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Before diving back to more complicated problems, it would make sense to take the simplest and most common example of isotropic bulk sample and look in detail on the whole process of determining the dielectric response from ellipsometric quantities. We have already calculated the Fresnel’s coefficients, so we can take the results and continue from there. As usual, we will work in Fresnel’s convention and use the geometrical wave-vectors, \(\boldsymbol\kappa\).
Let’s consider semi-infinite sample with planar interface in the \(x-y\) plane, at \(z=0\), having isotropic dielectric response \(\varepsilon\) and no magnetic response, \(\mu=1\). The ambient will be vacuum with \(\varepsilon_{a}=1\), and the plane of incidence will be the \(x-z\) plane. We define the angle of incidence \(\varphi\) and the incident, reflected and transmitted wave-vectors will have the same geometric x-component, \(\xi=\sin\varphi\).
The geometric z-components of the wave vectors are \(q=\cos{\varphi}\) for the incident wave, \(-q\) for the reflected, and \(\kappa = \sqrt{\varepsilon-\xi^{2}}\) for the transmitted wave, where we take the “positive” root in the first quadrant of the \(\mathbb{C}\) plane, that is, the one with non-negative real and imaginary parts. The angle of refraction \(\varphi’\) has no use for us.
We note that in isotropic ambient and sample media, propagation of the waves (direction, velocity, attenuation) will be independent on their polarization state. The only distinction between polarizations is given by the geometry needed to couple the incident, reflected and transmitted waves across the interface, where we observe two principal polarizations, p and s. The latter means that incident light linearly polarized in the plane of incidence, p-polarized, will reflect and transmit as p-polarized. The same applies for the incident light polarized perpendicular to the plane of incidence, s-polarized: the reflected and transmitted waves remain s-polarized. There will be no mixing between the p- and s-polarization states, only change in the complex amplitudes of the respective waves. We can write the Fresnel’s reflection coefficients for p- and s-polarized waves as:
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\( \begin{align}\label{eqsFresnelsRefIso} r_{p} = \frac{\kappa – \varepsilon q}{\kappa + \varepsilon q},\quad\quad r_{s} = \frac{q – \kappa}{q + \kappa}. \end{align} \) |
We can also write the transmission coefficients, but they will not be needed in this context:
\(
\begin{align}\label{eqsFresnelsTraIso}
t_{px} = \frac{2\kappa}{\kappa + \varepsilon q},\quad\quad
t_{s} = \frac{2q}{q + \kappa}.
\end{align}
\)
We remember that the the \(t_{px}\) is the ratio of the x-components of the transmitted and incident p-polarized amplitudes.
The main concept of ellipsometry is based on the intentional and controlled breaking of the symmetry between p- and s-polarizations. On normal incidence, there is no distinction between p and s. By changing the angle of incidence, we observe a difference between the amplitudes and also phase shifts of the p- and s-polarizations. As we will see below, the effect is stronger for p-polarization, while the s-polarization is quite unimpressed by the change of the angle of incidence. We can understand the ellipsometric experiment as a measurement of p-polarized reflectivity, while the s-polarized reflectivity is taken as a reference.
The trick is that we control the linear polarization state of the incident wave, and we send in a beam with a combination of p and s. Due to the different amplitude reflectivities and phase shifts of the p- and s- components, the reflected wave will have a different polarization state, generally elliptical. With an appropriate polarimetry method, we determine the outgoing polarization state. Generally speaking, we measure the ellipticity and tilt of the polarization ellipse. Since we know the incoming polarization state, we can express the change of the polarization state with a complex number, or two real numbers. These two quantities are directly related to the strength and phase delay of the material’s response on given frequency of the electromagnetic radiation, and the response is then written down as the complex refractive index, or dielectric permittivity, or optical conductivity.
In the following paragraphs, we will illustrate how the quantities introduced above behave as functions of the angle of incidence (AOI), for one fixed value of complex dielectric permittivity. We will take a realistic example, with \(\varepsilon=15+0.2i\), which is comparable to Silicon response at 2 eV.
Let’s just first look on the individual terms appearing in eqs. \eqref{eqsFresnelsRefIso}: \(\kappa\), \(q\) and also \(\varepsilon q\):
We see that in this example with large polarizability (real part of \(\varepsilon\)) and small absorption (imaginary part), the real parts of the terms dominate – the imaginary axis is 10x magnified.
The geometric z-component of the wave vector in the medium, \(\kappa\), does not change much with angle of incidence. Since the real part of \(\kappa\) is always larger than \(q\), the real part of the \(r_s\) will be negative for all angles of incidence, and \(r_s\) will show no anomaly. This is what we stated above, s-polarized reflectivity does not depend strongly on the incident angle.
On the other hand, the real part of \(\varepsilon q\) crosses real part of \(\kappa\) around 75 deg, which leads to a change of sign of \(r_p\) and a minimum in the absolute value of the p-polarized reflection at that angle of incidence, which is called the Brewster’s angle. The next plot shows real and imaginary values of the Fresnel’s coefficients \(r_p\) and \(r_s\):
Since we are using the Fresnel’s convention, the \(r_p\) and \(r_s\) are identical for normal incidence, \(\varphi=0\). For grazing incidence, that is \(\varphi\rightarrow 90^{\circ}\), the values of \(r_p\) and \(r_s\) are approaching real 1 and -1, respectively.
We can also write the complex numbers in polar coordinates, \(r_p = |r_p|\exp{(i\Delta_p)}\) and \(r_s = |r_s|\exp{(i\Delta_s)}\). The following plot can be found in any ellipsometry or optics textbook, where it may differ in Delta depending on the used convention. Here, we magnified the bottom part of the Delta graph to see better the behavior of \(\Delta_s\).
We see again that in Fresnel’s convention, for normal incidence, both the coefficients have the same argument Delta (about -179.8° in our example). For diminishing absorption, \(\varepsilon_2\rightarrow 0\), the argument will go towards -180°. The \(\Delta_s\) decreases smoothly towards -180° at grazing incidence while \(\Delta_p\) makes a sharp upturn at the Brewster’s angle and ends up near 0°. At the Brewster’s angle, the magnitude of \(r_p\) is minimal, reaching value about 0.003 in our example.
Now we can define the complex ellipsometry ratio, \(\rho\), and the ellipsometry angles \(\Psi\) and \(\Delta\):
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\( \begin{align}\label{eqsRhoPsiDel} \rho \equiv \frac{r_{p}}{r_{s}} = \frac{|r_{p}|}{|r_{s}|}\cdot e^{i(\Delta_p – \Delta_s)} \equiv \tan\Psi\cdot e^{i\Delta} \end{align} \) |
Let us plot \(\Psi\) and \(\Delta\) as a function of angle of incidence, again for the same material with \(\varepsilon=15+0.2i\).
The ellipsometric angle \(\Psi\) is restricted between 0° and 45°. This is a consequence of the fact that for any isotropic bulk sample, the magnitude of p-polarized reflectivity is smaller than the magnitude of s-polarized reflectivity. This means \(\tan\Psi\) will be smaller than 1. In the presence of anisotropy or in the case of layered sample, this restriction is lifted and \(\Psi\) might acquire any value between 0° and 90°. The minimum of \(\Psi\), at the Brewster’s angle of about 75.5°, dips down to the value of 0.2° for our example.
Angle \(\Delta\) is non-negative and covers values between 0° and 180°. Again, that is true for any isotropic bulk sample, and also holds for something like silicon with native oxide, that is, very thin film with weaker response than the substrate. Anisotropic or layered system might show any value of \(\Delta\) in the full 360° loop.
However, the sign and orientation of the \(\Delta\) curve plotted here corresponds to the used Fresnel’s convention. There lies a source of one notorious sign confusion – if written in Verdet’s convention, the \(r_p\) will be opposite to ours in \eqref{eqsFresnelsRefIso}, so \(\Delta_p\) will be than shifted by 180°. In that convention, angle \(\Delta\) will end up being always non-positive for common isotropic samples, starting at -180° and jumping up towards 0° at the Brewster’s angle. Nevertheless, people were using rotating analyzer ellipsometers for a long time, and rotating analyzer cannot determine the sign of \(\Delta\), as it measures only \(\cos\Delta\). Naturally, \(\Delta\) ended up being plotted as the positive solution of the arccosine function, that means starting at 180° and jumping down towards 0°.
In some near future blog post, I want to show that the traditional \(\Psi\) and \(\Delta\) can be seen as an odd choice for plotting ellipsometry data. It is more straightforward to look at data in the raw form, which are the elements of the Mueller matrix. For an isotropic sample, as well as for any configuration where Jones matrix is diagonal,
\(
\begin{align}
\mathbf{\hat{J}}=
\begin{bmatrix}
r_p & 0 \\
0 & r_s
\end{bmatrix},
\end{align}
\)
the Mueller matrix has simple form known as “NCS”. We define it here with a positive sign in front of \(N\) (this is a new convention that is otherwise inconsequential):
\(
\begin{align}
\mathbf{\hat{M}}=
\begin{bmatrix}
1 & N & 0 & 0 \\
N & 1 & 0 & 0 \\
0 & 0 & C & S \\
0 & 0 & -S & C
\end{bmatrix},
\end{align}
\)
where the elements \(N\), \(C\), and \(S\) are directly experimentally accessible with rotating analyzer or rotating compensator ellipsometers. Next figure shows the \(N\), \(C\), \(S\) dependence on the angle of incidence, calculated for the same situation as our graphs above:
The values on \(N\), \(C\), and \(S\) are related to \(\Psi\) and \(\Delta\) via \(N=-\cos2\Psi\), \(C=\sin2\Psi\cos\Delta\), and \(C=\sin2\Psi\sin\Delta\), so their behavior corresponds to what we’ve already seen on the graphs above.
Finally, we should discuss the inversion of the ellipsometry problem, that is, how to obtain the value of the dielectric function from the experimental quantity \(\rho\). It is just simple math, but for the sake of clarity, let’s show it almost step by step:
\(
\begin{align}\label{eqsRhoToEps}
\rho = \frac{r_{p}}{r_{s}} = \frac{\kappa – \varepsilon q}{\kappa + \varepsilon q}\cdot \frac{q + \kappa}{q – \kappa} =…=
\frac{\kappa q – \xi^2}{\kappa q + \xi^2}.
\end{align}
\)
We skipped the procedural steps, but note that the explicit value of \(\varepsilon\) that appeared in \(r_p\) has disappeared, specifically, factor \((\varepsilon-1)\) appeared both in numerator and denominator and was cancelled. Next we will invert the formula \eqref{eqsRhoToEps} to get \(\kappa\):
\(
\begin{align}\label{eqsRhoToEps2}
\kappa = \frac{\xi^2}{q}\cdot \frac{1+\rho}{1-\rho} = \frac{\xi^2}{q} \rho’ ,
\end{align}
\)
where we denoted the fraction \((1+\rho)/(1-\rho)\) as \(\rho’\). It still describes the change of polarization state measured by the ellipsometer, only in different coordinates – this will be discussed in the next blog. Mind you, the signs in front of \(\rho\) in the numerator and denominator are also given by the Fresnel’s convention and will be opposite if \(r_p\) is defined in Verdet’s convention. The last form in eq. \eqref{eqsRhoToEps2} is as direct as it gets. It shows that the change of polarization state \(\rho’\) is directly related to the geometric z-component of the wave vector in the medium, which has the role of “effective” refractive index: it describes the wave propagation (velocity and attenuation) along the z-axis. The prefactor \({\xi^2}/{q}\) is a smooth function of the angle of incidence starting at 0 for normal incidence and diverging for \(\varphi\rightarrow 90^{\circ}\).
To obtain the desired \(\varepsilon\), we square the last equation:
\(
\begin{align}\label{eqsRhoToEps3}
\varepsilon – \xi^2 = \frac{\xi^4}{q^2} {\rho’}^2 ,
\end{align}
\)
and reorder to arrive to:
\(
\begin{align}\label{eqsRhoToEps4}
\varepsilon = \sin^2\varphi \left( 1 + \tan^2\varphi \left(\frac{1+\rho}{1-\rho}\right)^2 \right) ,
\end{align}
\)
where we inserted back the familiar symbols. This last expression is a well known textbook formula. This tells us that for homogeneous isotropic bulk sample without any overlayers, we can calculate the complex optical response value from the directly measured change of polarization state.
It happens often that our sample does not exactly fit the assumptions for validity of the formula \eqref{eqsRhoToEps4}. The sample might be anisotropic or have an overlayer. Notorious example is again Silicon, that develops stable native oxide on its surfaces.1 Since our ellipsometer still measures \(\Psi\) and \(\Delta\), we can express \(\rho\) and apply the formula. What we get is something that looks like a dielectric function, so we call it a pseudo-dielectric function and denote it as \(\langle \varepsilon \rangle \).
1 D. E. Aspnes, Optical Properties of Thin Films, Thin Solid Films 89, 249-262, page 259 (1982).