Ellipsometry

Ellipsometry measures the change of polarisation of an electric field when reflected from a stack of materials. From this change, film parameters like the optical constants, ˜n=n₋ik, or the film thicknessdcan be determined, as schematically illustrated in Figure 3.3. This is possi- ble because the reflection and transmission coefficients of an electric field, which are a function of the complex optical constants, also depend on the plane of incidence (see Equation 3.7). As the components parallel and perpendicular to the electric field are changed in a different

Figure 3.3: Working principle of a spectrographic ellipsometer with rotating compensator. White light is elliptically polarised by means of a linear polariser and rotating compensator. It is incident on the sample at angleϕ. The polarisation of the reflected light,Erefl, is analysed with a second linear polariser and the spectral differentiation uses a spectrograph in combination with a CCD camera. An automated, iterative fit of a model of the sample to the measured ellipsometric parameters,Ψand∆, is used to determine e.g. the complex refractive index.

manner upon reflection and transmission, the polarisation of the light changes. This section will describe the theory behind ellipsometric measurements and is based on descriptions in Tompkins & Irene_[194_], J. A. Woollam Co., Inc._[195_], and Cobet_[196_].

In order to record the change induced by the probed material, the electric field before and after incidence onto the surface needs to be determined and compared. Figure 3.3 shows that this is done by placing one linear polariser between source and sample and another as analyser between sample and detector. Additionally, a compensator or retarder can be built in after the polariser or before the analyser. This setup creates elliptically polarised light because polarisa- tions along different axes of the compensator propagate at different speeds. The challenge of the method lies in determining the change of polarisation from a measurement of intensities on the detector. For this, the polarisation state of the light incident on the sample and of the anal- yser needs to be known. In order to obtain information on the properties of the sample, (i.e., to distinguish between absorption and suppression by different polarisations in analysed light and analyser) either of the polarisations needs to be varied, which is usually done by rotating one component continuously. An internal, instrument-dependent function then calculates the change in polarisation, from which the ellipsometric parameters can be determined. Depend- ing on the nature of the sample, the number of parameters that are necessary to fully describe the optical response ranges from two (simplest, assuming in-plane isotropy) to six or more. In

this thesis, only two parameters were necessary to represent the optical response of the stacks:

ρ= rk

r_⊥ :=tan(Ψ)·e

i∆_{. (3.12)}

Here, ρis the complex ratio of reflection coefficients for p-polarised and s-polarised light,r_k andr_⊥, respectively. Ψ and∆serve to facilitate the analysis by splittingρinto an amplitude and a phase term, respectively.

Similar to the transfer matrix method explained in Section 3.1, the change of the electric field is described with one matrix per optical component. The interaction with the sample is also described by matrices. Since the incident and reflected field are measured and the matrices for polariser, compensator and analyser are known, the reflectivity of p- and s-polarised light, and thusΨand∆, can be deduced.Ψ and∆are the output parameters of the measurement. However, it is not trivial to obtain the complex refractive index from this data—especially, when p- and s-polarisation are cross-correlated in anisotropic media.

In order to draw any conclusions about the properties of the sample from the knowledge of Ψ and ∆, a model of the sample needs to be built. This model represents all relevant interfaces with defined thickness and potential surface roughness, and all layers with their optical constants. The system is defined as well as possible (input of known parameters versus variables) to increase the specificity of the fit. All unknowns are set as variables, e.g. the thickness of the film and/or parameters in model functions for optical constants. In an iterative process, the variables in the model are then fitted to approximate the measured parameters. While the fitting process and the compliance with the Kramers–Kronig relations∗for the optical constants are software controlled, the different models are defined by the user. Thus, the quality of the data obtained by ellipsometry measurements strongly depend on the adequacy of the chosen model and thus on the user.

In practice, the number of variables in the system is held as small as possible. Experimen- tally, this means that thin films are analysed on substrates with established optical constants and films are intended to be homogeneous with little surface roughness. The number of vari- ables in the simulations is kept low by determining different parameters successively. For instance, a first fit to determine the thickness of the film only considers wavelengths far above

∗_{The Kramers–Kronig relations link the real and imaginary part of an arbitrary complex function (the only restric-}

tion being analyticity for positive imaginary numbers) via the Cauchy Principal Value. Physically, this connection makes sense because both dispersion and attenuation are caused by dipole interaction with the incident light.

the bandgap. This is useful because the extinction coefficientkcan be neglected in this range and the refractive index can be expected to follow a Cauchy model,n(λ) =A+B/λ2+C/λ4. Once the thickness of the film and the background refractive index are established, a fit over the entire range yields the full set of isotropic optical constants obeying the Kramers-Kronig relations. Depending on the expected micro-structure of the sample, an anisotropic fit is per- formed subsequently using the isotropic optical constants as starting point.

The data presented in this work were obtained from a variable-angle spectroscopic ellip- someter with rotating compensator, M-2000DI by J.A. Woollam Co., Inc. In the instrument, white light by a Deuterium and a quartz tungsten halogen lamp is used to cover the spectral range from 200 to 1700 nm. Angles can be varied between 45◦and 90◦in reflection and arbi- trarily in transmission mode, which is important for acquiring ellipsometric data close to the Brewster angle. The rotating-compensator configuration ensures high accuracy for the entire range of ellipsometric angles (∆) across the entire wavelength range and enables to determine the sign of∆. This is in contrast to rotating polariser or analyser configurations, where the accuracy drops for∆≈0,πand the sign of∆is not accessible. Ellipsometric data analysis is performed using the WVASE32 software by J.A. Woollam Co., Inc.