\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_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{Theory of Signal Formation in Total Current Spectroscopy Experiments} \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*}

\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}

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.

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. 

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*}

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:
\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}.


\subsection{Variable Angle Spectroscopic Ellipsometry}

Traditional ellipsometers were restricted to single angle and often single wavelength measurements \cite{}. With advances in computing technology in the 1970s, 

\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
Ellipsometry (VASE), Part I:
Basic Theory and Typical Applications}
\cite{woolam1999})}
	\label{ellipsometer.png}
\end{center}

\section{Transmission Spectroscopy}

\section{Scanning Electron Microscopy}

