Equally, it is not necessary that the lateral periodicity in (x,y) is the same as the bulk periodicity (a,b). On the other hand, because the surface layers are in close contact with the bulk, there is a strong tendency for the periodicity to be, if not the same, a simple mutiple, sub-multiple or rational fraction of a and b. This leads to Wood's notation for surface and adsorbate layers, which is described in all the books. Note that we don't have to have these 'commensurate' structures, they can be 'irrational' or 'incommensurate'.
But for now, let's get the basic notation straight. This can be somewhat confusing. For example, here I have used (a,b,c) for the lattice constants; but these are not necessarily the normal lattice constants of the crystal, since they were defined with respect to a particular (hkl) surface. Also, several books use a1,2,3 for the real lattice and b1,2,3 for the reciprocal lattice, which is undoubtedly more compact. Wood's notation originates in a (2x2) matrix M relating the surface parameters (a,b) or as to the bulk (a0,b0) or ab. But the full notation, e.g. Ni(110)c(2x2)O, complete with the matrix M, (diagram 23) is rather forbidding- this is its 'Sunday' name. If you were working on oxygen adsorbtion on Nickel you would simply refer to this as a c(2x2), or centered 2 by 2, structure.
Typical structures that you may encounter include the following:
(1x1): this is a 'bulk termination'. Note that it doesn't mean that the surface is similar to the bulk in all respects, but that the average lateral periodicity is the same as the bulk. It may also be referred to as '(1x1)', implying that 'we know it isn't really' but that is what the LEED pattern shows. An example is the high temperature Si and Ge(111) structures, which are thought to contain mobile adatoms which don't show up in the LEED pattern because they are not ordered.
(2x1), (2x2), (4x4), (6x6), c(2x2), c(2x4), c(2x8), etc. These occur frequently on semiconductor surfaces. We shall consider Si(100)2x1 in detail later. Note that the symmetry of the surface is often less than that of the bulk. Si(100) is 4-fold symmetric, but the 2-fold symmety of the 2x1 surface can be constructed in two ways (2x1) and (1x2). These form two domains on the surface.√3x√3R30o. This often occurs on a trigonal or hexagonal symmetry substrate, including a whole variety of metals adsorbed on Si or Ge(111), and adsorbed gases on graphite (0001). 'Root three by root three rotated thirty degrees' is a classic Sunday name. Anyone who works on these topics calls it the √3, or Root-3, structure. It can often be incommensurate (book figure 1.16).
So, read the corresponding chapters, take in that there are 5 Bravais Lattices in 2D, as against 14 in 3D (diagram 21); check that diagrams 22 and 23 make sense, and move on. The methods used to determine structures, especially LEED, will be covered later, but if you want to get ahead, you can usefully read about it now. Luth, Chap 4.1-4.4 has essentially all that we will need. The article by Van Hove and Somorjai has references to compilations of solved structures.
Metal (1x1) surface layers tend to relax inwards by several percent. This is a feature of metallic binding, where what counts primarily is the electron density around the atom, rather than the directionality of 'bonds'. We can return to this point, which is embodied in 'Effective Medium' theories of metals, at a later stage.
Rare Gas Solids (Ar, Kr, Xe, etc) are an opposite limit. These can be modelled fairly well by simple potentials, and very well with accurate potentials plus small many body corrections. Such potential calculations for (1x1) surfaces have been used to explore the spacings and lattice vibrations at the surface. The surface expands outwards by a few percent in the first 2-3 layers, more for the open surface (110) than the close packed (111), see diagram 24 (book figure 1.17). Diagrams 25 and 26 explore the vibrations calculated for the Lennard-Jones potential, and remind us that the lower symmetry at the surface means that the mean square displacements are not the same parallel and perpendicular to the surface; on (110) all 3 modes are different. Table 27 (book table 1.2) shows us that different lattice dynamical models have given rather different answers. This is because the vibrations are sufficiently large for anharmonicity to assume greater importance at the surface.
Listening to specialists in this area can tax your geometric imagination, because the dimers form into rows, perpendicular to the dimers themselves- dimer and dimer row directions are not the same. Moreover, there are two types of 'single height steps', referred to as SA and SB, which have different energies, and alternate domains as described above. There are also 'double height steps' DA and DB, which go with one particular domain type. Then you can worry about whether the step direction will run parallel, perpendicular or at an arbitary angle to the dimers (or dimer rows, if you want to get confused, or vice versa). The dimers can also be symmetric (in height) or unsymmetric, and these unsymmetric dimers can be arranged in ordered arrays, 2x2, c(2x4), c(2x8), whatever. Keep drawing, and don't let anyone fool you, they may not know themselves.
Have I put you off completely yet? The point is, with all this intrinsic and unavoidable complexity, to ask whether you need to know all this stuff. Semiconductor surface structures are a specialist topic, which we will return to later, assuming that several of you really do need to know about these structures; they are remarkably important! Meanwhile we should abstract three salient points.
a) the existence of a particular type of structure, eg 2x1, does not determine the actual atomic arrangement. This typically has been determined by a detailed analysis of LEED I-V curves, and an experiment theory comparison in the form of a reliabilty or R-factor. Thus, see diagrams 28, 29 and 30 (book figure 1.19), three different models of Si(100)2x1 were proposed before the dimer model (diagram 28, book figure 1.19(a)) became accepted.
b) the number of domains depends on the symmetry. For Si(111) with the metastable 2x1 structure which is produced by cleaving, there are three, with the p-bonded chain model finding favor. This (2x1) structure, and the corresponding electronic structure, is described by Lüth, section 3.2 (3rd edition, pages 80-81) and section 6.5.1 (3rd edition pages 292-5).
c) major contributions to calculations and explanations of the surface and step strucures have been made, as set out briefly by D.J. Chadi, Surface Sci 299/300 (1994) 311. The editor of this volume, C.B. Duke, has also set out ways of considering the binding at such reconstructed semiconductor surfaces, for example in Applied Surface Sci. 65 (1993) 543, but also in several other journals and book chapters. We will return to the details later, partly in response to your queries.
A recent development is to combine structural experiments (e.g. LEED) on ultra-thin films grown on conducting substrates, to avoid problems of charging, with ab-initio calculation. Some of these methods and results can be found in the atlas due to Watson et al. (1996), review chapters in King & Woodruff (1997) and a 1999 conference procedings also entitled The Surface Science of Metal Oxides, to be published as Faraday Disc. Chem. Soc. volume 114. Grazing incidence X-ray spectroscopy is also helping to determine structures (Renaud 1998).
A notable exception to the general rule is alumina (Al2O3), where the surface oxygen ion relaxations have been calculated to reach around 50% of the layer spacing on the hexagonal (0001) face (Verdozzi et al. 1999). I would welcome the opportunity to learn more myself, and so have offered the topic as a project, which also contains a list of the above references. But we are getting ahead of ourselves: you can see how soon we need to read the original literature, but we do need some more background first!
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