Commensurate superlattice

The diffraction pattern from a normal crystal is characterised by an array of spots separated by a distance /a = a that arise from the parent structure, together with a set of commensurate superlattice reflections that arise as a consequence of the additional ordering. In this case the spot spacing is 1 /na = a /n, where n is an integer, (Figure 8.21a, b). In modulated structures, the modulation might be in the position of the atoms, called a displacive modulation, (Figure 8.21c). Displacive modulations... 

The balance between these different types of bonds has a strong bearing on the resulting ordering or disordering of the surface. For adsorbates, the relative strength of adsorbate-substrate and adsorbate-adsorbate interactions is particularly important. Wlien adsorbate-substrate interactions dominate, well ordered overlayer structures are induced that are arranged in a superlattice, i.e. a periodicity which is closely related to that of the substrate lattice one then speaks of commensurate overlayers. This results from the tendency for each adsorbate to seek out the same type of adsorption site on the surface, which means that all adsorbates attempt to bond in the same maimer to substrate atoms. 

A superlattice is temied commensurate when all matrix elements uij j are integers. If at least one matrix element uij j is an irrational number (not a ratio of integers), then the superlattice is temied incommensurate. A superlattice can be inconnnensiirate in one surface dimension, while commensurate in the other surface dimension, or it could be mconmiensurate in both surface dimensions. 

FIGURE 3.2. (a) Chemical structure of octanethiol. (b) A constant current STM image of octanethiol SAM on Au(l 11). Au reconstruction is lifted and alkanethiols adopt commensurate crystalline lattice characteriized by a c(4 x 2) superlattice of a (a/3 x V3)R30°. (c) Model of commensuration condition between alkanethiol monolayer (large circles) and bulk-terminated Au surface (small circles). Diagonal slash in large circles represents azimuthal orientation of plane defined by all-trans hydrocarbon chain. (Reprint with permission from Ref.25 G. E. Poirier, Chem. Rev., 97, 1117-1127 (1997). Copyright 1997 American Chemical Society.)... 

The last example we would like to discuss is a lattice of holes formed in stoichiometric hexagonal (h) BN double layers on Rh(lll), see Fig. 5(c) and [99]. The lattice is composed of holes in the BN-bilayer with a diameter of 24 2 A, and an average distance of 32 2 A. The holes in the upper layer are offset with respect to the smaller holes in the lower layer. We note that well-ordered superstructures with a large period have already been observed some time ago by means of LEED for borazine adsorption onto Re(0001) [102], while borazine adsorption onto other close-packed metal surfaces, such as Pt(lll), Pd(lll), and Ni(lll), leads to the self-limiting growth of commensurate ABN monolayers [103,104]. For BN/Rh(lll) it is not clear at present whether the Rh(lll) substrate is exposed at the bottom of the holes. If this was the case the surface would not only be periodic in morphology but also in chemistry, and therefore would constitute a very useful template for the growth of ordered superlattices of metals, semiconductors, and molecules. 

An additional feature of the temperature dependent X-ray scattering is the persistence above Tc of intensity at the superlattice positions51). This is consistent with a soft phonon mode at a wave vector commensurate with the changes that occur on dimerization. It has been suggested that this low frequency lattice mode may be a requirement for the observation of a spin-Peierls transition. 

In Ni(pc)I and Ni(tbp)I, the iodine superlattice spacing, c, is a multiple of the Bragg spacing, c (c = 3/2 c), so the superlattice is commensurate with the Bragg lattice and the repeat unit contains three Bragg sites. The disorder in these two systems is not severe, and could be modeled satisfactorily2,132,116 No special treatment of the iodine atoms was necessary to solve the Bragg structures, and the only structural manifestation of the... 

Figure 29 STM image (120 A wide) of a TTF-TCNQ crystal at temperatures below 80 K. A one-dimensional superlattice commensurate with the underlying lattice, but with double the period, starts to appear. It is attributed by Pan et al. [209] to the commensurate 2kF CDW (0.5a, 0.296, 0c ) observed by x-ray diffraction study at 50 K [214]. However, the modulation in the b direction was never observed. (From Ref. 209.)...

Adsorption processes on crystallographically well-defined substrate surfaces lead to the formation of 2D Meads phases with well-ordered structures denoted as overlayers". Generally, three different types of overlayers, depending on the degree of registry between overlayer and substrate, can be distinguished commensurate, higher-order commensurate or incommensurate overlayers, as illustrated schematically in Fig. 3.14. TTie term superlattice stmcture" is frequently used for commensurate overlayers which can be characterized by either the Wood or the matrix notation [3.271-3.274]. 

Figure 3.15 Different expanded commensurate Meads overlayer (superlattice) structures.

When the modulation wavelength exactly fits a number of parent unit cells, (that is, is commensurate with it), it will be possible to index the reflections in terms of a normal superlattice. Recently a number of structures that were once described in this way have been restudied and found to be better described as modulated structures with commensurate modulation waves - commensurately modulated structures. 

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