Incommensurate overlayers

In practice the distinction between rational and irrational values of 1/6 is unimportant, because LEED cannot distinguish between unit cells larger than the coherence distance of the electron beam ( 100 A). It is customary to designate as incommensurate any overlayer that produces a coincidence unit cell larger than the LEED coherence distance. In fact, a truly incommensurate overlayer is impossible, since it could only occur in the limit of vanishing adsorbate-substrate forces parallel to the surface. 

Adsorption in many different adsorption sites simultaneously is expected for overlayers with an incommensurate lattice (cf. Sect. III). This has been confirmed by LEED intensity analyses for the case of an incommensurate overlayer of Xe on Ag(l 11), where both the substrate and the overlayer consist of hexagonally close-packed layers (with unrelated unit cells) parallel to the surface. 

A more recent study has examined the SH response from a Au(lll) electrode during UPD of a variety of metals, Ag, Cu, Pb, T1 and Sb [155]. In situ x-ray diffraction techniques have examined Ag, Cu and Pb on this substrate [159-161]. Silver is shown to form an epitaxial overlayer with Ag atoms sitting in 3-fold hollow sites forming a (1X1) commensurate overlayer. The lattice mismatch between lead and the substrate is shown to prevent formation of a commensurate overlayer but forms a hexagonal close-packed overlayer contracted by 0.7% from bulk lead. Although T1 and Sb on Au (111) have not been examined by x-ray diffraction, a close packed structure would necessitate an incommensurate overlayer due to the lattice mismatch. 

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 3.14 ID schematic representation of the degree of commensurability of Meads overlayers on single crystal substrates. Commensurate overlayers (a) and (b), higher order commensurate overlayer (c), and incommensurate overlayer (. ...

However, the experimental results do not allow one to decide whether the rotation of the Pb UPD overlayer (cf. eq. (3.22)) can be explained by the coincident site lattice" concept (involving higher order commensurate overlayers, Fig. 3.17) or the static distortion waves" (SDW) concept (dealing with incommensurate overlayers, Fig. 3.18). The estimation of d as a function of AE from a statistical analysis of in situ STM images coincides with GKS results illustrated in Fig. 3.27 [3.176]. 

Figure 2.36(a) Schematic representation of the incommensurate close-packed overlayer of Cu on Au formed in the perchlorate electrolyte The open circles are the gold atoms. Only part of the monolayer is shown in order to exhibit the overlayer-underlayer orientation, (b) Schematic representation of the more open lattice formed in the sulphuric acid electrolyte. From Manne... 

Figure 4.10. Schematic representation of the possible modes of registry. Overlayer and substrate lattice points are depicted as dark grey and light grey balls, respectively, and the primitive cell vectors s, and o, are also indicated. Ag(llO) has been selected as substrate (Fm3m, a = 0.409 nm). Examples of (a) commensurate registry, (b) coincidence-IA registry, (c) coincidence-IB registry, (d) coincidence-II registry and (e) incommensurate registry.

When 1/6 is an irrational number, the overlayer lattice bears in general no relationship to the substrate lattice the surface unit cell becomes infinite and the unit cell areas become incommensurate. This case corresponds to totally independent lattices, as is approximated by physisorbed systems. 

The commensurability of overlayers is characterized by the coincidence of reciprocal lattice vectors of adsorbate and substrate. Low or higher order coincidence leads to commensurate or higher order commensurate overlayer structures (Figs. 3.14a, b, c). K the reciprocal lattice vectors do not coincide, the overlayer is incommensurate (Fig. 3.14d). 

Me-S interactions leading to higher order commensurate and under certain conditions to incommensurate Meads overlayer structures give superstructures" and moire pattern" which can also be described by the Wood or matrk notation (cf. Figs. 3.17, 3.19). 

Figure 3.18 Rotated incommensurate 2D hep Meads overlayer in presence of significant Me-S misfit, (do,Me- o,s)/ o.S > 0.

Higher order phase transitions in systems with two differently ordered commensurate phases or with a commensurate and an incommensurate phase have been suggested in 2D Meads UPD overlayers [3.93, 3.94, 3.110-3.114, 3.223]. However, there has not been clear experimental evidence supporting this assumption. 

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