+
+ \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}
+