+\emph{NOTE: I need to draw some fine structure on this curve somehow. Or find an actual spectrum to reproduce.}
+
+Figure \ref{se_dist.pdf} shows a simplified model of an energy distribution of secondary electrons.
+
+The spectrum shown in Figure \ref{se_dist.pdf} may be divided into several regions based upon the originating processes of the secondary electrons in each region. However, it is important to note that these processes are determined by the primary electron energy $E_p$; each process has a threshold energy, below which it cannot occur. For example, Auger electrons are only produced if the primary electrons have sufficient energy to excite an inner level electron to above the Fermi level.
+
+The narrow peak centred at $E = E_p$ is largely due to elastically scattered primary electrons\footnote{A typical width is $0.5-1.0$eV; as a result, small energy losses due to phonon excitations ($10-50$meV) can not resolved from truly elastic reflections \cite{komolov}}; the width of this peak is determined by the distribution of primary electron energies, as well as the resolution of the detector.
+
+Fine structure due to low energy losses can be observed just below the primary peak. These energy losses are due to transitions between the valence and conduction bands, and plasma vibration excitation. This part of the spectrum is the focus of Energy Electron Loss Spectroscopy (EELS).
+
+The central region of the distribution is mostly due to inelastically scattered (or ``rediffused'') primary electrons. If the energy of primary electrons is sufficient, this region may also contain fine structure due to Auger excitations.
+
+The broad, asymetrical curve at low energy is due to inelastically scattered primary electrons which have undergone multiple scattering events. So called ``true'' secondary electrons, the direct result of secondary electron emission, also appear in this region for sufficiently large primary electron energies.
+
+For a more detailed discussion, refer to \cite{komolov}.
+
+
+Techniques of Secondary Electron Spectroscopy can be divided into two classes. Energy-resolved methods are based upon observation of the secondary electron distribution at a fixed primary electron energy. The angular distribution of emitted electrons is often also recorded. These methods aim to examine the properties of secondary electrons emitted in a particular energy interval.
+
+
+In contrast to Energy-resolved methods, Total Current (or Yield) methods measure the total current of secondary electrons whilst varying the primary electron energy. As the primary electron energy reaches the threshold for a particular mechanism of secondary electron scattering, the analysis of the total secondary electron current as a function of energy can give information about the threshold energies for processes of interest.
+
+Total Current methods are generally simpler to realise experimentally compared with Energy-resolved methods, as they do not require energy analysers, and current measurement may be performed external to the vacuum chamber, using a conventional low current ammeter. It is also simple to combine a Total Current methods with existing Energy-resolved methods.
+
+
+\subsection{Low Energy Total Current Spectroscopy}
+
+As the name suggests, low energy Total Current Specroscopy is based upon measurement of the total secondary electron current at low primary electron energies, typically in the range of $0-15$eV. At low primary energies, the secondary electron current is predominantly composed of inelastically reflected primary electrons which have lost energy in causing interband transitions.
+
+Figure \ref{tcs_simple.pdf} shows a simplified schematic for the Total Current Spectroscopy experiments conducted during this study. Electrons are produced via thermionic emission by heating a cathode. A series of electrodes are used to accelerate and focus a current $I_1$ onto the target. The energy of primary electrons is controlled by adjusting the power supply $U$, which determines the potential between the cathode and target. The transmitted current $I$ to flow through an ammeter external to the chamber. For a more detailed description of the experimental setup, Refer to Appendix B for a discussion of hardware to automate the measurement of $I$ and control of $U$. Refer to Appendix D for a discussion of the electron gun and its control circuit.
+
+\begin{center}
+ \includegraphics[scale=0.50]{figures/tcs_simple}
+ \captionof{figure}{Simplified Diagram of TCS Experiments}
+ \label{tcs_simple.pdf}
+\end{center}
+
+The goal of Total Current Spectroscopy is to measure variations in the secondary electron current, $I_2$. It can easily be demonstrated that this can be accomplished by measurement of $I$.
+
+In the following discussion, we will summarise the approach adopted by Komolov to relate measurement of $I(E_1)$ to characteristics of the sample under study \cite{komolov}.
+
+From the above, it is obvious that $I = I_1 - I_2$. Assuming that $I$ is a constant, independent of primary electron energy $E_1$, we define the Total Current Spectrum (TCS) as:
+\begin{align*}
+ S(E_1) &= \der{I}{E_1} = - \der{I_2}{E_1}
+\end{align*}
+This result also assumes that $I$ does not vary during the time taken to perform a measurement of $S(E_1)$ for a range of $E_1$ values. This is generally valid in the period after the cathode reaches thermal equilibrium.
+
+The energy of a single primary electron arriving at the sample is given by $E = e U + c$, where $e$ is the electron charge, $U$ is the potential difference between cathode and sample, and $c$ a constant including the contact potential between the cathode and sample.
+In reality, the cathode emits electrons with a distribution of energies, which is further altered by the focusing properties of the electrodes; as a result, the energy of the incident primary electrons is described by a distribution $f(E - E_1)$ about the mean value $E_1$, with the maximum of the distribution at $E = E_1$.
+
+The primary electron current $I_1$ for a mean energy $E_1 = e U$ can be written as:
+\begin{align*}
+ I_1(E_1) &= e A \int_{0}^{\infty} f(E - E_1) dE
+\end{align*}
+Where $A$ is the surface area irradiated by the beam.
+
+Introducing the secondary emission coefficient $\sigma(E)$, which gives the probability for a primary electron of energy $E$ to give rise to a secondary electron, we can write the secondary electron current as:
+\begin{align*}
+ I_2(E_1) &= e A \int_{-E_1}^{\infty} \sigma(E_1)f( E - E_1) dE
+\end{align*}
+
+The total current $I$ may then be written as:
+\begin{align*}
+ I(E_1) &= e A \left[ \int_{0}^{\infty} f(E - E_1) dE - \int_{-E_1}^{\infty} \sigma(E_1)f( E - E_1) dE \right]
+\end{align*}
+
+Differentiating, using the fundamental theorem of calculus, we can determine the total current spectrum:
+\begin{align*}
+ S(E_1) = \der{I}{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*}
+
+The first term in the above expression is determined solely by the distribution of primary electrons $f$. This term will be maximised when $E_1 = 0$; meaning that $U$ is equal to the contact potential $c$ between the cathode and sample.
+
+The second term contains all dependence of $S(E_1)$ on characteristics of the sample. At the threshold for a particular process, the secondary emission efficiency $\sigma(E_1)$ is expected to undergo a sharp change. This results in a well defined maxima or minima in the derivative $\der{\sigma(E_1)}{E_1}$, which can be seen as a corresponding maxima or minima in the total current spectrum $S(E_1)$. From the convolving function $f(E - E_1)$, it can be seen that the distribution of primary electron energy determines the degree to which $\der{\sigma(E_1)}{E_1}$ may be resolved from measurement of $S(E_1)$.
+
+The total current spectrum $S(E_1) = \der{I}{E_1}$ can be obtained from measurement of $I(E_1)$ using a finite difference approximation. Often, the conventional ammeter and DC power supply in Figure \ref{tcs_simple.pdf} are replaced with a lock-in amplifier and AC power supply, as in Komolov's description \cite{komolov}. Lock-in amplifier techniques have the advantage of measuring $S(E)$ directly. The lock-in amplifier also eliminates unwanted sources of noise. For this study, the lock-in amplifier approach was inpractical due to the limitations on available equipment. For future studies, it is suggested that the lock-in amplifier approach be adopted.
+