Surfaces single slab

Single slab. A number of recent calculations of surface electronic structures have shown that the essential electronic and structural features of the bulk material are recovered only a few atomic layers beneath a metal surface. Thus, it is possible to model a surface by a single slab consisting of 5-15 atomic layers with two-dimensional translational symmetry parallel to the surface and vacuum above and below the slab. Using the two-dimensional periodicity of the slab (or thin film), a band-structure approach with two-dimensional periodic boundary conditions can be applied to the surface electronic structure. 

The description of the molybdenum disulfide particles as regular hexagonal single slabs well dispersed over the alumina support surface permits calculations of the number of molybdenum ion per M0S2 slab and the M0S2 slab surface... 

Fig. 13. ER spectra generated theoretically for the Ex and E1 + A, transitions of p-GaP (Nk = 1018 cm 3) assuming (a) that the depletion layer can be modelled as a single slab of thickness IF in which the electric field is fixed at a constant value corresponding to half that calculated at the surface using the classical Schottky model and (b) that the depletion layer can be modelled by dividing the depletion layer into twelve slabs within each of which the electric field is fixed at the value obtained from the Schottky model for the mid-point of the slab.

Figure 5 Tjrpical periodic boundary conditions used for computer simulations of metal-water interfaces (a) and (b) geometries that are periodic in the directions parallel to the metal surface (c) geometry that is periodic in all three dimensions. The ome-tries illustrated correspond to (a) a single slab of water sandwiched between metal surfaces, (b) a slab of water that is on top of a metal slab and has a free surface, and (c) a system with an infinite number of parallel, alternating, metal and water slabs.

As was seen above in the slab model the surface plane is supposed to be fixed by its orientation (by the set of Miller indices) relative to the bulk structure and its symmetry elements. All the atoms of the bulk primitive unit cell are distributed within one or several atomic planes with the same 2D-translation periodicity, see Figures 11.2-11.4. Let these planes of atoms form a layer, being stoichiometric and neutral as by definition it contains all the atoms of the bulk primitive unit cell. Depending on the host-crystal structure the one-layer single slab can have zero (types-1 and type-2 surfaces) or nonzero (type-3 surfaces) dipole moment along the normal... 

The symmetry of the single slab corresponds to one of 80 diperiodic (layer) space groups. Fig. 11.5 shows a 3-layer single-slab model of (001) surface of MgO crystal (each layer consists of one atomic plane). The symmetry group of this slab DG61 PAjmmm) belongs to a square system. 

Table 11.4 shows the convergence of results for the single slab (2D) MgO (001) surface model depending on the slab thickness. 

Table 11.4. Convergence of results for the MgO (001) single slabs depending on the 2D slab thickness [Es - surface energy in J/m, Ef - Fermi energy in H), [776]...

The numerical value of Fermi energy Ep may be used for approximate estimation (according to the Koopmans theorem) of the surface ionization potential that defines the adsorption energies. Due to the correlation effects the DFT Fermi energy Ep differs essentially from that in HF calculations. Table 11.4 shows that Ep is inside the valence band of a perfect crystal, i. e. the resonance surface states are predicted both in HF and DFT single-slab calculations. 

Table 11.6. Convergence of the results for single-slab (2D) model of (110) rutile Ti02 surface, [777], (Es - surface energy, Ef - Fermi energy)...

The first two-periodic all-electron HP LCAO calculations of the rutile relaxed surfaces, made in [779], gave atomic displacements of surface atoms that did not differ significantly from the later results of DPT-PW investigations. Purther periodic LCAO studies of Ti02 bare surfaces have been made in [777,799,800]. Por studies of H2O adsorption on Ti02 the single-slab periodic HP-LCAO and DPT-LCAO methods were first applied in [790] and compared with PW-DPT results to test various methods with cyclic- and embedded-cluster calculations and resolve discrepancies between the methods. 

Single-slab LCAO Calculations of Bare and Hydroxylated Sn02 Surfaces... 

The first-principles methods have made an increasingly significant contribution to understanding the nature of clean Sn02 surfaces [806—808] and the interaction of these surfaces with adsorbed water [809-811], methanol [812], CO [813,814], and O2 [815] molecules. Many of these calculations use the DFT PW periodic-slab model. The LCAO single-slab approach has been successfully apphed for investigation of cassiterite [806,812-815] A review of both PW and LCAO studies of adsorption on the perfect and reduced surfaces of metal oxides can be found in [784]. 

The results of LCAO calculations of the bare surface slabs validate the admitted approach (including single-slab model, LCAO basis, and BSSE correction) and indicate that the use of hybrid functionals to describe Sn02 surfaces provides more reasonable results. 

We discuss here in more detail the results of a hybrid HF-DFT LCAO comparative study of cubic SrZrO and SrTiOa (001) surface properties in the single-slab model [825). As known from [824], the consideration of systems with 7 8 atomic layers is sufficient to reproduce the essential surface properties of cubic perovskites. Three different slab models have been used in [825]. The first (I) and the second (II) ones consist of 7 crystalline planes (either SrO- or MO2-terminated, respectively) being symmetrical with respect to the central mirror plane but nonstoichiometric (see Fig. 11.4). The central layer is composed of MO2 (M = Ti, Zr) units in the model 1 and SrO units in the model 11. Both models 1 and 11 have been apphed for studying the surface properties of titanates by ab-initio calculations [832]. The asymmetric model 111 is stoichiometric and includes 4 SrO and 4 MO2 atomic planes. Accordingly, it is terminated by a SrO plane on one side and by a MO2 plane on the other side and there is no central atomic layer. The model 111 has been included in the simulation to investigate the influence of the stoichiometry-violation in the symmetrical models 1 and II on the calculated surface properties. For all slabs a 1 x 1 surface unit cell has been taken. For the 2D translations in slabs the experimental bulk lattice constants of SrZrOs (4.154 A) and SrTiOs (3.900 A) were used that does not differ significantly from DFT B3PW LCAO theoretical values (4.165 A and 3.910 A respectively). 

The first ab-initio calculation of cubic LaMnOs surface properties [851] has been made by the HF LCAO method for (110) LMO surface in the single-slab model. The extension of these calculations to the (001) LMO surface was made in [852]. The bulk cubic rmit cell atoms are distributed over atomic planes (normal to the surface direction) in the following way for the (110) surface - 02-LaMnO-02-LaMnO - - - for the (001) surface - La0-Mn02-La0-Mn02 (see Fig. 11.4). In both cases the surfaces are polar. 

The LCAO single-slab calculations [852] of the electronic structure of the polar LnMnOs (001) and (110) surfaces clearly demonstrate that the stoichiometric slabs have considerably lower energies than the nonstoichiometric ones. It should be stressed that the structural oxygen vacancies are energetically required and hence are essential elements of the (110) polar surface structure. Their formation makes the (110) slabs stoichiometric and energetically more favorable than the stoichiometric slabs stabilized by the near-surface electronic density redistribution necessary to compensate the macroscopic dipole moment perpendicular to the asymmetric LaO Mn02 surfaces. 

An ideal single surface (a slab of material so thick that no radiation is bounced back from the far side) ... 

Figure 6-12. Model for Ihe Calculation of the van der Waals potential experienced by a single T6 molecule on a Tfi ordered surface. Each molecule is modeled as a chain of 6 polarizable spherical units, and the surface as 8-laycr slab, each layer containing 266 molecules (only pan of the cluster is shown). Tire model is based on X-ray diffraction and dielectric constant experimental data. The two configurations used for evaluating the corrugation of the surface potential are shown. Adapted with permission front Ref. [48].

In European laboratories, ceramic tile is very popular, but it is rarely seen here. The cost is well below that of monolithic slabs but considerably higher than that of plastic laminate. Its resistance to both chemicals and heat is outstanding. In case of damage, single tiles may be replaced. Modem materials used for grouting are also very resistant to chemicals. The slightly uneven surface does not seem to cause any complaints among users. 

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