Lecture notes by John A. Venables. Revised for Spring '03;

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 a_{1,2,3}
for the real lattice and b_{1,2,3} 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 a_{s} to the bulk (a_{0},b_{0})
or a_{b}. 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.

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 S_{A} and S_{B}, which have different energies, and
alternate domains as described above. There are also 'double height steps' D_{A}
and D_{B}, 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
(Al_{2}O_{3}), 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!