Ionic, Covalent, and Metallic Surface States of a One-Dimensional Mathieu Potential with Arbitrary Termination

Abstract

A model of surface states on a one-dimensional crystal with sinusoidal potential has been simply solved for arbitrary values of the sinusoidal amplitude $q$, lattice termination ${z}_{0}$, surface step height $\ensuremath{\xi}$, and band-gap index $m$. The Schr\"odinger equation inside the crystal is equivalent to the Mathieu equation, whose eigenfunctions and eigenvalues are well known. Surface states are constructed as hybrids of bulk states [incorporating the proper nearly free-electron (NFE) and linear-combinations-of-atomic-orbitals (LCAO) limits], and they move towards the centers of the band gaps the stronger the surface "perturbation." It is shown that ionic surface states occur for $m=2$, ${z}_{0}=\frac{1}{4}\ensuremath{\pi}$, and $m=2$, ${z}_{0}=\frac{3}{4}\ensuremath{\pi}$; covalent surface states occur for $m=1$, ${z}_{0}=\frac{1}{2}\ensuremath{\pi}$; and metallic (or virtual) surface states occur in the allowed band between $m=0$ and $m=1$. Shockley states (looked for at ${z}_{0}=0$, all $m$) do not appear. This work has been favorably related to the author's semiclassical ionic model, to previous NFE and LCAO approximation methods, to the theories of Shockley and Statz, and to real surfaces in accordance with low-energy-electron-diffraction studies. In addition, it has uncovered a new linkage between the NFE and LCAO terminologies.