\chapter{Techniques} \label{chapter_techniques}

\section{Total Current Spectroscopy} \label{tcs}

The optical properties of metals depend upon both conduction and valance band electrons. At the surface of a material, the electron spectrum (energy levels, band structure and densities of states) may differ greatly from the bulk. This is largely due to the sharp change in potential at the interface; rearrangement of lattice sites, ``dangling'' molecular bonds, and adsorbed defects are also contributing factors

Total Current Spectroscopy is an electronic spectroscopy technique which is extremely sensitive to the electronic properties of the surface region. The experimental setup for Total Current Spectroscopy can be easily integrated into existing apparatus for sample preparation, which allows for measurements to be made in situ. 

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

In Total Current Spectroscopy experiments (Figure \ref{tcs_simple.pdf}), 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 $I_2$. $I_2$ includes components formed from elastically and inelastically scattered primary electrons, as well as any ``true secondary'' electrons emitted by the surface.

For each 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 sensitive means to characterise properties of the sample under bombardment. The energy $E_1$ of primary electrons is controlled by adjustment of the potential $U$ applied between the sample and cathode.


Primary electrons are thermionically emitted at a cathode with a distribution in energies. A series of electrodes (an electron gun) focuses the emitted electrons into a beam and produces the current $I_1$ at the target. A feedthrough connected to the sample holder allows the transmitted current $I$ to be measured external to the vacuum chamber using an electrometer. Measurement of $I$ instead of $I_2$ is generally preferred, since changes in $I$ can be directly related to changes in $I_2$.


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. This assumption is generally valid over long time scales once the cathode has reached thermal equilibrium. \footnote{In the final experimental setup, the primary electron current was found to vary by about $2\%$ over the course of several days; far longer than the time taken for total current spectrum measurements (30s to 10min)}


\subsection{General form of $S(E_1)$} \label{tcs_theory1}

In the following, a generalised formula for $S(E_1)$ will be presented as derived by Komolov \cite{komolov}.

The current of primary electrons incident perpendicular to the surface can be written as:
\begin{align*}
	I_1(E_1) = e A \int_0^{\infty} f(E - E_1) dE
\end{align*}
where $f(E - E_1)$ gives the distribution for electrons of energy $E$ arriving at the surface. $f(0)$ (ie: $E = E_1$) is the maximum of $f$.

The total secondary current may be written as:
\begin{align*}
	I_2(E_1) &= e A \int_{0}^{\infty} f(E - E_1) \sigma(E) dE
\end{align*}

Where the cross section $\sigma(E)$ is the probability for a primary electron of energy $E$ to give rise to a secondary electron (of any energy $E_2 \leq E$).

Using $I = I_1 - I_2$, it is straight forward to arrive at a general expression for $S(E_1)$:
\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\} \label{tcs_full}
\end{align}

All $E_1$ dependence in the first term is due soley to the primary electron energy distribution. This term is responsible for the first peak in an $S(E_1)$ curve.

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 electron energies.

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} \label{tcs_simple}
\end{align}

A form of \eqref{tcs_simple} is usually assumed when constructing a more detailed theory of secondary electron current formation, as the location of peaks due to $\der{\sigma(E_1)}{E_1}$ is the same in both \eqref{tcs_simple} and \eqref{tcs_full}. Figure \ref{komolov1979} illustrates an idealised total current spectrum.

\begin{figure}[H]
	\centering
	\includegraphics[width=0.8\textwidth]{figures/tcs/komolov1979_tcs.png}
	\caption{An idealised total current spectrum (Komolov et al 1979 \cite{komolov1979})}
	\label{komolov1979}
\end{figure}

\subsection{Contact Potential and the Surface Peak}

The case $E_1 = 0$ does not in general correspond to $U = 0$. Figure \ref{contact.pdf} describes a simplified step potential model for the surfaces of the cathode and sample. In a classical approximation, primary electrons will be elastically scattered from the sample unless they have energy greater than its vacuum level. 

At the cathode, electrons are thermionically excited from the conduction band to the vacuum level. The vacuum levels of the cathode and surface are not generally equal. The contact potential is the difference in work functions of the materials. As $U$ approaches the contact potential, primary electrons are able to penetrate into the sample, and a sharp peak termed the surface peak is seen in $S(U)$. The maxima of this peak occurs at an applied potential $U$ such that $E_1 = 0$; its shape is indicative of $f(E - E_1)$.

\begin{figure}[H]
	\centering
	\includegraphics[width=0.80\textwidth]{figures/tcs/contact.pdf}
	\caption{Illustration of the step potential model for the surfaces}
	\label{contact.pdf}
\end{figure}




\subsection{Electron-Electron Interactions} 
\label{tcs_e}


In the Sommerfield free electron gas approximation, electrons are assumed to be contained within a rectangular potential well, with energies given by $E_k = \frac{\hbar^2 k^2}{2m}$, where $\vect{k}$ is the electron wave vector. Electrons fill states up to the Fermi level, with density of states $N(E_k)$ proportional to $\sqrt{E_k}$.

An incoming primary electron interacting with the free electron gas may either excite a single electron to above the Fermi level, or a collective vibration (plasmon). A plasmon may be excited if $E_1 \geq \hbar \omega_p$. 

The response of a free electron gas to an external electric field is characterised by $\phasor{\epsilon}$. The imaginary part of $\phasor{\epsilon}$ describes the attenuation of the electric field; it describes the process of energy transfer from the primary electron to the electron gas. Since the intensity of the electric field entering the solid decreases by a factor of $|\phasor{\epsilon}^2|$, the efficiency of energy transfer is given by:

\begin{align*}
	\frac{\epsilon_2}{|\phasor{\epsilon}|^2} = -\text{Im}\left(\frac{1}{\phasor{\epsilon}}\right)
\end{align*}

This function is often referred to as the ``Bulk loss''. A maxima in the bulk loss is indicative of a high probability for primary electrons to inelastically scatter from the free electron gas.




\subsection{Implementation of Total Current Spectroscopy Experiment}

As part of this project, we designed and built most of the electronics for an automated Total Current Spectroscopy setup. Figure \ref{tcs_block} shows a block diagram illustrating the key components of the setup. Figure \ref{electron_gun_circuit} shows the circuit diagram for control of the electron gun. Due to the limited range of available power supplies, our experiment has focused on the surface peak of the samples.

In order to automate TCS experiments, both Digital to Analogue and Analogue to Digital Convertors were required (DAC and ADC). To provide these, a custom DAC/ADC Box was designed and constructed. The box can be controlled by any conventional computer with available RS-232 serial communication (COM) ports. The full design of both hardware and software is available on request.

\begin{figure}[H]
	\centering
	\includegraphics[width=0.8\textwidth]{figures/presentation/tcs_block.png}
	\caption{Block diagram for TCS setup}
	\label{tcs_block}
\end{figure}

\begin{landscape}
\begin{figure}[H]
	\includegraphics[width=1.25\textwidth]{figures/egun/electron_gun.pdf}
	\caption{TCS circuit diagram}
	\label{electron_gun_circuit}
\end{figure}
\end{landscape}

\pagebreak

\section{Ellipsometry} \label{Ellipsometry}

Ellipsometry is a versatile optical technique which can be used to gain a great deal of information about optical and structural properties of thin films. In particular, Ellipsometry may be used to determine the pseudo-dielectric constant $\phasor{\epsilon}$ of a sample. This makes ellipsometry a useful complementary technique to secondary electron spectroscopy experiments, where $\phasor{\epsilon}$ is important for descriptions of electron-electron interactions.

An ellipsometer is designed to establish a beam of light with known polarisation, and measure the change in polarisation due to reflection of the light from a surface. This change in polarisation can be related to the optical properties of the surface, or the thicknesses of a multi-layered sample. 

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 polarised 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{Illustration of an ellipsometric measurement \cite{oates2011}}
	\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.

For an anisotropic sample, $\psi$ and $\Delta$ alone are not sufficient to characterise the sample; three seperate ratios of fresnel reflection coefficients are required. 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 formalism (which uses 4 component vectors to describe polarisation states) 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 polarised light, viewed along the wavevector of the light}
\end{table} 


\subsection{Relation of measurements to properties of the sample}

\subsubsection{Bulk Optical Constants}

The simplest sample is a flat substrate of infinite thickness. In this case, the optical constants of the substrate can be determined directly from $\rho$:
\begin{align*}
	\phasor{\epsilon} &= \epsilon_1 + i \epsilon_2 = \sin^2 (\phi) \left( 1 + \tan^2(\phi) \frac{1 - \rho}{1 +\rho}^2\right)
\end{align*}
Where $\phasor{\epsilon}$ is the psueodo-dielectric function. The complex refractive index of the material can be easily related to $\phasor{\epsilon}$:
\begin{align*}
	\phasor{\epsilon} &= \phasor{n}^2 = (n + i \kappa)^2
\end{align*}
Where $n$ is the refractive index, and $\kappa$ is the extinction coefficient.



\subsubsection{Multi-layered Samples}

At each layer in a multi-layered film, light is be both reflected and transmitted. The measured signal is the result of interference between reflections from each layer. The phase difference for each component of the beam is determined by both the optical constants and the thicknesses of the layers through which the component has passed. 

\begin{figure}[H]
	\centering
	\includegraphics[width=0.5\textwidth]{figures/ellipsometer/multilayered.pdf}
	\caption{Interaction of light with a multi-layered sample}
\end{figure}


\subsubsection*{A Note on Si Substrates}

A real substrate stored in laboratory air will generally have thin surface layers of oxides. For example, most of the thin film samples produced for this study were prepared on Si substrates. The inclusion of a layer of $\text{SiO}_2$ between the Si substrate and the thin film was found to have a significant effect on the accuracy with which the properties of the sample could be fit.


\pagebreak

\subsection{Variable Angle Spectroscopic Ellipsometry}

Although Ellipsometers have been in use for thin film analysis since the late 19th century, traditional instruments were usually limited to single angle and single wavelength measurements, due to the painstaking manual process involved in repeated measurements \cite{tompkins1992}. With advances both in software and hardware during the last half of the 20th century, Ellipsometric measurements have become largely automated, allowing for a huge number of measurements to be easily obtained from a sample \cite{woolam1999, woolam2000}.

Spectroscopic Ellipsometry obtains measurements for a range of wavelengths of the incident light. A Variable Angle Spectropic Ellipsometer (VASE) repeats measurements over a range of different angles of incidence. Because of the wealth of data aquired, these techniques are well suited to numerical analysis and fitting procedures to determine both the optical constants and thickness of unknown samples.

At CAMSP a Variable Angle Spectroscopic Ellipsometer (VASE)\footnote{J. A. Woolam and Co} and the associated software (WVASE32) have been employed in the analysis of thin film samples. Figure \ref{ellipsometer.png} shows a block diagram of the VASE.

The WVASE32 software allows for the construction of a model of a given sample, using tabulated literature values for optical constants, and allowing the user to specify film thicknesses. This model can then be adjusted using mean square error minimisation fitting algorithms to determine the thicknesses and/or optical constants for the model which best match the measured data. The modelling procedure is illustrated in Figure \ref{procedure.png}.

\begin{figure}[H]
	\centering
	\includegraphics[width=0.9\textwidth]{figures/ellipsometer/ellipsometer.png}
	\caption{Block diagram of the VASE \cite{woolam1999}}
	\label{ellipsometer.png}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[width=0.9\textwidth]{figures/ellipsometer/ellipsometry_block.png}
	\caption{Illustration of the VASE analysis and modelling procedure \cite{woolam1999}}
	\label{procedure.png}	
\end{figure}


\section{Vacuum Techniques and Sample Preparation}

All samples required for this study were prepared using available equipment at CAMSP. A small (approx. 25L) vacuum chamber, which has been used previously for thin film preparation at CAMSP \cite{kostylev2011} was repurposed to allow for both deposition of metallic thin films, and subsequent study using total current spectroscopy. A diagram of this setup is shown in Figure \ref{chamber}.

A rotatable stainless steel sample holder was positioned in the centre of the vacuum chamber. A flange containing evaporators was placed on one side of the chamber, whilst the electron gun required for total current spectroscopy was placed directly opposite. Shielding was used to ensure the electron gun was not coated during deposition. Two sample holders were attached to either side of the shielding to allow for preparation and study of two samples without the need for venting the vacuum chamber. 

The evaporators consisted of a tungsten wire filament (diameter 0.5mm) attached using spot welds between two feedthroughs. A piece of the desired metal was folded over the apex of the tungsten wire. The metal was then heated by passing a current through the filament; at sufficiently high currents, the metal piece began to evaporate, with a proportion of the flux of evaporated particles striking the target sample. To remove any contaminating layers from the metal, and ensure uniform evaporation, this procedure was first performed at low pressure (below $10^{-6}$ mbar) with no sample in the chamber, with the current increased until the metal piece began to melt and form a ball on the wire.

This study focused primarily on depositing Au and Ag films on Si substrates (approx 10x10x3mm wafers), at both high and low pressures. To remove any organic layers, the substrates and sample holders were cleaned in an acetone bath immediately prior to insertion in the vacuum chamber. The thickness of the deposited layer was controlled by varying both deposition time and the heating current.


	\begin{figure}[H]
		\centering
		\includegraphics[width=0.8\textwidth]{figures/pressure/chamber.pdf}
		\caption{Layout of apparatus within the vacuum chamber}
		\label{chamber}
	\end{figure}



