\begin{center}
- \includegraphics[scale=0.60]{figures/tcs_simple}
+ \includegraphics[scale=0.60]{figures/tcs/tcs_simple}
\captionof{figure}{ A simplified schematic of Total Current Spectroscopy Experiments }
\label{tcs_simple.pdf}
\end{center}
The experimental goal of Total Current Spectroscopy is the measurement of $S(E_1) \propto \der{I_2}{E_1}$. More information on the experimental setup and techniques are presented in Appendix \ref{}. The remainder of this section will give an overview of concepts needed for relating $S(E_1)$ to properties of a sample.
-\subsubsection{Theory of Signal Formation in Total Current Spectroscopy Experiments} \label{tcs_theory1}
+\subsubsection{Relationship between $S(E_1)$ and electron-surface interactions} \label{tcs_theory1}
Here we will summarise the approach of Komolov \cite{komolov} in constructing a theory relating $S(E_1)$ to scattering events within the target surface.
S(E_1) &= \der{I}{E_1} = e A \frac{d}{dE_1} \left( 1 - \sigma(E_1) \right) = e A \der{\sigma(E_1)}{E_1}
\end{align*}
+{\bf NOTE: Komolov goes on to find expressions for $\sigma(E_1)$ in terms of specific inelastic processes...}
+
\pagebreak
\section{Ellipsometry}
-Ellipsometry is an optical technique which measures the change in polarisation of light reflected from a surface. This change in polarisation can be related to properties of the surface; a common application of ellipsometry is determining the thickness of multilayered thin films.
-
-
-\begin{center}
- \includegraphics[scale=0.50]{/home/sam/Documents/University/honours/thesis/figures/ellipsometry_measurement}
- \captionof{figure}{Diagram of Ellipsometry Measurements}
- \label{ellipsometer_measurement.png}
-\end{center}
+\subsection{Overview}
-As shown in figure \ref{}, linearly polarised light incident upon a surface is in general reflected as elliptically polarised light. Ellipsometers are designed to produce a known linear polarisation of light incident on a sample, and to record the polarisation of reflected light.
+Ellipsometry is an optical technique which measures the change in polarisation of light reflected from a surface. This change in polarisation can be related to the optical properties of the surface, including optical constants or the thickness of thin film surface layers.
In the Jone's formalism, polarisation states may be represented by orthogonal electric field components $E_p$ and $E_s$, which are polarised parallel and perpendicular to the plane of incidence respectively. The reflection of a light ray from the surface is described by the matrix equation:
\begin{align*}
\left[\begin{array}{c} E_{rp} \\ E_{rs} \end{array}\right] &= \left[ \begin{array}{cc} r_{pp} & r_{ps} \\ r_{sp} & r_{ss} \end{array} \right] \left[\begin{array}{c} E_{ip} \\ E_{is} \end{array}\right]
\end{align*}
-Where $\vect{E}_i$ and $\vect{E}_r$ are the incident and reflected rays. Each element of the $2\times 2$ matrix $r_{ij}$ is the reflection coefficient for $i$ polarised light due to incident $j$ polarised light; these values are generally complex to include the phase change.
+Where $\vect{E}_i$ and $\vect{E}_r$ are the incident and reflected rays. Each element of the $2\times 2$ matrix $r_{ij}$ is the reflection coefficient for $i$ polarised light due to incident $j$ polarised light; these values are generally complex to include the phase change.
-Generalised Ellipsometry measures three ratios of $r_{ij}$ values:
-\begin{align*}
- \rho_{pp} &= \frac{r_{pp}}{r_{ss}} = \tan \psi_{pp} e^{-i \Delta_{pp}} \\
- \rho_{ps} &= \frac{r_{ps}}{r_{ss}} = \tan \psi_{ps} e^{-i \Delta_{ps}} \\
- \rho_{sp} &= \frac{r_{sp}}{r_{ss}} = \tan \psi_{sp} e^{-i \Delta_{sp}}
-\end{align*}
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.80\textwidth]{figures/ellipsometer/ellipsometer_measurement.png}
+ \caption{Diagram of an Ellipsometric Measurement \cite{woolam1999}}
+ \label{ellipsometer_measurement}
+\end{figure}
-If the sample is isotropic, then $r_{ps} = r_{sp} = 0$. In this case, only the ratio $\frac{r_p}{r_s} = \frac{r_{pp}}{r_{ss}}$ must be measured:
+As shown in figure \ref{ellipsometer_measurement}, linearly polarised light incident upon a surface is in general reflected as elliptically polarised light. For an isotropic sample, $r_{ps} = r_{sp} = 0$, and we can write $r_{pp} \equiv r_{p}$ and $r_{ss} \equiv r_{s}$. A standard ellipsometric measurement expresses the ratio of the $p$ and $s$ reflection coefficients in terms of the two parameters $\psi$ and $\Delta$
\begin{align}
\tan(\psi) e^{i \Delta} &= \rho = \frac{r_p}{r_s} \label{ellipso}
\end{align}
-The parameter $\psi$ gives the angle by which ...., whilst $\Delta$ gives the phase difference between $E_p$ and $E_s$ components.
-The Jone's formalism is not sufficient to describe the response of samples which de-polarise the incident beam; the more general Stoke's formalism must be used for such cases \cite{woolam2000} \cite{oates2011}.
+The value of $\tan(\psi)$ gives the ratio of amplitudes between the $p$ and $s$ components of the reflected electric field, whilst $\Delta$ gives the phase difference.
+
+In the case where the sample is anisotropic, $\psi$ and $\Delta$ alone are not sufficient to characterise the sample; three seperate ratios of fresnel reflection coefficients are required \cite{woolam2000}:
+
+Light may consist of two or more components of well defined polarisation states; as a result, the total beam cannot be described with a single well defined Jones vector. Ellipsometers are designed to establish a well defined polarisation state of light incident on the sample; however non-uniform films or backside reflection from a substrate may cause the reflected beam to be partially polarised. As a result, the Jone's formalism is not sufficient for characterisation of these samples, and the more general Stokes formalise (with 4 component vectors) must be employed \cite{woolam2000} \cite{oates2011}.
+
+\begin{table}[H]
+ \centering
+ \begin{tabular}{lll}
+ \includegraphics[scale=0.75]{figures/ellipsometer/circular_polarisation} &
+ \includegraphics[scale=0.75]{figures/ellipsometer/elliptical_polarisation} &
+ \includegraphics[scale=0.75]{figures/ellipsometer/linear_polarisation} \\
+ \end{tabular}
+
+ \captionof{figure}{From left to right, circular, elliptical and linearly polaristed light, viewed down the axis of beam propagation}
+\end{table}
\subsection{Variable Angle Spectroscopic Ellipsometry}
-Traditional ellipsometers were restricted to single angle and often single wavelength measurements \cite{}. Although the measurement of With advantages in computing technology in the 1970s,
+Although Ellipsometers have been in use for thin film analysis since the ??? \cite{}, traditional instruments were usually limited to single angle and single wavelength measurements, due to the painstaking manual process of repeated measurements. With advances both in software and hardware during the last half of the 20th century, Ellipsometric measurements have become largely automated, allowing for a wide range of optical information to be obtained from a sample. Spectroscopic Ellipsometry obtains measurements for a range of wavelengths, whilst Variable Angle Spectroscopic Ellipsometry repeats these measurements over a range of angles of incidence.
-\begin{center}
- \includegraphics[scale=0.50]{/home/sam/Documents/University/honours/thesis/figures/ellipsometer.png}
- \captionof{figure}{Block diagram of the VASE \\(taken from \emph{Overview of Variable Angle Spectroscopic
+The ease with which large amounts of data can be taken from an ellipsometer means that the method is well suited to numerical analysis and fitting procedures. At CAMSP a Variable Angle Spectroscopic Ellipsometer (VASE)\footnote{J. A. Woolam and Co} and the associated software\footnote{WVASE32} have been employed in the analysis of thin film samples.
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{figures/ellipsometer/ellipsometer.png}
+ \caption{Block diagram of the VASE \\(taken from \emph{Overview of Variable Angle Spectroscopic
Ellipsometry (VASE), Part I:
Basic Theory and Typical Applications}
\cite{woolam1999})}
\label{ellipsometer.png}
-\end{center}
+\end{figure}
+
+\subsection{Relation of $\psi$ and $\Delta$ to properties of the sample}
+
+\subsubsection{Bulk Substrates}
+
+The simplest sample is a bulk substrate. In this case,
+
+\subsubsection{Single Layered Thin Films}
+
+
+
+\subsubsection{Multi-layered Thin Films}
+
+Magical analysis
+
+\section{Optical Transmission and Reflection Spectroscopy}
-\section{Transmission Spectroscopy}
+Can be done with the VASE or with OceanOptics spectrometer.
\section{Scanning Electron Microscopy}
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