2 vtcmag@vtcmag.com November 2014฀฀•฀฀Vacuum Technology & Coating I n 1887, Paul Drude introduced ellipsometry to the world. 1 Ellipsometry is an optical tool that provides information about the interaction of light with materials. It is fast, gen- erally non-destructive, and non-invasive. It can be applied to characterize surfaces, interfaces, alloys, and multilayered films. It is a ratiometric technique so it (i) does not depend on the in- tensity of the light it employs, and (ii) provides highly sensi- tive, accurate, and reproducible results. Although, ellipsometry provides accurate information about film thicknesses, optical constants, roughnesses, and inhomogeneities, it is not capable of measuring any of these parameters directly. It only measures two quantities: ψ and ∆. The quantity ∆ represents the phase difference between the p- and s-components of polarized light that are reflected from a surface. Tan ψ represents the ratio of their amplitudes. To derive material properties from an ellipsometric measure- ment, one must in general build a model. Based on the physical laws that govern the interaction of light with matter, e.g., Fres- nel’s equations, predicted values of ψ and ∆ are generated from one’s model. In a subsequent regression analysis, the parameters of the model, e.g., film thickness or optical constants, are varied to minimize the difference between the predicted and experi- mental values of ψ and ∆. To become successful in model building in ellipsometry, the following are important. First, data fitting requires practice. Sec- ond, it is helpful to read the literature and consult with more ex- perienced colleagues. Third, it is useful to get to know some of the commonly used models that are used for different types of materials. These include the Cauchy, Sellmeier, Gaussian, Lo- rentzian, Drude, and Tauc-Lorentz models. Each is useful for a broad range of materials. There are also other models that one may wish to become familiar with. Fourth, when dealing with complex materials and building ellipsometric models for them, Models in Ellipsometry: The ‘No Model’ Model (Just Monitoring Psi and Delta) By Anubhav Diwan and Matthew R. Linford, Contributing Editors it is often very helpful to consider the information obtained from other analytical techniques. Information from atomic force mi- croscopy (AFM), transmission electron microscopy (TEM), scanning electron microscopy (SEM), X-ray diffraction (XRD), etc. can guide the creation of an ellipsometric model so that it stays grounded in reality, remaining consistent with other reli- able information. In this contribution the ellipsometric model we consider is no model at all – we referred to this situation in the title (with a little humor) as the ‘no model’ model. Here one simply plots ψ and/or ∆ as a function of some process variable, such as time, tempera- ture, or exposure of one’s material to a reagent. Alternatively, one may plot ψ and ∆ against each other. The fundamental premise behind this approach is very simple. As long as ψ and ∆ aren’t changing, one assumes that one’s material isn’t changing either. When ψ and ∆ do change, one assumes something has happened to one’s material. Thus, simply monitoring ψ and ∆ is a powerful way of studying many material processes. Examples from the Literature of the ‘No Model’ Model Example 1. Analysis of thin films of Cr on silicon. Tompkins and coworkers 2 studied different thicknesses of chromium on silicon, plotting the resulting ∆ and ψ values that were obtained at a single wavelength (632.8 nm), and at an angle of incidence of 70°. This is the wavelength of the HeNe laser that has been used in many single wavelength ellipsometers. Many people to- day use spectroscopic (multi-wavelength) ellipsometers. Figure 1 shows a plot of ∆ vs. ψ for various thicknesses of the metal. The sensitivity of the technique to the thinner films is apparent in this plot – notice how much ψ and ∆ change along the path between the film-free substrate and the 10 nm film. In contrast, Figure 1 teaches that ellipsometry has less and less power to differentiate between increasingly thick films of Cr – consider