## Ellipsometry

Morten Vesterager MadsenEllipsometry is a sensitive optical method proposed by Paul Drude (1863 - 1906) and has thus been used for over a hundred years to derive information about surface and bulk properties.Drude, P. Ueber Oberflächenschichten. I. Theil. Annalen der Physik 1889, 272, 532-560. It makes use of the fact that the polarization state of light changes when the light beam is reflected from a surface and the technique makes it possible to deduce information about the film properties, especially the film thickness. Generally optical measurement techniques are of great interest since they under normal circumstances are non-invasive and non-destructive. Ellipsometry is no exception. The basic principle of ellipsometry is, as mentioned, that upon reflection the polarization changes. The exact nature of the polarization change is determined by the properties of the sample including thickness and complex refractive index. The main advantage of ellipsometry is that, in opposition to other optical techniques that are inherently diffraction limited, ellipsometry exploits phase information and the polarization state of light, and can achieve angstrom resolution. In its simplest form, the technique is applicable to thin films with thickness less than a nanometer and up to thicknesses of several micrometers. An obvious application of ellipsometry is the use in the semiconductor industry, where thin layers of silicon dioxide are a central element throughout production. Ellipsometry enables process engineers to keep track of the thickness of the film.

In the field of organic solar cells several reports exist on applications of ellipsometry for determining optical constants and thickness, surface roughness, and morphology. While determination of thicknesses and optical constants are an important application of ellipsometry, this application is mostly used to augment other measurement and to optimize processes.DOI:10.1080/713738799DOI:10.1002/pip.1190 A more advanced use of ellipsometry is the use of ellipsometry to study the morphology of the bulk hetero junction. A number of approached to this exists in literature. Campoy-Quiles et al. have demonstrated work modeling the vertical composition profile of P3HT:PCBM films and reported a composition gradient varying from PCBM-rich near the PEDOT:PSS layer to P3HT-rich at the air interface.DOI:10.1038/nmat2102 This result is important in the understanding of the performance of solar cells made by spin coating. Germack et. al. have substantiated the results and proposed that changes in the surface energy significantly affects the vertical composition profile.DOI:10.1021/ma100027b Their analysis was based spectroscopic ellipsometry and near-edge X-ray absorption fine structure spectra.

### Ellipsometry theory

Ellipsometry, as mentioned, is designed to measure the change of polarization upon reflection or transmission. Calculations of the polarization state are therefore tied to the electric field vector, defining the direction of the polarization of the light wave. The electric field vector is decomposed into two components named p and s respectively, a tradition originating from their German names Parallel and Senkrect. Ellipsometry is primarily interested in how p- and s- components change upon reflection or transmission in relation to each other. The change in polarization is commonly written as $$\frac{r_p}{r_s} = \tan \left( \Psi \right) e^{i\Delta}$$ The right side of the equation is describing the measurement with $\tan \left( \Psi \right)$ representing the amplitude ratio upon reflection, and $e^{i\Delta}$ the phase shift. The left side of the equation describes the sample with rp and rs being the two components of the reflection coefficient. As ellipsometry measures a ratio of two values rather than an absolute value of either, the measurement is robust, accurate, and reproducible. For instance, ellipsometry is relatively insensitive to scattering and fluctuations, and requires no standard sample or reference beam. However, as ellipsometry is an indirect method, where the measured Ψ and Δ cannot be converted directly into the optical constants of the sample, a model analysis must be performed. Direct conversion into real data is only possible in simple cases of isotropic, homogeneous and infinitely thick films. In all other cases a layer model must be established, which considers the optical constant and thickness parameters of all individual layers of the sample including the correct layer sequence. Then using an iterative procedure unknown optical constants and / or thickness parameters are varied, and the right side of the equation is calculated using the Fresnel equations for $r_s$ and $r_p$. The best match provides the optical constants and thickness parameters of the sample. Roughness for example can be included in the model by using a effective medium approximation; effectively changing the optical constants in the model.

### Ellipsometry measurements

Obtaining $\Psi$ and $\Delta$ from the ellipsometric measurement is very dependent on the type of ellipsometer used. Rotating analyzer ellipsometry is probably the most widespread technique, but the technique has a weakness in that it is not capable of determining the phase $\Delta$, but rather $\cos \left( \Delta \right)$ . A rotating compensator ellipsometer is capable of overcoming this issue.

Rotating analyzer or rotating polarizer ellipsometry is a simple form of ellipsometry employed by many manufactures. See Figure 1 for a schematic representation of a rotating analyzer setup. The setup is based on an electromagnetic radiation emitted by a light source and linearly polarized by a polarizer. After reflection from the sample the radiation passes a second polarizer, used to analyze the polarization, and then falls into the detector. Using the Jones matrix formalism it is possible to describe the light passage through a rotating analyzer ellipsometer. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light $${\overrightarrow E _0} = {T_A}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over R} \left( {{\alpha _1}} \right){\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} _S}{\vec E_i} = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos \left( {{\alpha _1}} \right)}&{\sin \left( {{\alpha _1}} \right)}\\ { - \sin \left( {{\alpha _1}} \right)}&{\cos \left( {{\alpha _1}} \right)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{r_p}}&0\\ 0&{{r_s}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{E_i}\cos \left( {{\alpha _0}} \right)}\\ {{E_i}\sin \left( {{\alpha _0}} \right)} \end{array}} \right]$$ ${\vec E_i}$ is the Jones vector representation of the incident electric field after a linear polarizer. $\vec T$ is the sample reflection, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over R} \left( {{\alpha _1}} \right)$ is the rotation to match the coordinate system of the analyzer, and $T_A$ represents the analyzer. For a rotating analyzer ellipsometer $\alpha_1$ is changed to get several intensity readings. A rotating polarizer ellipsometer instead rotates the polarization of the incoming light. The intensity at the detector is the absolute value of the outgoing electric field ${_0} = {\vec E_0} \cdot \vec E_0^*$. By introducing the stokes parameters; ${s_0} = {\left| {{E_p}} \right|^2} + {\left| {{E_s}} \right|^2}$, ${s_1} = {\left| {{E_p}} \right|^2} - {\left| {{E_s}} \right|^2}$ , and ${s_2} = {E_p}E_s^* + {E_s}E_p^*$ the intensity can be written as $$I{_0}\left( {{\alpha _1}} \right) = \frac{1}{2}\left[ {{s_0} + {s_1}\cos \left( {2{\alpha _1}} \right) + {s_2}\sin \left( {2{\alpha _1}} \right)} \right]$$ The value of the Stokes parameters can be determined experimentally by conducting measurements of the intensity at a minimum of three rotations of the polarizer ( $\alpha$). Hereby three equations for $I_0$ is introduced with the three Stokes parameters as the only unknowns. It is possible to express the elliptical parameters by the Stokes parameters $$\Psi = \frac{1}{2}{\cos ^{ - 1}}\left[ { - \frac{{{I_0}\left( {0^\circ } \right) - {I_0}\left( {90^\circ } \right)}}{{{I_0}\left( {0^\circ } \right) + {I_0}\left( {90^\circ } \right)}}} \right]$$ and $$\Delta = {\cos ^{ - 1}}\left[ {\frac{{2{I_0}\left( {45^\circ } \right)}}{{\left( {{I_0}\left( {0^\circ } \right) + {I_0}\left( {90^\circ } \right)} \right)\sin \left( {\Psi '} \right)}} - \frac{1}{{\sin \left( {\Psi '} \right)}}} \right]$$ In the above example it was assumed that measurements of the intensity were conducted at 0°, 45°, and 90°. The value of $\Delta$ is an inverse cosine function. This means that the precision and accuracy is poor when $\Delta$ is near 0° or 180°. For applications not requiring several angles of incident to be measured this is not a big issue. It will then be possible to conduct the measurements near the Brewster angle and maintain good accuracy of $\Delta$. If several angles of incident is necessary in order to fit the model, poor accuracy in $\Delta$ can be problematic. This condition is encountered as an example when trying to model in-depth morphology; since multiple angles of incidence yields measurements at different optical path lengths providing valuable information. It is possible to install a compensator element into the beam path either before or after the sample. The compensator can convert the near linear polarization state near $\Delta = ^{\circ}$ or $180^{\circ}$ to a near circular polarization state ($\Delta = =90^{\circ}$), optimizing sensitivity for $\Delta$. Hereby both $\Psi$ and $\Delta$ can be accurately measured over their full ranges. However, a perfectly ideal spectroscopic compensator element does not exist and compensator elements which can be used spectroscopically are not achromatic. This means that the retardance of the compensator must be calibrated throughout the entire spectral range. Otherwise the accuracy of the ellipsometric data will be degraded by the introduction of the compensator element.

An alternative approach to introduce a compensator into the ellipsometer beam path is to implement the rotating compensator ellipsometer configuration. This setup is not restricted to measuring only $\cos \left( \Delta \right)$, since the rotating-compensator instrument provides all four Stokes vector components for the light beam reflected from the sample surface. In contrast, the rotating-polarizer instrument provides only three such components. This also means that this configuration is capable of measuring the depolarization which occurs from samples with non-uniform film thickness, roughness and other sample inhomoginities.

### Data analysis

After a sample is measured and the right side of the ellipsometry equation has been determined, a model must be constructed to describe the sample. The model is used to calculate the response from the Fresnel equations which describe each material with thickness and optical constants. When the values are not known they become fitting parameters for which a preliminary guess is applied. The calculated values from the left side of the equation are then compared to the experimental data. Any unknown parameter can be varied to improve the match between experiment and calculation. The best match between the model and the experiment is found through regression, where an estimator, like the Mean Squared Error (MSE), is used to quantify the difference between curves. The unknown parameters are allowed to change until the minimum MSE is reached. It important at this point to notice that the process of fitting can be complicated, and that many local minima may exist. It is very possible for the regression algorithm fall into a local minimum depending on the initial parameter guess.

The simplest example of an ellipsometry model comes in the form of a bulk sample (infinite film), see Figure 2.2. Following Equation 2.1 the model must describe the ratio of rp and rs. For the infinite film approximation $r_p$ and $r_s$ are simply given by the Fresnel reflection coefficients $${r_s} = \frac{{{n_0}\cos \left( {{\theta _0}} \right) - {n_1}\cos \left( {{\theta _1}} \right)}}{{{n_0}\cos \left( {{\theta _0}} \right) + {n_1}\cos \left( {{\theta _1}} \right)}}$$ and $${r_p} = \frac{{{n_0}\cos \left( {{\theta _1}} \right) - {n_1}\cos \left( {{\theta _0}} \right)}}{{{n_0}\cos \left( {{\theta _1}} \right) + {n_1}\cos \left( {{\theta _0}} \right)}}$$ where $n_0$ and $n_1$ is the index of refraction for medium 0 and 1 respectively. The refracted angle ( $\theta _1$) is related by Snells law to $n_0$, $n_1$, and $\theta _0$. Thereby the index of refraction for medium 1 remains the only unknown. Solving the ellipsometry equation with the simple Fresnel coefficients yields $${n_1} = {n_0}\sin \left( {{\theta _0}} \right){\left[ {{{\left( {\frac{{\tan \left( \Psi \right)\exp \left( {i\Delta } \right) - 1}}{{1 + \tan \left( \Psi \right)\exp \left( {i\Delta } \right)}}} \right)}^2}{{\tan }^2}\left( {{\theta _0}} \right) + 1} \right]^{\frac{1}{2}}}$$ A negative solution for the equation also exist, however, since the refractive index cannot be negative this solution is not shown. It follows that refractive index can directly be calculated from the ellipsometric parameters. No fitting is therefore necessary in this case.

A case of importance in ellipsometry is an optical system consisting of an ambient-film-substrate system as shown in Figure 3. When the refractive index of the film and the substrate is known it is possible to determine the thickness of the film in such a system by utilizing the Fresnel coefficients. For the single layer ontop of a substrate the coefficients are given by the Airy formula $${R_p} = \frac{{{r_{01,p}} + {r_{12,p}}\exp \left( { - i2\beta } \right)}}{{1 + {r_{01,p}}{r_{12,p}}\exp \left( { - i2\beta } \right)}}$$ and $${R_s} = \frac{{{r_{01,s}} + {r_{12,s}}\exp \left( { - i2\beta } \right)}}{{1 + {r_{01,s}}{r_{12,s}}\exp \left( { - i2\beta } \right)}}$$ where $r_{01}$ and $r_{12}$ are the reflection parameters for the ambient-film and the film-substrate system respectively. $\beta$ is the phase angle containing the thickness of the film and is given by $$\beta = \frac{{2\pi d}}{{\lambda {n_1}\cos \left( {{\theta _1}} \right)}}$$ where $d$ is the thickness, and $\lambda$ the wavelength. Inserting $R_p$ and $R_s$ into the ellipsometry equation yields a complex quadratic equation for $\exp \left( { - i2\beta } \right)$ which can be solved as $$\exp \left( { - i2\beta } \right) = \frac{{ - \left( {\frac{{{R_p}}}{{{R_s}}}E - B} \right) \pm {{\left[ {{{\left( {\frac{{{R_p}}}{{{R_s}}}E - B} \right)}^2} - 4\left( {\frac{{{R_p}}}{{{R_s}}}D - A} \right)\left( {\frac{{{R_p}}}{{{R_s}}}F - C} \right)} \right]}^{\frac{1}{2}}}}}{{2\left( {\frac{{{R_p}}}{{{R_s}}}D - A} \right)}}$$ where $A = {r_{01,s}}{r_{12,s}}{r_{12,p}}$, $B = {r_{12,p}} + {r_{01,p}}{r_{01,s}}{r_{12,s}}$, $C = {r_{01,p}}$ , $D = {r_{01,p}}{r_{12,p}}{r_{12,s}}$ , $E = {r_{12,s}} + {r_{01,s}}{r_{01,p}}{r_{12,p}}$, and $F = {r_{01,s}}$. This allows the thickness d to be calculated since the thickness is only represented in $\beta$. However, since the thickness is given in a complex exponential no single solution for the thickness exists. The thin film approximation deals with this issue by assuming that the lowest positive thickness value is the correct thickness. To determine the thickness for thicker films it is possible to conduct measurements for several wavelengths and thereby introduce more equations. For multiple isotropic layers, the calculation of the complex reflection coefficients is more complicated and performed using a matrix representation, where each layer is represented by two 2 X 2 complex matrices, one for the pp polarization and the other for the ss polarization.

### Optical coefficient parameterization

With ellipsometry the most typical situation with an ambient-film-substrate system is to know the complex refractive index of the substrate (either by previous measurement or from a table value), but not the thickness nor the complex refractive index of the film. In this case it is never possible to directly calculate all three unknowns. With spectroscopic ellipsometry the situation is better. However, since the refractive index is wavelength dependent the introduction of more wavelength add as many unknowns as equations. It is therefore necessary to model the optical dispersion by a simplified model to determine both optical constants and thickness. This parameterization of the optical components is done though a dispersion law simulating the optical indices and their variation according to the wavelength. A very common optical dispersion is the Cauchy optical dispersion where six parameters are used $$n = A + \frac{B}{{{\lambda ^2}}} + \frac{C}{{{\lambda ^4}}}$$ and $$k = \frac{D}{\lambda } + \frac{E}{{{\lambda ^3}}} + \frac{F}{{{\lambda ^5}}}$$ By using the dispersion relation the system becomes over determined making the fitting of the parameters more robust. The Cauchy dispersion is often used as a simple approach to determine the thickness of a film. If a wavelength range exist where the film has zero absorbance the $k$ component vanishes and only three fitting parameters remains beyond the thickness.

Another dispersion model often used is the the Tauc Lorentz model. This is typically used for the parameterization of the optical functions for amorphous semiconductors and insulators for which the imaginary part of the dielectric function $\varepsilon _i$ is determined by multiplying the Tauc joint density of states by the $\varepsilon _i$, as obtained from the Lorentz oscillator model. The real part of the dielectric function $\varepsilon _r$ is calculated from $\varepsilon _i$ using Kramers-Kronig integration, making the model Kramers-Kronig consistent.

### Effective medium approximation

Using an effective medium approximation (EMA), mixtures of materials with known refractive can be described. The EMA is a physical model that describes the macroscopic properties of a medium based on the properties and the relative fractions of its components. Based on the additive character of the polarizability, a generalization of the Claussius-Mossotti formula can be written as $$\frac{{\left\langle \varepsilon \right\rangle - {\varepsilon _h}}}{{\left\langle \varepsilon \right\rangle + 2{\varepsilon _h}}} = \left( {1 - f} \right)\frac{{{\varepsilon _1} - {\varepsilon _h}}}{{{\varepsilon _1} + 2{\varepsilon _h}}} + f\frac{{{\varepsilon _2} - {\varepsilon _h}}}{{{\varepsilon _2} + 2{\varepsilon _h}}}$$ where $\left\langle \varepsilon \right\rangle$ is the effective dielectric function, $\varepsilon _1$ and $\varepsilon _2$ are the dielectric functions of the two media subject to mixing, $\varepsilon _k$ the dielectric function of the host medium with the inclusions, and $f$ the volume ratio of material 2. The underlying assumptions of the equation are that it applies for spherical inclusions and dipole interactions only. In the Bruggeman model the effective medium itself act as the host material, so $\left\langle \varepsilon \right\rangle = {\varepsilon _h}$. The model is then self-consistent and the two phases play exactly the same role. The effective dielectric function of the mixture is given by the second order equation $$0 = \left( {1 - f} \right)\frac{{{\varepsilon _1} - \left\langle \varepsilon \right\rangle }}{{{\varepsilon _1} + 2\left\langle \varepsilon \right\rangle }} + f\frac{{{\varepsilon _2} - \left\langle \varepsilon \right\rangle }}{{{\varepsilon _2} + 2\left\langle \varepsilon \right\rangle }}$$ The validity of the Bruggeman effective medium approximation requires the sizes of the phases (dielectrics) in a composite material to be sufficiently greater than atomic sizes, but smaller than 1/10 of the wavelength, which indeed is true for the bulk heterojunction films. The effective medium approximation cannot represent non-additive features of the dielectric function, such as charge transfer absorption bands. Lastly the dielectric functions of the phases must be independent of size and shape.

### Practical considerations

There are a number of practical considerations to be familiar with in connection with ellipsometry measurements. The first major hurdle is backside reflections. Backside reflections occur when front surface and back surface reflections overlap and enter the detector. This happens for transparent substrates which are polished on both sides. This was the case for the glass substrates used during this thesis for modeling work. These unwanted backside reflections are incoherent with the desired reflection from the front side and can either be accounted for in the model or suppressed by experimental means. One approach is to roughen the backside so the light is effectively scattered.

Another effect encountered in connection with ellipsometry is depolarization. Depolarization occurs when totally polarized light used as a probe in ellipsometry is transformed into partially polarized light. The effect of depolarization is especially severe for a rotating angle ellipsometer as the instrument assumes that reflected light is totally polarized. Imagine a case where the reflected light of linear polarization is overlapped with circular polarization. For a rotating angle ellipsometer the polarization state of this reflected light will be interpreted as elliptical polarization, since this instrument assumes totally polarized light for reflected light. With a rotating compensator ellipsometer the depolarization can be measured and included in the model. The physical phenomena that generate partially polarized light upon light reflection are; surface light scattering caused by a large surface roughness, incident angle variation originating from the weak collimation of probe light, wavelength variation, thickness inhomogeneity in the film, and backside reflection. The measurement of the depolarization therefore gives a good indication of the quality of the sample.

Tompkins, H. G.; Irene, E. A.; Hill, C.; Carolina, N. Handbook of Ellipsometry; 2005