-\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}