A better account of the surface electronic structure

The above approach neglects the modifications of the electronic structure in the surface layers. In semi-conductors and insulators, as a result of the presence of dangling bonds or due to the reduction of the Madelung potential on under-coordinated atoms, surface states appear either at the gap edges or deep in the gap. Surface electron-hole pairs have thus lower energies than in the bulk. In addition, the reduction of the Madelung potential at the surface yields an increase of the matrix elements of y in (4.2.23). Both processes enhance the surface dielectric constant. 

Let us denote by Mi the cations which are located in the first coordination shell of the oxygen atom Oi on which the proton adsorbs, by O2 the oxygens located in the second coordination shell and so on. On 

a loss of electrons results from the O-H bond formation. It induces a strong lowering of the atomic levels (intra-atomic effects). On other substrate atoms, the proton electrostatic potential also shifts the levels towards lower energies, but the importance of the shift decreases with the distance. The energy separation between the Oi and Mi levels thus increases. The Oi-Mi bond is more ionic and electrons are transferred from Ml to Oi. On the M1-O2 bond, the level separation is smaller. The M1-O2 bond is more covalent and electrons are transferred from O2 to Ml, and so on. 

When a negatively charged species adsorbs on a surface, the electron motion has the opposite direction. Similar effects have been found in modelling tip-surface interactions (Castanier, 1995). 

As in the case of metals and semi-conductors, there exist specific surface excitations in insulating oxides. Three types of surface phonon modes may be distinguished the Rayleigh mode, the Fuchs and Kliewer modes and the microscopic surface modes. The first two modes have a long penetration length into the crystal. They are located below the bulk acoustic branches and in the optical modes, respectively. The latter are generally found in the gap of the bulk phonon spectrum. 

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