\chapter{Experimental Techniques}


\section{Total Current Spectroscopy}

In Total Current Spectroscopy experiments, a current of primary electrons $I_1$ is directed at a target surface. Upon interacting with the surface, the primary electron beam is split into two components; the transmitted current $I$, and the secondary electron current $I_2$. The current of secondary electrons includes all electrons emergent from the surface, regardless of origin. Generally $I_2$ includes components formed from elastically and inelastically scattered primary electrons, as well as electrons originating from bound states which have gained sufficient energy to leave the surface.

For any given mechanism behind the origin of an electron in $I_2$, there is an associated ``threshold'' primary electron energy which must be exceeded before the process can occur. As a result, measurement of changes in $I_2$ as a function of primary electron energy $E_1$ provides a very sensitive means to characterise properties of the sample under bombardment. The energy $E_1$ of primary electrons is controlled by adjustment of the potential $U$.



\begin{center}
	\includegraphics[scale=0.60]{figures/tcs/tcs_simple}
	\captionof{figure}{ A simplified schematic of Total Current Spectroscopy Experiments }
	\label{tcs_simple.pdf}
\end{center}

Figure \ref{tcs_simple.pdf} shows a simplified schematic for a Total Current Spectroscopy experiment
\footnote{For a more detailed description of the experimental setup, refer to Appendix \ref{}}.
When a current is passed through the cathode, electrons are thermionically emitted with a distribution in initial velocities. A series of electrodes (an electron gun) creates a potential which focuses the emitted electrons into a beam and accelerates them towards the target. The transmitted current $I$ can be detected external to the vacuum chamber using a conventional DC ammeter\footnote{It is also possible to use lock-in amplifier techniques for noise reduction \cite{komolov}. In this study, the DC ammeter has been used due to the relative simplicity of the measurement and control circuit.}.\

The total current spectrum (TCS) is defined as:
\begin{align*}
	S(E_1) &= \der{I}{E_1} = -\der{I_2}{E_1}
\end{align*}

This result assumes that the primary electron current $I_1$ is constant. Such an assumption is valid if the cathode has reached thermal equilibrium, and the potential due to the sample can be considered to have negligable effect on the focusing properties of the electron gun.

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{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. 

% Contact potential
A single electron arriving at the sample has energy $E = eU + c$, where $e$ is the electron charge, $U$ is the potential applied between the cathode and sample, and $c$ is a constant which includes the electron's energy relative to the sample when emitted. The minimum value for $c$ is the contact potential of the cathode relative to the sample.

At the cathode, electrons are emitted with a distribution in energies about some mean value. A realistic model should take into account this distribution.

If the primary electrons are incident perpendicular to the surface, then we can write $I$ as an integral over the whole distribution of energies:
\begin{align*}
	I(E_1) = e A \int_0^{\infty} f(E - E_1) dE
\end{align*}
where $f(E - E_1)$ is the distribution for an electron of energy $E$ arriving at the surface. Generally $f(0)$ (ie: $E = E_1$) is the maximum of $f$.

\emph{TODO: Discuss angular distribution of incident electrons, due to focusing of electron gun?}

To formulate a general expression for the secondary current, we introduce a cross section $\sigma(E)$, which gives the probability for a primary electron of energy $E$ to give rise to a secondary electron (of any energy $E_2 <= E$).

Then the total current of secondary electrons is:
\begin{align*}
	I_2(E_1) &= e A \int_{0}^{\infty} f(E - E_1) \sigma(E) dE
\end{align*}

Using $I = I_1 - I_2$, and $S(E_1) = \der{I}{E_1}$, it is straight forward to arrive at a general expression for the total current spectrum \cite{komolov}:
\begin{align*}
	S(E_1) &= e A \left\{ [ 1 - \sigma(0) ] f(-E_1) + \int_{0}^{\infty} f(E - E_1) \der{\sigma(E_1)}{E_1} dE \right\}
\end{align*}

All $E_1$ dependence in the first term is due soley to the distribution of primary electrons. It is clear that this term is maximised when $E_1 = 0$ with respect to the sample; ie: the contact potential between the cathode and sample is zero.

The second term contains dependence upon $\der{\sigma(E_1)}{E_1}$. As $E_1$ is increased past the threshold for a particular interaction, $\sigma(E_1)$ will undergo a sharp change. This corresponds to a narrow maxima or minima in the derivative $\der{\sigma(E_1)}{E_1}$. A corresponding maxima or minima will appear in $S(E_1)$, centred about the threshold for the interaction. The convolution with the primary electron distribution $f(E - E_1)$ has the effect of broadening and lowering these peaks; in other words, the resolution of Total Current Spectroscopy is limited by the distribution of primary electrons.

The (unphysical) case of a mono-energetic beam is equivelant to setting $f(E - E_1) = \delta(E - E_1)$. In this case, the integrals in the expressions for $I$ and $I_2$ collapse, and the resulting total current spectrum is:
\begin{align*}
	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}

\subsection{Overview}

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.

\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}

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 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}

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.

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{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}

Can be done with the VASE or with OceanOptics spectrometer.

\section{Scanning Electron Microscopy}

Done by CMCA, not me.
