\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
\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}
\end{itemize}
\end{itemize}
+\section*{Appendix A - Electron Gun Control and Current Measurement Circuit}
+
+
+\section*{Appendix B - Data Aquisition Box (DAQ)}
+
+\subsection*{Overview}
+In order to automate recording of TCS data, a data aquisition box (DAQ) was designed and constructed. The DAQ consists of the following major components:
+\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 RS-232 communications for control by a conventional PC or laptop
+ \item Seperate Power supplies for digital and analogue electronics
+\end{itemize}
+
+\subsection*{Microprocessor}
+The DAQ has been built around an Atmel AVR Butterfly; an inexpensive and simple demonstration board for Atmel's 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 of Butterfly
+
+
+
+
+\subsection*{Power Supplies}
+Two seperate power supplies are required for the DAQ, due to the presence of both digital and analogue electronics.
+
+\subsubsection*{Logic Power Supply}
+The AVR Butterfly runs off $3V < V_{cc} < 5V$ DC. Since $V_{cc}$ is also used as the reference voltage for the ADCs and DAC output, it is 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 DAQ 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.
+
+\subsubsection*{Op-amp Power Supply}
+The DAQ 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*{ADC Inputs}
+The AVR Butterfly offers easy access to four of the ATMega169's ADCs through PORTF. Each ADC is capable of measuring voltages (relative to ground) of $0 < V_{\text{adc}} < V_{cc}$ with 10 Bit resolution. A voltage divider constructed using a variable resistor allows manual adjustment of $V_{\text{adc}}$ so that voltages greater than $V_{cc}$ may still be measured. Diodes between the ADC input, and $V_{cc}$ and ground, ensure that the ADC is protected from exposure to voltages outside the acceptable range. Figure \ref{fig_ADC_normal} shows the typical input circuit, used for three of the four available ADCs.
+
+The voltage of the
+\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.
+
+The $V_{\text{in}}$ can be determined from the ADC counts by:
+\begin{align*}
+
+\end{align*}
+
+
+For differential or ``off-ground'' voltage measurements, the fourth ADC input is passed through an instrumentation amplifier. Low pass filters on the input are used to reduce AC noise on the inputs. The buffer amplifiers ensure that negligable current flows through the measurement circuit to ground.
+Figure \ref{fig_ADC5} shows the input circuit for off-ground voltage measurements. It is important to note that the circuit only functions when the input voltages are within the op-amp common-mode voltage range ($\pm15V$).
+
+\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 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 Total Current Spectroscopy, 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 $3.3V$ DAC output to $10V$. This output was then used to control a GW-Instek GPS-1850D power supply.
+
\pagebreak
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\bibliography{thesis}
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