Coincidence lattice

We assume in the following discussion that the solid surface under consideration is of the same chemical identity as the bulk, that is, free of any oxide film or passivation layer. Crystallization proceeds at the interfaces between a growing crystal and the surrounding phase(s), which may be solid, liquid, or vapor. Even what we normally refer to as a crystal surface is really an interface between the crystal and its surroundings (e.g., vapor, vacuum, solution). An ideal surface is one that is the perfect termination of the bulk crystal. Ideal crystal surfaces are, of course, highly ordered since the surface and bulk atoms are in coincident positions. In a similar fashion, a coincidence site lattice (CSL), defined as the number of coincident lattice sites, is used to describe the goodness of fit for the crystal-crystal interface between grains in a polycrystal. We ll return to that topic later in this section. 

The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice, or CCSL (Chen and King, 1988). In this book we treat only cubic systems. Interestingly, whenever an even value is obtained for E in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession until an odd value is obtained thus, E is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory (Bollman, 1967). The O-lattice takes into account all equivalence points between two neighboring crystal lattices. It includes as a subset not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would take us well beyond the scope of this book. The interested reader is referred to Bhadeshia (1987) or Bollman (1970). 

The STM contrast arises from Moire interference between the hexagonal substrate and a hexagonal overlayer rotated with respect to the substrate, as first proposed in [20]. The Moire coincidence lattice and corresponding STM images for the three (w, and w )-TiO on Pt(lll) phases are shown in Fig. 8.9. 

Fig. 8.9 Moire coincidence lattices (with Ti atoms shaded) of (a) the w-TiO structure I 5 I...

Those interested can examine the notes by Rovida and Pratesi, Huber and Oudar, and Orent and Hansen.Thus although O2 or NO can produce an apparent c(2 x 6) overlayer on Ru(lOTO) at 750 °C, Orent and Hansenpoint out that the overlayers do not account for the stability of the coincident structures, for the fact that transition from simple to coincident lattice was irreversible, or for the activation energy necessary to form these structures. For these reasons Orent and Hansen favour reconstructions, an example of which is shown in Figure 5, which is taken from a catalytic study of the NO + O2 reaction on a Ru(lOTO) surface. 

A clean silver surface will adsorb oxygen in a dissociated form. LEED studies show that on the Ag(l 11) face oxygen adsorbs to form a stable (4 x 4) superstructure. This has been interpreted as a coincidence lattice between the Ag(lll) plane and the (111) plane of silver(i) oxide. There is evidence that oxygen adsorption on faces other than Ag(lll) results in the formation of Ag(l 11) facets. " Incorporation of oxygen into the subsurface appears to be rather slow in the presence of molecular oxygen, but alternate cycles of oxidation and reduction with CO results in the build up of a thin subsurface layer of oxidized silver. ... 

Confusion can arise if the Park-Madden symbol is made to refer to an ad-layer mesh alone, rather than to the proper combined mesh of the surface structure which by definition inclvdes the substrate mesh (Section IIA). The temptation to describe just the adlayer structure arises because the symmetry of the combination mesh is commonly lower, and the combination mesh can be awkwardly large. In such a situation LEED patterns can be very complicated even though structure is basically simple. Suitable conventional notation to handle this has not been formulated, and in such cases it is appropriate and desirable to describe the overlayer mesh separately, in addition to describing the combination mesh. This is particularly useful for coincidence lattices (Section IVE). 

We first note that the coincident lattice concept includes simple structures, exhibiting registry, in which the adsorbate atoms are all located in positions of high symmetry on the substrate surface. For example, in the structure of Fig. 13a, the CO adsorbate net is in coincidence with the substrate one can properly consider that the one-quarter order beams of Fig. 13b arise from a (4 x 4) coincidence lattice. On the other hand, for a more general overlayer, most of the adatoms will often lie in positions of low symmetry. 

In its most primitive form the idea behind the coincidence lattice assumes that the forces within a monolayer are sufficiently strong to ignore the potential of the substrate, except for small adjustments of the lattice in the plane of the overlayer that may be required to achieve perfectly coincident registry. In a real situation it is likely that there will be small individual displacements of the overlayer atoms, not only parallel to the plane of the surface but also normal to it. Such relaxations will be expected to offset misfit strains, and should occur in regular fashion within the coincident mesh. Such displacements have been theoretically considered in papers by Tucker (329), Palmberg and Rhodin (165), Fedak and Gjostein (173) and Ducros (330). 

Study of epitaxial growth by the LEED method is often quite clear-cut and uncomplicated by formation of unexpected structures. One simply observes development of a characteristic crystal plane of the deposit bulk structure, and this is usually quite obvious from simple inspection of the pattern. Orientation of the film on the substrate is usually easily evident also. In some cases, the overlayer is coherent with the substrate as a coincidence lattice producing complicated LEED patterns. Yet it is usually relatively straightforward to decipher such patterns, because spacings in the overlayer structure are often easily assigned from known X-ray spacings of the substance being deposited [see, for example, Bauer (95)]. 

Crystal structure. A close match in the lattice parameters of the film and substrate is one important requirement for epitactic growth. There must also be a reasonable number of coincident lattice sites on either side of the interface. Frequently, although not always, this means that the film and substrate should have similar crystal structures. The higher the number of coincident sites, the better the chance of good epitaxy. 

Epitaxy is often described simply as an overlap between a pair of substrate lattice vectors and a pair of overlayer-substrate vectors, wherein the lengths of the coinciding lattice vectors of the opposing lattices are related by integral multiples. This generates a condition known as com-mmsurism, wherein each overlayer lattice point sits on one of a set of translationally... 

The value of det C thus characterizes the distortion of the ideal crystal structure due to cleavage. When det C is an integer, the superlattice is referred to as simple (Figs 2.2a, b). When detC is a rational number, the superstructure is called coincidence lattice. In such a case, if there is no angle between the... 

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