The next step in my project was to take ellipsometry data on the sapphire sample. The purpose of this experiment is to obtain the optical constants of the sample, so that we can use this information in later experiments. This is necessary because ellipsometry measurements characterize the entire system, so if we are interested in a film grown on a sapphire substrate, the ellipsometer will give us information about the film and the substrate together. Modeling the substrate independently is critical, so we can correctly isolate the optical properties of the film.
Ellipsometry is an optical procedure that measures the change in polarization state of a beam of light after it is reflected off of (or transmitted through) a sample. The ellipsometer first emits a beam of light of known polarization, and detects the polarization state of the beam after reflection. The ellipsometer actually measures two parameters called Δ (delta) and Ψ (psi). The tangent of psi is the ratio of the magnitudes of the total reflection coefficients, Rp and Rs. These are the reflection coefficients for the p- and s- polarized components of the light wave, respectively. The p-component is the component that is linearly polarized parallel to the plane of incidence, which is defined by the normal to the sample face and the incident beam, and the s-component is polarized perpendicular to the plane of incidence. Any polarized light can be described in terms of these two components. When a beam of light is reflected, the phase of the p- and s- components shift, but the shift is not necessarily the same for each component. The difference between these two shifts is delta. The reflection coefficients are the ratios of the amplitudes of the incident and reflected waves. Psi and delta characterize the polarization change of the reflected beam and can be modeled to obtain the optical constants n and k, which are the refraction index and extinction coefficient respectively of the substrate. In the set-up and data analysis for this experiment, I worked with Tyler Huffman and Peng Xu, who are graduate students in our lab. Tyler and I worked on experimental set up and procedure, while Peng helped me with the data analysis and modeling. We did reflection ellipsometry with angles of incidence of 55°, 65°, and 75°. In theory, we only needed one angle to characterize our sample, but using the multiple angles of incidence gives us more constraints to make the modeling more accurate. We chose these angles because they were far enough away from the Brewster angle (almost exactly 60° for sapphire) that they would not affect our data. The Brewster angle is a unique angle of incidence that polarizes any incident light by reflecting only s-polarized light. This would render our data useless because we would get no intensity for the p-polarized component, so we would have divide by zero errors in Psi, and Delta would not be correct. Once we were sure that we had avoided the Brewster angle, we needed to figure out how to model the anisotropy of the sample.
Once we obtained data for n and k for a wavelength range of .6 to 6.5 eV, we modeled the data with Cauchy functions to give us a curve that defines the value of n as a function of wavelength. The Cauchy function we used for fitting is given by the following:
n(λ) = An + (Bn/λ2) + (Cn/λ4)
where An, Bn, and Cn are fitting parameters that were adjusted for our data, and λ is the wavelength of the beam. Once we had values for n and k, we digitized Graph 2, taken from a paper on sapphire by H. Yao and C. H. Yan to check our data. We found that the n values for the ordinary ray were greater than those of the extraordinary ray by about .008, in accordance with Yao and Yan, and our results agreed with theirs to within 2% of the value.
H. Yao and C. H. Yan, Journal of Applied Physics. 85, 6717 (1999).