Electronic surface states on (111) aluminum

We have calculated the projection of the three-dimensional energy bands of aluminum onto the two-dimensional Brillouin zone (BZ) of the (111) face. Using a pseudopotential constructed in the same way as that in our publications on the (001) and (110) faces, we have calculated eigenvalues and eigenfunctions at high-symmetry points of the two-dimensional BZ for (111) thin films. Unlike Boudreaux who found surface states only in the gap around $\overline{\Gamma }$, we find surface states exist in all the energy gaps of the projected bands at the $\overline{\Gamma }$,$\overline{M}$, and $\overline{K}$ symmetry points of the two-dimensional BZ. We conjecture that surface states always exist in the projected energy gaps of low-index surfaces of simple metals. For the first time we find a surface eigenfunction which decays inwardly from one surface toward the other; heretofore all surface states we have found decay inwardly from both surfaces toward the center of the film.