## 4 Distribution of the Optical Properties  
The 25 mm \(\times\) 10 mm combinatorial sample was scanned by a M-2000 DI rotating compensator spectroscopic ellipsometer (Woolam Co. Inc., Lincoln, NE, USA). A focused spot has been moved parallel to the long edge along the center line of the wafer. The plane of incidence was parallel to the short edge. The angle of incidence was varied between 60\({}^{\circ}\) and 70\({}^{\circ}\), and the corresponding size of the focused spot was 0.3 mm wide and 0.6-0.9 mm long. The measurement time was a few seconds for one spot and one angle of incidence. That provided high-resolution and high-accuracy of the data library with a reasonable acquisition time of the entire map (Figure 4). In order to determine the parameters of the deposited a-Si\({}_{1-x}\)Ge\({}_{x}\) films, an optical model was constructed to calculate the ellipsometric angles from the model, and to fit the parameters of the model to obtain a good match between the measured and fitted ellipsometry spectra. Since the a-Si\({}_{1-x}\)Ge\({}_{x}\) film was deposited on a silicon wafer, the model uses a Si substrate assuming a vertically uniform structure, in which only a surface nanoroughness was taken into account by the usual model of mixing 50% layer material with 50% void. The dispersion of the layer was modeled using the Tauc-Lorentz (TL) approach [25]. In this parameterization, the dielectric function is described as a combination of a Lorentz oscillator with the three usual parameters (the broadening, the amplitude and the peak energy position) and an additional Tauc gap parameter. This approach allows that the total number of fitted parameters is kept at a reasonable value (thicknesses of the roughness layer and the film, and the TL parameters. The spectra measured at each position (each composition) have been fitted independently, by changing the above parameters to find the best match between the measured and calculated ellipsometry spectra. The spectra have been recorded at three angles of incidence providing six measured values at each wavelength, and a total of about 2000 measured values for each scanned position. Consequently, the fit was robust, avoiding local minima in the process, resulting in a smooth  
Figure 3: High resolution TEM image of a 10 nm thick self supporting combinatorial SiGe sample taken for the composition of \(x=0.5\). map of the optical spectra, as shown by Figure 4. Note the smooth changes of the spectra in Figure 4 in spite of the independent fit procedures at each position, which show the reliability of the fit procedure. The measured \(\Psi\) and \(\Delta\) spectra fitted by the TL parameterization provide us not only with the thickness of the layer (\(\approx\)100 nm) and the surface roughness (less than 3 nm) but also with \(n\) and \(k\) spectra at each spatial position of the sample. The maps in Figure 4 show those spectra as a function of the position along the center line of the wafer, with the zero position corresponding to the composition of \(x=0.5\). To check the crystallinity of the layer, High Resolution Transmission Electron Microscope (HRTEM) and selected area electron diffraction (SAED) investigations were carried out by a 300-kV JEOL 3010 HRTEM (Japan Electron Optics Laboratory, Tokyo, Japan) with 0.17 nm point resolution. Both the granular image contrast and the diffuse rings of the inserted selected area electron diffraction pattern represent an amorphous structure (Figure 3). The most important feature of the optical properties is that both \(n\) and \(k\) show a linear dependence on the position in the whole measured photon energy range, as shown by the color stripes with straight edges corresponding to large changes of the optical properties as a function of the photon energy. This is remarkable, because together with the RBS results of Figure 2 that indicates a linear dependence of the composition on the position (also shown in Ref.