# Ellipsometry

Ellipsometry
관련코스 현대광학
소분류 물리
선행 키워드
연관 키워드

Ellipsometry is a term refers the methodology of measuring complex refractive index (or dielectric constant) of materials by comparing differed polarization state between the incident and reflected light. It was coined by A. Rothen in 1945 by his literature review, from the fact that the light is observed as a elliptic phase. Requiring models including consideration of the structure or constituents of the materials, however, this technique is widely used in field where it is highly sensitive to both the refractive index and the thickness in a few micrometer range, for instance, semiconductor fabrication which is aimed to produce thinner films.

## Historical View

Even we cannot observe how light moves as well, one can imagine the light as an electromagnetic oscillation in its generic explanation, where electromagnetic wave propagates with its two fields perpendicular to each other. In such aspect of view, unpolarized light could be described as a stochastic (say, Gaussian) chunk of such waves with different direction of oscillation. We all know Maxwell enlightened the light by his convergent, mathematical viewpoint into four 'seems different but actually alike' equations. (Of course, the modern Maxwell's equation was attributed to Heaviside with his engineering insight. What Maxwell devised was in fact, had a form of Quaternion, which also enables 3-Dimensional vector space to have rotation operations as well.)

However, from since early 1800, when Young had advocated the diffraction using interference and Fresnel disproved what Huygens had insisted - the light is rather a wave than a particle - and this made the corpuscles concepts of Newton declined (Meanwhile, these 'corpuscles' sometimes referred electrons.), the term 'polarization' started being accepting through the scientists. According to I. An, Professor of Hanyang university, suggested that this concept of polarization and the blueprint of ellipsometry might be originated from Malus and Brewster, who are renowned for their excellent foresighted work of Malus's law and Brewster angle. In fact, it was Fresnel who coined the term linearly, circularly, and elliptically polarized.

It was Drude who applied ellipsometry in thin films and surface in 1889. (It was also Drude who introduced abbreviation term of speed of light as 'c'.) Until Photomultiplier tube and automatic analyzers like computers had been introduced into this field, this method inevitably involved painstaking analysis just like what optical experiment had been, for instance, it was bare eye of researcher that was used to observe the result of null ellipsometry, which, as stated below, needs to detect its absorption, and therefore its accuracy cannot be guaranteed.

## Basic Electromagnetics

Ellipsometry requires overall understanding of electromagnetics and optics. Since it is just a introductory paper of the ellipsometry, one can just refer the range that covers undergraduate knowledge. Below here is the summary brought from each pages. (Magnetic contribution is usually ignored.)

Maxwell's equations, especially in macroscopic (that is, accounting for the interactions between materials and the electromagnetic field) aspect, are described in differential forms.

$\nabla \cdot \mathbf{D} = \rho_\text{f}$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$

$\nabla \times \mathbf{H} = \mathbf{J}_\text{f} + \frac{\partial \mathbf{D}} {\partial t}$

Light as a electromagnetic plane wave is described as well, from Maxwell's equations to the Euler's formula.

$\tilde{f}(z,t) = \tilde{A}e^{i(kz-\omega t)}$

Using Kramers-Kronig relation, define refractive index where they both contain refractive index coefficient and the extinction coefficient term connected by causality. Also, define permittivity in similar way, where the phase difference of the components would give rise to the energy loss (say, dielectric loss.) These two optical constants have dispersion relation, so that they are function of (angular) frequency, or wavelength. Here they are functions of frequency.

$N(\omega)=n(\omega)+i\kappa(\omega)$

$\epsilon(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$

Surprisingly, they indicate the same situation; that is, how material reacts to the external electric field. This means the refractive index and (relative) permittivity are connected. (More precise explanation about this paragraph is that both permittivity and permeability are related to the refractive index. Thus, equation below just follows approximation that the relative permeability is almost 1. When it shows considerably high magnetic permeability, then, equation should be changed into $N=n+i\kappa=\sqrt{\epsilon\mu}$)

$N=\frac{c}{v}=\frac{\sqrt{\epsilon\mu}}{\sqrt{\epsilon_0\mu_0}}= \sqrt{\epsilon_r\mu_r} \cong \sqrt{\epsilon_r}$

$N=n+i\kappa=\sqrt{\epsilon}$

$\epsilon_{1}(\omega)=n^{2}-\kappa^{2}, \epsilon_{2}(\omega)=2n\kappa$

In aspect of permittivity, it is normal to use Lorentz harmonic oscillator model. This can show the response of material against the electric field with dynamic change, which means the light. (Note that $\omega_{p}=\frac{N_{m}e^{2}}{\epsilon_{0}m}$ is plasma frequency)

$\epsilon(\omega)=1+\frac{\omega_{p}^{2}}{\omega_{0}^{2}-\omega^{2}-i\gamma \omega}$

In films, consideration of multiple oscillators is inevitable. Introducing oscillator strength, $f_{j}=\frac{N_{j}}{N_{total}}$, where $N_{j}$ is the electrons in the j-th set.

$\epsilon(\omega)=1+\omega_{p}^{2}\sum_j\frac{f_{j}}{\omega_{0j}^{2}-\omega^{2}-i\gamma_{j} \omega}$

Lorentz harmonic oscillator is acceptable only when it is semiconductor. In metallic cases, there are no suitable models in dispersion relation. When it goes to the dielectrics, and especially when it is 'invisible', it is Sellmeir relation to use. This is in fact a special case of Lorentz harmonic oscillator, since there is no extinction. Say, $\kappa=0$ or $\epsilon_2=0$ and thus,

$n^2=\epsilon=1+\omega_{p}^{2}\sum_j\frac{f_{j}}{\omega_{0j}^{2}-\omega^{2}-i\gamma_{j} \omega}$ (Lorentz)

$n^2=A+\sum_j \frac{B_j\lambda^2}{\lambda^2-\lambda^2_{0j}}$ (Sellmeir)

For refractive index, there is no proper principle or model related to the electron motion. However, should it be taken account of Absorption when the light intensity becomes smaller so that it cannot provide information. Extinction effect of the electromagnetic wave and the absorption coefficient of the material might do the same role, which implies their formation with different variable(s) might indicate the same result. One can moreover define the optical penetration depth $D_{op}$, where its intensity drops to $e^{-1}$ of which its physical meaning is the 'indistinguishable depth' whether the material is the film or the bulk.

$I_0exp(-2k\kappa z)=I_0exp(-\alpha z)$

$D_{op}=\frac{1}{\alpha}=\frac{\lambda}{4\pi\kappa}$

We must take account of the light polarization, too. Jones calculus, which can also be applied into quantum physics with spin-1/2 case in a same way, is designed to explain polarized light in a matrix form. However, it has unpractical limit that the light in the given system should be perfectly polarized. Therefore, Mueller calculus, or Stokes-Mueller formalism, which has 4-by-4 matrix formula using Stokes parameters and Mueller calculus, is mainly used in order to explain its general state of light.

## Reflection

We already treated reflection and refraction (or in general form using Euler's formula here) with boundary conditions of Maxwell's equations, or Snell's law. Yet, here's difference from what we have learned from the class; assume that the only one film is stacked, phase difference of the light reflected from the air-film interface and the film-substrate interface is expressed by $2\beta$, where $\beta$ is

$\beta=2\pi\frac{d}{\lambda}Ncos(\theta)$

$r=r_{12}+t_{12}t_{21}r_{23}exp(-i2\beta)(1+(r_{21}r_{23}exp(-i2\beta))+((r_{21}r_{23}exp(-i2\beta))^2+...)=r_{12}+\frac{t_{12}t_{21}r_{23}exp(-i2\beta)}{1-(r_{21}r_{23}exp(-i2\beta))}=\frac{r_{12}+r_{23}exp(-i2\beta)}{1+(r_{12}r_{23}exp(-i2\beta))}$

$t=\frac{t_{12}t_{23}e^{-i\beta}}{1+r_{12}r_{23}e^{-2i\beta}}$

Although these reflections are just simultaneous equations at most, polarization is mainly expressed as the matrix form. Moreover, it goes painstaking when considering multi-layers. Thus, in order to apply information of polarization, it would be considerable choice to introduce transfer matrix. (Using matrix instead of equations is quite familiar concept. Refer anti-reflection film.) Scattering matrix S of nth multi-layers on the substrate is expressed by interface matrix I and layer matrix L, as following. Note that $I_{12}$ means the interface effect between air and the film, while $I_{n+1 n+2}$ means the interface between the last film and the substrate. each r and t means reflection coefficient and transmission coefficient.

$S=I_{12}L_{2}I_{23}L_{3}...I_{n}L_{n+1}I_{n+1 n+2}$

Where each I and L becomes

$I=\begin{pmatrix} \frac{1}{t_{jj+1}} & \frac{r_{jj+1}}{t_{jj+1}} \\ \frac{r_{jj+1}}{t_{jj+1}} & \frac{1}{t_{jj+1}} \end{pmatrix}$

$L=\begin{pmatrix} exp(i\beta_{j}) & 0 \\ 0 & exp(-i\beta_{j}) \end{pmatrix}$

$\beta_{j}$ means the phase difference due to the layer passing by, defined by

$\beta_{j}=\frac{2\pi d_{j}N_{j}cos(\theta_{j})}{\lambda}$

Thus, reflection coefficient of p-wave and s-wave can be expressed (i,j)-th element of the scattering matrix.

$r_{p}=\frac{S_{21p}}{S_{11p}}$

$r_{s}=\frac{S_{21s}}{S_{11s}}$

## Ellipsometry Parameter

Firstly, one can use new coordinate system that is comprised of p-wave and s-wave, where they are orthogonal. So to speak, they replace x-axis and y-axis instead.

$\vec{E_{i}}=\hat{p}E_{ip}+\hat{s}E_{is}=\hat{p}E_{i0p}e^{i\delta_{i}}+\hat{s}E_{i0s}e^{i\delta_{i}}$

$\vec{E_{r}}=\hat{p}E_{rp}+\hat{s}E_{rs}=\hat{p}E_{r0p}e^{i(\delta_{i}+\delta_{p})}+\hat{s}E_{r0s}e^{i(\delta_{i}+\delta_{s})}$

$r_{p}=\frac{E_{rp}}{E_{ip}}=\frac{E_{r0p}}{E_{rip}}e^{i\delta_{p}}=|r_{p}|e^{i\delta_{p}}$

$r_{s}=\frac{E_{rs}}{E_{is}}=\frac{E_{r0s}}{E_{i0s}}e^{i\delta_{s}}=|r_{s}|e^{i\delta_{s}}$

Here, ellipsometry parameters, which are named $\Delta, \Psi$ respectively, are fully-extracted information from the material, by defining complex reflection coefficient ratio,

$\rho=\frac{r_{p}}{r_{s}}=|\frac{r_{p}}{r_{s}}|e^{i(\delta_{p}-\delta_{s})}=\tan(\Psi)e^{i\Delta}$

Thus, $\Delta$ is the phase difference between reflected p-wave and s-wave (when they are incident, there is no such difference.) while $\Psi$ is the ratio of the absolute value of reflection coefficient. (It is also defined by the ratio of electric field between incident p-wave and reflected s-wave, in order to show the correlations of these two factors.)

isotropic film is expressed by both Jones matrix and Mueller matrix

$J=\begin{pmatrix} r_p & 0 \\ 0 & r_s \end{pmatrix}=r_{s}\begin{pmatrix} \tan(\Psi)e^{i\Delta} & 0 \\ 0 & 1 \end{pmatrix}$

$M=\begin{pmatrix} 1 & -N & 0 & 0 \\ -N & 1 & 0 & 0 \\ 0 & 0 & C & -S \\ 0 & 0 & S & C \end{pmatrix}$

Where $N=\cos2\Psi$, $S=\sin2\Psi \sin\Delta$ $C=\sin2\Psi \cos\Delta$, and $N^2+S^2+C^2=1$

Since most of the deposition is intended to fabricate the film in isotropic way, its anisotropic effect is usually ignored. (However, it is not impossible to express anisotropy.)

## Principles of Ellipsometry

There are a lot of ellipsometers, but their basics are nearly same, thus here I describe the null ellipsometry measurement.

'Null' means the full extinction of light so that no light interaction inside the detector exists. It must maintain the light linearly polarized and at last, must find the point that the light becomes extincted as detected. Schematic diagram below describes how it actually works.

by Guoliang Wang, Hans Arwin, and Roger Jansson; Reference retrieved from OSA

From the incident plane of the film, where it is absolutely the reference of the whole system, the polarizer should move to make the light linearly polarized, say, transmit the component that is on the polarization axis, while absorb the other direction. In compensator, its fast axis differs from that of the slow axis, due to the velocity difference. and surely, the film shows difference in its reflectivity of its incidence; whether it is horizontal or vertical. And most of all, the null point that we desired to find is where the analyzer make extinction of linear polarization. Then, its matrix form becomes, when assume the rotation angle of polarizer, compensator and analyzer from the incident plane are respectively P, C and A.

$E=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} cosA & sinA \\ -sinA & cosA \end{pmatrix}\begin{pmatrix} r_p & 0 \\ 0 & r_s \end{pmatrix}\begin{pmatrix} C_1 & C_2 \\ C_3 & C_4 \end{pmatrix}\begin{pmatrix} cosP & -sinP \\ sinP & cosP \end{pmatrix}\begin{pmatrix} E_0 \\ 0 \end{pmatrix}$

$\begin{pmatrix} E_0 \\ 0 \end{pmatrix}$ : linear polarization after the polarizer

$\begin{pmatrix} cosP & -sinP \\ sinP & cosP \end{pmatrix}$ : considering the rotation of the linear polarization by polarizer, where it is rotated by -P

$\begin{pmatrix} C_1 & C_2 \\ C_3 & C_4 \end{pmatrix}=\begin{pmatrix} cosC & -sinC \\ sinC &cosC \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & e^{-i\delta} \end{pmatrix}\begin{pmatrix} cosC & sinC \\ -sinC & cosC \end{pmatrix}$ : considering fast axis of compensator, which is tilted about C; thus, make a transformation matrix. (It can be translated into-rotate C to make phase difference $\delta$ and reverse-rotate toward -C to express the light by the incident plane. $\delta$, the retardation of the compensator, is known. For instance, quarter waveplate has 90° of retardation.)

$\begin{pmatrix} r_p & 0 \\ 0 & r_s \end{pmatrix}$ : the matrix above, \begin{pmatrix} C_1 & C_2 \\ C_3 & C_4 \end{pmatrix}, is explained by the incident plane; thus, use the reflection coefficient of p-wave and s-wave on the corresponding frame.

$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} cosA & sinA \\ -sinA & cosA \end{pmatrix}$ : to consider the analyzer, the reflected p-wave and s-wave must be rotated by the A and transmission occurred by the polarization-axis component only.

When the light is not perfectly polarized, then Mueller matrix must be used, while its expression becomes complicated. However, in both Jones and Mueller expression. One can solve the remained electric field by calculating the whole matrix, of which its intensity must be 0 due to the null condition.

$E=E_0r_s[pcosCcos(P-C)cosA-\rho\rho_c sinCsin(P-C)cosA+sinCcos(P-C)sinA+\rho_c cosCsin(P-C)sinA]$

$\rho_c=\frac{t_{cs}}{t_{cp}}=T_cexp(i\delta) \cong exp(i\delta)$ (approximation valid for ideal compensator)

$I_d ∝ |E*E|=0$ (null condition of detector)

$\rho=-tanA[\frac{tanC+e^{i\delta}tan(P-C)}{1-e^{i\delta}tanCtan(P-C)}]$ (complex reflection coefficient ratio)

From these eqations, it is easily derived that the ellipsometry angles, $\Delta$ and $\Psi$, are decided by P, C, and A. For instance, when using quarter waveplate (that is, $\delta=-\frac{\pi}{2}$) and $C=\frac{\pi}{4}$, (fixed C in $\frac{\pi}{4}$ is one of the conventions.) assume that it has null point at (P', A'). Then,

$\rho=-tanA'[\frac{tan(\frac{\pi}{4})+(-i)tan(P'-\frac{\pi}{4})}{1-(-i)tan(\frac{\pi}{4})tan(P'-\frac{\pi}{4})}]=-tanA'exp(-2i(P'-\frac{\pi}{4}))$

Then, there must be other null point, say, (P, A)=(P'+$\frac{\pi}{2}$, $\pi$-A'). Considering the case when $C=-\frac{\pi}{4}$, 4 null zones at most, still one null point is enough to find ellipsometry angles for ideal case. Say, when it is found in two zones of $C=\frac{\pi}{4}$, then,

$\Delta=-2P_2+\frac{\pi}{2}=\frac{3\pi}{2}-2P_1$

$\Psi=A_1=-A_2$

(Of course, it requires calibration - that is changed as the film is. - with several methods, but I skipped for convenience.)

## Measuring Thickness

It is the case when we know optical properties but not thickness. For monolayer film, one can use the reflection coefficient of p-wave and s-wave by Drude formula as stated above,

$r_p=\frac{r_{p,12}+r_{p,23}e^{-2i\beta}}{1+r_{p,12}r_{p,23}e^{-2i\beta}}$

$r_s=\frac{r_{s,12}+r_{s,23}e^{-2i\beta}}{1+r_{s,12}r_{s,23}e^{-2i\beta}}$

Where each Fresnel reflection coefficient of p-wave and s-wave between air(1) and the film(2), or film(2) and the substrate(3)

$r_{p,12}=\frac{N_2cos\theta_1-N_1cos\theta_2}{N_2cos\theta_1+N_1cos\theta_2}$

$r_{p,23}=\frac{N_3cos\theta_2-N_2cos\theta_3}{N_3cos\theta_2+N_2cos\theta_3}$

$r_{s,12}=\frac{N_1cos\theta_1-N_2cos\theta_2}{N_1cos\theta_1+N_2cos\theta_2}$

$r_{s,23}=\frac{N_2cos\theta_2-N_3cos\theta_3}{N_2cos\theta_2+N_3cos\theta_3}$

$\beta=2\pi\frac{d}{\lambda}N_2cos\theta_2=2\pi\frac{d}{\lambda}(N_2^2-N_1^2sin^2\theta_1)^{\frac{1}{2}}$

$\rho(d)_{cal}=\frac{r_p(d)}{r_s(d)}=tan\Psi (d)e^{i\Delta(d)}$

Such d, which must be the same in the experimental result, is what we want to know. However, when $N_2$ is real, not complex, for instance, 'transparent' films, it could make several solutions since the light makes several periods of light wave in the film unless it is short in length, or using 'spectroscopic' ellipsometry. (In this case, one must use dispersion relationship.) So to speak, there is solution set d' When refractive index is real.

$d'=d+\frac{m\lambda}{2}(N_2^2-N_1^2sin^2\theta_1)^{\frac{-1}{2}}$

P.S. Even transmission is not used, however, we can find the value from Fresnel equation.

$t_p=\frac{t_{p,12}t_{p,23}e^{-i\beta}}{1+r_{p,12}r_{p,23}e^{-2i\beta}}$

$t_s=\frac{t_{s,12}t_{s,23}e^{-i\beta}}{1+r_{s,12}r_{s,23}e^{-2i\beta}}$

$t_{p,12}=\frac{2N_1cos\theta_1}{N_2cos\theta_1+N_1cos\theta_2}$

$t_{p,23}=\frac{2N_2cos\theta_2}{N_3cos\theta_2+N_2cos\theta_3}$

$t_{s,12}=\frac{2N_1cos\theta_1}{N_1cos\theta_1+N_2cos\theta_2}$

$t_{s,23}=\frac{2N_2cos\theta_2}{N_2cos\theta_2+N_3cos\theta_3}$

## Measuring Optical Properties

It is the case when exploring new materials, or considering deviations of compounds composition by chemicals, changing temperature, existing considerable grain size effect, and also when dispersion should be taken account into the measurement. Here we can use complex reflect coefficient, $\rho$, as stated. When it is bulk, say, the depth is more than $D_{op}$,

$\rho=\frac{r_p}{r_s}=tan\Psi e^{i\Delta}=\frac{sin^2\theta-(\frac{\epsilon}{\epsilon_a}-sin^2\theta)^{\frac{1}{2}}cos\theta}{sin^2\theta+(\frac{\epsilon}{\epsilon_a}-sin^2\theta)^{\frac{1}{2}}cos\theta)}$

This can be switched into permittivity or refractive index

$\epsilon=\epsilon_1+i\epsilon_2=\epsilon_a sin^2\theta[1+tan^2\theta(\frac{1-\rho}{1+\rho})^2]$

$N=n+i\kappa=\sqrt{\epsilon}=sin\theta\sqrt{1+tan^2\theta(\frac{1-\rho}{1+\rho})^2}=tan\theta\sqrt{1-sin^2\theta\frac{4\rho}{(1+\rho)^2}}$

Should note that the surface be clean, unless it affects to the result; for instance, surface oxidation would make ~2nm film. If this is inevitable, use pseudo-dielectric constant, $\lt \epsilon\gt =\lt \epsilon_1\gt +i\lt \epsilon_2\gt$, or pseudo-refractive index, $\lt N\gt = \lt n\gt +i\lt \kappa\gt$, where their experimental value is similar to the desired one.

In thin-film case, our measurement contains 2 information $(\Delta, \Psi)$, while they contain depth(d) not only (n, k) as bulk does. This even contains substrate information, but we usually use some known materials as substrate, for reducing the complexity. It is normal to use spectroscopic ellipsometry for this conventional n-k-d problem, and several methods invented. For instance, direct inversion just subtract $\Delta$ of calculation value and the experiment value and choose the smallest one. Surely one can use cross-sectional SEM to find correct depth and use direct inversion. It is also a way that changing several parameter(s), such as incident angle or film depth, to obtain several data to compare.

Here I end the chapter by stating dispersion relation. This is actually using Lorentz oscillator(for semiconductors-but requires many coefficients to solve) or Sellmeir relation(for invisible dielectrics-relatively easier method), and other dispersion relations. Since they satisfy Kramers-Kronig relation, it shows smooth graph by the wavelength-axis, and its depth is not approximated, unlike direct inversion. (Regression analysis, or simply, data fitting using unbiased estimator, is adopted.) However, this is not suitable for metallic cases.

$n^2(\lambda)=A+\frac{B\lambda^2}{\lambda^2-\lambda^2_0}$ (In Sellmeir relation, has 4 variables)