-\subsection{Electron Spectra of a Surface}
-\begin{itemize}
- \item Description of the near surface region
- \begin{itemize}
- \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 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 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.
-
- \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 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}
- \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}
- \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}
- \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{Electron Spectra of Solids and Surface}
+
+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}
+
+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}
+
+\subsubsection{The Near-Surface Region}
+
+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.
+