Surface unit cell

The second important parameter to consider is the size of the surface unit cell. A surface unit cell caimot be smaller than the projection of the bulk cell onto the surface. However, the surface unit cell is often bigger than... 

In addition, the surface unit cell may be rotated with respect to the bulk cell. Such a rotated unit cell is notated as... 

Most metal surfaces have the same atomic structure as in the bulk, except that the interlayer spaciugs of the outenuost few atomic layers differ from the bulk values. In other words, entire atomic layers are shifted as a whole in a direction perpendicular to the surface. This is called relaxation, and it can be either inward or outward. Relaxation is usually reported as a percentage of the value of the bulk interlayer spacing. Relaxation does not affect the two-dimensional surface unit cell synuuetry, so surfaces that are purely relaxed have (1 x 1) synuuetry. 

The three-dimensional synnnetry that is present in the bulk of a crystalline solid is abruptly lost at the surface. In order to minimize the surface energy, the themiodynamically stable surface atomic structures of many materials differ considerably from the structure of the bulk. These materials are still crystalline at the surface, in that one can define a two-dimensional surface unit cell parallel to the surface, but the atomic positions in the unit cell differ from those of the bulk structure. Such a change in the local structure at the surface is called a reconstruction. 

Figure Al.7.5(a) shows a larger scale schematic of the Si(lOO) surface if it were to be biilk-tenninated, while figure Al.7.5(b) shows the arrangement after the dimers have been fonned. The dashed boxes outline the two-dimensional surface unit cells. The reconstructed Si(lOO) surface has a unit cell that is two times larger than the bulk unit cell in one direction and the same in the other. Thus, it has a (2 x 1) synnnetry and the surface is labelled as Si(100)-(2 x i). Note that in actuality, however, any real Si(lOO) surface is composed of a mixture of (2 X 1) and (1 x 2) domains. This is because the dimer direction rotates by 90° at each step edge.

Figure Al.7.5. Schematic illustration showing the top view of the Si(lOO) surface, (a) Bulk-tenninated structure. (b)Dimerized Si(100)-(2 x 1) structure. The dashed boxes show the two-dimensional surface unit cells.

The surface unit cell of a reconstructed surface is usually, but not necessarily, larger than the corresponding bulk-tenuiuated two-dimensional unit cell would be. The LEED pattern is therefore usually the first indication that a recoustnictiou exists. However, certain surfaces, such as GaAs(l 10), have a recoustnictiou with a surface unit cell that is still (1 x i). At the GaAs(l 10) surface, Ga atoms are moved inward perpendicular to the surface, while As atoms are moved outward. 

Although most metal surfaces exliibit only relaxation, some do have reconstmctions. For example, the fee metals, Pt(l 10), Au(l 10) and Ir(l 10), each have a (1 x 2) surface unit cell. The accepted stmeture of these surfaces is a... 

When atoms, molecules, or molecular fragments adsorb onto a single-crystal surface, they often arrange themselves into an ordered pattern. Generally, the size of the adsorbate-induced two-dimensional surface unit cell is larger than that of the clean surface. The same nomenclature is used to describe the surface unit cell of an adsorbate system as is used to describe a reconstructed surface, i.e. the synmietry is given with respect to the bulk tenninated (unreconstructed) two-dimensional surface unit cell. 

An example of the fomiation of a new reconstmction is given by certain fee (110) metal surfaces. The clean surfaces have (1x1) synunetry, but become (2x1) upon adsorption of oxygen [16, 38]. The (2x1) synuiietry is not just due to oxygen being adsorbed into a (2 x 1) surface unit cell, but also because the substrate atoms rearrange themselves... 

It is useful to define the tenns coverage and monolayer for adsorbed layers, since different conventions are used in the literature. The surface coverage measures the two-dimensional density of adsorbates. The most connnon definition of coverage sets it to be equal to one monolayer (1 ML) when each two-dimensional surface unit cell of the unreconstructed substrate is occupied by one adsorbate (the adsorbate may be an atom or a molecule). Thus, an overlayer with a coverage of 1 ML has as many atoms (or molecules) as does the outennost single atomic layer of the substrate. 

RHEED intensities cannot be explained using the kinematic theory. Dynamical scattering models of RHEED intensities are being developed. With them one will be able to obtain positions of the surface atoms within the surface unit cell. At this writing, such modeling has been done primarily for LEED. 

This equation also limits the set of observable LEED spots by the condition that the expression inside the brackets must be greater than zero. With increasing electron energy the number of LEED spots increases while the polar emission angle relative to the surface normal, 6 = arctan(k /kz), decreases for each spot except for the specular spot (0,0) which does not change. Eig. 2.47 shows examples of common surface unit cells and the corresponding LEED patterns. 

Whereas the spot positions carry information about the size of the surface unit cell, the shapes and widths of the spots, i.e. the spot profiles, are influenced by the long range arrangement and order of the unit cells at the surface. If vertical displacements (steps, facets) of the surface unit cells are involved, the spot profiles change as a function of electron energy. If all surface unit cells are in the same plane (within the transfer width of the LEED optics), the spot profile is constant with energy. 

Analysis of the LEED pattern or of spot profiles does not give any quantitative information about the position of the atoms within the surface unit cell. This type of information is hidden in the energy-dependence of the spot intensities, the so-called LEED 7-Vcurves. 

A special notation is used to describe surface reconstructions and surface overlayers and is described in books on surface crystallography (Clarke, 1985). The lattice vectors a and b of an overlayer are described in terms of the substrate lattice vectors a and b. If the lengths la I = mlal and Ib l = nibl, the overlayer is described as mXn. Thus, a commensurate layer in register with the underlying atoms is described as 1 X 1. The notation gives the dimension of the two-dimensional unit cell in terms of the dimensions of an ideally truncated surface unit cell. 

Figure 5.5 Top and side views of the unreconstructed Au(100)-(1 x 1) and the (5 x 1) hexagonal reconstructed Au(100)-hex surfaces. Unit cells are indicated by rectangles.

Figure 9.1 High symmetry adsorption sites for O and O2 on Pt(l 11). Large gray circles represent Pt atoms, and small open circles represent O atoms, t, b, h, and f stand for top, bridge, hep, and fee, respectively. The surface unit cell is dehneated. (Reproduced with permission from Xu et al. [2004].)...

All the work we describe in this chapter was carried out in UHV on the rutile Ti02(l 1 0)1 x 1 surface. Exposures to vapors and gases are given in Langmuirs (L) where 1 L = 1.333 x 10 s mbar s. Coverages of defects or molecules adsorbed at the surface will be given in monolayers (ML), where 1 ML corresponds to the density of primitive surface unit cells. 

There are many deposit-substrate combinations where the basic lattice mismatch is very large, such as when a compound is formed on an elemental substrate, but where excessive strain does not necessarily result. Frequently a non one-to-one lattice match can be formed. If a material can match up every two or three substrate surface unit cells, it may still form a reasonable film [16]. In many cases the depositing lattices are rotated from the substrate unit cells, as well. In a strict definition of epitaxy, these may not be considered, however, it is not clear why high quality devices and materials could not be formed. 

Stability of sputtered molecules. In order to further mimic experimental conditions, many trajectories are evaluated by choosing an ensemble of impact points for the energetic particle within the surface unit cell. The experiments with which the simulations are compared are performed so that the majority of the bombarded surface is undamaged This makes direct comparisons between the simulated and experimental results possible. 

Develop supercells suitable for performing calculations with the (100), (110), and (111) surfaces of an fee metal. What size surface unit cell is needed for each surface to examine surface relaxation of each surface ... 

Pt(l 10) is known experimentally to reconstruct into the so-called missing-row reconstruction. In this reconstruction, alternate rows from the top layer of the surface in a (2 x 1) surface unit cell are missing. Use a supercell defined by a (2 x 1) surface unit cell of Pt(110) to compute the surface energy of the unreconstructed and reconstructed surfaces. Why does this comparison need to be made based on surface energy rather than simply based on the total energy of the supercells Are your results consistent with the experimental observations Use similar calculations to predict whether a similar reconstruction would be expected to exist for Cu(110). 

Section 4.5 Surface relaxations were examined using asymmetric slab models of five, six, seven, or eight layers with the atoms in the two bottom layers fixed at bulk positions and all remaining atoms allowed to relax. For Cu(100), the supercell had c(2 x 2) surface symmetry, containing 2 atoms per layer. For Cu(l 11), (y/3 X /3)R30 surface unit cell with 3 atoms per layer was used. All slab models included a minimum of 23 A of vacuum along the direction of the surface normal. A 6x6x1 /c-point mesh was used for all calculations. 

Fig. 10.2 Surface hydroxyl configuration on the goethite 001, 101, 100 and 210 faces. Distances of O and Fe ions to the projection plane are indicated next to the corresponding row of ions. Rows of singly, doubly, and triply coordinated O ions are indicated as S, D, and T, respectively. Solid line rectangles represent the two-dimensional (surface) unit cell. Dotted-line rectangles show contiguous singly coordinated hydroxyls (Barron and Torrent, 1996, with permission).

Fig. 1. A step in the SOS model at temperature keT/s = 0.9. The concentration ofadatoms and of vacancies is about 0.02/Q, where Q is the area of the surface unit cell. From Bartelt et al. (1994a), with permission. 

In the majority of cases where adsorbates form ordered surface structures, the unit cells of those structures are larger than the unit cell of the substrate the surface lattice is then called a super lattice. The surface unit cell is the basic quantity in the description of the ordering of surfaces. It is necessary therefore to have a notation that allows the unique characterization of superlattices relative to the substrate lattice. 

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