absorption peak of Ag colloidal solution (Fig. 4b). In the spectra of hybrid samples Ag/DNA/APTES/Si, the peak at 4.5 eV originated from the contribution of DNA was additionally observed.

Fig. 4. (a) Experimental (points) and modelled (curves) of ellipsometric parameters T and A for the sample Ag/APTES/Si and (b) dielectric function spectra of the effective surface layer obtained from the fitting procedure as compared with the absorption spectrum of colloidal solution of Ag nanoparticles. 

The basic ellipsometric parameters are defined in Figure 17.1.8. The difference in phase angle between the leading and trailing components is given by A, and the ratio of electric field amplitudes defines the second parameter, ifj ... 

The effect of nanometer dielectric films on the ellipsometric parameters and reflectance of linearly polarized light is investigated within the framework of the perturbation theory. The novel approach is developed for simultaneous determining the thickness and dielectric constant of nanometer-scale films by the differential reflectance and ellipsometric measurements. 

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 ... 

Next, in the case of transparent substrate the pure ellipsometric method is not applicable for the determination of e on the basis of approximate Eqs. (6) and (7) whereas the relation is independent of. If ellipsometric parameter 5A is... 

The thickness, dispersion of refractive index, and absorption of the obtained films were measured on a spectral elUpsometer FIlipw developed at the AV.Rzhanov Institute of Semiconductor Physics SB RAS (http //www.isp.nscru/). The optical parameters of the film according to ellipsometric parameters delta (A) and p i ( P) were found by approximating the single-phase model of the Si-substrate/ absorbing film. 

At planar interfaces there are four main types of linear optical signals that are detected by different techniques. Three of them are related to reflection and one to attenuated total reflection. In reflection methods the basic measurable parameters are related to rp and rs - the complex am-phtudes of the reflection coefficients of the light polarized parallel p) and perpendicular s) to the incidence plane, respectively (Fig. 17). These are reflection coefficient = rs(p), phase of reflected light 5j(p) = (l/2/)lnK(p)/rf(p), ellipsometric parameters = tan rp/rs and A =... 

---