\section{Overview of Theory}
Summarise the literature, refer to past research etc
-\subsection{Electron structure of surface}
+\subsection{Electron Spectra of a Surface}
\begin{itemize}
- \item Overview of electron spectrum properties
+ \item Description of the near surface region
\begin{itemize}
- \item Density of states $n(E)$
- \item Energy band structure $E(\vect{k})$
- \end{itemize}
- \item Properties of surface region
- \begin{itemize}
- \item Difference between potential of surface and bulk
+ \item All real solids occupy finite volumes in space.
+ \item The surface of a solid is important because interactions between the solid and its surroundings occur in the near surface region.
+ \item Characterised physically by:
\begin{itemize}
- \item Change between the two limits in the ``near-surface'' region
+ \item Termination of periodic crystal lattice
+ \item Violation of geometric order
+ \item Distortion of interatomic distances and hence interaction forces
+ \item There is a transition ``near surface'' region between bulk and surface properties, roughly 5 atomic distances.
\end{itemize}
- \item Theoretical models for the potential, 1D vs 3D
+ \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 Simplest case is a step potential.
- \item Various improvements on this model, discussed in Komolov's book.
+ \item Metal & Semiconductor
+ \end{itemize}
+ \item In reality,
+
+ \end{itemize}
+
+ \item The Electron Spectra
+ \begin{itemize}
+ \item Electron Spectra describes the energy eigenstates for an electron in a Bulk or Surface potential
+ \item Characterised by
+ \begin{enumerate}
+ \item Energy dispersion $E(\vect{k})$
\begin{itemize}
- \item Possibly adapt CQM project to model these potentials, if I get time
+ \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 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}
- \end{itemize}
- \item Limitations of theoretical models
- \begin{itemize}
- \item Real surface is not a step potential
- \item Adsorption of foreign particles onto the surface also plays a large role in determining the electron spectrum.
- \end{itemize}
+ \item Density of States $N(E)$
+ \begin{itemize}
+ \item $N(E) = \frac{\Delta N}{\Delta E} = \frac{1}{4\pi^3}\int_S\left(\der{E}{k}\right)^{-1} dS$
+ \item Integral is in momentum space over the isoenergetic surface of energy $E$
+ \item For a free electron gas, $N(E) = $
+ \end{itemize}
+ \end{enumerate}
\end{itemize}
+
+ \item Surface states
+ \begin{itemize}
+ \begin{enumerate}
+ \item Tamm States: Periodic potential in solid, free space outside, jump at surface
+ \begin{itemize}
+ \item Energy eigenvalues lie in the forbidden band of the bulk spectra
+ \item Attenuation of eigenvalues from surface to vacuum, oscillation of state within surface
+ \item Max electron density occurs on the crystal surface
+ \end{itemize}
+ \item Shockley states: Potential of surface and bulk cells equal
+ \begin{itemize}
+ \item Corresond to free valences (dangling bonds) at the surface
+ \end{itemize}
+ \end{enumerate}
+ \item Tamm and Shockley states arise from two extreme models (large change and small change respectively between bulk and surface). In reality, a combination of Tamm and Shockley states appear.
+ \item These states arise from termination of the lattice; but the surface cells are assumed undistorted
+ \item In reality surface cells are distorted by relaxation and reconstruction of the surface
+ \end{itemize}
+
\item Main reference: Komolov "Total Current Spectroscopy"
\item "Solid State Physics" textbooks and "Electron Spectroscopy" textbooks
\end{itemize}
\subsection{Total Current Spectroscopy}
\begin{itemize}
- \item Overview of technique
+ \item
+ \item Total Current Spectroscopy methods measure the total current of secondary electrons as a function of primary electron energy.
+ \item These methods are distinguished from ``differential'' methods (such as Auger electron spectroscopy and energy loss spectroscopy) which measure the secondary electron spectrum at a fixed primary electron energy.
+ \item
\begin{itemize}
\item Low energy beam of electrons incident on sample
\item Measure slope of resulting I-V curve
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