d-Band Surface States on Transition-Metal Perovskite Crystals: I. Qualitative Features and Application to SrTiO3

The qualitative features of the bulk and surface energy bands of ionic transition-metal perovskite crystals such as SrTi${\mathrm{O}}_3$, BaTi${\mathrm{O}}_3$, KTa${\mathrm{O}}_3$, and BaZr${\mathrm{O}}_3$ are discussed. The linear-combination-of-atom orbitals (LCAO) method is used in combination with a generalized-Seitzionic model to derive parameters for calculating the bulk and surface energy bands. The model for the bulk bands differs in detail from that used by Kahn and Leyendecker for SrTi${\mathrm{O}}_3$ but is qualitatively similar. Surface energy bands are calculated utilizing the transfer integrals of Kahn and Leyendecker and also those corresponding to the recent results of Mattheiss. In the model, the electrostatic Madelung potentials and ionization potentials have the largest energies. The ($\mathrm{pd}\sigma$) interactions between the $d$ orbitals of transition-metal ions and the $p$ orbitals of the oxygen ions have the next largest energies. The ($\mathrm{pd}\pi$) interactions and the crystalline field splittings follow in magnitude. The ($\mathrm{pp}\sigma$) and ($\mathrm{pp}\pi$) interactions between adjacent oxygen atoms have somewhat higher energy than the spin-orbit interaction. The effective ionic charges are determined by fitting the observed energy gap which is 3-4 eV for these materials. The bulk bands near the energy gap consist of three $t_{2g}$ and two $e_g$ conduction bands and nine valence bands derived from the oxygen $p$ orbitals. The $d$-conduction-band widths are controlled by the ($\mathrm{pd}\pi$) and ($\mathrm{pd}\sigma$) integrals. The valence-band widths also depend on these parameters but the ($\mathrm{pp}\pi$) and ($\mathrm{pp}\sigma$) integrals add directly to the bandwidth. It is shown that the qualitative features of the energy bands are preserved if the $p-p$ interactions are neglected. In this approximation, analytic results are obtained for the energy bands and the wave functions. Analytic expressions are also derived for the surface-state energy bands for a (001) surface. The perturbations due to spatial variations in the Madelung potentials, change in the electrostatic splitting, variations in the layer spacing, and small rotations in the surface bond angles are included in these analytic expressions. Exact expressions for the spatial variations in the Madelung potentials are derived by a scheme similar to that of Hund for the bulk potentials. The potentials are changed only negligibly except on the surface where changes of the order of several eV occur. The effects of surface irregularities, impurities, and vacancies can easily be treated by the method. Two types of (001) surfaces occur for the $\mathrm{AB}{\mathrm{O}}_3$ perovskite structure. The type-I contains oxygen and $B$ ions (the transition-metal ions) while the type-II surface has $A$ and oxygen ions exposed. Formulas for the surface bands of both type surfaces are derived. The dependence of these surface bands on the surface perturbations is discussed. Multiple surface bands and truncated branches which exist only over a certain portion of the Brillouin zone are found to occur. The conduction surface bands follow approximately the dispersion of the bulk and edge. Valence surface bands behave differently for type-I and -II surfaces. The valence surface branch can have a small upward curvature on a type-II surface while for the type-I surface, the dispersion follows the bulk-valence band edge and has a downward curvature as a function of the wave vector parallel to the surface.