\newcommand{\curl}[1]{\nabla \times #1} % curl
\newcommand{\grad}[1]{\nabla #1} %gradient
\newcommand{\pd}[3][ ]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} %partial derivative
-%\newcommand{\d}[3][ ]{\frac{d^{#1} #2}{d #3^{#1}}} %full derivative
+\newcommand{\der}[3][ ]{\frac{d^{#1} #2}{d #3^{#1}}} %full derivative
\newcommand{\phasor}[1]{\tilde{#1}} % make a phasor
\newcommand{\laplacian}[1]{\nabla^2 {#1}} % The laplacian operator
%\tableofcontents
\section*{Acknowledgements}
+
\begin{itemize}
\item Sergey Samarin
\item Jim Williams
\section{Overview of Theory}
Summarise the literature, refer to past research etc
-\subsection{Electron Spectra of a Surface}
+\subsection{Electron Spectra of Solids and Surface}
+
+\begin{itemize}
+ \item Overview of section
+\end{itemize}
+
+In this section, we will first introduce the basic concepts needed to describe the electron spectra of solids. A short description of methods for calculating the electron spectra will be given, and the results shown by these calculations. We will then discuss the electron spectra for the near surface region of solids, compared to the ``bulk'' spectra far from the surface.
+
+
+\subsubsection{Description of Matter in the Solid State}
+
+\begin{itemize}
+ \item Define and describe solid geometrically
+ \begin{itemize}
+ \item Basis + Lattice
+ \item Basis groups are assumed to be fixed relative to the lattice
+ \end{itemize}
+ \item Describe general properties of the potentials seen by electrons in solids
+\end{itemize}
+
+In the simplest models, a solid is represented by an infinite crystalline lattice; a geometrically repeated arrangement of some basis group of atoms. The nuclei of atoms are assumed to remain in fixed positions.
+
+The potential seen by an electron in the lattice is periodic. For a single nuclei, the potential seen by an electron is
+
+\subsubsection{Calculation of Electron Spectra}
+
+\begin{itemize}
+ \item Define electron spectra
+ \begin{enumerate}
+ \item
+ \end{enumerate}
+ \item Free electron gas
+ \item Free electron entering a periodic potential
+ \item Potentials of real solids
+ \item Single vs Multiple electrons
+ \item Numerical calculations
+ \begin{itemize}
+ \item TODO: Actually research this
+ \item Find a comparison with a real experimental result
+ \end{itemize}
+ \item
+\end{itemize}
+
+
+\subsubsection{The Near-Surface Region}
+
+\begin{itemize}
+ \item Real solids have surfaces
+ \item Differences between surface region and bulk
+ \begin{itemize}
+ \item Resulting differences in potential
+ \end{itemize}
+ \item How the electron spectra may differ from bulk values
+ \begin{itemize}
+ \item Surface states
+ \begin{itemize}
+ \item Tamm \& Shockley States
+ \end{itemize}
+ \end{itemize}
+\end{itemize}
+
+In the preceeding sections, solids were assumed to have infinite spatial extent. In practice, any real solid occupies a finite volume in space. Any interactions between a solid and its environment take place at the surface of the solid. As the volume of the solid is decreased, the role of the surface region in determining the behaviour of the solid in its environment is increased.
+
+
+
+
\begin{itemize}
\item Description of the near surface region
\begin{itemize}
\end{itemize}
\item Potential seen by an electron at a surface can differ greatly from the bulk
\item $\implies$ the electron spectra of the near surface region differs from the bulk spectra
- \item Simplest case: Step potential at surface
- \begin{itemize}
- \item Metal & Semiconductor
- \end{itemize}
- \item In reality,
+ \item Simplest case: Step potential at surface.
\end{itemize}
\begin{itemize}
\item Dependence of Energy on electron wave vector
\item Obtained theoretically by solving Scrhrodinger's Equation
- \item For a free electron gas, $E = \frac{\hbar^2 k^2}{2m}
+ \item For a free electron gas, $E = \frac{\hbar^2 k^2}{2m}$
\item Periodic potential in bulk solid leads to band gap structure of $E(\vect{k})$
\item Periodic potential $\implies$ E is periodic. Only needs to be defined in first Brillouin zone.
\end{itemize}
\item Surface states
\begin{itemize}
+ \item Simplest model: Step potential
+ \item Two major models
\begin{enumerate}
\item Tamm States: Periodic potential in solid, free space outside, jump at surface
\begin{itemize}
\section{Experimental Techniques}
-\subsection{Preparation of samples}
+\subsection{Secondary Electron Spectroscopy}
+
+Secondary Electron Spectroscopy encompasses a large group of techniques used for studying the electron spectra of surfaces and solids. In these methods a beam of primary electrons is directed at the surface of a solid. The interactions between primary electrons and the surface give rise to an energy distribution of secondary electrons scattered from the surface. Analysis of this secondary electron distribution gives information about the interaction between primary electrons and the surface.
+
+%\subsubsection{Electron-Surface Interactions}
+
+
+\subsubsection{Methods of Secondary Electron Spectroscopy}
+
+Energy-resolved methods of Secondary Electron Spectroscopy are based upon observation of the secondary electron energy distribution at a fixed primary electron energy. The primary electron energy determines which processes are possible, whilst the observed distribution can be related to the probability distribution for the possible processes.
+
+In contrast to Energy-resolved methods, Total Current (or Yield) methods are based on observation of the total current of secondary electrons as a function of primary electron energy. As the primary electron energy is increased, the threshold energies for particular processes are passed. This
+
+Both Energy-resolved and Total Current methods can be performed
+
+\subsection{Total Current Secondary Electron Spectroscopy}
+
+Figure \ref{} shows a simplified schematic for the Total Current Spectroscopy experiments conducted during this study. Electrons are emitted from a cathode held at negative potential relative to the target. The electron beam is focused and accelerated onto the target by the electric field of an electron gun. A detector is used to measure the total current passing through the target.
+
+
+
+\subsubsection{Electron Optics}
+
+The electron gun used for this experiment was repurposed from an old Cathode-Ray Oscilloscope (CRO). Figure \ref{} shows a simplified diagram of the electron gun, whilst Figure \ref{} shows a photograph of the gun.
+
+The full circuit diagram for the electron gun control circuit is shown in Appendix A.
+
+\subsubsection{Automatic Data Acquisition}
+
+In order to collect data on the large number of planned samples for the study, some form of automation was required. The automated system needed to be able to incrementally set the initial energy by controlling a power supply, and record the total current measured by an ammeter.
+
+The available power supplies at CAMSP only featured analogue inputs for external control. This meant that a Digital to Analogue Convertor (DAC) card was needed to interface between the control computer and the power supply. In addition, the available instruments for current measurement at CAMSP produced analogue outputs. As a result, Analogue to Digital Convertors (ADCs) would be required to automate the recording of total current.
+
+Although an external DAC/ADC box was already available for these purposes, initial tests showed that the ADCs on the box did not function. The decision was made to design and construct a custom DAC/ADC box, rather than wait up to two months for a commercial box to arrive. The design of the custom DAC/ADC box is discussed in detail in Appendix B, and the software written for the on-board microprocessor and the controlling computer are presented in Appendix C.
+
+
\begin{itemize}
\item Black-Au - 1e-2 mbar vacuum
\item ``Shiny'' - 1e-6 / 1e-7
\end{itemize}
\end{itemize}
-\subsection{Total Current Spectroscopy}
+\subsection{Electron Spectroscopy}
+
+Secondary electron spectroscopy methods are a broad class of methods which investigate surface electron spectra through observing processes in which the surface electrons participate directly \cite{komolov}.
+
+Total Current Spectroscopy is a group of electron secondary
+
\begin{itemize}
\item
\item Total Current Spectroscopy methods measure the total current of secondary electrons as a function of primary electron energy.
\end{itemize}
\end{itemize}
+\section*{Appendix A - Electron Gun Control and Current Measurement Circuit}
+
+Figure \ref{} shows the complete electron gun control circuit. The circuit was designed and constructed as part of this project. The design is based upon examples found in \cite{komolov} and \cite{Moore}.
+
+s
+
+
+\section*{Appendix B - DAC/ADC Box - Hardware}
+
+\subsection*{Overview}
+
+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. Most modern computers no longer feature COM ports; a commercially available convertor can be used to interface between the box's RS-232 output and a standard Universal Serial Bus (USB) port.
+
+
+The key components of the DAC/ADC box hardware include:
+
+\begin{itemize}
+ \item Microprocessor (AVR Butterfly ATMega169)
+ \item Four Analogue to Digital Converter (ADC) inputs
+ \item Single Digital to Analogue Converter (DAC) output (Microchip MCP4922)
+ \item Analogue electronics for amplification at ADC inputs and DAC outputs
+ \item Seperate power supply circuitry for Digital and Analogue electronics
+ \item RS-232 communications for control by a conventional PC or laptop
+\end{itemize}
+
+\subsection*{Microprocessor}
+The DAC/ADC box has been based upon Atmel's AVR Butterfly; an inexpensive and simple demonstration board for the ATMega169 16 Bit microprocessor. The features of the AVR Butterfly include easily accessible ports for Analogue to Digital Convertor (ADC) inputs and digital input/output, an onboard Universal Asynchronous Reciever/Transmitter (USART) for RS-232 serial communications, and a 6 character Liquid Crystal Display (LCD). The AVR Butterfly can be programmed using a conventional computer over the USART using a RS-232 COM port. For modern computers (which do not usually posess COM ports), a RS-232 to USB converter may be used.
+
+Figure \ref{avr_butterfly.pdf} is a labelled photograph of the AVR Butterfly showing the use of the available ports for this project.
+
+
+%Figure of Butterfly
+\begin{center}
+ \includegraphics[scale=0.70]{figures/avr_butterfly.pdf}
+ \captionof{figure}{AVR Butterfly} \label{avr_butterfly.pdf}
+\end{center}
+
+Unless otherwise stated, all voltage differences are specified relative to the power supply ground of the AVR Butterfly.
+
+\subsection*{ADC Inputs}
+
+The AVR Butterfly offers easy access to four of the ATMega169's ADCs through PORTF. Each ADC is capable of measuring voltages of $0 < V_{\text{adc}} < V_{cc}$ with 10 Bit resolution. For measuring voltages outside this range, some circuitry is required between the input voltage and the ADC input. In addition, it is desirable to provide the ADC with some form of input protection against accidental overloading. Figure \ref{adc_normal.pdf} shows the input circuit which was used for three of the four available ADCs.
+
+\begin{center}
+ \includegraphics[scale=0.50]{figures/adc_normal.pdf}
+ \captionof{figure}{ADC4,6,7 Input} \label{adc_normal.pdf}
+\end{center}
+
+
+For making voltage measurements above $V_{cc}$, a voltage divider allows reduction of the voltage at the ADC. By constructing the voltage divider using a variable resistor, the range of measurable inputs could be manually adjusted.
+
+The diodes shown in Figure \ref{adc_normal.pdf} ensure that the ADC is protected from accidental exposure to voltages outside the acceptable range. In normal operation both diodes are off. If $V_{\text{adc}}$ were to become greater than the reference point $V_{cc}$, current would flow between the ADC input and the reference point, acting to reduce $V_{\text{adc}}$ until it reached $V_{cc}$. Similarly, if $V_{\text{adc}}$ fell below ground, current would flow from ground to the ADC input, acting to increase $V_{\text{adc}}$ until it reached ground.
+
+The voltage at the ADC input can be related to the input of the voltage divider using Kirchoff's Voltage Law and Ohm's Law:
+\begin{align*}
+ V_{\text{adc}} &= \frac{R_1}{R_1 + R_2} V_{\text{in}}
+\end{align*}
+Where $V_{\text{in}}$ is the voltage at the input of the circuit, $R_1$ is a fixed resistor, and $R_2$ is variable resistor.
+
+$V_{\text{in}}$ can be therefore be determined from the registered ADC counts by:
+\begin{align*}
+ V_{\text{in}} &= \left(\frac{\text{ADC counts}}{2^{10}}\right) \times \frac{R_1 + R_2}{R_1} V_{cc}
+\end{align*}
+
+\subsubsection*{Differential ADC Input}
+
+During the testing of the TCS experimental apparatus, it became desirable to measure the emission current of the electron gun. The electrometer used for this current measurement was capable of producing an analogue output in the range of $0-1V$. However, the negative terminal of this output was not at ground potential, but rather at the same terminal as the negative input terminal. Directly connecting the electrometer output to one of the ADC inputs discussed above would create a short circuit between the initial energy power supply, and ground (refer to Figures \ref{} and \ref{}). Therefore, it was decided to add a differential stage before the input of one of the ADCs.
+
+Figure \ref{adc5.pdf} shows the modification made to the input for ADC5 on the AVR Butterfly. The original voltage divider and input protection discussed above are still present. The modifications include the addition of an instrumentation amplifier, and low pass filters.
+
+\begin{center}
+ \includegraphics[scale=0.70]{figures/adc5}
+ \captionof{figure}{Differential Input stage for ADC5}
+ \label{adc5.pdf}
+\end{center}
+
+
+The instrumentation amplifier consists of two stages of operational amplifiers (op-amps); input buffers, and a difference amplifier.
+The difference amplifier can be shown using the ideal op-amp model to produce an output voltage proportional to the difference between its inputs:
+
+\begin{align*}
+ V_{out} &= \frac{R_2}{R_1} \left(V_{2} - V_{1}\right)
+\end{align*}
+
+The two op-amps at the inputs to the differential amplifier act as unity gain buffers. Although the output of the unity gain buffer is equal to the input on its positive terminal, the buffer prevents current from flowing from the positive terminal to ground. With the buffer amplifiers absent, a current of: would flow between each of the input terminals and ground.
+
+Instrumentation amplifiers are usually constructed in the schematic shown in Figure \ref{}. In this version, the gain of the amplifier can be changed by altering a single resistor. However, more resistors are required. The version actually constructed was designed based upon the small number of resistors available, within a short time frame. Although the design could have later been changed, this would have been of no real benefit, since there was no requirement to adjust the gain of the amplifier.
+
+In principle, two ADC channels could be used to record the positive and negative outputs of the electrometer seperately, with differencing done in software. However this would require modification to the output cable of the electrometer, which may prove inconvenient for future uses.It was decided that the modification of the cable and added complexity of the software required would be more time consuming than differencing the two inputs using the hardware methods described above.
+
+The low pass filters were added to the inputs of ADC5 after it was found that an unacceptable level of AC noise was being output by the electrometer. The level of noise was too high to be filtered in software, for reasons that will be discussed in Appendix D.
+
+\subsection*{Power Supplies}
+Due to the presence of both analogue and digital electronics in the DAC/ADC box, three seperate supply voltages were required:
+\begin{enumerate}
+ \item Digital logic in the range $3 \to 4.5$V
+ \item Positive op-amp supply in the range $10 \to 15$V
+ \item Negative op-amp supply in the range $-10 \to -15$V
+\end{enumerate}
+
+Circuitry was designed which allowed two seperate single pole power supplies to be used for Digital logic and the op-amps. A dual 0-30V DC power supply has been used for both digital and analogue circuitry.
+
+\subsubsection*{Logic Power Supply}
+The AVR Butterfly runs off $3V < V_{cc} < 4.5V$ DC. Since $V_{cc}$ was also used as the reference voltage for the ADCs and DAC output, it was desirable that $V_{cc}$ be kept constant, despite the absolute level of the power supply. A $3.3V$ voltage regulator has been used for this purpose. The capacitor further smooths the output by shorting high frequency fluctuations to ground.
+
+When the DAC/ADC box was first constructed $V_{cc}$ was supplied by three $1.5V$ batteries. However, due to higher than expected power usage, and the unreliability of the voltage regulator as the input voltage fell below $4V$, inputs for an external power supply were later added.
+
+\begin{center}
+ \includegraphics[scale=0.70]{figures/logic_ps}
+ \captionof{figure}{Logic Power Supply}
+ \label{logic_ps.pdf}
+\end{center}
+
+\subsubsection*{Op-amp Power Supply}
+The DAC/ADC box circuitry involves several operational amplifiers (LF356), which require dual $\pm 10-15V$ supplies. As there were no dual $\pm$ power supplies available, a single $30V$ power supply was used, with the circuit shown in figure \ref{fig_opamp_supply} used to produce $\pm 15V$ relative to ground.
+
+The buffer amplifier ensures that negligable current can flow from the power supply into the logic and ADC circuits, whilst the capacitor removes high frequency fluctuations of the power supply relative to ground.
+
+To simplify circuit diagrams, op-amps will be drawn with the power supply connections ommitted from this point onwards.
+
+\subsection*{DAC Output}
+A commercial DAC board was used to produce the DAC output. The Microchip MCP4922 ET-Mini DAC is controlled by the AVR Butterfly using Motorola's Serial Peripheral Interface (SPI) Bus. The software used to implement SPI between the MCP4922 and the AVR Butterfly is discussed in Appendix D.
+
+The ET-Mini DAC can only be powered off $3V$ to $5V$. Using $V_{cc} = 3.3V$ means that the DAC output cannot exceed $V_{cc} = 3.3V$. For TCS, energies of up to $15eV$ are required, so amplification of the DAC output was clearly necessary. A simple non-inverting amplifier with a manually adjustable gain was used to amplify the DAC output by a factor of three. This output was then used to control a laboratory power supply to produce the full range of initial energies.
+
+\subsection*{RS-232 Communications}
+
+The AVR Butterfly features an onboard USART, which can be used both for programming and communication with the ATMega169 processor. The RS-232 communications requires only three wires; Recieve (RX), Transmit (TX) and a common ground.
+
+The requirement that the AVR Butterfly share a common ground with the controlling computer lead to increased noise through ground loops. This is discussed in more detail in Appendix D.
+
+Although the RS-232 is relatively simple to implement, which makes it ideal for non-proprietry microprocessor applications, most modern computers no longer feature RS-232 COM ports. Although a computer with COM ports was available at CAMSP, due to the extreme unreliability of this computer, it was quickly replaced with a laptop that did not possess COM ports, and a commercial RS-232 to USB converter was used to interface with the laptop.
+
+
+
+\section*{Appendix C - Pressure Monitoring}
+
+Over time, it was noticed that the pressure in the chamber was variable.
+
+
+
\pagebreak
\bibliographystyle{unsrt}
\bibliography{thesis}
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